ENERGY DEPENDENCE OF THE 
EFFECTIVE INTERACTION FOR 
NUCLEON-NUCLEUS SCATTERING 
by 
Helmut Seifert 
Dissertation submitted to the Faculty of the Graduate School 
of The University of Maryland in parti al fulfillment 
of the requirements for th e degree of 
Doctor of Philosophy 
1990 
(I I \., L??_ I. 
J. J r { 
Advisory Committee: 
Dr. James J . Kelly, Advisor 
Dr . Chia-cheh Chang 
Dr. Nicholas S. Chant I t r/ ) ~ 
Dr . Stephen J . Walla ce 
Dr. Timothy M. Heckman Sc..'1frd 1 
/ l . 
Vu,? / 
,--I) __ ~) 
Abstract 
Title of Dissertation: Energy Dependence of the Effective Interaction 
for Nucleon-Nucleus Scattering 
Helmut Seifert, Doctor of Philosophy, 1990 
Dissertation directed by: J ames J . Kelly, Associate Professor 
Department of Physics and Astronomy 
We have measured cross sections and analyzing powers for 40 ,42 ?44 ?48Ca 
and 160 at IUCF using the new high-resolution K600 spectrometer for 100 
and 200 Me V protons . Measurements at 318 Me V for 40 ?42 ?44 ,48Ca and 32 ,34 S 
were done at LAMPF using the HRS spectrometer. 
In this work, we obtain empiri cal effective interactions by fittin g inelas-
tic scattering data for many low-lying normal-parity isoscalar excitations of 
the self-conjugate nuclei 16 0 and 4?Ca, assuming a local tp folding model. 
One-nucleon transition densities are from (e, e') . The fitt ed interactions are 
iterated to generate optical potentials self-consistently. We find that the 
fitt ed parameters are essentially target independent, which supports the va-
lidity of the local density hypothesis . Elastic scatte ring is predicted by ex-
tracting the rearrangement factor (1 + pd/ dp) from the fitt ed inelast ic inter-
actions. Below 300 Me V the stre11gth of the empirical int eraction is r .du ced 
at zero density and th e general density dependence is weaker compared to 
the theoretical interaction . Above 300 MeV we find the density dependence 
is stronger than expected. The empirical interactions provide better descrip-
tions of elastic and inelastic data than IA calculations or LDA calculations 
using theoretical G-matrices, and can be used for nuclear structure studies 
of other nuclei . Fitted optical potentials above 300 MeV are comparable to 
equivalent Schrodinger potentials from the relativistic IA2 model. 
Dedication 
To all the ones I love: 
my family, my fri ends 
and 
Tuleen Parveen 
11 
Acknowledgements 
I first would like to thank my ad visor , J ames J . Kelly, fo r guiding me 
through all these years of exciting research , and for suggesting this interesting 
dissertation topic . I also want to thank him for helping me through the for me 
as a foreign student sometimes difficult time at the beginning of m y graduate 
career . 
I would like to thank the Experimental Nuclear Physi cs gro up as a 
whole-faculty, staff, and all my collegues- for providing a very pleasant 
work atmosphere which I will always fondly remember. To my famil y goes, 
of course, my deepest gratitude for their neverending support. 
During my time here in Maryland I have also met some very special 
people who have become very good fri ends of mine, and whom I want to 
mention here . First, there are Raul and Steve, and especially Grace, who 
have shared all the good times and bad times with me, and who were always 
there for me . J ohn and Sabrina, George and Liz I want to thank for having 
invited me so many times to spend the holidays together with th eir wond erful 
family. Finally, last but certainly not least , there is Tulee 11 Parveen who will 
always be in my heart. 
llJ 
Contents 
List of Tables IX 
List of Figures XVI 
1 Introduction 1 
2 Theoretical Background 11 
2.1 The Transition Matrix as an Effective Interaction 11 
2.2 Elas tic Nucleon-Nucleus Scattering- T he Opti cal Model 13 
2.3 Multiple Scattering Approach 16 
2. 3.1 Introduction . . .. . . 16 
2.3.2 The Full Folding P otential and its Approxim ations 18 
2.3.3 Expressions for the Mixed Target Density 22 
2.3.4 A Local Pseudopotenti al 24 
2.3.5 Off-Shell T -Ma.trices .. 28 
2.3.6 NRIA Calculation Results 30 
2.4 Nuclear Matter Approach 48 
2.4 .1 Int roduction . . . . 48 
2.4. 2 Density Dependent Interactions 51 
2.4.3 The Local Density Approxi mation 62 
2.4 .4 Treatm ent of the Exchange In teraction 64 
2.4 .5 LDA Calculation Results .. 66 
2.5 Relativistic Impulse Approximat ion 86 
2.5 .1 Introduction . . . .. . .. . 86 
IV 
2.5 .2 The IA2 Model ..... . .. . 87 
2.5.3 The Virtual P air Interpretation 90 
2.5.4 IA2 Calculation Result s . 95 
2.6 Off-Shell Effects and Full-Folding 104 
2.7 Inelastic Nucleon-Nucleus Scattering 107 
2. 7.1 Distorted Wave Approximation 107 
2. 7. 2 The Perey Effect . . . . . 109 
2. 7.3 The Rearrangement Effect 111 
2.8 An Empirical Effective Interaction . 113 
2.8.1 Description of Theoretical Interactions 113 
2.8.2 Description of Data . . . . . . . . . . . 131 
2.8.3 Linear Expansion Analysis and Fitting P rocedure 135 
3 Experiment 140 
3.1 Introduction 140 
3.2 Accelerator and Beam 141 
3.3 Targets . 146 
3.4 Spectrometer 147 
3.4.1 The K600 Spectrometer 147 
3.4.2 The HRS Spectrometer . 150 
3.5 Detectors 150 
3.5. 1 Scinti lla.tors 15 J 
3.5.2 Wire Chambers 151 
3.6 Elect ronics . 152 
V 
3.6.1 Scintillator Electronics . . 152 
3.6.2 VDC Readout Electronics 153 
3.6.3 Coincidence Electronics 157 
3.6.4 Electronics for Integrators and BL 2 160 
3.6.5 Data Acquisition . .. . . ... . . . 162 
4 Calibrations 163 
4.1 Matching Between Beam Line and Spectrometer 163 
4.2 Scattering Angle 171 
4.3 Beam Polarization 184 
4.4 Other Calibrations and Tests 189 
5 Drift Chamber Detectors 193 
5.1 Introduction . .... . 193 
5.2 Classification of Events 198 
5.3 Calculation of Position and Angle 207 
5.4 Drift Cell Resolution- Straightness of Tracks . 214 
5.5 Summary and Conclusions 216 
5.6 Some Diagnostics .. . . . 218 
6 Data Reduction 221 
6.1 Replay .. .. 22 1 
6.2 Line-shap e Fitting 227 
6.3 Formulae For Cross Sect ions and Analyzing Powers 237 
6.4 Kinematic Formulae 240 
VJ 
6.5 Normalizations 241 
6.5 .1 100 MeV Cups 242 
6.5.2 200 MeV Cups 243 
6.5.3 Normalizations between Targets 243 
6.6 Cross Section and Analyzing Power Results 245 
6.6.1 Fitted Data 245 
6.6.2 Some General Observations 249 
6.7 Results and Treatm~nt of Errors . 252 
7 Results and Discussion 290 
7.1 Data Selection and Iteration Procedure 290 
7.2 Fit to Inelastic Data 293 
7.2.1 100 MeV . 293 
7.2.2 200 MeV. 306 
7.2.3 318 MeV . 317 
7.2.4 500 MeV. 330 
7.2.5 Interior Sensitivity 336 
7.3 Inclusion of the Perey Effect 339 
7.4 Inclusion of Elastic Data 342 
7.5 A Dependence of the Effective Interaction 351 
7.6 Comparison with IA2 and Dirac Phenomenology 367 
7.7 En ergy Dependence of the Effective lnternction 382 
7.8 Outlook .. 390 
8 Summary and Conclusions 392 
VII 
A Nuclear Densities 399 
B Differential Recoil Diagrams 408 
C RAYTRACE Simulations of the K600 412 
D Effective Interaction Grids 422 
E List of Runs 427 
F Data Tables 441 
F .1 9 Be(p,p') and 16O(p,p') at Ep = 100 MeV . 44 1 
F .2 4?Ca(p,p') at Ep = 100 MeV . .. . ... . . 468 
F .3 9 Be(p,p') and 16O(p, p') at Ep = 200 MeV 490 
F.4 4?Ca(p, p') at Ep = 200 MeV . 523 
References 545 
VIIJ 
List of Tables 
2.1 Reparametrization of the PH interaction 119 
2.2 Reparametrization of the NL Interaction 120 
2.3 Reparametrization of the LR Interaction 121 
2.4 Low-q Form of Retf0 . . . ? . ? . ? . 132 
2.5 Simple Reparam etrization of the PH and LR Interactions 134 
3.1 Beam Energies for Experiment 268 143 
3.2 Apertures used for Experiment 268 145 
3.3 Targets for Experiment 268 146 
3.4 Comparison of HRS and K600 Specifications 150 
5.1 Grid in x101 . . . . . 205 
6.1 Energy Levels of 9 Be 229 
6.2 Energy Levels of 16 0 230 
6.3 Energy Levels of 16 0 (cont .) 231 
6.4 Energy Levels of 4?Ca . . . 232 
6.5 Overall Normalization Factors for Experiment 268 . 244 
7 .1 Empirical Effective Interactions (100 Me V) 297 
7.2 Empirical Effective Interactions (200 MeV) 307 
7.3 Empirical Effective Interactions (318 MeV) 320 
7.4 Empirical Effective Interact ions (500 l\.IeV ) 332 
7.5 Empirical Interaction with P erey Effect (1 6 0 , lfl() MeV ) . 34J 
7.6 Empirical Interaction with P erey Effect (1 6 0 , 200 MeV) . 341 
7.7 Fit to Elastic and Inelastic Data. (1 6 0 , 100 MeV) 344 
IX 
7.8 Fit to Elastic and Inelastic Data (1 6 0, 200 Me V) . 344 
7.9 Interaction Parameters for Various Energies . 384 
A.I Expansion Coefficients for Pg and Ptr (1 6 0) . 403 
A.2 Expansion Coefficients for pg and Ptr (4?Ca.) . 404 
C.1 Polynomials for 0tgt and o from RAYTRACE . 416 
D.1 Interaction Grid: 160 (200 MeV ) . 423 
D.2 Interaction Grid: 4?Ca (200 MeV) . 424 
D.3 Interaction Grid: 160 (318 MeV) . 425 
D .4 Interaction Grid: 160 (500 MeV) . 426 
E .1 a) BeO Runs (100 MeV) . . . . . 428 
E .1 b) BeO Runs (100 MeV, cont.) . 429 
E .2 a) 4?Ca Runs (100 MeV) . . . . . 430 
E.2 b) 4?Ca Runs (100 MeV, cont.) . 431 
E.2 c) 4?Ca Runs (100 MeV, cont.). . 432 
E.2 d) 4?Ca Runs (100 MeV, cont.) . 433 
E.2 e) 4?Ca Runs (100 MeV, cont.). . 434 
E.3 a) BeO Runs (200 MeV) ... . . 435 
E .3 b) BeO Runs (200 MeV , cont.) . 436 
E .3 c) BeO Runs (200 MeV , cont.). . 437 
E .4 a) 4?Ca Runs (200 MeV) .... . 438 
E.4 b) 4?Ca. Runs (200 MeV, cont.) 439 
E.4 c) 4?Ca Runs (200 MeV, cont .) . . 440 
F .1 a) 9 Be(p,p)3/21 (elastic) ; Er = 100 MeV . . 442 
F.1 b) 9 Be(p, p')l / 2T ( 1.680 Me V) ... . . .. . 443 
X 
F.l c) 9 Be(p,p')5 / 21 (2.429 MeV ) . 444 
F.1 d) 9 Be(p,p')l / 21 (2 .78 MeV). . 445 
F .1 e) 9 Be(p,p')5 / 2't (3 .049 MeV) . 446 
F.1 f) 9 Be(f, p')3 / 2?t (4 .704 MeV) . 447 
F .1 g) 9Be(p,p') Lor (6 .5 MeV) . . . 448 
F .2 a ) 16O(p,p)0i (elastic) ; Ep = 100 MeV . 449 
F.2 b) 16O(p,p')0f (6 .0494 MeV). . 450 
F.2 c) 16O(p, p')31 (6 .1299 MeV ) . . 45 ] 
F.2 d) 16O(p, p')2i (6.9171 MeV) . . 452 
F .2 e) 16 O(p,p')l1 (7 .1169 MeV) . . 453 
F .2 f) 16 O(p,p')21 (8.8719 MeV) . . 454 
F .2 g) 16 O(p, p')l2 (9 .585 MeV ) . 455 
F.2 h) 16O(p, p')2f (9 .8445 MeV). . 456 
F.2 i) 16O(f, p')4?t (10.356 MeV) . 457 
F .2 j) 16 O(p,p')01 (10 .957 MeV) . 458 
F .2 k) 16 O(.r, p')4t (11.097 MeV) . . 459 
F .2 1) 16O(.r, p')2t (11.520 MeV) . . 460 
F.2 rn) 16 O(.r, p')0t (12.049 MeV) . 461 
F .2 n) 16O(f,p')l3 (12.440 MeV ) . . 462 
F.2 o) 16O(p, p')22 (12 .530 MeV). . 463 
F.2 p) 16 O(p, p')02 (12 .796 MeV) . . 46cl 
F .2 q) 16 O(p, p')23 (12.969 MeV ) . . 465 
F.2 r) 16 O(p,p')l3 .08 (13 .08 Me\/) . 466 
F .2 s) 16O(p, p')34 (1 3 .259 Me V) . . 467 
Xl 
F.3 a) 4?Ca(p,p)0t (elast ic) ; Ep = 100 MeV . 469 
F .3 b) 4?Ca(p,p')0t (3 .3521 MeV ) . 470 
F.3 c) 4?Ca(p,p')31 (3 .7364 MeV) . 471 
F .3 cl) 4?Ca(p,p')2t (3 .9041 MeV) . 472 
F .3 e) 4?Ca(p, p')51 (4.4915 MeV ) . 473 
F .3 f) 4?Ca(p, p')0f (5.213 MeV) . . . 474 
F.3 g) 4?Ca(p,p')2t (5 .249 MeV) . . 475 
F .3 h) 4?Ca(p, p' )4t (5.279 MeV) . . 476 
F .3 i) 4?Ca(p, p' )41 (5 .6143 Me V) . 477 
F.3 j) 4?Ca(p,p')2t (5 .6301 MeV) . 478 
F .3 k) 4?Ca(p, p' )l1 (5 .9033 MeV) . 479 
F .3 l) 4?Ca(p,p')6.028 (6.028 MeV) . 480 
F.3 m) 4?Ca(p,p')32 (6 .2858 MeV) . . 481 
F.3 n) 4?Ca(p, p')4t (6.5084 MeV) . 482 
F .3 o) 4?Ca(p,p' )4t (6 .5436 MeV) . 483 
F.3 p) 4?Ca(p, p')33 (6 .5833 Me V) . 484 
F.3 q) 4?Ca(p,p' )22 (6.7509 MeV) . 485 
F.3 r) 4?Ca(p, p' )6.909 (6 .909 MeV) . .. .. . 486 
F .3 s) 4?Ca(p, p')6 .931 (6.931 MeV) . 487 
F .3 t) 4?Ca(p,p' )12 (6 .951 MeV) . . 488 
F .3 u) 4?Ca(p, p')7 .11 (7 .11 MeV) ? 1l8D 
F.4 a) 9 Be(p,p)3 / 21 (elastic) ; Ep = 200 MeV . . 49 1 
F.4 b) 9 Be(p, p')l / 2i (1.680 MeV ) . 492 
F.4 c) 9 Be(p, p')5 / 21 (2.429 MeV) . 493 
Xll 
F.4 d) 9 Be(p,p')l/21 (2.78 MeV). . 494 
F.4 e) 9 Be(p, p')5 /2i (3 .049 MeV) . 495 
F .4 f) 9 Be(p,p')3 / 2i (4.704 MeV) . 496 
F.4 g) 9 Be(p,p') Lor (6 .5 MeV) . . . 497 
F .5 a) 16 O(p,p)0i (elastic); Er = 200 MeV . . 498 
F .5 b) 16O(p,p')0t (6 .0494 MeV) . - . 499 
F.5 c) 16 O(p, p')31 (6.1299 MeV) .. . 500 
F .5 d) 16 O(p, p' )2 i (6 .9171 MeV). . 501 
F .5 e) 16 O(p, p')l1 (7 .1169 MeV) . . 502 
F.5 f) 16O(p, p')21 (8 .8719 MeV) .. . 503 
F.5 g) 16O(p,p' )12 (9.585 MeV) .. . 504 
F.5 h) 16O(p, p' )2t (9 .8445 Me V) .. . 505 
F.5 i) 16O(p,p')4i (10 .356 MeV) . 506 
F .5 j) 16 O(p, p')01 (10 .957 MeV ) . 507 
F.5 k) 16 O(p, p')4t (11.097 MeV ) . . . 508 
F.5 1) 16O(p, p')2f (11 .520 MeV) . . . 509 
F .5 m) 16 O(p, p')0f (12.049 MeV ) . 510 
F .5 n) 16O(p, p' )13 (12.440 MeV ) . . 511 
F.5 o) 16 O(p, p')22 (1 2 .530 Me V) . . 512 
F .5 p) 16O(p,p')02 (12 .796 MeV). . 513 
F .5 q) 16O(p, p')23 (12 .969 MeV) . . 5 1-1 
F.5 r) 16 O(p', p')l3 .08 (13 .08 MeV) . 515 
F.5 s) 16O(p,p')3i (13 .259 MeV) . . 516 
F.5 t) 16O(p,p')l i (13.664 MeV) . . 517 
... 
Xlll 
F.5 u) 16 O(p, p')4f ( 13.869 Me V) . . . 518 
F .5 v) 16 O(f,p')25 (13.98 MeV) . 519 
F .5 w) 16O(f, p')0t ( 14.032 Me V) . 520 
F .5 x) 16 O(p, p')41 (14 .302 MeV) . . 521 
F .5 y) 16 O(p, p')5i ( 14.399 Me V) .... . 522 
F.6 a) 4?Ca(p,p)0i (elastic); Ep = 200 MeV . 524 
F.6 b) 4?Ca(f, p')0 f (3.3521 MeV) . 525 
F.6 c) 4?Ca(p, p')31 (3. 7364 Me V) . 526 
F.6 d) 4?Ca(f,p')2?t (2.9041 MeV) . 527 
F.6 e) 4?Ca(p,p')51 (4 .4915 MeV) . 528 
F.6 f) 4?Ca(f, p')0f (5 .213 MeV) . . 529 
F .6 g) 4?Ca(f, p')2t (5 .249 MeV) . . 530 
F.6 h) 4?Ca(f, p')4 ?t (5.279 MeV) . . 531 
F.6 i) 4?Ca(f, p')41 (5 .6143 MeV ) . 532 
F.6 j) 4?Ca(f, p')2f (5 .6301 MeV ) . 533 
F.6 k) 4?Ca(p, p')l1 (5 .9033 MeV ) . 534 
F .6 1) 4?Ca(f,p')6 .028 (6.028 MeV ) . 535 
F .6 m) 4?Ca(p, p')32 (6 .2858 MeV) . . 536 
F .6 n) 4?Ca(p, p')4t (6.5084 MeV ) . 537 
F.6 o) 4?Ca(p, p')4f (6 .5436 MeV) . 538 
F.6 p) 4?Ca(p, p')33 (6 .5833 MeV) . 53D 
F.6 q) 4?Ca.(p, p')22 (6 .7509 MeV) . 54() 
F.6 r) 4?Ca.(p,p')6 .909 (6 .909 MeV) . 54 1 
F .6 s) 4?Ca.(p, p')6 .931 (6 .931 Me\!) . 542 
XlV 
F.6 t) 4?Ca(p,p')l2 (6.951 MeV ) . . 543 
F.6 u) 4?Ca(p, p')7 .ll (7.11 MeV) . 544 
xv 
List of Figures 
1..1 Interior Sensitivity of Nucleon Inelastic Scattering 3 
1.2 Shell Structure of 160 and 4?Ca . . . . . . . . . . 6 
2.1 Comparison of PH , NL and FL t-Matrices (100 MeV) 33 
2.2 Comparison of PH, NL and FL t-Matrices (200 MeV ) 34 
2.3 Comparison of PH, NL and FL t-Matrices (318 MeV) 35 
2.4 NRIA Calculations for er(q) and Ay (1 6 0, 100 MeV) . 40 
2.5 NRIA Calculations for er(q) and A 411 ( ?Ca, 100 MeV) 41 
2.6 NRIA Calculations for cr(q) and Ay (1 60, 200 MeV) . 42 
2.7 NRIA Calculations for er( q) and Ay (4?Ca, 200 Me V) 43 
2.8 NRIA Calculations for Q (1 6 0 and 4?Ca, 200 MeV) 44 
2.9 NRIA Calculations for er(q) and Ay (1 6 0, 318 MeV) 45 
2.10 NRIA Calculations for er( q) and Ay (4?Ca, 318 Me V) 46 
2 .11 NRIA Calculations for Q (4?Ca, 318 MeV) 47 
2.12 LDA Optical Potentials (1 6 0, 100 MeV) 69 
2.13 LDA Optical Potentials (4?Ca, 100 MeV) 70 
2.14 LDA Optical Potentials (1 6 0, 200 MeV ) 71 
2.15 LDA Optical Potentials (4?Ca, 200 MeV) 72 
2.16 LDA Optical Potentials (1 6 0 , 318 MeV ) 73 
2.17 LDA Optical Potentials (4?Ca. , 318 MeV) 7,1 
2.18 LDA Calculations for er(q) and A 611 (1 0 , ]()() MeV) 78 
2.19 LDA Calculations for er(q) and Ay (4?Ca. , 100 MeV) 79 
2.20 LDA Calculations for er(q) and Ay (1 6 0 , 200 MeV) 80 
XVl 
2.21 LDA Calculations for a-(q) and Ay (4?Ca, 200 MeV) 
81 
2.22 LDA Calculations for Q (1 6 0 and 4?Ca, 200 MeV ) . 
82 
2.23 LDA Calculations for a-(q) and Ay (1 6 0, 318 MeV) 
83 
2.24 LDA Calculations for a-(q) and Ay (4?Ca, 318 MeV) 
84 
2.25 LDA Calculations for Q (4?Ca, 318 MeV) . 
85 
2.26 Z-Graph for Scattering of a Dirac Particle 
92 
2.27 IA2 , LDA , NP, and NRIA Optical Potentials (1 6 0 , 200 MeV ) . 97 
2.28 IA2 , LDA , NP, and NRJA Opt;caJ P otenUa!s ("Ca, 200 MeV) 98 
2.29 IA2, LDA, NP, and NRIA Optical Potentials (1 6 0, 318 MeV ) . 99 
2.30 IA2 , LDA , NP, and NRIA Opbcal Potentials ("Ca, 318 MeV) JOO 
2.31 Elastic Scattering with IA2 , LDA , NP, and NRIA Potenti als 
(200 MeV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 
2.32 Elastic Scattering with IA2 , LDA , NP, and NRIA Potential s 
(318 MeV) ...... ... .. . .... . . 
. 103 
2.33 CV Estimate of the Pauli Blocking Factor 
. 116 
2.34 Reparametrization of the PH Interaction ( 100 Me V) . 
. 122 
2.35 Reparametrization of th e PH Interaction (200 MeV) . 
. 123 
2.36 Reparametrization of the PH Interaction (318 MeV ) . 
. 124 
2.37 Reparametrization of the NL Interaction ( 100 Me V) . 
. 125 
2.38 Reparametrization of the NL Interaction (200 IVle V) 
. 126 
2.39 Reparametriza.tion of the NL Interacti on (318 l\ leV) 
127 
2.40 Repararn etriza.tion of the LR Interaction (21l0 l\ leV) . 
128 
2.41 Reparametrization of the LR Interaction (318 MeV) 
. 129 
2.42 Reparametrization of the LR Interac tion ( 500 Me V) 
. 130 
XVll 
3. 1 IU CF Cyclotron Floor P lan 142 
3.2 K600 Spect rometer .. 148 
3. 3 Sein tilla tor Electronics 154 
3.4 Wiring of LRS 2735b P rinted Board . 156 
3.5 V DC Logic Circuit 158 
3.6 Coincidence Circuit 159 
3. 7 Electroni cs for Integrators and BL 2 . 161 
4. 1 Kinematic Defocusing . . . . . . . . . 167 
4 .2 Kinematic Defocusing and Dispersion Matching 169 
4 .3 Kinemati c Factor for 160 and 4?Ca .. .. . 175 
4.4 Scattering Angle Offset- Cross Plot Method 177 
4.5 Angle Offset / Integrator Asymmetry (January 1987) 182 
4. 6 Angle Offset / Integrator Asymmetry (April 1987) 183 
4. 7 Beam P olarizations in Janu aryand April 1987 186 
4.8 Beam P olarizations in January and March 1988 187 
5.1 T ypical Drift Time Spectrum . . ... 194 
5.2 Drift Table- Graphical Representation 197 
5.3 Extremum Rays for n-Hit Events 199 
5.4 Efficiency and 2-Hit Events 206 
5.5 P osition and Angle for Various Event T ypes 208 
5.6 2-Hit Recovery- One-Plane Slope Metl10d 212 
5. 7 Calculated dM 1 (00110) Drift Length 213 
5.8 Diagnostics- Hot Wire . 219 
5.9 Diagnosti cs- Cross Talk 220 
XVJil 
6.1 Background due to Beam Halo ...... . ... .. . . . . . . 224 
6.2 0tgt-Spectrum for Run with Beam Halo and Sli t Scat tering . . 225 
6.3 Background due to Slit Scattering . 226 
6.4 Fitted BeO Spectrum . 235 
6.5 Fitted 4?Ca Spectrum . . 236 
6.6 Histograms of Fitted Peak Widths . . 250 
6. 7 Comparison with Previous 200 Me V Elastic Measurements . 251 
6.8 9 Be(p, p')3 / 21, 1/ 2t, and 5/21 (Ep = 100 MeV) . . 254 
6.9 9 Be(p, p')l / 21, 5/ 2t , and 3/ 2t (Ep = 100 MeV) . . 255 
6.10 9 Be(p, p')Lor (Ep = 100 Me V) .. .. . 256 
6.11 16O(p,p')0t, ot, 31 (Ep = 100 MeV) . 257 
6.12 16O(p, p' )2t , 11, 21 (Ep = 100 Me V) . 258 
6.13 16 O(p, p')l 2, 2i, 4i (Ep = 100 MeV) 259 
6.14 16 0(p, p')01, 4i , 2f (Ep = 100 Me V) 260 
6.15 16 0(p, p' )0t, 13, 22 (Ep = 100 Me V) 261 
6.16 16 0(p,p' )02, 23, 13 .08 (Ep = 100 MeV) . 262 
6.17 160(p,p' )34 (Ep = 100 MeV) . .. . . 263 
6.18 4?Ca(p,p')0t , 0t, 31 (Ep = 100 MeV) 264 
6.19 4?Ca(p, p' )2t , 51, Of (Ep = 100 Me V) 265 
6.20 4?Ca(p, p' )2i , 4t, 41 (Ep = 100 MeV ) 266 
6.21 4?Ca(p, p' )2f, 11, 6.028 (Ep = 100 MeV) 267 
6.22 4?Ca(p, p')32, 4f, 4f (Ep = 100 Me V) . 268 
6.23 4?Ca(p, p')33, 22, 6.909 (Ep = 100 MeV) 269 
6.24 4?Ca(p, p' )6 .931 , 12, 7.11 (Ep = 100 MeV) 270 
XIX 
6 .25 9 Be(p, p')3 /21, 1/2t, and 5/ 21 (Ep = 200 MeV) . . 271 
6.26 9 Be(p, p')l /21, 5/ 2t, and 3/ 2t (Ep = 200 MeV) . . 272 
6.27 9 Be(p, p')Lor (Ep = 200 MeV) .. . . . 273 
6.28 16 O(p, p')0t, 0t, 31 (Ep = 200 Me V) . 274 
6.29 16 O(.P, p')2i, 11, 21 (Ep = 200 Me V) . 275 
6.30 16 0(p, p')l2, 2t, 4t (Ep = 200 MeV) . 276 
6.31 16 O(p, p')01, 4t, 2f (Ep = 200 MeV) . 277 
6.32 16 O(p, p')0t, 13, 22 (Ep = 200 Me V) . 278 
6.33 16 O(p,p')02, 23, 13 .08 (Ep = 200 MeV) . . 279 
6.34 16 O(p, p')34, It, 4f (Ep = 200 MeV) . 280 
6.35 16 O(p,p')25 , ot, 41 (Ep = 200 MeV ) . 281 
6.36 16 0(p,p')5t (EP = 200 MeV) .... . 282 
6.37 4?Ca(p, p')0t , ot, 31 (Ep = 200 MeV) . 283 
6.38 4?Ca(p,p')2t, 51, Of (Ep = 200 MeV) . 284 
6.39 4?Ca(p,p')2t, 4i, 41 (Ep = 200 MeV) . 285 
6.40 4?Ca(p,p')2f, 11, 6.028 (EP = 200 MeV ) . 286 
6.41 4?Ca(p, p')32, 4t, 4! (Ep = 200 MeV ) . . 287 
6.42 4?Ca(p, p')33, 22, 6.909 (Ep = 200 Me V) . 288 
6.43 4?Ca(p, p')6 .931, 12, 7.11 (Ep = 200 MeV) . 289 
7.1 Comparison of EMP(1 6 O) and PH Interactions (100 MeV) . 298 
7.2 EMP Calculations for 16 0 31, 2?t , and 11 (lll0 l\,JeV) . 299 
7.3 EMP Calculations for 16 0 4;, 2j , and Oj (100 l\ leV) . 30(1 
7.4 EMP Calculations for 4?Ca. ot, 31, and 2{ (100 MeV) . . 301 
7.5 EMP Calculations for 4?Ca 51, 32, and 33 (100 MeV) . . 302 
xx 
7.6 EMP Elastic Calculations for 16 0 and 4?Ca ( 100 Me V ) . 303 
7.7 EMP Optical Potenti als (1 6 0 , 100 MeV) 304 
7.8 EMP Opti cal Potentials (4?Ca, 100 MeV) . 305 
7.9 Comparison of EMP(1 6 0) and PH Interactions (200 MeV) 308 
7.10 Comparison of EMP(1 6 0) and LR Interactions (200 MeV) . 309 
7.11 EMP Calculations for 160 31, 21", and 11 (200 MeV) . 31 0 
7.12 EMP Calculations for 160 41", 2j , and ot (200 Me V) 311 
7.13 EMP Calculations for 4?Ca. Of , 31, and 21" (200 Me V) . 31 2 
7.14 EMP Calculations for 4?Ca 51, 32, and 33 (200 Me V) . . 313 
7.15 EMP Elastic Calculations for 16 0 and 4?Ca (200 Me V) . 314 
7.16 EMP Optical Potentials (1 6 0 , 200 MeV) 31 5 
7.17 EMP Optical Potenti als (4?Ca , 200 Me V) . . 316 
7.18 Comparison of EMP(1 6 0) and LR Interactions (318 MeV) . 321 
7.19 Comparison of EMP(1 60) and PH Interactions (318 MeV) . 322 
7.20 EMP Calculations for 160 31, 2t? , and 11 (318 MeV) . 323 
7.21 EMP Calcula tions for 160 4t? , 2j , and 0j (318 Me V) . 324 
7.22 EMP Calculations for 4?Ca Of , 31, and 21" (318 MeV) . 325 
7.23 EMP Calculations for 4?Ca 51, 32, and 33 (318 MeV) . 326 
7.24 EMP Elastic Calculations for 16 0 and 4?Ca (318 MeV) . 327 
7.25 EMP Optical Potentials (1 60, 318 MeV) 328 
7.26 EMP Optical Potentials (4?Cn, 318 MeV) . '.32D 
7.27 EMP Elasti c Calculations fo r 16 0 and 4?Ca. (50 (1 MeV) . 333 
7.28 EMP,IA2 , LDA , and NRIA Opti cal Potentials (1 60 , 500 MeV) 334 
7.29 EMP,IA2 , LDA , and NRIA Optical Potentials (4?Ca. , 500 MeV )335 
XX! 
7.30 E nergy Dependence of the Interior Sensitivity- 16 0 31 337 
7.31 Energy Dependence of the Interior Sensitivity- 16 0 11 338 
7.32 Fit to 100 MeV 16 0 Elastic and Inelastic Data (I) . 345 
7.33 Fit to 100 MeV 16 0 Elastic and Inelastic Data (II) 346 
7.34 Fit to 100 MeV 16 0 Elastic and Inelastic Data (III) 347 
7.35 Fit to 200 MeV 16 0 Elastic and Inelastic Data (I) . 348 
7.36 Fit to 200 MeV 16 0 Elastic and Inelas tic Data (II) 349 
7 .37 Fit to 200 Me V 16 0 Elastic and Inelastic Data (III) 350 
7.38 A Dependence of the Effective Interaction (1 60, 100 MeV , I) 352 
7.39 A Dependence of the Effective Interaction (1 60, 100 MeV , II) 353 
7.40 A Dependence of the Effective Interaction (4?Ca, 100 Me V, I) 354 
7.41 A Dependence of the Effective Interaction (4?Ca., 100 MeV, II) 355 
7.42 A Dependence of the Effective Interaction (16 0 , 4?Ca., 100 MeV)356 
7.43 A Dependence of the Effective Interact ion (1 6 0 , 200 MeV, I) . 357 
7.44 A Dependence of the Effective Interaction (1 6 0 , 200 MeV, II) 358 
7.45 A Dependence of the Effective Interaction (4?Ca, 200 MeV , I) 359 
7.46 A Dependence of the Effective Interaction (4?Ca, 200 MeV , II) 360 
7.47 A Dependence of the Effective Interaction (1 6 0 , 4?Ca., 200 MeV)361 
7.48 A Dependence of the Effective Interaction (1 6 0 , 318 MeV, I) . 362 
7.49 A Dependence of the Effective Interaction (1 6 0 , 318 MeV, II) 363 
7.50 A Dependence of the Effective Interaction (4?C;i , 318 Me V, I ) 364 
7 .51 A Dependence of the Effective Interaction (4?C;:i , 3J 8 Mc V, I I) 365 
7.52 A Dependence of the Effective Interaction (16 0 , 4?Co , 318 MeV)366 
XXll 
7.53 Elastic Scattering with EMP, IA2, NP, and NRIA Potentials 
(200 MeV ) .... . .... . . .... .............. 371 
7.54 Elastic Scattering with EMP, IA2 , NP, and NRIA Potentials 
(318 MeV ) ... ....... .... .. .......... . .. 372 
7.55 EMP, IA2, NP, and NRIA Optical Potentials (1 6 0, 200 MeV) 373 
7.56 EMP, IA2 , NP, and NRIA Optical Potentials (4?Ca, 200 MeV ) 374 
7.57 EMP, IA2 , NP, and NRIA Optical Potentials (1 6 0 , 318 MeV) 375 
7.58 EMP, IA2 , NP, and NRIA Optical Potentials (4?Ca, 318 MeV) 376 
7.59 Comparison to Dirac Phenomenology (a-/ o-R , A 11 ) 377 
7.60 Comparison to Dirac Phenomenology (Q) . . . .. 378 
7.61 EMP( 4?Ca) , EMP(1 6 0) , and DP Optical Potentials for 4?Ca 
(100 MeV ) . . . . ......... . ... . .. . . .... ... 379 
7.62 EMP( 4?Ca), EMP(1 60) , and DP Optical Potentials for 4?Ca 
(200 Me V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 
7.63 EMP( 4?Ca), EMP(1 6 0) , and DP Optical Potentials for 4?Ca 
(318 MeV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 
7.64 Comparison of EMP(1 6 0) , PH, and LR Interact ions (Retf0 ) 385 
7.65 Comparison of EMP(1 6 0), PH , and LR Interactions (Imtf0 ) 386 
7.66 Comparison of EMP(160) , PH, and LR Interactions (Ret&'5 ) 387 
7.67 Energy Dependence of th e Empirical Intera.ct io11 388 
7.68 Difference of the PH and ], Lt-Ma.tri ces 389 
A.1 Ground State Charge Densiti es for 160 and 4?Ca . 405 
A.2 Transition Charge Densities for 160 . 406 
A.3 Transition Charge Densities for 4?Ca . 407 
XXlll 
B .1 Kinematics of an Inelastic Binary Reaction . 410 
B.2 Differential Recoil for 100 and 200 Me V Kinematics 411 
C.1 Design K600 Medium Dispersion Focal Plane . 417 
C.2 Effect of the K-Coil on the Focal Plane 418 
C.3 Effect of the K-Coil on Nuclear Lines in x f 0f  . 419 
C.4 Effect of the H-Coil on the Focal Plane 420 
C.5 Effect of the H-Coil on Nuclear Lines in x 10 f . 421 
XXJV 
1 Introduction 
Studies of the projectile-nucleon effective interaction in the nuclear me-
dium and of the internal structure of nuclei are among the dominant themes of 
intermediate energy nuclear physics. Electron scattering for a long time has 
been an extrem ely popular tool for nuclear structure studies. The electroex-
citation of discrete nuclear transitions can be described as a direct reaction 
which is driven by a local two-body interaction. We believe that both the 
reaction mechanism and the effective interaction are understood accurately. 
Together with the fact that nuclei are essentially transparent to high energy 
electrons , this permits us to extract the relevant aspects of the structure 
of the nucleus with very high precision . Since electromagnetic probes a.re 
sensitive to several one-body target densities, including charge, spin, cur-
rent, and spin-current degrees of freedoms , a wealth of important st ru ct ural 
information on nuclei can be obtained [Pet 81]. 
In spite of all these advantages, the versatility of the electron as a 
nuclear probe is limited . Electrons are largely insensitive to the bulk neu-
tron distribution and view neutrons almost exclusively through the magneti c 
m oment of a single unpaired valence neutron. Hadronic projectiles, on the 
other hand , are about equally sensitive to neutrons and to protons . They 
also sample transition densities which are not accessible to elec t ron scatter-
ing at all. Direct o+ - o- transitions are examples of a type of tra.11 sitio11 
which is forbidden in first order for electron scattering [DeF 66], which a.re 
however readily excited by proton scattering through th e longitudinal spin 
1 
response [Lov 83]. Therefore, a truly comprehensive investigation of all the 
one-body aspects of nuclear structure requires the inclusion of a complemen-
tary analysis of hadron scattering . To make hadrons a quantitative probe for 
nuclear structure , however , we must understand or at least calibrate the rele-
vant aspects of the reaction mechanism and of the projectile-nucleus effective 
interaction . 
Of the hadronic interactions, the nucleon-nucleon interaction is the one 
which has the most sound theoretical foundation . Furthermore, there is a 
roughly 300 Me V wide "window of visibility" within which the interior sen -
sitivity for nucleons becomes quite comparable to that of electrons . The 
good interior sensitivity is primarily due to the fact that the isosca.la.r pa.rt 
of the dominant matter interaction , which drives distortion effects , passes 
through a broad minimum between about 200 and 500 MeV [Car 85] . Also, 
at these energies the wavelength of the probe is short which provides for 
excellent radial resolution . Finally, in this region multistep processes are at 
a minimum which allows us to assume that direct , one-step excitations are 
the dominant reaction mechanism . The sensitivity of protons to the nuclear 
interior and the energy dependence of this sensitivity has been studied for 
the lowest lying m onopole transition in 4?Ca for energies between 60 and 
800 Me V with pseudodata [Kel 88] . The results of the pseudodata analysis 
a re shown in Figure 1. 1. Even though absorption in creases steadil y th rough-
out this energy range the interior sensitivity improves dramati cally between 
60 and about 200 MeV . The improving sensitivity is due to the decreasing 
importance of Pauli blocking in the nuclear interior . This causes the con-
2 
2 
0 
- 2 
- 4 4
....--.. 4?Ca (p ,p) o; ?Ca(p,p)o; 
r<) 
I - 6 EP = 60MeV EP = 400MeV 
E 
4- - 8 
r<) 2 
I 
0 
---- 0 
>, - 2 
-+-
. (/) 
C - 4 4 4
(1) ?Ca (p ,p )o; ?Ca(p ,p)o; 
0 - 6 EP = 1 00MeV EP = 51 SMeV 
C - 8 
0 2 
-+-
(/) 0 
C 
0 
~ -2 
- 4 4 4C ?Ca(p ,p)o; ?Ca (p ,p) o; 
0 
L - 6 EP = 150MeV EP = 650MeV -+-
::) 
(1) - 8 
z 2 
0 
- 2 
- 4 4 4?Ca(p,p)o; ?Ca (p ,p) o; 
- 6 EP 200MeV EP = 800MeV 
- 8 
0 2 4 6 8 0 2 4 6 8 
r (f m) r (f m) 
Figure 1.1: Energy dependence of the interior sensitivity of nucleon inelas-
tic scattering; the window of maximum transparency is between about 200 
and 500 MeV. The figure shows the results of a pseudodat a analysis fo r the 
ot state of 4?Ca [Kel 88]. 
3 
tribution of the interior to increase relative to that of t he surface, thereby 
improving the interior sensitivity. T he sensitivity is op timal between about 
200 and 500 Me V since in this energy region declining density dependence 
and increasing absorption balance. At higher energies absorp tion becomes 
dominant and because of the decreasing penetrability, the interior sensitvity 
deteriorates. 
Direct , one-step reactions can be conveniently treated within the single-
scattering framework by means of a local "tp" folding model, where the scat -
t ering amplitude is constructed from the effe ctive nucleon-nucleon interaction 
weighted by the nucleon distribution inside the target nucleus . Within this 
m odel, knowledge of either the interaction fact or t or the stucture factor p 
permits the systematic evaluation of the unknown fact or. In practice, the 
comple te amplitude for a certain transition contains not only an implied sum 
over charge indices, but may additionally involve a whole variety of differ-
ent densities and interaction components. The first problem can be ignored 
for the case of self-conjugate nuclei where transitions are purely isoscalar or 
isovector in character. Furthermore, for a few special classes of transitions the 
number of relevant tp terms can effectively be reduced . The mos t import ant 
class of transitions is comprised of natural parity transitions (61r = (- )t.J 
with 61r = 1ri1rJ) to low-lying collective states in the t arget nucleus. Not 
only is the matter density for this type of transition much larger th an eith er 
spin , current , or spin-current densiti es, in addition the matter interaction 
also dominates over the other interaction components [Car 85]. In proton 
scattering , the excitation of natural parity states is driven exclusively by 
4 
t he spin-indep endent central and spin-orbit components of the interact ion. 
For n a tural parity t ransitions, therefore , we will be able to study these two 
components of the nucleon-nucleus interaction, since both the reac tion mech-
anism is reason ably well known and all the relevant nuclear structure aspects 
can be reliably determined by independent means such as electron scattering. 
The self-conjugate nuclei 160 and 4?Ca are the ideal laboratory for effec-
tive interaction studies. Both nuclei have doubly closed shells (see Figure 1.2 ) 
and their spectra contain many strong well separated low-lying collecti ve 
sta tes fo r which also comprehensive elect ron scattering data are available 
[But 86 , Mis83 , Har 84] . Some of these states are surface excitations, such 
as the 31 states , and are sensiti ve to t he low-density apects of the effecti ve 
interaction . Other states, such as the inelasti c monopole transitions or the 
11 st at e of 16 0 , peak in the interior and thus probe the high-density pro per-
ti es of the effective interaction . Intermediate cases like the 32 state of 4?Ca, 
which has strong lobes of the transition density both in the interior and at 
the surface, will provide interesting challenges for any effective interaction . 
Finally, we find in these t argets states which give insights into different reac-
tion mechanisms , such as the 2f state of 160 which is a good candidate for 
a two-step excitation . For some states, such as the Of state in 16 0 , densiti es 
other the matter density might become import ant fo r cert ain energies and 
m omentum transfers [P et 85] . 
The quality of t heoretical interact ions and their usefulness in quanti -
t a tive studies of nuclear structure can be easily tes ted on the isosca.l a. r nat -
ural parity states of these targets . Unsuccessful nonrelati visti c impulse ap-
5 
....., ....., 
,...... 
N
0 .0...0..  ....
 .. 
.N... .. 
,-... ,-... ,-... 
N
,-... ,-... ,-... .N._ , .'.d_",  ._
 , 
.'<
N 
._:j",  .._, ."._?,  
~ 
JJ p.tj,, p.~ ..
t.i...)  
.J, ..... ..... ..... N.,... 
a; 
0 ::c: ~ 
u 
3 
? 
p?? gure 1.2: Shell structure of the doubly magi.c  self-conjugate nuclei "O 
and Ca for protons (or neutrons)? On the Je[t are the harmonic oscillato r 
levels40, to the right the actual )evels including the spin-orbit splitting. The 
'.'umbers in parentheses are the occupation numbers for the level, the numbers 
m brackets the sum of all particles up to and including the level 
6 
proximation calculations, especially for interior-peaked states ( e.g. [Kel 82]), 
suggested the necessity of strong medium modifications to the effective in-
teraction. The strongest density dependence is exhibited by the central spin-
independent isoscalar component. Pauli blocking manifests itself as damp-
ing of the imaginary part , while short-range correlations modify the real 
part of the central component by adding a short-range repulsive contribution 
which is proportional to density. All of the available theoretical models pro-
duce medium modifications which are qualitatively similar [Ger 79, Ger 83, 
Rik 84 , Nak 88, Ray 90]. However , their predictions of scattering data differ 
significantly [Kel 89a] . Unfortunately, however, the various models do not 
only use different nucleon-nucleon potentials but also use different numerical 
procedures and approximations to calculate the effective interactions . Ad-
ditionally, all the interactions are given in the form of enormous tables of 
amplitudes or tables of Yukawa expansion coefficients in different rep resen-
tations which do not allow much insight into the physics of the respective 
model. A comparison of one model to others based on these interaction tables 
is therefore extremely difficult, if not altogether impossible. 
The interactions presently available have been found not to be suffi -
ciently accurate for detailed nuclear structure applications [Kel 86c], raising 
th e ques tion whether this problem is due to defici encies of th e interaction or 
due to the local density prescription [J eu 74, J eu 76] whi ch is used to app ly 
th e effec tive interaction to nucleon-nucleus scattering. The prescripti on as-
sumes that the knowledge of only the local density of th e medium in which 
the particular reaction takes place is sufficient to evaluate project il e-nucleus 
7 
scattering. Within this framework it should not matter whether the inter-
act ion takes place in the interior or at the surface of the nucleus ; it should 
also not matter whether the reaction occurs in a small or a large nucleus . 
Although physically plausible, this prescription has never been established 
as an actual approximation to a m ore accurate theory whose corrections can, 
even in principle, be evaluated . Presently, the local density prescription 1s 
more of a convenient ansatz than an act ual approximation . 
In an alternate approach, we use a physically motivated phenomenology 
which can serve both as a means to compare theoretical models to each oth er, 
as well as a means to describe data [Kel 85a]. The effective interaction which 
emerges from such a. fit to data can then be considered a measured interaction 
for a finite nuclear system. A comparison to parameters which were obtained 
from a fit to theory will help assess differences and /or deficiencies of the 
theoretical model. Finally, we will have a means to assess the validity of the 
local density hypothesis in a consistent and more or less model independent 
fashion by systematically studying the dependence of the interaction on the 
local density, the target, and the energy. These studies, therefore, are the 
major goal of the present work , in which we will systematically address all 
the major premises of the local density prescription . 
* * * 
In Chapter 2 an attempt is made to rev iew more or less ca.rcfu ll y and 
completely all the theoretical background which is relevant fo r thi s work , 
including the relativistic impulse approximation and th e more recent full -
folding calculations . We present in this chapter various IA and LDA calcula-
8 
tions, as well as IA2 calculations, and compare them with our new 160 and 
4oc 
a data at 100, 200 and 318 Me V. Also, in this diapter we introduce the 
tools wl1ich will be used for our phenomenological analysis . 
Chapter 3 deals primarily witl1 the experimental setup, including the 
accelerator, t11e beam line optics, t11e apertures and targets used, tlie spec-
trometer and focal plane detectors, and t11e electronics and data acquisition . 
Chapter 4 explains the necessary calibrations for our experiment . Some of 
these calibrations were done on-line during tl1e experiment, sucl1 as vertical 
scans and the matching of the spatial and angular dispersion; otl1ers were 
done during data analysis based on measurements taken during the exper-
iment . This includes the determination of t11e beam polarization, the true 
scattering angle, and various tests addressing tl1e stability of the cross sec-
tions with respect to parameters like acceptance and event rate . Chapter 5 
is dedicated to our drift cliamber detectors . Here we discuss the observed 
event types and the methods used to accurately determine particle trajecto-
ries. We also will identify all the global corrections to t11e cross section which 
are required due to tlie efficiency of the chambers, as well as to certain back-
ground modified events which are not processed in the analyzer. Finally, we 
will mention some of the diagnostics that are available for finding problems 
1n the chamber performance. 
Ch apter 6 deals w1? t 1 tlie data reduction. This includes a discussion 1 
of the data replay and the cuts used , as well as a. brief discussion of th e 
line-shape fitting part of the analysis . Also covered in this chapter are the 
computation of the unpolarized cross section, the analyzing power, and their 
g 
errors, as well as the kinematic formulae which were used to express our 
results m c.m. quantities . Furthermore, the vanous normalizations that 
have to be applied to the data are explained . At the end of this chapter 
we present and discuss our experimental results. In Chapter 7 we show and 
discuss the res ults of our effective interaction analysis . We will compare the 
ability of our empirical interaction to accurately predict scattering data with 
that of the available theories, including some relativistic models . To test the 
:1 
validity of the local density prescription, we will study the dependence of the 
effective interaction on both the mass number A and the energy E. Finally, 
in Chapter 8 we will give a brief summary of our conclusions. 
10 
2 Theoretical Background 
2.1 The Transition Matrix as an Effective Interaction 
To motivate the concept of an effective interaction let us consider an 
example from elementary scattering theory. The Schrodinger equation for 
the scattering of a particle of mass ? and energy E by a potential v( r) can 
be written 
(E - K)x(r) = v(r)x(r), (2.1) 
where I{ = - '7 2 /2 ? is the kinetic energy operator1 . A solution of Eq. (2.1) 
can be obtained succinctly by dividing both sides by ( E - K) 
1 
X = (2 .2) E - K vx. 
Proper treatment of the singularity by insertion of an infinitesimal +it: in 
the denominator and addition of any solution of the homogeneous eq uation 
(E - K)</>(r) = 0 (2.3) 
to satisfy the required boundary conditions, incoming plane waves and out-
going spherical waves, yields the formal solution of the Schrodinger equation 
(2.4) 
with E+ = E + it:. This is known as Lippmann-Schwinger equation , wh ere 
Gt(E) is the free-particle Green functi on with outgoing boundary conditions. 
1 Alternatively, this problem can also be thought of as the scatter in g of two nucl eons 
where? is the reduced mass and v is the nucleon-nucleon (NN) potential. 
11 
The transition amplitude for elastic scattering from an incident momentum 
k to a final momentum k' is 
T(k',k) = (</>(k')/v/x(+) (k) ). (2 .5) 
If one assumes that the effect of the scattering potential on the incident 
wave is small, one can replace in the matrix element x(+) by?> and write the 
transition amplitude in the first Born approximation as 
.. 
,I 
T(k', k);:::: (</>(k')/v/<f>(k)). (2.6) 
It is now useful to introduce the transition matrix t by 
(2.7) 
Eq. (2.4) then gives a Lippmann-Schwinger equation fort 
t = v + vGiit (2.8) 
and the transition amplitude, Eq. (2.5), can be written as the plane wave 
matrix element 
T( k', k) = (?>(k')lt/?>(k )) . (2 .9 ) 
Thus, the significance of the transition matrix here is that it acts as an 
effective interaction for which the Born approximation is exact , provided an 
appropriate t can be found [Sat 83, Kel 85]. This feature will be appli ed later , 
when we treat the Born approximation as an exact model for sea.tiering to 
extract an effective interaction . 
12 
2.2 Elastic Nucleon-Nucleus Scattering- The Opti-
cal Model 
Since its introduction by Feshbach, P orter and Weisskopf [Fes 54] in 
the 1950s, the nuclear optical model has been a simple and yet powerful 
tool fo r describing the complex problem of elastic nucleon-nucleus scatter-
mg . The name "optical model" comes from the many similarities between 
nuclear elastic scattering and optical refraction and absorption of light . T he 
wave describing the incident particle experiences within the range of the nu-
clear potential refr action , very much like a light wave entering a medium with 
different index of refraction 2 . If the potential is not constant but has, for in-
st ance, a Woods-Saxon shape, the sca ttering process is comparable to a light 
wave incident on a crystal ball with a radially variable index of refracti on. 
The occurence of nuclear reactions due to interactions of the projectile with 
the target nucleons leads to absorption from the elastic channel. Here the 
optical analog is the absorption of a light wave in a "clouded" crystal ball 
[Fes 54] . Absorption increases with increasing incident energy since the prob-
ability for interaction with the nucleons in the target becomes larger ; in th e 
presence of very strong absorption one, therefore, observes cross section an-
gular distributions that are similar to those for diffraction of electromagneti c 
waves by an absorptive "black disk" . 
To account for both its refracti ve and absorptive feature th e opti cal 
2 For exa mple, the index of refraction n for a spuare-well potenti al of depth - Vi i a nd 
a n incident particle with energy E is simply n = J 1 + Vn/ E. 
13 
potential has to be complex 
U(r) = V(r) + iW(r), (2.10) 
where the imaginary part W of the potential must be negative to allow 
for particle absorption . Furthermore, the optical potential is composed of 
a central part (Uc ) and a t erm (UL5 ) which accounts for the spin-orbit 
coupling3 between target and projectile . We thus have the form 
:t 
I 
~I 
(2.11) ,, 
,, 
,', , 
where uz ,, , describes Coulomb scattering . 
r 
Macroscopically, the real central part of V ( r) usually is assumed to ,. 
I'" 
h ave the Woods-Saxon shape V( r) = - V0 f( r ), where the radial fun ction 
f( r) = [1 + e( r - R, 12 )/aJ- 1 reflects the density distribution of the nucleons4 . 
For the spin-orbit part one assumes tha t a particle in the inner , flat part of 
the p otential moves in a homogeneous medium without a center with respect 
to which an angular momentum could be defined . Therefore, only the nuclear 
surface should contribute to the real part of the spin-orbit interaction . T hi s 
radial dependence is phenomenologically oft en described by a Thomas-form, 
h(r) ~ r - 1 fJf(r) / fJr . It is less straightforward to define a phenomenological 
form for the imaginary part W of the optical potential. For energies below 
3 It is found experimentall y that unpolarized par ticles a re polcir ized by t he scatter in g 
potential; for polarized particles , one find s a left-right asymmet ry in the angu la r d istribu-
t ion of the cross section. 
4 For r = R 1; 2 , the density decreases by one ha lf; within the surface thickness l , related 
to the surface diffuseness a by t = 4.4a, the density decreases from 90% to 10%. 
14 
about 20 Me V a potential which features predominantly surface absorption 
has been found adequate , while for higher energies ( above 80 Me V) a poten-
tial with volume absorption appears to be more appropriate . Cross section 
angular distributions, however, have been shown to be not very sensitive to 
the exact details of the shape of the imaginary potential [May 84] . 
An important extension to the optical model are coupled channels cal-
culations, where certain reaction channels are treated explicitly, rather than 
being represented by simple absorption. The total wave functi on is expanded 
with respect to wave functions of the elastic and the prominent inelastic 
channels and only the remaining reaction channels are treated by simple ab-
sorption . The resulting system of coupled differential equations can be solved 
for certain simple cases, e.g . for collective excitations. 
On a microscopic level, construction of the optical potential is a more 
difficult problem due to the complexities of the interplay between target 
ground-state properties, the effective nucleon-nucleon interaction , and me-
dium modifications . The traditional nonrelativistic approaches to the op-
tical potential in nucleon-nucleus scattering are the multiple scattering for -
malisms of Watson [Wat 53], and of Kerman, McManus and Thaler (KMT) 
[Ker 59] and the nuclear matter self-energy formalism developed by Hiifner 
and Mahaux [Hiif 72] . More recently, a relativistic approach was initiated by 
McNeil, Shepard and Wallace [McN 83]. In the next sections th ese different 
approaches will be reviewed. 
15 
2.3 Multiple Scattering Approach 
2 .3.1 Introduction 
In the multiple scattering approach the optical potenti al is t he sum of 
interactions in which the incident projectile excites the t arget nucleus out of 
the ground state and does not de-excite it back until the la.st step. T he first 
order t erm of the potential , the so-called single-scattering term , is a coherent 
sum of terms representing the scattering of the incident projectile by each i11 -
divid ual t arget nucleon . The second order term , called the double-scatterin g 
term , involves a sum in which the incident projectile scatters sequenti ally 
from any pair of nucleons . Nucleon-nucleon correlati ons in the nuclear wave r 
' 
functi on are, therefore , represented by this t erm. At intermediate energies 
the single-scattering term of the optical potential dominates: the excita tion 
of a single nucleon in a two- body collision with the incident projectile is the 
m ost likely reaction process due to the short range of the interaction and the 
high velocity (and associated short transit time through the target nucleus) 
of the projectile [Wal 87] . 
The Watson multiple scattering series [Wat 53] for the optical potenti al 
operator appropriate to the elastic scattering of a projectile (0) from a bound 
sta te of A particles is 
A A 
u = L 'Di + L 'OiQ c +, oj (2.12) 
i = l j -j: i 
Here Q is th~ (A + 1)-particle projection operator onto the complete, anti -
16 
symmetric space of the t arget nucleons with exception of the ground state5 
and Q+ is the propagator with outgoing boundary conditions. r{QQ is the 
asymptotic Hamiltonian projected off the ground state and contains the sum 
of the projectile kinetic energy operator I<0 and the target Hamiltonian HA 
(2 .13) 
The scattering operator r0; is a solution of 
:i ~I I, 
~ I 'I 
(2 .14) 'I I 
:,  I 
I 
,,, ,,  
,,, 
where v 0 ; is the bare internucleon potential between the incident projectile 
and the ith target nucleon . 
r 
In the single-scattering approximation , the first t erm of the scatter-
mg series, Eq . (2.12), yields the momentum-space optical potenti al matrix 
element ' 
l'/ 
A /1 
U( k' , k) ~ L (k' ; <I>olro;lk; <I>o )A- (2 .15) i 
i = l 
Ii 
The scattering operator r can be related to the t -matrix , the solution of the ,1 
I 
scattering of two nucleons (see Section 2.1), by 
(2.16) 
where 
(2 . l. 7) 
5 Related operators are P which proj ects onto the ground state <P11 1 and A = P + Q 
which projects onto the complete , antisymmetric space includin g th e gro unrl state. 
17 
j 
The essence of the impulse approximation (NRIA) is now to approximate t 
in Eq. (2.16) by the free two-nucleon t -matrix and to calculate the op tical 
potential from Eq. (2 .15) , without any further approximations6 . 
There is an asymmetry between the propagation of the incident pro-
jectile and the bound target nucleon which is characteristic of the multiple 
scattering approach: the projectile propagates freely in intermediate states, 
while the target nucleon moves in the nuclear mean fi eld . In the nuclear mat-
ter many-body approach (se~ below ) the symmetry is res tored by including 
the self-energy of the incoming projectile. One of the shortcomings of the 
multiple scattering approach is the potentially very slow convergence rate of 
the scattering series at low energies. 
2.3.2 The Full Folding Potential and its Approximations 
For simplicity, in the following expressions we will neglect the fact that 
the optical potential is actually an operator in the spin and isospin spaces of 
the projectile and adopt a somewhat schematic, not always rigorous , nota-
tion . We will also omit the boost from the NN c.m . to the NA c.m. fr ame 
( e.g . via the M!ISller factor [M!ISl 45]) . 
If the target ground state is taken to be a Slater determinant of single-
particle states {y,0 } , the optical potental, Eq. (2 .15), can be compactly writ -
G T he KMT optical potentia l differs by a scale factor of AA! from the Watson opticil l 
potential. The difference a rises from a different organiza.t ion of the multipl e scat terin g 
series a nd becom es relevant only for higher order terms . In the sing le-scallering 11pproxi-
mati on, we do not make a mistake if we simpl y omit the KMT factor. 
18 
t en as ( we fo llow the not ation of [Are 90a]) 
U(k' , k ; E ) = Jd P L <p~ (p') t(q , Q ; Za)<pa (P), (2.18) 
<n < <p 
where the initial and final momenta of the incident projectile (bound nucleon) 
are denoted by k and k' (p and p') and the init ial and fin al relative momenta 
of the interacting pair by K- = ?( k - p) and ,._,, = ?(k' - p') , respectively. 
The m omentum transfer is defined as q = k - k' or 1K- - K- , and the exchange 
m omentum transfer as Q = K - P or"- + "-' ? The vectors K = ?(k + k') and 
P = ?(p+ p') are averages of the initial and fin al projectile and target nucleon 
m om enta. The total energy available in the NN center-of-mass sys tem is 
(P + K) 2 
Za = E + Ea - M , (2.19 ) I" 
2
where M is the total mass of the interacting pair and Ea the energy of the 
single-particle states. 
The reduced one- body t -matrix 
(2.20 ) 
is an antisymmetrized matrix element between states of initial ( K-) and fi -
nal ( K. 1 ) relative momenta of the interacting pair . This t -matrix satisfies a 
Lippmann-Schwinger equation of the form 
i( z) = v + vGt(z )t(z). (2 .21) 
In calculating the optical potential from Eq. (2. 18) , the required anti symmet-
ric structure is conserved by limiting the t-matrix to NN states which are 
19 
allowed by the Pauli principle. The kn ock-on exchange term for NN scatter-
ing is accounted for naturally with this prescription and it is not necessary 
to separate the interaction into direct and exchange components. At high er 
projectile energies the variation of the single-particle energy becomes rela-
tively unimportant and above about 100 MeV it is possible to replace E0 by 
its weighted average (E) in z. With the definition 
p(q , P) = L cp~(p')cpa(P), (2.22) 
Q 
for the mixed target density, the optical potential can then be written in the 
so-called full-folding form 
uJull(q, Q) = I dP t(q, Q; z)p(q , P), (2.23) 
a convenient starting point for several further approximations . Nonlocality 
enters into the full -folding optical potential through the interplay between off-
shell degrees of freedom of the interaction and the nonlocality of the ground-
state mixed density. In the subsequent approximations more or less off-shell 
and nonlocality information in the ingredients of the optical potential will be 
sacrificed for expediency. 
If one assumes that the nucleon-nucleon amplitude varies only slowly as 
a function of its arguments compared to the nuclear density ( or equivalently, 
that the range of the nuclear force is small compared to the nuclear size ) , one 
can expand i in a Taylor seri es in P about a fixed value Pa - The reference 
m omentum Pa is determined from the requirement that the contribution of 
th e first derivative term be minimized . In th e optimum factorization approx-
20 
imation for elastic scattering , P 0 is chosen to be zero7 . After integration 
over the mixed density matrix one obtains the optimum fact orized optical 
potential Ufac in t erms of the diagonal one-body density 
uJac(q ,Q) = t(q,Q; z)p(q) . (2 .24) 
Since q and Kand z are completely arbitrary and independent of each other, 
this is also known as "off-shell tp" approximation [Pie 84 , Els 90]. Although 
the off-shell t-matrix enters ( albeit over a restrict ed range of momentum), 
information about the nonlocal structure of the mixed density is los t in this 
approximation [Are 90b] . 
If for any Q the two- body t-matrix t(q, Q ; z) in Eq. (2.24) is equal to 
its on-shell value ?0 n( q; z ) , one has the "local tp" approximation . Here the 
exchange momentum transfer Q is completely determined by the argument 
q at a given energy z , via q ? Q = 0 and q 2 + Q2 = 4K2 = 4K12 = 4mz . The 
potential is independent of the variable Q and, therefore, local. Explicitly 
we have 
u 0 n(q) = i(q ; z)p(q) . (2 .25) 
This latter form has been rather popular due to its computational simplic-
ity and due to the fa.ct that a local NRIA optical potential is conveniently 
constructed from an on-shell NN t -matrix which can be derived from ex-
perimental scattering amplitudes and the nuclear point density, also directly 
7 For elastic scat tering this corresponds to Breit fram e kin ematics as defined in 
[M cN 83]; the reason that this factorization may be opt imal li es in th e time-reversa l 
property of the nuclear groud state [Pie 84] . 
21 
accessible experimentally8 . However, in a recent paper Redish and Stricker-
Bauer [Red 87] pointed out the possible importance of retaining the nonlo-
cality in the optical potential. In a later section we will mention some of the 
studies where the full off-shell characteristics of the NN interaction and the 
effects of full folding were explicitly treated. 
2.3.3 Expressions for the Mixed Target Density 
The mixed target density in momentum-space p(q , P) , Eq. (2.22) , can 
be Fouri er transformed to its coordinate-space representation , and it is found 
that p( r , r') can be adequately described by the Slater expansion 
p(r, r') = p(R)C(k, s), (2 .26) 
where the separation s between a projectile at r and a bound nucleon at r' 
is defined as r - r', and their midpoint R as ?( r + r') . The nuclear point 
density p(R) is given by the diagonal terms of the general mixed density, 
while the correlation function C measures the degree of nonlocali ty of the 
mixed density [Are 90b] . 
A simple choice for the correlation function is the Slater form . For 
elastic scattering it is obtained from a Fermi gas model [Sla 51], while for 
inelastic scattering an approximate form appropriate for excit ations near the 
8 Ground state charge densities can be obtained experimenta ll y fr om elas ti c scatter ing, 
transition charge densities from inelastic scattering. The nuclear point densities for fo lding 
calculations can be obtained from the charge densities by unfolding the nucleon form factor 
(see Appendix A). 
22 
Fermi surface is used [Lov 78] . We get the functions 
C(k' ,s) = { 3b~ 
elastic 
 (2 .27) 
io( ks) inelastic 
where k is a functi on of R and 
' 3~2 
k(R) = kp(R) = - p(R). (2.28) 
2 
Refinements to this model account for the nonuniformity of the density in a 
finit e system . Curvature corrections which consider the slope of the density 
,, I 
in the tail region have been studied by Negele and Vautherin [Neg 72] and ;,:, I 
,, ' 
by Campi and Bouyssy [Cam 78] . While Negele and Vautherin performed a (,,  
Taylor expansion of p( r, r') around the Slater expansion , Campi and Bouyssy ( 
suggested a form which maintains the functional form of the Slater expansion , 
but with 
5 1 
k(R) = --{T(R) - - v' 2p(R)} (2 .29) 
3p(R) 4 ' 
where 
(2 .30) 
The mixed density constructed with one of these correlation functi ons can be 
transformed into momentum-space and used in the full -folding integral given 
above [Are 90b], and has also been used in the local density approximation 
for the exchange part of the optical potential (see Sections 2.4 .3 and 2.4.4) 
[Bri77 / 8] . 
23 
2 .3.4 A Local Pseudopotential 
Before we di scuss general off-shell t-matrices from realistic NN poten-
tials or from inverse scattering , we first want to exami ne t he local t-matrix by 
Love and Franey (F L) which h as been extensively used in NRIA calcu lations 
of nucleon-nucleus scattering at energies up to 800 MeV [Lov 81 , Fra 85]. 
T his interaction has some appealing feat ures . Its on-shell values are reali st ic 
since they are fitt ed to NN dat a9 , and its off-shell extension is well defin ed 
with it s ass umed fun ctional fo rm . 
The Love and Franey (FL) t- matrix is constructed by fittin g the on-shell 
NN amplitude with the form 
(2.3 1) 
whi ch enforces exchange symm etry off the energy shell by having the proper 
symmetry in each two-body state (Q 0 n denotes the on-shell value of Q); the 
resulting functi on is then used for all values of q and Q. Since both the direct 
and the exchange term in Eq. (2 .31) are intrinsically local fun ctions, in thi s 
m odel off-shell and nonlocal effects only enter into t NN through the interplay 
of the two terms [Pi e 84 , Red 87]. A local pseudopotential with central , 
spin-orbit and tensor components can be represented as [Lov 81, Kel 89a] 
V = I.: vscr (1? )PsPr + L Vf: 5 (r)L ? S Pr + L V,f(r)S12( r )A?, (2 .32) 
S T T T 
!J In the NN c.m. system , the on- shell t - rn a.tr ix is related to t he NN a.mpli t ude viil 
lNN = 77/ltfNN, where 77 = - 41r(hc)2 / ENN ? W it h t he nucl eon ma.ss m and t he c.m . wave 
number k of either pa rticl e, E 1N = m 2 c~ + (hck)2 [Lov 81]. 
24 
where Ps and Pr are spin and isospin projection operators ( S and T are the 
t otal spin and isospin of the nucleon-nucleon system) 
P S =O = 1 (1 - lT1 ? lT2 ) 4 , pS= l = ?(3 + lT1 ? lT2) 
(2 .33) 
P T =O = 41  (1 - 7"1 . 7"2 ) , pT= l = ?(3 + 7"1 ? T2), 
and where 
L = ?( r 1 - r 2 ) x (p1 - pi) 
5 = ?(u1 + lT2) (2 .34) 
The antisymmetrized matrix elements of the pseudopotential are required to 
reproduce the on-shell t -matrix 
iNN(k', k) = (k'IV(l - X)lk ), (2 .35) 
where the exchange operator X = x rX 5 xr is the product of the space-
exchange operator 
(2 .36) 
the spin-exchange operator 
(2 .37) 
and the isospin-exchange operator 
(2 .38) 
For computational convenience, and to simulate to some ext ent the exchange 
of various mesons , the radial parts of the central and spi11 -orbit components 
of V are expressed as sums of Yukawas ; the tensor term includes an additional 
25 
factor r2 . With the Yukawa functi on Y;( r) = e- ?;r / r we have 
vc ( r) = Li lf;c Y; ( r) 
vLs(r) = Li 11;Ls}';(r) (2 .39) 
VT(r) = Li v/r2 }i (r) , 
where the ? are complex strengths. Several masses ?i were chosen to repre-
sent in some fashion the meson carriers of the interaction ( the smallest mass 
is that of the pion, responsible for the long-range part of the interaction) . 
Inserting \/ from Eq. (2.32) into Eq. (2 .35) , we can express iNN in terms 
of the momentum transfer q = k' - k, the exchange momentum transfer 
Q = k I + k' and n = q X Q' as 
The quantities i~T and i!;,5 are given by 
i~T = VscT(q) - (- )5+TvfT(Q), 
(2.41) 
[!;,S = ?qQ[VfS(q) + (- )l+TfifS(Q)] . 
It is also convenient to introduce the auxiliary quantity i-LS , such that 
(2.42) 
Finally, the Bessel transforms 1/ C, 1  LS and ,:-r are defin ed as 
ll c(q) = 471" J dr r2 j 0 (qr)Vc(r) 
1/ LS(q) = 471" Jdrr4]1~~r)vLs(r) (2.43) 
lf T(q) = 471" J drr 2j 2(qr)VT(r) , 
26 
with analogous expressions for the transforms that are functions of Q. 
To apply the above results to nucleon-nucleus scattering, one evaluates 
the knock-on exchange contributions to the NA amplitudes in a short -range 
approximation( e .g. [Pet 69]), which is well established and thought to be 
accurate for most natural-parity transitions at intermediate energies [Kel 89] . 
In its simplest form, the zero-range approximation (ZREA) , the exchange 
part of the interaction is reduced to a zero-range form by replacing Q in the 
exchange terms by kA, the projectile wave number in the nucleon-nucleus 
system10 . That is, to a good approximation the momentum transfer required 
for the exchange term in the nucleon-nucleus system is just that associated 
with stopping the incident nucleon [Lov 81]. Additionally, a transformation 
of the interaction from the NN to the NA system is necessary via the above 
m entioned M0ller factor [M0l 45]. More specifically, 
(2.44) 
where Ep (Et) is the total energy of the incident (target) nucleon in the NA 
system and Eo the total energy of the projectile in the NN system [Lov 81]. 
For the isoscalar natural-parity inelastic transitions (6.1r = (- )~J with 
Ll1r = 1ri1r 1 ) which we consider in this work , the effective interaction for a 
l(l This is based on a Born approximation argument for NN scattering, treated using 
NA kinematics where the motion of the struck nucleon is negligible and Q is alm ost a ngle 
independent. Thus, the momentum transfer for exchange scattering is sharpl y peaked at 
a value approximately equal to kA , justifying a n expansion about that point and retainin g 
only its first term [Pet 69, Ke! 81]. 
27 
projectile with spin u reduces to 
(2 .45) 
provided that all target spin and current densities may be neglected [Kel 89a] . 
Because u(q) ex IT1;(q)j2, in the plane wave limit the differential cross section 
for inelastic scattering of an unpolarized incident beam is 
(2.46) ,, , 
,,I 
where?(?') and k (k') are the reduced energies and projectile wave numbers ?,,:1,  
I 
in the initial (final) channel, and J is the target spin11 . The plane wave 
analyzing power is 
I" 
(2 .4 7) 
2.3.5 Off-Shell T-Matrices 
It has been criticized [Lov 81 , Red 87] that the form Eq. (2.31) involves 
a certain ambiguity in that [D( q) may differ for two fits of equivalent quality, 
with [X ( Q) compensating for the difference, to yield essentially th e same 
on-shell NN amplitude. In the NA sys tem , however , where [D(q) stays the 
same for a given q, lX(kA) obtained from different on-shell fits may well 
lead to different values for iNA ? The uncertainty in thi s off-shell extension 
prescription is a serious problem for reali st ic NA scat tering calculations sin ce 
an on-shell NN t -matrix exists only over a range of mome11tum transfer which 
11 The factor CJ"ou = lt,,,(q)/ 2 = /t c(q) /2 + /tL 5 (q)/ 2 in Eq. (2.46) is call ed matter inter-
action. 
28 
is signifi cantly less than the one accessible in NA scat tering. With t he NN 
energy fixed at the value determined from the beam energy, the on-shell NN 
m omentum k is approximat ely half of the on-shell NA momentum kA for a 
heavy t arget . Therefore, beyond a nucleon-nucleus scattering angle of abou t 
60? , which corresponds to a NN scattering angle of 180? , the interaction t 
must be extrapolated off the energy shell. 
Because of the aforementioned problems and their import ance for both 
full -folding and factorized c1:1,lculations, there has been renewed interest in 
fully off-shell t-matrices and their properties . Starting with realisti c poten-
ti als from meson exchange theory like the Pari s [Lac 80] or Bonn [Mac 87] 
pot entials, or , m ore recently, from invers e scattering [Kir 89], fully off-shell t-
m atrices have been constructed , with various methods for the solution of th e 
Lippmann-Schwinger equation , and their properties have been investigated 
in det ail [Red 87 , Amo 88 , Ger 90] . 
Motivated by the paper of Redish and Stricker-Bauer [Red 87], one of 
the m ost important questions then posed was, to what degree the off-shell 
behaviour of the resulting t -matrices would depend on the different input 
NN potentials. All of the potentials listed above reproduce the on-shell NN 
observables, usually in the form of phase shifts , with comparable quality. 
However , while the P aris and Bonn potenti als include explicit momentum 
and energy dependencies , potenti als from inverse scatterin g are constrn cted 
to be local and energy independent in all channels. Because of the way th ey 
were generated , the inversion potentials yield exact on-shell NN t -matri ces. 
On the other hand , one might think th at the st rong energy and momentum 
29 
dependencies of the meson exchange theory would produce unique off-shell 
behaviour which a local and energy independent inversion potenti al could 
not rep roduce [Ger 90] . 
Amos et .al. [Arno 88] compared off-shell interactions based on the Reid 
[Rei 68] and Paris potentials with a new phenomenological potenti al, while 
von Geramb and Amos [Ger 90] compared half-off-shell interactions based 
on the Paris , the Bonn, and the inversion potential from experimental phase 
shifts [A rn 87] . Both groups came to the same conclusion : off-shell inter-
actions are very similar over any physically significant region of momenta 
whenever they give essentially the same phase shifts at all energi es, i .e . if 
they a re phase equivalent . Any differences off the energy shell can be t raced 
to differences on the energy shell. This result allows , therefore, comparison of 
full -folding calculations utilizing different potentials , provided th e same ap-
p roximations ( e.g . treatment of the energy in the interaction matrix element 
and the mixed particle density) are being used . 
2.3.6 NRIA Calculation Results 
In Figures 2.1- 2.3 we compare the PH and NL interactions ( see Sec-
tion 2.4 .2) at zero density with the FL t-matrix at 100, 200 , and 318 MeV. 
To illustrate some of the effects which the various free t-matri ces will have on 
NRIA predictions for natural-parity transitions, we also di splay the matter 
interactions u 00 ( or ltml 2 ) and analyzing powers Ay in the plane wave ap-
proximation (see Section 2.7.1) . As can be seen , the differences between the 
interactions become progressively bigger with increasing energy. 
30 
At 100 Me V all the Retf0 components are very nearly equal for q < 
2 fm - 1 , but then separate at larger momentum transfer . The lmtf and Ret?-5 0 
components of the PH and NL interactions are essentially equal over the 
whole momentum transfer range of 3 fm - I, but the FL t-matrix starts to differ 
already at small momentum transfers for the lmtf0 component and it begins 
to differ for the Ret{;5 component for q > 1.5 fm - 1 . The Imt?-5 component 
of the PH interaction resembles very much that of the NL interaction at 
momentum transfers below 0.5 fm - 1 . Above about 1.5 fm - 1 , on the other 
hand, the NL interaction much more resembles the FL interaction . The 
matter interactions begin to differ beyond about 2 fm - 1 , and the analyzing 
( 
powers beyond about 1.2 fm - 1 ,,, . 
At 200 MeV the Retf0 component of the PH interaction is about 20% 
stronger12 than the FL interaction and about 30% stronger than the NL 
interaction at q = 0. At 3 fm - 1 it is twice as strong as the FL interaction and 
about 50% stronger than the NL interaction. Also, because of a more rapid 
q dependence it changes sign first. Although at q = 0 the Imtf0 component 
for PH is about 20% stronger than for NL, the NL component is about 88% 
stronger than PH at 3 fm - 1 . FL is intermediate between PH and NL . The 
resulting matter interaction for PH is 25% (35%) stronger than FL (NL) 
at q = 0 and about 35% (15%) stronger than FL (NL) at q = 3 fm - 1 . 
For the Ret~5 component the NL and FL interactions are very similar for 
q < 1 fm - 1 , for Imt~ 5 they are very similar both below 0.5 fm - 1 and agarn 
12 The relative strengths which are quoted here are defined as l00(t - t') / t' . 
31 
above 2 fm - 1 5 . For Ret~ PH is somewhat stronger than the other interactions 
below 1.5 fm - 1 , but it becomes weaker at large momentum transfers . For the 
lmt~5 component PH is stronger than the other interactions over the whole 
momentum transfer region . As for 100 MeV, the FL analyzing power begins 
to differ from around 1.2 fm - 1 , while PH and NL analyzing powers remain 
almost equal up to 1. 7 fm - 1 . At q = 3 fm - 1 the FL and NL analyzing powers 
become again almost equal, but both are more than twice as large as the PH 
analyzing power . 
At 318 MeV the Ret~0 component of the PH interaction at q = 0 is 
about twice as strong as the NL interaction, but only about 10% stronger 
than FL. Again, the q dependence of PH is much stronger and at 3 frn - 1 
PH is now a bout 65 % stronger than NL and about twi ce as strong as FL . 
At q = 0 the PH matter interaction is about 70% (25%) stronger than NL 
(FL) , at 3 fm - 1 about 65% (50%) stronger than NL (FL) . For the spin-orbit 
components the situation is very similar to 200 MeV, albeit amplified. The 
main difference is that all the analyzing powers converge again at 3 fm - 1 . 
In the Figures 2.4- 2.11 we show NRIA calculations at 100 , 200, and 
318 Me V for elastic and inelastic scattering data from 16 0 and 4?Ca. Elastic 
cross sections are presented as ratios to the point-charge Ru th er ford cross 
section <TR to enhance detail. The bands represent the range of vari at ion 
due to the different free interactions. For both targets , the 31 states a.re 
transitions with surface-peaked transition densities. The 11 state in 160 
peaks in the high-density interior and has only a small lobe in the low-density 
region , while the 32 state in 4?Ca has a lobe in the high -density interior which 
32 
N ,,....., 
I 
./ ( ..8..  
~-. ? 
/ . - O" .. 
8 
b 
0 
0 
"0-'  "0' 
.. 
 0 - "0' 0 0   0 "0'  I I 
(") ,,, 
~.. ., )) ,,I 
~ ?' I .......  _/' 
N ,,....., '''  I 
?--~ _/ 
,,,,  
I 
~ ---/ ..8...  ( 
?--~ ,-
--/ ," 
O" 
0 
u..5' -
--~ 
-~ 
0 
0- 0 0 0 0 0- - 0 
0 0 0 
0 0 N "..'.  0 "' 0 "I' 0  "...'  0 "' "' 0 "' N "N ' (') - I I I 
I I I I I (") 
-------~ 
N ,,....., 
, _ I ..8..  
O" 
u8 -
+> '--
~ 
~ 
0 
0 0 0 0 0 
0- 0- 0 0 0 0- 0 0 
0 0 0 0 0 0 
"' 0 "' "' 0 I "- N <O a:, 0 0 N ' 0 N "N ' (') "(')'  I ""I ' I I - - -""' I I I I I I I I I 
(cwJ J\aW) 
Figure 2.1: Comparison of the free t-matrices for the PH (solid line), 
NL (short dashes), and FL (long dashes) interactions at 100 MeV, including 
knock-on exchange. 
33 
0 \\. .... 
0 
b 
I) C\I I 
..8..  
,,?? 
- O" 
., .. 0 0- ll'l 0 ll'l 0 0- -0 ci ci ci I -I 
C'l 
/ 
_/ 
__ / 1:1 C\I 
__ y ?,,',, I , 
???. .....~  I 
.._.8_...  . 
??- .. ~ /1/ ( 
?----~ - O" ?",,  ?? .. ---~ 
-------~-
0 0 0 0 0 0 0 0 ll'l 0 ll'l 0 ll'l 
0 
0... .. ll'l ll'l 0- "....'.  0 C\I C\I I C\I "C\'I  - -I I I I 
\,..\ 
..,. . 
I 
..8..  
u_o0,   ~ ~_ , 
(I) 
~ 
0 
0 0 0 
0 0- 0 0 
0 0 0 0 0 00 0 0.., . 0 0 0 a:, ..,. <O 
0 ll'l 0- 0 C\I C\l "' "I' 0 ll'l C\I <O   ~ "' C\l C\I I I I I - - -I -I I -I I I I I 
(c UlJ N~W) 
Figure 2.2: Comparison of the free t-matri ces fo r the PH (solid line), 
NL (short dashes) , and FL (long dashes ) interactions at 200 MeV, includin g 
knock-on exchange. 
34 
I 
I Cl! I 
..8...  / ----
g 
b 1/ 
.. 0 0 I() 0 I() 0 
"-0'  0 0 0 0 I I 
("') 
: 
: 
?-~ 
~ Cl! ,......._ 
~ I 
?----~ -8 
0 ?------~ .... 
----
O' 
uo 
,..; ~ 
,..; 
..~...  i 
0 
I() 
0 0 0 0 0 0 I() 0 I() 0 .... 0... . I() 0 
I() 0
I .. .. I..(.).  0 
I() ("') ("') N N 
C\1 C\1 
I I I I ("') 
._ 
._ 
\...,\ 
\._\ \ Cl! ,......._ 
I 
?-- ~ 8 
.... -
----
O' 
uo0  ~ 
,..; ,..; 
~ 
e:: 
0 
0 0 0 0.. , 0 0 0 0 0.. 0 0 0 0 0 0 0 0 , 
0 I..(.).  0... . I() I() 0.. .. I() Cl! C\1 I .... I I "I' 
co 0 C\1 
I .... .... .... "... '.  
I I I I I 
I 
(cWJ J\,)W) 
Figure 2.3: Comparison of the free t-matrices for the PH (solid lin e), 
NL (short dashes ), and FL (long dashes) interactions at 318 MeV , including 
knock-on exchange. 
35 
is twice as strong as the exterior lobe in the low-density surface region ( see 
Appendix A). The 100 and 200 MeV cross sections and analyzing powers for 
160 and for 4?Ca are from this work (see Chapter 6 and Appendix F). The 
318 Me V cross sections and analyzing powers for 4?Ca are taken from [Fel 90], 
those for 160 from [Kel 90b]; the 318 Me V data for the spin rotation function 
are from [Ble 88] . The 200 Me V spin rotation function data for both 16 0 and 
4?Ca are taken from [Ste 85] and P. Schwandt (private communication) . 
Although all the elastic scattering calculations used optical potentials 
based on the free interactions, for the inelastic states we generated distorted 
waves with a density dependent interaction . We used the PH g-matrix to 
calculate distorted waves for NRIA calculations based upon the PH and FL 
interaction, but we chose to use the NL g-matrix for distortion of the NRIA 
calculations using the NL interaction. The inelasti c transitions are driven 
by the respective free interaction . It was pointed out in [Kel 89a] that the 
use of the NRIA potential for inelastic scattering , although more consistent, 
would be misleading since in the absence of Pauli blocking we would have an 
unrealistically strong absorption . This absorption would both overly suppress 
inelastic cross sections and at the same time distort the analyzing powers . 
The first observation is that for all the energies NRIA calculations based 
on the various free interactions are reasonably consistent with each other . At 
100 MeV the predicted analyzing powers are much too positi ve and display 
a structure which does not resemble that of the data at all. Elast ic cross 
sections for both targets are reasonable well described fo r q < 1 fm - I and fo r 
160 also again above 2 fm - 1 . The cross sections of both the 31 surface states 
36 
are slightly overestimated at forward angles, the cross section for the interior 
11 state of 16 0 is overestimated by up to a factor of 10. The forward cross 
section of the 32 in 4?Ca, on the other hand , looks reasonable . A prediction 
of the cross section that is too large at forward angles is due to the omission 
of Pauli blocking and the effect is especially grave for a state that is almost 
entirely peaked in the interior, such as the 11 state . For states that peak 
predominantly at the surface the omission of Pauli blocking appears to have 
,, , 
less impact . The 32 is an interesting case in the sense that, because of the two 
? ?' I 
strong lobes in both the high- and low-density of the nucleus , a cancellation ?' I 
,,,,'   I ,. 
of various density dependent effects seems to take place, making the forward 
angle cross section prediction correct. At large momentum transfers the 
predicted cross section for the 31 state of 4?Ca is slightly too high . The cross 
sec ti ons for the 31 and 11 states in 160 and for the 32 state of 4?Ca, on 
the other hand , fall well below the data, indicating a need for a short-range 
repulsive correction to the free interaction . 
At 200 Me V the analyzing powers for all the states improve somewhat , 
becoming more negative, and the ones for th e 31 states of 160 and 4?Ca look 
quite reasonable . The overestimation of the cross section at small momen-
tum transfers due to the omission of Pauli blocking , and the underes timation 
at large momentum transfers due to the absence of short-range repulsion is 
evident clearly for all the states, including the 32 state of 4?Ca at this en-
ergy. Also, the predicted cross sections appear to be shifted slightly towards 
smaller momentum transfers for all the inelastic st ates. The addition of 
short-range repulsion would cause a shift towards larger momentum trans-
37 
fers and alleviate this problem . The spin rotation functi ons Q at this energy 
look not unreasonable for momentum transfers beyond about 1.2 fm - 1 for 
160 and about 1.0 fm - 1 for 4?Ca, although they should become slightly more 
negative to describe all the data points . At small momentum transfers, the 
NRIA predictions fall well below the data. 
At 318 MeV the overall description of the data for the two targets, 
both elastic and inelastic, has improved dramatically over the lower energies, 
although again better agreement could be attained by a slight shift of the 
patterns towards a larger momentum transfer . At small momentum transfers 
the cross section calculations now appear to fall below the data, a fact that 
has also been noted by Kelly et al. [Kel 90b] . It seems that while Pauli block-
ing so far has suppressed the absorptive potential in the interior , we now find 
an enhancement . At 500 MeV this effect has been reported to be even larger 
[Fla 90]. With exception of the 32 state of 4?Ca, the predicted analyzing 
p owers at forward angles are substantially larger than the measured data. 
Since for small momentum transfers the lmtg0 component of the interaction 
is much larger than the Ret~5 component , Eq. (2 .47) in Section 2.7.1 sug-
gests that a suppression of these analyzing powers could be achieved either 
by an enhancement of lmtg0 or by suppression of Ret~
5 [Kel 906] . The 4?Ca 
spin rotation function falls well below the data for momentum transfers in 
the range between 0.5 fm - 1 and 1.0 fm - 1 ; the oscillat ion between l.0 fm - 1 
and 1.5 fm - 1 is not reproduced adequately and above 1.5 fm - 1 the pattern 
is shifted too much towards smaller momentum transfers. 
At 500 Me V the trends observed for the 318 Me V data become more 
38 
pronounced . For states peaking in the high-density interior, the forward 
angle NRIA predictions fall even more below the data, and the low-q ana-
lyzing powers are still systematically too large . The oscillatory structure of 
the analyzing powers is both too strong and too rapid to describe the dat a 
adequately. For a more thorough discussion of the 500 Me V data and in-
teractions we refer to the paper by Flanders et al. [Fla 90] . At even higher 
energies ( 2'. 800 Me V), inclusion of second-order terms of the optical potenti al 
which account for target nucleon correlations, and electromagneti c spin-orbit 
(EMSO) corrections appear to improve the NRIA [Ray 85] . For 650 and 
800 Me V , the beneficial impact of the EMSO potential on the forward angle 
spin observables for nonrelativistic calculations has been reported by Coker 
and Ray [Cok 90] . 
39 
? 
... -+-1-
I 
0.. ..s....  
._0_.,.  
0 
"' 
.., 0 
i N I I 0 I() 0 I() 0 
0 0 0 0 0 0 0 0 0 
I -I 
(") 
. ? 
? ? 
1- ? N ,-._ 
("') 
,-._ ? I 
0. ..s....  
._0_.,.  .__, 
0 er 
~ 
0 
i N 0 I 0 I() 0 I() 0 0 0 0 0- 0 0 0 I -I 
(.1s/ qw) UP/ DP ?v 
N 
+- ,-._ 
0 
,-... I 
0.. ..s....  
._0_.,.  
0 er 
~ 
.. 0 i 0 I() 0 I() 0 0 
0 0 0- 0- 0 0 0 -I I 
'IJn / n ?v 
Figure 2.4: 16NRIA calculations of (J'(q) and Ay for selected states of 0 a.t 
100 MeV; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section ( (TR) to enhance detail. The bands represent the 
range of variation of the calculation due to the different free interactions. 
The data are from this work. 
40 
IN 
C N ,....'.)._  ,......_ 
c.. I 
c.. s 
--a-j- .....  
c..> ----c,' 
~ 
.., 0 
lO 0 lO 0 o 0- i "I'  
I ~ 
0 0- 0.. .. 0 6 0 6 ...; I I 
C'? 
.. 
,_ 
C') N ,......_ 
0.. I 
c.. ..s...  
--a-- ____, j 
c..> c,' 
0.. . -
0 
o 0 i "I'  0 lO 0 
lO 0 
N- 0 0 0 0 .... 6 6 6 I -I 
(.1s / qw) UP/ DP l.v 
+- N ,......_ 
,0.... .._ 
c.. I 
__c_._.,  _..s_._._.  , 
aj 
c..> c,' 
0.. . 
0 
lO 0 lO 0 o 0 i 0 0 0 .... 6 0 .... 0 
I I 
'/j.D / .D l.v 
4
Figure 2.5: NRIA calculations of a-(q) and Ay for selected states of ?Ca 
at 100 M e V; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section ( a-R) to enhance detail. The bands represent the 
range of variation of the calculation due to the different free interactions . 
The data are from this work . 
41 
? 
-- ? 
I 
a 
.._.._.., , 
0 
N 
0 I I lO 0 lO 
.0. .. 0 0 .0.. . ~ 0 0 0 
I 
('l 
? 
N ,-.... 
I 
..a...  
0-
0 
N 
0 0 I I ~ lO 0 lO 0 0 0 0- 0 0 0 I 
(.1s/ qw) UP/ np 'v 
+- N ,-.... 
.0... .._ I 
Q. 
Q. ? ..
a...  
.__,, 
0 0-
~ 
0 
"' 0 i 0 lO 0 lO 0 0 0 .0. .. 0 0 0 0 
I 
'llIJ / D 'v 
Figure 2.6: NRIA calculations of <T( q) and Ay for selected states of 16 0 at 
200 MeV; the elastic cross section is presented as ratio to the point -charge 
Rutherford cross section ( <TR) to enhance detail. The bands represent th e 
range of variation of the calculation due to the different free interactions . 
The data are from this work . 
42 
. (') 
? . 
IN 
,C..."..)_  N ,....__ 
Cl. I 
._P.,.  ..s...  
?:S 
u ------O" 
~ 
0 
IO 0 IO 0 
0 00. . . I 0 0 0 0 ...; I I 
(') 
? 
?? 
N ,....__ 
. ? ? I ? ..s...  
... ------O" 
u.u..a..U...J-...l-llil.L.J...J...L_l_llil.L.J...J...L_l.=llWL.LL.L..L-N_J L~~~-1--~~~--'-~--==-:.L....,~~_.____J 0 
IO 0 IO 0 
0.. . 0 b ~...  6... ~ 0 0 0 ... I I 
(.ts/ qw) UP/ np l.v 
? ? 
? 
. 
0 
i N I ~ IO 0 
\() 0 
0 0 0 0.. . 0 0 0 0 ... I I 
'lln / D l.v 
Figure 2. 7: 4NRIA calculations of <T( q) and Ay for selected states of ?Ca 
at 200 Me V; the elastic cross section is presented as ratio to the point -charge 
Rutherford cross section ( <TR) to enhance detail. The bands represent the 
range of variation of the calculation due t o the different free interactions . 
The data are from this work. 
43 
+ ..... 
? 0 ,...--..._ C\1 ..,...... --..._ 
~ I 
~ 
--..__.., s 
--l+..-_--_-4 ? CIJ .., u .--i 
0 CT 
"<t' 
0 
0 l() 0 l() 0 
.--i 0 0 0 ,_. 
I 
C"J 
+ ..... C\1 ,...--..._ 
0 ..... 
? ,...--..._ I ~ 
~ s 
~ --l+..-_---_4. ? --..__.., ., ? 0 ,_. 
c..:.o..  CT 
0 
0 l() 0 l() 0 
,_. 0 0 0 ,_. 
I 
Figure 2.8: NRIA calcula tions of Q fo r elas ti c scat terin g of 160 and 4?Cn. 
at 200 Me V . The bands represent the range of vari at ion of the calcula tion 
due t o the different free interactions. T he dat a are from [Ste 85] a nd P. 
Schwandt , private communication . 
44 
1....- ----
-0---
I 
. s
0. ..... 
 
'-' 
'-' 
0 c,' 
~ 
,., 0 N 
0 i I I ... ~ 
U"l 0 U"l ~
0 0 0 0 0 0 0 0 ...  
I I 
(") 
11 
I ' 
,_ N 
,(.."..)_  ----I 
0. ..s...  ,,I  
0. 
'-' 
0 c,' 
~ 
0 00. . . .I 0. . 
(.rs / qw) up/ op 
? 
I 
....s._.._.  , 
.. 0 U"l 0 U"l 0 0 i 0 
0 0- 0 0 0 0 0 ... I I 
8n / D r.v 
Figure 2.9: NRIA calculations of a-(q) and Ay for selected states of 16 0 at 
318 Me V; the elastic cross section is presented as ratio to the point -charge 
Rutherford cross section ( <TR) to enhance detail. The bands represent the 
range of variation of the calculation due to the different free interactions . 
The data are from [Kel 90b] . 
45 
IN 
C") 
,,.-... N ,,.-... 
0. I 
._0_.,  .._.
8....  
(U _, 
C> 
~ CT' 
N .., 0 
0 i I I C? ll) 0 ll) 0- 0 0- 0 0- 0 0 -C? 0 I I 
M 
1' I 
,, I 
N ,,.-... 
? ... 'I I 
..8 : I ....  
- CT' 
0 
N 
N 0 i I 0 ll) 0 ll) 0 
0 0- 0 0 0 0 0 0 I -I 
(.1s/ qur) UP/ DP ?v 
I 
..8....  
0 
i N I 0 ll) 0 0 ll) 0 0 0 0 0 0 0 0 ..; 
I I 
HD / D ?v 
Figure 2.10: NRIA calculations of a-(q) and Ay for selected states of 4?Ca. 
at 318 Me V; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section ( a-n) to enhan ce detail. The bands represent the 
range of variation of the calculation due to the different free interactions. 
The data are from [Fel 90] . 
46 
> 
Q) ..... C\1 ~ 
~ * ....... I 
co s 
~ ? ~
CV) --..__
 ., 
II ? ?? ..---i 
0.. ~ 
w 
I I 
0 i , I 
0 I.{) 0 . I.{) 0 
..---i 0 0 0 ~ , I 
I ; I 
I 
Figure 2.11: NRIA calculations of Q fo r elas tic scattering of 4?Ca at 
318 MeV. The bands represent the range of variation of the calcul ation due 
to the different free interactions . The data are from [Ble 88]) . 
47 
2.4 Nuclear Matter Approach 
2.4.1 Introduction 
At low and intermediate energies the nuclear matter approach [Hiif 72] 
has proven to be an alternative and very fruitful approach to nucleon- nucleus 
scattering. The optical model in infinite nuclear matter is identified with 
t he lowest order term in the hole-line expansion of the self-energy or mass 
operator of the projectile nucleon propagating through the target medium . I I , I 
Density dependence enters into the model through the effective interaction , 'I 
I I 
'I 
the so-called reaction or g-matrix . This interaction satisfies the analog of 
the Lippmann-Schwinger equation, the Bethe-Goldstone equation , and in -
cludes medium effects via Pauli blocking and energy shifts in the intermedi-
ate projectile- target-nucleon scattering states . Further density dependence 
enters through the the various averaging procedures which are used to reduce 
the energy and momentum dependent interactions to simple local forms . 
For a nucleon with energy E and momentum k moving in infinite nu-
clear matter of density13 (i.e. Fermi momentum) kF one can write down , 
following Hiifner and Mahaux, the average complex potential felt by the 
incident projectile as the diagonal , plane wave, antisymmetrized two- body 
Brueckner reaction matrix element 
U(k,E;kF) = L (pklg(w)lpk )A, ( 2 .48) 
IPl<lcp 
13 The Fermi momentum kF and the nuclear matter density PNM of an iso t ro pic, sym-
metric and noninteracting Fermi gas are related by PNM = 3;, ki . 
48 
Where lpk)A = l</>(p)</>(k)) - l</>(k)</>(p) ) is the antisymmetrized two- body 
plane wave function, and pis again the momentum of the bound nucleon14 . 
To account for off-shell effects ( e. g. the effects of short-range correlations) , 
the starting energy w = E + 1:(p) is defined self-consistently for all particles 
above and below the Fermi level. This is done by chasing the single-pa rticle 
energy 1:(p) as 
1:(p) ~ p 2 / 2m + Re[U(p, 1:(p) ; kF )], (2.49) 
I I Ii,;, 
with the nucleon mass m and the (self-consistent) optical potential U; in , I I 1:11 ! lj li 
I ;11: 
I I 1:, 
this equation we have also approximated the full potential1 6 by its real part I 
'. I 
!1 1: 
[Bri77 / 8] . 11 1,1 
l'I 
The reaction matrix satisfies the Bethe-Goldstone equation I 1, 
g(w) = v + vQpG+(w)g(w) (2 .50) 
14 It is a fundamental result of many-body theory tha t in the cluster expansion of th e 
total energy all self-energy terms to all orders cancel against the terms involving the single-
particle potential U, provided the above form for U, also known as Brueckner-Hartree- Fock 
condition , is satisfied (for finite nuclei additional conditions for the off-diagona l matrix 
elements between hole states exist). Also, it is easy to see that U indeed represents the 
average effect of all particles in the medium on the particle in a state k . With g being the 
effective interaction between particles in the medium, the potential energy for a pa rticl e 
in state k is just the interaction energy of a pa ir of particles in states k and p , summed 
over all occupied states p [Pre 75] . 
15 Nakayama and Love [Nak 88] discuss to some extent the sensitivity of their g-matrix 
to the choice of the single-particle energy. Specifically, they set the potentia l at la rge 
momenta, where the real part of the full y self-consistent potenti a l becomes repulsive , to 
zero . They find, however , only sma ll sensiti vity even fo r the s trong spin-ind epend ent 
isoscalar central component which should be pa rti cularl y sensitive to the effe cts of short-
range correlations . 
49 
where v 1s the internucleon potential , and the Green fun ction 
(2.51 ) 
includes outgoing boundary conditions and medium effects through the influ-
ence of the average potential on the single-particle states 16 . In this equ ation , 
q1 and q2 are the m omenta of nucleons in intermediate sta tes. The projection 
operator Q from Section 2.3.1 becomes for nuclear matter the Pauli operator 
I I 
Q p with the properties 
I I 
(2.52 ) 
otherwise. 
This ope ra tor , therefore, restricts intermediate scattering states to outside 
the Fermi sphere. 
To calculate the reaction matrix g, one can define the correla ted two-
nucleon wave function 7./; by 
g<p = v7./;, (2.53) 
where ?( K) is a plane wave with the relative momentum of the interact-
ing pair17 . From Eqs . (2 .50) and (2 .50) follows a Bethe- Goldstone integral 
16 In comparison, the sing le-particle energies in the energy denomin a tor of th e 
Lippmann-Schwinger equation for two free nucleons contain on ly the kinetic energy. 
17 In this form the na ture of g as a n effect ive in teract ion becomes pla in : we replace 
the actual potential v by a quant ity g , so tha t g ac ting 0 11 t he m odel (un correla t ed ) two-
bod y wave fun ct ion rf> is the sam e as v a ppl ied to t he actua l (co rrela ted) wave functi on. 
This supposition , the independent pair m odel, is j ust ified if t he healing dis tance fo r t he 
co rrelated wave function is small , which is to a good approximation t he case for low density 
[Pre 75] . 
50 
equation for the correlated wave function 
(2 .54) 
These developments are very similar to those in Section 2.1 for the free NN 
scattering case and the t-matrix . 
2.4.2 Density Dependent Interactions 
Starting from realistic NN potentials, several complex , energy and den-
sity dependent interactions have been constructed for nuclear matter for me- , I : ) 
dium energies below about 400 MeV . Brieva, Rook and von Geramb [Ger 79] 
constructed an interaction (BRG) based on the Hamada-Johnston potential 
[Ham 62], the Hamburg group under von Geramb [Ger 83, Rik 84] an inter-
action (PH) based on the Paris potential [Lac 80], and another interaction 
(NL) based on the Bonn potential [Mac 87] was constructed by Nakayama 
and Love [Nak 88]. More recently, Ray [Ray 90] evaluated medium modifi-
cations to the Watson optical model for energies above the pion threshold . 
The resulting interaction is based on a nucleon-isobar coupled channels po-
tential model [Ray 87] which explicitly includes NN inelasticities in the form 
of NN, N ~(1232) and NN*(1440) channels . In the following we want to 
outline briefly, how the PH [Ger 83 , Rik 84] NL [Nak 88] and LR [Ray 90] 
interactions were generated and then discuss some of their important prop-
erties. 
* * * 
51 
The Bethe-Goldstone equations, Eqs . (2 .50) or (2.51), are the starting 
points from which all of the abovementioned effective interactions are com-
puted . The first step in these calculations is a partial wave decomposition of 
the respective integral equation to be solved . For an operator F( k' , k) we 
can write fo r each isospin channel 
(S'IF(k', k)IS) = 41r L iL- L'ytJ5 ,J(k')Fff,5 'T(k', k)Yfs~(k) Pr, (2.55) 
LL'JM 
1' 
where 
I I I 
Fft,5 'T(k', k)Pr = 2- j dk' dk Yff; ,i(k')F(k' , k)YfsJ(k)Pr , (2 .56) 
41r 
and the spin-spherical harmonics are defin ed as 
Yfsi(k) = L (LML;Slvfsl J M)Yt L(k) x'ff5 . (2.57) 
ML,Ms 
It will also be convenient to defin e Fff,ST = Ff{T . 
Solving the full Bethe-Goldstone equation for the scattering problem in 
nuclear matter is a very complex and numerically difficult problem and one I 
I : 
employs, therefore, certain approximations to simplify the propagator. The 
first approximation, spherical angle averaging of the Pauli blocking operator 
Qp, is a standard procedure which has been shown to be a reasonable and 
quite accurate approximation in nuclear matter18 . If we define K = k + p to 
be the total m om ent um of the interacting nucleon pair , K = ?( k - p) to be 
18 Nakayama a nd Love discuss in their paper [Nak 88] the uncertainty in t he in ternct io11 
due to the use of an angle averaged Pauli operator Q p and fi nd for the largest interact ion 
com ponent , the isoscalar, spin-independent central com ponent, tha t a 20% cha nge in Qr 
corresponds roughly to a 10% change in the m odulus of the interaction . 
52 
th
eir relative momentum, and q = Hq 1 - q2) to be the relative momentum of 
the intermediate states (momentum conservation also yields Q = q + _ 
l q2 -
K for the intermediate scattering states), Eq. (2.52) a11d ang1 e  averaging 
combined give tl1e nonrelativistic Pauli function [Bri77 / 8) 
0, a < 0 
a, 0 :::; a :::; 1 (2 .58) 
0, a > 1, 
where a = (? 1 { 2 + q2 - k} )/ J{ q. Here, and in all the following integrations 
over tl1e angle between two vectors, we always choose one of the vectors to be 
oriented along the z-axis . Ray evaluates for his interaction an angle averaged 
Pauli blocking operator for each of tl1e three clrnnnels NN, Nb. and NN? 
using relativistic kinematics to relate two-body c.m. and laboratory frames 
[Ray 90) . 
The energy denominator in the propagator can be averaged in several 
Ways. First we observe that tl1e single-particle energies t:( q) in Eq . (2 .51) 
contain implicitly tlie angle between the total momentum Kand tl1e relative 
momenta ,,._ and q. The NL [Nak 88) interaction is constructed using tli e 
method proposed by Brueckiier and Gammel [Bru 58} w11ich treats the angle 
averaging procedure in an approximate way. The metl10d consists of writing 
a polynomial expansion of t:( qi) + t:( q2) and expressing q 1 and q 2 as ?K  ? q ' 
respectively. Then, in tl1e quartic term , ( K. q )2 is replaced by its spheri cal 
average 
_!:_JdiJ (K ? q) 2Qp(K, q; kF)- (2 .59) 
41r 
53 
To fourth order in the energy expansion, this is equivalent to replacing in 
the single-particle energies q1,2 ---t q? = ?K 2 + q2 ? [f ?Q p(K,q;kF)Kq . 
The same procedure is applied to the starting energy w, when expressed as 
e:(k) + e:(p); one only has to replace q by"' [Bri77/ 8] . 
The PH [Rik 84] interaction uses a slightly different approximation 
scheme. First, the kinetic energy terms are collected in terms of the relative 
momenta"' and q, then angle averaging is performed on the individual poten-
tials, which bear the full angular dependence, without any approximations. 
Thus , for instance, for the projectile with momentum k we have 
J dK U(l?K + Kl)Qp(K , K; kF) 
(2 .60) 
JdKQp(K,K ;kF) 
and similar t erms for the nucleons with momenta p , q 1 and q 2 . 
