ABSTRACT
Title of dissertation: Measurement of the Electric Form Factor
of the Neutron
at High Momentum Transfer
Jonathan Miller, Doctor of Philosophy, 2009
Dissertation directed by: Professor Elizabeth Beise
Department of Physics
The electric form factor of the neutron, GnE, provides key understanding of
the structure of one of the basic building blocks of visible matter in the universe.
Recent interest in this quantity is the result of the improved quality of data provided
by double polarization experiments, which have substantially improved in the last
decade.
This thesis presents precision measurements of GnE by the E02-013 collabora-
tion at Q2 of 1.7, 2.5, and 3.5 GeV2. This measurement used a double polarization
technique, a highly polarized 3He target, a polarized electron beam, a large accep-
tance spectrometer to detect the scattered electrons, and a large neutron detector
to detect the recoiling hadrons in the reaction 3 vectorHe(vectore,eprimen). These measurements will
be compared to a variety of models of the nucleon?s internal structure, as well as
used to extract individual contributions of the up and down quarks to the nucleon
form factors.
Measurement of the Electric Form Factor
of the Neutron at High Momentum Transfer
by
Jonathan Miller
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2009
Advisory Committee:
Professor Elizabeth Beise, Chair/Advisor
Dr. Bogdan Wojtsekhowski
Professor Nick Hadley
Professor Michael Coplan
Professor Steve Wallace
c? Copyright by
Jonathan Miller
2009
Dedication
In memory of my father
Robert Lewis Miller
1957-2003
and to my mother
Jeanene K Priddy
ii
Acknowledgments
Many people have inspired me, supported me, mentored me, advised me, in-
fluenced me, challenged me, befriended me, and otherwise helped me come to where
I am today, successfully completing my graduate career. It has been a long career,
totalling over seven years, and I am unable to thank everyone completely for their
contribution. I also would like to thank those who caused me to enter the physics
program, from former professors to my parents. In particular, I would like to thank
Betsy Beise, Nick Chant, and Jim Kelly, without whom I would not be graduating.
Without Jim Kelly, I would have left graduate school years ago. He was my
professor, mentor, adviser, and a friend. He believed in me, even when I was strug-
gling to become a productive scientist. I very much wish that he could have been
alive to see me become a productive experimentalist and the successful completion
of my dissertation and defense.
I would like to thank my mentor at the Jefferson Lab, Bogdan Wojtsekhowski,
for his invaluable direction over these last two years. Many of the ideas and tech-
niques that I developed and that are found in the analysis presented in this thesis
were formed in discussions with him or where suggested to me by him. His love of
physics has been an inspiration to me.
From my candidacy, to my defense, my adviser Betsy Beise has supported,
mentored, advised, and guided me. I would like to thank her for providing feed
back so quickly as I struggled to write this thesis, and for her expert direction on
scientific communication. I would also like to especially thank her for accepting me
iii
as a student and for her patience as I worked towards graduation.
When I first entered into graduate school, Nick Chant was the chair of the
physics graduate program. His advising, kindness, and support were crucial, both
in the years before I joined a research group, and during the period of Jim Kelly?s
illness. I would also like to thank Jane Hessing. Without her ability to take care
of almost any administrative issue, as well as timely and frequent reminders to take
care of what was needed, I am sure I would have been unable to complete this
program.
I would like to thank my committee members for reading this thesis and agree-
ing to be on my committee. Specifically Steve Wallace for providing detailed com-
ments and for being a pleasure to TA for. And Nick Hadley for agreeing to be on
my committee on short notice. In particular I would like to thank Mike Coplan for
his advice, understanding, and detailed comments.
I came down to Newport News at the beginning of 2006 to become the expert
of the Neutron Detector with no experience with no experience with how to do so.
There are a number of people who worked with me on the neutron detector to get the
software and hardware working so that the detector could begin taking data when
the experiment started. These people were Rob Feuerbach, Shigey Tajima, Alex
Camsonne, Igor Rachek, Albert Shahinyan, and Nerses Gevorgyan. I also would
like to thank Bodo Reitz for taking a bit of time to direct me in 2005 when I joined
the E02-013 experiment. I would also like to thank Rob Feuerbach for mentoring
early in my E02-013 career.
I would like to thank Gordon Cates and Nilanga Liyanage, spokespeople of the
iv
E02-013 experiment, for their support and for the opportunity to take part in this
experiment. I would also like to thank the E02-013 collaboration for the opportunity
in taking part in this experiment, for the work accomplished, and for the opportunity
to present results. In particular, I thank Bob Michaels, Brad Sawatzky, and Steve
Wood for the parts they played during the early part of E02-013.
There were many students contributed to this experiment. Thanks to Tim Ngo
for working with me on the Neutron Arm geometry, to Aidan Kelleher for advice and
for teaching me about the target. I would like to thank Ameya Kolarkar, Brandon
Craver, and Sergey Abrahamyan for working with me on making this experiment
a success and in working with me to get results. Also I thank Seamus Riordan for
providing feedback on the Big Bite sections of this thesis and for his hard work in
completing the analysis.
In Newport News there were many people who provided me friendship and
support while I studied and worked there. These include my former roommates
Alex Coppens, Aji Daniel, and Nuruzzaman. Also my friends Phil Carter, Fatiha
Benmokhtar, John McNabb, Amrendra Narayan, Micah Veilleux, Megan Freind,
Kalyan Allada, Chiranjib Dutta, and many more provided me with support, dinner
partners, and at times an audience.
I thank the nuclear physics students here at University of Maryland for be-
friending someone who some had not met before the semester of my defense. I would
particularly like to thank Tanya Horn for help with the University of Maryland com-
puter systems at JLab, Jainglai Liu for taking time to answer ROOT questions when
I joined the group, and especially Colleen Ellis for assistance with the ENP groups
v
printer?s and for being friendly when I imposed upon her office. Thanks to Judy
Myrick for taking care of me from far away as the secretary for the experimental
nuclear physics group. She made sure that all the administrative things were cared
for without my presence. I would also like to thank Herbert Breuer for guiding my
investigation of nuclear instrumentation.
I spent more years down at Jefferson Lab than I did at the University of
Maryland. I still needed to return to the University periodically, and I owe a lot
to Buckley and Beth Hopper for the use of their guest room over the years. Also,
during these long years they have provided friendship and a home away from home.
They have helped me when I needed assistance with the necessities of living alone.
Buckley has been my friend since we first entered the University of Maryland physics
program, almost 7.5 years ago. Beth?s parents also provided me with a family to
share holidays with.
I would like to thank my friends at the University of Maryland for provid-
ing me with friendship and company and often a place to stay. In particular my
former roommates: Greg Morrison, Tommy Willis, John Polastro, Scott Fletcher,
Wayne Witzel, and my friends Ben Cooper, Andy York, Tracy Moore, Matt Reames,
Margaret Cheung, and Chris Fleming.
There are many professors who I worked for and learned at the University of
Maryland who took some personal time to advise me or instruct me. Just a few of
these were Prof. Dorfman, Prof. Pati, Prof. Anderson, Prof. Greenberg, and Prof.
Gates.
I have many professors at Gustavus Adolphus College to thank for teaching,
vi
mentoring, and advising me before I enrolled in graduate school. I would particularly
like to thank my mathematics adviser Jeff Rosof, and my physics adviser Paul
Saulnier. I would also like to thank Chuck, DC Henry, Steve Melema, and Thomas
Huber who invited the physics students into their homes and made the physics
department more a group of friends than just a place to learn physics.
Thanks to my friends at Gustavus Adolphus College, Matt Miller, Jonathan
Jennings, Stephen Bacques, Andrew Ohrt, Mike Bland, Mandy Havnen, for being
my friends at Gustavus Adolphus College and entertaining me when on breaks from
home work.
I have many relatives who have supported me, provided me with a place to stay
and to holiday with, who have helped me move, and other necessities. In particular,
I would like to thank my grandpa and grandma Priddy, my grandpa and grandma
Miller, my aunt Diane and her family, and my uncle Ken and his family.
A heartfelt thank to my father, Robert, and mother, Jeanene for providing me
an environment where education was valued. One of my earliest memories is going
to a NASA museum, and how as a youngster I wore out many of the astrophysics
books we had. I would particularly like to thank my mother for sending me care
packages, providing me support, and giving me advice during my education. My dad
was always interested in physics and collected many general physics books which
provided inspiration for me to go into physics myself. I miss my father, and wish he
was still with us.
Besides my parents, my siblings have also provided me with campionship dur-
ing my childhood. Thanks to Ross Miller, for providing me with interesting educa-
vii
tion in humanities in highschool, when the science classes were not as interesting.
Most especially, I would like to thank God for giving me the intelligence to
become a scientist, and for giving me the insight necessary. Also I am thankful for
my continued health, and for all the wonderful people I have met and interacted
with during these years.
viii
Table of Contents
List of Tables xii
List of Figures xvi
List of Abbreviations xxvi
1 Physics of the Nucleon 1
1.1 Introduction to the Nucleon . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Perturbative QCD . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Nucleon Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Partons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 The Constituent Quark Model . . . . . . . . . . . . . . . . . . 8
1.3.3 Generalized Parton Distributions . . . . . . . . . . . . . . . . 9
1.3.4 Vector meson dominance models . . . . . . . . . . . . . . . . . 10
1.3.5 Chiral effective field theory . . . . . . . . . . . . . . . . . . . . 11
1.3.6 Dyson-Schwinger Equations . . . . . . . . . . . . . . . . . . . 12
1.4 Experimental Investigation . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Physics of Electron and Nucleon Scattering 14
2.1 Introduction to electron scattering . . . . . . . . . . . . . . . . . . . . 14
2.2 Definition of kinematic variables . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Current operator . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Dirac and Pauli form factors . . . . . . . . . . . . . . . . . . . 16
2.3 Breit frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Quasi-elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Impulse approximation . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 PWIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Polarization observables . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Recoil polarization . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Target polarization . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Cross section with unpolarized observables . . . . . . . . . . . . . . . 25
2.7 Cross section with polarized observables . . . . . . . . . . . . . . . . 26
3 Form Factors 29
3.1 Models and fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 Vector meson dominance model . . . . . . . . . . . . . . . . . 30
3.1.3 Pion cloud and CQM . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.4 Model of Generalized Parton Distributions . . . . . . . . . . . 31
3.1.5 pQCD predictions . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Measuring GEn via the Rosenbluth technique . . . . . . . . . . . . . 32
3.2.1 Elastic e-D scattering . . . . . . . . . . . . . . . . . . . . . . . 34
ix
3.2.2 Quasi-elastic scattering and the Rosenbluth Technique . . . . 35
3.2.3 Difficulties with Rosenbluth method . . . . . . . . . . . . . . . 36
3.3 Double polarization techniques . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Quasi-elastic scattering and (e,e?n) . . . . . . . . . . . . . . . 38
3.3.2 Recoil polarimetry . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3 Polarized target measurement . . . . . . . . . . . . . . . . . . 43
3.3.4 Experiments with polarized deuterium . . . . . . . . . . . . . 44
3.3.5 Experiments with polarized helium-3 . . . . . . . . . . . . . . 45
3.4 Summary of past measurements . . . . . . . . . . . . . . . . . . . . . 48
3.5 E02-013 at Jefferson Lab . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 GnE experiment E02-013 51
4.1 Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Principle and Experiment setup . . . . . . . . . . . . . . . . . . . . . 52
4.3 The Electron Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 The Helium-3 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.1 Target System . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 Polarization Measurement . . . . . . . . . . . . . . . . . . . . 68
4.5 Big Bite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.1 The Big Bite Apparatus . . . . . . . . . . . . . . . . . . . . . 73
4.5.2 The Big Bite Scintillators and Shower . . . . . . . . . . . . . . 75
4.5.3 The Wire Chambers and Tracking . . . . . . . . . . . . . . . . 76
4.5.4 Big Bite Electronics . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Big Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6.1 Neutron Geometry . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6.2 Veto and Marker Bars . . . . . . . . . . . . . . . . . . . . . . 83
4.6.3 Neutron Arm Electronics . . . . . . . . . . . . . . . . . . . . . 84
4.7 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7.1 Encoding and Decoding . . . . . . . . . . . . . . . . . . . . . 88
5 Analysis 93
5.1 Event filtering and scintillator timing . . . . . . . . . . . . . . . . . . 94
5.2 Electron parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Hadron identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Identification of the hadron charge . . . . . . . . . . . . . . . . . . . 113
5.5 Dilution of the neutron sample by protons . . . . . . . . . . . . . . . 119
5.5.1 Single Track Analysis and Background Correction . . . . . . . 130
5.5.2 Nitrogen Dilution . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5.3 RF correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6 Asymmetry calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.7.1 FSI Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.7.2 Inelastic Contribution . . . . . . . . . . . . . . . . . . . . . . 146
x
6 Results and Discussion 149
6.1 Framework Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3 Pauli and Dirac Form Factors . . . . . . . . . . . . . . . . . . . . . . 153
6.3.1 Flavor Form Factors . . . . . . . . . . . . . . . . . . . . . . . 154
6.4 GPD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4.1 Impact Parameter Space . . . . . . . . . . . . . . . . . . . . . 158
6.4.2 Fit to our data . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.4.3 Quark Orbital Angular Momentum . . . . . . . . . . . . . . . 161
6.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A Neutron Arm Analysis 165
A.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.2 Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.2.1 High Voltage Tune . . . . . . . . . . . . . . . . . . . . . . . . 169
A.2.2 Time Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.2.3 Amplitude Correction . . . . . . . . . . . . . . . . . . . . . . 178
A.2.4 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . 183
A.2.5 Veto Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 183
A.3 Neutron Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.4 Veto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
B Error Analysis 194
C Calculations 206
C.1 Asymmetry Best Value . . . . . . . . . . . . . . . . . . . . . . . . . . 206
D Jefferson Laboratory E02-013 Collaboration 208
xi
List of Tables
1.1 The three light quarks and their respective charges. Other quarks are
the c (charm), b (beauty), and t (top) quark, but they are too heavy
to play a large role in the experiments described here. These three
light quarks form the SU(3) group. . . . . . . . . . . . . . . . . . . . 4
3.1 Double polarization experiments for measuring the electric form fac-
tor of the neutron that have been carried out to date [77]. . . . . . . 39
4.1 A table showing the running conditions, time, accumulated charge,
and kinematics of the different measurements referred to here and
elsewhere as kinematics or for short kin. Kinematic one includes both
the commissioning time and the first measurement. The measurement
of Q2 = 1.3 and 1.7 GeV2 provide a comparison of the polarized 3He
technique with previous recoil polarization data while the other two
measurements extend the measured range of Q2. Not shown is the
periods for commisioning and kinematic 1 which won?t be covered in
this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 A table of the beam polarization as measured by different means at
different times. The M?ller measurements were done in the Hall A
M?ller. The Mott measurements were done using the injector Mott
measurement. Only selected Compton data is presented, and the
Compton data isn?t used in this analysis. . . . . . . . . . . . . . . . . 92
5.1 A table of the initial event selection for analysis. Sanity cut 1 is the
requirement that Big Bite has a track, Sanity cut 2 is the requirement
that at least one scintillator paddle in Big Bite has an event, Sanity
cut 3 is that the coincidence time (L1A) was recorded. Sanity cut
4 is the requirment that there is a coincidence event. Negative is a
requirement that only negative particles in the drift chambers can be
considered as electrons. . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 A table of selection criteria used to determine which veto hits to
consider when doing charge identification. The spatial cuts of Y1,
Y2, and X determine which bars are to be included, events were only
differentiated by time and amplitude within a veto detector which
was within range of the hadron hit. The subscript V signifies veto,
while C signifies the cluster in the neutron arm. . . . . . . . . . . . 119
xii
5.3 A table of the ratios for different criteria. Over the course of a kine-
matic setting, runs varied in rate. This had a small effect on the
purity factor, the results presented in this table is for a normal run. . 126
5.4 The effective ratio of ZN for the target and cuts. The ratio for 3He is
from simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Table of quasi-elastic event selection cuts. The Single QE cut is that
there is one and only one hit in the associated region of time and
q? < 0.15. This is in addition to the region of time and q? which
are selected as quasi-elastic event candidates. The vertical cut and
horizontal cut are neutron arm fidicual cuts. . . . . . . . . . . . . . . 138
5.6 Determination of the physical asymmetry from the measured exper-
imental asymmetry. A full discussion and analysis of errors is pre-
sented in Appendix B. Systematic uncertainties are listed second. . . 139
5.7 Expansion Coefficients in the determination of ? from the physical
asymmetry. Uncertainty in ? calculated as discussed in Appendix B. . 142
5.8 Displayed is the calculated values for ? and Q2, and the value of GnM
as arrived at from a parameterization [59]. From this the value of
GnE can be determined. Determination of uncertainty is discussed in
Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.9 Shown are preliminary inelastic asymmetries and dilutions for the
various kinematic [82]. The * row relates to an expanded selection
of quasi-elastic events, with a maximum invariant mass of 1.15 GeV
rather than the usual maximum of 1.05 GeV and a maximum missing
mass of 2.2 GeV rather than 2 GeV (see Table 5.5). . . . . . . . . . . 148
6.1 Given here are previous fit parameters [59] and a new fit presented in
this thesis. The fit only includes data from experiments using double
polarization techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Given here are the fit parameters from the previous work [45] and a
new fit presented in this thesis. . . . . . . . . . . . . . . . . . . . . . 161
6.3 Presented is a calculation of the quark orbital angular momentum
using Ji?s Sum Rule [54] in the framework of [45]. Compared is this
previous work (**), with a new fit (*) using the form factor results
presented in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.1 A table showing the proton and neutron efficiency as calculated using
the Monte Carlo Simulation. . . . . . . . . . . . . . . . . . . . . . . . 167
xiii
A.2 Table of the list of high voltage changes (HV) for the neutron arm in
the E02-013 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.3 A table showing the deadtime region for a set of veto detectors. This
was selected as the maximum for that set of detectors. . . . . . . . . 193
A.4 A table of the correlation beetween hits in Veto plane 1 and Veto
plane 2. The is for all events within a single run in kinematic 2A
(3190). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
B.1 A table showing the measurements and error calculations for the ni-
trogen dilution of the uncharged sample, for all three kinematics.
N3He is the number of background corrected counts for the selected
3He runs, ?N
2 is the nitrogen density, Q3He is the total charge of the
selected 3He runs. This calculation is as described in section 5.5.2.
Densities are determined by target measurements [62][57][81]. . . . . . 200
B.2 A table of the errors for the calculation of Aproton. The calculation is
calculated according to equation 5.54. The uncertainty in GpM and GpE
is determined using a parameterization [59] and the propagation of
the error is according to equation B.1. Parenthesis is used to identify
the systematic uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . 201
B.3 A table for all three kinematics showing the measurements for the
charged and uncharged background and the uncertainty in the back-
ground ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
B.4 A table of the errors for the calculation of GnE. Uncertainty in the
ratio of form factors was calculated from the physical asymmetry as
expressed in section 5.6 and Table 5.8 and Table 5.7. The uncertainty
in GnM is determined from a parameterization [59] and propagation
of the error is according to equation B.1. . . . . . . . . . . . . . . . . 202
B.5 A table of the uncertainties that contribute to the physical asymme-
try (Aphys). The contributions are calculated and contribute to the
physical asymmetry precision as determined by equation B.1. This
table shows the errors for Q2 = 2.5 GeV2. . . . . . . . . . . . . . . . 203
B.6 A table of the uncertainties that contribute to the physical asymme-
try (Aphys). The contributions are calculated and contribute to the
physical asymmetry precision as determined by equation B.1. This
table shows the errors for Q2 = 3.5 GeV2. . . . . . . . . . . . . . . . 204
xiv
B.7 A table of the uncertainties that contribute to the physical asymme-
try (Aphys). The contributions are calculated and contribute to the
physical asymmetry precision as determined by equation B.1. This
table shows the errors for Q2 = 1.7 GeV2. . . . . . . . . . . . . . . . 205
xv
List of Figures
1.1 Shown is a schematic of the DVCS process, in the framework of gen-
eralized parton distributions (GPDs). Here ? is the skewness, x is
the longitudinal momentum fraction, N is the nucleon, ?? is the vir-
tual photon, and ? is the detected photon. The skewness gives the
longitudinal momentum asymmetry, as shown in the figure. . . . . . . 8
2.1 Feynman diagram for one interaction between an electron and a nu-
cleon. The electron has four-momentum k? and the nucleon has four-
momentum p?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Feynman diagram for interaction between an electron and the nu-
cleus. The electron has four-momentum k? and the ejected nucleon
has momentum p?f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Polarization exchange in the one photon exchange approximation for
polarized-electron, polarized nucleon scattering. ? and ? define the
polarization of the recoiling nucleon. . . . . . . . . . . . . . . . . . . 22
3.1 Various fits discussed in the text. The points plotted are from [66],
which was used to fit the pQCD curve. These data are only repre-
sentative, see figure 3.2 for the full set. The Bijker curve is a rep-
resentative Vector Meson Dominance model, the Guidal curve is a
representative GPD based model, the Miller curves are from a CQM. 33
3.2 PreviousGnE datafromexperimentsinvolvingpolarizationtechniques.
The curve is the Galster parametrization [40]. Herberg provided new
calculations of the Ostrick data, while Golak provided a FSI corrected
analysis of the Becker data. For references see Table 3.3. . . . . . . . 37
4.1 Image of Jefferson Laboratory. Shown is the accelerator and the
mounds over the three experimental halls. The mound to the left is
Hall A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 General layout, not to scale, of the experiment. Shown is the beam-
line, the target, the Big Bite detector stack, and the Neutron Arm
detector. Also shown are the two polarimeters in the hall before the
target, and the approximate distances to the detectors. The compo-
nents of the Big Bite (BB) detector stack are also shown and consist
of the Big Bite dipole magnet, multi-wire drift chambers (MWDCs),
pre-shower, shower, and timing plane. . . . . . . . . . . . . . . . . . . 55
xvi
4.3 Coordinate systems for Big Hand, Target, and Lab. The lab z is along
the beam line while the target z is toward Big Bite. All coordinate
systems are right handed. . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Schematic of the accelerator as it existed in 2006. Shown are the
three halls, injector, and LINACs. . . . . . . . . . . . . . . . . . . . 57
4.5 Diagram of the 3He nucleus. Shown is the most common state with
the proton spins aligned anti-parallel. The neutron carries ? 86% of
the polarization of the nucleus while the protons carry ? 3%. . . . . . 61
4.6 Picture of an empty 3He cell. On top is the spherical pumping cham-
ber, and below is the cylindrical target chamber. The beam passes
through 40cm of target material [41]. . . . . . . . . . . . . . . . . . . 62
4.7 Measurement of the holding field as a function of position along the
beamline in the target. The origin is the target center. . . . . . . . . 65
4.8 Magnet box for Target [57]. Shown is the coils that create the mag-
netic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Figure showing the polarization process [57]. Optical pumping excites
the electron to an excited state, which can exchange (50%) to a state
that decays to a spin state that is not optically pumped by the lasers. 67
4.10 In a traditionally polarized helium cell, one that contains only Rb and
3He, the Rb exchanges spin in one of three ways. Spin is exchanged
either by changing the spin of the He nucleus, or by just rotating
the atoms it interacts with. This rotation is not useful for polarizing
the He, so by adding K, which is likely to engage in ?spin-rotation?
with the Rb, but likely to engage in ?spin-exchange? with He, the
polarization efficiency of the cell is increased. Only the spin-exchange
collision process causes polarization of the 3He [57] [93]. . . . . . . . . 69
4.11 The coils used for the NMR measurement of target polarization.
Shown is the holding field coils (iron box), the RF coils (which can be
adjusted), and the pick-up coils. Two sets of pick-up coils are down at
the target chamber, one more set is up above the pumping chamber.
One of the RF coils could be adjusted, as could the pick-up coils, to
maintain the transverse relationships needed for an NMR measurement. 70
4.12 Measured polarization in the 3He cells during the experiment [56].
Shown is all cells, including Dolly, Edna, and Barbara used during
Kinematics 2, 3, and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
xvii
4.13 Schematic of the Big Bite detector. Shown is the large dipole which
provides a magnetic field integral of 1.2 T?m, the 15 planes in three
chambers that made up the wire chamber, and the lead glass shower
and preshower separated by a layer of plastic scintillator. . . . . . . . 74
4.14 This diagram, which is not to scale, shows Big Bite, the location of
the dipole, target, drift chambers, and shower, and the quantities
known as the deflection angle (?def), ?tgt, and the back and front
tracks. Shown on the upper left is the coordinate system for the
target coordinates. The shower is shown providing a fourth location
for the particle, to anchor the track reconstruction. Big Bite detector
coordinates have their origin at the center of the first plane. The z
direction is perpendicular to the first chamber, and x is the magnetic
dispersion direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.15 Momentum resolution achieved with the latest optics model. Shown
is the momentum resolution for all four kinematics. . . . . . . . . . . 80
4.16 Resolution of the carbon foils after vertex corrections. The foils
should be 6.7 cm apart with the BeO foil at 0 cm. This histogram is
of data from a foil target run in kinematic 3. These foils were used
for the optics calibration. . . . . . . . . . . . . . . . . . . . . . . . . 80
4.17 The electronics schematic for Big Bite. Shown are the modules and
the wire lengths. Two neighboring preshower blocks, and two neigh-
boring sets of shower blocks, which have a summed amplitude that
passes some threshold provides the trigger for the electron arm. . . . 81
4.18 Neutron Arm electronics schematics. Included in the schematic is the
length of the cables. From the PMTs of the various tubes, the signal
goesintobothaTDCandasummodulewiththenon-summedoutput
going to an ADC. The different summed signals are added together
and sent to both an ADC and TDC, with an OR of the TDCs forming
the trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.19 Neutron arm (NA) schematic for the sums. In front are the two veto
planes. Shown is the relative offset. Additionally, color coded, the
sums are shown. Each sum was made up of two neighboring color
bands. Shown on top are the bars from Glasgow (GLA), in the front
are four planes of bars from Carnegie Mellon University (CMU), and
inthe backare three planes of bars from Universityof Virginia (UVA).
The colorless bars in the final plane are bars that were not connected
to the electronics. The first two thin planes are the veto planes, the
figure shows the vertical shift between the two planes. . . . . . . . . . 87
xviii
4.20 Plot showing the reference time and coincidence counts in relation
to the Big Bite trigger time [37]. Best coincident events are in the
circled region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 The flowchart for the analysis. This is after the preliminary replay
of events that is described in section 4.7. The resulting ROOT files
contain the subset of events necessary to undertake the remaining
analysis, also assigns charge, and identifies quasi-elastic event candi-
dates. Two separate scripts are run as part of this process for each run
before this post processing. This is to determine the RF correction
(see section 5.5.3) and the hadron time-of-flight. . . . . . . . . . . . . 95
5.2 Flowchart for the event processing. Events in the analysis are ignored
if no tracks or Big Bite scintillator hits exist, or if the event does not
have a well defined trigger time as demonstrated by the existence of
the Level 1 Accept (L1A) Time. The optics can be updated here if
necessary, and the kinematics of the event are calculated to ease post
processing. All event cuts are detailed in Table 5.1. . . . . . . . . . . 96
5.3 Flowchart associating clusters, both with each other and as quasi-
elastic event candidates. Some clusters should be associated together,
where some neutron detector did not fire to make the cluster continu-
ous. Charged candidates are identified, and momentum is calculated
for the hit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Amplitude distribution for all events showing pions and electrons in
Big Bite. In blue is the distribution after cuts have been applied to
remove events which are lacking the information needed for analysis,
shaded is the region selected. See Table 5.1 for cuts, in black is
all events and in blue is with just the ?sanity? cuts applied. This
distribution is for Q2 = 3.5 GeV2. . . . . . . . . . . . . . . . . . . . . 98
5.5 Electron momentum for all tracks in Big Bite. In blue are just those
tracks for which the preshower energy is greater than 450 channels,
shaded is the selected momentum. See Table 5.1 for cuts, in black
is all events and in blue is with the ?sanity? and pre-shower energy
cuts applied. This distribution is for Q2 = 3.5 GeV2. . . . . . . . . . 101
5.6 X position (m) within the Big Bite scintillator detector. This is for
before cuts (black), after electron and sanity cuts (blue), with the
selected region of X shaded. Variation is due to the lack of calibration
for the total shower in the Big Bite detector. See Table 5.1 for cuts,
in black is all events and in blue is with the ?sanity?, pre-shower
energy, and electron momentum cuts applied. This distribution is for
Q2 = 3.5 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xix
5.7 Vertex, this shows the target z position reconstructed by the Big
Bite wire chambers for all events with tracks (black). In blue is after
cuts on particle ID. Shaded is the region selected as being within the
target. See Table 5.1 for cuts, in black is all events and in blue is
with the ?sanity?, pre-shower energy, Big Bite fidicual, and electron
momentum cuts applied. This distribution is for Q2 = 3.5 GeV2. . . . 103
5.8 q? versus W for neutral particles in the various kinematics. Shown
is the selection of quasi-elastic events in the square. As shown, at
higher Q2 the inelastic tail becomes stronger. . . . . . . . . . . . . . . 105
5.9 q? versus W for charged particles in the various kinematics. In the
red box is the selection of quasi-elastic events. . . . . . . . . . . . . . 106
5.10 qbardbl versus W for charged particles in kinematic 3. . . . . . . . . . . . . 107
5.11 Distribution of q? in the various kinematics. Shown in blue is the
neutral spectrum, in black is the total spectrum (That passed the
event based selection). The majority of neutral quasi-elastic events
fall under 0.15 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.12 ?T (ns) for the various kinematics. In blue is the neutral time-of-
flight spectrum, in black is the total time-of-flight spectrum. The
photon peak is shown at ? ?3ns. Quasi-elastic events are selected
with time between -1 and 1 ns. . . . . . . . . . . . . . . . . . . . . . 110
5.13 Missing mass (GeV) for the various kinematics. In blue is the neutral
scaled up by a factor of 20 for Q2 = 3.5 GeV2 and 10 for Q2 = 2.5
and 1.7 GeV2, in black is the total spectrum. . . . . . . . . . . . . . . 111
5.14 ?T (ns) for the various kinematics. In blue is the charged time-of-
flight spectrum, in blue is the charged time-of-flight spectrum. The
charged spectrum is scaled down to the neutral spectrum. . . . . . . 112
5.15 Shown in these figures is the time versus q? for various kinematics.
Also shown is the box showing which cuts were used for identifying
quasi-elastic clusters within the neutron arm. As is shown, the ability
to select quasi-elastic clusters is much better for the lower kinematics. 114
5.16 Shown in these figures is the time versus invariant mass for various
kinematics. Also shown is the box showing which cuts were used for
identifying quasi-elastic clusters within the neutron arm. As is shown,
the ability to select quasi-elastic clusters is much better for the lower
kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
xx
5.17 Shown in these figures is the time versus missing mass for various
kinematics. Also shown is the box showing which cuts were used for
identifying quasi-elastic clusters within the neutron arm. As is shown,
the ability to select quasi-elastic clusters is much better for the lower
kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.18 Shown in these figures is the invariant mass versus missing mass for
various kinematics. Also shown is the box showing which cuts were
used for identifying quasi-elastic clusters within the neutron arm. As
is shown, the ability to select quasi-elastic clusters is much better for
the lower kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.19 Vertical position (m) within the Big Hand detector. This is for before
cuts (black), after event selection cuts (blue), with the selected region
of x shaded. In the vertical direction the neutron arm was segmented,
giving the segmented structure observed. GLA bars are removed, as
seen in the most negative region. See Table 5.5 for cuts.This distri-
bution is for Q2 = 3.5 GeV2. . . . . . . . . . . . . . . . . . . . . . . . 120
5.20 Horizontal position (m) within the Big Hand detector. This is for
before cuts (black), after event selection cuts (blue), with the selected
region of y shaded. In the horizontal direction the neutron arm was
not entirely covered by the veto planes, giving the peaks on the two
edges. See Table 5.5 for cuts.This distribution is for Q2 = 3.5 GeV2. . 120
5.21 Presented is the ratio of protons to neutrons within the target for
nitrogen. Shown are two models, one a simple extension to the plane
wave impulse approximation (in blue), and the other is a model by
Udias (in black) [96]. Shown is the ratio for various cuts of p?. A cut
of 150 MeV was used in this experiment. This is for Q2 = 3.5 GeV2. . 122
5.22 Presented is the ratio of protons to neutrons within the target for
3He. Shown are from cuts on p?. The applied cut on p? was 0.15
GeV with Pbardbl cuts of 400, 250, and 200 MeV for kinematics of Q2 =
3.5, 2.5, and 1.7 GeV2. Shown is the ratio for various cuts of inverse
distance. This study was done by Aidan Kelleher. . . . . . . . . . . . 123
5.23 Shown in these figures is the measured uncharged to charged ratio, the
uncharged to charged ratio within the model, and the fit line of the
model in nitrogen. These are plotted against the natural dependence
?
T . In green are the points of the model. . . . . . . . . . . . . . . . . 127
5.24 Shown in these figures is the measured uncharged to charged ratio, the
uncharged to charged ratio within the model, and the fit line of the
model in hydrogen. These are plotted against the natural dependence
?
