ABSTRACT
Title of dissertation: BENCHMARKING CHARGE EXCHANGE
THEORY IN THE DAWNING ERA OF
SPACE-BORNE HIGH-RESOLUTION
X-RAY SPECTROMETERS
Gabriele Latif Betancourt-Martinez,
Doctor of Philosophy, 2017
Dissertation directed by: Professor Christopher Reynolds
Department of Astronomy
Dr. F. Scott Porter
NASA/Goddard Space Flight Center
Charge Exchange (CX) is a process in which a highly charged ion captures
one or more electrons from a neutral atom or molecule into an excited state during
a close interaction. The electron’s subsequent radiative cascade to the ground state
produces diagnostic line emission in the X-ray band. CX with solar wind ions occurs
frequently in the solar system, and CX may also occur astrophysically.
In order to properly identify CX in astrophysical spectra and make use of
its diagnostic properties, we must be able to model the emission. Theoretical treat-
ments of CX are often computationally expensive, experimental benchmarks at high
resolution are fairly scarce, and there is often poor agreement between the two.
This dissertation seeks to build a better understanding of the mechanics and
spectral signatures of CX through high-resolution experimental data paired with the-
oretical calculations of CX. Chapter 1 outlines the necessary ingredients for modeling
and identifying CX spectra, describes several astrophysical environments in which
CX has been observed or postulated to occur, and presents some of the challenges
we are facing in identifying and understanding this emission. Chapter 2 describes
the theoretical and computational tools used in this work. Chapter 3 discusses
the experimental tools and facilities we use, namely an Electron Beam Ion Trap
(EBIT) and an X-ray microcalorimeter. Chapter 4 presents experimental K-shell
data that highlights both the subtle nature of the CX interaction and the difficulty
in including those nuances in spectral synthesis codes. Chapter 5 presents the first
high-resolution L-shell CX spectra of Ne-like Ni and describes what we can learn
from these results. In Chapter 6, we take these data a step further and present a
pipeline to calculate relative state-selective capture cross sections, previously only
available from theoretical modeling. We then compare some of our results to theory.
In Chapter 7, we discuss several future steps for our work.
BENCHMARKING CHARGE EXCHANGE THEORY
IN THE DAWNING ERA OF SPACE-BORNE
HIGH-RESOLUTION X-RAY SPECTROMETERS
by
Gabriele Latif Betancourt-Martinez
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
2017
Advisory Committee:
Professor Christopher Reynolds, Chair/Advisor
Dr. F. Scott Porter, Co-Chair/Advisor
Dr. Maurice Leutenegger, Co-Advisor
Professor Richard Mushotzky
Professor Alberto Bolatto
Professor Jordan Goodman, Graduate Dean’s Representative
c© Copyright by
Gabriele Latif Betancourt-Martinez
2017
Preface
Some of the work presented in this thesis has been published in peer-reviewed
journals or presented at national and international conferences. Portions of Chapter
3 were presented at the conference SPIE Astronomical Instrumentation & Telescopes
as a poster in June 2014 [Betancourt-Martinez et al., 2014b] and as an oral presenta-
tion in June 2016. Much of chapter 4 was published in the journal Physical Review
A as “Observation of highly disparate K shell x-ray spectra produed by charge
exchange with bare mid-Z ions” [Betancourt-Martinez et al., 2014c]. Preliminary
results from Chapter 4 were presented at the American Astronomical Society winter
meeting in 2014 [Betancourt-Martinez et al., 2014a]. Some of this work was also
presented at the conference of the High Energy Astrophysics Division of the Amer-
ican Astronomical Society in 2016 [Betancourt-Martinez et al., 2016]. The work
presented in Chapters 5 and 6 is in preparation for submission to peer-reviewed
journals.
The majority of this research was supported by a NASA Space Technology
Research Fellowship.
Work at the Lawrence Livermore National Laboratory was performed under
the auspices of the U.S. Department of Energy under Contract DE-AC52-07NA27344
and supported by NASA APRA grants to LLNL and NASA/GSFC.
ii
To Rosella Clemmons Washington and Dr. Margaret Anne Cyrus Mills: two
strong, brilliant women who I know have been cheering me on from afar.
iii
Acknowledgments
Well, I wasn’t always sure that I’d actually get to this point, but here I am,
and with an exciting new adventure ahead of me to boot. As cliche´ as it sounds,
I could not have gotten here without many wonderful people on my side to help
encourage, guide, and support me.
Maurice: I realized early on that I got really lucky to have you as an advisor.
Thank you for being so generous with your time and your vast sea of knowledge,
which you always share with extreme patience, genuine excitement, and good humor.
I’ve appreciated the guidance, mentorship, feedback, sharing of data and codes,
company at conferences, explanations of random pop culture references that I’ve
missed, and your being solid company (along with Renata) at a Livermore bar on
that most dismal night in November 2016.
Chris, thank you for always having my back and making sure that I’m not only
productive, but happy. You’ve always been willing to step in when the situation
merits it, and offer a kind and sympathetic ear along with good advice when I’ve
had a rough patch. It’s meant so much to know that I can always count on your
support.
Scott: you’ve always looked out for my professional development, and for that
I’ve always been grateful. Thank you for sending me to White Sands for the XQC
rocket launch, for making sure I can attend conferences when I want to, for your
succinct answers to my science questions, and for opening up research doors and
career paths, even when I’ve been unsure of what direction I want to take.
iv
To the TES team: I’m really glad I got to learn and grow in such a friendly
environment. Steve, Joe, Megan, and Simon: special shout-out to you guys for
showing me the ropes in the lab. Megan, thank you for always being there for
advice and editing help, and for your friendship.
Renata, my fellow nature lover: I so appreciate you for busting your butt to
get those MCLZ results in time for my thesis, for being available any time of the
day or night for discussions about CX theory, and for the moral support during
this process. And from dance classes to random adventures in which we hugged
redwoods and did not fall off the side of a mountain—I’m glad to count you not just
as a colleague, but also as a friend.
Now onto the EBIT crew. To Greg: thanks for allowing me the opportunity
to spend a small chunk of my life at EBIT. I learned a ton, got interesting data,
and even managed to have a good amount of fun. I also appreciate the nudges to
think more deeply about my analysis, to make my proposals more rigorous, and to
finish this thesis when I said I would. Tom: thanks for tag-teaming the CX data
acquisition with me, and for staying later than you really needed to just to get me a
few more hours of counts. Also thanks for late night lab dance parties, the peanut
butter that I stole, the song suggestions and Instagram memes, and for being just
as weird as I am. Natalie, thank you for all the figures, the EBIT facts, and the lab
gossip. And thanks, Ed and David, for keeping things running smoothly at EBIT
while I was there, and for helping us to make whatever weird ideas we had to make
the atomic H source work better a reality.
To my fellow “second years:” thanks for showing up for my Fall Into Falls and
v
Spring Into Springs, for the pie days, salad bowls, and fancy dress parties, and for
not hating me when I sent what seemed like a total jerk email about the quals.
Krista and Ashlee, or the other 2/3 of the Peanut Butter Patties/Paddies/Padis:
you both are rockstars, and I’m really lucky to have had you around as officemates,
advice-givers, cheerleaders, workout buddies, travel companions, puppy/kitty-time
givers, and obviously, friends.
Rafael and the other Gabriele: thanks for being such fun officemates at God-
dard, and no thanks for all the pressure to drink more coffee!
I’m lucky to have found (in undergrad!) smart, caring, and inspiring astro-
physicists who also happen to be boricuas. He´ctor Arce: thank you for believing in
me as a wet-behind-the-ears sophomore, taking me under your wing, and giving me
the chance to start my research career with you. I’m honored and proud to still call
you my mentor. Tali Figueroa: thank you for seeking me out to join your group
for a summer, introducing me to the exciting world of calorimeters, and providing
a warm and enriching environment in which to learn.
To my non-grad school friends, who probably won’t read this: thanks for
keeping me grounded, traveling, dancing, and laughing.
Thank you to my extended family, both related by blood and chosen, for
celebrating my victories with me as if they were your own, for pushing me to be
better, and for your unconditional love.
Lastly, to my parents, Mamita and Papito: Thank you for supporting me
throughout this long haul of a journey, and for sacrificing for and always prioritiz-
ing my education—from Kindermusik, piano, and horseback riding, to Montessori,
vi
Westtown, and Yale, plus all the camps and activities in between. Thank you for
helping me get here, and for being there for all that I’ve been through in the last six
years, grad-school related or otherwise. You were around to help me buy my first
car, you celebrated with me when I won fellowships or acquired jobs, you helped
me heal from heartbreak (twice), and you are always around to give me a quick
hug from the train platform as I’m passing through Wilmington on the way up or
down the East Coast. You came to cheer me on at my first triathlon, you gave me
advice on personality clashes, and Mamita, you didn’t even hesitate to rush down
to Silver Spring with the ingredients for chicken soup when I got super sick. You
guys remind me where I came from, who I am, and who I can be. Thank you for
being an endless source of love, support, wisdom, and a deep sense of home. I love
and appreciate you more than you’ll ever know.