The LR interaction which is based on Watson multiple scattering ig-
nores the self-energy of the incoming projectile and applies binding energy 
corrections to the target nucleons only. Here , a quadratic expression of the 
form U( 2 2 1t ) = A+ B 1t is being used . For the NN channel, with exception of 
the A parameter of the initial state, the parameters are chosen to reproduce 
the well depths of fitted Schrodinger equivalent real, central opti cal poten-
tials from Dirac phenomenology. The binding potentials for Nb,,. and NN " 
were assumed to be 60% that of the NN channel. Instead of the above aver-
aging schemes Ray makes an effective mass approximation in combining th e 
kinetic energy terms with the velocity dependent B terms. He thus arrives at 
a simple form for the energy denominator which includes the reduced effec-
tive mass of the target nucleon and its effective momentum in the entran ce 
54 
channel. For the calculation of this momentum , the correct c.m. relativis-
tic value was used and finally, for nucleon-isobar channels, also the energy 
dependent isobar widths f were included [Ray 90] . 
In the following , we want to briefly discuss the gen eralized ref erence 
spectrum method [Bri77 / 8] as one representative example of how the Bethe-
Goldstone of the scattering problem in nuclear matter can be solved . This 
t echnique transforms the integral equation for the correlated wave functi on 
,/; into a system of two coupled differential equations, which are then read-
ily solved by numerical quadrature and partial wave decomposition . Let us 
adopt a simplified notation and write QpG+ in Eq. (2 .54) as Gt. The gener-
alized reference spectrum method is effectively an interpolation between the 
two limiting cases of 
? free NN scattering : here Gt becomes the free propagator Gt and 
Eq. (2.54) reduces to the Lippmann-Schwinger equation. The pole 
in the Green function introduces a phase shift in the asymptotic wave 
function. 
? the bound state problem: w < 2E( kF) and the kernel of Ct is always 
non-singular and positive due t o the Pauli principle . Asymptotically, 
no phase shifts occur and the correlated wave function ,/; (which is 
purely real) heals towards the uncorrelated plane wave cp. 
To achieve this interpolation Brieva and Rook separated the Green functi on 
ct into two parts 
(2 .61) 
55 
where the two components have t he fo rm 
(2.62) 
G+ - Cp F - m Q2 + 2+ ? ? 
0 q ?~ 
Here q is agam the relative momentum in the intermediate stat es , Q0 is 
the pole position in Gt, and the other parameters are optimized to sati sfy 
Eq . (2 .61) as closely as possible . T hat this is indeed a good approximation 
has been shown in the paper by Brieva and Rook [Bri77 / 8] . ,, 
I I 
Now the_i ntegral equation for 'I/;, Eq . (2 .54) , can be written as a sys tem 
of two coupled differential equations 
A = q; - GRv 1./; 
(2.63) 
'lj; = A - Gt v'f/; 
where 'I/; should be a good approximation to the true wave fun ction . The 
quantity A is an auxiliary wave functi on . This system of differential equations 
can be also written as 
(\72 - -y2)[A - ?] = mcRv'lj; 
(2.64) 
(\7 2 + Q~)['lj; - A] = mcFv'f/; , 
which can be solved numerically by means of partial wave expansions . How-
ever , one can easily recognize the following limiting cases 
? for free NN scattering with th e free propagator Gt one has CR = 0, 
cF = 1 and A = q; . One is left with th e Schrodinger equ a tion fo r free 
N N scattering 
(\72 + QG )'I/; = mv'I/;, (2.65 ) 
where Q 0 is the asymptoti c relative momentum . 
56 
? for the bound state problem one has CF = 0 because of the non-singular 
kernel. Hence, one is left with the traditional reference spectrum me-
thod equation where A = 'lj; , i. e. 
(2 .66) 
* * * 
Although the generalized reference spectrum method was used by von 
Geramb, Brieva and Rook for the BRG interaction [Ger 79]) , another pro-
cedure was chosen for the PH interaction. With the above approximations 
for the numerator and the denominator of the propagator, Eqs . (2.58) and 
(2 .60), the integral equation for the correlated wave function in coordinate-
space, ui,ff (r, a), was solved i~ each partial wave J :S: 10 with standard 
matrix techniques for Fredholm integral equations of the second kind ( a 
represents the dependence of the wave function on E, k, p and kF ). The 
self-consistency process was initiated with the choice U = 0 and was iterated 
once [Ger 83, Rik 84]. 
The prescription of Siemens [Sie 70] was then used to construct a local 
energy and density dependent effective interaction in coordinate-space. The 
method simulates, in an average way, all the properties of the correlated 
wave functions, including the behaviour at small r . In this way, im portant 
off-shell amplitudes which arise from short-range, repulsive anti correlation 
effects which tend to keep identical nucleons apart are retained at least on 
the average [Kel 89a] . The basic idea is that with the correlated wave functi on 
'lj; the plane wave matrix element of g can be deduced from (cplg lcp) = (f~,~~). 
57 
Explicitly, one has 
(2. 67) 
where the vf,f!: (r ) are the two-body NN potential radial form fact ors of the 
Paris potential in the channel ( J ST; LL") [Rik 84] . The angular momentum 
indices are such that the unprimed L refers to the entrance channel, while 
the primed ones refer to subsidiary channels . 
An interaction in the form of a local pseudopotential, Eq. (2 .32) , can 
be obtained if one expresses g( r) as a scalar product of the rank k tensor 
opera.tors Rk(r), acting only in coordinate-space, and Sk , acting only in spin 
and isospin space 
2 
g(r) = I: Rk(r) ? Sk- (2 .68) 
k=O 
Using partial wave analysis and Racah algebra, one can isolate the t ensor 
amplitudes which are related to the g-matrix elements of Eq . (2 .67) . On e 
finds for instance for the central component (k = 0) that S0 = PsPr and 
(2.69) 
In the here adopted notation we have :i; = 2x + 1. The remaining L depen-
dence in the Rk can be eliminated by performing a weighted average over 
the Fermi sphere. The appropriate weighting fact or for the central interac-
tion is the sta.tistical fact or L times the probability that the relativ angul ar 
momentum of projectile and target nucleon is L [Ke! 81], i.e. 
WL = r dp lh(~r)l2 . (2. 70) 
l1p1 -s_ kp 
58 
With this result we get for the cent ral interaction 
"' L' R(oL L ,ST) C L, ? C:W? 
9sr = (2 .71) 
where the fact ors c: restrict the sum to allowed antisymmetrized states . Anal-
ogous expressions for the spin-orbit (k = 1) and tensor (k = 2) components 
a re given in [Ger 79 , Ger 83 , Rik 84]. 
Finally, following the procedure in Section 2.3.4, a parametrization in 
I I 
t erms of Yukawas is done. The fin al interaction is available for densities 
between 0.5 and 1.4 fm - 1. However , for kF - 0, the PH interac tion is 
only required to reduce to the free NN t matrix for the two momentum 
transfers q = 0 and q = k , which might explain the large differences th at 
have been found , particularly at intermediate momentum transfers , fo r the 
low-density PH tensor interaction and the tensor components of the FL t-
matrix [Kel 89a] . 
* * * 
The NL interaction is calculated in momentum-space on th e energy shell 
by solving the operator equation , Eq . (2 .50) , directly for the gf{T (K , "- ,"- )in 
each Pauli allowed partial wave channel (J ::; 10 , 15 , 18 for TL(MeV) ::; 100, 
100 < TL(MeV) ::; 270 and 270 < TL(MeV) ::; 425 , res pectively ) vi a matrix-
inversion19 [Nak 88]. The calculation is fully self-consistent . Apart from the 
approximations for the propagator described above , a simple Fe rmi average 
l !J The method which is generall y used , is the one described by Hafte l and Taba kin 
[Haf 70] . 
59 
for the m agnitude of the initial mom entum of the struck nucleon was chosen, 
namely (IPI ) = ?kp . Subsequently, fo r simplicity, the magnitudes of t he total 
m om entum K and the initial rela tive momentum K- were determined from k 
and p by an unweighted average over the direction of p with respect to the 
local m om entum k of the incident projectile 
(K) (2. 72) 
where k> (k<) is the larger (si:naller) of k and ?kp . The advantage of thi s sim-
ple averaging procedure is that nonrelativistically one recovers the free NN 
t m atrix in the limit kp --> 0, independent of momentum transfer [N ak 88] . 
Simply substituting the gfJ!'(K, K , K) back into the partial wave expansion 
for the g-matrix, gives a local and angular momentum independent interac-
tion for a particular projectile energy, without further averaging. F in ally, 
the interaction is cast in the operator form , Eq. (2 .40) , using cert ain rela-
tionships between the g-matrix and the coeffi cients fif r , g~5 and gj, [Nak 88, 
Nak 84] . Subsequently, the interaction components are fitt ed for several en-
ergies and densities to sums of antisymmetrized momentum-space Yukawa 
forms, similar to Section 2.3.4, using only on-shell momenta q < 2k . 
Nakayama and Love have studied , to a certain extent , the off-shell and 
nonlocal behaviour of the gf0 component of their interaction . The auth ors 
justify the on-shell prescription which was used to constrn ct their local inter-
action by observing that at moderate excita ti on energies the on-shell or near 
on-shell matrix elements will be the dominant ones due to th e limited spread 
in relative collisi on m oment a allowed by the momentum di stributions of the 
60 
initial and final target state wave functions . To show the off-shell effect 
' 
Nakayama and Love compare at kp = 0 tl1e parametrized (i .e. local) form 
of the g-matrix with the exact g-matrix, evaluated under special kinematic 
conditions: while t11ey vary the magnitude of the final relative momentum 
K', both K and ,,.,, are taken to be parallel (i.e. the on-shell point is at K' = K 
' 
where qQ = 0) . They observe that large differences between the exact and 
local interactions only exist for such significantly different relative momenta 
which are unphysical and not allowed by the nuclear wave functions . 
To study nonlocality, they make use of tl1e fact that in momentum-
space a purely local interaction depends solely on the momentum transfer 
q but not on the exchange momentum transfer Q. Thus, examining the 
Q dependence at a fixed value of q samples the overall non1oca1ity of an 
interaction . Considering Eq. (2 .50) , they observe that for a purely real NN 
input potential v the nonlocality of the imaginary part of the interaction 
will be entirely due to the correlation term G"f,g . A comparison with the Q 
dependence of the bare NN potential, in this case the Bonn potential [Mac 87) 
which itself is strongly nonlocal due to the exchange of er and w mesons , 
seems to indicate that the nonlocality contributions from the potential greatly 
outweigh those due to correlations . 
* * * 
For the LR interaction, the evaluation of the Watson t-matrix wu.s 
? ?JST ( ) ? 
achieved by introducing a correlated wave funct10n , u/3LL' r, a ' rn coordi -
1l a t e-space and solvm. g equa.t1?0 n E q. ('2 ? 54) in each partial wave and for 
eacli 1 . b l I a with matrix techniques . The treatment of nuc eon-1so ar c ianne I-' 
61 
the propagator has been discussed earlier. With the correlated wave func-
tions, medium modified NN amplitudes f are generated on the energy shell 
with f = (</>lvl'l/i ) for specific initial and final two-nucleon spin projections 
[Ray 90]. Following the notation of [Ray 85], these amplitudes are then 
transformed into Wolfenstein amplitudes for (pp) and (pn) channels 
(2. 73) 
where <Tiz = <Ti ? x, and q, Q and it are defined as before in Section 2.3.4 . 
From this equation , a medium corrected t -matrix in the NN system can be 
directly obtained by multiplying the scattering amplitudes by the factor 1J 
which was also given in Section 2.3.4 . 
2.4.3 The Local Density Approximation 
Effective interactions are conveniently applied to nucleon-nucleus scat-
t ering using the local density approximation (LDA), in which the effect ive 
interaction between an incident projectile and a target nucleon bound in a 
finit e system is assumed to be the essentially the same as that approp ri -
ate for infinite nuclear matter with the same local density [J eu 74 , J eu 76 , 
Bri77 / 8] . For natural-parity isoscalar transitions , the scattering potenti al 
can be expressed as 
(2 .7 4) 
where uz is the potenti al obtained from folding th e Coulomb interact ion 
with either the ground-state or the transition charge density. The elas ti c 
62 
optical pot ential has the form Eq. (2.11) , where 
ULS (r) = ~ fJpLS (2.75) 
r or 
The central and spin-orbit potenti als uc and pLS are expanded in t erms of 
their multipoles as 
uc ( r) = L J UJ ( r) ? YJ ( r) 
(2.76) 
pL5 (r) = L J F} 5 (r) . YJ(r) . 
The multipoles are obtained by folding the matter transition density PJ with 
I I:: 
1 
central and spin-orbit components of the effective interaction ic and i-Ls I I 1I I 
I I 
I' 
according to20 
UJ (r) = ; J dq q2jJ(qr)tc (q)iiJ(q) , 
(2.77 ) 
F}s(r) = ; J dq q2jJ(qr)i-LS(q)pJ(q) , I I 
where the point transition densities PJ( q) will be taken directly from ( e, e') 
inelastic scattering. The essence of the local density approximation is that the 
interaction is assumed to be the one calculated in infinite nuclear matter. The 
density dependence of the effective interaction is evaluated at the projectile 
position r , i. e . i( q) -+ i( q, pg( r) ), where pg( r) is the local ground-state density 
[Kel 89a.] . 
This approach to the LDA is the numeri cally most simple one, but is not 
unique. Other forms can be employed in the coordinate-space representation 
of the optical potential described , for example, by Brieva. and Rook [Bri77 / 8]. 
For instance, the central potenti al in coordinate-space reads 
(2. 78 ) 
2? From now on, [C and :;-L s can a lso deno te com ponents of t he density de pendent 
interaction . 
63 
where the mixed transition density and the relative coordinate s are defined 
as in Section 2.3 .3 and where k is the local wave number. Again , both 
the direct ( tD) and the exchange ( tx ) interactions are app roximated by the 
corresponding effective interactions in nuclear matter with the appropria te 
density. However , in the coordinate-space fo rm the ground state density, and 
thus the interaction, can now be evaluated a t either r , r' , or the mid poin t 
R = ?lr + r'\ . 
2.4.4 Treatment of the Exchange Interaction 
Eq. (2 .78) is derived from a more general formulation of the NA op-
ti cal potenti al which is nonlocal in coordinat e-space and energy dependent 
[B ri 77 / 8] . This potential can be approximated by the sum of a local di rect 
t erm and a nonlocal exchange term . For the exchange term it is convenient 
t o define a local equivalent potential via 
ux(r)'lj;(r) = j dr' Ux(r , r')VJ(r') , (2. 79) 
where 'lj; is the scattering wave function of the incident projectile nucleon . If 
the potential is small compared to the total energy of the projectile and if fu r-
thermore its spatial variation is small compared to the projectile wavelength , 
it is reasonable to approximate the momentum in the vicinity of r by its 
(self-consistent) local value k(r) = 2m[Ep - ReU(r) ]. T he wave fun ct ion 
'lj; ( r') can then be written , using the t ranslati on opera.tor from elementary 
quantum mechanics , as 
(2.80 ) 
64 
where s = r - r' was used . T11is leads to 
(2 .81) 
If U X ( r, r ') 1. s su ffi C.i ently s110rt-ranged so tliat all non-sp11erical contributions 
can be neglected , integration over the angle between k( r) and s yields with 
Jo(x) = sinx 
"' 
(2 .82) 
where a factor of 41r was put back into t11e volume element [Kel 81) . If 
ux (r , r') is expressed in t11e folding model approach as tX ( ks )p( r , r') , we 
get the above expression, Eq. (2. 78) . 
* * * 
At this point several further approximations to the excl1ange term 
can be discussed . For tliis we recall tl1e expression for tJ1e mixed density, 
Eq. (2.29), as product of a point density and a correlation functi on , here 
P(r, r') = p(r')C(k, s) . 
? ZREA {zero-range exchange approximation): since s ~ 0, both j 0 and 
C(k, s) go to unity (see also Section 2.3.4) . 
? FREA (finit e- range exchange approximation) : Jo is retained , while th e 
correlation functi on goes to unity. 
? SEA (Slat er exchange approximation): both t11 e spherical Bessel fun c-
t . an d the correl  a t? f un ct1?011 are retai11ed . TJ1e correlation funct 1?0 11 1011 1011 
is assumed to 1rnve the Slater form. 
65 
? AEA ( Asymptotic en ergy approximation): the local wave number k is 
calculated with the asymptotic energy Ep, i.e. k = J2mEp, rather 
than with the fully self-consistent local energy (LEA) which was used 
in the previous paragraph . Although the choice of the local energy can 
also affect the direct part through the intrinsic energy dependence of the 
g-matrix , this contribution is usually much less important . However, 
Kelly et al. find for 135 Me V that AEA and LEA scattering calculations 
using the PH interaction are almost indistinguish able , and that the 
largest effect is a 6% reduction in the real central part of the optical 
potenti al [Kel 89a] . 
2.4.5 LDA Calculation Results 
Optical potentials for elastic scattering of 100, 200 , and 318 MeV pro-
tons from 160 and 4?Ca are shown in Figures 2.12- 2.17. In these figures we 
compare the optical potentials for the PH (solid line) , NL (short dashes), 
LR (dashes), and FL (long dashes ) interactions . For 100 Me V only the PH 
and NL interactions are available. In the following calculations the LDA was 
evaluated at the projectile position and exchange was treated using ZREA 
and AEA . As anticipated, all the LDA potentials approach the NRIA in the 
low-density surface region . While all the central potentials are subject to 
strong medium modifications , the effects on the real pa.rt of the spin-orbit 
interaction are only modes t. Th e medium modifi cations of the imagin ary 
spin-orbit p otential are quite significant , but have no large impact on the 
66 
calculations because this potential is very small21 . 
Relative to the NRIA, represented by the FL potential , all the LDA 
imaginary central potentials are smoothly damped in the interior due to P auli 
blocking. The PH potential remains for all energies the most absorptive. By 
contrast , the NL interaction produces the weakest absorption. The potential 
generated with the LR interaction is intermediate between the two others. 
The PH p otential is always about 50% more absorptive than the NL potential , 
the LR potential always between about 5 and 10% more absorptive than the 
NL p otential. 
The real central potentials are generally more repulsive than the NRIA 
potential and for 100 and 200 MeV show more radial structure. The PH 
p otential is m ore repulsive and has more structure than either the NL or LR 
p otentials. At 100 Me V the NL potential is very similar to the FL potenti al 
and only slightly more repulsive in the center . For 200 Me V the LR potential 
has the same strength in the center as the PH potential , but becomes vir-
tually identical with the NL potential for r > l.5fm. For 318 Me V even the 
NL and LR interactions produce potentials with significant radial structure. 
21 The "wiggles" in the small ImULS poten t ia ls are due to the fact that for these LDA 
ca lculations , we evaluate the density dependence of the in teraction using q-folding a t each 
projectile position and by interpolating expansion coefficients of a given Yukawa tabl e. The 
density dependence of the interaction which is represented by such a table is not entirely 
smooth . Evalua ting the interaction in coordina te-space instead , compri ses effect ively a 
sm oothing process because the potentials a re first ca.lculated on a radi a l mes h and then 
interpo lated by a sm ooth polynomia l. On th e other ha nd , reparamet ri zed t heo ret ica l 
interac tions, as well as the empirica l interact ions in Chap te r 7, do have smooth density 
dependence and, hence, produce smooth poten t ia ls. In either case, the wigg les a re too 
small t o affect the observables . 
67 
The very characteristic radial shape of the LDA potenti als, commonly called 
"wine- bottle" shape, is due to the delicate competition between short- range 
repulsion and attraction in the interaction. The details of the balance depend 
upon the difference between the gradient of the density and the gradient of 
the repulsive component [Kel 89a]. The energy dependence of the shape is 
du e to the different energy dependencies of the direct and exchange parts 
of the folding potential , the exchange part having a slightly sharper energy 
dep enden ce than the direct part . As we will see later the wine-bottl e shape 
also arises from our empirical interaction , the IA2 [Ott 88], and Dirac phe-
nomenology [Ham 90]. Finally, the equivalent local potentials which will be 
dis cussed in Section 2.7.2 also show this shape. 
68 
<O 
U') 
> 
V 
-;:::::s '<I' 
O:o 
'--'O
0 ....  
-E 
(') ...._..,. , 
~ II ... I< 
w s... / N 
--~ / 
-~ 
...... 
:::---
,,-:;.:--- 0 
N 0 
0 U ex, <O '<I' 
Cli -') 0- U') 0 ti') 0- t-i') 00 ci 6 ci N ~ 6 6 6 6 c::i I I I I 
(N~W) s,O W,f 
(.i\aw) s,O 8cf 
<O 
U') 
,._ '<I' -..E...  (') .._, , 
s... 
~ --.. N 
~- ~ 
... ... ' 
-.. ,, ~ 
0 
0 
0 I() 0 I() 0 I() 0 ti') 0.. ,. t../',).  00 U') 0.. . "-' 0 (IJ " 0 (IJ'  (') "(') ..,. I (\I N (') (') t/') I ' 
I I I I I I I I I I I I I I 
I I 
(.i\aw) il 8<f (N~W) ::ill W,f 
Figure 2.12: Optical potentials for elastic scattering of 100 MeV protons 
by 160 for the PH (solid line) , NL (short dashes ), and FL (long dashes ) 
interactions. 
69 
(0 
> I[) 
<V 
'o: ::::i1 
.,,. 
O..o 
'-' O 
~ al ..... -o..u.  11 ("') .:s=:,  
t.J" ' 
J.-, 
N 
: 
' 
~ 
0 
0 I[) 0- I[)"' N - 0 - N 0 0 _; 0 ci ci ci 0 ci I I I I I 
(i\ aw) s,fl ocJ (N1N) s,fl WJ 
(0 
I[) 
~ -- .,,. 
~..... .......... -
~ ~ ("') ..
s...  
'-' 
' 
~ ~ J.-, N 
-~ \ 
0 
0 I() 0- I-[) 0 I() 0 If) 0.. .. I..f.).  0 I[) 0 I[) 0 If) 0 If) 0 I N N ("') ("') I N (\) (") .... I I I I I I I I I -I I I I "I'  I 
(i\aN) il ac1 (i\c)N) :,fl WJ 
Figure 2.13: Optical potenti als fo r elas tic scattering of 100 MeV protons 
by 4?Ca for the PH (solid line), NL (short dashes ), and FL (long dashes ) 
interactions . 
70 
i 
~? ~ ,.., / 
?~. .. .,-,-/ : 
--~-~ .,, ' ;., 
,, 
N 
\ , 
,---- ---- ------- -??\ ,.,_ ' 
L-~--'-'--~--'---~-"--'-'-----'-'-~--' 0 
l() 0 l() 0 l() l() ?I- "' N 0 ~ ~ ci ci ci ..; ~ 0 0 ci 0 ci ci 
I I I 
(J\ d}'{) s,O Wf 
(i\ ~w) s,O oef 
I() 
'\  ?,? , -~?,  .---. ?.".\\ a (") ............ , 
\ .~ N 
' ?. .... , .. ~ 
-.::-~---:. : -,. 
L---'----'----'-------"--...U..----'L--- 0 
0 l() 0- l-() 0 l() 0 l() 0 l() 0I N N (") I I I I "I'  I - 
l-() 0 I() 0 l() N (IJ I I I I "I'  "I'  I 
Figure 2.14: Optical potentials for elastic scattering of 200 MeV protons 
by 160 for the PH (solid line), NL (short dashes) , LR (dashes) , and FL (long 
dashes) interactions . 
71 
<.O 
~ ?, '/ 
... / ' - --.. ,f' I() /,' 
-~---::--:_ .:-::-___ / 
/ 
?- ---:--... ::._~ / : sf' 
:.--
', --- \ 
> ,/ "/ (, ' 
--8-- 
(") ..... 
<l) 
--;;:~ __ .-:,;?-:..???;:~ ----' s... _.,y,,;.:?-~ 
P.o ' 
-o ' 
C\I 
<d Cl2 (~- ., ~ \ " 
U II ?---~ - I 
$ ... -- -..: :_ -
~ ( 11 
' 0 
0 N sf' <.O '? 0 C\I 0 0 0 0 0 0 0 0- 0 0 
I I I I -I I 
(A an) s,O ay (N~W) s,O 'W.J 
<.O 
\ 
"- ., ,., \ ... \ I() 
?-.,~ ' ", -?:,~,_ ~ sf' 
\' 
\ .,,.~  
-._, ~ ----
\\ '-----...._ 8 (") ..... 
'-??,  .. '-----...._ \' ----, .. -. \ ~ s... ,?., \' C\I 
'??, \ 
' ------------ I \ 
,:,
:,:  "" I  
0 
0 If) 0.. . ,..n.  0 ti') 0 0 ti') 0.. . t.i.'.)  0 ti') 0 ti') I (IJ (IJ (") I (IJ (IJ (") (") 
I I I I I I I I I I I 
(Aa:N) ::ll ay (11aw) ::,fl W.J 
Figure 2.15: Optical potentials for elastic scattering of 200 MeV protons 
by 400 for the PH (solid line) , NL (short dashes), LR (dashes) , and FL (long 
dashes) interactions . 
72 
<O 
I[) 
> 
<l) 1 
a: ~ -~ _ti 
ci. \:: 
"<I' 
co ~- .---, ,__,, ..... 
1/,,? E 
0 (") ' .... .... 
.. " '?"::- 1/ 
(") 
~ II ? ,,,_,,, . ....... 1/ 
??.:--,. 
w V ??-~' s... \ N 
.// 
.,.::..??, ~ 
?.- :--~- -- - ? 
....,_ 
/ 0 
I[) 0 I[) 0- I-() ('") N - 0 
N 
"'. ~ 0 0 0 0 0 0 0 0 0 
I I I I I 
(L\ ::rw) s,O ocf (A~W) s,11 UJ,J 
<O 
I[) 
\~ ~ 
\\ "( ,., "<I' 
-~ .---, _)) --~ E -..~  (") ,.,.,_..,., , 
?.-::-,- .
( ?
,- -~,:--'-- s... 
,,' / ??.-'. _, N ~ 
:_ I 
?-~ ?.' ~ '-' 
'.' 
----------- \ 
0 
I() I() 0 I() 
0 I[) 0 I[) 0- I-() 0 0 I() 0 0 N I - - N N (") (") I I I I I I I I I I 
(L\ ~w) oD 8cf (L\ ~w) :,11 'Ulf 
Figure 2.16: Opt ical potent ials for elastic scattering of 318 MeV protons 
by 160 fo r the P H (solid line), NL (short dashes), LR (dashes ), and FL (long 
dashes ) interactions. 
73 
' 
0 C\I ?I' a:, a:, N 0 
c:i c:i 0 c:i c:i c:i c:i 0 
I I I I 
, ,..,....--,----,~--,-~---~---~a:, ... 
, I ... \ 
'. ' \; 
'-. ' '," '? 
'.,/ ?-~ ' ?~ ??--:-.,-:- ' ??-:-.., :::,...,_ 
, ' / ???\, "-......-........... 
: I 
. ?..  ' ? ?\ . . ' -.., ~ ??. ' ... \ (\) 
\' \ I 
\ I \ I \ 
, .. _: I / i I 
.',' I 
~~~--'----'----'-~...L.L--L-L-.....U.--' 0 
0 '? 0- ,no ,no ,no ,no ,n o I I - - N N M M ~ I I I I I I I I I 
(A~W) ::>fl W.J 
Figure 2.17: Optical potenti als for elast ic scattering of 318 MeV protons 
by 4?Ca for the PH (solid line ), NL (short dashes), LR (dashes) , and FL (long 
dashes ) interactions . 
74 
In F igures 2.18- 2.25 we show LDA calculations fo r the same data and 
m the same format as in Section 2. 3.6. The line code is the same as in the 
previous figures for the optical potenti als: PH (solid) , NL (short dashes) , and 
LR (dashes) . All of the calculations employ self-consistent distorted waves 
and transition densities from electron scattering ( see Appendix A) . 
While the cross section calculations based on the PH interaction look 
reasonable for elastic scattering and the 31 states at 100 MeV, neither the 
11 state of 16 0 , nor the 32 state of 4?Ca are particularly well described . 
The NL forward cross sections are generally too high , a consequence of the 
weak absorption of the NL potenti al. The PH interaction provides an overall 
better description of the analyzing powers than the NL interaction which 
gives much too positive analyzing powers at large momentum transfers, and 
which for 4?Ca produces rapid oscillations in the analyzing power which are 
not present in the data. Nevertheless , neither interaction describes the data 
particularly well . 
At 200 MeV, although both the NL and the LR interaction describe the 
elastic cross sections slightly better than the PH interaction , their predi ctions 
for the forward angle elastic analyzing powers and spin rotation functi ons fall 
well below the data . The PH interaction, on the other hand , describes these 
data quite well. Inclusion of Pauli blocking and short-range anti correlations 
lower the inelastic forward cross sections and fill iu strength at large momen-
tum transfers . The analyzing powers become more negative and are closer 
to the data. On the other hand , if we compare the LDA calculations for the 
32 state of 4?Ca with the corresponding NRIA calculations in Figure 2. 7 we 
75 
do not see much improvement : on the contrary, t he second maximum in the 
cross section appears to be slightly to high . The reason for this behaviour 
might be the special shape of the transition density which has a st rong lobe 
both in the high-density interior and the low-density surface region . It is 
also striking that all the LDA models overestimate the forward cross sections 
of the surface-peaked 31 peaks . This sugges ts that the effec tive interact ion 
is m odified even for low densities. The spin rotation functions are clearly 
improved over the NRIA in Figure 2.8, especially at forward angles . Finally, 
the LOA calculation follows more closely the large amplitude of the oscilla-
tory pattern which is described by the analyzing power data. While the PH 
interaction is clearly superior at small momentum transfers , it appears that 
the NL and LR interactions become slightly superior at larger momentum 
transfers . Overall , the PH interaction provides the best description of all 
the data. While the cross section predictions for the LR interaction are very 
similar to those for the NL interaction , the LR analyzing powers are clearly 
superior to those of the NL interaction. Analyzing powers for energies at or 
below 135 Me V are especially poor with the NL interaction . 
At 318 Me V the PH interaction provides a clearly superior description 
of the 160 cross section data, while the NL and LR predictions fall well 
below the data. The PH analyzing powers , on the other hand , appear to be 
shifted a little bit too much towards lower momentum transfers while th e 
two other interactions follow the data more closely. For 4?Ca the NL aud 
LR interactions perform better on the cross sections and approach the P H 
predictions . Their predictions of the analyzing powers and the spin ro tation 
76 
fun ction at larger m omentum t ransfers are st ill slightly superior to the PH 
predictions . It is significant , however , that even the LDA is not able to 
suppress enough the forward angle analyzing powers at this energy. 
Kelly et al. attribute the superior performance of the PH interaction 
over the NL interaction for isoscalar natural-parity transitions to the special 
procedures used in reducing the g-matrices to a local form . As discussed 
above , the PH interaction is believed to reproduce more realisti cally the 
short-range correla tions of the interaction , thus giving rise to off-shell be-
haviour that may be m ore realistic . Since the on-shell behaviour of the two 
interactions , as judged from the the Pauli suppression of the forward scat -
t ering cross section , appears to be comparable, the off-shell performance of 
an interaction seems to be indeed important . In spite of the careful justifi-
cations for their on-shell and momentum averaging prescriptions leading to 
a local pseudopotential form , the off-shell behaviour of the NL interac tion 
by Nakayama and Love might be inadequate . On the other hand , it has 
also b een observed that the NL interaction , in some respects is more sound 
than the PH interaction , especially with regard to self-consistency and the 
low density limit . And indeed , the NL tensor interaction, being in better 
agreem ent with the FL t-matrix , gives a better description of the excitation 
of stretched states than does the PH interac tion [Kel 89a.]. A comparison of 
the PH and NL interactions to the LR interact ion is less st raight fo rw ard aud 
we defer it to a la ter section . 
77 
? ?. .. ? 
. ..  -? 1 N ....-. I ,,??? 
.. s~--??' .....  -
??? ----
:_-_. 
., 0 
i N 0 \() 0 \() 0 I I ~ 
0.. .. 0.. .. 0.. .. 0 _; 0 0 0 I 
("l 
. ?-? ,_ ? N .. . - (") I __ -----
---? ..s...  ----
r::1' 
0 
N 
0 \() 0 \() 0 
0 0 
i I 
0- 0 0 - 0 0 0 I -I 
(.1s / qw) UP/ DP l.v 
_,? . N +-
? 0 j
-'-- --
--~--- ..s...  
0.. ? 
--- - -?- ----0 -
~ ? 
r::1' 
: 
0 
i 0 \() 0 \() 0 
N 0 0- 0 0 0- 0 0 0 I 
Hn / n Av 
16
Figure 2.18: LDA calculations of <T(q) and Ay for selected states of 0 at 
100 Me V; the elastic cross section is presented as ratio to the point -charge 
Rutherford cross section (<TR) to enhance detail. Line code: PH (solid) , NL 
(short dashes) . The data are from this work . 
78 
-
I 
..s...  
.., 0 
0 i N I I ~ I() 0 I() ~ 
0- 0 0 0- 0 0 0 I -I 
C') 
... 
? ? 
Cl! 
? -0--.-. i- ---
..i_::i,.  ,.s .._.._. , 
a:l 
C, 
0.. . - O" 
0 
N 
N- 0 0 i -I 0 
I() 0 I() 
0 0 0 0 ..; ~ 0 0 0 
I -I 
(.1s/ qm) up/ .op I. .V  
I 
.._, s ,..._.._. , 
? a:l C, 
0.. . 
0 I 0 0 I() 0 0 0 0- 0 0 I -I 
Figure 2.19: LDA calculations of o-(q) and Ay for selected states of 4?Ca 
at 100 MeV; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section ( o-R) to enhance detail. Line code: PH (solid) , NL 
(short dashes) . The data are from this work . 
79 
-
Cl) -I 
..8...  
N ., 
0 
i I I lO 0 lO 0 ~ 
0 0 0- 0 ~ 0 0 0 I -I 
(") 
,_ 
- Cl) (") -I 0. ..8...  
._0_.,  
0 O' 
'.2 -
0 
i N I 0 lO 0 lO 0 
- 00 0-  ~ 0 .... 0 0 0 I -I 
(.1s/ qur) OP/ DP ,.v 
(") 
,:-:.:-:--?- .. 
? ? 
Cl) 
+--0 i 0. 8 
._0_.,  .
._.._.. , 
0 O' 
'.2 -
0 
N 0 i 0 lO 0 lO 0 
0- 0 0- 0 0 0 0 I 
MD / D ,.v 
Figure 2.20: LDA calculations of o-(q) and Ay for selected states of 16 0 at 
200 Me V; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section (o-R) to enhance detail. Line code: PH (solid) , NL 
(short dashes), LR (dashes) . The data. are from this work . 
80 
? 
C') 
., 
r---- ? 
IN 
(") N ,......_ ,......_ 
I 
0. s 
,_0_.,  -,,___ ..... 
<O 
u O' 
0.. . 
'--'~~~. ......... ~~~,.1.._...__.:=....~-'-~~ _._Jo 
"' l() 0 l() 0 I I 0 0 
0 0 ... 0 0 0 ... 0 I I 
?? 
N ,......_ 
? ? I ? ..s...  
O' 
0 
l() 0 l() 0 ':' 0 
"0'  0.. . 0 I 0 0 0 ...; 0 0 0 ...; I I 
(.1s / qur) UP/ Dp "v 
? 
N ,......_ 
I 
..s...  
'--' 
O' 
0 
"' l() 0 l() 0 0 0 
0.. . 0.. . 
i I 
0 0 0 0 0 
I 
8 D / D "v 
4
Figure 2.21: LDA calculations of o-(q) and Ay for selected states of ?Ca 
at 200 MeV; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section (o-R) to enhance detail. Line code: PH (solid) , NL 
( short dashes), LR (dashes) . The data are from this work . 
81 
-- --;. 
- --.=---.:;_-:_:::.:--
+ ..... 
,0,-- ..... C\1 
,---..,_ 
~ 
0.-. I 
0... s 
'--' 
? co ..
~.._,, , 
u 
0 ....-i 
--- -.:t' ry 
) ----
0 
0 I.{) 0 I.{) 0 
....-i 0 0 0 ....-i 
I 
.. Cl") ~ 
' ' 
+ ..... 
0 C\1 
,---..,_ 
,,--..... ~ 
I 
0.-. 
0.. 8 
'--' ..~.._,, , 
:.:-:,,_??- -- ... ? ?-. 0 co 
' ..... ....-i ??: ry 
I ,: 
? / .. ?? /,?' 
~-,,."_(???? 
0 
0 I.{) 0 I.{) 0 
0 0 0 
I 
4
Figure 2.22: LDA calculations of Q for elastic scattering of 16 0 and ?Ca 
at 200 MeV . Line code: PH (solid) , NL (short dashes) , LR (dashes) . The 
data are from [Ste 85] and P. Schwandt , private communication . 
82 
N 
N .., 
0 
0 i I I 0 I() 0 I() ~ 0...  0 0.. . 0 0 ...; 0 0 0 I -I 
("') 
,_ N 
-(") i- 
~ ....8...... 0.. ,  .._, 
0 c:r' 
~ 
. ..- - 0 
i N 0 I() 0 I() I ~ 
0 0 0 0 0 0 0 0 
I 
(.rn / qm) up/ .op 'v 
, , .- -=---"'::"-
-?? -~ 
N 
+--0 -I 0.. -- ..8...  ....., 
..0_.,.  
0 c:r' 
~ ( 
0 
I() 0 I() 0 
N. .. 0 0 
i ~ 
0 0 0 0 0 0 
I 
ll.o / D 'v 
16
Figure 2.23: LDA calculations of o-( q) and Ay for selected states of 0 at 
318 MeV; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section (o-R) to enhance detail. Line code: PH (solid) , NL 
(short dashes), LR (dashes) . The data are from [Ke] 90b]. 
83 
N 
I 
..s...  
7 o~.~~~-~..,__~__.:_._:_...1-~~~--'--~~~~o 
I 
0 0 a a- 0 ~ 0 6 6 :3 I 
, ':: 
',  ,, 
I I ' 
I,:: 
N I 
I 
.s._._.._.,  , 
O" 
I I 
N '--'~~~L.....,.=...,___.__:,:;_:,.i~~~.........i.~~~~ 0 
I O ~ 0 ~ 
N 0
a- 0 a- I  a ~ _; 0 0 0 I 
(.xs/qur) UP/ Dp 
' ''
' < ' I  
? j 
I I 
'
____ j' 
t?  
+- N 
,0... 
.__, 
.._ 
0. I s 
._0_..  ..... 
aJ 
u 
~ l ? 
0 
I "I 0 ~ 0 ~ 0 
0 0 a a a 0 0 0 
I 
4
Figure 2.24: LDA calculations of a-( q) and Ay for selected states of ?Ca 
at 318 MeV; the elastic cross section is presented as ratio to the point-charge 
Rutherford cross section (o-n) to enhance detail. Line code: PH (solid) , NL 
(short dashes), LR (dashes) . The data a.re from [Fel 90] . 
84 
p-
l 
____--_-:: .:: -----.- .-: .......: : _,_ ____ -
---=:.=--==-==-=--==-==-:::r==:.=::.:.:::.:.:---
C\1 ,.--..... 
...... 
I s 
_t+-_--1 , 
- o-4 
0 
0- l[) 0 l[) 0 0 0 0 I -
Figure 2.25: LDA calculations of Q fo r elas ti c scattering of 4?Ca. at 
318 MeV . Line code: PH (solid) , NL (short dashes ), LR (dashes). T he 
data are from [Ble 88] . 
85 
2.5 Relativistic Impulse Approximation 
2.5.1 Introduction 
The failure of nonrelativistic calculations to reproduce elastic scattering 
data at 500 MeV, especially the forward angle spin observables, and the 
problems of the NRIA even at higher energies lead to the question whether 
the underlying theory of nuclei is in fact relativistic. Clark et al. showed a. 
phenomenological fit to the 500 MeV a-(q) and Ay data which was obtained 
with the Dirac equation using large scalar (S ~ - 300 MeV) and vector 
(V ~ +250 MeV) potentials [Cla 83] . They showed that without further 
variations of the fitted parameters the third observable for elastic scattering, 
the spin rotation function Q, was predicted correctly. 
Spin observable measurements in intermediate energy proton scattering 
provide information about the spin-orbit interaction which, at a fundamental 
level, is due to the strong spin dependence of the NN interaction. Spin 
phenomena are difficult to understand within a nonrelativistic framework. 
They arise however very naturally within the relativistic Dirac theory. The 
fact that Dirac phenomenology appeared to be so successful in linking the 
spin observables provided a major incentive to develop a rigorous relativistic 
theory for proton-nucleus scattering. In 1983, therefore , a relativistic version 
of the impulse approximation was developed by McNeil , Shepard and Wallace 
[McN 83] . The success of these relativistic calculations , especially of th e 
recent IA2 [Ott 88], in describing all the elastic scattering observables above 
300 MeV is remarkable . The essential difference between the Schr "dinger and 
86 
the Dirac approach has been identified as the implicit inclusion of virtual N N 
pair effects . 
Some of the other topics in relativistic nuclear physics, which are be-
yond the scope of this work , include the bound state problem of nuclear mat-
ter. Here , t he combination of attractive scalar and repulsive vector potentials 
provides a simple mechanism for nuclear saturation. A field of research which , 
in analogy to quantum electrodynamics (QED), is called "quantum hadro dy-
namics" or QHD, provides a complete and consistent theoretical treatment 
of nuclei and includes a complete description of all the baryonic and mesonic 
degrees of freedom in the nuclear system [Ser 86] . Finally, relativistic theories 
also allow t o unify the theoretical descriptions of nucleon-nucleus scattering 
with that of NN scattering and , to a certain extent , with that of meson 
exchange currents . 
In the following we will give a brief outline of the relativistic IA2 model 
and its interpret ation, and define the Schrodinger equivalent potenti als which 
will be compared in Chapter 7 to potentials formed from our empiri cal non-
relativistic m odel. Most of the presented material is taken from the following 
sources, and references therein : [Che 86, Wal 87a, Wal 87b , Ott 88 , Ott 90] . 
2.5.2 The IA2 Model 
In the relativistic theory, wave p ro pagation of a nucleon with mass m , 
m omentum p and energy E = p 2 + m 2 is described by the Dirac equ ation 
[E--y? - 1 ? p - m - U]?j; (r) = 0, (2.83 ) 
87 
where the optical potenti al U contains apart from the att ractive scalar poten-
tial and the repulsive vector potential , which were used in the phenomenol-
ogy, a small t ensor potential and two even smaller scalar and vector spin-orbit 
potentials 
U(r) = S(r) + , 0 V(r) - io ? r T (r) - [SLs(r) + , 0 VLs(r)]o- ? L . (2.84) 
This form of the optical potential is derived from a optimum fact orized 
, :: : 
tp approximation in mom~ntum-space, very similar t o the nonrela ti visti c ,, ? f ' 11 , 
'I  ' I ~ ' 
case, Eq. (2 .24) , when t is replaced by the Feynman amplitude J,.J and t he ' . 
r ' I 
nonrelativistic form factor by a rela tivistic one , p( q ) . More specifically, one ! 't i I I 
I' 
' 
can write 
(2.85 ) 
In this expression , the trace is over the Dirac indices of particle 2, conven-
tionally designated to be the target nucleon. The nuclear form factor , when ''''' 
?'
 II  
 .I  
based on the Dirac-Hartree model of nuclear shell structure, is compri sed of I ' 
? I 
I I 
three distinct form factors : scalar , vector and t ensor. These form fact ors are , 
determined in the IA2 from the wave functions of Horowitz and Serot which 
were obtained by solving the Dirac equation with large scalar and vector 
potentials from meson exchange models . 
The correct construction of the Feynman amplitude l'vf is criti cal fo r 
the quality of the optical potenti al and has been shown to explain t he supe-
rior description of da ta by the present IA2 model, as compared to previous 
versions of the relativistic impulse app roximation . In simple terms, th e NN 
amplitude in the full Dirac space of two nucleons can be expanded in terms 
88 
of a complete set of Lorentz invariant amplitudes times kinematical covari-
ant s fo rmed out of Dirac matri ces and the four-momenta which characterize 
initial and fin al scattering states . The kinematical covariants must properly 
take into account parity invariance, time-reversal invari ance , charge symme-
try and the generalized Pauli principle. They must be linearly independent 
and the associated amplitudes must be free of kinemati cal singulariti es . 
For on-shell kinematics , 44 independent amplitudes are needed in each 
isospin channel. Of these, only the five Fermi amplitudes which are part of 
the positi ve-energy sector of Dirac space are determined directly from phase 
shift an alysis of NN scattering data . Their contribution to the full Feynman 
amplitude is 
5 
AI (p , q) = - ry L FJ { i , (2.86) 
i = l 
where T/ is a kinematical factor , and the Fermi covariants are K 1 = 1 (scalar ), 
K2 = 11 /2 (vector) , K3 = a-rv <T2? v (tensor) , K4 = ,h~ (pseudoscalar) and 
K 6 = ,h~1 112 (axial vector) ; the Fi are the corresponding amplitudes . T he 
remaining 39 amplitudes have vanishing matrix elements in the positive-
energy states and must be predicted from an adequate model which allows 
an unambiguous extension to all sectors of the Dirac space . T he method 
adopted by the IA2 as a dynamical basis for this t ask is a relativistic me-
son exchange model with appropriate vertex cutoffs. In elas ti city in th e NN 
channel is included vi a couplings between NN , N 6 and 66 states. T he cou-
89 
pied channel integral equations are solved in a Bethe- Salpeter framework22 
with pseudovector 71" N coupling (important for maintaining reasonable be-
haviour a t low energies and large momenta ) and are easily extended to in-
clude negative-energy intermediate stat es. 
In order to treat nucleon exchange contributions to the optical poten-
ti al, each of the 44 amplitudes is fitt ed by a direct and exchange part , very 
similar to Eq. (2 .31), and thus each amplitude is explicitly symmet ric or an-
tisymmetric with respect to particle exchange. The exchange term s are then 
localized, as was done for the nonrela tivistic optical potential in Section 2.4 .4, 
t o yield a local optical potential suitable for coordinate-space analysis. 
2.5.3 The Virtual Pair Interpretation 
The basic new ingredient of the Dirac approach is the implicit coupling 
to virtual nucleon-antinucleon states through the Dirac optical potenti al U. 
This contribution introduces the additional density dependence which ap-
pears to be necessary to describe data above 300 MeV. 
In the relativistic theory the central scalar and vector potenti als are of 
considerable strength and therefore can allow the projectile nucleon to scatter 
2 2 The B ethe- Salpeter equation is the four-dimensional covariant version of the 
Lippma nn-Schwinger equation , the NN amplitude here being the sum of a ll di st inct Feyn-
man diagrams from field theory in which the two particles a re inte racting through the 
exchange of fi eld bosons . 
90 
to a virtual negative-energy state23 at time, say, t2 . In this state the nucleon 
propagates backward in time until an earlier time t 1 < t 2 when it is scattered 
back into a positive-energy state which then propagates forward in time. The 
backward going negative-energy state is equivalent to a positive antiparticle 
state propagating forward in time. Therefore, in the time interval between t 1 
and t 2 in addition to the incident nucleon a virtual nucleon-antinucleon pair 
is present . Figure 2.26 shows the time-ordered process [Wal 85] . 
In the following, we will try to quantify the differences between the 
nonrelativistic and the relativistic approaches to the optical potential. For 
this purpose it is very instructive to find an equivalent Schrodinger potential 
which yields the same scattering as the Dirac potential. This can be accom-
plished by writing the Dirac equation as coupled equations for the upper-
and lower-component wave functions , <I> and x (for simplicity, we consider 
here only the scalar and vector potential fields) 
(E - m - S - V)<I> - u ? p x = 0 
(2.87) 
u ? p <I> - (E + m + S - V)x = 0. 
The second equation yields for the lower component 
U?p 
X = -----<I> . (2.88 ) 
E+m+S - V 
Substituting back into the equation for th e upper component , one gets 
1 
(E - m - S - V - u ? p-----u ? p)<T> = 0. (2 .89) 
E+ m +S - V 
23 The amplitudes for these negative-energy states vanish when the scattering nu cleon 
is far away from the nucleus ; they are virtuall y present, however , in t he in teraction reg ion 
where the optical potenti a l is strong, i.e . in the immediate vicinity of the nucleus. 
91 
-+-- z 
Figure 2.26: Z-graph for a Dirac particle scattering from an external 
potential at times t 1 and t2, where t i < t2. 
92 
Assuming the potentials S and V are almost constant with respect tor, allows 
us to neglect the spin-orbit and Darwin terms. Rescaling the wavefunction by 
the substitution 1/; = E+m~S - V <I> leads with k2 = E 2 m 2 - to an equation in 
the form of a Schrodinger equation for the central potential with relativistic 
kinematics: 
(2 .90) 
where the equivalent Schrodinger potential is 
(2.91) 
If one assumes a local tp model for the optical potential, one can set S = sp 
and V = vp and write Eq. (2 .91) as 
m 1 2 2 2 
UNR = tNRP = ( Es+ v)p + E(s - v )p . (2 .92) 
2
Since only the first term of tNR remains in the limit p ---, 0 limit , the first term 
of the potential U NR comprises the nonrelativistic impulse approximation . 
The second term represents the relativistic, density dependent correction 
due to virtual nucleon-antinucleon pairs . Since in actual nuclei the absolute 
value of s is larger than that of v, the relativistic correction manifes ts itself as 
a short-range repulsion which becomes stronger in the nuclear high-densi ty 
interior region . 
Other form s for the two terms in Eq . (2 .92) can be fo un d by expanding 
the Dirac wave fun ction in terms of positive and negative-energy states 1/; 1 = 
A ?1/; of the potential-free Dirac equation24 . Elimination of negative-energy 
24 The projection operators are defined as A? = ?{~,"', where jJ = -y"Er - 1 ? p . 
93 
components from the Dirac equation then yields an equation for the positive-
energy component. At the level of approximations that has been used before, 
one gets 
(2.93) 
where (;?? = A ?(JA ? and where the pair potential is 
{;pair = {J+ - [E + Jp 2 + m 2- {J--J-l {J - + . (2.94) 
The potentials(;+ - and(; - + are responsible for the scattering into and out of 
the negative energy states. In Figure 2.26 these potentials are represented by 
the little bubbles at end of the wavy lines. Thus, the equivalent Schrodinger 
central optical potential is essentially the combination of a part (;++, which is 
the sum of strong scalar ( ~ - 400 Me V) and vector ( ~ +300 Me V) potentials , 
and a pair term which includes coupling to the other energy sectors of the 
Dirac space. N onrelativistivcally, the 2 22~ ( S - V ) term is linked to the spin-
orbit force . 
If one includes additionally medium effects like Pauli blocking etc., a 
density and energy dependent interaction for elastic scattering at intermedi -
ate energies is well described by [Che 86] 
(2 .95 ) 
or , with the Brueckner g-matrix by 
t(p) ~ g(p) + 5tnc(P), (2 .96) 
where 
5tnc(P) = -1 (s 2 - v 2) p. (2.97) 
2E 
94 
The 1/ 2E factor in the relativistic correction term suggests that in the high 
energy region the Dirac and Schrodinger approaches might converge. 
2.5.4 IA2 Calculation Results 
In Figures 2.27-2.30 we compare the full IA2 op tical potential (solid) 
for elastic scattering of 200, and 318 MeV protons from 160 and 4?Ca with 
a nonrelativistic LDA potential based on the PH interaction (long dashes). 
In the "no-pair" analysis of the IA2, all couplings to negative-energy states 
are eliminated and all the relativistic nuclear densities are taken to be equal 
to the measured nonrelativistic matter density. The Schrodinger potenti al 
equivalent to the Dirac potential from the "no-pair" analysis is called the NP 
potential (dashes) . Finally, for comparison we also display an NRIA potenti al 
based on the PH t-matrix. Comparisons of the IA2 with our empirical model 
and a recent , very successful Dirac phenomenology by Rama et al. [Ham 90] 
will be made in Chapter 7. 
At 200 MeV the real central IA2 potential is , although on the whole still 
attractive, much more repulsive in the nuclear center than any of the other 
potentials , including the LDA potential which is about intermedia te between 
the IA2 and the NP and NRIA potentials. Like the LDA potential , the 
IA2 potential has a "wine- bottl e" shape. The NP potenti al is slightl y more 
repulsive in the center than th e NRIA potenti al but , like th e NIUA potc11ti al, 
does not have the surface depression of the .LDA or IA 2 pote11ti a.l. Fn r I.h e 
imaginary central component , the IA 2 potenti al is virtually identi cal to th e 
NP and NRIA potentials , being only sli ghtly less absorptive for 4?Ca. This 
95 
... 
fact is due to the absence of Pauli blocking . T he real spin-orbit potentials 
are essentially all the same for 160; for 4?Ca the IA2 potential resembles 
somewhat the LDA potential. The imaginary spin-orbit potentials differ 
significantly, but are again too small to affect calculations significantly. 
At 318 Me V both the IA2 and the LDA potentials become repulsive in 
the nuclear interior , while the NP and NRIA potentials stay attractive. The 
IA2 potential stays repulsive even at the center where the LDA potenti al is 
again attractive . The strong repulsion in the real part is due to the virt ual 
N N pairs . In the imaginary central part of the optical potential for 16 0 we 
see an interesting effect whose discussion we will pick up again in Chapter 7 
in context with our empirical effective interaction: the IA2 imaginary central 
potential at this energy becomes more absorptive relative to the NRIA po-
tential. This effect, which becomes very pronounced for 500 Me V, is in thi s 
work often called "anti-Pauli" blocking. 
96 
N 
'---~--'-- ~....L.:..l-_:_.____---'---~~- 0 
0- I() 0 I() 0 IO '<I' C') N 0 0 0 0 - o o o o o I I I 
(N~W) Slf1 Ulf f If 
' ' 
" 
I ' 
I() ',, 
' 
''  ' 
"\,,
. ), ,,,  
( '\. 
,, 
\', 
'?~ N 
--,?-\:-.-. ,?:..??.:.?::-~-;----- \ 
?~, 
' '----'-----''------'----'--1...l.-..-i:~_j 0 
0 I() 0- I..(.)  0 10 0 1()0 IO 0 I..(.)  0 I() 0 I() I C\l C\l C') C') I C\l C\l C') C') I I I I I I I I I I I I 
Figure 2.27: Comparison of the full IA2 (solid) , LDA (long dashes ), NP 
(dashes), and NRIA ( short dashes ) optical potentials for 200 Me V elastic 
scattering by 160 . Both LDA and NRIA are based on the PH interaction . 
97 
<O 
'- > l{) <l) 
--;:::.: 
P.o 
...__, 0 "' 
<O C\l -
U II M ..s 
'; " ..._.._. , w 
I,., 
C\I 
/' --
~ 
0 
0 ~ <I' <O a:, 0 C\I M C\I 0 
ci 0 ci ci ci _; 0 ci c
-i ci 
I I I I I I 
(i\8 W) ill/ 
(1\ s,fl a w) s,O <Jcf 
<O 
l{) 
"' 
, .., ., ~ -~ .... a , .. ..... ,._ .... M ...__, 
'-?--.,:_. _, 
:--
- ..,  ~ ?-:" I,., __ ?? ~ C\I 
"\ 
\ "'?< .,, ?,._ '<-I. \ I \ 
! \ ', 
( 'l 
0 
0 l{) 0- I-ll 0 Ill 0 0 II') 0 II') 0 I(:, 0 II') I (\J (\J M I (\J (\J M M I I I I I I -I I I I I 
(N~W) :>{). <J cf (i\~W) il W,J 
Figure 2.28: Comparison of the full IA2 (solid) , LDA (long dashes ), NP 
(dashes), and NRIA (short dashes) optical potentials for 200 MeV elasti c 
scattering by 4?Ca. Both LDA and NRIA a.re based on t he PH interacii n . 
98 
<O 
!> ll) 
o::::V;  
__0__: c..o...  "" 
0 C") ~- -s
~ II ~ - (") ..... 
 
a. '-' 
t.J )'\ 1-.. 
N 
1/ 
_-pJ--~ --
V 
0 
0.. . ll) 0 IO 0 I.O..  (") C\I 0 0 0 0 0 0 0- 0 0- C\I  0 
I I I I I 
(Aaw) s,O ocf (Aaw) s,O 'Ulf 
<O 
ll) 
"" -s 
(") ..... 
'-' 
1-.. 
N 
' 
---' ~ ... .. .. :..--:. ...... 
?. ..'  
0 
0.. .. IO 0 IO 0.. .. I-(} 0 0 I(} 0- I-(} 0 I(} 0 I(} I C\I I C\I C\I (") (") I I I I I I I I I 
(/\aw) ::>(l ocJ (/\aw) :)fl W./ 
Figure 2.29: Comparison of t he full IA2 (solid) , LDA (long dash es) , NP 
(dashes), and NRIA (short dashes) optical potentials for 318 MeV elasti c 
scat t ering by 160 . Both LDA and NRIA are based on the PH interaction . 
99 
ID 
N 
0 
0 
C\I 
o"' 6 0 
<O ~ 0 
0 C\I s1' 
c:i c:i 0 0 I 
I I 
I 
(0 
\ ID .. 
""? 
---~---._ 
\--~ 
N 
\, 
' i-. ' I I 
I ' 0 
I 
I() 0 I() 0 
0... . I..D..  
0 
C\I "' "' "It' ID C\I I I I..(.).  oO I I I I 0
0... . ll') 0 ID ...
 . C\I I I I 
I I I 
F"i gure 2.30: Comparison of the full IA2 (soli.d ) , LDA (long dashes) , NP 
(dashes), and NRIA (short dashes) optical potentials for 318 MeV elastic 
scattering by 40Ca. Both LDA and NRIA are based on the PH interaction 
100 
Finally, in Figures 2.31 and 2.32 we compare IA2 calculations (solid) for 
elastic scattering of 200 and 318 MeV protons from 160 and 4?Ca with LDA 
calculations (long dashes ) based on the PH interaction, with NP calculations 
(dashes ), and with NRIA calculations (short dashes) which are based on the 
free PH interaction . 
For 200 MeV we observe that although the LDA provides a better over-
all description of the cross section data and does slightly better on the analyz-
ing powers at small momentum transfers, at larger momentum transfers the 
IA2 follows more closely the large oscillations prescribed by the data . The 
IA2 is superior for the spin rotation functions , especially at forward angles. 
For 318 Me V the biggest improvement achieved by the IA2 is the su ppres-
sion of the forward angle analyzing powers. LDA cross section predictions 
are slightly better overall than the IA2 predictions. LDA predictions for the 
4?Ca spin rotation function are equivalent in quality to IA2 predictions. At 
both energies, the inclusion of density dependence , relativistic or nonrela-
tivistic, leads to a dramatic improvement in the description of the data. As 
the figures show, the NP and the NRIA predictions are very similar. The 
slight differences are due to the fact that the NP potentials were const ructed 
with Dirac-Hartree wave functi ons and phase shifts, while the NRIA was 
constructed from measured ( e, e') charge densities and the PH t-matrix . 
101 
?~-
+-
+- ,0
N 
,... ..._ ,,......_ ? ,0,... ..._ 0.. I 
0.. 
- 0.. 8 0.. - ..... <tl 0 {._) 
~ 0..  --O" 
0 l() 
0 ci 
I 
.?  ? 
+-
,0,... ..._ 
0.. I 
-0.. <tl 
.{.. _) -
..8...  
0 
0 
N 
0 l() 0 l() 0 
0 Q- 
i I 
0 0- 0- ci ci ci I -I 
.. M 
? 
+- N ,,......_ 
,0,... ..._ I 
0.. 
- 8 0.. ..... 0 --O" ~ 
0 
l() 0 l() 0 
N 0 Q 
i 0 
0.. .. 0 0 - ci ci ci ..; I I 
Hn / [) 'v 
Figure 2.31: Comparison of IA2 (solid), LDA (long dashes), NP (dash es ), 
and NRIA (short dashes) calculations for elasti c scattering at 200 MeV. Elas-
tic cross sections are presented as ratios to the point-charge Rutherford cross 
section (o-R) to enhance detail. Both LDA and NRIA are based on the PH 
interaction ( data for a- and Ay from this work, data for Q from [Ste 85] and 
P . Schwandt, private communication) . 
102 
??????=-=?=? 
+-
,.0... ..._ 
0.. I 
..8...  ..__, 
0- ll') 0 ll') 0 0 0 ci I I 
+-
0 
,-._ 
0.. . N ,-._ I 
..0_.,.  ..8 .._.._. , 
crJ 
l.) 
0.. . - O" 
0 
i N 0 I 0- ll') - 0 ll') 0 0 0 0 0 0 0 0 I -I 
(") 
- -, 
+- N ,-._ 
0 
,-._ I 
0. 8 
.._0_.,  ..... 
0 
~ - O" 
0 
0 i 0 ll') 0 ll') 0 "0'  0- 0 0 0 0 0 -I I 
Hn / n ?v 
Figure 2.32: Comparison of IA2 (solid) , LDA (long dashes ), NP (dash es ), 
and NRIA (short dashes) calculations for elast ic scat tering at 318 MeV. Elas-
tic cross sections are presented as ratios to the point-charge Rutherford cross 
section ( o-R) to enhance detail. Both LDA and NRIA are based on the PH in -
teraction (data for o- and Ay from [Fe] 90 , Ke! 906], data for Q from [Ble 88]) . 
103 
2.6 Off-Shell Effects and Full-Folding 
The optical potentials introduced in the last two sections, the ones 
based on nonrelativistic density-dependent interactions as well as the ones 
based on a relativisic impulse approximation model (RIA), give very good 
descriptions of nucleon-nucleus scattering data at intermediate energies. In 
contrast, the NRIA seems to have failed in this task. 
It was noted , however , by Redish and Stricker-Bauer [Red 87] that the 
g-matrix and RIA approaches are local models which disregard the effect 
of the full off-shell interaction . It was found that an optical model which 
retains this nonlocality can reproduce some of the characteristics of elas tic 
spin observables which have been generally attributed to medium effects such 
as Pauli blocking and/or relativistic effects . 
In an earlier study, Pickelsimer et al. [Pie 84] tried to es timate the role 
of nonlocality, off-shell, and relativistic effects using the free NN t-matrix 
of Love and Franey [Lov 81, Fra 85] . In particular , they compared various 
factorizations with the optimum factorized potential. It was found that non-
locality only plays a role at relatively large angles above about 60? and / or 
energies lower than about 150 Me V. At 500 Me V nonlocal and off-shell ef-
fects were found to be unsufficient, by themselves , to explain the data; the 
relativistic description was generally superior. However, as was explain ed in 
Section 2.3 .4, the method used to extrapolate the on-s li ell t -matrix off th e 
energy shell probably has serious ambiguities and problems and , hence , the 
resulting off-shell behaviour is probably not realistic. 
Cheon [Che 87], on the other hand , suggested in his paper that a non-
104 
relativistic theory with a realistic off-shell t-matrix and proper t reatment of 
the nonlocality can indeed successfully describe intermediate energy nucleon-
nucleus scattering . Subsequently, other authors studied off-shell and full -
fo lding effects using interactions based on the Bonn potenti al [Els 89, Els 90] 
a nd on the Paris potenti al [Are 89, Are 90a] . 
Elster and Tandy [Els 89] compare optimum factorized off-shell and 
on-shell tp calcula tions and find substantial effects for the spin observables, 
in particular an improvement in A y, of 4?Ca at 200 MeV. At 500 MeV they 
also find a strong off-shell effect on the spin observables; the cross section , 
however , is only being m oderately affected in its diffraction minima. Im-
provem ent with the off-shell potential is only found for the spin rotation 
fun ction Q, but an impairment is seen for the analyzing power Ay. 
It was suggested by Pickelsimer et al. [Pie 84] that uncertainties are 
being introduced in the optical potential by uncertainties in the NN t-matrix , 
as well as by the nuclear density and the applied fact orization method ; the 
uncertainties being largest for very light nuclei and back angles. It was ., 
argued , therefore, that these cases require a complete folding calculation 
involving a realistic off-shell t -matrix. 
Taking up this point, full-folding studies were done by Elster et al. 
[Els 90] and by Arellano et al. [Are 89 , Are 90]. The form er paper compares 
local on-shell, optimum fact orized off-shell , and full -fold .ing ca.lculatio11 s fo r 
160 . It is found that the optimum fact orized off-shell potential is a very good 
approximation t o the full-folding potenti al for energies as low as 100 Me V. 
The main source of nonlocality in the optical potenti al was identifi ed to be 
105 
the off-shell structure of t he t -matrix . In this calculation , the energy variation 
of the interaction was ignored and t he energy z in Eq . (2.18) was fixed at 
the two-body c.m . energy corresponding to free N N scattering at the beam 
energy. 
In the calculations by Arellano et al. the full energy-dependence of 
the t -m atrix is included . They find , in cont rast to Elster et al., significant 
differences between calculations using full-folding potenti als and even the 
off-shell optimum factorized potential. The full -folding provides a superior " ? .? f  
,, , 
description of the data at both 200 MeV and 300 MeV . These latter calcula- . " 
. 
tions might be consistent with the former ones by Elster et al. if the energy ' ~ 
vari a tion of the interaction indeed plays an import ant role in the full-folding. I ..,  I  
The studies which we have just described have caused new interest in ..,   
the nonrelativistic impulse approximation and they have certainly motivated 
a critical review of density-dependent interactions and /or relativistic models. ' 
However , it has been observed recently by Ottenstein [Ott 90] that a correct ..  
1, 
treatment of the Coulomb interaction , as well as a correct treatment of the 
.;
?' 
,'  
boost from the NN c.m . frame of the t-matrix to the frame in which the op-
tical potential is evaluated , is very crucial for a meaningful interpretation of 
the results and a judgement of the size and importance of off-shell effects . All 
of the nonrelativistic off-shell and full-folding studies mentioned above use a 
nonrelativistic Galilean boost ( the lVfo ller factor) instead of th e correct rela.-
ti vistic boost which includes Wigner rot ati on effects [O tt 90] . Furthermore, 
the above calcula tions used various nonequivalent methods fo r treating the 
Coulomb interaction . Ottenstein et al. conclude that th e observed off-shell 
106 
effects in [Els 89] are probably too large. 
2. 7 Inelastic Nucleon-Nucleus Scattering 
2.7.1 Distorted Wave Approximation 
Since it is the main purpose of the present work to extract an effective 
interaction from inelastic scattering observables, in the following we will give 
a brief and more or less schematic summary of inelastic scattering. For our 
calculations we will assume that the transition potential is sufficiently weak 
so that its effect can be calculated in perturbation theory and we will further 
assume that we are dealing with direct 26 , one-step excitations which can be 
described accurately in the distort ed wave approximation (DWA) when using 
an effective interaction . 
In the DWA the total scattering potential U is decomposed into a dom-
inant term which is responsible for elastic scattering (the optical potential 
U0 Pt) and a perturbation which induces inelastic transitions (the transition 
potential D.U) 
U = Uopt + tiU. (2 .98) 
The optical for elastic scattering was discussed earlier. Because of the refrac-
tive and absorptive features of the optical potential , the solutions x to the 
Schrodinger equation 
( 2. 99) 
25 A very thorough treatment of direct reactions can be found in the book of Satchler 
[Sat 83). 
107 
are not plane waves, but so-called distort ed waves . Asymptotically, they have 
the form of an incident plane wave plus outgoing scattered spherical waves , 
hen ce the ( + ) superscript 26 . 
The transition amplitude for inelastic scattering in the DWA is the 
matrix element of the transition potential, evaluated between the distorted 
waves of the initial and final channels 
(2.100 ) 
Analogously to elastic scattering , the transition potential can be described by 
an effective interaction , at- or g-matrix, fold ed with the transition density /:).p 
which carries all the nuclear structure information , i. e. !:).U(q) = t(q)!:).p(q) . 
In the plane wave limit , the differential cross section and analyzing 
power for natural-parity transitions are simply given by Eqs . (2.46) and (2 .47) 
from Section 2.3.4. For the general case of distorted waves, one constructs 
instead the quadratic forms 
.,,,  
(2 .101) .'?, '  
which are traces over the spm projections of both the projectile and the 
target [Kel 896] . One can identify 
(2 .102) 
Here, D00 = 1 and the analyzing power Ay = Dyo are simply elements of th e 
depolarization matrix D013 . The cross section can now be compactly written 
26 The distorted wave x< - l( k , r) = x<+r( - k , r ) has incoming spherical waves [Sat 83] . 
108 
as 
??' k' I0(q) 
a-(q) = (21r) 2 I 21 + 1 ? (2 .103) 
* * * 
There are certain corrections which have been proposed to be important 
specifically for inelastic scattering : first , the effect of nonlocality in the optical 
p otential on the interior wave function and, second, the effect of certain 
diagrams in the hole-line expansion of the effective interaction t which are 
not present for the elastic interaction. 
2. 7.2 The Perey Effect 
Elastic scattering is m ore or less insensitive to the interior of the nucleus 
and the fine details of the optical potential , only measuring the aymptoti c 
properties of the scattering wave function . On the other hand , inelastic 
scattering, transfer, and knockout reactions require in the DWA treatment 
the elastic nucleon-nucleus wave function in the nuclear interior. Thus, al-
.:?''   
though potentials with different nonlocal structure may produce equivalent ?" 
elastic scattering observables results , they still may yield different interior 
wave functions. 
Nonlocal optical potentials were first studied in the effective mass ap-
proximation [Fra 57 , Wya 60 , Lem 60]. T he Schrodinger equation with a 
nonlocal kernel 
(E - K) x(r) = j d1?' v( r , r') x( r') , (2.104) 
was reduced t o the usual form, Eq . (2.1) , but with K = - 'v 2 / 2? replaced 
109 
by K = - ?{V 2 2:. + V 22:. V + 2:. V } . Here? ? denotes a spatially variable 
effective mass which is defin ed as 
(2.105) 
In this expression, the parameter /3 is of the order of 1 fm and characterizes 
the range of nonlocali ty in the potential. It is defined such that f3 = 0 
corresponds t o the local case. In the literature, the ratio P(r) = ?? (r) /? is 
oft en called the Perey factor . ,, . ? 
The effective mass approximation 1s based on the assumption that "??.'   
k/3 << l , where k is the wave number of the nucleon . Since f3 has been 
'I 
found empirically to be of the order 1 fm , however , k/3 is expected to be 
.,  
small only for very low energies . Perey and Buck [Per 62, Per 63], therefore, .. . ;  
chose a different approach. They found that over a wide range of energy the ,, 
results predicted by a nonlocal model could always be reproduced by using 
1 I 
an equivalent local optical potential. For a given nonlocal potential depth . ..'I '  ' ? I 
J I 
UN, they find that at a given energy E the corresponding local potential . .I I.=,?  ' 
 
 
: I 
depth UL should be chosen such that 
(2 .106) 
This implies that for a fixed nonlocal potential depth the equivalent local one 
decreases with energy [Per 62] . For very small energies E ~ UL , an ex pausion 
of the exp onential function yields to fir st order an expression which includ es 
the P erey factor from the effective mass approximation 
(2 .107) 
110 
M ore general treatments of the nonlocali ty which were applied to energies 
up to 500 MeV can be found in [Che 87, Els 90] . 
The equi valent local potenti al by dint of it s construction gives t he same 
asymptoti c wave fun ction as the nonlocal one. However , Perey [Per 63] find s 
tha t the wave functions differ in the region of the potential. More specifically, 
he finds a reduction of the wave functi on in the nuclear interior by a facto r 
of j P( r) due to the nonlocality of the original potential. While for surface 
excitations the effect of this reduction should be small, only moving the 
interaction to a slightly larger radius , he generally expects , however , the 
nonlocali t y to reduce the contributions of interior wave functions to transition 
ma trix elements for DWBA calculations . 
Several authors find the Perey fact or in the nuclear center , P(O) , to be 
approximately 0.8 for medium energies ( e. g. [Els 90]) . With this the radial 
dep endence of the P erey factor can be parametrized by 
P(r) ;:::: A + Bp(r) / p0 , (2 .108) 
where A = 1, B :=::: - 0.2, and the saturation density is p0 = 0.16 fm - 3 . 
2. 7 .3 The Rearrangement Effect 
In the past it had been observed that the BRG interaction with it s 
st ronger density dependence described inelas tic scattering bet ter than th e 
PH interaction . The PH interaction , on the other hand , was superior to th e 
BRG interaction in the description of elastic scattering data . A consistent 
description of both elastic and inelastic scattering with the PH interact ion 
111 
( a nd also the disqualifi cat ion of the BRG interaction) was reached in t he mid 
1980s. It was recognized that an import ant rearrangement cont ribution in 
the di agrammati c expansion of the inelas tic interaction , not present in the 
elas ti c interact ion, had been overlooked [Che 85] . 
T he effect is m ost easily explained in the collective model and with 
the tp folding approach to the optical potential , i .e. U = t(p )p ; here, t (p) 
is the Brueckner g-matrix and p is the ground state density. In the simple 
case tha t an inelastic t ransition can be considered as the excitation of a spin-
and isospin independent collective mode of the t arget nucleus, the transition 
density 6.p can be described as a vari ation of the ground state density p 
with respect to the collective variable implicitly included in p . With thi s 
interpre tation one can express the transition potential 6. U as th e functi onal 
derivative 
6.U = 5U !).p . (2. 109 ) 
5p 
With the folding prescription , the consistent microscopic transition potent ial 
becom es 
5
6. U = { t(p) + p ~~)} 6.p = { [1  + p : pl t(p)} 6.p , (2.110) 
a result which also holds for transition potenti als that include relativisti c cor-
rec ti ons [Che 86] . The density deri vati ve of th e g-matrix introduces terms of 
second order in the number of g-matrices . T he di agrams associa ted with th ese 
deriva tives are, therefore, intimately rela ted to the Brueckner rearrangement 
t erm in the two- hole line contribution to the self-energy [Hiif 72] . 
The (1 + po/ op) fact or causes an elas tic medium modifi cation which 
112 
1s proportion al t o k';., t o be enhanced for inelast ic scattering by a factor of 
(l + n / 3) . As we will see later , this leads to medium modifications for inelastic 
scattering which are a bout twice as st rong as those fo r elas ti c scat tering. 
2.8 An Empirical Effective Interaction 
2.8.1 Description of Theoretical Interactions 
The effective interactions which were discussed in Section 2.4 .1 include 
m edium effects due to phase space blocking and binding energy corrections 
of intermedi ate scattering states, as well as due to the various averagmg 
procedures which were used to construct convenient local forms . However , 
although the general features of the medium modifications are common to all 
the interactions, we saw in Section 2.4 .5 that their quantit a tive predictions 
for nucleon-nucleus scattering may differ , the differences arising from both 
the different input NN potentials and the different approximations schemes 
used in the construction of the local form of the interaction . In order to 
compa re the various effective interactions more easily, it is convenient to 
search for an empirical parametrization which is guided by nuclear m a tt er 
theory. Another useful feature of such a parametrization is that parameters 
from fits t o data can easily be compared to those obtained from theoreti cal 
m odels . 
To gain som e insight into the nature of t he medium modifi cations to the 
effective interaction , we want to consider the dominant isoscalar spin -inde-
113 
p endent central component 27 , t~0 . Qualitatively, t he medium modifications 
can be interpret ed with help of the correlated pair wave function 7/; in nuclear 
m a tte r . R ela tive to the uncorrelated wave pair fun ction ?, the correlated 
p a ir wave functi on 7/; in a low-density environment is reduced in amplitude, 
p ossesses a wound 28 in the repulsive core region , and is shifted inwards by 
the over all a tt raction of the interaction . In a high-density environment , on 
the other hand , Pauli blocking inhibits scattering and absorption , enhances 
the amplitude relative to the unblocked wave fun ction and also enhances the .'  
' -.J  I 
wound as it keeps the nucleons apart . Finally, the overall attractiveness of ' ..  
.?  I 
the interaction is reduced , shifting the wave function to a phase intermediate 
I '':  l 
I 
b e tween the noninteracting and the unblocked interacting case [Kel 82] . ' ,?  'I  
These effects can be clarified by examining separately the imaginary 
a bsorptive, and real part of t~0 (see for instance Figure 2.34) . It appears 
tha t the density dependence of the absorptive part is best described by a 
',  'I  
multiplicative damping fact or which suppresses the total cross section . A ...  ' 
' t 
simple phase space estimate of this blocking fact or , due to Clemente! and Villi ,, I I 
:,,,
I  . ?  ' I 
' 
[C le 55], evaluates the effective laboratory cross section ((J" ) for scattering of 
a nucleon with kinetic energy E0 from a Fermi gas with Fermi momentum kF 
27 Density dependence is less important for the spin-orbit interaction because relati ve 
s-states a re absent [Ke! 89a] . 
28 Th e def ect wave fun ction is defin ed as ( = rp - if; a nd goes t,o zero for lm ge valu es 
of separation 8 as t he wave funct ion heals. Th e in tegral w = J drdr ' 1(1 2 is oft en call ed 
t he wound in the wave function a nd is the probabi lity of findin g the actua l system exc ited 
o ut of t he noninteracting Fermi sea, re la tive t o the pro babili ty of find ing it in the ground 
st a t e [Pre 75] . 
114 
,. 
under the simplifying ass umption t hat the free two- nucleon cross section cro 
is ind ep endent of energy and isotropic in the center of mass [Ke] 82]. With 
dimensionless quantities which will be convenient later , "-F = kF / kFn and 
E = E0 /c F (a t satura tion density kF,, :::::: 1.33 fm - 1 and ?Fn :::::: 37 MeV), t he 0 
Pauli blocking fact or can be written as 
( er ) 1 - ?s .5,: . 
(2. 111) 
2
cro { 1 _ ?.'J
 _+~'J_2 ( - ' )s/2 
5 , 5 , 2 ~ 
Invoking the optical theorem, which connects the total cross section with 
the imaginary part of the interaction at forward angles, we expect Imtf0 for 
E o > 2cF0 to be damped by a fact or (1 - dK-} ), where d decreases with energy 
as E 10 . T he dam ping fact or is shown as a functi on of the (dimensionless) 
energy E and Fermi m omentum "-F in Figure 2.33 . 
The density dependence of the real part of the central interaction , Retf0 , 
1s best described by the addition of a short-ranged repulsive core whose am-
plitude is p rop ortional to density, i.e. K-} . The repulsive core is due to the 
anticorrelation between identical nucleons and its effect is to enhance the 
differential cross section for large m omentum transfer . As can be seen for 
example in Figure 2. 34 where the different symbols and curves represent dif-
ferent densities , with increasing density the curves are displaced upwards by 
an am ount p roporti onal to density and almost independent of mome11t11m 
transfer . The cur ves are not quit e parallel because the finit e range of t he 
repulsive core redu ces its effectiveness and thereby causes the cur ves to draw 
t oge ther fo r large m omentum transfers [Kel 89a]. 
With these results , a param etrized form for the vanous component s 
115 
I.() 
N 
..' ' ' 
' 1 ' 
CJ t,.. ' .' 
le! ? 
C\1 . :'  I 
II 
\J,) I 
N \J) '', I 
? I 
I 
I : I 
I 
I 
I ; I 
I 
I ? I 
I 0 
co I 
0) 
r- .... f I I 0 I 
0 0 II i I 
II II 
I II I.() 
I ,:.. le"!"  
,:.. "" I le! ~ .'I .I  le! 
le! I , . 
I 
I 'I 
I f,  .I  
I , I 
0 
C\1 
c.o 0 
cD 0 
0 
0 
Figure 2.33, Clemente! and Vilh estimate of the Pauli blocking facloc 
116 
of the isoscalar effective interact ion can be developed. Following the (new) 
n otation of Kelly et al. [Kel 90b], all the central and spin-orbit components 
of the interaction can be parametrized by separate fun ctions of the generic 
form 
N 
t i ( q , "'F ) -- (Si  - di KFo:? )t(i J)( q ) + "'-FYi q 6, '"' 0 a ;nY /3 ,( q I ?;n )?  (2.112) 
n = l 
H ere, tUl(q) is a free interaction , y( x) = (l + x 2t 1 is a Yukawafunction , and 
the ?n are various mass p arameters whi ch were chosen to optimize the fit . . 
' , ..'   
If the individual components of the interaction are considered to be appro- '' . 
pri a te Fourier transforms of Yukawa expansions , one finds th at the natural I' '?  
exponents /3 are 1 for central , 2 for spin-orbit, and 3 for tensor interactions . ..,  
Similarly, for the tensor interaction 5 = 2 is used and 5 = 0 otherwise . As . .;  
we have already discussed , the Clemente! and Villi model predicts for the 
imaginary part of the central interaction a = 2 and d ex E 10 . In order to 
recover the free interaction at zero density, the scale factors S are required 
t o be unity for theoretical effective interactions . This restriction can and , as ?'  : 
I 
we will see, must be relaxed for the phenomenological analysis of data. 
We performed empirical interaction fit s for the PH, NL and LR effective 
interactions for energies between 100 MeV and 500 MeV. We used the free 
PH t -rna.trix for the PH fits and the FL t-matrix for the LR fits ; the NL 
interaction contains the t -rnatrix as the kF ----> 0 limit. The PH interact ion 
was fitt ed in the ranges O S q S 3 fm - 1 and 0.6 S kF $ ] .4 fn, - 1 with a step 
size of 0.1 fm - 1 for the momentum transfer and a. step size of 0. 2 fm - 1 fo r 
the Fermi m om entum. The NL interaction was fitt ed in the same momentum 
117 
transfer range, and with the same step size for q, for Fermi momenta in the 
range 0.6 :S kF :S 1.4 fm - 1 (step size 0.2 fm - 1 ), excep t for 100 MeV where 
only kF = 0.0 and 1.36 fm - 1 were used because the density dependence is 
not monotonic at that energy. The LR interaction was fitted for kF = 0.0, 
0.7, and 1.4 fm - 1; the momentum transfer ranges were 0 ~ q ~ 2.5 fm - 1 
(200 MeV ), and 0 :S q :S 3 fm - 1 (318 MeV and 500 MeV ). The c.m . wave 
number was evaluated for 160 . 
The fitt ed parameters are listed in Tables 2.1- 2.3, and both the theo-
re ti cal interactions and the empirical fits are shown in Figures 2.34-2.42 (the 
PH and NL parameters for 135 MeV are from [Kel 89b], the PH parameters 
for 180 MeV from [Ke! 90a]) . Considering that only two parameters are be-
ing used for each component, the fits are remarkable over the whole energy 
range. Also, apart from 100 MeV and 318 MeV where some modifications 
t o th e mass parameters and exponents are necessary, it appears that one 
fun ctional form is able to accommodate all the interactions: PH, NL , and 
LR. 
118 
R et~ Imt~ R erLS ImrLS 0 0 0 0 
TL p au a12 d2 I a21 a31 I a 3 2 d4 I a4 1 
C -123 .67 201.13 0.561 -2.29 -20 .09 15.73 0.603 -4. 61 
100 ? 0 3.0 - 0 3.0 6.0 - 3.0 
e [3310] [2210] [3320] [3220] 
C -94. 71 181.29 0.443 -6.36 -13 .80 10.71 0.571 -5.26 
135 ? 0 3.0 - 0 3.0 6.0 - 3.0 
e [3310] [2210] [3320] [2220] 
C -74 .33 157.32 0.326 -10.27 - 8. 73 7.21 0.586 -5.64 
180 ? 0 3.0 - 0 3.0 6.0 - 3.0 
e [3310] [2210] [3320] [2220] 
C -64.15 141.31 0.287 -10 .49 -7 .39 6.40 0.384 -3.36 
200 ? 0 3.0 - 0 3.0 6.0 - 4.0 
e [3310] [2210] [3320] [3220] 
C -45 .90 123 .23 0.173 -24.17 -4.61 3.69 0.324 -2. 78 
318 ? 0 3.0 - 0 3.0 6.0 - 7.0 
e [2210] [2210] [3320] [1140] 
Table 2.1: Reparametrization of the PH interact ion. Units: Te, (MeV) , 
d (1) , a (MeV fm 3 ) for central and (MeV fm 5 ) fo r spin -orbi t components, 
? (fm - 1 ) ; an entry of 0 for ? is to be interpreted as a delta fun ct ion with 
? - 1 = 0 . The exponents for ea.ch component a.re give n in th e fo rm [a,,85]. 
119 
Ret~ Imt~ RerLS lmT'.LS 0 0 0 0 
TL p au I a12 d2 I a21 a31 I a32 d4 I a41 
C -186.80 200 .54 0.700 -4. 06 -24 .56 14.20 0.705 -0.69 
100 ? 0 3.0 - 0 3.0 6.0 - 3.0 i 
e [3310] [2210] [3320] [2220] ' . 
? 
C -62. 78 85.15 0.671 -6.64 -18.89 12 .00 0.728 -1.30 
' .f ' '  
' 
135 ? 0 3.0 - 0 3.0 6.0 - 3.0 
e [3310] [2210] [3320] [2220] 
C 0.34 24 .93 0.568 -4 .96 -14.14 9.18 0.663 -1.30 
... 180 ? 0 3.0 - 0 3.0 6.0 - 3.0 I  
e [3310] [2210] [3320] [2220] 
C 15.00 9.63 0.458 -5.34 -10. 72 7.13 0.610 -1.61 
200 ? 0 3.0 - 0 3.0 6.0 - 3.0 
' ,' 
e [3310] [2210] [3320] [2220] 
, 
' ( 
C 9.73 18.53 0.280 -9.46 -5.00 3.45 0.052 1.64 
318 ? 0 3.0 - 0 3.0 6.0 - 3.0 
e [3310] [2210] [3320] [2240] 
Table 2 .2: Reparametrization of th e NL interacti on . Un it s: Ti (MeV), 
d (1) , a (MeV fm 3 ) for central and (MeV fm 5 ) for spin -orbit compo 11 e11ts , 
? (fm - 1 ); an entry of 0 for ? is to be interpret ed as a. delta functi on wit h 
? - 1 = 0. The exp onents for each component are given in the form [a,,85]. 
120 
Ret~ Imt~ Re-rLs lm-rLS 0 0 0 0 
TL p au I a12 d2 I a21 a31 I a32 d4 I a41 
C -67 .84 108.10 0.443 -5.46 -8.93 7.54 -0.027 -1.17 
200 ? 0 3.0 - 0 3.0 6.0 - 4.0 
e [3310] [2210] [3320] [3220] 
C -38 .89 100 .03 0.263 0.83 -3.31 3.76 -0 .020 -1.53 
318 ? 0 3.0 - 0 3.0 6.0 - 2.0 
e [3310] [2210] [3320] [3220] 
C -35 .08 108 .28 0.174 -4.65 -1.03 1.92 -0.007 -1.44 
500 ? 0 3.0 - 0 3.0 6.0 - 2.0 
e [3310] [2210] [3320] [3220] 
Table 2.3: Reparametrization of the LR interaction . Units : TL (MeV) , 
d (1), a (MeV fm 3 ) for central and (MeV fm 5 ) for spin-orbit components , 
? (fm- 1 ); an entry of 0 for ? is to be interpreted as a delta function with 
? - 1 = 0. The exponents for each component are given in the form [a1,B5]. 
121 
g 
b 
.... ., '? 0 0 . 0 ci ... I I 
("'l 
+ 
+ .? ?+ .  
 
? 
N 
I 
-..8....  
0
0..  . 0 0 0 0 0 00 U"l 0 U"l 0 U"l 0 U0 U"l O ION I ... ..".l  '?I  _. N N I I 
r--1?-1r---,----,-----.---,--I- ~~--1_ _1  
N 
I 
-..s....  
..0 uo,  
0 
0.. . 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 
0 
0 '? '? 0.. . U..".l  0 U"l 0 l() N ... t.O co 0.. . N...  ......  t...O.  I N N (") (") I I I I 
I I I I I I I I 
(clUJ J\dW) 
Figure 2.34: Reparametrization of the PH interaction (100 MeV) . The 
symbols show the original interaction for kF = 0.6 fm - 1 (triangles) , kF = 
1.0 fm - 1 (crosses), and kF = 1.4 fm - 1 (diamonds) . The curves show two-
parameter fits. 
122 
N 
I 
..E .._.., , 
. 0 I[') 0 I[') 0 0 
"0'  0 - 0 0 0 I -I 
C'l 
' ? ' 
~.. ., .'  1 ' 
..~....  . N ,......._ '1 
I 
-E I .._,, ,: ,I  
- er ; I 
?' 
, ; I 
? I 
0 
I[') I[') 0 I[') 0 
0 0 0 0 0 0 0 0 0 
I[') I[') 0
I ...
 .. I-[') 0 I[') C'l C\l C\l C\l C\l - -I I I I C'l 
.''I,   ''   
I I 
N ,......._ I I 
'. 
I 
E : l 
...._..,.  , 
..... er 
0 
0 0 0 0 0 0 0 0 00 0 0.., . 0 0 0 0 00 .., . 0 0 C\I 
0 I..[.'.).  0... 
co <O 
.. I[') l() 0 l() 0 I[') N ..... ..... '.? N I - - C\l C\l I I I I - I I I I I I 
(c WJ N:iw) 
Figure 2.35: Reparametrization of the PH interaction (200 MeV) . T he 
symbols show the original interaction for kF = 0.6 fm - 1 (triangles ), kF = 
1.0 fm - 1 (crosses ), and kF = 1.4 fm - 1 (diamonds). T he curves show two-
parameter fits . 
123 
N 
I 
-8 
0 
., .. ~ I() 0 I() 0 
0- 0- 0 0 0 I -I 
(") 
' .' 
.?
,,'  
?.' .  N 
,--. '1 
I ;1 
-8 : I ? I 
o' ; I 
I ; I 
I 
0 
0 0 0.. . 0.. . 0 0 0 0 0 I() 0 L/) 0 ...,. I() 0 
I() 0 I() 0 I() 0 
I() (") (") N N 
I N N C") 
I I I I I (") 
..'.,I ..    ,  
N 
,--. ...'   
I 
8 :i 
... -o' 
0 
0 0 0
I() 0 I.. . 0.. . 0 0 0 0 00 0 0 0 0 0 () 0 I() \() 0.. . I() N ...,. <O 0:, 0 I ... 
0 0..... ,~ ..  0<.. O.  
N N I I I I 
I I 
(cWJ A<)W ) 
Figure 2.36: Reparametrization of the PH interaction (318 MeV ). T he 
symbols show the original interaction for kF = 0.6 fm - 1 (triangles ), kF = 
1.0 frn - 1 (crosses), and kF = 1.4 fm - 1 (diamonds) . Th e curves show two-
parameter fits . 
124 
8 
b 
., . 0 0 I() 0 I() 0 
0 "0'  0 - 0 0 0 I -I 
C".l 
N 
I 
-8 ..__, 
I() 0 
0 0 0 0 0 0 0 00 I() 0 
0 
0... .. I() I() 0I ...
 .. I I ..(..).  0 I() C\I -C\I C\I - - I 
I I I I 
N 
I 
8 
.-.__, 
.... er 
0 
0 0 0 0 0 0 00 0 0 0 0 0 
0 0 
0 0 0 
I() 0 I() 0 I() 0 N '<of' <O (0 0... . NI..(..).  0 I() ..... ..... I I ...
 . .'<...I'.  
I N C\I C".l I I 
I I I I I I I 
I 
(cWJ AdW) 
Figure 2.37: Reparametrization of the NL interaction (100 MeV) . The 
symbols show the original interaction for kF = 0.0 fm - 1 (triangles ), and 
kF = 1.36 fm - 1 (diamonds) . The curves show two-parameter fits . 
125 
,....__ 
I 
-8 
0 
0 
b >-
< 
0 
., o 0- I() 0 I() 00- 0 0 0 I -
 
I 
(") 
~.. ., 
..~....  
N ,....__ 
I 
--8 - er 
0 0 0 0 0 0 00 0 II') 
0 I() I() .... .. 0.... .. ".....'.  ON I I N -
I I I 
N 
I 
..8....  
- er 
0 
0 0 0 0 0 0 0 00 0., ,. 0 0 0 0 0., ,. 0 
' I() <O ..?....  0... ... l{) '? 0... ... ...... 0 C\I <O co 0 C\I I N I I I I 
I I I -I -I -I -I 
(cWJ J\dW) 
Figure 2.38: Reparametrization of the NL interaction (200 MeV) . Th e 
symbols show the original interaction for kF = 0.6 fm - 1 (triangles) , kF = 
1.0 fm - 1 (crosses) , and kF = 1.4 fm - 1 (diamonds) . The curves show two-
parameter fits . 
126 
N 
I 
-..8..  
0 
If) 0 
.. If) 0 ., 0 ci ci -
- 00 ... -
ci I I  (") 
N 
I 
-..8..  
.... O' 
~_ , 
..~....  
0 
0... . ll'.) 0 If) 
oo If) 
0 .... 
C\l 
0 0 (') 
C\l 
(\) 
I 
N 
I 
-..8..  
.... O' 
0 
0 0.., . 0 
0 0 C...\.l  .... <D .., .... oO 0 . 0 a, <D s I I 
0 0 0 0... 
I 
C\l I I I I 
l..O 
0 . 
... I l(.) "I'  I 
(cU1J A;}~) 
sp Y? m b ? Reparamet riza tion of the NL inter action ( 3 I 8 Me V) . The 
.tOg ufmroel-s  l 2sh.3o9w?  tl1 e or1? g?m a 1 m? terac t?1 0n f or .k F  = O? 6  f m - t (t r,?a ngl e s ) , kF = 
Para (crosses), and kF = 1.4 frn "1 ( dia1nonds) . The curves show two-
1 
meter fit s . 
127 
N 
I 
..E...  
----
- O" 
.. 0 0- ll') 0 ll') 0- "0-'  0 ci ci ci I I M 
~_ , 
N 
,,-... 
I 
..E...  
.... --e-r- 
0 0 0 0 0 0 00 0 0 
ll') 0 l,l..'),  
ll"l ll"l 0 l
I .... .l."..l  0 ll) C\1 
,.., I 
N N I I 
I I I I 
N 
I 
..E...  
.... er 
0 
0 0 0 0 0 00 0 0 0 0 0 0 0 
0 
ll"l 0- ~ 0 ll) C\1 ~ "' (() 0- C-\I ~ I N N I I I I - "' I I I I I I I -I 
(c UJ J AdW) 
Figure 2.40: Repararnetrization of the LR interaction (200 Me V) . The 
symbols show the original interaction for kF = 0.0 fm - 1 (dots) , kF = 0.7 
fm - 1 (triangles), and kF = 1.4 fm - 1 (diamonds) . The curves show two-
parameter fits . 
128 
8 
b 
I 
..a...  
-----
.. 0 ., 0- tO 0 tO 0 0- 0 0 0 0 I -I 
(') 
~_ , ? 
1 
? 
l 
N ,....._ 
I 
-a 
.... o' 
0 0 0 0 0 0 0 tO 0 tO
tO II') 0... . I.I..').  0 ll'l N N ....
  0.. .. 0 
I N N 
I I I I 
N 
I 
a 
------
0 
0 0 0 0 0 00 0 0 0 0 0 0 
0 0 
0- II') II') 0- I.I..').  N ""' <O co 0 N .... <..O..  I I I I I .... .... ""' I I I I I I 
(cWJ J\aW) 
Figure 2.41: Reparametrization of the LR interaction (318 MeV ). The 
symbols show the original interaction for kF = 0.0 fm - 1 (dots), kF = 0.7 
fm - 1 (triangles), and kF = 1.4 fm - 1 (diamonds) . The curves show two-
parameter fits . 
129 
N 
I 
..8...  
- O' 
., .. 0 .,., 0 .,., 0 0 
0 0 0 0 0 
I -I 
(") 
.? . ..?   
N ,,....._ 
I 
-8 
- O' ,. .  
: I 
?1 
0 
0 .0, ., 0 0 .0,., 0.,., 
.,., 0 .,., 0 .,., .,.,  0 0 0 0 0 0 
0 .,., 0 l()..,. ..,. (") (") N N ..... ..... 
I - N N (") M I - I I I I (") 
.',I .  
?,,'  
N ,,....._ 
I .' I'  
-8 ~) 
- O' 
u_o0,   
(\) 
~ 
0 
0 0 0 0 0 s0  0 0 0 
0 0 0 0 0.. ,. 0 
l-{) 0 l() l() N ..,. <O (X) (\J ~ I I I I I s -I -
(cWJ N~W) 
Fig ure 2.42: Reparam etrization of the LR interaction ( 500 Me V). T he 
symbols show the original interact ion for kF = 0.0 fm - 1 (dots) , kF = 0.7 
fm - 1 (triangles ), and kF = 1.4 fm - 1 (diamonds) . T he curves show two-
parameter fit s . 
130 
The strength Vn and the range R of the repulsive core of Ret~0 can be 
identified from the above parametrization more easily in the low-q expansion 
2 2
Reti0( q , "'F) :::::: Reti0( q , 0) + K.} Vn [ 1 - q : ] , (2.113) 
where Vn = a 1 + a 2 and R 2 = ~0J . Table 2.4 lists these two quantities for 
?I II. 
the three interactions . For the PH interaction Vn increases between 100 and 
135 MeV and then decreases until about 200 MeV where it goes through 
a m11umum . Above 200 MeV, the repulsive core becomes again slightly 
stronger with _increasing energy. For the NL interaction Vn increases until 
about 180 Me V, decreases slightly and also goes through a minimum around 
200 MeV . Above this energy the strength of the repulsive core increases 
rapidly with energy. The strength Vn of the LR real central interaction 
(whi ch is only available for energies above 200 Me V) increases monotoni-
cally with energy. For all the interactions the range R of the repulsive core 
decreases monotonically with increasing energy. The table also shows that 
Vn for the NL interaction is much weaker than for the PH interaction. The 
shorter range of the NL repulsive core above 180 Me V makes the density 
dependence of the real central component almost independent of momentum 
transfer (see also Figures 2.38 and 2.39) . The strength of the LR repulsive 
core is intermediate between the ones for the PH and NL interactions , its 
range however is slightly larger than for the PH interaction. 
2.8.2 Description of Data 
It has been previously establish ed that an empiri cal effective interact ion 
provid es an excellent description of both elastic and inelas tic scattering data. 
131 
II I I 
TL PH NL LR 
I I 
(MeV) VR R VR R VR R 
100 77.5 1.32 13. 7 3.12 - -
135 86.6 1.18 22.4 1.59 - -
180 83.0 1.12 25.3 0.81 - -
200 77.2 1.11 24 .6 0.51 40.3 1.34 
318 79 .2 1.03 28 .3 0.66 61 .1 1.04 
500 - - - I - 73.2 I 0.99 
, ' 
I 1 , ' 
I I 
Table ' 2.4: Low-q form of R et?i : strengt11 of tlie repulsive core VR 
(MeV fm 3 ), and range R (fm). 
I 
; I 
that is superi or to any prediction based on a t11eoretical model . Examples I , II ,,  
can be found in {Kel 89b , Kel 90a, Kel 90b , Fla 90} . The resulting parameters 
; I 
represent the effective interaction in a finit e nuclear system , and a compari- 'I 
, ' 
son witl1 parameters obtained from t11eory allows us to critically review tl1e 
f' 
JI 
various models and approximation scl1emes for tl1e nucleon-nucleus scattering , 
, ' 
problem . The parametrization itself provides greater insig11t into tl1e p11ysics ,J . ,I  ,  
of tl1e effective interaction tlian do t11eoretica1 interactions presented in the 
form of voluminous tables of Yukawa amplitudes. 
Apart from its use in the study of tl1e nuclear effective interaction 
proper, another equally important use for tl1 e effective interaction is it s ap-
plication t o phenomenological nuclear structure studies. Tlz e ex traction of 
neutron transition densities from proton scattering da ta requires a rath er 
accurate description of the two-nucleon effective interaction and its density 
dependence. Kelly demonstrated that nucleon inelasti c scattering for energi es 
132 
between roughly 200 and 500 possesses very good intrinsic radial sensitivity 
a nd p ermits neutron densities to be determined even in the nuclear interior, 
particularly for targets with moderate A [Kel 88]. However, residual errors in 
the effective interaction can induce artificial distortion of the fitt ed densities 
and thereby limit the accuracy which can be achieved in practice [Kel 89b] . 
While the accuracy of presently available theoretical effective interactions is 
not sufficient for this task , empirical interactions have already been success-
fully applied [Kel 86b , Kha 89 , Kel 90a, Fel 90]. ,,.?,'   
If we want to use the empirical interaction, Eq. (2.112) , to fit data we ' 
can not use exactly the same form which was used in the previous section to 
:I1  
fit theoretical interactions . First , it appears that data for states with surface- ?, 'I 
,' , 
peaked transition densities cannot be fitt ed unless one allows for a. scaling of 
the free interaction. The interpretation of this is that the interaction does not 
,, 
heal t o the free interaction at low densities . Second, the strong correlations 
1' 
i I 
among terms with various range parameters ? complicate the analysis of I 
,?.'  
,, 
scattering data and it is difficult to obtain stable fits with more than two free J I 
?? 
paramet ers p er interaction component . Kelly, therefore, proposed simpler :I 
parametrizations to interpret scattering data involving only a scale fact or 
and one m ore free parameter for each component [Kel 89b]. In this work we 
found it convenient to use for the real central part of the interaction 
(2. ll 4) 
for the imaginary central part 
(2 .115) 
133 
PH 
100 72 .74 0.551 3.70 - - -
135 89 .99 0.415 2.57 - - -
180 89.38 0.272 2.11 - - -
200 83.99 0.229 2.10 38 .10 0.410 1.83 
318 80 .52a 0.035 1.03 61.19a 0.267 1.90 
500 - - - 76.02a 0.153 1.34 
I 
Tahle 2.5: Simple reparametrization of the PH and LR interactions 
(100 MeV- 500 MeV) with the same units as above . 
1
a?1 = 2.0 fm - . 
and for the real spin-orbit part 
(2 .116) 
Usually, the imaginary part of the spin-orbit interaction is too weak to be 
fitted and is fixed to its nuclear matter value from the previous section . 
Here, b1 plays the role of VR in the low-q expansion of Retf0 and ?1 provides 
a finite range comparable to R (in the low-q approximation ?1 ~ v6/ R) . 
The parameters b1 , d2 , and b3 resulting from a fit of this simpler form to 
the PH and LR theoretical interactions are listed in Table 2.5. The scale 
factors for this fit are of course all unity. Th e ranges for q and kF , th e 
calculation of the wave number , and th e free interactions used , are th e same 
1 
as in the previous section . The masses used were ?, 1 = 1.5 fm - for all th e 
1
energies up to 200 MeV; above 200 MeV we used ?, 1 = 2.0 fm - . Th e 
value ?, 1 3 = 6.0 fm - proved to be acceptable for all energies. We found that 
134 
these fits for the PH and, especially, for the LR interactions are almost as 
good as the two-parameter fits in the previous section . The NL interaction 
is not as easily parametrized in terms of the simple form . However, we 
will later compare the parameters from the fit to data only to the PH and 
LR parameters, since those two theoretical interactions describe the data 
best . The PH interaction is available for energies up to 400 MeV and gives 
superior results over all the other interactions at 100, 200 , and 318 MeV (see 
Section 2.4 .5) . For higher energies , the LR interaction is the only available 
nonrelativistic effective interaction with medium modifications. 
2.8.3 Linear Expansion Analysis and Fitting Procedure 
An empirical interaction can be obtained from a fit to the scattering 
observables of isoscalar natural-parity transition in a self-conjugate nucleus . 
One assumes that, in the absence of significant transverse form factors, all 
densities with the exception of the matter density are negligible ( e.g. spin and 
current densities) and that this density is accurately determined from elec-
troexcitation data . Charge symmetry ensures that the neutron and proton 
transition densities are approximately equal. Thus, all the nuclear structure 
information required to interpret proton scattering data within the tp fold-
ing model for the scattering potential is specified through electron scattering 
measurements . This enables us to isolate the effective interaction with littl e 
residual ambiguity due to nuclear structure . 
In the following we want to outline very briefly the linear expansion 
analysis (LEA) method . A more complete and thorough discussion of the 
135 
m ethod can be found in the paper by Carr et al. [Car 85], in the papers by 
Kelly [Kel 85a, Kel 86a, Kel 86c , Kel 87] , and in t he paper by Kelly et al. 
[Kel 86b]. 
* * * 
In the linear expansion analysis , the transition amplitude Tis expanded 
as a linear series 
(2.117) 
' ' 
n ' I 
I 
where the basis amplitudes Tn(q) are expressed in the folding model as the 
: I 
convolution of the effective interaction with the transition density. With 
: I 
'I 
Eq. (2 .100) we get the overlap integrals ? I 
' 
'I 
(2 .118) I I 
where 6.Un = tnb.p is a component of the transition potenti al, and tn the cor-
responding component of the interaction t which is expanded as t = I:n antn , 
For example, in our simple parametrization the total interaction has six com- ,I :' '  
, 
ponents whose parameters an can be adjusted , plus the components of the 
imaginary spin-orbit interaction which are generally kept fixed . With the ba-
sis amplitudes Tn(q) one then constructs the quadratic forms x;:;j' (q) , anal-
ogous to Eq. (2.101) . Evaluating contractions of these quadrati c fo rm s leads 
to 
(2 . .l19) 
rn place of Eq . (2 .102) . The connection with the scatterin g observables was 
discussed in Section 2. 7 .1. Fitting of the expansion coeffi cient s an to data is 
136 
done using a rel atively simple bu t effi cient search proced ure: in t he first step , 
the distorted waves X are generated wi th a good approximation to the opti-
cal p otential , such as provided by the PH interaction or a previous empirical 
interaction . Then for each term tn in the linear expansion of the effective in-
teraction the overlap integrals Tn are computed . Subsequently, the quadrati c 
forms X are constructed and stored for the entire dat a set . Since both the 
overlap integrals and the quadratic forms are independent of the expansion 
coeffici ents an , it is then straightforward to optimize the coefficient s with 
a very simple search algorithm . To achieve consistency between elastic and 
inelastic scattering , the fitt ed interaction is iterated and the optical potential 
in each step is computed self-consistently after the ( 1 + po/ op) contibution 
has been rem oved from the interaction . Removing the rearrangement con-
tribution is crucial for self-consistency, and self-consistency cannot achieved 
without it . 
As a recent development, we have now also the option to include elastic 
scattering in the fit . This provides an independent means to ch eck both 
self-consistency and the ( 1 + po/ op) prescription. In order to treat elas tic 
scattering within the same perturbative framework as inelasti c scattering, we 
decompose the optical potential U or,t into a dominant reference potential Ure! 
and a small perturbation 5U which serves as a "correction" 
Uo pt = Ure f + OU, (2.120 ) 
where Ure! tpg and 5U 5tpg . The correction to the elas ti c scattering 
137 
amplitude, t herefore, is 
(2 .121) 
The correction to the interaction , 8t, is expanded as 8t = L n 8antn. Hence, 
for a reference interaction t which may be a theoretical interac tion or an 
interac tion from inelastic scattering, the elas ti c fit finds the corrections to 
the interaction parameters rather than the interaction parameters proper. 
The new interaction parame~ers a~ are then a~ = an + 8an . 
Finally, we want to make some further remarks to the data select ion for 
the empirical interaction fits . It is the fundamental premise of the local den-
sity approximation that the medium modifications depend solely upon the 
local density in which the interaction occurs, but not upon the specific stru c-
ture of each t arget . The interaction should not depend upon the fin al st ate 
of the nucleus . If this concept is sound , therefore, as a minimum require-
m ent the empirical interaction has to reproduce the data for all the states 
of a single target . For this analysis it is important to include both surface-
peaked states which are sensitive to the low-density properties of the effecti ve 
interaction , as well as interior-peaked states which sample the medium mod-
ifications at high density. For a finit e nucleus, the low-density properties of 
the effective interaction need not necessarily be the same as those of the free 
interaction . Finall y, intermediate cases will determin e th e rate at whi ch th e 
m edium effects depend upon density. Anoth er , more .strin gent test of th e 
local density hypothesis , is to t est wheth er the interaction is independent 
of t a rget . In this work , therefore, we will extract and compare interact ions 
138 
from single-target fits t o 16 0 and 4?Ca, as well as from combined fits to 160 
and 4?Ca. Since many data between 100 MeV and 500 MeV are avai lable, 
th e smooth dependence of the param eters upon the projectile energy can be 
tested as well . 