T . In green are the points of the model. . . . . . . . . . . . . . . . . 128
xxi
5.25 Shown in these figures is the measured uncharged to charged ratio,
the uncharged to charged ratio within the model, and the fit line of
the model in 3He. These are plotted against the natural dependence
?
T . In blue are the points of the model. . . . . . . . . . . . . . . . . . 129
5.26 Plot q? for a region in time. This is 5 ns removed from the peak region.131
5.27 Plot q? for a region in time. This is at the quasi-elastic region. . . . . 131
5.28 Plot of the ratio of uncharged to charged for the region of q? between
0.55 and 0.60 GeV. The region near 3 ns is removed due to the photon
peak being there. Add axis information time (ns), update description
(changed plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.29 Plot of q? versus time-of-flight for events with only a single hit in the
|t| < 7 and q? < 0.15 region. The area in the spectrum of events
coming from k photons via pi0 production was removed (between 2
and 4 ns). This is for run 4090. . . . . . . . . . . . . . . . . . . . . . 132
5.30 Spectrum used to determine the RF correction. This is for run 3888.
The peak location is determined from the Mid parameter, which is
trfshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.31 RF correction versus run number for Q2 = 2.5 GeV2. Note that since
the plotted function is Fmod ? 10.499?that the values of 2 ns and 0 ns
are almost equivalent. Run 4146 with a shift of ? 1 ns demonstrates
that the shift likely is not adequately determined for ?junk? runs. . . 137
5.32 Shown is the run number versus raw asymmetry for Q2 = 2.5 GeV2. . 137
5.33 The four dominant lowest order diagrams in single nucleon scattering.
Shown is (a) impulse approximation, (b) meson exchange currents, (c)
isobar contributions, and (d) FSI [88]. . . . . . . . . . . . . . . . . . . 144
5.34 In nuclei with more than a single nucleon, the plane wave impulse
approximation can be used to give the description of the interaction.
Shown are the leading order diagrams for 3He which is the PWIA dia-
gram, single re-scattering diagram, and double re-scattering diagram
[88]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.35 The asymmetry as a function of invariant mass. The plotted curve
is the expected change due to the acceptance of the detectors. This
shows that the invariant mass doesn?t vary much over the region
selected (see Table 5.5). . . . . . . . . . . . . . . . . . . . . . . . . . 147
xxii
6.1 The electric form factor of the neutron with values presented for var-
ious experiments including the one presented in this thesis. Also
included are representative models of the CQM [71], VMD [15][65],
GPD [28][45], and diQuark variety [26]. In black is the traditional
Galster parametrization. Circled is the measurement at Q2 = 1.3
GeV2 which is not included in this thesis. . . . . . . . . . . . . . . . . 150
6.2 Shown is the transverse density in the neutron. This is with the
magnetic form factor described as a dipole, and the electric form
factor described by the fits in Table 6.1. Here + is the previous fit
and ? is the new fit, the transverse density for these fits is almost
identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3 The Dirac form factor, Fn1 , using measured values of GnE and the
dipoleapproximationforGnM. ShowninblackistheGalsterparametriza-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.4 The Pauli form factor, Fn2 , using measured values of GnE and the
dipoleapproximationforGnM. ShowninblackistheGalsterparametriza-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.5 Ratio of the down and up quark flavor Dirac form factors. The form
factors not dependent on GnE are just dipole approximations. The
black curve is for GnE in the Galster parametrization, the dotted curve
is for GnE = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.6 Ratio of the down and up quark flavor Pauli form factors. The form
factors not dependent on GnE are just dipole approximations. The
dotted curve is for GnE in the Galster parametrization, the blue curve
is for GnE = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.7 The combined form factors of the proton (?GpEGn
M
) with values presented
for various experiments. Included are the calculations of the electric
form factor in the GPD framework with the fits as described in Table
6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.8 The electric form factor of the neutron with values presented for var-
ious experiments including those presented in this thesis. Included
are the calculations of the electric form factor in the GPD framework
with the fits as described in Table 6.2. . . . . . . . . . . . . . . . . . 162
6.9 Shown are the upper bounds of the value for b. These bounds are
determined by applying the condition in equation 6.17 [45]. The
solid line is for the Up quark while the dotted is for the down quark. 162
xxiii
A.1 Calibration of the mean time for plane 1 bar 20. This shows the
difference between the time of flight and the expected time of flight
peak for quasi-elastic protons between the BigBite scintillators and
the front plane of the neutron arm. . . . . . . . . . . . . . . . . . . . 175
A.2 Calibration of the horizontal position for plane 1 bar 20. This his-
togram shows the peak in the horizontal bars time difference, with
coincidence with a hit in the ?marker? counters. . . . . . . . . . . . . 176
A.3 Calibration of the mean time difference for plane 2 bar 12. This is the
difference between neighboring bars in different planes. In particular,
this is the difference between plane 1 bar 12 and plane 2 bar 12. . . . 179
A.4 Calibration of the relative horizontal position for plane 2 bar 12. This
is the horizontal position (transformed to time) of the hits in plane 1
bar 12 and plane 2 bar 12 for events with well timed hits within the
two bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.5 Calibration summary plot for run 3975 after pass 3 calibration. In
this plot, X is the result, + is the before calibration. On the x axis,
0 is for plane 1 and 1-6 show the differences between planes 1 and
planes 2-7). At 7 on the x axis is the horizontal absolute position
calibration for plane 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.6 Calibration of the veto time for the left veto plane 1 bar 16. . . . . . 185
A.7 Plot of X position in the veto subtracted by the X position in the
neutron arm versus ?t. These are for quasi-elastic hits in the neutron
arm. Plotted on the x axis is the time difference between the neutron
arm and the veto arm. This identifies a region of 0.3-0.5 m to consider
in the veto detector to identify charge hits. . . . . . . . . . . . . . . . 189
A.8 Plotted is the time difference between the veto hit and the neutron
arm hit for a single veto plane versus the amplitude in the veto bar.
This figure shows all bars in the plane. The gate is seen to start
arround 40 ns before the coincidence hit in the neutron arm. The
deadtime of the various veto detectors is between 40 and 110 ns.
During this dead time, the hits are identified as charged if the am-
plitude is above 200. For a region of ?10 ns around 0, all events are
identified as charged. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.9 Plot showing the rate both with and without a 200 channel ADC
cut for the veto detector L1-14. The ADC cut rate is in blue. The
window between 300 and 350 ns was used to calculate the rate. This
rate is in kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
xxiv
A.10 Plot of hits within the Veto detector. Colors are given to show coin-
cidence between the two separate halves of the V1 veto detector. . . . 191
B.1 Thephysicalasymmetryastheaccidentalbackgroundratioofcharged
to uncharged used in the calculation is changed. This figure shows
that the dependence on this ratio is very small, with an estimated
fractional uncertainty on the accidental background charge ratio of
20%. This esimated fractional uncertainty corresponds to a fractional
uncertainty of less than 0.2%. Plotted is a linear fit, showing the small
dependence. This is for an analysis of Kinematic 2(b). . . . . . . . . 196
xxv
List of Abbreviations
? alpha
? beta
? gamma
? eta
? lambda
QED quantum electro-dynamics
QCD quantum chromo-dynamics
SRF superconducting radio frequency
RF radio frequency
RTD resistive temperature device
CEBAF continuous electron beam facility
EPR electron paramagnetic resonance
BB Big Bite
BH Big Hand
NA Neutron Arm
Rb Rubidium
He Helium
K Potassium
SEOP Spin Exchange Optical Pumping
NMR nuclear magnetic resonance
xxvi
Chapter 1
Physics of the Nucleon
1.1 Introduction to the Nucleon
In 1933 with the discovery that the magnetic moment of the proton (having
mass MN and charge e) was not simply that of a point particle, e2M
N
, nucleons were
known to have internal structure [52]. This structure has been the object of interest
to physicists for over 50 years. Nucleon structure may be examined with scattering
experiments utilizing electromagnetic, hadronic or ?weak? probes. The electromag-
netic interaction, by scattering real or virtual photons from the nucleon, can be used
to parametrize the unknown electromagnetic structure into effective form factors.
One set of parametrizations for the electromagnetic form factors is the Sachs form
factors, GE and GM, which relate to the charge and magnetic distributions of the
nucleons. These form factors only depend on the square of the four-momentum
transfer carried by the photon, Q2.
However, the electromagnetic form factors provide more than just a relation to
thechargeandmagnetization distributions ofthenucleon. Anytheoryofthenucleon
must explain the nucleon?s structure, and the form factors provide an experimental
test of this structure. The form factors are also important for the analysis of many
other experiments, so having precise measurements can enable better determination
of other quantities.
1
This thesis reports on a measurement of the neutron?s electric (otherwise
known as charge) form factor extracted from the asymmetry resulting from po-
larized electron scattering from a polarized 3He target. The structure of this thesis
is as follows. In Chapter 1, the nucleon structure is introduced and discussed within
the framework of various models. In Chapter 2, the formalism required for describ-
ing the form factors and the discussion of the measurement is developed. In Chapter
3, previous measurements of the electric form factor are presented in addition to se-
lected fits within the framework of the models discussed in Chapter 1. In Chapter
4, the experiment is introduced, along with a description of the experimental appa-
ratus. In Chapter 5, the analysis is presented along with the experimental values for
GnE. Finally, in Chapter 6 new results are discussed in the context of the presented
models, the flavor form factors are decoupled, and the quark orbital angular mo-
mentum is presented. The remainder of this chapter will have a short introduction
to quantum chromo-dynamics (QCD) presented in section 1.2 and an introduction
to the various models of nucleon structure in section 1.3.
1.2 QCD
The ground state structure of the nucleon can be investigated using momentum
transfer in an elastic scattering experiment, where the initial state of the nucleon is
not changed. In electron scattering, the electromagnetic properties of the nucleon
are probed, and they are characterized by the aforementioned form factors. These
form factors express the difference between scattering from an object with internal
2
structure and scattering from a point particle.
In the very successful theory of the strong interaction, quantum chromo-
dynamics (QCD), the nucleon is the lowest energy three-quark bound state. QCD
is similar in structure to quantum electro-dynamics (QED), but where in QED the
strength of interaction is governed by electric charge and the exchange particle is
photon, in QCD the exchange particles are gluons and the interaction strength is
due to color charge. The nucleons are then made up of these gluons and quarks,
with the individual quarks being resolved in the Bjorken limit. The three lightest
quarks are identified in Table 1.1.
The notion of valence versus ?sea? quarks, where it is the valence quarks that
define the electromagnetic attributes of the hadron, is useful. For the proton, the
valence quarks are two up quarks and one down quark, for the neutron the valence
quarks are two down quarks and one up quark.
Protons and neutrons, as has been implied here, share many similar proper-
ties. Models and theories have been developed to explain both proton and neutron
properties together, in terms of their shared properties as nucleons. Similar prop-
erties are expected because both nucleons are made up of the same constituents,
and have the same types of valence quarks. Both also have the same spin, are very
close in mass, and are long lived. This allows one to discuss the two nucleons as two
states of a single object, differentiated by the additional quantum number ?isospin?.
When the next lightest quark, the strange quark (with strangeness charge ?1), is
included, an SU(3) symmetry group can be formed, although this symmetry is only
approximate due to the larger mass of the strange quark.
3
The charge in QCD is color, with three colors instead of the single electro-
magnetic charge which is familiar from classical physics. A significant distinction
of QCD is that, unlike the photon in QED, the exchange particles (gluons) hold
the property of color themselves. As a result, the gluons can interact with each
other, and a quark of a particular color will be ?anti-screened?, causing the force to
increase as distance increases. This phenomenon results in confinement. The three
color charges of the quark form a SU(3) group as well. This confinement and the
creation of the QCD vacuum implies the existence of ?sea quarks? (?uu, ?dd, ?ss) and
gluons within the nucleon [46].
name electric charge isospin strangeness current quark mass [35]
u (up) +23 12 0 1.5 to 4 MeV
d (down) ?13 ?12 0 4 to 8 MeV
s (strange) ?13 0 ?1 80 to 130 MeV
Table 1.1: The three light quarks and their respective charges. Other quarks are the
c (charm), b (beauty), and t (top) quark, but they are too heavy to play a large role
in the experiments described here. These three light quarks form the SU(3) group.
1.2.1 Perturbative QCD
Quantum electrodynamics is a highly successful and calculable theory, in part
because it is perturbative. This is not the case for QCD. At large distances from
the bare charge, the QED coupling is the familiar ?(Q2) ? 1137 while the QCD
coupling, ?s(Q2), is ? 1 at low momentum transfer making an expansion in ?s(Q2)
impossible. However, the coupling decreases with increasing Q2, the theory is said
4
to be ?asymptotically free?. The renormalization scale, ?, defines the energy at
which the effective coupling becomes large. When Q2 is well below this, quarks and
gluons are bound into hadrons [46].
At sufficiently high momentum transfers (Q2 > ?2), we would expect to be
able to understand the nucleon structure using perturbative QCD. A virtual photon
of high enough transferred momentum will see the nucleon as consisting of three
massless quarks moving collinearly. In (quasi-)elastic scattering the momentum
of the virtual photon is shared among the three quarks through two hard gluon
exchanges, with each gluon?s momentum being proportional to Q. This gives the
dominant scaling of 1/Q4 for the helicity conserving form factor (known as F1(Q2)).
This power counting is justified by QCD factorization theorems [12].
1.3 Nucleon Structure
When the transferred momentum is below the scale ?, the QCD interaction
is strong and quarks are confined. In this parameter space there is no clear way
to calculate quantities using QCD analytically. There is much promise for future
calculations using direct computational techniques such as Lattice QCD. In the
meanwhile, various models have been developed to describe the dominant features
of nucleon structure.
5
1.3.1 Partons
In deeply inelastic scattering (DIS), where the proton breaks up, the final state
of the interaction cannot be described by a particle in a single final state. Because
of this, a tensor W?? is used to parametrize the unknown final state. In the Bjorken
limit (Q2 ??), apparent point particles known as quarks are resolved inside of the
nucleon, each of which is carrying some fraction of the nucleon?s four-momentum.
The nature of the strong interaction is such that individual quarks are never seen;
instead these ?partons? exit the nucleon in jets of colorless hadrons; either qqq
baryons or q?q pairs (mesons). At lower momentum transfers where deep inelastic
scattering is not applicable, mesons can also be produced through the production
and decay of a nucleon excited state.
In the infinite momentum frame, where the momentum of the hadron is large
and all masses can be neglected, and in the case that the invariant mass of the system
is large, the response tensor W?? can be represented in terms of dimensionless
structure functions. It is useful for these to be expressed in terms of Bjorken x,
a dimensionless kinematic variable of the virtual photon. This variable gives the
component (xE) of the initial energy/momentum carried by the resulting ?parton?.
In the limit x ? 1, all of the momentum is being carried by the parton, and so
the valence quarks dominate the interaction. As x approaches 0, the sea quarks
dominate. The momentum which is not carried by the quarks and antiquarks is
carried by the gluons [46].
The analysis of partons and the constituents of the nucleon is made more
6
complicated because many energetic terms must be included. This means that
relativistic effects should be considered. However, this makes the interpretation of
the wave function unclear, because a Lorentz boost mixes the momentum states. If
QCD is quantized at a fixed light-cone time, ?, where ? = t + zc, the challenges of
mixing due to Lorentz boosts is removed along with the complexity of the vacuum
in relativistic quantum field theory. The following models and interpretations of
nucleon structure are mostly considered in such a quantization. Further discussion
on this is in Section 6.1 [19][20][21][64].
In elastic scattering, where the nucleon neither breaks up nor is excited, there
is a single final state. The tensor W?? then can be reduced to a form factor. Ob-
viously, a complete model of the internal structure of the nucleon must include a
description of form factors. In the case of a spin-12 object like a nucleon, there are
two independent form factors, which can be expressed in the form of Sachs electric
and magnetic form factors (GE and GM) or as the Pauli and Dirac form factors,
F1 (chirality conserving) and F2 (chirality-flip) (The relationship between these will
be described in Chapter 2). In addition to elastic and deeply inelastic scattering
other hard exclusive processes such as deeply virtual Compton scattering (DVCS),
can also used to investigate nucleon structure (for a pictorial description see figure
1.1) [23] [46].
7
Figure 1.1: Shown is a schematic of the DVCS process, in the framework of gen-
eralized parton distributions (GPDs). Here ? is the skewness, x is the longitudinal
momentum fraction, N is the nucleon, ?? is the virtual photon, and ? is the detected
photon. The skewness gives the longitudinal momentum asymmetry, as shown in
the figure.
1.3.2 The Constituent Quark Model
An early model of nucleon structure is the constituent quark model (CQM).
In this case, the nucleon is a ground state of three massive quarks in a confining
potential. The masses of these quarks are determined by SU(3) flavor symmetry and
it is assumed that the mass of the hadron is held by just the valence quarks. While
relativistic modifications of this model have had some descriptive success, the model
does not satisfy the chiral symmetry of the QCD Lagrangian. Some modifications,
such as goldstone-boson-exchange (GBE) where there is an additional quark-quark
interaction, and the one-gluon exchange (OGE), provide a general fit to the data
for the form factors and the radii of the nucleons.
The long distance behavior of hadronic wave functions should be able to be
described by the exchange of the lightest of the q?q states, the pions. As a result,
pion signatures should be seen in the low momentum behavior of hadrons, and the
8
addition of this pion cloud to the constituent quark model is used to form a better
model of the nucleon. An early model of this type is the cloudy bag model. Here the
pion cloud interacts with the confined quarks such that chiral symmetry is restored.
This cloudy bag model provides a good fit of low Q2 nucleon form factors [76][69].
1.3.3 Generalized Parton Distributions
In a number of reactions, the scattering amplitudes factorize such that they
can be described by a simple diagram known as the handbag diagram. Here there is
one valence quark interacting with the virtual photon, with the rest of the structure
contained within a generalized parton distribution (GPD). An appealing feature of
GPDs is that, depending on the kinematic limit selected, they can be related to
a wide variety of scattering processes, such as deeply inelastic scattering, deeply
virtual Compton scattering, wide angle compton scattering, and elastic scattering
[29] [45].
Currently, GPDs cannot be measured explicitly, and are instead expressed
by models which are generally simple parametrizations constrained by the GPD
relationship to the form factors, properties of the nucleon, and by the fact that the
GPDs become parton densities in vanishing momentum transfer.
In this framework, hard exclusive reactions (like DVCS) result in an interaction
between the probe and a single quark within the nucleon (see Figure 1.1). At leading
order, the interaction that is described by the traditional form factors can also be
described by four Generalized Parton Distributions (GPDs), these GPDs depend
9
on the longitudinal momentum fractions (light-cone and skewness), and the squared
four-momentum transfer (Q2) to the nucleon. The four GPDs are Hv and Ev, where
v indicates either an up quark or a down quark. 1
These GPDs depend on Bjorken x, momentum transfer Q2, skewness (?), and
a scale parameter ?. The following sum rules, which are independent of ?, relate
the GPDs and the form factors:
Z 1
?1
dxHv(x,?,Q2) = Fv1(Q2) , (1.1)
Z 1
?1
dxEv(x,?,Q2) = Fv2(Q2) . (1.2)
Fv1,2(Q2) are the flavor form factors, which describe the distribution of that flavor of
quark within the nucleon. These can be expressed in terms of the electromagnetic
form factors as Fu1 = 2Fp1 + Fn1 + Fs1 and Fd1 = 2Fn1 + Fp1 + Fs1, where Fs1 is the
strangeness form factor of the nucleon. It is assumed, here as in most analyses, that
the strange form factor is negligible. Similar relationships exist for the Pauli form
factor, F2. The GPDs must still be modeled in order to predict form factors, and
usually this is done using the sum rules [29][28][45].
1.3.4 Vector meson dominance models
Some of the earliest models coupled a virtual photon to the nucleon both
through its internal structure and through exchange of intermediate vector mesons
with a meson cloud. These vector mesons (the ?0, ?, and ?) have the same quantum
1There are other GPDs which aren?t related to the electromagnetic form factors such as one to
describe the polarized distribution.
10
numbers as the virtual photon and are the lightest hadrons in the time-like region
(Q2 < 0 in the standard definition), and so could potentially play a major role in
nucleon structure at low Q2. The form factors are commonly described by a dipole
form, resulting from two nearby vector meson resonances (poles) that have opposite
residue. This structure can be written in terms of a photon-meson coupling strength
(C?Vi) and a meson-nucleon vertex (FViN):
Fiv,is =
X miC?V
i
Q2 +m2i FViN(Q
2). (1.3)
These isovector and isoscalar form factors (Fiv and Fis) are then written as linear
combinations of the electromagnetic form factors [76][77].
The internal structure coupling is sometimes identified with the three valence
quarks. The initial models had only F1 coupling with the internal structure, but
some of those considered in this thesis include F2 as well [15][65].
1.3.5 Chiral effective field theory
In chiral effective field theory, the electromagnetic form factors are computed
using a chiral effective Lagrangian with pion, nucleon, and ? fields. The short
distance physics is determined by low-energy constants, which are fit using the
measured nucleon charge radius and moments. While this technique has had some
success at low Q2, by about 0.4 GeV2 the predictions break down, due to an in-
creasing role of vector mesons. Since this thesis deals with GnE at intermediate Q2,
such models can not reliably provide a description of the discussed data. Thus, the
focus will be on models that are more appropriate to the energy regime investigated
11
in this experiment [76].
1.3.6 Dyson-Schwinger Equations
As has been expressed earlier, there is a significant difference between the bare
quark and the dressed (or ?anti-screened?) quark. All models of nucleon structure
at intermediate Q2 try to describe this difference. In a framework that uses Dyson-
Schwinger equations, the primary cause of this parton behavior is a dense cloud
of gluons which surround the quark at low momentum transfer (Q2 < M2N). This
is a manifestation of dynamical chiral symmetry breaking (DCSB), and becomes a
source for the nucleon?s mass [26].
Describing the relativistic region in a framework that is consistent with this
low energy description is difficult. In terms of these DSE, the nucleon appears as a
pole in a six-point Green?s function. Here, the three dressed quarks are described
in terms of a ?bystander? quark and a diquark pseudoparticle in a single color-?3
channel. The binding depends on the exchange of the bystander quark and quarks
in the diquark pseudoparticle. This framework combines the descriptions of mesons
and nucleons. It is the observation of this diquark description of the nucleon which
is used to truncate the DSE and allow calculations using DSE.
An interesting prediction of models in this framework is that the radius of
the ?dressed? u-quark in the neutron is greater than that of the d-quark [14] [83]
[26]. Another is that since the up and down current quark masses are thought to be
small, the dominant mechanism for the helicity flip behavior in QCD comes from
12
quark orbital angular momentum and an additional polarized gluon [26].
1.4 Experimental Investigation
These various models described above provide a description of the nucleon
which is more complete than the simple constituent quark model or dynamical chiral
symmetry breaking. Most of these models share, as part of their description, a non-
zero current quark orbital angular momentum. In this regime, the constituents of
the nucleus are dominated by the up and down quarks (with possibly a small strange
quark component). The simplest probe available of this structure is the electron.
The common experimental techniques used to study this behavior and to investigate
the models are deep inelastic scattering (DIS), deeply virtual Compton scattering
(DVCS), and (quasi-)elastic scattering. By expanding our knowledge of the nucleon
form factors, the pieces needed to complete our picture of the neutron are brought
together. The measurement described in this thesis is one important piece. The
physics of electron scattering will be developed in the next section.
13
Chapter 2
Physics of Electron and Nucleon Scattering
2.1 Introduction to electron scattering
Scattering is the process in which a particle is deflected during an interaction.
There are two main categories of scattering, elastic, where the particles are left
intact and only the momentum changes, and inelastic, where the scattered particles
are modified, excited, or destroyed. The differential scattering cross section is often
written as the product of a cross section for scattering from a spin-0 point particle
and a ?form factor? which characterizes the structure of the target
d?
d? =
 d?
d?
?
Mott
frec|F(q2,?e)|2 . (2.1)
Here F is the form factor, q is the momentum transfer, the Mott cross section is
that of a spin-12 electron scattered from a point-like particle, and frec is a kinematic
factor (to be defined later). For a spin 12 particle, F(q2,?e) can be expressed in terms
of the form factors GE and GM, which will be developed later [46].
The neutron is known to have no net charge. With high energy electrons,
distances smaller than the size of a nucleon can be probed, allowing the structure
to be described at different scales. Nonetheless, the neutron electric form factor
is small compared to the other nucleon form factors. Various models exist that
provide detailed descriptions of the nucleon over different kinematic ranges; specific
14
k = (E ,
k = (E ,
k )
p = (? ,p )
q = (? ,q )
i
f ff?
?
?
?
p = (?i ? ,p )
f
i i i
fk )f
i
Figure 2.1: Feynman diagram for one interaction between an electron and a nucleon.
The electron has four-momentum k? and the nucleon has four-momentum p?.
examples of models giving GnE at intermediate Q2 will be given in more detail in
section 3.1.
2.2 Definition of kinematic variables
The interactions of electrons and photons are well understood within the the-
ory of quantum electrodynamics (QED). In this interaction, the properties of the
incident and scattered electron and recoiling nucleon can be measured.
In the Feynman diagram in figure 2.1, the following kinematic variables are
defined. The nucleon four-momentum is p? = (epsilon1,p), k? = (E,k) is the electron
four-momentum, and q? = (?,q) is the virtual photon four-momentum. The four-
momentum transferred to the nucleon is q2 = q?q? = ?2 ?q2, and is carried by the
virtual photon. If evaluated in the laboratory frame, pi = 0 and epsilon1i = MN, where
15
MN is the recoil nucleon mass. Since q2 in electron scattering is space-like, Q2 is
best defined as Q2 = ?q2 = 4EiEf sin2 ?e2 (in the approximation me = 0), where
?e is the angle the electron deviates from its initial direction. Q2 is typically the
quantity used to parametrize the form factors.
2.2.1 Current operator
The current of the electron in Figure 2.1 is arrived at from QED [46]
j?(kf,ki) = ?u(kf)(?ie??)u(ki) . (2.2)
Here ?? is the standard Dirac matrix and e is the electric charge. In this expression
u is the Dirac spin 12 spinor. For the spinor the following completeness relation holds
X
s=1,2
u(s)(p)?u(s)(p) = ??p? +m . (2.3)
Similarly, the current for a structureless, spin 12 charged particle can be expressed
as
J?(pf,pi) = ?u(pf)(?ie??)u(pi) . (2.4)
These provide the invariant scattering amplitude M using the rules for Feynman
diagrams. Following standard conventions, repeated indices are summed over
iM = J?(pf,pi)?ig??Q2 j?(kf,ki) . (2.5)
2.2.2 Dirac and Pauli form factors
The vertex factor for the nucleon-photon interaction is not exactly known,
but rather parametrized in terms of form factors. These form factors must be
16
functions of q2 since it is the only independent Lorentz scalar at the nucleon vertex in
elastic scattering. Generally, the form factors can be constructed with any Lorentz-
invariant constituent; ??, ?5, or ???. However, the choices of i???q? and ?? are
made to conserve parity. Other possible forms are restricted due to current and
parity conservation. The two terms, ?? and i???q?, give a current of
J?(pf,pi) = ?u(pf)[e(??F1(q2)+ i?
??q?
2MN F2(q
2))]u(pi) , (2.6)
where F1 and F2 are known as the Dirac and Pauli form factors. They are normalized
to give the proper charge and magnetic moment at q2 = 0, so that
Fp1(0) = 1, Fp2(0) = ?p
Fn1 (0) = 0, Fn2 (0) = ?n .
(2.7)
where ? is anomalous contribution to the nucleon?s magnetic moment.
The Dirac and Pauli form factors are a parametrization of the electromagnetic
structure in a different basis than the Sachs form factors. The Sachs form factors
can be expressed in terms of F1 and F2 as [58]
GE = F1 ??F2 , GM = F1 +F2 , (2.8)
where ? = Q24M2
N
.
2.3 Breit frame
As mentioned previously, the Sachs form factors can be related to the charge
and magnetic moment distribution, in a particular reference frame called the Breit
frame. No energy is transferred, only momentum (pi = ?pf), therefore
Q2 = q2B . (2.9)
17
An important aspect of the Breit frame is that the Fourier transform of the
electric form factor gives the charge density distribution, and that of the magnetic
form factor gives the magnetic current density distribution.
The four components of the hadronic current in this frame are [46] [76]:
J0 = ie2MN?uf(F1 ??F2)ui = ie2MN?ufGEui , (2.10)
??J = ?e?u
f(??? ?qB)(F1 +F2)ui = ?e(??? ?qB)?ufGMui . (2.11)
Here ??? is the Pauli spin matrices. Using this, Sachs [87] showed that GE and GM
can be expressed as the Fourier transforms of the nucleon charge and magnetization
densities
GE(Q2) = 4piQ
Z
rdr?ch(r)sinQr (2.12)
GM(Q2) = 4piQ
Z
rdr??mag(r)sinQr. (2.13)
At low Q, equation 2.12 can be expanded such that
GE(Q2) = 1? 16Q2?r2E?+ Q
4
120?r
4
E?+... (2.14)
From this we can see a simple formula for the charge radius of the neutron. At very
low momentum transfer, the slope of the form factor with respect to Q2 defines the
mean square radius of the distribution [60],
< r2n >= ?6d(G
n
E)
dQ2
flfl
Q2?0. (2.15)
This is obviously only true in the non-relativistic limit. Recently, Miller et al. have
developed a formalism that allows a representation of the transverse charge and
magnetization densities that are not reference frame dependent [70].
18
k
q
i
f
f
?
??
?
?
N
N ,p
p
k
A
B
i
Figure 2.2: Feynman diagram for interaction between an electron and the nucleus.
The electron has four-momentum k? and the ejected nucleon has momentum p?f.
2.4 Quasi-elastic scattering
Rarely in scattering experiments is there an opportunity to scatter from a
single nucleon. This is especially true of the neutron, which is short lived in its
free state. There are many different approximations that can be used to put the
nucleus in terms of elements that are better understood. Some of these are refer-
enced in sections 5.5 and 5.7.1. While discussing all of them is out of the scope
of this thesis, two which allow the final state nucleus to be understood in terms
of a remainder nucleus and a scattered nucleon merit development. These are the
impulse approximation (IA), and the plane wave impulse approximation (PWIA).
Here and throughout much of this thesis, discussion will be formulated in terms of
the one-photon exchange approximation.
19
2.4.1 Impulse approximation
In the impulse approximation the virtual (off-shell) photon interacts with a
single nucleon. Afterward, the struck nucleon leaves the nucleus without any further
interaction with the remaining ?spectator? nucleons. This approximation can also
be described as the assumption that the current of the nucleus is given by the sum
of the currents of the individual nucleons, where the nucleons are treated as free
particles [27].
The simplest nucleus that can be discussed in such terms is the deuteron.
In the IA, the differential cross section for scattering from a neutron or proton
embedded in the deuteron can be factorized. This allows the contributions of the
neutron and proton to be separated. For an electron to scatter from a nucleon inside
the deuteron, the contributions from the proton and the neutron can be separated
d?ed
d? ?
d?ep
d? +
d?en
d? . (2.16)
This approximation, however, completely neglects nuclear binding and the inherent
Fermi motion of the bound nucleons. It also neglects that the nucleon might re-
scatter, the virtual photon might couple to a virtual meson exchanged between the
nucleons, or the virtual photon might couple to an excited state of a nucleon (known
as Isobar Configuration or IC). The quasi-elastically scattered nucleon interactions
with the other ?spectator? nucleons are known as final state interactions (FSI).
20
2.4.2 PWIA
A more detailed simple approximation is the Plane Wave Impulse Approxima-
tion (PWIA). This approximation is a quasi-free one, and is important in discussions
of neutron double polarization experiments, where either the target or the recoiling
nucleon is polarized. In PWIA the polarization of the recoiling nucleon or of the
target is restricted to the scattering plane. The longitudinal component is paral-
lel to the recoiling nucleon?s momentum vector, and the transverse component is
perpendicular [66]. In this approximation the initial and final state of the target
nucleus are represented as products of plane waves for a nucleus and nucleon, where
the nucleon is bound (with momentum pi). The nucleon absorbs the virtual pho-
ton, while the nucleus remains as a spectator. The nucleon then leaves the nucleus,
which is assumed to be free after the absorption of the virtual photon, without any
further interaction.
2.5 Polarization observables
A convenient coordinate system to work with observables when the beam,
target, or recoiling nucleon are polarized is defined by [77]
?z bardbl q,
?y bardbl ki ?kf,
?x = ?y ? ?z.
(2.17)
It is useful to refer to the polarization vector ??P , which can serve as either
the polarization of the nucleon inside the target in beam-target asymmetry mea-
21
y xzk
? q p? * ? ?k
i *
*ef
Figure 2.3: Polarization exchange in the one photon exchange approximation for
polarized-electron, polarized nucleon scattering. ? and ? define the polarization of
the recoiling nucleon.
surements, or the polarization of the ejected nucleon in beam-recoil polarization
experiments.