vii
Table of Contents
Preface ii
Dedication iii
Acknowledgements iv
List of Tables xi
List of Figures xii
List of Abbreviations xviii
1 Charge Exchange: an Introduction 1
1.1 Charge Exchange Basics . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Introduction to Charge Exchange Theory . . . . . . . . . . . . . . . . 7
1.3 Theoretical Approximations . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Spectroscopic Observables . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Charge Exchange in Astronomy . . . . . . . . . . . . . . . . . . . . . 25
1.5.1 Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.2 Astrophysical and Other CX . . . . . . . . . . . . . . . . . . . 30
1.6 Diagnostic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 The Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Theoretical and Computational Tools for Simulating and Understanding CX 42
2.1 fac, the Flexible Atomic Code . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Charge Exchange Models: spex-cx and acx . . . . . . . . . . . . . . 44
2.3 Multi-Channel Landau Zener Theory . . . . . . . . . . . . . . . . . . 48
3 The Role of Laboratory Astrophysics in Improving Charge Exchange Models 57
3.1 The LLNL Electron Beam Ion Trap . . . . . . . . . . . . . . . . . . . 57
3.2 The EBIT Calorimeter Spectrometer . . . . . . . . . . . . . . . . . . 64
3.2.1 X-ray Microcalorimeter Basics . . . . . . . . . . . . . . . . . . 65
3.2.2 The EBIT Calorimeter Spectrometer . . . . . . . . . . . . . . 78
viii
3.2.3 ECS Data Acquisition and Analysis . . . . . . . . . . . . . . . 80
3.2.3.1 ECS Filters and Windows . . . . . . . . . . . . . . . 83
3.2.3.2 Absorber . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.3.3 Amplification, Filtering, and Digitization . . . . . . . 86
3.2.3.4 The Software Calorimeter Digital Processor . . . . . 88
3.2.3.5 Triggering . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2.3.6 Optimal Filtering . . . . . . . . . . . . . . . . . . . . 89
3.2.3.7 Event Grades and Instrumental Response . . . . . . 91
3.2.3.8 Gain Scale . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.3.9 Final Analysis . . . . . . . . . . . . . . . . . . . . . 94
3.3 The Transition-Edge EBIT Calorimeter Spectrometer . . . . . . . . . 94
3.3.1 Technical Specifications . . . . . . . . . . . . . . . . . . . . . 97
3.3.2 Device Readout . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.3.3 TEMS Cooling System . . . . . . . . . . . . . . . . . . . . . . 103
3.3.4 Current Status of TEMS and Outlook . . . . . . . . . . . . . 103
4 Observation of Highly Disparate K Shell X-ray Spectra Produced by Charge
Exchange with Bare Mid-Z ions 107
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 Analysis and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 High-Resolution Spectra of L-Shell Charge Exchange with Ne-like Ni 125
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6 L-shell Ni CX: Comparisons to Models and Relative Capture Cross Section
Calculations 152
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Comparisons to Spectral Models . . . . . . . . . . . . . . . . . . . . . 153
6.3 A Pipeline to Extract State-Selective Relative Cross Sections from
Spectra: Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.4 Cross Section Calculations: Result for S+He . . . . . . . . . . . . . . 180
6.5 K-shell S+He Cross Sections: Pipeline Model Comparison to MCLZ
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.6 Cross Section Calculations: Result for Ni19++H2 . . . . . . . . . . . . 190
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7 Outlook 205
ix
A Spectral Basis Set Elements for Model Fit to S+He CX Data 208
B Spectral Basis Set Elements for Model Fit to Ni+H2, Ni+He CX Data 219
C X-Ray Emission Mechanisms 232
Bibliography 241
x
List of Tables
3.1 Comparison between the ECS and the TEMS. . . . . . . . . . . . . . 97
4.1 Summary of the normalized H-like series line strengths and hardness
ratios for the experiments presented in Chapter 4 . . . . . . . . . . . 118
4.2 Summary of the normalized He-like series line strengths for the ex-
periments presented in Chapter 4 . . . . . . . . . . . . . . . . . . . . 119
5.1 CX lines identified in this work. . . . . . . . . . . . . . . . . . . . . . 146
5.2 DE and model lines identified in this work. . . . . . . . . . . . . . . . 147
xi
List of Figures
1.1 Illustration of the CX interaction. . . . . . . . . . . . . . . . . . . . . 5
1.2 Radiative decay paths from electron capture into a bare ion at low
collision energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Radiative decay paths from electron capture into a bare ion at high
collision energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 CTMC-calculated spectra for CX between bare O and H at various
collision velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Radiative decay paths from electron capture into a H-like ion assum-
ing capture into a singlet state. . . . . . . . . . . . . . . . . . . . . . 21
1.6 Radiative decay paths from electron capture into a H-like ion assum-
ing capture into triplet states. . . . . . . . . . . . . . . . . . . . . . . 22
1.7 EBIT spectra resulting from collisional (direct) excitation (DE, top)
and CX (bottom) of bare and H-like S with He. . . . . . . . . . . . . 23
1.8 Simulated spectra using the xspec apec model for an ionized ther-
mal plasma, and the AtomDB CX (acx) model to represent the CX
emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 CX spectra with the same ion and different neutrals. . . . . . . . . . 35
2.1 Schematic of the CX collision along diabatic and adiabatic potential
energy curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 One pseudocrossing p between initial potential state 1 and final po-
tential state 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 N-1 pseudocrossings between N potential states, where each pn repre-
sents a pseudocrossing happening between the initial state and final
state n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Schematic of the LLNL EBIT-I. . . . . . . . . . . . . . . . . . . . . . 60
3.2 Sketch of the trap region of the LLNL EBIT-I. . . . . . . . . . . . . . 61
3.3 Schematic of an ideal microcalorimeter. . . . . . . . . . . . . . . . . . 68
3.4 Schematic of an X-ray pulse from an ideal microcalorimeter. . . . . . 70
3.5 A simplified calorimeter bias circuit diagram for a Si-thermistor device. 76
3.6 The ECS cryogenic package. . . . . . . . . . . . . . . . . . . . . . . . 81
xii
3.7 Image of the ECS detector chip. . . . . . . . . . . . . . . . . . . . . . 82
3.8 Filter transmission for the ECS optical and thermal blocking filters,
and optional windows along the ECS line of sight. . . . . . . . . . . . 84
3.9 Quantum efficiency for the absorbers on the ECS. . . . . . . . . . . . 87
3.10 Sketch of the gain calibration procedure and gain curve for the ECS. 93
3.11 The resistance vs. temperature curve of five pixels in the TEMS array. 98
3.12 A kilopixel array like the one currently used in the TEMS. . . . . . . 99
3.13 Circuit diagram for a 2-row × 2-column time-division SQUID multi-
plexer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.14 The current NIST-developed TEMS focal plane assembly. . . . . . . . 102
3.15 Spectral resolution of all multiplexed pixels in TEMS. . . . . . . . . . 104
3.16 FWHM energy resolution of 2.35 eV at 6 keV for one pixel in the
current TEMS detector array. . . . . . . . . . . . . . . . . . . . . . . 105
4.1 Spectrum of bare argon undergoing CX with neutral argon. . . . . . . 116
4.2 Spectra from bare Mg undergoing CX with four different neutral gases.117
4.3 S16+ and He charge exchange. . . . . . . . . . . . . . . . . . . . . . . 117
4.4 Hardness ratio as a function of the atomic number of the ion. . . . . . 121
4.5 Hardness ratio as a function of the ionization potential and the num-
ber of valence electrons of the neutral gas. . . . . . . . . . . . . . . . 122
5.1 Measured X-ray energy as a function of the EBIT cycle. . . . . . . . 130
5.2 X-ray filter transmission. . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Signal and background signals during a CX experiment. . . . . . . . . 137
5.4 Spectra of Ni19+ created in DE and CX of Ni19++H2, from 850–1175
eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5 Spectra of Ni19+ created in DE and CX of Ni19++H2, from 1175–1550
eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6 Spectra of Ni19+ created in DE and CX of Ni19++He, from 850–1175
eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.7 Spectra of Ni19+ created in DE and CX of Ni19++He, from 1175–1550
eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.8 CX spectra of Ni19++H2 and Ni
19++He. . . . . . . . . . . . . . . . . 144
6.1 spex-cx model and acx models plotted against Ni19++H2 EBIT CX
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2 spex-cx model and acx models assuming CX with atomic H plotted
against Ni19++H2 EBIT CX data, from 888 eV–1048 eV. . . . . . . . 157
6.3 Same as Figure 6.2, but for 1048 eV–1210 eV. . . . . . . . . . . . . . 158
6.4 Same as Figure 6.2, but for 1210 eV–1368 eV. . . . . . . . . . . . . . 159
6.5 Same as Figure 6.2, but for 1368 eV–1528 eV. . . . . . . . . . . . . . 160
6.6 Identifications of high-n CX lines assuming energies as calculated by
fac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.7 Comparison of high-n lines in the Ni19++H2 CX data and the two
acx models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
xiii
6.8 spex-cx model and acx models assuming CX with atomic H plotted
against Ni19++He EBIT CX data. . . . . . . . . . . . . . . . . . . . . 164
6.9 spex-cx model and acx models assuming CX with atomic H plotted
against Ni19++He EBIT CX data, from 888 eV–1048 eV. . . . . . . . 166
6.10 Same as Figure 6.9, but for 1048 eV–1210 eV. . . . . . . . . . . . . . 167
6.11 Same as Figure 6.9, but for 1210 eV–1368 eV. . . . . . . . . . . . . . 168
6.12 Same as Figure 6.9, but for 1368 eV–1528 eV. . . . . . . . . . . . . . 169
6.13 acx models assuming CX with He plotted against Ni19++He EBIT
CX data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.14 acx models assuming CX with He plotted against Ni19++He EBIT
CX data, from 888 eV–1048 eV. . . . . . . . . . . . . . . . . . . . . . 172
6.15 Same as Figure 6.14, but for 1048 eV–1210 eV. . . . . . . . . . . . . . 173
6.16 Same as Figure 6.14, but for 1210 eV–1368 eV. . . . . . . . . . . . . . 174
6.17 Same as Figure 6.14, but for 1368 eV–1528 eV. . . . . . . . . . . . . . 175
6.18 Simulated L-shell spectra for capture into Ne-like Ni configurations
2p1/29s1/2(J = 0) and 2p1/29s1/2(J = 1). . . . . . . . . . . . . . . . . 178
6.19 Model fit to the data for H- and He-like S+He CX. . . . . . . . . . . 182
6.20 Relative n, l, j-resolved capture cross sections for H-like S+He CX. . . 183
6.21 Relative n, l, j, J-resolved capture cross sections for He-like S+He CX. 184
6.22 Normalized relative cross sections as obtained with our pipeline and
fitting procedure and MCLZ calculations for CX between H-like S
and neutral He. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.23 Simulated spectrum from MCLZ calculations compared to data. . . . 189
6.24 Data, model fit to the data, and fit residuals for CX between Ni19+
and H2, for the M2 and 3G lines assuming only a 2p3/2 core hole. . . 192
6.25 Data, model fit to the data, and fit residuals for CX between Ni19+
and H2, for lines with energies between 890–1530 eV, assuming only
a 2p3/2 core hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.26 Data, model fit to the data, and fit residuals for CX between Ni19+
and H2, for the M2 and 3G lines, including all three core hole config-
urations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.27 Data, model fit to the data, and fit residuals for CX between Ni19+
and H2, for lines with energies between 890–1530 eV, including states
with all three core hole configurations. . . . . . . . . . . . . . . . . . 197
6.28 State-selective cross sections extracted from the model fit to Ni19+
and H2 data, for states with a 2p3/2 core hole. . . . . . . . . . . . . . 199
6.29 State-selective cross sections extracted from the model fit to Ni19+
and H2 data, for states with a 2p1/2 core hole. . . . . . . . . . . . . . 200
6.30 State-selective cross sections extracted from the model fit to Ni19+
and H2 data, for states with a 2s1/2 core hole. . . . . . . . . . . . . . 201
A.1 Cascade spectrum of H-like S with initial electron configuration 8s1/2. 209
A.2 Cascade spectrum of H-like S with initial electron configuration 7s1/2. 209
A.3 Cascade spectrum of H-like S with initial electron configuration 8p3/2. 210
A.4 Cascade spectrum of H-like S with initial electron configuration 7p3/2. 210
xiv
A.5 Cascade spectrum of H-like S with initial electron configuration 8d3/2. 211
A.6 Cascade spectrum of H-like S with initial electron configuration 8f5/2. 211
A.7 Cascade spectrum of H-like S with initial electron configuration 8f7/2. 212