139 
3 Experiment 
3.1 Introduction 
Experiment 268 was conducted at the IUCF cyclotron facility in four 
run periods between January 1987 and March 1988 and it was the first exper-
iment to utilize the new K600 spectrometer. The objective was to measure 
cross sections and analyzing powers of the low-lying states of the Ca isotopes 
4?Ca, 42 Ca, 44 Ca and 48 Ca at the two energies 100 MeV and 200 MeV. A BeO 
target was used to get scattering angle information from differential recoil. 
We obtained excellent measurements of some of the lower 9 Be states, as well 
as of all the important 16 0 states up to about 14 Me V excitation energy. 
In January 1987 we collected data at 100 Me V for BeO and all the 
Ca isotopes, with exception of 44 Ca which was destroyed early in the run . 
200 Me V large angle measurements for the same targets were done i11 April 
1987. In January 1988 we completed the forward angle measurements at 
200 Me V and also took data for the new 44 Ca target over the whole angular 
range . In March 1988 we concentrated on 100 MeV measurements for 44 Ca 
and 48 Ca. Finally, in all our run periods a lot of diagnostic and development 
work was done. In parallel to the experiment at IUCF, we took 318 MeV data 
for all Ca isotopes (EXP 919) and for 32 S and 34 S (EXP 956) at LAMPF , 
using the HRS spectrometer. 
This chapter describes the IUCF experime11t in detail ; rnai11 re fere11 ces 
are the IUCF K600 Lectures [K600 L] and several IUCF Scientific and Tech -
nical Reports ([Ber 86, Ber 88, Ber 89]) . A brief comparison of th e HRS 
140 
and K600 spectrometers can be found in this chapter ; a more detailed and 
comprehensive discussion of the LAMPF experiments is presented in [Fel 90] . 
3.2 Accelerator and Beam 
A floor plan of the IUCF cyclotron laboratory is shown in Figure 3.1. 
[IU CF 86] . Beam from the polarized ion source first passes three preprocess-
ing units: a chopper, a buncl1er and a pulse selector. Acceleration to the 
final energy is achieved in two isochronous cyclotrons . The smaller injector 
cyclotron accelerates the 0.5 MeV protons from the source up to 15.2 MeV, 
the field limit . Before entering the bigger mainstage cyclotron the beam is 
guided through a short beam line section which contains a polarimeter. The 
polarization was generally about 75 percent . More about the polarizati on 
measurements and their analysis will be reported in Chapter 4. Beam split -
ting and momentum analysis is done after extraction from the mainstage in 
beam line 3 (BL 3). 
The average energy of the beam delivered by the dual cyclotron was 
determined from several readings of the BL 3 bending magnet NMR. In 
January 1987 the average proton energy varied between 100.5 MeV and 
100.7 MeV . In April 1987 the average beam energy was 201.53 MeV and in 
January 1988 it vari ed between 201.24 MeV and 201.4 MeV . In March 1988 
the average proton energy was found to be 100.5 MeV . This is summarized 
in Table 3.1. 
Beam line 8 (BL 8) is the section between the bending magnet at the 
end of BL 6 and the beam dump in the K600 experimental area. It con-
141 
Figure 3 .1: IUCF Cyclotron Floor Plan 
142 
B eam Energy Run Period Beam Energy Runs 
(nominal) (measured) 
January 1987 100 .5 MeV 1- 145 
100 MeV 100.7 MeV 146- 416 
March 1988 100.5 MeV 2221- 2353 
April 1987 201.53 MeV 501- 651 
200 MeV January 1988 201.24 MeV 761- 979 
201.4 Mev 982- 1023 
Table 3.1: Beam Energies for Experiment 268 . 
tains among other beam optical components the old QDDM spectrometer 
as a bending m agnet and momentum analyzer . The size of the horizontal 
object slit upstream of the QDDM determin es the resolving power of the sys-
t em (R = 5- 1 = p/ dp = Dx / !:::.x 0 lvfa:) and the en ergy disp ersion on the target 
( dT / T = (2 - T / mp) / R) . With D"' = 10.3 m and M"' = 0.9 for the QDDM 
one gets for the 1.1 mm slit ( transmission generally better than 90%) a re-
solving power of 10,404 which corresponds to about 18 keV at 100 MeV and 
about 35 keV at 200 MeV. These values also correspond roughly to the width 
of the image in the K600 focal plane, and thus provide a lower bound to the 
focal plane resolution. An available 0.55 mm slit improves the resolution by 
a fact or of two, however at the price of lower intensity since only a.bout 60% 
of the beam can be focussed through thi s slit. The thi ckness of the brass 
object slit is chosen to degrade the beam in the slit jaws by about 1 % in 5, 
corresponding to a deviation of about 3 inches in the QDDM focal plane. The 
143 
degraded beam is stopped in a flange ahead of the first of four quadrupoles 
between the QDDM and the scattering chamber while the main beam con-
tiirnes to the target . The four quadrupoles after the QDDM serve to adjust 
the momentum dispersion on the target and to provide a monochromatic 
focus on the target ( or slightly beyond it for reactions with a large kinematic 
factor), as well as the correct angular dispersion so that the scattering angle 
remains constant across the target. 
The bottom of the scattering chamber holds a rotating platter with 
a three-hole hodoscope used for dispersion matching. An internal Faraday 
cup necessary for forward angle measurements can also be mounted on this 
platter. There are two beam exits at the scattering chamber. One connects 
with the K600 spectrometer and contains a cassette holding several aper-
tures of different geometrical shapes ( circular and rectangular) and different 
horizontal aud vertical acceptances ( c.f. Table 3.2) 1 . The other leads to the 
beam dump with the external Faraday cup. Measurements down to 18 deg 
were possible with the external cup and the old dump pipe. In March 1988 
a small pipe was available, making measurements with the external cup pos-
sible down to about 15 deg . 
1 The acceptances were calculated with dfl = A/ R 2 where A is the aperture area and R 
the distance from the target ( the rounded corners were taken into consideration). An exact 
formula for circular apertures (radius a) is dfl = 21r(l - J l - (a / R) 2 ) . For rectangula r 
apertures with half-height a and half-width b the exact express ion is dfl = 4 a rclan( ;~1 ) 
where d = ?a2 + b2 + R 2 [Cra 53]. 
144 
Aperture Ap.# I Dimensions (h x v,? r) Acceptance 
1/ 2 in diam . Al 1.27 cm 0.2500 msr 
1.6 in x 1.6 in A7 4.06 cm x 4.06 cm; 1/ 8 in 3.2345 msr 
3/8 in diam. Bl 0.95 cm 1.4000 msr 
1/2 in X 1 in B2 1.27 cm x 2.54 cm ; 1/ 4 in 0.5681 msr 
1/ 2 in X 2 in B3 1.27 cm x 5.08 cm; 1/ 8 in 1.2556 msr 
2 in X 1 in B4 5.08 cm x 2.54 cm; 1/ 8 in 2.5282 msr 
1.6 in X 1.6 in B5 4.06 cm x 4.06 cm; 1/ 4 in 3.1834 msr 
.56 in x 1.48 in B7 1.43 cm x 3.74 cm; 9/ 32 in 0.9875 msr 
Table 3.2: Apertures used for Experiment 268 : the A apertures are ma-
chined from 1/ 2 inch thick brass , the B apertures are 3/ 4 inch thick . Horizon-
tal (h} and vertical (v} acceptances can be calculated from the corresponding 
dimensions and the distance from the target (71.2 cm). For the rectangular 
apertures the radius of curvature ( r) of the corners is given in inches. 
145 
Tgt 40Ca 4oca 42 Ca 44 Ca ,o ld I 44Ca ) new 4sc a 
t 15 .2( 4) 5.2(2) 9.6(2) 28 .5(1.0) I 9.85(50) 15 .2(2) 
40Ca 99 .965(10) 5.08 1.25(2) 2.20(5) 
42ca 0.009(3) 93.71 0.04( 1) 0.03(1) 
43Ca 0.002(1) 0.334 0.02(1) < 0.01 
44Ca 0.022(3) 0.869 98.68(2) 0.07(1) 
46Ca < 0.001 < 0.003 < 0.005 < 0.005 
4sca 0.002(1) 0.014 0.008 97 .69(5) 
Table 3.3: Targets for Experiment 268 . Thicknesses (t) are given m 
mg/ cm2, abundances in% . 
3.3 Targets 
Table 3.3 lists the thicknesses (in mg/ cm2) and compositions of the 
Ca targets ( abundances in % ) as determined by Oak Ridge National Lab . 
Uncertainties are given where available . The old 44 Ca (28.5 mg/ cm2) and 
BeO (21.58 mg / cm2) targets were destroyed early in the January 1987 run 
period . The thicknesses of the new 44 Ca and BeO targets were determined 
at IUCF. The 44 Ca thickness was measured with a micrometer ; the BeO 
thickness {22.26(7) mg/cm2) was calculated from the target mass and the 
known areal density of BeO ( the error is likely to be greater than the above 
quoted one but probably less than 5 percent) . 
A previous measurement of the BeO thickness with a mi crometer had 
yielded 21.99 mg/ cm 2 and all the 9 Be and 160 cross sections were ca.lcula.tecl 
wi th this value . An overall normali zation fact or of 0.988 was therefore appli ed 
146 
to the fin al da ta to reflect the new measurement. 
3.4 Spectrometer 
3.4.1 The K600 Spectrometer 
A schematic drawing of the K600 spectrometer is shown in Figure 3.2 . 
The spectrometer is of the QDD type and operates in HRH (Horizontal Dis-
persion, H orizontal Scattering, H orizontal Analysis) momentum-loss config-
ura tion . The main components of the spectrometer are an entrance qua-
drupole (Q) and two dipoles (D) . The K600 entrance quad provides most 
of the vertical focusing of the spectrometer. It should be adjusted such 
that the cross-over of the vertical and horizontal focal planes (which are in -
clined steeply relative to each other ) occurs near the central momentum ray. 
This ensures that the vertical image size at the extreme ends of the focal 
plane is within the vertical acceptance of the detectors . In the absence of 
any y-information, for the duration of our experiment the quad current was 
set according to ion-optics calculations based on measured quad properties. 
A triangular shaped paleface winding within the second dipole, the K-coil , 
acts effectively as a combination of a dipole and quadrupole lens . It mod-
ifies the ( x I 0) transport coefficient and thereby corrects kinematic image 
b roadening by keeping the image plane location fixed (fork = p- 1 dp / d0 .,;, 0 
the image plane m oves downstream and for k < 0 upst ream by a di sta nce 
lk = - 1~~i~x;"k) . The H-coil is another paleface winding which is located in 
the first dipole . It is of elliptical shape and provides a. combination of dipole 
147 
Figure 3.2: K600 Spectrometer 
148 
... 
and hexapole lens . This coil helps correct kinematic image broadening by 
m odifying ( x I 05) and therefore maintaining the tilt angle of the image line 
(since the dispersion is a function of k, the image line in the x z plane rotates 
by tan 1/; = - 1;'1~) with respect to the central ray) . Modifying second-order 
9 T. 
transport coefficients such as (x I 02 ), the H-coil helps correct aberrations-
in Appendix C we show with RAYTRACE simulations the impact of the H-
and K-coils on the focal plane and on nuclear lines in x 101. Finally, for large 
vertical acceptances, a sharp line in the x-position (i .e. momentum) spec-
trum may exhibit noticeable shape asymmetry due to ( x I y 2 ) aberration . In 
the absence of y-position information this can be approximately corrected 
on-line by adjustment of the K600 entrance hexapole magnet . 
The maximum spectrometer acceptance is circular and is horizontally 
and vertically ? 44 nu; the maximum solid angle acceptance is 6.1 msr. The 
maximum horizontal acceptance is determined by the quadrupole aperture, 
the dipole-gap spacers, the dipole anti-scatter baffles and the H- and K-coil 
supports . The maximum vertical acceptance is determined by the quadrupole 
aperture, the dipole gaps, the H- and K-coil gaps and the vertical openings 
of the dipole vacuum chambers ( constrained by main coils) . 
Some of the important first-order design transport coefficients for the 
K600 spectrometer in "normal" dispersion mode are th e horizontal ( vertical) 
linear magnification Mx = (x I x) = - 0.53 (My = (y I y) = - 5.7 ), th e 
horizontal (vertical) angular magnification Me = (0 I 0) = - 1.90 ~ Mx- 1 
(l\1cp = (<PI </J) = - 0.17 ~ !vfy- 1 ), and the spatial dispersion Dx = (x 15) = 
8.4 m . An astigmatic double focus implies that to first order both (x I 0) 
149 
1 ? 
II HRS I K600 
Maximum Proton Energy 800 MeV 493 MeV 
Bend Radius 3.5 111 2.1 111 
Solid Angle Acceptance 3.6 msr 6.1 msr 
Resolving Power (R = p/ dp) 104 ~ 3.5 X 104 
Angular Resolution 0.8 mrad ~3 mrad 
Dispersion 18.2 111 8.4 111 
Table 3.4: Comparison of HRS and K600 specifications . The HRS resolving 
power is given for a 20 MeV bite. 
and (y I </>) must vanish . 
3.4.2 The HRS Spectrometer 
The HRS spectrometer, as the K600 spectrometer, is of the QDD type 
and utilizes the momentum-loss principle . The mode of operation is VHV 
(Vertical Dispersion, Horizontal Scattering, Vertical Analysis) , a configu-
ration which effectively decouples scattering angle and energy. Table 3.4 
compares some of the important design specifications of HRS with those of 
the K600 (normal dispersion mode) [Zei 71, K600 P , K600 L] . 
3.5 Detectors 
We used two types of detectors in the focal plane: plasti c scintillators 
and multi-wire vertical drift chambers (VDCs) which served as horizontal 
( x, 0) detectors . At the time of our experiment the proposed horizontal drift 
chambers (HDCs) which would have provided vertical (y , ?) information were 
150 
not yet available. 
3.5.1 Scintillators 
T wo plastic scintillators paddles with photomultiplier (PM) tubes on ei-
ther ends were used in all our run periods ( the two scintilla tion det ectors used 
in January 1987 were originally designed for pion detection with the QDD M 
and covered only slightly more than half of the available fo cal plane). Both 
paddles were m ounted behind the wire chambers . While the front scintill a tor 
(Sl) ha d a thickness of 1/ 4 in , the rear scintilla tor (S2) had a thickness of 
1/ 2 in . During the 100 MeV runs a 1/ 16 inch aluminum absorber sheet was 
inserted between Sl and S2. For the 200 MeV run periods a combination 
of two 1/ 16 inch and one 0.09 inch aluminum sheets was used . To find the 
correct op erating voltages for the photomultipli er tubes the stand ard plateau 
m ethod was used . This involves measuring the count rate as a fun ction of the 
applied PM voltage and then fixing the voltage in the middle of the plateau . 
The scintillator stack has several functions . Sl-S2 coincidence provides an 
event trigger and also a position-independent start signal for the VD C drift 
time m easurement . Plotting 6.E1 versus 6.E2 allows particle identification 
and background rej ection . 
3.5.2 Wire Chambers 
The front VD C plane (X l) is positioned along the focal plane. T he rear 
plane (X2) is parallel to Xl and is mounted at a di stance of 103 .2 mm from 
X l. The horizontal offs et between the chambers is 79.2 mm , 1. e. channel 0 
151 
of Xl coincides with channel 792 of X2 (10 channels = 1 mm) . Each cham-
ber consists of 160 gold-plated 20 ?,m tungsten sense wires with a spacing 
of 6 mm. Located between each pair of sense wires are two 50 ?,m guard 
wires (Be-Cu) such that the overall wire spacing is 2 mm . All the wires are 
held at ground potential. The active detector area is approximately 100 cm 
horizontally and 7 cm vertically. Two stretched 1 mil graphite-coated alu-
minum mylars constitute the cathode planes ( cathode-to-cathode spacing: 
12 .7 mm, operating voltage :_ :::::: -4.5 kV) and serve at the same time as gas 
barriers . The filling gas is a mixture of 50% argon gas and 50% isobutan e 
quencher gas . This gas mixture is bubbled through refrigerated n-propanol, 
an antipolymerizing agent. During the experiment a steady gas flow was 
maintained. At the operating voltage the electron drift velocity saturates to 
46 .1 mm/ ?,s . As in the case of the scintillators , the correct operating volt-
ages were found by a plateau measurement in which for a given threshold 
the efficiency is plotted as a function of the applied chamber voltage . For 
this measurement all the wires should be illuminated uniformly by the beam. 
The definition of the efficiency usually includes the requirement of good hit 
patterns in the chamber . 
3.6 Electronics 
3.6.1 Scintillator Electronics 
Signals from the four scintilla.tor photomultipli er tu bes ( th e high -mo-
men tun.1 side tubes are denoted by P , the low-momentum side tubes by N) 
152 
are divided by linear fans (PS 744) into signals gomg to charge sensitive 
analog-digital converting units (LRS 2249 QDCs) for pulse-height analysis 
and into signals going to constant-fraction discriminators (EG&G 934 CFDs) 
( c. f. Figure 3.3) . CFD output signals are fed into time-digital converters 
(LRS 2228A TD Cs) and each pair of P and N signals is also collected in a 
mean-timer unit (LRS 624) to generate a position-independent timing signal 
for each of the two scintillators . 
The rear scintillator timing signal is the signal we want to use to define 
a START for the wire TDCs . Due to its thickness , the energy loss in S2 is 
larger than in S1 and, hence, the signals are stronger. Also, the front scinti l-
lator and the aluminum plate(s) eliminate most of the noise2 . Therefore, the 
Sl mean -timer signal should be delayed with respect to the corresponding S2 
signal. Pulse shaping and delay of the S 1 signal are done with a computer 
programmable delay (IUCF) , the S2 mean-timer pulse is shaped in a dis-
criminator unit (LRS 821). Figure 3.3 schematically shows the pulse shapes 
and their relative timing . Finally, the S1 and S2 signals are collected in a 
PS 755 module with coincidence level two. The coincidence unit is vetoed by 
the run gate and the CAMAC read from the output register (JORWAY 41) . 
3.6.2 VDC Readout Electronics 
For ea.ch VDC there a.re ten printed boards (LRS 2735B) with four am-
plifi er chips per boa.rd . Each board thus amplifies and di scriminates signals 
2 The fast circuit in Ma rch 1988 used Sl as a basis for timing. 
153 
S2N S2P SIN SIP 
~....  I! I I!  I I!  I I! 
oq 
.C.,:  
('I) 
~ 
~ 11 11 ~ l~I 
en .. . . s s~ f I I u t-s s 
0. 
c.....,... .;.,  = ,,-__?..   
.;,i..i. . 
.0..  
t".l 
-(D () ... ~ ~ 1 " t-s s 
0 I i 
2. I i 
() i i i 
"' I 
F.'a .rt Gat.e Fas't  Start rrr-u 
Clear Clear S2 ~ If  
veto (rom 
SltS2t4NH Output 
Register 
of sixteen consecutive wires . This is schematically shown in Figure 3.4. At 
some stage of the experiment we were concerned about cross talk between 
wires within a chip. IUCF technicians ascertained that either the first wire 
or the fourth wire of each chip picked up noise from the other three wires, 
i .e. either wires 1, 5, 9 and 13 or wires 4, 8, 12 and 16 on each card showed 
most cross talk . It was also determined that the cross talk signal followed 
the original signal by about 25 ns . In the March 1988 run period we were 
able to control the cross talk problem by setting the threshold3 values of each 
card individually. 
Signals from consecutive wires are fed to five consecutive multiplexers 
(IUCF MUX) via a sixteen-line ribbon cable (total delay::::::: 210 ns), such that 
every fifth wire is connected to the same MUX. The IUCF Multiplexer is a 
single width CAMAC module which accepts 32 ECL-level differential inputs 
from two LRS 2735B wire chamber preamplifier cards or their equivalents, 
and a gate input (NIM) . During a gate, a fast OR output is generated and 
latched for the first event ( available at a front panel connector) . Events are 
also written into registers for CAMAC readout . Two of the three registers , 
A and B, are simple 16 bit registers , corresponding to the upper and lower 
34-pin connectors on the module, respectively. Into these, events are latched 
directly during a gate . The third register , register E , provides encoded infor-
mation about single or multiple events (bit 16) , module address (bits 6- lJ) 
and the address of th e high es t number wire being hit (bits 1- 5) . If no valid 
3 Another indicator for insufficiently high thresholds is an increase in the observed 
number of so-call ed multiple hits , see Chapter 5. 
155 
V) - - -
~ - - -
ft') - - -
N - - -
- - - -
~ 
i,.; 
I Q.. (.I 
~f;; ~S~ ?~ .. ~~ ?"4 ~ 0 -~
\,\~ ~i~ 
 -!. C 
nf&J' i ! 
.. - - - -
Figure 3.4: W iring of LRS 2735b Printed Board 
156 
events have occured , all the bits are set to zero. To limit multiple events, the 
gate is internally shortened to a maximum of 33 ns following the first event. 
The m odule is capable to operate both in a normal or a fast 400 ns CAMAC 
cycle . 
All MUXs are gated by a Sl-S2-BUSY coincidence which is distributed 
by a 16-fold fan (Figure 3.5). The fan also provides a common START 
signal for the ten wire TDCs (LRS 4210) . An OR output from a MUX is 
discriminated (LRS 821) and serves as a STOP for its TDC. In March 1988 
a sum of the five front ( and rear VDC discriminators, respectively) was used 
to form a coincidence ( see next section) . 
3.6.3 Coincidence Electronics 
The coincidence circuit is shown in Figure 3.6. There are several inputs 
t o the coincidence circuit. One is the scintillator coincidence which has al-
ready been discussed. The output register has two outputs: bit 1 is the run 
gate and bit 2 is set while the CAMAC is being read . When the run gate is 
off or the CAMAC is being read or the spin valid bit is not set (i. e. during 
spin flip) or in the case of a B-Hive alarm, an inhibit (INH) is generated in 
a logic fan (LRS 429A) which inhibits all the scalers (Event 10) , and via the 
scintillator coincidence all of the further regular event (Event 4) processing 
( e .g . gates on the MUXs and STARTs on the wire TD Cs ). Also , a sci ntill a-
t or coincidence via a gate and delay generator (EG&G GG8000) generates 
a BUSY signal and vetoes accepting any new events until processing of the 
old events has been completed. A BUSY signal can also be generated via 
157 
,, ,  _.. -? -' I-+ 
t ~ ~ _.. - ~ 
~ --+ -' f-+ -? 
ti} 
--
-
?- -- Q. - - 0 - tl - tl - -
= --., __ - ~ -?- --? 1 1 ~ .. ctS 
~ ~ Q ~ Q 
"-'Cl 0 8 8 
c.-c 
I .... ~ 
'~? Cl> 
+-I 
('IS ~ ~ 
00 
~ ?_. .. ~ ... 
Figure 3.5: VDC Logic Circuit 
158 
veto 
St?S2 >< 1NH 
TT L 
r NIM 
DIS 
i 
s 
------, C I I .....,_ I I ......_ I ..__. I -- Lo... G 
~....  g, 11 1.!.11 I L!J 11 Fan R= "rn I I Va1. oq Pre-
.::: Seal. 
"1 
(!) 
w 
.O.'l  
...... 0
c.n g_
  
co .... 
o. 
ar DC 0... 
~ ~,___~ s ~t 
n <Wt 
~ .i;, _,,i.~:::;  II  rnI) =? I 
9. __ 1 i \ ...., ___ ...._ ___ C _____ _\ n ..._ ....._ \  
i:: 
.,.... -...._ 
---? .s 
16-fold Fan 
clear 
TDC TDC X 10 
,., 
the LAMPF Event Trigger Module by a scaler event or a beam line 2 (BL 2) 
event (Event 8) . The spin bits (valid , up , down) are collected in the coinci-
dence register (LRS 2341) which is gated by either an INH , a spin not valid, 
an Event 8 or Event 10 . The B-Hive alarm was temporarily disabled during 
some of the run periods . The connection between the clock and the coincin-
dence circuit was disabled early in the experiment since we decided to read 
scalers during the spin flip . In March 1988, a fast coincidence between Sl 
and S2 triggered the STARTs for the TDCs and opened the MUX and QDC 
gates; a second coincidence tested for hits in both chambers (PS 755) . A fast 
CLEAR halted focal plane processing and issued a crate CLEAR in the case 
of one-chamber-only hits; also, the event trigger to the CAMAC crate was 
suppressed. When the level 2 coincidence is set to I -fold (all the events from 
Sl -S2-BUSY pass) and the fast clear coincidence to 3-fold, the fast CLEAR 
is effectively disabled and the circuit identical to the previousl y used one. 
3.6.4 Electronics for Integrators and BL 2 
Signals from the left and right integrators are passed through gate and 
delay generators and are discriminated before they are fed into coincidence 
units and scalers ( c.f. Figure 3. 7). A BUSY vetoes updating of the corre-
sponding scalers (INH also inhibit s all scalers) . Both BUSY and INH gate 
through a logic fan and a 2-fold coincidence with the BL 2 strobe any BL 2 
signal processing in the LRS 35 11 ADCs . 
160 
.... L. Int. ""rj GDG I Disc t--+ICoincl I Fan t+-INH 
(Jq 
C..,:  s veto f I ~ s I l..:!.d SLR; s13 
C;J 
~ 
BUSY I Fan I 'I Fan 
M 
~ 
n 
,~..,  .s veto t .s 
0 
2. 
...... n 
a ,:n ......,.  o' IR. lnt.l-+I GDG 1-+1 Disc .,_..Coinc ,.., 
...... 
::: 
~ gate 
(b 
(),.Q
veto 
.,  
"e+,0.. ', - BL2 
,:n Strobe -I Fan ~ Coinct--+I GOG I I ADC I I ADC 
~ 
0- I 
tJ:j s s 6 I BL2 r Left Right tv 
3.6.5 Data Acquisition 
With exception of the integrator and BL 2 circuit components which 
could be found in CA MAC crate 1 ( data acquisition area) all the other elec-
tronics was gathered in crates 2 and 3 in the K600 cave. Data. from these 
crates reached crate 1 and the MBD through a Branch Highway. A UNIBUS 
fed data to a VAX 11 / 750 . 
162 
4 Calibrations 
4.1 Matching Between Beam Line and Spectrometer 
In principle , the energy spread of the beam determines the available 
energy resolution for an experiment. As a result, in order to achieve a highly 
monochromatic beam for high resolution , the greater part of the primary 
beam from the accelerator has to be discarded after the energy analysis on en-
ergy defining slits . Of course., this severely limits the available beam intensity. 
In contrast, dispersion-matching systems are able to make use of essentially 
all the beam, at the price that the beam has to be dispersed spatially a.cross 
th e target . In this way, at each point on the target a highly monochromati c 
beam can be provided. In the case of energy-loss spectroscopy the desired 
information gained from a measurement is the scattering angle and the state 
of the residual nucleus . If, as in the case of our experiment, the cross sec-
tion varies very slowly with energy, e.g. there are no sharp resonances, only 
the momentum difference between incident and outgoing particles need to 
be determined. This may be accomplished by properly matching the beam 
to the spectrometer . It is then not necessary, or even possible, to determine 
the point on the target from which an event originated. In energy-loss con-
figuration the accelerator, beam transport , and spectrometer form a.n a.chro-
matic system that images the beam onto the spectrometer focal plane. Each 
m onochromatic component is focuss ed on the target (or slightly bey nd th 
target), but the beam is dispersed spatially across the target to match the 
spectrometer [Ber 79 , Zei 71] . 
163 
On the way from the cyclotron extraction to the detector plane the co-
ordinates of a particle undergo several transformations. If we denote the (hor-
izontal) coordinates at the entrance slit of the beam line system by (x 0 , 00 , 50 ) 
and the ones in the detector plane with (x, 0, 5) the transformation in the 
bending plane can be described as 
X Xo 
0 = S -T?B ? ( 4.1) 
5 
where S is the transport matrix representing the spectrometer , T = T ? 1 
is the target transformation which takes care of the scattering angle a and 
the target angle <I> , and B is the beam line transport matrix. In terms of 
the targe t transformation T = cos( a - <I>) / cos <I>, the kinematic broadenin g 
fact or I< = ~ t governing the variation of momentum with scattering angle1 , 
and the dispersion matching fact or C = 8 Ell. .!!l!.... , the transformation Eq . ( 4.1) 
P 8 Po 
can be writt en m ore explicitly as 
( 4.2) 
In this equation the s;1 and b;1 are elements of the transport matri ces S and B , 
resp ectively. The ij transport matrix elements are associated with the matrix 
1 K is identical to the k factor fr om Section 3.4 .1. 
164 
elem ents of Section 3.4.1 as follows: 11 ---t (x I x) = M"' (magnification), 
22 ---t (0 I 0) = Me (angular magnification) , 12 ---t (x I 0) (point-to- point 
fo cus) , 21 ---t (0 I x) (parallel-to-parallel focus) , 16 ---t (x / 5) = D"' (spatial 
dispersion), and 26 ---t (0 / 5) = De (angular dispersion) . T he angle 02 
is defined by the angular acceptance of the spectrometer . When properly 
m a t ched , the coefficients of 00 , 50 and 02 vanish . The (momentum) resolving 
p ower of the matched system is then 
( 4. 3) 
where the overall magnification is given as M 0 v = s11 b11 T - s16b21 I<. There-
fore, the resolution of the total system is a functi on of J( and will usually 
becom e worse for large values of J( . Tuning the spectrometer such that the 
coeffi cient of 02 vanishes is done in the K600 with the K-coil ; this so-called 
kin ematic displa cem ent of the focal plane has been described earlier in Sec-
tion 3.4 .1. Tuning the beam to eliminate the influence of the momentum 
spread in the beam on the image width restricts the dispersion of the beam 
line . From Eq. ( 4.2) we see that the coefficient of 50 vanishes if 
( 4.4 ) 
For J( = 0, i .e. light particles on heavy targets , one recovers the well known 
disp ersion matching condition 
Dbcorn = - (D"' ) C ( 4.5) 
]if"' spec T 
B ecause C and T are always close to uni ty 2 , thi s implies that the dispersion 
2 In t ransmission configuration where <I> = o. / 2, the ta rget transform a ti on T is exact ly 
un ity. 
165 
of the beam has to be equalized by the resolution 8 16 / 8 11 of the spectrometer. 
Finally, for a finite K value the beam line tune has to fulfill the condition 
-b12 = -S16 -J{ = L 
b22 ( 4.6) S11 T - ' 
called the kinematic defocusing of the beam at the target. The quantity L 
represents the defocusing distance . This step eliminates kinematic broaden-
ing due to the angular spread of the incident beam by forming a correlation 
at the target between position and incident angle (see Figure 4.1) [Mar 83 , 
K600 L]. A prerequisite for kinematic defocusing to work is a convergent 
beam . Convergence or divergence of the beam, and also the beam angle 
and its temporal fluctuations , can be monitored to a certain extent with the 
left-right integrator asymmetry. 
Quadrupoles Q8, Q9 and Ql0 m BL 8 between the QDDM and the 
target are used to control both kinematic defocusing and dispersion match-
ing. Also, the vertical spot size can be varied with Q9. For 120 Me V and 
200 Me V beam energies extensive grid searches covering the space of relative 
changes 5Qi = !:iQ;/Q; ( i = 8, 9, 10) in the quadrupole strengths were per-
formed . The result of these investigations was that in this space there is a 
more or less well defined "valley" where the beam is correctly ( de )focussed. 
Subsequently, one moves along this valley until the dispersion is matched. 
For this task we used a full target ( usually either a Au or Ca target) and th e 
three-hole hodoscope which was mentioned in Chapter 3. First , the correct 
K- and H-coil settings were verified by inspecting the slope and the curva-
ture of the elastic line in an x 101 histogram without the hodoscope. Then, 
166 
OJ 
~ <D 
f II 
OJ 
0... ~ ~ 
< II 
N N 
~ ~ 
Figure 4.1: Kinem atic defocusing; if a convergent beam ( the extreme rays 
of the beam define a cone with opening angle 0 ) is defocused by the correct 
distance L , the scattering angle a becomes independent from the position on 
the target . 
167 
the center hole of the hodoscop e was positioned such that a section of the 
elastic line with b.01 ;::::: 50 mrad appeared in x 10 J between approximately 
550 and 650 mrad (i .e. in the good region of the focal plane). Subsequently, 
we subdi vided thi s section by two or three software windows which defined 
different p ositions on the t arget and gated 1-d x J histograms on t hese win-
dows . With the help of the 1-d histograms we determined both (de)focusing 
and dispersion m a tching: the form er was t ested by examining the F WHM 
line widths , the la tter was evaluated from the difference of the cent roid po-
sitions found in the histograms gated on the low-0 f and high-0 f windows, 
respectively. Figure 4 .2, t aken from the March 1988 run book, shows sample 
histograms. The procedure is to first find the fo cus valley by varying Q8 and 
then adjusting Q9 and Ql0 according to 
( 4. 7) 
where a = 4.23 (3 .24) for 120 MeV (200 MeV) ; this ensures that we approach 
the focus valley perpendicularly. The beam is correctly ( de )focussed when 
the FWHM line widths are minimized . Then, for dispersion matching we 
vary Ql0 and adjust Q8 and Q9 according to 
( 4.8) 
This ensures that we now go parallel to the bottom of the focus valley and 
thus maintain the correct fo cus . The dispersion is m at ched when the low -01 
168 
I ffl 
, : I' I 
. .. 
?, 
?,:. 
?, . .... ?. 
. :.1i ;1 ?i? ,,; .. 
. ? I'"? 11 , 
.' .. ? ! :,? ,: :: :?? 
: i ,'? .. i.: . 
'? f ' .?? 
"' iQ 'en . . 
1-?r" .:. . -- . 
, I'? I;?:?: I .. ?? ? 
I':";? 'I'., . I, ,'' .. ' . . . 
, l?,~"l1'?i ,, ,??1 ??,? ?, :,' 
. <'':; 1::! ?; 1,?.: 1 1'j,: 1::, ?: ... . , .. ? 
. :,L,;!1~1, .. :,l,,1,1.,,,,,,,. . ,:,:;?, 
? ? : ?? ?.; ~ l' JI  HI'j  l.?l ~ l'  ~.~.~ ~?I  'I'  ?.  . .. --,_;il!l 'I ?' :"!.i!~li11 
, I ?'1 I , , 
., __ .J_-: _J 
G 
~----???:?_?,,-:_.: ?_ __- --...L.1~;~ 
Figure 4.2: Kinematic defocusing and dispersion matching . The beam is 
correctly ( de )focussed when the FWHMs are minimum; when the dispersion 
is matched correctly, additionally the centroid positions are the same . 
169 
and high-0 f centroid positions are equal. Finally, in order to maintain disper-
sion matching once it was achieved for a particular k for different kinematic 
values, the K-coil and the quadrupoles were corrected with dK / dk = +1586, 
dQ 8 / dk = - 2138, dQ 9 /dk ~ 0, and dQ 10 / dk = +1466 (K600 Development 
Logbook #9, page 123) . With this prescription, 16 0 peak widths of ~120 keV 
(dispersion matched for a gold target) could be reduced to ~60 keV. 
Expanding the angle 0 in the fashion of Eq. ( 4.2) for x, reveals the 
dependence of the scattering angle on the incident m omentum . Therefore, 
the sometimes large spread of the dispersed beam on target could lead to 
significant uncertainties in the scattering angle, especially if the illumina tion 
of the target varies with different beam tunes, unless the angular dispersion 
of the beam is properly adjusted. Correct angular dispersion will also ensure 
that the cross section will be stable against small fluctuati ons in the cyclotron 
beam energy. The goal, therefore, is to tune the beam such that , for different 
beam rays across the target (for a dispersion matched beam these different 
rays correspond to different beam momenta) , the incident angle changes in 
a way that leaves the scattering angle the same . In other words, if R is the 
distance from slit 2 in BL 8 to the target and T is the target transformation , 
one has to adjust the angular dispersion b 26 of the beam such that b26 = 
b16T / R. 
To t est whether the angular dispersion is correc tly matched one ca.11 
shift the beam across the target with the QDDM bending magnet and record 
the cross sections for each beam position . Usually, this t es t is performed 
with a BeO target because the ben.m creates a clearly visible lumin escent 
170 
spot on the target, thus allowing an accurate position determination. The 
spectrometer angle is chosen to be in a steep region of the cross section 
angular distribution to optimize the sensitivity to changes in the scattering 
angle ( for example, at 22? the slope of the 100 Me V 160 elastic cross section 
is about 25%/ deg). A (0 / o) correlation for a dispersion matched beam 
will translate scattering angle dependence of the cross section into a position 
dependence. 
This position dependence can be removed if the angular dispersion is 
properly matched. As described in [Ber 89], the spatial and angular disper-
sions on the target can be decoupled by shifting the focus of the QDDM ana-
lyzing system into the middle of quadrupole Q7 of BL 8. It was demonstrated 
that with appropriate settings of the kinematic coils within the QDDM the 
angular dispersion can then be varied using Q7 until the scattering angle 
into the spectrometer is nearly independent of the beam ray on target . The 
spatial dispersion on target , the dispersion matching , and resolution are not 
a ffected by this procedure. This procedure was not available, however, at the 
time of our experiment. Therefore, we are not sure whether our beam was 
indeed prop erly angular dispersion matched , although the aforementioned 
"slop e" test was performed and at the time seemed to indicate so. 
4.2 Scattering Angle 
During the experiment the spectrometer angle was set with help of an 
electronic encoder . The calibration which related the encoder reading to the 
171 
spectrometer angle was 
0.pcc = Bene - 0.115(8) . (4 .9) 
For ea 1
c 1
  scattered event an angle 01 in the focal plane is determined . This 
angle i 1 . 
s re ated with the angle at tl1e target (0tgt) and with the scattering 
angle3 (0 
~ scatt) according to 
0scntt = 0spcc - 0tgt + L\0 , (4 .10) 
Where /\0 ? 
u ~ Is the scattering angle offset . The ability to reconstruct the scat-
t . 
ermg angles from the focal plane quantities is especially important when the 
acceptance is to be divided into smaller bins 4 . Although we only analyzed 
unbinn e d pos1?t 1? 0n spectra and employed t 11 e ang1 e  at t 11 e target almost ex-
clusively only for background rejection , we want to outline in the following 
our calibration of 0 tgt ? For the calibration we used a multislit aperture which 
Was mounted on a 1/ 2 inch thick brass frame. T11e widtl1 of the slits was 
?1. 2 Inrad, with exception of the center slit which had a width of ? 2.3 mrad . 
The centers of the seven slits defined angles of -41.587, -27.681 , -13.925, 0. 792 
( center slit), 15.555, and 29.296 mrad at the target. For tl1e calibration we 
J 
While the scattering angle is counted positive to the left of the beam (counter-
clockwl? se ) , the angle at the target is counted pos1?t 1? ve tot h e ri?g  h to f t h e b earn ( clockwi?s e) . 
4 B? 
inned spectra are x spectra gated on 0ti; t ; the observed 0ti; t spectrum shape 
Y(x) f; J 
or an angular acceptance 2a results from the convoluti on of the cross section an-
gular d 1?s tn?b ution D( x) an acceptance function A ( x ) w h.1 c1  ? 1 1s 11rn? ty f or /x  / :S a a nd 
Zero ' . , = 1 _ 1 ( ~ ) ' ( otherwise, and a (Gaussian) resolution functwn R(x - x) rr ?21r e , " : 
Y_ x) :::: D(x) ? A(x) ? R(x) . For the simplest case where D(x) ~ D, a short calculation 
Yields Y(x) :::: Q {erf( tl.!,1; ) _ erf( 'E..=.l! )}, where erf is the error functi on; the quantities 
o- and ? 2 (T ?2 " ./2 . ( 
' if unknown, D can be obtained from fitting a 0tgt spectrum by y x) . 
172 
used a BeO target (18?, 100 MeV, k = p- 1dp/d0 = 0.022) and a 197 Au target 
(24?, 100 MeV, k = 0.002) . The slits generate for each nuclear line distinct 
"splotches" in a x 10 I histogram . Gating on these splotches allows to asso-
ciate unambiguously the x I and 0I  coordinates of each event with its 0tgt 
value . The three values x I , 0I  and 0tgt for each event were stored in a file 
as "rays" . The angle at the target , 0tgt , was then fitted as a polynomial in 
terms of the focal plane quantities, xi and 01 . Using the form 
( 4.11) 
for the BeO target, we found the following values for the coefficients and 
their relative uncertainties : 
Coefficient Value ?fl (%) 
a - 3.01048E+ 02 0.7 
b - 2.27857? - 02 7.0 
C +5. 78602?- 01 1.3 
d - 6.23202? - 05 4.3 
e - 3.24431?- 05 22.6 
The uncertainties in this table were estimated from the error matrix. In order 
to achieve uniform weighting, the same number of rays for each 0tgt were used 
in the fit . For BeO, the centroid angles of the slits could be reproduced with 
this polynomial to better than 0.27 mrad , corresponding to about 0.015?. For 
gold, with its ten times smaller k-fact or , we fo und that th e reprodu cibility 
became correspondingly worse by a fact or of t en . The reason for this worse 
reproducibility are , as we will show in Appendix C, the different kinematical 
coil settings for targets with different k. 
173 
In our experiment, the k-factor varied between about 0.002 (forward 
angles , h eavy t argets ) an d 0.065 ( large ang1 e s, li.g ht targets), corresponding 
to uncertainty contributions to the calculated scattering angle of about 0.150 
and 0.05?, respectively. The k-factor is plotted for 160 and 4?Ca at both 100 
and 200 Me Vin Figure 4.3. We see tha t for 4?Ca at 100 Me V we should get 
reasonably small scattering angle uncertainty contributions which are less 
than 0.05? for lab angles above about 14? (for 48Ca above about 17?). A 
Inore accurate 0tgt calibration would find tl1e coefficients in Eq . (4.11) for 
a variety of k values. Each coefficient will then be a function of k and the 
value of the coefficient appropriate for a specific reaction can be gained by 
interpolation. 
For the angle calibration to succeed, it is important that the H- and 
R -coils be set correctly (i .e. all tJie lines are straight and not tilted in x101) 
for both the experiment and the calibration. This is crucial since, as the 
RAYTRACE calculations in Appendix C demonstrate, the dipole contribu-
tions and the associated shifts in both x f and Bf produced by the kinematic 
coils are quite significant . In Appendix C we show the results for a 0tgt and 
a 5 or m z?s sz?n g mass cal 'b ra t ? ?th the H- and K-coi1 to dipole ratios as 1 1011 WI 
explicit v ari? a bl es m.  t l1 e cal ib ra t ? polynomial. The simulation was done 10 11 
With rays wlii'ch were generate d bY  tl1 e code RAYTRACE. The rays were 
then t ll d D-coordinate system of RA YTRA CE 
ransformed from the so-ca e 
into the . f r front drift cliamber wire plan e. Th e 
x f , 0 f coordmate system o ou 
fair} . e describe the input rays witl1 an 
Y complicated polynomials wJuch emerg 
accu b t O Ol o) for 0i t and about 0.03 % for racy of about 0.17 mrad ( or a ou ? g 
174 
0 
I I l'-
I I 
I I 
I I 
I I 
I ro \ 
0 I 
\ u I co 0 I 
\ '<I' I 0 
\ I c.o 
\ I 
\ I 
\ I 
\ I 
\ 
\ I 
\ I 
\ I 
I 0 \ 
\ I l() 
\ I 
\ I 
\ I 
\ I 
\ \ 
\ 
\ I 
\ \ 0 
\ \ 
\ 'tj' 
\ \ 
\ I 
\ I ,.........___ 
\ I 
\ \ Q() 
\ \ 
\ Q) 
\ \ 
\ I 0 '
I (;) ..,
U__, , 
\ 
\ I 
\ I --:J 
\ I 
\ CJ:) 
\ \ 
\ I 
\ I 
\ \ 
\ 0 
\ N 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
-0 
0 
co 'tj' N 0 
0 '0?  0 0 0 
0 0 0 0 0 
Figure 4.3: Kinematic factor k = p- 1 dp / d0 for 16 0 and 4?Ca; dash ed lines 
are for 100 Me V , solid lines for 200 Me V protons. 
175 
5. 
Tlie scattering angle offset was determined, where possible, with two 
metliods . First, elastic cross sections were measured at several angles on 
either side of tl1e beam . Second, the differentail recoil of 9Be witl1 respect to 
160 was used. Tl1e first metlwd was only used for redundancy and as a cl1eck 
for tlie 100 Me V runs from January 1987. The April 1987 runs were at large 
angles, and a problem witl1 the integrator scale in January 1988 rendered 
many of tlie elastic forward angle runs useless . An analysis of Mardi 1988 
data is underway. Elastic measurements for the first method were performed 
in 2? steps between 10? and 16? and - 10? and - 16?, respectively. Tli e 
simplest way to grapl1ically extract the scattering angle offset is to plot the 
unpolarized cross sections for tl1ese measurements , as shown in Figure 4.4 , 
symmetrically about tl1e midpoint at ?13? and find tl1e intersection point , 
where tl1e cross sections are the same. The distance of tl1e intersection p oint 
to tlie midpoint is tl1e scattering angle offset; if tl1ere were no offset tl, e 
cross plot would be perfectly symmetric about tl1e midpoint . In our sign 
convention tl1e scattering angle offset is positz've if the intersection point is to 
the left of the midpoint and is negative if it is to the rigl1t of tl1e midpoint . 
Instead of doing the plot by 11and, we fitted tl1e data points with a quadratic 
function wl1ich takes into account the cross plot geometry. Witl, 
0spcc + .60 - 13 , for 0 > 0 
(4 .12) 
X = ( 0spcc + .60 + 13, for 0 < 0 
176 
c...o....o..  
I 
'\. I I 
'\. I 
'\. I I 
'\. I 
I 
I 
I I..DI ..
.......,.  
I 
I 
0.0 I 
Q.) I 
-0 I '\. I 
c.o I I 
C\l I 
0 f 
II \ I I 
CD \ I 
<] \ I I 
\ I 
I 
\ I 
I 
\ I 
I 
\ I 
t 
I \ 
/ \ 
I 
I \ 
I 
I \ 
C\1-.:t' 
.I -+-
/ \ --
I 
I \ 
I 
I \ 
II / / 
0.. \ I 
t:?1 / \ I 
/ \ 
I 
I \ 
/ 
I \ 
/ 
/ \ 
/ 
I \ 
/ 
/ 
/ \ 
/ o co 
0 0 0 0 0 --
0 0 0 0 0 
ID 0 I-D 0- l{) C\l C\l 
(..r s / qw) UP/ DP 
Figure 4.4: Determination of t he scatterin g angle? with the cross plot 
m ethod ; 0 ? is the spectrometer angle. T he dashed (positi ve angles ) and 
dot -dashed (negative angles) lines are fit s to the data points using the same 
fitting functi on . 
177 
it can be dem onstrated that the fittin g functi on fo r both posi tive and negative 
angles can be accomodated by 
y = a + b(/0,pcc + Ll0/ - 13) + c(/0spcc + fi0/ - 13) 2 . (4.13) 
The fitting parameters are a, b, fi0 and c. Imposing the same shape fo r 
positive and negative angles makes the fit robust even with only a fe w data 
points . A simple grid search program (GRIDLS , [Bev 69]) is used to optimize 
the p arameters . The fit to 4?Ca yields a scattering angle offset of 0.26?. A 
fit to 16 0 elastic data, on the other hand , yields a value of 0.40? which is 
probably too large. It is possible , however, that the asymmetry in angle of 
th e da ta set used in this fit is responsible for the discrepancy. The forward 
angle measurements for negative spectrometer angles were between - 10 and 
- 16?, while th ose on the positive angle side of the beam were between + 14? 
and + 18?. 
For the differential recoil method we first read the peak positions Pxa 
and Pxb of any convenient pair a and b of calibration peaks ( e.g. 16 0 O't 
and 9 Be 3/ 2~ ) from the summary file created by ALLFIT [Kel ALL] . Since 
the peak positions for our experiment are stored as momentum losses, the 
experimental differential recoil Llcxp can be easily calculated . It is simply the 
difference of the outgoing projectile energies E30 and E3b or , with the mass 
m and m omentum p1 of the incoming projectile, 
( 4.14) 
Using the kinematical formulae in Appendix B , a computer code calculates 
differential recoils Ll( 0) as a functi on of the scattering angle and then mini-
178 
mizes l.6.cxp - .6.(0)1 using a grid search. The method was used for many runs 
and a wide range of spectrometer angles. Subsequently, the scattering angle 
offset was determined as weighted average of the individual measurements . 
We found 0.14(7) 0 for the 100 MeV run in January 1987, and for the 200 MeV 
runs scattering angle offsets of 0.19(7)0 (April 1987) and 0.06(9) 0 (January 
1988), respectively. Thus, it appears that the scattering angle offs et for Jan -
uary 1987 found with the differential recoil method is smaller than the one 
found with the cross plot method. The reason for this is as foll ows: since 
the spectrometer magnetic fields are set for 160 kinematics and due to the 
finite horizontal angular acceptance , the Be lines appear slanted in x 101 ( the 
slope is a function of scattering angle and can be estimated from Figure B .2 
in Appendix B) . The projection of a Be-peak onto Xf is weighted by its cross 
section angular distribution within the acceptance and, therefore, yields a 
systematically smaller peak position . Hence , the differential recoil method 
systematically underestimates the scattering angle. 
Over a wide range of angles the 9Be 3/2~ cross section angular dis-
tribution can be approximately described by an exponential u(q) = ce - aq _ 
For 100 MeV one finds a = 4.26 and c = 1480.3, whereas for 200 MeV th e 
corresponding parameters are a = 3.88 and c = 735.1. To estimate the nec-
essary correction to the scattering angle from differential recoil we evaluate 
the weighted average 
(q) = J qu(q)dq ( 4 .15) 
Ju( q)dq 
over the acceptance q0 ? .6., where 2b. = b.q is the angular acceptance ex-
pressed in terms of momentum transfer and q0 is the central momentum 
179 
transfer . W 1'  tl1  the approximations 
3 
sinh a~~ a~ + ?(a~)
2 
e?all ~ 1 ? a~+ ?(a~)
(l + xf1 ~ 1 - x, 
Eq. (4.15) becomes 
~ 1 2 ~ 2) 1 2 1 3 q qo ( 1 - -6 a - -2 a~ + -12 a ~ 
4 (4.16) 
( ) . 
Evalua ti ng tlu .s  expression for the parameters quoted above, we find 
2 
(q) ~ { qo(l - 0.7562~q2 ) - 0.5325~q + 0.4026~q4, (100 MeV) 
qo(l - 0.6273~q2 ) - 0.4850~q2 + 0.3042~q4, (200 MeV) . 
( 4.17) 
menturn transfer is a function of the scattering angle and one can 
The mo 
94 
approximate this function for 100 Me V by q ~ 0.05 ? 0?? and for 200 Me V 
The quantity ~q in Eq. ( 4.17) can then be approximated 
by q ~ 0.06 . 00 .01 _ 
by 
(100 MeV) (4.18) 
(200 MeV) , 
ts m units of nuad . For q == l.Sfm - 1 and an aperture with a width 
where ~0 .. 0 
of 1/ 2 I. ncl 1 , corre s ponding to a 17. 84 mrad hori.z ontal angular acceptance, 
1 
We gel from Eq. ( 4.17) (q) = 1.4978fm- 1 for 100 MeV ( (q) = 1.4957fm- for 
200 MeV). Thus, the average correction to the scattering angle from dilferen? 
tial recoi.l  should be 0.06' {or 100 MeV and 0.08' for 200 MeV, respecb vely 
With tl e correction we obtain therefore from dI' ff erenti:a 1  reco1?1  a scat term. g 
1
al\g)e offset of t.0 = 0.20(7)' for the 100 MeV run in January 1987, which 
4 1
is cons1?s  t ent wi? th the cross plot value o f 0 ?2 6 f rom ?C a. Fo r tl e 200 M e V 
180 
runs w bt ? 
e O arn angle offsets of 0.27(7) 0 (April 1987) and 0.14(9) 0 (January 
1988) . 
It was pointed out (G . P. Berg, IUCF, private communication) that 
3 2 
? 111111 steering in the last quadrupole before tl1e target, QIO, causes an 
angle cl tange at t 11 e target of 0.1 ? (d.1 stanceq10- target = 183.88 cm). Tlie 
scattering angle offset is cliaracteristic for a particular beam tune and sl10uld 
not vary in time. However, steering in tl1e quadrupoles could make the offset 
time dependent. To investigate a possible time dependence we plotted tl1e 
original individual angle offsets (wl1icl1 were measured at different times) 
Versus tl1eir integrator left-right asymmetries ( which are a measure of tl1e 
beam angle on the Faraday cup). Tliis is shown in Figures 4.5 and 4.6 for 
t
lle 100 MeV run in January 1987 and tl1e 200 MeV run in April 1987. The 
figur es s1 1 0w that in January 1987 tl1e beam was fairly evenly distributed 
over the left and right lialf of the split cup, implying a small beam angle 
at the target, while in April 1987 the beam was centered more on the rigl1t 
half (looking upstream), implying a larger beam angle at the target . This 
is consistent witli the larger angle offset observed in April 1987. However, 
th
ere appears to be 110 obvious correlation between the scattering offset and 
th
e integrator asymmetry, and 110 time dependence of tl1e angle offset. We 
believe, therefore, that it is justified to use an averaged offset. 
* * * 
At the end of the 100 MeV run in January 1988 (runs 946- 991) the 
encoder had an e1 e c t rom.c s pro bl e m causing a systematic offset of 3.4?. Until 
the encoder  pro bl em was fi xe d , tl 1e spectrometer angle was set witl1 l1elp of 
181 
0 
I- 0 p 
~00 
a 0 
I o 0 
LD 
a 0 
41) 
0 a 0 
0 
I- ~ ? 00 
,,--...._ 
0:: 
a + 0 
0 -~.._.. , 
0 a 
D 0 ",,----..-..-_ 
D 0 D 0 
D o of 8 DO 0:: 
0 9 0 h 0 I 
-~.._.. , 
["-
co D o LD 
0) 0 
~ I 
:>---. 
H 
(iJ 
::i 
~ 
(iJ 
I-:, 
...-4 
0 
r--- (0 LD tj< (:) C\1 ...-4 0 
0 0 0 0 0 0 0 0 
Figure 4.5: Correlation betweeen the scattering angle offset and th e in -
tegrator left-right asymmetry (January 1987). The diamonds denote angle 
offsets from spectra with spin-up, the squares angle offsets extracted from 
spectra with spin-down. 
182 
lD 
0 
,,.--.... 
~ 
+ 
0 i--.::i '--" 
~ 
,,.--.... 
~ 
0 0 I 
i--.::i 
'--" 
0 0 ? 
~ boo o0 o 
? C lD 
['- ? 6 ? 
co ? 0 0 0 0 e 0 0 0 0 I 
0) 1 ? IJQO I 
,-4 ? ? 
?~- 0 0 00 ~ 
0.. ? ? 
<r: D 0 
,.--l 
co ['- (0 lD -tj< ("') N ,.--l 0 
0 0 0 0 0 0 0 0 0 
('.aa p) ev 
Figure 4.6: Correla tion betweeen the scat terin g angle offset and the inte-
grator left -right asymmetry ( April 198 7) . T he di amonds denote angle offsets 
from spectra with spin-up , the squares angle offsets extracted from spectra 
with spin-down. 
183 
the mechanical scale on t he K600 rail and a tape meas ure . T he angles of 
runs with the encoder problem were corrected by the appropri ate amount (in 
the li st of runs , Appendix E, runs with encoder problem are marked with an 
asterix ) . 
4. 3 Beam Polarization 
The polarization of the beam was determined in the same way as has 
been described in ref. [Ke! 89a]. In absence of a continuous on-line polarime-
t er , the beam polarization was measured every few hours using a 4 He cell 
which was lowered into the beam line segment connecting the injector and 
main-stage cyclotrons (BL 2) . Surface barrier detectors placed symmetri -
cally on either side of the beam line at scattering angles ? 122? were used to 
m easure the left -right asymmetry for both beam orientations. The intrinsic 
de tector asymmetry, due to differences in detector geometry or effi ciency, was 
m easured on two occasions (January 1987 and March 1988) with an unpolar-
ized b eam obtained from the outgassing of the walls of the source when the 
atomic beam valve was closed ; both times it differed from unity by about 2%. 
The transfer line energy is 8.6 MeV for 100 MeV protons and around 15 MeV 
(field limit) for 200 MeV protons. Over this energy range, the 4 He analyzing 
p ower is known to be essenti ally unity at the chosen scatterin g angle [Dod 76, 
Dod 77]. More specifically, we a.re using Ay = 0.985 for 100 Me V protons and 
Ay = 0.991 for 200 MeV pro tons (H . Petri , IUCF, private communi cation) . 
To eliminate the effect of detector effi ciencies, the left and right yields 
Yi ,r for each beam orientation were normalized with respect to corresponding 
184 
pulser signals N 
l,r ? In terms of the ratios R1,r = Yi ,r / N1,r for each beam 
0 rientati on, we o b tam?  for spm. -up and spm. -down polan.z at1.0 ns 
p _ _!:_ aR1u.,d - Rru.,d 
u.,d - A R + R , (4 .19) 
y D'. lu.,d ru. ,d 
where a == R ro / R .10 1s  t l1 e intrr.n s1.c  d etector asymmetry; the subscript "O" 
denotes ? ? 
quantities for unpolarized beam . Using Poisson statistics, the (sta-
t.1 stical) u1 1cer t am?  t?1 es o f spm? -up an d spm? -d own po1 a n?z a t1? 0ns are gi?v en as 
The beam poJ a n?z ab?o n f,o r eac1 1  on?e ntatr?n n was measured  t o a statistical 
Precisi on o f generally about 1 % or better. 
The measured and interpolated beam polarizations for all the run pe-
riods a 1 
re s 1own in Figures 4. 7 and 4.8. The appropriate beam polarizations 
for ind ? .d . 
lVJ ua1 runs were determined by interpolatrnn of the measured polar-
ization s. We  assi.g n errors of up to 2% to the interpo1 a te d po1 a rizations in 
the J 
anuary 1987 and January 1988 "dip regions" of the polarization . Tl1e 
interpol a t ed beam polarization values and their uncertam? b?e s 1r 0r each run 
are st0red in a file which is read by a computer sort code that calculates un-
Polarized cross sections and analyzing powers fr om ALLFIT summary fil es 
fI<el ALL]. 
In January 1987 (100 MeV) the spin-up polarization varied be tween 71 
aud 75 .5%, while the spin-down polarization varied between 73 and 77.5 %. 
We can not explain the observed wide dip in the polarizations, with excep-
tion of th e very deep excursion at about 250 11 ours w 11 en tl1 e fil amen t;  o f 
th
e ion source burned out . In April 1987 (200 Me V) the beam polarizations 
185 
0 
""" 
0 
C\l 
0 
0 
r-- c0o  --
co ..c
-:- 
'-" 
0) 
-....... (I.) 0 ..... <D a 
~ ~ 
0.. 
~ 0 
""" 
0 
C\I 
0 
0 0 0 0 0 0 0 r-- 0 
0 0 
lf) (") C\l (0 lf) (") 
r-- r"-"-" r-- r-- r-- r-- r-- r-- z""-.".  r--
0 
lf) 
(") 
-~ - --:::=EE. 0 =- 0 (") 
-=- = -- 0 lf) 
C\l 
- --
r-- 0 
- co 0 ------=- 0) - C\I ..c: ....... ~ '-" 
--- :>., -- 0 (I.) = lf) ........ ~ 8 <t1 
= ::l --- E?--c:: - 0 
- ..r.o,,  0 
=-= - -= --= --- 0 l{) 
- -
0 
0 0 0 0 0 0 0 0 0 0 
co 0 0 0 0 0 (0 C\I 0 co CD 0 co (0 C\I 0 co 
r-- r-- r"-"-" r-- z-.. <D <D (""0"  co r-- z-.. r"-"-" r-- r-- (0 
nd X 0001 Pd X 0001-
Figure 4.7: Beam polarizations in January 1987 (100 MeV) and April 1987 
(200 MeV ). 
186 
0 
r--
0 
l[) 
co 
-co (;) 
0 
C\l 
0... .. 
0 
0 
0 
0 0 0
'<I' 
r-- C\l r-- -
C'J 
 l() r--r--
r-- 0 - tj' -
~ 0 C\l 
-= co -co 
- -
0) ~ 0- 0 = I 0 ,..-.. 
-~ co ..d 
~ 
= --~- 
0 
-.I' 
n d x 0001 
p19?i 8g8u r(IeO  4 ? 8 : Beam polarizations Ill January 1988 (200 MeV) and March 
0 MeV) . 
187 
were fairly constan t tluouglw u t the wlw le run: the polarization for spin-up 
was 73.2(5) %, tli e p olarization for spin-down was 75.4(5 )% . T J1e quoted un -
certainties a re the s tandard devia tions a.bout the m ean . In J anuary 1988 
(200 M e V) we o bser ved another deep dip in tlie p olarizations be tween tl1 e 
50tl1 and 60tli hours , then a recovery to almost the original p olariza ti ons. 
However , subsequently both p olarizations deteriorate again . The spin-up p o-
larization varied be tween 71 .5 and 77%, the spin-down p olarization between 
73 .5 and 79 %. Finally , in M arch 1988 (100 Me V) the beam p olariza tions were 
again stable tluougl10ut tl1e entire run and we determined average p olariza-
tions o[ 72 .8(5) % for spin-up and 74.8(5) % for spin-down . The uncert ainties 
are, as before, the standard deviations a bout the mean . It appears tliat dur-
ing all our runs the spin-down p olarization was better by about 2% tha t the 
spin-up p olarization . 
The technique used in our experiment measures tlie beam p olariza tion 
between tl1e two cyclotrons . Scl1wandt et al. verified by remeasuring 12 C 
elastic analyzing p owers at several energies that the depolarization during 
main-stage acceleration is n egligible {Scl1 82}. Tl1ese measurem ents rep ro-
duced tlie results of earlier double-scattering experiments . An upper limit of 
3% for the depolarization during main-stage acceleration Jias been previously 
estimated based on elastic analyzing p owers {Kel 89a} , but may not apply for 
200 M e V because of a dep olarizing resonance near 180 MeV (J . J. K elly , 
private communication) . 
188 
Other Calibrations and Tests 
Because of tl1e large vertical magnification of the K600 spectrometer 
(of the order 6 ? ? " 
111 normaJ " d1. Spers1? 0n mo d e, see Cl iapter 3) , correct vertical 
centering of the beam on target is critical for proper vertical centering of tl1e 
una 
ge 1
?1 1 the focal plane5 ? Tl1is is especially important at tl1e extreme ends of 
tl1e focal pl b . 1 . 1 . . r 
ane ecause of tl1e potentially arge vert1ca image size 10r a large 
</>-acceptance. In the absence of a y-position chamber, vertical centeri11g of 
the im ? 
age 111 the focal plane was first checked by means of a pair of small, 
t
liin "top" an d "b ottom" diagnostic scintillators w1 u ?c 1 1  over1 a ppe d ver t1? ca 11 y 
by about ?5 
nun centered about the K600 median plane. Measurements of 
the scint"1l l ator coincidence yield versus verti?c a1  pos1?t 1?0 n o f be arn on t arge t 
(determined by the current of vertical steerer VT 5 in BL 8) for a small 
eutran ce aperture were used as an indicator for correc t ver t1? ca1  cen t err?n g 6 . 
In tl1 e course of these measurements, 110wever, i?t  became apparent tl ia t tl ie 
optical alignment of the wire cliambers and the scintillators relative to tl1e 
llledian plane was off by about 1 cm and tliat the center of tl1e magnet 
acceptance did not coincide witli tlie center defined by llie detectors . For 
s
ubsequent v er t1? ca l scans, tl1 ere ?o re, 1? t wa s decided to use a different method 
Where the ero ss sec t1? 011 o f t 11 e 12c 2+ s t a t e was measured as a function of the 1 
5 
p ro-~per -ve-rti-cal- ce-nt-eri-ng- on- t-he target is also i. mportan tr,o r ac curate char. ge measure-
11lents in th 
Vert" e Faraday?  cups since ' like the spectrometer m 
a g nets ' t'h e dump pipe defin es a 
ical acceptance. 
6 
d Res u It  of these measurements is a trapezoi.d al h ape,,  the correct VT 5 current is then 5 
efined b 
Y the middle of the plateau . 
189 
VT 5 current . B tl 
or 1ese measurements the spectrometer angle was cliosen to 
be 220 
' an angle w11 ere t11e 2! cross section is known to be fla t . 
*** 
th 
Wi li elp of a CH2 target we checked t11e stability of t11e 12C 2t cross 
sectio11  wi? t 11  respec t to 
? acceptance 
' 
? computer live time 
. ' 
? beam current (i. e. accuracy of charge integration), and 
? elastic/ inelastic run (i .e. position on focal plane and rate dependence) . 
While the . .. 
sens1tiv1ty of cross sections to t11 e acceptance was tested in dedi-
cated ru1 f, 
ls or both 100 and 200 Me V, dedicated rate and cliarge integration 
st udies 
Were done only at 100 Me V . At 100 Me V, t11e 1/ 2 incl1 diameter 
aperture (Al 
, 0.25 msr) was t11 e smallest aperture used for t11e acceptance 
test and 2 . . . . ) 
a lllch diameter aperture ( 4.0 msr, not 11sted rn Table 3.2 was the 
largest A . 
? t 200 Me V the smallest aperture used for the test was the 1/ 2 rncl1 
diainet . . 
er aperture ( A 1) and the largest was t11e 1. 6 111cl1 x 1. 6 111cl1 aperture 
(B5 ' 3 ? 18 l11sr ) . We found at both energies tJiat cross s.e ct1? 0ns measure d ? 1 w1t 1 
Various . 1 
acceptances were consistent within 2%. Therefore, no specia ac-
Ceptanc e correctw. ns to our cross sectw? n results a.re necessary prov1'd  e d tl 1a l 
Proper v t? 
er ical centering has been achieved. 
First, we found that 12c 2t cross sections from a sample of elas tic and 
111elastJ?c  spectra agreed within 3%. The elas tic spectrum was t a k en ?t1w1 1  a 
190 
l nA beam tl t . 
, le wo melastic spectra with an 11 nA beam. Furthermore, 
one inelastic 
spectrum contained tl1e elastic peak (higl1 rate) while in tl1e 
other the 1 . 
e astic peak was shifted bel1ind a copper block (low rate, different 
focal pla .. 
ne pos1t10n) . Second, we studied the cross section dependence of 
the 2+ 
1 state as a function of both computer live time and beam current. 
For this t es t we used m. elasti.c  spectra. In one samp1 e  o f runs the live time 
Varied between 21 and 96% while tl1e beam current 7 for these same runs 
' 
varied b t 
e ween 2.5 and 16.3. nA . Here we observed a total spread of 1.8% 
in the c . 
ross sect10n. In the otlier sample the live time varied between 51 
and 76~0 1 . , w ule the beam current varied between 2.8 and 29.4 nA . rDo r t 1u .s  
sample , a smaller total spread of 1.1 % in the cross section was o bs erve d . 
It ap 
pears from these results tlrnt the accuracy of tl1e live time correction 
might b e s 11? g h t.l y more important.  for tl1e correct cross sect10? 11 t 1r nn tl ie b earn 
current ? Cll arge 1. 11tegrat1. 0n seems to be faJ.r ly accurate iro r a 'd WJ e range 0 f 
beam 
currents. Off-line analysis of the 100 and 200 Me V production runs 
confirm d 
e these findings . 
Although the data rate for tl1e production runs was usually lower than 
about 300 Hz , our group studi.e d for a ?e w cases tl ie eJalle ct of the fast clears 
(run Period in March 1988) on the dead time for runs with high data rates 
(B. S. Fla1 1 d ers, Amen.c an Urn.v ers1. ty, pn.v at  e com munication) ? It was found 
that the live time seems to be independent of the fast clear rate for rates 
Up to l kHz. In this region the live time correction wl1ich is aplied to tl, e 
7 Avera____ _____ m of the left and right integrator 
s I ge beam currents are calculated from the su 
ca ers . 
191 
cross sect . 1 1 
Ion s 1ou d be therefore reasonably accurate. However, fast clears 
do app 
ear to reduce the live time for rates in excess of 1 kHz. Based on 
some test cases, for 3- 3.5 kHz the dead time increases between 1 and 5%, for 
3 4 
- kHz by about 5%, for 4- 5 kHz between 7 and 10%, and for 8-10 kHz tlie 
dead tin . 
le lllcreases between 16 and 19%. 
192 
5 Drift Chamber Detectors 
5.1 Introduction 
fhe K600 drift chambers detectors allow the accurate determination 
of an ncident particle trajectory by measuring the drift times of ionization 
electr ns with the help of appropriate electronics (see Section 3.6 .2 ). As an 
energe .ic incident particle ionizes the chamber gas along its track, it gen-
erates positive ions which travel to the cathode planes and electrons which 
drift to the anode wires in the mid -plane of the chamber. The electrons -. 
.. . 
which are collected at the anode create a signal which is used as a STOP for -. 
the TDC associated with the wire ( we remember that the START signal is 
provided by a scintillator coincidence) . Figure 5.1 shows a typical drift time 
?r 
spectrum for one of the multiplexers. On the software level , drift times are 
first converted into drift distances . Then, after the character of the chamber 
event has been identified, an appropriate algorithm computes the exact po-
sition and angle of the trajectory in the focal plane (which coincides fairly 
well with the wire plane of the front VDC) . Finally, since certain classes 
of events (e .g. multiple hits), while not being analyzed, may contain good 
events, appropriate corrections of the final cross sections are necessary. 
The configuration of the (guard and sense) wires and the cathode planes 
is such that each sense wire defines a drift cell of 2.0 mm width and 12. 7 nnn 
length . Consecutive sense wires are separated by 6 mm . The electric fi eld is 
very uniform across the whole drift cell and only in the immediate vicinity of 
the anode wires and very close to the cathode plane do we find nonlinearities . 
193 
0 
c.o 
~ 
0 
,q< 
~ 
0 
C\1 
~ 
0 
0 
~ 
--lfl ~ J: -? 0 
o:J , .. Q) 
s .-.?.  
0 ?b- 
c.o .+-) 
?~- '( ~ 
0 Q 
,q< 
0 
C\1 
0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
r-- c.o LO tj' C'J C\1 ~ 
+P / NP 
Figure 5 .1: Typical drift time spectrum; ea.ch multiplexer is associated 
with one such spectrum . The spectrum shown was ta.ken by illuminating the 
whole drift chamber uniformly. 
194 
If the electric field were exactly the same everywhere, we could just form 
the product of the drift velocity ( 46 .1 mm/ ? s) and the measured drift time 
(1 channel in the drift time spectrum corresponds to 0.317 ns) to get the drift 
dist ance. However , because of the nonlinear behaviour at both very short and 
very long drift distances, this method may lead to errors in the determination 
of the correct particle trajectory. Also, although for each chamber the relative 
timing of the drift times is easily adjusted by simply aligning the leading edges 
of the drift time spectra , there remains an unknown overall timing offset . For 
algorithms which involve the differences of drift distances ( e.g. three or more 
consecutive hit s) this absolute offset does not mat ter , but algorithms which 
include the sums of drift distances ( e.g . two-hit events) will be sensitive to 
the correct absolute timing offs et . 
An approach which implicitly takes care of the problem of absolute 
timing, as well as of any nonlinearities and inhomogenities in the chamber 
fields is the construction of a lookup-table or drift table which transforms drift 
times into drift distances. This table is based on the knowledge of the mea-
surable maximum drift length , Smnx, which is equal to the distance between 
the wire and cathode planes, the total charge collected , Q, and the drift time 
distribution , dJ;' . After the drift time to drift distance transfo rm ation has 
been performed for an internally consi stent set of multiplexers, an absolu te 
timing offs et no longer matters because the shortest possible drift length is 
zero and the longest possible drift length is the di stance from t he cath ode 
plane to the wire. The details of the drift time spectrum in corporate all the 
non-linearities and inhomogeniti es of the chamber fields. For example, th e 
195 
st rong fi eld in the immediate vicinity of a wire accelerates a drift electron 
until it creates by ionization secondary elect rons which in turn may create 
more elect rons . This avalanche effect and the associated increase in particles 
collected at the wire is responsible fo r the spike at the leading edge of the 
drift time spectrum . The flat region of the spectrum represents the linear 
region of the drift cell . Finally, the tr ailing edge is not abrup t but is smeared 
out , in part because electrons with long drift distances may be lost (e.g . due 
to recombination , et c.) and not be collected a t the anode wire. 
If l : we assume that the ionization occurs uniformly along the track, the ... . 
verti cal axis of the drift time spectrum, Figure 5.1, represents a ( time de- ,. . 
... . 
pendent) velocity d::. = d:. ~; .. . = cv(t) . Thus, integration of the drift time 
spectrum up to a time twill yield the corresponding drift dist ance s(t) ac-
?r 
cording to 
?r.' 
t dN ' 
s(t) = c 1- dt' . (5.1) o dt' . 
I I'  
The constant c in this equation is c = Brno.x i Ntot , where Smo.x = 12 .7 mm , and 
where Ntot = Q / e is the total number of counts in the drift time spectrum . 
In Figure 5.2 we display graphically a drift t able, in this case for the front 
chamber in January 1987. Experimentally, the drift time spectra which are 
the input for a drift table are generated by illuminating the entire chamber 
uniformly using the high excitation energy continuum of some t arget . 
196 
0 
c.o 
.-t 
0 
C\2 
.-t 
0 ,.----..._ 
0 rJ} 
.-t 
.__~_., , 
Q) l : 
0 
OJ ?-6 ,.., ~ -..  
+.-) n, 
~ 
0 
c.o ?-~ 
Q 
?r 
'i 
0 
'tj< 
i 
I : 
0 
C\2 
0 
I.[) C\2 0 
Figure 5.2: Graphical Representation of a drift table; note the deviation 
of the drift distance s(t) from linearity for drift times t below about 40 ns 
and above about 130 ns. 
197 
5.2 Classification of Events 
Before we can calculate the position and angle of a particle track in the 
focal plane we need to characterize the chamber event. Then the appropriate 
algorithm can be applied to calculate the desired information. To this end, 
one examines the MUX register E data words . As stated in Section 3.6.2, 
information is encoded as follows : 
? the data word is zero if the MUX did not fire; 
) : 
? the data word is negative, i.e . bit 15 is set, if the MUX fired at least .. . 
,. . 
twice; .. 
.. ..  
? the data word is positive if the MUX fired exactly once; then the first 
five bits (bits 0- 4) contain the encoded wire number, the next five bits ?r: 
(bits 5- 9) the encoded MUX number, and the remaining bits (bits 10-
15) are all zero. I :' 
If any MUX register E data word is negative the event is classified as a 
multiple hit and rejected. Otherwise, the MUX ( and associated wire) with 
the shortest drift distance is determined, and a hit pattern is established by 
examining the adjacent MUXs on either side . We usually denote the MUX 
with the shortest wire by "O", the two MUXs on the left by "Ml" and "M2" , 
and the two MUXs on the right by "Pl " and "P2" . If n of the five MUX 
register E data words are nonzero , the event is called an n-hit event . T he 
number of hits a ray can make, i.e . the number of drift cells traversed by 
the ray, is a functi on of angle. As we can see in Figure 5.3, n decreases 
198 
) 
... 
+. ._ ________ :a ~ 
t t 
+-- -=-
Figure 5.3: Extremum rays for n -hit events ; d = 12. 7 mm is the spacing 
be tween the cathode planes , s = 2.0 mm is the width of a single drift cell. 
Both the minimum and maximum angles increase with decreasing n . 
199 
with increasing angle . With the cathode spacmg, d = 12 .7 mm, and the 
width of a single drift cell , s = 2.0 mm, the minimum and maximum angles 
for an n -hit trajectory with the wire plane are 0~~ = arctan ( n! ), and 
3 2
0t2x = arctan ( n~ ), , respectively. This is also summarized in the following 
3 4
table [K600 L] . 
n 2 3 4 5 
0min 38.4? 30 .0? 24 .4? 20 .5? 
0max 72.5? 51.8? 32 .4? 30.0? 
J 
Our calibration for 0tgt gives an angular magnification of about 1. 7 (see 
Section 4.2), and RAYTRACE calculations (see Appendix C) yield for central 
(in 0tgt) rays with various o's the first order relationship 
(5.2) 
where x I is in units of mm , and 0I  is in units of mrad. Because of the slope 
in 01(x1 ), apertures with 0.5, 1.0 , 1.6, and 2.0 inch width, corresponding 
to horizontal angular acceptances of 17.8, 35. 7, 60 .0, and 71.4 mrad , define 
slanted windows in an x 10 I histogram of 30 .2, 60. 7, 102.0, and 121.4 mrad 
width in 01 , respectively. If we look at the focal plane region of 200 mm < 
x 1 < 800 mm, where most of our data are taken , we find that the various 
acceptances cover the following angles: 
Aperture 0.5 in 1.0 in 1.6 in 2.0 in 
0min 32 . 7? 31.8? 30.6? 30. J. O 
0mox 38.5? 39.4? 40.6? 41.2? 
200 
From this t able it follows that while 5-hits should be relat ively ummpor-
t ant, 2-hit events will become increasingly important for large x 1 , i.e . high 
excitation energies, and large scattering angles. Events that make only a 
scintillator coincidence (0-hits) or one hit in either chamber (I -hit s) are most 
likely due t o background and are not analyzed . Indeed, in March 1988 0-hits 
were eliminated altogether on the hardware level by the fast clear coincidence 
circuit which was described in Section 3.6.3. 
The particles which traverse a drift chamber a t an angle to the anode 
plane usually produce a cluster of firings as they cross over several wires . l 
The low-energetic drift electrons involved are produced in "grazing" atomic 
collisions (primary ionization) . Additionally, high-energy electrons, so-called 
o-rays , can be form ed in "knock-on" atomic collisions. These high-energy 
electrons traverse the chamber and can produce fresh electrons and ions as 
well (secondary ionization) . Those events cause multiple firings elsewhere in 
the chamber and have t o be distinguished from the original cluster which 
carries the real position and angle information for the incident track1 . In 
hardware, additional multiple firings can be limited to a certain extent by 
choosing the gate width of the MUXs appropriately. As mentioned in Sec-
tion 3.6.2, for this we internally shortened the gate width to a maximum of 
1 The proba bility for another valid event to occur simultaneously to the fir st event. 
is sma ll : with the usua l cyclot ron pu lse period of 33 ns and with a typical event rate 
of 300 Hz we find a proba bi lity of 9 x 10- G fo r one event per pul se. T he probabi li ty 
for two real events to occur in one pulse is thus 8.1 x 10 - 11 , corresponding lo a rate of 
2. 7 x 10 - 3 Hz. The rate of true simulta neous events is fi ve orders of mag ni t ude sma ller 
and therefore negligible. 
201 
33 ns following the first event . Other, less important mechanisms to generate 
firings of chamber wires , which are unrelated to those from the primary ion-
ization, are pick-up of noise by wires ( e.g. discriminator threshold too low) 
and , to some extent, ionization or pair production by background photons . 
For a regular event which causes only one cluster of wires to fire, all the 
wire numbers in the cluster should be consistent with that of the shortest 
wire . Additional wire firings due to 5-rays, pick-up, or photons that occur 
somewhere else in the chamber but at the same time, can cause a MUX to 
) : 
fire twice. This kind of event is called a multiple hit and bit 15 in MUX 
... 
register E is set; we don't process these events further and only correct the 
.. ..  
final cross section by a factor which represents the probable fracti on of good 
events among the multiples . Since the probability for multiple hits is propor-
tional to the amount of background present, we call this correction a clutter 
correction . In the case of an accidental coincidence where no MUX is firing 
I' 
more than once, the wire numbers of the extra cluster are inconsistent with 
the ones of the good cluster . We call these events extra hits and we do cal-
culate position and angle for most of them , assuming that the shortest wire 
is part of the good cluster. This assumption is good for the majority of the 
extra hits 2 . 
In the analyzer code the character of an event is represented by a hit 
pattern. The hit pattern is a ten bit word (bit s 0- 9) , where bit 2 always 
2 Most extra hits a re caused by 6-rays. Since the extra cluster is due to electrons fr om 
the secondary ionization, it occurs later in time than the good cluster which is du e to 
electrons from the primary ionization. 
202 
represents the shortest wue and is unity, and where bit 7 is always zero. 
While bits 0- 4 are reserved for MUXs which have "consistent,, wire num-
bers , i .e. wire numbers which are within ?2 of the shortest wire, bits 5- 9 
are reserved for MDXs with "inconsistene1 wire numbers . As an example, 
let us consider a 3-hit event for which MDX n ( and wire m) is the MDX 
with the shortest short est drift dist ance, and where the adj acent multiplex-
ers MDX (n - 1) and MDX (n + 1) fired also. Further , let us assume that 
the encoded wire number of MDX ( n + I) is m + 1 and thus consistent , and 
l : 
that the encoded wire number of MDX ( n - I) is l = m - 5k - 1 ( k 2: 1 
and integer , l 2: 0) and thus inconsistent . The hit pattern for such an event 
would then be 
~9 ~8~ ~7 ~6~ ~5 .,4. ..3 .._2 _1 ~~0  
0 1 0 0 0 0 0 1 1 0 . ? r 
~~ ~~~~~~~
M2 Ml Pl P2 M2 Ml O Pl P2 
accidental hits regular hits 
This very comprehensive pattern word was, however, used only for diagnosti c . I: 
purposes . The streamlined analyzer code which was employed for production 
data replay utilized a simpler pattern word in which only the Ml and P l 
wires and the lower five bits of the pattern were used , and no bit was set for 
inconsist ent wires. The resulting pattern word was thus always of the fo rm 
0xlx0, where x = 1, 0. The above example would be represented by the 
pa ttern 00110 , which means that this 3-hit event with an in consistent wire 
would be treated like a regular consecuti ve 2- hi t event . 
With a diagnostic analyzer and tes t package we examined several runs 
from the various run periods. We found that the great majority of event s 
(usually about 95%) were always events that made 3 hits or more. However , 
203 
we also found that generally over 80% of all the 5-hi ts were comprised of a 
4-hit and an accidental coincidence with an inconsistent wire . T he number of 
0-hits and 1-hits was observed to increase with background . But, while the 
fracti on of 0-hits in some cases could be up to about 70% of all scintillator 
coincidences ( one of the reasons for developing the fast clear) , the fraction 
of 1-hits stayed always below about 1.5%. The fraction of non-consecutive 
hits (e.g . patterns 10101 , 01101, 00101, etc .) was found to be generally 
less than 0.5% . Between 1 and 2% of all events were 2-hit events, becoming 
) : I 
more important at large x f and 0f . This was found by a grid in x 101 ( see ? I 
_,.,, I 
Table 5.1) . Omission of the 2-hit events (Figure 5.4c) leaves "holes" in the .- . 'I  
., : : 
x f spectrum ( see Figure 5.4b ). Because of the periodicity of the holes , a 
r I 
global correction of the effici ency is not possible. We also found that the 
?r: 
probability of losing a wire, thus making a 2-hit out of a 3-hit or a 1-hit out 
of a 2-hit event, is very small . Therefore, 2-hits are real events with a. steep 
I ''  
' 
angle and should be included in the efficiency. On the other hand , sin ce 1-hit 
events are not generated by 2-hit events, they are true background events 
( e.g. due to the beam or the Faraday cup) and should not be included in the 
efficiency. The fraction of multiple events generally stayed below 1.5%, while 
the fraction of extra hits was often between 2 and 3%. This is a somewhat 
surprising result since most of the hits are 3-hits or more. The probability for 
an accidental coincidence to fire th e same MUX twice, and th erefore make 
a multiple hit rather than an ex tra hit , should be more than 60% for th ese 
types of events. As mentioned before, extra n -hit events usually have only 
one inconsistent wire and therefore can be treated like an (n - 1)-hit . 
204 
::::i ~ -
si:::  ;-:?.  r~: i ~e:..  
O"" ..., '-' 
,(D;  -::::i ? 
(l i-~- (0 ::;- c;, .0c.. ... ,; r!~; ;c...n..  ,~,  
ct, --~ 
:.::,:.i ..c:,::q:i  ~~  C) 
u, ,; 
<+ H C~./"J. -'-1 ..... c..?  Thetaf/ xf 2000 - 3500 3500 - 5000 5000 - 650 0 6500 - 8000 8000 - 9500 channels 
Ill =~  ~o?  =- ?  ===---===================-------==============-===-==----==-=====================----------------R-- 665 - 725 96.47 +- 7.19 9 7 . 8 6 +- 1. 8 6 98.44 +- 0 . 84 
u,0-,::::,H 60 5 - 665 98.24 +- 2.17 97.90 +- 1.11 97 .35 +- 0 . 81 97.30 +- 0.61 efficiency 
~ 545 - 605 98 . 28 +- 1.69 98.31 +- 1.12 97.86 +- 1.10 97. 78 +- 1.17 ( % ) Q:)~~ 
~-':: .__ 485 - 5 45 97.54 +- 5.29 97.98 +- 4.04 96.30 +- 8.84 ~ 
::,- "::;--' ...... ===========================-===-=======================--====================== 66 5 - 725 1 .0 9 +- 0 . 61 0.45 +- 0.09 0 . 33 +- 0.03 
- ;..;.:, :. tv 0,;  605 - 665 0.12 +- 0.06 0.29 +- 0.04 0.32 +- 0.03 0.36 +- 0.03 bent hits 
Cb (b ... ' 1-1 545 - 605 0.04 +- 0 . 03 0 . 12 +- 0 . 03 0.17 +- 0.03 0.23 +- 0 . 04 ( % ) 
.... ~ w i::: 48 5 - 545 0 . 14 +- 0 . 2-1 
~ ~ ... ? ~ 
66 5 - 725 13.32 +- 2.04 13 . 0 7 +- 0 . 51 17.19 +- 0.27 
tv ::::,S~c.n 
o :::..cr::::i c.n 60 5 - 665 1.84 +- 0 . 21 1 . 67 +- 0.10 2.95 +- 0.10 3.39 +- 0.08 2-hits 
CJ1 v ~ C..G) 545 - 605 1.81 +- 0 . 17 0 . 55 +- 0.06 0.94 +- 0.08 1.01 +- 0.09 (%) 
~ ,; ~ ,.-._ 485 - 545 1. 01 +- 0. 41 0.17 +- 0.15 
0 2..;. ~ ===========================--=----------=====-==-=--------------==-================================ 66 5 - 725 82.34 +- 6 . 40 84.83 +- 1.67 80 . 98 +- 0.73 
...... - ? 0 605 - 665 93.01 +- 2 . 08 86.33 +- 1.01 83.86 +- 0.73 83 . 52 +- 0.55 3-hits 
tv ~~Ill 
~ :=- .... 545 - 605 7 8 . 4 6 +- 1. 4 4 58.97 +- 0 .78 55.92 +- 0.74 56.58 +- 0.79 ( % ) 
:::. ~...:it--:> 485 - 545 44.49 +- 3 . 06 27.97 +- 1.74 25.93 +- 3.69 
t:""1"- en i--- 0 ============================--------=-=======-==-=---=--=====================--=========-===== 
u, ..., 0 66 5 - 725 2.72 +- 0 . 91 1. 60 +- 0. 1 7 1.43 +- 0.07 
~ ~ 60 5 - 665 4 . 6 4 +- 0.34 11.50 +- 0 . 29 12 .6 7 +- 0.22 ~ := ~ 12.57 +- 0.17 4-hits 
c.. 54 5 - 605 19.05 +- 0.58 39.60 +- 0.60 42.2 3 +- 0.61 41.43 +- 0.64 ( % ) ...,.. si:::  ~ < 485 - 5 45 52.90 +- 3.43 70.18 +- 3.17 72 . 8 4 +- 7 . 2 2 --- - - --- - - -
=' - O"" - Thetaf/ xf 2000 - 3500 3500 - 5000 5000 - 6500 6500 - 8000 8000 - 9500 c hannels .... ~ ~ ==========================---=== 
00 o II 
c.. ;:; ~ 
~ e-' ~ -: ~co  ~~  u, 
C~J'J  ~-  ? (b 3, 
,; (l 
~ c'ii" 
...:i ~ = 
- u, (l 
(b ~ :< 
.......  
0 0 0 0 0 0 0 
0 
0 0 0 N00 0 0 0 
0 0 0 oo n 0 0 0 
('I') N rl ?II'\Z II_, ('I') N .... 
O\o-Cle2: 
r-'--;;~e::=!:::=---7 ~~ 
ol-0 
oHC"'l ? . 
0 -rl 
?'~ 
Z~11 
~ 
C0::>0 
0rlll)ct 
0 
II'\00 
COZO\\O 
Hf('I') 
I-? ? 
XUCO 
O'<I' 
orl I?? 
0 rlO 
0 N 
co::.:: 
> -'\Ori 
Cl~ 
xO 
00 
.JJ 0~ II'la......::xn "ti 
L _ ___- ---=:!!l!!~=-------Lr--xco- "?? I 
0 o 0 0 0 0 
0 00 0 0 0 0 
0 0 N00 
.0.. . 0 00z 0 0 0 0 Ir\ ?0Hll 11 co \0 ~ N 
.-----L----__.J'---------t-rl 0-' I _J<l: ' ' ' r, 
w_,> <.u.......  
t-~ ?r. 
~CO ? . 
~ 
~ 
0~ I:'  
0 
000 
II'\ZO\\O 
>HIN 
Cl t- ?? 
XUO 
X OII'\ 
.-4 I .. 
rlO 
N 
::.:: 
-'\Ori ~ 
CilLf'C) 
Ll"1fN ~ 
Cl 
0 
0 - ~ t:lZI- u 
L__c_ ___________. .L.o~ -=! 
Figure 5.4: For run 556 we display in a) and b) the Xf spectrum for 3 or 
more consecutive hits . In c) we display the same spectrum for 2-hits only. 
In d) we show the spectrum for 2 or more consecutive hits . 
206 
5.3 Calculation of Position and Angle 
In the following, we will denote the interpolated fractional wire spacing 
by f and the (one-plane) slope by a. With the wire spacing s, and the 
shortest wire number n , the position and angle in the focal plane can be 
expressed as 
x s(n + f) 
0 arctan a . ( 5.3 ) 
::t : I 
.... . j 
The track position in the front chamber is always the position which we quote 
-..?. . .,i .I '  
as x I. The focal plane angle can also be calculated as two- plane angle from 
. ' 
the track positions in the front and rear planes (xF and xR), the relative r, 
offset between the two planes (L'.ix) , and the distance between the two planes . r, 
(z). We get '' '' 
= z 01 arctan ------+ (5.4) 
' 
I' 
XF llX - XR 
It is this two-plane angle which is always quoted as 01 . For a good event , 
however, we require consistency between the two- plane and one-plane angles. 
The most important classes of events are those which make three of 
m ore consecutive hits . All of these events have the pattern xlllx, where 
x = 1, 0. The calculation of position and one-plane angle for such a track 
is based on the drift distances dM 1 , d0 and dp 1. As Figure 5.5a shows , th e 
trajectory can pass either between the Ml and the 0 (xl.llx) , or between 
the 0 and the Pl wire (xll.lx) . We find for the xl.llx case 
f - _l dr1 - dM1 a = ~ - 2 dr - do ' . ( 5.5) ) 1 
207 
~ . ! ~ . 1 -..-, :'\ ?, M.,?  u > \ I: I: 0 - 0 ~ 0 ~ ~ 0 0 0 i 8 0 0 0 _-; -
,,---.."' - -
() :a ~~\ ...... :a 
:! : I 
~ . i ? I 
;;: I \ '\ 
M -;-;;: I ...,?   ...,  .~ .a ? I u 1:1 1:1 0  0 . ' 
:~ 0 ~ ~ 0 ~ ~ ~ . 
I 
: : 
;; ;; :a . -- -0 0 r, 
:o-~ 0 0 '? ~ . r, 
I ':  
.,, 
I 
0 
;-; ___ .., __ _ 
Figure 5.5: Event types: a) consecutive 3-hit events; b,c) consecutive 2-hit 
events; d) recovery of 2-hit events via two-plane consistency; e ,f) asymmetric 
3-hit events (bent rays) . 
208 
and for the xll.lx case 
a = ~- ( 5.6) ? 
Another very important class of events are consecutive 2-hits (patterns 
00110 and 01100) which have to be included to ensure a flat effici ency in 
x 101 . As before , for each case we have the two possibilities that the track 
passes left or right of the shortest wire (see Figures 5.5b and 5.5c). In con-
trast to 3-hits, however, the positive identification of the left- or right-case is 
less straightforward since two driftlengths are insufficient for unambiguously :;[ : I 
? ? ? I 
determining their relative sign . Intuitively, it is obvious that most consecu-
-?II I 
. ' 
,,- ? I 
tive 2-hits will be of the type 001.10 or 01.100, rather than of type 00.110 ?? If I 
or 011.00 because of the steep angle required for a true 2-hit . Events for r, 
. r, 
which the track passes outside the cluster feature smaller angles and should 
have been 3-hit rather than 2-hit events. Indeed, it was found that the ma-
. 
jority of 2-hits are are of the interior type . We believe that the 2-hits of I ''  
the exterior type are probably due to the loss of either the Ml or Pl wire 
of a potential consecutive 3-hit event. We calculate position and angle for 
consecutive 2-hits as follows: for the 001.10 case we use 
a = ~ . ( 5. 7) ) 
and for the 00.110 case 
f - ___ql[__ a = ~ - ( 5.8) - dp 1 - do ' ? 
For the 0 1.100 case we use 
a = ~ . ( 5.9) ) 
209 
and for t he 011.00 case 
f - ~ a = ~ - (5 .10) - dM1- du ' ? 
We can identify the appropriate 2-hit case in two ways: t he fi rst way 
uses two- plane consistency while t he second method employs the one- plane 
slope . In the fi rst method one calculates the projected posit ion in t he other 
plane for both the left and the right case of a gi ven 2-hit pattern (rays I and 
II in Figure 5. 5d) . The correct ray will mini mize the difference bet ween the 
crude position (i. e. the shortes t wi re ) and the projected position in the other ;t : I 
? ? I 
plane . For example, let us consider the 01100 pa ttern from Figure 5.5d . 
.-? ? I ' 
?? ? I 
If we assume typical values for the dM 1 and d0 drift dist ances of 6 mm and .-? .?  I ' 
r, 
2 mm , respectively, we find for ray I 
?r 'I  
m? ? 
= i. e . m - ~~ 13 (5.11 ) dM1 + z dM1 + do ' - d Ml + d fl ~ ' 
and for ray II 
m? = ? i. e . m = ~ ~ 26 (5.1 2) dM1 +z dM1 - du ' dM 1 - do ' 
i .e . the two projected positions differ by 13 wire spacings. The correct ray 
can therefore be easily identified, even if only the crude position in the other 
plane is available for comparison . 
The other method for identifying whether t he track passes l ft or right 
of the shortest wire m akes use of the fact that if, fo r example, a 00.110 
type event is treated like a 001.10 event , the calculated one- plane slope will 
increase . That thi s is indeed the case can be seen if 011 e compares the slope 
in Eq . (5 .8) , which involves the difference of d p1 and d0 , with the one in 
210 
Eq. ( 5. 7) , which involves the sum of the same drift dist ances . Assuming 
again 2 mm for a typical d0 drift length , the difference of the two slopes, 
Eqs. ( 5. 7) and ( 5.8) , is ~ ~ 0.67 which corresponds to 588 mrad or 33. 7?. 
Figure 5.6 shows the one-plane slope (times a fact or 1000 ) calculated from 
Eq. (5 .7) . The 001.10 events comprise the large peak. The 00.110 events 
which are treated like 001. 10 events are responsible for the long t ail on the 
large-slope side of the peak (beyond about channel 875) . It is evident that 
here the interior type constitutes the majority of all 00110 events . The same 
can be shown for the 01100 case: for most of the 2-hit events the track lies 
between the two firing wires. The same conclusion can be drawn from the 
next figure, Figure 5. 7, where we compare the calculated drift length dM 1 for 
00110 events with the corresponding measured drift length for consecuti ve 
3- hit events . Most of the calculated drift lengths are very large, whi ch is 
consistent with the steep angle of the 001.10 case. The 00.110 case which 
features a smaller slope would require small dM1 drift lengths . 
211 
0 
0 
0 
N 
0 
0 
l.() 
~ 
(/) 
......,...J.  ,--.._ 
..c: :t : I 
l.() ,c 0 .. 
0 ['-- (.) 
~ ro 0 ..__ 0 
~ ~ 
Q) ,... ." : 
~ . : 
0 
..---( r, 
(/) 
. r : 
0 
0 
LD 
I ':  
0 
0 0 0 0 0 0 0 0 
0 l.() 0 l.() 0 0 l.() 
-tj< (:J (:J N N ~ 
s1uno;) 
Figure 5.6: 2-Hit recovery by examination of the one- plane slope; if a 
00.110 event is treated like 001.10 the slope increases by a significant 
amount . The slope shown in this histogram is the slope from Eq. (5 .7 ), 
multiplied by a factor 1000. 
212 
CX) 
,.....-...._ 
lD s 
s 
-.._.,, ::.t: ., . I 
tj< Q) 
() .. . 
~ ...  ' co . ''  
...? ' 
if) 
C"J . ........ r, 
Q . r : 
...? 
4.-...-....?.  
C\l S--! 
Q 
I ''  
' 
0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
lD 0 lD 0 lD 0 lD 
("') ("') C\l C\l ...-i ...-i 
s 1-unoJ 
Figure 5 . 7: Comparison of the calculated drv1 1 drift length for a 00110 
event with the corresponding measured drift length for consecuti ve 3- hi ts . 
Most of the reconstructed 2-hit drift lengths are very large , indicative of the 
steep angle which is characteristic fo r the 001.10 case . 
213 
5.4 Drift Cell Resolution- Straightness of Tracks 
For the case of symmetric consecutive 3-hit events (see Figure 5.5a) 
two interesting quantities can be derived : first, as mentioned before, if no 
drift table is being used the knowledge of the absolute timing offset ~ is 
important for any position and angle calculation which involves sums , rather 
than differences of drift times t . One finds 
(01.110) 
(5.13) 
(011.10) . '..t.:  I , 
When one histograms the quantity ~ one finds a Gaussi an-shape distribution , .
, ...  : 
with a certain width, the centroid of which represents the absolute timing ' ..  : 
offs et. The width of the distribution , on the other hand , is proportional r , 
. r : 
to the intrinsic drift cell resolution . This resolution is found to be about 
0.25 mm . 
For the case of asymmetric consecutive 3-hit events (pattern 001. 11 or I :'  
11.100) one can study the "straightness" of the track (see Figures 5.5e and 
5.5f) . Testing the straightness is equivalent to testing the internal consistency 
of the measured drift lengths . Especially, we want to study the reliability 
of the very long drift lengths . For a straight 001.11 (11.100) track, typical 
values for dp 2 , dp 1 , and d0 (dM2, dM1 , and d0 ) drift lengths are 6, 2, and 2 mm , 
respectively. These values yield a typical focal plan e angle of 0f  = 588 mrnd. 
Th e working assumptions in the following analysis are that 
? the drift length d0 associated with the shortest wire is cor rect; 
? the particle track crosses the wire plane within the cluster , i.e. we have 
214 
the cases 001.11 or 11.100; 
? together with the shortest wire a. consis tent wire reproduces the "bench 
mark" angle of 588 mra.d , while an inco nsistent wire does not . 
For the case of a. straight 001. 11 track the quantity o = dp 2 - 2dp 1 - d0 
vanishes. If o < 0 the track is bent and convex, while for o > 0 it is bent and 
concave (see Figure 5.5e) . The convex case a.rises if either dp 2 is too short or 
if dp1 is too large . The concave case, on the other hand , occurs if either dp 2 
is too large or If dp1 is too short . In order to identify the inconsistent wire 
?? ? I 
we have to form , as we have indicated above, separately the one-plane angles ,. . ' 
:. I: 
tt ? I 
of dp 2 with d0 , and of dp1 with d0 . We do this by evaluating the quantities 11 1 I 
. r, 
a = 18 - dp2 - dol and /3 = 14 - dp1 - dol - In the case that a > /3 we can ,. . ' 
identify dp 2 to be the inconsistent drift distance; from a < /3 it would follow 
that dp1 is the inconsistent drift length . 
' 
Analogously, for the case of a. straight 11.100 track the quantity o = I : 
dM2 - 2dM1 - d0 must vanish . For negative o the track is bent and concave, 
for positive o it is bent and convex (see Figure 5.5f). The concave case arises 
for either a dM 2 that is too short or for a. dM1 which is too large. The convex 
case occurs if either dM2 is too large or if dM 1 is too short. Here we form the 
quantities a = 18 - dM2 - dol and /3 = 14 - dM1 - dol- For a > /3 the drift 
length dM2 is inconsistent, whil e for a < /3 it foll ows that dM 1 is in co11 sistent. 
We histogramed the quantity o for a.symmet ri c 3-hit events and found 
it to be predominantly positive for both 001.11 (i .e . concave) and 11.100 
(i.e . convex) events . Additionally, for these cases it was found that genera.Uy 
215 
the outer wires P2 and M2 were the inconsistent wires and that their drift 
lengths were longer than they were supposed to be. It was also observed that 
the frequency of occurence for bent rays increased with x 1 and that it showed 
a certain periodicity which sugges ts that there is a possible connection with 
2-hit events . If, for example, the P2 or M2 signal were due to pickup , the 
associated drift length will be inconsistent with the ones that are caused by 
the incident particle track. A similar analysis for symmetric consecutive 3-hit 
events demonstrated that the Pl and Ml drift lengths produce reasonably 
t : I 
straight tracks . ? ? I 
.. ' 
' 
5.5 Summary and Conclusions 
r, 
We can summarize the major results of this chapter as follows: ?r : 
? it is sufficient to consider only consecutive hits. 
? the long drift lengths associated with the M2 and P2 wires are oft en I :'  
inconsistent and , therefore , should not be used for position and angle 
calculations . The Ml and Pl drift lengths, on the other hand , yield 
reasonably straight tracks . 
? if 5-rays are the predominant mechanism for accidental coincidences 
with an inconsistent wire, we can assume that the shortest wire is part 
of the good cluster. We can then treat the n -hit event with the in con-
sistent wire just like an (n - 1)-hit event where all wires are consistent . 
? we correct the final cross section for multiple events by a global fact or 
(1 + /multi) , where fmulti is the fraction of multiple events . Sin ce th e 
216 
number of multiple events seems to track with the overall background 
we call this correction a "clutter" correction to distinguish it from the 
intrinsic efficiency of the chamber. 
? a similar global correction for 2-hit events is not possible because of the 
x J and 0 f dependence . Without the ( significant number of) 2-hit events 
the efficiency surface in x1B1 is not flat and has holes . V,/e have means 
to resolve the left-right problem which is characteristic for 2-hit events 
and are, therefore, able to include them in the efficiency explicitly. .t:.  
? we require from a good event that it 
...  : 
1. passes the particle identification test; r I 
2 . is not a multiple event or background event ( usually 0-hit or 1-hit ); . r : 
3. is not an event due to beam halo or slit scattering which can be 
' 
discerned by examining the angle a.t the target ( see next chapter); I : 
4 . allows the accurate reconstruction of the focal plane angle. Here 
we require that the two one-plane angles as well as the two-plane 
angle are all internally consistent . 
? We define our efficiency c to be the fracti on of good events that have 
additionall y a good hit pattern , i.e. events which make at lea.s t two 
consecutive hits . T he contribution to the effici ency of events which are 
not included in this definition, e.g. certain classes of non-consecutive 
or extra hits, is negligible. With this definition , the effici ency was 
generally between 95 and 100%. 
217 
5.6 Some Diagnostics 
We developed some useful diagnostics to identify problems with the 
discriminator thresholds on the preamp cards ( see Section 3.6.2): 
? if all the discriminator thresholds are zero, every chamber hit becomes 
a multiple hit ( example in March 1988). 
? during the development run in September 1988 the threshold of one 
single preamp card was too low, producing what was long believed to 
t : I 
be a "hot" wire . Certain wires of the card were picking up noise and 
appeared, therefore, for most events as the shortest wire . Additionally, . : 
. : 
however, the adjacent wires were misidentified . When the chamber is 
r. 
illuminated uniformly, this feature can be seen in a plot of the the short - . (: 
est wire versus the M2, Ml , Pl, and P2 (extra) wires (see Figure 5.8) . 
Normally, shortest and extra wires are distributed uniformly among . 
I : 
all the wires , yielding the diagonal lines in the figure. The fact that 
certain wires fire more often than others, together with their adjacent 
wires being misidentified , explains the off-diagonal "hot spot" . 
? cross talk between Wifes (example March 1988) manifests itself, de-
pending on the individual thresholds , on all the cards . This leads to a 
more or less pronounced peri odic st ructure whi ch can be observed in 
any spectrum for inconsistent wires (see Figure 5.9) . The cross talk 
can be removed by adjusting the individual thresholds of the preamp 
cards appropriately. 
218 
--
0 > ... ... Cl 
...... 0 0 Do 0\ II\ >- ('11 ...... ...... ...... 1-.-.- -----~------ -~-------~---------i- N 
....... ..  . - ~ I 
?CJD ?o 
?II H z . . 
?u.J'IIO ?? 
0 ? ? 0
?o ~. s: ~......  . ?co 
0: ? II 
?"' . ??  ?? ? .. ~I  V I ? 
N ?? 
V ??? .. ~ :::'cQ' l I 0 ??I I? 0 Cl(_). ..... 0  II? 
+ ??.. . O\~ w 
II x';:F 0 
?II .. '--(/'\ 0 0 II? ...... ??
0 I. C ll . . ... .. ?? 0 . 
II ?? 
.z... . ??. . . 
_J . 
II ?? ??I. I ... .. ?.? . 
I ??
I ??
  
.. . .. 
I 
I.I . ?I??.  .. ..  
??. . ......  
II ?? 
? II
I. I . ??
? ?
 ? 
?? .. ?II 
0 ? I.I. . ??. 
co . ' - ... .. ...
.  . fl II ~o \0 0 
II I . .. 0.--1(/)<! I ? II\ II 00 
???? ? 
ZCN'-
 .,.... 1(11 I ..."..  II ~- -II ??..  I ? x:::r--< I I ? c:.O >- ?? I? .-I I .. ?.  ... .. . 0 O\r---II\ . '<t N..--1 II\ 
...... ?I . ?? >::.:: . I I ? 0~ 
?? . .,.   ? x~'<t I?ll ? 
?? a.. 
0 " ? > 
.,..,___,:_;_J  ____________________________??_? I I  _ 01..&...Zf->- ('11~~ 
Figure 5.8: Diagnostics for a "hot wire" which is caused by the corres pond-
ing preamp threshold being set too low. The figure displays the shortest wire 
versus its ( extra) adjacent wires . Since the adjacent wires of th e shortest 
wires with the noise problem are misidentified, they appear off-diagonally. 
219 
I ' 
0 
CD 
,--l 
....., 
I 
0 
'tj< 
,--l 
I 
::r: 0 C\1 
I ,--l 
I 
c.:, 0 
I 0 h 
,--l 
Q) 
[:,;.. ,D 
I 
0 6 
co ;:j 
Ci] z 
I 
- Q) 
t 0 h 
Q CD 
I ?~- 
u 0 
I 'tj< 
0 
C\1 
0 
0 0 0 0 0 0 
0 0 0 0 0 0 0 
C\1 0 co CD 'tj< C\1 
,--l ,--l 
Figure 5.9: Diagnostics for cross talk; the wire spectrum of any ( extra) 
wire will display a periodicity due to the individual preamp cards (A- J) . 