??P = P(sin?cos??x+sin?sin??y +cos??z) (2.18)
In the general case of parity conserving polarized scattering there are 18 in-
dependent response functions. Under the approximations of one photon exchange,
PWIA, and parallel kinematics, only four terms survive. The cross sections of both
a recoiling polarized nucleon and a polarized target share similar forms, and can
be written in terms of products of kinematic quantities and response functions [58].
The kinematic variables (following the development in [58]) for these reduced terms
are
V primeTL = (1??)?1 tan ?e2 , (2.19)
VT = 12(1??) , (2.20)
22
VL = 1(1??)2 , (2.21)
V primeTT =
r
(1??)?1 +tan2 ?e2 tan ?e2 . (2.22)
The response functions depend on the polarization of the recoiling nucleon and
the target nucleus. These kinematic variables arise from the electron tensor in an
alternate basis, where the coefficients refer to the nuclear response function with
which the kinematic variable is associated (L, TL, T or TT). The response functions
arise from the nuclear response tensor (W??). The nuclear response tensor serves to
parametrize the increased generality of the nucleus, where the same current doesn?t
exist in the initial and final states [46]. The four reduced response functions are
RprimeTT(?z), RprimeLT(?x), RL, and RT, the first two of which reflect components associated
with the polarized nature of the cross section. These subscripts L and T correspond
to ?longitudinal? and ?transverse?.
Two additional response functions exist for nuclei with more than one nu-
cleon. These are RTT and RTL and only appear when the change in total angular
momentum is greater than two units [31] [80].
2.5.1 Recoil polarization
From PWIA it can be shown that the unpolarized nuclear response tensor is
symmetric, W?? = W??. This means that the response tensor for recoil polarization
is
W??(?a) = ?W??(?a) = W??(??a) . (2.23)
In double-polarization experiments parallel kinematics are often used; under
23
thesekinematics thenucleon?sfinal momentumisparalleltothemomentumtransfer.
Also, at the electron beam energies in the experiments described here, only the
longitudinal component of the electron polarization is relevant.
Inthelaboratoryreferenceframetherearetworesponsefunctionsthatdescribe
the polarization (in PWIA), RprimeTT(?z) and RprimeLT(?x). These are the longitudinal and
transverse response functions from which the transverse (Px) and longitudinal (Pz)
components of recoil polarization arise [58].
The general form of the nuclear structure tensor, with a polarized target or
recoiling nucleon, can be written as [76]:
W?? = J?J?? = W0?? +W??(??P f)+W??(??P i)+W??(??P i,??P f) . (2.24)
In recoil polarization experiments, typically only the final nucleon polarization is
measured, and W?? reduces to (in the Briet frame)
W?? = 12TrF?F????? ???P . (2.25)
Here F0 = 2MNGE and Fj = iMNGMvector? ? qB ? ?j, where j = x,y,z. This gives
W??(Px) ? 2MNGEGM and W??(Pz) ? G2M.
As an example, this leads, after transformation back to the lab frame, to the
following components of the recoil polarization of a free neutron [47][92]:
Px = ?Pe
p2?epsilon1(1?epsilon1)Gn
EG
n
M
epsilon1(GnE)2 +?(GnM)2 , (2.26)
Py = 0 , (2.27)
Pz = Pe ?
?1?epsilon12(Gn
M)
2
epsilon1(GnE)2 +?(GnM)2 , (2.28)
where epsilon1 = [1+2(1+?)tan2 ?e2 ]?1, and Pe is the polarization of the electron beam.
24
2.5.2 Target polarization
The arguments and results are similar for a polarized target, where the de-
tected nucleon is either unpolarized or not detected. Once more, in parallel kine-
matics, the one photon interaction approximation, and PWIA, only the ?x and ?z
components of the target polarization vector remain.
In the response function formulation, and with spin-12 nucleons, to within
a multiplicative constant, the response functions are functions of the components
FL = (1 + ?)GE and FT = p2?(1+?)GM. The general expression for such a
response function is
X
j,jprime
Aj,jprime(k)FjFjprime (2.29)
where the Aj,jprime(k) has only two terms left for the polarized response, the F2T and
FLFT terms. This formulation is developed in general for more complicated systems
than elastic scattering with a polarized nucleon by Donnelly and Raskin [31] [80].
The response functions for the target are then [58]:
RTLT(?x) = 2
p
?(1+?)GEGM , (2.30)
RTTT(?z) = 2?G2M . (2.31)
2.6 Cross section with unpolarized observables
The unpolarized differential cross section is dependent on the square of the
scattering amplitude (spin averaged since it is unpolarized),
d?
d? =
m2e
4pi2
Ef
Ei frec
1
4
X
|M|2, (2.32)
25
where frec = [1 + 2EiM sin2 ?e2 ]?1. It is useful to define the Mott cross section, which
corresonds to a relativistic electron scattering from an unstructured target
d?Mott
d? =
?2 cos2 ?e2
4E2i sin4 ?e2 . (2.33)
Here ? is the fine structure constant.
Using the equations 2.2, 2.6, 2.32, and the Feynman rules the following rela-
tionship between the four-momenta, angles, and other elements of the electron and
interacting fermion give the differential cross section. This relationship between the
form factors and the cross section is also known as the Rosenbluth formula.
d?
d? =
?
?2
4E2 sin4 ?e2
!
Ef
Ei
? 
F21 ? q
2
4M2F
2
2
?
cos2 ?e2 ? q
2
2M2 (F1 +F2)
2 sin2 ?e
2
?
(2.34)
Putting this in terms of the Sachs form factors gives [58]:
d?
d? = frec
 d?
d?
?
Mott
[G
2
E +?G
2
M
1+? +2?G
2
M tan
2 ?e
2 ]. (2.35)
This is much simpler than equation 2.34 and readily suggests a method of measure-
ment of the form factors. This method of measurement is known as the Rosenbluth
Technique.
2.7 Cross section with polarized observables
The general form for the cross section when the beam of electrons is polarized
and the polarization of the ejected particle is detected is:
d6?h,s
dEfd?kfdepsilon1fd?pf = ?unpol
1
2(1+
??P
i ???? +h(A+
??P
T ???? )). (2.36)
26
Here Pi is the induced polarization in the proton, h is the helicity of the incident
electron, A is the beam analyzing power, s is the nucleon spin projection on ?, ?unpol
is the unpolarized cross section, and PT is the polarization transfer coefficient [58].
In coplanar kinematics A = 0, Pi???? = Pi? ?N, and PT ? ?N = 0. Since Pi? ?N = 0 in
PWIA, two terms survive to provide the nucleon polarization, and from equations
2.26 and 2.28 an elegant measurement of GnE is possible.
This is similar in form to the cross section of polarized electrons on a polarized
target where the recoil polarization is not measured:
d6?
dEfd?kfdepsilon1fd?pf = ?unpol
1
2(1+
??P
t ?
??A
T +h(A+
??P
t ?
??Aprime
T)). (2.37)
The new variables here are Pt which is the target polarization vector, AT which is
the target analyzing power, and AprimeT which is the correlation vector [58].
Using a longitudinally polarized electron beam and a polarized target, the
cross section can be written as:
?pol = ?+h?, (2.38)
where ? is the unpolarized cross section from equation 2.35, and ? is the helicity
dependent part [31]. Under the approximations of PWIA and one photon exchange,
and in the case of parallel kinematics and relativistic electron beam, the cross section
simplifies to [58]:
?pol = ?Mottfrec[VLRL +VTRT +hPtarget(V primeTRprimeTT(?z)+V primeTLRprimeTL(?x))]. (2.39)
27
Using equations 2.19, 2.22, 2.31, 2.30, ? is given by
? = ?2?Mottfrec tan ?e2
r ?
1+?(
r
?(1+(1+?)tan2 ?e2 )cos?G2M+GEGM sin?cos?).
(2.40)
28
Chapter 3
Form Factors
3.1 Models and fits
While the theory for strong interactions, QCD, is well known, because of
confinement it is impossible to use perturbative techniques to do calculations at low
Q2. Since it is low Q2 that is pertinent to the structure of hadrons, approximate
models must be used to understand this physics. Understanding the form factors,
especially GnE since it is relatively unknown, provides assistance in understanding
nuclear structure. Models have been developed to describe the structure of hadrons,
and data is needed to determine which models provide a better description.
There are a large number of models of nuclear form factors, a few classes of
which will be discussed here. Although lattice QCD has the most long term promise
for understanding the physics of strong interactions, since the comparison of the
results with data will show the current limitation in computing power, comparisons
with lattice QCD will not be included.
3.1.1 Fits
In the early studies of nucleon form factors, a phenomenological fit was per-
formed to determine the behavior of GnE, based upon unpolarized data and the
Rosenbluth technique. This fit of the data is known as the Galster parameterization
29
[40], and is
GnE(Q2) = ? ?n1+5.6?GD(Q2), (3.1)
where GD is the dipole form factor
GD = 1(1+ Q2
0.71)
2 . (3.2)
The other three form factors (GpE, GpM?p , GnM?p ) all show good agreement with GD using
data acquired from the Rosenbluth method. As new data have become available,
this simple fit has been updated. See for example the fit by Kelly [59].
3.1.2 Vector meson dominance model
A representative vector dominance model which is in reasonable agreement
with the low Q2 data is one by Bijker et al. [15] This model has pQCD scaling rela-
tions built into it, and a phenomenological contribution attributed to the nucleon?s
qqq structure. The added term for FV2 is g(Q2)1+?Q2 introduced to reflect pQCD. The
intrinsic form factor used was g(Q2) = (1+?Q2)?2.
Another representative model, by Lomon et al., also included meson pole terms
and a term with pQCD behavior. This model also includes the additional vector
mesons of ?prime and ?prime compared to the previously described one, requiring a total of
14 free parameters [65].
3.1.3 Pion cloud and CQM
Miller, et al., expanded the constituent quark model into the light-front cloudy
bag model [69]. Here the three relativistic constituent quarks are surrounded by a
30
nonrelativistic cloud of pions. Poincar?e invariance provides an additional constraint
needed to fit the GnE data. This model has recentlybeen expanded intothe lightfront
cloudy bag model, with the pion cloud contributing through a relativistic pi-nucleon
form factor. The corrected form factors are in terms of the form factors without
relativistic effects, and the virtual pion four momentum. An additional parameter
is introduced in the pi-nucleon form factor, for the relativitistic correction [69].
3.1.4 Model of Generalized Parton Distributions
The interest in GPDs recently has led to several models being developed.
Among these are models by Guidal et al. [45] [76] and Diehl et al. [28]. In these
models, the GPD framework is used to describe observables that are independent of
skewness ? (see Figure 1.1). Guidal et al. parameterize the GPDs in the following
way
HqR2(x,0,Q2) = qv(x)x?prime(1?x)Q2 , (3.3)
EqR2(x,0,Q2) = ?
q
Nq(1?x)
?qq?(x)x?prime(1?x)Q2 . (3.4)
Here ?prime is the universal Regge slope, while the ?q govern the behavior for the helicity-
flip GPDs as x approaches 1. This was added to produce a faster falloff at large Q2
in the x ? 1 limit. The model by Diehl et al. is described in Section 6.4.2.
3.1.5 pQCD predictions
As stated in section 1.2.1, perturbative QCD gives the prediction that [12]
F1 ? 1Q4 . (3.5)
31
This relationship is also expected to be relevant at moderate Q2 when higher order
QCD terms are included, which treat small-x partons better. The small quark
masses mean that the dominant mechanism for helicity flip in QCD comes from
the quark orbital angular momentum. Generalized power counting including this
orbital angular momentum gives a scaling of
F2 ? 1Q6 . (3.6)
This behavior depends on the leading order and next to leading order light-cone
wave functions, the latter dominated by the probability amplitude for one quark to
carry one unit of orbital angular momentum. It is suggested that the higher order
resummation suppresses the low x contribution, providing an effective cut off for
the integrals at x ? ln2 (?2/Q2) [12].
F2
F1 ?
ln2 Q2?2
Q2 . (3.7)
Here ? is the QCD renormalization scale. While this scaling relation is accurate for
the proton, a rough calculation of Q6F2(Q2) gives about 13 the value of experimental
data for Q2 < 5 GeV2 [12]. This scaling, scaled to the previous measurement at
Q2 = 1.5 GeV2, is presented in Figure 3.1 in addition to the other models just
discussed.
3.2 Measuring GEn via the Rosenbluth technique
As mentioned earlier, equation 2.35 suggests one technique to determine the
electric and magnetic form factors. By carrying out multiple measurements at dif-
32
]2  [GeV2Q
0 2 4
n EG
0.00
0.05
0.10
Bijker
Miller
Miller (q-only)
Guidal
2)/Q2?/2(Q2 ln? 
1/F2FGalster fit
Figure 3.1: Various fits discussed in the text. The points plotted are from [66], which
was used to fit the pQCD curve. These data are only representative, see figure 3.2
for the full set. The Bijker curve is a representative Vector Meson Dominance model,
the Guidal curve is a representative GPD based model, the Miller curves are from
a CQM.
33
ferent angles for a given Q2 the two components of the cross section may be in-
dependently determined, allowing a separation of the electric and magnetic form
factors. This technique is known as Rosenbluth separation. Experiments to mea-
sure the electric form factor of the neutron that use this Rosenbluth method fall into
two categories: those in which the impulse approximation for quasi-elastic electron
deuteron scattering with the Rosenbluth method is applied directly to the neutron
component, and those where the Rosenbluth method is applied to the deuteron
through elastic scattering, and an NN model is use to extract a value for GnE [58].
The latter is described first.
3.2.1 Elastic e-D scattering
The deuteron, containing two nucleons, is spin-1. Because of this, the equation
for the elastic cross section has three form factors, the magnetic GM, the charged
GC, and the quadrupole term GQ [77]:
d?
d? =
d?Mott
d? [G
2
C +
8
9?
2G2
Q +
2
3?(1+?)G
2
M +
4
3?(1+?)
2G2
M tan
2 ?e
2 ] . (3.8)
While the ?e dependence allows GM to be separated out, the charged and quadrupole
terms cannot be separated using the Rosenbluth technique. In addition, it is the
coherent sum of proton and neutron electric form factors which is extracted (GpE +
GnE). Both of these complications add to the uncertainty in the determination of
GnE. The electric form factors must be ?unfolded? from GC and GQ using a model,
which makes the results of this technique dependent on the model of the deuteron?s
wave function [76] [84]. The most recent extraction uses high precision data for GC,
34
and the Arenhovel model of the deuteron [6] [36].
3.2.2 Quasi-elastic scattering and the Rosenbluth Technique
Recalling equation 2.16, a technique using quasi-elastic scattering from a nu-
cleon rather than elastic scattering from a deuteron and the Rosenbluth formula
(equation 2.35) directly presents itself. By subtracting the cross section of the pro-
ton from that of the deuteron, or by measuring the neutron in coincidence with the
electron, at the quasi-elastic peak, the cross section of the neutron remains. Then
the Rosenbluth formula can be used to arrive at a value for GnE.
Most commonly, the neutron is detected in coincidence with the scattered elec-
tron, to identify that the interaction was with the neutron. However, this technique
has also been performed with anti-coincidence measurements, where ?no proton? is
required in coincidence with the electron [77]. Another technique is to measure the
ratio of the neutron to proton production cross sections in the electro-disintegration
the deuteron. This method has worked very well for the neutron magnetic form
factor.
There have been some serious difficulties with these types of experiments.
Among these are a heavy dependence on the proton form factors and the afore-
mentioned problems with the Impulse Approximation (section 2.4.1). Additionally,
coincidence and anti-coincidence experiments depend on the absolute detection ef-
ficiency, which is difficult to determine for neutrons [77].
35
3.2.3 Difficulties with Rosenbluth method
The uncertainties for these measurements based upon unpolarized quasi-elastic
scattering from deuterium are too large to give a definitive nonzero measurement of
GnE. The dependence on the proton form factors and the uncertainty directly from
IA dominate the measurement.
The Rosenbluth technique has another inherent problem: the magnetic form
factor is much larger than the electric form factor for the neutron, and the ratio
( GE?G
M
)2 becomes smaller as the transferred momentum increases. This means that
the electric form factor is difficult to decouple from the magnetic form factor, since
both the tangential and constant (relative to angle) portions of the cross section are
proportional to G2M [91].
3.3 Double polarization techniques
With the difficulties in measuring the electric form factor of the neutron using
the Rosenbluth method, new techniques have been sought. Following the suggestion
of Arnold, Carlson, and Gross [10] a technique using a longitudinally polarized beam
and a recoil polarized neutron has been investigated. The most obvious reaction for
such an experiment would be 2H(??e ,eprime??n)p.
The most important advantage is that the measured term is proportional to
GnEGnM instead of (GnE)2. Additionally, only one measurement is needed for a given
Q2 if all interesting components can be measured.
As mentioned earlier, polarized targets can allow the same physics to be ac-
36
]2  [GeV2Q
0.0 0.5 1.0 1.5
n EG
0.00
0.05
0.10
PasschierHerberg
Madey
Warren
Golak
Bermuth
Glazier
Figure 3.2: Previous GnE data from experiments involving polarization techniques.
The curve is the Galster parametrization [40]. Herberg provided new calculations of
the Ostrick data, while Golak provided a FSI corrected analysis of the Becker data.
For references see Table 3.3.
37
cessed as detecting a polarized recoil neutron. Experiments have been carried out
with either recoil polarization, where the polarized neutron is detected in a po-
larimeter, or with a polarized target, either polarized deuterium or polarized 3He,
with a detected unpolarized neutron [66]. Table 3.3 contains a summary of recent
experiments using all of these techniques.
All experiments using the double polarization technique are dependent on a
helicity dependent experimental asymmetry,
Aexp = N+ ?N?N
+ +N?
, (3.9)
which can then be related to GnE. Here N+ gives the number of events with helicity
aligned with the nucleon polarization or with the beam direction and N? gives
the number of events with helicity with the opposite sign compared to nucleon
polarization or against the beam direction.
3.3.1 Quasi-elastic scattering and (e,e?n)
For PWIA to be applicable, quasi-elastic events must be selected. Even though
the cross section is not being measured, the process of selection of these events
is important, as double polarization techniques depend on asymmetries, which are
diluted by events which are not part of the desired asymmetry. Two useful quantities
for selecting quasi-elastic events in coincidence experiments are the missing energy
(defined by Em = mf +mB ?mA+i with A, B, i, and f as shown in figure 2.2) and
the missing momentum (defined by pm = pi ?q).
In the experiments that have been carried out to date, the experimental trig-
38
Q2 Date Reference target
0.16 1991 Jones-Woodward [55] 3??He
0.2 1992 Thompson [95] 3He
0.31 1994 Meyerhoff [68] 3??He
0.255 1994 Eden [34] 2H
0.15, .34 1999 Ostrick [74] 2H
0.4 1999 Becker [11] 3??He
0.21 1999 Passchier [75] 2??H
0.67 1999 Rohe [84] 3??He
0.45, 1.13, 1.45 2003 Madey [66] 2H
0.5, 1 2003 Warren [97] 2??H
0.3,0.6,0.8 2003 Glazier [43] 2H
0.67 2003 Bermuth [13] 3??He
Table 3.1: Double polarization experiments for measuring the electric form factor
of the neutron that have been carried out to date [77].
39
ger was typically formed by coincidence between the scattered electron and recoiling
neutron. The electrons were detected in a spectrometer or granulated calorimeter.
The measured electron kinematics, in concert with large solid angle neutron detec-
tors, allow for event-by-event reconstruction of the particle tracks. For the neutron
detector, in higher Q2 experiments a multi-plane neutron detector was used, with a
layer or two of veto detectors in front. In other experiments, two separated layers
of scintillator were used, with each layer having veto detectors in front.
Recoil polarimetry experiments required a different detection scheme in order
to detect the polarization of the neutron, as it leaves the target. Polarimeters were
set up to measure the ?up-down? scattering asymmetry due to the transverse com-
ponent of the recoil polarization. The polarimeters included a plastic or mineral oil
scintillator which determined the neutron time of flight (TOF), and also served as a
scatterer, with the polarized scattering exhibiting an asymmetry that was measured
in a second set of scintillators. Since the spin vector of the neutron could have
any combination of longitudinal and transverse polarizations, a dipole magnet was
sometimes used to precess the neutron?s spin vector.
These detectors provided scattering angles for both neutrons and electrons.
The momentum of the particle(s) provides the needed fifth quantity to select quasi-
free (e,n) scattering, and is determined by considering the time of flight (TOF) for
the particle(s). A missing signal in the veto detectors along with the determination
of the flight time of the particle from the target to the detector provided neutron
detection. Additionally, pulse heights in the neutron detector scintillators can be
used to filter out accidental and inelastic events. Quasi-elastic events can also be
40
selected by comparing the expected time of flight (from the electron energy and the
angles) to the measured time of flight.
Other events, which dilute the quasi-elastic neutron events, were one of the
main sources of uncertainty in these measurements. A major source of background is
proton to neutron conversion in the shielding in front of the neutron detectors. This
can be minimized with a tight time of flight cut at low hadron momentum. Good
time resolution also allows removal of pion events, another source of dilution. Monte
Carlo simulations have been used to determine the proper dilution factor from the
uncertainties and to average out the theoretical asymmetries and the effects of finite
acceptance.
3.3.2 Recoil polarimetry
Four experiments used recoil polarization techniques to measure GnE. These
experiments were Madey et al. [66] in Hall C at Jefferson Lab, Glazier et al. [43]
at the Mainz Micotron, Ostrick et al. [74] at the Mainz Microtron, and Eden et al.
[34] at the MIT-Bates laboratory. The first three named experiments (Eden et al.
was a proof of concept experiment) produced published data.
In these experiments, neutrons were produced through electrodisintegration of
deuterons in an unpolarized liquid deuterium target using longitudinally polarized
electrons. Contamination from hydrogen within such targets was found to be small
[34] [66].
From equations 2.26 and 2.28 we can develop a relationship between the ratio
41
of the components of polarization (see discussion in section 2.5), and the ratio of
the form factors
GE
GM = ?
Px
Pz
Ei +Ef
2M tan
?e
2 . (3.10)
Here Px and Pz are the x and z components of polarization. In order to determine
the up-down asymmetry Ameas must be observed. It follows a sinusoidal dependence
Ameas = Ay|??P|sin(?+?0) . (3.11)
Here Ay is the beam analyzing power, and ? is the spin precision angle, of the
detected particle as it travels through a magnetic field.
tan?0 = PxP
z
= ?GEG
M
cos ?e2q
? +?2 sin2 ?e2
. (3.12)
Using this spin precision technique removes the need for absolute calibration of the
electron beam polarization and effective analyzing power [78].
Arenhovel [7] [8] showed that for a deuteron target, corrections due to meson
exchange currents and isobar configurations are small, and these have little depen-
dence on Q2. It was found, however, that at lower values of Q2, effects from final
state interactions (FSI) can play a large role [74].
Ostrick et al. found at Q2 = 0.35 GeV2 that the neutron polarization was
reduced by less than 4% due to FSI effects, but at Q2 = 0.12 GeV2 the transverse
polarization is reduced by 50%. These final state interactions were calculated to
mostly arise from p?n charge exchange via pion exchange. A model was used to
calculate the change to the polarization by FSI, and a correction was applied [74].
Glazier et al. found that at neutron energies of a few hundred MeV the
scattering in the polarimeter was dominated by quasi-elastic scattering, which added
42
to the effective analyzing power of the polarimeter. This means that Ameas, the
effective asymmetry, could not be computed accurately, and the statistical precision
was not exactly known [43].
Madey et al. also used a cross ratio technique, which makes the neutron
polarimeter results independent of any luminosity change based on beam helicity,
as well as of the efficiencies and acceptances of the two halves of the polarimeter.
This experiment used a different technique to account for FSI effects. By using
possible GnE values in a simulation, the neutron polarization was calculated and
then compared to experiment [66].
3.3.3 Polarized target measurement
To measure GnE using a polarized target, we measure the asymmetry A
A = ?+ ????
+ +??
= ?? , (3.13)
where ?? are the cross sections (Equation 2.38) with electron helicity ?1. If the
polarization direction of the target is flipped, this changes the sign of the asymmetry.
This asymmetry is not explicitly measured, but rather
Ameas = PePtA , (3.14)
where Pe is the polarization of the electron, Pt is the polarization of the target, and
A = ?
2p?(1+?)
h
tan ?e2 sin?cos?GnEGnM +
q
?[1+(1+?)tan2 ?e2 ]cos?(GnM)2
i
(GnE)2 +?[1+2(1+?)tan2 ?e2 ](GnM)2 .
(3.15)
43
Here ? and ? are the angles defined in Figure 2.3. Setting the angles properly
(? = 90? and ? = 0?) greatly simplifies things, giving the perpendicular asymmetry:
A? = ?2
p?(1+?)tan ?
e
2
GE
GM
(GEG
M
)2 +?[1+2(1+?)tan2 ?e2 ]. (3.16)
Since the ratio of GEG
M
is small, the perpendicular asymmetry is roughly proportional
to the ratio GEG
M
. The parallel asymmetry can be used to normalize the value, so
that
GnE = abGnM (PePtV)bardblA?(P
ePtV)?Abardbl
, (3.17)
where a = 2p?(1+?)
q
?[1+(1+?)tan2 ?e2 ] and b = 2p?(1??)tan ?e2 .
3.3.4 Experiments with polarized deuterium
Two experiments used polarized deuterium targets to measure the value of
GnE, one in Hall C at Jefferson Lab (Warren et al.[97]) and the other at NIKHEF in
Amsterdam, the Netherlands (Passchier et al. [75]).
Under PWIA, for deuteron targets with polarization in the scattering plane
and after proper averaging of the asymmetry (symmetrically around q), the mea-
sured asymmetry is
Ameas = V PeP
d
1A
V
ed
1+Pd2ATd , (3.18)
where V is the dilution factor, Pd1 is the vector polarization, Pd2 is the tensor po-
larization, AVed is the deuteron vector beam-target asymmetry, and ATd is the tensor
deuteron target asymmetry. For most targets the tensor polarization is small, and
that term can be ignored.
44
IntheseexperimentsdatawerecomparedtopredictionsbyArenhovel[7][8][9],
which accounted for nuclear and FSI effects. This physics model is non-relativistic,
and includes meson exchange, isobar configuration currents, and relativistic correc-
tions. The comparison was done by simulating different observables, like pm, and
plotting them versus the asymmetry, all using different values of GnE within the
confines of the model [75] [97].
In the experiment performed by Warren et al., polarized deuterated ammonia
was used as the target. The ammonia granules were submerged in liquid helium and
aligned with 5 T magnetic field resulting in a typical polarization of 24%. Good
agreement with MC predictions showed that quasi-elastic scattering dominated the
scattering reaction. This experiment had a narrow range of acceptance, so it was
fairly insensitive to Q2 dependence [97].
In the experiment performed by Passchier et al., polarized electrons were in-
jected into a recirculating storage ring, allowing for large beam currents. An atomic
beam source injected a flux of polarized deuterium atoms into a cell in the storage
ring with an electromagnet used to orient the polarization axis. This was created
by deuteron atoms in two hyperfine states, with an electromagnet to orient the
polarization axis [75].
3.3.5 Experiments with polarized helium-3
Another class of experiments to measure GnE have used 3??He as the target. Var-
ious experiments were carried out at the Mainz Microtron (MAMI) [11][84][13][68]
45
and at the MIT-Bates laboratory[55][95]. The results presented in [13] include those
from the experiment reported in [84]. Additionally, experiment E02-013 is of this
type and in early 2006 ran at Jefferson Lab in Hall A. The analysis of this experi-
ment, with data up to Q2 = 3.5 GeV2, is being presented in this thesis.
Opticalpumpingtechniquesareusedtopolarizethe 3Heeitherthroughmetasta-
bility exchange where the 3He is pumped directly, or through spin exchange. In the
latter, the valence electron in rubidium is optically pumped and the polarization is
transferred to the 3He nucleus through collisions. The polarization process for 3He
generally does not take place in the target chamber, rather a pumping chamber is
used and the polarized helium-3 diffuses into the target cell or is forced in with a
compressor. The polarization vector inside of the 3He target can then be measured
with NMR.
One of the advantages of a polarized 3He target is that in the ground state the
spins of the protons are to a large extent aligned antiparallel to each other. This
means that most of the spin of the helium nucleus is carried by the neutron; the
polarization of the neutron is approximately 86% of the polarization of the nucleus
[16]. Further details about polarized 3He targets will be presented in section 4.4.
At low Q2, the complications due to the nuclear wave function can be ac-
counted for with calculations. However, at Q2 > 0.6 GeV2 this proves difficult,
and extrapolation was used in these earlier experiments. Data for GnM is used, or
a parametrization, in order to arrive at a value of GnE from the experimental asym-
metry (equation 3.16). A Monte Carlo simulation was used to determine radiative
loss and neutron energy loss before the detector, as well as other corrections.
46
It is possible to select the kinematics such that the measured asymmetry is
expressed as:
Aexp = PbeamPtargetDneutronVotherA?. (3.19)
Here D accounts for dilution from sources related to the target (such as it not being
a free neutron), and V is the dilution caused from other sources and from reactions
other than quasi-elastic scattering. This formula assumes that the angles in equation
3.15 are exactly such that Abardbl = 0; in practice this isn?t possible.
The experiment performed by Becker et al. used circulated 3He which was
compressed into the target cell by a Toepler compressor. An experiment using
deuterium provided a parallel measurement, using the same equipment, pointed to
a significant role for FSI that decreases as Q2 increases. The work by Golak et al.
expanded upon the analysis of Becker et al. by summing the cross sections before
forming asymmetries using Faddeev calculations. It was discovered that even at
the quasi-elastic peak FSI effects played a significant role. Using theoretical ratios
determined from models and comparing to experimental values, GnE was extracted
from the FSI corrected results [11][44].
In the experiments performed by Rohe et al. and Bermuth et al., the data
were accumulated by rotating the target spin so that both A? and Abardbl could be mea-
sured. The measurement of both parallel and perpendicular asymmetries allowed
the measurement of the target analyzing power (Ay) which provided a check on the
understanding of FSI effects. These were compared to the calculation by Golak et
al. and FSI effects caused a 3.4% decrease in the reported value of GnE. These FSI
47
effects are primarily caused by the photon coupling to one of the protons followed
by charge exchange [13][84].
3.4 Summary of past measurements
These three different double polarization techniques have a variety of advan-
tages over the Rosenbluth method, as described in this chapter. While models are
used to make corrections, and for polarized deuterium for the measurement itself, the
general method is model independent. The three different, independent, techniques
used in double polarization measurements provides a check on the understanding of
the different types of corrections required. The data from the double polarizations
is presented in Figure 3.2 while the experiments are listed in Table 3.3.
Because 3??He behaves similarly to a polarized neutron, and because of the low
detection efficiency neutron polarimeters, polarized 3??He experiments provide the
best statistical precision for measuring GnE. FSI effects are taken into account by
models for polarized deuterium and calculations for 3??He. Because of the robustness
of the deuterium model, FSI effects have traditionally played less of a role in deu-
terium experiments compared to those using 3??He. For the discussion and calculation
fo FSI effects in polarized 3He for this experiment, at intermediate Q2, please see
the section 5.7.1.
The completed experiments provided precise enough data to clearly identify a
small enhancement in GnE at Q2 = 0.4 GeV2. This enhancement is larger in value
than the measurements from the Rosenbluth data. At the highest measured values
48
of GnE a noticeable difference between the Rosenbluth and polarized data is seen.
This is speculated to be due to incomplete radiative corrections for the Rosenbluth
data.
In the past decade, precise measurements of the neutron electric form factor
have become possible and have provided precise knowledge of the structure of nucle-
ons for Q2 below 1.5 GeV2. In this region, several models provide a good agreement
with the data. The majority of the recent measurements have used the double polar-
ization method, which has the advantage of having greater precision and less model
dependence compared to the historic Rosenbluth method.
3.5 E02-013 at Jefferson Lab
Asintroduced, thephysicsofnucleonsmovefromtheboundbehaviordescribed
by massive partons, into the behavior described by bare quarks in QCD. There is a
complicated transitive region, in the models presented in Section 1.3, that starts at
less than 1 GeV and continues to many GeV. The actual point where this behavior
changes, and the description of the change, is unknown. In the experiment presented
here, the key quantity GnE needed to understand nucleon stucture, and behavior such
as quark orbital angular momentum, is measured at three separate points in the Q2
region between 1 and 4 GeV2.
In this experiment, E02-013, numerous improvements were made to increase
the figure of merit (effective statistics) for a measurement in this region. Advances
in both instrumentation and in the polarized 3He target technology have been ac-
49
complished to improve this measurement. By injecting a small amount of potas-
sium along with the rubidium, the polarization transfer efficiency from the optically
pumped rubidium was improved to provide polarizations as high as 50% in the tar-
get chamber. The experimental apparatus and target will be explained in detail in
Chapter 4. The value of the asymmetry will be analyzed as a function of pm,?, the
perpendicular component of the missing momentum, to study FSI effects. A tight
cut on pm,? is needed to select low nucleon momenta in the 3He wave function,
but also will suppress dilution from proton polarization and FSI interactions. The
high Q2 at which this measurement is undertaken should suppress meson exchange
currents. Additionally, meson exchange currents, delta isobar contributions, and
finite acceptance effects should be suppressed by a tight cut on pm,? [25]. This will
be described in detail in Section 5.7.1.