A.8 Cascade spectrum of He-like S with initial electron configuration
1s1/26s1/2(J = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.9 Cascade spectrum of He-like S with initial electron configuration
1s1/27s1/2(J = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.10 Cascade spectrum of He-like S with initial electron configuration
1s1/26s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.11 Cascade spectrum of He-like S with initial electron configuration
1s1/27p1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.12 Cascade spectrum of He-like S with initial electron configuration
1s1/26p3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.13 Cascade spectrum of He-like S with initial electron configuration
1s1/27p3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.14 Cascade spectrum of He-like S with initial electron configuration
1s1/27d3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.15 Cascade spectrum of He-like S with initial electron configuration
1s1/27d3/2(J = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
A.16 Cascade spectrum of He-like S with initial electron configuration
1s1/27d5/2(J = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
A.17 Cascade spectrum of He-like S with initial electron configuration
1s1/27d5/2(J = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.18 Cascade spectrum of He-like S with initial electron configuration
1s1/24p3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.19 Cascade spectrum of He-like S with initial electron configuration
1s1/25p3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
A.20 Cascade spectrum of He-like S with initial electron configuration
1s1/27f7/2(J = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
B.1 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/28s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.2 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/29s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.3 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/210s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.4 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/211s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.5 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/211p1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.6 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/211p3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.7 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/29d5/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
xv
B.8 Cascade spectrum of Ne-like Ni with initial electron configuration
as one or a combination of 2p3/28f5/2(J = 2), 2p3/29f5/2(J = 2),
2p3/210f7/2(J = 2), 2p3/211f7/2(J = 2), 2p3/28g7/2(J = 3), 2p3/29g7/2(J =
3), 2p3/210g7/2(J = 3), or/and 2p3/211g7/2(J = 3). . . . . . . . . . . . 223
B.9 Cascade spectrum of Ne-like Ni with initial electron configuration
as one or a combination of 2p3/28f7/2(J = 2), 2p3/29f7/2(J = 2),
2p3/210f5/2(J = 2), 2p3/211f5/2(J = 2), 2p3/28g9/2(J = 3), 2p3/29g9/2(J =
3), 2p3/210g9/2(J = 3), or/and 2p3/211g9/2(J = 3). . . . . . . . . . . . 224
B.10 Cascade spectrum of Ne-like Ni with initial electron configuration
as one or a combination of 2p3/28f5/2(J = 4), 2p3/29f5/2(J = 4),
2p3/210f5/2(J = 4), 2p3/211f5/2(J = 4), 2p3/28g7/2(J = 5), 2p3/29g7/2(J =
5), 2p3/210g7/2(J = 5), or/and 2p3/211g7/2(J = 5). . . . . . . . . . . . 225
B.11 Cascade spectrum of Ne-like Ni with initial electron configuration
as one or a combination of 2p3/28f5/2(J = 4), 2p3/29f5/2(J = 4),
2p3/210f5/2(J = 4), 2p3/211f5/2(J = 4), 2p3/28g7/2(J = 5), 2p3/29g7/2(J =
5), 2p3/210g7/2(J = 5), or/and 2p3/211g7/2(J = 5). . . . . . . . . . . . 226
B.12 Cascade spectrum of Ne-like Ni with initial electron configuration
2p3/27s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
B.13 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/28s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.14 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/28s1/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.15 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/28d5/2(J = 3), 2p1/29d5/2(J = 3), 2p1/210d5/2(J = 3), 2p1/211d5/2(J =
3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.16 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/29f5/2(J = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.17 Cascade spectrum of Ne-like Ni with initial electron configuration one
or a combination of 2p1/28d5/2(J = 3), 2p1/29d5/2(J = 3), 2p1/210d5/2(J =
3), 2p1/211d5/2(J = 3), 2p1/28f7/2(J = 4), 2p1/29f7/2(J = 4), 2p1/210f7/2(J =
4), 2p1/211f7/2(J = 4), 2p1/28g9/2(J = 5), 2p1/29g9/2(J = 5), 2p1/210g9/2(J =
5), or/and 2p1/211g9/2(J = 5). . . . . . . . . . . . . . . . . . . . . . . 229
B.18 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/26d3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.19 Cascade spectrum of Ne-like Ni with initial electron configuration
2p1/29d3/2(J = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.20 Cascade spectrum of Ne-like Ni with initial electron configuration
2s1/211s1/2(J = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
B.21 Cascade spectrum of Ne-like Ni with initial electron configuration
2s1/211d5/2(J = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
C.1 Radiative cascade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
C.2 Scattering of X-ray light by a neutral. . . . . . . . . . . . . . . . . . . 233
C.3 Bremmstrahlung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
C.4 Collisional excitation by electron impact. . . . . . . . . . . . . . . . . 235
xvi
C.5 Photoexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
C.6 Collisional ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
C.7 Photoionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
C.8 Autoionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
C.9 Auger ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
C.10 X-ray fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
C.11 Radiative recombination. . . . . . . . . . . . . . . . . . . . . . . . . . 239
C.12 Dielectronic recombination. . . . . . . . . . . . . . . . . . . . . . . . 240
xvii
List of Abbreviations
AOCC Atomic Orbital Close Coupling
CAP Calorimeter Analog Processor
COB Classical Over-the-Barrier
CTMC Classical Trajectory Monte Carlo
CX Charge Exchange
DE Direct Excitation
EBIT Electron Beam Ion Trap
ECS EBIT Calorimeter Spectrometer
ESA European Space Agency
FAC Flexible Atomic Code
GSFC Goddard Space Flight Center
ICM Intracluster Medium
ISM Interstellar Medium
JAXA Japan Aerospace eXploration Agency
JFET Junction gate Field-Effect Transistor
LLNL Lawrence Livermore National Laboratory
LTE Long Term Enhancement
MCLZ Multi-Channel Landau Zener
MEC Multi-Electron Capture
NASA National Aeronautics and Space Administration
NEP Noise Equivalent Power
NIST National Institute of Standards and Technology
QE Quantum Efficiency
QMOCC Quantum Molecular Orbital Close Coupling
SCDP Software Calorimeter Data Processor
SEC Single Electron Capture
SQUID Superconduting QUantum Interference Device
SWCX Solar Wind Charge Exchange
TDM Time Domain Multiplexing
TEMS Transition-edge EBIT Microcalorimeter Spectrometer
TES Transition Edge Sensor
XARM X-ray Astrophysics Recovery Mission
xviii
Chapter 1: Charge Exchange: an Introduction
Around 3am on a late-January night, the Japanese amateur astronomer Yuji
Hyakutake decided to trudge back up the mountain where he made his first discovery
as a comet-hunter. The weather was supposed to get better soon, and he wanted
to re-photograph his comet. But when he looked up with his plate-sized binoculars,
he saw an object in the sky that he had not seen before. Late the next morning,
he faxed a report to experts in order to confirm his observation. A cometary expert
verified the sighting, and the media latched on; Hyakutake’s accidental discovery of
the comet that would soon bear his name would bring him so much fame that for
some time, he and his wife could not make telephone calls due to the number of
incoming press calls to their home. [Telescope, 2004, Times, 2002, Lab, 1996]
Three months later, Comet Hyakutake, shining as bright as a star and leaving
a 62,000-mile long tail in its wake, made its closest approach to Earth. It was near
enough that patient watchers could detect its motion against the stars using just
their eyes. But Comet Hyakutake would not only be remembered for its dramatic
show in the night sky.
Years before, scientists had predicted the generation of X-rays from comets
due to collisions between particles released by the comet as it approached the sun
1
and dust in the solar system [Ibadov, 1990]. Even before that, scientists detected
high-energy electrons in the atmosphere surrounding a comet when they performed
a flyby observation, suggesting that a process energetic enough to create X-rays
may be occurring in the coma [Kirsch et al., 1988]. Armed with this background
and motivated by the close approach of comet Hyakutake, Carey Lisse and his team
proposed to observe the comet with the ROSAT X-ray satellite. And with this
observation, comet Hyakutake made the news again: it was the first comet caught
emitting X-rays [James, 1998, Lisse et al., 1996].
The discovery came as a surprise to the astronomical community, and prompted
a surge of interest in determining the cause of the emission (e.g., Krasnopolsky
[1997], Bingham et al. [1997], Northrop et al. [1997], Haberli et al. [1997]). X-rays
are typically observed from hot and extreme objects, like black hole accretion disks
or shock-heated gas in supernova remnants—objects and environments that are mil-
lions of degrees. Comets, however, have temperatures that are well below freezing
[Krankowsky et al., 1986]. How could an object this cold produce such a large
amount of energetic light?
About a year after the 1996 ROSAT observation, Cravens [1997] identified the
answer. In contrast to a million-degree object producing thermal X-rays, cometary
X-rays are produced by a process well-known to plasma physicists but relatively
unfamiliar to astronomers called charge exchange (CX). CX is a close interaction
between highly charged ions and neutral atoms or molecules wherein the ions capture
electrons from the neutral into an excited quantum state. The electrons then cascade
down to the ground state, generating X-rays and other radiation. In the case of
2
comets, the ions are supplied by the solar wind, and the neutral species begin frozen
in the surface of the comet, and sublimate away as the comet approaches the sun.
Since this surprising discovery, and as we learn more about its unique spectral
signatures and likelihood of occurrence throughout the universe, CX has become
a popular explanation for anomalous X-ray emission in astrophysical environments
(see Section 1.5). However, upon closer inspection, it is becoming clear that we can-
not blindly trust even the best CX models to accurately represent our observational
data. There are many open questions regarding the underlying atomic physics be-
hind CX that lead to its unique spectral features, described in the following chapters,
that we must address through further experimentation and improved modeling.
This dissertation seeks to build a better understanding of the mechanics and
diagnostics of CX through the marriage of these two subfields: high-resolution ex-
perimental data together with theoretical calculations of CX. In the rest of Chapter
1, we outline the necessary ingredients for modeling and identifying CX spectra,
describe several astrophysical environments in which CX has been observed or pos-
tulated to occur, and present some of the challenges we are facing in identifying
and understanding this emission. In Chapter 2, we describe the theoretical and
computational tools used in this work. Chapter 3 dives into the experimental tools
and facilities we use, namely an Electron Beam Ion Trap (EBIT) and an X-ray mi-
crocalorimeter. We also describe an instrument in development that will allow us
to perform these measurements faster and with better resolution than previously
possible. Chapter 4 presents experimental K-shell (n = x → 1 transitions) data
that highlight both the subtle nature of the CX interaction and the difficulty in
3
including those nuances in spectral synthesis codes. Chapter 5 presents the first
high-resolution L-shell (n = x → 2 transitions) CX spectra of Ne-like Ni and de-
scribes what we can learn from these results. In Chapter 6, we take these data a
step further and present a pipeline to calculate relative state-selective capture cross
sections, previously only available from intensive theoretical modeling. We then
compare some of our results to existing theory. In Chapter 7, we discuss where we
will go next. Finally, the appendices present supplemental information: Appendies
A and B present the individual spectra used in the models described in Chapter 6
for S16++He and N19++H2, respectively, and Appendix C serves as a refresher on
various X-ray emission mechanisms discussed in this work.