220 
6 Data Reduction 
6.1 Replay 
Our data were replayed under the Q-system. The analyzer, as well as 
the test- and histogram-setup files were streamlined for speed and incorpo-
rated many of the results from the previous chapter . Several important cuts 
were applied to the data during replay. The particle identification (PID) cut 
and tests for multiple, zero- and one-hits in each VDC plane were hardwired 
t : I 
in the analyzer. ? ? I 
.. 
The PID cut is made on a two-dimensional histogram of the geometric . 'I  
? .? I ' 
mean pulse heights J SlN ? SlP and JS2N ? S2P and we eliminate with it 
r, 
deuterons, reaction tail from the scintillators, and general backgound. The . ( ; 
analyzer also examines the MUX register E data words for multiple hits 
(bit 16 set, i .e. the data word has a negative value) and for zero-hits (all bits 
J :'  
zero, i.e . the data word has a value of zero). If an event does not have a 
good PID or if it is a multiple hit or if it does not make at least two hits 
in either of the two planes , further processing halts and the next event is 
processed . The cut on the difference of the two one-plane angles ens ures 
consistency and correctness of the position and angle calculations for each 
chamber. Finally, a cut on the reconstructed angle at the target 1 , 0tgt , was 
necessary to remove from position spectra backgound du e to slit scattering 
(mainly for elastic spectra) and to beam-halo (mainly for inelastic spectra at 
1 In the convention used here 0tgt = - 0,catt ? 
221 
forward angles) . 
During the early 1987 runs , the long low-momentum tail of the beam 
extended out to the beam pipe walls and caused there a halo which, because 
of the special geometric configuration at forward angles, reached the spec-
tromet er acceptance and appeared as significant background in the spectra 
( c.f. Figure 6 .1 ). Due t o the large dist ance from the beam pipe p ort to the 
spectrometer entrance, halo events generally make very small angles with 
the beam axis and, therefore, always appear under a smaller (less negati ve) 
scattering angle than real events from the target at p ositive (negative) spec- ? ? I 
trometer angles 2 . Halo events can be seen as a very distinct feature ( c. f. ' ..' 
' .. ''  
Figure 6 .2) in the 0tgt spectrum, and can easily be separated from good 
r, 
events . At p ositive (negative) spectrometer angles they will have, with our r . 
sign convention, negative (positive) 0tgt values. Slit scattering can oft en also , . 
' ' 
be easily recognized ( c.f. Figure 6.2). Assuming the cross sectional angular 
I ''  
distribution of the aperture material (brass) falls very steeply over the angu-
lar range of the aperture at forward angles, the intensity of slit scattered rays 
coming from the right (left) edge of the aperture will be much higher than of 
rays scattered from the left (right) edge for p ositive (negative) spectrometer 
angles ( c.f. Figure 6 .3) . Scattering changes the direction of the rays t oward s 
larger (more negative ) scattering angles for positive (negative ) sp ectromete r 
angles . Hence, they appear as a pedes t al or tail on the negative (positive ) 
2 A simple geometric argument shows that th e ha lo a ngle would a lways be exactl y ha lf 
of the spectrometer a ngle if the ha lo originated on the beam-axis at the entran ce to th e 
scattering chamber. 
222 
- ~--
side of the center peak, the good events, in 0tgt ? 
223 
Positive Spectrometer Angles Negative Spectrometer Angles 
+- I --? Beam 
.... axis Halo events ~ 
(Jq 
.:: 
'1 
/'ti 
0) 
I..- ' 
IJ:l 
Pl 
(l Scattering 
::,;" 
O,.Q.,  Chamber 
tv 
tv 0 
~ C =0..  
0.. 
.:: 
(..b. . 
0 
cr" 
(b Counts Counts 
sPl  
::r- Uk low momentum Beam Pl --? 0 Pipe kL 
- 0 + THTGT Beam - 0 + THTGT 
0 
c.o 
0 
'tj< 
~ 
II 
Cl) 
> 0 
Q) tj' 
::E 
0 
-0 
II 
0.. 
l:::il 
...--.. 0 
p. - 01 
---0. : I ro u J ,,,.--.__ ? I 
~ s~  .. '  ....__,.. . ''  ' 
0 _, _Q,l ) r, 
@ 
r , 
0 ' 
01 I ''  
I 
?_-, -U) 
0 
tj' 
0 0 0 0 0 0 0 o' 
0 0 0 0 0 0 0 
r-- c.o l[) tj' C"1 01 ......-4 
s1unoJ 
Figure 6.2 : 0tgt- Spect rurn for Run with Beam Halo and Sli t Scattering 
225 
~ 
cjq. Slit Slit 
= Scattering ~~::=ro-J Scattering "'1 Events Events ~ Negative Spectro-0) meter Angles ~ 
? ---? 
to Dutt.lion o11ncru,r.re Dirtt.tion ol lncru,r. 
?,l 
(") 
';>;""' Aperture 
~...,  \ 
0 
t-.:) p 
t-.:) i:l 
cr, p_. 
p_. 
p 
(1) Counts Low High Low Counts 
.,.... Intensity Intensitiy Intensity 
0 
r:J'J I t Ii\ WI 
t,:,:."..:  do 
r:J'J 
(") \& \ilR 
?, ,l .,,.......  -0+ THTGT Target Target - 0 + THTGT 
(1) 
':!. 
i:l 
~ 
6.2 Line-shape Fitting 
Fitting of the replayed spectra was performed with the line-shape fitting 
code ALLFIT [Kel ALJ. Tl1e searcl1 code employs a Poisson (E:2) rather tlian 
a Gaussian (x2 ) goodness-of-fit criterion and thus avoids the under-fitting of 
low-statistics data, which is a problem with the standard Gaussian criterion 
[Kel 81, Hyn 81) . The spectrum is descibed by a model function which is 
constructed from a background and a sum of individual peaks 
N 
y(x) = B(x) + LY;(x) . (6 .1 ) 
i = l 
Each peak can be described as the convolution of an intrinsic line-shape 
1( x) and a resolution function R( x) which empirically considers spectrometer 
aberrations, target thickness effects, kinematic broadening ( dependent on 
t . 
arget mass and scattering angle) and beam properties 
y;(x) = l;(x) @R;(x) . (6.2) 
Peaks, whose natural width is negligible are described by just the resolution 
functi on. 
The most often employed resolution function shape is an asymmetric 
hyper-Gaussian in the central region 3 with left (L) and right (R) exponential 
3 The widths of the left and right central regions might differ and the exponent of the 
argument o f tl1 e G auss1?a n nee d no t be 2?,  exponents less (greater) than 2 produce pointed 
(flat) tops). 
227 
tails I 
"/ 
~ 
He IK .~1UL I
= "/ I.  R(x) (6 .3) 
...!:..=J!_ 
He IK ,_,wn I I 
H and p are the height and the posi tion of the peak, >.L ,R the left and I 
right slopes of the exponential tails , and W? ,R = w(l =i= a) the left and right 
I 
widths; herein , w and a are the total width and the asymmetry of the peak , 
resp ectively. The exponent of the Gaussian is 1 , K-y = 0.5(ln 2t 1h, and the : I I 
? I 1; 
I 
left and right match points , mL,R = K-yWL ,R( - ln !L,R) 1h, include the left and ,I I 
ii I 
right match fracti ons , fL and /R . Finally, ? represents the curvature of the ? I ? I 
right tail. 
The standard shape for the 9Be and 160 peaks which have intrinsic 
width is a Lorentzian with height H, position p, width f and threshold Xt 
H(:x - :x,)2r2 
I( X) = ~(:x - :x,)2+(p- :x,)2]+(:x - :x ,)2r2 (6 .4 ) 
{ 
X > Xt, 
appropriate when the resonance position is within a few widths of the reac-
tion threshold (here all the positions are to be interpreted as Q-values ) . The 
threshold for all of the low-lying 9 Be peaks is at 1.665 MeV (9 Be - 8 Be + 11) . 
7.162 MeV, corresponding to the decay 160 - 12 C + a, is appropriate for 
m os t of the 160 p eaks ; for the 02 state a thres hold of 12. 128 Me V, c rre-
sponding to (1 6 0 - 15 N + p ), is assumed to apply. Excitation energy, width 
and threshold of each state in the fit are stored in a "levels" fil e which is read 
by ALLFIT. These data are taken from [Ajz8G/ 8, Dix 86] for 9 Be, from 
228 
1-.r E., r c.m. Decay t Comments n 
(MeV) (MeV) (MeV) 
3/21 0 stable 
1/2t 1.680 0.217 11 1.665 
5/21 2.429 
1/21 2.78 1.08 n 1.665 
5/2t 3.049 0.282 n 1.665 
3/2t 4.704 0.743 11 1.665 
: I 
Lor 6.5 1.25 n 1.665 7/ 21 and 9 / 2i . 
Table 6.1: Energy Levels of 9 Be; data from [Ajz86 / 8, Dix 86] . . : 
. : 
[Ajz86/8] for 160; the data. for 4?Ca are from [End 78] . Tables 6.1- 6.4 a.re r, 
a summary ( comments to some of the states appearing in these tables are r : 
made in Section 6.6.1). 
' :'  
229 
I > -
]Tr E,, r c.m. Decay t Comments n 
(MeV ) (MeV) (MeV) 
ot 0 stable 
ot 6.0494 
31 6.12989 
2+ 
1 6.9171 
11 7.11685 
r1  8 .8719 
1-
2 9.585 0.420 a 7.162 
2+ 
2 9.8445 
4+ 10.356 0.026 a 7.162 1 
01 10.957 
3+ 
1 11 .080 unresolved 
4+ 
2 11.0967 
2+ 
3 11 .520 0.069 a 7.162 
Table 6.2: Energy Levels of 16 0 ; data from [Ajz86 / 8]. 
230 
-
j1f 
n Ex rc.rn. Deen t Comments 
(MeV) (M eV) (MeV) 
ot 12 .049 
1-
3 12.440 0.091 /)' 7.162 
: I I 
r2 
?I I 
 12.530 
I'  
o;- 12. 796 0.040 p 12 .128 . : " 
.?  f ' 
2-
3 12.969 I 
1' 
I 
13 .08 13 .08 0. 13 0 7.162 2+ and 1- r , II' 
3- 11 
4 13 .259 0.021 a 7.162 I ' 
1+ 
1 13 .664 0.06l1 Cl 7.162 I 
43+  13 .869 0.089 Cl 7.162 f 
r6  13.98 0.02 /)' 7.162 
ot 14.032 0.185 Q 7.162 
4-
1 14.302 
5+ 
1 14.399 
-
Table 6 .3 : En e rgy Le ve ls nf 16 0 :( c ,rnt.) data from [Ajz86 / 8] . 
23 1 
J'" o+ n 1 ot 31 2+ 2+ 1 51 ot 2 
: 11 I 
Ex 0 3.3521 3. 7364 3.9041 4.4915 5.213 5.249 
., ' 
I 
? I 
, I I 
J'n"  4+ 4- 2+ 1-1 1 3 1 6.028 32 4t 
, I I 
Ex 5.279 5.6143 5.6301 5.9033 6.028 6.2858 6.5084 
, I 
2- and 3+ .'  ''  
pr 4+ r '
'  
n 3 33 2  6.909 6.931 1-2 7.11 I 
I I 
Ex 6.5436 6.5833 6. 7509 6.909 6.931 6.951 7.11 
2+ triplet 1- and 4-
Table 6.4: Energy Levels of 4?Ca; all states have negligible intrinsic width 
( data from [End 78]) . 
232 
Figure 6.4 shows a representative fitt ed spectrum for Be0 . The abrupt 
rise in the backgound is due to the aforementioned decay threshold for 9Be. 
Noticeable are the strong asymmetry of the 1/2i state, which is only 15 keV 
above the threshold , and the broad Lorentzian peak4 which is prominent in 
the continuum region. The 2t state (11.52 MeV) of 160 with I' = 69 keV 
is a good example of a strong state with non-negligible intrinsic width . A 
fit to this state with the resolution function alone would be too narrow and 
underestimate the true cros~ section . 
Small peaks or unresolved peaks in multiplets can be extracted by lock-
ing their positions and shapes together or to those of a strong peak nearby. 
Examples are the at of 160 which was always locked to its strong neighbour, 
the 31 peak, and the doublet and triplet states in 
4?Ca ( e.g. at, 2; , 4t or 
41, 2t ). Figure 6.5 shows a representative spectrum for this target . The 
pedestals visible to the left and right of the strong 31, 2t and 51 states can -
not be attributed to (x / 02 )- or (x / 03 )-type aberrations (which would be 
visible in an x 101 plot). We believe they are either due to correlations with 
the vertical coordinates (y, <P) via matrix elements like ( x / <P) or ( x / y) or to 
the beam itself ( tails, split beam, etc. )5 . 
There was some small 12C contamination in the Be0 target , and the 
4 A combination of the 7/ 2""; state at 6.384(59) MeV (I' = 1.207(231) MeV) and the 
9/ 2t state at 6.76(6) MeV (I' = I.332(90) MeV) in ?Be [Dix 86 , Gli 89]. 
5 We examined the effect of the pedestals on the measurem ent of the Of state. Com-
paring the ratios of 31 and at heights with a typical peak-pedesta l ratio shows th a t th e 
Pedestal is generally much smaller than the at sta.te. 
233 
ground state and 2f state of carbon a.re clearly visible in many of the spectra. 
The 4?Ca target had traces of 12 C, 160 and 27 Al. The elastic peaks of these 
contaminants are present in most of the elastic spectra. For some of the 
inelastic spectra the 2f state of 12 C and the 31 state of 160 also have to be 
included . Some of the other background and structure might be attributed 
to some of the excited states of 27 Al, but a clear identification is impossible. 
I : 
234 
""1" 
....-1 
go ?c1 0 
""1" 
C\1 
CO II C\1 
+ CD ....-1 
f2 
Zt > <l) ----
+ > 
::E Q) 
Ii, ::E 
+ 0 ..._., 
0 0 
f2 C\1 ....-1 >--, 
II 
p.. till 
(iJ ~ 
Q) 
,..--... ' ~ 
ea6 0.. 
~ 
ClJ w '" 
,._0..,.,.  
0 ~ .. ,'  
~1 .c..o..  -0~  .'  ,,'  !2 _, 
~ 
.c ege -
-
<l) 
~8 _(D,  o::i 
!o / en co -~ u 
~ 
w 
~ ea6 
- <; 
~ ege 
. 1 
C\1 
0 
0 
....-1 
Figure 6.4: Fitted BeO Spect rum 
235 
u ?l 
~1 
?6 "9 
0 06 '9 
.....--i 
C') ~2 
II ~? 
CD Zi, 
+ 
> 
(l) ~? 
::% 
0 
0 
C\1 ~l ->--- -
II Q) 
0.. ::g 
~ C2 + ~t 
,,--... -----
0.. >, : I I - bl) I ,I l ,II 
...0__.,.,  't + z H 
ro +2 
'Ij  
Q) 
11 
u 
LD ~ 0 
~ lVlZ ?J Ii' 11 
I 
~ 
0 
?...-i 
,.....) 
~g ro 
?,.-....) () 
~ 
-tj< ?J 
!2 
~? 
!o 
.....--i 0 
-tj-< (:) C\2 0 0 
0 0 0 
.....--i .....--i .....--i .....--i .....--i 
s1unoJ 
Fig ure 6 .5: Fitted 4?Ca Spectrum 
236 
6.3 Formulae For Cross Sections and A naly zing Pow-
ers 
In the following we denote the differential cross sections for up- and 
down-beam ori entations as CTu(B) = (dCT/dO)u and CTd(0) = (dCT / dO)d - T he 
unpolarized cross section is , analogously, denoted by CT0 (0) = (dCT / d0) 0 . For 
each beam orientation the differential cross section is given as 
where CTu ,d(0) are the differenti al cross sections (in mb / sr) , Nu,d the peak 
sums of the state of interest , LT u,d the scintilla tor ( or computer) li ve times, 
E u ,d the chamber effici encies, dO the acceptance solid angle (in sr ), t the areal 
number thickness of the target (in nuclei/ mb ), and nu,d the tot al numbers of 
incident protons . 
The scintillator or computer live times can be calculated from the ratios 
of the scalers S1 ? S2 ? BUSY and S1 ? S2 for each beam orientation . The 
efficiencies for each orientation are defined as the ratio of events that make 
at least two consecutive hits in both chambers and satisfy an extended PID 
test (includes particle identification , good one-plane angles and the cut in 
0tgt) and all the events that pass the extended PID (multiple hits and 0- or 
1-hit events in either chamber are automati call y excluded by the anal yzer) . 
Finally, the "clutter" factor (1 + J,:~:/ti), applied to the fin al cross sec ti ons, 
accounts glo bally for those events whi ch fire a single M UX twice (multiple 
hits) and which are not processed by the analyzer. T he total number of 
incoming protons can be calculated from the ratio of the total charge and 
237 
the elementary charge (1 .602 x 10- 19 C). T he total charge is the product of 
the sum of the left and right integrators, the integrator full scale and a device 
ch aracteristic scale fact or 
n., ,d = Q.,,d/e = [(INT(L) + JNT(R)) x FS/ 1000] /e. (6.6) 
For a composite target with n components of atomic masses A; (in g) the areal 
number thickness of the relevant species ( i) is the number of scattering centers 
for the beam and is given with the above units for charge and acceptance as 
(6.7) 
where NA is Avogadro's number (6 .022 x 1023 1111 0 }- ), Ji the stoichometric 
fraction of species (i) , d the target areal density in g/ mb and 0t the target 
angle ( angle of the normal on the target with the beam axis) . 
The target orientation was chosen to be in t ransmission geometry and 
the target angle usually to be half of the scattering angle. In this geometry the 
path length for the projectile in the target is independent of the position of the 
scattering center . Losses in resolution due to straggling a.re thus minimized . 
However , the cross sections may reflect nonuniformities in target composition 
and thickness, since the position where the beam strikes the target depends 
on the scattering angle. 
The spin-up and spin -down cross sections are given with th e unpo-
la rized cross section, u0 ( 0) , the beam polarizations Pu,d , and the analyzing 
power of the target Ay as 
<Tu,d(0) = <To(0)[1 ? Pu,d ? Ay] , (6. 8) 
238 
Conversely, both the unpolarized cross .section and t he analyzing power can 
be obtained from the spin -up and spin-down cross sections 
(6.9) 
and 
(6 .10) 
If we assume that all the errors are uncorrela ted , we can write the uncer t ain-
ties of cross sections and analyzing powers as 
b.CTo = ..j..:(.._P _db_.a_-u_) 2 _+ _(P,__.b._CTd_) 2 _+_ [(P_db_.P_u)2_ +_ (P_ub_.Pd_)2_](C_To_A 2 )~11  (6 .11) 
Pu + Pd 
and 
b. A CJ'db.CTu) 2 11 {[( + ( 2C Tu ~CTd) ](Pu + Pd)2 
+ [( CTd~Pu)2 + (C Tu~Pd)2]( (J'd - CJ',_. )2} 112 
(6.12) 
* * * 
For a number of runs t here was a problem with either the spin-up or 
the spin-down integrator scalers . Assuming the same current for both beam 
ori entations , the correct integrator scaler value was reconstru ct ed from the 
clock scaler and the calculated current from the other beam orientation . The 
correspond ing spin-up or spin-down cross sect ion was th en multipli ed with 
a n a ppropriate correction factor . The multipli cation code generated a new 
ALLFIT summ ary fi le with the corrected polarized cross section . T he runs 
with a scaler problem are m arked in the li st of runs, A ppendix E , and the 
applied scale fact or for the polarized cross section is given . 
239 
6.4 Kinematic Formulae 
The final data will be expressed in the center-of-mass system . In the 
following, all the lab quantities are unprimed and the center-of-mass quanti -
ties primed . The notation is the same as in the code ALLFIT [Kel AL] . In 
the binary reaction m 1 + m 2 ----+ m 3 + m 4 we choose m 1 to be the incoming 
projectile and m 2 to be the target. The (detected) outgoing projectile parti-
cle is m 3 and the (undetected) outgoing target particle m 4 . The target may 
be excited and, hence, m 4 = m 3 - Q where Q is the Q-value of the reaction . 
Utilizing conservation of four-momenta, the lab-energy and momentum of 
the outgoing particle can be expressed as 
A + u cos 0 ? A 2 - m5 ( l - u 2 cos 0) 
c3 (6 .13) 
1 - u 2 cos 2 0 
(6 .14) 
In these equations, with ct = c1 + m 2 (the total energy of the system), the 
velocity of the center-of-mass is u = pif ct, and the auxiliary quantity A 
is A = (c: - p~ + m~ - m!) / 2ct . Transformation of the incoming (pi) and 
outgoing (p3 ) projectile momenta is achieved by the Lorentz boosts 
(6 .15) 
240 
where , = 1/?1 - u 2 . With tan 8' = P~1- / P~II and the geometrical identity 
I+ tan 2 8' = I/ cos 2 8' we find (r = uc3/ p3) 
I COS 8 - r 
cos 8 = ? 2 2 + 2 . 1 - u sin 8 r - 2r cos 8 (6 .16) 
The momentum transfer q' in the center-of-mass is defined as q' = p' _ p' 
1 3 , 
and its magnitude is 
q f = ?P if  2 + p3I  2 - 2P 1I  p3I  cos 8' . (6 .17) 
For the calculation of the Jacobian fcc:;:, we replace in the equation for cos 8' 
all unprimed quantities by primed ones and observe that r -t - r' . With 
c' = m5 + p~ 2 and r' = uc~/p~ we can write 
cos 8' + r' 
cos 8 = 2 . 
?1 - u 2 sin 2 81 + r' + 2r' cos 81 (6 .18) 
The Jacobian is then easily evaluated to be 
dcos 8 (1 - u 2 )(1 + r' cos 8') 
dcos8' = (1 - u 2 sin 2 8'+r'2 +2r'cos8')312 (6.19) . 
Finally, to enhance detail, we will display for elastic scattering always the 
ratio of the differential cross section and the point-charge Rutherford cross 
section <TR which is 
(6.20) 
Here ? is the reduced energy in the center-of-mass, ? 
6.5 Normalizations 
Depending on the spectrometer angle, during our experim ent various 
Faraday cups lrnd to be used. As mentioned before in Chapter 3, the external 
241 
cup was used at angles above 18 deg (March 1988, above 15 deg) together with 
a dump pipe . At smaller angles, various internal cups which were mounted 
on the rotating platter in the scattering chamber were used . We believe the 
external cup to be accurate to about 1 percent and to be more accurate 
than the internal cups which have energy-dependent normalization problems 
[N ad 77] . All data are , therefore , normalized to the external cup standard . 
6.5.1 100 MeV Cups 
For the normalization of the 100 MeV January 1987 long internal cup 
we had overlapping measurements with the external cup at 18 deg for 4?Ca 
and at 18 deg and 22 deg for 160. Cross section ratios for the 0i , 31, 2i 
and 11 states in 16 0 and for the O't , 31, 2i and 51 states in 4?Ca were 
computed and weighted averages calculated . For the 4?Ca states we get 
1~:;) = 1.008(55) and for the 16 0 states /Es7) = 1.004(21) . Th e 
\ wgt \ 187 wgt 
overall normalization is thus /~:;) = 1.005(20) . 
\ wgt 
For the normalization of the short internal cup in January 1987 we had 
overlap points at - 10 deg for 4?Ca and at - 10 deg and - 12 deg for 160. For 
4?Ca we have only data for the O't state . For 160 again the four lowest states 
were available at - 10 deg and the three lowest states at - 12 deg . We find 
for 4?Ca /~:;) = 0.999(55) and for the 16 0 states (~!;) = l.022(23) . 
\ wgl wgl 
This gives an overall normalization fact or of 1~;;) = 1.019(21) . 
\ wgl 
242 
6.5.2 200 MeV Cups 
For the cup normalizations a t 200 MeV it was necessary to compare 
cross sec tions of different run periods. In April 1987 we used exclusively the 
external cup ; in January 1988 both the internal and the external cup were 
used. Beca use of the different scattering angle offs ets in different run periods 
cross sections at the same spect rometer angles are not immediately compa-
rable . We found comparable cross sections by first - or , where necessary, by 
second-order i1~terpolation of the data points in the vicinity of the overlap . 
Overlap p oints with the internal cup for 4?Ca and 16 0 were a t 18 deg 
and at 19.5 deg. Cross section ratios were computed for the same four st a tes 
as in the 100 Me V case. For the 160 states the weighted average for the 
normalization factor is ( 87 4E ) = 0.873(22) ; for the ?Ca states we find 
188 wgt 
( f:; ) = 0.900(32) . The overall normalization is ( f:;) = 0.884(29) . 
wgt wgt 
There was also a difference in normalization between the external cup runs 
in January 1988 (after the encoder problem was fixed) and April 1987. Based 
on interpolated cross sections from external cup measurements in 1988 a t 
18 deg, 19.5 deg , 21 deg and 25 deg for the four 4?Ca states at, 31, 2t aud 
51 we find a normalization fact or of ( ~!: ) = 0.931(27) . It appears tha t 
wgt 
the runs with the encoder problem do not differ in normalization from the 
external cup runs in April 1987. 
6.5.3 Normalizations between Targe ts 
For the normalization between the old (BeO I) and th e ne w BeO 
(BeO II) targets in January 1987 we had cross sections for both t arge ts at 
243 
Energy Item I Normalization I Normalization Factor 
Cups E87 / 187 1.005(20) 
E87 / S87 1.019(21) 
100 MeV BeO BeO II / BeO I 1.105(41) 
Targets new / old meas . 0.988 
Ca Targets thick / thin 0.959(15) 
200 MeV Cups E87 / 188 0.884(29) 
E87 / EBB 0.931(27) 
Table 6.5: Overall Normalization Factors for Experiment 268 . 
angles between 18 deg and 70 deg ( all of these runs used the external cup) . 
The weighted average for the cross section ratios is ( t~ \1 tgt = 1.105( 41) . 
All the BeO cross sections were calculated assuming a target thickness 
of 21. 99 mg/ cm 2 . Recently, a new measurement of the target thickness at 
IUCF yielded a value of 22.26 mg/cm 2 (see Chapter 3) . Therefore, all BeO 
cross sections were additionally multiplied by a factor of 0.988 . 
In January 1987 we tried to obtain better computer live times for for-
ward angle measurements by using a thin 4?Ca target . Normalization points 
for the ot, 31 and 2i states were at angles between - 16 deg and +1 8 deg. 
The weighted average for the ot normalization is ( ;!~\~~) wgt = 0.976(29) , 
for the 31 normalization ( ~~-ck ) = 0.952(19) and for th e 2't normal -
in wgt 
ization ( Thi.ck) = 0.956( 48) . The overall normalization factor is th en 
Th111 wgt 
( Thi.ck) = 0.959(15) . All the overall normalization fact ors are summarized Thin wgt 
in Table 6.5. 
244 
..iii! !S: SE !A! 
6 -6 Cross Section and Analyzing Power Results 
6 ?6-1 Fitted Data 
We fitted 4?Ca states below about 7 Me Vin excitation energy. In this 
excitation energy range the peaks are in a good region of tl1e focal plane 
and are still relatively well separated . Also, all tl1e states of interest with 
available complementary electron excitation data are in this range . Some of 
the states were named in our own notation as follows : the state at 6.028 Me V 
is comprised of a 2- (6 .026 MeV) and a 3+ (6.029 MeV) state [End 78] ; 
the 6. 909 Me V state is in tl1e same reference quoted to be a 2+ state. The 
6.931 MeV state is a triplet: a state at 6.927 MeV, at 6.930 MeV (6+ ), and at 
6.938 MeV. Finally, the 7.11 MeV state is a combination of a 1- (7.113 MeV) 
and a 4- (7.114 MeV) state. 
100 MeV spectra for 9Be and 160 were fitted, whenever possible, up 
to the 160 3- state at 13.259 Me V. Higher excitation energies were not ac-
cessible mainly because of the smaller effective focal plane in January 1987, 
when the short scintillators were used . The 200 MeV spectra we tried to fit 
' 
only partially successfully though, up to the 160 5- state at 14.399 Me V. 
The problems that occur for many of the high-lying small states, especially 
reflected in their analyzing powers , come from the fact that they appear 
above a large background, due to th e beryllium continuum and resonances, 
whicI1 has a very significant analyzing power of its own . As was mentioned 
on page 233, tlie broad 9Be Lorentzian peak is the combination of a 7/ 21 and 
9/2t state ([Dix 86, Gli 891). The 160 state at 13.08 MeV is the combination 
245 
of a 2+ (13 .02 MeV ) and 1- (1 3.09 MeV) state . Finally, we did not succeed 
in isolating the nearby ( :=:::: 10 ke V) 3+ state from the 4t of 160. Since t he ana-
lyzing powers fo r abnormal-parity t ransitions are generally small , one would 
expect the cross section of the doublet to be enhanced and the analyzing 
power to be somewha t diluted [Kel 89a]. T he quality of our measurement of 
this doublet at both energies was so poor , however, that we can not see thi s 
effect . 
Several fitt ed data points had to be excluded from the fin al dat a t ables 
and figures- the list of runs in Appendix E makes a note of every sta te that 
was omitted for a particular run . There are various reasons which can warrant 
an exclusion . Generally, when a state was excluded , there was sufficient 
redundancy available , however . 
? P roblems in added spectra: during the experiment prospect ive ex-
tremely long runs, typically inelastic runs where good sta tistics fo r the 
small states was required , were oft en broken down into several shorter 
runs with identical magnet settings , etc . Off-line, the spectra were 
added to improve both the stati sti cs of the small st a tes and to facili-
tate the fitting of problematic line shape parameters ( t ails, e tc. ). Due 
to small drifts of the magneti c fields in the spect rometer or , in the case 
of non-consecutive runs, du e to imperfect reprodu cibility of th e fi elds , 
slight vari a ti ons in the already highly non-linear energy-pos ition cnJi -
bration occured . Affected by thi s are oft en , but not exclu sively only, the 
unresolved 4?Ca doublets and triplet s in whi ch a random redi stribution 
of strength among the constituents can be observed. The result s a re 
246 
then totally incompatible with other measurements at t11e same angle 
or with t11e overall angular distribution. In hindsig11t , spectra should 
probably not be added . 
? Peaks at the edge of the effective focal plane: although we found no 
evidence for a position-dependent efficiency beyond the one which was 
resolved by the recovery of the 2-hit events , for some January 1987 runs 
with the short scintillators we find a more or less abrupt loss of strength 
in peaks at the edge of the effective area. Because of redundancy we 
did not try to recover these particular states which , in the first place, I 
II 
I 
were often only put in t11e fit to constrain the background as in the case 
I 
of some high-lying 160 states on top of t11e beryllium continuum. 
? Obstructed peaks: becaus~ of differential recoil the beryllium states 
move witl1 respect to tl1e oxygen states as one goes to larger angles . 
Some of the 9Be states are very strong and narrow such as tl1e 5/2-
1 ' 
others are moderately strong but with a large intrinsic width such as 
the Lorentzian (I' ~ 1.25 Me V) . These peaks cross at certain angles 
over some of the 160 states, making a reliable measurement either dif-
ficult or even impossible. In Appendix B we provide recoil diagrams 
appropriate for 100 and 200 MeV kinematics where this situation is 
displayed graphically. 
? NormaLzation with respect to current: during some of tli e runs we 
noticed that the cup or the associated electronics seemed to leak cur-
rent to some small extent , thereby introducing some uncertainty i11 the 
247 
normalization . Evidence for a sys tematic dependence of this effect on 
the beam current or the integrator full scale was not fo und, however . 
Runs in January 1988 with the 2 nA integrator scale had a very seri-
ous charge integration problem of this kind and had to be discarded . 
Apart from leakage, correctly zeroing the integrators during the run 
sometimes posed a problem . This may also have contributed to errors 
in the absolute normalization of certain runs which were found to have 
m any of their sta tes inconsistent. 
? Non-reproducibility of the scattering angle: although we found gener-
ally good reproducibility for our measurements (2- 3% for the 200 Me V 
m easurements) , we find some cases (most notably among the 16 0 mea- 'I I 
I 
surements at 100 Me V) which differ systemati cally from earlier mea-
surements by about 10- 20%. A different scattering angle is one expla-
nation for the observed systematics ( the uncertainty in the laboratory 
scattering angle may be up to 0.2? depending on the beam tune), al-
though normalization problems due to charge integration are also pos-
sible. 
? Background at forward angles: certain elastic measurements at fo rward 
angles were only done at positive spectrometer angles or a t negative 
angles but with a useless intgrator scal e ( example: 4?Ca. elasti c mea-
surements with the 2 nA scale in J anu ary 1988 ). In the off-line anal ys is, 
the very strong background due to t he internal Faraday cup , whi ch is 
present in spectra at positive angles, proved to be unm anageable and 
248 
the respective runs had to be discarded . 
6.6.2 Some General Observations 
In Figure 6.6 the fitted widths for 9Be, 16 0 and 4?Ca are summarized . 
The states used to generate the histograms did not have any intrinsic width . 
Hence, the fitted widths can be regarded as a measure of the spatial resolution 
that was achieved in our experiment. Because resolution depends critically 
on all the beam and spectrometer parameters , and since all these parameters 
were set for either 16 0 or 4?Ca kinematics, we are not surprised to find large 
widths for 9 Be . However, as the histograms for 160 and 4?Ca also show, for 
most of our 4?Ca runs the resolution was between about 30 and 50 keV, while 
for 16 0 we find resolutions between about 50 and 80 keV. The reason for this 
better 4?Ca resolution is the smaller target thickness . 
As a cross-check for the accuracy of the overall normalization of our 
200 Me V data, we compare in Figure 6. 7 elastic cross sections and analyzing 
powers for 160 and 4?Ca with previous data taken at IUCF ([Glo 85]) and 
TRIUMF ([Ste 85] and P. Schwandt, private communication), respectively. 
We find consistency, with the exception of the 160 cross sections at momen-
tum transfers between about 0.7 and 1.7 fm - 1 where we believe our data to 
be better than the previous measurements . In this momentum transfer range 
the 160 analyzing powers of the prev ious measurement agree with ours , while 
the corresponding cross sections appear to be too high . This poss ibly points 
to a normalization problem in the IUCF data. . 
249 
":' 
20 50 80 
9Be(p,p') 160(p,p') ,oCa(p,p' ) 
100 MeV 
f 40 ~ 100 MeV r 11 100 MeV 15 60 
1..-.t.j  I-. 
oq Q) 30 
..:.:,  ,..Cl a 
~ ::, 10 ~ f I I 1 40 
O') z llnnn l 20 
O') 
5 t I u I n l 10 L I 111 l 20 ~ 
C.,./.l. . 
0 
().Q..,  0 
p 0 
~ a 15 0 c:.;, 40 40 
0 C/l 
.0.. 
9Be(p,p') 
.... 200 MeV "O(p,p') ~ [ 40Ca(p.p') 200 MeV 
.~ .,,........  30 
10 n 
11 200 MeV 
30 
(b 
0... I-. 
Q) 
"rj .a 
(b 
p E 
;,, ::, ~ ~ In j 20 r lln 7 20 
z 
~ 5 
0.,.......  
::;-- ~ ) u] In ] 10 r I LIL, n 10 C/l 7 
0 
0 100 200 300 400 500 0 100 200 300 0 50 
Width (keV) 100 150 Width (keV) Width (keV) 
~I I 
N 
I 
..a..  .___, 
? ? 
- 0 
I 
0 0- 0 
N 
,-.. 
+-
0 I 
,-.. 
p. a 
.._._._. , 
._0__..,  
? 0 .... er 
'.? 
? 
0 
u"l 0 
..-. 
0 
0- i. 
"I'  0 ll') 
0 0- 0 0. . 0- - 0 0 0 I -I 
l.y 
'lliJ / D 
Figure 6.7: Comparison with Previous 200 MeV Elasti c Measurements. 
4
E268 data are displayed as symbols , 16 0 IUCF data ( [Glo 85]) and ?Ca. 
TRIUMF data ([Ste 85] and P. Schwandt , private communication) as bands. 
251 
6. 7 Results and Treatment of Errors 
The data for all the 9Be, 160 and 4?Ca states at 100 and 200 Me V are 
plotted in Figures 6.8- 6.43 of this chapter, and are also tabulated in Ap-
pendix F. The quoted values and associated uncertainties for cross sections 
and analyzing powers have been obtained as follows: for single measurements 
they are simply calculated from Eqs . (6 .9)-(6.12). Input for these equations 
are the fitted spin-up and spin-down cross sections cru,d , their uncertainties 
t)..o-u ,d , the spin-up and spin-down beam polarizations Pu,d, and their uncer-
tainties t)..Pu,d? The quantities /j,cru,d are the larger of the estimated fitting 
error or the statistical error . The values for Pu,d and jj,Pu,d were estimated 
in Chapter 4. Although the presented data contain all the cup and target 
normalizations listed in Table 6.5, the normalization w1certainties were not 
included. Finally, in the case of repeated measurements a weighted average 
is quoted; the associated uncertainty is tl1e larger of the uncertainty of the 
mean or the standard deviation about the mean [Bev 69]. 
For 200 Me V we did a detailed study on the case of the three lowest 
strong states of both 160 and 4?Ca. We found that with the exception of some 
odd points the reproducibility of both our cross section and analyzing power 
measurements was between 2 and 3% (which is consistent with the earlier 
findings described in Chapter 4) . The fluctuations in tl1 e cross sec tions a.re 
mainly due to normalization un certainties from variations in uniformity of 
the target and from variations in the charge integration , dead time correc-
tion , effi c1? ency, e t c. Utlcertainties in the polarization should not have much 
impact on the cross section if tlie difference of the spin-up and spin-down 
252 
beam polarizations is small-for our experiment this difference was about 
0.02. It is striking, however, that for the aforementioned January 1987 runs 
which were inconsistent by 10-20% with previous runs, the beam polariza-
tion was also consistently very low . Uncertainties and fluctuations in the 
beam polarization have a bigger impact on the analyzing powers; the effect 
of large backgrounds with analyzing power on the analyzing powers of small 
states has been discussed above. The uncertainty in overall normalization is 
estimated to be about 5%. 
253 
. ? ,_ ... .
?  
? ? ? ? N ? C\I 
. . ? 
? 'I-{--)- ? ? I 
0. . ? a 
? ? -0.. ... . . -.... ? <1) ??  o:i ? O' -?
? 0, ..-. ??? ? ~ ? --+- ..... 
0 
i N 0 0 I 0- u'l 0 u'l 0 0 0 0 ~ ci 0 I 
-?-
... -+-- ? 
+- ,_-_.- 
N ? 
'.-.-.-.- ? ...... ... 
C\I 
--.-..-. --
------ I ? a 
?--
..
 ..... --
. ...., 
-?- ---- -
0 
i N 
.., 
0 I I ~ u'l 0 u'l 0 ~ 0 0 0 0 0 0 
I 
M 
.?  ... 
? . ? ? . I ? ? N ? C\I 
? 'C--'-)- ... 
. 
? ,....._ ?? 
I 
. ? 0..  ? - ..
a..  
0.. 
? <1) -?- .? -?  O' ? o:i -?-
-- ? 
0, ? 
??  ':?  
0 
.., i N N 0 I 0 u'l 0 u'l 0 
0 0 0- 0- 0 0 0 0 0 I 
(.1s / qw) UP/ .op 'v 
Figure 6.8: 9Be(p, p') at Ep 100 MeV: 3/ 21 1/ 2?t (1.680 MeV), and 
5/21 (2 .429 MeV) . 
254 
? --
..... - N I 
.-s _., 
-+-
0 
0 
'i' 0 l() 0 l() _; 
0 I 0 _; 0 0 0 
0 0 I 
.... I-
? -
...... -+-
N 
- ....... I ..... -?---- --- -
8 
._., 
? -
0 
0 l/') 0 l() 
0 I - 0 
0 0 
0 0 I 
-+- N 
-- I .... ..... -s ._., 
---
0 
.., l() 0 
N 
i I I ~0 0 0 ... 
l() 0 
 
0 0 0 
0 0 I 
(Js / qw) up/ .op 'v 
Figure 6.9: 9 Be(p, p') 100 MeV : 1/ 2 (2 .78 MeV), 5/ 2; at Ep 1 
(3 .049 MeV ) and 3/ 2; (4.704 MeV) . 
255 
-
ll i .j J 1 
-?-- - --
-+-
.? N  
....... .
-.-.-.-.-.-. --- I 
? 
? ... ..s...  
? 
-+- ..?.  ---
? 
? ---- ..... 
- ? ........ ? 
0 
j' 0 II') 0 
II') 0 
0. .. 0 0 0 0 
... 
0 0 I I 
I.V 
(.is/ qw) UP/ op 
Figure 6.10: 9 Be(p, p') at Ep 100 MeV: Lor (6.5 MeV) 
256 
- ---- - - -
M 
? 
. ? ? 
? . ? ? 
? N ,,......_ 
? 1-? (I") 
? 
.--.. ? I 
? 0. 8 ? .. .___, ? ? ._0__.,  ?  ? -ct 
? 0 ~ ??
\ . .  ? 
? -? ? ? 
0 
N \() \() 0 ~ 
i I ~ 0 0 0 
0 0 0- 0 0 I 
- I>-
? --<t--
-?-
- -?-?- N +N ------
..... .... 
,0-. .. -- I --
.... -?- ----- - -8 -----
.__, 
.. ----?--
-?-
0 
0 \() N " \() ~ i I 0 0
0- 0- 0 -
I'  ~ 
 0 0 0 I 
M 
? ? ? ? 
' 
?? ? . N ?   ? ,,......_ +-0 ? I 
? .--
 .. ? 
? ~ ? 8 
? ? .__, 
? ._0__.,  .. ? -
? 0 ct ? 
? ? ? 
~ ? .. 
? 
?? - - ? '??   
0 
N ~ 
.., i I 0 
\() 0 ~ 
N 0 0 - 0 0 0 0 0 -0 
_; 0 0 I 
(.rs / qw) UP/ DP "v 
160 (p, p') 100 MeV: OT , ot (6 .0494 MeV) , and 3~ Figure 6.11: at Er 
(6.1299 MeV) . 
257 
M 
? ? 
--- ? ... 
? ......  ,_ .. N 
? ? C\l ..... 
. 
.. I ,::--
? ? 0. ....... 
 
?  ,._.
8._..  , 
._0_.,  ....... 
??? 
 
 0 ............ ~ ..... 
l --..... ?.  
? --
I 0 
.., 0 I() 0 I() ,...; 
N 
0 i .I .. 0.
I .. 0 0  0 I 0 0 0 M 
? 
? ... ? 
? ? 
? ? ,_ 
... N 
? _.. ... 
? 
,,...._ I ? 
? ? ? ,.8 _.._.. , 
? ? 
? -?.- .-. 
? 
??  ?--? 
? . ---
~ 0 
0 I() 0... I() 0 0 . 0 
i I 
0.. .. 
? 
? .... ~ 
N 
....?...  
? 
? ---- .. I . ___._ ? +- .. ..8 ? C\l ? ... --::--
0. ... ? ? 
? ._0_.,  ?. ... 
? ? 0 ? 
??  ~ ----~---- ? ? ? ? L..,_,._.__.._..1---~...__._.__,___.._~~.___J__._~~~ 0 
I() 
0 
I() 
t.t. ~ 
0 0 0 I 
I 0 
0 0 
. 
0... . 0.. . ?v 
(.1s/ qw) UP/ DP 
MeV), 
100 MeV: (6.9171 
Figure 6.12: 160(ji,p') at Er == 
(7.1159 MeV), and 2~ (8 .8719 MeV) . 
258 
I . 
. ? 
? ? ..
??  
? +- - ...
  ? N 
...- ,.-... ??  -.:I' -- I ? 0--. ? 8 
? 0.. ? .... '--' 
? '--' ?. 0 ...... ? -
? ?? 
~ 
 -+----+-- O" ?- -
.., 0 
N I 0 U') i 0 
U') 0 
I 
0 0.. . 0 0 ~ 0 0 0 ... 0 I 
M 
? ? --... --
1 1 I, ,1.", 
? ? ...... 
j ,, 
+N --+- N ' I ? ,.-... 
.?  ? C\2 It . ---- --+-- I 1, . 0. ...- +- 8 I' 0.. ('1 
? '--' -+- -0 : ? ~ ....... O" ,,,  \ 
? .. ,, 1 ... -- I ~ I 
0 
N U') 
i "I  ~ U') 0 I ~ 0. .. 0 0 0 0 0 0 0 I 
-?--
-- ? ...-... -------I..N...  -- N - - -- I 0-.-. ..8..  0.. '--' 
'--' 
0 O" 
-- ~ 
.., 0 
N I 0 U') 0 
U') 0 
I I ... 0.. . .0. . 0 0 0 
~ 
0 I 
(.rn / qw) UP/ op "v 
Figure 6 .13: 16O(ji,p' ) at Ep 100 MeV: (9.585 MeV) , 
(9.8445 MeV), and 4i (10 .356 MeV) . 
259 
(") 
. . 
?..   ? 
? N -------- ,....._ ? -?-? +-M ... ? C\l ...? . . I ..a...  ? 0.. . 
. ? 
? ---
? ..0,_.,.  er ? ? 0 ?.  
? ? '.!'! ? ..
?.?   
0 
N 
i I 0 I() 0 I() ~ 0 
0 0 0 0 0 0 0 I 
---- -+-N N """ I 
.....
--.. 
- -- 8 -- - -
---
0 
N 
I I() 0 I() 0 i
0. .. ~  0 0 0 0 ...; I 
(") 
... ..... 
? 
.. -+- ,_ - N 0 -I 
??  -0.. ---- -- a ?.? , 0..  ..  -- -?- -0 - ---
...+. - '.!'! 
.. ? ?.  --
er 1 
--
-?-
., 0 .., 
N I() 
i I I I 0 I() 0 ~ 
0 0 0 0 0 0 0 I 
(.1s / qw) UP/ DP ?v 
Figure 6.14: 160 (p, p') at Ep 100 MeV: 01 (10 .957 MeV) , 4f 
(1 1.097 MeV), and 2f (11 .520 MeV) . 
260 
--
1 
... 
? ? 
----+-
---+- N 
? ? IN ? ,C-\2
- I 
.. . ---? 8 
? 0:- -- - ..... --+-
? ? 
,.0_.,.  -----+- ----0 . ? ~ ?? .~.. - ---
?? ---?  
0 
.., 0 0 
N 
0 j I 
0 .... 0.. . .
I . 0 -0 . 
0 
..-
.. .. 
-?-- ------? ------ N ,-... 
I. ...". ' -- I 
+ --:---
? 0:- ..
8.....  
? ,.0_.,.  -- ----? O' 
?. ? .. -- 0 ~ 
0 
.., 0 \I') 
0 
\I') 
.0. . 0 
_; 
N .I 0. . 0 0 j I I 0 0 0.. . 0.. . n 
? .... 
+ 
? ......... N ? ,-... ??  +"' 0 
? ,-... - -- ... I 
? ?o.. ..
8 
? .
...  ....... ----
? ,.0_.,.  ..... O' 
0 
,? ? 
? ~
~ ......  
? + 
0 
\I') 
.., 0 ~ \I') 0 
N .I .. ~...  0 
0 I 
0 j .I .. 0 0 0... 0  r.v 
(.ts / qUI) OP/ DP 
100 MeV : ot (12.049 MeV) , 
Figure 6.15: 160(ji,p') at EP === 
(12.440 MeV), and 2~ (12.530 MeV) . 
261 
... 
? ? 
--
? co 
--+--
--+-
? 0 - >-- N ? (,."..).  I 
? ? 
? ... 
,,-.. 
?. .?.  ..
a..  
??  
0. 
.? ._0_.,  O' ? 0 ~ 
??  '.e ---+ 
0 
I') 
i N II') I ~ II') 0 0 0 I 
0 0 0 0 0 0 0 _; 
I 
n 
... 
? ? 
. ? ? IM N N ,,-.. 
? --? I ~ a? ? 0. ....  ? .__, -+-
. ? 0 .... 
--+-
O' 
? ? '.e :::-. 
??  --
N .., 
0 
i II') 0. .. . .. I .I 0 
II') 0 
0 0 0 0. . ~ _; 0 0 0 I 
n 
---?--? 
.. --.....
. . IN --- N ,-... 
.. 0 ,,-.. I 0. a 
?. . ... --- .... ._0_.,  0 
- '.e O' 
N .., . 0 i I I I 0 II') ~ II') 0 
0 0.. . 0 0 0 0 0 I 
(..1s/qw) UP/ DP 'v 
Figure 6.16: 160(p, p') at Ep 100 MeV: 0~ (12.796 MeV) , 23 
(12 .969 MeV), and 13 .08 (13.08 MeV ). 
262 
..... . -+-
? ? 
? 
I.., C\I 
,C...'.)_  
? ? 0.. ......- ..-
-
.. I 8 
? ? -.A....,  . --....., ? 0 
I ~ 
? .... 
.., 0 
0 i N I I 0... 
l() 
. 0 l() 0 0 0 0.. .. 0 0 0 0 .... I I 
(.rs / qw) UP/ DP ?v 
Figure 6.17: 160(p, p') at Ep = 100 MeV: 34 (13.259 MeV) . 
263 
), 
. ... 
? ? 
?.?   . ? I (') ? N 
? -::-- ? 
? 0. ? I 
. ? ? -0. ? ? <O . -a 
? u ? 0.. . ? O' 
??
?
  
 ??  
?? ? -.. \? ,_ ? 
0 
0 i "' 0 l() 0 l() I 0 "0'  0 0 0 0 ...; 0 0 0 ...; I 
(") 
.. .. -------.?.. .. -+N 
? 0 b- N ,--. ? ,--. ....... ? 0.. ....... ? ....... 
I 
p. 
..... ? - --+-- a (U 
? u ----....... 
-
- O' "1: ~ --..... -
---- --
0 
,., .. 
i "' l() 0 
l() 
I I I ~ ~ 
0 0 0 0 0 0 0 I 
(") 
. ? ? . ? ? . ? 
? +- ? N 0 ? ,--. ? ,--. ? 0. ? I 
? - ? 
 
0. ? a ? 
? <O ? -
? u ? ... ? O' ? 0 
-- ?
 
?  ? ? 
? ? 
?? J 
0 
.. 0 i "' 0 l() 0 l() I ~ 0 .. 0 0 0 0 0 0 0 0 0 I 
(.xs/ qw) OP/ DP 'v 
Figure 6 .18: 4?Ca(p,p') at Ep 100 MeV: Oi , Of (3 .3521 MeV) , and 31 
(3 .7364 MeV ). 
264 
--
+n --
,0.... .._ N ,......_ 
0.. I 
-0.. -- 8 al 
0..u .  -- --+- O' 
-+-
.., .. .., 0 i N I I I I ~ I() 0 I() 0. .. ~ 0 0 0 0 0.. . 0 0 0 0 I 
(') 
. .. 
? . ? , j"' 
. . ? ,_ l() . . ? .,. ... N ? ,......_ ' ?i ,f,  0. ? I l ;:
. ? 
? 
- . ? 8 ,, 
? 
? 0. ..... ' 1\ 
al 
? u ? 
? 0? .. . I 
O' 
? 
? 
~ ...... - L ? ......... 
-+---
0 
i N I ~ I() 0 I() 0 0 0 0.. . 0 0 0 0 0 I 
(') 
? ... ..... 
. ? ? 
? ......
+- .
C\2 ..
   
? ,......_ 
N 
? ,......_ 
? ?o.. ? I 
? ? -0.. ? ? 8 ? al . 
? u ? 
? 0.. . . -O' 
? ? ?? ? ?  
\. ~ ??  ...:.... 
.., 0 
0 i N I I 0 I() 0 I() 0 0 0 0 0 0 0 0 0 
I 
(.1s / qw) UP/ DP 'v 
Figure 6.19: 4?Ca(p,p') at Ep 100 MeV: 2t (3.9041 MeV ), 51 
(4 .4915 MeV) , and ot (5 .213 MeV) . 
265 
--+-
?- --
-+-- - ~ N 
-+- I 
-+- 8
-+-- .....  
? -+-
---+-- ..___ ~ -+-
? .... ----?---
-+-
.,., 0 .,., 
0 
0 I 
0 0 0 0 0 
I 
(') 
... -
.....- - I,...?-.- N 
-+- I 
----
-+- --------- .
---?- -
8 
._., 
+- -+-
1--- --........ ---+------+-
0 
N .,., 0 .,., 
I I 
0 0 0 0 0 
I 
(') 
-- ? .. -?- - ----?..... .. .. 0.. ----- - N 
..p_...,  ----------
I 
-8 ..... <d 
...  -+--
l.) 
-+- 0.., . -
? ... 
'-:-..... ~ -
N .. .,., .,., 0 0 0 
0 I I .I 0 0 0.. . ~ 0. . 0 0 0 _; I 
(.1s / qw) up/ .op !.v 
Figure 6.20: 4?Ca(p, p') at Ep 100 MeV: (5 .249 MeV) , 
(5 .279 MeV), and 41 (5 .6143 MeV ). 
266 
.. ---? ? co -----
??  
C\l -
0 --- - - N ? <O ..... I ? 
? ,-.. 
? 0.. .-.- ... ,._.8._.  , 
? ..0_.,.  ? - .....- - er 
I ? 
al 
u - ~ 
$ 
..._...,.: ?- ----?--
0 
.. U') 0 U') 0 c:, 0 _; N "' I 
I I. 
 
.. 0.. . 0 
0 
- 0 I 0 
[ _.
? .
_..  
? ? N 
??  --...... 
..... 
. ?  
I a 
? ,._..._.. , 
? ? 
? .?..  
,? --..-....-  
? ..-. -? I 0 
-?-
U') 
1' U') 
0 
0 
... 0 0 
0 
N 
0 0 _; I 
-I I 0 0 0- 
--?-
+n .....- ? 
-- - -- N 
..-.. 
,C-..\.l - I _ a 
0.. I-+- - ,._.._.,  
-Cl, ..-.- --- -- i-(1j 
u -- - -
0... . .... ...... 
_ ? ? . ..... ---+-
---... - 0 .. c:, U') 0 
I c:, ~ 0 0 
N "I'  .0. . ... 0 I i I 
0 0 
0.. . 
'"v 
(.1s / qlll) up/ nr 
Figure 6.21: '?Ca(ji,p') at Ep - 100 MeV, 2j (5.6301 MeV) , 1~ 
(S .9033 MeV), and 6.028 (6 .028 MeV) . 
267 
--
C\I 
I 
-..s..  .... 
........... 
--
0 
0 \() 0 
I 0 0 
0 I 
---
? ? - ---
?-
C\I 
? ? ------ ... I ? ..a..  ? 
? ...?...  
~~ 
.. 0 .., I() 0 \() N 0 ~ 
0 i I I I 
0 0-  0- 0- =! ~ 0 0 0 I (") 
? ? ? ? 
--
? ? IN le ? C\I 
? C') ,....._ ,....._ ? ? 
? 0. ..?  I 
? -p. ? .. -..a..  ? ? <O ? 
? (._) r::r 
? 0.. . ..  
?. 
. 
.. ?? ? .  
?? ? .. -~"'! 
' 0 
N M \() 0 I() 0 
i I 0
0-  0- 0 .
I 
0. . 0 0 0 0 I 
( .IS / qur) OP/ DP !.';/ 
Figure 6.22: 4?Ca(p, p') at Ep 100 MeV : 3; (6.2858 MeV) , 4f 
(6.5084 MeV), and 4t (6 .5436 MeV) . 
268 
. ? 0) -- -
? 0 --? 0) ------ -- C\I ? ? <D I ? ,......_ -+-? 0. 8 
? ? 
 -- ,__, 
,_0_..,  
? -
--+-
? <O ?..? .. .. o' ? ? u 
? 0.. . ..". ~..
....  
? . ---- -
.., 0 
o 0 i 
N 
I I ~ l/) ~ l/) ~ 
0 0- 0 0 0 0 0 I 
C"l 
? --...... 
? ? 
? --+-
IN --+-
C
? ? ,....
\..l_  --
? 0. -- C\I -
? .
+..- I 
? ,_0_..,  --+-? ,<O ? u ? ... -
8 
__, 
? 
: 0.. . ? - - - -- o' ? ~ .-L....... ?l -? ---?--
N .., .. 0 
0 i I I. .. I 0 
l/) 0 l/) ~ 
0 0 0 0 0- 0 0 0 I 
C"l 
? ... .... . ? ? ? IM 
? ,C....'.)._  ? 
? 
C\I ,......_ 
??  0. ? 
? 
I 
? ? ,_0_.,.  
? ? ... -8 ? ..... ,__, <O ? u 
0 o' 
? ? ? 
... l ? 
? ?- .. ? _?-_ ? I----+-
.., 0 
i N I 0 l/) 0 l/) 0 0 I 
0 D D 0 0 0 0 
I 
(.1s/ qw) UP/ DP f.v 
Figure 6.23: 4?Ca(p, p') at Ep 100 MeV: 33 (6.5833 MeV) , 22 
(6.7509 MeV), and 6.909 (6.909 MeV ). 
269 
. ? ... 
? ? .........
-+-
?  
 .. 
? ~ .. ...?.  (\/ ? I 
??  0.. .... ..a...  
? ? ,_0_.,  . ? 
cu .
.. .. O' 
\. ..... --+-u 
.:?= . ~ - -------?= := ---
... .. 0 
0. .. i 
N 
I I I ~ ll') 0 ll') ~ 
0 0 0 0 0.. . 0 0 ci I 
(") 
. ? -
? ? .... ,'...."...... 
-?-
. _'  - --- (\/ 0. -- I 
? --0. a 
... ? ? --c--
- ---?- - ..... u ... u ---+--0
-l- ... ..
 . O' 
r----
.., 0 
i N I I ~ ll') 0 ll') 0 0 
0 0 0 .0. . c:i ci ci I 
(") 
- .. --... -- ----C"') 
0) ... ------- (\/ co -..+. - -----+--- ,....._ I 
....-.?. - ..aC. - - ... 
 
--+-
0. 
--c-u- 
u ---..- 
? 
O' 
0.. . 
--+- --
... 0 N ll') 0 ll') 
0 i I I 0 ~ 
0 0 0.. . 0 0 0 ci I 
(.1s/ qw) UP/ DP !.v 
Figure 6.24: 4?Ca(p, p') at Ep 100 MeV: 6.931 (6 .931 MeV), 12 
(6.951 MeV), and 7.11 (7 .11 MeV ). 
270 
---- (') ... , ? ......_. _._ 
? ? 
? ? .._ 1-
?? N # N 
?? ? 
?' "I{)  ? ? ? ??? I 
??  0.. ? ? ? ? ..
8..  
??  .._
C_l,.  ? 
? Q) ,,,. . O' 
? m ?  0, ? ? ? ? 
0 
N II') 
0.. . 0 i I 0 II') 0 0 0 0 0 0 0 0 ..; I 
---.. 
+-
N -- N 
.~. "?'- . ".....  ---Pt-- ---- I -?------- -- ..8..  
...-.....  
.. -+---- ..__, 
--+------- ~i.----'---- -+-
N .., 0 
0 i. .. I I 0 II') 0 '? ~ 0 0 0 0 0 0 0 I 
.. . 
(') 
.... 
It. 
? .. 
? ? 
, ? ? 1-
, , N ~ N (")
. ? ",-.._ ?
? 
 I 
 
? Cl. . . ? ?  ...8._._.  , 
? ? .._C_l? ,
.  
?.. . ? ? Q) O' 
. ? ? 
m  
0, ? 
??  ? ? . 
??  ??  
0 
.., i N I ~ II') 0 N 0 0 
0 
0 0 0 0 0 0 0 '0?  
I 
(.1s / qw) up/ op 'v 
Figure 6.25: 9 Be(p, p') at Ep 200 MeV: 3/ 2;, 1/ 2;+- (1.680 MeV) , and 
5/2; (2 .429 MeV) . 
271 
-+--
N 
I 
.. -- -+- 8 ? ------+-? --+- .-__, .:. ::.:.=.- +.---?.?. ?   -+- --+-
., 0 
j' N l() 0 l() 0 I I ~ ~- 0 0 0 0- 0 0 0 0 I I 
(') 
----- N 
I 
--------- ----------+------ --+- -
8 
-+- O' 
_::;:!:--
0 
N 0 l() ~ 
0 I I 
0 0- 0 ci ci -I I 
N 
-~--.--.-- - I 
--+--- ..... - --
-------
..8....  
.. -+- O' .. .... -- ---+---+-
-+-
0 
"' '7 0 
l() 0 
0 I- I - C! 0 0 0 0 0 0 ci I 
(.ts / qw) UP/ DP 
Figure 6.26: 9 Be(p,p') at Ep = 200 MeV: 1/ 21 (2.78 MeV) , 5/ 2{ 
(3 .049 MeV) and 3/2i (4 .704 MeV) . 
272 
? - ~-- --+- s.. ... N -# 0 ,......._ ,...........l_  ? I 
? ? 0. 
--
-+----- ..8...  
-.--~ ... ?  .._0_.,  ?- --<I> 
- o:i ...
......  - O' ~?  O> ? -
--
0 
7 0 I() 
0 0 0 
I() 
0 0 c.i c.i ~ c.i 
I -I 
(.1s/qur) UP/ .op AV 
Figure 6.27: 9 Be(p, p' ) at Er 200 MeV: Lor (6.5 MeV ) 
273 
C'l 
? ? + ? ? . ?
.. , ? , . 
 
., , 
,_ ?? N 
C') 
. ,.-._ ?? I 0. . ??? 
??  ? 8 ? ..0_.,  ? ? 
? .-._, 
?? ? 
0 
?  . ~ i.. ? O' ? . ?
  
? ? + . 
? ? ..... ? ---
0 
N 
i I 0 I() 0 0 
0.. .. 0.. .. 0 0 ci ci 'c?i 
~ 
 
I 
C'l 
...... ? - ??,,".' !,
+ ? +N ----?------- N , ,,  
0 ,--. I?? 
~ I .... 0.. ,
,,,. 
--.::: -8 ,,,
  
-..-.. - ..
0_.,  1 ' ..- -- ---
0 
~ O' .,;
;',  
....... ,, 
??  
?- -- ',,'\  ---
.., 0 
N 
i I I ~ I() 0 0 0 
0.. .. 0 0.. .. 0 ci ci 'c?i  
. I 
, ? . 
C'l 
?? 
? ?? ? 
? 
? 
, , .. ?? N 
.? +- , ? 0 ,.-._ ? . I 
. . 
. 0. 
? ? . 8 
..0_.,  . . -
? 0 ? . --
??  O' 
~ 
I 
? ? 
? 
? ? ? ? . ??  ? ? 
0 
.., i N I() I 0 I() 0 0 
0 "0.'.  .. 0 
0 
0 0.. .. 0 ci 0 ci I 
(.1s/ qru) UP/ .op /.v 
Figure 6.28: 16O(fi, p') at Ep 200 MeV : Oi, at (6 .0494 MeV ), and 3~ 
(6 .1299 MeV) . 
274 
.. ?. . -----
-.- ? -.. ----# ,_ I =-c-- (\J C\l ~ 
?? ........ ~ ? 0. ---?-
? ? 0 -?--- I - ..a..  ? .___.,  ?... .. -...- 0 --.. O" ~ ...-..- --?-
..
+
......
-
. .... .....  
N ., .. 0 
i. .. I. . I 
I ~ 0 ~ 
0 0 . 0.. . 0 "0'  0 "0'  ... I (") 
. ... ?
?  ---- . . If fl ll ? 
?? ,-_ . .v- (\J ?? ........ .,  ........ ? ? I .? ? 0. . ? ? ?? a 
? ._0__.,  ? .
._._._. , 
-? 0 r? ? O" ~ ? . .... . ? ... -+- ....... ?. .. . -------
N ., 
0 
i I I 0 0 0 0 
0 0 0 0 "0'  0 "0'  
I 
(") 
.,. . --
...p--
,, ? 
?? . 
-#-
,~  +- (\J . --+-
........ 
, .C...\..l. . -- ? --q.... I 
? 0.. ? ? a 
? ? ._0__.,  ? ? -- ...... 
.? ? ? 0 ?  O" ~ ? 
: . ? ? ? . ? ? ?- -+? ---
0 
i N 0. .. I. .. "I'  
0 U') ~ 
0.. ~  . 0 0 0 0 "0'  0 0 I 
(Js / qw) UP/ DP 'v 
Figure 6.29: 160(p, p') at Ep 200 MeV: (6.9171 MeV), 
(7 .1169 MeV ), and 21 (8 .8719 MeV). 
275 
.. ... ? -- --
? .....- ... , ?? -+- ~ N 
?? .s.:-t.'.  ~ 
I ......  -.?..
.... 
-
. - ..... 
I 
?= E ? .__, 
? --- ..... - --?---------?..-.. ---
--+--- ,_--_ 
0 
i N I "I'  0 0 
0 0 0 0 '0?  0 0 '0?  
0 
I -
C'l 
...... .. -#c- -- --?-
., ... -#- +N C\l - -?- N ..-.. I 
"'"+ "o. 
-----+-
.. 0.. ..~.  -- ..E._._..  , -....-... ? - '--' --0 .........  .. '.? --...-.. .-..... O' 
-+-
-?- -
0 
i N I "I'  ~ '? 0 0 0 0 0 0 0 '0?  
I 
C'l 
....-. -
N 
- --+- I..N...  
I 
-E 
O' 
0 
N 
I I "I'  0 0 0 
0- 0 0 '0?  0 '0?  I 
(.1s / qw) up / op 'v 
Figure 6.30: 160(p, p') at Ep 200 MeV: (9. 585 MeV ), 
(9 .8445 MeV ), and 4t? (10 .356 MeV) . 
276 
,. . 
? ...- ------
.. 
? 
? ----- ---- N - ,,....._ ,. ? 
? ? +M ('J . .. ? -- ..
~.  I 8 
,-._ -
? ? 
? 
Q. ... ? r::r ? Q
? .......
.,  -?- 
? ? ? 0 ? . ? ? ? ? ~ ? ? -+-
' 0 
I() 0 I() 0 
0 I 
"I'  0 
- 0 0 0 
0 
0 0 0 - I 
... --
- - N 
I 
--8- 
~ r::r 
0 
0 I() 0 "I'  ..; 
0-I 0 0  0- I 
- -,_ N 
0 
,-._ I 
--8- 
r::r 
... . --- -+--.- .. -- -+---+--
? -? - .. 0 I() 0 I() 0 
I 0 
I 0 ci ci ci 
0 I 
~v 
(Js/ qw) UP/ DP 
o- MeV), 
Figure 6.31: 16O(ji,p') at Ep 200 MeV: 1 (10.957 
(11.097 MeV ), and 2t (11 .520 MeV) . 
277 
N 
-+-
I 
- ..8--- .._.., 
 
... ? , 
??
-
  -- --
??  ti-
? ....... ? i--
., 0 
N IO 0 IO 
0 i I I ~ 
0 0 0 0 
~ 0 0 0 
I 
(") 
---+'---
...... -_____ 
__~ .......  I- t"l N ,,-... ? ,,-... . ?o.. -- -- I . ~?  s 0
-- - .._,
.,.  
-..-... -
-- -
-- 0 -- - o' ~ 
--------
., 0 
i N I I 0 IO 0 IO 0 ~ 
0 0 0 0 0 0 0 
I 
(") 
- - --? 
.~
-- N 
. ..... 
 
---+--
.- ~ 
? - ....... I ... t ..8. --?- ..  I-?--+-
??? .-. .- --?--? ... . -~ 
? .. ~ --?? -- - - .__ 
., 0 
N 
I I 0 IO 0 IO 0 0 I 
0 0 0 0 0 0 0 
I 
(.1s / qw) up/ op !.v 
Figure 6.32: 160(p, p') at Ep 200 MeV : at (12 .049 MeV ), 
(12.440 MeV), and 22 (12 .530 MeV). 
278 
... ... -----+---
.., ? ~ co -+-0 ---+-- N 
,. r C.."..)  ... ,....._ I 
..? ... 
? -- -?...- --? -?-0. ...... -? -?- -
a 
.?.... 0..    '-' --- O' -+-..  0 --.- ? --?----'.? ... --???  ------+-
N .., 0 
0 i I I 0 I() 0 I() 
0 0 0 0 ,..; 0 ~ 0 0 
I 
(") 
--?- ,, 
' 
N ,....._ 
-
__,;:_ I 
-a -+-
? . 
. 
? ?? ---- O' --+--
? -.  .
-+-
+.. -
? ?.. . 
N .., 0 
0 I I I I() 0 I() -0 0- 0- 0- c:i c:i 0 I 
l 
-
-- -- IN ,0.- ... N 0. 
.. ._0_..,  0 -...... - --
I 
-a +-
- - ~ 
-+- -
I 
N .., 0 
I I I 0 I() 0 I() 0 
0 0 0- 0 0 0 I 
(.rs / qw) UP/ np ?v 
Figure 6.33: 16O(p,p') at Ep 200 MeV: 02 (12.796 MeV) , 23 
(12 .969 MeV), and 13.08 (13.08 MeV) . 
279 
' 
? 
? 
? 
? ---- N +M 
?'?  ,-,s.-t'_  --:=ta,: ~ I 
? ? 0. .-.-..... -.- 8.. .....  '- ? ._0_.,  ...-. . . ~ -----0 ---- ~ -----+--...... --.-......--+- -... 
0 
.., 0 IO 0 IO 
N I 0 0 ... 
0 i .I .. 00 .. . 0 I 0 0 
l 
I 
... ,___ N 
-- I ..8+- ...  
-----
~.... -- __::::!::- -- ------ 0 
0 IO 0 IO ..,; 
0 0 0 I 
.........  --
N 
.....~  .. 
--- I -4.-=-~..... _.._. .~_  ..... ..8...  
...... ? ? ~..-..
.. .-.. --
-- ---?-?--+-...... 
+?- - -
-------
--?-...... 
...... 0 
.., IO 0 IO 0 
.I 
0 
0. . ..,; 0 
ci 0 
I 
0 I 
0 0 
,v 
(.ts/ qm) op/ op 
Figure 6.34: i?o(JJ,p') at Ep ~ 200 MeV, 3_; (13.259 MeV) , 
(l3.664 MeV), and 4f (13 .869 MeV) . 
280 
- '--- N 
I 
- ..8...  
N .., 0 
I I I IO 0 I/') 
0 0 0 c:i c:i ~ c:i .... 
I I 
I 
...... N 
.0... .... -+-
0.. 
._0_.,.  - - ..._ I 8-..-...-.  0 ---+- I---
~ --- O' 
----
1---
- --
'' I ' 0 
0 I IO 0 I/') 
0 0 c:i c:i c:i 
I 
(") 
I 
____, 
I IO N 
.N.... ... ----'---
I--- I 
0.. 8
._0_.,.  - ...... 
 
0 
"' --------------- - O' 
I .., 0 
N 
I I I 0 IO 0 I/') 0 
0 0 0 0 c:i c:i .... 
I I 
(.rn / qw) op/ op ?v 
Figure 6 .35: 16O(ji, p') at Ep 200 MeV: 2i (13 .98 MeV), 01 
(14 .032 MeV ), and 41 (14 .302 MeV) . 
281 
+-
l() 
,.-.._ 
.-......- --+-- --
N 
I 
..a..  
---:_=,3~~~~~-------.i -
0 
N 
I .I .. "I'  -0 I() 0 I() 0- 0 0 0 0 0 0 I I 
(.1s/ qur) UP/ op l.v 
Figure 6.36: 16O(p,p') at Ep = 200 MeV: and st (14 .399 Me V) . 
282 
M 
? ? 
? ? , ?? 
? ? 
? 
,_ ? ? 
. ? N ?  C") ? ,--.. ? ? I 
? 0. ? 
? ? -p. - ?? ? -..8...  ? ? (0 ? ? 
? ? 
u C er 0 
?? 
... ? 
? 
? 
 .... ? . ?.  ? ? ? ? ? .. ? 
0 
0 1/) 0 N 1/) 
i I ~ ..; 
0 0.. . 0.. . ci ci ci 0 0 I 
M 
..... ... -
? ---- - ,,11? 'I, ,~  ... -?-
+N -?--- - N ~ ,., ? 0 ,--.. .. . ? ,--.. I?? I ?" ---... 
0. 
-p. -..8...  -- --+- (0 u er --?-....-.-. ...  ~ --
--....... -?--?----- .. 0 1/) 0 1/) 0 
N 
i "I'
_; 
  I 
.0. . 0.I . . 0.. . 0.. . 
0 
0 ci ci I 
M 
.. . ? lo ? 
? ? ? .?  ? ? . +- ? N ? ,--.. ,0-- .. 
. ? .. . ? I ? 0. - , ? p. ?? -..8...  '? ? .?  ? (0 , 
??  u 
?? ?? er 
? 0. ? ..
 . 
? . 
? ? 
? ? 
? ? ? 
? 
0 
.. 0 1/) 0 .. N 0 1/) i I ... 0 0.. . 0 ..; ci ci ci "0.'.  . 0 0 0 I 0 
f.v 
(.1s / qw) up/ .op 
4 200 MeV: 07? , Of (3 .3521 MeV ), nnd 31 Figure 6.37: ?Ca(p, p') at Ep 
(3 .7364 MeV) . 
283 
-
+ M N ,-.. 
,0,.. ..._ I 
0.. 8 
0.. 
--(U-- ----- 
u o' 
0.. . --
.. 0 .., "' ll') 0 ll') 0 N I
i I -I 
I ~ 
- 0 0- 00 0 - 0 
0 0 
I 
C') 
? 
, ? 
? , 
? ? 
. ? ? . 1 - . ? N l{) ~ 
? 
? 0.. 
I 
? .- ? ? 
? 
? 8 ?  ,._0__....  -? ,-.___.. \ ?- (U  - u 0.. . --... O" ? .. ?  ? ... ? -?-- --~ 
--?--?---- 0 
.., 0 ll') " ll') ~ i I'  I 0 0 0 
0- -0 0- 0- 0- ..; 0 0 I C') 
? -- -+-, ? 
... ??  ---
+- .... N ? C\2 ,-.. 
? 'c' ? I 
? 0.. ... ? 
??
?  ... s 0.. ,.___.. 
:?  .__, --- -(0 
-? ? u ..... -- o' 0? .. . ..  -
??
?  -- .? 
? ?  ? ? --?- ? 
0 
., .. ll') 0 
N 0 ll') 0 i I I 
0 0 0 - I 0 0 0- 0- 0 0 0 I 
(Js / qrn ) UP/ DP /4v 
(3 .9041 MeV) , 5~ 
Fig ure 6.38: 4?Ca(p, p') at Ep 200 MeV: 
(4.4915 MeV) , and ot (5 .213 MeV) . 
284 
-- N 
...?..
? 
   - --- I ..8...  ?. .. . -- '- --- ----..+  
-------
0 
,., .. l/'l 0 \() ~ 
N ci 
i I I 
"I'  ~ 0 
I ... 0.. . 0 0 I 0 0 0 C") 
... .. --
.. .. . 
? 
? -- N 
+- ,-... 
,q' 
.... . ,-... ---0. -- --- I ..8...  . -+ ? ? A . ? --(-0- .... ........ -- ----er c..> 
??. . 0 .. . ----- --
0 
.. ~ 
<') l/'l ~ 'I':! N 
i I I .I .. ... ~ ... 0 0 0 
0 0 
I 
0 0 
C") 
.. .... .. ?  -- --
. .... -- +N - N ,-... C\l 
.. ? ,-... --? 0. -- I ..8...  
? ? A -
??  --(-0- ? c..> - - ----er 
? ? .
0. . 
? 
??  -....-. --
+- -+- -
N .., .. 
0 
l/'l 0 
i 'I':! 
~ 
... I 
I I ~ ci 
0 
0.. . 0 0 0 0 0 0.. . I 
(.1s/ qur) up/ .op 
/4v 
4?Ca(p, p') at Ep 200 MeV: (5 .249 
MeV) , 4?t 
Figure 6.39: 
(5.279 MeV), and 41 (5 .6143 MeV ). 
285 
..... ---
...... - --- co 
.. ----- C\l 0 N 
. +.. ? <D I - ,,-... --
? ? 0. - ..8..  ....?..   ..p._, ? , a,j ---------
----
.-.-.. 
.?... . c..> 0 
----- "' -?-
.., 
N .. 0 "' \() 0 \() i- 
I I I I ~ 
0 0- 0 -0 0- 0 0 0 I 
? ..... ? ..... -+-
. ? ? ... ? ,_ -?--+- ' 
. ? ...... - N '( , ,,-... ... . ,,-... ,' . 
.. ?:  
p, I 
 -?- --?-
..p._,,  -- -+- ...8..._..  I ' ? -+- , a,j ? ... t' .? c..> ? 0 ...... ... . O' I I, ? "' ? ,, ?  . -+-- -...-.. - --?-
.. 0 
i N "' \() 0 \() 0 I 0 I - I ~ 0 0 0 0- 0- 0 0 0 I .,  
-...... 
-+- J 
', 
~-- i 
+M 
.. C\l 
N 
---+--- ,,-... 
..... 0. I 
? -?-+ .-.... ..
p._,,  ..8..  
a,j 
.. ?. ... c..> 0 ? "' ~
..._, 
O' 
? -
--??  -
-----+-
...... 
--+--
0 
N \() 0 \() 
0 
0 0-I I  0 0 0 0 I 
(.1s / qw) UP/ op r.v 
Fig ure 6 .40 : 4?Ca(p, p' ) at Ep 200 MeV: (5 .6301 MeV ), 1~ 
(5 .9033 MeV) , and 6.028 (6 .028 MeV). 
286 
-- ---- N 
I 
-a -----+-- ==== 
-+-
N 
I I 
0 0 
-----?----
? - - f--?-.... ..?  ,, .. --- -- N 
? I-+- --- I 
? ? -?-
.. . .  ..... . .... --
-?- -a 
? . O' ? -?-... .. ----..._ ------+-
.. .., 0 N 
.I. . I "' 
0 ti') 0 ti') I I. .. .I 
~ 
0 0 0.. . 0 0. . 0 0 0 ' I I' 
M 
? ? 
? ? . ? ., 
. ? . 
? . 
? 
? 'C"")'  N .?  ,....._ ? . 0.. ? ? I ? , ? ._p.., ? . ? , ... a ,,  
? ? 
.-__., ccs 
? u ? 
? ., 
? 0.. . .. O' ? ? ? ? ? ? ? .. ? ? .......? -- .
 .. ... 
0 
j' N I "I' ti')   ti') 0 0 0. 0. . 0... ~  0 0 0 0 0 I 
(.rs / qw) UP/ DP ?v 
Figure 6.41: 4?Ca(ji,p') at Er 200 MeV: 32 (6 .2858 MeV) , 41 
(6.5084 MeV), and 4f (6 .5436 MeV). 
287 
? 
-?--
? ? -?- -------
? ...... 
-
-- N 
??  -- ---+- I 
??
?  .
?  ..
.... . 8 
- + -----. ??  ? .? : -?  '- ... 
? .. ----
0 I 
0 0 
-+-
N 
? i----------
.. 
I 
? ? I ? 
? 
-? --
---------- -
8 
? --+-
?.  -+--+-- ,,I  ---+-:. ?.. . -
?. .... -f-+
----- ,' 
0 
N .., U"l 0 U"l 
I I I 
0 0 0 0 0 0 
~ ~ I 
? 
? 
? --
? ? .. ? 
. IM ? (;) ? N 0 ? 
??.   0.. . + I + 
? . 0.. ? ---- -- ..8....  ; <O 
? u + -- ,1 .? ? ? 0 ,+-'1' .? ? . ... ? .... --+-? ? -- ----+-
.., 
N .. 0 i I I ~ U"l 0 U"l 0 0 I 0 
0 0 0 0 0 0 0 
_; 
I 
(.1s / qw) up/ .op r.v 
Figure 6.42: 4?Ca(p, p') at Ep 200 MeV: (6 .5833 MeV) , r2  
(6 .7509 MeV) , and 6.909 (6.909 MeV) . 
288 
- M ~ ? -
. ? ? -- -~ -
? -.-.... ... C\I ? 
??  .... .... ....... 
... I 
??  ...... 8 ??  ... .-._, 
? ..... ........ ?? . ?
-.
.  .  - ------ -
0 
N "6'  I() 0 I() 0 I I 0 
0 0 0 0 0 
I 
M 
... ? --
? ---
? ? 
-- .... ,,
? - .... ...- - .,
'4 ..'  
? IN ;C\I ',,,  
? ,-.._ ... I,, 
? C. 
? ? ..C_,.  --- --- I If' ........ ...... ..8? ..._... ,  <U 
? ? u ...... .. ? .??.   0.. . ..... -. - ' 'I 
. I f~ 
---
0 ' ., 
i N I() 0 I() ,I 0 I 
0 0 0 0 0 0 
I 
.. - M ', ?? J   , 
..?...  
(- ----") i 
? 0) .?  <D 
i - ,-.._ --
C\I 
- -+-- --
0.. - - I --+-I- ..8....  
... ..?.  ..C_,.  <U '----
..... -+- u 
.. ~ -
0 
N I() 0 I() 0 
0 I I 
0 0 0 0 0 0 
I 
(.is/ qw) up/ op 
Figure 6.43: 4?Ca(p, p') at Ep 200 MeV: 6.931 (6 .931 MeV) , 
(6.951 MeV), and 7.11 (7 .11 MeV) . 
289 
7 Results and Discussion 
7.1 Data Selection and Iteration Procedure 
The 160 and 4?Ca nuclei provide an excellent laboratory fo r the study of 
m edium m odifications to the effective interaction . Inelasti c t ransitions whose 
amplitude is concentrated in the nuclear surface are sensiti ve to t he low den-
sity p roperti es of the effective interaction , while amplitudes concent rated in 
the interior are sensitive to the high density properties of the interaction . In-
., ,., 
termediate cases determine the rate at which the effec ts depend upon densi ty. ' ~' l ! 
I' 'f"  
Inelastic monopole and isoscalar electric dipole t ransitions t end to be good ' . 
I ,' 
interior states . Inelastic transition densities involving large orbit al angular I' 
' ' ' 
' l1j 
m omentum transfers , on the other hand , tend to peak at the surface. l'.  
States with various multipolarities and with both interior and surface 
character can be easily resolved for the two targets, even if only modest res- ., 
., . 
olution is available . For our interaction analysis we choose the 31 sta te a t 
' 
i 
6.130 MeV , the 2f state at 6.917 MeV, the 11 state at 7.117 MeV , the 4i 
state at 10.356 MeV , and the 2J state at 11.520 MeV from 160 (see Tables 6.2 
and 6.3 in Chapter 6) . The data for the 4f are omitted because of contam-
ination by an unresolved 3+ state ; the 2t is also omitted because it is very 
weakly excited and because of evidence of multistep contributions [Ke! 89a] . 
Finally, the Of state at 6.049 MeV is small and is diffi cult to resolve from 
the neighboring 31 state . The ot state is spec ial and was omitted because of 
the possible spin-convection density contributions near q = 1. 5 fm - 1 , wh ere 
the electroexcitation form fact or passes th rough a. minimum [Pet 85]. The 
290 
bulge in the cross section is prominent at 100 Me V, becomes less noticeable 
at 200 MeV , and has essentially vanished at 318 MeV. Kelly et al. performed 
at 135 Me V fits both with and without this state and found essentially no 
differences in their fitting results . None of the fits can reproduce the cross 
section for momentum transfers in the region between 1.2 and 2.0 fm - 1 . How-
ever, the analyzing power data and the remainder of the cross section data 
are fitted well either way. Furthermore, they find that the other data are not 
affected in any noticeable way by the presence or absence of this state in the 
fit [Kel 89b] . Therefore , although we show calculations for the ot state in '.' ....  
i,  ,~.  
the subsequent sections, the effective interactions were obtained without it . I ~ 
For 4?Ca we choose, with the exception of the ot state at 3.352 Me V, 
, , 
which is very weak and which possibly may also have multistep contributions, 
all the states for which transition densities from (e, e') are available [Mis 83] . 
Those states are the 31 state at 3.736 MeV, the 2f state at 3.904 MeV, the , 
51 state at 4.492 MeV, the 32 state at 6.286 MeV, and the 33 state at 6.583 ,' 
i 
Me V. Both the 32 and the 33 states have strong interior lobes and should , 
therefore, probe the high density aspects of the effective interaction. Finally, 
we also included the elastic scattering data in some of the fits . The results 
which include elastic data are discussed in Section 7.4 . 
For all the energies we performed separate fits to the states of each of 
the two targets , and also a simultaneous fit to all the stntes of both targets. 
This should help us to detect an A dependence, if prese11t . For the analysis , 
we folded additional uncertainties into th e fitt ed data. This assures that all 
states have a similar influence upon the analysis and that the data are equall y 
291 
weighted for all momentum transfers . Furthermore, we achieve smaller val-
ues for X~ in this fashion , and also equalize the quality of the cross section 
and analyzing power data, as measured by x~ per datum. Because mos t of 
the electron scattering data is limited to q ::; 2.7 fm - 1 , and because of un-
certainties in our reaction model at large momentum transfers ( e.g. coupled 
channels effects become important) , our analysis includes only momentum 
transfers up to 2.7 fm - 1 . 
All our fits employed a self-consistency cycle . The first iteration usually 
'' , 
utilized distorted waves based either upon the PH interaction (in th e case :'  .,  
,' ,,.  
of 160), or a previously fitt ed empirical interaction (interaction fitt ed to 160 ?-
for 4?Ca). After rem oval of the (1 + po/op) contribution (see Sec tion 2.8 .3) 
the fitt ed interaction is then used to compute a new optical potential and 
distorted waves. The fit is repeated and the distortion iterated until con- ' . ' I 
f' 
vergence is achieved . We usually observe that good convergence is obtained 
,,
,,   
within about seven iterations . The results in the following sections represent 
.i ,  
the endpoint of the self-consistency cycle. One reason that we generally ex- ,,,  
cluded elastic scattering from the fit is that it serves as an independent check , 
for this self-consistency procedure. ' 
We found that the final fitting results are stable and independent from 
the starting parameters. This was verifi ed by gridd ing each of the six pa-
rameters of th e empirical effective interaction from Chap ter 2, Eqs. (2. 1 H )-
(2 .116) , while letting all the oth er param eters vary. Each parameter grid 
consisted generally of four or five more or less wid ely spaced values for th e 
para.meter. The range of param eter values was chosen to allow interpolation 
292 
of the parameter with the minimum x~- Grids were done at 200 MeV for 
both 160 and 4?Ca, and for 160 at 318 and 500 MeV . The grids demonstrate 
that there is a unique solution in the space of x~ for each target at 200 and 
318 MeV. At 500 MeV, on the other hand, we fou nd two true solutions (see 
also [Fla 90]) . The results for the interaction grids are compiled in the tables 
of Appendix D. 
7 .2 Fit to Inelastic Data 
,.,,  fll' ,_, 
7.2.1 100 MeV l~ 
I #" 
I(' 
The parameters for the empirical effective interaction resulting from the 
fits to inelastic data of 160 and 4?Ca at 100 MeV are collected in Table 7.1. 
The data for the interaction fit are from this work . For the results shown here, :, 
I , 
the interactions were modeled upon the PH free interaction which gave results 
.,  
superior to fits using the FL t-matrix at this energy. For all the fit s it was j, 
necessary to constrain the b3 parameter which, as a grid search proves, is not 
strongly determined by the data. Unless elastic scattering is included , for 16 0 , 
.,;  
we additionally had to constrain d2 to prevent that parameter from getting 
unreasonably large. For both parameters we chose the PH values, a choice 
which is later justified by the combined fit to both elastic and inelastic data. 
In the case of 4?Ca and also the combined fit we fo und that no reasouable 
and stable fit could be ob tained unless both b1 and b3 were co 11 strai11 ed . Tf 
we constrain d and b as in the case of 162 3 , 0 , b1 cannot be determined and 
the fit does not converge. With b1 and b3 constrai ned, th e free d2 param eter 
293 
assumes a rather large value in the final result which yields opposite signs of 
(S2 - d2) at saturation density for elastic and inelastic scattering (we saw in 
Section 2. 7.3 that for inelastic scattering the d2 parameter is to multiplied 
by a factor of 5/3 to account for the rearrangement contribution) . Our self-
consistency procedure may be affected by this effect . 
In Figure 7.1 we compare the effective interaction (EMP) from the fit to 
160 wi.t  11  the PH m. teract1.0 n . T 11 e R etC an d R e tLS 00 O components both appear 
to have less density dependence, quite in contrast to the Imtf0 component . At 
large momentum transfers or for large density, the real central part resembles 
Very closely the nuclear matter interaction. However, we also see that the 
strength is reduced at low density. In accordance with previous analyses 
([Kel 89b, Kel 90aJ) it was found, that in order to obtain a good fit to the 
surface peaked inelastic states, the scale factors S; cannot be constrained to 
unity . However, after having reduced the strength at low density to fit the 
surface states, the subsequent density dependence has to be reduced as well 
to fit the interior states . The fact that the effective interaction does not heal 
to the free interaction at low densities, and the fact that the interaction is 
less effective in the interior relative to the nuclear matter interaction , I1as 
been interpreted as nonlocal density dependence of the effective interaction 
beyond the local density approximation. In a finite nucleus, nuclear orbits at 
the surface sense due to tl,eir finit e spread , some of the high -density nu clear 
' 
interior; by the same token , interior orbits extend to less dense regions. 
In Figures 7.2- 7.6 we show our fits to the various states of 160 and 
4
?Ca. In these figures we compare calculations based on our empirical effec-
294 
tive interactions (EMP) for 16 0 and 4?Ca with LDA and NRIA calculations 
based on the PH interaction. For the NRIA we used the PH g-matrix to 
generate distorted waves and the PH t-matrix for the inelastic transition . 
Although none of the analyzing powers are described particularly well, we 
see improvements in the cross section for almost all the states. The effects a.re 
most dramatic for the 160 states : for the surface-type states like the 31 and 
the 4j states, the forward cross section is lowered by the reduced strength of 
the interaction at low densities. For the 11 interior state, on the other hand , 
we see that the reduced density dependence of our empirical interaction in- '''  .-.. 
~ ~: 
creases the cross section relative to the LDA . In the case of the 2t state the 
reduction of the interaction strength proves to be slightly too severe. Finally, 
although the ot state was not included in the fit , our interaction provides a 
description of this state which is about equal to the one by the LDA . The 11, 
ot, and the 2+ analyzing powers are still significantly improved by the EMP. 
The major improvements provided by the empirical interaction for 4?Ca are 
in the cross section of the 51 state and the second cross section maxima of 
the 32 and 33 states . The effect of the density dependence is, however, m ore ,,  
1 
clearly seen in the 31 and 2t analyzing powers where the NRIA prediction 
is too high . Although these states peak at the surface , significant medium 
modifications to the NRIA are necessary to adequately describe th e analyzing 
powers . 
Elastic scattering was not expli citl y includ ed in th e fit s of thi s sect ion 
and was simply predicted from in elasti c scatterin g using the (1 + pf) / op) pre-
scription from Sections 2. 7.3 and 2.8.3 . For elastic scattering at thi s energy, 
295 
it appears that the LDA based on the PH interaction provides the best de-
scription of both the cross section and analyzing power data . The fact that 
the PH interaction gives better results for elastic scattering suggests t hat 
there might be a problem in the self-consistency procedure which m ay be 
related to the opposite signs of ( S2 - d2 ) for elastic and inelastic scattering 
at saturation density. It might be that second order terms in the ( 1 +po/ op) 
prescription are needed at this energy to restore the consistency between elas-
tic and inelas tic scattering . _The empirical interaction is sti ll quite successful 
for 160 at 135 Me V [Kel 89b], but seems to deteriorate rapidly between 135 
and 100 MeV . 
Figures 7. 7 and 7.8 show the optical potentials for elastic scatteri ng. 
While for 160 only the imaginary central potenti als vary significantly, for 
4?Ca also th e real central potentials vary in the center by about 17% (the 
empirical potential being more attractive). For the imaginary central part , 
the empirical potential is less absorptive than either the LDA or NRIA po-
tentials. Finally, the empirical and LDA spin-orbit interactions are virtually 
identical. 
296 
Retg0 Imtg0 RerL0 S [1mTl
5
] 
Data set S1 b1 S2 d2 S3 b3 t<n d4 U41 
160 0.86 41.8 0.69 [0.55] 0.82 [3.7] PH 0.60 -4.61 
40Ca 0.95 [41.8] 1.10 0.89 0.89 [3 .7] PH ?4 = 3.0 rm - 1 
rno+4oca 0.92 [41.8] 0.97 0.82 0.82 [3 .7] PH [o:-,-,86] = [3220] 
Th eory 0 [1.0] 72.7 [1.0] 0.55 [1.0] 3.7 PH ref.: Table 2.1 
Table 7.1: Empirical effective interactions for 100 MeV protons . We use 
the following exponents [a,,85] and masses? : [3310], ? 1 = 1.5 fm - 1 (Retg0 ); 
[2210] (Imtg0); [3320], ? 3 = 6.0 fm - 1 (ReTt 5 ). Units : Si and di (1) , b 1 (MeV .','    
fm3), b3 and a (MeV fm 541 ) . Square brackets indicate that the parameter or , 
component is fixed . .,,  
0 Values from Table 2.5. ' 
297 
...,.  4:,<> ,' ? d 1': p / ?? ."' ? ? ,; 4 ? + 4 0 ....  </ ,'+ Cl 0 ... 1\ 0 ,. ? 0 
0 ..
0 ....
.
....
  I\ "", , .". 
0 ..   1 N 0 ,:\  
?-
? j 
I 
?+o , 8 
<I+ o,? 
Cl+ , 0 1 
? + ," IJ -
,c1 + , ?'; o 
4 + ?? ?.,. 0 
4
d + ~ ,,// 0 O 
,q + ?? / 0 
0 ? + ,,.? / 0 
0 d ? ,? / 0 
b <I + ,. ? / 0 .. + , 0 
4 ? ,' / 0 
<I + ,' 0 
0 
., \I') 0 \I') 0 
0- "0-'  c:i c:i c:i . I ... ".,.  \ <I+\ 0 .~ .d, '  ' , 0 ? 4 I 
....... t, 0 ? / (I I .... 0 + <I , ?~"  0 + / .. ,, " ... 0 O ? _../ Cl ~ ,. , ., ,.. , 
0 O ? ? / d ,,? ,.," N ~ ,1 
0 + / 4 ... .... 
o! + / 4 ... 4.. -- ' .,.,,. I I 
? / <I ...... 
"'?id":? .. .,.,, 8 I. 
+ A ~ ... -- -- ....._..,  
+ ,if I . 
uo0 ~.p O' _,  ,.t 
. ~! .,.,;. + ..  .. ... ,:-,. + .. --"I:--+- . .... -~ 
0 
0 0 0 0 0 0 00 \I') 0 \I') 0 \I') 0 0 -0 \I') \I') 0- \I') 0 \I') C\l I I N C\l ~-----~-~-~~~~-~-~I --I '- :-, ? + 0 ? + 0 
'-: ? + 0 ? + 0 
\\ ? + 0 ? + 0 
',\ ? + 0 ?+ 0 
\ ?+ 0 \ ?? 0 N 
4' 0 
:' .. 0 ? 0 // .+oo I , 
,?/ ..,?o ..8...  ' ; 
,,;
,,;; 
u...o.0,   .. ,.;: .. 
 ... . J 
_ ...,1:."0N ... 
~ ... ~ 
0 
0 0 0 0 0 0 0 0 0.. ,. 0 0 0 0 0 0 0.. 0 0 - \I') C\l co 0 N ,. 
0 
0 \I') \I') 0 \I') 0 \I') 0 
-' C\l C\l M M I I "I'  I -' "-'' I  
I I I I I 
(CUJJ J\dW) 
Figure 7.1: Comparison of the EMP(1 6 O) and PH interact ions at 100 MeV 
for kF = 0.6 fm - 1 (triangles, short dashes ), kF = l.0 fm - 1 (crosses, dashes ), 
and kF = 1.4 fm - 1 ( diamonds, solid) . 
298 
? ?? f 
,?: 
-::-??'-'?? ?
?-..... ... .
 .. ? 
. 1- ? N / 
I ?? ? ~ ... I 
) ???. :' a 
'?? ... .._....,  
??-. 
0 
N 
0- I I 0 0 0- 
,:. ... 
??--:-:..~ ."' -~ ""?I?.:  ? 
.,. ,., N  ,...._ ,. 
+- I a ' 
,. 
,? 
,C...\.l_  
0. ?.? -... 'I' 
I -
._C.,.  '?, ' O" 
0 
~ 
I .,  
0 
0 0 I 
I() 0 I() 0 
0 0 0 0 0 ..; 
I 
(") 
??--:: __ _ 
?? ... . 
? 
N 
I a
.._...  ..,  
O" 
0 
"I'  0 I() 0 I() 0 
0- 0 i  0 0 0- 0 0 0 I 
(.ts / qur) UP/ DP f.v 
Figure 7.2: Comparison of EMP (solid) , LDA (dashes), and NRIA (short 
dashes) calculations for the 31, 2{, and 11 inelasti c states of 16 0 at 100 MeV. 
The data are from this work. EMP based on PH t-matrix, LDA on PH g-
matrix , and NRIA on PH t-matrix ( the distorted waves are genera.ted with 
the PH g-matrix) . 
299 
- ---- ------ -
1 ---
N 
I 
??  
..8.  ..__.. . 
,,.,.--?-?-?;,??;,:?-;,??" 
~ ~- -
0 
" l() 0 l() 0 "I'  .I  0 0 I 0 0. . 0 0 0 ...; 0 0 I 
(") 
.?.. ?j ? I  
_; N 
~---~ ... ... I " .-?? + (") 
? ' N .8 
? .._
.._... . I I , 
0 
l() 0 l() .I, ' . 0 I  0. . 0 0 0 0 0 I 
,' 
.. .?  ... t N 
I 
..8..  ?,  ..__.. , 
J 
?, 
0 
j N "I  0 l() 0 \{) 0 0
0. . . 0.. . I 0 0 0 0 0 I 
(.rs / qw) op/ op ?v 
Figure 7.3: Comparison of EMP (solid) , LDA (dashes), and NRIA (short 
16
dashes) calculations for the 4{, 2j , and 0j in elastic states of 0 at 100 Me V. 
The data are from this work. EMP based on PH t -matrix , LDA on PH g-
matrix , and NRIA on PH t-matrix ( the distor ted waves are generated with 
the PH g-matrix) . 
300 
. ? 
... . ? +- N C\l ,--, ,--,  
I 
0.. 
-p. ..8..  ro 
0 '? O' 
0.. . 
l_,~~~.L-~~~-'-~~~ --'-~~~-1 0 
'? ~ lt) 0 lt) 0 
0 I - - 0- 0 ci ci 
... 
0 0 0 I 
~~~-.--r--~~.,,...-.--~~~~~~~-C'l 
/ ....? ? 
t?'? " . . - .. 
..... .  
.; ~ ' I , ,, ,..,  
-8 
., 
0 
+N 
,0- -, , -+- _.:-,,? 
0.. I 
-p. ..8..  ,'.  ro ' 
0 -- ?:---:-: 
-
. .- :-.. ~ - --- ,; 
0.. . 
N lt) 0 lt) 0 
I I 
0 0 ci ci ci 
... 
I 
(.ts / qw) UP/ np 
Figure 7.4: Comparison of EMP (solid) , LDA (dashes ), and NRlA 
4
( short dashes) calculations for the ot , 31, and 21' inelastic states of ?Ca 
at 100 MeV. The data are from this work. EMP bas d on PH t-matrix, 
LDA on PH g-matrix, and NRIA on PH t-matrix (the distorted waves are 
generated with the PH g-matrix) . 
301 
(\) 
I 
..8..  
~~~~_._,~~~_._~~~..J.._~~~_J 0 
N 0 \() 
0 I I 7 ~ ~ 
0 0 0.. . 0 0 0 ... I I 
-.,... ,......,~~~....-,r.:-T~--c,--~~~~~~~~ C"l ,, _,_ ______ ,.. _,.._,,_,;.? 
.,...  -. -+-? - .:-:.::. . 7 . -
~ ,. ,-' 
.. I:--~- :-; -:"' ? IN (\) C') ,--- --0---.- I ? . 
0.. ..8..  
--'?--u  . c:r 0.. . \ 
.. .__.~~~.L......,~~~_._~~~..J.._~~~_J 0 7 ~ \() 0 \() 0 0 I I 
0 0 0 0.. . 0 0 0 I 
1-
1{) 
I 
0.. 
..8 0.. .. 
? I --'?--u  c:r 
0.. . 
0 
N 
j' I 0 lt"l 0 \() 0 
. 0 0. . 0.. . 0 0 - 0 0 0 I 
(.ts / qw) UP/ DP ?v 
Figure 7.5: Comparison of EMP (solid) , LDA (dash es ), and NRlA 
( short dashes) calculations for the 5~ , 3~ , and 33 inelasti c states of 4?Ca 
at 100 MeV. The data are from this work . EMP based 0 11 PH t-matrix , 
LDA on PH g-matrix, and NRIA on PH /.-matrix (the distorted waves a.re 
generated with the PH g-matrix) . 
302 
( ... 
.. ? '- -
\. 
\ , - .,,,,J ? / 
N 
,-... 
I 
+-
0 8 
,-... ,._.._. , 
0.. 
,_0_.,.  O" 
<O 
u :::, \ 
0.. . ??t ~ 
0 
i 0 If) 0 If) 0 0 
0.. . 0 0.. . 0 0 0 ... I I 
(") 
_:',, 
,, ~. .: :?,. ,., I .,,.., ..I ?.... / I I 
N I ,.,
  
+- ,-... , .,..  
,0-. .. I I'  
0.. -8 , ,__, I ,_C_i,.  ,' 
0 t ... O" 
~ I ',  
0 
i ~ If) 0 If) ~ "0' 
0 
 0.. . 0 0 0 0 0 I 
H.o / .o ?v ,, 
,- 
''',   
J 
Figure 7.6: Comparison of EMP (solid) , LDA (dashes ), and NRIA (short 
dashes) elastic calculations for 160 and 4?Ca at 100 Me V. The data are from 
this work . EMP based on PH t-matrix , LDA on PH g-ma.trix , and NRIA on 
PH t -matrix. 
303 
- - - - ---- -
r--- --,r----.-- -.---- -- "' 
N 
0 
N 0 
I(:, 0 I(:
I.(.:.,  "' 0 ...; ...
, 0 0 a:, <O  N _; 0 0 0"" ' 0 0 0 ~ 
C\i ci 0 ci I I I I 
. (.i\dW) s,fl W.J 
(A<~W) s,fl acJ ., .. 
I _., 
i , ) 
'I,,..,,  
l() If' 
I 
\ 
\ 
\ 
?,._ ,_ 
"? <-,_ ..::'-.? '-  .. 
\ ??, 
', 
?..__?, ... \ ?? ... 
'? .. I I 
' N \ .., 
? I 
I r 
' ' 0 
,r., I() 0 
0 I() 0 I() 0- "-' 0 0 - I() N N (".) (".) 0- I() 0 I() 0 0 I() (".) (".) ""' ""' I I I I I I I ""I ' I N N I I I I I I I I 
(.i\dW) :iD W.J ,.  
(N~W) ::P a(! 
Figure 7. 7: Optical potentials for elastic scatteri11g of 100 Me V protons 
by 160 for the EMP (solid), LDA (dashes ), and NRlA (short dash es ) inter-
actions . EMP based on PH t-matrix , LDA on PH g-ma trix, and NRIA on 
PH t -matrix . 
304 
I ? 
<O 
I 
> I 
V I "' --;;:~ I 
P.o 
-o "" ,-... 
' (I) -J-) -..s /4 ... II n .. Ci]"  s... 
C\I 
~ 
0 
- I-() n C\I 0 N 0 " ~ 0 0'  0 0 0 0 0 0 I I I I I 
(/\aw) s,O ocf (A aw) s,O W,J ... 
<O , .. , r 
.:'  ._"'I,?.  "'  ,  
I 
' ' ' "" I 
I -
' s \ I n -.... ' I ' \ s... 
I 
C\I 
' I ' ' I 
I 
I 
I , 
0 
- - 0 I() 0 I() 0... . I..(.).  0 I() 0- I-() 0 I() 0 I() 0 0 I() 0 I() n n I N N n n .... I C\I N ',  I I I I I I I I I I I I I I I 
(N~N) :){) ocf (N~W) 
I 
:){) W,J ';  
?,,  
 
) 
Figure 7 .8: Optical potenti als for elastic scatterin g of 100 Me V protons by 
4?Ca for EMP (solid), LDA (dashes ), and NRIA (short dashes) interactions. 
EMP based on PH t-matrix , LDA on PH g-matrix , and NRlA on PH t-
matrix. 
305 
7.2.2 200 MeV 
In Table 7. 2 we show the results for 200 Me V protons . As in the case 
for 100 Me V, the data for this interaction fit are from this work. The fit was 
very stable and none of the parameters had to be constrained . Again the PH 
free interaction proved superior to the FL t-matrix as a model for the free 
interaction . In Figure 7.9 we compare the interaction for 160 with the PH 
interaction . Very noticeable at this energy, and especially in the Ret~0 com-
ponent, is now the overall reduction in strength of the empirical interaction . 
.,..,  , 
Figure 7.10 compares our empirical interaction with the LR interaction . The ,,.,,, ,,..  
 
real central component seems to be shifted upward and has about the same I 
t ? 
density dependence as the LR interaction. The imaginary central component I ~ 
of the empirical interaction, however, displays much less density dependence I .,  
,, 
than the corresponding LR component. Figures 7.11- 7.15 shows the fits to , 
160, and 4?Ca and Figures 7.16 and 7.17 show the optical potentials for elas-
tic scattering. ?with the exception of the forward angle cross section of the 
. 
2t ,, state of 160 which is still too low, and some of the analyzing powers be- ,,,  
,,. 
low about 1.0 fm - 1 which are slightly too high , the empirical interaction at 
~ 
this energy provides a description of all the data which is much superior to 
either LDA or NRIA calculations. In particular, the empirical model follows 
the deep excursi ons of the analyzing power data above 1.0 fm - 1 much more 
closely than the other calculat ions. Again , although neither the nt state 
of 160 nor the Of state of 4?Ca were included in the fit , mos t of the cross 
section data and essentially all of the analyzing power data are desc ribed 
very well. We see that for the Of cross section data , the minimum predicted 
306 
Retf Imtf RerLS 0 0 0 [ImroLS] 
Data. set Si bi S2 d2 S3 b3 tU) d4 U4i 
160 0.70 65.5 0.77 0.14 0.84 2.03 PH 0.38 -3.36 
4oca 0.61 60 .5 0.84 0.15 0.85 5.73 PH ? 4 = 4.0 fm - 1 
160+ 4?Ca 0.66 59 .8 0.80 0.16 0.85 3.27 PH [cx-y/36] = [3220] 
Th eorya [1.0] 84 .0 [1.0] 0.23 [1.0] 2.10 PH ref.: Table 2.1 
Table 7.2: Empirical effective interactions for 200 MeV protons. We use 
the following exponents [a,,85] and masses? : [3310], ?i = 1.5 fm - i (Retf0 ) ; 
[2210] (Imt~0 ); [3320], ?3 = 6.0 fm - i (Rerf5 ) . Units : S; and d; (l), bi (MeV , ,. 
fm 3 ), b3 and a
5
4 i (MeV fm ) . Square brackets indicate that the parameter or 
'.,.,,  
: ... 
component is fixed . ' .. 