50
Chapter 4
GnE experiment E02-013
Experiment E02-013 was carried out at Jefferson Lab from February 28 to
May 11, 2006 to measure the electric form factor of the neutron. In this chapter the
experiment setup and instrumentation needed to measure the interaction 3??He(vectore,eprimen)
are described. First the experimental overview and relevant coordinate systems will
be presented. Then a short summary of the beam and accelerator will be presented
in section 4.3. The target and method of polarization is described in section 4.4. In
section 4.5 the spectrometer (Big Bite) used to detect the electron and measure its
momentum is portrayed. The neutron arm (Big Hand), used to detect the coincident
hadron, is described in section 4.6. Finally, the data acquisition, software used for
decoding and initial physics analysis, is presented in section 4.7.
4.1 Experimental Overview
During the running period of E02-013, data were collected in four kinematic
settings which were spaced over six time intervals, not including the time spent on
commissioning the apparatus. Three separate polarized 3He target cells were used
for the measured asymmetry. For calibration purposes, a foil target (containing
six carbon foils and one BeO foil), a reference cell (providing an empty cell target,
hydrogen target, and a nitrogen target), and a setting with no target cell were used.
51
Big Bite optics calibrations and beam spot check were acheived using the foil
target. The nitrogen target is used to determine the contamination in scattering
events from N2 in the 3He cells. The hydrogen target was used to calibrate the
momentum and yields of the elastic events for Big Bite and Big Hand. Additionally,
the nitrogen and hydrogen targets are used to determine the dilution of the neutral
sample by protons empirically, which is an important correction for the physics
asymmetry.
Data taking took place in this experiment for several different settings depen-
dent on target, beam energy, neutron arm location, and electron scattering angle.
These are described in Table 4.1. Kinematic 1 will not be presented in this the-
sis. Kinematic 2(a) used two different settings for the neutron arm threshold. This
change in threshold creates a difference in the neutron and proton detection effi-
ciencies which are a key component in the empirical measurement of the dilution of
the neutral sample by protons. This difference in efficiencies is calculated using a
Monte Carlo simulation.
4.2 Principle and Experiment setup
ThisexperimenttookplaceinHallAofJeffersonLaboratoryinNewportNews,
Virginia, USA. Jefferson Laboratory is the home of CEBAF, a continuous electron
accelerator that can provide beam energies of 0.6 - 6 GeV [3] [63]. The three exper-
imental halls, A, B, and C, can all receive beam simultaneously. Each hall contains
various standard equipment detector systems that facilitate the types of experiments
52
Figure 4.1: Image of Jefferson Laboratory. Shown is the accelerator and the mounds
over the three experimental halls. The mound to the left is Hall A.
53
Kinematics 2 (a) 3 (a) 2 (b) 3 (b) 4
Target Cell Dolly Edna Edna Edna Edna
Beam E. (MeV) 2637 3291 2641 3290 2079
Q2 GeV2 2.5 3.5 2.5 3.5 1.7
BB Angle (?) 51.59 51.59 51.59 51.59 51.59
NA Distance (m) 10 10 10 10 8
Time Frame 3/09-3/24 3/24-4/17 4/17-4/24 4/24-5/02 5/02-5/09
Table 4.1: A table showing the running conditions, time, accumulated charge, and
kinematics of the different measurements referred to here and elsewhere as kinemat-
ics or for short kin. Kinematic one includes both the commissioning time and the
first measurement. The measurement of Q2 = 1.3 and 1.7 GeV2 provide a compar-
ison of the polarized 3He technique with previous recoil polarization data while the
other two measurements extend the measured range of Q2. Not shown is the periods
for commisioning and kinematic 1 which won?t be covered in this thesis.
the halls were designed to accomplish. While the beam was provided similarly to
other experiments in Hall A, the equipment used to make this measurement was a
custom installation that included a large momentum acceptance spectrometer, Big
Bite, a high efficiency segmented neutron detector, Big Hand, and a polarized 3He
target, which serves as a source of highly polarized neutrons. A general layout of
the apparatus is shown in Figure 4.2.
The requirements for Big Bite and Big Hand were developed using simulations.
To match Big Hand to Big Bite, the required acceptance for elastics was simulated,
and then expanded to insure the acceptance of all events with at least pm,? = 150
MeV. Because at Q2 = 3.5 GeV2 the desired hadron momentum is 2.6 GeV, the
thresholds, shielding, and depth of the neutron arm were developed to maximize
54
MollerPolarimeterComptonPolarimeter
He target3
BB Dipole
MWDC?s
Timing PlanePre?showerShower
Neutron Arm(Big HAND)
~2.5 m
~9 m
Figure 4.2: General layout, not to scale, of the experiment. Shown is the beamline,
the target, the Big Bite detector stack, and the Neutron Arm detector. Also shown
are the two polarimeters in the hall before the target, and the approximate distances
to the detectors. The components of the Big Bite (BB) detector stack are also shown
and consist of the Big Bite dipole magnet, multi-wire drift chambers (MWDCs), pre-
shower, shower, and timing plane.
statistics. The desired Big Bite momentum resolution was 1-1.5% and the desired
vertex resolution was 6 mm [25].
There were four major coordinate systems in this experiment: one for the
lab, one for Big Bite, one for the target, and one for the Neutron Apparatus. In
the hall system, the origin is at the center of the target, y is vertical with + as
?up?, z is nominally along the direction of the beam, and x forms a right handed
coordinate system. For the target coordinate system, the origin coincides with the
hall origin, x is vertical with + as ?down?, z is parallel to hall floor along the Big
Bite central ray, and y completes the right-handed coordinate system. For the Big
Bite detectors the origin is at the center of the first plane of the drift chambers.
The +z direction is defined in respect to the direction of particles perpendicular
to the first drift chamber (and so is at a ? 10? angle with the x ? z plane in the
55
(a) Lab Coordinate System
(b) Neutron Arm (NA) Coordinate System (c) Target Coordinate System
Figure 4.3: Coordinate systems for Big Hand, Target, and Lab. The lab z is along
the beam line while the target z is toward Big Bite. All coordinate systems are right
handed.
56
Figure 4.4: Schematic of the accelerator as it existed in 2006. Shown are the three
halls, injector, and LINACs.
lab frame), while +x is in the magnetic dispersion direction, and y completes the
right-handed coordinate system. In Big Hand, the depth is z, the height is x, and
the horizontal position is y. In this coordinate system positive x is associated with
?down?, positive z is associated with deeper into the Neutron Apparatus (NA), and
positive y is away from the beam line. The origin was located in the center of the
active area at the front of the shielding. Figure 4.3(c) shows the target, lab, and
Neutron Arm coordinate systems and their relation to each other.
4.3 The Electron Beam
During 2006, the accelerator was capable of delivering beams with energy up to
5.7 GeV at currents of up to 150 ?A semi-simultaneously to all three halls. The two
main components of the accelerator are the injector and the two recirculated linear
accelerators (LINACs). These delivered beams are not quite continuous, rather the
beam was pulsed with 2 ns wide bunches at a rate of 499 MHz. Polarized electrons
are produced from a strained GaAs photo-cathode via the photo-electric effect. A
57
maximum polarization of approximately 85% can be achieved. Inside the injector,
the lasers for each hall are 120 degrees out of phase relative to each other. This allows
for the energy and current to be determined for each hall semi-independently. After
traveling through 18 superconducting radio frequency (SRF) cavities, the electrons
are injected into the main accelerator at 23-68 MeV. After the initial acceleration,
a prebuncher and chopper ensure that the three individual beams are separated in
time and longitudinal spread. See Figure 4.4 for a schematic of the accelerator.
The polarization of the high energy beam as it enters Hall A can be measured
with a Compton polarimeter. This is done by scattering electrons from polarized
photons, and measuring the asymmetry in the cross section due to the beam he-
licity change. The scattering takes place when the electron beam interacts with a
photon beam in a Fabry-Perot cavity (which enhances the yields). This method of
measurement was used at the same time as data collection during kinematics two
and three. On four separate occasions the beam polarization was measured by a
M?ller polarimeter in Hall A. This was done using magnetized foils which provide
a target of polarized atomic electrons. The cross section was measured for two dif-
ferent orientations of the foil, in order to separate the longitudinal and transverse
components of the beam polarization. At various times during the experiment both
of these two methods along with a Mott polarimeter near the injector, were used
to measure the polarization. This allowed for a good understanding of systematic
errors for the beam polarization. An abridged summary of the beam polarization
measurements is presented in Table 4.2.
The sign of the electron?s helicity was put into the CODA [48] data stream with
58
a copy sent to the E02-013 trigger supervisor. The algorithm responsible produces
helicity states (that last for a duration of 33.3 ms) in sets of four (+??+ or?++?).
A 105 kHz clock was used to reconstruct the helicity if there was a problem with a
missing helicity. After every transition there is a short unknown time during which
the helicity is unknown (given by 0) in the data.
In the Hall A beamline before the target, a set of field coils move the beam to
create a uniform pattern across a section of the target cell. This is done so that a
small portion of the target is not overheated and the target cell is not compromised.
The beam sweeps through its pattern at a rate of 17 to 24 kHz in a process known
as rastering. The current used to do this is read into the data stream.
Beam position monitors (BPMs) are located upstream of the target (at 7.52
m and 1.29 m). The average position of the BPM is recorded at a rate of 1 Hz
in EPICS. The fast rastering system is located 23 m upsteam of the BPMs. The
BPMs were connected to an ADC for readout, such data could then be converted to
a position for the beam. Harp scans in which a thin set of wires are swept through
the beam, were carried out to provide absolute position information which could
be used to calibrate the positions. The BPM and raster calibrations were carried
out by Brandon Craver and discussed in greater detail in another thesis [81]. No
scraping of the beam on the sides of the target was observed.
The two LINACs each accelerate the beam by up to 600 MeV during each pass.
The beam can be bent and returned in RF cavities for up to five passes through
the LINACs. The beam is accelerated in the east and west and bent 180 degrees
in the arcs. See Figure 4.4 for details of the accelerator setup and beam cavities.
59
Because different electron energies need to travel through different magnetic fields
to have the proper bend angle, the beam is split in a spreader and then recombined
after traveling through the arc. Each cavity is also at 499 MHz to keep the beam
properly bunched. The beam can be sent into the switch yard after any pass, where
it is sent to hall A, B, or C. Beam energies were determined using the Tiefenbach
method and recorded in EPICs. This method relates the beam energy to the current
applied to one of the arc magnets [53].
4.4 The Helium-3 Target
Polarized targets and polarized 3He have been used in previous experiments,
as described in Section 3.3.3. To review and expand upon this description, polarized
3He targets are one of the preferred methods of realizing a polarized neutron target.
As described, nuclear targets are used rather than free neutrons as the latter only
survives with a halflife of 885.7?0.8 s [4]. For polarization experiments 3He is ideal,
because its polarization is almost entirely carried by the neutron [4]. Additionally,
the relatively simple nature of the 3He nucleus allows final state interactions to be
understood. These final state interactions are thought to be small at high momen-
tum transfer (see section 5.7.1). In polarized 3He, the neutron carries ? 86% of the
spin of the nucleus (Figure 4.5) [4]. Most of the time, the spins of the protons are
in opposite directions. About 3% of the time, the protons are polarized, and this
proton asymmetry is corrected for when considering proton contamination of the
neutral hadron candidates (see Section 5.5 and 5.6).
60
Figure 4.5: Diagram of the 3He nucleus. Shown is the most common state with the
proton spins aligned anti-parallel. The neutron carries ? 86% of the polarization of
the nucleus while the protons carry ? 3%.
The process of spin-exchange optical pumping (SEOP) was used to polarize
the 3He nucleus. In this process, alkali atoms are optically pumped and, through
spin exchange, polarize the 3He nucleus. An image of the target cell is provided
in Figure 4.6. A unique feature of this target was the use of potassium (K) and
rubidium (Rb) for the optical pumping, rather than the more traditional rubidium
only. The process of SEOP will be described in Section 4.4.2.
61
Figure 4.6: Picture of an empty 3He cell. On top is the spherical pumping chamber,
and below is the cylindrical target chamber. The beam passes through 40cm of
target material [41].
62
4.4.1 Target System
The goal of the target system was to create a polarized 3He target. To have a
polarized target, a magnetic field is required to provide an axis with which to orient
the target spin. To do this, the target system consists of a ladder holding the targets
including a cell to hold the 3He gas, a laser system to polarize the gas, and a system
to create the uniform magnetic field.
The target ladder contained four different targets, and was enclosed in an
iron box to provide a ?constant? magnetic field in the target that is immune from
interference due to the Big Bite dipole. The targets used were the 3??He, a set of optics
(carbon and beryllium oxide) foils for calibration, and a reference cell (containing
N2 or H2) targets. The reference cell was similar to the 3He cell and approximately
40 cm in length.
An oven was used to keep the pumping chamber at a constant temperature
of 240? C in order to maintain a sufficient density of potassium vapor for the spin
exchange process.
The target cell was made from hand-blown glass, with two chambers, a pump-
ing chamber and a target chamber, and a transfer tube between the two. The po-
larized gas in the heated top chamber would diffuse into the bottom target chamber
where the polarized 3He served as the target. The cell was constructed of alumine-
silicate glass and was filled with 8 atm at room temperature of 3He in addition to
N2 and sealed.
To optically pump the Rb, five 30 W lasers were used (totalling 150 W). In
63
order to decrease radiation damage and ease safety concerns, the lasers were not
kept in the hall, rather they were kept in a shed. To pipe the light to the target, 75
m of optical fiber carried the laser light to the target where it was combined in the
pumping chamber.
All components of the target system were held in an iron box, except the lasers.
Around the target were coils to drive the RF field for the polarization measurements,
sets of pickup coils to measure the NMR signals, and a coil to provide the excitation
for the EPR measurement (see Figure 4.11). The magnitude of the field provided by
the iron box was limited to 25 G in order to keep the beam from bending away from
the beam dump. The iron box provided shielding from Big Bite, with eight coils
arranged to create a ?uniform? field in the target region (Figure 4.8). The holding
field was not completely uniform, and since the experiment was very sensitive to
polarization direction, this was measured using a custom built compass [61]. Figure
4.7 shows the relationship between position along the target and direction of the
magnetic field.
4.4.2 Polarization
The 3He target used in E02-013 was the first implementation of a hybrid
target at Jefferson laboratory. The target?s hybrid nature is due to containing two
alkali metal vapors to reach a higher sustained polarization in less time than the
previous mono-alkali targets. This target system was developed by Alan Gavalya
and the polarized 3He groups at the University of Virginia, the College of William
64
Figure 4.7: Measurement of the holding field as a function of position along the
beamline in the target. The origin is the target center.
65
Figure 4.8: Magnet box for Target [57]. Shown is the coils that create the magnetic
field.
66
Zeeman Splitting
 795 nm?+
Collisional Mixing
5P
5S
1/2
1/2
B Field (25G)
M=?1/2 M=+1/2j = +1/2,?1/2
j=?3/2,?1/2,+1/2,+3/2
Figure 4.9: Figure showing the polarization process [57]. Optical pumping excites
the electron to an excited state, which can exchange (50%) to a state that decays
to a spin state that is not optically pumped by the lasers.
and Mary, the University of Kentucky, and the Hall A staff. By using a potassium
and rubidium mixture in the cells, a polarization higher than 50% was acheived (see
Figure 4.12).
In the process of spin exchange optical pumping (SEOP), rubidium is put in
a magnetic field and exposed to the circularly polarized light, this pumps up the
valence electron to a spin up sublevel in a P state. This excited state will either
directly decay, or exchange spin through collision and then decay (see Figure 4.10).
The photon released during decays can depolarize; the nitrogen in the cells (roughly
2% of 3He volume) serves to quench this, allowing the transition back to a bound
state through kinetic collisions rather than radiation. After decay there is a 50%
chance to end in a spin state. This spin state is not excited by light from the
67
lasers and so becomes densely populated (see Figure 4.9). Angular momentum can
be transferred from the polarized alkali valence electron to 3He via the hyperfine
interaction. A description of these processes is presented in Figure 4.10.
The spin exchange efficiency for K-3He interaction is 10 times more efficient
than Rb-3He. Commercial lasers for optically pumping Rb are more readily available
than those for potassium. Potassium (K) and other alkali metals have a high spin
exchange cross section with rubidium and so the polarization of two such gases in a
cell will be the same. This means that including the potassium into the cell enhances
the transfer of polarization into the helium-3, decreasing the time to polarize the cell
by a factor of two and causing a corresponding increase in sustained polarization in
beam.
4.4.3 Polarization Measurement
The polarization of the target was measured by a combined use of EPR (elec-
tron paramagnetic resonance) and NMR (nuclear magnetic resonance) techniques.
EPR provides an absolute measurement of 3He polarization, while NMR provides a
relative measurement. Both techniques cause depolarization of the target, but NMR
disrupts the experiment less due to causing less depolarization. Because of this and
because it could be carried out directly in the scattering chamber, NMR was done
more frequently. These two techniques used two separate locations in the target
cell. Due to using the alkali gases, EPR measured the polarization in the pumping
chamber while NMR could be used to measure polarization in the target chamber.
68
Figure 4.10: In a traditionally polarized helium cell, one that contains only Rb
and 3He, the Rb exchanges spin in one of three ways. Spin is exchanged either by
changing the spin of the He nucleus, or by just rotating the atoms it interacts with.
This rotation is not useful for polarizing the He, so by adding K, which is likely
to engage in ?spin-rotation? with the Rb, but likely to engage in ?spin-exchange?
with He, the polarization efficiency of the cell is increased. Only the spin-exchange
collision process causes polarization of the 3He [57] [93].
69
Primary NMR pickup coils
Upperchamber 
pickup coil
RF coils
Coils
Holding Field
Coils
Holding Field
Figure 4.11: The coils used for the NMR measurement of target polarization. Shown
is the holding field coils (iron box), the RF coils (which can be adjusted), and the
pick-up coils. Two sets of pick-up coils are down at the target chamber, one more
set is up above the pumping chamber. One of the RF coils could be adjusted, as
could the pick-up coils, to maintain the transverse relationships needed for an NMR
measurement.
The diffusion of polarized gas in the cell was modeled to relate the two techniques
and properly calibrate the NMR signal.
To perform a NMR measurement, control of the magnetic field of the target
is necessary. After an application of a radio frequency (RF) field, when resonance
conditions are met, a signal is measured that is proportional to the polarization of
the target.
The resonance was found using adiabatic fast passage (AFP). In adiabatic fast
passage the magnetic field is changed with a perpendicular radio frequency field (91
kHz) held constant to find the resonance. At resonance, 3He undergoes spin reversal.
This spin reversal produces an EMF signal that was detected in a separate set of
coils known as pick-up coils. See Figure 4.11 for a description of the coils involved
in the NMR measurement.
This measurement technique switches the direction of the spins, so the mag-
70
netic holding field was swept back to give the original target polarization direction
during each measurement. The depolarization caused by this process was found to
be on the order of 1%. This measurement technique is relative due to the EMF
signal depending on the magnetic flux through the coils, the amplification of the
electronics, and the density of 3He. The calibration of this measurement technique
is presented in another thesis [56].
In an EPR measurement of the polarization of the 3He, it is light from the
polarization of the alkali metals in the cell that is measured. This is due to the alkali
atoms being very sensitive magnetometers. This energy shift is due to the Zeeman
effect. Obviously this means that there are two shifts in the Zeeman responses of
the K and Rb, one due to spin exchange and another due to the magnetic field
experienced by the metal including that of the polarized 3He.
There are numerous magnetic fields which cause this shift, such as the holding
field and interactions with the other atoms. However, these other shifts are not
dependent on the polarization of the target, and by flipping the polarization the
effect due to the polarized 3He on the magnetic field creating slight differences in
the K energy levels can be determined. For a spherical sample, combining shifts due
to collision and the classical magnetic field, the following relationship for the signal
shift is obtained [51]
??EPR = 8pi3 d?EPR(F,M)dB ?0?HePHe . (4.1)
Here B is the magnet field, PHe is the polarization of the 3He, and ?0 is a parameter.
To flip the target polarization, the holding field is kept constant and the applied
71
RF field is swept (which is in resonance with the 3He). The EPR resonance causes
depolarization and the fluorescence caused by that repolarization can be tracked as
a function of RF frequency. This decay is caused by a shift in electron energy levels
in the potassium atoms.
To understand the two measurement techniques relative to one another, NMR
measurements were performed both before and after each EPR measurement. The
EPR measurements had the disadvantage of causing significant target depolariza-
tion. This is because the EPR transition causes the 3He spins to be anti-aligned
with that of the spin state the alkali is being pumped into. To minimize the depo-
larization, this transition was done twice for every EPR polarization measurement.
The EPR measurement depends on ?0, which is different for every alkali and noble
gas combination and has temperature dependence [33] [85].
The measurements required to relate the absolute EPR measurements and the
relative NMR measurements and the model used to relate the polarization at the
different locations of the two measurements is presented in detail in another thesis
[56]. The corrected polarization is shown in Figure 4.12. From equation 4.1 it is
obvious that the target density needs to be known well. Resistive temperature de-
vices (RTDs) were placed at eight locations on the cell to measure the temperature,
from which the density was computed using the ideal gas law. These RTDs did
not measure the internal temperature, and so a series of NMR measurements were
carried out under various conditions to determine the true internal temperature [56].
Many corrections are needed to provide the needed polarization and direc-
tion of polarization to properly determine the physical asymmetry. The material
72
Time (days)70 80 90 100 110 120 130
Polarization (%)
0
10
20
30
40
50
Polarization for E02-013
Kin 2a(Mar 10 - Mar 21)Kin 3 (Mar 21 - Apr 17)
Kin 2b(Apr 17 - Apr 24)
Kin 3 Kin 4(May 5- May 10)
Figure 4.12: Measured polarization in the 3He cells during the experiment [56].
Shown is all cells, including Dolly, Edna, and Barbara used during Kinematics 2, 3,
and 4.
surrounding the helium-3 had to be understood to build a proper Monte Carlo
simulation, while the unpolarized nitrogen within the cell caused a dilution in the
asymmetry. Details of these corrections are presented in section 5.5.2.
4.5 Big Bite
4.5.1 The Big Bite Apparatus
Big Bite is the name of the electron spectrometer, and of the large 1.2 T
dipole magnet that provides the magnetic field for the spectrometer. It consists of
the large magnet, a plane of 13 scintillators, a calorimeter, and three multi-wire
drift chambers (MWDC) containing a total of 15 wire planes. The calorimeter is
73
Figure 4.13: Schematic of the Big Bite detector. Shown is the large dipole which
provides a magnetic field integral of 1.2 T?m, the 15 planes in three chambers that
made up the wire chamber, and the lead glass shower and preshower separated by
a layer of plastic scintillator.
74
split into a shower and preshower, and is constructed of 250 lead glass blocks. A
schematic of the Big Bite detector is shown in Figure 4.13.
4.5.2 The Big Bite Scintillators and Shower
The Big Bite scintillators are located between the preshower and shower and
provide the time of the event within Big Bite tracker. Each of the thirteen paddles
has a photomultiplier tube on each end, with the signal split going to both a TDC
via a discriminator and an ADC. The scintillators are used to calculate the time of
the particle at the drift chambers, which are about 1 m away. The scintillator time,
with resolution ? = 300 ps, is required for reconstructing the hadron time of flight
due to its use in providing the reference time. These scintillators are 64 cm by 220
cm.
The front plane of the total shower, known as the pre-shower, is located 1 m
behind the drift chambers. The preshower consists of 54 lead glass blocks in two
columns of 27 rows. Behind the preshower, the shower consists of seven columns
and 27 rows. Its lead glass blocks are 8.5 cm by 8.5 cm, while the preshower has
lead glass blocks that are 35 cm by 8.5 cm. The total height of the structure was
230 cm. Each block is connected to a single photomultiplier tube, which collects the
Cerenkov light. The signal is then sent to both an ADC and summation module,
with the summed signals going both to an ADC and a TDC. The sum of the shower
and preshower gives a signal roughly proportional to the energy of the particle. The
total shower provides the trigger for Big Bite. A final set of information provided by
75
the total shower is the rough location of the particle. This aids track reconstruction
in the multi-wire drift chambers and reduces the the search space by a factor of 10.
The energy resolution provided by the total shower is approximately ?dE
E
= 10%.
4.5.3 The Wire Chambers and Tracking
The drift chambers were constructed by the University of Virginia for this ex-
periment in order to reconstruct the trajectory of the electron as it travels through
the detector. To do this, the planes which make up the drift chambers were con-
structed with three different orientations, known as U, X, and V. The chambers
were roughly 35 cm apart. In the Big Bite detector coordinate system, the X plane
has wires running parallel to the y axis, while the U and V wires are rotated by
?30? with respect to that axis. The sense wires were 1 cm apart from each other
within the plane. Cathode planes were placed 3 mm above and below each wire
plane and field shaping wires were placed between each pair of sense wires to create
a roughly symmetric potential around the sensing wires.
Charged particles which pass through the chamber release electrons by ioniz-
ing the gas within the chamber. This gas is a 50% argon and 50% ethane mixture
that had been bubbled through ethyl alcohol and is kept slightly above atmospheric
pressure. Since a voltage difference exists between the sense wires, the field shaping
wires, and the cathode planes, the charges are attracted to the detector wires and
interact with them forming a signal which is then detected in a TDC (after amplifi-
cation and discrimination). The time it takes to drift to the wire can be determined
76
and used to calculate the distance between the wire and the particle track. Each
plane was set at a different voltage to create the desired field surrounding the sensing
wires.
The detector determines more than just the track that the particle traveled, it
is also determines the particle momentum and its point of origin within the target.
An effective bend plane model was used to determine this, where the interaction
of the magnet is treated as occurring at the magnetic mid-plane. If we assume
that there is only dispersion along Big Bite detector x, then the complete track
between the target and the total shower can be reconstructed. By finding the point
in the magnetic mid-plane that the observed track (back track) in the drift chambers
points to, and assuming that dispersion happens only in the x (Big Bite) direction,
the origin of the track along the beam (forward track) can be determined (V0). All
coordinates in this section are in the Big Bite detector coordinate system unless
otherwise indicated.
In this model, the vertex is (after corrections cxprime, cy, cyprime, and cx)
VLAB = c0V0 +cx0x0 +cxprime0xprime0 +cy0y0 +cyprime0yprime0 +f(xbend,ybend) . (4.2)
Here the V is the z vertex location in the lab coordinate system, and f(xbend,ybend)
is a parameter to determine deviations outside of this model. The x0 and y0 describe
the location of the intersection of the track with the plane z = 0 in the detector
coordinate system. The variables ?tgt = xprime0 and ?tgt = yprime0 describe the track between
the bend plane and the target and are found by xprime0 = dxdz and yprime0 = dydz in the detector
coordinate system. The bend coordinates (xbend,ybend) are the detector coordinates
77
Figure 4.14: This diagram, which is not to scale, shows Big Bite, the location of
the dipole, target, drift chambers, and shower, and the quantities known as the
deflection angle (?def), ?tgt, and the back and front tracks. Shown on the upper left
is the coordinate system for the target coordinates. The shower is shown providing a
fourth location for the particle, to anchor the track reconstruction. Big Bite detector
coordinates have their origin at the center of the first plane. The z direction is
perpendicular to the first chamber, and x is the magnetic dispersion direction.
78
where the track intersects the bend plane. A diagram of the effective bend plane
model is in Figure 4.14. A histogram showing the foil target vertex reconstruction is
shown in Figure 4.16. This shows that the position along the target is reconstructed
properly.
The momentum can also be determined, as
pBB = c0(xbend,ybend)+cxxbend?
def
+c??tgt +cyy +cyprimeyprime +f(xbend,ybend) . (4.3)
This ?def is the deflection angle, and defined using the vectors vectorxf and vectorxb which
describe the tracks the particle takes between the target and the magnetic mid-
plane and between the magnetic mid-plane and the wire chambers, and is defined
to be
?def = cos?1
 vectorx
f ?vectorxb
|vectorxf||vectorxb|
?
. (4.4)
The energy for the electron in elastic events can also be determined as
Eelastic = mpEem
p +Ee (1?cos?e)
, (4.5)
where mp is the mass of the proton and Ee is the energy of the electron, and ?e is
the electron scattering angle. The momentum calibration was done using hydrogen
data. The momentum resolution is demonstrated in Figure 4.15.
4.5.4 Big Bite Electronics
Signals from each wire in the multi-wire drift chamber travel through an ampli-
fier and discriminator before terminating at a LeCroy 1877 multi-hit TDC running
in common-stop mode. Information is read out from these after each trigger. The
79
p/p?-0.1-0.08-0.06-0.04-0.02 0 0.020.040.060.08 0.1
0
200
400
600
800
1000
1200
1400
1600
1800
 = 0.85%?Kin 4, 
 = 0.76%?Kin 2, 
 = 0.92%?Kin 3, 
 = 0.80%?Kin 3, 
 = -0.3 MeV0?Kin 4, 
 = -0.9 MeV0?Kin 2, 
 = -1.2 MeV0?Kin 3, 
 =  0.5 MeV0?Kin 3, 
p/p - Fittable Area?
Figure 4.15: Momentum resolution achieved with the latest optics model. Shown is
the momentum resolution for all four kinematics.
 (m)zv-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.150.2 0.25
0
200
400
600
800
1000
1200
1400
1600
1800
2000
 = 4.9 mmBeO?
zv
Figure 4.16: Resolution of the carbon foils after vertex corrections. The foils should
be 6.7 cm apart with the BeO foil at 0 cm. This histogram is of data from a foil
target run in kinematic 3. These foils were used for the optics calibration.
80
FI/O D
FI/O
L428F
D
PS706
Sh1/1Sh1/2
Sh1/3Sh1/4
Sh1/5Sh1/6
Sh1/7
Sh2/1Sh2/2
Sh2/3Sh2/4
Sh2/5Sh2/6
Sh2/7
Sh3/1Sh3/2
Sh3/3Sh3/4
Sh3/5Sh3/6
Sh3/7
L428F
FI/O
L428F
FI/O
SPL
0.5/0.5
SPL
0.5/0.5
SPL
0.5/0.5
SPL
0.5/0.5
50?
50?1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
4
4
8
8
PS776Amp     680
LRS428FFI/FO 52 13
PS706Discr 452
8ns 104
6ns 26
8ns 52
11ns 52
1
2
3
4
5
5 6
D
PS706
5
5 6
D
PS706
5 16ns 5268 4ns 2
13
10
11
12 3ns 2
80ns 1
1ns 28
1ns 28
TOTAL 44
PS755Logic 12
PS757OR 26 2
Sum7 1427
Splitter 54 4
S8
S8
OR
OR
PS757
PS755
L428F PS706
S8
BigBite  Trigger  Logic   for    GEN
A
A
PS776
PS776
SPL
0.5/0.5
SPL
0.5/0.5
PS3L
PS3R
A
A
PS1L
PS1R
PS776
PS776
A
APS2R
PS2L
PS776
PS776
chan modules
Lengthamount1ns 54Group
ADC "TS01A"TDC "TS01T"
ADC "PS01A"
ADC "PS02A"
ADC "TS02A"TDC "TS02T"
ADC "P01L"
ADC "P01R"
ADC "P02L"
ADC "P02R"
ADC "P03R"
ADC "P03L"
ADC "S1.1?S1.7"
ADC "S2.1?S2.7"
ADC "S3.1?S3.7"
ShowerPreshower
TDC "PS01T"
TDC "PS02T"
"electron"
Implemented on Feb 24 2006
Figure 4.17: The electronics schematic for Big Bite. Shown are the modules and
the wire lengths. Two neighboring preshower blocks, and two neighboring sets of
shower blocks, which have a summed amplitude that passes some threshold provides
the trigger for the electron arm.
scintillators have their signal split between a LeCroy 1881 ADC and a discriminator,
with the discriminated signal terminating on a CAEN 775 TDC. The shower and
preshower have all their signals (individual and summed) put into a LeCroy 1881
ADC, with the sums also having the signal discriminated and put into a LeCroy
1877 TDC.
The sums of the preshower and shower that are over some predefined threshold
provide the Big Bite trigger.
81
4.6 Big Hand
While the subject of Big Hand is a major component of this thesis, the cali-
bration and detailed analysis of the neutron arm?s behavior is not needed directly
for the analysis. As a result, the geometry was described in detail in a document
put together by Tim Ngo [73] and the software, detector description, analysis, and
calibration are presented in Appendix A. Neutron detection took place through
hadronic interaction in iron converter layers in front of neutron detector counters
organized in seven planes behind two veto layers and lead shielding. Discussion of
neutron detection is also the subject of Chapter 5 and Appendix A.