1.1 Charge Exchange Basics
Charge exchange (CX), sometimes referred to as charge transfer, is a semi-
resonant process in which a highly charged ion captures one or more electrons from
a nearby neutral atom or molecule during a close interaction. After being captured,
the electron(s) relax to the ground state through a radiative cascade, emitting one
or more photons. It can be represented as
Aq+ +B → A(q−m)+ +Bm+, (1.1)
where A is the highly charged ion with charge q, B is the neutral species, and m
is the number of captured electrons. A pictorial representation of this interaction
assuming single electron capture (SEC) is shown in Figure 1.1.
4
Figure 1.1 Illustration of the CX interaction. The process can generally
be described in three steps: an ion and neutral approach each other,
the ion captures one or more electrons from the neutral into a highly
excited state, and then those electrons radiatively cascade to the ground
state. Figure drawn in the style of the CX figure in http://chandra.
harvard.edu/xray_sources/solar_system.html.
5
During the CX interaction, as the ion and neutral approach, their electric
fields are superposed, distorting the individual energy levels. At certain internuclear
distances, the energy levels of the ion and neutral overlap. It is at this so-called curve
crossing that the electron(s) from the neutral will non-radiatively transfer to the ion
[Wargelin et al., 2008]. There are several curve crossings at different internuclear
distances, so electron(s) can be captured into different quantum states. Because
the electron(s) keep approximately the same binding energy during the transfer,
capture is often into a highly excited quantum state of the ion [Smith et al., 2014b]
(see Equation 1.2). The details of the electron capture depend on the ion and
neutral species, the collision energy, and the number of electrons transferred [Janev
and Winter, 1985]; other quantities may also be important, but their effects are
difficult to disentangle (see Chapter 4).
Typical total cross sections for CX can be approximated by σCX ∼ q × 10−15,
where q is the charge of the ion [Wargelin et al., 2008]; for astrophysical plasmas,
they are often on the order of 10-14 to 10-15 cm2, depending on the collision energy
and reaction pair [Gu et al., 2016]. This is very large compared to, for example,
collisional ionization, which, using Fe K-shell excitation as an example, is on the
order of 10-22 cm2 [Krasnopolsky et al., 2004, Stancil et al., 2002, Cravens, 2002,
Gilbody, 1986]. For this reason, it is vital to consider the contribution of CX in
any astrophysical spectrum measured from an environment in which hot plasmas
interact with cool gas. See Section 1.5 for more information on the astrophysical
relevance of CX.
In order to properly identify the CX component in astrophysical and laboratory
6
spectra, we must be able to model (1) the state-selective (i.e., n, l, j, J-resolved in
jj-coupling or n, L, S-resolved in LS coupling) cross section of the captured electron
(what is its initial quantum state?), as well as (2) its subsequent radiative decay
to the ground state (what are the energies of the X-rays that the electron emits
during its downward cascade?). (2) is relatively straightforward given the initial
distribution of n and l capture states, requiring only accurate atomic data and
quantum mechanical selection rules (though obtaining accurate atomic data is not
necessarily trivial). The bigger challenge lies in (1): though CX is a conceptually
simple process, a detailed theoretical understanding of the subtleties involved can be
daunting. Several theoretical methods have been developed to calculate total and
state-selective cross sections, with variations in accuracy and utility when applied to
different environments. The modeling efforts used in this dissertation are described
more completely in Chapter 2; here, I give a brief overview of some of the most
common theoretical methods applied to CX.
1.2 Introduction to Charge Exchange Theory
Several parameters influence the aforementioned variations in accuracy and
utility between theoretical methods. A primary parameter is the collision velocity:
the details of the close interaction between the ion and the neutral as they approach
just before, during, and after electron capture are highly velocity-dependent. This
strongly affects the distribution of angular momentum state(s) l for the captured
electron(s).
7
Low velocity collisions, where v  v0 (where v is the collision velocity and
v0 is the classical velocity of the bound electron) are in the “adiabatic region”
[Janev, 1983]. This region is applicable to the experimental CX data presented
in this dissertation, and is explained in more detail in Section 2.3. In these slow
collisions, the ion-neutral system can be seen as a pseudomolecule that shares the
electron during the collision. In this case, the key parameters are the various initial
(pre-electron transfer) and final (post-electron transfer) potential energy curves of
the pseudomolecule and how they interact [Janev et al., 1985]. At intermediate
velocities (v ∼ v0), the number of potential final states (also called reaction channels)
increases. This is significant because to determine the final reaction channel, one
must consider the interaction of all possible states. At these velocities, the fractional
amount of time that the electron is in a shared state decreases as compared to
collisions at low velocities; the electron is mostly bound to one particle. It is thus
more accurate to represent this collision as an ensemble of atomic states. At high
velocities (v/v0 ∼ 3−4), neighboring potential final states couple more strongly, and
in some cases, the electron can be transferred to the continuum [Janev et al., 1985].
In general, theories that consider the pseudomolecular state of the ion-neutral pair
better describe slow CX collisions. The most accurate codes for the higher energy
ranges consider how discrete atomic and/or continuum states interact [Janev et al.,
1985].
Besides the relative velocity of the collision, two other parameters affect the
detailed interaction between the ion and neutral during CX: the charge of the ion
(q) and the number of electrons in the ion. The charge scales the velocity ranges
8
for which various theoretical methods are valid, and more electrons in the system
makes the interaction between reaction channels more complex [Janev et al., 1985].
The number of electrons captured from the neutral also significantly affects the
resulting CX spectrum. Single electron capture (SEC) dominates in most astrophys-
ical environments due to the high cosmic abundance of atomic H. If multi-electron
neutrals are present, however, multi-electron capture (MEC) is a non-negligible pro-
cess in slow CX collisions [Ali et al., 1994, 2005, Hasan et al., 2001]. This is likely
present in the data presented in Chapter 5. Unfortunately, only a small fraction of
theoretical treatments have incorporated multi-electron capture (MEC) into their
calculations (just the Classical Over the Barrier method and the Classical Trajec-
tory Monte Carlo method [Cariatore and Otranto, 2013, Ali et al., 2005]). Section
1.7 discusses this further.
The simplest theoretical treatment of CX is with the Classical Over-the-Barrier
method (COB), applicable to collisions between 100 eV/amu–10 keV/amu [Bode-
wits, 2007]. COB is based off the classical estimations of the joint potential well of
the ion and neutral. In COB, the electron is transfered from the neutral to the ion
if the potential barrier between the two nuclei is lower than the binding energy of
the electrons. One can estimate the total capture cross section and the most likely
n level for electron capture with COB, but it does not give a n or l distribution
[Bodewits, 2007].
The Classical Trajectory Monte Carlo method (CMTC) is slightly more so-
phisticated than the COB method. It traces the three-body dynamics between the
ion, neutral, and one active electron. It determines the time-dependent position and
9
momentum of each of these particles for a large ensemble of configurations to sim-
ulate a collision [Olson, 2006, Abrines and Percival, 1966, Olson and Salop, 1977].
CTMC is best for high energy collisions (above ∼5 keV/amu), but does not ade-
quately describe experimental data taken at low collision energies [He et al., 2009,
Olson, 2006, Janev, 1983, Beiersdorfer et al., 2000a].
The Multichannel Landau-Zener approximation (MCLZ) is best for low-energy
collisions (from several tens of eV/amu to ∼5 keV/amu) [Mullen et al., 2016]. It
considers the quasimolecular energy configurations of a slow collision between an
ion and a neutral. It calculates diabatic potentials of pre- and post-capture states
and calculates the probability of a pseudocrossing, or a location where the quasi-
molecule can move from the pre-capture potential curve to the post-capture potential
curve, representing the electron transfer process. MCLZ can estimate state-selective
capture cross sections and can generally predict large-scale trends in capture cross
section, but results when compared to experiment are mixed [Wu et al., 2011]. We
use MCLZ calculations to compare to our data in Chapter 6. More details on MCLZ
can be found in Section 2.3.
The most accurate methods are close coupling methods, such as Quantum
Mechanical Molecular Orbital Close Coupling (QMOCC) and Atomic Orbital Close
Coupling (AOCC). These methods calculate the quasi-molecular or atomic structure
of the electron transfer, respectively, and can yield state-selective capture cross sec-
tions. QMOCC is most accurate for low-energy collisions (below ∼1 keV/amu), and
AOCC is most accurate for high-energy collisions (up to ∼500 keV/amu) [Cumbee,
2016]. However, few of these close-coupling calculations exist due to their compu-
10
tational intensity.
1.3 Theoretical Approximations
In the absence of a full state-selective cross section calculation via one of the
aforementioned methods, one can approximate the initial distribution of n and l
capture states. The peak in the n capture distribution can be approximated as:
nmax ∼ q
√
IH
In
(
1 +
q − 1√
2q
)−1/2
∼ q3/4, (1.2)
where IH and In are the ionization potentials of hydrogen and the neutral target,
respectively, and q is the ion charge [Janev and Winter, 1985]. This relationship
follows [Janev and Winter, 1985] from considering the diabatic potentials of the
initial (Vn0, covalent) and final (Vn, excited ionic) states of the reaction partners
in the adiabatic collision velocity regime, at large internuclear distances [Janev and
Presnyakov, 1981]:
Vn0(R) = − 1
2n20
+O(R−4) (1.3)
and
Vn(R) = − q
2
2n2
+
q − 1
R
+O(R−2). (1.4)
The internuclear distances at which the strongest interaction between the ini-
tial and final potentials occurs when the initial and final potential energies become
degenerate (i.e. equating Equations 1.3 and 1.4). Solving for Rn gives the internu-
11
clear distance where electron transfer occurs:
Rn ' q − 1q2
2n2
− 1
2n20
. (1.5)
It should be noted that this approximation results from ignoring the higher
order terms in Equations 1.3 and 1.4. For states that have the same initial and
final symmetry, one cannot ignore these terms. In this case, electron exchange
effects become important close to Rn, which add additional terms to the initial and
final potentials. This leads to several so-called “avoided curve crossings,” which are
discussed further in Section 2.3.
Solving Equation 1.5 for n, setting n0 = 1 (assuming an initial ground state
of the transferred electron) and considering the distance R0 where electron capture
dominates in the decay model [Janev and Presnyakov, 1981, Janev, 1983],
R0 ' 2n20
√
2q, (1.6)
yields Equation 1.2.
The scaling of nmax ∼ q3/4 in Equation 1.2 also follows from other approxi-
mations, such as the COB method and the distorted-wave approximation [Janev,
1983].
Though this commonly-used approximation for nmax is independent of velocity,
it is more accurate (though not straightforward) to consider how nmax behaves in
the three collision velocity regimes described above [Janev et al., 1985]. The shape
of the distribution of n values around nmax also changes as a function of velocity:
12
for low Z (Z. 10) and very low collision velocities, the n distribution is narrowly
peaked around nmax. This distribution widens with increasing collision velocity.