I f 
avalues from Table 2.5 . ..  I' 
( : 
between about 1.2 and 2.0 frn - i is filled up ; for the ot state, this region is ,/ :,  
I, 
around 1.0 fm - i . The elastic state of 16 0 features an enhancement of its 
cross section in just about the same region as the ot state. At 200 Me V the 
empirical interaction is clearly superior in its description of all the scattering 
observables . It should be pointed out again , that the elastic data were not .,  
included in the fits and that the (truly impressive) Figure 7.15 shows pre-
,'',   
dictions based on the effective interaction determined solely from inelast ic J 
scattering. Most notably, the spin rotation functi ons at small momentum 
transfers , which are underestimated by both the LDA and the NRIA , are 
correctly reproduced by the empiri cal effect ive .iuteraction . As iii th e case of 
100 MeV , the spin-orbit potential s are not very different for EMP and 1D !\ . 
At 200 MeV the EMP and LDA real central potentials are more distinct than 
for 100 Me V, the EMP potential being slightly more repulsive. 
307 
0 + 
0 .. 
0 +? 
0 ?? 
0 ?? 
g 0 +. 0 + ? 
0 + ? N 
b 0 ... 
0 + ? 
0 + ? I 
0 + ? 
0 ?? E 
0 +
0 .+.. ?  .._,, 0. .....,  -
? + 0 
? + 0 
.. + 0 ,? / 
+ ,,., ? / 
+ ,.o / 
+ _.?o,,, 
+ ,,--:~ /0 / 
.?  + ,? 0 / 0 , 0 
I(') 0 I(') 0 
0 ..,; ., 0 0 
I 
0 
.,.,  
f 
N 
' " I  .,.?  
I 
..E...  1
'  .,  
.._,, I? 
CT , 
I I 
,..  
+ . 
+ ..  0 ' . I 
0 
I(') 0 I(') 0 
I(') 
0 0 0 
0 0 0 0 
0 
0 I(') (") N N 
I(') I(') 0... . I(') N N I I I 
? + 0 
\\ ? + 0 
? + 0 
\\ ? + 0 
? + 0 
'.\ ? + 0 ... 0 
.. \ ?? 0 ?? 0 
?..?.. ..0   N , ', \ ?? 0 ' }I ... I 
/1 ::;' s .._,, 
,1/ ." -
?. . _ ~-. ,,; .. CT \ '?? .? ? 
u,._o0,   0 ' ... <Cl ? 0 , ? ? ? d 
0 '??t <I 
0 b ' ? ,+  .. + c:i 
' ~..0.- .. + ?.4  <d(I 0 
0 ?? + 
0.. ,. 0 0 0 0 0.. ,. 
0 
0 0 0 0 0 
0 0 
0 0 N (() CX) 0 
N 
0 .... ~ 0 0 I(') 0 
0 I(') 0.. .. I(') .... 
I.(..').  0 I(') .... .... N N I I I I 
N I I I 
(cwJ J\c}W) 
Figure 7.9: Comparison of the EMP(1 6 O) and PH interactions at 200 MeV 
1 
for kF = 0.6 fm - 1 (tri angles , short dashes) , kF = 1.0 fn, - (crosses, dashes), 
and kF = 1.4 fm - 1 (diamonds, solid) . 
308 
0 
0 
0 
0 
0 
0 
0 0 
0 0 
b 0 
0 N 
0 
0 
O...<.I   I 
4+0 8 
? + 0 ._,, 
? + 0 -
<I + 0 , 
+ Q. ... / 
4 + + _. .. ?0"?/  / 
4 +-+ __ ..... ",,,./ 
. + + . .. ?? / 00 + ,,? / 0 
??  
? ?? / 0 
+ ' 0 
0 
I() 0 l() 
"' ~ 0..  .. 0 0 0 I . (") 0 
/ 0 + . 
' ' 0 ? . / 0 . . 
' 0 .? / '0 +  . ? , ,. 
/ 0 ., 6 + ~ 
(l, .. + ... 
/ o,., 4 4 N ' "f"  
0., + <I ,,. 
/0,' + ? 
0 ... + <I I 
(",,-:? .. ~~ 
, + ? 8 
0., + 4 .._...,.,  
, + ? 
(,1 .. ~ O" 
+ ? 
+ -? .... + ? .. , + ? --.. .--- .:: 
0 
0 0 0 0 0 00 
I() 0 I() 0 I() 0 
"' "I'  0... .. "....'.  
0 (") Cl) N 
Cl) 
I 
,~ (") ~ ',\ ?? 0 ; 
,\ -?? 00  ~ \\ ?? 0 ,~ ,\ ?o~ \  \ -- 00  
,-. 00~ \\ -4+ 00  ,,, O +d .. 0 
, .... I 4+ 0 N ?,:'" 
',\ 
 .. 0 ', ,, 0 +4  :' ?+ O ..... ~ :' :, .. 0 I .. 0 , .... 0 .... .. 0 ? 
,,.> 8 0 +4 4' 0 ..... f 
", 0 +Cl ,'/ .. 0 ,. 
._: .. ,!>. ?? .. ? 4 ,; .. o 4' 0 
0 ''? .. o +4 ,;/, ? o O" uo ' ? .. 0 ? <I 
+-' ' .. , .. o ? <I ?.: 
''. . 9 + <I 
' .. ~. . o +: <I 
' 'Q + ci 
\ ~- ???<I 
0, + 4 
0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 "' 0... .. "' 0 Cl) 
.,,, .,,, 
"' ..... "' <O co 0 I Cl) I I I I ..... ~ <..O...  c...o..  I 
(c WJ J\dW) 
Figure 7.10: Comparison of the EMP(1
6O) and LR interactions at 
0,6 fm - 1 (triangles, short dash es ), kF = 1.0 1 200 Me V for kF fn1 - (crosses, 
dashes), and kF 1.4 fm - 1 (diamonds , solid) . LR medium modifications are 
modeled upon the FL t -matrix 
309 
I 
.....a.....,,  , 
0 I 
0- ~ 
.,,..,   
.,.. 
I 
a .,? 
 
.  
..........,., , ' 
, 
', 
,..  
.,  
I 
,_ 
-("') I 
.a ? .........,., , ,'.  
O' , ) 
__ ~:..:-t.:.~. -~? . .,  
L.....o.~~~L...,~~~.L..--~~~J_~~_._J 0 
I I() 0 0 
I() 0 
0- 0-  0 0 0 0 I 
(.is / qw) UP/ DP 
Figure 7.11: Comparison of EMP (solid) , LDA (dashes ), and NRIA (short 
dashes) calculations for the 31, 2i, and 11 inelastic states of 
16 0 at 200 M V. 
The data are from this work . EMP based on PH t-rnatrix, LDA on PH g-
matrix , and NRIA on PH t -matrix ( the distorted waves are generated with 
the PH g-matrix) . 
310 
N 
I 
-a .._, ?-
0 I 
0 0- 
... - _-z,-_ --r- -- --- -- -""T" -
' .,,.. , N ,.  
I ' ',".  
-a ,, 
I 
-a ,,.  
O" ,? 
N ll) 0 ll) 0 
0 I I 
0 0- 0 0 0 0 I 
(.is / qw) UP/ np 
Figure 7.12: Comparison of EMP (solid ), LD A (d as hes ), and NRIA (short 
dashes) calculations for the 4i, 2j, and Oj inelas ti c states of 16 0 at 200 Me V. 
The data are from this work . EMP based on PH t-matrix, LDA on PH g-
matrix , and NRIA on PH t-matrix ( the distorted waves a.re generated with 
th e PH g-ma.trix ). 
311 
+- N 
Cl1 ,,-... 
,,-... 
0.. I 
-p.. -..8...  aj 
0 O' 
~ 
0 
j" N "I'  .. I 0 lO 0 lO 0 
a I Q 0 0 0 0 0 0 0 0 
I 
(") 
,. . 
,_ .,,.. ,  
N 
("') ,,-... ,. 
,,-... 
p.. . I "' 
-p.. aj -..8..  ' 
0 O' 
0.. . 
0 
j" N 
a C I 
lO 0 lO ~ 
N ~ - 0 0- 0 0- 0 0 0 I -I (") 
? 
-, 
1 
? +N N 
0 ,,-... ' 
,,-... -~ 
p.. I 
-p.. -..8...  ?,  aj " 
0 O' 
0.. . 
0 
j" N I "I' 
.. 
 I ~ lO 0 lO 0 
0 0- 0 0 0 0 0 I 
(.1s / qw) op/ op 'v 
Figure 7.13: Comparison of EMP (solid) , LDA (dashes) , and NRIA 
( short dashes) calculations for the Of , 31, and 2i 4inelasti c states of ?Ca. 
at 200 MeV . The data are from this work . EMP based on PH t-matrix , 
LDA on PH g-matrix, and NRIA on PH t-rnatrix (the distorted waves are 
generated with the PH g-matrix). 
312 
M 
-+-
? 
IM N 
C'? ---
C. I 
A ..8...  
--(-0- ----
0 er 
,1' 0 
:J. <I' 
'" -.:; 
----_"':';, 
.., 0 
i N 
.. 
I I 0 l() 0 l() I 0 Q- 0- 0 0- 0 0 ci ci ci I -I 
. M 
- - - ? 
.. , 
..
IN N --- ,,.,
. ,  
C'? 
,-._ ..  
0. I ,. 
0. 8 
--(-0- -
0 ~ er , 
0 . 
<I' 
--~--~ ..  
.., 0 
N 
Q- i 
I 0 l() I 0 l() 0 
0 0 0 0- ~ ci ci ci I - ~ I 
:::::-- M -..;..:. ;~__ _ __ 
1' 
,_ 
N I 
l() 
,-._ --- .~.  
C. I ' 
A ..8...  ' . 
---- f,. (0  
0 ~ er 
0 1 
<I' 
,: ) 
_.,.; 
.., 0 
N 
i I 0 l() I 0 l() 0 - Q 0 0 0- -0 0 ci 0 ci I -I 
(..rs/ qw) UP/ DP f.v 
Figure 7.14: Comparison of EMP (solid ), LDA (dashes) , and NRIA 
( short dashes ) calculations for the 5~ , 32 , a11d 3-3 inelasti c states of 4?Ca 
at 200 MeV. The data are from this work . EMP based O il PH t-m atri x, 
LDA on PH g-ma.trix, and NRIA on PH t-matrix ( the distorted waves a.re 
generated with the PH g-ma.trix) . 
313 
+--0 N Cl.. 7- 
._C,l.,.  .._.a..,.  , 
? <O u 
$ O' 
0 
\() 0 \() 0 0 
\() 0 \() 0 
0 
ci ci ci ci 0 ci 
I I 
0 0 
(") 
? ? ,,
,.
..
 
,  
,.. 
+- . N - ",?"  ,0... __ .. .. . 7 
0., 
._0.
a 
,,.  ..... ,,  
ro 
u ... O' 
0.. . .'  
"
0 .,IJ  
i "I'  ~ \() 0 
\() ~ 
... 00 0. . . 0 0 ci ci ci I 
.. (") 
? 
N - 't ,- +--  0 I ? 0.. ..a...  ,,.  
__0_._.,  
0 O' 
~ ' 
0 
i \() 0 \() 0 0 
N 
0 0.. . 0 0.
 . 0  . 0 ci 0 I 
8n/ n /.v 
Figure 7.15: Comparison of EMP (solid) , LDA (dashes), and NRIA (short 
dashes) elastic calculations for 160 and 4?Ca at 200 MeV. The data for o- and 
Ay are from this work , the data for Q are from [Ste 85] and P. Schwandt , 
private communication . EMP based on PH t-matrix , LDA on PH g-rnatrix , 
and NRIA on PH t-matrix. 
314 
<O 
ll"l 
> 
Q) 
a:~ ~- ../' 
ci,o ~ 
.__, 0 ~ -
0 C\l ::--... - s 
~ II -- - C"l ..... 
"' -- ...... ------Ci] s... ' __, -- C\I 
: \ 
---
--- \ 
I 
0 
0 ll"l 0- l-l"l C"l Cll 
0 
0 ll"l 
...; 0 0 0 0 0 
0 0 
I I I 
(/\aw) s,n 'ULJ 
(/\aw) s,O <J 'cf 
,., 
<O 
?,,,'.".,   
ll"l ,,..  
~ ; 
' ../' ' , '-,, , \::. .. - , 
\', -..: .,. ? .. s C"l ..... 
1'? -. ,?. . 
'?-. ------
' ,, s... ' ' 
I ' C\I -
I ' 
' ' -, ' ' I 
-- , ' ' \ 
\ 
0 
... ... I() 0 I() 0.. . I.(..)  0 I() 0 I() 0 ll"l 0 ll"l 0 I() 0 C"l I C\I Cll C"l C"l I Cll C\l C"l I I I I I I I I I I I I J' 
(/\~N) oD c!'cJ (/\aw) on 'ULJ '-I .,  
? 
f 
Figure 7.16: Optical potentials for elastic scattering of 200 MeV protons 
by 160 for the EMP (solid) , LDA (dash es ), and NRIA (short dash es ) inter-
actions . EMP based on PH t-mat rix, LDA on PH g-matrix , and NRIA on 
PH t -matrix . 
315 
r-------~- --- --- <Cl 
I[) 
' ' ' 
" N ' \ 
I 
.._~_.__;c__.__-~l'--L-~-~~-~ _J 0 0 N <t' <Cl <Xl 0.. . N C\l 0 0 0 0 0 0 _; 0 0 0 
I I I I I 
,. 
rr-.----r----r--~-~----<D .... ,f,'  
,,,,..,  
I[) 
,,  
, 
' N -? 
'? ' \ "11 
' I ., I I 
I 
I 
---'-----'-----''---.L-1.---1.-'----'-.:..__I 0 
0- I-[) 0 II') 0 0 II') 0- I-I') I() 0 II') II') 0 0 C\l N C") I C\l C\l C") C") I I I I I I I I I I I I 
(.N~N) :i) 8cf (N~N) ....  
Figure 7 .1 7: Optical potentials for elastic scattering of 200 Me V pro tons by 
4?Ca for EMP (solid), LDA (dashes) , and NRIA (short dashes) interactions . 
EMP based on PH t-matrix, LDA on PH g-matrix, and NRIA on PH t-matrix 
( the distorted waves are generated with the PH g-matrix) . 
316 
7.2.3 318 MeV 
Below 300 MeV the PH free interaction was used fo r the effective in-
teraction analysis since it provided somewhat bett er fits , as measured by x~, 
than the FL t-matrix (e.g. [Kel 896]) . At 318 MeV, however , due to the very 
large amplitude of the Ret~0 component at large momentum transfer , the fit s 
based upon the PH t -matrix yield unstable results which oscillate between 
two or three solutions ( J. J. Kelly, private communication) unless some of 
the parameters are constrained . The FL free interaction, on the oth er hand 
,, 
.,., 
has been shown to give stable fits at 318 MeV [Kel 906], and is also used in ,. 
..,,.
.
..
 
  
this work . 
Above 300 MeV the most natural thing is to compare the effective 
, 
interaction to the LR interaction which is specifically designed for energi es -
above the pion threshold and which , for energi es above 400 MeV, is indeed 
the only available interaction . However, for the sake of consistency with the 
analyses at 100 and 200 MeV, we will compare the empirical interaction for 
318 Me V also with the corresponding PH theoretical interaction . Finally, 
., 
, 
elastic and inelastic scattering calculations with the empirical interaction are ) 
compared to LDA and NRIA calculations which , like for 100 and 200 MeV, 
are based on the PH interaction . In the paper of Kelly et al. [Ke] 90b] 
scattering calculations with the empiri cal interaction are compared to LD A 
calculati ons based on the LR interaction , and to NRIA calculations based 0 11 
the FL free interaction. 
The fitted parameters for th e 318 MeV interactions are li sted in Ta-
ble 7.3. The 160 data for this fit are from [Kel 90b], while the data for 4?Ca. 
317 
are from [Fel 90] . We see 111 the table that the scale factors of the central 
components become equal to unity or even slightly larger than unity, while 5 3 
retains approximately the same value as for the lower energies . Figure 7 .18 
shows a comparison of our empirical interaction with the LR medium modifi-
cations modeled upon the FL t-matrix, while Figure 7.19 shows a comparison 
with the PH interaction at 318 MeV. In contrast to the lower energies, it ap-
pears that the real parts of both the EMP central and the EMP spin-orbit 
interaction show more density dependence than either the corresponding PH 
or LR components . In the table this stronger density dependence is reflected 
by the larger values for b1 and b3 . For the spin-orbit interaction , this result 
should be taken with a grain of salt since , as the effective interaction grid for 
318 MeV in Appendix D shows, the.b3 parameter is not very well determined 
by the data at this energy. For the imaginary central component it seems 
that the density dependence is reversed compared to the theoretical interac-
tions . This is also reflected in the parameters where d2 < 0 and S2 - d2 > 1. 
The magnitude of the density dependence of the imaginary central compo-
nent for our empirical interaction is comparable to the PH interaction , but is 
much less than that of the LR interaction and is of opposite sign . While at 
low densities the empirical interaction is very similar to the LR interaction , 
at high densities it is much larger . On th e other hand , compared to th e PH 
interaction , the empirical interaction is larger at small 111 omentu111 tran sfers 
and smaller at large momentum transfers . 
In Figures 7.20- 7.24 we compare calculations based 0 11 the empiri cal in -
teractions EMP from the fits to 160 and 4?Ca to LDA and NRIA calcu lations 
318 
which are both based on the PH interaction . Distorted waves for the NRIA 
calculations were generated with the PH g-matrix. The EMP description of 
the data is superior for all the states and all the observables. In particular , 
we want t o point out the analyzing powers at small momentum transfers 
which are overes timated by both the LDA and the NRIA . For the 160 elastic 
cross sections we observe that although the first maximum is described well 
by the empirical interaction, for m omentum transfers above about 1.5 fm - 1 
the cross section calculation appears to be shifted outward a little bit too 
much. For 4?Ca this shift does not occur until about 2.2 fm - 1 . Finally, in 
Figures 7.25 and 7.26 we show the optical potentials for elastic scattering 
at this energy. The EMP potentials are based on the interactions fitt ed to 
16 0 and 4?Ca, repectively. The LDA and NRIA potentials are based on the 
PH g-matrix and PH t-matrix, resp ect ively. The EMP real central potentials 
are much m ore repulsive in the nuclear center than either the LDA or the 
NRIA potentials. For 4?Ca the EMP potential assumes a positive value of 
about +11 MeV . In comparison, the NRIA potential has a strength of about 
- 13 Me Vin the center , while the LDA potential almost vanishes . The EMP 
imaginary central potentials are less damped than the corresponding LDA 
p otentials. 
319 
Retf Imtf Re,Ls [I m,cf50 0 0 ] 
,,, 
Data set S1 b1 S2 d2 S3 b3 tU) d4 a41 .. , 
f..'.  
160 ... 1.07 144.7 1.02 -0.07 0.83 7.14 FL 0.0 -1.58 ,, ,,,,  
40Ca 1.04 140.0 1.00 -0.01 0.78 5.25 FL ? 1 4 = 1.0fm -
16O + 4oca 1.07 142.2 1.00 -0.04 0.78 5.88 FL [a:-y,66] = [3320] .,  
PH-Theory0 [1.0] 80.5 [1.0] 0.04 [1.0] 1.03 PH ref. : [Kel 90b] ..,  
LR-Theory0 [1.0] 61.2 [1.0] 0.27 [l.O] 1.90 FL 
Table 7.3: Empirical effective interactions for 318 MeV protons ([Kel 906], 
and J. J. Kelly, private communication) . We use the following exponents .2  
[a,,Bo] and masses ?: [3310], ?1 = 2.0 fm - 1 (Retf0 ) ; [2210] (Imtf0 ) ; [3320], ,. 
? = 6.0 fm - 1 3 (Rercf5 ). Units: Si and d; (1) , b1 (MeV fm 3), b3 and a41 (MeV f 
fm 5 ) . Square brackets indicate that the parameter or component is fixed . ,. 
0 Values from Table 2.5 . ' 
320 
6 /+. 
g ,: : 
0 / + d 
0 +. 
:0  ,/:+C:l  
0 J ... N 
~Cf?  !.:.  ' I 
f'o ! / 8 
I 04- ,' __, 
I :: -
1 ,l.!?o 
I /4 + o 
, .-? + 0 
r ,' d ? 0 4 + o 
,.I .. + 0 
,'/ d + 0 
,..?, .. ? : ~ 
~ ': : ~ 0 
., If) 0 If) 0 ci ci ci ..; 0 I 
,-----.-----,------r-- --,-----,,--- --, ("l 
,,,...,.   
0 +? N 
0 + .,. .. '"'ii:: ,--.. .,. 
0 +Cl ,.. ,., 
0 + .." 'i:::. .. I ,? 
0 + d "'\,::: 
0 + 4 ' .. -:::;::: 8 . 
0 0 + ?? ; ,-::-;. . __, 
0 + 4 .. ~ -
0 + Cl ',-::,.;: 
0 O + ? 4 ., .... ~ O' 
u_o0,   0 + 4 '~ 
0 + ',~ ~_ , 
0 + 4 ' 
0 + ? '0-
00 ...... 44 ', ,~ 
?o ~ ?... ..4 ',\ ..... 
0 ? 0 
0 0 0 0 0 If) 0 
If) 0 If) 0 
0 0 
'?I  0 ~ '? 0 '? 
("l N N 
N N 
I I ("l 
\,4+ 0 ... ... 0 ? +   0 0 . 0 , I ?.+. 0 4+I .   0 0 0 
N t 
I ....  0 0 ,. 
I ?+ -o 
.. 0 I 
I <+ o 1 
4+ 0 8 
f 
.. / .. 0 .. 0 - ~ ~_ , / .. 0 ,,?? ?. ... O' 
u_o0,   ,? ?-~ . ??/ ? 
~ , ?'/ ,..1/ 
_,,,_ .:,,.-
0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 <C> co 0 
0 
N '~? 0  '? '? 
0 
I ~ '? N ""' <C> <O 0 ~ ~ ~ N I I I I ""' I 
(c WJ A~W ) 
Figure 7.18: Comparison of the EMP(1 6O) and LR intera ctions at 
1 
318 MeV for kF 0.6 fm - 1 (triangles, short dashes) , kF = 1.0 fm - (crosses, 
dashes ) , and kF 1.4 fm - 1 (diamonds , solid) . LR medium modifications a.re 
modeled upon the FL t-matrix . 
321 
.
0 ??
0 ....  . I  I 0 ,,,', 
0 ?? ,-;, 
0 ? ? 
0 + ? ,'/ 
0 ? ? ? 
0 + ? ,'/ .. 
+ ? ,-> ?o C\I 0 
+ ? 
+ ? ,--.?  ??"O" ' + ? 4+ 0 / I 0 0 + 0 ?? 0 
0 + ? I . 0 ..8..  
0 +? 
0 + 4 / 
0 o+?c:I / 
: ': +o 
 ,, , .. 
oe / ,_. .<..1 ~o!0,?   O" 
~f( ?O 4) 0 .'  ,.s_,0 .  -...,_ >-
b <l! 
,4' :+1' .( i 0 
? 'I 0 
4,+ 0 0 
., 0- ~ 
lO 0 lO ~ 
0 0 0 0 
I 
(") 
0 ? ? 
0 .? ? 0 + ?  0 + ? 0 ? 
0 ? ? 
0 + ? .. , 
0 + ? .., 
0 ?? 
o +? N ,., 
0 +. ,, 
O ?? 
0.?..4..   I .,.?.  -.. t. ., , ...8..  ? ~ __, ?I 
?o. . \, O" ,' 
?O 
~. , .. ..,  .... , ?<> ,;:,,. . ., ' 
4> ..._, ... -... _. " .. _ 
..~....  O?? "" -
0 
II') 
0 0 0 0 0 0 0 lO 0 
I{') 0 II') 0 0 -, 
I{') 0 I{') "" (") (") C\I (\J I{') 0 I I C\I C\I 
I I 
4+ o'. 
0 .. 
0 .. .. " ?? Q 
0O+..<.,  4+,o \ 0
0 .. ?i. 0 I , 
0 +? ... 0 
0 +4 
0 ?? : ~10 
0 +? N ? 
0 + : t..?. 
0 ? ., 
0 / ~ I 
I ?O 8 ,, ++ ?O . . ' ' / . .......  , 0 0+ cl ' / o?:4, ? O" 
u.,8  0 + ll- '?, 0 ? ... ' .. ,, 0 + 
0 
--: <d ?- ?? , 
0 
\.+ ,+4  ?? 0 4 ?._ 
0 ? ? ? 
0 \ + 4 ', 
0 -+ 4~ 0 
0 0 0 0 
0 0 0 0- 0 
0 0 0 0 0 0 0 0 0 0 0 
I{') 0 II') 0 I{') I{') 0- I-I') (\J C\I C\I I I "I"  <P co 0- I I ~ "" <O co 0 C\I I 
(cWJ N~W) 
6
Figure 7 .19: Comparison of the EMP(1 O) and PH interactions at 
0.6 1 kF = 1.0 1 318 MeV for kF fm - (triangles, short dashes), fm - (crosses, 
dashes ) , and kF 1.4 fm - 1 (diamonds , solid) . 
322 
,_ 
-..... N ,-... I 0. 
0. .._.s..  __, 
'--' 
0 
~ - O' 
.., 0 
i N 0 - I I 
I() 0 I() 
0 ~ 0 0 0 0 ~ 0 0 0 
I -I 
(") 
.. , 
+-- N N ,-... ,'".,  .. ::: .:-..- - - .-: - - - " ??-????? -~ __-_._-. ....::.- -- ---~?- I .. , 8 '" 
0. .._._.._ , 
'--' 
0 
~ :,.1 - O' 
" ' ?'  '--<t-
'? '~., -~ ,..  
.., 0 
i N I I I() 0 I() 0- ., - 0 0 0 0 0 0 ~ f 0 0 0 I I 
(") 
? 
/, "'?"' 
-I N I (".) ...... ,,-.. ;:-..:.''"' 'I -..  0. .--~ -- .:. -
._0__.,  .._.8.  ? __. , '
?? 
0 - '..
 
:~   O' 
~ "?~ -~ ' 
0 
0 i N I ~ I() 0 I() ~
0- 0- 0 0 0 0 0 I - I 
(.1s / qw) UP/ op f.v 
Figure 7.20 : Comparison of EMP (solid) , LDA (dashes) , and NRIA (short 
dashes) calculations for the 31, 2i , and 11 inelasti c states of 16 0 at 318 Me V ? 
The data a.re from [Ke! 906]. EMP based on FL t -rnatrix , LDA on PH g-
matrix , and NRIA on PH t-matrix ( the distorted waves a.re generated wit h 
the PH g-matrix) . 
323 
----------~-
N 
I 
..a...  
N 
Q I I 
0 0 0 
N 
???-???- -~ -- I 
-? _,_.-- a
.._.._..  , 
-' ,, 
Q I lO 0 U'l 0 ~ 0- 0 ci ci ci I I 
,--,~-~-r-~-.-.~--,-~~-~~~~--,M 
. __ .... 
- ~ -- - - - ?_,. . .. -~,<-~ 
+-- N -.:!' ? '-,'   I 0. , 
0. ..a...  ,-....., ,.  
0 
U) J 
Q I 
0 0- 
(.rs / qm) UP/ np 
Figure 7.21: Comparison of EMP (solid) , LDA (das hes) , and NRIA (sh ort 
dashes) calculations for the 4{, 2f , and Of inelastic states of 16 0 a.t 318 Me V. 
The data are from [Kel 90b] . EMP based on FL t-matrix, LOA on PH g-
matrix , and NRIA on PH t-matrix ( the distorted waves are generated with 
the PH g-matrix) . 
324 
--~ -':"'"---=- - -~ --"' N 
-I 
-..8..  
N 
... 0 I I 0 0 0 0.. . 
~
,_ ,,
'  
N ,., 
('".) , 
..--.. 
0. I ,'",  
- 8P. ....  <d u 
0.. . 
, 
~ 
., 
')' 0 , 
N 
0 0 0. 0. . I 0 0 ...; 
-- ------
N ''  
I 
..8..  
..... O' 
-..:..:..---- (-~-
L...,.~~....-_-J~-~-.-.-...:_.........L~~~..__L~~~_, 0 
N ., 
I I I T o lt') o lt') o 
0 0.. . 0 0 ...; 0 0 0 ... I I 
(.is / qw) UP/ np 
Figure 7.22: Comparison of EMP (solid) , LDA (dashes ), and NRIA 
4
( short dashes) calculations for the Of, 3~ , and 2t inelasti c states of ?Ca 
at 318 MeV . T he data are from [Fe) 90] . EMP based on FL t-matrix , LDA 
on PH g-matrix , and NRIA on PH t-matrix ( the distorted waves are gener-
a ted with the PH g-rnatrix) . 
325 
N 
I"' 
C") I 
'C' 
-A A .. __ .. ?------? -
8 
(0 
0 
0.. . '.!.: 
0 
0 
" "' 
0 
..: 
I'  ~ "ci'  ci ci 0... . I I I 0.. .. (".) 
N 
I 
8 
- ------- - - - - -'+-I 
O' 
0 
0 
0 "' ..: 
- "' "I'  0 "' 0 
0 
0 i I 
I 
0_ 0 0... . , 
_, 0 I 
0 0.. .. (".) .... 
N 
,-.. 
1-
ID I 
'C' 
-~ -
8 
A ..... O' 
(0 
0 
0.. . 
0 
0 
0 
"' "0'  ..... "' "I'  0 0 ..... 0 I I I 0 
0 0 i 0_ , _, 0 0_ , 
?v 
(.ts/ qur) op/ .op 
r.~:tre 7.23: Comparison of EM_P (:olid), L?A (dashes) , and N,RIA 
rt dashes) calculations for the 51 , 32 , and 3, rnelas\Jc states of ?Ca 
at 318 M eV. The data are from [Fel 90] . EMP base d on F? L t-matn?x , LDA 
PH g-matrix and NRIA on PH t-matrix ( the distorted waves are gener-
011 
ated w.i th the PH'  g-matrix) . 
326 
+- N 
0 ,-._ ,.... .._ 
0.. -I 
0.. ..s....  
'-' '-' 
a:l 
(..) 
... - Ci' 0 
0- IO 0 IO 0 0 0 0 I 
,, 
+- ., 
0 ?' 
,-._ 
0.. I ?1 
0.. s
'-' ...... 
 
a:l 
(..) 
~ 
0 I 
i N I 0- IO 0 IO 0 0 0- 0 0 0- 0 0 0 I -
' 
I 
C'1 
N 
+-
0 -----
,-._ I 
0.. s 
._0_.
...... 
..,  '--' ? 
0 Ci' 
~ - ' 
.. 0 i 0 IO 0 IO 0 0 
0 0 0 0- - 0 0 0 I I 
Mn / n ?v 
Figure 7.24: Comparison of EMP (solid) , LDA ( dashes) , and NRIA (sh ort 
dashes) elastic calculations for 160 and 4?Ca. at 318 Me V. The data f r er and 
Ay are from [Kel 90b] (1 6 0) and fr m [F I 90] (4?Ca); the data. for Qare from 
[Ble 88] . EMP based on FL t-mat ix, LD n PH g-nrntrix , and NRIA on 
PH t-matrix. 
327 
.---- .----.----~ --~ - ~CD 
> "' 
cl) 
0: ::s 
O:c-o ,___, 
OC'J 
~ II ... 
w 
(\J 
0 
0 ~ 0- "' 0 II) 0- II) "' N - 0 - N C\i 0 0 0 _; 0 0 0 0 0 0 I I I I I 
(N~W) s,O ay (J\aw) s,O Ulf 
., 
CD ..,, .  
"' .
,
,,.
 
.  
~ ~- -.1' 
----
(') ,.8_._.._.  , 
s... 
..::..~ .. (\J 
...... :.: ?.,  .. .... 
' I ' \ ' 
I 
' 0 
"' 0 1/) 0 1/) 0- 1-/) 00 1/) 0 I C\I I - ~ 0 II) 0 1/) C\I C\I (') I I I I I I I I "I'  
(J\aw) ::,fl ay (Aaw) ::,0 Ulf 
Figure 7.25: Optical potentials for elastic scattering of 318 MeV pro tons 
by 16 0 for the EMP (solid), LDA (dashes), and NRIA (short dashes) inter-
actions . EMP based on FL t-matrix , LDA on PH g-ma.trix , and NRIA 0 11 
PH t -matrix . 
328 
...._ _ 
---............................ _____ 
-..... -,, 
;,' 
,,,,,;"' 
.-, --:::-,""' 
.,,,,.,., .... 
.,,,,.,,"' 
/,--
0 
0 C\l '<I' <O co 0 C\l 0 
0 0 0 0 0 ...; ci ci ci 
I I I I I 
(J\~W) S'Jo (J\~W) otf S'Jo WJ 
\ <O \ " 
\ 
\. ~ 
" 
~-,... , ~ ~ ID 
I \ "'-
I 
/ ) '<I' ,......._ 
/ t E
I M .....  '. 
I 
?, ' -----' ' I.. 
? ?, ' ' N \ 
/; 
0 
1/) 0 L(') 0 1/) 0- 1-/) 0 1/) 0 1/) 0 1/) 0 1/) 0 I I - - (\I C\l M M ..,. I I I I I I I I I 
(J\~W) ::ifl  o'<J (J\~W) :,0 WJ 
I 
Figure 7 .26: Optical potentials for elastic scattering of 318 Me V protons by 
4?Ca for EMP (solid), LDA (dashes) , and NRIA (sh ort dashes ) interactions. 
EMP based on FL t-matrix , LDA on PH g-matrix , and NRIA on PH t -matrix . 
329 
7.2.4 500 M eV 
In this work we only want to discuss elastic scattering at 500 Me V. 
Inelastic scattering is discussed in detail in the paper by Flanders et al. 
[Fla 90] . Like in the case of 318 Me V, the empiri cal interaction is based 
upon the FL t-matrix. The data for 160 are from [Fla 90] , the 4?Ca data 
from [Set 85, Lis 89] . 
The parameters for the various fits to inelastic 16 0 and 4?Ca data are 
collected in Table 7.4. We show calculations for elastic scattering fo r the first 
effective interaction from Table 7.4 ( called "EI-1" in [Fla 90]) in Figure 7.27 
(soli<l curves) . In this figure we also show for comparison an LDA calculation 
based on the re parametrized LR interaction from Table 2.3 ( dashed curves) , 
and an NRIA calculation based on the FL t-matrix ( short dashes ). Th e 
improvements of the EMP predictions over the LDA and NRIA predictions 
are remarkable, especially for the spin-observables at low-q. These observ-
ables were in the past only correctly described by the relativistic models . In 
Figures 7.28 and 7.29 we display the EMP(EI-1) (solid) , LDA (dashes ), and 
NRIA ( short dashes) potentials , and additionally the IA2 potenti als (long 
dashes) . Th e EMP and IA2 central potentials are indeed very similar over 
the whole region of the nucleus. The imaginary central potentials are more 
absorptive than either the LDA or th e NRIA potentials . This trend wa.s 
already observed in the previous sect ion for 318 MeV, albeit much less prn-
nounced . 
We want to point out that the EMP elastic scattering calcula tions for 
4?Ca, which are shown in Figure 7.27 , were done with an interaction from 
330 
a fi t t o inelasti c 16 0 da ta . T he excellent result confirms both the consis-
t ency between elastic and inelastic scat tering, and t he A independence of 
the effective interaction . 
In Table 7.4 both the b1 and the b3 parameter are again larger than 
the corresponding LR parameter but, as in the case of 318 MeV, b3 also is 
not strongly determined by the data. Like fo r 318 MeV, d2 is larger than 
the theoretical parameter and has the opposite sign . We find that S2 -
d2 > 1 which leads to a reverse density dependence of the imaginary cent ral 
component of the interaction . This result confirms the corresponding findings 
a t 318 MeV . 
Finally, we want to mention that for the case of 16 0 we used a grid on 
the interaction parameters and found a second solution in the space of x~-
The grid is shown in Appendix D. We find that if S1 is restri cted to the 
vicinity of 0.88 , a slightly better value of X~ is obtained with a larger value of 
b1 , and values for the other parameters which are similar to the first solution . 
For a m ore detailed discussion of this , we refer to the paper of Flanders et 
al. [Fla 90] . 
331 
Retg0 Imtg0 ReT:0L S [Im,;
5
] 
Data set S1 b1 S2 d2 S3 b3 tU) d4 a41 
160 1.26 121.1 0.84 -0.28 0.72 4.65 FL 0.0 -1.92 
l6Qb 0.88 152.2 0.80 -0.30 0.73 5.14 FL ?4 = 1.0 rm - 1 
40Ca 0.82 236 .1 0.93 -0.18 0. 71 6.95 FL [o-y/36] = [3320] 
16Q+4oca 0 .87 199.5 0.88 -0.25 0.73 5.80 FL ref. : [Fla 90] 
Theory 0 [I.OJ 76 .0 [I.OJ 0.15 [I.OJ 1.34 FL 
Table 7.4: Empirical effective interactions for 500 Me V protons [Fla 90J . 
We use the following exponents [a,,B5J and masses ? : [3310], ? 1 = 2.0 fm - 1 
(Retg0); [2210J (Imtg0); [3320], ?,3 = 6.0 fm - 1 (Re,;5 ) . Units: Si and di 
(1), b1 (MeV fm3), b3 and a 41 (MeV fm 6 ) . Square brackets indicate that the 
para.meter or component is fixed . 
0 Values from Table 2.5; &Second solution. 
332 
I() 0 I() 0- 0 I() 0 I() 0 ci ci ci ci ci ci I I I -I 
r,,~_,.,--.-,..,..,,..,........,~~~...-,--~~~~~-- C'l 
~~::~.,~ 
I 
....s_..,.  , 
N L....~~.........JL.....,..~-'-'~~~.......,_~-~~0 
I I() 
0 0 0 0 ~ 
0 
0 ci ~ 0 
I -I 
,-,...,.....~,--.-~~-~-~~~~-- C') 
______: _-___- -+-....,,......,~-
,,,,. / -; - C\I ----
' 
..s...  
- O' 
0 
~ 
.........._~ ~_._.~ ~~.........c~ - ~ --'-~~~_Jo 
0 I 0 I() 0 I() 0 N 
0 0 0 0 0 0 0 
I -I 
Figure 7.27: Comparison of EMP(1 6O ) (solid) , LDA (dashes) , and NRlA 
(short dashes) elastic calculations for 16 0 and 4?Ca at 500 MeV. The <la.ta. 
for 16 0 are from [Fla 90], the data. for 4?Ca are from [Set 85, Lis 89] . EMP 
based on FL t-matrix , LDA on LR interracti on , and NRlA on FL t-matrix . 
333 
,----,,----,-----,-- ----,-- --.----.-----. (0 
I[) 
#' 
' 1'( 
"'?, 
0 
... co ..., ..., <D '? (") N - 0 N (") ..., 0 <D N 0 N 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 
I I I I I I I I 
(A aw) $10 8cf (N~W) $10 W,/ 
(0 
I[) 
..., 
,..__ 
..s(') .,_...  ,. , 
1-. 
N 
' \ 
\ 
?, ?-. 'I  
?, 0 
0 1../..)  0... . I[) 0 II') 0-0 I() 0- I-[) 0 I[) 0 I[) 0N N (") (') .., . I..[,).  0 (\! I l() I I I I I I I I I I I 
(/\aw} :,fl atJ (A~W) :,fl W,J 
Figure 7.28: Optical potentials for elastic scattering of 500 MeV pr tons 
by 160 for the EMP(1 6 0) (solid) , IA2 (long dashes) , LDA (dash es) , and 
NRIA (short dashes) interactions . EMP based on FL t-rnatrix , LDA on LR 
interaction , and NRIA on FL t-matrix . 
334 
<O 
fl l{) > 
-~ <I) --;;:~ ~ 
~ -.t' o.o 
~-,  -co o 
~ -
l{) r. s 
,' ) o....u  11 .. 
(") ,._.._.. , 
w ~ - J... 
0') N 
y 
0 
0 
- N (") -.t' II') <O r-- (") 0- ci 0 N  0 0 0 0 0 0 ci o ci ci 
I I I I I I 
I 
(N~W) S1fl 11..LJ 
(N~W) s,fl ac{ 
<O 
l{) 
-.t' -s 
(") ,._.._.. , 
/ J... 
I 
I N 
\ 
\ 
) I I 
0 
0 II') 0 II') 0 II') II') 0 II') l{) 
.... - If) 0 II') 0 
I
0 If) 0 I - -f) 0 II') 0 (") (") ..... ..... II') I - - C\l C\l I I I I I I I I I I C\l I I 
(A~W) :,11 11..LJ 
(N~W) :,fl ac{ 
Figure 7 .29: Op ti ca.I potentials for elastic scattering of 500 Me V pro tons 
by 4?Ca for the EMP(1 6 O) (solid) , IA2 (long dashes), LDA (dash es) , and 
NRIA (short dashes) interactions. EMP based on FL t-matrix, LDA on LR 
interaction , and NRIA on FL t-matrix. 
335 
7.2.5 Interior Sensitivity 
We investigated to a certain extent the energy dependence of the inte-
rior sensitivity to the effective interaction . The results of this analysis confirm 
and complement the ones from Figure 1.1 of Chapter 1. T he computer code 
LEA [Kel LEA] has the capability to eliminate the effect of the scat tering 
potential within a certain radius r. This "notch" test calculates on a grid 
of values for the "notch" radius r the scattering observables for an arbit rary 
inelastic state. Then it measures the difference between the calculati on and 
the data in terms of x~- We have done this tes t for the 31 surface state 
(Figure 7.30) and the 11 interior state (Figure 7.31) of 
16 0 . In these figures 
a small value of x~ signifies small sensitivity to the effective interaction . The 
31 state, as expected , is sensitive to the effective interaction onl y at the sur-
face and shows essentially no sensitivity at all inside a.bout 1.5 fm. At large 
radii we can discern the energy dependence of the interior sensitivity from 
the spread of the three calculations for 100 (dashes ), 200 (dot-dashes ), and 
318 MeV (solid) . The increase in x~ with increasing energy and the associ-
ated increasing sensitivity to the effective interaction is due to the increasing 
transparency of the nucleus as we go from 100 to 200 MeV. This "wind ow 
of visibility" is open up to about 500 Me V where the sensitivity decreases 
again due to the growing absorption. For the interior-peaked 11 state we see 
sensitivity at smaller radii for all the energies. T herefo re, t he 11 data are 
sensitive t o the high density properties of the effective interaction . 
336 
"? 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ I 
I 
\ I 
\ 
\ 
\ 
\ LO 
' ' C\) ,...--.,__ 
....._ ' ' 
....._ ~........ ' ', s 
""' ' ..4._-_1,. . ' ' C\) ' s..... 
'( 
\ 
\ 
\ 
\ 
I LO 
I I 
I, ~ 
I I 
I j 
~ 
LO 
0 
0 
0 0 
I.() 0 0 0 
N 0 
0 
N LO r-l 
\ 
2 
Figure 7.30: Energy depen cl cncc n I li e interior sensiti vity to the effective 
interaction for the 31 st a te of 16 0 : l fl [ I ~leV (dashes), 200 MeV (dot-das hes) , 
318 MeV (solid) . 
LO 
I tj< 
I 
I 
I 
I 
I 
\ I 
\ I I 
\ I 
\ I 
I 
\ I LO 
\ I C") 
'\ I \ 
'\ \ 
'\ \ 
I 
\ I 
\ I I 
I 
I 
I I 
I LO 
I I N ,,-... I 
I I 
I s 
I _4_-1_ f , 
I 
I N 
\ \ h 
\ 
\ 
\ ' ' 
" ' ' \ LO " \ "? ,-i 2\ \. 
\ 
\ '\ 
\ 
\ .--. \ 
I 
I \ 
I 
\ 
\ 
\ LO 
I 
I 0 
I 
I 
I 
I 
0 
0 0 0 
0 
0 0 lD 
0 0 0 0 0 lD ,...-i ,...-i 
0 LO 0 lD (\] 
"1'1 C") C0 N 
Figure 7.31: Energy dependence of the interior sensitivity to the effeclive 
mteraction for the 11 state of 160, JOO MeV (dashes) , 200 MeV (dot-dashes) 
318 MeV (solid). ' 
338 
7 .3 Inclusion of the Perey Effect 
In Section 2.7.2 we found that the NA wave function is damped in the 
nuclear interior by the Perey factor /PH due to the nonlocality of the 
original optical potential. This damping of the wave functi on is equivalent 
t o making an effective mass approximation . 
To t est sensitivity to the Perey effect [Per 62, Per 63], we multiplied all 
the scattering potentials by the same damping factor P(r) = A+ Bp(r) / p0 , 
where p0 = 0.16 fm - 1 . This damping factor is expected to be less than 
one in the high density interior and to approach unity in the low density 
surface region . For medium energies the Perey factor in the nuclear center 
is approximately 0.8 which yields A = 1.0 and B ~ - 0.2. Subsequently, we 
fitt ed the 100 and 200 MeV inelastic scattering data for 16 0 . However , as 
Tables 7.5 and 7.6 show, inclusion of the Perey effect does not qualitatively 
change the results which were obtained earlier without the P erey effect . In 
particular , the scale factors are unaffected . Indeed, if we constrain the scale 
factors to unity, and vary only the b-parameters , we get a fit with a very bad 
x~ (see Table 7.6) . Thus , it appears that inclusion of the Perey effect does 
not affect the consistency between elastic and inelastic scattering . 
Conversely, we also tried to fit the Perey parameters A and B directly, 
using the PH t-matrix and the usual fi ve inelas ti c states of 160. For 100 MeV 
we find A = 0.88 and B = 0.59 (x~ = 11 .7 ), while for 200 MeV we find 
A = 0. 76 and B = 0.28 (x~ = 7.3 ). These para.meters c rres pond to an 
interaction which is enhanced in the interior and suppressed in the surface 
relative to the LDA. This is in good qualitative agreement with our more 
339 
sophisticated empirical interaction analysis, but is in complete disagreement 
with the conventional Perey effect . 
340 
Retf0 Imtf0 Re-rLs 0 
Fit S1 bi S2 d2 S3 b3 tU) x~ 
No Pereya 0.86 41.8 0.69 [0 .55J 0.82 [3.7J PH 10.2 
P erey 0.85 39.2 0.68 [0.54] 0.84 6.6 PH 10.2 
Theorl [I.OJ 72.7 [I.OJ 0.55 [I.OJ 3.7 PH 
Table 7.5: Empirical effective interaction with Perey effect for 16 0 and 
100 MeV protons (A = 1.0, B = - 0.15). We use the following exponents 
[a,,.85] and masses ?: [3310], ?1 = 1.5 fm - 1 (Retf0 ); [2210J (Imtf ) . Units: 0
S; and d2 (1), bi (MeV fm 3 ), b3 (MeV fm 5 ) . Square brackets indicate that 
the parameter or component is fixed. 
avalues from Table 7.1; bValues from Table 2.5. 
Retf0 Imtf Re-rLs 0 0 
Fit S1 bi S2 d2 S3 b3 tU) x~ 
No Pereya 0.70 65 .5 0.77 0.14 0.84 2.03 PH 4.3 
Perey 0.68 84 .5 0.79 0.13 0.86 6.64 PH 4.4 
Perey [I.OJ 31.4 [I.OJ 0.52 [I.OJ 1.88 PH 25.9 
Theoryb [1.0] 84.0 [l.OJ 0.23 [1.0] 2.10 PH 
Table 7.6 : Empirical effective interaction with Perey effect for 16 0 and 
200 MeV protons (A = 1.0, B = - 0.20). We use the following exponent s 
[a,,.85] and masses ?: [3310], ?, = 1.5 fm - 1 (Retf0 ); [2210] (Imt~,) - Units: 
S; and d 3 52 (1), b1 (MeV fm ) , b3 (Me\/ fm ) . Square bra ckets indi cate that 
t he parameter or component is fix ed. 
avalues from Table 7.2; bValues from Table 2.5 . 
341 
7.4 Inclusion of Elastic Data 
We tested the effect of including elastic data expli citly in the fit of the 
effective interaction. Some results of this analysis for 100 and 200 Me V are 
shown in the following tables and figures . In Tables 7.7 and 7.8 we compare 
the parameters from fits to inelastic data only, from combined fits to elastic 
and inelastic data, and from fits to elastic data only. We also include the PH 
parameters for reference and convenience. 
First we consider the 100 Me V fits . In the case of 16 0 the inclusion 
of elastic scattering data renders constraining d2 unnecessary. The damping 
factor S2 - d2 increases by a factor of about 2.2. The ma.in effect on the other 
parameters is an increase in the scale factors and in b1 . We did also fits to 
elastic data alone and found that, relative to the fit to inelastic data, the 
damping factor is three times larger while b1 is smaller than for the inelastic 
fit. The damping factor for the elastic fit is 0.42 and is therefore very close 
to the PH value of 0.45. We gridded the b3 parameter and found it to be not 
very well determined by the data. The grid finds the minimum in x~ to be 
around 6.0 for elastic scattering alone; a similar grid for inelastic scattering 
alone yields a value close to the PH value of 3. 7. Therefore , we tried combined 
elastic and inelastic fits with both values for b3 . We see in the table that the 
values of the other parameters are indeed relatively insensitive t the choice 
of b3 . Inclusion of elastic data does not help th e fit to 
4?Cc1 or the combin ed fit 
to 16 0 and 4?Ca: we still get unstable fits for these cases , unless we constrain 
b1 . Figures 7.32- 7.34 show inelast ic and elastic scatterin g for the various 
interactions which exclude (solid) and include (dashes, long dashes ) elast ic 
342 
data. For the inelastic data no improvement of either interaction over the 
others can be discerned from inspection of the figures. For the elastic state, 
the combined fit to elastic and inelastic data tries to fill in some strength 
around 1.5 fm - 1 . Thus, although the parameters are slightly different for the 
various fits, the predictions for elastic and inelastic scattering are essentially 
indistinguishable. 
The differences between the various interactions are very small for the 
case for 200 MeV, as Table 7.8 and Figures 7.35- 7.37 show. For the fit to 
elastic data only, the best result is obtained when b1 is fixed to the inelastic 
value. The elastic fit yields a slight improvement in the cross section and 
gives the same analyzing power as the combined fit to elastic and inelastic 
data. 
The results for the fits including elastic scattering at 100 Me V, and 
the fact that the PH interaction describes elastic scattering better suggest 
that the self-consistency procedure might have problems at this energy. The 
dam ping factor ( S2 - d2 ) changes from 0 .14 for the inelastic fit to 0. 31 for 
the combined fit , and to 0.42 for the elastic fit . The PH value is 0.45 . It has 
been speculated that at 100 MeV second order terms in the (1 + p8/ 8p) pre-
scription might be necessary for the consistency between elastic and inelastic 
scattering. At 200 MeV, on the other hand , self-consistency works well with 
th e (1 + p8/ 8p) prescription. In the following , the empiri cal i11t.cra.ctio11 s 
which are used in calculations are always those from the fit s to the i11clasti c 
data. 
343 
Data set S1 
m 0.86 41.8 0.69 [0 .55] 0.82 [3 . 7] PH 0.14 
el+in 0.90 53 .6 0.87 0.56 0.98 [3 . 7] PH 0.31 
0.91 51.9 0.91 0 .59 0.96 [6 .0] PH 0.32 
el 0.90 40.2 0.82 0.39 0.92 [3 . 7] PH 0.43 
el 0.90 34 .3 0.81 0.39 0.86 [6 .0] PH 0.42 
Theorya [1.0] 72. 7 [1.0] 0.55 [1.0] 3. 7 PH 0.45 
Tahle 7.7: Empirical effective interactions for 100 MeV protons from fits 
to elastic and inelastic 16 0 data. We use the following exponents [cry,85] and 
masses? : [3310], ? 1 = l.5 fm - 1 (Retg0 ); [2210] (Imtg0 ); [3320], ? 3 = 6.0 fm - 1 
(ReTl 5 ). For ImTl 5 see Table 7.1. Units : Si and d, (1), b 11 (MeV fm: ) , b3 and 
a 541 (MeV fm ) . Square brackets indicate that the parameter or component 
is fixed . 
avalues from Table 2.5; bFit with b3 from grid on elastic data . 
Retg0 Imtg RerLS 0 0 
Data. set S1 b1 S2 d2 S3 b3 tU) S2 - d2 
Ill 0 .70 65.5 0.77 0.14 0.84 2.03 PH 0.63 
in+el 0.63 69.0 0.77 0.12 0.86 2.27 PH 0.65 
el 0.57 [65.5] 0.73 0.11 0.90 1.55 PH 0.62 
Theorya [1.0] 84.0 [1.0] 0 .23 [1.0] 2.10 PH 0.77 
Table 7.8: Empirical effective interactions for 200 MeV protons from fit s 
to elasti c and inelastic 16 0 data. We use the following expone11ts [a,,65] a 11d 
masses?: [3310], ? = 1.5 fm - 1 1 (Retf:0 ); [2210] (Imtf0 ); [3320], ?3 = 6.0 frn - 1 
(ReTl 5 ) . For ImTl5 see Table 7.2. Units: S, and d, (1), b, (MeV fm 3 ) , b3 and 
a 41 (Me V fm 5 ). Square brackets indicate that the para.m eter or component 
is fixed. 
avalues from Table 2.5 . 
344 
? ? ? 
-+-
? 
1
,....._.
-. N ,..__ _ 
I 
0.. 8 
.._0_..,  -
0 
~ - er 
0 
i N "I'  0- I() 0 I() 0 0 I 0- 0 0- 0- 0 0 0 I I (") 
. 
..... 
-+- N ,..__ 
# +- I 
N 
,-.... 8 
?o.. -
.._0_..,  - er 
0 
~ 
0 
N I() 0 I() 0 
- 0 
i I 0 
0 0 0- 0 - 0 0 0 I I 
(") 
. 
1- N ,..__ 
(") 
,-.... I 
0.. 8 
.._0_..,  -
0 - er ~ 
0 
i N I 0- I() 0 I() 0 0 00-  0 0- 0 0 0 I 
(.1s / qm) UP/ DP ?v 
Figure 7.32: Comparison of EMP s from fit to inelasti c data onl y (solid), 
and from com bined fit s to bot h elastic and inelasti c data.: b3 = 6.0 (clashes) 1 
and b3 = 3.7 (long dashes). Shown are t he 31, 21", and 11 inelasti c states of 
16 0 a t 100 MeV . The data are from this work . 
345 
N 
. - -I 0. E 
0. ,.._._.._ , 
'-' 
0 .... er 
~ 
.. 0 
j' "' 0 It) 0 \I) 0 I I 0- 0 0.. . 0 0- 0 0 0 I -I (') 
?.?   
? N -
? -~+ M I 
? -C\2 ,.._.E._..  _, 0. 
,._0__.,  .... er 
0 
~ 
0 
j' N It) 0 \I) 0 
0 I ~ 
0- 0 0 -0 0 0 0 I -I (') 
? 
??  
? +- N ? -'<t' -I 0. 
0. ,..
8
._._._.  ,.___, , 
0 er 
~ 
0 
j.' 
.. "' \I) 0 \I) 0 
0 I I 
0 
0 0. . 0- 0 0 0 0 I -I 
(.1s / qw) UP/ DP !.v 
Figure 7.33 : Comparison of EMPs from fit to inelasti c data onl y (soli d) , 
and from combi ned fits to both elasti c and inelastic data: b3 = 6.0 (dashes) , 
and b = 3. 7 (long dashes) . Shown are the 4j, 2j, and 0! inelastic states of 
3 
160 at 100 MeV . The data are from this work. 
346 
- O' 
0 
I() 0 
I() 0 d ... i 0 
0- 0"-' - 0 .
 . . ..; d d I I 
0 0 
,.. v  
Hn / IJ 
Figure 7.34: Comparison of EMPs from fit to inelastic data only (solid) 
au from combined fits to both elastic and inelastic datac b, = 6.0 ( dashes )' 
anOdd   b'  = 3. 7 ( long dash es ). Sh own are cak ulat1.0 ns for the ground stale of'  
16 at 100 MeV. The <la.ta. are {ro111 this work. 
347 
. 
I. ...-. Cll ,--... 
,--... 
0. I 
- 8 0. ..... 0 
~ - er 
---+-
.., 0 
i N I I ~ II) 0 II) 0 0 
0 0 0 0 0 0 0 ...; 
I I 
(") 
-1-- Cll 
? N ,--... ,--... 
?o. I 
-0. ..8...  0 -
~ er 
.., 0 
0 i 
N 
I I 0- II) 0 II) 0 7:, 0- 0 0 0 0 0 0 I -I 
(") 
1- Cll 
C"') ,-... ,......,_ 
I 
0.. 
- 8 0. ? ..... 0 
~ - er 
' 0 N 
0 i I 0 II) 0 II) 0 -0 0 0 0 0 0 0 I I 
(.1s / qw) UP/ DP lV 
Figure 7 .35: Compari son of EMPs from fit to in ela.sti c data onl y (solid) , 
and from a combined fit to both elastic and inelastic data (dashes) . Shown 
are the 31, 2't, and 11 inelast ic states of 
16 0 at 200 MeV . T he data are from 
this work . 
348 
N 
I 
.. .-. 8 
? .._.._.. , ... ??  
....?.  
? 
? I +<'> 
N .-8 __, 
'C' 
0. 
._0_.,  
? 0 ??  '.? "" ? ? 
0 
I() 
0 ~ 
I() 
i "I'  
0 0 ci 0 -I I 
0 0 0 0... 0- ...;  -- ....... 
N 
I 
? -8 .__, 
0 
I() 0 
0 _; 
I() 
7 0 0 
0.. . ~ 0 I  
r.v 
(.rs/ qw) UP/ op 
Figure 7.36: Comparison of EMPs from fit to inelasti c data only (solid) , 
and from a combined fit to both elastic and inelastic data (dashes) Shown 
are the 4f, 2+ and o+ inelastic states of 16 0 at 200 Me V . The data. are from 
3 3 
tl 11? 5 work . ' 
349 
+-- N 0 --I 0.. s 
...0__..,  ..... 
0 
~ - O' 
0 
0.. . I() 0 I() 0.. 0 0 0 . 
I I 
b 
. (') 
? ..  ? 
+-- N 0 -I 0.. s 
...0__..,  .
...._.._. , 
0 - O' ~ 
.. 0 Q i 0 I() 0 I() 0 
0 0 .0. . 0 0 0 0 
I -I 
'd.o / .o "v 
Figure 7.37: Comparison of EMPs from fit to inelasti c data onl y (solid) , 
and from a combined fit to both elastic and inelastic dat a. (dashes) . Shown 
are calculations for the ground state of 16 0 at 200 Me V. The data are from 
this work . 
350 
7.5 A Dependence of the Effective Interaction 
If the LDA prescription is to be considered sound, the empirical inter-
action must satisfy the following three conditions . The first condition is that 
the effective interaction must be capable of describing all the relevant data 
for a given target at a given energy. We have demonstrated in the previous 
sections in accordance with other work [Kel 89b, Kel 90a Kel 90b Fla 90] 
' ' ' J 
that we can indeed find interactions which describe all the states for partic-
ular nuclei at particular energies. As a second and more stringent condition, 
the effective interaction should be independent of the target nucleus in which 
the reaction occurs. In Figures 7.38- 7.52 we show for each energy and for 
each target calculations based on the interactions fitt ed to 160 (solid) and 
to 4?Ca (dashes) . The interactions from the combined fit s produce scatter-
ing results intermediate between the two single-target fits and are omitted 
to reduce clutter. We see in these figures that , although we extract numer-
ically different values 1 for the interact ion parameters from the two targets , 
the differences in the description of data are minute , especially for 200 and 
318 MeV. The minor differences at 100 MeV are probably related to our 
difficulty in extraction of a stable interaction for this energy. Our analysis 
extends the A independence of the effective interaction , previously shown for 
the case of 16 0 and 28Si at 180 MeV [Kel 90a], to energies between JOO and 
318 MeV and to a larger range of A. 
1 In general we found that lGO constrained the pmameters of the empirical interaction 
better than 4?Ca which is a la rger nucleus and is overall less transpa rent th a n oxyge n. 
351 
1/? ? ? 
1- ? --...... N ,-, 
0.. I 
0.. ..8...  
'-" 
0 
~ - O' 
r 
0 
i N I "I'  0 \() 0 \() 0 0 
0 0 0 0 0 0 0 
I -I 
(') 
? 
N 
,-, 
+ I 
,-, 8 
> 
~ -
0.. O' 
'-" 
0 -
~ 
0 
0 i 
N 
I \() 0 \() 
0- ~ ~  0- 0 0- 0 0 0 I -I 
(') 
. . 
. . ? N 
? ,-, 
I 
8 
0.. 
'-" -
0 
~ ..... O' 
0 
i N I ~ \() 0 \() 0 ~ 0 0 0 0 0 0 0 
I -I 
(.is/ qm) UP/ DP ?v 
Figure 7.38: Compari son of EMP(1 6 O) (solid) and EMP( 4?Ca) (dashes ). 
We show calculations for the 31, 2t , and 11 inelastic states of 160 at 
100 Me V . The data are from this work . 
352 
-.. 
N 
I 
? 8 
? .
._....... . 
?  1/ 
,? 
? ? 
--- 0 
., 0 II') 0 II') 
N I ~
j I .0.. . .... 
 0 0 0 I -I 
0- 0... . (") 
?.?   N \ 
? ? ...
? H ..
..  .. I ..8...  ? + r> 
,C\2 !' ._.... -... 
~ ? 
._0,.,  ?.?  
?? 0 ? ? ? 
? ~ ? 
0 
II') 0 
N II') 
0 
0 0 
j .I0.  . ...
 . 0 0 I 
0 0- 0 0... . 
? -- .-. 
?? ?  
? c;,- N . ,,, 
? +~ I -d' 
?? 
?/  ,-... 
 ~ / .. .._.8.....  .. 
_A__,  
0 
l ? ~ 
~ 
-+-
-== 0 
II') 0 
., 0 0 ... II') 0 
N I ~ 0 I 
0 j I 0.. . ... 0 0.. .. 0.. .. "v 
(.1s/qlll) up/ op 
40
~gure 7.39: Comparison of EMP(1'O) (solid) and EMP( Ca) (dashes) 
lO~ show calculations for the 4{, 2,i, and Oj? inelastic states of "O at 
Me V The data are from this work . 
353 
? 
---- . 
? ? 
? +- C\l 
C\1 ~ 
/ 0.. I 
.a ._p.,,  .-8 _., a:i 
u O' 
~ -
,., 0 
i N I I ~ I() 0 I() 0 0 0 0- 0 0 0 ~ 0 0 I -I 
M 
? -- . 
1/.?  ? 
? 1- C\l (;) ~ 
~ ~ 
0. I 
E: .
a:i -8 _., 
u 
... ??  - O' 0 ? 
? 
i. 
::,.. 
.. 0 - N 0 i- I 0 
I() 0 I() 
- ~ 0 0 0 0 0 ...; 0 0 0 I -I 
M 
.... 
+N C\l 
0 ~ 
~ 
0.. I 
._p.,,  -- .-8 _., 
? (0 u O' 
~ -
,., 
N .. 0 i I I 0- I() 0 I() 0 I 0- 0 ~ 0 - 0 0 0 I -I 
(.is / qw ) up/ .op ?v 
Figure 7 .40: Compari son of EMP(1 6 0) (solid) and EMP( 4?Ca) (dashes). 
We show calculations for the Oi , 31, and 2t inelasti c states of 4?Ca at 
100 MeV . T he dat a are from thi s work. 
354 
. 
? IM N C'? ,......_ 
.? ( ,......_ 0.. I 
0. -a ___. ___. co 
0 
0.. . O" 
0 
0 i 
N 
I "I'  ~ If) 0 If) ~ 
0 0 0 0 0 0 0 
I 
(") 
.... 
? IN N 
? I ,C....'?.._  
,......_ 
0.. I 
_0__..  
8 
_-__. co 
0 O" 
~ 
0 
i N I "I'  ~ If) 0 If) 0 
0 0 0.. . 0.. . ~ 0 ci 0 I 
(") 
. .. ? 
., ,> ,_ ? ? N ,l..{...)._  ,......_ 
? 0.. 
I 
_0__..  . a ___. 
co 
0 ? 
-
0.. . : 
O" 
0 
N 
i I 0 If) 0 If) 
0 0 
~ 
0 0.. . 0 ci ci 0 I 
(.1s / qw) OP/ DP ?v 
Figure 7.41: Comparison of EMP(1 6 0) (solid) and EMP( 4?Ca) (dash es) . 
4
We show calculations for the 5~, 32, and 3~ inelastic states of ?Ca at 
100 MeV. The data are from this work . 
355 
I 
..E --..- 
?r  ? ? 
??  ? +--' I 
? 0 E 
? ) 
? --0--
...... 
. ----
? /2 ._0_.,  
? 0 
/ ~ 
0 
I{) 0 
0 
I{) 
.. i 0.. . ci ci 
..; 
... ci I I 0 
0 0.. . "0.'. . ?v 
Hn / D 
:gure 7 .42: Comparison of EMP("O) ( solid) and EMP('?Ca) (dashes). 
16 
d e show calculations for the ground st ates of 0 and '?Ca at I 00 MeV . The 
ata are from this work . 
356 
C\I 
I 
a 
-----
N 
0 I I 
0 0 0 
C\I 
? +- ,......._ ,C...\.1._  
I 
0. 
0. 
---- -a 0 
~ - ----O' 
0 
i N 
., 
0 I I 0 l/') 0 l/') 0 0 0- 0 0 0 - ci ci ci I -I 
C'1 
. 
1/ 
1- C\I ,-.._ 
,(.."...)_  
I 
0. ...., ? 
0. ? a 
-0--- ~ - O' 
~ ??  
I, \ 
0 
i N 0 I 0 l/') 0 l/') ~ -0 0 0 0 ci 0 0 I -I 
(.1s / qm) OP/ DP 'v 
Fig u re 7 .43 : Comparison of EMP(1 6 0) (solid) and EMP( 4?Ca) (das hes) . 
16
We show calculations for the 31, 2{ , and 11 inelasti c states of 0 at 
200 MeV . T he data are from this work . 
357 
? 
N 
+-"' 0 .r: I ? 0. ??  -0. --8 0 er 
~ 
.., 0 
N 
i I I ~ In 0 In 0 0 
0.. . 0.. . 0 0.. . 0 0 0 ... I 
(") 
? N -I 
+-C\"1 ' 8 0. -
. ? -0. .... er ? 0 ? ? ? ~ 
0 
0 i 
N 
0 In 0 In 0 I 
0 0 0 0 ..; 0 0 0 ... I I 
(") 
----+--- --
N 
,, - -I 0. 8 
??  0. ? -- - -0 er ~ 
.., 0 N 
i I I 0 In 0 In 0 0
0. . . 0 0 0 0 0 0 I 
(.rs / qw) UP/ DP ?v 
4
Figure 7.44: Comparison of EMP(1 6 O) (solid) and EMP( ?Ca) (dashes) . 
16
We show calculations for the 4t, 2j , and 0! inelast ic states of 0 at 
200 Me V . The data are from this work . 
358 
C'l 
? 
+-- N C\1 -0. I 
-~ crj -8 
0 O' 
0.., . 
0 
M .. 
0 i 
N 
-I I I 
0 If) 0 If) ~ 
-0 0 0- 0 0- 0 - 0 0 0 I -I 
C'l 
,_ 
- N C'? -0. I 
8 
8 
crj -
0 O' 
~ 
0 
0 i 
N 
I ~ If) 0 If) "-0'  0- 0- ~  0 0 0 0 0 I -I 
C'l 
... .... 
? +-N N 0 -+- -0.. I 
- 8 ~ crj -
0 - O' ~ 
N .., .. 0 
I 0- If) 0 If) 0 0-I I  0 0 0 0 0 I -I 
(.ts / qw) UP/ op 'v 
4
Figure 7.45: Comparison of EMP(1 6 O) (solid) and EMP( ?Ca) (d ashes). 
4
We show calculations for the 0!, 3~, and 2t inelasti c states of ?Ca at 
200 Me V . The data are from this work . 
359 
? 
IM N ..--. 
.(."-'-).  
I 
0.. 
- 8 P. aj -
u O' 
? ~ 
? 
? 
.. 0 <'I 
i N \() 0 \() 0 0 I I I ~ 
0 0 0.. .. 0 0 0 0 0 I 
n 
( ? ? 
IN N 
("') ,-.. 
,-.. 
I 
0.. 
- -..8P. ...  aj 
..u,  O' 0 ? ? ? .. 
0 
N <'I 
0 i 0 \() 0 
\() 
I I ~ 
0 0.. .. 0 .0.. . ..; 0 0 0 I 
n 
,_ N 
l{) ..--. 
~ I 
0.. 
- -8 P. aj 
O' 
0..0  , ? ---
0 
i N <'I I I ~ \() 0 
\() 0 
0 0 0.. .. 0 0 0 0 0 0 I 
(.1s / qw) UP/ DP ?v 
4
Figure 7 .46: Comparison of EMP(16 0) (solid) and EMP( ?Ca) (dashes ). 
4
We show calculat ions for t he 5~, 3~, and 33 inelastic states of ?Ca at 
200 MeV . T he data are from t his work. 
360 
+-
+-- 0 N ? 0 ,---, -~ I Q. 
,_Q. 
.._Q_.,  ? _.. ..
8....  
? <ti '-' 
0 u 
~ 0.., . c,' 
0 
~ '? 0 ci ci 'c? ~ ~ '? 
0 0 
i ci ci 'c?:i  
I I -I 
b b 
7 ?? 
. ? +- N 
? -0 -----Q. I 
.._Q_.? , 
 ..8...  
<ti '-' 
u c,' 
0.., . 
0 
i "I'  0 0 0 Q 
0- 0 0- 0 'c?i  ci 'c?i  I -I 
(") 
N 
+--0 -I Q. 
..8....  
.._Q_.,  '-' 
0 c,' 
~ 
0 
Q i 0 
"0'  0- 0- 0 'c?i 
0 0 
 ci '0?  
I -I 
8n/ n 'v 
4
Figure 7.47: Comparison of EMP(1 6 0) (solid) and EMP( ?Ca) (dash es) . 
4
We show calculations for the ground states of 16 0 and ?Ca at 200 Me V. The 
data for <T and Ay are from this work , the data for Q are fr m [Ste 85] and 
P . Schwandt , private communication . 
361 
......  1/ 
+ 
----
1 N .....-. ,-... 
--::--- I 
0.. 
- ..a0.. ...  0 
~ O' 
~ 
.., 0 N 0 l() 0 lO 0 
... 0 i 
I I 
0 0 0- 0- 0- - 0 0 0 ..; I I 
M 
+- N ,-... 
C\2 
--::--- I 
0.. 
- a A -0 - O' ~ 
? 
0 
N .., l() lO 
I 0 0 ~ 
0 0 
i I 
0 0 0 0 0 
~ 0- - - - I -I M 
" 
N 
1- ,,......_ 
,(,.'.."..)._  I 
~ --8 -0.. 0 O' 
~ 
+ 
0 
N lO 0 
i l() 0 I 
0 ~ .. 0 0 0 0 0 -0 . 0 I I 
(Js / qw) up/ .op 'V 
4
Figure 7.48: Comparison of EMP(160) (solid) and EMP( ?Ca) (dashes ). 
16
We show calculations fo r the 3~ 2i , and 1~ inelas ti c states of 0 at 
1 
318 Me V . The data a.re from [Kel 90b] . 
362 
( ? 
+M N ,-... 
0 
,-... I 
-0.. -8 0.. ? ? 0 ? ct ~ ? 
? ? ..~.  
., 0 
N l{) j' I I 
0 ~ 
l{) 0 ~....  
0 0.. .. 0 0 0 c:i c:i I I 
(") 
+M N ,-... 
C\l 
,-... I 
-~ 8 0. -0 ct 
~ 
0 
N ., j' I I ~ l{) 0 
l{) 0 
0 _; 0 0 0 0 0 0 0 0 I I 
... (") 
N 
+- ,-... 
? "'1' ,-... 
? I 0.. 
- 8 0. ? ? -0 ? ct 
~ 
~ 0 
., l{) 0 
j' N I I 0 l{) 0 0 
0 0 0 0 _; 0 c:i 0 I -I 
(.IS / qru) UP/ DP 'v 
4
Fig ure 7 .49 : Comparison of EMP(1 6 O) (solid) and EMP( ?Ca) (da16shes). 
We show calculations for the 4i , 2j , and 0j inelastic states of 0 at 
318 MeV. T he data are from [Kel 90b] . 
363 
N ,......_ 
,......_ 
I 
0. 8 
...0__.,  ....._.._. , 
aj 
0 ~? - c:r ~ 
? 
t 
N ., 
0 
i I() 0 0 I I 0 I() 0 
0- 0 0 0 0 ci ci ci I -I (") 
<... 
~ 
1 - N ,......_ 
C"') 
"'.'"' I 
0. 8 
...0__.,  .-..__, 
a:s 
0 - c:r 0.. . -~ 
0 
.... . 0 i 
N 
I 0 I() 0 
I() 0 
0 0 0 0 0.. . - ci 6 ci I -I 
1c-.=_ _/,_,. 
-
c::::" N 
- I ~ 8
....._..  _. , 
~~ 
- c:r 
C: -
...__ 
~ c:::=-
(" 
.. 0 N 
I ~...  I() 0 ~ ~ I I 0 0.. . ... 0- ci ci 0 I I 
(.1s/ qur) UP/ DP f.'<f 
Figure 7 .50: Comparison of EMP(1 6 O) (solid) and EMP( 4?Ca) (dashes ). 
4
We show calculations for t he at , 3;, and 2i inelastic states of ?Ca at 
318 MeV. The data are from [Fel 90] . 
364 
L 
IM N ,......... 
,(..."...).. . 
I 
0.. a 
.._0_..,  .-.__, ro 
u c:r 
$ 
.. 0 
i I "I'  0 0 ...; '? 0 '? ~ 0 
0- 0 0 0 0 0 I -I C'l 
IN N ,......... 
,(.."...)._  
I 
0.. E
0. ...._.... 
 
..__, _, 
ro 
u er 
$ 
.. 0 
i I "I'  0 ~
0 0 0 0- ~ '?  0 0 '0?  0 0 I -
 
I 
C'l 
,_ N ,......... 
,l..{...)._  I 
0.. E 
.._0_.,  .-.__, 
ro 
u er 
0.. . 
(.is/ qw) UP/ DP 
4
Figure 7.51: Comparison of EMP(1 6 O) (solid) and EfvIP( ?Ca.) (dash es) . 
4
We show calculations for the 51, 32, and 33 inelastic states of ?Ca at 
318 MeV. The data are from [Fe! 90] . 
365 
l 
+- N ,......_ 
0 
I 
- ..s...  Ill 
u ... -O' 
~ 
0 
0.. . I() 0 I() 0 ci ci 0 ... 
I I 
0 
? ? + 
+- N ,......_ 
,0... ..._ 
I 
C. 
-C. ..s...  cd 
u ... -O' 
~ 
0 
j' "' 0 I() 0 I() 0 
.. 0 
I 
0 . 0.. . 0 0 ... ci ci ci ... I I 
(") 
\ 
N ,......_ 
+-
,0.... .._ I 
C. 
- ..
s...  
C. 
0 ... -O' 
~ 
0 
j' 
N 0 
0.. . 0.. . 0 .0. . 
0.. . I() 0 I() ~ 0 ci ci ... 
I I 
H.o / .o l.v 
Figure 7.52: Comparison of EMP(1 6 0) (solid) and EMP( 4?Ca) (dash es) . 
We show calculations for the ground states of 16 0 and 4?Ca at 318 Me V. Th e 
data for u and Ay are from [Kel 906] (1 6 0) , and from [Fel 90] (4?Ca) ; the 
data for Q are from [Ble 88] . 
366 
7 .6 Comparison with IA2 and Dirac Phenomenology 
In Figures 7.53 and 7.54 we compare the empirical interaction EMP 
from fits to 160 (solid) with the full IA2 (long dashes) , the relativistic NP 
(dashes), and the NRIA ( short dashes ) models for elastic scattering of 200 and 
318 Me V protons from 16 0 and 4?Ca. The corresponding optical potentials 
are displayed in Figures 7.55- 7.58. The EMP for 200 MeV is based on the 
PH t-matrix, while for 318 MeV the FL t-matrix was used . The NRIA 
calculations for both 200 and 318 Me V are based on the PH t-matrix . 
At 200 MeV the empirical interaction is superior in its description of the 
data. In particular, the IA2 produces cross sections which are too large. The 
reason for this can be seen in the figures for the 200 Me V optical potentials . 
If we compare the central optical potentials we notice that although the real 
parts of the EMP and IA2 compare quite well (both show the same repulsive 
behaviour), the EMP imaginary central potential is damped relative to the 
corresponding IA2 potential due to Pauli blocking which is still important 
at this energy. Thus, the deficiencies in the IA2 can be reduced by applying 
ad hoc Pauli blocking to the imaginary central potential. This was done by 
Murdock and Horowitz for a related RIA model [Mur 87] . 
At 318 MeV , this Pauli blocking effect should be less important due 
to its E - 1 dependence. Indeed , the empirical and IA2 imaginary central 
potentials are almost identical at this energy. Nevertheless, th e empirical 
m odel still provides a slightly better description of the data. Although th e 
160 calculation is shifted outward a little bit too much at large momentum 
transfers, the empirical description of the second and the third cross section 
367 
maximum of 4?Ca is much improved over the IA2 , for which predictions are 
too high . Also, the low-q analyzing powers are slightly better for the empirical 
interaction . 
The principal difference between the relativistic and nonrelativistic ap-
proaches to elastic scattering is the contribution of virtual N N pairs in the 
nuclear medium (see Chapter 2). In a Schrodinger equivalent formulati on , 
this contribution manifests itself as an effective density dependence which can 
be described as primarily a short -ranged repulsive interaction . The figures 
show that the the pair contributions to the spin-orbit and the imaginary cen-
tral potential are small (the differences between the NP and IA2 are small) . 
By contrast, the repulsive pair contdbution has a strong impact on the real 
central potential, a behaviour which is also exhibited by our effective inter-
action . 
Finally, we compare elastic scattering calculations with our effective 
interaction to calculations based on a recent phenomenological global Dirac 
optical model by Hama et al. [Harn 90]. A phenomenological Lorentz-scalar-
Lorentz-vector model is fitted to a large set of elastic scattering data for a 
wide variety of targets and energies , and the resulting fitted Dirac optical 
potential parameters bear both energy (E) and mass number (A) dependence. 
The targets which are used for this analysis a.re 4?Ca., 48 Ca. , 56Fe, 60 Ni , 90 Zr , 
and 208 Pb, i .e . the mass number ranges between 40 :'S A :'S 208 . The energy 
range covered by the data set is between GS and 1044 MeV. The authors 
show that their global optical potential interpolates quite well in E and A . 
An extrapolation is not recommended , however . Since the data set did not 
368 
include data for q > 3 fm - 1 , the authors do not expect their model to perform 
well for higher momentum transfers . 
Figures 7.59 and 7.60 show calculations for elastic scattering from 4?Ca 
for 100 , 200, and 318 Me V based on the empirical interactions EMP (solid) 
which were fitted to 4?Ca and the global Dirac op tical potential model DP 
(dashes) . Considering that both 100 and 318 MeV are interpolated energies , 
the excellent performance of the Dirac phenomenology is extremely remark-
a ble a t all the energies and _for all the observables. The only exception is 
the cross section at 200 MeV, an energy for which 4?Ca data [Ste 85] were 
available and included the fit . 
In Figures 7.61- 7.63 we compare the empirical optical potentials which 
are based on interactions fitt ed to inelastic 4?Ca (solid) and 16 0 (dashes ) 
data with the global optical potenti al (short dashes ) for the three energies. 
The parameters for the empirical interactions are those from Tables 7.1, 7.2, 
and 7.3. Since the spin-orbit potentials are small , we want to concentrate 
on the central potentials . At 318 MeV, where our empirical description of 
the data is m ost similar to the DP prediction, also the potentials are m os t 
similar. The DP imaginary central potential is about 17- 20% less absorptive 
than the empirical potentials, and the DP real central potential does not dis-
play the dip in the interior which is present for the empiri cal potentials . At 
200 Me V where th e qu ality of the data descripti on by th e empiri ca l interac-
ti on s is still comparable to the Dirac phenomenology, we find th at the (large) 
imaginary central potentials are almost identical, while the real central po-
t entials display differences whose impact on the scattering data a.re diffi cult 
369 
to judge . T he surface depression of the DP potential is more pronounced 
than for our empirical potentials. T hese, however , have a depression deeper 
m the interior whi ch is absent fo r the DP potential. Since elastic scat tering 
m some fashi on samples an average of the whole nucleus , it is conceivable 
that on the average the different real central potentials have the same effect . 
Comparing our empiri cal potentials to the DP potential is most interesting 
for 100 Me V , the energy where our model does very poorly and the DP does 
very well . We noti ce that our real central potenti als are between about 11 
and 22 % m ore attractive than the DP potenti al , but the main difference is 
undoubtedly to be found in the imaginary central potentials. The DP poten-
tial is much m ore absorptive within a radius of about 4 fm than the empirical 
potentials. For r < 4 fm the absorptive potential for the empirical interaction 
is only - 3 to - 4 Me V , whereas the DP potential assumes a depth of about 
- 11 MeV at the nuclear center . However , as we recall from Section 7.4, a t 
100 Me V the damping factor S2 - d2 for the interactions from inelastic fits is 
much smaller than the corresponding damping factor for interactions which 
include elastic scattering (Table 7. 7) . Potentials based on the interactions 
from the combined fit to elastic and inelastic data, or to elastic data alone, 
would be much m ore absorptive in the center . 
370 
+--0 0. 
._0_.,  
0 
~ 
0 
0 lO 0- ~ lO 0 lO ~ lO ~- 0 0 0 0 0 0 I I I I 
0 0 
+-- N 0 -I 0. 
._0_.,  .....s_.._.,  , 
a:S 
u ?? - c,' 0.. . 
0 
N 
i I ~ lO 0 lO ~ 0 0 0 0 0 0 0 0 
I -I 
.. (") 
N 
- -I ~ s 
._0_.,  ? .
._.._..,  
0 - c,' 
~ 
0 
i 0 lO 0 lO 0 
N 0 0 0 0- 0 0 0 ci I -I 
'/J..o I .o ?v 
Figure 7.53: Comparison of calculations for elasti c scattering at 200 MeV 
based on the empirical interaction EMP(1 6 0) (solid) , IA2 (long dashes) , NP 
(dashes), and NRIA (short dashes) . Both EMP and NRIA are based on the 
PH t -matrix . The data for u and Ay are from this work , the data for Q are 
from [Ste 85] and P . Schwandt , private communication. 
371 
~~~~(') 
--: --~ --...... ..................= -~::::? 
..... O" 
...._._~~..._,_~~~---1-.~~~_.__~~_.__,o 
0 I() 0 I() 0 
..... ci ci ci 
I I 
+- N 
0 ,-.. ,-.. 
0.. I s 
.._0_..,  ...._.._. , 
<d 
~ u ..... 
... O" 0 
0 
i "I'  0- I() 0 I() ~ 0 0 0 0- -0 ci ci ci I -I 
C') 
+-
,0- .. I 
0.. .s..._...  
.._0_.. 
_, 
, 
0 ..... O" 
~ 
0 
i 0 I() 0 I() 0 
"' 0 0 0 0 0- - ci ci c:i I -I 
Hn / n 'v 
Figure 7.54: Comparison of calculations for elastic scattering at 318 MeV 
based on the empirical interaction EMP(16 O) (solid), IA 2 (long dashes ), NP 
(dashes), and NRIA ( short dashes) . EMP is based on the FL , N RIA on the 
PH t-matrix. The data for <T and Ay are from [Fe! 90 , Kel 90b]; the data for 
Q are from [Ble 88] . 
372 
(0 
l() 
I 
fl '</' 
1/ ----
~ ., s C') ...... 
-~ - ----
s... 
N 
----- 0 
0- tO 0 tO 0 C') N 0 0 0 0 0 0 -0 0 I I 
(A;;}J'i) s,fl W/ 
tO 
\~\-\-.~ -,,., , 
\-. ~--
<-,. , .. ~ 
,?., ------? N ( ----,-.. -----~-~ , 
, ?::. .> ..: .. ::?- .. . . 
' ..... . .. ... .. -- \ 
' \-._ '-----''----'----'----"---'---'"---' 0 
0 l() 0 l() 0 l() 0 l() 0 l() 0 
I - ... C\l C\l M M I ... l() 0 l() 0 l() ... C\l C\l M M I I I I I I I I I I I I 
Figure 7.55: Comparison of the EMP(1 6 O) (solid) , full IA2 (long dashes), 
NP (dashes) , and NRIA (short dashes) optical potentials for 200 MeV elasti c 
scattering by 160. Both EMP and NRIA a.re based on th e PH interaction. 
373 
,----------.-----~---~ <:O 
ll) 
,? ,. 
, ? ,. 
( ( ; 
\ I 
~?.  ' N 
) / 
'----~__,___,__ -_-1. :. .__..J..._...J...__:..,__ 0 
0 C'J <:O <O 
c::i c::i c::i c::i -0 N7M0  N - 0 c::i c::i c::i I I I I 
rr.--.----.---~-~----- ~<:O 
~"~' - ll) ----' ?, ?,. 
\ , ~--,. .,  .. 
, . .__,_"-? :.:.-._,_._ 
,._ N 
~-~. . . 
.... 
? \ 
f 
{ 
0 \() 0 l-l) 0 00 I N 'C?\l  M '? '? 0 0 I 0 C\l C\l M I I I I I I I I '? I I 
Figure 7.56: Comparison of the EMP(1 60) (solid) , full IA2 (long dashes) , 
NP (dashes), and NRIA ( short dashes) optical potentials for 200 Me V elastic 
scattering by 4?Ca. Both EMP and NRIA are based on the PH interaction. 
374 
- - - - ~---- --- - -<O 
> lf) 
o:~V  
._t_i., c..o...  \ .. 
0 C'? .. 
~ II ?? ... 
N 
0 
0 l-() 0- l() 0 l() 0 l() M N 0 N C\i 0 0 0 - 0 0 0- 0 0- 0 I I I I I 
(Aaw) s,O ocf (Aaw) s,O W,J 
<O 
lf) 
\, ""' 
\~ 
/ ::' 
-
.s M 
 ' ._.._..._. , 
,',' I 
7 , ~ I N 
( .. -.. ..- -?-...... ' , \ ?? ...... 
??. .'  \ 1 
0 
l-() 0- ll') 0 ll') 0- l-() 0 0 lf) 0 l-f) 0 l() 0 lf) I N I N (\) M M I I I I I I I I I 
(A~W) :,0 W.J 
Figure 7.57: Comparison of the EMP(16 O) (solid) , full IA2 (long dashes ), 
NP (dashes ), and NRIA (short dashes) optical potentials for 318 MeV elastic 
scat tering by 160 . EMP is based on the FL , NRIA on th e PH interaction. 
375 
/. /' 
'(, ~\ '. ' -...--... 
N 
0 N -,j- <D a:, 0
0 0 0 6 0 .. . NC".! N 0 ... 0 0 0 0 
I I I I I I 
,,,-----,.- ---T'"-~-~-~----- <D 
',-:,l;'\\ N 
' ' \\ I i-_ 
I 
I 
I 
~-'----'----'---'--'----'---"--'----' 0 
0 I/) 0 I/) 0- U-'l 0 0 U'l 0- I../.).  0 I/) 0 U'l 0 I ')2 I N C\l (") (") ..,. I I I I I I I I I 
(A~W) :){) lli/ 
Figure 7.58: Comparison of the EMP(1 6 O) (solid) , fu11 IA2 (long dashes) , 
NP (dashes) , and NRIA (short dashes) optical potenti als for 318 MeV elasti c 
scattering by 4?Ca. EMP is based on the FL, NRIA on the PH interaction . 
376 
? 
I 
8 
.-.__, 
0 0 
I 
0 0 
> 
(l) 
I 
- ;:;E 
...0_,...,  .-8 .__, 0 
ro 0 
u N 
0.. , 
0 
i N II') 0 II') 0 0 I ~ 
0.. . 0 0 0 ci ci ci I -I 
("') 
... -- -
~ 
\ -.. - -?'' ?"' 
> N ,......_ 
(l) I 
;:;E 
. ..
8...  
0 ,
0 o' 
........ 
0 
i 0 II') 0 II') 0 0 
0 0.. . 0 ci 0 ci I -I 
H.o / .o ?v 
Figure 7.59: Comparison of EMP( 4?Ca) (solid) and Dirac phenomenology 
DP (dashes) calculations for r:, / r:,R and Ay) ? The data for 100 and 200 MeV 
are from this work, the data for 318 MeV are from [Fel 90]. 
377 
J 
(") 
> 
---------- <l) ::E 
C-lJ (") C\1 ..-...-. ----
I s 
4---i 
'--"' 
--- ....--l 
~ 
0 
0 l.() 0 l.() 0 
0 0 0 ....--l ....--l 
I 
(") 
C\1 ...-..-. ----
I s 
---,. 
> 4---i 
<l) '--"' 
::E ....--l 
- - - - - - ~ 0 
0 
' C\l 
0 
0 l.() 0 l.() 0 
0 0 0 ..---l ....--l 
I 
F igure 7.60: Comparison of EMP( 4?Ca) (solid) and Dirac phenomenology 
DP (dashes) calculations for Q. The 200 MeV data for Qare from [Ste 85] 
and P . Schwandt, private communication ; the 318 MeV Q data are from 
[Ble 88] . 
378 
- ~?~ ?.??: -- ?:-?? ?:?~?A-?-;? ?h O': ?~. ?.  .., ?.; ~-- ~;?:? ? ?:.?. ?,: - : ?~: ?: ? - ?'..':'::-~-,- ?,?,,?'," 
<0 
> l() 
,..__<ll 
0.. ::g 
O..o 
.._.., 0 "' 
<ll ...... -
/ U II ..8...  
~ w"' 
M .._.., 
 
r... 
C\l 
l()N~-~~-~~-o~~~~--~~ 0No 
0 l() 
ci ci 7? ci ci O 0 
I I I 
~ ~-- ... 
' .-..-. -... 
'?-'?.? 
' 
'- . ? ' .. \ 
' ' , ?- I ,.-. _, . I -, ._, ,I  
-- ' \ 
\ C\l 
--... ' ' 
-.\  ' ' 
I 
I 
~~-1...__.1._J.L~-1...~--'--~-'-__;.- 0 
0 l() 0.. I .. l..(.).  0 l() 0 l() 0 C\l <0 cc 0 C\l C\l C\l M M I "I'  I I .... .... 
I I I I I I I I 
6
Figure 7.61: Comparison of the EMP( 4?Ca.) (solid) , EMP(1 O) (dashes) , 
and DP ( short dashes) optical potentials for 4?Ca at 100 Me V. 
379 
<O 
_v>  I{) 
0.::; 
O. o '<I' 
-o 
aj C'll ----
U II _s,  
~ ("') wa . 
.._., 
s.... 
N 
---( 
' 
' ''  0 
0 C\I '<I' <D a:, ~ C\IN 0 
0 0 0 0 0 _; 0 ci ci 
I I I I I 
(Aaw) s,O Ulf (N~W) s,O <lcf 
<O 
" " ..... .. ..... . . ~ 
~ I{) ..... ..... ~ 
' \ ' ' '<I' I --- ' ....? -.... . / ' ... . ----' ("') ._s._,  ' . , 
s.... 
N 
' 'I  
/ 
/ 
0 ... 0 C\I ("') ?t' I{) <D ['- a:, 0, 0- 0 I{) 0I I I I I I I I I I I - I-{) 0 IO (\I (\I I I I I 
(Aaw) 30 <J cf (Aaw) i1 W.J 
6
Figure 7.62: Comparison of t he EMP( 4?Ca.) (solid) , EMP(1 O) (dashes) , 
and DP (short dashes ) optical potentials for 4?Ca at 200 MeV. 
380 
<O 
> ID 
<l) 
o;:::; 
::,... _ .,,,.,_ 
.. ~ a.. co st' 
-- .. ::-:-,. ,....._ 
?-> -(-l:l- -C--') -U II \ s
-? ~ n ,._.._... 
 
0. ., 
-- -:. ....-. --- w s... 0--
/' C\I 
.,.:.: 
' 
0 
0 C\I st' <O co 0 C\I 0 
0 0 0 ci 0 ci c-i 0 
I I I I -I 
(N~W) lli/ 
(A aw) sin ay S1n 
<O 
ID 
st' 
,....._ 
?- .... :-: ..s..... n ,_....  _. , 
_.---;. ' ' ' s... \ ' N 
\ 
' ,,,, 
/. 
' ' I 
' "" I 
0 
C\I 0... . co <O st' C\I 0 C\I st' 0 I() 0I I I ... . l..D..  0 lD 0 lD C\I Cll n n 0 "'I' 
I I I I I I I 
(Aaw) :,f1 ay (A~W) :,fl W,J 
Figure 7.63: Comparison of the EMP( 4?Ca) (soli<l) , EMP(1 6 O) (das hes ), 
and DP (short dashes) optical potentials for 4?Ca at 318 MeV. 
381 
-
7. 7 Energy Dependence of the Effective Interaction 
Finally, we want to address the last topic of this work, the energy de-
pendence of the effective interaction. The interaction parameters for energies 
between 100 and 500 MeV are collected in Table 7.9 . In Figures 7.64- 7.66 we 
compare our empirical interactions for 100, 200, and 318 Me V with both the 
density dependent PH and LR interactions . The figures show that the overall 
density dependence of the real central component for 100 and 200 Me V is 
smaller than the PH interaction and that the interaction is suppressed for low 
densities . Relative to the LR interaction at 200 MeV, we find that our EMP 
is still suppressed at low density, but retains similar density dependence. At 
318 MeV, the real central component is more density dependent than either 
interaction. With the exception of 100 MeV, the imaginary central EMP 
component is less density dependent than either the PH or LR interactions . 
For 100 and 200 MeV we see again the suppression at low densities . At 
318 MeV, however, the density dependence appears to flip around and is 
opposite to that from theory. This behaviour makes the imaginary central 
component more absorptive with increasing density which is in contrast to 
the lower energies . The effect is even larger at 500 Me V [Fla. 90] . These 
results may be associated with a new absorption mechanism , such as the 
opening of the pion channel. Finally, for the real spin-o rbit component we 
o bserve suppression at low density for all the energies. However , alth ough 
the overall density dependence is also reduced for 100 a11d 200 MeV, at 318 
and 500 MeV we observe an enhanced density dependence which is much 
larger than that of either the PH or LR interactions. 
382 
? 
In F igures 7.67 an d 7.68 we graphically display the energy dependence 
of the parameters. The PH (solid) and LR (dashes ) parameters are also given 
for comparison . One problem is that below 318 MeV we used the P H free 
interaction for the fit , while for 318 MeV and above we used the FL t-matrix. 
Figure 7.68 shows the difference between the two free interactions at the var-
ious energies. For 318 Me V the Retf0 component of th e PH interaction is 
much stronger at q = 2.0 fm - 1 than the corresponding FL component , thus 
m aking the b1 parameter mqdeled upon FL larger than it would be with th e 
PH free interaction . Therefore, the apparent discontinuity in the energy de-
pendence of this parameter at 318 Me V might be to a certain extent artificial. 
A fit with the PH free interaction at this energy had proved to be difficult 
and unstable in the past . Fits with the FL free interaction at all energies 
are planned or in progress. The real and imaginary central scale factors are 
multiplied with their respective free interaction at zero momentum transfe r 
to remove the impact of the different t-matrices . For the d2 parameter we 
see the smooth E - 1 behaviour which has been predicted from theory, but 
also the change of sign around 300 Me V which is not predicted by the the-
ory. An interpretation of the real spin-orbit results is more diffi cult and will 
need more work . It is possible, however , that large correlations between b3 
and other parameters confuse the results. For example, it is striking that 
a t 180 MeV the scale factor appears to be too higl1 while at the same t im e 
the b-parameter is too small. It was also fo und that b3 fo r 100 Me V and fo r 
energies 2 300 Me V is not a very strongly determined parameter , and th at 
for instance at 100 Me V both a value of 6.0 and 3. 7 sati sfi es the data. 
383 
--
Retf0 lmtf0 (elas .) (inel .) 
(MeV) S1 
100 0.86 41.8 0.69 0.55 0.82 3. 79 0.14 -0.23 
135a 0.84 56 .8 0.81 0.44 0.84 3.14 0.37 0.08 
180b 0. 75 62 .9 0.76 0.23 0.88 0.51 0.53 0.38 
200 0. 70 65.5 0. 77 0.14 0.84 2.03 0.63 0.54 
318c 1.07 144.7 1.02 -0.07 0.83 7.14 1.09 1.14 
1.26 121.1 0.84 -0.28 0. 72 4.65 1.12 1.31 
500d,e 0.88 152 .2 0.80 -0. 30 0.73 5.14 1.10 1.30 
Table 7.9: Interactions parameters for 160 at various energies. Both the 
interactions at 318 and 500 Me V are based on the FL t-matrix, while the 
interactions at the lower energies employ the free PH interaction . Mass 
parameters ? 1 = 1.5 fm - 1 (2 .0 fm - 1 ) were used for energies up to (above) 
200 MeV. Units: S; and d; (1) , b1 (MeV fm 3 6), b3 (MeV fm ) . 
aRef. [Kel 89b] ; bRef. [Kel 90a]; cRef. [Kel 90b]; dRef. [Fla 90] ; "Second 
solution . 
If it can be shown that the discontinuity in b1 is indeed due to the dif-
ferent free interactions and that the energy variation of b3 can be explained 
by correlations and/or insensitivity of the data to this parameter, the effec-
tive interaction could then be said to display a reasonably smooth energy 
dependence . 
384 
M 
.,.. 
0-,,0 
0 .. 
o ..o 
0 .. 
0 4<0 s:-
o .. <1) 
0 ... :::s 
0 ... C\l 
0 c-o -I- ----C'-1 ..8....  
fJ --
O" 
0 + 
0 0 \ ... .... 
'lit- ? .. 
00 , ......? . .. 
o I+ .. ; 
0 + '?' 0 
.. M r..  .. 
1: ?" 
1:: 
1\. ?."..  ->--- -<1) 
\',, .,O.oM; . :::s N 
0 \ '. (HI 0 , , o-M 
0 ------- \0. . 
0 I 
N \\,o.  ... . C\l ..8....  '. 
',o ... 
'-,o .,. fJ -
<,? .... O" 
,o .. 
\ '-o ,... 
, ??p .... 
,?~o+:.. 
\ 'o ..,. 
\ \ ?:. . 
\ ~ +? 
<I +? 
0 
0 0 0 0 0 0 0 0 0 0 0 0 0 
I() 0 I() 0- I() I() 0 I() 0 I() 0 I() C\l C\l I - C\l C\l (') (') I I I I 
->--- -
<I) 
:::s 
0 
0 0,.. 0 
0 \",._, 
0 '<.., 
0 ~ 
0 0 ~ ''? ~? 
" -t? 
''d-4 
'\?t," . . 
''?,,+?. +... (I 
\?,+ ? 
' + ? 
\ \ ,++ 4 . 
0 0 0 0 0 0 0 0 0 0 0 0 0 
I() 0 I-f) -0 I() I() 0- I() 0 I() 0 I() C\l C\l I C\l C\l (') (') I I I I 
(cWJ J\<~W) 
Figure 7.64: Comparison of the EMP(1 6 O) , PH , and LR interactions for 
1 
the Retg component; kF = 0.6 fm - 1 (triangles, short dashes) , kF = 1.0 fm -
0 
(crosses, dashes), and kF = 1.4 fm - 1 (diamonds, solid) . 
385 
0 + ..?  
o<:::~ co 
0 o+o +..- -:~: ~, C") I a 
0 + .. ~ ..... 
0 + 4 ',~ 
0 0 + .....- ,~ ----
0 + ? -~ 
0 + ? --~ 
0 + .. ' 
0 + .. '!\ 
0 + ? ._,\ 
0 + .... 
0 + .. '.\ 
0 ..... 
0 + ? 
0 + ... ! 
0 
(") 
,--._ 
> 
V 
~ 
N 
0 0 ,......_ 
'l? ~?-.  0 0 C\1 I C\1 8
\:,~ QM -- .....  
~,o"" 
'\.?.,0-?+4 
----
_;: .. 5 O' 
"'?, ~  ..+.  .. 
,'<l"O ? : <Cl 
' 'Q + .. 
<~o + + .. " 
\ b + "' 
<\ + ? 
0 
0 0 0 0 0 0 0 0 
0 I() I() 0
I ... . 
I() 0 I() 
N N 
I I 
,......_ 
> 
V 
~ 
0 
'11' 0 
W" ... 0 u 0 ~~ 
'(> \ +4 . -- +J \ '?-~: -::r: 
'o- ', ..... 
~ ?., + 0... 4 
0 ', + .. 
~ ' , + 
~ ?,. + 
\ .. ?? ?? + + ~ + 
\ + + 
+ . ~ 
I + . 
0 0 0 0 0 0 0 0 
0 t() t() 0... . t() 0 I() I N N 
I I 
Figure 7.65: Comparison of the EMP(1 6 O), PH , and LR interactions for 
the lm.tf component; kF = 0.6 fm - 1 (triangles, short dashes) , kF = 1.0 fm - 1 
0 
(crosses, dashes), and kF = 1.4 fm - 1 (diamonds, solid) . 
386 
? + \,'4 ? 0 0 \ -+ <I -.
-?~ .
? + 10 I ,,......_ . 
?.
...
    0 0 
---- ?{ 0 . 
0 
0 
0 
> 'I,:.. I 
Cl) 
:::E '' -~?o  I  > .. 0 Cl) I ::::;J -?- 0 ..?  0 + 0 0 N 
:' t.oo I ? + 0 
C...D...  1-
co I .. 0 
?+ 0 I 
.C__'),. , I -40 (-Tj ' I <>-+ 0 0 
40 
I I - o .._.
8._..,.  ., 
:J:: / .. ?  ~ ,'. . / -~ 0 0.. // , .. ' / * O: O" / 
...... ;.. / . .. ,./'/? 
,,, .. . , ,,,,,. .. .. ,,,,,,,. ..,:..--
, / 
.... :.:- . . ? , .. ;.--: 
,.,Y. 
,.,.-; ? 
0 
('? 
0 
',\ ? + ',\ 
?+ 0 
.. 0 
? +. 0 \ \ - 0 \\ ? + 0 - 0 ? +.  0 .. 0 ---- ',\ 0 ',\ ?? 0 > <+ 0 
Cl) ',\ ?? 0 
> 
Cl) \1 .. 0 .. o 
?? 0 
:::E ::::;J ',I .. 0 ',\ <>? 0 .. o N 
- 0 :1 .. 0 0 : -? O 0 .. 0 ?+ 0 0 ;,'  - 0 :, .. 0 ----I .C__\,l .. 0 ., ?o C\l ?+ 0 
40 8 ,:1 
:J:: .. . ,
.. ., :1 4t 0 .._.._..,.. , 4t 0 
,1/ ,1/ ?t- 0 
0.. ~ 
.,7
, ,/ _,? o O" 
j,   . ? j, ~ 
4""'-J.._-L--L--''---''----'---'---'---'--' 0 
0 0 0 "0 0 0 0 0 
0 0 0 
N ...,. ' CO O N sto <D CO 0 
I I I I N I 
0 
0
.._._.
 .,.. , 
0 0 0... ,. 0 0 0 0 0 0 0 C\I <O CO 0 ,to <e CO 0 
I I I I ,..... 
,..... N 
I 
Figure 7.66: Comparison of the EMP(1 6 O), PH, and LR interactions for 
1 
the Rett5 component; kF 0,6 fm - 1 (triangles, short dashes) , kF = 1.0 fm -
(crosses, dashes), and kF = 1-4 fm - 1 (diamonds , solid). 
387 
0 
0 
I (0 
? ? ? ? I 
I 
- I 00..  . ->-----I  
<I) 
I 
+ ,-.. .:_::? "' _
:s_ , 
.? J >. - > 8 tlO ? <I> 0 N 1-, ? <I) 0 ~., ..,  C 0 w en .D 
I 0 
a:, I'- <O "' ... M N 0 O> a:, I'-
0 0 0 
0 
0 
(0 
++-
+ ? I 
I 
I 
I O0 -:-;->-- -
I ... <I) 
> I ::::s I 11) I 
:::i; / >-. 
,,. / 0 tlO 0 
0 0 ff, N 1-, 0 0 <I) 
? -:., C 
0 w 
en" ' -0" ' 
... 0 
0 0 0 0 0 0 0 N 0 <O ... N 
I() 0 I() 0 C! 0 0 
M M N N "' ~ "' 0 0 0 0 I I 
0 
0 
Ill 
+ .. ? + I 
\ 
\ 
\ 00..  . ->---- -
\ <I) 
> \ ,,...._ .:_::_:s ? \ ('l _, 
Cl) 
l ' .? ~ ' >-. ' > 0 tlO 0 0 ? ff, 0 ' <I> N 1-, ? <I) 0 ~ 0 -_, C 0 
0 w en .D 
0 
.0, ., 0 .0,. , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N ~ a:, 0 Ill ~ N 
M M N N "' "' ~ ! 
16
Figure 7.67: E nergy dependence of the interaction parameters for 0 . 
100- 200 MeV (diamonds) wit h PH free interaction , 318 and 500 MeV 
(crosses) with FL free interaction. For comparison , the energy dependence 
of the PH (solid) and LR (dashes) interaction are also given. 
388 
0 
0 
"" 
0 
0 
., -> 
(1J 
!R o:::S ..., o-N 
Q) >, 
etc: 
I tlO 1-. 
'.'..!. .i,l : 8 (1J 
Q) i:: 
etc: CxJ 
0 
0 
0 
~ 
0 
0 
., -> 
(1J 
0 :::s 
~...,  O'-' N 
.? >, 
tlO 
I 
:,: 1-. 0 
UC,. (1J 
~ ~ c 
.? CxJ 
.,., 0 
0 I() 0 
(") N N ~ 
0 
0 
0 
"" 
/ 
/ 
/ 
/ 
/ 
I 
I 
\ 
I 
I 
~_.____.__~_.__~~-'-~-L-_._......J 0 
0 0 0 0 0 0 
0 co <O "" N 
Figure 7 .68: Difference of the PH and FL t-Matrices. Plotted is the ab-
1
solute value of the difference. In the first pan el, dashes denote q = 0.0 fn, - , 
solid q = 2.0 fm - 1 . In the second panel the difference is plotted for 
1
q = 0.0 fm - 1 , in the third panel for q = 2.0 fm - . 
389 
.. 
7.8 Outlook 
In order to resolve the question whether there is indeed a discontinuity 
in the energy dep endence of our model near 300 Me V, it would be helpful to 
complem en t our d a ta for 200 and 318 MeV by measurements a t an interme-
di a te energy, say 260 Me V. Also, t he empirical interaction analysis should 
be done with the FL t-m a trix fo r all the energies. An independent check 
for the a bsorption m echanism a bove 300 Me V would be for example ( e, e'p ) 
m easurem ents where the pro ton is generated deep inside the high-densi ty 
interior of the nucleus. 
M easurem ents which would be useful to extend the empiri cal effective 
interaction to higher energies include inelastic measurements above 500 Me V 
where data for the low-lying collective states are so far sparse . Measurements 
of the inelastic D a.(3 observables for energies below 300 Me V would provide a 
challenging t est for our present interac tions . 
To extend the m om entum transfer range of the effective interaction , the 
corresponding range for electroexcitation measurements and analyses should 
be extended . It should be interesting to see whether or not coupled channels 
effects become important at large mom entum transfers . The present analysis 
of the 4?Ca( e, e') data should be improved and extended to strong states such 
as the 11 sta t e a t 5.903 MeV. This also holds for the analysi s of a vail able 
electroexcita ti on data of oth er t a rge ts such a.s 28 Si . 
System a ti c fit s to the elastic scat tering data, which are abund ant fo r 
many of the interesting target s a t many energies, should help to illumin ate 
further the A and E dependence of the effective interaction. On t he theoret-
390 
ical side, tl1e density dependence of tl1e IA2 slwuld be clarified. To acl1ieve a 
clear separation of all the different contributions to tl1e density dependence, 
relativistic full-folding calculations wl1icl1 include medium modifications due 
to Pauli blocking, dispersion , isobars, etc ., are necessary. 