4.6.1 Neutron Geometry
The neutron apparatus (NA), sometimes referred to as Big Hand, is a large
hadron detector, designed to match Big Bite?s acceptance at the highest kinematic
with Q2 = 3.5 GeV2. The dimensions of Big Hand are 4.2?2.0?6.2 m3 and it has a
100 msr solid angle at a distance of 8 meters. The neutron detector is made up of
244 neutron bars in seven planes and 192 segmented, single sided veto bars in two
planes. These counters were provided by University of Virginia (UVA), University
of Glasgow (GLA), and Carnegie Mellon University (CMU). For the UVA counters
PMT type XP 2282B were used, for the CMU counters PMT type XP 2262B, while
for the GLA EMI 5? were used [98]. In order to properly identify quasi-elastic
events at the highest kinematic, a time resolution of 0.3 ns was desired, which
corresponds to neutron momentum resolution of 250 MeV/c at a hadron momentum
82
of 2.58 GeV [25]. The scintillation material in the counters was the standard organic
plastic scintillation material [1]. To increase the detection efficiency of neutrons, a
thin (one inch) layer of iron was placed in front of each counter to cause some
portion of the incident neutrons to undergo hadronic interactions. A high degree of
segmentation was important so that the vertical (X) position of the incident particle
could be measured with the required resolution. This allows the selection of quasi-
elastic events for which it is necessary to have a good measurement of momentum
perpendicular to the virtual photon. Four ?marker? counters were included for
calibration purposes.
4.6.2 Veto and Marker Bars
The veto counters were divided into two so that they would provide the needed
time resolution without problems with attenuation. The two veto planes were offset
from each other both horizontally and vertically, for complete coverage of the active
area of the neutron bars. This, in addition to the height of veto bars (11 cm)
compared to the neutron bars (15 cm for CMU bars), meant that multiple veto bars
could possibly fire to define the charge for an event within the neutron bars.
The marker counters were included late in the construction to calibrate the
horizontal position reconstruction of the neutron detectors. These are long vertical
scintillators placed before the first plane of the neutron detector, but after the
shielding and veto detectors. They were placed within two constructed channels,
with the same amount of space between the top marker bars and the bottom marker
83
bars for each channel [72]. Even with two marker bars to cover the whole height
of the neutron arm, most coincident hits between a marker bar and a neutron bar
were only measured in a single PMT in the marker bar. Using these marker bars
assisted in the time offset calibration of the neutron arm.
4.6.3 Neutron Arm Electronics
The initial discriminators, amplifiers, and summing modules, were all placed
in an electronics hut behind the neutron detector. The ADCs and TDCs were put
behind shielding some 100 meters away, where the signals were discriminated again
to provide a good signal. Signals from the detectors were combined into sums. This
was done to increase the neutron detection efficiency at Q2 = 2.5 GeV2. Figure 4.19
shows the neutron bars and veto bars, a sum is made up of the left or right PMTs
for two neighboring color coded sections.
The ADCs used for the detectors within the neutron arm were all Lecroy 1881,
while the TDCs used for the veto detectors and sums were LeCroy 1877 TDCs. The
F1 TDCs were specially developed electronics for Jefferson Lab and were used for
the neutron counters. These TDCs were used in a common-stop mode and provided
a resolution of 118 ps. The F1 TDC required that a reference signal be used, this was
a delayed signal from the trigger and the F1 TDC signals could be reconstructed
relative to it. Summing modules, amplifiers, fan in/out modules, discriminators,
ADCs, TDCs are all presented in Figure 4.18 (and logic). This gives the NA side
of the trigger as arising from any sum channel being past threshold on the left or
84
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
LRS428F PP(ADC)
PS706D
LRS428F PP(ADC)
PS706D
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
LRS428F PP(ADC)
PS706D
LRS428F PP(ADC)
PS706D
1
1 1
1
1
1
1
1
1
1 1
1
1
1
1
1
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
706PS
706PS
706PS
706PS
706PS
PS776
PS776
PS776
PS776
PS776
PS776
PS776
PS776
706PS
706PS
1
1 1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1 1
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1
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10
2
SUM8
706PS
PP (ADC)PP(TDC) SUM8
706PS
PP (ADC)PP(TDC)
SUM8
706PS
PP (ADC)PP(TDC)SUM8
706PS
PP (ADC)PP(TDC)
PS757
PS755N?trigger
right  end
SUM8
706PS
PP (ADC)PP(TDC) SUM8
706PS
PP (ADC)PP(TDC)
nseclength two endsone endamount
LEMO ? LEMO    Cables
SUM8
706PS
PP (ADC)PP(TDC) SUM8
706PS
PP (ADC)PP(TDC)
From  PMs at  CMU bars
From  PMs at  CMU bars
From  PMs at  CMU bars
From  PMs at GU bars
From  PMs at UVA/JLAB bars
From  PMs at UVA/JLAB bars
From  PMs at UVA/JLAB bars
From  PMs at GU bars
2 232 4648 232 464
10 24 4814 24 48
4 6 126 15 30
1 15 308 1 2
18 1 222 1 2
Figure 4.18: Neutron Arm electronics schematics. Included in the schematic is the
length of the cables. From the PMTs of the various tubes, the signal goes into both
a TDC and a sum module with the non-summed output going to an ADC. The
different summed signals are added together and sent to both an ADC and TDC,
with an OR of the TDCs forming the trigger.
85
right.
4.7 Data Acquisition
Six triggers were used in this experiment. These were the neutron arm trigger
(1) known as T1, the Big Bite trigger (2) known as T2, the coincidence trigger (3)
known as T3, the 8.5 Hz pulser (7), the 105 kHz helicity synchronization signal (8),
and the 30 Hz helicity quad duration signal (9).
The T3 trigger was formed by coincidence between the neutron arm and Big
Bite. This coincidence was formed between any single sum channel on the neutron
arm (left or right) and detection beyond a certain threshold in the preshower and
shower. In Figure 4.2 the grouping of neutron bars is presented. Two neighboring
colors would be linked together to form one sum, with the left and right PMTs being
considered separately. This was done to increase neutron detection efficiency at a
higher threshold [79].
The time of the BigBite Shower signal was used for the time of the T2 trigger.
The Big Bite scintillator time associated with this was then sent to the neutron arm
to provide both the coincidence for the trigger and the reference time of the neutron
arm events.
The neutron detector real time was recorded relative to a readout time or
reference time. Thus the neutron arm recorded time was given as trecorded = treal ?
treadout. The Big Bite trigger time relative to this same read out time was also
recorded as tL1A = treadout ?ttrig known as the level one accept time and recorded
86
1
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07 0707
08080808
17
1818
171717
1818
Figure 4.19: Neutron arm (NA) schematic for the sums. In front are the two veto
planes. Shown is the relative offset. Additionally, color coded, the sums are shown.
Each sum was made up of two neighboring color bands. Shown on top are the bars
from Glasgow (GLA), in the front are four planes of bars from Carnegie Mellon
University (CMU), and in the back are three planes of bars from University of
Virginia(UVA).Thecolorlessbarsinthefinalplanearebarsthatwerenotconnected
to the electronics. The first two thin planes are the veto planes, the figure shows
the vertical shift between the two planes.
87
in the TDC. The neutron arm time relative to the Big Bite trigger time is then
treal = trecorded +tL1A.
A trigger from both detectors, Big Hand and Big Bite, created a T3, the
coincidence trigger. The treadout time was based on this T3 time. The T1 came 35
ns later than the T2 trigger due to delays added in. If the T1 came too early, treadout
and T3 were based on on the T2 trigger. If the T2 trigger came very early, then the
Big Bite readout was determined by treadout and not the T2 trigger. This is known
as forced retiming. A plot showing the data with the different types of events is
shown in Figure 4.20.
Both detectors provided information at each coincidence event. Single trigger
events were also taken but prescaled to provide a minimum number of events in
the experiment. All three multi-wire drift chambers provide TDC information, the
scintillators provided both ADC and TDC, and the preshower and shower provided
ADC information (with the preshower and shower sums providing TDC in addition
to ADC information). All the neutron arm detectors provided both TDC and ADC
information, with the neutron detectors also grouped into sums which provided ADC
and TDC information as well. The neutron arm reference time was recorded in both
the Neutron Arm TDCs and the Big Bite TDCs.
4.7.1 Encoding and Decoding
There are two main data acquisition systems. One is EPICS, which provides
updates occasionally (no more often then every few ms) and provides information
88
Figure 4.20: Plot showing the reference time and coincidence counts in relation to
the Big Bite trigger time [37]. Best coincident events are in the circled region.
89
about the target, accelerator, electronics, and scalars. This is included in the data
stream with the events from the detectors. The events from the detectors are ac-
quired using CODA, which can monitor, acquire, record, and decode data taken
during experiments [48]. Some target information was stored separately.
To do the analysis and turn the raw data into usable information, a modifica-
tion of the ROOT software package called the Hall A Analyzer was used [5]. This
package has features which make it easier to deal with large amounts of data and to
make histograms and graphs. Special classes for the analysis of the different detec-
tors have been developed for the analysis of the neutron arm data and for tracking
and momentum reconstruction in Big Bite and are discussed in other sections. The
library which contains these classes is the AGen library. The E02-013 experiment
used a modification of this Hall A Analyzer. This was done so that in the decoding
and initial calculation of physics quantities, Big Bite tracking code could take ad-
vantage of other detectors. The scalar data was put in separate trees in this initial
decoding and analysis of the data. During the initial decoding and analysis steps,
the first 1000 events of a run were discarded (due to being helicity diagnostic events)
as were non-coincidence events. During decoding and processing, the data is put
into trees, which is a data format commonly used in ROOT and are referred to as
root files [86].
These root files were then processed an additional time and then the quasi-
elastic events were selected using the Big Bite preshower, the reconstructed momen-
tum, the reconstructed vertex hit location, and the hit locations and times of the
coincident hadron and electron. This post processing and analysis is described in
90
Chapter 5, with the final value of GnE calculated for the kinematics Q2 = 3.5, 2.5,
and 1.5 GeV2.
91
Date Method Polarization Statistical Systematic
March 30, 2006 Mott 83.32 % 1.45 1.00
March 30, 2006 Mott 81.62 % 1.45 0.98
April 13, 2006 Mott 84.12 % 1.11 1.01
April 13, 2006 Mott 83.25 % 1.11 1.00
Average Mott 83.08 % 0.65 1.00
February 28, 2006 M?ller 88.8 % 0.2 3.0
February 28, 2006 M?ller 86.8 % 0.2 3.0
March 4, 2006 M?ller 88.2 % 0.14 3.0
March 9, 2006 M?ller 86.5 % 0.15 3.0
March 25, 2006 M?ller 82.2 % 0.3 3.0
Average M?ller 86.5 % 0.09 3.0
April 18, 2006 Compton 82.92 % 2.31 2.40
April 20, 2006 Compton 85.63 % 3.87 1.55
April 22, 2006 Compton 86.47 % 1.63 3.55
April 24, 2006 Compton 82.65 % 1.64 3.20
Table 4.2: A table of the beam polarization as measured by different means at
different times. The M?ller measurements were done in the Hall A M?ller. The
Mott measurements were done using the injector Mott measurement. Only selected
Compton data is presented, and the Compton data isn?t used in this analysis.
92
Chapter 5
Analysis
In experiment E02-013 the electric form factor of the neutron was measured at
four kinematic points. Three of these kinematic points are presented in this thesis,
with the fourth still to be analyzed by another collaborator. The analysis procedure,
excluding the initial decoding and analysis steps, is shown in flow charts in Figures
5.1, 5.2, and 5.3. The initial decoding and analysis passes provide the electron arm
physics information, as described in section 4.7, in addition to neutron arm cluster
information. The first step of this procedure is the event based filtering, as described
in Table 5.1, which uses the time-of-flight of the electron, momentum, invariant
mass of the recoiling hadron, and transferred momentum vectors which were already
calculated from the electron arm (in section 5.1 and 5.2) information to select hadron
event candidates. Next, hadrons are identified using the transferred momentum
vectors and the constructed time-of-flight of the hadron, which is discussed in more
detail in section 5.3. A charge is assigned to these hadrons as described in section
5.4. Next, various dilutions of the asymmetry (section 3.3), and corrections to
it, are described and calculated in section 5.5. The neutron physical asymmetry
determined from the neutral quasi-elastic events is then calculated in section 5.6,
using the Table 5.5. In this section preliminary results for the electric form factor
are also presented with corrections. The final state interaction corrections which
93
will be included in the final value of GnE are discussed in section 5.7. Discussion of
error propagation can be found in Appendix B.
Electron event selection in the Big Bite spectrometer was achieved by means of
the shower detector and scattering angle. Selection of events from within the target
and by a coarse cut on transferred momentum were made using the multi-wire drift
chamber (MWDC). Hadron identification was achieved using the location and time
of the hit in the neutron detectors, and by using the veto detectors to differentiate
the protons from the neutrons. Finally missing mass, missing momentum, and
invariant mass were used to select the quasi-elastic process.
5.1 Event filtering and scintillator timing
The total shower detector, as described in section 4.5.2, is divided into two
parts, called the shower and the pre-shower, separated by a plane of scintillator
detectors. Using only the pre-shower amplitude, pions and electrons could be iden-
tified and clearly differentiated, (as shown in Figure 5.4). For any selection based
on preshower energy which doesn?t remove an unreasonable portion of electrons,
some portion of pions will still exist in the selected sample. These pions will create
a dilution of the (e,e?h) interaction. An emperical study was done to determine the
contamination of the neutral hadron candidate sample by (ep,eprimepi0n) events. This
contamination was shown to be less than 2% [2].
While the total shower was the hardware trigger for the electron arm, the
time based on the calorimeter alone would not provide the needed resolution for the
94
Figure 5.1: The flowchart for the analysis. This is after the preliminary replay of
events that is described in section 4.7. The resulting ROOT files contain the subset
of events necessary to undertake the remaining analysis, also assigns charge, and
identifies quasi-elastic event candidates. Two separate scripts are run as part of
this process for each run before this post processing. This is to determine the RF
correction (see section 5.5.3) and the hadron time-of-flight.
95
Figure 5.2: Flowchart for the event processing. Events in the analysis are ignored
if no tracks or Big Bite scintillator hits exist, or if the event does not have a well
defined trigger time as demonstrated by the existence of the Level 1 Accept (L1A)
Time. The optics can be updated here if necessary, and the kinematics of the event
are calculated to ease post processing. All event cuts are detailed in Table 5.1.
96
Figure 5.3: Flowchart associating clusters, both with each other and as quasi-elastic
event candidates. Some clusters should be associated together, where some neu-
tron detector did not fire to make the cluster continuous. Charged candidates are
identified, and momentum is calculated for the hit.
97
Channels0 500 1000 1500 20002500 30003500 4000
Counts
0
500
1000
1500
2000
2500
3000
Figure 5.4: Amplitude distribution for all events showing pions and electrons in Big
Bite. In blue is the distribution after cuts have been applied to remove events which
are lacking the information needed for analysis, shaded is the region selected. See
Table 5.1 for cuts, in black is all events and in blue is with just the ?sanity? cuts
applied. This distribution is for Q2 = 3.5 GeV2.
98
event. This function is provided by the plane of scintillators directly behind the
pre-shower, which served as the primary time reference for both the electron and
neutron arms. The path length for the electron was emperically determined using
the optics with respect to this plane. This reference time was corrected for the
variations in path length coming from differences in vertex location and corrections
for vertical and horizontal position in the scintillator stack.
The time of the event within the electron arm, te, can be presented as the time
in the scintillator ts, with corrections for the vertical path length tx, the horizontal
pathlengthty, andthedifferencebetweenthescintillatorplaneandthewirechamber
tc.
te = ts ?tx ?ty ?tc (5.1)
tx = 1c1.2|Px,BB +0.25| (5.2)
ty = 1c [2.2532sin(??BB)+YBB cos(??BB)]sin[tan?1(P
y,BB)??BB]
?2.2 (5.3)
tc = 1c ?LD2S(1+P2x,BB +P2y,BB?2 (5.4)
Here, LD2S = 0.95 m is the time from the detector to the scintillator, Px,BB is the
track slope tan? = dxdz in Big Bite detector coordinates, Py,BB is the track slope
tan? = dydz in Big Bite detector coordinates, YBB is the reconstructed horizontal
position vector y in Big Bite detector coordinates, and ?BB = 51.59? is the central
angle at which the detector was placed for the Q2 of 1.7, 2.5, and 3.5 GeV2. The
other constants were empirically determined using the optics data.
99
5.2 Electron parameters
Quasi-elastic (from 3He) and elastic events (from hydrogen) were selected using
the momentum of the electron as measured by the Big Bite spectrometer in addition
to deposited energy in the preshower (see Figure 5.5). Additional event selection
was performed by removing events that were reconstructed to start from outside
of the target (see Figure 5.7), or that travelled through a region of Big Bite where
the magnetic field is not well understood (see Figure 5.6). Among the quantities
computed from the identified electron tracks are the invariant mass of the hadron,
the transferred momentum of the virtual photon, the electron momentum, and the
electron scattering angle.
The transferred momentum, vectorq, calculated from the beam energy and scattered
electron momentum, was used in the identification of quasi-elastic hits within the
neutron arm. The transferred momentum is entirely independent of the neutron arm
time-of-flight. The parallel and perpendicular components of transferred momentum
with respect to the hadron momentum vector can be defined as
qbardbl = vectorq ? ?ph , (5.5)
q? = vectorq ? ?ph . (5.6)
Here ?ph is the unit vector of the hadron momentum as determined by the neutron
arm. This is determined purely from the hit location (x, y, and z) in the neutron
arm, and is independent of time-of-flight.
100
Electron Momentum (GeV)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Counts
0
500
1000
1500
2000
2500
Figure 5.5: Electron momentum for all tracks in Big Bite. In blue are just those
tracks for which the preshower energy is greater than 450 channels, shaded is the
selected momentum. See Table 5.1 for cuts, in black is all events and in blue is with
the ?sanity? and pre-shower energy cuts applied. This distribution is for Q2 = 3.5
GeV2.
The invariant mass of the hadron is
W2 = 2?MN +M2N ?Q2 (5.7)
Here MN is the nucleon mass, Q2 is the transferred four-momentum squared, and
? is the virtual photon energy (see section 2.2).
The invariant mass and perpendicular transferred momentum (q?) are the two
time-of-flight independent variables which are used for quasi-elastic event selection.
The selection of quasi-elastic events are shown for Q2 = 2.5 GeV2 in Figure 5.8(a)
and 5.9(a). Both of these figures include fidicual cuts in the neutron arm. The
final quasi-elastic neutral candidates selection for kinematic 2 as well as the overall
spectrum is presented in perpendicular transferred momentum is presented in Figure
101
X position (m)-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Counts
0
200
400
600
800
1000
1200
1400
Figure 5.6: X position (m) within the Big Bite scintillator detector. This is for
before cuts (black), after electron and sanity cuts (blue), with the selected region
of X shaded. Variation is due to the lack of calibration for the total shower in
the Big Bite detector. See Table 5.1 for cuts, in black is all events and in blue is
with the ?sanity?, pre-shower energy, and electron momentum cuts applied. This
distribution is for Q2 = 3.5 GeV2.
102
Z position (m)-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Counts
0
200
400
600
800
1000
Figure 5.7: Vertex, this shows the target z position reconstructed by the Big Bite
wire chambers for all events with tracks (black). In blue is after cuts on particle
ID. Shaded is the region selected as being within the target. See Table 5.1 for cuts,
in black is all events and in blue is with the ?sanity?, pre-shower energy, Big Bite
fidicual, and electron momentum cuts applied. This distribution is for Q2 = 3.5
GeV2.
103
name Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
W (GeV) |W ?0.95| < 0.6 |W ?0.95| < 0.6 |W ?0.95| < 0.6
E. E. (Ch) Eps > 450 Eps > 450 Eps > 450
E. Mom.(GeV) |pe ?1.1| < 0.9 |pe ?1.1| < 0.9 |pe ?1.1| < 0.9
Vertex (m) |Vtgt| < 0.16 |Vtgt| < 0.16 |Vtgt| < 0.16
X Fid. (m) ?0.4 < XBB < 0.45 ?0.4 < XBB < 0.45 ?0.4 < XBB < 0.45
Sanity 1 NTracks > 0 NTracks > 0 NTracks > 0
Sanity 2 NPaddles ? 0 NPaddles ? 0 NPaddles ? 0
Sanity 3 (Ch) tL1A < 4000 tL1A < 4000 tL1A < 4000
Sanity 4 NType3 > 0 NType3 > 0 NType3 > 0
Negative Px,BB ?0.3XBB < 0 Px,BB ?0.3XBB < 0 Px,BB ?0.3XBB < 0
Table 5.1: A table of the initial event selection for analysis. Sanity cut 1 is the
requirement that Big Bite has a track, Sanity cut 2 is the requirement that at least
one scintillator paddle in Big Bite has an event, Sanity cut 3 is that the coincidence
time (L1A) was recorded. Sanity cut 4 is the requirment that there is a coincidence
event. Negative is a requirement that only negative particles in the drift chambers
can be considered as electrons.
5.11(a). The expected invariant mass is about 0.94 GeV, the mass of a nucleon.
5.3 Hadron identification
Once electron events are selected and the invariant mass and the direction and
magnitude of vectorq are determined from information from the Big Bite spectrometer, the
neutron arm is used to determine the three-momentum and charge of the hadron.
Hits in the neutron arm were joined together into clusters as described in Appendix
A. Typically, multiple bars recorded hits in the same event in the neutron apparatus
due to particles with high energy traveling deep into the neutron apparatus, and to
the segmented nature of the neutron apparatus.
The time of the counter with the earliest hit (of those nearby in time and
space) was used as the time of the cluster. This time, the time of the electron
104
in the Big Bite scintillator, and the relevant path lengths allow the calculation of
the momentum of the particle, assuming it is a hadron. Only those hits where the
calculated velocity is less than c are included in the analysis.
 W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV) q
0.00
0.05
0.10
0.15
0.20
0.25
0
50
100
150
200
(a) Q2 = 2.5 GeV2 q?
 W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV) q
0.00
0.05
0.10
0.15
0.20
0.25
0
50
100
(b) Q2 = 3.5 GeV2 q?
 W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV) q
0.00
0.05
0.10
0.15
0.20
0.25
0
200
400
 Vs W for Neutralq
(c) Q2 = 1.7 GeV2 q?
Figure 5.8: q? versus W for neutral particles in the various kinematics. Shown is the
selection of quasi-elastic events in the square. As shown, at higher Q2 the inelastic
tail becomes stronger.
The clusters were combined if they were nearby in time and space. The time
and position of only the first hits were used to determine whether the later cluster(s)
should be considered part of the earlier cluster. Multiple clusters in a single event
were considered as a single cluster if ?X (vertical position) and ?Z (depth) were
within 0.4 m, and ?Y (horizontal position) was within 0.2 m. Additionally, the time
105
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
0.00
0.05
0.10
0.15
0.20
0.25
0
200
400
600
800
(a) Q2 = 2.5 GeV2 q?
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
0.00
0.05
0.10
0.15
0.20
0.25
0
500
1000
(b) Q2 = 3.5 GeV2 q?
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
0.00
0.05
0.10
0.15
0.20
0.25
0
2000
4000
Perpendicular Momentum (GeV) versus Invariant Mass (GeV)
(c) Q2 = 1.7 GeV2 q?
Figure 5.9: q? versus W for charged particles in the various kinematics. In the red
box is the selection of quasi-elastic events.
106
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
1.5
2.0
2.5
0
1000
2000
(a) Q2 = 2.5 GeV2 qbardbl
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
2.0
2.5
3.0
0
500
1000
1500
(b) Q2 = 3.5 GeV2 qbardbl
W (GeV)0.4 0.6 0.8 1.0 1.2 1.4
 (GeV)q
1.0
1.5
2.0
0
2000
4000
6000
8000
(c) Q2 = 1.7 GeV2 qbardbl
Figure 5.10: qbardbl versus W for charged particles in kinematic 3.
107
 (GeV)q0.0 0.2 0.4 0.6 0.8
0
50
100
310?
(a) Q2 = 2.5 GeV2 q?
 (GeV)q0.0 0.2 0.4 0.6 0.80
100
200
300310?
(b) Q2 = 3.5 GeV2 q?
 (GeV)q0.0 0.2 0.4 0.6 0.80
100
200
300
400
310?
(c) Q2 = 1.7 GeV2 q?
Figure 5.11: Distribution of q? in the various kinematics. Shown in blue is the neu-
tral spectrum, in black is the total spectrum (That passed the event based selection).
The majority of neutral quasi-elastic events fall under 0.15 GeV.
108
difference had to be within 10 ns. See Appendix A for more discussion of clusters
and neutron arm software.
The expected time for hadron events within the neutron arm Th is given by
Th =
q
L2NA +Z2tgt ?2LNAZtgt?ph ? ?z
c q?m2
h+q2
. (5.8)
Here LNA is the path measured from the center of the target to the neutron arm hit,
Ztgt is position of the interaction in the target as reconstructed by Big Bite, vectorph is
the hadron momentum vector (in the laboratory coordinate system). Since at this
point the charge of the hadron is unknown, its mass is nominally set to the nucleon
mass mN = 0.939 GeV.
This is compared to the actual time-of-flight which is constructed from the
measured location of the time-of-flight peak measured using the neutron arm and
Big Bite scintillators. This actual time is constructed as
TTOF = tNA ?te +trf +tL1A ?tpeak . (5.9)
Here, the tNA is the time measured in the neutron arm, trf is the correction that
can be done using the beam information (to be discussed in section 5.5.3), tL1A is
the level 1 accept time (see section 4.7), and tpeak is the peak location in the total
time-of-flight spectrum. To select quasi-elastic events, the expected time-of-flight is
compared to the actual time-of-flight, giving ?T. The time-of-flight spectrum for
total events and quasi-elastic neutral candidates in kinematic 2 is shown in Figure
5.12(a).
From the time-of-flight, the hadron velocity, ?, and momentum, ph, can be
109
Time (ns)-4 -2 0 2 4
5000
10000
15000
(a) Q2 = 2.5 GeV2 time
Time (ns)-4 -2 0 2 4
10000
20000
(b) Q2 = 3.5 GeV2 time
Time (ns)-4 -2 0 2 4
20000
40000
Time for Raw Events
(c) Q2 = 1.7 GeV2 time
Figure 5.12: ?T (ns) for the various kinematics. In blue is the neutral time-of-flight
spectrum, in black is the total time-of-flight spectrum. The photon peak is shown
at ??3ns. Quasi-elastic events are selected with time between -1 and 1 ns.
110
Missing Mass (GeV)1.6 1.8 2.0 2.2
0
10000
20000
30000
40000
(a) Q2 = 2.5 GeV2 missing mass
Missing Mass (GeV)1.6 1.8 2.0 2.20
10000
20000
30000
40000
(b) Q2 = 3.5 GeV2 missing mass
Missing Mass (GeV)1.6 1.8 2.0 2.20
10000
20000
30000
40000
(c) Q2 = 1.7 GeV2 missing mass
Figure 5.13: Missing mass (GeV) for the various kinematics. In blue is the neutral
scaled up by a factor of 20 for Q2 = 3.5 GeV2 and 10 for Q2 = 2.5 and 1.7 GeV2, in
black is the total spectrum.
111
Time (ns)-4 -2 0 2 4
0
200
400
(a) Q2 = 2.5 GeV2 time
Time (ns)-4 -2 0 2 40
200
400
(b) Q2 = 3.5 GeV2 time
Time (ns)-4 -2 0 2 40
500
1000
1500
2000
Time for Neutral Events
(c) Q2 = 1.7 GeV2 time
Figure 5.14: ?T (ns) for the various kinematics. In blue is the charged time-of-flight
spectrum, in blue is the charged time-of-flight spectrum. The charged spectrum is
scaled down to the neutral spectrum.
112
constructed (all in laboratory coordinates):
? =
q
L2NA +V 2tgt ?2LNAVtgt?ph ? ?z
cTTOF , (5.10)
ph = mhp1??2 . (5.11)
Also calculated is the unit vector relating this hit and the central location in the
target. Additionally, the perpendicular and parallel momenta can be calculated:
ph,bardbl = vectorph ? ?q (5.12)
ph,? = vectorph ? ?q . (5.13)
An additional quantity used to select quasi-elastic events which is dependent on
hadron identification is the missing mass. This is defined as
M2miss = (Pi +q?ph)2 = (EHe +Eq ?Ep)2 ?(vectorq ?vectorph)2 , (5.14)
where Pi is the initial 3He momentum four-vector, q is the transferred momentum
four-vector and ph is the nucleon four-vector.
The selection of quasi-elastic neutral events using time, invariant mass, and q?
is demonstrated in Figures 5.15(a) and 5.16(a). The inelastic contribution becomes
larger at higher Q2 and so more aggressive cuts in invariant mass must be used.
5.4 Identification of the hadron charge
While a particle can be determined to be a recoil hadron candidate by hit
location and time-of-flight, it is not easy to differentiate between protons and neu-
trons using only this information from within the neutron counters. For the purpose
113
Time (ns)-5 0 5
 (GeV)q
0.0
0.1
0.2
0.3
0.4
1000
2000
(a) Q2 = 2.5 GeV2 time versus q?
Time (ns)-5 0 5
 (GeV)q
0.0
0.1
0.2
0.3
0.4
500
1000
(b) Q2 = 3.5 GeV2 time versus q?
Time (ns)-5 0 5
 (GeV)q
0.0
0.1
0.2
0.3
0.4
2000
4000
Time versus Perpindicular Momentum for Neutrals
(c) Q2 = 1.7 GeV2 time versus q?
Figure 5.15: Shown in these figures is the time versus q? for various kinematics. Also
shown is the box showing which cuts were used for identifying quasi-elastic clusters
within the neutron arm. As is shown, the ability to select quasi-elastic clusters is
much better for the lower kinematics.
114
Invariant Mass (GeV)0.4 0.6 0.8 1.0 1.2 1.4
Time (ns)
-2
-1
0
1
2
0
50
100
150
(a) Q2 = 2.5 GeV2 time versus invariant mass
Invariant Mass (GeV)0.4 0.6 0.8 1.0 1.2 1.4
Time (ns)
-2
-1
0
1
2
0
50
100
(b) Q2 = 3.5 GeV2 time versus in-
variant mass
Invariant Mass (GeV)0.4 0.6 0.8 1.0 1.2 1.4
Time (ns)
-2
-1
0
1
2
0
100
200
300
Time versus Invariant Mass for Neutrals
(c) Q2 = 1.7 GeV2 time versus in-
variant mass
Figure 5.16: Shown in these figures is the time versus invariant mass for various
kinematics. Also shown is the box showing which cuts were used for identifying
quasi-elastic clusters within the neutron arm. As is shown, the ability to select
quasi-elastic clusters is much better for the lower kinematics.
115
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Time (ns)
-2
-1
0
1
2
0
1000
2000
3000
4000
(a) Q2 = 2.5 GeV2 invariant mass versus missing mass
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Time (ns)
-2
-1
0
1
2
0
2000
4000
6000
(b) Q2 = 3.5 GeV2 invariant mass
versus missing mass
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Time (ns)
-2
-1
0
1
2
0
5000
10000
15000
20000Invariant Mass versus Missing Mass
(c) Q2 = 1.7 GeV2 invariant mass
versus missing mass
Figure 5.17: Shown in these figures is the time versus missing mass for various
kinematics. Also shown is the box showing which cuts were used for identifying
quasi-elastic clusters within the neutron arm. As is shown, the ability to select
quasi-elastic clusters is much better for the lower kinematics.
116
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Invariant Mass (GeV)
0.6
0.8
1.0
1.2
1.4
0
1000
2000
3000
(a) Q2 = 2.5 GeV2 invariant mass versus missing mass
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Invariant Mass (GeV)
0.6
0.8
1.0
1.2
1.4
0
500
1000
(b) Q2 = 3.5 GeV2 invariant mass
versus missing mass
 (GeV)missM1.6 1.8 2.0 2.2 2.4
Invariant Mass (GeV)
0.6
0.8
1.0
1.2
1.4
0
5000
10000
Invariant Mass versus Missing Mass
(c) Q2 = 1.7 GeV2 invariant mass
versus missing mass
Figure 5.18: Shown in these figures is the invariant mass versus missing mass for
various kinematics. Also shown is the box showing which cuts were used for identi-
fying quasi-elastic clusters within the neutron arm. As is shown, the ability to select
quasi-elastic clusters is much better for the lower kinematics.
117
of charge identification, two layers of veto counters were positioned in front of the
neutron counters but behind some of the lead shielding (see section 4.6.1).
The charge of these hadrons was determined by an analysis of hits in the veto
in a neighborhood around the hadron hit position in the neutron counters. The
veto detectors were segmented vertically similarly to the neutron detectors. Due
to scattering in the shielding and detectors, the relevant veto hit position could be
greater than a few centimeters away from the identified hadron hit location. The
factors used to select relevant veto detectors are listed in Table 5.4.
Determination of whether the hadron is charged or not was accomplished using
the time between the neutron bar hit and the veto hit and the segmentation of the
detectors. This segmentation defined the x and y position of each veto hit. The
individual veto detectors had high rates ( 1 MHz) and high dead-time (up to 110
ns). This indicates that there exists a large number of possible hits where an earlier
accidental hit masked the hit from the quasi-elastic event candidate. For coincident
events, the hit in the veto bars varied in veto amplitude in addition to time, the
veto amplitude was used in addition to time when selecting veto hits to associate
to quasi-elastic events.