The shape of this distribution is often asymmetric around nmax due to the fact
that for n < nmax, the energy spacing between levels is larger than those with
n > nmax, so the capture cross section into those (n < nmax) states decreases more
rapidly [Janev et al., 1985]. For intermediate to high collision velocities, electron
momentum transfer begins to contribute to the electron transfer process, and nmax
strays from Equation 1.2 and becomes velocity-dependent [Janev and Winter, 1985,
Janev et al., 1985].
The l-state distribution, which is highly dependent on the relative velocity
between the ion and neutral, is more difficult to calculate theoretically. This is es-
pecially problematic in the low collision energy regime (below 1 keV/amu), where
collisions are slow enough that electron-electron interactions and pseudo-molecular
effects become important. Two spectral synthesis codes for CX, spex-cx and acx
[Smith et al., 2014a, Gu et al., 2016], discussed in Section 2.2, employ certain ap-
proximations for the l distribution based on work by Janev and Winter [1985]. These
approximations lead to four different distribution functions Wn(l), for a given value
of n:
Landau-Zener:
Wn(l) = l(l + 1)(2l + 1)
(n− 1)!(n− 2)!
(n+ l)!(n− l − 1)! (1.7)
Separable:
Wn(l) =
2l + l
q
e
−l(l+1)
q (1.8)
13
Even:
Wn(l) =
1
n
(1.9)
Statistical:
Wn(l) =
2l + 1
n2
(1.10)
The Landau-Zener and Separable approximations are applicable to the low
collision energy regime [Janev, 1983, Janev and Winter, 1985]. In this regime,
there is not enough momentum in the system to populate a higher orbital angular
momentum state. As collision energy increases, the distribution moves towards
statistical weighting, where higher l values are preferred [Janev, 1983, Janev and
Winter, 1985]. This can be explained through a classical description of the CX
process, where momentum from the initial collision is transferred into the electron’s
orbital angular momentum. The even distribution does not have a physical basis,
but instead represents a regime between the low- and high-velocity extremes [Smith
et al., 2014a]. Though these approximations have utility as starting points for
calculating CX spectra, they do not incorporate the quantum mechanical or quasi-
molecular effects of the CX interaction, and should be used with great care (see
Chapter 6).
1.4 Spectroscopic Observables
Though modeling CX emission can be challenging, we have learned that there
are spectral features that can be generally diagnostic of the occurrence of CX.
For SEC onto bare ions, if the electron populates a low angular momentum
14
state, the resulting cascade has a characteristic spectrum of one or several strong
high-n Lyman lines from direct decay from n = nc, l = 1 to the ground state,
where nc is the principle quantum number of the captured electron. These lines are
much stronger in CX than from collisional excitation [Smith et al., 2014b]. If the
electron initially populates a high angular momentum state, it will decay along the
Yrast chain in steps of ∆n = −1 and ∆l = −1, giving off low-energy photons, and
eventually leading to a strong Lyman α line from the n = 2 → 1 transition. In
this case, the high-n line(s) is suppressed [Wargelin et al., 2008]. These scenarios
are shown in Figures 1.2 and 1.3. As the l distribution moves towards statistical
weighting at high collision energy, a higher collision energy usually results in capture
into a high l state. Because of this, the ratio of high-n → n = 1 lines to Lyman α
can be used as a diagnostic of the collision energy. Chapter 4 uses this ratio, called
the hardness ratio, to perform spectral diagnostics of experimental K-shell data.
Figure 1.4 illustrates this spectral dependence on the collision energy, as calculated
with a CTMC model.
For SEC onto H-like ions (creating He-like ions), the angular momentum cou-
pling between the already-present electron with the newly captured one results in
singlet and triplet states in an LS coupling scheme. Singlet and triplet states have
distinct decay patterns.
Singlet np can directly decay to the 1s state, so singlet states can have high-
n = nc → 1 emission as in the H-like case. However, most capture states decay
to singlet 2p, which subsequently decays to the ground state to yield a strong w
(resonance) line.
15
Lyman-δ	
Lyman-α	
l	value	
0	 1	 2	 3	 4	
1	
2	
3	
4	
5	
n	
va
lu
e	
Figure 1.2 Radiative decay paths from initial electron capture into a bare
ion at low collision energy, which preferentially populates low angular
momentum states. In this case, capture primarily into 5p leads to a
strong Lyman-δ line (shown in purple). Not all electron transitions are
shown, and energy spacing is not to scale.
16
1	
2	
3	
4	
5	
n	
va
lu
e	
l	value	
0	 1	 2	 3	 4	
Lyman-α	
Low	energy	photons	
Figure 1.3 Radiative decay paths from initial capture into a bare ion
at high collision energy, where the l states are populated statistically.
In this case, capture into mostly high angular momentum states leads
to significant Yrast cascades. In this case, decay along the Yrast chain
after capture primarily into 5g leads to the low-energy photons shown
in purple. These cascades primarily populate the 2p state, which then
decays to the ground state, yielding a strong Lyman α line (shown in
yellow). Not all electron transitions are shown, and energy spacing is
not to scale.
17
Figure 1.4 Theoretical spectra as calculated with the CTMC method for
CX between bare O and H, at several collision velocities. This shows the
suppression of the high-n Lyman line with increasing velocity, indicating
a statistical population of l states. Reproduced from Wargelin et al.
[2008].
18
In low-Z ions, which are well-described by an LS coupling scheme, triplet states
are spin-forbidden to decay to the 1s2 1S0 ground level via an electric dipole tran-
sition. This means that electrons will cascade downward from their initial capture
state to populate the 1s2s 3S1 level or the 1s2p
3P0,1,2 levels. From there, they
undergo a magnetic dipole transition to the singlet 1s2 1S0 ground state. This can
be via a z (forbidden) line (1s2s 3S1 → 1s2 1S0) or an x or y (intercombination) line
(1s2p 3P2,1 → 1s2 1S0). Triplet states are thus dominated by emission from these
lines, and high-n→ 1 transitions are suppressed.
Schematics of these transitions for He-like ions are shown in Figures 1.5 and
1.6.
The Kα complex, or the combination of the w, x, y, and z lines, thus dominates
CX spectra from He-like ions. He-like CX spectra typically show relatively weak
high-n→ 1 lines in comparison to H-like spectra, though these lines are still stronger
in CX than in collisional excitation. With increasing Z, spin-changing transitions
become allowed, and emission from high-n → 1 lines increases [Beiersdorfer et al.,
2000a].
Recombination typically preferentially populates triplet states due to their
larger statistical weight [198, 1989] (although some CX experiments have shown
departures from this, e.g. Leutenegger et al. [2010], Bliek et al. [1998]). This
means that for CX emission from He-like ions, the sum of the intercombination
and forbidden lines is typically much stronger than the w line.
It is important to note that enhanced forbidden and intercombination lines
can also be created with other processes such as radiative recombination of H-like
19
to He-like ions or inner-shell ionization of Li-like ions. To distinguish radiative
recombination from CX, one may look for a continuum feature that will be present
in spectra from a radiatively recombining plasma (see Appendix C), but not CX.
The challenge, however, is that these and other processes are often occurring in
parallel and are thus not straightforward to disentangle.
Figure 1.7 shows a K-shell laboratory spectrum resulting from collisional exci-
tation (or direct excitation, DE) compared to CX, for the case of bare and H-like S
interacting with He. One can see the enhanced forbidden line in the He-like species,
as well as the strong high-n Lyman line in the H-like species.
Specific and widely accepted spectral diagnostics for L-shell ions at high-
resolution are scarce in the literature; Chapter 5 presents experimental data for
Ne-like Ni to begin to rectify this.
In order to obtain a global view of the differences between a collisional and
CX-dominated plasma, it is necessary to consider spectra from more than one ion.
The xspec spectral fitting package [Arnaud, 1996] is useful for comparing various
emission mechanisms to note spectral differences, and can incorporate a CX model
called acx. acx uses approximations to calculate the n, l electron capture state
rather than theoretical or experimental cross sections (see Section 2.2) and thus has
limited accuracy and applicability, but it can give a general sense of CX spectral
signatures and diagnostics. Figure 1.8 shows an acx spectral model compared to
a collisional thermal plasma model (apec). Both focus on the energy range 0–2
keV and assume an electron temperature of 4 keV and a solar abundance of metals.
One can notice the stark differences in spectral shape between the two models. This
20
1	
2	
3	
4	
5	
n	
va
lu
e	
l	value	
0	 1	 2	 3	 4	
1S0	
1D2	
1F3	
1G4	
w	
1P1	
Singlet	states	
Figure 1.5 Radiative decay paths from initial capture into a H-like ion
assuming capture into a singlet state and an even l-distribution. Elec-
trons that initially populate l=1 can decay directly to ground (in this
case, the blue K-δ line), but electron capture into other states primar-
ily decay to 1P1, which decays to the singlet ground state
1S0 via the w
(resonance) line, shown in yellow. Not all electron transitions are shown,
and energy spacing is not to scale.
21
1	
2	
3	
4	
5	
n	
va
lu
e	
l	value	
0	 1	 2	 3	 4	
1S0	
x	
3P2	
3S1	
3P1	
3P0	
y	z	
3D3	
3D2	
3D1	
3F4	
3F3	
3F2	
3G5	
3G4	
3G3	
Singlet	ground	
Triplet	states	
Figure 1.6 Radiative decay paths from initial capture into a H-like ion
assuming capture into triplet states and an even l-distribution. Direct
decay from a high-n state is forbidden; transitions to the 1S0 ground
state require a magnetic dipole transition. Transitions from 1s2p 3P2
and 1s2p 3P1 to ground result in an x or y (intercombination) line,
respectively (shown in yellow). Electrons in the 1s2p 3P0 state transition
to 1s2s 3S1. Transitions from 1s2s
3S1 to ground lead to a z (forbidden)
line, shown in red. Not all electron transitions are shown, and energy
spacing is not to scale. In particular, the spacing between the 3P0,1,2
states is exaggerated for clarity.
22
DE
CX
Figure 1.7 Spectra resulting from collisional (direct) excitation (DE, top)
and CX (bottom) of bare and H-like S with He, as measured with an
X-ray microcalorimeter at an Electron Beam Ion Trap. Note the highly
enhanced Ly-η line and the He-like z line in CX.
demonstrates the importance of considering CX in astrophysical observations: if CX
is present but not accounted for, the derived abundances of, for example, a purely
thermal model may have to be significantly enhanced in order to reproduce strong
line emission that actually stems from CX.
Though it is vital to consider the contribution of CX to our observations, it
is perhaps even more important to tackle the challenges of detailed modeling: if we
apply inaccurate CX spectral models to our data, we run the risk of misinterpreting
the physical attributes of the objects and environments we observe. This will become
especially obvious thanks to recent advancements in spectrometers: we will soon be
able to resolve astrophysical and laboratory plasmas, especially extended sources, at
higher resolution than ever before. This makes the need for accurate models more
23
0.0 0.5 1.0 1.5 2.0
Energy (keV)
APEC
AtomDB CX (ACX)
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure 1.8 Simulated spectra under 2 keV resulting from a 4 keV plasma
using the AtomDB CX (acx) model to represent the CX emission (top)
and the xspec apec model for an ionized thermal plasma (bottom).