The nonlocal effects wliich where found for the empirical effective in-
teraction should stimulate tl1e development of a tl1eory of tl1e finite nucleus 
which predicts the necessary corrections to tl1e LDA . The (1 + p8 / 8p) pre-
scription slwuld be extended to low energies. In particular, tl1e role of second 
order terms slwuld be explored. Finally, a tl1eory wl1icl1 explains both tl1e 
increasing short-range repulsion and increasing absorption above 300 Me V 
sl10uld be developed . 
391 
8 Summary and Conclusions 
We have measured the cross sections and analyzing powers for many 
states of 160 and of the calcium isotopes 40 ?42 ?44 ?48 Ca at 100 and 200 Me V. We 
also made measurements at 318 Me V for 32 ?34 S and the calcium isotopes. The 
data for 16 0 and 4?Ca are the ones which are discussed in this work and are 
used for the study of effective interactions . Data for 160 at 318 Me V for these 
studies were taken from [Kel 90b]; the 500 MeV data for 160 are from [Fla 90], 
the ones for 4?Ca from [Set 85 , Lis 89] . We study effective interactions for 
natural-parity isoscalar transitions in self-conjugate nuclei within a local tp 
folding m odel. We assume that for this type of transition we have full and ac-
cura te knowledge of both the reaction mechanism and the nuclear structure. 
We believe that direct one-step processes which can be treated within the 
single-scattering approximation are the dominant reaction process and that 
channel coupling effects are negligible . All the nuclear structure information 
which is relevant for the extraction of an effective interaction is assumed to 
be accurately known from complementary electron scattering measurements. 
Charge symmetry ensures that for our self-conjugate targets neutron and 
proton densities are very nearly equal. 
If proton scattering is to be used for nuclear structure studies as a 
probe complem entary to electron scatterin g, we must assure ourselves of th e 
accuracy of the effective interaction for pro ton-nucleus scattering . It has 
b een found that even at high energies medium modifi cations to th e effective 
interaction still remain important . Several theoreti cal models are available 
392 
-
which try to es timate these medium modifications . However, although these 
models all contain the basic ingredients of Pauli blocking and binding energy 
corrections and predict qualitatively similar medium effec ts , their quantita-
tive predictions differ substantially when implemented via the local density 
approximation (LDA) for nucleon-nucleus scattering. Since all of the mod-
els are based on different NN potentials and employ different approximations 
and numerical procedures to arrive at a local effective interaction , an estimate 
of their accuracy and a comparison of the individual models is very difficult . 
The accuracy of these theoretical models is inadequate for the use in nuclear 
structure studies where we want to minimize the residual uncertainties in the 
transition densities due to un cert ainties in the reaction mechanism and the 
effective interaction . 
We have, therefore, developed a phenomenology which is guided by 
theory and which can be used to fit various theoretical models , as well as 
data. With a physically motivated analytical form which involves only six free 
parameters , this empirical effective interaction not only facilitates the task of 
comparing theories to each other , but also allows a quantitative comparison 
of the effective interaction parameters fitted to data with those from theory. 
Finally, the empirical interaction proves to be much superior in its description 
of the data , making it an accurate tool for nuclear structure studies. Th e 
phenom enology also permits study of th e validity of th e LDA prescription 
in a manner which is largely independent of th e accuracy of nuclear matter 
calculations . If the LDA is to be considered sound , the empirical interaction 
will satisfy three criteria. First , a unique interaction must be capable of 
393 
J ltJT\I nr ..... ? 
describing all relevant data for a given t arget and energy. Second , a more 
stringent condition is that the effective interaction be independent of t arget . 
Third , the param eters should vary smoothly with energy and hop efully be 
comparable with the p arameters from theory. 
In this work we have dem onstrated for the case of both surface and 
' 
interior sta t es of 160 and 4?Ca that we can indeed find a. unique interaction 
which describes all the states of a. particular target at a given energy. Elastic 
scattering can be either expl~citly included in the fit or just predicted based 
on the effective interaction from inelastic scattering . Either way, we find no 
significant differences in the quality of the description of elastic scattering for 
energi es above 100 Me V . Our work shows, therefore , that proper treatment 
of the rearrangem ent contribution ensures consistency between elastic and 
inelastic scattering . We furtherm ore established the A independence of the 
empirical effective interaction by dem onstrating that , although the numerical 
values of the parameters from fits to different targets might vary slightly, fits 
to different data sets nevertheless represent the same interaction . Data for 
16
0 are described nearly as well by the interaction fitted to 4?Ca. data, and 
data for 4?Ca are described well by the interaction from 16 0 . Thus , we find 
that the effective interaction , although it is strongly dependent on the local 
density, is largely insensitive to the target. The LDA appears to provide a 
reasonable and accurate m odel for the density dependence of the effecti ve 
interaction in nuclei . 
As we go from 135 to 100 MeV, the ability of the empirlcal effective 
interaction t o accurately describe th e data, deteriorates rapidly. This appli es 
394 
especially to analyzing power data at 100 Me V which look dramatically differ-
ent from the ones above 135 MeV . While the analyzing powers above 135 MeV 
have essentially the same features , albeit sharper and more pronounced for 
the higher energies, most of the structure in the 100 Me V analyzing powers 
has vanished or is very suppressed . The fact that at 100 Me V the PH inter-
action is superior to our empirical effective interaction in the description of 
elastic scattering, and is nearly as good for inelastic scattering, suggests that 
the consistency requirement between elastic and inelastic scattering might 
have broken down . One possible source that could cause such a breakdown 
is the rearrangement contribution which might require second order terms at 
this energy in order to describe the density dependence adequately. Another 
p ossibility is that coupled channel effects might become significant at this 
energy. Finally, interior contributions might be stronger than at higher en-
ergies . To clarify all these problems, further investigations at lower energies 
are needed . 
Our empirical effective interaction differs from theoretical interactions 
m several important ways. Between 100 and 200 Me V we find that the in-
teraction at low densities is suppressed relative to the free interaction- the 
effective interaction for finite nuclei never heals to the free interaction, even 
at the surface . Second, the subsequent density dependence of the empirical 
interaction is substantially smaller than predicted by nuclear matter calcu-
lations. This means that modifications of the interaction are less effective in 
the high-density interior of a finite nucleus than expected for infinite nuclear 
matter of the same density. These results have been interpreted as a nonlocal 
395 
density dependence beyond the LDA which is characteristic of finite nuclei. 
Evidently, the nonlocal effect is due to the finit e spread of the nucleon wave 
functions . Nevertheless, the consist ency of the empirical interaction suggests 
that a local approximation to the effective interaction for finit e nuclei is still 
possible and reasonable . 
For 318 MeV we find that the nature of the medium modifications to 
the effective interaction appear to change . At this energy the damping fac-
tor applied to the imaginary central component of the interaction is smaller 
in magnitude and of opposite sign than predicted by theory. In contrast to 
the b ehaviour at lower energies, where Pauli blocking is responsible for the 
damping of the absorption, the phenomenology suggests that for energies near 
318 M eV absorption is enhanced . A similar effect with greater magnitude 
has b een reported for 500 Me V [Fla 90]. Therefore, a new absorption mech-
anism may b e important for energies above 300 MeV which is not present in 
the existing theories . This new mechanism may be associated with the pion 
threshold . The second observation is that the density dependence of the real 
central interaction is much stronger than theoretical predictions. This is also 
in contrast t o the case at lower energies . Finally, although the density depen-
dence of the spin-orbit interaction is much more difficult to determine above 
200 M e V , the fits suggest enhanced density dependence above 300 Me V . 
For all the energies that we have examined , the energy dependence 
of the Pauli blocking parameter appears to be consistent with the E - 1 be-
havio ur which is predicted by a simple phase space argument and which is 
also supported by more sophisticated nuclear matter calculations . Sin ce we 
396 
II N J\I f\r ...... --
interpret the slight discontinuities in t he energy dependence of some of t he 
o ther p aram eters to be m ainly due to the use of very different free inter-
actions in the fit we believe that t he effective interaction varies reasonably 
) 
smoothly with energy. 
Finally, we have compared our results with the rela tivistic IA2 model for 
elastic scattering , as well as with a recent global optical potential from Dirac 
phenom enology (DP) . We found tha t the Dirac phenomenology provides an 
exceptionally good descript~on fo r our elasti c data and that , therefore, we 
almos t can consider it as a benchmark for optical potentials generated by 
o ur empiri cal interactions . At 200 and 318 Me V, where our empirical results 
are of n early the same quality as the DP results , we also find that our em-
piri cal p o tentials are very similar to the ones from Dirac phenomenology. At 
100 MeV , where our empirical interaction performs poorly, we find significant 
differences in the imaginary central potentials and slight differences in the 
real central p o tentials . This comparison will give new impetus to reexamine 
the 100 MeV interaction. 
We find that the optical potentials which emerge from our phenomeno-
logical analysis are very similar to the Schrodinger equivalent potentials from 
the relativistic IA2. The contribution of virtual N N pairs , which distin-
guishes the relativistic approach from the nonrelativisti c impulse a.pprox1-
m a tion , is evid ently simula ted by an effective density dependence wh ich is 
compara ble to that of our phenomenology. It wo uld pro bably be very prof-
it a ble and instructive, therefore, to ca.s t relativistic calculations into a fo rm 
similar to our phenom enology so that theoreti cal predi ctions of th e density 
397 
-
dependent parameters could be quantitatively compared to their fitted val-
ues . New relativistic calculations probably should also try to elucidate the 
role of full-folding and additionally incorporate medium effects such as Pauli 
blocking and binding energy corrections . 
398 
-
A Nuclear Densities 
The nuclear structure information needed for the extraction of an effec-
tive interaction from proton scattering data within a tp folding model comes 
from electron scattering. In the plane-wave Born approximation (PWBA) 
the unpolarized cross section for the electroexcitation of a nucleus from a 
state of angular momentum J; to a state of angular momentum 11 can be 
written in terms of longitudinal and transverse contributions as [DeF 66] 
upwsA( q) 2 2 7JC7MottZ [:~IFL(q)l
+ [~ l:~I + 2 tan i] IFT(q)l2] , (A.1) 
where q 2 and qi are the squares of the three- and four-momentum transfers , 
respectively, Z is the atomic number of the target nucleus, and 0 is the 
scattering angle. The cross section for a point-charge , O'Mott, is given by 
a ] 2 cos2 (0/2) 
O'Mott = [ (A.2) 2Ep sin 4 ( 0 / 2) ' 
where Ep is the incident energy and a 1s the fine structure constant. T he 
density of final states factor is 
k 1 - k; cos 0 l-i 
= l + ET ' (A .3) 1J [ 
where k1 (k;) is the exit (incident) elec tron momentum and Er is the total 
energy, including the rest mass , of the recoil nucleus . Factors of !i and c have 
been suppressed. All the structure information is contained in the longitud i-
nal and transverse form factors FL(q) and FT(q) , respect ively. The simplest 
399 
class of t ransitions for whi ch the relevant structure information can be com-
ple tely determined from elect ron scattering consist s of those natural-parity 
isoscalar transitions of self-conjugate nuclei whose transverse form fact ors 
are negligible . In the PWIA the Jth multi pole Ff( q) of the longitudinal 
form fact or is related to the Fourier-Bessel transform of the transition charge 
density PJ( r) by 
(A .4) 
where x = ?2x + 1. The results of electromagnetic structure analyses are 
oft en presented in terms of rnultipole moments of the transition density. The 
transition amplitude for a CJ transition is directly proportional to the matrix 
elem ent 
(A .5) 
where k = 0 ( k = 2) for J > O ( J = 0). For the ground state one recovers 
with 41r M.x=o = Z the nuclear charge. Finally, the transition radius is the 
rms radius for a specific excitation 
(A .6) 
The plane-wave approximation , Eq . (A .1) , does not account for the disto-
ti on of the incoming and outgoing waves by th e Coulomb field of th e target 
nucleus. For the actual analysis of the data, the di stortions due to the spl1 er-
ical component of the ground-st ate charge distribution are included in t he 
distort ed-wave Born approximation (DWBA) [But 86] . 
Ground state as well as transition charge densiti es are usually expressed 
400 
as a linear series of the form 
N 
pi(r) = L anfni(r) , (A .7) 
n = l 
where the radial basis functions fn1(r) in principle can be drawn from any 
convenient comple te se t. Most oft en used in the analysis of electron scattering 
data, however , are certain variations of the Fourier-Bessel expansion , e. g. 
(FBE) 
(A .8) 
These basis fun ctions are all defined t o vanish for r > R, the cutoff radius . 
An alternative expansion, used for the 160 11 state , is the Laguerre-Gaussian 
expansion (LGE) 
(A .9) 
where x = r / b and L~ ( z ) is a generalized Laguerre polynomial of the form 
La( z ) _ ~ (- )m f(a + n + 1) zm 
n - ~o f ( n - m + 1 )f( a + m + (A .10) 1) m! 
The expansion is complete for any value for the harmonic oscillator parameter 
b. For a thorough discussion of the properties and the application of the 
various available expansions we refer t o [Kel 88] . 
In the following we tabulate (Tables A .1 and A.2) and graphically dis-
play (Figures A .1- A.3) th e ground stat e and transition charge densities used 
in our analysis . P oint -nucleon densities were obtained from th e charge densi-
ti es by num erically unfolding the nucleon form factor f(q) = [1 + (q / A)2J-2, 
where A = 4 .33 fm - 1 . Source for th e ground state density of 160 is [Lah 82], 
for 4?Ca it is [Emr 83] . Source for the transition charge densities of 160 
401 
is [But 86). 
Reference for tl1e transition charge densities of the ot state of 
4oc 
a is [Har 84), for all the otlier states [Mis 83). It sl10uld be pointed out 
th
at the ele c t ron scattering analysis of tl1e 40 Ca data 1? s cons1? d ere d to be  st1? 11 
Preliminary (J ? J ? K elly, . . . ) private commumcat10n . 
402 
.. 
I 
- 4t 2t o+ 3 
-J7r 
1-
n o+ 3- 2+ 1 1 1 
1 
8.0 8.0 
R,b 8.0 1.8 
8.0 
-'----- 8.0 8.0 E FB E FB E FB 
exp E FB E FB E FBE LG 
~ ~ 26.944 50.4 71 
-8.974 
619 .386 
a 1 202 .38 99 .542 73 .808 
119.461 -82 .744 
136.488 69 .711 
a2 447.93 303.073 158.471 -147 .279 
75.343 95 .774 
a3 335 .33 402.446 108.520 
-57.730 
7.200 -100.967 
2.520 43 .729 
a4 35.030 293.928 -0.544 
-58. 751 21. 704 
-55 .323 2.033 
13.465 
as -122.93 125.227 83 .182 
1.693 3.114 
-35.584 
aa -103.29 32.413 -34.005 -9.535 
-1.839 -1.504 
13.435 
a7 -34 .036 4.905 -5 .508 
0.620 -4.283 
-0. 713 
1.143 0.582 aa -4.1627 -0.450 
-0.254 1.488 
0.376 
ag -9.4435 -0.229 -0.181 
0.107 -0.556 
0.248 
a10 -2.5771 0.210 0.021 
-0.047 0.221 
-0.952 
all 2.3759 -0.125 -0 .023 -0 .092 3.769 0.021 
a12 -1.0603 0.067 0.019 
a13 0.41480 1.843 1.062 
.0195 20 .5 
lvl., ., -0 0.6367 14.20 2.791 
16.48 4.48 
3.98 I 
I- R-tr 2.737 4.049 I 4.026 st1an'adb li ? A ?. ' Expansion coefli ci en ts ( without uncertain ti es ) for ground st e . rans11t ion ell d ?t ? ?r I6Q [L l 82 But 86] . The coefficients n.re 
n un ? arge enSJ res o a, , . . . 
Ie?fmJ~! of 10- 4 fm - ' . R, b and R" are in units of fin ; MJJ " n? units of 
, see Eq. (A .S). 
403 
I 
0.25461 
1.5917 
3.450 
Table A.2: Expansion coeffi cients (without un certainties) for ground st,at e 
and transition charge densities of 4?Ca fE mr 83, Mis 83, Har 84}. Th e coef-
fici ents are in units of 10 - 4 fm - 3 . R, band Rir a.re in units of fm ; llfJ.J is in 
units of e-fm J+k, see Eq. ( A .5 ). 
404 
r-
+.-. 
0 
+ .... ,,......_ 
0 v 
0 Q) 
(1) 
...(_1_) , -ro- u 
<-C>o .
0., . 
(\j 
,-, 
0 
0 
-d' N 
a:) <O 
0... . 00 a:, <O N ,-, ?s8d 
(/llJ/ ~) 001 X 
16
Grnund stale charge densities {or 0 [Lah 82] and '?Ca 
p?i gure A.I: 
[E mr 83] . 
405 
(0 
- +M I.. ... 0 <D 
--:---- -Q) 
V v 
.._v_?,  
c-o0 
-0 
 ~ 
C\l 
0 
C'l 
C\l I 
I 
0 l() .... 
0 
It') 
I 
sutUJ.d 
X 
(cUlJ/ a) 001 (0 
+- I'-
C\l 
--:---- +
V -M C\l <D .._~_,  <l) .n -
coo -v s ""' .._...,.,  0 
~ C'l I-< 
C\l 
..-< 
0 
0 .... 
C'l C\l 
.... I 
l() -t' 
.., 0 0) a:, t-- st' C'l Cll .... .... 0, cO 
I'- <D 
ID It') 0 I suuJJ.d 
X 
(cUlJ/ a) 000 1 co 
,_ I'-
C'1 
--:---- +--' <D 
V 
.._(I
""' .n ,,
_) , -Q)  -s
..._  
~o ""' ..... 
C'l I-< 
C\l 
..-< 
0 
0 l() l() 
l() 0... . I 
l() 0 .... Cl) 
ll') l() l() 0 
(\j 0 l(') 0 C'l C'l 
C\l 
C\l lt'l 0 I 
suuJJ.d 
)( 
(cUlJ/ a) 00001 
SU'BJJ.d 
(clUJ/ ~) OOO T X 
86
Figure A. 2: Transition charge densities for "0 [Bui 1? 
406 
----
.... 
20 7 
.~..  
oq 
= I \40 40Ca {e, e ')3~ a t (\ Ca{e,e?)2t .., ........ : r 15 ~ M 0 ~ 1 1 5 (1) E 
> .... 
.".V ~ _ , -1 r (\ I i r I \ 1 4 10 ~ I \ i 3 
0 -2 
,r-;3  
0 
0- ~ I V f I \ j 2 "' - 3 1 5 ::l 
.~...- X 
0=  ..
"<.,:':   0 0 
E--
n Cl.. 
:::-- ~: t "'ca( e, e?Jo; j "' _J l -1 ~ - C z ,; j OQ 
(n -2 
:< 
0,,  
~ 
0 0.. l I I 3 [ I s: ~ (n 1 : t (\ ~ A l ::, =~...-.  (\ " ca(e,e?Js; 2 2 
(n ........ 
en M 
..6..  7 o' 
,; 
..,. '-V-- 6 0 .._ , 0 
0 
0 5 -l 0 
"' 0 ~ \ I 1 -1 0 4 -2 
~ -
en 3 -3 I- \ I -I - 2 X 
00 
w ., 2 <:: -4 L \ I l -3 
E 
0::: E-- l -5 
.., Cl.. Pl 0 -6 l V 40Ca(e ,e?)3i j 4-4 ~ V ?Ca(e,e?)33 
00 
~ -1 -7 -5 
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 
r (fm ) r (fm ) r (fm ) 
~ 
B Differential Recoil Diagrams 
Consider inelastic scattering in the laboratory frame as shown in Fig-
ure B .1. Four-vector conservation requires that 
(B .1) 
Using Lorentz invariance one can eliminate the unknown momentum of the 
recoiling (excited) target nucleus by 
(B.2) 
where Mis the target nucleon mass and X the excitation energy. After some 
algebraic manipulations one obtains with the auxiliary quantity A = E1 + M 
( the total energy of the system) ? 
(B .3) 
In this equation m, p1 and E1 are the mass, three-momentum and (total) 
energy of the incoming projectile, p3 and ? 3 the corresponding quantiti es 
of the scattered (i. e. outgoing) projectile, and 0 is the scattering angle in 
the laboratory frame . Expressing ? 3 in terms of m and p3 and introdu ci ng 
B = 2m 2 + E 1 M - M X - '~ , yields a quadratic equation for p3 
(A 2 2 - Pi cos 0)p; + ( - 2Bp1 cos0)p3 + (A 2m 2 B 2- ) = 0. (13 .4) 
'--v----' 
a C 
The solution of this equation is 
- b+ b2 - 4ac 
p3 = (B .5) 
2a 
408 
which yields the energy of the scattered projectile 
(B.6) 
If E 03 denotes the energy of any reference state and Eb3 the one of any other 
state ( e.g. a different excited state of the same nucleus or any excited state 
of any other nucleus) , we can calculate the differential recoil, defined as 
(B. 7) 
Figure B .2 shows the differential recoil (here expressed as Q value, Q = - ~) 
at 100 and 200 Me V, with the gound state of 160 as reference state, for the 
measured excited states of oxygen and beryllium . Since at some angles we 
also observed the lowest two states of carbon , they have been included as 
well. 
409 
s -~ 
w~  
'-" 
wrt'i' 
II 
 
'-" "Q"..  ,i< 
II + 
~ ~ 
II 
s 5:: 
Figure B.1: Kinematics of an inelasti c binary reaction. 
410 
> I.{) ' I 
V I.{) 
:::E / 
/ 0 I[) 
0 I 
0 / I[) 
C\] I , 'tj< ~ , , 
I 
I , , f ' , 0 'tj< 
I 
I I[) ,........_,_ 
,,! 
C') tlJ) 
/ ,1 ; 
,,',,1,' 0 QJ 
,' ,/ C") "d 
I , ,'1 ,' 
I.{) 
I 
I C\l .Q 
I I 0 j 
I I C\1 ? ' 
I I ' I.{) 
I ,' I 
,: I 
,,  ,,  , ., '  .0.. .. 
., , I I[) , ' . 
0 
?- ?(\]- '-.. h 0 0 
C') -J 
> 
V I 
:::E !? 
0 J I 
0 ?1 
0 
..... I <D 
-:---
0.. I 0 
1??, I.{) 
0.. I 
,.__, I ,?!/ 
<DO I / / 
/ / 1 
I 
I 
I 
I 
I 
I 0 
I I C") 
I 
I 
0 
I C\l 
I 
I 
.0.. .. 
o ..... N M -.::/' ID <0 ~ ~ m o ..... 0 C\l C":l -.::/' I.{) 
I I I I I I I I ..... ...... ...... ..... ..., ...... 
Figure B.2: Differential recoil fo r 100 and 200 MeV kin em ati cs. Refe rence 
state is the ground state of 160 . T he dashed cur ves denote 12C , th e dotted 
curves 9 Be. 
411 
C RAYTRACE Simulations of the K600 
We investigated the impact of the H- and K-coils on the 0tgt calib ra-
tion . The kinematic coils are used to compensate for the k appropriate to 
a particular reaction and to compensate for aberrations (important for large 
acceptances) . The polynomial for 0tgt which was given in Chapter 4 is based 
on a reaction with a specific value of k and corresponds to one specific set of 
H- and K-coil settings. For these studies we used an improved IUCF version 
of the RAYTRACE code originally developed by Spencer and Enge , and by 
Enge and Kowalski [Spe 67 , Eng 70] . The output was transformed , where 
necessary, to the focal plane coordinates ( x f , 0f ) . Figure C .1 shows the vari -
ation of rays with different 5's across the focal plane; the locations of the 
wire plane and the VDC windows are based on design values . Note that 
the D-coordinate system is a lefthanded system . In the following figures we 
show the effect of the kinematic coils on both the focal plane and on nuclear 
lines (k = 0) as observed in x101; we used the angles which are defin ed by 
the multislit in Chapter 4 as input for the angles at the target . Calcula-
tions were done for seven values of 5 between ? 4.65% (c .f. Figure C.l) . The 
strength of the kinematic coils is here represented as the ratio of the K-coil 
and dipole 2, rK = BK(0) / Bv(0) , and the ratio of the H-coil and dipole 1, 
rH = BH(0) / Bv(0) , respectively. The B(0) are th e fi eld strength s ex peri -
412 
enced by the center ray1 . The K-coil both shifts and, since rk is usually a 
few percents only, slightly deforms the focal plane ; the effect in x 10 I is a 
shift and a tilt of a nuclear line. Conversely, the K-coil can therefore be used 
to correct the shift of the focal plane and the associated tilt in x 101 caused 
by a reaction with k i- 0. The magnitude of the tilt is focal plane position 
dependent . The main effect of the H-coil is a slight rotation of the focal plane 
and a shift and bend of nuclear lines in x101 . The H-coil can therefore be 
used to correct the focal plane rotation due to ( x I 05) i- 0 and to straighten 
out aberrations due to (x I 02 ) i- 0. 
To estimate the effect of the kinematic coils on our calibration poly-
nomial for 0tgt, we consider the central ray (5 = 0) with xi = 670 .13 mm 
and 01 = 641.31 mrad (from RAYTRACE) . For this ray an H-coil variation 
of b.rH = 0.01 causes variations of b.x1 = 5.38 mm and 6.01 = 0.40 mrad, 
respectively. A K-coil variation of 6.rK = 0.01 causes variations of 6.x 1 = 
6.44 mm and 6.01 = 1.45 mrad, respectively. In Chapter 4, the measure-
ments for the gold target and the BeO target were taken with kinematic coil 
1 For the K-coil, we have with the effective length L(0) = 47.5 cm, the nominal field at 
the center B(0) = 5.66 GA - 1 , and the ratio Q / D = - 0.013 (ratio of the quadrupole and 
dipole components of the K-coil) 
JB (x, z )dz = B(0)L(0)[l + (Q/ D)x] ; (C. l) 
for the H-coil, with L(0) = 24.5 cm, B(0) = 4.70 GA - 1 4 , and H/ D = - 3.6 ? 10 - (ra.t.io of 
the haxapole and dipole components of the H-coi l) we have 
JB (x ,z )dz = B(0)L(0)[ l + (H/ D)x 2 ] (C.2) 
(P. Schwandt, IUCF, private communication). 
413 
setting that differed by ~ry = 0.01 and ~rK = 0.02. We fou nd that the 
reproducibility of 0tgt for gold was worse by about 2.4 mrad than for BeO 
when the polynomial based on the BeO dat a was used . T he effect on the 
angle at the target is evaluated for the central ray with 
(C.3) 
and the coefficients from Eq. ( 4.11 ) . For our case of gold and BeO we find tha t 
the H-coil makes a contribution of 0.39 mrad and the K-coil a contribution 
of 1.65 mrad to the value of 0tgt ; both together change 0tgt by 2.03 mrad . 
Considering that the H-coil does not vary much during a run period and 
that the K-coil setting depends strongly on k , this calculation shows that the 
angle inaccuracies for targets with different k are caused mainly by vari a tions 
in the kinematic coil settings. 
Finally, we want to present calibration polynomials for 0tgt and fo r 
5 (used for missing mass spectra ) which we found for rays generated with 
RAYTRACE . The computed rays sample a grid in rKrH where ry and rK 
vary betweeen ? 0.25 , include eight ( with the central ray ) angles at the target , 
and sample seven values of 5 between ? 4.65 % (we fit a total of 1524 rays ). 
For 0tgt and 5 we find 
+ 2 02 mrK X 1 + nrH 1, (C.4 ) 
and 
414 
O a + bx f + c0 f + dr K + er H 
+ fx} + g0} + hx101 + irKXf + jrHxf + krK0f + lrH0f 
+ mrHTK + nrKx} + orK0} + prKx/0! 
+ qrHx} + STH0} + trHx101 , (C.5) 
respectively. The values of the parameters are listed in Table C.1. The uncer-
tainties are estimated from the error matrix and reflect the large correlations 
among some of the coefficients . The 0tgt polynomial reproduces the angle at 
the target for the input rays to better than 0.17 mrad . In the case of the o 
polynomial, in the final fit some of the coefficients were held fixed to values 
from a previous fit . We find that this polynomial reproduces o for the input 
rays to about 0.03% (absolute value-5 has the units %). 
415 
Polyn . 0tgt 5 
Coeff. Value ?~ (%) Value ?~ (%) 
a -3.04985E+02 1.1 7.33331E+oo 3.7 
b -5.41538E- 03 34 .4 -l.09068E - 02 6.8 
C 4.99095E- 01 2.3 1.63160E- 03 -
d -1.30807E+02 1.9 4.86517E+0 1 -
e -5. 79378E+01 26.3 -3.88741E+Ol -
f -9.38733E- 06 7.3 -1.05095E- 06 52 .8 
g 5. 70454E- 05 17.7 -1.84326E- 06 22.4 
h -7.12595E-05 4.6 3.99000E- 07 -
1 -2.61264E - 02 16.2 1.88963E- 03 -
J -4.38560E - 03 20 .0 -2.52274E- 02 35.1 
k 1.90439E- Ol 1.8 -9.05960E- 02 3.3 
1 1.96344E- 01 24.7 1.68374E- 01 5.8 
m -1.92327E - 05 16. 7 -6.77000E- Ol -
11 -1.27212E-04 30.2 -3.18000E- 07 -
0 - - 2.60058E - 05 19.4 
p - - 1.26675E- 05 10.8 
q - - -3.82000E- 06 -
s - - -1.56582E- 04 9.9 
t - - 4. 7470] E - 05 29 .6 
Table C.1: Coeffi cient s for t he 0tgt and 5 polynomials from RAYTRA CE 
simulat ions. T he quoted uncertainti es for the coefficients a.re relat ive errors 
in units of %. Coeffi cient s without listed error were held fixed to values from 
another fi t . 
416 
0 
0.... .. 
-,}- ~ 
0 
Ill 
~ (rJ 0 
aJ 
I() 
.?co 
~ 
II 
'O 
~ 0 
.? ,,; co 
II 
'O 
I() ,' 
.?~- 
'O"  
o .? 0 
O' 
.?o 'St' 
II ,,--._ 
'O 
ll') 
~ --? 8 
.T .._0_ ,.. 
'O"  
0 N 
o ? N 
.. r' ?,,; 
I ,....., .-...,..-....-...,,,-...,.-... 
'O"  N.... . 0 O O ON -~? 0 0) N aJ ~ l'-.? ... ~
I .Il.l  ~ N  r .? 1-, ..--.. 'Sf' 0) ...... C\i l'- 6 _<I,)  ..... l'- 0) l'-0 
'O"  Iii / . 0 
+ P'. V ...... 0 ~ /,?, .? A s II 0 0) N /,' U".l 0 0 0 0 < ca. >< aJ aJ C') aJ co ?,.? 0 
_I-,,  s lO 'Sf' aJ ..... 6 
co 
N ""1' N 'Sf' 'St' 
..... 
C) 
+? V C) .._I_ ., ..__., .._I_ ., ..__., .__.. .__.. 
0. 
< r:n 
0 
U..."...l. 
0 
 c...o... 
0 
i::: N 0  I 
-~ II .I._I_l.,  Cl) Cl) f/l - 1-, I-. Cl) Cl! (\I Cl Cl ct IIl ct IIl i i 
0 
'St' 
I 
0 0 0 
C') N ..... 0 0... ... 0 0 0 0 C') U".l 
I I I N 'Sf' 
(w0) X 
F igure C .1: Design K600 medium dispersion focal plane from RAYTRACE 
calculations in the D-coordinate system of dipole 2. T he dott ed lines a.re th e 
front and rear window of t he front VDC, the dash ed line is th e focal plan e, 
and the solid line denotes the wire plan e. The rays represent trajectori es of 
specific 5 wit h respect to the central ray. 
417 
IO 
N 
0 
+ 
0 
N 
0 
+ 
0 
~ 0 
0 ..... 
+ 
3 
0 
+ 
1/) 
0 
0 
+ 
0 
0 
,._o 
/ 
/ / '? 0 ,-s-.._ / 0 ;;/ 0 l{)  
/ I 
/ 0 
/ 
N 
'N0?   
I 
0 
0 
l{) 
0 0 0 0... .. 0 -0 0 0 ol "'1' (") (\} (\} (") "'1' I I I 
(wo) x 
Figure C.2: Effect of the K-coil on the fo cal plane from RAYTRACE 
calculations in the D-coordinate system of dipole 2; the numbers represent 
the strength of the K-coil field in the center as fr action of the correspondin g 
dipole 2 field . The K-coil both shift s and deforms the fo cal plane . 
418 
0 
0 
C\l 
~ 
~---~---?---~ - --~---~------0 0 
\\ 0 0 
~ 
~ - - -<>- - ----0-- --+.. ---~- -- ..,_ -- ---0 
\\,? . ...  
0 
~---~-----o----~---~--- ? ------0 0 
,? .. co 
\., 
\ ?. 
---? - -- ~---~--,-..-.  --0------0------0 
\'? 
\'-,. ...  
?--- ?-- - -4- - - -4,)- -- -<>-- - ~- -- -<, ' ... 
,? 
\\ 
, ... 8 8 --0 
<)--- a 8 e ,\ 0 
0-- - - -~ - - --0- - - -~ - - --0- - - -0- - - --0 
\'- 0 ,..  'tj' 
,?....  . 
o-- - - .. - - -0- - - -~ - - -0-- - - -0-- - - -0 
0 
0 
C\l 
0 
0 0 0 0 0 0 0 
0 I.[) 0 I.[) 0 I.[) 0 
co ['- ['- c.o c.o I.[) I.[) 
Figure C.3: Effect of the K-coil on nuclear lines in x 10 f (fr nt) from 
RAYTRACE calculations. The dashed lines are without K-coil ; for the solid 
lines the K-coil strength is 5% of the dipole 2 strength . Th e di amond symbols 
represent the seven 0tgt angles defin ed by the multi slit from Chapte r 4. The 
K-coil shifts and tilts lines; the strength of the tilt varies with x 1 . 
419 
0 
cD 
0 
l() 
0 
C') 
0 
C\l 
u 
S N 
0 
0... ... 
0 
C\l 
I 
0 
C') 
0 l() 0 l() 0 I.{) I.{) 0 I.{) 0 I.{) 0 I.{) I 
C') C\l C\l ...... ...... 0 I ..... ..... C\l C\l C') C') 
I I I 
(ruo) X 
Figure C.4: Effect of the I-I -coil on the focal pla ne from RAY T RACE 
calculations in the D-coordinate sys tem of dipole 2; the numbers represent 
the strength of the I-I -coil field in the center as fr action of the corres ponding 
dipole 1 field . The main effect of the I-I -coil is a (slight) ro t a tion of the focal 
plane. 
420 
0 
0 
C\1 
,--i 
\ 
\ 
~-- - ~--- ?--- -o-;-- --~- - -~- - -~ 
'..\ 0 0 
??.\ 0 
~ 
??, \ 
<>- - - --?-- - --?- --~- -- .,. _ _ _ .., __ _ --0 
' . 
'?.\ 
0 
<r- - -<r - --<>- - - - ~.-- - ?- -- ?- ---(> 0 
\.\  co ~ 
_- ?, \ e ---., 
\\ 
- - - ? - - - ? - - - i '~ - - -<>- - - -(> - - - -(> 
'-. \ 
\\ 
~'.'> ?- - - ?- - - -4- - - .: - - --?- - - --0- - - -(> 
\\ 
?., \ 
\\ 
<r - - -o- - - --?- - - - ~ - - --?- - - ---0- - - 0 -(> 
, , 0 
tj< 
<r-- _.,.. __ -o- - - - ~ - - --?- - - --?- -- --0 
0 
0 
C\1 
0 
0 0 0 0 0 0 0 
0 I.[) 0 I.[) 0 I.[) 0 
co r-- r-- c.o c.o I.[) I.[) 
Figure C.5: Effect of t he H-coil on nu clear lines in x 101 (fro11 t) from R AY-
TRAC E calculations . T he dashed lin es are with ou t H-coil ; for t he solid lines 
the H-coil strength is 15% of th e dip ole 1 st rength . T he di amond symbols 
represent the seven 0tgt angles defin ed by t he multi slit from Chapter 4 . T he 
H-coil shifts and bends lines ; th e strengt h of the bend varies with x 1 . 
42 1 
D Effective Interaction Grids 
!'  !'  
422 
~--=========================-===============--===-==-====-------------------------------- --
I sl bl s2 d2 s3 b3 it chir chig/ std parg I 
~ ----====-====================== ==-=-========-=--====-=-------------- -------------- . ----
~- I 0 . 680 \ 11 ) 65.1 ( 1.9) 0.769 ( 9) 0 . 165 ( 10 ) 0.844(7 ) 1.67 ( 24 ) 4 .28 4 . 28 i:::r ~ \ 0. 500 83 . 2 ( 1.8 ) 0.803 ( 8 ) 0 . 180 ( 10 ) 0 . 885( 7) 2.08 ( 2 4) 9 5.95 0. 600 79.5(1.7 ) 0 . 795(9 ) 0.164 ( 10 ) 0 . 871(7 ) 1.95(25 ) 6 4 .53 4.15 0 . 6 44 
tj 0 . 700 64.4(1.8) 0 . 770(9) 0.169(10 ) 0.846(7 ) 1.96 ( 24) 3 4. 22 3.E-1 ( 6) 
0.800 33.6 ( 1.5 ) 0.751 ( 8) 0 . 257 ( 10 ) 0 . 785 ( 6 ) 2.50(17 ) 10 6.4 3 I 
I--' 
\ 0.690 ( 10 ) 60 . 0 0.767 ( 9 ) 0 . 181 ( 10 ) 0 . 846(6 ) 2.19(20 ) 3 4 .27 
...... 0 . 689 ( 10 ) 63.0 0.770 ( 9 ) 0 . 174 ( 10) 0.848(7 ) 2.07 ( 21 ) 3 4.22 4 . 11 78. 4 
= 0 . 688 ( 10 ) 66 . 0 0.773 ( 9 ) 0.168 ( 10 ) 0.850(7 ) .,..,.. 1.95(22 ) 3 4. 18 3.E-4 ( 2. 6 l 0 . 687 ( 10 ) 69 . 0 0.776 ( 9 ) 0.162 ( 10 ) 0.852 ( 7 ) 1.83(22 ) 3 4. 15 
('t) \ 
,; 
I" \ 0. 697 ( 9 ) 49.2 ( 1.7 ) 0 . 600 0. 100 ( 8 ) 0.839 ( 6 ) 0. 21 ( 20 ) 7 5 . 59 n 
;::-_ 0.694 ( 9 ) 59.3 ( 1.8 ) 0 . 700 0 . 137 ( 8 l 0.848 ( 6 ) 1.10 ( 22 ) 6 4. 5 0 4.14 0.801 
~ 0 .684 ( 10 ) 70 . 0 ( 1.9) 0.800 0. 177 ( 8) 0 . 850 ( 6 ) 2 . 23 ( 2 4) 4 4. 1 4 l . E- 2 ( 10) 
t,,j 0=  0 . 667 ( 11 ) 80 . 2 ( 2.0 ) 0 . 900 0.221 ( 9) 0 . 846 ( 7 ) w 4. 26 ( 25 ) 2 4 . 48 -\ 
()Q \ 0.706 ( 10 ) 75 . 5 ( 1.8 ) 0.718 ( 7 ) 0 . 050 0 . 845 ( 7 ) 1.62 (24) 6 4. 58 ::! . 0 . 693(10) 70 . 6(1.9 ) 0 . 755(7 ) 0 . 125 0 . 849(7) 1.83 ( 23 ) 6 4. 19 4.17 o .1 55 I 
0.. 0.684 ( 11 ) 63 . 7 ( 1.9 ) 0.788 ( 7 ) 0.200 0 . 851 ( 7 ) 1.86 ( 22 ) 3 4. 25 l . E- 2 l 9) 
0.682 ( 11 ) 50 . 7 ( 1.8 ) 0 . 805 (7) 0. 275 0.8 46 (6) 2. 12 ( 20 ) 7 4. 7 0 I 
0) 
0 I o . 712 ( 12 ) 87 . 9 ( 2.0 ) 0.923 ( 9 ) 0 . 180 ( 11 ) 0.700 7 . 17 ( 21 ) 7 5 . 7 4 
I 0. 697 ( 11 ) 72 . 2 ( 2 .0) 0 . 825 (9) 0.17 6( 10 ) 0.800 ,---._ 3 . 82 ( 21 ) 6 4 .32 4. 08 0.827 o .67 4 ( 10 ) 62 . 7 ( 1.8 ) 0 . 720 ( 9 ) 0.152 ( 10 ) t,,j 0 . 900 0.02 ( 1 8) 6 4.4 8 2 . E- 1 ( 6 l 
0 I0 .625 ( 10 ) 53.1 ( 1 . 6 ) 0.610 ( 8 ) 0.125 ( 9) 1 . 000 -3.37 ( 1 4) 8 7.29 I 0 -s: 0.702 ( 10 ) 75.2 ( 1.6 ) 0.756 ( 9 ) 0 . 132 ( 10 ) 0 . 881 (6) 0.00 4 4 .32 0.695 ( 1 0 ) 71.5 ( 1. 7 ) 0 . 766 ( 9 ) 0.1 49 ( 10 ) 0 . 865 ( 5 ) 1. 00 4 4 .19 4. 16 1. 60 
('t) 0 . 687 ( 11 ) 66. 7 ( 1.7 ) 0.774 ( 9 ) 0.167 ( 10 ) 0 . 849 ( 5 ) 
< 2.00 4 4. 17 3.E- 3 ( 23 ) -- 0.679 ( 11 ) 60.2 ( 1.8 ) 0.781 ( 9 ) 0 .188 ( 10 ) 0 . 831 ( 5 ) 3.00 4 4. 28 =======-=-====--=-=======---======-=== - - - - I -
... --- - --- .- ,.,_........... --- ... ,. __ 
======----==========-==-===== 
I sl bl s2 d2 s3 b3 it chir chig/std parg 
~ ======================--=====-==-===============--====---------=------------==== 
r-:::r I o.582 ( 13 ) 72.5(2.8) 0.895 (8) 0.198(13) 0.842(7) 5.63 (2 6 ) 9 .11 9 .11 ~ I 0.400 80.2(2.6) 0.880(8) 0.146(13) 0.843(6) 5.80(21) 6 9.96 0.500 71.7(2.9) 0.897(8 ) 0.198(13) 0.852(6) 6.19(24) 2 9.74 8.90 0.501 
0 0.600 71.5(2.6) 0 . 893 ( 8) 0.200(13) 0.839(6) 5.65 (2 5 ) 3 9.06 1.E+O ( 5) 
1" 0.700 -19.7 (1. 7) 0 . 833(7) 0.386(10) 0.759(5 ) 5.06(12) 10 13 . 51 -
...... I -0.582(12) 69.0 0.893 (3) 0.204(12) 0.842(6) 5.87 (2 3) 3 9.10 - 0.581(12) 71.0 
0.895(8) 0.201(12) 0.842(6) 5.85(24 ) 3 9 .11 9.10 68.3 
=: 0.581(12) 73 .0 0.897 (8) 0.197(12) 0.841(6) 5.84(24) 3 9 . 14 2.E-3 ( 1.3) 
.(.1.),  0.579(12) 75.0 0.899 (8) 0.194(12) 0.841(6) 5.87(24) 4 9.18 -I ---?-P-l n 0.607(10) 45 . 5 ( 2.2) 0 . 700 0.140(10) 0.857(5) ----------------1.35(20) 5 10.84 0.597(11) 58.5 ( 2.5) 0.800 0 . 170(10) 0.850(6 ) ~ 0-. 3 . 72(23) 8 9.43 9 .13 0.865 0.579(12) 74.1(2.8) 0.900 0 . 196(11) 0.841(6 ) - 5.93(26) 5 9.17 5.E-2 ( 7) tv 0.555(14) 87.7(3.0) 1. 000 0.236(11) 0.8 32 (6) 7 . 83(27) 9 10.29 ~ 
O..Q.,  -I -o.57 5 (13) 78.7(2.6) 0.898(7) 0.170 0.836(7 ) 6.3 2 (2 6 ) 6 9.36 
p... I 0.579(13) 74.9(2.5) 0.898(7) 0.190 0.840(7) 5 . 98(25) 5 9.19 9.06 0.228 o.580(13) 70.9(2.5) 0.897(7) 0.210 0.843(6) 5.57(25) 
... 2 9.08 5.E-3 ( 6) 0.580(12) 67.3(2 .5) 0.898(7) 0 . 230 0.844(6) 5.26 (24) 3 9.06 
0 -I -?--?--0 0.547(15) 121.5(3.0) 1.066(8 ) 0.143 ( 15) 0.700 12.07(29) 10 14.94 
-Pl 0.605(12) 70.4(2 . 7) 0.930(8) 0 . 215(13) 0 . 800 5.64(23) 3 9.24 8.89 0.852 0.579 ( 12) 54.3(2 . 3) 0.802(8) 0.211 ( 12) 0 .900 1.95(19) 5 9.83 5.E-1 ( 3) tv I o.532(11) 32.2(2.0) 0 .634(7) 0.190(11) 1.000 -3.22(12) 6 14.45 
0 
0 I 0.611(11) 79 .7(2.4) 0.878(8) 0.176(13) 0.860(6) 4.00 7 9.25 
'7 
,-:;a., 0. 59 5 ( 12 ) 77 . 0(2.5) 0.889(8) 0.186 ( 13) 0.850(6) 5.00 5 9.17 9.14 5.56 
(1) 0.577(12) 72.2(2.5) 0.898(8 ) 0.199 ( 13) 0.839(6) 6.00 3 9 .14 3.E-2 ( 26) 
..<__, 0.555(12) 66.7 ( 2.7) 0.905(8 ) 0 . 210 ( 12 ) 0.826(6) 7.00 2 9.24  ==-=======--= - - -- - - - ------------==-===================I  
=====-=================================-=======--=-=-==--====------------- . ------- ---
I sl bl s2 d2 s3 b3 it chir chig/ std parg I 
~ --=============--=======================--===-=========-------======---------------------------
Ill 1 1.000 144 . 7(1.6) 1.000 - 0 . 071 ( 10) 0.849(10) 6.81(23) 5.65 5.65 
-O" -I- -----------------------143.4(1.7) 0 . 989(10) - 0.033(13) 0.900 ( 11) 6.43(24) 0 6 6 . 61 ('ti 0 ..790500  146.8(1.7) 1.009(10) - 0. 0 50(13) 0.8 6 2(11) 6 . 8 8 (24) 5 5.98 5.63 1.102 
tj 1 . 050 145.6(1.7) 1.022(10 ) - 0. 0 64 ( 13) 0.83 5 (11) 7 . 14(2 5 ) 4 5.62 4.E-2 ( 21) 
l. 200 1 42.8(1.7) 1.034(10) - 0. 0 7 4 ( 14) 0.81 0 (1 1 ) 7.42(26) 7 5.71 
(;;J -------
- 11.079( 3 3) 120.0 0.993(10) - 0.038(13) 0.903(14) 6.86 ( 25) 8 5 .65 1.08 0 (32) 13 5 .0 1.0 13 (10) -0. 0 5 6 ( 13) 0.858(13 ) . 7 . 12(2 5 ) 4 5.52 5.49 133 . 3 I ;:: 1.062(30) 1 5 0 . 0 1 . 028(10 ) -0 . 069(14) 0.819(13) 7 . 22(25) 5 5 . 72 5.E-2 ( 2. 0 ) .,.... 1. 0 34(28) 165.0 1. 0 43( 9) - 0 . 073(1 5 ) 0.793(13) 7.02(25 ) 3 6 .4 6 I 
.(.b.,  - -?-----------
Pl I1- 0.583(27> 4 5 .3(1.5) 0.750 -0. 3 08 ( 6) 1.27 5 ( 6 ) -1 . 81 ( 13 ) 10 7 . 54 n.,.. .. 0.980(36) 129.9(1 . 6) 0.900 -0 . 111 ( 12) 0.915(13 ) 5.02(25 ) 6 5.82 5.57 0 .982 I 
.;:.. 1.080(31 ) 148.6(1.8 ) 1.0 5 0 - 0.0 51(11) 0.817(13 ) 7 .6 5(24 ) 8 5.73 1.E-2 ( 10) 
tv 0 1.153(30 ) 156 . 3(2.1 ) 1. 200 0.037(10) 0.764(13) 10 . 26 ( 24 ) 9 7 . 32 I 
c.,, :::l - ---?-
OQ 11.11 5 ( 26 ) 174.0 ( 1.7 ) 0.946( 7 ) -0. 3 00 0.74 5 (12) 5 . 80 ( 22 ) 5 8 . 43 
:l . 1.108(27 ) 164.3(1.7 ) 0.983( 7) -0.200 0.770(12) 
I 
6.77(22) 6 6.78 5.63 -0 .028 
C. .L  1.081 ( 29 ) 151.6 ( 1 . 7 ) 1.016 ( 7) - 0.100 0 . 810 ( 13 ) 
I 
7.24 (2 3 ) 8 5.79 4. E-2 ( 10) 
I 1.022(32! 134.1 ( 1.8) 1.032 ( 8) 0.000 0.882(13) I 
.... 6.84(24) 4 5.67 I 
a, I -1.297(23) 161.8(1.8) 1.103( 9) -0.090 ( 13 ) 0 0.600 10.43(22) 8 7.99 
I 1.154(2 5 ) 153.6 ( 1.8 ) 1.056 ( 9) - 0 . 076(14) 0. 750 8.40 ( 24 ) 6 6.02 5.57 0.873 ,.-.._ o.982(28) 137.6 ( 1.7 ) 0.991 ( 9) - 0.054(14) 0.900 6.20(26) 8 5.63 l.E-2 ( 11 ) 
<:,.;) 
...... 0 . 737(35 ) 111.7 ( 1.7 ) 0.887( 9) - 0. 021 ( 13) 1.050 - 3.73 (2 5 ) 10 6 . 58 00 -
1 1.111 ( 30 ) 147.6 ( 1.6 ) 1.002( 9) - 0.102(13) 0.867(13) 5.00 2 5.84 
3: 1.088(30 ) 146.4(1.7) 1.011 ( 9 ) - 0 . 083 ( 13 ) I 0.849(13) 6.0 0 2 5 .68 5.63 7.04 
(b 1.069(31) 146.3 ( 1.7) 1 . 021 ( 9) - 0. 069 ( 13) 0.835(13) I 7.00 5 5 .6 3 3 . E-3 ( 27) 
< 1.058 ( 31 ) 1 44 .5 (1.8) 1.034( 9) -0 . 054 ( 12 ) 0.813(13) 8.00 5 .___, 5 . 67 ------------------------ ------------- -------=-=---=-===-----==------~-------~--=======---I 
==-=========----===-===------- ----
I sl bl s2 d2 s3 b3 it chi r chig/ std parg I 1-3 ==-===-----=-====-----------------------------=-=------- =====-===-------------------- -
~ 
-er II 1 . 000 190 . 3 (2.4) 1. 000 - 0.069 ( 9 ) 0.701 ( 15 ) 7.12 ( 24 ) 4 . 90 4 . 90 ------?- ---c'D 0.7 50 173.7 ( 3 . 1 ) 0.8 10(9 ) -0 . 274 ( 13 ) 0. 73 6 ( 1 4 ) 5.05 ( 25 ) 9 4. 50 0 . 900 1 58 .6 ( 3 . 1 ) 0.803 ( 9 ) -0 . 287 ( 13 ) 0.73 3 ( 14 ) 5. 16 ( 24 ) 9 4. 4 7 4 . 46 
tj 0 . 85 6 I 1.050 141. 9(3.0) 0 . 794(9 ) - 0 . 302 (12 ) 0.73 1 ( 14 ) 5 . 21 ( 24) 9 4.58 4 . E- 4 ( 33) 
1. 200 123.7 (2.9 ) 0.78 5 (9 ) - 0 . 317 ( 12 ) 0.7 30 ( 14 ) 5.20 ( 23 ) 9 4 . 84 
~ 
I- --- -------------------- ----- -- 0 . 860 ( 2 5 ) 180 . 0 0.8 48 ( 9 ) -0 . 193 ( 13 ) 0.713 (14 ) 5 . 74(23) 8 4.35 0 . 819 ( 2 5 ) 18 6 . 0 0.8 56(9 ) -0.178 ( 13 ) 0.711 ( 14 ) 5 . 77(23) 8 4 . 40 4 .3 4 173. 4 I .=...  0 . 77 5 ( 2 4) 19 2 . 0 0 . 864 ( 9 ) - 0.162 ( 13 ) 0.709 (1 4) 5.79 ( 24) 9 4 .4 6 2.E- 4 ( 3 . 1 ) ..., 0 . 731 ( 24 ) 198 .0 0 . 872 ( 9 ) - 0 . 146 ( 13 ) 0 . 707 ( 14 ) 5.81 ( 24 ) (1) 9 4 .5 6 - - --- -------------- I 
(:ll 
.... / 1.217 ( 37 ) 114.4 (4 .0 ) 0 . 75 0 - 0.350 ( 9 ) 0.729 (14) 4 . 97 (2 1 ) 1 0 5 .08 (l ,.,._ -? 1.1 5 7 ( 37 ) 1 56.9 ( 3 . 9 ) 0 . 900 -0 . 1 8 9 ( 9 ) 0.717 ( 14 ) 6. 33(23) 1 0 4 . 53 4 .5 9 0 . 88 0 I 0 1 . 203 ( 3 5 ) l 79 . 1 ( 3 . 7 ) 1 . 050 -0 . 018 ( 9 ) 0 .6 90 ( 15 ) 7 . 77 ( 2 4) 10 5. 47 1 .E-1 ( 11 ) ts., 
0) = 1 . 239 ( 35 ) 200.5 ( 3. 6) 1. 20 0 0. 155 ( 9 ) 0 . 649 ( 16 ) 9 . 10 ( 25 ) 10 7 .4 1 ------?-()Q -?--?-----
~- / 1.176 ( 36 ) 123 . 9 (4 . 1 ) 0.768 ( 7 ) - 0. 350 0 . 73 4( 1 4) 5 . 02 ( 20 ) 1 0 4 .89 1 . 221 ( 36 ) 122 . 3 ( 4. 0) 0 . 793 ( 7 ) -0 . 300 0.727 ( 1 4) 
0.. 5 . 29 ( 20 ) 9 4. 8 4 4 . 84 1.278 ( 35 ) 118.6 ( 3 . 9 ) 0 . 820 ( 7 ) - 0 .30 8 I -0.250 0 . 721 ( 14 ) 5. 56 ( 20 ) 7 4 . 9 3 l .E-3 ( 11 ) 
;;; 1.348 ( 3 5 ) 113 . 0 ( 3 . 8 ) 0 . 848 ( 7 ) - 0. 20 0 0 .71 4( 14 ) 5.84 ( 20 ) 10 5 . 1 6 -
0 / 1.11 3 ( 36 ) 154 . 1 ( 3 . 5 ) 0. 862 ( 9 ) - 0 . 185 ( 12 ) 0.4 50 7 . 67 ( 14 ) 9 6. 21 
--- 1.186 ( 36 ) 131.5 ( 3 . 9 ) 0 . 803 (9) - 0.2 82 ( 12 ) 0 .650 5 . 94 ( 17 ) 10 4. 8 7 4 . 87 1 . 218 ( 38 ) 111.7 (4. 2 ) 0. 762 ( 9 ) -0.358 ( 12 ) 0 . 850 4. 03 ( 19 ) 10 5 .2 5 l. E-1 0.( 17422CJl ) 1 
0 1.13 7( 39 ) 10 7 . 9 (4. 5 ) 0 . 738 ( 9 ) - 0 .38 4( 12 ) 1. 050 1. 99 ( 20 ) 
- 1 0 6.75 0 ---?--------- -
/ 1.207 ( 37 ) 122.0 ( 3.8 ) 0. 783 (9) ~ - 0.3 21 (11 ) 0 . 734 ( 11 ) 5.10 1 0 4 . 86 1 . 20 4( 3 7) 128 .4 ( 3 . 7 ) 0. 80 1(9 ) -0 .2 80 ( 11 ) 0.6 95 ( 11 ) 6.0 0 
(1) 10 4 . 77 4 . 77 6.07 I 
< 1 . 202 ( 36 ) 133 . 7 ( 3 . 7 ) 0 .821 ( 9 ) - 0 .2 40 ( 11 ) 0.6 56 (11 ) 6. 90 10 4. 8 4 6. E- 5 ( 19) 1.202 ( 36 ) 137 . 9 ( 3.7 ) 0 .8 4 3 (9) - 0.2 01 ( 11 ) 0 .617 ( 11 ) 7 . 80 
'--' 9 5 .0 5 -- ------------? ---- ------------------- ----=-==--======-=====--=======================I  
E List of Runs 
The notation in the following list of runs is as follo ws: t he fir st of added 
runs is denoted by ?>", each subsequent run is denot ed by "+"; excluded 
states following a ""'" belong t o the previous line . An "*" denotes a run 
with encoder problem and the quoted angle is the co rrected angle . An ?!" 
denotes a run with a scaler problem; the necessary scale fact or f for the 
a ffect ed polarized cross section ( u for spin-up , d for spin-down) is given in 
the last column . March 1988 runs show an "all" in tbe column for excluded 
states since they were not used at all in the present work , and are just listed ,,.   \ 
here for completeness . 
1 
'1  
' 
~ 
427 
C. - oo ? H - --- --======================= ===========-=======-===--- ---- ------------ ----- ---- ---- -----
;;; ~ 0.. ~ j Run Angle By Evts Tangle Tgt _ Ap Ifs ~u~ Polup Poldn Ener~y Exc_:uded States I 
(1) r::r --------- - - ------- --- --------- -
~ O"' :=: ;" 348 -16.00 el 145 -8.00 new Bl 20 187 0.748 0.748 100.70 160:lml 
i:o s~  0 349 - 16.00 in 100 -8 . 00 new Bl 20 187 0.749 0.749 100.70 (b ~ trj 
,....,._  O~Q  o.. 358 -14.00 el 116 -7.00 new 82 20 187 0.757 0.761 100.70 ~ 
~.,.... O"' 363 - 12.00 el 186 -7.00 new 82 20 187 0.757 0.761 100.70 
O O '-< i:,, 364 -12.00 in 57 -7.00 new B2 20 187 0.757 0.761 100 . 70 l60:4pl 
H C. ~----- 397 -12.00 el 93 -6.00 new 82 6 S87 0.757 0.761 100 . 70 
.......,(b v o, 
,..._ ~ (1) 365! - 10.00 el 185 -5.00 new 82 6 187 0.757 0.761 100.70 fu=6216.195 
0 "d - O 390 - 10.00 el 154 -5.00 new 82 6 S87 0.757 0.761 100.70 
H ~ (1) 
"d ::: . =~ - C 404 14.00 el 87 7.00 new Al 20 S87 0.757 0.761 100.70 160:3ml,2pl 7-- 0 = C 00 00 408 16.00 el 72 8.00 new Bl 20 S87 0.757 0.761 100.70 160:lml 
C 00 C i:,, 
"d ::;-: O"'.,..,.. 137 18.00 el 283 9.00 old Al 60 E87 0.727 0.750 100.50 
,...--... ::: ~ tr1 138 18.00 in 245 9.00 old Al 60 E87 0.727 0.750 100.50 160:3ml 
~ ~ ,.c ..,, 224 18.00 el 208 9.00 new Al 60 187 0. 736 0. 748 100 . 70 
'---' C II > 228 18.00 el 11 9.00 new Al 20 187 0.736 0.748 100.70 
0 ? ~ + 229 18.00 el 168 9.00 new Al 20 187 0.736 0.748 100.70 
H ::: ~ ,_. 230 18.00 el 83 9.00 new Al 20 E87 0.736 0.748 100 . 70 
00 
t; "?- -~_  c C > 262 18 . 00 el 61 9.00 new Bl 60 187 0.733 0.753 100.70 O + 263 18.00 el 68 9.00 new Bl 60 187 0.734 0.753 100.70 
00 'o1. . = ~ 0.. .:I_(1).  > 5 22.00 el 93 11.00 old Al 20 E87 0.756 0.774 100.50 160:3ml,2ml ; O (1) + 6 22 . 00 el 215 11.00 old Al 20 E87 0.756 0.774 100.50 9Be:lpl,3pl 
::f: ::: --.., 7 22.00 in 202 11.00 old Al 20 E87 0.756 0.774 100 . 50 9Be:Lor 
::: S?,_ +~  Z 212! 22.00 el 111 11.00 new B2 60 E87 0 . 736 0 . 748 100.70 fd=l0380.102 ,........_ (1) o 220 22.00 el 75 11.00 new Al 60 E87 0.736 0 . 748 100.70 
00 
i;:,... ~ .,..,.. > 222 22.00 el 87 11.00 new Al 60 187 0.736 0.748 100.70 
'---' ~ - ? ~ + 223 22 . 00 el 95 11.00 new Al 60 187 0.736 0.748 100 . 70 9Be:5pl 
-~?  H (~1)  -~ ? C 139 24.00 el 132 12.00 old Al 60 E87 0.727 0 . 750 100.50 
~-::: C ~ 140 24.00 in 174 12.00 old Al 60 E87 0.727 0.750 100.50 160:3ml;9Be:Lor 
~ ~ - ~;. 18 26.00 el 183 13.00 o ld Al 60 E87 0.756 0.774 100.50 
:. ==- Cl.. (1) 19 26.00 in 232 13.00 old Al 60 E87 0.756 0 . 774 100.50 160 : 3ml;9Be:Lor 
::: 00 ::ri 160 26.00 el 104 13.00 new Al 60 E87 0 . 736 0.748 100.70 
.,.... i:,, g- ~ 161 26 . 00 in 51 13.00 new Al 60 E87 0.736 0.748 100.70 160:2p2,2ml,Oml;9Be : 3pl 
:::--- r.n c-- c:--
(1) 8 (1) o 28 ! 30 . 00 el 85 15.00 old Al 6 E87 0.756 0.774 100.50 fu=69.313 
p,? Cb 00 -. 30 ! 30.00 in 258 15.00 old Al 20 E87 0.756 0.774 100 . 50 160:3ml; fd=41.878 
00 H Q' ~ 162 30.00 el 109 15.00 new Al 60 E87 0.736 0.748 100.70 
~1; O ~ 38 3 4 .00 el 127 17.00 old 82 20 E87 0.756 0.774 100 . 50 
?. 0 =::f: ? o.. > 39 34.00 in 22 17.00 old Al 60 E87 0.756 0.774 100.50 C 2:: =~  --+ ---48 ~ ()Q -0 ----3-4-.0--0 ---in 438 17.00 old Al 60 E87 0 . 756 0 .774 100.50 160:0p2 , 3ml --- ------------ -----------------------------------=----===------==== 
::::, ,.J ;., ~ 
:::.~ o"pl H 
... pl_o..lll 
(1) - 0 0.. O"' 
pj" ~ s~?  g__ ;- ==-===============-==--====-===============-===============-------------------------------~ "Cl r; I Run Angle By Evts Tangle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States I 
8 >; (}q .:: t'.fj - - - - - - - ---- - ---0 := 155 34 . 00 el 228 17 . 00 new B2 60 E87 0.736 0.748 100.70 
- g: pl oo >--' 156 34 . 00 in 178 17.00 n ew B2 60 E87 0.736 0.748 100.70 
.:: (1) - ..... 
3 S > 00 47 38.00 el 114 19.00 old Al 60 E87 0.756 0.774 100.50 
::: -? ~ 0.. O"' 48 38 . 00 in 376 19.00 old Al 60 E87 0 . 756 0 . 774 100 . 50 
,.... O"' ~ ____. 168 38 . 00 el 65 19.00 ne w Al 60 E87 0.736 0.748 100.70 9Be:3pl 
::;-- (1) 0 tI:l 
(1) O,.... (1) 56 42.00 el 107 21.00 old Al 60 E87 0.750 0.766 100.50 9Be:lml 
~ ..,. g__ Q 57 42.00 in 258 21.00 old Al 60 E87 0 . 749 0.765 100.50 
pl~ 174 42.00 el 82 21.00 new B2 60 E87 0.736 0.748 100.70 
- O"' r; 
(1) ,,... '< S 58 46.00 in 212 23.00 old Al 60 E87 0 . 748 0.764 100.50 
~ O ~ ~ 59 46.00 el 133 23.00 old Al 60 E87 0.748 0.763 100.50 9Be:lml 
~:::. V ~ 179 46.00 el 99 23 . 00 new B2 20 E87 0.736 0.748 100.70 
0 ~ :::: e-+-
..r.;.. _,-(o1)  - r 80 50.00 el 77 25 . 00 old B2 200 E87 0.727 0.750 100.50 (1) 81 50.00 in 133 25.00 old B2 200 E87 0.727 0.750 100 . 50 160:3m4 ;9Be:3ml ,5p l 
_, ~ ~ 186 50 .0 0 el 100 25.00 new B2 60 E87 0.736 0.748 100.70 
o < - II 
r; o? - 82 54 .00 in 204 27.00 old B2 200 E87 0.727 0.750 100.50 160:0ml;9Be:5pl 
"'" oo C ~ f-' 83 54 .00 el 83 27 . 00 old B2 200 E87 0.727 0.750 100 .50 
~ '2 . oo erg 190 54.00 el 125 17.00 new B2 60 E87 0.736 0.748 100.70 
.... 00 
~ 5'] :;;.,o 91 58 .00 el 58 29 . 00 old B2 200 E87 0.727 0.750 100 . 50 160:2pl 
-c:, (1) c ~<  92 58 .00 el 55 29.00 old B3 200 E87 0.727 0.750 100.50 160:2pl;9Be:lpl ? ?., .,_ :(1::)  93 58.00 el 55 29.00 old A7 200 E87 0.727 0.750 100.50 160:2pl;9Be:lpl > 94 58.00 in 111 29.00 old B3 200 E87 0.727 0.750 100 . 50 160:0p2,2pl;9Be:3ml,5ml 
__,, ,,...- ,,... ,..-... + 95 58.00 in 103 29.00 old B3 200 E87 0.727 0.750 100.50 
0 := C =8  194 58.00 el 100 19 .0 0 new B2 60 E87 0.736 0.748 100 . 70 9Be:lpl ,..., .... I-' 
00 ?-:; - : 124 62.00 el 90 31.00 old A7 60 E87 0.727 0.750 100.50 
"C=l ? - O"'____. 125 62.00 in 315 31.00 old A7 60 E87 0.727 0.750 100 . 50 9Be:3ml,lp l 0.. '< ~ 201 62.00 el 72 31.00 new A7 60 E87 0.736 0.748 100 .70 
I (l) ~ ~ 
0.. ~ + ~ 114 66.00 el 100 33.00 old A7 60 E87 0.727 0.750 100.50 160:0p2 
~ 0 ~ p, 116 66.00 el 140 33.00 old A7 60 E87 0.727 0.750 100 . 50 160:0p2 
..,. CD-? ;::. > 117 66.00 in 338 33 . 00 old A7 60 E87 0.727 0.750 100.50 
- oo (1) O + 118 66.00 in 209 33.00 old A7 60 E87 0.727 0.750 100.50 
--. p, -~ := 206 66.00 el 81 33.00 new A7 60 E87 0.736 0.748 100.70 160:0p2;9Be:5ml ~ 
":':- =~  g__:::, 111 70.00 el 94 35 . 00 old A7 60 E8 7 0.727 0.750 100.50 00 (1) ~ 112 70.00 in 539 35.00 old A7 60 E87 0.7 27 0.750 100 . 50 
oq ... o,. 209 70.00 el 70 35 . 00 new A7 60 E87 0.736 0.748 100.70 
<. ::;; 00 ;::h - - - - - - '====== ------=----============ 
(l) - ? ~ -1 
~ C
- ... 
 ~ ~ 
~ 
~ ~ ~ 
{/, 
/1 
I. 
:-::- 00. i-3 
('ti > 0.. ~ Run Angle By Evts Ta ngle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States II l 
(e'n)  o- ('ti g: =======================-==========--===========-=--=-=---- --=---==-===-==--=======-----------=------- :::i C'D 303 - 16.00 el 52 - 8.00 thn Bl 6 !87 0 . 740 0.744 100.70 p, ('ti 304 -16.00 el 56 -8.00 thn Bl 6 I87 0 . 740 0.744 100.70 ('ti -0 ~t_:!:j 305 -16.00 e l 12 -8 . 00 thn Bl 6 I87 0.740 0.744 100.70 
...... :::i ('ti ? o..t-,,) 306 - 16.00 el 59 - 8.00 thn Bl 20 I87 0.740 0.744 100.70 p., ()-q 307 - 16.00 el 49 - 8.00 thk Bl 6 I87 0 . 740 0.744 100.70 
~ .,..... 0- 308 - 16 . 0 0 e l 70 - 8.00 t h k Bl 6 I87 0.740 0.744 100.70 
0 0 '-< 
~ 344 -16.00 el 53 - 8 . 00 thk Bl 20 I87 0.744 0 . 741 100.70 H .,..... _.._p._l ~ ~ . 345! -16.00 in 18 -8.00 thk Bl 20 I87 0 . 744 0 . 742 100.70 fu=382.577 
(I) V ,.. 346 - 1 6.00 i n 226 - 8.00 thk Bl 20 I87 0.74 5 0 . 744 10 0 .70 
o''"O _,, 0 347 - 16.00 in 66 -8 .00 thn Bl 20 I87 0.747 0.747 100.70 
'"1 H Q )2289 - 16 . 00 el 60 0.00 thk Al 2 588 0. 728 0.748 100.50 all 
"' (I) (I) p., \ +2290 
-16.00 e l 86 0 . 00 thk Al 2 588 0 .7 28 0.748 100.50 all 
'"O < P., H 
s? o? & e 311 -14.00 el 38 - 7.00 thn Bl 6 I87 0.740 0. 7 44 100.70 
' C ~ ,.. 312 - 14.00 el 43 -7.00 thn Bl 6 !87 0 . 740 0 . 744 100.70 C en en 313 - 14.00 el 51 - 7 . 00 thn Bl 6 !87 0.740 0.744 100.70 
'"O t=-:0-~ 314 -14.00 e l 41 -7.00 thn Bl 6 I 87 0.740 0.744 100 . 70 I' 
.-~-. . 1;:: en trj \ 315 - 14.00 el 56 - 7 . 00 thk Bl 6 !87 0.740 0.744 100.70 I? (I) (I) ..__, ? ,-Cl '"0 316 -14 . 00 el 50 -7.00 thk Bl 6 187 0.740 0.744 100.70 3 50 - 14 . 00 el 2 55 -7.00 thk Bl 20 187 0. 7 51 0.752 100.70 Op2 ~ ~; II ) 3 51 - 14 . 00 in 202 - 7 . 00 thk Bl 20 187 0.7 5 2 0.754 100.70 \ r\l  
en ~ ~ \ + 3 5 2 - 14.00 ,-.J in 136 - 7.00 thk Bl 20 187 0.7 5 3 0 . 756 100 . 70 '"O ::: 0 353 -14 . 00 in 102 - 7 . 00 thk Al 20 !87 0.754 0 . 757 100.70 
~ 
w - ? ? - "='"'1  0 - 14 . 00 in 49 -7 . 00 thn B2 20 187 0.7 56 0.759 100.70 0 ~ ~ ~ - 14 . 00 in 266 -7.00 thn B2 20 !87 0.757 0.761 I 100.70 0.. 0..,.. '\: n35~7  -14 . 00 in 182 - 7.00 thn B4 20 187 0.757 0.761 100.70 0 (I) 0- (<I)  2286 - 14.00 ~ el 84 0 . 00 thk Al 2 588 0. 728 0.748 100 . 50 all ::; '-< 
.,....  0- ::: .z  317 -12.00 el 51 - 6.00 thn Bl 6 187 0.740 0.744 100 . 70 
~ ~ +_ 0 318 - 12.00 el 49 -6.00 thn Bl 6 !87 0.740 0.744 100.70 
'---?l _.- g- 319 -12.00 el 61 -6.00 thn Bl 6 !87 0.740 0.74 4 100.70 320 -12.00 el 64 -6 . 00 thn Bl 6 187 0 . 740 I 
'ij;" ,.., (b ~ 0 . 744 100.70 361 - 12 . 00 in 67 -7.00 thk B4 20 !87 0.757 0 . 761 
o-q C X 0 ? 100.70 362 - 12.00 in 254 -7.00 thk B4 20 !87 0 . 757 0.761 100 . 70 
-< ? ~ n-  i,-..- \.  2282 -12.00 el 93 0 . 00 thk Al 2 588 0. 728 0.748 100.50 all 
(I) ~ C 
~ ~~;:. 321 - 10.00 el 54 -5 .0 0 thn Bl 6 !87 0.740 0.744 100.70 
~- ~ 0.. (t) 322! -10.00 el 58 -5.00 thn Bl 6 !87 0.740 0.744 100.70 fu=507. 527 
...., ~ en ~ 323 -10 . 00 el 47 -5 .00 thn Bl 6 !87 0.740 0 . 744 100.70 
<=+- .,.... H 324 -10.00 el 45 -5.00 thk Bl 6 !87 0.740 0.744 100 .7 0 Cn/l  ~ C/l ~ 325 -10.00 el 51 -5 . 00 thk CT) Bl 6 !87 0 . 740 0.744 (.""'f-- 100 . 70 
_PJ (I) 0... .., 366 -10.00 el 106 -5. 00 thn B2 6 187 0 . 757 0.761 100 . 70 p., - en 367 -10.00 in 129 -5 .00 thn B2 6 !87 0.757 0.761 100 . 70 
e.,n.. .. (..I.),  o' CJ 368 -10 . 00 in 151 -5.00 thn B4 6 !87 0.. 0.757 0.761 100.70 
:= 378 -10.00 n el 211 - 5.00 '"O thk Al 6 587 0 . 757 0.. 0 .76 1 100.70 
0 ..., 0 379 -10.00 (I) in 98 -5 . 00 thk Al 6 587 0.757 0 . 761 100.70 
- 0 ~ 0.. 380 -10.00 in 26 4 -5.00 thk B4 6 587 0.757 0 .761 100.70 
C3  ~ :::? 381 -10 . 00 in 156 -5.00 thk B2 6 587 0. 757 0.761 100.70 (I) ,.. '"1 
~ 3 = ------- -OQ 
p., ; 
L - -- - __ ... ... .. .. ~ ....... _ --- .... --
e+tll ...... p) L...) 
::,-n ?..o..~ 
(1) Pl ..- 0.. ------- ============--===-==--=======-======-=--------
~ Run Angle By Evt s Tangle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States - 0 (1) O"" 
~ ,; ?. 0.. ;- ----------==============--2279 -10 . 00 el 1 48 0 .00 thk Al 2 S88 0.728 0.748 100.50 all 
- '"Cl ::::: ,; 
n80Q.;:t:,:j 384 -8.00 in 53 - 4.00 thn 82 6 S87 0.757 0.761 100 . 70 
s..~~i;;N 
= 300 10.00 el 61 6 .00 thn Bl 6 187 0.740 0 . 745 100.70 3  3(1)  ~"-'?  301 10.00 el 70 6.00 thn Bl 6 187 0.740 0 . 745 100.70 
!:3--? :: O..cr- 302 10 . 00 el 57 6.00 thn Bl 6 187 0.740 0.745 100.70 -(1) ..__, .,,. 329 ..., 10.00 el 164 5 . 00 thn Bl 6 187 0.740 0.744 100 . 70 ::,' -O"" 332 10.00 el 108 5 .00 thk Bl 6 187 0.740 0.744 100 . 70 (1) (1) 0 o ~o 333 10 .00 in 25 5.00 thk Bl 6 !87 0.740 0.744 100.70 2pl,5ml,2p2,lml,2m2,6.91 Cll Q 2267 10.00 el 
n ::::: 111 5 . 00 thk Al 20 S88 0.728 0.748 100.50 all 0.. ~ >2293 10.00 el 87 0.00 
~ thk Al ()q 2 S88 0 .728 0 . 748 100.50 all 
O"" ,; +2294 ~ 10.00 el 80 0 .00 thk Al 2 S88 0.728 0.748 100.50 all 
0-  '< = ~ ::::: 29 4 12 .00 el 43 6.00 thn Bl ::: tll 6 187 0.740 0.746 - 100.70 n 295 12.00 el 57 6.00 thn Bl :- V ~ 6 187 0 .7 40 0.746 100.70 296 12.00 el 54 6.00 0 thk Bl 6 187 0 .7 40 0 . 746 100 . 70 
,; (1) :: - 297 12.00 el 51 6 .00 thk Bl 6 !87 0.740 0.746 100.70 
........ -0 (1) M 299 12 .0 0 el 53 6.00 t h k Bl 6 187 0.740 0. 746 100 . 70 ,; Pl '"O 33 4 12.00 el 97 6.00 thk Bl 6 !87 0 .7 40 0 . 741 100.70 
o' ~ g. II 33 5 12.00 in 152 6.00 ,.,. thk Bl 6 187 0.740 0.741 100.70 5ml,4pl -? 336 12.00 in 173 6.00 thk Bl 6 !87 0.740 0.741 ,; 100.70 5ml , 6 . 0 28 , 7 . 11 
w Cl) =0  Cl) >-' >2297 12.00 el 76 0.00 thk Al 2 S88 0.728 0.748 100.50 i:: all 0 +2298 >-' -0 - 12.00 el 267 0.00 thk Al 2 S88 0.728 0.748 100.50 all === ? -r.n O"" 0 ..., ?Cl) >2 323 12 . 00 in 277 6.00 thk B2 20 S88 0.728 0.748 100.50 all ~ ::::: ]  a;:: +2324 12.00 in 212 6.00 thk 82 20 S88 0 . 728 0.748 100.50 all 
-0 (1) i:: (1) 288 
.,......._ . g < 14.00 el 32 7 .0 0 thn Bl 6 187 0 .7 40 0.747 100.70 289 14.00 el 32 7.00 thn Bl 6 !87 0.740 0.747 
.:! 100.70 290 Op2 > - ----- 14.00 el 45 7.00 thn Bl 6 I87 0 .7 40 0.747 100.70 291 Op2 - ,; n 14 . 00 el 178 7.00 thn Bl 6 I87 0.740 
0 
,; -' ~- 8 0.747 100.70 292 14.00 el 56 7.00 thk Bl 6 187 0 . 740 0.747 100 .70 ...., .c...-, - 293 14.00 el 53 7.00 thk Bl 
Cl) 6 187 0.740 0.746 100 .70 
"8. --- r:r ,,:_, 
337 14.00 el 102 7.00 thk Bl 20 187 0.740 0.740 100 . 70 
'< z > 339 14.00 in 32 7.00 thk Bl 20 I87 0 .7 40 0 . 740 ::, 0.. +-  + 340 
100.70 
14.00 in 154 7.00 thk Bl 20 I87 0.740 0.739 
2301 100 . 70 g 14.00 el 79 lml,lm2 0.. - ~ 0.00 thk Al 2 S88 0.728 0.748 0 O :: ~ 2317 100 . 50 14.00 in all 420 7 . 00 .... thk 82 20 S88 0. 728 0 . 748 100.50 all ~ -? 
~ (1) , D -~ ? 280 16.00 el ~- I ., 75 8.00 thn Bl 20 187 0.740 0.750 
281 100.70 >< 16.00 el 56 8.00 thn Bl 6 187 0.740 0.749 
282 100.70 ~~ Q... 16 . 00 el 58 8 . 00 thn Bl 6 187 0 . 740 0.749 100.70 
,; i:: - 283 16.00 el 67 8 . 00 thn Bl 6 I87 0.740 0.749 :=- 284 16.00 el 100.70 Cl) i:: 0.. 63 8.00 thk Bl 6 187 0.740 ::::: 0 .7 49 28 5 100.70 (1) (1) 16.00 el 
()q 31 8.00 thn Bl 6 187 0.740 
~ 0.. :::n 
0.748 
286 100.70 16 . 00 el 46 8.00 thn Bl 6 187 0.740 0.748 
::;? Cl) ..., 287 16.00 el 100.70 39 8.00 thk Bl 6 I 87 0.740 
('O - 0 .7 48 100 . 70 
::, ;:.?  ~ Ul ~ ~ 2304 16.00 el 79 0.00 thk Al 2 S88 0. 728 0.74 8 100.50 all 
::, (1) 0 ' ' ' ' . -------------------------================------=================== 
"' CJ') ,...... 
:- :;- ((/'ll 
-. Ill H 
 
(n !:... 0 0.. ~ := 0.. c::r Run Angle By Evts Tangle Tgt Ap Ifs Cup Pol up Poldn Energy Excluded States 
I- ll .(.n.,  0? . (n -0.. ~ 2311 16 . 00 in 216 0.00 ..., thk B2 20 588 0. 728 0.748 100.50 all ~ -.0..,  ;:: 
0 OQ c::: ('l ~ ;:: 132 18 . 00 el 123 9.00 thk Al 60 E87 0. 727 0 .7 50 100.50 
0 er Ill t-' 136 18.00 in 357 9 .0 0 (/l thk Al 60 E87 0.727 0.750 100.50 
~~ 266 18.00 el 5 9.00 thn Bl 60 I87 0 . 740 0.757 100 . 70 s s (/l 267 18 . 00 el 26 9.00 thn Bl 60 I87 0 . 740 0 . 756 100.70 > 
;:: 0.. ('l 268 18.00 el 26 9.00 thn Bl 60 I87 0.740 0 . 756 100.70 
269 18.00 el 17 9.00 thk Bl 60 I87 0.740 0.756 100.70 
:-:;-- er ;(:n:  ----.,-. 270 18 . 00 el 52 9.00 thk Bl 60 I87 0.740 0.756 100.70 
(n t'n O o 271 18.00 el 62 9.00 thk Bl 60 I87 0.740 0 . 756 100.70 
(/l o Cb o 272 18.00 el 58 9.00 thk Bl 6 I87 0.740 0 . 753 100.70 
~ ~ 0.. Ill 273 18.00 el 35 9.00 thn Bl 6 I87 0.740 0.752 100.70 
- -?e.r:.:.., 
274 18.00 el 34 9 . 00 thn Bl 6 I87 0.740 0.752 100. 70 
(n : 275 18 .0 0 el 20 9 . 00 thn Bl 6 I87 0.740 0.752 100.70 
~ 0 ~ 
p :: 276 18.00 el 34 9.00 thn Bl 20 I87 0. 740 0.751 100 . 70 (/l 277 18.00 el 29 9.00 thn Bl 20 I87 0.740 0.750 100.70 
~ ;:. V Ill 278 18 . 00 el 31 9.00 thk Bl 20 I87 0.740 0.750 100.70 
.0.. , (n :: ,... 279 18.00 el 31 9.00 thk Bl 20 I87 0.740 0 . 750 100 .70 
..., (n M 341 18.00 el 79 9.00 thk Bl 20 I87 0.740 ......... -0 0.737 100 . 70 Ill -0 > 342 18.00 in 215 9.00 thk Bl 20 I87 0 . 740 0.737 100 . 70 
-. (n 
-? n + 343 18.00 in 52 9.00 thk Bl 20 I87 0.740 0.736 II 100.70 0 <! ..., ~ 2335 18.00 in 902 9.00 thk B2 20 588 0. 728 0 . 748 100.50 all 
~ (/l 0 (/l ,_. 
c,;, -0 .::: C o 2221 20 . 00 el 484 10.00 thk B4 60 E88 0 . 728 0.748 100.50 all 
t-.:l 
:-;:?  (/l er o (/l 2228 20.00 in 612 10 .00 t h k Al 60 EBB 0. 728 0.748 100.50 all ;:: 2 2229 20.00 el 41 10.00 thk Al ~ 60 E88 0. 728 0.748 100 . 50 all C 2264 20.00 el 169 10 .0 0 thk Al 60 588 0.728 0.748 100 . 50 all 
-0 t'n C t'n 
. (n < 1 22 .0 0 el 329 11. 00 thk Al 6 E87 0.756 0.774 100 . 50 
? ? ~n' 2 22.00 el 263 11 . 00 t h k Al 20 E87 0.756 0 .774 100 . 50 
'-"'":: 1=-1  0;::  8 22.00 in 150 11. 00 thk Al 20 E87 0.756 0 . 774 100 . 50 ..., 2235 22.00 el 36 ... 11 . 00 thk Al 60 EBB 0.728 0.74 8 0 - 100 . 50 all -' ~ 2241 22.00 in 209 11. 00 thk Al 60 EBB 0. 728 0.748 100 . 50 all 
en -~ er-
~ - '< z 141 24.00 el 104 12. 00 thk Al 60 E87 0.727 0 . 750 10 0 . 50 
::: 0... -- > 144 24.00 in 224 12.00 thk Al 60 E87 0. 727 0. 750 100.50 
6.. g + ~ + 145 24.00 in 546 12.00 thk Al 60 E87 0 . 727 0.750 100.50 
0 O :: P 2244 24.00 el 70 11. 00 thk Al 60 E88 0 . 728 0.748 100.50 all 
~ .,..... - ;::- 2247 24.00 in 9 34 11. 00 thk Al 60 EBB 0. 728 0.748 100.50 all 
- (n (n 0 2248 24.00 in - Cl)>< ~ 1209 11. 00 thk B4 60 EBB 0. 728 0. 748 100.50 all 
~ P n 11 26.00 el 120 11. 00 t h k Al 20 E87 0.756 0.774 100.50 
----- ..., C .,..... 12 26.00 el 363 11. 00 thk Al 20 E87 0.7 56 0 . 77 4 100.50 
C/l C 0... :::- + 13 26.00 el 589 11 . 00 thk Al 20 E87 0.756 0.774 100.50 
oq t:: ~ (n 21 26.00 in 100 13. 00 thk Al 60 E87 0.7 56 0. 774 100.50 
22 26.00 i n 103 13. 00 thk Al 60 E87 0.7 56 0.774 100.50 
:;;? ~ C/l ~ 
(b :- -- ~ C/l 23 30.00 el ~ ~ ~ 259 15 . 00 thk Al 60 E87 0.756 0 .77 4 100.50 
;:: (") 0 p en ~ 
I 
;. ~ o' ~ r-3 ------------------------------------- ----------------- - -====- - ---------
~ ~ ?= ~ ~ l===!~~===~~!===!;==!~~!=~!~~:==-~gt Ap Ifs _cup _Polup Poldn Ener gy Exclude~ State!_=====~--
~ '""' ~- 0.. ~ + 24 30.00 el 320 15.00 thk Al 60 E87 0.756 0.774 100.50 
.,..... 'O ~ >-i 27 30 . 00 in 251 15.00 thk Al 60 E87 0.756 0.774 100.50 
n '""'o oq C t'.lj 32 30.00 in 204 15.00 thk B2 60 E87 0.756 0.774 100.50 
0 0- ~ ? 
~ ~ 00 - N 33 34.00 el 147 17.00 thk Al 20 E87 0.756 0.774 100 . 50 
~ ~ ?~  ~- > 36 34.00 in 262 17.00 thk B2 60 E87 0.7 56 0.774 100 . 50 E: ~ p_. p_. + 37 34.00 in 356 17.00 thk B2 60 E87 0.756 0 .77 4 100.50 
(1) '-' 
;. 0- ~ ~ > 41 38.00 el 92 19.00 thk Al 60 E87 0.7 56 0 . 774 100.50 
(1) (1) 0 o + 42 38.00 el 95 19.00 thk Al 60 E87 0.7 56 0.774 100.50 
oo O ~ Q 45 38 .00 in 246 19.00 thk B2 60 E87 0.756 0.774 100 . 50 
n ::: 0.. ~ 46 38.00 in 198 19 . 00 thk B2 60 E87 0.7 56 0.774 100.50 
e:_ oq 0- 1-1 49 38. 00 in 189 19. 00 thk B2 60 E87 0. 756 0. 774 100. 50 
(1) .,..... '-< C 
--. 0 ::: 52 42.00 el 187 21 . 00 thk B2 60 E87 0.753 0.771 100.50 
~ 00 .,..._ ~ > 53 42.00 in 277 21.00 thk B2 60 E87 0 . 752 0.769 100.50 
~ ::;- V ~ + 54 42.00 in 205 21.00 thk B2 60 E87 0.7 52 0.769 100.50 4ml 
0 (1) :: .,..... 
'""'"Cl -(1) trj 60 46.00 el 82 23.00 thk B2 60 E87 0.743 0.757 100.50 
........ >-i ~ ~ > 66 46.00 in 45 23.00 thk B3 60 E87 0.737 0.750 100 . 50 
,-, (1) n + 67 46.00 in 536 23.00 thk B3 60 E87 0.729 0.750 100.50 
~ :':: . ::;- II 68 46 .00 in 61 23.00 thk B2 60 E87 0.728 0 .750 100.50 
,.,._ 0 00 
W 'rj C00  C S 72 50.00 el 119 25.00 thk B2 60 E87 0.727 0.750 100.50 
W ~ - 0- O 73 50.00 in 27 25.00 thk B2 60 E87 0.727 0.750 100.50 
7 t="'. ~ 74 50.00 in 57 25.00 thk B3 60 E87 0.727 0.750 100.50 
C ::: ..C ~ > 75 50.00 in 155 25.00 thk B3 200 E87 0 . 727 0.750 100.50 
.".C.--l .. ~ C (<1)  + 76 50.00 in 5 25.00 thk B3 200 E87 0.727 0.750 100.50 ,~,- ,.-... + 77 50 .00 in 403 25 . 00 thk B3 200 E87 0.727 0.750 100.50 I=:! ? 
._..,.::: 1-1 n 85 54 . 00 el 42 27.00 thk B2 200 E87 0 . 727 0 .750 100.50 
0 C O > 89 54.00 in 218 27.00 thk B3 200 E87 0.727 0.750 100.50 
>-i . ..:: :-:: ;::_ + 90 54. 00 in 301 27. 00 thk B3 200 E8 7 0. 727 0. 750 100. SO ~'-- ::: '< '.- ' > 96 58 . 00 el 19 29. 00 thk B3 200 ES 7 0 . 727 0. 750 100 . 50 
o... 0.. +~ 'Z + 97 58.00 el 69 29 . 00 thk B3 200 E87 0.727 0.750 100.50 O~  :: 0 > 101 58.00 in 383 29.00 thk A7 200 E87 0.727 0.750 100.50 O ~ + 102 58.00 in 18 4 29.00 thk A7 200 E87 0 . 727 0.750 100.50 
:;; ,,... - . -
::: (1) ,,... 127 62.00 el 99 31.00 thk A7 60 E87 0.727 0 .750 100.50 
00 
...--.. ~ 8 > 128 62.00 in 53 31.00 thk A7 60 E87 0.727 0.750 100.50 
R.. ~ Q.--' + 129 62.00 in 410 31.00 thk A7 60 E87 0.727 0.750 100 .50 
'-' >-i C 
~- C 0... :- 119 66.00 el 82 33.00 thk A7 60 E8 7 0.727 0 . 750 100.50 
::: (1) (1) > 122 66.00 1n 267 33.00 thk A7 60 E87 0.727 0. 750 100.50 
~< - ::s 0.. ::n + 123 66.00 in 203 33.00 thk A7 60 E87 0 . 727 0.750 100 . 50 00 ,.., 
g 2::: ;- ~ 103 70.00 el 75 35 . 00 thk A7 200 E87 0 . 727 0.750 100.50 
- - e-- s.. - ----------------------------- . ------- ----------- -------------====--================= ~ ~ 
11-11---11 
II II II 
II II II 
II II II 
11 II 11 
11 II 11 
11 II 11 
II II 
II II 
II II 
II Ul II 
II Ql II 
11.._. II 
II ,u II 
II .._. II 
II Vl II 
II II 
'tl II 
Ql II 
'tl II 
..:.:.,.  II 
u 
""'  
>,. 0000 
0, 1./Hl")l.f'ILl"I 
"(I)  0000 
I C: 0000 II w .-t.-t..-4....-i 
II 
11 C: 0000 
II -o Lf'I i.n"' Ln 
II ,..... r- r- r- r-
11 0 
IIP. 0000 
II 
II I 
II a.11r-r-r-r-
ll :, IJNNNN 
II ,..... II r- r- r- r-
11 o II ? ? ? ? 
11<><110000 
II II 
II II 
II a.ur-r-r-r-
11:, 11 coco coco 
II U II r.:ir.:ir.:ir.:i 
II 11 
II II 
U1 1\0000 
'+-I II o\0\0\0 
H II N 
II 
II 
a, II r- r- r- r-
.0: II<<<< I 
II 
II 
.? II .!(.!IC~.::..:: 
D'II J::.<:.<:.<: 
E-< II.._..._..._..._. 
II 
II 
41110000 
r-t l! OOOO 
D'II ???? 
C: tlt.nt.ntl"ILl"I 
!U IIMMMM 
E-< II 
II 
t11 I ?' i.n co r-
.? r-\0~\0 
> ..... ..... 
"' 
I >< <: <: <: <: 
'"' ?rf ?.-f ?.-f?..-4 
Q) 
..... 0000 0000 
~"'  0000 r- r- r- r- I 
II 
II 
II 
<: r- co?' o 11 
;l 000..-4 11 
pc; .-t..-4..-4.-f II 
II 
I " ++ II 
II II 
II - --- 11 
Table E.2: e) 4?Ca runs at Ep = 100 MeV (cont.) Notation: the fir st of 
added runs is denoted by ">", each subsequent run by "+"; excl uded states 
following a "" " belong to the previous line. An " i>' denotes a. run with a 
scaler problem; the scale factor J for spin-up ( u) or spin-down ( d) is given in 
the last column. 
434 
CJl =?>  ::: cii" t-3 ===============================================================--===----------- ------ --- ----'E=.  - ~- ~ I Ru n Angl e By Evt s Tangle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States 0.. -~ ~ ,;;;r --------------- ------------------- -------- - - - - ==== ===========---------- -O"':;:: ;;" 867 - 6.00 in 546 0 . 00 new B2 6 I88 0.717 0.733 201.24 
O - ~ 0t =~  8  t_'.l:j 8 5 3 6.00 e l 118 6. 5 0 n e w Bl 20 I88 0 . 717 0 . 733 201.24 ?> ..., 0.. ? 
,_, OQ W 840 7 .5 0 el 15 2 6. 5 0 new Bl 20 I88 0 . 7 50 0.766 201.2 4 
~ i:: .,.... O"' 846 7. 5 0 in 42 6. 5 0 new B2 20 I88 0.735 0.751 201.24 
--......- :=:: O,... .. '-< - ?> 847 7.50 in 546 6.50 n ew B2 6 I88 0 . 733 0. . 749 20 1.24 
cii" ~- ::;- V---;; 8 3 4 9.00 el 12 5 6.50 new Bl 20 I88 0.768 0.784 201.24 
O-Q? :,,..-.. ~Cb '-I :: Cb 8 39! 9 . 00 i n 511 6 . 5 0 n ew B2 20 I88 0.75 5 0.771 201.24 160:2m2 ,2m3 ; fd=1003.315 
~ c., ~ Cb Q 821 10.50 el 108 6. 5 0 n e w Bl 20 I 88 0.770 0.786 20 1 .24 
:=:: ? ;::. ~ ; 831 10. 5 0 in 3 1 6 6.5 0 new B2 20 I88 0 . 770 0 . 786 201.24 160:2m2,2m3 
::: ? ~ o ::;- :=:: 832 10.50 in 201 6. 5 0 n e w 8 2 20 I8 8 0.770 0.786 201.24 160:2m2,2m3 
- ?> i:: C/l C/l 
.,.... cii" u, c ?> 767 12 . 00 el 308 5.00 n e w B2 20 I88 0. 766 0. 784 201.24 
::;- ,_, t::": O"',.... 9 15 12.00 in 60 0 .00 new B2 20 I88 0.765 0.788 201.24 
~'O ~ ~ M 916 12.00 in 489 0 . 00 n e w B2 20 I88 0.765 0 . 789 201.24 
Cl ,_., . .0 "Cl 
i.n O c 773 13.50 el 233 6.50 new B2 20 I88 0.770 0.786 201.24 
,..... ?: ? Cb II 907 13.50 in 478 0.00 new B2 20 I88 0.764 0.785 201.24 9Be:Lor 
n('t) = =r:v 
2... ~ *-- .,,_..,. . 0 780 15 . 00 el 279 6.50 n e w B2 20 I88 0.770 0 . 786 201.24 ,.,._ i:: .:-'. " 5 Ca; : 786 15.00 in 266 6.50 new B2 20 I88 0.770 0.786 201.24 ~ ;3 ,..... 787 15 . 00 el 138 6.50 n e w B2 20 I88 0 . 770 0.786 201.24 
:=:: ::;- O.. C<b  > 788 15.00 el 224 6 . 50 n e w 82 20 I88 0.770 0 . 786 201.24 Cb Cb O"' + 789 15 . 00 el 91 6 . 50 new B2 20 I88 0.770 0.786 201.24 
u,:::: '-< 790 15 . 00 in 181 6 . 50 new 82 20 I88 0.770 0.786 201.24 
n :;?,. +:::  z > 792 15.00 in 435 6 .50 new B2 20 I88 0.770 0.786 201.24 E.. Cb 0 + 793 15.00 in 300 6.50 new B2 20 I88 0.770 0.786 201.24 (t) r:n ~ ~ 
-, ?> -? ?> 801 16.50 in 97 6.50 new 8 2 20 I88 0.770 0.786 201.24 160 :2 pl ,lml,2p2, 2p3;9 Be : 
Cl =,_,  Cb g,.....  802 16 . 50 in 359 6.50 new B2 20 I88 0.770 0.786 201.24 160:2pl,lml,2p3. ? 5m l ~ ~ 803 16 . 50 el 150 6.50 n e w B2 20 I88 0.770 0.786 201.24 !60:0pl,3ml,2pl,lml,2p3; 
0 -' - 9Be: 3ml , 5ml 
,..., ~ i:: ,..... 604 1 8 .00 el 72 9.00 new B2 60 E87 0.732 0.754 201.53 
'--,.., ;:.: ~ ::,- 60 5 18. 00 in 20 3 9. 00 new B2 60 E87 0. 732 0 . 75 4 201. 53 
::,- Q... Cb 809 18.00 el 167 6.50 new B2 20 I88 0.770 0.786 201.24 9Be:3ml 
o' u, ::n 815 18. 00 el 23 6. 50 new 82 20 I88 0. 770 0 . 786 20 1. 24 9Be: 3ml 
,_, n~  ,,... ,..., 816 18.00 in 152 6.50 n e w 82 20 I88 0 . 770 0. 786 201.24 u, ~ ~ 818 18 .00 in 4 97 6.50 new 82 20 I88 0 . 770 0 .786 201.24 
'O O (l) 0 
:: ? 0.. u, -, 597 19.50 el 92 9.75 new B2 60 E87 0 .732 0.754 201.53 
~ ~ 0 ?> 598 19.50 in 245 9.7 5 new B2 60 E87 0.732 0.754 201.53 
i:: o-  0.. 6 10 19.50 el 66 9.75 new 82 60 E87 0.732 0.754 201.53 'O ~ ~ 819 19.50 el 100 6.50 new B2 20 I88 0.770 0.786 201.24 160:2ml ,Oml,2m3,4 p 3 
----- 0? : :~: ? 0....,. . ~ 590 21.00 el 106 10 . 50 new B2 60 E87 0 .732 0.754 201.53 
__. ~ Qq C ------------------ ------
0 ;j = 
..., - ?> :fl 
>og~S.,H 
.... . n ?> Ill --
~ 0 ~ b,. ~ \ Run Angle By Evts Tangle Tgt Ap If;--C~p Polup Poldn Energy Excluded States 
i:: 0.. "' 0.. ('D --- - - --- =======-=================--===--
"O ~ ...., ro 591 21.00 in 233 10.50 new B2 60 E87 0.732 0.754 201.53 
----."O ~ 0.. t:rj 613 21.00 el 103 10.50 new B2 60 E87 0 .732 0 . 754 201.53 
.:: ..., 0 ..., ~ 
.___,, 0 ~ i:: 584 23.50 el 65 11.75 new B2 60 E87 0.732 0.754 201.53 
0 c:;- ~-::: 585 23.50 in 550 11.75 new B2 60 E87 0.732 0.754 201.53 
..., Ca boq"..'.. . 617 23.50 el 51 11.75 new B2 60 E87 0.732 0.754 201.53 <J'J o-
'8. - ?> <J"J .___,, 501 25.00 in 210 12.50 new B2 20 E87 0.732 0 . 754 201.53 
::: ?> ~ O.. tJj 507 25.00 in 156 12 .50 new B2 60 E87 0.732 0.754 201.53 
6.. ::: ~ ro ro 
O - ::: 0 577 26.50 el 104 13.25 new B2 60 E87 0.732 0.754 201.53 
~ . ..:: O'" ~ 578 26.50 in 230 13.25 new B2 60 E87 0.732 0.754 201.53 
::: ::< ro ro ..., 
n, 0 0.. e 627 27.25 el 130 13 .6 3 new B5 60 E87 0 .73 2 0 . 754 201.53 all 
..-.. ~ ~ ,-J 
._R__.,.,  2 ...... 0-<J'J ()Q ? 508 28.00 el 159 14.00 new B2 60 E87 0.732 0.754 201.53 
........ ,0,.. ... ::: P 515 28.00 in 552 14.00 new B2 60 E87 0.732 0 . 754 201.53 C/l ,...,, ~ 
0<.S . .~... . ;..:..:.. :. V M 571 29.50 el 78 14 . 75 new 82 60 E87 0.732 0.754 201.53 ::< '"Cl 
ro ;::. ro - 633 30 . 25 in 156 15.13 new B5 60 E87 0.732 0.754 201.53 16O:4p3;9Be:lrnl 
.i::.. ::: -o ro II 
W ~ - ?> @ ~ t0 516 31.00 in 541 15.50 new B2 60 E87 0.732 0.754 201.53 
~ ,..... ~ ~- ::;- o 517 31.00 el 250 15.50 new B2 60 E87 0.732 0.754 201.53 
;:::_ pi O 0 
Cb Cb i:: C 634! 33.25 el 124 16 .6 3 new B2 60 E87 0.732 0.754 201.53 fd=5182.256 
_ ... <J'J o-2= 
?> :=-: <J"J <ro  526 34.00 el 121 17 . 00 new B2 60 E87 0.732 0.754 201.53 16O:2rn3;9Be:lpl <.,J.".J. . ~o  ::: ~ 528 34.00 in 546 17 . 00 new B2 60 E87 0.732 0.754 201.53 16O:3rn4,4p3;9Be:lpl ro -
8 g: ~--;:;' 6 37 36.25 el 68 18.13 new B2 60 E87 0.732 0.754 201.53 16O:2p3 
- ro ? ::: o 
~ 3 ::: .,.- ::: 532 37.00 el 294 18 . 50 ne w B2 60 E87 0. 732 0.754 201.53 
3 - ? ~ 1-1 ,,-,.. 534 37.00 in 287 18.50 new B2 60 E87 0.732 0.754 201.53 
=c-+, * C...__, 
- - ... 6 39 39 .2 5 el 55 19.63 new B2 60 E87 0.7 32 0.754 201.53 16O:lml,0ml,4p2,2p3,0p3, 
ro o.. ?O '" Z ?3m4,4p3,4m1;9Be:3ml Q<J"J  gro  _ 0 540 40.00 el 111 20.00 new B2 60 E87 0.732 0 .7 54 201.53 ~ 541 40.00 in 353 20 . 00 new B2 60 E87 0.732 0.754 201.53 
- ~ +.... e+ ro ro  o? 546 43.00 in 326 21.50 new B2 60 E87 0 .732 0.754 201.53 
~: __: :< ::: 547 43 .00 el 172 21.50 new B2 60 E87 0.7 32 0.754 201.53 
(') . 
0 C ~ ;:::_ 64 5 45 .2 5 el 84 22.60 new B5 60 E87 0 . 732 0 .75 4 201.53 16O:3ml;9Be:3ml 
1-1 ::: Q.. ro 552 46.00 el 107 23.00 new B5 60 E87 0 .732 0.754 201.53 
'--, ~ ;=; ;n 553 46.00 el 454 23.00 new B5 60 E87 0.7 32 0 .75 4 201.53 
o';:: (b ~ 
t-1 =- a.. e-
11 - 11 II 
II II II 
II II II 
II II II 
II II II 
II II II 
II II r-- II 
II II .... II 
II II r-- II 
II II II 
II u, II .a.:..,  II II Cl> II II 
II .._, II N II 
II"' II II II 
II.._, II '1J II 
II Vl II .... II 
II II II 
111' II II 
II Cl> II.-< N I 
111' II la C. 
I' I' ... N f .:-:I <
I u II 0 0 
>< 11 \.0 
"' II ,.... "...'.  
II 
II 
:>, IIM M M M 
t,, 11'1 11'1 11'1 11'1 
"Q)  .... .... .... .... 
C: 0 0 0 Of 
"' N N N NII II 
C: ..,. ..,. .,. II ..,. II 
.'1..J.  11'1 11'1 11'1 "'II r-- r-- r-- r-- II 
1 0 ? II 
II o. 0 0 0 o II 
II II 
II II 
II C. II N N N NII 
II ::1 1 M M M M II 
II.-< r-- r-- r-- r-- II 
II o ? II 
II o. 0 0 0 o II 
II II 
II 
II c. r-- r-- r-- ,... II 
II ::I ., ., ., a:, II 
11 u 
II "' "' "' "'II II 
II II 
II u, 0 0 0 o 11 
11-... 
II H "' "' "' "'II II 
II II II 
II II II 
II C. II in 11'1 11'1 11'1 II 
II < II Ill Ill Ill Ill II 
II II II 
II II 
11.._, II :l ) ) ) II 
II 0>11 Cl> Q) Q) Cl> II 
II E-< II C: C: C: C: II 
II II II 
II 11 II 
0 0 o 11 
I'I  .C-lt>  If o0  
I o, , "' "' ~ II 
II c: ..,. ..,. N r-- II 
II"' N N N NII 
II E-< II 
II ..,. II 
II" 11'1 M II .._',  a, "' .0 ~ II II > "' ... ..,. II II"' II 
II II 
II II 
II :,, .... .... .... ,.... II 
II Ill I CO .. .. Cl> II 
II II II 
II II II 
II Cl> lllll 0 Ill o II 
If.-< II N 0 N o 11 
II O>II ? II 
II~ II : a..,,.  .... 11'1 II 11'1 11'1 If 
II II II 
II II II 
II II II 
11 C: 11r-- r-- 0 
II ::I II ..,. '? II "' "' "'II ""' " 11'1 11'1 If II II "' "' II II II II 
II II II 
11 -11 II 
Table E.3: c) BeO runs at Ep = 200 MeV (cont .) Notation: the fi.r st 
of added runs is denoted by ">", each subsequent run by "+"; excluded 
states following a ?A>> belong to the previous line. An "*" denotes a run with 
encoder problem, an "!" a run with a scaler problem ; tl1 e scale factor J for 
spin-up ( u) or spin-down ( d) is given in the last column . 