While it would be nice to have a detector with uniform neutron detection
efficiency, this was not the case for Big Hand. The GLA detectors did not have the
resolution to identify quasi-elastic neutrons as the rest of the neutron arm could,
and the horizontal edges were not adequently covered by the veto detectors. Two
additional cuts, on the neutron arm?s x and y fidicual regions were used to select the
region of the detector that had a high neutron detection efficiency. This selection is
118
name Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
Amplitude AV < 200 AV < 200 AV < 200
Spatial (m) |XV ?XC| < 0.3 |XV ?XC| < 0.3 |XV ?XC| < 0.3
Time (ns) |TV ?TC| < 10 |TV ?TC| < 10 |TV ?TC| < 10
Y1 Cut (m) YC < 0.076&YV < 0 YC < 0.076&YV < 0 YC < 0.076&YV < 0
Y2 Cut (m) YC > ?0.774&YV > 0 YC > ?0.774&YV > 0 YC > ?0.774&YV > 0
Table 5.2: A table of selection criteria used to determine which veto hits to consider
when doing charge identification. The spatial cuts of Y1, Y2, and X determine which
bars are to be included, events were only differentiated by time and amplitude within
a veto detector which was within range of the hadron hit. The subscript V signifies
veto, while C signifies the cluster in the neutron arm.
shown in Figures 5.19 and 5.20.
The area of the veto used to determine the charge of the hadron is determined
by looking at the coincidence between the hadron hit and the veto hit, see Appendix
A.4 for more details.
5.5 Dilution of the neutron sample by protons
Between the target and the veto detectors, there were several centimeters of
lead and other material [73]. Hadrons that leave the target could be observed as
if they were in the other isospin state, through hadronic interactions. This would
create a dilution in the neutral candidate sample due to quasi-elastic events still
being detected as hadrons, but with the wrong charge. This is especially important
in the case of protons observed as neutrons, which creates a significant dilution (see
Table 5.5) in the neutron sample. This conversion can be understood, and accounted
for, using the ratios of observed protons and neutrons for different targets. The three
targets used were hydrogen, which is a single proton, 3He, which is the target of
119
Vertical Position (m)-2 0 2
0
500
1000
310?
Figure 5.19: Vertical position (m) within the Big Hand detector. This is for before
cuts (black), after event selection cuts (blue), with the selected region of x shaded.
In the vertical direction the neutron arm was segmented, giving the segmented
structure observed. GLA bars are removed, as seen in the most negative region. See
Table 5.5 for cuts.This distribution is for Q2 = 3.5 GeV2.
Horizontal Position (m)-1.5 -1.0 -0.5 0.0 0.5 1.0
0
1000
2000
3000
310?
Figure 5.20: Horizontal position (m) within the Big Hand detector. This is for
before cuts (black), after event selection cuts (blue), with the selected region of y
shaded. In the horizontal direction the neutron arm was not entirely covered by
the veto planes, giving the peaks on the two edges. See Table 5.5 for cuts.This
distribution is for Q2 = 3.5 GeV2.
120
interest, and nitrogen, which has equal numbers of protons and neutrons. These
ratios are referred to as Rtgt,
Rtgt = fnf
p
. (5.15)
Here the subscript tgt stands for target, and is replaced by N for nitrogen, H for
hydrogen, and He for 3He. The fraction fp(n) is that of protons(neutrons) to the
total number of hadrons.
Thepurityfactor, Dn, istheratioofobservedneutronstotheneutronsknocked
out of the target at the interaction point with the electron beam. Similarly, Dp is
the ratio of observed protons to those starting from the target. For the observed
Rtgt, the fp(n) are observed fractions. However, as mentioned, it is possible that
interactions between the target and detector can cause a conversion in the observed
iso-spin state. To account for this, the ratio can be expressed in terms of mixing
coefficients. These provide the numbers of observed Np(n) resulting from an initial
target population Np(n). Expressed in this manner
Rtgt = N
n
n +N
n
p
Npp +Npn =
?n?nn + ZtgtNtgt?p?np
?n?pn + ZtgtNtgt?p?pp . (5.16)
As an example, Nnp is the number of observed neutrons from initial protons. The
above can be presented more compactly as:
Rtgt =
?n
?p
?nn
?pp +
Ztgt
Ntgt
?np
?pp
?n
?p
?pn
?pp +
Ztgt
Ntgt
. (5.17)
Here ?p(n) are the single nucleon cross sections, ? are the mixing coefficients, and ZtgtNtgt
is the ratio of protons to neutrons (unique for each target, and may be dependent
on perpendicular and parallel missing momentum) within the target. Obviously
121
 (GeV)m,p0 50 100 150 200 250
0.8
0.9
1
1.1
1.2
Figure 5.21: Presented is the ratio of protons to neutrons within the target for
nitrogen. Shown are two models, one a simple extension to the plane wave impulse
approximation (in blue), and the other is a model by Udias (in black) [96]. Shown
is the ratio for various cuts of p?. A cut of 150 MeV was used in this experiment.
This is for Q2 = 3.5 GeV2.
Ntgt = 0 for hydrogen, ZNN
N
? 1 for nitrogen, and ZHe3N
He3
? 2 for the 3He. In particular
models [96][90], the ratio was investigated for the applied cuts and kinematics (see
Figure 5.21 and 5.22). This provided values for these two coefficients (see Table
5.5). In terms of these mixing coefficients, the purity factor is then
Dn = N
n
n
Nnn +Nnp =
?n
?p
?nn
?pp
Ztgt
Ntgt
?np
?pp +
?n
?p
?nn
?pp
. (5.18)
Using this information, it is possible to solve for the mixing ratios from the
ratios of hydrogen (H2), 3He, and nitrogen (N2)
?np
?pp = RH , (5.19)
?pn
?pp =
?p
?n
2
4RHe
ZHe3
NHe3 ?
ZN
NN RN +
?
ZN
NN ?
ZHe3
NHe3
?
RH
RN ?RHe
3
5 , (5.20)
122
Figure 5.22: Presented is the ratio of protons to neutrons within the target for 3He.
Shown are from cuts on p?. The applied cut on p? was 0.15 GeV with Pbardbl cuts of
400, 250, and 200 MeV for kinematics of Q2 = 3.5, 2.5, and 1.7 GeV2. Shown is the
ratio for various cuts of inverse distance. This study was done by Aidan Kelleher.
?nn
?pp =
?p
?n
?
RN
 Z
He3
NHe3 ?
ZN
NN
?R
He ?RH
RN ?RHe ?
ZN
NN RH
?
. (5.21)
Therefore the dilution due to the proton to neutron conversion in the sample,
Dn = RHe[(
ZHe3
NHe3 ?
ZN
NN )RN +
ZN
NN RH]?
ZHe3
NHe3RNRH
RHe
?
ZHe3
NHe3 ?
ZN
NN
?
(RN ?RH)
. (5.22)
This accounts for mixing due to conversions in the air and other material between
the target and the veto detectors. However, there are additional considerations due
to the veto detectors before the final dilution factor, due to protons observed as
neutrons, can be determined.
As mentioned in section 5.4, the veto detector efficiency was related to large
dead-time in the veto detector TDCs and high rate within the veto detectors. The
dead-time could be as high as 110 ns for some paddles, while the rate in the indi-
123
vidual veto detectors was on the order of 1 MHz. Additionally, the hydrogen and
nitrogen data were acquired at different rates from the 3He data. Because of this, it
is desirable to account for the rate dependence in the charge identification. This is
complicated for a couple of different reasons. First, the veto was segmented into left
and right and into two planes, which made the veto difficult to model. Also, because
of the high activity in the veto, every event had multiple random hits within the
veto planes.
The observed protons and neutrons detected in the veto can be represented in
the following equations, where the fp(n) stands for the fraction of protons(neutrons)
incident on the veto detectors over the total number of hadrons incident on the veto
detectors; which is equivalent to the fraction when there is no deadtime. In these
equations r represents the rate in the detectors.
Npn(r)+Npp(r) = fpT(r)+fnT(r)Pvetobusy(r) (5.23)
Nnn(r)+Nnp (r) = fnT(r)(1?Pvetobusy(r)) (5.24)
A charged hit is determined by a hit in the proper range of either veto plane
(see Table 5.4). So the rate dependent number of observed neutrons is the rate de-
pendent total number of hadrons, T(r), incident on the neutron detector multiplied
by the fraction, fn, of the number of neutrons to hadrons multiplied by the proba-
bility that there was no accidental background hit in the veto detector to make it
appear charged (1?Pvetobusy). The rate dependent number of observed protons is the
total number of hadrons multiplied by the fraction of protons over hadrons, fp, plus
124
the fraction of neutrons misidentified by the accidental background.
These probabilities can be determined by modeling them using the Poisson
distribution, and then adding up all the probabilities for each hadron event. This
gives real numbers for everything but the fractions incident on the veto detectors.
The observed ratios, at a given frequency, are
Rtgt(r) = N
n
n(r)+N
n
p (r)
Npn(r)+Npp(r) . (5.25)
Using this formula twice, once for the observed rate r and once for the desired rate
d, gives a formula to relate every observed ratio Rtgt(r) to a desired ratio Rtgt(d):
Rtgt(r) = Rtgt(d)T(d)[F
r(?)]
Rtgt(d)T(d)[T(r)?Fr(?)]+[Fd(?)?Rtgt(d)(T(d)?Fd(?))]T(r) .
(5.26)
The fitting parameter ? accounts for correlations in hits in different veto bars in a
given event. The function F depends on ? and ?, where ? is the total rate dependent
probability for veto detectors to be busy in an event:
Fr(?) =
T(r)X
0
Pbusy(?,r) . (5.27)
here ? is a useful quantity to define for the calculation and fit of
Fr(1) = ?(r) =
T(r)X
0
Pbusy(? = 1,r) , (5.28)
which is just Fr(?) assuming no correlation between veto hits. The probability of
an accidental hit, Pbusy(?) with r being the rate and ? being the dead-time (or busy
time) of the bar, in a given event is
Pbusy(?,r) = (e?
Pr?
)? . (5.29)
125
The functional form of Fr(?) can be approximated by assuming that the rate r
multiplied by the deadtime ?d is approximately the same for all bars, so the quantity
A = e?Pr?d is the same for each bar. This is a good assumption, since the rate
multiplied by dead-time is the same order of magnitude for all veto bars. From this,
Fr(1) = ?(r) ? T(r)A(r) . (5.30)
Therefore a good approximation for F(?,?) is
Fr(?) ? T(r)
 ?(r)
T(r)
??
. (5.31)
This allows the ratio at a specific rate of the different targets to be calculated
using the fractions of uncharged and charged particles incident on the veto. Using
these ratios in equation 5.22 gives the rate independent dilution correction for proton
to neutron conversion:
Q2 = Ratio for 3He Ratio for Hydrogen Ratio for Nitrogen Purity Factor
2.5 GeV2 0.105 ? 0.001 0.020 ? 0.002 0.195 ? 0.010 0.806 ? 0.028
3.5 GeV2 0.077 ? 0.001 0.015 ? 0.002 0.106 ? 0.024 0.885 ? 0.053
1.7 GeV2 0.107 ? 0.001 0.035 ? 0.001 0.191 ? 0.008 0.697 ? 0.019
Table 5.3: A table of the ratios for different criteria. Over the course of a kinematic
setting, runs varied in rate. This had a small effect on the purity factor, the results
presented in this table is for a normal run.
Q2 = Helium Ratio Nitrogen Ratio
2.5 GeV2 2.15 1.1
3.5 GeV2 2.31 1.07
1.7 GeV2 2.15 0.9
Table 5.4: The effective ratio of ZN for the target and cuts. The ratio for 3He is from
simulation.
126
T(r)(r)?
0.7 0.75 0.8 0.85 0.9 0.95 10.15
0.2
0.25
0.3
0.35
0.4
0.45
(a) uncharged to charged ratio for Q2 = 2.5 GeV2
T(r)(r)?0.4 0.6 0.8 1 1.2
00.05
0.10.15
0.20.25
0.30.35
(b) uncharged to charged ratio for Q2 =
3.5 GeV2
T(r)(r)?0.750.8 0.850.9 0.951
0.15
0.2
0.25
0.3
0.35
0.4
(c) uncharged to charged ratio for Q2 =
1.7 GeV2
Figure 5.23: Shown in these figures is the measured uncharged to charged ratio,
the uncharged to charged ratio within the model, and the fit line of the model in
nitrogen. These are plotted against the natural dependence ?T . In green are the
points of the model.
127
T(r)(r)?
0.8 0.82 0.84 0.86 0.88 0.90.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
(a) uncharged to charged ratio for Q2 = 2.5 GeV2
T(r)(r)?0.5 0.6 0.7 0.8 0.9 1
0.01
0.02
0.03
0.04
0.05
(b) uncharged to charged ratio for Q2 =
3.5 GeV2
T(r)(r)?0.780.80.820.840.860.880.90.920.940.026
0.0280.03
0.0320.034
0.0360.038
0.040.042
0.0440.046
0.048
(c) uncharged to charged ratio for Q2 =
1.7 GeV2
Figure 5.24: Shown in these figures is the measured uncharged to charged ratio,
the uncharged to charged ratio within the model, and the fit line of the model in
hydrogen. These are plotted against the natural dependence ?T . In green are the
points of the model.
128
T(r)(r)?
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(a) uncharged to charged ratio for Q2 = 2.5 GeV2
T(r)(r)?0 0.2 0.4 0.6 0.8 1
00.1
0.20.3
0.40.5
0.60.7
0.8
(b) uncharged to charged ratio for Q2 =
3.5 GeV2
T(r)(r)?0.760.780.80.820.840.860.880.90.92
0.08
0.1
0.12
0.14
0.16
(c) uncharged to charged ratio for Q2 =
1.7 GeV2
Figure 5.25: Shown in these figures is the measured uncharged to charged ratio, the
uncharged to charged ratio within the model, and the fit line of the model in 3He.
These are plotted against the natural dependence ?T . In blue are the points of the
model.
129
Dn = RHe(d)[(
ZHe3
NHe3 ?
ZN
NN )RN(d)+
ZN
NN RH(d)]?
ZHe3
NHe3RN(d)RH(d)
RHe(d)
?
ZHe3
NHe3 ?
ZN
NN
?
(RN(d)?RH(d))
. (5.32)
5.5.1 Single Track Analysis and Background Correction
For approximately 10% of the detected events, the hadron interacted in the
lead resulting in multiple events in the neutron apparatus or an accidental back-
ground hit that could not be differentiated from the hadron. If there were two or
more hits in the region of parameter space used to select quasi-elastic events (here-
after known as the quasi-elastic region), the charge becomes impossible to determine
since just one ?hit? in the veto detectors associated with the quasi-elastic region will
often set all hits in the quasi-elastic region as charged.
By restricting the analysis to only those events which have a single hit identi-
fied as quasi-elastic, the problem of charge identity and multiple hits in the quasi-
elastic region is solved. Using this single track analysis (SQE), however, causes the
determination of the background to become more complicated.
When multi-hit events are eliminated, the spectrum far from the quasi-elastic
hit is clean (see Figure 5.29). Using the perpendicular transferred momentum, q?,
the distance far from the quasi-elastic hit can be parametrized in the same variable
which is used to select quasi-elastic events. In this band of q?, the ratio of charged to
uncharged background is relatively constant (see Figure 5.28). This gives the ratio
of charged to uncharged for the background which is unrelated to the quasi-elastic
events. The only determination that is made in this region is the charge ratio of the
130
 (GeV)q0.0 0.2 0.4 0.6 0.8
0
20
40
Figure 5.26: Plot q? for a region in time. This is 5 ns removed from the peak region.
 (GeV) q0.0 0.2 0.4 0.6 0.8
0
10000
20000
Figure 5.27: Plot q? for a region in time. This is at the quasi-elastic region.
131
 Time(ns)-10 -5 0 5 10
0
2
4
6
Figure 5.28: Plot of the ratio of uncharged to charged for the region of q? between
0.55 and 0.60 GeV. The region near 3 ns is removed due to the photon peak being
there. Add axis information time (ns), update description (changed plot)
 Time(ns)-10 -5 0 5 10
 (GeV) q
0
0.2
0.4
0.6
0.8
1
Figure 5.29: Plot of q? versus time-of-flight for events with only a single hit in the
|t| < 7 and q? < 0.15 region. The area in the spectrum of events coming from k
photons via pi0 production was removed (between 2 and 4 ns). This is for run 4090.
132
background. The total background count is determined via a different mechanism.
Since it is required that every hit in the quasi-elastic region is either a single
quasi-elastic hit, or a single background hit, the total number of counts (and events)
is:
NA+BSQE = NAback +NAQE +NBback +NBQE , (5.33)
where A and B are two regions within the SQE region. The SQE region is chosen
so as to include all of the quasi-elastic events within it. It is useful to define region
B as being a region where NBQE = 0.
This is a lot simpler than that of the full events, which can be expressed as
the following:
NA+Bfull = NAback +NAQE +NAQE|back +NAback|back +NBback|back +NBQE|back +
NAQE|QE +NAback|QE +NBback +NBback|QE ,
here |back is the number of hits which also have at least one background hit, |QE is
the number of hits with at least one QE hit. Another class of hits are NAQE|QE which
are the number of QE hits with at least one other QE hit; these would be from
fragmentation caused by interaction between the target and the final hit location in
the neutron arm. Here A is the SQE region with the quasi-elastic hits within it and
B is a region in the SQE region without any quasi-elastic hits.
If the region is restricted to being outside the QE region, the expression is left
as:
NBfull = NBback +NBback|back +NBback|QE . (5.34)
Obviously it is just desired to remove the NBback from the QE sample in region A.
133
Since equation 5.33 is true for SQE analysis, outside of the QE region the expression
for the counts is
NBSQE = NBback , (5.35)
due to NBback|back and NBback|QE events not occuring because only one hit can occur
in the SQE region. Once the regions A and B are scaled to be the same size, the
number of quasi-elastic events in SQE analysis can be determined
NQE = NA,SQE ?NB,SQE , (5.36)
here B is a sub region of the SQE region where the QE events don?t reside, and A
is the sub region of the SQE region where QE events do reside.
For times earlier than the quasi-elastic window, as described above, the hits
are all background, coming from an accidental coincidence between the electron
and hadron (see Figure 5.29). This is in the -4 ns to -13 ns window. Using the
counts (scaled properly) from this background region provides the background in
single quasi-elastic hit analysis. The same area in q? is used to make scaling easier.
To determine the neutral accidental background, the earlier determined background
ratio term is used:
Nneut,back = Nbackfback , (5.37)
here f is the fraction of neutrals to hadrons in the background.
These background counts, neutral or charged, are used to determine the acci-
dental background dilution and asymmetry used in the determination of the physical
asymmetry. The neutral background dilution of a sample (N) can be expressed as
Dback = 1? Nneut,backN . (5.38)
134
Similarly the background asymmetry can be defined in the standard way
Aback = N
+
back ?N
?
back
P3HeN . (5.39)
5.5.2 Nitrogen Dilution
The 3He inside the target cell is diluted by unpolarized nitrogen, used as a
buffer gas to decrease interactions between the polarized 3He and glass cell walls
and to aid in the polarization process (Section 4.4.2). This contamination creates a
dilution of the 3He asymmetry, and is measured by looking at events which appear
to be quasi-elastic in a N2 target and scaling by the relative densities and total
charge on target. This dilution factor is
DN2 = 1? NN2 ?Nback,N2N ?N
back
Q3He?3He
QN2?N2 . (5.40)
The values used to calculate this dilution factor are provided in Table B.1.
5.5.3 RF correction
While the timing of the coincidence event relative to the beam crossing cannot
be determined absolutely, it can be determined up to an arbitrary constant. Event
by event variations about this constant can be determined from the 499 MHz RF
signal, which can be used to improve the hadron time-of-flight measurement between
the target and detectors.
The signal from the RF is put into the F1 TDC, which wraps (starts counting
again at 0) at W = 65526 channels, and has tres = 0.1183 ns resolution. The
RF signal repeats in a Twin = 2 ns window. In the following formula tRF = RF
135
Figure 5.30: Spectrum used to determine the RF correction. This is for run 3888.
The peak location is determined from the Mid parameter, which is trfshift.
time, tref = reference time, tL1A = level 1 accept time, and the aforementioned
time-of-flight (equation 5.9).
The RF correction is calculated using the following formula
RFcorr = Vtgtc +tres
 
tRF ?tref ?
 ?t
RF ?tref
0.5W
?
Fmod 1
?
W
?
, (5.41)
trf =
?
(te ?tL1A ?RFcorr ?trfshift) Fmod
 1
0.499
?
+ 20.499
?
Fmod
 1
0.499
?
.
(5.42)
This trf is the shift used for correcting the time-of-flight in equation 5.9. The trfshift
is the peak location shown in Figure 5.30. This shift is plotted versus run number
in Figure 5.31. This correction is due time of the scattering within the target.
136
Run Number4050 4100 4150 4200
RF shift (ns)
0.0
0.5
1.0
1.5
2.0
Figure 5.31: RF correction versus run number for Q2 = 2.5 GeV2. Note that since
the plotted function is Fmod ? 10.499? that the values of 2 ns and 0 ns are almost
equivalent. Run 4146 with a shift of ? 1 ns demonstrates that the shift likely is not
adequately determined for ?junk? runs.
Run Number4050 4100 4150
Raw Asymmetry 0.0
0.5
Figure 5.32: Shown is the run number versus raw asymmetry for Q2 = 2.5 GeV2.
137
Name Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
Time (ns) |t| < 1 |t| < 1 |t| < 1
Invariant Mass (GeV) |W ?0.9| < 0.2 |W ?0.875| < 0.175 |W ?0.9| < 0.2
Q Perp (GeV) q? < 0.15 q? < 0.15 q? < 0.15
SQE region B (ns) |t?8.5| < 4.5 |t?8.5| < 4.5 |t?8.5| < 4.5
Missing Mass (GeV) Mmiss < 2 Mmiss < 2 Mmiss < 2
Vertical Cut (m) |X ?0.2| < 1.7 |X ?0.2| < 1.7 |X ?0.2| < 1.7
Horizontal Cut (m) |Y +0.183| < 0.6 |Y +0.183| < 0.6 |Y +0.183| < 0.6
Table 5.5: Table of quasi-elastic event selection cuts. The Single QE cut is that
there is one and only one hit in the associated region of time and q? < 0.15. This
is in addition to the region of time and q? which are selected as quasi-elastic event
candidates. The vertical cut and horizontal cut are neutron arm fidicual cuts.
5.6 Asymmetry calculation
After the selection of quasi-elastic neutron events (Table 5.5), the measured
asymmetry is corrected to give the physical asymmetry, which gives GnE as presented
in section 3.3.3. This is the general expression, as mentioned in the previous sections
there are corrections which need to be done for nitrogen dilution in the target cell
(section 5.5.2), proton to neutron coversion between the target and scintillator bars
(section 5.5), accidental background asymmetry and dilution (section 5.5.1), neutron
polarization in the nucleus [39], beam polarization (section 4.3), target polarization
(section 4.4), and final state interactions (section 5.7.1). The corrected expression
is
APhys = Asum ?Aproton ?AbackP
3HePbeamPnDN2DnDFSIDback
. (5.43)
Since the proton contamination of the neutron sample (Dn), proton asymmetry
(Aproton), nitrogen dilution (DN2) , background asymmetry (Aback), and background
dilution (Dback) all depend on similar quantities, corrections need to be applied
as presented in sections 5.7 and 5.5. The nitrogen dilution is corrected for the
138
background dilution and the proton contamination dilution is corrected for both
the nitrogen dilution and the background dilution.
The physical asymmetry, corrected for the discussed dilutions and asymme-
tries, is presented in Table 5.6. The dilutions can be expressed in terms of those
due to polarization (P), those due to theoretical calculations like FSI (Dfsi) and
neutron polarization (Pn), and experimental corrections not due to polarization (V).
Expressed this way
P = P3HePbeam (5.44)
and
V = DnDN2Dback . (5.45)
Q2 = 1.7 GeV2 2.5 GeV2 3.5 GeV2
Ameas ?0.0569?0.0031 ?0.0446?0.0055 ?0.0358?0.0081
N 105529 32990 15319
Pnucl 0.488?0.02 0.436?0.02 0.477?0.02
Pbeam 0.835?0.03 0.835?0.03 0.835?0.03
Pn 0.86?0.02 0.86?0.02 0.86?0.02
Dback 0.9697?0.0002 0.9790?0.0003 0.9731?0.0005
Aback 0.00004?0.0000003 0.0013?0.00002 ?0.0013?0.00003
Dn 0.696?0.019?0.035 0.806?0.023?0.040 0.885?0.053?0.044
Aproton ?0.0054?0.0007 ?0.0031?0.0005 ?0.0018?0.0003
DN2 0.947?0.002?(0.005) 0.949?0.004?(0.005) 0.978?0.005?(0.002)
P 0.408 0.364 0.398
V 0.640 0.749 0.842
Aphys ?0.254?0.015?0.027 ?0.191?0.024?0.022 ?0.126?0.030?0.013
Table 5.6: Determination of the physical asymmetry from the measured experimen-
tal asymmetry. A full discussion and analysis of errors is presented in Appendix B.
Systematic uncertainties are listed second.
At times during the measurements at Q2 = 2.5 and 3.5 GeV2, the target
cell and beam were changed. This meant that the Pbeam, P3He, and Dn could
139
change during the run. To account for this, the target polarization corrected the
asymmetry on a run by run basis. Using standard error analysis (and Appendix B),
the summation was
Asum =
PA
i 1?2iP
1
?2i
, (5.46)
?sum = 1qP
1
?2i
, (5.47)
where the above equation was used for statistical uncertainties and the Ai are the
measured asymmetries for each run. After this summation, the physical asymmetry
is calculated using
APhys = Asum ?Aproton ?AbackP
beamPnDN2DnDFSIDback
. (5.48)
Many runs were taken over the course of a measurement at a given Q2. The stability
of the asymmetry over the course of the run, for a given running condition, was stable
as shown in Figure 5.32.
While the other asymmetries and dilutions can be corrected for without using
acceptance information, the FSI dilution and proton asymmetry require knowledge
of the kinematics to be properly applied.
The framework using knowledge of the kinematics to determine the ratio of
the electric and magnetic form factors, and so determine GnE, is based on equation
3.15. This provides the relationship between kinematic variables, the asymmetry,
and the form factor ratio. The relation between the physical asymmetry and the
ratio, ?, can be expressed as:
A = B?+C?2 +D , (5.49)
where ? = GEG
M
is the form factor ratio, and B, C, and D are the following kinematic
140
quantities
B = ?2
p
?(1+?)tan(?e2 )sin?cos? , (5.50)
C = ?2?
r
1+? +(1+?)2 tan2(?e2 )tan(?e)cos? , (5.51)
D = ?? +2?(1+?)tan2(?e2 ) . (5.52)
and the quantities ?, ?e, ?, and ? were defined in section 4.4.2. These quantities
can be averaged from all events, and are used in the determination of the proton
asymmetry correction, the determination of Q2, and the determination of ? from
the physical asymmetry.
The dilution corrections are presented explicitly in section 5.5. The asymmetry
corrections will be presented explicitly here. To make the best use of the good proton
form factor measurements [76], values for the proton form factor ratio were used to
determine the Aproton. The equations 5.50, 5.51, and 5.52 were used to calculate the
proton physical asymmetry
Aphys,proton = (B?proton +C)?2
proton +D
. (5.53)
This can then be corrected for the nitrogen dilution of the charged events (Dch,N2),
the proton polarization (Pp), background dilution (Dback), and the beam polarization
(Pbeam) to give the proton asymmetry correction
Aproton = 1?DND
backDN2
PbeamPpAphys,proton . (5.54)
By using the plot of this ratio at different values of the physical asymmetry,
the error and value for the ratio can be determined. However, if instead of being
141
plotted, ? is solved for, an inversion must be used. This can be done using the
following method [38]. Using equation 5.50, 5.51 and 5.52, we can express 5.49 as:
?(?,?)
?(?,?) = Aphys =
B?+C
D +?2 =
C
D +
B
D??
C
D2?
2 ? B
D2?
3 + C
D3?
4 + B
D3?
5 . (5.55)
Q2 = 1.7 GeV2 2.5 GeV2 3.5 GeV2
Aphys ?0.254?0.015?0.027 ?0.191?0.024?0.022 ?0.125?0.030?0.013
T0 -0.0693 -0.0129 0.0433
T1 0.9877 0.8350 0.7112
T2 0.0863 0.0115 -0.0223
T3 -1.2175 -0.6804 -0.3842
T4 -0.1091 -0.0102 0.0116
T5 1.5253 0.5630 0.2107
? ?0.199?0.018?0.033 ?0.222?0.033?0.029 ?0.243?0.046?00.020
Table 5.7: Expansion Coefficients in the determination of ? from the physical asym-
metry. Uncertainty in ? calculated as discussed in Appendix B.
These expansion coefficients can be expressed as T0 = CD, T1 = BD, T2 = CD2,
T3 = BD2, T4 = CD3, and T5 = BD3. In this expansion, these provide all the acceptance
and kinematic information for a given event. This allows the expression of Aphys as a
polynomial in ? with coefficients being equal to the averaged Tn values. For specific
averaged values, ? can be solved for giving the ratio of the electric and magnetic
form factors for that specific kinematic. To first order, this ? is
?0 = Aphys ??T0??T
1?
. (5.56)
Using these terms, the value for Q2 was also corrected for finite acceptance:
Q2 =
PT
1,iQ2i
T1 . (5.57)
The value for ?,
? = G
n
E
GnM , (5.58)
142
provides the electric form factor of the neutron as a ratio to the magnetic form factor.
Using this corrected value for Q2 and using the value of the magnetic form factor
as determined from a common parameterization [59], the value of the electric form
factor was determined. This calculation is presented in Table 5.8. The calculation
of the uncertainty is presented in appendix B.
The results and discussion of them in relation to previous measurements and
selected models are presented in Chapter 6.
Q2 = 1.7 GeV2 2.5 GeV2 3.5 GeV2
? ?0.199?0.018?0.033 ?0.222?0.033?0.029 ?0.243?0.046?0.020
Q2 1.73 (GeV2) 2.49 (GeV2) 3.48 (GeV2)
GnM ?0.164?0.003 ?0.0954?0.0019 ?0.0556?0.0011
GnE 0.033?0.003?0.005 0.021?0.003?0.003 0.014?0.003?0.001
Table 5.8: Displayed is the calculated values for ? and Q2, and the value of GnM
as arrived at from a parameterization [59]. From this the value of GnE can be
determined. Determination of uncertainty is discussed in Appendix B.
5.7 Corrections
5.7.1 FSI Corrections
As mentioned previously, in section 3.5, to first order there are four types
of interactions at lower momentum transfer. Shown in Figure 5.33, these are the
impulse approximation (IA), isobar current (IC), meson exchange currents (MEC),
and final state interactions (FSI). As has already been argued, meson exchange
currents and isobar currents are small for Q2 > 1 GeV2. This is due to their
scaling as 1Q4. This leaves just final state interactions as a necessary correction
143
Figure 5.33: The four dominant lowest order diagrams in single nucleon scatter-
ing. Shown is (a) impulse approximation, (b) meson exchange currents, (c) isobar
contributions, and (d) FSI [88].
Figure 5.34: In nuclei with more than a single nucleon, the plane wave impulse
approximation can be used to give the description of the interaction. Shown are
the leading order diagrams for 3He which is the PWIA diagram, single re-scattering
diagram, and double re-scattering diagram [88].
144
to the physical asymmetry, which was arrived at from the impulse approximation.
The final state interactions are based on nucleon-nucleon cross sections, which are
flat above 2 GeV [88]. To lowest order, these FSI will be either single or double
re-scattering between the spectator hadrons and the detected hadron (see Figure
5.34).
There are two similar approximations made in computing the effects of these
FSI. These are the Glauber approximation and the generalized eikonal approxima-
tion (GEA). In the Glauber approximation the incident particle?s energy is assumed
to be much larger than that of the potential, and is large enough so that the wave-
length of the incident particle is smaller than the potential width. This means that
the Glauber approximation truncates the terms after double scattering due to the
spectator hadrons being considered at rest. In the generalized eikonal approxima-
tion, the high momentum particle can not interact with a slow hadron a second time
after interacting with another bound hadron.
The eikonal approximation reduces to the Glauber approximation in the limit
of zero longitudinal momentum transfer, ? = q0|q|(Es ?m) ? 0. Here q is transfered
momentum, the subscript s refers to spectator, and m is the recoiling nucleon mass.
This is due to the Glauber approximation assuming that all nucleons are stationary
while the GEA allows for nonzero initial momentum of the recoiling nucleon [88].
145
5.7.2 Inelastic Contribution
The inelastic contribution to the quasi-elastic asymmetry is the final correction
to be developed. This contribution can be limited by removing some of the quasi-
elastic region which is closest to the delta resonance (at W = 1.2 GeV). The observed
width of the quasi-elastic peak in the invariant mass spectrum is increased due to
the motion of the nucleon within 3He and due to the resolution of the detectors. The
inelastic resonance width is similarly increased. This inelastic contribution increases
the tail in the quasi-elastic sample towards greater invariant mass as demonstrated
in Figures 5.10(a), 5.9(a), 5.8(a), and 5.16(a) for Q2 = 2.5 GeV2.