24
acute. Luckily, attention on CX has recently increased due to a a wider recognition
of its unusual spectral signature and large total cross section, which may lead to
greater modeling effort. The next section describes several environments where CX
has been observed or postulated to occur.
1.5 Charge Exchange in Astronomy
CX occurs ubiquitously in our solar system due to interactions between neu-
trals in various locations and the ions in the solar wind (called solar wind CX,
or SWCX), and most likely in many other astrophysical environments where hot
plasma comes in contact with cool gas. While only SWCX has widespread accep-
tance, it is worthwhile to give a sampling of the myriad objects and environments
where CX may occur in order to understand the importance of having an accurate
and widely applicable spectral theory for CX.
1.5.1 Solar System
The solar wind is the source of the highly charged ions for most instances of
CX in the solar system. Its composition is mostly protons, electrons, and alpha
particles, but 0.1% of the composition consists of highly charged ions. These are
mostly bare, H-like, and He-like C and O, though bare and H-like Ne and Mg may
be present during coronal mass ejections, as well as some L-shell Fe ions and bare,
H-like, and He-like N [Schwadron and Cravens, 2000, von Steiger et al., 2000]. The
wind velocity ranges from 300–800 km/s; the fast wind stems from the polar regions,
25
and the slow wind comes from the equatorial regions and has a higher abundance
of highly charged ions [Schwadron and Cravens, 2000, von Steiger et al., 2000].
As described earlier in this chapter, CX was discovered in our solar system in
1996 with the unexpected detection of X-rays from the comet C/Hyakutake 1996
B2 via observations with the ROSAT satellite by Lisse et al. [1996]. After Cravens
[1997] determined that SWCX was the X-ray emission mechanism in Hyakutake,
more than 20 comets were subsequently observed in the EUV and soft X-ray bands
(e.g., Lisse et al. [1999], Krasnopolsky et al. [2000], Dennerl et al. [2003], Bodewits
et al. [2007], Christian et al. [2010]). More recently, Beiersdorfer et al. [2003b] verified
experimentally using an Electron Beam Ion Trap (see Section 3.1) that cometary
spectra can be extremely well-described solely via CX X-ray emission. Because
cometary X-ray emission stems almost completely from CX, comets are one of the
cleanest astrophysical laboratories for theoretical benchmarking and comparison to
experiment.
SWCX has also been observed in planetary atmospheres. In nonmagnetic
planets such as Venus and Mars, the solar wind interacts with the exospheres much
like it does with cometary comae; the main difference is that the exospheric density
is higher than in a coma, so the ions more quickly lower their charge state as they
undergo multiple CX reactions in a thin layer. Planets also exhibit a smaller CX flux
than nearby comets due to their smaller spatial extent on the sky. The first instance
of planetary CX was observed in Mars; both Chandra and XMM reported X-ray
emission from the halo [Dennerl, 2002, Gunell et al., 2004, Dennerl et al., 2006].
While the spectra were dominated by X-ray fluorescence, there was line emission
26
from highly charged C, N, O, and Ne ions in the halo, similar to that of cometary
spectra. The forbidden line was also enhanced in the He-like O lines, confirming CX
as the origin of the emission.
Given CX being detected in Mars, Venus was expected to show even stronger
signs of CX due to its larger size and closer proximity to the sun, placing it in a
denser region of the solar wind. However, initial observations of Venus with Chandra
showed that the X-ray emission is dominated by fluorescence [Dennerl et al., 2002].
Later observations performed at solar minimum showed two spatially distinct origins
of X-rays: the troposphere and the exosphere. The X-rays from the exosphere were
brightest in the He-like O spectral line band, which are also the brightest lines
in cometary X-ray spectra resulting from CX. It was thus determined that the
exospheric X-ray emission resulted from SWCX with neutrals [Dennerl, 2008].
Emission lines below 2 keV from the polar region in Jupiter have been at-
tributed to CX [Gladstone et al., 2002, Elsner et al., 2005, Branduardi-Raymont
et al., 2007], however it is not clear if the CX is due to interactions with ions
from the solar wind, those accelerated in Jupiter’s magnetosphere, or both [Bunce
et al., 2004, Cravens et al., 2003]. Ions are accelerated by the magnetosphere to
velocities of at least ∼5000 km/s (∼ 130 keV/amu), so in principle, with sufficient
understanding of the spectral dependence on velocity (i.e., better models) and high
enough spectral resolution, one could determine the origin of the ions with further
high-resolution observations.
Although comets are one of the brightest sources of SWCX, CX in the Earth’s
exosphere and the heliosphere has the most pervasive effect on X-ray observations
27
we make. Anomalous diffuse X-ray emission was discovered during the ROSAT All-
Sky Survey and dubbed “long term enhancements” (LTEs) [Snowden et al., 1994],
as it varied on time scales from several hours to days. The connection between solar
wind variation and the long term enhancements was made by Cravens et al. [2001],
suggesting that SWCX was being produced locally with neutrals in the Earth’s ex-
osphere. This was confirmed with Chandra spectra of the dark moon [Wargelin
et al., 2004]: as the moon blocked out the cosmic X-ray background, only CX pro-
duced locally was left, and the observed emission matched models for X-ray emission
via SWCX. Similarly, SWCX with the stream of neutral ISM flowing through the
heliosphere leads to variable X-ray emission on longer time scales than the LTEs.
This local CX emission means that any observation we make has a spatially,
temporally, and spectrally variable X-ray foreground: exospheric CX affects all low-
Earth orbit observations, and heliospheric CX affects any observation made from
within the solar system. This presents a subtle challenge: this local X-ray emission
is superposed on more distant objects, and thus can lead to misinterpretations of
observations [Snowden et al., 2004]. Background subtraction is complicated due to
the fact that the emission is ubiquitous and temporally variable on many scales. If
the emission varies over time, we might hope to identify the CX component, but
the X-ray flux does not always change during the observation. Even if it does, it is
unlikely that we can identify all of it, especially the heliospheric contribution that
may vary on long time scales. This foreground CX will continue to affect future
observations until a more satisfactory way of identifying or modeling the foreground
CX is found.
28
Observations of the soft X-ray background is one example of the complication
of exospheric and heliospheric SWCX. It was thought that the Local Hot Bubble,
a non-symmetric cavity around the sun about 60 pc in radius thought to contain a
106 K plasma, was responsible for the soft X-ray background [Sanders et al., 1977,
Cox and Anderson, 1982, Snowden et al., 1990]. However, it was realized that this
diffuse emission has a signature that is similar to CX, both morphologically and
spectrally [Cox, 1998, Cravens, 2000, Lallement, 2004a, Welsh and Lallement, 2005,
Koutroumpa et al., 2009b]. Observations by Chandra [Smith et al., 2005, Hickox
and Markevitch, 2006], XMM [Snowden et al., 2004] and Suzaku [Fujimoto et al.,
2007, Henley and Shelton, 2007] measured temporal variations in the soft X-ray
background flux, adding credence to the idea of its exospheric/heliospheric origin.
Various authors made estimates of the CX contribution to the soft X-ray
background that ranged from 5 percent to nearly all of it, depending on direction and
energy band [Cravens, 2000, Wargelin et al., 2004, Cravens et al., 2001, Lallement,
2004a, Koutroumpa et al., 2007, Robertson et al., 2009, Lallement, 2004b]. Finally,
the DXL sounding rocket flight made an observation of the helium focusing cone,
an area within the heliosphere of a greater than average density of neutral He. The
2012 observation led to a firm measurement that up to 40% of the diffuse X-ray
background stems from heliospheric SWCX [Galeazzi et al., 2014]. An accurate
measurement of the SWCX contribution to the diffuse X-ray background directly
affects our assumptions of the conditions of our local ISM, as well as astrophysical
objects.
29
1.5.2 Astrophysical and Other CX
CX may also be very important in setting the ionization balance and for cool-
ing via line formation in nonequilibrium plasmas such as supernova remnants, in
interactions between the ISM and stellar winds, and in galaxies and galaxy clusters.
It has been postulated for some time that CX can occur in supernova remnants
[Chamberlain, 1956, Serlemitsos et al., 1973], at the interface between the shock-
heated hot plasma and the neutral ISM beyond the shock front. However, Wise and
Sarazin [1989] found that CX emission should contribute less than 10% via He-like C
and N lines, and even less from other elements. Observations by Lallement [2004b]
showed that emission from CX at the shock front is negligible, save possibly having
an effect on limb brightening if CX is occurring along our line of sight. However,
several authors continue to suggest CX as a likely mechanism for emission in the
Cygnus loop [Katsuda et al., 2011, Cumbee et al., 2014, Roberts and Wang, 2015].
Thus, currently, the picture of CX occurring in supernova remnants remains unclear,
though this could be due to limitations in detector resolution for extended sources
and inaccuracies in theoretical modeling.
CX likely plays a part in X-ray emission around winds, ranging in scale from
stellar to galactic. Pollock [2007] found that CX is likely to be as important in
stellar winds interacting with neutral ISM as it is for SWCX around comets in the
solar system. On the galactic scale, it is possible that ionized galactic wind could
collide with cool ambient gas in halos of starburst galaxies and result in CX. M82 is
a potential example of this: Tsuru et al. [2007] did a deep observation with Suzaku
30
and found that CX may contribute significantly to emission in the K-shell O lines
in the extended X-ray halo. Ranalli et al. [2008] observed the central region of the
galaxy and found unusual emission from H-like O lines that did not fit a solely
thermal model, but could be explained by CX emission. Liu et al. [2010] observed
the bulge and found unusually strong emission from the He-like triplet. Though
these results are intriguing, none lead to a definite answer regarding the occurrence
of CX in these regions.
Closer to home, the Galactic Ridge is an area of the Milky Way that extends
a few degrees above and below the Galactic Plane, and around the Galactic Center,
extending about 45 degrees in longitude. Diffuse X-ray emission was observed in
Worrall et al. [1982] as both continuum radiation and emission lines from H- and
He-like Mg, Si, S, Ar, Ca, and Fe [Koyama et al., 1996, Ebisawa et al., 2001, Muno
et al., 2004]. Tanaka et al. [1999] and Tanaka [2002] postulated that this could
come from CX between low energy cosmic rays and neutral gas in the Galactic
Plane. However, others [Bussard et al., 1978, Rule and Omidvar, 1979] calculate
that there are not enough cosmic rays of the right energy to produce the H- and
He-like lines observed, and that the cross section for CX at those high energies is
too small to lead to the observed line intensities.
CX is also very likely to take place in galaxy clusters, when rarified hot intra-
cluster medium gas encounters cold interstellar clouds or filaments at high velocity.
Walker et al. [2015] showed that CX may contribute a fraction of the X-ray flux in
the 0.5–1.0 keV band in galaxy clusters like Perseus, where the hot ICM interacts
with Hα filaments.