437 
-a P> ::: en ? .....:i 
.... ::: > r-:l 
7 - ~o..~ Run Angle By Evts Tangle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States 
0.. . ..:: cr- (1) C"' ==== 
0 ~ (1) ::: ;- > 868 -7.50 el 38 0.00 thk Bl 6 I88 0. 717 0.733 201. 2 4 
~ ..... 0 + 869 -7.50 el 34 0.00 thk Bl 6 I88 0.717 0. 733 201.24 
=<=;  P., :0:::: : (-1) -L. .. J 
_____ .... o-qo..;i:,. 862 -6.00 in 427 6.50 thk 82 6 I88 0. 717 0.733 201.24 2m2 
863 -6.00 in 708 0.00 thk 82 6 I88 0.717 0. 733 201.24 
R. C ,,... 0-
,___, ::: 0 ? 
en . ~ c. ::: 841 7.50 el 42 6 . 50 thk Bl 20 I88 0.749 0.765 201. 24 Opl -?:.., 842 7.50 el 244 6 . 50 thk Bl 20 I88 0.747 0.763 201.24 
()-q ~Cb V ... 845 7.50 in 454 6.50 thk 82 20 I88 0.739 0.755 201. 24 <. ::i--'O -~ 0 > 848 7. 50 in 263 6.50 thk 82 6 I88 0 . 729 0.745 201.24 
+ 849 7 . 50 in 233 6.50 thk 82 6 I88 0.727 0.743 201.24 6 .91 
gp,~<( 'l)Pl Pl 850 7. 50 in 500 6.50 thk 82 20 I88 0. 725 0. 741 201.24 6.91 >; 
s? ~ oc:: ? 8- :: 835 9.00 el 306 6.50 thk Bl 20 I88 0.767 0.783 201.24 Opl ?,l ~ 
~i--cnC/l(/l 838 9.00 in 449 6.50 =-~ thk 82 20 I88 0.759 0.775 201.24 i---~ p., 
(1) ... . cr- - 822 10.50 el 94 6 . 50 thk Bl 20 I88 0 . 770 0.786 201.2 4 
..... 'O ~ ~ tr:1 828 10.50 in 327 6.50 thk Bl 20 I88 0.770 0.786 201 . 24 4ml ,lml,6.03 ,4 p2 ,6. 91 , 
~ .... . ..c -0 829 10.50 in 189 6.50 thk 82 20 I88 0.770 0.786 201.24 ? 1m2,7.11 
,,... 0 C 
~ ?'. ? (1) II 766 12.00 in 352 5.00 thk 82 20 I88 0.765 0.782 201.2 4 6.9 3 
0u := 
('t) ~ :::: 
- ::l ,,... tv >l'>- - -i-J?  *= :: .... a 774 13.50 el 147 6.50 thk 82 20 I88 0 . 770 0 . 786 201. 24 c:,.,; ;3 .,.... 0 ~ 775 13. 50 in 62 6. so thk 82 20 I88 C 0 . 770 0 . 786 201. 24 
00 ~ :=-o...~ ~ + 776 13.50 in 241 6.50 thk 82 20 I88 0 . 770 0. 786 201. 24 cr- + 777 13.50 in 202 6.50 thk 82 (1) (1) (1) 20 I88 0. 770 0 . 786 201.24 6 .91 , 7 . 11 
en = '--< < 
0 781 15.00 el 243 6 . 50 thk Bl 20 I88 0 . 770 ("l 0.786 201. 24 
Pl .,.... + Z > 785 15.00 in 278 6 . 50 thk 82 20 I88 0. 770 0.786 201 . 24 ..... (1) 
- 0 + 794 15.00 in 334 6.50 thk 82 20 I88 0. 770 0 . 786 201 . 24 (1) "?''  >--) _ _-- ~ + 795 15.00 in 34 6 6.50 thk 82 20 I88 0. 770 0.786 201.24 lm 2 
Pl ,; (1) ,,... 
~ c:: >< --,. 797 16.50 in 537 6.50 thk 82 20 I88 0.770 0.786 201.24 o = n... .. 8 + 798 16.50 in 326 6. 50 thk 82 20 I88 0.770 .... 0.786 201.24 > 804 16.50 el 53 6 . 50 thk 82 20 I88 0. 770 
~ 0.786 201. 24 oC ..- + 805 16 . 50 el 78 6 . 50 thk 82 20 I88 0. 770 0.786 201.24 
'--;., ~ ("O ==-
o' ;:;-- 0.. (1) 603 18.00 el 71 9.00 thk 82 60 E87 0.732 0 . 754 201 . 53 
.... =(1)  "' ::ti 607 18.00 in 208 9 . 00 thk 82 60 E87 0.732 0.754 201. 53 c-+- f"""1 810 18.00 el 157 6.50 thk 82 20 I88 0.770 
cnn?'cn 0 . 786 201. 2 4 
'O O .,.... .,.... 811 18.00 in 675 6.50 thk 82 20 I88 0 . 770 0. 786 201.24 2p2 ,4 pl , 3m 2 ,3m 3 
5? ~ 1012 18.00 el 69 9.00 thk 82 20 E88 0 . 743 0.. 0 0 . 763 201 . 40 
' (1) ...... 
C '""' Q' Pl 595 19.50 el 99 9.75 thk 82 60 E87 0.732 0. 754 201.53 
'O'"d :::::0.. > 600 19.50 in 152 9 . 75 thk 87 60 E87 0.732 0 . 754 201. 53 
.... 0 0.. + 601 19.50 in 404 9.75 thk 87 60 E87 0. 732 0.754 
g.. 2.. 201 . 53 2p2 ,4pl ? - -~ ? 820 19.50 el 99 6.50 thk 82 20 I88 0 . 770 0.786 201.2 4 ----- 1007 19 . 50 el 83 9 . 75 thk 82 60 EBB 0. 74 4 0.76 4 (1) - .., 201. 40 o 3o--qC 
..., -- ~ f;; 
~.... . ;~:;  ~ .0.. ... L...J ~ ~ 
=?o;;~C"' -------Run Angle By Evts Tangle Tgt Ap Ifs Cup Polup Poldn Energy Excluded States 
i;::O..
'O (b ..c...n,oo..(.1.)- 
...., 0 (b 589 21. 00 el 112 10.50 thk B2 60 E87 0.732 0.754 201. 53 
? '"Cl :::: 0.. t:fj 593 21. 00 in 540 10.50 thk B7 60 E87 0.732 0.754 201.53 6.93,7.11 
...., 0 ? 1004 21. 00 el 82 10.50 thk B2 60 E88 0 . 744 0.765 201.40 
...__.. 0 ::a: ...., ~ 
o lcor-----":::-::, ? e 583 23.50 el 101 11.75 thk B2 60 E87 0 . 732 0.754 201.53 1-1 ',-1 
s(b, ..... CJl 587 23.50 in 566 11. 75 thk B2 60 E87 0.732 0.754 201.53 lm2 ~ V'< O"' 
CJl ...__.. 
..... - ?' > 503 25.00 el 13 12.50 thk B2 20 E8 7 0.7 32 0.754 201.53 
::: .... + 504 25.00 el 70 12.50 thk B2 20 E87 0 . 732 0.754 201.53 
' ?' 0 
0.. ::: > ~ 505 25.00 in 481 12.50 thk B2 20 E87 0.732 0.754 201.53 6.93 :::: :::, 
0 Q 999 25.00 el 62 12.50 thk B2 60 EBB 0.745 0.766 201.40 ?' 
::a: O"' 0 
::: - (b .,..... - (b ...., 574 26.50 el 69 13.25 thk B2 60 E87 0.732 0 . 754 201.53 
?' 0 0.. C > 580 26.50 in 213 13.25 thk B2 60 E87 0.732 = 0.754 201. 53 A... = + 581 26.50 in 301 13.25 thk B2 60 E87 0.732 0.754 201. 53 ...__.. ...., ()q O"' CJl 
..... B ~ ~ .?,..'. . 946* 26.60 el 115 9 .00 thk B2 60 E88 0.760 0.783 201.24 CJl ,..., 0 ::: 
()q ::a: ~ v_ M 512 28.00 el 60 14 . 00 thk B~ 60 E87 0 . 732 0 . 754 201.53 
-<?  .-,.....?  ,- ..... ""Cl 513 28.00 in 498 14.00 thk B2 60 E87 0.732 0.754 201.53 (b -
~ :=- "d (b II 
~ ~ p., 568 29 . 50 el 151 14.75 thk B2 1-1 60 E87 0.732 0.754 201.53 w ',-1 t--.:l 569 29.50 in 334 14.75 thk B2 60 E87 0 . 732 0 . 754 201.53 CD - ? (b (l =::::, C/l < ~ 0 
.,.... n?'  -0 ? o > 520 CJl 31.00 el 240 15.50 thk B2 60 E87 0.732 0.754 201 . 53 
::;- ::;- i:: C ~ + 521 31.00 el 567 15.50 thk B2 60 E87 0.732 0.754 201.53 lrn2 
(b ' " CJl O"' '""' 
- l'--1 - C/l (<1')  > 524 34 . 00 el 569 17.00 thk BS 60 E87 0.732 0 . 754 201.53 ~~ 8"d ;~::;? -(gb  __ + 525 34.00 el 437 17.00 thk BS 60 E87 0.732 0 . 75 4 201.53 
(l O"' (b ("l > 530 37 . 00 el 577 18.50 thk BS 60 E87 0.732 0.754 201.53 
2..CO~~g + 531 37.00 el 480 18.50 thk BS 60 E87 0.732 0 . 754 201.53 
C ::, ,..., .,.... 
s: ~2'-' 538 40.00 el 535 20.00 thk BS 60 E87 0.732 0 . 754 201.53 
~ *:::: 539 40 . 00 el 97 20.00 thk B2 60 E87 0.732 0 . 754 201.53 t:"+- :j 
0.. O"' ~ > 967* 41 . 60 el 155 22.50 thk B4 60 EBB 0 . 757 0 . 780 201.24 (b 
CJl (b '< .,.... + 968* 41 . 60 el 112 22 . 50 thk B4 60 EBB 0 . 757 0 . 780 201.24 
n ::= p., 
?' 0 ::: .,.... > 54 4 43.00 el 548 21.50 thk BS 60 E87 0 . 732 0.754 201.53 
~ ~ + 0 + 548 43.00 el 551 21.50 thk BS 60 E87 0 . 732 0 .75 4 201.53 
C/l :::: ::: 
~ ~ - ? 972* 45.60 el 298 24.50 thk B4 6 0 EBB 0.757 0 .779 201.24 
~ i-1 ~ c--
0 =C  >< ;::;-' > 550 46. 00 in 562 23.00 thk BS 60 E87 0 . 732 0 . 75 4 201.53 ~ ~ (b + 551 46 .00 el 566 23.00 th k BS 
'--, :;; g_ 60 E87 0. 732 0. 75 4 201.53 - ? ::n ..., 
o..., ' .~...,  ~ ~ 
11 - 11 II 
II II II 
II II II 
II II 
II II 
II II 
II II 
II II 
II II 
II II 
t/l II II 
a, II II 
.? II II 
lt1 II II 
.? II II 
1/l II II 
II II 
,:, II II 
I a, II II 
1J II II 
::l 11 II 
..-, II II 
U II II 
" II II 
~ II II 
II II 
II II 
>, II MM MM MMM II 
O'l II U"I Lf'I 
"'II "'"' "'"'"'? II 
a, II .-4.-4 .......... .-4.-1.-4 II 
t:: 1100 00 ooo 11 
t&J !I NN NN NNN 
II 
II 
c:: II <!'<I' <!'<I' <!'<!'<!' 
,.1..J,  IIII  "r-'-r"--' "r-'-r"--' "r-'-r"--'r"--' 
o II 
"- fl oe 00 000 
II 
II 
0. NN NN NNN 
..:.:.l  MM MM MMM r--r-- r--r-- r--r--r--
0 
"' 00 00 000 
0. r--r-- r--r-- r--r--r--
::l co co co co co cO cO 
u ~~ ~~ ~~~ 
I ..t./.l  00 00 000 
II' "'"' "'\0 H "'"'"' 
II 
II 
a, II II) Ll"I 
< II ai ai "ai' "a'i  "ai' "a'i "a'i 
II 
II 
.? II -"-" 
0'11..C::..C:: "..c:':"..c:':  "..c:':"..c:':"..c:':  
E-< II .? .? .? .? -LJ ,? ,? II 
II II 
II II 
Q.I 11 00 00 ooo 11 
..-4 lllOtn 00 u, LI) LI) II 
o, II ? II 
,::: 1\ -.:;J+'Q' \0"' r- r- r- II 
rU II NN NN NNN II 
E-< II II 
II II 
tn llr---O'I .nao \Or-0\ II 
-L.I II .._,. 11"1 r- \0 \0 II 
:> II tn a.n ""''""''  U') U"'I ..n II ~ II II 
II II 
II II 
I >, II .-< .-< .......... ............... 
ai II a, a, ., ., ., ., ., 
II 
II 
muoo 00 000 
..-4 II OO 00 000 
o, II 
!i! 1II " NN ! -.;t"<-.';f''  
II "'"' ""''""''""''  
II 
II 
c:: Ii"'"' 0\0 ::l llU'"\U'"\ 
P: II ,n ,n "'"' \
""0""'r'"--' 
II "'"' "'"'"' 
II ,... + ,-, + ,-, ++ 
II 
- II 
Table E.4: c) 4?Ca. runs a.t Ep = 200 MeV (cont.) Notation: th e first 
of added runs 1s denoted by ">", ea.ch subsequent run by "+"; exclud ed 
states following a "" " belong to the previous lin e. An "*" denotes a. run with 
encoder problem, an "!" a run with a scaler problem ; th e scale factor J for 
spin-up (u) or spin-down (d) is given in the last column . 
440 
F Data Tables 
The notation for these tables is as follows: angles are listed in de-
grees . Center-of-mass momentum transfers qc.m. are listed in units of fm - 1 . 
Center-of-mass differential cross sections a-c.m. ( q) are quoted in mb / sr and 
their uncertainties 6.a- in percent; the uncertainties in the analyzing pow-
ers are absolute quantities . The tabulated uncertainties include statistical 
errors, fitting errors and uncertainties from the polarization . In the case 
of multiple measurements a weighted average is given, with the uncertainty 
being the larger of the uncertainty of the mean or the standard deviation 
about the mean . The uncertainty in the laboratory angle may be as large as 
0.2? . The overall reproducibility of the data is estimated to be within 3% , 
the uncertainty in overall normalization is estimated to be about 5%. 
F.1 9Be(p,p') and 16 0(p,p') at Ep 100 MeV 
441 
ec.m. I- qc.m. I O"c.m .(q) I l'io-(%) I Ay I L\Ay I 
-15.80 17 .71 0.620 l.654E+02 0.5 0.260 0.007 
-13 .80 15.47 0.542 2.181E+ 02 0.6 0.222 0.007 
-11.80 13.24 0.464 2.756E+02 0.8 0.220 0.005 
-9.80 10.99 0.386 3.170E+02 1.0 0.201 0.005 
14.20 15 .92 0.558 l.963E+02 0.6 0.253 0.008 
16.20 18.16 0.636 l.467E+02 0.7 0.261 0.009 
18 .20 20 .39 0.713 9.974E+0l 1.3 0.296 0.005 
22.20 24.85 0.867 4.314E+01 3.1 0.307 0.037 
24 .20 27 .08 0.942 2.566E+01 0.5 0.383 0.007 
26 .20 29 .30 1.019 l.555E+0l 3.2 0.414 0.010 
30.20 33.73 1.169 5.214E+ 00 3.7 0.489 0.048 
34.20 38 .14 1.316 2.185E+oo 2.3 0.476 0.019 
38.20 42 .53 1.461 l.436E+oo 5.4 0.441 0.017 
42.20 46 .91 1.604 1.056E+00 5.1 0.424 0.024 
46.20 51.26 1.743 6.917E - 01 2.8 0.384 0.030 
50.20 55.58 1.878 4.689E- 01 4.9 0.288 0.026 
54.20 59.88 2.011 3.093E- 01 1.2 0.186 0.025 
58.20 64 .15 2.139 l.935E- 01 0.6 0.018 0.010 
62.20 68.39 2.264 l.180E- 01 2.8 -0.117 0.012 
66 .20 72.59 2.385 7.587E - 02 2.5 -0.230 0.008 
70.20 76.77 2.502 5.359E- 02 5.0 -0.273 0.028 
Table F .1: a) 9 Be(f, p)3 / 2~; Er = 100 MeV 
442 
O"c.m.(q) I 60-(%) I Ay I 6Ay I 
18 .20 20.41 0.711 4. 105E- 02 94 .2 0.765 1.159 
22.20 24 .87 0.864 7.244E - 02 16.5 0.165 0.406 
24.20 27 .10 0.939 l.030E - 01 12 .7 -0.087 0.190 
26 .20 29 .32 1.015 1.129E- 01 6.7 0.243 0.137 
30.20 33 .76 1. 164 9.965E - 02 15 .3 0.385 0.066 
34.20 38.17 1.311 5.806E- 02 12.8 0.390 0.037 
38.20 42 .57 1.456 3.175E- 02 20 .6 0.208 0.086 
42 .20 46.95 1. 597 2.267E- 02 7.7 0.210 0.082 
46 .20 51.30 1. 736 l.449E- 02 15.5 0.041 0.056 
50.20 55 .62 1.871 l.045E - 02 12.4 0.085 0.076 
54.20 59.92 2. 002 l.205E- 02 8.2 -0.188 0.046 
58.20 64.19 2. 128 l.244E- 02 6.5 -0.099 0.095 
62 .20 68 .44 2.255 l.093E- 02 10.9 0.009 0.868 
66.20 72.65 2. 375 2.335E- 03 17.1 -0.174 0.11 9 
70.20 76 .83 2.489 l.552E- 03 5.0 -0.216 0.072 
Table F.1: b) 9Be(_p,p' )l / 2i (1.680 MeV) ; Ep = 100 MeV 
443 
0-c .m.(q) I Lio-(%) I Ay I LiA11 I 
-15 .80 17.73 0.617 4.119E+oo 3.2 0.101 0.042 
-13 .80 15.49 0.540 3.479E+ oo 4.4 0.131 0.057 
-11.80 13.25 0.463 2.409E+oo 3.9 0.048 0.051 
-9.80 11 .01 0.385 1.658E+oo 4.8 0.196 0.068 
14.20 15.94 0.555 3.596E+00 4.7 0.144 0.063 
16.20 18 .18 0.633 4.837E+oo 3.9 0.076 0.051 
18 .20 20.42 0.710 5.488E+00 4.1 0.189 0.032 
22 .20 24 .88 0.862 6.361E+oo 8.4 0.199 0.038 
24.20 27.11 0.937 5.961E+00 1.0 0.258 0.014 
26 .20 29 .33 1.013 5.143E+ oo 6.1 0.289 0.007 
30.20 33 .77 1.162 3.835E+oo 1.8 0.345 0.035 
34.20 38 .19 1.309 2.334E+00 4.2 0.282 0.008 
38 .20 42 .59 1.453 1.538E+oo 4.9 0.176 0.009 
42.20 46.96 1.594 1.051E+oo 2.7 0.024 0.011 
46.20 51.32 1.732 7.419E - 01 2.2 -0.104 0.023 
50.20 55 .64 1.867 5.617E- 01 2.9 -0.188 0.010 
54.20 59.94 1.999 4.123E - 01 1.9 -0.270 0.016 
58 .20 64 .22 2.127 2.470E - 01 3.6 -0.325 0.040 
62.20 68 .46 2.251 1.442E- 01 3.0 -0.390 0.017 
66.20 72.67 2.368 7.922E - 02 1. 7 -0.435 0.017 
70 .20 76.85 2.484 3.911E- 02 0.8 -0.488 0.010 
Table F .1: c) 9 Be(p,p')5 / 21 (2 .429 MeV); Ep -= 100 MeV 
444 
24.20 27 .11 0.936 1.622E- 01 71.4 -1.358 1.658 
26 .20 29.34 1.013 1.313E- 01 73 .3 0.371 0.931 
30.20 33.78 1.161 1.158E- 01 39 .1 0.148 0.526 
34.20 38.20 1.308 5.268E- 02 34.9 0.138 0.630 
38.20 42 .59 1.450 4.729E- 02 62.7 0.545 0.864 
42.20 46 .97 1.591 7.669E- 02 13.4 0.104 0.191 
46 .20 51.32 1.729 5.917E- 02 17.4 0.280 0.256 
50.20 55.65 1.866 5.749E- 02 38.9 0.183 0.202 
58 .20 64.23 2. 123 2.082E - 02 15 .4 -0.561 0.217 
70 .20 76 .86 2.482 8.412E- 03 7.0 -0.262 0.103 
Table F.1 : d) 9 Be(p,p' )l / 21 (2.78 MeV) ; Ep = 100 MeV 
445 
18.20 20.42 0.709 1.732E- 01 25 .6 0.347 0.566 
22 .20 . 24 .89 0.861 2.274E- 01 14.0 0.550 0.216 
24.20 27.12 0.936 l. 545E- 01 22.8 0.659 0.334 
26.20 29 .34 1.012 1.698E- 01 17.3 0.419 0.168 
30.20 33.78 1. 161 1.131E-01 11.1 0.663 0.142 
34.20 38.20 1.307 9.732E - 02 6.9 0.319 0.146 
38.20 42.60 1.451 7.706E- 02 12.2 0.307 0.113 
42 .20 46.98 1.592 4.813E- 02 19.1 0.467 0.073 
46.20 51.33 1. 730 3.200E- 02 41.9 0.189 0.383 
50.20 55 .66 1.865 1.540E- 02 20.3 -0.106 0.378 
54 .20 59.96 1.996 2.031E- 02 17.4 -0.442 0.255 
58. 20 64.24 2.121 6.535E - 03 15.3 0.189 0.236 
62.20 68 .48 2.245 5.755E- 03 7.3 0.036 0.109 
66 .20 72.69 2.364 5.300E-03 14.4 -0.095 0.213 
70 .20 76 .87 2.480 1.357E- 03 59 .6 -0.263 0.810 
Table F .1: e) 9 Be(p, p' )5 / 2t (3 .049 MeV); Ep = 100 MeV 
446 
O"c.m.(q) I ~u(%) I Ay I ~Ay I 
18 .20 20.44 0.707 6.190E- 0l 22 .1 0.479 0.366 
22 .20 24.91 0.859 2.982E- 0l 21.6 0.557 0.272 
24.20 27.14 0.933 l.697E - 0l 44.2 -0.268 0.688 
26 .20 29.37 1.008 l.358E- 01 33.6 -0.214 0.510 
34.20 38 .23 1.302 l.059E- 01 15.8 0.631 0.200 
42.20 47.02 1.584 3.006E - 02 23 .3 0.643 0.356 
46 .20 51.38 1.721 8.871E- 03 54.7 -0.463 0.847 
50.20 55.71 1.855 6.220E- 03 50.6 -0 .288 0.765 
54.20 60.01 1.986 l.052E- 02 19.8 0.394 0.313 
58.20 64.29 2.113 l.142E- 02 16.0 -0.292 0.229 
70 .20 76.93 2.470 l.154E- 02 7.2 -0.391 0.103 
Table F .1: f) 9 Be(p,p')3 / 2f (4 .704 MeV); Ep = 100 MeV 
447 
O'c.m.(q) I ~u( %) I A y I .6 A y I 
-15 .80 17.77 0.615 5.733E+00 35 .7 0.190 0.474 
-11 .80 13.28 0.463 3.021E+ oo 16.9 0.120 0.222 
18 .20 20.46 0.705 4.l0lE+oo 4.6 0.250 0.062 
22 .20 24 .94 0.856 4.229E+oo 5.1 0.037 0.073 
26.20 29 .40 1.006 3.S72E+oo 5.2 0.336 0.067 
30.20 33.84 1.152 2.101E+ 00 2.8 0.305 0.036 
34.20 38 .27 1.298 I.792E+ oo 12.1 0.274 0.023 
38.20 42 .68 1.439 1.159E+ oo 2.8 0.187 0.040 
42 .20 47.06 1.578 7.190E- 0l 4.2 0.171 0.060 
46.20 51 .42 1.714 5.531E- 01 3.8 -0.068 0.056 
50 .20 55 .76 1.848 4.810E- 01 5.2 -0.100 0.076 
54 .20 60 .07 1.977 4.253E- 0l 4.8 -0.108 0.072 
58 .20 64.34 2.103 3.179E- 01 3.8 -0.319 0.054 
62 .20 68 .59 2.226 1.864E- 01 3.1 -0.299 0.045 
66 .20 72 .81 2.344 1.853E- 01 11 .4 -0.457 0.165 
70 .20 76 .99 2.458 1.037E- 01 10 . 7 -0. 799 0.083 
Table F .1: g) 9 Be(p,p') Lor (6 .5 MeV) ; Ep = 100 MeV 
448 
0-c.m.(q) I ~a-( %) I Ay I ~ Ay I 
-15.80 16.88 0.621 3.369E+ 02 0.4 0. 251 0.005 
-13.80 14.75 0.543 4. 672E+02 0.4 0.240 0.005 
-11.80 12.61 0.465 5.914E+ 02 0.3 0.227 0.005 
-9.80 10 .48 0.386 6.770E+ 02 0.3 0.219 0.01 4 
14.20 15.1 7 0.558 4.156E+ 02 0.5 0.252 0.006 
16 .20 17.31 0.636 2.847E+ 02 0.6 0.276 0.007 
18.20 19.44 0.714 l.829E+ 02 1.0 0.293 0.002 
22 .20 23 .70 0.869 6.185E+ Ol 1.1 0.332 0. 037 
24 .20 25 .83 0.944 3.234E+ Ol 0.5 0.437 0.006 
26 .20 27 .95 1.021 l.698E+ Ol 4.0 0.527 0.005 
30.20 32.20 1.173 5.860E+oo 8.8 0.717 0.031 
34 .20 36.43 1.322 4.595E+oo 2. 4 0.666 0.016 
38 .20 40 .65 l.469 3.459E+ oo 1.3 0.673 0.012 
42 .20 44 .86 1.614 2.212E+ oo 6.9 0.734 0.018 
46 .20 49.06 1.756 1.041E+ oo 2.1 0.820 0.012 
50.20 53 .25 1. 895 5.870E - 01 4.2 0.733 0.015 
54.20 57.41 2.032 4.377E- 01 1.1 0.486 0.018 
58.20 61.57 2.165 3.370E- 01 2.0 0.309 0.01 4 
62.20 65 .70 2.294 2.363E- 01 1.9 0.187 0.018 
66 .20 69 .82 2.421 1.402E- 01 3.0 0.117 0.016 
70 .20 73 .92 2.543 6.473E - 02 4.4 0.023 0.011 
Table F.2 : a. ) 16O(p,p)0?t; Ep = 100 MeV 
449 
O"c.m.(q) I ~o-( %) I Ay I ~ Ay I 
-15.80 16.91 0.616 1.275E- 01 10.2 0.309 0.134 
-13.80 14.77 0.539 7.844E- 02 68 .1 -1.300 1.257 
-11.80 12 .64 0.463 7.987E - 02 19.3 0.438 0.326 
14.20 15.20 0.555 1. 744E- 01 43.5 0.018 0. 577 
16.20 17.34 0.631 1.062E- 0l 66 .6 1. 296 1.074 
18.20 19.48 0.707 1.346E- 01 13.3 0.179 0.203 
22 .20 23 .74 0.858 1.657E- 01 11.6 0.242 0.160 
24.20 25 .87 0.933 8.654E- 02 8.6 0.106 0.190 
26 .20 28 .00 1.008 8.378E- 02 10 .1 0.212 0.136 
30.20 32 .25 1.157 6.275E- 02 11 .9 0.163 0.093 
34.20 36 .50 1.304 5.913E- 02 15.3 0.164 0.074 
38.20 40 .73 1.448 3.909E- 02 13 .2 0.268 0.162 
42.20 44 .94 1.591 3.819E- 02 15.3 0.393 0.092 
46 .20 49 .15 1.730 2.370E- 02 15.8 0.600 0.142 
54 .20 57 .51 2.001 1.478E- 02 22.0 0.191 0.094 
58 .20 61.67 2.132 1.388E- 02 7.8 0.208 0.057 
62 .20 65 .81 2.259 1.122E- 02 12.5 0.016 0.064 
66 .20 69 .93 2.381 9.047E- 03 5.5 -0.013 0.081 
Table F.2: b) 16O(p,p')0f (6 .0494 MeV) ; Ep = 100 MeV 
450 
Bc.rn. qc.rn. O"c.Tn.(q) I ~u(%) I Ay I ~ Ay I 
-15 .80 16.91 0.616 1.628E+oo 1.7 0.061 0.036 
-13 .80 14.78 0.539 1.368E+00 8.2 0.019 0.107 
-11 .80 12 .64 0.463 8.053E- 01 3.0 -0.090 0.035 
-9.80 10.50 0.387 5.922E - 0l 9.6 -0.060 0.127 
16 .20 17.34 0.631 l .763E+00 7.5 0.017 0.099 
18.20 19.48 0.707 2.408E+ 00 4.6 0.083 0.028 
22 .20 23 .74 0.858 3.492E+oo 1.0 0.129 0.033 
24 .20 25 .88 0.932 4.077E+00 1.3 0.166 0.019 
26 .20 28 .00 1.008 4.231E+oo 3.2 0.147 0.012 
30.20 32.26 1.157 4.140E+00 0.6 0.210 0.031 
34.20 36 .50 1.303 3.161E+00 3.3 0.168 0.007 
38.20 40.73 1.448 2.455E+ 00 1. 7 0.097 0.005 
42 .20 44.94 1.590 l.809E+00 2.8 -0.013 0.005 
46 .20 49 .15 1. 730 l.323E+oo 2.4 -0.138 0.019 
50 .20 53 .34 1.867 l.069E+oo 2.5 -0.255 0.019 
54.20 57.51 2.001 7.913E- 0l 3.0 -0.347 0.008 
58 .20 61.67 2.132 5.339E- 01 1.2 -0.421 0.004 
62 .20 65.81 2.259 3.093E- 0l 2.3 -0.515 0.007 
66.20 69 .93 2.383 l.662E- 0l 3.5 -0.651 0.013 
70.20 74 .03 2.503 7.605E- 02 5.0 -0.785 0.009 
Table F.2: c) 16O(p, p')31 (6.1299 MeV) ; Er = 100 MeV 
451 
-15 .80 16.92 0.615 2.462E+ 00 1.3 0.164 0.018 
-13 .80 14.78 0.540 l. 986E+ oo 6.6 0.082 0.086 
-11 .80 12.64 0.464 1.416E+oo 7.7 0.117 0.043 
-9.80 10 .50 0.388 9.810E- 0l 7.9 0.055 0.092 
16.20 17.35 0.631 2.450E+ oo 6.3 0.193 0.079 
18 .20 19 .48 0.706 2.743E+oo 1.1 0.202 0.011 
22 .20 23.75 0.857 2.510E+ oo 1.4 0.230 0.052 
24 .20 25.88 0.931 2.022E+ oo 1. 7 0.295 0.019 
26.20 28.01 1.007 l.581E+ oo 2.7 0.286 0.020 
30.20 32.26 1.155 7.207E - 01 4.5 0.308 0.035 
34.20 36 .51 1.301 3.414E- 0l 4.5 0.351 0.034 
38.20 40 .74 1.446 1.803E- 0l 2.0 0.330 0.027 
42 .20 44.95 1.587 1.285E- 0l 5.9 0.440 0.067 
46 .20 49.16 1. 727 8.527E- 02 2. 2 0.630 0.031 
50 .20 53 .35 1.863 5.318E- 02 7.7 0.810 0.049 
54.20 57 .52 1.997 3.329E- 02 10.3 0.778 0.068 
58 .20 61.68 2.1 27 2.723E- 02 19 .1 0.601 0.283 
62 .20 65 .82 2.254 2.527E- 02 2. 4 0.150 0.045 
66 .20 69 .94 2.378 2.418E- 02 1.5 -0.028 0.031 
70 .20 74.05 2.498 1.937E- 02 2. 8 -0.155 0.01 5 
Table F.2 : d) 160(p,p')2t (6.9171 MeV ); Ep = 100 MeV 
452 
Bc.m. qc.m. <Tc.m.(q) I ~<T( %) I Ay I 6. Ay I 
-15 .80 16 .92 0.615 2.048E- 0l 6.5 0.046 0.086 
-11 .80 12.64 0.464 9.898E- 02 11 .2 0.065 0.198 
-9.80 10.50 0.388 1.191E- 01 28.6 -0.343 0.408 
14. 20 15.21 0.555 8.307E- 02 69 .8 -0.266 0.958 
18 .20 19 .48 0.706 3.039E- 0l 5.8 0.122 0.091 
22 .20 23 .75 0.857 5.658E- 01 4.5 0.087 0.085 
24. 20 25 .88 0.931 6.406E - 01 3.4 0.185 0.025 
26.20 28 .01 1.007 7.404E- 01 6.7 0.180 0.037 
30.20 32 .27 1.155 7.420E- 01 5.2 0.196 0.058 
34.20 36.51 1.301 7.024E- 01 4.7 0.224 0.018 
38.20 40 .74 1.445 5.695E - 01 1.3 0.152 0.008 
42 .20 44.96 1.587 4.498E - 01 2.3 0.075 0.015 
46 .20 49 .16 1.726 3.262E- 01 2.8 -0.018 0.036 
50 .20 53.35 1.862 2.415E - 0l 4.1 -0.126 0.015 
54.20 57 .53 1.996 1.476E- 01 5.1 -0.225 0.039 
58.20 61.68 2.126 8.423E- 02 9.8 -0.348 0.027 
62 .20 65 .83 2.253 3.360E- 02 4. 7 -0.520 0.045 
66.20 69.95 2.377 1.515E- 02 2.1 -0.609 0.014 
70 .20 74 .05 2.497 6.951E- 03 4. 0 -0.465 0.027 
Table F.2: e) 16 0(p,p')l ~ (7.1169 MeV) ; Ep = 100 MeV 
453 
Bc.m. qc.m. O"c .m .( q) I fio-(%) I Ay I ~Ay I 
-15 .80 16.93 0.616 l.271E - 0l 7.2 0.394 0.095 
-11 .80 12.65 0.466 1.330E- 01 7.6 0.423 0.097 
18 .20 19.49 0.705 l.469E- 01 3.4 0.332 0.048 
22.20 23 .77 0.854 1.227E- 01 4.6 0.019 0.065 
24.20 25 .90 0.929 l.256E- 0l 4.4 -0 .029 0.066 
26 .20 28.03 1.003 1.244E- 01 3.7 -0.111 0.053 
30 .20 32.28 1.150 9.404E - 02 3.5 -0. 390 0.053 
34.20 36 .53 1.296 l.062E- 0l 5.4 -0.388 0.041 
38.20 40 .76 1.438 9. 793E - 02 2.3 -0. 393 0.034 
42 .20 44.98 1.579 8.192E- 02 2.7 -0. 393 0.039 
46.20 49 .19 1. 717 6.474E- 02 2.5 -0.333 0.036 
50.20 53 .38 1.853 4.918E- 02 2.9 -0.352 0.043 
54.20 57.56 1.985 3.109E- 02 2.5 -0.372 0.036 
58 .20 61.72 2.115 2.045E - 02 2.4 -0.384 0.034 
62 .20 65 .86 2.241 1.074E- 02 2.2 -0.222 0.032 
66 .20 69.98 2.364 6.612E - 03 2.1 -0. 168 0.031 
70.20 74.09 2.483 2.725E- 03 4.6 0.160 0.073 
Table F. 2: f) 16O(p,p')21 (8 .8719 MeV); Ep = 100 MeV 
454 
0c .m. qc.m. O"c.m.(q) I LiO"(%) I Ay I ~Ay I 
-15 .80 16.93 0.616 5.511E- 02 49.6 0.027 0.657 
18.20 19 .50 0.705 4.812E- 02 31.2 1.125 0.513 
42.20 44 .99 1.576 3.892E- 02 24 .1 -0.083 0.350 
50 .20 53 .39 1.849 l.606E- 02 49.3 -1.297 0.921 
54.20 57.57 1.982 8.294E- 03 28.0 -1.039 0.506 
58.20 61.73 2.111 l.047E - 02 11.8 -0.676 0.172 
62 .20 65.87 2.237 l.449E-02 10.1 -0.360 0.143 
66.20 70 .00 2.359 8.679E- 03 6.2 -0 .562 0.087 
70.20 74.10 2.478 l.885E- 03 51.1 -1.374 0.735 
Table F .2: g) 16O(f,p')12 (9 .585 MeV); Er = 100 MeV 
455 
G"c m (q) I ~a-(%) I Ay I ~Ay I 
-15.80 16 .93 0.617 9.041E-02 9.2 0.104 0.121 
-11.80 12.65 0.468 6.465E - 02 12 .6 -0.225 0.166 
18.20 19.50 0.705 1.152E- 01 4.0 -0.033 0.060 
22 .20 23 .77 0.854 1.321E- 0l 4.3 -0.052 0.061 
24.20 25 .91 0.928 1.365E- 0l 4.0 -0.027 0.060 
26 .20 28.04 1.002 1.418E- 0l 3.3 0.077 0.046 
30.20 32.29 1.148 1.039E- 0l 3.2 0.013 0.045 
34.20 36.54 1.294 8.546E- 02 6.6 0.126 0.077 
38.20 40 .77 1.435 6.333E- 02 2.6 0.173 0.036 
42.20 45.00 1.575 4.390E- 02 3.6 0.105 0.051 
46.20 49.20 1. 713 3.541E- 02 4.3 0.077 0.062 
50.20 53.40 1.848 3.046E-02 5.2 -0.052 0.077 
54.20 57.57 1.981 1.718E-02 4.3 -0.189 0.063 
58 .20 61.73 2.110 l.268E- 02 3.6 -0.329 0.051 
62.20 65.88 2.235 7.215E- 03 3.7 -0.491 0.050 
66 .20 70 .00 2.358 3.995E- 03 2.7 -0.578 0.037 
70 .20 74 .11 2.477 1.654E- 03 3.6 -0.860 0.049 
Table F.2: h) 160(p,p')2t (9.8445 MeV); Ep = 100 MeV 
456 
Clem (q) I ~(!{%) I Ay I ~A ' !/ 
. . . . 
. . 
16.1 -0 .045 0.213 
-15.80 16.94 0.617 
5.510E-02 
7.3 0.301 0.104 6.634E - 02 
18.20 19.50 0.705 0.093 
0.854 9.031E- 02 
6.6 0.057 
22 .20 23 .78 6.1 0.189 0.088 9.684E- 02 
24.20 25.91 0.927 3.8 0.161 0.064 1.217E- 01 
26.20 28.04 1.002 0.048 
1.147 1.114E- 01 
3.4 0.093 
30.20 32.30 0.157 0.022 
1.278E- 01 3.1 
34.20 36.55 1.293 2.0 0.098 0.029 l.043E- 01 
38.20 40.78 1.434 -0.015 0.032 
9.185E- 02 2.3 
42.20 45.00 1.574 2.4 -0.131 0.034 8.038E- 02 
46 .20 49.21 1.711 3.2 -0 .162 0.047 
50.20 53.40 1.846 
6.937E- 02 
2.5 -0.400 0.037 5.075E- 02 
54 .20 57 .58 1.978 2.1 -0 .386 0.029 
58.20 61 .74 2.107 
3.766E- 02 
-0.464 0.022 
2.356E- 02 1.6 
62.20 65.89 2.233 -0 .504 0.018 
2.355 1.454E- 02 
1.3 
66.20 70 .01 -0 .491 0.018 
2.473 7.915E- 03 
1.3 
70.20 74 .12 
Table F.2: i) 160(p,p')4i (10 .356 MeV); Ep = 100 MeV 
457 
. . c.m . c . 0.152 
6.730E- 02 11.4 
-0.113 
-15.80 16 .94 0.617 0.204 
5.482E- 02 15.2 
-0.254 
-11 .80 12.66 0.470 
6.3 -0.302 
0.097 
18.20 19.51 0.706 6.384E- 02 -0.351 0.134 
22.20 23.78 0.853 5.043E - 02 
8.8 
-0.264 0.201 13.0 
24 .20 25.92 0.927 3.060E - 02 0.191 
2.579E- 02 12.0 
-0.559 
26.20 28 .05 1.000 -0.042 0.175 
30.20 32 .31 1.146 l.790E - 02 
12.2 
0.076 
2.074E- 02 5.6 
0.169 
34.20 36.55 1.291 0.388 0.088 
40 .79 1.432 1.809E- 02 
6.4 
38 .20 0.093 
1.708E- 02 6.5 
0.544 
42 .20 45 .01 1.572 0.140 0.095 
46.20 49 .22 1.709 1.513E- 02 
6.6 
0.478 0.156 
53.41 1.844 1.118E- 02 
10.2 
50 .20 
14.2 0.408 0.232 
58 .20 61.75 2.104 4.139E- 03 
7.9 -0.026 0.116 
62 .20 65 .90 2.229 
3.578E- 03 
11.7 0.056 0.176 
66 .20 70.02 2.351 
1.037E - 03 
18 .9 0.050 0.283 3.666E- 04 
70 .20 74.13 2.470 
Table F.2: j) 16O(_p,p')01 (10 .957 MeV) ; EP === 100 MeV 
458 
D"c.m.(q) I ~o-(%) I Ay I ~Ay I 
-15.80 16 .94 0.618 3.823E- 02 74.0 1.076 1.060 
18.20 19 .51 0.706 3.008E- 02 61.8 -0.296 1.036 
22 .20 23 .78 0.853 4.484E - 02 77 .6 -0.344 1.353 
24.20 25 .92 0.927 3.924E - 02 66 .8 0.677 1.214 
26 .20 28 .05 1.001 6.959E- 02 16.6 0.067 0.234 ','  ' 
30.20 32.31 1.146 8.522E- 02 11 .6 0.161 0.164 
34.20 36 .56 1.291 7.401E - 02 15.9 0.007 0.218 
38 .20 40 .79 1.431 8.887E- 02 7.7 0.121 0.109 
42 .20 45 .01 1.571 5.446E- 02 28 .8 0.316 0.501 
46 .20 49 .22 1.708 5.742E- 02 17.9 -0.280 0.283 
50.20 53.42 1.843 3.867E- 02 31.1 -1.228 0.546 
58 .20 61.76 2.103 3.357E- 02 29.2 -0.248 0.392 
62 .20 65 .90 2.228 2.671E - 02 17.3 -0.266 0.238 
66 .20 70 .03 2.350 2.108E- 02 18 .6 -0.260 0.264 
70 .20 74 .13 2.469 1.196E- 02 18 .3 -0.318 0.285 
Table F .2: k) 16 0(p,p')4f (11.097 MeV) ; Ep = 100 MeV 
459 
-
0c.m . %.m. <Tc.m .(q) I ~a-( %) I Ay I ~Ay I 
-15.80 16.94 0.618 1.392E+00 1.9 0.166 0.025 
-11.80 12 .66 0.472 7.839E- 0l 3.0 0.222 0.039 
18 .20 19.51 0.706 1.644E+ oo 0.9 0.197 0.013 
22 .20 23 .79 0.853 1.503E+00 1.1 0.248 0.015 
24 .20 25.92 0.926 1.254E+00 1.2 0.280 0.017 
26 .20 28 .05 1.000 9.787E- 01 1.2 0.241 0.016 
30 .20 32 .31 1.145 4.468E- 0l 1.5 0.166 0.021 
34.20 36 .56 1.289 2.356E- 01 1.4 0.198 0.019 
38.20 40.80 1.430 1.336E- 0l 1.9 0.156 0.027 
42.20 45 .02 1.570 1.032E- 01 2.3 0.200 0.033 
46.20 49 .23 1. 707 8.197E- 02 2.5 0.399 0.035 
50.20 53. 42 1.841 5.964E- 02 3.4 0.437 0.050 
54.20 57.60 1.973 3.438E- 02 3.8 0.648 0.060 
58 .20 61.76 2. 101 2.518E- 02 4.3 0.625 0.072 
62.20 65 .91 2.226 2.612E- 02 2.8 0.258 0.044 
66 .20 70.03 2.348 2.688E- 02 1.2 0.042 0.018 
70.20 74 .14 2.466 2.386E- 02 0.9 -0.044 0.013 
Table F .2: l) 16O(p,p')2f (11.520 MeV); Er = 100 MeV 
460 
CTc.m.(q) I flo-( %) I Ay I fl Ay I 
-15 .80 16 .95 0.619 9.621E- 02 8.8 0.071 0.117 
18.20 19 .52 0.706 1.456E- 01 3.4 0.080 0.050 
22 .20 23 .79 0.853 2.051E - 01 3.2 0.070 0.045 
24 .20 25 .93 0.926 2.000E- 01 3.2 0.062 0.047 
26 .20 28 .06 0.999 1.672E- 01 3.0 0.065 0.043 
30.20 32.32 1.144 9.963E - 02 3.2 -0 .107 0.047 
34.20 36.57 1.287 8.134E- 02 2.3 -0.120 0.033 
38.20 40 .80 1.429 6.320E- 02 2.5 -0.142 0.036 
42.20 45 .03 1.568 5.463E- 02 2.8 0.061 0.040 
46 .20 49 .24 1.705 2.958E - 02 4.0 0.309 0.057 
50.20 53.43 1.839 1.551E- 02 7.2 0.533 0.108 
54.20 57.61 1.970 7.318E- 03 8.4 1.193 0.154 
58 .20 61.77 2.098 7.772E- 03 6.5 0.898 0.115 
62 .20 65 .92 2.223 8.438E- 03 4.1 0.631 0.069 
66 .20 70 .05 2.345 7.362E - 03 2.7 0.387 0.044 
70.20 74 .15 2.463 5.484E- 03 2.0 0.242 0.030 
Table F.2: m) 16O(p,p')0t (12 .049 MeV) ; Ep = 100 MeV 
461 
O"c.m .(q) I ~er(%) I Ay I ~Ay I 
18 .20 19.52 0.706 9.331E- 03 82 .6 -1.376 1.982 
24.20 25.93 0.926 1.645E- 02 51.5 0.467 0.753 
26 .20 28 .06 0.999 6.207E- 02 12.1 0.060 0.170 
30.20 32 .32 1.143 1.331E- 0l 4.8 0.020 0.068 
34.20 36.57 1.287 7.560E- 02 4.6 0.316 0.062 
38.20 40.81 1.428 5.644E - 02 5.1 0.269 0.071 
42.20 45 .03 1.567 5.640E - 02 4.6 0.111 0.066 
46 .20 49.24 1.703 4.126E- 02 5.7 0.026 0.082 
50.20 53.44 1.837 2.616E - 02 9.2 0.136 0.137 
54 .20 57.62 1.968 l.600E- 02 9.4 -0.268 0.136 
58 .20 61.78 2.096 1.122E- 02 9.7 -0.433 0.135 
62 .20 65 .93 2.221 l.040E - 02 6.9 -0.450 0.095 
66 .20 70.05 2.342 2.983E - 03 19.0 -0.463 0.251 
70.20 74 .16 2.460 2.076E- 03 18.2 -1. 331 0.258 
Table F.2: n) 16 O(p, p')l3 (12.440 MeV) ; Ep = 100 MeV 
462 
. . 
-15.80 16 .95 0.619 2.445E- 01 4.8 0.012 0.064 
-11 .80 12 .66 0.474 2.143E- 01 5.8 -0.107 0.076 
18.20 19 .52 0.706 2.147E- 01 2.7 0.052 0.040 
22.20 23 .80 0.853 1.536E- 01 3.8 -0.173 0.056 
24.20 25 .93 0.926 1.302E- Ol 4.4 -0.124 0.065 
26 .20 28 .06 0.999 1.052E- 01 4.3 -0.204 0.063 
30.20 32.32 1.143 6.679E- 02 5.2 -0.328 0.077 
34.20 36.57 1.286 4.374E- 02 4.5 -0.366 0.069 
38 .20 40 .81 1.428 2.806E- 02 5.7 -0.339 0.086 
42.20 45 .04 1.566 2.419E - 02 6.2 0.109 0.088 
46.20 49.25 1. 703 1.991E- 02 7.0 0.461 0.104 
50 .20 53.44 1.837 1.869E- 02 7.9 0.285 0.118 
54 .20 57 .62 1.968 1.343E- 02 7.0 0.594 0.120 
58 .20 61.78 2.096 1.061E- 02 6.3 0.390 0.104 
62 .20 65 .93 2.220 5. 759E- 03 7.7 0.377 0.133 
66 .20 70 .06 2.342 3. 790E - 03 6.9 0.158 0.109 
70.20 74.16 2.459 1.647E- 03 12 .0 0.303 0.209 
Table F .2: o) 160(p,p' )22 (12.530 MeV) ; Ep = 100 MeV 
463 
O'c.m.(q) I ~a-(%) I Ay I ~Ay I 
18.20 19.52 0.707 7.847E- 03 75 .6 1.325 1.254 
22.20 23 .80 0.853 1.501E- 02 42 .6 0.781 0.637 
30.20 32.33 1.143 4.017E - 02 10.0 0.024 0.142 
34.20 36.58 1.286 4.682E- 02 5.4 0.016 0.077 
38.20 40 .81 1.427 2.035E- 02 9.4 -0.021 0.135 
42 .20 45 .04 1.566 1.503E- 02 10.9 -0.254 0.162 
46.20 49 .25 1. 702 1.360E- 02 10.4 -0.134 0.151 
50.20 53.45 1.836 8.895E- 03 15.9 -0 .121 0.235 
54.20 57 .63 1.967 6.313E- 03 13.7 -0.247 0.203 
58 .20 61.79 2.094 4.948E- 03 12.4 -0.517 0.180 
62 .20 65 .93 2.219 3.652E- 03 9.5 -0.128 0.137 
66.20 70.06 2.340 6.387E - 04 46 .1 0.065 0.696 
70.20 74.17 2.458 8.651E- 04 28.3 0.424 0.474 
Table F.2 : p) 16O(p,p' )02 (12.796 MeV); Ep = 100 MeV 
464 
CTc.m.(q) I ~a-(%) I Ay I ~Ay I 
-15.80 16 .95 0.620 7.014E- 0l 3.5 0.332 0.047 
18.20 19.52 0.707 6.420E- 0l 1. 7 0.303 0.025 
22.20 23.80 0.853 4.283E - 0l 2.6 0.363 0.037 
24.20 25 .94 0.926 3.277E- 0l 3.3 0.360 0.049 
26 .20 28 .07 0.998 2.590E- 0l 2.9 0.298 0.039 
30.20 32 .33 1.143 l.161E - 0l 4.2 0.165 0.059 
34.20 36.58 1.285 7.590E- 02 3.6 0.065 0.051 
38.20 40 .82 1.426 4.557E-02 4.5 -0.038 0.064 
42.20 45.04 1.565 2.994E - 02 6.1 -0.282 0.091 
46.20 49.25 1. 701 2.703E- 02 5.8 0.Oll 0.084 
50.20 53 .45 1.835 2.284E- 02 8.2 -0.121 0.1 23 
54.20 57 .63 1.966 1.312E- 02 8.1 -0.204 0.123 
58.20 61.79 2.094 9.296E- 03 8.1 -0.222 0.121 
62 .20 65.94 2.218 6.604E- 03 6.0 -0.167 0.087 
66 .20 70 .07 2.339 3.730E - 03 7.6 -0.043 0.ll2 
70.20 74 .17 2.457 2.221E- 03 10.1 0.084 0.155 
Table F .2: q) 16 O(p,p')23 (12 .969 MeV); Ep = 100 MeV 
465 
O'c.m.(q) I ~u(%) I A y j fi A y I 
-15.80 16.95 0.620 5.627E- 0l 6.8 0.192 0.092 
18.20 19.52 0.707 5.574E- 01 3.2 0.032 0.047 
22 .20 23 .80 0.853 5.658E- 01 3.3 0.101 0.047 
24 .20 25 .94 0.926 5.199E- 01 3.6 0.120 0.053 
26 .20 28 .07 0.998 4.339E- 0l 3.2 0.154 0.044 
30.20 32.33 1.142 3.656E- 0l 2.7 0.124 0.038 
34.20 36 .58 1. 285 3.389E- 01 1. 7 0.185 0.024 
38.20 40 .82 1.426 2.389E- 0l 2. 0 0.101 0.028 
42.20 45.04 1.565 l.586E- 01 2.5 0.068 0.036 
46.20 49 .25 1.701 l.050E - 01 3.0 0.005 0.044 
50 .20 53.45 1.835 6.925E - 02 4.8 -0.001 0.071 
54.20 57.63 1.965 4.527E- 02 4.3 0.002 0.063 
58 .20 61 .79 2.093 3.005E- 02 4.4 -0.186 0.065 
62.20 65.94 2.217 l.917E - 02 3.6 -0.219 0.052 
66. 20 70 .07 2.338 1.038E- 02 5.3 -0 .274 0.075 
70 .20 74.18 2.456 6.604E- 03 9.7 0.185 0.157 
Table F.2: r) 16O(p, p' )13 .08 (13 .08 MeV) ; Ep = 100 MeV 
466 
<Tc .m.(q) I ~a-(%) I Ay I ~Ay I 
-15.80 16.95 0.620 8.633E- 02 15.6 -0.165 0.209 
18.20 19 .52 0.707 l.312E- 01 5.6 0.077 0.082 
22.20 23.80 0.853 l.726E- 01 4.9 0.062 0.069 
24 .20 25 .94 0.926 l.676E- 01 4.9 0.113 0.071 
26 .20 28.07 0.998 l.585E - 0l 4.3 0.105 0.061 
30.20 32.33 1.142 l.522E- 01 3.2 0.041 0.046 
34.20 36.58 1.285 l.384E- 01 2.2 0.163 0.031 
38 .20 40.82 1.426 ? 1.0lOE- 01 2.6 0.163 0.037 
42.20 45.05 1.564 6.143E- 02 3.5 0.307 0.051 
46.20 49 .26 1. 700 3.125E- 02 5.2 0.193 0.075 
54 .20 57.63 1.964 l.082E- 02 8.8 0.035 0.131 
58.20 61.80 2.092 8.559E- 03 7.7 -0.224 0.111 
62.20 65 .94 2.216 6.320E- 03 6.1 -0 .287 0.088 
66 .20 70 .07 2.337 3.296E- 03 10.5 -0. 368 0.150 
70.20 74.18 2.455 2.931E- 03 11.2 -0.113 0.158 
Table F.2 : s) 16 0(p,p')34 (13 .259 MeV ); Ep = 100 MeV 
467 
F.2 4?Ca(p,p') at Ep = 100 MeV 
468 
Bc.m . %.m. O"c.m.(q) I ~o-( %) I Ay I ~ A y I 
-15 .80 16.23 0.621 5.352E+02 1.4 0.199 0.011 
-13 .80 14.18 0.543 1.015E+ 03 5.3 0.200 0.014 
-11.80 12.13 0.465 1.648E+ 03 2.9 0.207 0.010 
-9 .80 10.07 0.386 2.414E+ 03 2.7 0.226 0.019 
10 .20 10 .48 0.402 2.171E+ 03 2.6 0.224 0.003 
12.20 12.54 0.480 1.562E+03 6.8 0.201 0.018 
14.20 14.59 0.559 8. 720E+ 02 1.2 0.200 0.008 
16.20 16 .64 0.637 4.229E+ 02 0.7 0.193 0.004 
18 .20 18.70 0.715 1.687E+ 02 3.3 0.201 0.007 
22.20 22 .80 0.869 2.055E+ 01 0.5 0.730 0.006 
24.20 24 .85 0.946 2.168E+ 0l 0.5 0.821 0.006 
26 .20 26.90 1.023 2.879E+0l 2.5 0.710 0.003 
30 .20 31.00 1.175 2.782E+0l 0.3 0.614 0.004 
34.20 35 .10 1.325 1.250E+0l 0.4 0.643 0.005 
38.20 39.19 1.474 4.301E+oo 0.4 0.705 0.005 
42.20 43 .27 1.621 2.362E+ oo 0.6 0.709 0.007 
46 .20 47 .35 1.765 1.883E+ 00 8.7 0.668 0.009 
50 .20 51.42 1.907 1.04oE+oo 0.6 0.770 0.007 
54 .20 55.49 2.046 4.803E- 01 1.1 0.868 0.011 
58.20 59 .55 2.183 2.830E - 01 0.8 0.784 0.009 
62.20 63.61 2.317 2.326E- 01 0.6 0.596 0.008 
66.20 67 .66 2.447 1.665E- 01 0.6 0.509 0.008 
70 .20 71.70 2.574 8.274E- 02 0.7 0.512 0.008 
Table F .3: a) 4?Ca(p, p)Ot ; Er = 100 MeV 
469 
O"c.m.(q) I L\o-(%) I A ll I L\ A y I 
-15 .80 16 .24 0.617 2.323E- 02 56.9 0.114 0. 431 
-13 .80 14.19 0.540 3.630E- 02 28.4 -0.038 0.188 
-11 .80 12.13 0.462 3.590E- 02 16.6 -0.179 0.143 
-9.80 10.08 0.385 3.404E- 02 31.8 -0.235 0.184 
-7.80 8.02 0.307 2.868E- 02 49.5 1.302 0.639 
12.20 12.54 0.478 4.076E- 02 90 .2 0.830 1.319 
14.20 14.60 0.555 7.497E- 02 58 .8 -0. 755 1.008 
16.20 16 .65 0.633 5.140E- 02 39 .7 -0.095 1.119 
18 .20 18 .71 0.710 2.237E - 02 17.2 -0.176 0.297 
22 .20 22 .81 0.862 9.176E- 03 45 .1 0.028 0.437 
24 .20 24.86 0.939 1.161E- 02 8.3 0.340 0.220 
26 .20 26 .91 1.014 l.208E - 02 13.0 0.453 0.093 
30.20 31.01 1.165 l.458E - 02 5.2 0.511 0.061 
34.20 35.11 1.314 9.959E- 03 14.9 0.608 0.066 
38.20 39 .20 1.462 4.023E- 03 6.6 0.844 0.077 
42.20 43.29 1.607 4.195E- 03 8.4 0.713 0.056 
46 .20 47 .37 1.750 5.050E- 03 4.0 0.501 0.048 
50.20 51 .44 1.891 3. 474E- 03 5.4 0.677 0.053 
54 .20 55 .51 2.029 1.449E- 03 6.1 1.040 0.080 
58 .20 59 .57 2.1 64 1.185E- 03 17.6 1.117 0.048 
62 .20 63 .63 2.297 l.229E- 03 8.9 0.693 0.091 
66.20 67.68 2.426 l.115E- 03 11.7 0.608 0.064 
70.20 71. 72 2.552 6.384E- 04 9.9 0.581 0.084 
Table F.3: b) 4?Ca(p, p')0t (3 .3521 MeV ); Er = 100 MeV 
470 
17c.m.(q) I ~u(%) I Ay I ~Ay I 
-15.80 16.24 0.617 8.216E+oo 2.7 0.111 0.009 
-13.80 14.19 0.540 6.085E+oo 1.5 0.099 0.005 
-11 .80 12.13 0.462 4.556E+00 0.9 0.062 0.009 
-9.80 10.08 0. 385 3.257E+oo 4.6 0.044 0.007 
-7.80 8.02 0.308 2.345E+oo 3.1 0.036 0.040 
10.20 10.49 0.401 3.907E+oo 6.7 0.023 0.091 
12.20 12.54 0.478 5.232E+oo 5.6 0.095 0.027 
14.20 14.60 0.555 6.765E+oo 1.4 0.103 0.023 
16 .20 16 .65 0.632 8.415E+oo 3.7 0.136 0.036 
18.20 18. 71 0.709 1.042E+0l 1. 7 0.127 0.016 
22.20 22 .81 0.862 1.142E+01 0.4 0.142 0.005 
24.20 24.86 0.938 1.084E+0l 0.4 0.161 0.006 
26.20 26 .92 1.014 8.858E+oo 1.8 0.172 0.004 
30.20 31.02 1. 164 4.81 3E+oo 1.2 0.187 0.007 
34.20 35.11 1.313 2.226E+00 1.3 0.228 0.006 
38.20 39.20 1.460 1.304E+00 0.6 0.309 0.013 
42 .20 43.29 1.605 8.805E- 0l 0.4 0.433 0.008 
46 .20 47.37 1.748 4.877E- 01 0.4 0.594 0.016 
50.20 51.45 1.889 2.333E- 01 1.3 0.779 0.011 
54.20 55.51 2.027 1.661E- 01 1.0 0.648 0.008 
58.20 59.58 2.162 1.759E- 01 1.2 0.351 0.009 
62. 20 63 .63 2.294 1.554E- 01 0.9 0.173 0.008 
66. 20 67.68 2.423 9.515E- 02 0.5 0.027 0.004 
70.20 71. 72 2.549 3. 750E - 02 0.8 -0.134 0.030 
Table F .3: c) 4?Ca(p, p')31 (3 .7364 MeV) ; Ep = 100 MeV 
471 
O"c.m.(q) I L\o-( %) I Ay I L\Ay I 
-15.80 16 .24 0.617 2.673E+oo 1.9 0.155 0.031 
-13.80 14.19 0.540 2.383E+oo 2.3 0.146 0.018 
-11.80 12.13 0.462 1.885E+oo 0.9 0.144 0.011 
-9.80 10 .08 0.385 1.333E+ oo 3.5 0.130 0.028 
-7.80 8.02 0.308 1.075E+oo 4.7 0.130 0.060 
12.20 12 .54 0.478 2.235E+ oo 4.8 0.124 0.020 
14.20 14.60 0.555 2.635E+00 3.1 0.136 0.041 
16.20 16.65 0.632 2.715E+00 8.7 0.165 0.053 
18.20 18.71 0.709 2.436E+oo 2.6 0.180 0.023 
22.20 22 .81 0.861 1.311E+oo 0.9 0.200 0.020 
24 .20 24 .87 0.937 8.020E-01 0.8 0.239 0.011 
26.20 26.92 1.013 4.471E- 0l 1.4 0.303 0.017 
30.20 31.02 1.164 2.083E- 01 5.7 0.632 0.023 
34.20 35.11 1.313 2.106E- 01 1.5 0.698 0.011 
38.20 39 .20 1.460 1. 709E- 01 3.9 0.697 0.011 
42.20 43 .29 1.605 9.425E- 02 1.1 0.757 0.013 
46.20 47.37 1.748 4.784E- 02 1.4 0.743 0.015 
50 .20 51.45 1.888 3.472E- 02 3.8 0.668 0.014 
54.20 55.52 2.026 2.469E- 02 1.4 0.675 0.035 
58 .20 59 .58 2.161 1.341E- 02 1. 7 0.689 0.020 
62 .20 63.63 2.293 5.785E- 03 2.7 0.720 0.057 
66 .20 67.68 2.422 3.434E- 03 7.6 0.737 0.039 
70.20 71. 72 2.548 3.489E- 03 4.7 0.454 0.049 
Table F .3 : cl) 4?Ca.(p, p' )2i (3.9041 MeV) ; Ep = 100 MeV 
472 
-15 .80 16.24 0.616 2.075E - 01 12 .4 -0 .012 0.050 
-13 .80 14.19 0.539 1.314E- 01 21.7 -0.031 0.120 
-11.80 12 .13 0.462 9.633E- 02 8.7 -0.225 0.065 
-9.80 10.08 0.385 6.335E - 02 19.4 -0.205 0.229 
-7.80 8.02 0.309 7.670E- 02 22 .6 -0.673 0.300 
14.20 14.60 0.555 1.996E- 01 6.0 -0.162 0.082 
18 .20 18.71 0.708 2.638E- 01 4.2 -0.021 0.036 
22.20 22 .81 0.860 5.092E - 0l 2.8 0.000 0.013 
24.20 24 .87 0.936 6.664E- 0l 0.7 0.045 0.009 
26 .20 26 .92 1.012 8.367E- 01 2.3 0.048 0.014 
30 .20 31.02 1.162 1.142E+oo 0.5 0.065 0.012 
34.20 35 .11 1.311 1.248E+ oo 0.6 0.049 0.004 
38.20 39.21 1.458 1.142E+ 00 0.6 -0.006 0.017 
42 .20 43 .29 1.602 9.599E- 01 1.9 -0.113 0.004 
46.20 47.37 1.745 7. 743E- 0l 0.5 -0 .212 0.006 
50 .20 51.45 1.885 6.007E- 0l 0.9 -0.311 0.005 
54.20 55 .52 2.023 4.269E- 01 0.6 -0.388 0.005 
58 .20 59 .58 2.158 2.646E- 01 0.5 -0.472 0.004 
62 .20 63.64 2.290 1.405E- 01 0.4 -0.564 0.010 
66 .20 67 .69 2.419 6.329E- 02 0.8 -0.663 0.007 
70 .20 71.73 2.544 2.193E- 02 1. 7 -0. 742 0.031 
Table F.3: e) 4?Ca.(p, p')51 (4.4 915 MeV) ; Ep = 100 MeV 
473 
0-c .m.(q) I ~o-( %) I Ay I ~ Ay I 
-15 .80 16 .24 0.616 l.036E - 01 35 .8 1. 133 0.589 
26 .20 26 .92 1.011 3.995E- 03 66 .3 -0.388 0.789 
30.20 31.02 1.160 1.636E- 03 43 .5 0.481 0.577 
34.20 35.12 1.309 3.071E - 04 81.7 0.335 0.974 
42 .20 43 .30 1.600 4.187E- 04 64 .8 -0.408 0.903 
50.20 51. 45 1.882 l.568E- 04 229 .9 0.003 1.015 
62.20 63.64 2.286 l.204E- 04 74.5 1.240 0.967 
66 .20 67.69 2.414 6.051E - 05 101. 7 0.589 1.130 
70 .20 71.73 2.540 1.165E- 04 30 .9 0.727 0.437 
Table F.3: f) 4?Ca(p,p')0f (5 .213 MeV); Ep = 100 MeV 
474 
-15 .80 16.24 0.616 l.663E - 01 9.0 0.086 0.191 
-13.80 14 .19 0.539 l.654E- 01 7.1 -0.013 0.039 
-11 .80 12 .13 0.463 1.444E- 01 7.4 0.073 0.096 
-9.80 10.08 0.386 l.165E - 0l 11 .3 0.108 0.104 
-7.80 8.02 0.310 8.680E - 02 21.7 0.083 0.278 
12.20 12.55 0.478 l.220E- 01 15. 7 0.471 0.208 
14.20 14.60 0.555 l.679E - 01 9.2 0.018 0.124 
18 .20 18 . 71 0.708 l.802E- 01 3.0 0.055 0.040 
22 .20 22.82 0.859 l.192E - 01 14.0 -0.015 0.242 
24 .20 24 .87 0.935 9.600E- 02 3.4 0.053 0.046 
26.20 26 .92 1.010 5.969E- 02 24.5 0.052 0.211 
30.20 31.02 1.160 l.948E- 02 32 .3 0.485 0.112 
34.20 35 .12 1.309 l.768E- 02 23 .2 0.690 0.076 
38.20 39.21 1.455 l.669E - 02 15.5 0.403 0.093 
42.20 43.30 1.600 l.315E - 02 17.9 0.391 0.097 
46.20 47.38 1.742 5.476E- 03 19.6 0.716 0.223 
50.20 51.45 1.882 3.846E- 03 22.9 0.804 0.120 
54.20 55 .52 2.019 4.352E- 03 8.9 0.489 0.098 
58.20 59 .59 2.154 3.252E- 03 13.4 0.186 0.224 
62 .20 63 .64 2.285 2.880E- 03 31.5 0.474 0.153 
66.20 67.69 2.414 l.575E- 03 35 .6 0.118 0.235 
70.20 71.73 2.539 5.338E- 04 10.3 1.037 0.192 
Table F .3: g) 4?Ca(p, p')2f (5 .249 MeV) ; Ep = 10() McV 
475 
O"c.~.{q) I ~a-{ %) I Ay I ~ Ay I 
-15 .80 16.24 0.616 3.704E- 02 30 .0 0.183 0.404 
-13.80 14.19 0.539 l.955E- 02 20 .7 0.221 0.268 
-9.80 10 .08 0.386 2.187E- 02 93 .8 0.084 0.524 
-7.80 8.02 0.310 3.204E- 02 50 .7 -0.248 0.674 
18 .20 18.71 0.708 3.835E- 02 34.8 0.331 0.168 
22.20 22 .82 0.859 6.807E - 02 29.7 0.289 0.336 
24.20 24.87 0.935 6.908E- 02 9.4 0.223 0.062 
26.20 26.92 1.010 7.302E- 02 17.1 0.282 0.183 
30.20 31 .02 1.160 4.991E- 02 11 .5 0.137 0.052 
34.20 35.12 1.309 2.639E - 02 20 .0 0.059 0.102 
38 .20 39.21 1.455 l.738E- 02 18.5 0.384 0.095 
42 .20 43.30 1.599 l.081E- 02 20 .8 0.546 0.118 
46 .20 47 .38 1. 742 7.319E- 03 20 .2 0.486 0.162 
50 .20 51 .45 1.882 3.905E- 03 24.1 0.930 0.449 
54.20 55.52 2.019 2.355E- 03 19.5 0.911 0.184 
58 .20 59 .59 2.154 3.147E- 03 14.8 0.445 0.192 
62 .20 63 .64 2.285 l.908E- 03 51.0 -0.248 0.272 
66 .20 67.69 2.414 l.420E- 03 17.1 -0.263 0.246 
70.20 71 .73 2.539 5.717E- 04 13 .5 -1.178 0.204 
Table F.3: h) 4?Ca(p, p' )4i (5 .279 MeV) ; Ep = 100 MeV 
476 
CTc. ;,, ,(q) I ~o-( %) I Ay I D. Ay I 
-15.80 16.25 0.616 l.369E - 0l 19 .1 -0.256 1.280 
-13 .80 14.19 0.539 4.339E- 02 26 .7 0.278 0.330 
-11 .80 12 .13 0.463 2.123E- 0l 36.0 0.372 0.458 
-7.80 8.02 0.310 7.682E- 02 45 .1 -0.698 0.692 
18 .20 18 .71 0.707 2.337E- 0l 11.1 0.293 0.146 
22 .20 22 .82 0.859 1.181E- 0l 25 .6 0.019 0.135 
24.20 24.87 0.934 4.418E- 02 25.4 0.210 0.223 
26 .20 26.92 1.010 4.120E - 02 25 .7 0.268 0.277 
30.20 31.02 1.159 5.479E- 02 24.6 0.207 0.100 
34.20 35 .12 1.308 4.488E- 02 13.0 -0.060 0.201 
38 .20 39.21 1.454 2.559E- 02 33 .8 -0.598 0.536 
42.20 43 .30 1.598 2.394E - 02 28.5 0.125 0.388 
46 .20 47.38 1.740 2.457E - 02 26.9 -0 .124 0.306 
50.20 51 .46 1.880 2.181E- 02 9.0 -0.323 0.199 
54 .20 55.53 2.017 1.489E- 02 30 .6 0.071 0.160 
58.20 59 .59 2.152 5.689E- 03 33 .3 -0.934 0.635 
62 .20 63 .65 2.283 2.440E- 03 65.3 -1.375 0.920 
66.20 67.69 2.412 1.406E- 03 87.9 -0.244 1.186 
70 .20 71.74 2.537 8.217E- 04 32 .3 -0.094 0.452 
Table F.3: i) 4?Ca(p, p')4~ (5 .6143 MeV) ; Ep = 100 MeV 
477 
Bc.m. qc.m . 
-15 .80 16.25 0.616 1.325E- 01 22 .5 0.684 1.090 
-13.80 14.19 0.539 2.197E- 01 10.8 0.152 0.106 
-11 .80 12.13 0.463 1. 741E- 01 15 .9 0.188 0.214 
-9.80 10 .08 0.386 l.204E- 01 15.1 0.104 0.337 
-7.80 8.02 0.310 4.018E- 02 97 .1 1.093 1.502 
12.20 12.55 0.478 1.965E- 01 18 .3 0.087 0.459 
14.20 14.60 0.555 2.598E- 0l 11.8 0.084 0.161 
18.20 18 .71 0.707 2.334E- 0l 4.0 0.068 0.053 
22.20 22.82 0.859 4.884E- 02 39 .5 0.412 0.488 
24 .20 24.87 0.934 6.184E- 02 12.1 0.139 0.162 
26 .20 26.92 1.010 4.878E- 02 18. 7 0.061 0.249 
30.20 31.02 1.159 8.233E- 03 89 .7 -1.268 1.318 
34.20 35.12 1.307 1.248E- 02 32 .1 1.116 0.355 
38.20 39.21 1.454 4.709E - 02 17.4 -0.076 0.116 
42 .20 43 .30 1.598 3.631E- 02 31. 7 -0.428 0.113 
46 .20 47.38 1. 740 l.437E- 02 28 .0 -0.196 0.521 
50.20 51.46 1.880 6.498E - 03 98 .3 0.058 0.642 
54.20 55.53 2.017 5.008E - 03 37 .5 -0.967 0.418 
58.20 59 .59 2.152 5.811E- 03 29 .5 0.417 0.675 
62.20 63.65 2.283 4.737E- 03 18.6 -0.024 0.129 
66.20 67.69 2.412 2.419E - 03 11 .7 0.120 0.158 
70.20 71.74 2.537 9.154E- 04 49.2 0.263 0.734 
Table F .3: j) 4?Ca(p, p')2f (5.6301 MeV ); Er = 1nn MeV 
478 
0c.m. qc.m. <Tc.m.(q) I ~a-( %) I Ay I ~ Ay I 
-15 .80 16.25 0.616 3.345E- 02 12.6 0.119 0.300 
-13 .80 14.19 0.539 4.l0lE- 02 30 .1 0.072 0.202 
-11.80 12.14 0.463 4.891E- 02 8.5 0.043 0.312 
-9 .80 10 .08 0.387 5.828E- 02 13.4 0.149 0.192 
-7.80 8.02 0.311 5.248E- 02 29 .5 0.387 0.369 
18 .20 18 .71 0.707 8.193E- 02 10.6 0.063 0.039 
22 .20 22 .82 0.858 1.373E- 01 3.6 0.062 0.042 
24 .20 24 .87 0.934 1.338E- 01 1.5 0.081 0.070 
26 .20 26.92 1.009 l.l0lE- 01 2.1 0.063 0.036 
30.20 31.02 1.159 5.936E- 02 3.0 0.056 0.024 
34.20 35.12 1.307 3.361E- 02 2.9 -0.001 0.026 
38.20 39.21 1.453 2.140E- 02 9.0 0.214 0.028 
42 .20 43 .30 1.597 l.944E - 02 2.2 0.479 0.026 
46 .20 47 .38 1.739 l.363E- 02 6.4 0.649 0.073 
50 .20 51.46 1.879 l.232E- 02 4.6 0.537 0.047 
54.20 55.53 2.016 l.200E- 02 1.9 0.318 0.048 
58.20 59 .59 2.150 9.521E- 03 1.6 0.156 0.028 
62 .20 63 .65 2.282 4.925E- 03 4.6 0.016 0.026 
66.20 67.70 2.410 2.040E- 03 4.8 -0.176 0.043 
70.20 71. 74 2.535 6.366E- 04 5.0 -0.120 0.081 
Table F.3: k) 4?Ca(p, p')11 (5 .9033 MeV) ; Ep = 100 MeV 
479 
0-c.rn.(q) I ~o-(%) I A y I ~Ay I 
-15 .80 16.25 0.616 4.896E- 02 17.6 0.347 0.118 
-13 .80 14.19 0.539 6.319E- 02 27 .2 0.293 0.056 
-11 .80 12.14 0.463 5.403E - 02 8.3 0.204 0.112 
-9.80 10 .08 0.387 6.791E- 02 9.4 0.185 0.123 
-7.80 8.02 0.311 9.548E- 02 19. 7 0.569 0.236 
14.20 14.60 0.555 5.580E- 02 15 . 7 0.125 0.213 
18 .20 18. 71 0.707 4.293E- 02 11.1 -0.016 0.069 
22.20 22 .82 0.858 3.800E- 02 4.3 -0.272 0.114 
24 .20 24.87 0.934 3.690E- 02 3.2 -0.432 0.044 
26 .20 26 .92 1.009 3.115E- 02 9.8 -0.557 0.090 
30 .20 31.02 1.158 3.734E - 02 4.9 -0.373 0.032 
34.20 35.12 1.306 2.319E- 02 2.5 -0.566 0.033 
38.20 39.21 1.452 1.571E- 02 4.4 -0.403 0.053 
42 .20 43 .30 1.597 8.944E- 03 3.4 -0.358 0.047 
46.20 47 .38 1.739 4.645E- 03 8.0 -0.253 0.132 
50.20 51.46 1.878 2.382E- 03 6.4 0.108 0.131 
54 .20 55 .53 2.015 1.421E- 03 7.4 0.544 0.096 
58.20 59 .59 2.149 1.222E- 03 5.8 0.577 0.163 
62.20 63 .65 2.281 9.562E- 04 14.2 0.346 0.173 
66.20 67.70 2.409 5.017E- 04 5.7 0.407 0.100 
70 .20 71. 74 2.534 3.346E- 04 7.3 0.383 0.073 
Table F.3: 1) 4?Ca(p, p')6 .028 (6.028 MeV ); Ep = 100 MeV 
480 
O"cm.(q) I ~u(%) I Ay I ~Ay I 
-15.80 16.25 0.616 6.799E- 01 9.2 0.053 0.030 
-13.80 14.19 0.539 5.190E- 01 1.3 0.099 0.040 
-11 .80 12.14 0.463 3.853E- 01 2.5 0.042 0.036 
-9.80 10 .08 0.387 2.844E- 0l 7.3 0.010 0.030 
-7.80 8.02 0.312 2.124E- 01 11.8 0.126 0.150 
12.20 12.55 0.478 4.542E- 01 3.9 0.137 0.052 
14.20 14.60 0.555 6.021E- 01 2.7 0.168 0.037 
18.20 18.71 0.707 8.383E- 01 5.6 0.113 0.011 
22 .20 22 .82 0.858 7.473E- 01 2.3 0.099 0.011 
24.20 24 .87 0.933 6.075E- 01 1.5 0.111 0.013 
26 .20 26 .92 1.008 3.909E- 01 4.6 0.068 0.023 
30 .20 31.03 1.158 1.133E- 01 2.3 -0.224 0.038 
34.20 35 .12 1.306 8.566E- 02 1.2 -0.223 0.016 
38.20 39.21 1.452 1.652E- 01 2.2 0.107 0.028 
42 .20 43 .30 1.596 2.074E- 01 1.5 0.214 0.031 
46 .20 47 .38 1.737 l.991E- 01 1.1 0.169 0.009 
50.20 51.46 1.877 l.697E- 01 1.5 0.084 0.009 
54.20 55 .53 2.014 l.309E - 01 0.8 -0.024 0.007 
58.20 59 .59 2.148 8.677E- 02 0.5 -0.102 0.007 
62 .20 63.65 2.279 4.473E- 02 0.9 -0.203 0.017 
66.20 67 .70 2.407 1.834E- 02 0.7 -0.402 0.022 
70.20 71. 74 2.532 5.334E- 03 2.7 -0.479 0.043 
Table F.3: m) 4?Ca(p, p')3; (6.2858 MeV) ; Ep = 100 Me\/ 
481 
O'cm(q) I ~0'(%) I A y I ~A y 
.. . . 
. . 
35.5 0.2 78 0.2 69 
-15.80 16.25 0.616 
2.511E- 02 
0.480 
14.19 0.539 1.379E- 02 
46.0 0.198 
-13.80 
1.528E- 02 39.2 -0.414 
0.769 
-11.80 12.14 0.463 22 .0 -0.017 0.375 
-9.80 10.08 0.388 
1.543E- 02 
94 .2 -0.078 1.293 1.299E- 02 
12 .20 12.55 0.479 32.0 0.274 0.438 2.908E- 02 
14.20 14.60 0.555 -0.346 0.395 
0.706 5.236E- 02 
28.1 
18 .20 18.71 11.8 0.063 0.139 
22 .20 22.82 0.858 
9.873E- 02 
1.047E- 01 2.2 
0.081 0.074 
24.20 24.87 0.933 0.124 0.066 
1.065E- 01 4.8 
26 .20 26 .93 1.008 0.152 0.055 
1.157 9.744E- 02 
4.7 
30.20 31.03 0.138 0.025 
1.305 6.393E- 02 
1.9 
34 .20 35 .12 2.6 0.068 0.087 3.98GE- 02 
38.20 39.22 1.451 -0.077 0.037 
1.595 2.186E- 02 
2.8 
42 .20 43.30 0.056 0.049 
1.737 1.199E- 02 
3.6 
46.20 47.39 4.7 0.162 0.072 5.707E- 03 
50.20 51.46 1.876 0.422 0.117 
2.013 2.172E- 03 
10.8 
54.20 55.53 28 .1 0.406 0.197 1.084E- 03 
58 .20 59 .59 2.147 9.6 -0.182 0.128 
62.20 63.65 2.278 
1.029E- 03 
-0 .194 0.089 
2.406 9.672E- 04 
6.7 
66.20 67.70 4.6 -0.196 0.076 7.596E- 04 
70.20 71.74 2.531 
Table F.3: n) 4?Ca(p, p')4i (6 .5084 MeV) ; Ep == 100 MeV 
482 
O'c.m.( q) l ~tr(%) I ~ Ay I I Ay I fh Bc.m. qc ,m, 41.9 -0.687 0.594 l 
-13.80 14.19 0.539 
1.754E- 02 
-1.352 1.150 
0.706 6.602E- 03 
59 .6 
18.20 18.71 78 .0 0.352 0.803 3.452E- 03 
24.20 24.87 0.933 33 .1 0.068 0.525 1.940E- 02 
26 .20 26 .93 1.008 18.6 0.104 0.198 
34.20 35 .12 1.305 
7.752E- 03 
97.1 1.332 1.787 3.548E- 04 
54.20 55 .53 2.013 
7.l 70E- 04 103 .1 
-0.234 0.434 
58 .20 59 .60 2.147 110.3 -0.188 0.855 5.744E- 04 
62.20 63 .65 2.278 -0.994 0.503 
2.406 3.304E- 04 
37 .6 
66.20 67.70 80 .5 0.625 0.692 I.539E- 04 
70 .20 71 .74 2.531 
Table F .3: o) 4?Ca(f, p')4f (6 .5436 MeV); Ep = 100 MeV 
483 
<T (q) I ~<r( %) I A y I ~ A y 
. . . . c .m . 
7.6 0.15 7 0.040 
-15.80 16.25 0.616 
3.766E- 01 
0.161 0.065 
14.19 0.540 2.926E- 01 
1.7 
-13.80 8.0 0.037 0.049 
-11.80 12.14 0.463 
2.l00E- 01 
1.630E- 01 4.3 
0.014 0.128 
-9.80 10.08 0.388 21.2 0.051 0.272 
-7.80 8.02 0.312 
8.768E- 02 
6.6 0.185 0.125 
12 .20 12.55 0.479 
2.353E- 01 
4.3 0.165 0.058 3.258E- 01 
14.20 14.60 0.555 1.4 0.173 0.026 
18 .20 18 .71 0.707 
4.693E- 01 
4.240E- 01 1.3 
0.196 0.021 
22 .20 22.82 0.858 1.9 0.200 0.014 3.375E- 01 
24 .20 24.87 0.933 0.153 0.019 
1.008 2.255E- 01 
5.1 
26 .20 26.93 5.6 -0.141 0.045 
30.20 31.03 1.157 
6.218E- 02 
2.3 -0.493 0.032 
34.20 35 .12 1.305 
4.170E- 02 
3.2 -0.066 0.017 7.782E- 02 
38.20 39 .22 1.451 0.076 0.013 
1.594 1.002E- 01 
1.5 
42 .20 43. 30 
9.417E- 02 1.8 
0.054 0.013 
46 .20 47 .39 1.736 -0.005 0.0ll 
8.187E- 02 1.5 
50 .20 51.46 1.876 -0.131 0.013 
2.012 6.228E- 02 
1.4 
54.20 55 .53 
4.00lE- 02 0.9 
-0.207 0.029 
58.20 59 .60 2.146 3.2 -0.383 0.017 1.980E- 02 
62 .20 63 .65 2.278 -0.472 0.021 
2.406 7.908E- 03 
1. 7 
66 .20 67 .70 2.0 -0.458 0.056 
2. 561E- 03 
70 .20 71 .74 2.530 
Table F.3: p) 4?Ca.(p,p')33 (6 .5833 MeV); Ep == 1()0 MeV 
484 
CTc. m .( q) I ~er(%) I A y I ~A y - I . . . . 
13.5 I 0.115 0.103 
-15.80 16.25 0.616 
8.261E- 02 
20.9 0.180 0.105 
-13.80 14.19 0.540 
9.151E- 02 
9.6 0.282 0.066 
-11 .80 12.14 0.464 
1.051E- 01 
1.367E- 01 7.9 0.139 
0.081 
-9.80 10.08 0.388 
1.286E- 01 15.7 
0.279 0.198 
-7.80 8.02 0.313 8.2 0.181 0.168 
12.20 12.55 0.479 
1.659E- 01 
9.7 0.141 0.131 
14.20 14.60 0.555 
9.967E- 02 
6.894E- 02 7.8 
0.037 0.115 
18.20 18.71 0.707 5.5 -0.250 0.132 6.752E- 02 
22.20 22.82 0.857 8.1 -0.337 0.047 4.611E- 02 
24.20 24.87 0.933 -0.479 0.039 
1.008 4.385E- 02 
6.0 
26.20 26 .93 4.7 -0.381 0.029 4.221E-02 
30.20 31.03 1.157 4.8 -0.261 0.025 3.635E- 02 
34.20 35. 12 1.304 6.4 -0.169 0.058 
38.20 39.22 1.450 
2.846E-02 
1.697E- 02 2.5 
-0.109 0.033 
42 .20 43.30 1.594 7.0 -0.107 0.043 8.613E- 03 
46.20 47.39 1.735 7.7 -0.056 0.108 
50 .20 51.46 1.875 
4.432E- 03 
0.240 0.073 
2.011 2.711E- 03 
10.9 
54 .20 55 .53 4.7 0.351 0.064 1.951E- 03 
58.20 59.60 2.145 4.8 0.308 0.067 
62.20 63 .65 2.277 
1.320E- 03 
0.227 0.057 
2.405 9.289E- 04 
4.1 
66.20 67.70 7.0 0.328 0.114 4.429E- 04 
70 .20 71 .75 2.529 
Table F.3: q) 4?Ca.(f,p')22 (6 .7509 MeV ); Ep == 100 MeV 
485 
I qc.m. 
0.138 0.052 
-15.80 16 .25 0.616 2.245E+OO l 4.2 
0.149 0.020 
-13.80 14.19 0.540 1.924E+OO 9.3 
0.081 0.048 
-11.80 12.14 0.464 1.367E+OO 4.8 
0.122 0.046 
-9.80 10.08 0.388 1.039E+OO 7.9 
0.181 0.088 
-7.80 8.02 0.313 7.589E- 01 6.9 
0.064 0.113 
12.20 12.55 0.479 1.896E+OO 9.0 0.107 0.036 
14.20 14.60 0.555 2.357E+OO 2. 7 0.134 0.039 
18 .20 18.71 0.707 2.240E+OO 3.3 0.127 0.057 
22.20 22.82 0.857 1.612E+OO 1.4 0.143 0.020 
24 .20 24.87 0.932 1.241E+OO 1.0 0.179 0.018 
26.20 26.93 1.007 8.407E- 01 4.2 
0.352 0.067 
30 .20 31.03 1.156 3.976E- 01 3.1 0.571 0.021 
34.20 35.12 1.304 2.607E - 01 1.9 0.580 0.048 
38.20 39.22 1.450 2.400E- 01 7.9 0.673 0.043 
42.20 43.31 1.593 1.452E- 01 2.2 0.846 0.140 
46.20 47.39 1.735 6.455E - 02 6.5 0.857 0.115 
50 .20 51.46 1.874 4.531E- 02 5.1 0.683 0.070 
54.20 55 .53 2.011 4.853E- 02 2.9 0.493 0.047 
58.20 59.60 2.145 3.738E- 02 3.3 0.368 0.064 
62.20 63 .65 2.276 2.272E- 02 6.8 0.056 0.121 
66.20 67.70 2.404 9.022E- 03 8.8 0.258 0.354 
70.20 71.75 2.528 3.767E- 03 35 .7 
Table F.3: r) 4?Ca(p, p')6 .909 (6 .909 MeV ); Er = JOO MeV 
486 
... 
__________ _,,.......,,TrL.,.,, .. . . 
. . O"c.m.(q) I LiCT(%) I Ay I Li A y I 
-15.80 16.25 0.616 l.598E- Ol 37 .0 1.039 0.508 
-13.80 14.19 0.540 1.267E- O1 15.7 0.211 0.197 
-11 .80 12.14 0.464 7. 265E - O2 98 .9 1.316 1.657 
-9 .80 10.08 0.388 6.H5E - O2 47 .1 0.029 0.805 
-7 .80 8.02 0.313 1.102E- Ol 42 .8 -0.713 0.659 
18 .20 18 .71 0.707 2.157E- Ol 39.5 0.282 0.190 
22 .20 22 .82 0.857 1.l 79E- Ol 139.2 -0.176 0.627 
24.20 24 .87 0.932 1.471E - Ol 10.2 0.068 0.137 
26 .20 26 .93 1.007 1.673E- Ol 48.4 0.137 0.272 
30.20 31.03 1.156 l.063E- Ol 29 .6 0.430 0.143 
34.20 35 .13 1.304 l.O09E- O1 9.1 0.131 0.071 
38 .20 39.22 1.449 4.4 l5E- O2 19 .0 0.010 0.412 
42.20 43 .31 1.593 3.631E- O2 17.6 -0.199 0.549 
46.20 47.39 1.735 3.7~5E- O2 14.0 0.022 0.106 
50 .20 51.46 1.874 3.570E - O2 22 .6 -0.178 0.137 
54.20 55.53 2.011 2.505E- O2 10.9 -0.448 0.131 
58.20 59.60 2.144 2.O6OE - O2 11.4 -0.440 0.106 
62 .20 63.65 2.275 1.334E - O2 24.7 -0.573 0.149 
66 .20 67.70 2.403 9.O62E - O3 13.6 -0.4 70 0.095 
70.20 71.75 2.528 4.222E - O3 22.4 -0. 722 0.181 
Table F .3: s) 4?Ca.(p, p')6.931 (6 .931 MeV) ; EP = 100 MeV 
487 
Bc.m. %.m. O"c.m.(q) I ~o-(%) I Ay I ~Ay I 
-15.80 16 .25 0.616 3.177E- 0l 39 .2 0.103 0.257 
-13.80 14.19 0.540 2.349E- 01 37.0 0.156 0.089 
-11.80 12.14 0.464 4.550E - 0l 6.7 0.068 0.148 
-9.80 10.08 0.388 4.432E - 0l 12.3 0.172 0.106 
-7.80 8.02 0.313 3.512E- 01 12.0 0.118 0.153 
12.20 12.55 0.479 4.052E - 0l 23.9 0.073 0.132 
18.20 18.71 0.707 4.306E- 01 13.2 -0.035 0.097 
22 .20 22 .82 0.857 7.075E - 0l 16.4 0.015 0.029 
24 .20 24.87 0.932 7.125E- 01 4.6 0.057 0.019 
26.20 26 .93 1.007 5.539E- 0l 11.2 -0.008 0.058 
30.20 31.03 1.156 2.898E- 01 13.4 -0.170 0.118 
34.20 35 .13 1.304 l.643E- 01 8.4 -0.127 0.104 
38.20 39.22 1.449 l.404E- 01 6.6 0.142 0.071 
42.20 43 .31 1.593 l.185E - 01 7.8 0.453 0.106 
46 .20 47 .39 1. 735 9.490E- 02 8.8 0.596 0.084 
50 .20 51 .46 1.874 7.772E- 02 13.2 0.587 0.171 
54.20 55 .53 2.010 6.911E- 02 6.4 0.424 0.144 
58.20 59 .60 2.144 5.072E- 02 3.8 0.294 0.045 
62 .20 63.65 2.275 2.629E- 02 4.2 0.145 0.059 
66.20 67.70 2.403 9.284E- 03 8.1 -0.019 0.109 
70 .20 71 .75 2.528 3.116E- 03 31.5 0.027 0.425 
Table F.3 : t) 4?Ca(p, p')l2 (6 .951 MeV); Er = 100 Me\! 
488 
O"c.m.(q) I b.o-(%) I Ay I b.Ay I 
-15 .80 16.25 0.616 5.340E- 02 26 .9 0.553 0.135 
-13 .80 14.19 0.540 5.420E- 02 35.4 0.139 0.187 
-11 .80 12.14 0.464 4.124E - 02 12.2 0.244 0.157 
-9.80 10.08 0.388 3.039E - 02 42.2 0.095 0.252 
-7.80 8.02 0.314 2.947E- 02 45.8 0.281 0.573 
14.20 14.60 0.555 6.427E - 02 15.1 0.045 0.205 
18 .20 18.71 0.707 5.299E- 02 7.6 0.213 0.120 
22.20 22 .82 0.857 5.605E- 02 23.7 0.286 0.147 
24 .20 24 .88 0.932 6.428E - 02 2.9 0.172 0.069 
26 .20 26.93 1.007 6.787E- 02 3.8 0.201 0.042 
30.20 31.03 1.156 8.567E- 02 2.0 0.235 0.032 
34.20 35.13 1.303 6.461E - 02 3.1 0.174 0.020 
38 .20 39.22 1.449 6.266E - 02 3.7 0.104 0.041 
42.20 43 .31 1.592 5.179E- 02 2.3 0.001 0.018 
46.20 47 .39 1.734 3.703E- 02 3.9 -0.069 0.034 
50 .20 51.46 1.873 2.501E- 02 2.3 -0.171 0.044 
54.20 55 .54 2.010 1.519E- 02 3.6 -0.215 0.026 
58.20 59.60 2.143 7.495E- 03 4.3 -0.303 0.029 
62.20 63 .66 2.274 3.329E- 03 3.0 -0.44 7 0.046 
66.20 67 .71 2.402 1.638E- 03 4.2 -0.437 0.046 
70.20 71. 75 2.527 8.634E- 04 7.3 -0.232 0.041 
Table F.3: u) 4?Ca(p, p')7 .ll (7 .11 MeV) ; Ep = 100 MeV 
489 
y-ryc,??? ... .. .... . 
F.3 9Be(p,p') and 16 O(p,p') at Ep == 200 MeV 
490 
0-c.m.  (q) I ~o-( %) I A y 
6.14 6.96 0.351 1.794E+02 1. 7 0.542 0.016 
7.64 8.65 0.436 l .667E+02 1.0 0.555 0.011 
9.14 10 .35 0.522 1.219E+ 02 1.0 0.594 0.011 
10.64 12.05 0.607 1.122E+02 0.9 0.711 0.008 
12 .14 13 .74 0.692 8.704E+ 0l 0.8 0.782 0.009 
13 .64 15.44 0.777 5.227E+0l 0.7 0.854 0.007 
15.14 17.13 0.861 3.376E+ 0l 1.2 0.931 0.007 
18.27 20 .66 1.038 l.251E+0l 1.0 0.951 0.008 
19.64 22 .20 1.114 8.676E+ oo 1.1 0.894 0.010 
19.77 22 .35 1.122 8.438E+00 2.0 0.901 0.034 
21.27 24.03 1.205 5.244E+00 0.6 0.817 0.005 
23 .77 26 .84 1.343 2.631E+ 00 0.6 0.542 0.018 
25 .27 28.52 1.426 l.837E+oo 3.0 0.367 0.007 
26 .77 30 .20 1.508 l.366E+oo 2.8 0.223 0.009 
28 .27 31.88 1.589 1.019E+oo 3.5 0.140 0.007 
29.77 33 .55 1.671 8.209E - 0l 1.4 0.109 0.019 
30 .52 34 .39 1.711 7.187E- 01 1.1 0.114 0.014 
31.27 35.22 1. 751 6.605E - 01 1.2 0.111 0.016 
33.52 37 .72 1.871 4.584E - 01 1.3 0.087 0.018 
34.27 38 .55 1.911 4.116E- 01 3.1 0.089 0.010 
36.52 41.05 2.029 3.038E- 01 1.9 0.036 0.025 
37.27 41.87 2.068 2.636E - 0l 1.6 0.049 0.011 
40.27 45 .18 2.224 1.595E- 01 1.2 0.010 0.013 
43 .27 48.48 2.376 8.513E- 02 1.0 -0.014 0.023 
46 .27 51.75 2.526 5.272E- 02 4.6 0.019 0.041 
48 .52 54 .20 2.637 3.234E - 02 3.2 -0.!l03 0.043 
49.27 55.02 2.674 2.651E- 02 1. 9 -0.010 ll .02G 
51.52 57 .46 2.782 l .372E- 02 3. 9 0.fl 57 0.05.l 
Table F.4: a) 9 Be(p, p) 3/ 2~ ; Er = 200 MeV 
491 
0c.m. I qc.m. I O"cm.(q) I fi o-( %) I A y I fi Ay I 
12 .14 13.75 0.691 7.197E- 02 49.9 0.625 0.364 
13 .64 15.44 0.775 l.648E- 0l 18.2 1.035 0.095 
15 .14 17.14 0.860 l.297E - 0l 39 .8 0.934 0.123 
16.64 18.83 0.944 l.052E- 0l 22.3 0.867 0.088 
18.14 20 .52 1.028 8.879E- 02 12 .8 0.662 0.097 
18.27 20 .67 1.036 9.987E- 02 14.2 0.969 0.099 
19.64 22 .21 1.111 9.295E- 02 14.2 0.910 0.103 
19.77 22 .36 1.119 1.242E- 0l 14. 4 0.829 0.198 
21.27 24 .04 1.203 8.821E- 02 9.7 0.805 0.119 
23 .77 26 .85 1.341 6.003E- 02 9.3 0.678 0.115 
25 .27 28 .53 1.423 3.980E- 02 12.1 0.652 0.063 
26 .77 30 .21 1.505 3.592E- 02 13 .6 0.325 0.100 
27 .52 31.05 1.546 3.915E- 02 12.9 0.636 0.167 
28 .27 31.89 1.586 2.546E- 02 8.9 0. 260 0.255 
29 .77 33 .56 1.667 2.634E- 02 11.4 0.391 0.148 
30.52 34.40 1. 707 2.910E- 02 11.0 0.095 0.141 
31.27 35 .23 1.748 2.048E- 02 4.6 0.042 0.064 
33.52 37.74 1.867 l.734E - 02 10.0 -0.380 0.132 
36 .52 41.06 2.025 l.399E - 02 14.6 -0.654 0.180 
37.27 41.89 2.064 l.249E- 02 24 .9 -0.482 0.432 
39.52 44 .37 2.180 8.680E- 03 25 .5 -0.841 0.267 
40 .27 45 .20 2.219 4.710E- 03 11 .7 -0.358 0.145 
43 .27 48 .49 2.371 2.651E- 03 9.6 -0. 773 0.211 
45.52 50 .96 2.484 3.870E- 03 25.4 -0.310 0.33 1 
Table F.4: 9b) Be(p, p')l / 2i (1. 680 MeV) ; Er = 200 MeV 
492 
Uc.m.(q) I ~u(%) I Ay I ~ Ay I 
12.14 13 .75 0.690 3.271E+00 2.4 0.674 0.019 
13 .64 15 .45 0.775 3.518E+ 00 2. 2 0.768 0.016 
15.14 17.14 0.859 3.886E+ 00 2.0 0.845 0.007 
18 .14 20 .53 1.027 3.953E+00 1.4 0.891 0.012 
18 .27 20.67 1.035 3.964E+oo 1.8 0.969 0.013 
19 .64 22 .21 1.110 3.667E+ 00 1.7 0.942 0.013 
19 .77 22 .36 1.119 3.817E+ 00 1.6 0.950 0.019 
21.27 24 .05 1.202 3.365E+ oo 3.5 0.939 0.007 
23.77 26 .86 1.340 2.670E+ 00 1.5 0.829 0.010 
25 .27 28 .54 1.422 2.231E+ oo 5.4 0.710 0.008 
26.77 30 .22 1.503 2.014E+ 00 0.7 0.582 0.014 
28 .27 31 .90 1.585 1.546E+ 00 4.4 0.420 0.006 
29.77 33.57 1.666 l.368E+ 00 1.3 0.253 0.016 
30 .52 34.41 1.706 1.118E+ 00 1.3 0.180 0.016 
31.27 35.24 1.746 l.136E+ 00 1.2 0.102 0.006 
33.52 37 .74 1.866 8.134E- 0l 1.5 -0.084 0.020 
34 .27 38 .58 1.905 6.958E- 01 4.2 -0.198 0.021 
36 .52 41.07 2.023 4.677E- 01 1.7 -0.427 0.022 
37 .27 41.90 2.062 4.142E- 0l 0.8 -0.437 0.014 
39.52 44 .38 2.178 2.696E - 01 3.8 -0.688 0.044 
40 .27 45 .21 2.217 2.122E - 01 2.2 -0.652 0.017 
43 .27 48 .50 2.369 9.734E - 02 1.0 -0.813 0.012 
45 .52 50 .97 2.481 5.589E- 02 7.3 -0.849 0.074 
46 .27 51 .78 2.519 5.167E- 02 3.6 -0.846 0.016 
49.27 55 .05 2.665 2.650E- 02 4.7 -0. 740 0.055 
51.52 57.49 2.773 1.356E - 02 12,3 -0.574 0.135 
Table F.4: c) 9 Be(p, p')5 / 21 (2.429 MeV ); Er = 200 MeV 
493 
. . 
12.14 13 .75 0.690 7.095E- 0l 28 .6 0.529 0.229 
13.64 15.45 0.775 l.451E+ oo 13.8 0.808 0.090 
15 .14 17.14 0.859 6.281E- 01 21.8 0.893 0.076 
16.64 18.84 0.943 6.978E- 0l 13. 7 0.770 0.253 
18.14. 20 .53 1.027 3.704E - 0l 23 .5 0.691 0.183 
18.27 20.67 1.035 3.226E- 0l 33.5 0.768 0.266 
19.64 22 .22 1.110 2.924E - 01 41.5 0.454 0.357 
19.77 22 .36 1.118 3.562E- 01 19.3 0.629 0.101 
21.27 24.05 1.201 2.294E- 0l 28.8 0.610 0.211 
23 .77 26 .86 1.339 l.168E- 0l 19.0 0.566 0.324 
25.27 28 .54 1.421 l.272E- 01 30 .0 0.706 0.179 
26.77 30.22 1.503 6.230E- 02 46 .9 1.202 0.608 
28 .27 31.90 1.584 8.692E- 02 15 .7 0.197 0.198 
29.77 33.57 1.665 9.329E - 02 45 .2 0.835 0.631 
31.27 35 .24 1.745 l.306E- 01 51.1 0.103 0.148 
33.52 37.75 1.865 l.404E- 01 24 .0 -0 .320 0.313 
34.27 38 .58 1.904 l.258E- 01 50 .5 -0.092 0.260 
36 .52 41.07 2.022 7.220E- 02 25.2 0.153 0.342 
37 .27 41.90 2.061 l.803E - 02 92 .1 -1.366 1.639 
40 .27 45 .21 2.216 6.290E- 03 99.5 1.032 2.030 
43.27 48.51 2.368 4.749E- 03 75 .4 -1.366 1.116 
Table F.4: d) 9 Be(p, p')l / 2~ (2.78 MeV ); Ep = 200 Me\! 
494 
O"c.m.(q) I L\o-(%) I Ay I L\ Ay I 
12 .14 13 .75 0.690 7.752E- 02 91.3 0.763 0.682 
15 .14 17.14 0.859 1.014E- 0l 31.1 0.270 0. 288 
16.64 18 .84 0.943 8.317E- 02 54 .7 0.680 0.488 
18.14 20 .53 1.026 7.050E- 02 61.2 0.395 0.359 
18.27 20 .68 1.034 7.868E- 02 51.2 0.675 0.436 
19 .64 22 .22 1.110 1.107E- 0l 38 .3 0.955 0.277 
19.77 22 .37 1.118 8.759E- 02 18.8 0.732 0.611 
21.27 24 .05 1.201 6.844E - 02 22 .1 0.635 0.543 
23.77 26.86 1.339 8. 721E- 02 18.7 0.608 0.153 
25 .27 28.54 1.421 4.629E - 02 33.7 0.298 0.176 
26 .77 30.22 1.502 7.223E- 02 30.7 0.306 0.144 
28.27 31 .90 1.584 3.459E- 02 14. 5 0.502 0.180 
29 .77 33.57 1.664 3.316E- 02 36.0 0.075 0.473 
31.27 35.25 1. 745 2.500E- 02 45 .9 0.309 0.613 
33.52 37.75 1.864 2.000E- 02 44 .3 -0.020 0.589 
37.27 41.91 2.061 1.479E- 02 45 .5 -0.161 0.412 
40 .27 45.21 2.215 7.356E- 03 30.8 -0.120 0.413 
43.27 48 .51 2.367 3.728E- 03 20 .8 -0.186 0.271 
Table F.4: e) 9 Be(p,p' )5 / 2i (3 .049 MeV) ; Er = 200 MeV 
495 
I (h O"c.m.(q) I ~o-( %) I Ay I ~Ay I 
12.14 13 .76 0.689 9.662E - 0l 13 .0 0.671 0.090 
13 .64 15.46 0.773 1.002E+oo 9.9 0.887 0.066 
15.14 17.15 0.857 1.035E+ oo 7.2 0.889 0.041 
16 .64 18.85 0.941 8.215E- 01 7.5 0.799 0.069 
18 .14 20 .54 1.025 6.657E- 01 16.3 0.726 0.060 
18 .27 20 .68 1.033 5. 708E - 0l 11 .6 0.913 0.083 
19 .64 22 .23 1.108 6.492E- 01 10.8 0.986 0.076 
19.77 22.37 1.116 6.321E- 01 5.0 0.905 0.058 
21.27 24.06 1.199 5.808E- 0l 13. 7 0.850 0.087 
23 .77 26.87 1.336 3.126E- 0l 4.2 0.821 0.071 
25 .27 28.56 1.418 2.261E- 0l 10.0 0.829 0.145 
26 .77 30.24 1.500 1.875E- 0l 22 .9 0.461 0.096 
27 .52 31.07 1.540 2.026E - 0l 14.2 0.255 0.187 
28 .27 31.91 1.581 7.942E- 02 20 .5 0.301 0.164 
29.77 33.59 1.661 4.029E- 02 41.9 0.459 0.560 
30.52 34.43 1.701 3.177E- 02 44 .1 -0.547 0.684 
31.27 35.26 1.741 5.441E- 02 15.9 -0.270 0.130 
33 .52 37.76 1.860 2.811E- 02 32 .2 -0.366 0.439 
34.27 38.60 1.900 2.964E- 02 40.4 -0.237 0.186 
36.52 41.09 2.018 2.398E- 02 42.0 0.186 0.572 
37.27 41.92 2.056 1.720E- 02 63.1 -0.061 0.456 
39.52 44.41 2.172 1.731E- 02 71.9 -1.366 1.391 
43 .27 48.53 2.362 4.371E- 03 54. 5 -1. 366 0.775 
Table F.4: f) 9 Be(p, p')3 / 2j'" (4.704 MeV ); Ep = 200 Me \/ 
496 
O"c.m.{q) I ~o-{%) I Ay I ~Ay I 
-5.86 6.65 0.337 2.661E+00 21.0 1.075 0.447 
7.64 8.67 0.436 1.456E+00 63 .8 0.546 0.705 
9.14 10.37 0.521 2.090E+00 40.4 0.820 0.297 
10 .64 12.07 0.605 3.860E+00 24.5 0.650 0.226 
12 .14 13.77 0.689 3.466E+00 3.2 0.813 0.023 
13.64 15.46 0.773 3.338E+oo 11.8 0.739 0.096 
15.14 17.16 0.856 3.325E+00 9.7 0.920 0.051 
16 .64 18 .85 0.940 3.641E+oo 3.2 0.885 0.040 
18 .14 20 .55 1.023 3.627E+oo 11 .8 0.922 0.070 
18.27 20.69 1.031 2.802E+oo 10.4 0.936 0.068 
19.64 22.24 1.106 2.812E+00 8.2 0.937 0.061 
19.77 22 .38 1.114 3.053E+oo 6.8 0.977 0.111 
21.27 24.07 1.197 2.562E+oo 8.2 0.908 0.022 
23 .77 26 .88 1.334 2.157E+00 2.9 0.817 0.092 
25 .27 28.57 1.415 1.697E+oo 3.1 0. 715 0.049 
26.77 30.25 1.497 1.454E+oo 3.9 0.624 0.057 
28.27 31.93 1.577 1.023E+oo 3.8 0.445 0.126 
29 .77 33 .60 1.658 9.167E-01 8.7 0.411 0.111 
30 .52 34.44 1.698 8.758E- 01 7.3 0.088 0.096 
31.27 35.28 1.738 8.954E- 01 5.8 0.081 0.039 
33 .52 37.78 1.857 7.449E - 01 8.2 -0.172 0.109 
34.27 38 .62 1.896 6.969E - 01 5.2 -0.119 0.132 
36 .52 41.11 2.013 5.283E - 01 10.0 -0.078 0.135 
37.27 41.94 2.052 3.718E- 01 5.3 -0.339 0.092 
39.52 44.43 2.168 2.384E - 0l 10.6 -0.602 0.113 
40 .27 45 .25 2.206 2.138E- 01 5.1 -0.522 0.138 
43.27 48 .55 2.357 1.191E- 01 14.9 -0.718 0.137 
Table F.4 : g) 9 Be(p, p') Lor (6 .5 MeV ); Ep = 200 MeV 
497 
CTc .m.(q) I ~er(%) I Ay I ~Ay I 
6.14 6.60 0.351 3.551E+ 02 1.1 0.494 0.011 
7.64 8.22 0.436 3.607E+02 0.9 0.607 0.010 
9.14 9.83 0.522 2.766E+ 02 0.9 0.656 0.011 
10 .64 11.44 0.607 2.466E+02 0.7 0.778 0.007 
12 .14 13 .05 0.692 l.817E+ 02 0.8 0.859 0.010 
13.64 14.66 0.777 l.033E+ 02 0.7 0.931 0.008 
15.14 16.27 0.862 6.155E+ 0l 1.3 0.991 0.007 
18 .14 19.49 1.031 l .948E+01 1. 7 0.777 0.035 
18 .27 19.63 1.039 l.793E+ 0l 0.9 0.803 0.008 
19.64 21.10 1.115 l.136E+0l 0.9 0.5 77 0.009 
19.77 21.23 1.124 l.064E+ 0l 3.4 0.532 0.009 
21.27 22 .84 1.207 6.475E+ 00 0.9 0.126 0.007 
23.77 25 .51 1.347 3.121E+ 00 0.9 -0. 309 0.021 
26 .77 28 .72 1.512 1.926E+00 1.0 0.060 0.014 
28 .27 30 .32 1.595 1.511E+00 0.9 0.428 0.011 
29 .77 31.92 1.677 1.311E+00 1.3 0.736 0.014 
31.27 33 .52 1. 758 1.099E+00 0.8 0.897 0.009 
33.52 35 .91 1.880 8.115E- 0l 1.1 0.912 0.011 
34 .27 36.71 1.920 7.812E- 0l 1.1 0.886 0.011 
36 .52 39.09 2.040 5.569E- 01 1.5 0.627 0.017 
37.27 39.89 2.080 4.952E- 01 0.9 0.567 0.010 
39 .52 42 .27 2.199 3.378E- 01 1.7 0.244 0.022 
40.27 43 .06 2.238 2.839E- 0l 1.3 0.189 0.017 
43 .27 46 .23 2.394 l.214E - 0l 1.2 -0.303 0.016 
45.52 48 .60 2.509 7.179E- 02 1.6 -0.573 0.020 
46 .27 49 .39 2.548 5.752E - 02 1.9 -0.741 0.024 
48 .52 51. 75 2.661 3.046E- 02 1.9 -0. 792 0.02 J 
49 .27 52 .54 2.699 2.550E - U2 1.1 -0 . 721 0.013 
51 .52 54 .90 2.811 1. 712E - 02 2.0 -0.228 0.026 
55.27 58.81 2.994 l.408E - 02 1.1 0.505 0.013 
Table F. 5: a) 160(p, p)Oi\ Ep = 200 MeV 
498 
.. 