The inelastic contribution will be deconvoluted using a Monte Carlo (MC).
This Monte Carlo is described in section A.1 and simulates the quasi-elastic and
inelastic spectrum near the quasi-elastic spectrum. Using the inelastic counts un-
der the quasi-elastic peak and the inelastic asymmetry from the simulation, the
measured quasi-elastic asymmetry can have the contribution from inelastic events
removed [82]. This Monte Carlo is being analyzed by another collaborator. When
this is completed, the invariant mass cuts in Table 5.5 can be widened to 1.2 GeV.
The contribution of the inelastic sample can be empirically estimated by con-
sidering the asymmetry as a function of invariant mass. This shows the asymmetry
both in the inelastic region, the quasi-elastic region, and the region with a sig-
nificant inelastic contribution to the quasi-elastic events. This is demonstrated in
Figure 5.35.
Preliminary values for the inelastic asymmetry contribution and dilution are
146
Figure 5.35: The asymmetry as a function of invariant mass. The plotted curve is
the expected change due to the acceptance of the detectors. This shows that the
invariant mass doesn?t vary much over the region selected (see Table 5.5).
presented in Table 5.9. The presented values are calculated in the framework of a
separate analysis with a different set of cuts and charge selection [81]. These are
not included in the analysis presented here, but give an estimate of the contribution
in the quasi-elastic event sample. The correction to the physical asymmetry due to
the inelastic asymmetry and dilution is
Acorr = Aphys ?AinelasD
inelas
. (5.59)
As seen, with the cuts presented in Table 5.5 the contribution of the inelas-
tic events is small. Effects due to two photon exchange are also expected to be
small, with less than a 5% suppression [17]. The likeliest source of a change in the
results presented in Table 5.8 is due to increased theoretical understanding of 3He
nucleus. The results presented and discussed in Chapter 6 will be based on the
results presented in Table 5.8.
147
Q2 Dinelas Ainelas Aphys
1.7 GeV2 0.9839 -0.0038 -0.254
2.5 GeV2 0.9834 -0.0030 -0.191
3.5 GeV2 0.9726 -0.0050 -0.125
3.5 GeV2* 0.905 -0.0143 -
Table 5.9: Shown are preliminary inelastic asymmetries and dilutions for the various
kinematic [82]. The * row relates to an expanded selection of quasi-elastic events,
with a maximum invariant mass of 1.15 GeV rather than the usual maximum of
1.05 GeV and a maximum missing mass of 2.2 GeV rather than 2 GeV (see Table
5.5).
148
Chapter 6
Results and Discussion
The results presented in Table 5.8 are shown in Figure 6.1 along with selected
theoretical curves and previous measurements using polarized techniques. Several
models, including ones in the GPD, VMD, diQuark, and CQM frameworks, show
agreement with the new measurements presented in this thesis. Further analysis
of the Q2 = 1.3 GeV2 point, and the measurement near 1.5 GeV2 at Mainz, will
hopefully shed light on the possible discrepancy of the measurements near 1.5 GeV2.
The form factors provide inputs for interpolations, including that by Guidal
et. al. This model, which will be described below, can be refit using the new data
for the electric form factor of the neutron in addition to other form factor data (in
section 6.4.2). Within the GPD framework, the quark orbital angular momentum
can be calculated using Ji?s Sum Rule [54]. This is done in section 6.4.3.
It is common in the literature to use t = ?Q2 = ??2 rather than Q2 when
talking about fits and models. This practice will be followed in this chapter. Addi-
tionally, unless otherwise mentioned, the QCD scale evolution parameter is ?2 = 1
GeV2.
The form factors have been fit many times [18] [59] [76]. For the magnetic
form factors, the simplest fit is proportional to the dipole form factor (equation
3.2). As described in section 3.1.1 the electric form factor of the neutron has long
149
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
n EG
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1 Ostrick
MadeyWarren
BeckerBermuth
RoheGeis
E02-013
PRELIMINARY
VMD - Bijker and IachelloVMD - Lomon (2002)
q(qq) Faddeev -  I. Cloet, ANLCQM - G. Miller
Miller (q-only)
GPD - Guidal2)/Q2?/2(Q2 ln? 
1/F2FGalster fit (1971)
Figure 6.1: The electric form factor of the neutron with values presented for various
experiments including the one presented in this thesis. Also included are represen-
tative models of the CQM [71], VMD [15][65], GPD [28][45], and diQuark variety
[26]. In black is the traditional Galster parametrization. Circled is the measurement
at Q2 = 1.3 GeV2 which is not included in this thesis.
150
been parametrized by the Galster fit. This fit was recently modified [59] to
GnE = A?1+B?GD . (6.1)
Parameter Previous Work[59] This Work
A 1.7 1.4?0.1
B 3.6 1.6?0.5
Table 6.1: Given here are previous fit parameters [59] and a new fit presented in
this thesis. The fit only includes data from experiments using double polarization
techniques.
6.1 Framework Formalism
As noted in section 1.3, QCD cannot be solved analytically in the regime
where the quarks can be described as confined. The theory is also difficult to discuss
analytically since Lorentz boosts of such a wavefunction is as complicated as solving
the theory. By putting the problem into the Light Front gauge, a simpler description
can be discussed. In Light Front Quantization, QCD is quantized at a fixed light
front time ? = t+ zc giving the following quantities of
P? = P0 ?PZ (6.2)
and vectorP? as the new four-momentum. Here P+ is the longitudinal momentum and
vectorP? is the transverse momentum. The wavefunctions of this formulation describe
the nucleon in an arbitrary moving frame. To describe the quarks and gluons within
the nucleon
p+i = xiP+ , (6.3)
151
and
vectorp?i = xiP? +vectork?i , (6.4)
are used where xi = p+iP+ and vectork?i represent the relative momentum coordinates. The
parton model is an approximation in a general reference frame, but is exact in the
Light Front quantization. The transferred momentum in this frame is q+ = 0,
q? = 2q?PP+ and vectorq?. Obviously Q2 = ?t = vectorq2? and is often referred to as ?2?.
In non-relativistic physics the natural framework for discussion of quantities
is the center of mass; in the relativistic framework of light front QCD the natural
framework is the center of transverse momentum, vectorR? [22]. The obvious, parton,
definition is
vectorR? = X
i
xivectorr?i . (6.5)
The conjugate position to the momentum vectork?i is vectorb?i . It is in terms of this position
that the distribution of the quarks can be described. Obviously Pivectorb?i = 0 . In this
mixed reference frame with longitudinal momentum and transverse position, the
GPDs have a density interpretation. The probability to find a quark with x andvectorb?
minus the probability to find an antiquark is given by
qv(x,vectorb?) =
Z d2?
4pi2 e
?vectorb? ??Hq
v(x,t = ??
2) . (6.6)
Using equations 1.1, 1.2 and 6.6, the parton charge density in this mixed
reference frame is [70]
?0(b) =
X
q
eq
Z
dxq(x,vectorb) =
Z d2q
4pi2F1(Q
2 = ?2)ei??vectorb . (6.7)
152
b (fm)0 0.2 0.4 0.6 0.8 1
?
-1
-0.8
-0.6
-0.4
-0.2
0
Figure 6.2: Shown is the transverse density in the neutron. This is with the magnetic
form factor described as a dipole, and the electric form factor described by the fits
in Table 6.1. Here + is the previous fit and ? is the new fit, the transverse density
for these fits is almost identical.
This integral can also be expressed in terms of the Sachs form factors
?(vectorb) =
Z ?
0
dQQ
2pi J0(Qb)
GE(Q2)+?GM(Q2)
1+? . (6.8)
Using the dipole parametrization of the magnetic form factor, and the electric form
factor parametrized in equation 6.1, the transverse density in this mixed frame is
shown in Figure 6.2.
6.2 Results
6.3 Pauli and Dirac Form Factors
As presented in equation 2.8, the Pauli and Dirac form factors can be related
to the Sachs electromagnetic form factors. To calculate Fn1 and Fn2 , only GnE needs
153
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
n 1F
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Previous Data
E02013PRELIMINARY
Galster fit (1971)
Figure 6.3: The Dirac form factor, Fn1 , using measured values of GnE and the dipole
approximation for GnM. Shown in black is the Galster parametrization.
to be determined due to knowing GnM at Q2 < 4 GeV2. As can be seen in equation
2.8, F1 has a stronger dependence on the electric form factor than does F2. Using a
dipole approximation for GnM
GnM = ?nGD , (6.9)
it is possible to determine the Pauli and Dirac form factors using only electric form
factor data. These are shown in figures 6.3 and 6.4.
-
6.3.1 Flavor Form Factors
If the following assumptions are made, the flavor form factors can be deconvo-
luted from the nucleon form factors. These flavor form factors are the form factors
of a flavor within the nucleon, not of a specific quark. Assuming that Iso-spin sym-
154
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
n 2F
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Previous Data
E02013
PRELIMINARY
Galster fit (1971)
Figure 6.4: The Pauli form factor, Fn2 , using measured values of GnE and the dipole
approximation for GnM. Shown in black is the Galster parametrization.
metry is conserved at Q2 > 0, the neutron can be considered to be a proton in which
the down quarks switch places with the up quarks. For the proton, since the charge
of the proton is defined to be 1 (see equation 2.7), the equation for the proton form
factors in terms of the flavor form factors is
Fp1,2 = 23Fu1,2 ? 13Fd1,2 . (6.10)
The formula for the neutron is similar,
Fn1,2 = ?13Fu1,2 + 23Fd1,2 . (6.11)
For both the magnetic form factors of the neutron and proton, the dipole
approximation is used. For the electric form factor of the proton, a modified dipole
is used [76]
GpE = (1.06+0.14t)GD . (6.12)
155
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
u 1/F
d 1F
0
0.1
0.2
0.3
0.4
0.5
0.6 Previous Data
E02013
PRELIMINARY
Galster fit (1971)
GEn = 0
Figure 6.5: Ratio of the down and up quark flavor Dirac form factors. The form
factors not dependent on GnE are just dipole approximations. The black curve is for
GnE in the Galster parametrization, the dotted curve is for GnE = 0.
This allows the Pauli and Dirac form factors for both the proton and the neutron to
be calculated, and the flavor form factors deconvoluted, with uncertainties assumed
to be coming only from the neutron electric form factor data. Theorists predict the
strange form factor is not different from 0 for these nucleons. There are no data
available beyond Q2 = 1 GeV2. There are many theoretical estimates, the majority
of which predicts a strange contribution that is negligible relative to the contribu-
tions of the up and down quarks. It is therefore ignored in this deconvolution.
The deconvolution into the flavor form factors is presented in Figures 6.5 and
6.6.
156
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
u 2/F
d 2F
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
Previous Data
E02013
PRELIMINARY
Galster fit (1971)
GEn = 0
Figure 6.6: Ratio of the down and up quark flavor Pauli form factors. The form
factors not dependent on GnE are just dipole approximations. The dotted curve is
for GnE in the Galster parametrization, the blue curve is for GnE = 0.
6.4 GPD Models
Generalized parton distributions are a popular new framework in which to
consider nuclear interactions and structure. These were discussed earlier in sections
3.1.4 and 1.3.3. The GPDs parametrize the non-forward pieces of the light-front
operators. They have the advantage in being able to describe, in part, elastic pro-
cesses, Deep Inelastic Electron Scattering, Deeply Virtual Compton Scattering, and
physical properties like quark orbital angular momentum. In terms of the nucleon
form factors, the GPDs give the quark contribution at a given x to the flavor form
factors:
Fq1(t) =
Z 1
0
dxHq(x,?,t) (6.13)
Fq2(t) =
Z 1
0
dxEq(x,?,t) . (6.14)
157
There are two other GPDs, which are related to the pseudoscalar and axial
form factors. These are not really affected by electric form factor measurements,
and are more difficult to constrain by experiments.
6.4.1 Impact Parameter Space
Because GPDs divide the momentum into longitudinal and transverse compo-
nents, t can be transformed (similar to a Fourier transform) into position space to
provide a description of the nucleon in terms of an impact parameter. This impact
parameter is the distance from the center of momentum of the nucleon. From this,
the distribution of the flavor can be determined for different longitudinal momentum
fractions (x). The impact parameter space versions of the GPDs are:
Hq(x,vectorb) =
Z d2vector?
(2pi)2e
?ivectorb?vector?Hq(x,?vector?2) (6.15)
Eq(x,vectorb) =
Z d2vector?
(2pi)2e
?ivectorb?vector?Eq(x,?vector?2) . (6.16)
This Hq(x,vectorb) is equivalent to q(x,vectorb) the distribution of partons as a function of
the transverse position vectorb from the center of momentum and light-cone momentum
fraction x.
Becausetheimpactparameterspacecorrespondstolocationswithinthehadron,
the density interpretation of GPDs is used to create conditions that must be satisfied
by the GPDs. For example, giving a relation between the impact parameter and the
longitudinal momentum fraction, any nucleon is going to have the allowed region of
impact parameter space go to 0 as the longitudinal momentum fraction goes to 1
158
[45] [24]. It has been shown that H and E in impact parameter space satisfy [45]
1
2|?vectorbE
q(x,vectorb)|? Hq(x,vectorb) . (6.17)
This requirement provides an important constraint to possible GPDs.
The probability interpretation of Eq(x,vectorb) is in the transverse basis rather than
the longitudinal basis. The distribution is then
qXv (x,vectorb) = qv(x,vectorb)? b
y
m
?
?vectorb2E
q
v(x,vectorb) . (6.18)
where qXv (x,vectorb) is the probability to find an unpolarized quark with momentum
fraction x and impact parameter vectorb in a proton polarized in the x-direction minus
the probability to find an antiquark.
6.4.2 Fit to our data
As mentioned previously (section 3.1.4), many of the models in the GPD
framework[28][45]haveparameterswhicharedependentonthenucleonformfactors.
Since 2005, new data on the nucleon form factors, in particular the neutron electric
form factor presented in this thesis. This allows for new fits.
The models for the helicity conserving GPD take the form of
H = qv(x)efq(x)t (6.19)
where qv(x) is the quark density function. This function is acquired from global
fits to DIS data and other data, the fits used by these models were the MRST2002
(Guidal et al.[45]) and the CRET (Diehl et al.[28]). New data are also available for
these fits [67].
159
The form of fq(x) is another parameter. Diehl, et al. fit the nucleon data
and use the Regge phenomenology to get a class of fits. Guidal, et al. use Regge
phenomenology, which shows behavior that is close to correct both as x ? 1 and
x ? 0. Guidal et al. use
fq(x) = ?primeln
 1
x
?
(6.20)
while Diehl et al. use
fq(x) = ?prime(1?x)2ln
 1
x
?
+Aq(1?x)2 +Bq(1?x)3 . (6.21)
Diehl et al. argue that expecting fq(x) to be described at high t by ? does not work,
since Regge phenomenology does not apply at that regime.
The helicity flip GPD has an additional component meant to change its be-
havior as a function of t compared to H. It should drop as 1t or faster, and so Guidal
et al. solve this by putting in a term (1?x)?q. This gives the E GPD as
E = N?qqv(x)(1?x)?q efq(x)t (6.22)
where N?q is a normalization constant to account for the added term.
Diehl et al. use a different profile function,
gq(x) = ?prime(1?x)2ln
 1
x
?
+Cq(1?x)2 +Dq(1?x)3 (6.23)
which gives the E GPD as
E = ?(2?? +?q)?(1??)?(1+?
q)
?qx?? (1?x)?q etgq(x) . (6.24)
The previous fit of Guidal et al. and a new fit using the measurement of the
electric form factor of the neutron presented in this thesis are presented in Table
160
]2  [GeV2Q1 2 3 4 5 6
p M/G
p E G
p?
0
0.2
0.4
0.6
0.8
1 GPD - Guidal
GPD - New Fit
Figure 6.7: The combined form factors of the proton (?GpEGn
M
) with values presented
for various experiments. Included are the calculations of the electric form factor in
the GPD framework with the fits as described in Table 6.2.
6.2. In figures 6.7, 6.8, and 6.9 the success of the new fit is demonstrated. This new
fit uses the MRST2008 NNLO fit for the valence quark densities [67].
Parameter Previous Work [45] This Work
?u 1.34 1.68
?d 1.34 0.60
Table 6.2: Given here are the fit parameters from the previous work [45] and a new
fit presented in this thesis.
6.4.3 Quark Orbital Angular Momentum
The quark orbital angular momentum can be calculated in the GPD framework
using Ji?s Sum Rule [54]:
2Jq = ?q +2Lq =
Z 1
?1
xdx[Hq(x,0,0)+Eq(x,0,0)] . (6.25)
161
]2  [GeV2Q0.5 1 1.5 2 2.5 3 3.5 4
n EG
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1 Ostrick
MadeyWarren
BeckerBermuth
RoheGeis
E02-013
PRELIMINARY
GPD - Guidal
GPD - New Fit
Figure 6.8: The electric form factor of the neutron with values presented for various
experiments including those presented in this thesis. Included are the calculations
of the electric form factor in the GPD framework with the fits as described in Table
6.2.
x0 0.2 0.4 0.6 0.8 1
)-1
 (GeVb
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 6.9: Shown are the upper bounds of the value for b. These bounds are
determined by applying the condition in equation 6.17 [45]. The solid line is for the
Up quark while the dotted is for the down quark.
162
In Table 6.3 these quantities are tabulated, including the integrals that depend on
the GPDs. In this table, the values for the original fit are presented along with
values for the new fit.
Quark ?q R H R E 2Jq ** 2Jq * 2Lq ** 2Lq *
u 0.6 0.37 0.238 0.595 0.583 -0.01 -0.017
d -0.25 0.20 -0.207 -0.031 -0.059 0.219 0.191
s - 0.04 0 0.04 0.04 - -
total - 0.61 - 0.568 0.524 - -
Table 6.3: Presented is a calculation of the quark orbital angular momentum using
Ji?s Sum Rule [54] in the framework of [45]. Compared is this previous work (**),
with a new fit (*) using the form factor results presented in this thesis.
6.4.4 Conclusions
With the advent of double polarization measurements, the structure of the
nucleon is being probed in fine detail. The neutron electric form factor, presented
in this thesis, provides not just a parametrization of the nucleon structure but, with
a few assumptions, provides knowledge about the flavor structure and quark orbital
angular momentum. The new measurements of this form factor more than double
the range of Q2 covered. These new measurements show agreement with many of
the models, but not with pQCD scaling scaled to the previous data.
A fit to GnE using these new data shows a decrease in the density of the neutron
at small vectorb compared to previous work. Using the Sachs form factor data and the
dipole approximation of the Sachs magnetic form factor, gives the Pauli and Dirac
form factors which shows the expected behavior. Deconvoluting these into the flavor
163
form factors gives
fd1
fu1 ? Constant (6.26)
and
fd2
fu2 ? Q
2 (6.27)
behavior. Refitting a model to determine GPDs allows the calculation of quark
orbital angular momentum. This shows a slight increase in the contribution to the
total angular momentum of the nucleon from the down quark.
New analysis and experiments are needed, both to increase precision at low
Q2 and to show pQCD scaling at higher Q2. The small effect of FSI and the near
100% neutron polarization seen in polarized helium-3 makes double polarization
experiments using polarized helium-3 ideal for future measurements of this form
factor.
164
Appendix A
Neutron Arm Analysis
The neutron arm was developed to detect the outgoing neutron in coincidence
with the electron in the interaction 3??He(e,eprimen). The neutron arm?s position, active
area, and sheilding were designed to match the electron spectrometer, Big Bite, at
the highest measured value of Q2 and to provide the highest possible neutron detec-
tion efficiency at this value [25]. See Table 4.1 for the kinematics and positioning of
Big Bite and the neutron arm.
Building upon the discussion in section 4.6, the neutron detector, known as
Big Hand, was a large detector made up of 244 neutron bars totaling 976 ADC and
TDC channels and an additional 192 single sided veto bars. In addition, there were
four ?marker? counters which were placed between the neutron detectors and the
veto detectors and used for calibration. Neutrons are identified using the vertical
position, horizontal position, time of flight, and the existence of a nearby hit in the
veto detectors. The neutron detector was highly segmented to improve the vertical
(x) resolution. The horizontal resolution and the time of flight resolution of the
neutron arm was improved by a time calibration of all the bars, with the ?marker?
counters playing a crucial role in the horizontal calibration.
165
A.1 Simulations
A detailed simulation was undertaken of the neutron arm using Geant 4 [49].
Its goals were to understand the inelastic contribution in the invariant mass and
time spectra, to simulate a value for the proton to neutron conversion between
the target and neutron bars, and to determine the result of the threshold change
for the on this conversion. Additionally, the simulation provides neutron detection
efficiency, hadron detection efficiency, and veto efficiency numbers that are needed
to understand the neutron arm.
The simulation was designed to include all the material in between the target
and the neutron bars in order to properly simulate the events within the neutron
detector. This included the iron target box, the target cell, and the material for
the target ladder. A document was prepared that shows the components of the
target system [57]. On the neutron arm side, all bars, PMT light guides, shielding,
cassettes, as well as detector response were included. Details of the geometry were
documented by Tim Ngo [73]. The Big Bite acceptance was modeled using Big Bite
geometry, a model for the vectorB field, and the total shower threshold and resolution,
and is in good agreement with observed data.
Quasi-elastic scattering was simulated in the impulse approximation, with 3He
momentum distributions from [90]. Parameterizations of the nucleon form factors
[59] were used to calculate elastic cross sections and asymmetries. Simple pion
production is the dominant inelastic contribution for these kinematics, with cross
sections and asymmetries provided by the MAID parameterization [32]. External
166
radiative effects caused by objects in the vicinity like the beamline were included
and internal radiative effects were modeled 1 [82] [50].
The neutron arm response was simulated using the events above on an event-
by-event basis with a GEANT4 simulation. Additional software was developed to
reproduce PMT signals and electronics responses, and mimic the NA trigger. The
simulated data then had the virtual thresholds changed to match the real data, and
the virtual data were processed similarly to the real data [82]. This Monte Carlo
was developed by Seamus Riordan.
The simulation was used primarily as method to understand the purity factor
and the inelastic contribution to the quasi-elastic sample. The purity factor (Dn) is
discussed in section 5.5 and the inelastic contribution in section 5.7.2. The simula-
tion also gives proton and neutron detection efficiencies, shown in Table A.1. These
are lower than what was predicted.
Q2 = 1.7 GeV2 2.5 GeV2 3.5 GeV2
proton detection efficiency 0.411 0.270 0.210
neutron detection efficiency 0.345 0.250 0.165
Table A.1: A table showing the proton and neutron efficiency as calculated using
the Monte Carlo Simulation.
A.2 Calibrations
To achieve the desired resolution needed to identify quasi-elastic events within
the neutron arm, all bars within the neutron arm (NA) were calibrated relative to
1Model by Mo and Tsai using the peaking approximation.
167
each other. For the measurement at Q2 = 3.5 GeV2 the required parallel momentum
resolution was 0.250 GeV which corresponds to a time of flight resolution of less than
500 ps [25].
During the commissioning period, the descriminators were set, the cables were
emplaced, and the high voltage (HV) for the detector bars was adjusted. The
threshold for the NA and the HV for the individual PMTs were adjusted over the
course of the experiment. This created a number of different ?periods? with different
neutron arm detection efficiencies. Changes in high voltage are shown in Table A.2.1.
The remainder of this section will be a description of the calibrations necessary to
get the desired NA resolution.
The desired situation was that the neutron arm provided a precise time of flight
resolution. As explained, this was necessary for quasi-elastic event selection and
neutron identification. The time calibration of the neutron arm not only acheived
a time resolution of less than 500 ps, but also acheived an accurate and precise
horizontal position. The tune of the high voltage, done numerous times over the
course of the experiment, provided ADC signals that were the same within 10% for
cosmic type events. Differences in the electronics and cables cause the signal read
out of the TDC to have a delay unique for each PMT.
The signals used for the HV tune, those from cosmic sources, were similar
to each other. However, signals from quasi-elastic events were different from these
signals used for HV tune. Due to this, events taken during running conditions
created dissimilar signals within the ADC and caused the discriminator to trigger at
a different point in the amplitude spectrum compared to events from cosmic sources.
168
Due to these dissimilar signals, for the best resolution an amplitude correction was
required in addition to the time offset calibration. A difficulty with using particles
from cosmic events was that the path they travel through the neutron arm was not
the same as events caused by interactions in the target. This caused a differently
shaped peak which was broader than the peak due to hadron events.
To account for the differences in delay and electronics, a time offset calibration
was done for each PMT. An amplitude correction to the mean time and horizontal
position within the bar corrected differences caused by differences in signal shape.
A.2.1 High Voltage Tune
The HV for the counters was set so that each neutron counter had similar
behavior during operation. The signals should be well behaved, meaning above
threshold but below saturation, for hadronic events, and for other events which are
expected within the detector during normal operation. This allowed the analysis to
treat all bars similarly when calculating time of flight, identifying charged particles,
and calculating perpendicular missing momentum (due to the calculated hit position
within the neutron arm), and the other quantities needed for quasi-elastic neutron
identification. A few detectors stopped being well behaved during the course of the
experiment. These were mostly noted and disconnected during the experiment.
The HV tune was done prior to the start of the experiment using cosmic
particles. All cosmic rays have similar signatures within the detector, but their
signals were not so uniform as those from protons and neutrons from the target.
169
Because of high voltage (HV) drift and the difficulties in calibrating all photo-
multiplier tubes (PMTs), a threshold of a 10% difference from the average was used
to determine whether the PMT?s HV needed adjustment. During the experiment
cosmic events continued to be used to adjust the HV when beam was not available,
if a PMT drifted outside of this 10% threshold.
A script was developed for online monitoring and adjustment of the high volt-
age. This allowed for particular HV settings to be saved and for the high voltage
of individual bars to be turned on and off as needed. It also allowed the resetting
of HV during a trip or other event. Each colored set of bars in figure 4.19 used the
same modules for the high voltage, amplifier, discriminator, and other electronics.
This high voltage script was written by Eugene Chudakov.
A script was created to realize the HV calibration2. The resulting files were
fed into the previously described script to change the high voltage. Diagnostic files
showing the fitted spectra for the PMTs were produced, with spectra divided into
files based on counter type, plane, and side of the counter.
In this HV calibration script, long cosmic events were selected. These were
events which were detected in a bar and its vertical neighbors. The amplitudes were
tuned towards a common goal (G), which was determined based on the height of
the bars relative to each other. This was 1000 channels for the UVA and GLA bars,
and 1500 channels for the CMU bars. Two different techniques were used to set
2Written by Igor Rachek, and modified by Jonathan Miller, Rob Feuerbach, and Pavel Deg-
tiarenko.
170
the gain, the first 3 using simply the amplitude of the bars. The peak for minimum
ionizing particles was directly compared to the goal to determine the gain to apply.
A second technique 4 used logorithms. This used two pieces of information
determined from amplitude, a mean (AM = lnAL+lnAR2 ) and a difference (AD =
lnAR ? lnAL). Here L is used for left, R for right, and A is the amplitude while
T is the time. In this case the gain was determined by fitting AD and AM, and
was given by e?lnG+AM?0.5AD where +(?) was for the right(left) gain. This second
technique was used during data collection.
A.2.2 Time Offsets
The resolution of the time of an event in the neutron arm (NA) is a convolution
of the alignment of the bars with the internal resolution of the bars. The internal
resolution of the bars provides a minimum resolution for the experiment (individual
PMT amplitudes can be used to improve the internal resolution of the bars). The
time of an event in a neutron bar or counter is just the mean time as determined
from the PMTs.
The core concept for the time offset calibration of the neutron arm involved
an ?alignment? of the times internal to the neutron arm. To call two times aligned,
the spectra of two (often neighboring) bars would be shifted to coincide after the
relevant corrections were applied. Alignment means that the spectrum in a bar,
after corrections for position within the NA and after selecting hadronic events, will
3Developed by Igor Rachek.
4Developed by Pavel Degtiarenko.
171
be similair to others bars after such corrections. For example, if a particle passes
through two neighboring bars, they are aligned if the time that the particle passed
through the bar is the same when corrected for the distance between the two bars
that the particle traveled.
Three steps were used to acheive the offset calibration of the PMTs. First, the
left times and right times were roughly aligned together. This allowed the left and
right times to be associated properly together to form a mean time and time differ-
ence of the bar (left time minus right time) which was used to give the horizontal
position of the hit. In the second phase of the calibration, the peaks for the mean
time and the time difference were aligned for neighboring bars. This allowed clusters
of hits to be created (known as clusters in the data), which are hits in neighboring
bars with similar mean times and are identified as coming from the same event.
In the final calibration, these clusters were used to improve the calibration using
charged events (in 3He). This only was used to do the final calibration of the first
plane, later planes had their mean time and horizontal position calibration improved
using events which travel deep into the neutron detector. The first plane also had
its horizontal position calibrated using the known information of the ?marker? bars.
The best understood particles that create events within the neutron arm
are protons (from H2), although muons will not deviate noticeably while travel-
ing through the detector. Initially the calibration of the neutron detector was kept
entirely internal to the neutron detector. In the final calibration the front plane was
calibrated relative to Big Bite using quasi-elastic protons (from 3He), with the later
planes calibrated relative to the first plane (N1).
172
The PMTs in each neutron bar determine two main quantities. These are
the mean time of the hit tmean and the horizontal position of the hit ypos. The
horizontal position of the hit is determined by the difference in time, tdiff, between
the measured signals in the two PMTs for a hit. These two quantities, only corrected
for offsets, are
tmean = A((tl,raw ?tl,offset)+(tr,raw ?tr,offset)) , (A.1)
tdiff = B((tl,raw ?tl,offset)?(tr,raw ?tr,offset)) . (A.2)
In these relationships tl,raw is the left raw time (from the TDC) and tl,offset is the
software offset applied to the left TDC value. Here A and B are constants that are
about 12 and defined to be so in all calculations.
During the course of the experiment the high voltage was changed many times.
Although this was the case, high voltage changes did not change values of the
amplitude by much more than 10% and so rough values for the mean time and time
difference for a bar will be independent of the high voltage setting. Because of this,
the initial calibration was done once, with later calibrations fine tuning the initial
calibration after each high voltage change.
The first improvement in calibration was acheived by comparing two bars in
neighboring planes and subtracting their mean times [37]. This created a peak for
events when there was a hit in both planes at that vertical (x) position. This peak
was then adjusted to give the proper time of flight for near c particles and so all
later planes were calibrated relative to the first plane (N1). The horizontal position
calibration was realized by using the time difference in neighboring bars. This was
173
done similarly to the mean time calibration. The vertical position within a plane
was calibrated using cosmic events.
The central problem with this technique is that cosmic events might not travel
through the entire neutron bar; the shower from the cosmic might start or end within
the bar. This is different than during running conditions, because the converter layer
in front of each neutron scintillator bar results in most detected hadrons beginning
interaction directly in front of the scintillator paddle where detection occurs.
Thenextimprovementtothecalibrationwasdonebyusingabetterselectionof
events. These better events were proton and muon events and selected by selecting
long events that pass through the neutron arm, where a particle lost little of its
momentum prior to being observed in the NA.
The high detection efficiency of protons and muons results in a cleaner spec-
trum for individual bars, and calibration using these events was better. This im-
proved internal calibration calibrated the later planes relative to the first, by shifting
tcentral to 0:
tcentral = t1st ?tNth ?t1st?Nth . (A.3)
Here t1st refers to the mean time of the first plane the particle passed through, and
tNth refers to the mean time of the nth plane the particle passed through (at the
same x as the first hit). Subtracted from this difference is the time for the particle
to travel between the first and nth planes (t1st?Nth). The time between planes was
calculated using the distance between the planes, and cn to approximate the speed
of a relativistic particle (since the particle is unknown). These well behaved events
174
p1b20tof
Entries  319
Mean   -0.03704
RMS      3.22
 Time(ns)-20 -15 -10 -5 0 5 10 15 20
0
5
10
15
20
25
30
35
40
Plane 1 Bar 20 TOF
Figure A.1: Calibration of the mean time for plane 1 bar 20. This shows the
difference between the time of flight and the expected time of flight peak for quasi-
elastic protons between the BigBite scintillators and the front plane of the neutron
arm.
were also used to calibrate the horizontal position of the hit within the Neutron Arm,
aligning all later planes to the first plane. This completes the internal calibration.
Figures A.3 and A.4 demonstrate this relative horizontal position and mean time
calibration.
The final improvements to the neutron arm calibration were to achieve an
accurate horizontal (y) position and to determine the overall offsets for the vertical
position (x) using particles. The best understood particles within the neutron arm
(NA) are elastic protons from hydrogen, which form well defined peaks in the mean
175
p1b21lr
Entries  665
Mean   -3.393
RMS     3.634
 Time(ns) -20 -15 -10 -5 0 5 10 15 20
0
5
10
15
20
25
30
35
40
Plane 1 Bar 21 LR
Figure A.2: Calibration of the horizontal position for plane 1 bar 20. This histogram
shows the peak in the horizontal bars time difference, with coincidence with a hit
in the ?marker? counters.