31
CX may even be confused with dark matter. Bulbul et al. [2014] and Boyarsky
et al. [2014] both independently discovered an unexpected emission line at ∼3.5
keV in stacked XMM-Newton data of 73 galaxy clusters. They suggested that since
there are no expected atomic transitions of a thermal plasma at that energy, it
was possible that this was a signature of a sterile neutrino decay, a proposed dark
matter particle. However, Gu et al. [2015] proposed that line was not due to a sterile
neutrino decay, but rather due to CX between bare S ions from the hot intracluster
medium that interacted with neutral H in cold, dense clouds. This was supported
by laboratory work done by Shah et al. [2016]. However, Hitomi observations of the
Perseus cluster with a X-ray microcalorimeter did not show any excess flux at this
energy [Aharonian et al., 2017], though it should be noted that the observation was
not long enough to compare to the stacked XMM-Newton data. In addition, the
authors of the Hitomi study mention that there was a slight excess at the 1.5σ level
at 3.44 keV that could be due to a high-n to n = 1 transition of H-like S following
CX.
1.6 Diagnostic Utility
With the wide variety of possible astrophysical sources of CX, it would be ben-
eficial to be able to use its observed spectrum to probe the conditions of the emitting
region. In many cases, the very factors that make modeling CX complex (e.g., ve-
locity dependence, number of electrons in the system) can be used as diagnostic
tools, provided we understand the underlying physics behind these relationships.
32
The flux that we observe from CX can be written as
FCX =
1
4piD2
∫
nionnneutralvσtot(v)dV, (1.11)
where nion and nneutral are the number densities of the ion and neutral species, v is
the relative velocity between them, and σtot is the total cross section for CX between
that particular ion and neutral pair. The number of photons that we observe from
CX thus holds keys to information about these parameters, though the challenge is
that typically, most of these variables are unknown.
We may use CX spectra to learn about the ions present in the emitting region.
We can deduce the ion temperature through comparing the relative strength of lines
from differing charge states of a particular ion. By comparing the strength of lines
from different ions across the entire spectrum, one could possibly learn about the
relative elemental abundances. This, however, requires information about the total
cross section for CX for each ion species.
For CX to occur, a hot plasma must come in contact with a cool cloud of
neutrals. Although these two ingredients may exist in many environments (as dis-
cussed in Section 1.5), their presence alone does not guarantee CX; the ions may be
shielded from the neutral gas by magnetic fields, for instance. Thus the occurrence
of CX is indicative of the interaction of these two temperature regions. Further,
the spectral signature of CX can tell us at what velocity these two temperature re-
gions are interacting. Again, Figure 1.4 shows an example of how the CX spectrum
may change as a function of relative velocity between the ion and neutral species.
33
Accurate models that can predict this behavior for a wide variety of astrophysical
ions and neutrals can aid us in better determining the interaction velocity in regions
where we currently only have approximate information.
According to Equation 1.11, if we have information on the relative collision
velocity and an estimate on the total CX cross section, we can then extract infor-
mation on the ion and neutral densities along the line of sight. However, even if this
proves too challenging, we can at least hope to diagnose the neutral species involved
in the reaction. It has been shown experimentally [Beiersdorfer et al., 2003b] that
the spectral signature of CX changes as a function of neutral species; this is shown
in Figure 1.9. Understanding why this happens requires a detailed understanding
of the atomic physics of CX, but in the meantime, we can use these empirical mea-
surements to identify the neutral species involved in our astrophysical observations.
This is especially valuable for atoms and molecules that are hard to detect via other
means, such as neutral He and H2.
Unfortunately, we are often limited in our ability to use these diagnostic tools
in our astrophysical observations, both for a lack of accurate theory and a lack of
photons. This dissertation focuses on the need for more accurate theory, but it is
worthwhile to mention that except for the case of SWCX (where CX may contribute
up to nearly all of the X-ray emission, as was found in Beiersdorfer et al. [2003b]), CX
may often be just a small perturbation on the larger X-ray signal. Take, for example,
the case of CX occurring in the Perseus cluster between the Hα filaments and the
hot ICM, as investigated by Walker et al. [2015] and as mentioned in Section 1.5.
A quick back-of-the-envelope calculation of the fraction of X-rays coming from CX
34
Figure 1.9 Experimental EBIT spectra from Beiersdorfer et al. [2003b]
showing how CX between the same ion but differing neutrals yields dra-
matic spectral differences.
35
versus other sources in the observation by Walker et al. [2015] leads to an estimate
of 1-10%.1 This is thus barely at the detection limit of most current instruments
without a significant amount of observing time.
Not only does CX contribute a small fraction of photons in many astrophysical
cases, but the spectral resolution in many of our observations is not sufficient to
actually resolve the spectral lines that would be most diagnostic. This can be seen,
again, in Walker et al. [2015], where there are hints of the presence of CX, but the
∼100-eV resolution of the Chandra CCD limited a clearer answer.2
1.7 The Challenge
We have seen that CX may be an important emission mechanism in a wide
variety of astrophysical plasmas, and that we need a better understanding of the
complex underlying atomic theory. The work detailed in this dissertation addresses
this challenge by presenting measurements of high-resolution CX spectra in order
to benchmark and compare to theory.
This research fits into a growing body of work that highlights the challenge
of developing accurate CX theory. Calculations with the most accurate models are
1This is done by assuming that CX would occur on the surface of the observed Hα filaments,
that the ions are moving at thermal velocities, and that most of the CX flux comes from oxygen
ions.
2On the other hand, as was mentioned previously, Aharonian et al. [2017] possibly detected
signatures of CX with the ∼ 5 eV resolution of the microcalorimeter on Hitomi. With a similar
detector and longer observation time, it would likely be possible to determine the origin of this
signal with more certainty.
36
few and far between, the most common models often produce results that disagree
with experiments, and experimental benchmarks for theory often do not exist at
high enough resolution, or at all. Selected examples of this follow.
Accurate calculations of the total cross section are vital for estimates of the
total flux we should expect from environments with CX. Though experimental and
theoretical results can diverge in the low collision velocity regime [Wargelin et al.,
2008], many experiments agree with theoretical calculations of the total cross section
(e.g., Ali et al. [1994], Bruhns et al. [2008], Greenwood et al. [2001], Wu et al. [2012]).
However, in order to model spectra, it is necessary to calculate state-selective cross
sections. Only certain theoretical methods can calculate such information (CTMC,
MCLZ in some cases, and AOCC/QMOCC, for example), and of the ones that can,
they often show discrepancies when compared to experiments.
For example, CTMC predictions do not match experimental results well at low
collision energies [Beiersdorfer et al., 2000a], such as those produced by an EBIT.
Otranto et al. [2006] demonstrated that while CTMC calculations qualitatively agree
with EBIT experiments showing a decrease of the n capture state with increasing
ionization potential, their CTMC model overestimates the flux in high-n Lyman
transitions following CX onto O8+. Beiersdorfer et al. [2000a] performed experiments
with an EBIT showing that, contrary to CTMC calculations, the so-called hardness
ratio (a proxy for the l capture state) following CX with bare Ne, Ar, Kr and Xe was
always nearly unity. Further, the disagreement between calculated and measured
hardness ratios worsened at higher atomic numbers.
Mullen et al. [2016] performed MCLZ calculations of CX with K-shell Fe, and
37
showed that in order to match data from EBIT experiments at low collision energy, it
was necessary to adjust the applied l-distribution by hand. Hasan et al. [2001] com-
pared various theoretical calculations to experiments performing cold-target recoil-
ion momentum spectroscopy, which had collision energies of a few keV/amu. The
authors found that the QMOCC model agreed well with the data, but the COB,
MCLZ, and CTMC methods were less accurate. Mancˇev et al. [2013] used another
theoretical technique called the Born approximation to calculate state-selective cross
sections, and compared them to experimental data points at a wide range of ener-
gies. In all cases, the two methods agreed well at high velocities, but diverged at
energies below . 100 eV/amu. Finally, Chapters 4 and 6 provide further evidence
of the mismatch between theory and experiments when calculating state-selective
cross sections or predicting spectra.
There has been a focus on obtaining theoretical results in the high collision
energy regime, applicable to, for example, the fast solar wind. These velocities can
be attained in the lab with, for instance, merged-beam experiments, or by colliding
an ion beam with a neutral target, as in Bodewits [2007]. However, as we have
just seen, there are many more discrepancies between experiment and theory in the
lower energy regime (. 100 eV/amu). This regime is relevant to, for example, solar
wind CX inside the bow shock of a comet, which may happen at velocities as low as
50 km s-1 and be responsible for up to 50% of the X-ray emission in comets [Allen
et al., 2008]. CX experiments with EBITs (described in Chapter 3) can reach these
low velocities, though the capability to vary this velocity is fairly limited.
There is a dearth of theoretical calculations for the more complex L-shell ions,
38
and experimental measurements are few and far between. Only Bliek et al. [1998],
Lubinski et al. [2000], Ehrenreich et al. [2005] measured EUV spectra of Li- and Be-
like C, N, and O, Tawara et al. [2002, 2003] performed measurements of Kr at high
collision energy, and more recently, Frankel et al. [2009a] measured L-shell spectra
of Fe and S. Chapter 5 presents the first high-resolution spectra of L-shell Ni CX.
Finally, obtaining a detailed understanding of CX following MEC has been
challenging. MEC likely plays an important role in SWCX, as the multi-electron
neutrals CO2, H2O, and H2 are abundant in cometary comae and planetary atmo-
spheres [Ali et al., 2005]. MEC has been shown to be important in the low-collision
energy regime [Ali et al., 1994, Otranto et al., 2006]. Perhaps unsurprisingly, we find
that MEC is likely a significant factor in the laboratory data presented here (see,
e.g., Chapters 4 and Chapter 5). Since MEC leads to a doubly excited state that is
unstable against autoionization, it can change the shape of the resulting spectrum
as compared to SEC [Tawara et al., 2006, Ali et al., 2005, Otranto et al., 2006]. In
order to extract reliable information from spectra involving both SEC and MEC, we
must thus understand how each process affects the spectrum. Although it is gen-
erally recognized that MEC is an important factor in astrophysical and laboratory
spectra, very few theories that incorporate MEC exist due to the computational
complexity required.
This is a challenge that we can begin to address from the experimental side.
To mimic most astrophysical CX and to ensure SEC, one may use atomic H as the
neutral partner in CX experiments. However, this is itself a challenge. Work by
Leutenegger et al. [2016] has shown that it is possible to ballistically inject atomic
39
H that has been thermally dissociated from molecular H2. However, it is difficult
to fully characterize the dissociation fraction, i.e. the relative amount of H versus
H2, and the reported fraction of H is an upper limit. Other methods are possible:
for instance, one can create an atomic H beam by passing a beam of H− through a
laser cavity, so the ions are neutralized via photodetachment [Bruhns et al., 2008].
An advantage to this method is that it is possible to vary the collision energy by
several orders of magnitude. However, in this setup, it is only possible to measure
total cross sections, not state-selective cross sections or spectral measurements.