-5.86 6.31 0.337 5.491E- 03 76 .2 0.787 1.027 
7.64 8.22 0.436 4.892E- 02 16.8 0.795 0.320 
9.14 9.84 0.521 9.247E- 02 8.3 0.833 0.081 
10.64 11.45 0.605 7.694E - 02 23 .7 0.907 0.233 
12.14 13 .06 0.689 1.723E- 01 5.2 0.870 0.047 
13.64 14.67 0.773 1.578E- 0l 5.5 0.897 0.124 
15.14 16.29 0.857 l.635E- 01 18.0 0.967 0.026 
16.64 17.90 0.941 l.568E- 0l 9.3 0.922 0.089 
18.14 19.50 1.024 1.191E- 01 22.5 0.724 0.064 
18 .27 19.64 1.033 l.l0lE- 01 8.0 0.763 0.265 
19.64 21.11 1.108 6.102E- 02 46 .3 0.248 0.448 
19.77 21.25 1.116 6.808E - 02 21.0 0.440 0.241 
21.27 22 .86 1.199 6.987E- 02 20 .8 0.253 0.336 
23.77 25.54 1.337 6.241E- 02 13.4 -0.024 0.059 
25 .27 27 .14 1.419 4.789E- 02 49 .9 0.074 0.447 
26.77 28.74 1.501 3.386E- 02 16.1 -0 .006 0.220 
27.52 29.54 1.542 4.473E- 02 14.4 -0 .609 0.197 
28 .27 30.35 1.583 3.277E- 02 12.6 0.101 0.064 
29 .77 31.95 1.664 3.136E- 02 12.2 0.383 0.156 
30.52 32.74 1.705 3.029E- 02 13 .9 0.167 0.177 
31.27 33.54 1.745 2.532E- 02 8.6 0.643 0.134 
36.52 39.13 2.025 2.302E- 02 12.3 0.395 0.161 
37.27 39.92 2.064 1.522E- 02 7.0 0.594 0.094 
39.52 42 .31 2.182 1.180E- 02 16.0 0.306 0.217 
40 .27 43 .10 2.221 1.664E- 02 5.0 -0.098 0.119 
43.27 46.27 2.375 8.072E- 03 5. 8 -0.J57 fl . 134 
Table F .5: b) 16O(p, p' )Of (G.0494 MeV ); Ep = 200 MeV 
499 
1 
Oi I 0c.m. I Qc.m. l O"c.m.(q) I ~o-( %) I Ay I ~ A 
I 1 1 !,' 
-5.86 6.31 0.337 4.315E- 02 16.1 -0.125 0.184 
7.64 8.22 0.436 1.487E- 01 4.3 0.162 0.049 
9.14 9.84 0.521 3.965E- 01 2.5 0.393 o0..0 27 0.4 23 020 
10.64 11.45 0.605 6.817E- 01 3.2 
0.50 1 0.0 27 
12.14 13.06 0.689 1.096E+ OO 3.1 
0.59 7 0.0 15 
13.64 14.67 0. 773 1.558E+ OO 2.5 
0.71 7 0.0 13 
15 .14 16.29 0.857 2.013E+ OO 1.9 0.731 0.0 09 
16.64 17.90 0.941 2.519E+OO 3.1 0.844 0.043 
18 .14 19.51 1.024 2.724E+ OO 3.3 0.851 0.009 
18.27 19.64 1.032 2.909E+ OO 1.8 0.854 0.015 
19 .64 21.11 1.108 3.127E+ OO 2.0 0.867 0.022 
19 .77 21.25 1.116 3.190E+OO 1.6 0.922 0.007 
21.27 22 .86 1.199 3.136E+OO 2.5 0.862 0.010 
23 . 77 25 .54 1.337 3.066E+OO 1.0 0.784 0.009 
25 .27 27.14 1.419 2.748E+OO 6.8 0.657 0.008 
26 .77 28 .74 1.501 2.719E+OO 0.9 0.529 0.009 
28 .27 30.35 1.583 2.365E+OO 1.2 0.355 0.012 
29 . 77 31.95 1.664 2.133E+OO 1.0 0.284 0.009 
30.52 32. 75 1. 704 I.880E+OO O. 7 0.201 0.006 
31.27 33 .54 1.745 I.911E+ OO 11..08  -0.045 0.013 
33 .52 35 .94 1.865 I.432E+ OO 1.9 -0.107 0.022 
34.27 36.74 1.905 I.342E+ OO 1.2 -0.345 0.015 
36.52 39.13 2.024 9.635E- 01 0.6 -0.411 0.007 
2.064 8.506E- 01 
37.27 39 .92 1.4 -0 .647 
0.017 
2.182 5.298E- 01 0.009 
39.52 42 .31 0.8 -0 .675 
2.221 4.818E- 01 0.009 
40 .27 43 .10 0.6 -0.9 10 
2.375 2.267E- 01 
43 .27 46 .27 1.2 -0.9 77 
0.023 
2.528 1.229E- 01 
46 .27 49 .43 1.4 -0.8 68 
0.0 l6 
2.640 6.301E- 02 0.009 
48 . 52 51 .80 0.8 -0.77 6 
2.678 6.099E- 02 
49 .2 7 52 .5 8 2.1 -0.4
7 5 0.032 
2.788 3.682E- 02 0.011 
51 .5 2 54 . 94 0.9 o.n3 1 
2.970 2.088E- 02 
I 1 55.27 58 . 86 
Table F .5, c) 160(f,p')3J (6.1299 MeV); E, ~ 200 MeV 
500 
-5.86 6.31 0.337 l. 528E- Ol 6.7 0.484 0.095 
7.64 8.22 0.437 6.351E- Ol 1.6 0.487 0.055 
9.14 9.84 0.521 1.ID5E+oo 1.4 0.606 0.015 
10 .64 11.45 0.605 l.44:9E+oo 1.1 0.659 0.011 
12.14 13.06 0.689 2.O22E+oo 6.4 0.718 0.020 
13.64 14.68 0.773 2.o:isE+oo 6.3 0.804 0.025 
15.14 16 .29 0.857 1.725E+ oo 4.1 0.902 0.015 
18.14 19.51 l.O24 1. 2~9E+oo 3.5 0.944 0.060 
18.27 19 .65 l.O32 1.2nE+oo 3.5 0.961 0.041 
19 .64 21.12 1.107 1.1 ,)8E+oo 3.4 0.926 0.023 
19 .77 21.26 l.115 1. O76E+oo 5.3 0.916 0.015 
21.27 22 .86 l.198 7.891E- Ol 4.2 0.842 0.027 
23 .77 25 .54 1.336 4 .3!0E-Ol 2.8 0.517 0.018 
25 .27 27 .14 1.418 2.772E - Ol 2.0 0.200 0.024 
26.77 28 .75 l.5OO J .870E - Ol 4.7 -0.110 0.086 
28 .27 30 .35 l.581 1.289E-Ol 4.7 -0.342 0.025 
29. 77 31.95 l.663 8.512E- O2 5.3 -0.599 0.068 
30.52 32.75 l. 703 6 835E - O2 4.2 -0.548 0.058 
31.27 33.55 l. 743 6.179E - O2 2.2 -0.375 0.029 
33.52 35.94 l.864 3.897E - O2 5.8 0.292 0.077 
34.27 36 .74 l.9O3 3. 76OE-O2 6.0 0.541 0.062 
36 .52 39 .13 2.022 3.107E - O2 9.0 1.023 0.116 
37 .27 39.93 2.062 3.3L2E - O2 10 .6 1.096 0.283 
39 .52 42.3 1 2.179 3.824E - O2 5.9 0.753 0.064 
40 .27 43.10 2.218 3.486E- O2 2.2 0.742 0.024 
43 .27 46 .27 2.373 2 509E - O2 7.1 0.467 0.023 
45.52 48.65 2.487 J . 797E - O2 4.9 0.285 0.065 
46.27 49.44 2.525 l A36E - O2 8.3 0.222 0. 069 
48 .52 51. 80 2.638 l (l08E - O2 6.5 -0.037 0.087 
49.27 52.59 2.675 8. 738E - U3 3.7 -0.(1 28 U.O5U 
51. 52 54.95 2.786 3 919E - O3 6.5 -0.350 0.092 
55.27 58.87 2.967 I .5t3 9E - O3 5.4 -0.598 0.069 
Table F .5 : cl) 16 0(7/, p' )2( (6.9171 MeV ); Er = 200 MeV 
501 
?.1 ?,.? 
O"c .m.(q) I Llo-(%) I Ay I LlAy I 
-5.86 6.31 0.338 4.897E- 02 10.4 0.549 0.127 
7.64 8.22 0.437 4.281E- 02 10.1 0.178 0.278 
9.14 9.84 0.521 5.670E- 02 9.7 0.309 0.103 
10.64 11.45 0.605 1.053E- 01 6.9 0.610 0.066 
12.14 13.06 0.689 1.615E- 01 18.7 0.687 0.052 
13 .64 14.68 0.773 2.480E - 01 4.0 0.787 0.039 
15 .14 16.29 0.857 3.228E- 01 8.5 0.916 0.044 
18.14 19.51 1.024 5.652E - 01 4.1 0.932 0.031 
18.27 19.65 1.032 5.889E- 01 2.8 0.946 0.020 
19.64 21.12 1.107 7.823E - 01 4.5 0.940 0.030 
19.77 21.26 1.115 7.228E- 01 2.5 0.907 0.015 
21.27 22 .86 1.198 7.690E- 0l 2.6 0.886 0.019 
23.77 25 .54 1.335 7.412E - 01 2.7 0.724 0.025 
25.27 27.15 1.418 6.866E - 01 1.1 0.576 0.013 
26.77 28 .75 1.500 6.264E - 01 1.6 0.435 0.029 
28.27 30.35 1.581 5.337E- 01 1.9 0.248 0.012 
29.77 31.95 1.662 4.767E- 01 2.1 0.060 0.028 
30.52 32.75 1.703 4.067E - 01 1.5 -0.005 0.020 
31.27 33 .55 1.743 3.980E-01 0.8 -0.112 0.011 
33 .52 35 .94 1.863 2.674E- 01 1.9 -0.382 0.025 
34.27 36.74 1.903 2.202E-01 1.1 -0.491 0.013 
36.52 39.13 2.022 l.398E - 01 3.3 -0. 702 0.040 
37.27 39.93 2.061 1.189E- 01 1.7 -0. 740 0.063 
40.27 43.11 2.218 5.411E- 02 6.5 -0.944 0.027 
43.27 46.28 2.372 2.199E- 02 3.5 -0. 727 0.026 
45.52 48.65 2.487 l.295E- 02 5.5 -0.177 0.073 
46.27 49.44 2.524 l.108E- 02 10.8 -0.145 0.065 
49.27 52 .59 2.674 6. 758E - 03 5.2 0.(313 0.071 
51.52 54.95 2.785 5.761E- 03 6.8 0.695 0.090 
55.27 58 .87 2.966 2. 798E - 03 3.4 0.-196 0.048 
Table F.5: e) 16 0(p,p')l~ (7 .1169 MeV) ; Ep = 200 MeV 
502 
~er( %) Ay I ~Ay I 
I fh 0c.m. I qc.m, -, 
O'c.m.( q) 
-0 .0 75 0. 203 
I 6.31 I 0 . 340 2.399E-
02 16.8 
-5 . 86 -0.2 73 0.15 4 
8.23 o.4 38 2.815E- 02 
12.4 
7.64 
2.421E- 02 15.4 
0.0 20 0.1 74 
9.14 9.8 4 0.5 22 -0.0 40 0.19 0 
0.6 05 2.542E- 02 
16.8 
10. 64 11 . 45 16.1 -0 .019 0.1 81 
12. 14 13 . 07 0.6 89 
3.545E- 02 
-0.053 0.204 
0.7 72 2.886E- 02 
17.9 
13.64 14. 68 -0.406 0.238 
0.856 2.073E- 02 
23.1 
15.14 16.29 -0.464 0.206 
0.939 3.0l7E- 02 
15.3 
16.64 17.90 -0.204 0.648 
1.022 3.088E- 02 
29.3 
18.14 19.51 -0.606 0.269 
1.030 3.579E- 02 
15.3 
18.27 19.65 -0.190 0.430 
1.113 5.114E- 02 
16.0 
19.77 21.26 -0.147 0.217 
1.196 5.672E- 02 
7.6 
21 .27 22.87 -0.177 0.076 
1.333 5.752E- 02 
5.4 
23.77 25 .55 -0.104 0.081 
1.415 6.501E- 02 
6.0 
25 .27 27.15 -0.263 0.106 
1.497 5.362E- 02 
9.8 
26.77 28.76 -0.268 0.072 
1.578 4.400E- 02 
5.2 
28.27 30 .36 0.183 0.122 
1.659 5.040E- 02 
9.3 
29.77 31.96 0.146 0.106 
1.699 3.775E- 02 
8.2 
30.52 32.76 3.6 -0.240 0.049 
33.56 1.739 
3.844E- 02 
31.27 -0.340 0.118 
1.859 2.273E- 02 
9.0 
33 .52 35 .95 -0.323 0.068 
1.899 2.384E- 02 
4.4 
34.27 36 .75 14.8 -0.336 0.197 
39 .14 2.018 
1.342E- 02 
36 .52 9.5 -0.501 0.073 1.246E- 02 
39 .94 2.057 19.4 -1 .366 0.287 37.27 
42.32 2.174 
6.107E- 03 
-0 .274 o.0 75 39.52 
2.213 7.076E- 03 
5.8 
087 
40 .27 43 .12 3.305E- 03 9.1 
-0. 236 o. 
43 .27 46.29 
2.367 (). 11 .8 -0. 143 149 
49.45 2.519 
3.225E- 03 
1. 165 0.2 17 46 .27 
668 9.8 77E- 04 
11 .4 
2 . 0.528 O.J 40 
49 .27 52 . 60 9.9 I 
2 . 779 1.111E- 03 1 
I 51.52 54 . 96 
Table F.5: !) I 6Q(f',r')2J (8 .8719 MeV); EP ~ 200 MeV 
503 
~<1( %)  l A1,1 I ~Ay l 
1 
I 0L 0c.m . I 
<Tc.m . ( q) 
qc.m . 
1.453E- 02 83 .5 
0.282 1.230 
[ 7.64 8.23 0.439 82.3 0.240 0.855 
10.64 11 .46 0.606 
2.953E- 02 
75 .2 0.535 0.683 7.998E- 02 
12 .14 13.07 0.689 
3.433E- 02 70 .1 
0.869 0.874 
26 .77 28 .76 1.496 -0.488 1.052 
1.536 3.828E- 02 
69 .3 
27.52 29 .56 18.3 0.423 0.361 7.003E - 02 
28 .27 30 .36 1.577 0.334 0.390 
1.738 7.571E- 02 
30.8 
31.27 33 .56 29 .7 -0.346 0.400 2.357E- 02 
36 .52 39 .14 2.016 1.429E- 02 15.8 
-0.288 0.206 
37.27 39 .94 2.055 0.588 1.124 
2.172 9.546E- 03 
78 .0 
39 .52 42 .33 -0.822 0.421 
2.211 1.544E- 02 
32.4 
40 .27 43 .12 -0.839 0.138 
2.365 8.508E- 03 
11 .2 
46 .29 -0.663 0.428 43 .27 
2.479 5.835E- 03 
39 .5 
0.879 
45 .52 48 .66 I - 0.341 
2.517 1.889E- 03 
75 .3 
46.27 49.45 
Table F.5: g) 160(f,p')l2 (9.585 MeV) ; Ep = 200 MeV 
504 
O"c.m.(q) I ~o-(%) I Ay I ~Ay I 
-5.86 6.31 0.342 4.525E- 03 74 .2 0.364 0.916 
7.64 8.23 0.440 9.675E - 03 31.0 -0.410 0.403 
9.14 9.84 0.522 2.507E - 02 14.8 0.095 0.165 
10.64 11 .46 0.606 3.569E- 02 44.1 0.146 0.327 
12.14 13.07 0.689 5.399E- 02 21.0 0.318 0.108 
13 .64 14.68 0.772 5.920E- 02 8.2 0.357 0.105 
15 .14 16 .29 0.856 6.986E- 02 8.1 0.419 0.059 
16.64 17.91 0.939 6.881E- 02 9.3 0.573 0.1 92 
18 .14 19.52 1.022 8.876E- 02 7.2 0.795 0.167 
18 .27 . 19.66 1.030 9.234E - 02 7.7 0.810 0.080 
19.64 21.13 1.104 7.146E- 02 19.2 0.883 0.137 
19.77 21.26 1.112 8.512E- 02 8.2 0.671 0.062 
21.27 22 .87 1.195 8.235E- 02 11 .8 0.807 0.046 
23.77 25.55 1.332 7.605E - 02 8.7 0.847 0.035 
25.27 27.16 1.414 6.702E- 02 4.8 0.911 0.053 
26. 77 28 .76 1.495 6. 735E- 02 4.9 0.710 0.139 
27.52 29 .56 1.536 6.904E - 02 7.9 0.393 0.101 
28.27 30.36 1.576 5.497E- 02 6.7 0.583 0.047 
29.77 31.96 1.657 6.228E - 02 8.4 0.451 0.107 
30.52 32.76 1.697 5.943E- 02 6.5 0.090 0.084 
31.27 33.56 1. 737 5.687E- 02 5.6 0.228 0.071 
33.52 35 .96 1.857 3.604E- 02 8.2 -0.075 0.109 
34.27 36.76 1.897 3.309E- 02 4.8 -0.126 0.064 
36.52 39.15 2.015 2.378E - 02 10 .6 -0.211 0.141 
37.27 39.94 2.055 2.308E- 02 4.0 -0.495 0.065 
39.52 42.33 2.172 l.128E- 02 13. 7 -1.296 0.190 
40 .27 43 .12 2.210 1.250E- 02 12.4 -0.697 0.055 
43.27 46 .29 2.364 6.695E - 03 7.4 -0.84 2 0.068 
45 .52 48.67 2.478 3.123E - 03 16 .5 -1.1 )42 0.192 
46.27 49.46 2.516 3.019E- 03 6.4 -1.l !33 0 101 
49.27 52.61 2.665 l.160E - 03 21.1 -1.34 7 0.297 
Table F .5: h) 16O(p, p')2 t (9 .8445 MeV) ; Ep = 200 MeV 
505 
O"cm.(q) I fla-( %) I Ay I ~ Ay I 
7.64 8.23 0.440 6.062E- 03 60 .9 1.312 0.789 
9.14 9.84 0.523 7.495E- 03 54 .1 0.304 0.580 
10.64 11 .46 0.606 l.368E - 02 43 .8 0.871 1.080 
12.14 13 .07 0.689 1.614E- 02 31.1 1.154 0.309 
13.64 14.68 0.772 3.290E- 02 15.4 0.867 0.139 
15.14 16.30 0.855 3.823E- 02 22 .6 0.777 0.127 
16.64 17.91 0.938 5.526E- 02 14.4 0.814 0.094 
18 .14 19 .52 1.021 9.228E- 02 10.6 0.903 0.047 
18.27 19.66 1.029 7.611E - 02 9.8 0.775 0.081 
19 .64 21.13 1.104 8.579E- 02 18.8 0.753 0.145 
19.77 21.27 1.112 9.240E- 02 11. 7 0.849 0.056 
21.27 22 .87 1.194 l.170E- 0l 5.3 0.897 0.057 
23 .77 25 .55 1.331 1.297E- 0l 3.8 0.876 0.113 
25.27 27.16 1.413 1.245E- 0l 7.6 0.667 0.036 
26.77 28.76 1.494 1.284E- 0l 3.4 0.647 0.038 
27.52 29.56 1.535 1.447E- 0l 4.8 0.343 0.062 
28 .27 30 .36 1.576 l.258E- 0l 2.2 0.430 0.026 
29 .77 31.97 1.656 1.355E- 0l 5.2 0.422 0.066 
30.52 32 .77 1.696 l.288E- 0l 3.8 0.201 0.048 
31.27 33.56 1. 737 1.225E- 0l 2.0 0.165 0.025 
33 .52 35.96 1.856 9.726E- 02 4.6 0.066 0.061 
34.27 36.76 1.896 8.406E- 02 2.8 -0.085 0.155 
36 .52 39.15 2.014 6.089E - 02 7.1 -0.364 0.091 
37.27 39.95 2.053 5.710E- 02 7.7 -0.449 0.045 
39.52 42.33 2.170 2.864E- 02 8.3 -1.005 0.105 
40 .27 43 .12 2.209 3.348E- 02 5.4 -0.605 0.051 
43.27 46.30 2.363 1.514E- 02 9.9 -0.821 0.034 
45 .52 48 .67 2.476 l.033E - 02 7.0 -0. 726 0.078 
46 .27 49.46 2.514 7. 709E- 03 3.6 -0.811 0.042 
49 .27 52 .61 2.663 3. 732E- 03 4.4 -0.678 0.053 
51 .52 54 .97 2.773 3.696E - 03 11 .0 -0.3 14 0.147 
Table F.5: i) 16 0(p,p')4t (10 .356 MeV) ; Ep = 200 MeV 
506 
O"c.m.(q) I ~o-(%) I Ay I ~Ay I 
-5.86 6.31 0.344 3.361E- 02 21.4 0.121 0.271 
7.64 8.23 0.441 3.642E- 02 9.6 0.207 0.394 
9.14 9.84 0.523 4.439E- 02 9.2 0.222 0.101 
10.64 11.46 0.606 4.037E- 02 11.3 0.456 0.256 
12.14 13.07 0.689 5.826E- 02 10.2 0.391 0.098 
13.64 14.68 0.772 4.818E- 02 10.6 0.398 0.101 
15.14 16.30 0.855 3.194E- 02 34.1 0.469 0.207 
16 .64 17.91 0.938 2.376E- 02 22.0 0.569 0.307 
18.14 19.52 1.021 l.626E- 02 36 .0 -0.193 0.335 
18 .27 19.66 1.029 l.363E- 02 39 .0 0.468 0.387 
19.77 21.27 1.112 1.189E- 02 35 .8 -0 .556 0.619 
21.27 22 .88 1.194 7.540E- 03 88 .8 -0.499 0.510 
23 .77 25.56 1.331 8.042E- 03 23 .0 -1.366 0.521 
25.27 27.16 1.412 8.822E- 03 27.3 -0.544 0.739 
26 .77 28.76 1.494 9.298E - 03 22 .7 -0 .543 0.354 
27.52 29.57 1.534 l.369E - 02 26.0 -0.699 0.393 
28.27 30.37 1.575 8.380E- 03 14.5 -0.526 0.220 
29.77 31.97 1.655 5.409E- 03 49.4 -0.170 0.676 
30.52 32.77 1.695 1.148E- 02 18. 7 -0.500 0.283 
31.27 33.57 1. 735 l.193E- 02 11. 7 -0.286 0.116 
33 .52 35.96 1.855 6.016E- 03 31.3 -0.399 0.423 
34.27 36 .76 1.895 l.055E- 02 19 .8 -0.224 0.268 
36.52 39 .15 2.013 5.906E- 03 48.6 -0.220 0.638 
37.27 39.95 2.052 3.818E- 03 27.2 -0.918 0.386 
40.27 43.13 2.208 2.028E- 03 45 .9 -0.138 0.456 
43 .27 46 .30 2.361 l.350E - 03 21.5 -1.072 0.269 
46 .27 49 .46 2.512 6. 738E- 04 56 .0 -1.363 0.796 
51.52 54 .98 2.771 5.892E - 04 47.9 -J.366 0.925 
Table F.5: j) 16O(p,p')0~ (10.957 MeV) ; Er = 200 MeV 
507 
O'c ,;,, _(q) I ~0'(%) I Ay I ~Ay I 
15.14 16.30 0.855 2.649E- 02 108.3 0.665 0.649 
18.27 19.66 1.029 5.545E- 02 56.4 1.006 0.477 
19.77 21.27 1.111 3.779E - 02 37 .0 0.909 0.212 
21.27 . 22.88 1.194 5.230E- 02 60.6 0.961 0.350 
23 .77 25 .56 1.330 6.134E - 02 21. 7 0.803 0.465 
25. 27 27.16 1.412 7.876E- 02 25.3 0.860 0.283 
26 .77 28.77 1.493 6.400E- 02 32 .5 0.958 0.649 
27.52 29.57 1.534 8.406E- 02 62 .5 0.462 0.873 
28.27 30.37 1.574 7.228E- 02 9.6 0.525 0.108 
29. 77 31.97 1.655 7. 712E- 02 21.4 0.555 0.254 
30.52 32.77 1.695 7.202E - 02 42 .6 0.504 0.598 
31. 27 33.57 1. 735 6.063E- 02 26 .6 0.607 0.241 
36.52 39.15 2.013 5.056E- 02 52.3 0.363 0.784 
37.27 39.95 2.052 3.318E- 02 45 .9 -0.305 0.804 
40 .27 43.13 2.207 2.693E - 02 34 .5 0.133 0.446 
43.27 46.30 2.361 l.452E- 02 31.2 -0.325 0.492 
45.52 48.67 2.474 1.445E- 02 57.1 0.124 0.728 
46.27 49.46 2.512 l.296E- 02 67.6 0.041 0.891 
51.52 54 .98 2.771 2.932E- 03 8.9 0.555 0.124 
Table F.5: k ) 160(p,p' )4t (11.097 MeV) ; Ep = 200 MeV 
508 
O"c. m (q) I ~a-(%) I Ay I ~Ay I 
-5 .86 6.31 0.345 l.115E- 0l 6.7 0.469 0.079 
7.64 8.23 0.442 3.551E- 0l 2.4 0.505 0.026 
9.14 9.84 0.524 6.260E- 0l 2.0 0.583 0.021 
10.64 11.46 0.607 8.098E- 0l 2.9 0.646 0.016 
12.14 13.07 0.689 1.155E+oo 4.4 0.715 0.026 
13.64 14.69 0.772 1.181E+oo 6.4 0.772 0.021 
15.14 16.30 0.855 l.037E+ oo 2.1 0.861 0.021 
18.14 19.52 1.021 8.l 72E- 0l 3.6 0.912 0.041 
18.27 19.66 1.029 8.207E- 0l 2.2 0.908 0.027 
19.64 21.13 1.103 6. 761E- 0l 4.6 0.847 0.034 
19.77 21.27 1.111 6.885E- 0l 5.4 0.875 0.051 
21.27 22 .88 1.193 5.191E- 0l 2.8 0.767 0.017 
23 .77 25.56 1.330 3.325E- 0l 4.4 0.468 0.025 
25.27 27.16 1.412 2.l0lE- 01 9.1 0.154 0.032 
26.77 28.77 1.493 1. 718E- 0l 3.0 -0.170 0.098 
28.27 30.37 1.574 1. l 78E- 0l 2.4 -0.363 0.034 
29 .77 31.97 1.654 8.878E- 02 7.0 -0.519 0.095 
30.52 32.77 1.694 8.397E- 02 4.9 -0.433 0.070 
31.27 33.57 1.734 6.700E- 02 3.3 -0.480 0.044 
33 .52 35.97 1.854 4.324E- 02 7.9 0.021 0.105 
34.27 36.76 1.893 4.288E - 02 4.3 0.175 0.066 
37.27 39.95 2.051 3.745E- 02 4.7 0.469 0.140 
40.27 43.13 2.206 2.981E- 02 5.3 0.779 0.175 
43 .27 46 .30 2.359 2.594E- 02 2.8 0.453 0.040 
45.52 48 .68 2.473 2.120E- 02 4.8 0.366 0.069 
46.27 49.47 2.510 1.944E- 02 8.2 O. l 77 0.140 
49.27 52.62 2.659 l.148E- 02 2.3 -0,()85 0.031 
Table F.5 : 1) 16 O(p, p')2f (11.520 MeV) ; Ep =- 200 MeV 
509 
I fh 0-c.m(q) lfia-(%)1 Ay I fiAy I 
-5.86 6.31 0.346 4.274E- 02 10.4 -0.002 0.126 
7.64 8.23 0.443 3.475E- 02 10.4 0.723 0.138 
9.14 9.85 0.525 6.045E- 02 7.4 0.840 0.074 
10.64 11 .46 0.607 7.976E- 02 6.6 0.932 0.062 
12.14 13.07 0.690 1. 701E - 01 10.4 0.930 0.035 
13.64 14.69 0.772 l.686E- 01 11.4 0.952 0.046 
15.14 16.30 0.855 2.104E- 01 6.1 0.950 0.026 
16.64 17.91 0.938 2.142E- 01 6.9 0.820 0.074 
18.14 19.52 1.020 2.162E- 01 6.5 0.749 0.026 
18.27 19.66 1.028 1.832E- 0l 4.6 0.732 0.041 
19.64 21.13 1.103 1.565E- 01 10.2 0.522 0.092 
19 .77 21.27 1.111 1.687E- 0l 6.1 0.584 0.116 
21.27 22 .88 1.193 1.458E- 01 6.3 0.341 0.053 
23 .77 25 .56 1.329 1.065E- 01 13.0 -0.071 0.070 
25.27 27.17 1.411 7.250E- 02 4.0 -0.326 0.075 
26 .77 28 .77 1.492 7.240E- 02 10.0 -0.425 0.120 
28 .27 30.37 1.573 4.640E - 02 6.8 -0.610 0.050 
29.77 31.97 1.653 3.879E- 02 9.6 -0.350 0.130 
30.52 32.77 1.693 3.196E- 02 7.6 -0.405 0.109 
31.27 33 .57 1.733 2.597E - 02 17.6 -0.271 0.060 
33.52 35.97 1.853 1.317E- 02 13.9 0.236 0.187 
34.27 36.77 1.892 1.460E- 02 13 .3 0.529 0.169 
36.52 39.16 2.010 1.411E- 02 15.8 0.670 0.222 
37.27 39.96 2.049 1.159E- 02 15.8 0.749 0.271 
40.27 43.14 2.205 1.013E- 02 7.2 0.688 0.112 
43.27 46. 31 2.358 6. 709E - 03 9.5 0.290 0.105 
45 .52 48.68 2.471 5.757E- 03 12 .0 0. 081 0.16 '.l 
46.27 49.47 2.509 4.430E- 03 24.9 -0.1138 0.089 
49.27 52.63 2.658 1.293E- 03 12.0 -0.~126 0.148 
Table F .5: m) 16O(p,p')0t (12 .049 MeV) ; Ep = 200 MeV 
510 
0-c.m.(q) I L\o-(%) I A y I L\ Ay I 
-5.86 6.31 0.347 l.592E - 02 45.8 -1.314 0.814 
7.64 8.23 0.443 2.061E- 02 32.7 -0.375 0.423 
9.14 9.85 0.525 3.923E - 02 20 .2 -0.666 0.275 
10 .64 11.46 0.607 l.375E- 02 67 .5 -1.299 1.248 
12.14 13.07 0.690 2.301E- 02 34 .7 -0.869 0.523 
13.64 14.69 0.773 1.213E- 02 68 .1 -1.309 1.321 
16.64 17.91 0.938 3.234E- 02 27 .4 0.565 0.261 
18.14 19.52 1.020 8.256E - 02 14.6 0.653 0.172 
18.27 19.66 1.028 7.508E- 02 22 .8 0.616 0.131 
19.64 21.13 1.103 1.059E- 01 31.9 0.646 0.246 
19 .77 21.27 1.111 9.800E - 02 9.9 0.752 0.146 
21.27 22.88 1.193 7.072E - 02 13.0 1.076 0.079 
23.77 25 .56 1.329 5.400E- 02 16.2 0.696 0.129 
25.27 27 .17 1.410 6.315E- 02 7.3 0.673 0.084 
26 .77 28 .77 1.491 5.675E- 02 8.2 0.573 0.098 
28.27 30.37 1.572 4.305E- 02 6.3 0.209 0.085 
29.77 31.98 1.653 4.459E- 02 14.5 0.443 0.190 
30.52 32.78 1.693 4.717E - 02 10.5 0.117 0.137 
31.27 33.58 1. 733 2.930E- 02 9.3 -0.046 0.093 
33.52 35.97 1.852 2.439E- 02 15 .5 -0.383 0.207 
34.27 36 .77 1.891 1.515E- 02 21.9 -0.168 0.291 
36 .52 39 .16 2.009 1.544E- 02 35 .8 -0.151 0.481 
37.27 39.96 2.048 l.108E- 02 13 .8 -0.499 0.179 
39.52 42.34 2.165 5.640E- 03 52 .3 -0.737 0.692 
40.27 43 .14 2.204 l.069E- 02 15 .4 -0.358 0.149 
43.27 46 .31 2.357 6.304E- 03 34 .8 -Cl. 130 0.35 1 
46.27 49.47 2.508 4.303E- 03 47.0 -0.751 0.533 
Table F.5: n) 16O(p,p')l3 (1 2.440 MeV) ; Ep -= 200 MeV 
511 
O"c.m.(q) I ~er( %) I Ay I ~Ay I 
-5.86 6.31 0.348 1.044E- 0l 5.6 -0.0 74 0.067 
7.64 8.23 0.443 1.785E- 0l 3.2 -0.108 0.051 
12 .14 13.08 0.690 2.857E - 0l 5.4 -0.063 0.096 
13 .64 14.69 0.773 2.465E - 0l 4.3 -0.052 0.045 
15 .14 16 .30 0.855 1.974E- 0l 5.3 0.018 0.068 
16 .64 17.91 0.938 1.709E- 0l 5.9 -0.007 0.178 
18.14 19.52 1.020 1.451E- 0l 14.4 0.099 0.147 
18.27 19.66 1.028 1.301E- 0l 5.8 0.056 0.144 
19 .64 21.13 1.102 1.208E- 0l 19.2 0.028 0.210 
19.77 21.27 1.110 l.107E- 0l 10.0 0.077 0.108 
21.27 22 .88 1.193 9.535E- 02 10.3 0.113 0.090 
23 .77 25 .56 1.329 4.915E - 02 10.1 0.002 0.100 
25 .27 27 .17 1.410 l.665E - 02 29 .7 -0. 739 0.638 
26 .77 28.77 1.491 1.591E- 02 17.6 -0.448 0.405 
27 .52 29 .57 1.532 l.532E - 02 62.0 -0.428 0.943 
28.27 30.37 1.572 9.710E- 03 58 .1 -0.452 0.224 
29 .77 31.98 1.652 7.506E - 03 48 .5 -0.692 0.762 
30.52 32.78 1.693 7.120E- 03 46 .3 -1.122 0.874 
31.27 33.58 1.732 6.618E - 03 35 .7 -0.152 0.427 
33.52 35.97 1.852 3.277E- 03 71 .8 -0.362 0.953 
34.27 36 .77 1.891 1.181E- 02 17.8 -0.131 0.239 
36.52 39.16 2.009 7.395E- 03 52 .5 0.839 0.871 
37.27 39.96 2.048 5.679E- 03 16.8 0.166 0.234 
39.52 42.34 2.165 2.625E - 03 69 .0 -1.366 1.106 
40 .27 43.14 2.203 4.164E- 03 34 .5 0.279 0.302 
43 .27 46 .31 2.356 3.207E - 03 16.4 -0.227 o.2 4 7 
46 .27 49 .47 2.507 1.924E- 03 51.6 o.:n1 U.846 
49.27 52.63 2.656 1.033? - 03 29.6 -0.263 0.389 
Table F.5: o) 160(p, p')22 (12 .530 MeV ); Ep = 200 MeV 
512 
0-c.m.(q) I ~a-(%) I Ay I ~Ay I 
-5.86 6.31 0.348 3.928E-02 14.9 0.114 0.178 
7.64 8.23 0.444 l.164E- 02 38.1 -0.432 0.501 
12.14 13.08 0.690 l.191E - 02 122.6 0.528 0.562 
13.64 14.69 0.773 5.890E- 03 99.4 -1.309 1.945 
15.14 16.30 0.855 2.796E- 02 37.3 1.047 0.510 
16.64 17.91 0.938 1.620E- 02 36.4 0.111 0.764 
18. 14 19.52 1.020 1.508E- 02 75 .7 1.089 0.934 
18.27 19.66 1.028 7.659E- 03 96.3 -1.205 2.239 
19.64 21.13 1.102 4.734E-02 70.0 -0.037 0.799 
19.77 21.27 1.110 2.928E- 02 49 .8 0.972 0.279 
21.27 22.88 1.192 5.129E-02 10.0 0.848 0.222 
23.77 25.56 1.329 4.734E - 02 7.2 0.793 0.104 
25 .27 27.17 1.410 2.627E- 02 16.6 0.718 0.131 
26 .77 28.77 1.491 2.251E- 02 30 .8 0.699 0.244 
27.52 29.57 1.531 2.444E- 02 22.7 0.159 0.292 
28.27 30.38 1.572 1.899E- 02 10. 3 0.403 0.221 
29.77 31.98 1.652 2.329E- 02 19 .4 -0 .194 0.264 
30.52 32.78 1.692 2.708E- 02 12.5 0.274 0.161 
31.27 33 .58 1. 732 1.454E-02 10 .0 0.010 0.234 
33.52 35.97 1.851 l.104E- 02 22.9 -0.295 0.306 
34.27 36.77 1.891 8.049E- 03 28.2 0.202 0.386 
36 .52 39.16 2.009 1.332E- 02 21.4 0.128 0.290 
37.27 39.96 2.048 4.768E- 03 25 .3 -0.202 0.416 
39.52 42. 35 2.164 6.993E- 03 30.4 1.107 0.499 
40.27 43.14 2.203 3.064E- 03 23 .6 -0.624 0.276 
43.27 46 .31 2.356 l.996E - 03 66 .0 -0.551 0.513 
46.27 49 .48 2.507 2.808E- 03 31.6 -(Ul94 0. 45 7 
Table F .5 : p) 16 0(_p, p')02 (12 . 796 MeV) ; Ep = 200 MeV 
513 
O"c.m.(q) I l'1o-(%) I Ay j l'1Ay I 
-5.86 6.31 0.349 5.019E- 0l 2.8 0.229 0.035 
7.64 8.23 0.444 6.650E- 0l 2.4 0.253 0.026 
12 .14 13.08 0.690 7.501E- 0l 1.9 0.277 0.034 
13.64 14.69 0.773 6.087E - 0l 2.7 0.266 0.060 
15 .14 16.30 0.855 3.988E- 0l 12.1 0.265 0.068 
16 .64 17.91 0.938 2.953E - 0l 3.8 0.204 0.078 
18.14 19.53 1.020 2.151E - 0l 5.3 0.342 0.086 
18 .27 19.67 1.028 1. 781E- 0l 5.6 0.248 0.065 
19.77 21.28 1.110 1.387E- 0l 13.l 0.277 0.093 
21.27 22.88 1.192 6.497E - 02 18.2 0.119 0.121 
23 .77 25.56 1.328 3.997E- 02 16 .1 0.297 0.105 
25 .27 27 .17 1.410 3.128E- 02 11.3 0.209 0.203 
26 .77 28 .77 1.491 2.521E- 02 14.4 0.090 0.215 
28.27 30.38 1.571 2.844E- 02 23 .6 0.120 0.116 
29.77 31.98 1.652 3.398E- 02 13 .3 0.067 0.175 
30.52 32.78 1.692 2.035E- 02 20 .8 0.037 0.275 
31.27 33.58 1.732 2.253E - 02 7.6 0.256 0.098 
33.52 35.97 1.851 1.921E- 02 13 .7 0.361 0.196 
34.27 36.77 1.890 2.102E- 02 47 .1 0.683 0.875 
36.52 39.16 2.008 1.933E- 02 26.0 0.442 0.427 
37.27 39.96 2.047 8.771E - 03 17.1 0.461 0.212 
39.52 42.35 2.164 3.725E- 03 54.7 -0.387 0.680 
40.27 43.14 2.202 6.863E- 03 49 .2 0.182 0.381 
43.27 46.31 2.355 2.267E- 03 32 .9 0.306 0.288 
46 .27 49.48 2 506 l.054E - 03 89 .0 -0.545 1.()3'1 
49.27 52.63 2.654 8.052E- 04 43 .1 -0. 1. 54 0.559 
Table F.5: q) 16 0(p, p')23 (12.969 MeV) ; Ep = 200 MeV 
514 
-5 .86 6.31 0.349 2.525E- 0l 5.7 0.203 
7.64 0.068 8.23 0.444 2.812E- 0l 5.1 0.300 
9.14 0.059 9.85 0.526 3.627E- 0l 4.5 0.356 
10.64 0.048 11.46 0.608 3.336E- 0l 7.9 0.344 
12.14 0.065 13.08 0.690 4.879E- 0l 4.5 0.432 
13.64 0.036 14.69 0.773 3.953E- 0l 11.1 0.555 
15 .14 0.098 16.30 0.855 3. 717E- 01 10.2 0.733 
16.64 0.026 17.91 0.938 3.951E- 01 4.2 0.741 
18.14 0.071 19.53 1.020 3.598E- 0l 5.3 0.753 
18.27 0.055 19.67 1.028 3.340E- 01 6.3 0.789 
19.64 0.069 21.14 1.102 4.294E- 0l 8.4 
19.77 0.816 21.28 0.081 1.110 3.405E - 01 9.2 0.692 
21.27 22.88 0.112 1.192 3.441E- 0l 3.7 0.817 
23 .77 0.042 25 .56 1.328 3.472E- 0l 2.5 
25.27 0.625 27.17 0.046 1.410 2.761E- 0l 12.1 
26 .77 0.522 28.77 0.058 1.491 2.401E- 0l 3.3 
28 .27 0.421 30.38 0.041 1.571 1. 775E- 0l 4.3 0.237 
29.77 0.032 31.98 1.652 l.513E - 0l 6.6 0.237 
30.52 0.085 32.78 1.692 l.376E- 0l 5.4 
31.27 0.137 33.58 0.070 1.731 l.200E- 0l 7.6 -0.140 
33 .52 0.036 35 .98 1.851 6.969E- 02 7.5 -0.343 
34.27 36.77 0.099 1.890 6.233E- 02 14.3 -0.098 
36 .52 0.568 39.17 2.008 4.812E- 02 16 .l -0.300 
37.27 0.191 39.96 2.047 4.073E- 02 5.3 -0.510 
39 .52 0.114 42 .35 2.163 2.042E- 02 19 .1 -1.366 
40.27 0.294 43 .14 2.202 2.317E- 02 7.4 -0.393 
43 .27 46.31 0.100 2.355 l.124E-02 7.8 -0.2 15 
46.27 49.48 0.205 2.506 l.148E - 02 11 .5 -0.279 
49 .27 O. l clll 52 .63 2.654 4.756E- 03 11 .9 -0.268 0. l 5G 
16
Table F.5 : r) O(p, p')l3.08 (13 .08 MeV); Ep = 200 MeV 
515 
I ~o-( %) l Ay I ~Ay l 
I fh 0c .m . qc.m. I 
O"c.m. ( q) 
-0.16 8 o. 222 l 1 
31 o.3 49 3.046E- 02 
18.1 
-5. 86 6. 0.13 5 o.1 88 
o.4 45 2.989E- 02 
37.7 
7. 64 8.23 
3.282E- 02 18.6 
0.14 7 0.2 03 
9.14 9.85 0.5 26 0.28 8 0.15 1 
0.6 08 5.311E-
02 28.2 
10.64 11 .46 8.3 0.268 0. 0 86 
12.14 13.08 0.6 90 
9.310E- 02 
9.156E- 02 13 .5 
0.393 0.1 10 
13 .64 14.69 0.77 3 0.443 0.063 
0.855 8.979E- 02 
19.9 
15.14 16.30 0.378 0.138 
0.938 I.090E- 01 
32.8 
16.64 17.92 0.245 0.110 
1.020 I.163E- 01 
22 .6 
18.14 19.53 0.566 0.078 
I.028 1.477E- 01 
6.8 
18.27 19.67 0.666 0.212 
1.102 I.672E- 01 
17.6 
19.64 21.14 0.356 0.091 
1.110 I.350E- 01 
5.8 
19.77 21.28 
I.257E- 01 11.1 
0.276 0.081 
21.27 22 .89 I.192 0.366 0.046 
I.328 I.158E- 01 
7.9 
23.77 25 .56 4.3 0.437 0.052 
25 .27 27.17 1.409 
9.616E - 02 
0.372 0.083 
1.490 7.891E- 02 
5.9 
26 .77 28.77 5.978E- 02 14.0 
0.107 0.184 
27 .52 29.58 
I.531 0.358 0.113 
I.571 4.823E- 02 
5.1 
30 .38 0.411 0.181 28 .27 
I.651 4.128E- 02 
13.6 
31.98 0.054 0.141 29.77 
I.691 3.509E- 02 
10.8 
32 .78 0.109 0.126 30.52 
1.731 2.401E- 02 
10.9 
0.241 
31.27 33.58 -0.423 
I.850 1.498E- 02 
18.7 
0.517 
33 .52 35.98 -0.244 
008 7.920?- 03 
38 .9 
2 . o.1 88 
36 .52 39 .1 7 7.656?- 03 13.8 
-0 .434 
.96 2 . 047 -0. 665 0. 182 37 . 27 39 13 .7 6.404E - 03 
43 .14 2.20 2 12.4 -0 562 0.4 
2n 
40 .27 
54 3. 905E- 03 2.3 455 0.2 2 (1 
43 .27 46. 32 18.1 -0. 
2. 50 5 4. 
196E- 03 
48 22 .7 -0. 799 0.3
2 3 
46. 27 49 . I 
53 2. 00 1? - 03 
49 . 27 I 52. 63 
2.6 
Table F .5: s) "O(f,p')3; (!3.259 MeV) ; Er - 200 M V 
516 
0-c.m.(q) I ~er(%) I Ay I ~Ay I 
-5.86 6.32 0.350 1.728E- 02 38.4 0.046 0.461 
7.64 8.23 0.445 1.119E- 02 69 .3 0.551 0.470 
9.14 9.85 0.527 1.822E- 02 33.3 -0.357 0.425 
10.64 11.46 0.608 1.262E- 02 57.1 1.195 0.609 
12.14 13.08 0.691 3.270E- 02 43.7 0.717 0.176 
13.64 14.69 0.773 2.369E-02 29 .7 0.808 0.287 
15.14 16.30 0.855 3.369E- 02 15.4 0.752 0.240 
16.64 17.92 0.938 3.125E- 02 35.9 0.823 0.278 
18.14 19.53 1.020 2.025E- 02 40 .5 1.268 0.353 
18.27 19.67 1.028 2.316E- 02 35.4 0.971 0.268 
19 .64 21.14 1.102 1.964E- 02 87 .8 -0.160 1.063 
19.77 21.28 1.110 1.493E- 02 96 .9 0.714 0.558 
23 .77 25.57 1.328 1.028E- 02 33.6 -1.365 0.775 
25 .27 27 .17 1.409 2.000E- 02 17.2 -0.071 0.235 
26 .77 28.78 1.490 2.442E- 02 19.1 -0.062 0.330 
27.52 29.58 1.530 3.516E- 02 18.6 -0.154 0.253 
28 .27 30.38 1.570 3.339E- 02 9.7 0.283 0.095 
29.77 31.98 1.651 2.771E- 02 20.5 0.061 0.269 
30 .52 32 .78 1.691 1.020E- 02 34 .4 -0.268 0.484 
33.52 35.98 1.849 2.826E- 03 99 .1 0.477 1.389 
36 .52 39.17 2.007 4.032E- 03 81.7 -0. 164 1.086 
39. 52 42 .35 2.162 2.135E- 02 14.1 1.325 0.235 
43.27 46 .32 2.353 6.919E- 04 92 .6 1.326 2.090 
49.27 52 .64 2.652 7.655E- 04 66 .0 -0.422 0.771 
Table F.5 : t) 160(p, p')r( (13 .664 MeV) ; Ep -- 20U Me\/ 
517 
I BL I 8,.m q,m I <T, m(q) I M(%) A, I LiA~ l 
-5.86 6.32 0.351 1 1.248E- 02 79 .5 -1.305 1.308 
73.4 -0.061 0.815 
7.64 8.23 0.446 9.068E- 03 103. 7 1.272 1.252 
10.64 11.46 0.609 1.249E- 02 26.7 0.475 0.266 
12.14 13.08 0.691 3.384E- 02 64 .9 1.237 0.468 
13.64 14.69 0.773 4.614E- 02 17.9 0.822 0.173 
15.14 16.30 0.855 5.573E- 02 37.1 0.843 0.158 
16.64 17.92 0.938 5.792E- 02 15.2 0.754 0.312 
18.14 19.53 1.020 7.058E - 02 24.7 0.981 0.115 
18.27 19.67 1.028 7.045E- 02 40.2 0.599 0.182 
19. 77 21.28 1.110 5.115E-02 11.2 o. 717 0.090 
21.27 22 .89 1.192 7.695E - 02 11 .8 0.782 0.186 
23.77 25 .57 1.327 6.371E- 02 7.1 0.539 0.131 
25 .27 27.17 1.409 7.202E- 02 9.8 0.594 0.145 
26. 77 28. 78 1.490 9.415E - 02 0.406 0.118 
27.52 29.58 1.530 9.889E- 02 9.0 0.513 0.038 
28 .27 30 .38 1.570 1.102E- 01 5.4 0.154 0.092 
29 . 77 31.98 1.650 1.184E- 01 7.0 
2.9 0.257 
0.036 
31.27 33 .58 1.730 9.823E- 02 6.1 0.153 
0.081 
33 .52 35 .98 1.849 7.757E- 02 7.5 0.110 
0.099 
34.27 36 .78 1.889 9.388?- 02 -0 .132 0.103 7.7 
36.52 39 .17 2.006 7.204? - 02 -0 .187 o.1 01 2.6 
37 .27 39.97 2.045 7.893E- 02 o.0 50 6.7 -0 .351 
40.27 43.15 2.200 5.839?- 02 5.3 -0. 455 
0.1 35 
43.27 46 .32 2.353 3.302E- 02 7.4 -0. 696 
0.1 16 
46.27 49 .48 2.503 2.oooE- 02 -0. G96 0.06 7 5.7 
I 49.27 52.64 ?,.651 1 1. 400E - 02 1 
Table F.5: u) "0(f,p')4j (13.869 MeV) ; Er ~ 200 MeV 
518 
O"c.m.(q) I ?la-(%) I Ay I fiA!I I 
7.64 8.23 0.446 3.596E- 02 61.2 0.198 0.733 
13.64 14.69 0.773 1.245E- 02 64 .1 0.058 0.702 
16.64 17.92 0.938 2.618E- 02 102.0 1.254 0.936 
18.14 19.53 1.020 3.217E- 02 43 .5 1.220 0.326 
19.77 21.28 1.110 2.444E- 02 34 .7 0.309 0.413 
21.27 22.89 1.191 2.787E- 02 41.9 0.510 0.447 
23.77 25.57 1.327 1.699E- 02 68.4 -1.366 1.439 
26 .77 28 .78 1.489 9.642E- 03 70.1 0.397 0.856 
27 .52 29.58 1.530 1.644E- 02 55.2 -0 .356 0.772 
28.27 30.38 1.570 4.210E - 03 71.1 -1.366 1.540 
30.52 32.78 1.690 1.896E- 02 18 .8 1.326 0.339 
34.27 36 .78 1.888 3.842E- 02 13.2 0.116 0.178 
43.27 46.32 2.352 4.898E- 03 19.6 -0.974 0.242 
46.27 49.48 2.503 2.054E- 03 84 .1 0.774 1.524 
Table F.5: v) 16 O(p, p')25 (13 .98 MeV); Ep = 200 MeV 
519 
~er(%) Ay ~A!/ 
<Tc.m.( q) 
0L 0c.m. qc.m. 
I.136E- 01 22 .0 
-0.327 0.274 
-5.86 6.32 0.3 51 0.736 0.425 
0.446 4.789E- 02 
36 .5 
7.64 8.23 -1.137 1.233 
0.527 2.562E- 02 
71.5 
9.14 9.85 0.429 0.179 
0.691 I.032E- 01 
18.4 
0.429 
12.14 13.08 49.1 0.997 
13.64 14.69 0.773 
4.704E- 02 
38.6 1.041 
0.211 
8.410E- 02 0.259 
15.14 16 .30 0.855 0.844 
0.938 I.288E- 01 
20.1 
0.220 
16.64 17.92 29.3 0.844 
1.020 1.209E- 01 18 .14 19.53 28 .9 1.058 
0.216 
18.27 19.67 1.028 
8.484?- 02 0.705 0.340 41.1 
1.530E- 01 0.197 
19.64 21.14 1.102 98 .8 0.896 5.875E- 02 0.2 12 
19.77 21.28 1.110 20 .9 1.012 
I.191 9.139E- 02 0.217 0.245 21.27 22 .89 32.6 
I.327 5.379E- 02 0.303 
23.77 25.57 39.2 
-0.208 
4.l91E- 02 
1.408 0.739 25.27 27 .17 45.6 
-0.604 
2.684E- 02 0.047 0.165 
26.77 28.78 1.489 108.l 3.949E- 02 
28 .27 30.38 I.570 13.8 
-0.445 0.197 
3.533?- 02 
1.730 1.345 
31.27 33.58 73 .9 
I.326 
345?- 02 
2.006 1. 
36.52 39 .17 
Table F.5: w) 160(f,p')0i (14.032 MeV); EP ~ 200 MeV 
520 
....... 
<Tc.m.(q) I ~CT(%) I Ay I ~ Ay I 
7.64 8.23 0.447 2.095E- 02 57.2 -0.321 0.713 
12.14 13.08 0.691 2.148E - 02 83.2 1.277 0.627 
13.64 14.69 0.773 3.162E- 02 53 .8 1.263 0.484 
18.27 19.67 1.028 1.200E- 02 71.9 0.792 0.614 
19 .77 21.28 1.109 l.579E - 02 56.0 0.516 0.559 
23.77 25 .57 1.327 8.073E- 03 34 .7 -0 .138 0.321 
25.27 27.17 1.408 9.165E- 03 56 .0 0.120 0.333 
26.77 28 .78 1.489 5.782E- 03 41.3 -0.459 0.609 
27.52 29 .58 1.529 4.290E- 03 94.9 -0. 753 1.528 
28 .27 30.38 1.569 1.102E- 02 21.7 0.059 0.157 
31.27 33.58 1. 729 1.674E- 02 8.6 0.171 0.201 
33.52 35.98 1.848 1.018E- 02 19.9 0.279 0.268 
34.27 36.78 1.888 5.571E- 03 35.8 0.046 0.477 
36.52 39.17 2.005 7.082E - 03 33 .7 -0.508 0.474 
37.27 39.97 2.044 7.716E- 03 11.0 -0 .144 0.142 
40.27 43.15 2.199 4. 786E- 03 12.2 -0.323 0.225 
43 .27 46.32 2.351 1.848E- 03 17.6 0.042 0.445 
49 .27 52 .64 2.650 3.751E- 04 80 .7 0.345 1.263 
Table F.5: x) 16 O(p, p')41 (14 .302 MeV) ; Ep = 200 MeV 
521 
O"c.m.(q) I ~a-(%) I Ay I ~Ay I 
7.64 8.23 0.447 4.244E- 03 89 .0 0.721 1.040 
12.14 13 .08 0.691 4.478E- 03 93.4 -1.307 1.626 
15.14 , 16 .31 0.855 l.999E - 02 68 .9 0.804 0.236 
16.64 17.92 0.938 2.446E- 02 56 .1 0.324 0.542 
18.14 19.53 1.020 5.542E- 02 58 .9 1.172 0.354 
18 .27 19.67 1.027 4.945E- 03 100.2 1.326 0.950 
19 .64 21.14 1.101 2.236E- 02 54 .0 0.708 0.455 
28 .27 30.38 1.569 5.590E- 03 27 .6 0.436 0.339 
31.27 33.59 1.729 l.009E - 02 47.4 0.264 0.142 
33.52 35.98 1.848 5.116E- 03 37.5 -0.336 0.509 
34.27 36.78 1.887 l.356E- 02 13 .2 -0 .186 0.177 
36.52 39.17 2.005 l.222E- 02 21.2 0.375 0.296 
37.27 39 .97 2.044 5.950E- 03 16.2 -0.463 0.277 
39.52 42 .36 2.160 4.119E- 03 44 .7 -0.585 0.620 
40.27 43 .15 2.199 4.200E - 03 15 .1 -0.631 0.204 
43.27 46 .32 2.351 2.410E - 03 15.4 -0.287 0.366 
49.27 52 .64 2.650 l.283E - 03 29 .8 -1.046 0.344 
Table F.5 : y) 16O(p,p')5't (14 .399 MeV) ; Ep = 200 MeV 
522 
F.4 4?Ca(p,p') at E p = 200 MeV 
523 
-? 
Uc.m.(q) I ~u(%) I Ay I ~Ay I 
-7.36 7.58 0.421 1. 778E+03 0.6 0.561 0.007 
10.64 10.96 0.608 4.613E+02 0.7 0.834 0.007 
13.64 14.05 0.778 7.187E+0l 0.7 0.905 0.008 
15.14 15 .60 0.863 2.278E+ 0l 1.4 0.604 0.021 
16 .64 17.14 0.948 1.446E+0l 0.7 -0.219 0.008 
18.14 18.68 1.033 1.482E+0l 0.5 -0 .145 0.020 
18.27 18.82 1.041 1.544E+0l 0.7 -0.030 0.009 
19.64 20.23 1.117 1.716E+0l 0.5 0.293 0.006 
19 .77 20.36 1.125 1. 787E+0l 0.6 0.356 0.007 
21.14 21.77 1.202 1.647E+0l 0.7 0.633 0.007 
21.27 21.90 1.209 1.638E+0l 0.6 0.677 0.007 
23 .77 24.47 1.349 l.056E+0l 0.7 0.950 0.008 
25 .14 25.88 1.425 7.244E+oo 1.0 0.963 0.010 
25.27 26 .02 1.433 6.828E+oo 0.7 0.924 0.008 
26 .74 27 .53 1.513 4.337E+00 0.8 0.794 0.009 
26.77 27.56 1.516 4.276E+oo 0.9 0.753 0.010 
28.27 29.10 1.599 2.418E+oo 1.1 0.444 0.014 
29 .77 30 .64 1.682 1.367E+oo 0.9 0.002 0.012 
31.27 32 .18 1.764 9.446E - 0l 0.4 -0.261 0.006 
34.27 35.25 1.927 5.306E- 0l 0.4 0.273 0.006 
37.27 38.33 2.089 3.927E- 01 0.5 0.866 0.006 
40.27 41.40 2.250 2.698E- 0l 1.7 0.819 0.018 
41.74 42.90 2.326 l.813E - 0l 1.0 0.611 0.011 
43.27 44.47 2.408 1.330E- 01 0.6 0.445 0.007 
45.74 46 .99 2.535 5.413E- 02 1.4 -0 .023 0.018 
46 .27 47 .53 2.565 4.259E - 02 0.7 -0.109 0.010 
49.27 50 .59 2.720 1.525E- 02 1.0 -0. 188 0.013 
52 .27 53 .65 2.872 1.146E- 02 1.0 0.635 O.Oll 
55.27 56.70 3.023 l.014E - 02 0.7 0.844 0.008 
Table F.6: a.) 4?Ca.(p, p)Ot ; Ep = 200 MeV 
524 
<Tc.m.(q) I ~er(%) I Ay I ~Ay I 
-5.86 6.04 0.335 2.060E- 02 31.9 0.774 0.441 
7.64 7.87 0.436 2.393E- 02 32.1 0.445 0.431 
9.14 9.42 0.521 3.833E- 02 14.8 1.267 0.129 
10.64 10.96 0.606 3.973E- 02 18 .3 1.160 0.271 
12 .14 12.51 0.690 2.758E- 02 18.8 0.962 0.153 
13 .64 14.05 0.775 1.905E- 02 14.7 0.763 0.122 
15 .14 15 .60 0.860 l.518E- 02 10.7 0.100 0.119 
16.64 17.14 0.944 l.344E- 02 12 .2 -0.127 0.253 
18 .14 18.69 1.029 2.127E- 02 19.2 0.249 0.088 
18 .27 18.82 1.036 1.350E- 02 13. 7 -0.371 0.281 
19.64 20.23 1.113 1.413E- 02 20 .4 -0.170 0.244 
19 .77 20 .36 1.120 l.640E - 02 10.4 0.061 0.337 
21.14 21.77 1.196 l.345E - 02 25 .3 0.614 0.244 
21.27 21.91 1.204 1.500E- 02 40.0 0.396 0.212 
23 .77 24.48 1.343 8.191E- 03 20.7 0.959 0.239 
25.14 25.89 1.419 8.945E- 03 25 .9 0.811 0.241 
25.27 26 .02 1.427 9.413E - 03 19.4 0.849 0.327 
26.74 27.53 1.507 3.591E- 03 30 .0 0.642 0.356 
26. 77 27.56 1.509 4.741E- 03 29.0 0.779 0.231 
28.27 29 .10 1.592 3.565E- 03 12 .2 0.113 0.162 
29 .77 30.64 1.674 4.869E- 03 11.3 -0.365 0.305 
31.27 32.18 1. 756 2.809E - 03 9.4 -0.372 0.126 
34.27 35.26 1.919 1. 809E - 03 9.1 -0.125 0.123 
37.27 38.33 2.080 2.022E- 03 7.0 0.740 0.082 
40.27 41.40 2.240 8.050E- 04 15 .7 1.174 0.224 
41.74 42.9 1 2.316 1.189E- 03 12.5 0.467 0.150 
43.27 44.47 2.398 6.190E- 04 9.4 0.641 0.122 
45 .74 47.00 2.524 5.009E - 04 15 .0 -0. 129 0.19.5 
46.27 47 .54 2.554 3.972E- 04 8.6 -0.075 0.] 15 
49.27 50 .60 2.708 1.261 E - 04 13 .8 0.1123 0.187 
52 .27 53 .66 2.859 1.143E- 04 11.U 0.660 0.133 
55 .27 56 .71 3.009 l.ll0E- 04 7.2 0.842 0.077 
Table F.6: b) 4?Ca(p, p')0f (3 .3521 MeV ); Ep = 200 MeV 
525 
O"c .m.(q) I ~a-(%) I Ay I ~A!/ I 
-5.86 6.04 0.335 2.448E- 01 10.0 0.043 0.047 
7.64 7.87 0.436 9.541E- 01 3.6 0.369 0.027 
9.14 9.42 0.521 2.343E+oo 3.7 0.475 0.013 
10.64 10.96 0.605 3.845E+00 1.2 0.578 0.025 
12.14 12.51 0.690 5.654E+oo 1.0 0.629 0.010 
13.64 14.05 0.775 7.073E+00 6.7 0.698 0.045 
15.14 15.60 0.859 8.120E+00 2.5 0.814 0.015 
16.64 17.14 0.944 8.623E+oo 1. 7 0.891 0.024 
18.14 18.69 1.028 7.725E+00 0.9 0.904 0.046 
18.27 18.82 1.036 7.735E+oo 0.8 0.949 0.010 
19.64 20.23 1.112 6.743E+oo 3.7 0.948 0.025 
19.77 20 .36 1.120 6.547E+oo 0.8 0.939 0.005 
21.14 21. 77 1.196 5.437E+oo 1.2 0.901 0.011 
21.27 21.91 1.204 5.172E+oo 0.8 0.861 0.005 
23 .77 24.48 1.343 3.116E+00 1.3 0.452 0.006 
25.14 25.89 1.418 2.157E+oo 1.5 0.133 0.018 
25.27 26 .02 1.426 2.174E+oo 2.3 0.096 0.028 
26.74 27.53 1.506 1.469E+oo 1.2 -0.270 0.015 
26.77 27.56 1.509 l.501E+00 1.7 -0 .316 0.022 
28. 27 29.10 1.591 l.0 l 0E+oo 1.9 -0 .614 0.009 
29.77 30.64 1.673 6.599E-01 0.8 -0.681 0.011 
31. 27 32.18 1.755 4.939E- 01 0.6 -0.406 0.008 
34.27 35.26 1.918 3.469E-01 0.6 0.556 0.007 
37.27 38 .33 2.079 2.776E- 0l 0.6 0.735 0.007 
40.27 41.41 2.239 2.0 12E- 01 0.7 0.312 0.024 
41. 74 42.9 1 2.314 l.430E- 01 1.1 -0.008 0.014 
43.27 44.47 2.396 l.093E- 01 0.6 -0.295 0.008 
45 .74 47.00 2.523 5.169E- 02 1.5 -0. 776 0.016 
46.27 47.54 2.552 4.805E - 02 0.8 -0.829 0.009 
49.27 50.60 2.706 2.549E - 02 0.8 -0.483 0.0 1() 
52.27 53.66 2.858 l.808E - 02 0.7 0.278 0.010 
55.27 56.71 3.007 l.225E- 02 0.7 0.622 0.008 
Table F.6 : c) 4?Ca.(p, p')31 (3.7364 MeV); Er = 200 MeV 
526 
eC.Tn. %.m. 0-c.m.(q) I ~a-( %) I Ay I ~ A y I 
-5.86 6.04 0.335 3.409E- 0l 3.1 0.547 0.123 
7.64 7.87 0.436 1.088E+oo 2.1 0.520 0.028 
9.14 9.42 0.521 1.588E+oo 2.3 0.616 0.018 
10.64 10.96 0.605 1.870E+oo 1.5 0.748 0.014 
12.14 12.51 0.690 2.206E+oo 1.5 0.749 0.015 
13 .64 14.05 0.775 1.637E+oo 5.7 0.859 0.056 
15 .14 15 .60 0.859 1.325E+oo 5.7 0.951 0.010 
16.64 17.14 0.943 1.018E+oo 1.3 0.970 0.015 
18.14 18 .69 1.028 5.566E- 01 2.3 0.824 0.061 
18.27 18.82 1.036 5.026E- 01 11.1 0.755 0.033 
19.64 20 .23 1.112 2.934E- 01 12 .6 0.463 0.048 
19.77 20 .36 1.120 3.084E- 0l 8.8 0.472 0.022 
21.14 21.77 1.196 1.580E- 01 8.0 -0 .346 0.108 
21.27 21.91 1.203 l.528E- 01 4.0 -0.387 0.092 
23 .77 24.48 1.342 1.481E- 01 8.4 0.013 0.090 
25 .14 25 .89 1.418 1.380E- 01 6.5 0.032 0.079 
25 .27 26.02 1.426 1.090E- 01 2.7 0.119 0.103 
26 .74 27.53 1.506 1.059E- 01 5.7 0.589 0.073 
26.77 27 .56 1.508 1.105E- 0l 2.3 0.653 0.051 
28 .27 29 .10 1.591 8. 762E- 02 3.8 0.892 0.039 
29 .77 30.64 1.673 7.171E- 02 2.7 0.951 0.032 
31.27 32.18 1. 755 6.148E - 02 1.9 0.832 0.023 
34.27 35.26 1.918 2.929E- 02 2.2 0.323 0.028 
37.27 38.33 2.079 1.134E- 02 3.2 -0.205 0.045 
40 .27 41.41 2.238 5.727E- 03 20.9 0.006 0.088 
41.74 42.91 2.314 3.611E- 03 9.7 0.702 0.122 
43.27 44.47 2.396 3.370E- 03 5.0 1.088 0.071 
45 .74 47 .00 2.522 3.621E - 03 6.6 0.900 0.089 
46.27 47 .54 2.552 2.991E- 03 3.5 0.983 fl . 049 
49 .27 50.60 2.706 1.64 7E - 03 3.8 0.G64 0.052 
52.27 53 .66 2.857 7.022E - 04 4.7 0.229 0.063 
55.27 56.71 3.007 2.031E - 04 9.1 0.375 0.113 
Table F.6: cl) 4?Ca(p,p')2f (2.9041 MeV) ; Ep = 200 MeV 
527 
O' (q) I Liu(%) I A y I LiA y 
.rn . qc .m. c.m. 
02 50 .5 0.0 34 0.6 
13 l 
-5. 86 6.04 0.3 35 1.500E- 30 0.34 5 
7. 64 7.87 0.4 36 l.284E-
02 29.4 -0.2 
48 .2 0.0 55 0.5 40 
9.14 9.42 0.520 9.365E-
03 
27 .3 -0.4 14 0.3 72 
10 .64 10.97 0.605 l.915E- 02 
14.7 0.103 0.1 63 
12.14 12.51 0.690 3.646E- 02 
14.2 0.432 0.072 
13.64 14.06 0.774 9.058E- 02 0.535 0.027 
15.14 15 .60 0.859 l.634E- 01 
2.7 
2.0 0.640 0.019 
16.64 17.14 0.943 2.971E- 01 
5.8 0.658 0.017 
18.14 18 .69 1.027 4.350E- 01 
2.3 0.704 0.036 
18.27 18.82 1.035 4.877E- 01 
2.6 0.777 0.075 
19.64 20 .23 1.111 5.897E- 01 0.782 0.011 
19 .77 20 .37 1.119 6.734E- 01 
4.3 
2.9 0.854 0.024 
21.14 21.77 1.195 7.566E- 01 
1.0 0.836 0.011 
21.27 21.91 1.203 8.192E- 01 
0.8 0.827 0.008 
23 .77 24.48 1.342 l.080E+00 0.801 0.019 
25 .14 25 .89 1.417 1.139E+00 
2.1 
0.9 0.784 0.018 
25.27 26 .02 1.425 1.157E+00 
1.4 0.711 0.015 
26.74 27 .53 1.504 1.112E+00 0.709 0.011 
27.56 1.507 l.204E+00 
0.6 
26.77 
1.188E+00 0.6 
0.610 0.007 
28.27 29.10 1.590 1.4 0.470 0.008 l.lOlE+00 
29.77 30.64 1.672 0.4 0.301 0.006 
31.27 32.18 1.754 
1.039E+00 
0.4 -0.016 0.005 7.838E- 01 
34.27 35 .26 1.916 0.4 -0.333 0.005 
37.27 38.34 2.077 
4.871E- 01 
-0.626 0.022 
2.622E- 01 0.7 
40 .27 41.41 2.236 1.0 -0.714 0.012 
41.74 42.91 2.312 
l.728E- 01 
0.6 -0.825 0.008 
43.27 44 .48 2.394 
l.246E- 01 
1.4 -0 804 0.015 
45 .74 47 .00 2.520 
5.835E- 02 
0.7 -0.833 (l.008 293E- 02 
46 .27 4 7.54 2.5 50 5. 0.8 -0 .5Jl n.o1 0 
49 .27 50 .60 2. 703 2. 
430E- 02 
0.9 -0 .1130 o. 012 
52 .27 53 .66 2. 855 1.1 
47E- 02 
0.9 0.195 o.0 12 
55 .27 56 . 71 3.0 04 
5.6 21E- 03 
' 
Table F.6: e) 40Ca(p,p')51 (4.4915 MeV) ; Er = 200 MeV 
528 
O"c.m.(q) I ~o-(%) I Ay I ~Ay I 
7.64 7.87 0.436 2.143E- 02 102.9 1.313 1.265 
10.64 10.97 0.605 1.146E- 02 110.8 -0. 780 1.802 
13 .64 14.06 0.774 l.004E - 02 57.1 -0.620 0.813 
19.77 20.37 1.118 l.324E- 03 79 .8 1.075 0.721 
28 .27 29 .10 1.588 8.662E - 04 48 .6 -0.489 0.686 
31.27 32.18 1. 752 3.574E- 04 71.4 0.430 0.911 
46 .27 47 .54 2.547 7.559E- 05 42 .9 -0.615 0.535 
Table F.6: f) 4?Ca(p, p')0f (5.213 MeV) ; Ep = 200 MeV 
529 
O"c. rn .(q) I ~a-( %) I Ay I ~ Ay I 
7.64 7.87 0.436 4.219E - 02 27 .6 0.521 0.333 
9.14 9.42 0.521 1.173E- 0l 15 .2 0.550 0.149 
10.64 10.97 0.605 9.885E- 02 77.3 0.848 0.308 
12.14 12.51 0.689 1.333E- 0l 9.1 0.712 0.085 
13.64 14.06 0.774 1.256E- 01 11 .1 0.898 0.094 
15.14 15 .60 0.858 1.190E- 01 6.6 0.911 0.052 
16.64 17 .14 0.942 8.308E- 02 8.1 0.932 0.063 
18.27 18.82 1.034 5.597E- 02 10 .2 0.956 0.086 
21.27 21.91 1.202 2.610E - 02 9.4 0.412 0.107 
23 .77 24.48 1.340 1.666E- 02 10.3 -0.393 0.147 
25 .27 26 .02 1.423 1.531E- 02 10.7 -0.177 0.147 
26 .77 27 .56 1.506 1.426E- 02 9.8 -0.191 0.132 
28 .27 29 .10 1.588 1.066E- 02 10.5 0.035 0.142 
29 .77 30.65 1.670 1.103E- 02 13 . 7 0.458 0.205 
31.27 32.18 1. 752 9.883E - 03 7.1 0.538 0.092 
34.27 35 .26 1.914 6.610E- 03 7.4 0.695 0.087 
37.27 38 .34 2.075 4.776E- 03 8.4 0.180 0.112 
40 .27 41.41 2.234 2.046E - 03 19.7 0.065 0.267 
41.74 42.91 2.310 1.152E- 03 25 .3 -0.615 0.335 
43 .27 44.48 2.392 1.006E- 03 14.9 -0.749 0.184 
46 .27 47 .54 2.547 5.00lE- 04 16 .6 -0.240 0.212 
49.27 50 .60 2.701 4.971E- 04 10 .8 0.504 0.155 
52.27 53 .66 2.852 2.346E- 04 18 .2 0.998 0.227 
55.27 56.72 3.001 1.297E- 04 18.4 0.834 0 184 
Table F.6: g) 4?Ca(p, p')2f (5 .249 MeV ); Ep =- 200 MeV 
530 
0-c.m.(q) I ~o-(%) I Ay I ~Ay I 
7.64 7.87 0.436 1.126E- 02 110.5 1.342 1.436 
10.64 10.97 0.605 5.148E- 02 45 .1 -0.339 0.603 
12 .14 12.51 0.689 3.066E- 02 34.2 0.184 0.363 
13 .64 14.06 0.774 3.564E- 02 34 .8 0.467 0.310 
15 .14 15 .60 0.858 6.255E - 02 11.1 0.799 0.088 
16 .64 17.14 0.942 6.495E- 02 9.8 0.919 0.075 
18 .27 18 .82 1.034 6.803E- 02 8.8 0.967 0.075 
21.27 21.91 1.202 6.323E- 02 4.7 0.881 0.044 
23 .77 24.48 1.340 3.774E- 02 5.3 0.475 0.068 
25 .27 26 .02 1.423 2.991E- 02 6.3 0.145 0.082 
26 .77 27.56 1.506 1.894E- 02 7.5 -0.142 0.101 
28 .27 29 .10 1.588 1.245E- 02 8.9 -0.652 0.121 
29 .77 30.65 1.670 8.744E- 03 13 .7 -0. 953 0.203 
31.27 32.19 1. 752 4.771E- 03 12.4 -0.536 0.181 
34.27 35.26 1.914 3.954E- 03 10.7 0.650 0.126 
37.27 38.34 2.075 4.338E - 03 7.9 0.871 0.100 
40 .27 41.41 2.234 3.512E- 03 11 .0 0.023 0.148 
41.74 42 .91 2.310 2.885E- 03 11 .1 0.098 0.146 
43 .27 44.48 2.392 2.209E- 03 6.9 -0.249 0.089 
46 .27 47.54 2.547 1.067E- 03 7.8 -0.740 0.094 
49.27 50 .60 2.701 4.064E- 04 12.9 -0.926 0.185 
52 .27 53 .66 2.852 3.126E- 04 11. 7 0.130 0.152 
55 .27 56 .72 3.001 2.126E - 04 10.1 0.613 0.109 
Table F .6: h) 4?Ca.(p, p')4i (5.279 MeV); Ep = 2no MeV 
531 
O"c.m.(q) I ~a-( %) I A y I ~ A ll I 
-5.86 6.04 0.336 3.515E- 02 81.6 0.440 0.892 
7.64 7.87 0.436 3.904E- 02 107.4 -1.363 1.950 
12 .14 12 .51 0.689 1.840E- 02 109 .0 1.253 1.047 
15.14 15 .60 0.858 2.519E- 02 34 .6 0.147 0.362 
16 .64 17.15 0.942 l.660E - 02 42 .5 0.564 0.378 
19 .77 20 .37 1.118 2.134E- 02 13. 7 0.366 0.171 
21.27 21.91 1.201 l.176E - 02 19.2 -0.032 0.259 
23 .77 24.48 1.340 l.626E - 02 13 .0 -0.286 0.177 
25 .27 26 .02 1.423 l.377E - 02 15.7 -0.467 0.228 
26.77 27 .56 1.505 1.105E- 02 16 .1 -0. 751 0.228 
28.27 29 .11 1.588 l.692E - 02 10.3 -0.201 0.140 
29.77 30 .65 1.670 1.339E- 02 15.1 -0.222 0.201 
31.27 32 .19 1.751 l.058E- 02 11.3 -0.424 0.146 
34.27 35 .26 1.913 1.098E- 02 7.9 -0.466 0.098 
37.27 38.34 2.074 8.420E- 03 7.4 -0.388 0.095 
40.27 41.41 2.233 l.809E- 03 25 .9 0.089 0.346 
41.74 42 .91 2.309 l.394E - 03 29 .1 -0.595 0.394 
43.27 44 .48 2.391 8.809E- 04 21.2 -0.232 0.295 
46 .27 47 .54 2.546 6.270E- 04 17.2 0.187 0.223 
49 .27 50 .61 2.699 2.285E- 04 25 .7 0.638 0.333 
52.27 53 .66 2.851 3. 795E- 05 79 .6 -0.237 1.098 
55.27 56.72 3.000 2.295E - 05 63 .1 -1.362 1.288 
Table F .6 : i) 4?Ca(p, p')41 (5 .6 143 McV ); Ep = 200 MeV 
532 
CTc .m.(q) I Liu(%) I Ay I L\Ay I 
-5.86 6.04 0.336 4.833E- 02 42 .5 0.855 0.486 
7.64 7.87 0.436 9.339E- 02 39 .7 0.577 0.370 
9.14 9.42 0.521 1.536E- 0l 25 .6 0.406 0.259 
10.64 10 .97 0.605 2.095E- 01 25.4 0.252 0.254 
12 .14 12.51 0.689 1.766E- 01 11 .7 0.671 0.104 
13 .64 14.06 0.774 1.655E- 01 12.4 0.796 0.112 
15.14 15 .60 0.858 9.147E - 02 9.9 1.085 0.082 
16.64 17.15 0.942 7.520E- 02 10.0 0.912 0.084 
18 .14 18.69 1.026 5.925E- 02 6.5 0.752 0.080 
18 .27 18 .82 1.034 5.405E- 02 10.0 0.702 0.112 
19.77 20.37 1.118 1.265E- 02 22 .4 0.042 0.296 
21.27 21.91 1.201 1. 731E- 02 13.5 -0.401 0.194 
23 .77 24.48 1.340 8.469E- 03 23.8 -0 .748 0.357 
25.27 26 .02 1.423 l.152E- 02 18.6 0.217 0.248 
26.77 27 .56 1.505 l.275E - 02 14.2 0.396 0.202 
28 .27 29.11 1.588 4.518E- 03 35 .0 1.023 0.565 
29.77 30.65 1.670 7.325E- 03 26 .7 0.206 0.372 
31.27 32.19 1. 751 7.054E- 03 16.7 0.461 0.255 
34.27 35.26 1.913 l.046E- 03 78 .1 1.154 1.638 
40 .27 41.41 2.233 l.172E- 03 39 .8 -0.107 0.542 
41.74 42.91 2.309 l.120E- 03 36 .3 0.558 0.520 
43.27 44.48 2.391 7.465E- 04 24 .8 0.812 0.349 
46 .27 47.54 2.546 2.276E- 04 45.6 1.299 0.681 
49 .27 50.61 2.699 1.908E- 04 30 .5 0.447 0.391 
52 .27 53 .66 2.851 8.388E- 05 36 .3 0.107 0.487 
55.27 56.72 3.000 3.559E- 05 40 .6 1.:323 0.6 6 7 
Table F.6: j) 4?Ca(p, p')23 (5.6301 MeV ); Er = 200 MeV 
533 
<ic.m.(q) I ~<7(%) I Ay I ~All I 
-5.86 6.04 0.336 2.640E- 02 29 .1 -0 .034 0.358 
7.64 7.87 0.436 1.837E- 02 35.3 0.491 0.197 
9.14 9.42 0.521 2.619E- 02 20 .0 0.269 0.212 
10.64 10.97 0.605 3.454E - 02 19.0 0.567 0.166 
12.14 12 .51 0.689 4.755E- 02 12.0 0.664 0.109 
13.64 14.06 0.774 8.926E- 02 5.9 0.857 0.048 
15.14 15.60 0.858 1.137E- 01 3.4 0.888 0.027 
16.64 17.15 0.942 1.345E-01 3.4 0.951 0.025 
18.14 18.69 1.026 1.182E- 01 3.8 0.802 0.032 
18.27 18.82 1.034 l.171E- 01 4.9 0.885 0.051 
19.77 20.37 1.117 1.141E- 01 2.7 0.649 0.029 
21.27 21.91 1.201 8.077E- 02 3.1 0.381 0.038 
23.77 24.48 1. 339 3.986E- 02 3.9 -0.340 0.053 
25 .27 26 .02 1.422 2.853E- 02 4.5 -0.806 0.053 
26.77 27.57 1.505 2.198E- 02 4.9 -0.744 0.060 
28 .27 29 .11 1.587 1.773E- 02 5.3 -0 .134 0.072 
29.77 30. 65 1.669 1.764E- 02 6.2 0.622 0.074 
31.27 32 .19 1.751 l.880E - 02 3.3 0.803 0.036 
34.27 35.26 1.913 l.859E-02 2.4 0.637 0.028 
37.27 38.34 2.073 l.290E- 02 2.3 0.035 0.031 
40 .27 41.41 2.232 7.403E- 03 4.3 -0.518 0.054 
41.74 42 .91 2.308 4.773E- 03 6.0 -0.699 0.069 
43.27 44.48 2.390 3.333E- 03 3.7 -0.961 0.037 
46.27 47.54 2.545 1.868E- 03 3.6 -0.254 0.048 
49.27 50 .61 2.698 1.265E- 03 3.7 0.519 0.046 
52 .27 53 .66 2.850 7.625E - 04 3.8 0.G80 n. 04 4 
55 .27 56.72 2.999 2.456E- 04 4.7 0.597 0.05G 
Table F.6 : k) 4?Ca.(p, p' )l ~ (5 .9033 MeV); Ep = 200 MeV 
534 
O"c.m.(q) I ~o-(%) I Ay I ~Ay I 
-5.86 6.04 0.337 2.262E- 02 33 .4 0.022 0.407 
7.64 7.87 0.436 1.147E- 02 27 .8 0.180 0.397 
9.14 9.42 0.521 2.305E - 02 22.5 0.018 0.257 
10.64 10.97 0.605 2.458E- 02 24.6 0.570 0.194 
12 .14 12.51 0.689 2.548E- 02 19 .0 0.068 0.211 
13 .64 14.06 0.773 l.707E- 02 18 .7 -0.266 0.231 
15.14 15.60 0.858 1.769E- 02 10.6 -0.093 0.122 
16 .64 17.15 0.942 1.539E- 02 13.2 -0.316 0.164 
18 .14 18.69 1.025 l.513E - 02 13 .3 -0.218 0.157 
18 .27 18 .82 1.033 2.102E- 02 12. 7 -0.240 0.175 
19 .77 20.37 1.117 2.359E- 02 6.6 -0.094 0.090 
21.27 21.9 1 1.201 l.560E - 02 7.9 -0.355 0.114 
23 .77 24.48 1.339 l.090E - 02 8.1 -0.628 0.113 
25.27 26 .02 1.422 8.098E - 03 9.6 -0.456 0.126 
26 .77 27.57 1.505 6.029E- 03 10 .6 -0.683 0.138 
28.27 29 .11 1.587 3.522E- 03 14.1 -0.349 0.191 
29 .77 30.65 1.669 3.579E- 03 16.2 -0.130 0.218 
31.27 32.19 1.750 2.700E- 03 10.7 0.069 0.143 
34.27 35 .26 1.912 l.312E - 03 13 .2 0.175 0.173 
37.27 38. 34 2.073 2.915E- 04 34 .6 0.469 0.458 
40 .27 41.41 2.232 1.545E- 04 72.3 0.116 0.987 
41.74 42.91 2.307 4.359E- 04 25.4 -0.147 0.327 
43 .27 44.48 2.389 2.019E - 04 23 .1 0.608 0.358 
46 .27 47 .54 2.545 l.040E - 04 21.9 0.508 0.329 
49.27 50 .61 2.698 8.403E- 05 21.2 0.751 0.285 
52.27 53 .66 2.849 5.007E- 05 22.3 0.G16 0.286 
55 .27 56 .72 2.998 1.118E- 05 41. l 0.G02 0.5J 3 
Table F .6: I) 4?Ca(p, p')G .028 (6 .028 MeV); Ep = 200 MeV 
535 
O"c.m.(q) I ~o-( %) I Ay I ~Ay I 
-5.86 6.04 0.337 3.589E- 02 22.1 0.310 0.261 
7.64 7.87 0.436 1.059E- 01 6.0 0.364 0.048 
9.14 9.42 0.521 2.l00E- 01 4.4 0.420 0.046 
10.64 10.97 0.605 4.440E- 0l 4.9 0.429 0.032 
12.14 12.51 0.689 5.199E- 0l 2.9 0.550 0.029 
13.64 14.06 0.773 5.824E- 0l 2.4 0.716 0.022 
15.14 15.60 0.857 5.652E - 0l 1.5 0.820 0.013 
16.64 17.15 0.941 5.154E- 0l 1. 7 0.886 0.014 
18.27 18.82 1.033 3.401E- 0l 3.0 0.930 0.036 
19 .77 20.37 1.117 2.222E - 0l 1.9 0.765 0.019 
21.27 21.91 1.200 1.360E- 0l 2.3 0.380 0.028 
23 .77 24 .48 1.339 7.529E- 02 2.8 -0.463 0.037 
25.27 26 .02 1.422 9.187E- 02 2.4 -0.008 0.033 
26 .77 27 .57 1.504 l.226E - 0l 2.0 0.478 0.025 
28.27 29.11 1.586 l.612E- 0l 1. 7 0.762 0.019 
29.77 30 .65 1.668 1.881E- 0l 1.8 0.804 0.019 
31.27 32.19 1. 750 2.057E- 0l 1.0 0.713 0.011 
34.27 35.26 1.912 1.901E- 0l 0.7 0.417 0.009 
37.27 38 .34 2.072 l.281E - Ol 0.7 -0.029 0.009 
40.27 41.41 2.231 6.945E- 02 1.3 -0.499 0.017 
41.74 42 .92 2.307 4.471E - 02 1.9 -0.676 0.022 
43 .27 44.48 2.388 3.099E - 02 1.2 -0.877 0.013 
46.27 47 .55 2.544 1.313E- 02 1.4 -0.715 0.016 
49 .27 50 .61 2.697 7.565E- 03 1.4 0.048 0.019 
52.27 53 .67 2.848 4. 629E - 03 1.5 0..t76 0.018 
55 .27 56.72 2.997 2. 249E- 03 1. 5 0.560 (l .018 
Table F.6 : m) 4?Ca(p, p' )32 (6 .2858 MeV) ; Ep = 200 MeV 
536 
<Tc.m.(q) I il<T( %) I A y 
10 .64 10 .97 0.605 1.733E- 02 62 .8 -0.064 0.732 
12 .14 12.51 0.689 2.114E- 02 26 .5 0.190 0.282 
13.64 14. 06 0.773 3.562E- 02 18.6 0.571 0.172 
15 .14 15.60 0.857 4.413E - 02 8.6 0.806 0.076 
16.64 17.15 0.941 6.903E- 02 6.7 0.828 0.056 
18 .14 18 .69 1.025 7.494E - 02 5.7 0.824 0.054 
18.27 18.82 1.033 7.932E- 02 7.2 0.849 0.073 
19 .77 20.37 1.117 8.339E- 02 4.1 0.891 0.036 
21.27 21.91 1.200 8.562E- 02 3.9 0.926 0.031 
23 .77 24.48 1.338 6.928E- 02 3.8 0.769 0.040 
25 .27 26.02 1.421 5.835E- 02 4.1 0.641 0.047 
26.77 27 .57 1.504 4.589E- 02 4.5 0.401 0.056 
28 .27 29 .11 1.586 3.158E- 02 4.6 0.298 0.061 
29 .77 30 .65 1.668 l .859E- 02 7.3 -0.097 0.098 
31.27 32 .19 1.749 1.378E- 02 5.2 -0.399 0.069 
34.27 35.26 1.911 4.668E - 03 8.5 -0.572 0.122 
37.27 38 .34 2.072 1.594E- 03 14.3 -0.454 0.194 
40.27 41.41 2.231 9.567E- 04 29 .0 -0.052 0.389 
41.74 42 .92 2.306 1.199E- 03 21.5 0.431 0.305 
43.27 44.48 2.388 7.056E- 04 14.5 0.518 0.210 
46.27 47 .55 2.543 6.465E- 04 8.4 0.214 0.115 
49 .27 50 .61 2.696 3.128E- 04 13.0 -0.101 0.175 
52 .27 53 .67 2.847 1.845E- 04 13 .5 -0.216 0.184 
55.27 56.72 2.996 2. 781E- 05 34. 7 -0.526 0.511 
Table F .6 : n) 4?Ca(p, p' )4t (6 .5084 MeV) ; Ep = 200 Me V 
537 
O"c.m.(q) I ~er( %) I Ay I ~ Ay I 
7.64 7.88 0.437 l. 711E- 02 82 .6 0.731 0.791 
9.14 9.42 0.521 l.453E- 02 101.5 1.287 1.154 
12.14 12.51 0.689 3.478E- 02 23 .9 0.676 0.224 
13 .64 14.06 0.773 1.655E- 02 75 .3 0.779 0.648 
15 .14 15.60 0.857 3.206E- 02 22 .1 0.498 0.201 
16 .64 17 .15 0.941 6.132E- 03 97 .1 1.272 0.831 
18 .27 18 .82 1.033 9.492E- 03 50 .1 0.627 0.509 
19 .77 20 .37 1.117 9.745E- 03 33.0 1.284 0.310 
21.27 21.91 1.200 9.061E- 03 33.1 1.226 0.298 
23 .77 24.48 1.338 2.346E- 03 88 .8 1.112 1.097 
25.27 26 .02 1.421 2.211E- 03 95 .2 -0.241 1.386 
31.27 32.19 1. 749 1.005E- 03 82 .9 1.274 1.329 
40.27 41.41 2.231 1.024E- 03 53 .9 1.307 1.103 
43.27 44.48 2.388 6.571E- 04 27 .3 0.627 0.455 
49.27 50 .61 2.696 l.718E- 04 42 .6 0.217 0.568 
52.27 53.67 2.847 4.871E- 05 96 .5 -0.912 1. 701 
55.27 56 .72 2.996 6.821E- 05 34.5 -0.300 0.493 
Table F .6 : o) 4?Ca(p, p')4! (6 .5436 MeV) ; Ep = 200 MeV 
538 
CTc.m.{q) I ~a-{%) I Ay 
-5.86 6.04 0.337 2.135E- 02 39 .6 0.568 0.703 
7.64 7.88 0.437 6.047E - 02 10.0 0.356 0.110 
9.14 9.42 0.521 1.515E- 01 8.2 0.477 0.084 
10 .64 10.97 0.605 2.440E- 01 6.4 0.515 0.060 
12 .14 12.51 0.689 2.854E- 01 4.3 0.664 0.041 
13.64 14.06 0.773 3.526E- 0l 3.9 0.749 0.034 
15 .14 15.60 0.857 3.201E- 01 2.6 0.886 0.022 
16 .64 17.15 0.941 2.942E- 01 2.6 0.889 0.021 
18.27 18 .82 1.033 2.082E- 01 3.9 0.933 0.042 
19 .77 20 .37 1.117 1.307E- 01 2.9 0.724 0.029 
21.27 21.91 1.200 8.088E- 02 3.6 0.379 0.042 
23.77 24.48 1.338 4.492E-02 4.3 -0.479 0.061 
25 .27 26.02 1.421 4.413E- 02 4.3 -0.037 0.058 
26.77 27.57 1.504 6.163E- 02 3.3 0.526 0.041 
28.27 29 .11 1.586 7.943E- 02 2.6 0.838 0.028 
29.77 30.65 1.668 9.875E- 02 2.6 0.856 0.029 
31.27 32.19 1.749 9.857E- 02 1.5 0.714 0.017 
34.27 35.27 1.911 9.808E- 02 1.1 0.324 0.015 
37.27 38 .34 2.072 6.537E- 02 1.1 -0.180 0.015 
40.27 41.41 2.230 3.525E- 02 2.2 -0 .667 0.028 
41.74 42.92 2.306 2.451E- 02 3.0 -0. 799 0.032 
43.27 44.48 2. 388 1.538E- 02 1.9 -0.958 0.021 
46.27 47.55 2.543 6.377E- 03 2.0 -0.612 0.024 
49.27 50.61 2.696 3. l0lE- 03 2.9 0.077 0.039 
52.27 53.67 2.847 l.700E - 03 3.2 0.472 0.041 
55.27 56.72 2.996 7.064E- 04 3.7 0.488 0 04 8 
Table F .6: p) 4?Ca(p,p' )33 (6.5833 MeV) ; Er = 200 MeV 
539 
lTc.rn.(q) I ~a-( %) I A y I ~ Ay I 
-5.86 6.04 0.337 3.810E- 02 22.2 -0.067 0.274 
7.64 7.88 0.437 5.058E - 02 8.1 -0.085 0.132 
9.14 9.42 0.521 4.028E- 02 15.9 0.080 0.179 
10 .64 10.97 0.605 6.466E- 02 14.0 -0.187 0.165 
12.14 12.51 0.689 6.025E- 02 10.3 0.229 0.112 
13.64 14.06 0.773 4.313E- 02 9.8 0.203 0.106 
15.14 15.60 0.857 4.524E- 02 5.9 0.373 0.061 
16.64 17.15 0.941 4.721E- 02 6.4 0.404 0.065 
18.14 18 .69 1.025 3.049E- 02 8.2 0.043 0.092 
18.27 18.82 1.033 3.392E- 02 9.7 0.201 0.126 
19 .77 20 .37 1.116 2.171E- 02 6.6 -0.342 0.093 
21.27 21.91 1.200 1.960E- 02 6.6 -0.482 0.095 
23 .77 24.48 1.338 1.935E- 02 6.1 -0 .188 0.083 
25 .27 26 .02 1.421 ? 1.674E- 02 6.4 -0.494 0.085 
26 .77 27 .57 1.503 1.320E- 02 6.9 -0.282 0.094 
28.27 29.11 1.585 9.564E- 03 8.1 -0.077 0.110 
29.77 30.65 1.667 6.676E- 03 12 .0 -0 .116 0.163 
31.27 32.19 1.749 5.423E- 03 7.3 -0.083 0.099 
34.27 35 .27 1.911 1.884E- 03 12 .2 -0.352 0.172 
37.27 38.34 2.071 2.044E - 04 59 .9 0.601 0.864 
40 .27 41.41 2.230 1.897E- 04 73 .1 0.492 1.117 
41.74 42 .92 2.305 1.266E- 04 92 .8 1.256 1.880 
43.27 44.48 2.387 3.356E- 04 18 .2 0.801 0.298 
46 .27 47 .55 2.542 2.369E- 04 14.7 0.378 0.209 
49.27 50 .61 2.695 6.505E- 05 27 .7 0.735 0.396 
52 .27 53 .67 2.847 3.745E- 05 29.0 0.684 0.370 
55 .27 56 .72 2. 995 1.779E - 05 29.4 0.082 0.389 
Table F.6: q) 4?Ca(p, p')22 (6 .7509 MeV ); Ep = 200 MeV 
540 
<7c.m.(q) I ~<7(%) I Ay 
-5.86 6.04 0.338 3.228E- 01 15.1 0.544 0.168 
9.14 9.42 0.521 1.294E+oo 11.2 0.593 0.112 
10.64 10 .97 0.605 1.549E+oo 15.3 1.272 0.063 
12 .14 12.51 0.689 1.821E+ oo 3.7 0.803 0.033 
15 .14 15.60 0.857 1.469E+oo 3.3 0.931 0.026 
16.64 17.15 0.941 1.261E+oo 3.7 0.930 0.024 
18 .14 18 .69 1.025 1.102E+00 2.2 0.832 0.023 
18.27 18 .82 1.033 9.604E- 01 3.4 0.871 0.041 
19 .77 20 .37 1.116 6.330E- 0l 2.4 0.682 0.026 
21.27 21.91 1.200 5.130E- 0l 2.4 0.288 0.029 
23 .77 24.48 1.338 3.152E- 01 2.8 -0.281 0.038 
25 .27 26 .03 1.421 2.465E- 01 3.1 -0.484 0.041 
26 .77 27 .57 1.503 2.003E- 0l 3.5 -0.304 0.046 
28.27 29 .11 1.585 1.552E- 0l 3.7 -0.057 0.050 
29.77 30 .65 1.667 9.213E- 02 5.4 0.296 0.071 
31.27 32.19 1. 748 7.683E- 02 3.4 0.675 0.038 
34.27 35 .27 1.910 6.146E- 02 3.5 0.730 0.040 
37.27 38 .34 2.071 3.485E - 02 4.1 0.313 0.054 
40.27 41.41 2.230 2.263E- 02 5.4 0.303 0.077 
41.74 42.92 2.305 1.409E- 02 9.4 -0.365 0.117 
43.27 44.48 2.387 9.209E- 03 6.7 -0.493 0.085 
46.27 47.55 2.542 4.627E- 03 8.4 0.088 0.117 
49 .27 50 .61 2.695 2.590E- 03 12 .5 0.739 0.191 
52.27 53 .67 2.846 1.4 72E- 03 17 .9 0.115 0.256 
55 .27 56 .72 2.995 6.382E- 04 Jl .9 0. t4 7 0.169 
Table F .6 : r) 4?Ca.(p, p' )G .909 (6.909 Me V) ; Ep = 200 Me V 
541 
0-c.m.(q) I Llo-(%) I Ay I LlAy I 
7.64 7.88 0.437 5.538E- 01 13.1 0.714 0.528 
10.64 10.97 0.605 4.116E - 01 83 .3 -1.283 1.844 
13.64 14.06 0.773 5. 733E- 01 21.0 0.880 0.148 
16.64 17.15 0.941 2.652E- 01 21.8 1.260 0.150 
18.27 18.82 1.033 2.085E- 01 17.3 0.800 0.159 
19 .77 20.37 1.116 1.145E- 01 16.8 0.722 0.182 
23 .77 24 .48 1.338 9.789E- 02 11.2 0.062 0.151 
26.77 27 .57 1.503 4.915E- 02 18.5 -0.324 0.245 
28.27 29 .11 1.585 7.211E- 02 10.3 0.019 0.138 
29 .77 30.65 1.667 7.356E- 02 10. 7 0.279 0.139 
31.27 32.19 1.748 3.881E- 02 9.9 0.366 0.118 
34.27 35.27 1.910 3.840E- 02 8.3 0.135 0.107 
37.27 38.34 2.071 2.864E- 02 8.5 -0.196 0.116 
41.74 42.92 2.305 1.124E- 02 19.8 -0.512 0.219 
43 .27 44.48 2.387 1.085E- 02 9.5 -0.665 0.099 
46 .27 47.55 2.542 6.863E- 03 10.3 -0.851 0.119 
49 .27 50.61 2.695 5.496E- 03 10.7 -0.277 0.138 
52.27 53.67 2.846 2.532E- 03 13.9 -0.206 0.203 
55.27 56.72 2.995 6.818E- 04 19.8 -0.315 0.293 
Table F .6: s) 4?Ca(p,p')6.931 (6 .931 MeV); Ep = 200 MeV 
542 
ec.m. qc,m. O"c.m.(q) I 60-( %) I A y I 6 Ay I 
-5.86 6.04 0.338 2.087E- 01 18.7 0.391 0.218 
7.64 7.88 0.437 2.653E - 01 79.4 0.442 0.781 
9.14 9.42 0.521 6.983E- 02 92 .3 -0.281 1.185 
12 .14 12.51 0.689 3.215E- 01 12 .1 0.730 0.106 
13.64 14.06 0.773 6.220E- 01 10.3 0.802 0.085 
16.64 17.15 0.941 6.207E- 0l 5.0 0.875 0.036 
18.14 18 .69 1.025 6.140E- 01 3.6 0.732 0.037 
18.27 18 .82 1.033 6.066E- 01 4.3 0.822 0.051 
19 .77 20.37 1.116 5.735E- 0l 2.5 0.631 0.028 
21.27 21.91 1.199 4.735E- 0l 2.4 0.368 0.029 
25 .27 26.03 1.421 1.885E- 01 3.6 -0.627 0.048 
26.77 27.57 1.503 1.165E- 01 5.1 -0.480 0.066 
28 .27 29 .11 1.585 8.297E- 02 6.0 0.084 0.080 
29 .77 30.65 1.667 9.303E- 02 6.3 0.692 0.079 
34.27 35 .27 1.910 1.040E- 01 2.8 0.649 0.034 
37 .27 38.34 2.071 7.896E- 02 3.1 0.085 0.042 
40 .27 41.41 2.229 5.311E- 02 3.4 -0.653 0.038 
41.74 42.92 2.305 3.086E - 02 7.2 -0.759 0.071 
43 .27 44.48 2.386 2.184E- 02 5.0 -0.940 0.044 
46 .27 47 .55 2.542 1.151E- 02 7.0 -0.387 0.079 
49.27 50 .61 2.695 6.511E- 03 11.3 0.309 0.173 
52.27 53 .67 2.846 4.255E- 03 9.6 0.843 0.125 
55 .27 56.72 2.995 1.921E- 03 7. 7 0.582 0. 108 
Table F.6: t) 4?Ca(p, p')l ~ (6 .95 1 Me\! ); Ep = 2()() !\kV 
543 
I (h O"c. m (q) I L.\<7(%) I Ay I L.\A!I I 
-5.86 6.04 0.338 l.692E- 02 45 .8 1.388 0.637 
7.64 7.88 0.437 l.885E- 02 41.0 0.299 0.785 
9.14 9.42 0.521 l.170E- 02 68 .0 1.279 0.747 
10 .64 10.97 0.605 4.206E - 02 29.4 -0.311 0.388 
12 .14 12.51 0.689 4. . 337E- 02 13.9 0.577 0.132 
15.14 15 .60 0.857 4.327E- 02 7.0 0.825 0.057 
16.64 17.15 0.941 4.904E- 02 7.1 0.971 0.052 
18.14 18.69 1.024 4.656E - 02 7.3 0.840 0.063 
18 .27 18 .82 1.032 5.666E- 02 7.5 0.852 0.074 
19.77 20 .37 1.116 6.501E- 02 4.2 0.802 0.042 
23 .77 24.48 1.338 4.555E- 02 4.0 0.626 0.049 
25 .27 26 .03 1.420 4.724E- 02 3.8 0.545 0.049 
26.77 27 .57 1.503 5.052E- 02 3.4 0.445 0.043 
28.27 29 .11 1.585 4.263E- 02 3.6 0.286 0.046 
29 .77 30.65 1.667 3.479E - 02 4.6 0.122 0.061 
31.27 32.19 1.748 3.085E - 02 2.7 -0 .140 0.036 
34.27 35.27 1.910 1.737E- 02 2.8 -0 .355 0.039 
37.27 38.34 2.070 8.560E- 03 3.3 -0 .422 0.044 
40.2 7 41.41 2.229 4.201E - 03 6.8 -0.327 0.090 
41.74 42 .92 2.304 2.946E - 03 9.0 -0 .266 0.113 
43 .27 44.48 2.386 2.013E - 03 5.7 0.005 0.077 
46 .27 47.55 2.541 9.802E- 04 6.5 0.285 0.089 
49.27 50 .61 2.694 4.058E - 04 9.0 0.393 0.118 
52.27 53 .67 2.845 1.970E- 04 9.5 0.063 0.127 
55 .27 56 .72 2.994 9.229E - 05 10.2 0.266 0.134 
Table F .6 : u) 4?Ca(f, p' )7 .11 (7 .11 MeV) ; Er -= 200 MeV 
544 
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