176
time spectrum of bars in the first plane (N1). After corrections (detailed in equation
5.9) based on the position of the hit within the neutron arm and the time of the
event within Big Bite, these spectra were aligned:
Tc,TOF = t1st +tL1A ?te ?th ?tpeak +trf . (A.4)
Here Tc,TOF refers to the centered time of flight, t1st refers to the mean time as
measured in plane 1 (N1), tL1A refers to the level one accept time (discussed in
section 4.7), te refers to the time for the electron (discussed in section 5.1), th is the
expected time for the hadron to travel to the neutron bar (discussed in section 5.3),
tpeak is the time for the hadron to travel to the neutron bar, and trf is a correction
to the known time of interaction due to RF information about the beam (discussed
in section 5.5.3). The spectrum for the time of flight relative to the expected time
of flight for protons in 3He is presented in Figure A.2.2.
The neutron arm was placed to measure the maximum number of quasi-elastic
events as possible in coincidence with Big Bite, not to measure as many elastics as
possible. Because of this, elastic events did not cover the entire face of the neutron
arm, but only a portion of it. To calibrate the entire detector, quasi-elastic events
were used rather than the cleaner elastic sample. These still gave a tight time of
flight peak and have events which pass through the detector.
Four ?marker? counters were installed vertically within the neutron detector.
These counters had poor resolution and were not meant to be used to determine
the vertical position of a hit within the neutron bar. These counters were used to
perform an absolute calibration of the horizontal position for all bars in the first
177
plane. This was done by using the coincidence between a hit in the marker counters
and a hit in the neutron arm for a given event. For a single marker PMT the number
of counts was small. By using the geometry information for the marker counters
[72], the coincident peaks for all four marker counters were combined (ycenter =
ydiff ?ymarker), providing the necessary statistics. This ycenter is the position within
the neutron arm for hits where there is a nearby well-timed marker bar hit. Here
ycenter is the peak location, ydiff is the original position along the bar, and ymarker
is the geometric position of the marker counter. In Figure A.2.2 is an example of
the resulting spectrum.
The results of the calibration are summarized in Figure A.5 using a single
plot showing the main points of calibration: the mean times of planes N2 through
N7, the time of flight for N1, and the ycenter position for the absolute horizontal
calibration. In this plot, some of the bars did not have high enough statistics to
form a proper peak and are shown as poorly calibrated. These bars have greater
uncertainty, and so for these the respective histograms are checked by eye.
A.2.3 Amplitude Correction
The amplitude spectrum was discriminated to provide a clear time for the
TDC. However, since there was a threshold for this discrimination, the time mea-
sured by the TDC is depended on the shape and size of the amplitude. The mean
time and horizontal position for an event were corrected for this amplitude depe-
178
p2b12diff
Entries  1938
Mean   0.03566
RMS    0.9385
 Time(ns)-20 -15 -10 -5 0 5 10 15 20
0
50
100
150
200
250
300
350
400
450
Plane 2 Bar 12 Diff
Figure A.3: Calibration of the mean time difference for plane 2 bar 12. This is the
difference between neighboring bars in different planes. In particular, this is the
difference between plane 1 bar 12 and plane 2 bar 12.
179
p2b12lr
Entries  1938
Mean   -0.0006288
RMS    0.7316
 Time(ns) -20 -15 -10 -5 0 5 10 15 20
0
100
200
300
400
500
Plane 2 Bar 12 LR
Figure A.4: Calibration of the relative horizontal position for plane 2 bar 12. This
is the horizontal position (transformed to time) of the hits in plane 1 bar 12 and
plane 2 bar 12 for events with well timed hits within the two bars.
180
Time Difference (ns)0 1 2 3 4 5 6 7 8
Bar
5
10
15
20
25
30
35
40
45
Summary of Calibration Results
Figure A.5: Calibration summary plot for run 3975 after pass 3 calibration. In this
plot, X is the result, + is the before calibration. On the x axis, 0 is for plane 1 and
1-6 show the differences between planes 1 and planes 2-7). At 7 on the x axis is the
horizontal absolute position calibration for plane 1.
181
dence. This relationship for the initial amplitude calibration was
tcorr = ti ?Cpar(Aadc)?1 +Cpar(Aref)?1 . (A.5)
In the initial calibration the reference paramters, Aref, were defined to be 0. The
values for Cpar were determined individually for each PMT.
This initial calibration was improved using a model where the amplitude cor-
rection was a function of the hit location (?Y) within the bar. In this model, the
amplitude dependent behavior should be the same for both sides of the bar. Ad-
ditionally, because the threshold was set to be the same for all bars, this behavior
should also be independent of which particular bar was hit. The correction can be
approximated as (with a A and ?Y dependence as information from the PMT):
(C1 +D1?Y)2
A ?
C2 +D2?Y
A . (A.6)
The desire is to determine the constants which make a narrow peak in the
time of flight between the Neutron Arm and Big-Bite. To do this Minuit [30] was
used to minimize the following:
Tmean + C22
 1
AL +
1
AR
?
+ D22
?
Y +YL + YL+YR2
AL +
Y +YR + YL+YR2
AR
!
(A.7)
here Y is the horizontal hit location in the neutron bar, YR is the position of right
PMT, and YL is the position of the left PMT. Y is obviously dependent on both C
and D but this is ignored as higher order.
While there were many quasi-elastic events in which the interaction was in
the first plane, and some where it was in the second plane, the later planes were
less often the plane of first interaction. By using quasi-elastic events the first two
182
planes were corrected for amplitude dependence. These bars were compared to bars
in later planes to fit their amplitude dependence. To do this, the following formula
was used, where TN+2 is the time of flight for a hadron in the desired plane,
TN+2 = 2TN+1 ?TN . (A.8)
A.2.4 Energy Calibration
The energy calibration of the neutron arm was not used in this analysis. Initial
efforts to do this are presented in Tim Ngo?s thesis [72].
A.2.5 Veto Calibration
The veto was calibrated relative to the neutron arm first plane (N1). This
calibration was acheived by using events with coincident hits in two veto bars with
similar x position in both veto planes. For example, a left(right) veto bar was
calibrated relative to a left(right) bar in the other plane. Doing this step by step
the entire veto was internally calibrated, using the relative shift between the two
veto planes. Figure 4.19 shows that each veto bar covers vertical space with two bars
on the other veto plane. These planes were also offset horizontally relative to each
other, the coincident region allowed left times and right times to be aligned. This
provided an overall, intra-veto calibration and alignment. A central veto detector
provided an overall offset compared to the neutron plane one (N1) using coincident
events between it and a neutron bar, completing the initial veto offset calibration.
This initial calibration was improved by looking at long events and aligning
183
the resultant peaks in the veto to those in the neutron detectors at corresponding
x position. The horizontal position within the bar can be corrected for, but this
would provide resolution beyond that needed due to the interactions before the veto
in the shielding, and was not included in this analysis.
These long tracks have hits in many planes within the neutron arm and almost
always have hits in the veto as well. These charged tracks were used to align the
veto peak with the observed peak in plane one (N1). The relationship between veto
amplitude and time was used to correct the time in the veto for this alignment. The
relationship in equation A.3 was used, except the veto planes were substituted in
for the Nth neutron plane. An example histogram used in the calibration is shown
in Figure A.6.
A.3 Neutron Software
The neutron arm software takes the raw data encoded in coda files and pro-
cesses it through the decoding, coarse processing, and fine processing to form clusters
with position, corrected times, and a calculated track to the target.
The data is first pulled from raw coda files in the process of decoding and
analysis. The reference hit, which was read into a specific channel of the F1 TDC,
must exist for the event to be decoded. The decoding process loops through all
TDC and ADC channels, and creates independent arrays of hits for raw analysis.
These include the corrections to change the PMT values from channels into time,
but does not allow the determination of mean time or horizontal position. An offset
184
l1b16tof
Entries  420
Mean   0.3273
RMS     3.205
 Time(ns)-20 -15 -10 -5 0 5 10 15 20
0
5
10
15
20
25
30
Plane 1 Bar 16 TOF
Figure A.6: Calibration of the veto time for the left veto plane 1 bar 16.
185
is applied to the PMT time (toff) to correct the time relative to the reference time
toff = tref ?W ?tres . (A.9)
In this formula +(?) is selected when the difference between time and reference
time is greater(less) than +(?) half of W, the maximum number of channels in
the F1 TDC. The trigger provides a delayed signal to serve as this reference time
(tref). Because of this all neutron arm times are given compared to the trigger time
(see NA trigger discussion in section 4.7). tres is the resolution, which was 0.118 ns
for the F1 TDC. The F1 TDC was constantly acquiring hits, keeping track of the
information in each channel relative to the reference time which was determined by
the trigger time. The trigger also determined when the TDC was read. The total
number of channels in the F1 TDC (W) is 135558. The PMT time also included
the time offsets, read in from the database. All indices are separate at this stage,
even for ADC and TDC information for the same PMT.
In the coarse processing, the hits were sorted by bar number and value. The
hits were looped through, each PMT time was checked if it was within a limited
range, and the existence of ADC and TDC hits for both PMTs for a bar was checked.
Hits that pass these checks were considered ?complete?. At this stage the mean time
was corrected for the amplitude correction. This is also where the mean time and
the horizontal position of a hit were calculated. The equations for the quantities
186
calculated during this processing are
tdiff = (tr ?tl)2
tmean = (tr +tl)2
yt = tdiffcn
Abar =
p
Al ?Ar .
(A.10)
Here tmean is the mean time of the hit (and tl and tr are corrected left and right
times), while tdiff is the difference between times of hits in the left and right PMTs.
This difference allows the calculation of the horizontal position (y = yt + yoffset)
of the hit. The constant cn is the propagation speed of the signal within the bar.
Also, Al and Ar are the left and right amplitudes, respectively, while Abar gives the
?energy? for the full bar.
The last step of the neutron arm decoding was the fine processing. The hori-
zontal position of the hit is checked to see if it exists within the neutron arm. After
this check, the clusters are created. This is done by sorting the hits and then looping
through hits with each hit placed in only one cluster. In this loop, the first hit is put
into a cluster, the AddNeigbors method is called (recursively) which collects (and
removes from further consideration) neighboring (in time and position) hits. A hit
is classified as a neighbor if it is one bar away and within 10 ns. These clusters
are then sorted by ?energy?. A track is created from the cluster (using only the
initial hit information) back to the target, and this is used further in analysis. The
path length and momentum direction variables (but momentum is not constructed
at this stage) are also created. The time window of 10 ns is large relative to the ?
187
0.5 ns resolution. Various improvements to the cluster algorithm were investigated,
but none of them significantly changed the number of clusters within 20 ns of the
expected neutron time of flight, and thus were not included in the analysis.
A.4 Veto Analysis
As previously described (section 4.6.2), the veto was made up of two planes
of detectors with 48 detectors per plane. These detectors were segmented into left
and right veto counters, providing a total of 192 separate PMTs which form the
veto detector. The two veto planes were also shifted vertically and horizontally
relative to eachother [72]. This vertical and horizontal segmentation improved the
localization of veto detector hits for association with neutron detector hits. Because
of the shift between the two planes, the edges of the neutron detector were only
covered by a single plane of veto, which reduced the detection efficiency of the veto
and hence provided different charge identification efficiency in these edges compared
to the central region of the neutron arm. These edges were removed in the analysis
presented in this thesis.
The horizontal segmentation of the veto detectors allowed for the identification
of charged hits in the neutron detectors based upon horizontal position of the cluster.
If the hadron hit was far to one side of the detector, than a charged identification
in the veto on the other side of the detector would not identify a charged hadron.
Figure A.10 shows this, where the time of the hit relative to the neutron detectors
and the horizontal position of the hit within the neutron detector are plotted. Only
188
 Time(ns)-100 -50 0 50 100
 X (m)
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
Figure A.7: Plot of X position in the veto subtracted by the X position in the
neutron arm versus ?t. These are for quasi-elastic hits in the neutron arm. Plotted
on the x axis is the time difference between the neutron arm and the veto arm. This
identifies a region of 0.3-0.5 m to consider in the veto detector to identify charge
hits.
veto bars within a vertical region were considered as hits to identify a charged hadron
(see figure A.7).
The veto dead-time was different for the individual veto counters. This was
determined by measuring the least amount of time between sequential hits. For
analysis, only a single value was used for a set of bars, this is shown in Table A.4.
The deadtime for individual bars could be 40-120 ns.
The veto detection efficiency depended on the horizontal, time, and amplitude
cuts used to identify charged particles. Many events in this deadtime region were low
amplitude. Since charged particles arrived within the coincidence time region, when
theADCgatewasactive, theselowamplitudeeventswerenoteventswhereanearlier
event masked the veto hit from the charged particle. Figure A.8 shows the time
189
 Time(ns)-100 -50 0 50 100
 ADC
0
100
200
300
400
500
600
700
800
900
1000
Figure A.8: Plotted is the time difference between the veto hit and the neutron arm
hit for a single veto plane versus the amplitude in the veto bar. This figure shows all
bars in the plane. The gate is seen to start arround 40 ns before the coincidence hit
in the neutron arm. The deadtime of the various veto detectors is between 40 and
110 ns. During this dead time, the hits are identified as charged if the amplitude is
above 200. For a region of ?10 ns around 0, all events are identified as charged.
Time (ns)0 100 200 300 400 500
0
50
100
150
200
250
300
Figure A.9: Plot showing the rate both with and without a 200 channel ADC cut
for the veto detector L1-14. The ADC cut rate is in blue. The window between 300
and 350 ns was used to calculate the rate. This rate is in kHz.
190
 Time(ns)-10 -8 -6 -4 -2 0 2 4 6 8 10
 Position(m)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2T difference versus Y position for V1
Figure A.10: Plot of hits within the Veto detector. Colors are given to show coin-
cidence between the two separate halves of the V1 veto detector.
versus amplitude, and the charge identification selection mechanism. By requiring
an amplitude for the hit, the effective rate in the veto detector was decreased (Figure
A.9). Using this technique, the combined veto detector efficiency was measured to
be 95%.
191
Identifier Date Comment Change
new crate19 May 6 Large, same as 19
new crate23 May 3 hatsv22:2002 S0
new crate23 May 3 Same
new crate22 May 3 Large
new crate21 May 3 Large
new crate20 May 3 hatsv22:2002 S0
new crate19 Apr 29 PS1:2004 S5 + S9 and PS1:2004 S9 + S11
new crate18 Apr 29 4335-4338 PS1:2004 S5 + S9 + S11 + PS1:2004 S9
new crate17 Apr 29 No Change
new crate16 Apr 23 4151 PS1:2002 S9 and PS1:2004 S9
new crate15 Apr 23 Small?
new crate14 Apr 19 4068 Large
new crate13 Apr 19 PS1:2002 S9
new crate12 Apr 18 4050? Small
new crate11 Apr 17 4016 Small
new crate10 Apr 17 Large
new crate9 Apr 17 PS1:2004 S11 + S15 + S9 +S10
045.set Apr 12 PS1:2002 S14 and PS1:2004 S11 + S15
new crate8 Apr 6 3714 S1:2002 S14 + S11
new crate7 Apr 4 PS1:2002 S14
new crate6 Apr 4 Large
new crate6.set Apr 4 3665 Same
new crate5.set Apr 4 Large
new crate5 Apr 4 Large
new crate4 Apr 4
new crate4.set Apr 4 Large
new crate3.set Apr 3 Large
044 new.set Apr 3 3621 Large
006 new.set Apr 3 Large
006 crate3.set Apr 3 Large
006.set Apr 3 Large
044 Apr 1 3570 Same
043 Mar 24 3320s/3330s Large
042 Mar 23 Large
041 Mar 23 3290/3270 Large
040 Mar 22 3268 Large
039 Mar 22 Only PS1:2002 S14
03-04-2006 Mar 15 3016? Large
038 Mar 1 2477 Large
Table A.2: Table of the list of high voltage changes (HV) for the neutron arm in the
E02-013 experiment.
192
Bar Number Veto Plane 1 Veto Plane 2
Right 120 ns 100 ns
Left 65 ns 65 ns
Table A.3: A table showing the deadtime region for a set of veto detectors. This
was selected as the maximum for that set of detectors.
Number of Events V1 only V1 with > 1 V1 V2 with > 1 V1
0 35.5% 92.7% 08.5%
At least 1 59.7% 06.8% 84.0%
At least 2 04.4% 00.4% 06.8%
Table A.4: A table of the correlation beetween hits in Veto plane 1 and Veto plane
2. The is for all events within a single run in kinematic 2A (3190).
193
Appendix B
Error Analysis
In the measurement of the electric form factor of the neutron at Q2 = 3.5 GeV2
statistical uncertainty is the main component of the error. The double polarization
technique removes some sources of systematic uncertainty (as described in section
3.3). Due to the small asymmetry, a large statistical sample is required. The primary
cause of the small asymmetry is the small size of GnE.
The error analysis for E02-013 entails propagation of correlated and uncorre-
lated systematic and statistical uncertainties for the equations used to calculate GnE
from the measured quantites. Equation 5.43 gives the factors that contribute to GnE.
The uncertainty in a few of these factors give the most important contributions to
the final precision of the GnE measurement.
For the propagation of the uncertainties, standard error techniques are used
[94]. Assuming the errors are uncorrelated the expression to determine the uncer-
tainty in a function q which depends on the components x,...,z is
?q =
s 
?q
?x?x
?2
+...+
 ?q
?z?z
?2
. (B.1)
When q is proportional to the product or quotient of the components,
?q
q =
s 
?x
x
?2
+...+
 ?z
z
?2
. (B.2)
The correlated systematic errors also contribute to the final uncertainty. As an
example, there were not independent measurements of polarization for every run, so
194
the systematic error was defined to be the same fractional error as the polarization
after the summation, rather than being included in the summation error calculation
(equation 5.47). Explicit discussion of the error analysis using formula B.1 follows.
For statistical error (with large statistics), assuming a Poisson distribution,
the formula
?N
N =
1?
N (B.3)
is used, where N is the pertinent statistics. For small asymmetries and large statis-
tics method of determination of the uncertainty is valid [81]
?A ? 1?N ? (N
+ ?N?)2
N 52 ?
1?
N . (B.4)
The formula B.4 is used to determine the statistical error in empirically measured
asymmetries like the raw accidental background asymmetry and raw neutron asym-
metry. The statistical uncertainty for quasi-elastic event counts, background counts,
and other counts is determined using equation B.3.
The following dilution factors, polarizations, and asymmetry corrections are
used in the calculation of the physical asymmetry (equation 5.43): Dback, DN2,
Aback, DpN2, Asum, Dp, Dpn, Pn, Pbeam, P3He (for description see below). Some of
these depend on the same sources of uncertainty. Common sources of uncertainty in
these factors and corrections are a systematic uncertainty due to the charge deter-
mination in the accidental background technique (see section 5.5.1) and statistical
uncertainties due to the raw neutral counts and the raw background counts. While
the systematic uncertainty due to the charged to uncharged background ratio is a
component of almost all dilutions and corrections, the small number of raw back-
195
Background Ratio (un/ch)0.2 0.4 0.6
physA
-0.1963
-0.1962
-0.1961
-0.1960
 as a function of Beam PolarizationphysA
Figure B.1: The physical asymmetry as the accidental background ratio of charged
to uncharged used in the calculation is changed. This figure shows that the depen-
dence on this ratio is very small, with an estimated fractional uncertainty on the
accidental background charge ratio of 20%. This esimated fractional uncertainty
corresponds to a fractional uncertainty of less than 0.2%. Plotted is a linear fit,
showing the small dependence. This is for an analysis of Kinematic 2(b).
ground makes it of minimal importance. This is observed in Figure B where the
change of neutron physical asymmetry as a function of the charge ratio of the back-
ground is plotted for kinematic 2. This ratio will be treated as a constant for the
rest of this error analysis. The value for this ratio and accidental background counts
is presented in Table B.3.
As described in section 4.4.2 the polarization changes run to run, and a modi-
fied asymmetry is calculated on a run per run basis, and then used in a weighted sum
for the entire kinematic (as derived in Appendix C). The only components which
change significantly on a per run basis are the target polarization and neutron purity
factor. The neutron purity factor and the rest of the dilutions are corrected for after
196
the summation (see equation 5.48)
Asum =
P
i
Araw,i
P3He,iNiP
2
3He,iP
i NiP
2
3He,i
. (B.5)
The factor Aback can be found by using equation
Aback = N
+
back ?N
?
back
N ?P3He? = Aback,raw
Nneut,back
N ?P3He? . (B.6)
The accidental background asymmetry correction contributes to the statisical uncer-
taintythroughthecountsintherawaccidentalbackgroundand quasi-elastic samples
and through the accidental background asymmetry. As described in section 5.5.1,
the neutral accidental background is related to the raw accidental background by
Nneut,back = fbackNback. Additionally, it contributes to the systematic uncertainty
through the target polarization.
The polarization of the target and nucleus and beam are presented in sections
4.3 and 4.4.2 and in [39]. The target polarization (P3He) uncertainty is correlated
between kinematics 3, 4, and 2(b). As mentioned in 5.7.1, the values for the po-
larizations of the protons and neutron within the helium nucleus depend on the
theoretical model used to calculate the nuclear corrections. Changes in these values
will be correlated. The value used for the beam polarization (Pbeam) and systematic
uncertainty is arrived at from inspecting Table 4.2.
The dilution caused by the accidental background (equation 5.38) is written
as
Dback = 1? Nneut,backN . (B.7)
The sources of statistical uncertainty are the counts of the raw accidental back-
ground and quasi-elastic samples, similar to the accidental background asymmetry
197
correction.
The dilution from the nitrogen in the neutron sample (equation 5.40 gives the
method to properly calculate this) can be written as
DN2 = 1? NN2N ?N
neut,back
, (B.8)
with a dependence on the uncertainties in the constants shown in Table B.1. The
dominant sources of systematic uncertainty in this factor are from the densities for
N2 in the 3He target and in the reference cell. The dominant source of statistical
uncertainty is due to the nitrogen quasi-elastic events and background. The uncer-
tainties for this dilution were propagated using equation B.1. The results for this
error analysis are also presented in Table B.1. This uncertainty is treated as inde-
pendent from the other uncertainties, although there is a small contribution from
the 3He quasi-elastic sample.
The dilution of the protons in the neutron sample (equation 5.32) can be
expressed as
Dn = 1? NprotonN ?N
neut,back ?NN2
. (B.9)
Here the sources of uncertainty are statistical from the background, charged quasi-
elastic, neutral quasi-elastic in H2, N2 and 3He, and systematic from the technique
and model. It is assumed that the statistical uncertainty from Minuit and the
systematic uncertainty from the technique dominates all other sources of error. By
considering other analyses, a fractional systematic error of 5% was arrived at. The
cited statistical uncertainty is that given by Minuit fitting [30].
Finally, the dilution caused by other final state interactions within the neutron
198
sample can be expressed as
DFSI = 1? NFSIN ?N
neut,back ?NN2 ?Nproton
. (B.10)
Because the dilution due to final state interactions is undetermined but guided
by theory [42][88], an estimate is determined for the systematic uncertainty that
accounts for the likely values of this dilution.
The proton asymmetry has similar dilutions as the neutral asymmetry, with a
couple of important differences. This formula 5.54 is
Aproton = 1?DnD
backDN2
PbeamPprotAp,phys . (B.11)
The polarization of protons within the nucleus, Pprot is provided by theory [89].
The proton physical asymmetry is calculated from the proton form factor ratio as
described in equation 5.49:
Ap,phys = B?proton +C?2
proton +D
. (B.12)
Here the uncertainty in ?proton is determined from the parameterization [59]. The
calculation and uncertainties for the quantities used in the determination of Aproton
are in Table B.2.
A reminder that the expression for the physical asymmetry is (equation 5.43)
Aphys = Asum ?Aback ?AprotonP
nPbeamDbackDN2DpDfsi
. (B.13)
By applying equation B.1 to equation 5.43, the statistical and systematic uncertainty
of the physical asymmetry are calculated. These results, along with the individual
contributions to the uncertainties, are presented in Table ?? for kinematic 2a, Table
B.5 for kinematic 2b, Table B.6 for kinematic 3, and Table B.7 for kinematic 4.
199
DN2 DcN2
Q2 = 2.5 GeV2 3.5 GeV2 1.5 GeV2 2.5 GeV2 3.5 GeV2 1.5 GeV2
?3He 0.164 0.164 0.164 0.164 0.164 0.164
?N2 4.4 3.6 9.1 4.4 3.6 9.1
Q3He 184399 228766 167206 184399 228766 167206
QN2 14020 37920 5489 14020 37920 5489
N3He 2089.1 252.9 6969.8 19076.7 3044.2 72525
NN2 220.5 20.3 668.8 1123.2 175.6 3760.3
DN2 0.9493 0.9781 0.9473 0.9714 0.9843 0.9715
Statistical 0.0036 0.0051 0.0022 0.0009 0.0012 0.0048
Systematic 0.0047 0.0021 0.0051 0.0027 0.0015 0.0028
Table B.1: A table showing the measurements and error calculations for the nitrogen
dilution of the uncharged sample, for all three kinematics. N3He is the number of
background corrected counts for the selected 3He runs, ?N2 is the nitrogen density,
Q3He is the total charge of the selected 3He runs. This calculation is as described
in section 5.5.2. Densities are determined by target measurements [62][57][81].
200
parameter Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
GpE 0.0330?0.0051 0.018?0.004 0.077?0.007
GpM 0.1378?0.0049 0.089?0.004 0.262?0.006
Ap,phys 0.177?0.028 0.185?0.030 0.197?0.021
Pprot ?0.1?0.004 ?0.1?0.004 ?0.1?0.004
Pbeam 0.835?0.03 0.835?0.03 0.835?0.03
DN2 0.949?0.004?0.005 0.978?0.005?0.002 0.947?0.002?0.005
Dback 0.9702?0.0003 0.9731?0.0005 0.9697?0.0002
Dn 0.806?0.023?0.04 0.885?0.047?0.044 0.697?0.019
Aproton ?0.0009?0.0002 ?0.0018?0.0003 ?0.0054?0.0007
Table B.2: A table of the errors for the calculation of Aproton. The calculation is
calculated according to equation 5.54. The uncertainty in GpM and GpE is deter-
mined using a parameterization [59] and the propagation of the error is according
to equation B.1. Parenthesis is used to identify the systematic uncertainty.
The uncertainty in the ratio of neutron form factors, or ?, is determined using
equation B.1 and 5.55. This is a straightforward calculation [38], but the value for
? is only determined for the central value of Q2. To determine the value for the
electric form factor of the neutron, equations B.2 and 5.58 were used. The results
are displayed in Table B.4.
201
parameter Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
Hadron Ratio 0.325?0.065 0.26?0.04 0.33?0.05
Hadons 2495 4201 2762
Neutral 1883.2 1092.3 911.5
Charged 611.8 3108.7 1850.5
TableB.3: Atableforallthreekinematicsshowingthemeasurementsforthecharged
and uncharged background and the uncertainty in the background ratio.
parameter Q2 = 2.5 GeV2 Q2 = 3.5 GeV2 Q2 = 1.7 GeV2
GEn/GMn ?0.2219 ?0.2434 ?0.1991
Error ?0.0325?0.0293 ?0.0457?0.0202 ?0.0183?0.0330
GMn ?0.0954?0.0019 ?0.0556?0.0011 ?0.1636?0.0032
GEn 0.0212 0.0135 0.0326
Error ?0.0031?0.0028 ?0.0025?0.0012 ?0.0030?0.0054
Table B.4: A table of the errors for the calculation of GnE. Uncertainty in the ratio
of form factors was calculated from the physical asymmetry as expressed in section
5.6 and Table 5.8 and Table 5.7. The uncertainty in GnM is determined from a
parameterization [59] and propagation of the error is according to equation B.1.
202
quantity value ?stat contrib. Aphys ?syst contrib. Aphys
Ameas ?0.0446 ?0.0055 0.024 ? -
P3He 0.4359 ? - ?0.02 0.018
Aback,raw 0.0279 ?0.0004 0.00004 ? -
DFSI 0.95 ? - ?0.01 0.002
Dn 0.806 ?0.023 0.005 ?0.040 0.008
Ap,phys 0.177 ? - ?0.028 0.0004
Pbeam 0.835 ? - ?0.03 0.007
Ntotal 33001 ?182 0.0001 ? -
Nback 4201 ?65 0.0003 ? -
Pn 0.86 ? - ?0.02 0.004
Pprot ?0.100 ? - ?0.004 0.0002
DN2 0.949 ?0.0036 0.0007 ?0.0047 0.0009
Aphys ?0.191 - ?0.024 - ?0.022
Table B.5: A table of the uncertainties that contribute to the physical asymmetry
(Aphys). The contributions are calculated and contribute to the physical asymmetry
precision as determined by equation B.1. This table shows the errors for Q2 = 2.5
GeV2.
203
quantity value ?stat Aphys contrib. ?syst Aphys contrib.
Ameas ?0.0358 ?0.0080 0.029 ? -
P3He 0.477 ? - ?0.02 0.0109
Aback,raw ?0.0233 ?0.0004 0.00004 ? -
DFSI 0.95 ? - ?0.01 0.0013
Dn 0.885 ?0.047 0.0054 ?0.044 0.0051
Ap,phys 0.182 ? - ?0.030 0.0005
Pbeam 0.835 ? - ?0.03 0.0046
Ntotal 15319 ?124 0.0001 ? -
Nback 2748 ?52 0.0003 ? -
Pn 0.86 ? - ?0.02 0.0030
Pprot ?0.1 ? - ?0.004 0.0001
DN2 0.978 ?0.0051 0.0006 ?0.0021 0.0003
Aphys ?0.132 - ?0.030 - ?0.013
Table B.6: A table of the uncertainties that contribute to the physical asymmetry
(Aphys). The contributions are calculated and contribute to the physical asymmetry
precision as determined by equation B.1. This table shows the errors for Q2 = 3.5
GeV2.
204
quantity value ?stat Aphys contrib. ?syst Aphys contrib.
Ameas ?0.0569 ?0.0031 0.0144 ? -
P3He 0.4883 ? - ?0.02 0.0218
Aback,raw 0.0007 ?0.00001 0.0000 ? -
DFSI 0.95 ? - ?0.01 0.0027
Dn 0.696 ?0.018 0.0043 ?0.035 0.0112
Ap,phys 0.1969 ? - ?0.0207 0.0013
Pbeam 0.835 ? - ?0.03 0.0096
Ntotal 105529 ?325 0.0001 ? -
Nback 17834 ?134 0.0003 ? -
Pn 0.86 ? - ?0.02 0.0059
Pprot ?0.1 ? - ?0.004 0.0005
DN2 0.9473 ?0.0022 0.0006 ?0.0051 0.0013
Aphys ?0.2537 - ?0.0150 - ?0.0272
Table B.7: A table of the uncertainties that contribute to the physical asymmetry
(Aphys). The contributions are calculated and contribute to the physical asymmetry
precision as determined by equation B.1. This table shows the errors for Q2 = 1.7
GeV2.
205
Appendix C
Calculations
C.1 Asymmetry Best Value
The formula to determine the best value of a set of gaussian measurements
is the one presented in undergraduate textbooks, and the method to determine the
best value of a distribution is also presented, but the formula for an asymmetry is
not. The method used here is the maximum likelihood method. For this asymmetry,
there are two Poisson distributions of N+ and N?.
A Poisson distribution is decribed by[94]
P?(?) = e???
?
?! , (C.1)
where ? is the number of occurrences and ? is the expected number of occurrences.
The asymmetry is completely described by the number of positive counts with pos-
itive and negative helicity, Ni+ and Ni? (i is the ith experiment). This gives the
likelihood
L = P?+(Ni+)?P??(Ni?)???? . (C.2)
The ?, or expected number of occurrences, for some run, is identified as
?? = ?Ne+ +Ne??1?A
e
2 =
Ne+ +Ne? ?(Ne+ ?Ne?)
2 = N
e
? . (C.3)
Here the index e gives the expected value for that given run period, while i is the
206
ith measured value. The expected value of counts for + and ? give the best value
for the asymmetry.
There are many measurements (given by index i), all having their own best
value for the counts to give the best asymmetry. This allows us to have
??,i = ??,i = ?i1?A
e
i
2 , (C.4)
where ? is the correct N to get the best asymmetry. Following the previous work
for the Poisson distribution
X?lnL
?A = (?1+
N?i
??i )(?
?iPi
2 ) =
X Ni+Pi
1?APi ?
Ni?Pi
1+APi = 0 . (C.5)
Using the standard expansion
(1?APi)?1 = 1?PiA , (C.6)
the relationship
X
Pi(Ni+ ?Ni?) = ?
X
P2i NiA (C.7)
is arrived at, which gives
A = ?
PP
i(Ni+ ?Ni?)P
P2i Ni . (C.8)
Putting it into a more recognizable form
A = ?
PP2
i Ni
Ai
PiP
P2i Ni , (C.9)
with the corresponding statistical uncertainty.
207
Appendix D
Jefferson Laboratory E02-013 Collaboration
The collaboration list for the E02-013 experiment is on the next page.
208
209
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