It is important to put the study of CX and its theoretical, observational, and
experimental challenges in context with the wider field. The ESA-led XMM-Newton
[Jansen et al., 2001] and NASA-led Chandra [Weisskopf, 1999] satellites have made
significant advances in many areas of astrophysics, including numerous observations
of CX in our solar system and possibly beyond. However, for the spectroscopic study
of CX, we are beginning to reach the limits of our current capabilities with data from
the instruments onboard these missions. Since the high-resolution slitless grating
spectrometers on XMM and Chandra are dispersive instruments that confuse spec-
tral and spatial information, they are not suitable for extended sources—ruling out
many targets that emit X-rays following CX. Chandra and XMM also have charge
coupled devices (CCDs). These CCDs provide exceptional imaging capabilities and
do not produce spatial-spectral confusion of extended sources, but they only deliver
moderate resolving power (R= λ/∆λ ∼20–50). Moving past simply (possibly) iden-
tifying the presence of CX in astrophysical observations and beginning to harness
its diagnostic power requires a non-dispersive spectrometer with higher resolution.
40
This need can be met with microcalorimeters: high-resolution, wide-band,
non-dispersive X-ray spectrometers whose physics is detailed in Section 3.2. Mi-
crocalorimeters on satellites have had a relatively unlucky history—most recently
with the JAXA-led Astro-H/Hitomi mission [Takahashi et al., 2016], which tragi-
cally broke apart after excessive rotation due to an erroneous signal. However, as
evidenced by Hitomi’s groundbreaking observation of the Perseus cluster of galax-
ies, which led to a Nature paper within its first month after launch [Collaboration,
2016], microcalorimeter data have the potential to revolutionize the field of X-ray
astronomy. There are several future planned (X-ray Astrophysics Recovery Mission
[Porter et al., 2010], Athena [Barret et al., 2013]) and proposed (Lynx [Bandler
et al., 2016]) observatories that include microcalorimeters.
These missions hold promise for improved observations and a clearer under-
standing of CX locally as well as across the universe. However, this impending
era of routine high spectral resolution measurements presents a new time-critical
challenge. With lower-resolution data, it is easier to ignore details of CX that are
not quite correct. High-resolution data does not allow for this margin of error. We
must be able to verify the accuracy of the underlying atomic physics of CX in our
spectral models at the same or higher resolution than these instruments, or we risk
misinterpreting our observations.
Laboratory studies at high resolution, such as the ones presented in this dis-
sertation, will help us benchmark and improve our theory in order to meet these new
challenges in this dawning era of space-borne high-resolution X-ray spectrometers.
41
Chapter 2: Theoretical and Computational Tools for Simulating and
Understanding CX
Experiment and theory go hand in hand for improving our understanding of the
complicated and subtle nature of CX. It is impossible to understand experimental
results without a theoretical basis, but without experiments to benchmark and guide
theory, we risk misinterpreting the atomic physics encoded in a given spectrum. In
this chapter, we present the computational tools used in this dissertation to analyze
our experimental spectra, and the theoretical models we use to simulate CX spectra
and compare to our experimental results.
2.1 fac, the Flexible Atomic Code
We use the Flexible Atomic Code (fac, [Gu, 2008]) in Chapters 4, 5, and
6 in order to calculate the energy levels and radiative transition rates for a given
ion and charge state. fac is a software package that calculates several atomic ra-
diative and collisional processes, including radiative cascades, collisional excitation,
collisional ionization, photoionization, autoionization, radiative recombination, and
dielectronic recombination (see Appendix C for a description of these processes).
fac is beneficial because of its accuracy when compared to experimental data [Chen
42
et al., 2006, 2007] and its relative speed and ease of use. Calculations in fac are
based on the relativistic Dirac equation and use jj-coupling, thus making the code
applicable to ions with large Z. The physics of how the code calculates atomic struc-
ture, including energy levels and transition rates, along with other atomic data
related to electron impact excitation, ionization, and other processes, can be found
in Gu [2008]. For demos on how to use fac, along with sample atomic datasets
generated by fac, see https://www-amdis.iaea.org/FAC/.
To derive energy levels and transition rates, we interact with fac via the
Python library pfac, which allows us to write a code for each ion and set of ion
structures we wish to calculate. We specify the atom (e.g., S), which electron shells
are closed (e.g., none for H-like S), and any desired electron configurations we wish to
consider. We typically specify the ground configuration along with several excited
states that consist of one excited electron above the ground configuration, with
increasing n values up to a limit we determine. This limit is often n = 15. fac
then considers the central potential of the specified configurations interacting with
one another, and diagonalizes the Hamiltonian to calculate the atomic structure.
Its output is a list of all possible configurations with fully specified n, l, j, and total
J values, along with the energy of that configuration with respect to the ground
state. For every transition between each of these configurations, it also calculates
the energy, the oscillator strength, the Einstein A coefficient, and the multipole
expansion, for all multipoles.
If it is necessary for better accuracy, one can manually adjust the values of the
energy levels. For example, for the S16+ and S15+ transition energies used in Section
43
6.4, we corrected the ionization energy required to make those ions according to
Johnson and Soff [1985] and Artemyev et al. [2005], respectively. This then corrects
the ground state energy level. For the Ni19+ data presented in Chapters 4 and 5,
the ionization energy was corrected according to calculations by Scofield [private
communication], and the 3 → 2 transition energies were corrected to match those
in Gu et al. [2007a].
2.2 Charge Exchange Models: spex-cx and acx
In section 6.2, we use the commonly available CX spectral synthesis codes
spex-cx and acx to compare to our experimental spectra. spex-cx is a model
available for use with the spex X-ray fitting package [Kaastra et al., 1996], and
acx is a standalone package that can be used with the xspec spectral fitting code
[Arnaud, 1996].
These codes first calculate the initial n and l distributions of electrons in
the ion after electron capture, then perform a radiative cascade using energy and
transition information from their respective atomic databases or atomic structure
calculations to generate a spectrum. For this second step, spex-cx uses fac as a
baseline tool, a second atomic structure code [Cowan, 1981], and measurements from
the National Institute of Standards and Technology (NIST). acx uses the AtomDB
database developed by Foster et al. [2011]. Though the radiative cascade from each
method may produce slightly different results even with the same initial conditions,
here we are more interested in the first step, which will more strongly influence
44
the resulting spectra: determining the initial distributions of electron states after
capture. It is from this distribution that we can begin to trace the detailed physics
of the CX reaction. Though the spex-cx and acx codes calculate this in similar
ways, there are key differences both in their methods of calculation and applicability
to experimental data.
The creators of spex-cx compiled available theoretical and experimental total
and state-selective (n-, nl-, or nlS-resolved) cross section data from the literature
for collisions between several ions and atomic H at specific collision energies. These
sources are shown in Table 1 of Gu et al. [2016]. For cases where these data do not
exist, scaling laws were applied to infer this information.
To determine the n distribution scaling, the authors collected n-resolved cross
sections for discrete values of n around nmax (from Equation 1.2) for nine different
ions (C5+, N6+, N7+, O6+, O7+, O8+, Ne10+, Fe25+, and Fe26+) undergoing CX with
atomic H at 14 different collision velocities. The authors then fit a third-order
polynomial to these values to derive a smooth distribution function for n at each of
these velocities. They assume that the n distribution around nmax and its velocity
dependence will be the same for all ion species. When using spex-cx, it is possible
to select a desired collision velocity between 50–5000 km s−1, which correspond to
the minimum and maximum values for which these n distributions were calculated
or extrapolated. The authors use a linear interpolation to calculate n distributions
for the velocities between the 14 discrete points [Liyi Gu, private communication].
The spex-cx authors determined an l distribution with a similar combination
of experimental and theoretical datapoints and scaling laws. They have seven ions in
45
their database for which theoretical or experimentally-measured l distributions exist,
presumably at a range of collision velocities. In order to compute an l distribution
for the rest of the ions, they compared the seven available l distributions to one of
five distribution approximations to determine the best fit. Four of these distributions
are presented in Chapter 1 as Equations 1.7 (which the spex-cx authors refer to as
“low energy II”), 1.8 (separable), 1.9 (even), and 1.10 (statistical); the fifth equation,
which they call “low energy I” is a variant of Equation 1.7, and is given as
Wn(l) = (2l + 1)
[(n− 1)!]2
(n+ l)!(n− l − 1)! . (2.1)
They determined which approximation best matches the theoretical or exper-
imental distribution over a velocity grid of 20 points (somewhat finer sampling than
in the n distribution calculation), spread between 50–5000 km s−1. They did this
for two sets of n values: n ≤ nmax and n > nmax. For example, for most ions,
for n ≤ nmax, they determine that the separable distribution should be applied
from 50–500 km s−1, and the statistical distribution is valid from 500–2000 km s−1.
When using the spex-cx code, it is possible to specify which distribution function
should be used, but the default (and recommended) option is to let the specified
CX collision velocity determine this distribution.
Finally, the user can specify the relative abundance of each ion, but in spex-
cx, the distribution of the ion charge states can only be specified by the ionization
temperature parameter. spex-cx currently cannot be used to simulate CX with
neutrals other than atomic H, and assumes only SEC.
46
The acx package similarly uses estimations for the n and l distributions, but
bases these purely from theoretical estimates without comparison to data.
The nmax in acx is calculated with Equation 1.2, and offers two possibilities
for the distribution of electrons around nmax if nmax is not an integer: assuming that
all electron capture is into the rounded nmax channel, or weighting the two nearest
n shells around nmax according to the integer fraction. After the n distribution
calculation, the code determines the LSJ coupled states that would result in a
photon up to n ≤ nmax, and uses the AtomDB database (and in some cases, the
autostructure code [Badnell, 1986]) to determine the energies and transition
probabilities between each of these levels. ACX only calculates the energies of shells
up to n = 10.
acx allows the user to determine the l distribution to be one of the four
equations presented in Chapter 1 (Equations 1.7–1.10), and also allows the user to
choose whether this distribution is applied to the total L or the l of the captured
electron. This is specified via the acx model parameter. The authors recognize
that using these equations ignores many important considerations such as velocity
dependence, and can only be used as rough approximations until more accurate
calculations or measurements become available [Smith et al., 2014a].
The acx package includes several models. The main model is acx, in which
the user can set an ion temperature distribution in keV, an abundance of metals in
the plasma compared to solar, and other parameters. vacx is the variable abun-
dance version of acx, where the user can specify the relative elemental abundances.
acxion allows for the calculation of a spectrum from a single ion at a specific charge
47
state after it undergoes CX with H, He, or a mix of the two. All the acx mod-
els can incorporate both H and He as the neutral partner by adjusting the model
parameters, but like spex-cx, it only considers SEC.
2.3 Multi-Channel Landau Zener Theory
In Chapter 6, we present comparisons of our experimental data to Multi-
Channel Landau Zener (MCLZ) theory. MCLZ was briefly introduced in section
1.2, but we now go deeper into its physical basis. Recall from Section 1.1 that CX
at low energies resembles a molecular collision process. This is because in the limit
of v  v0, where v is the relative velocity between the collision partners and v0 is
the classical orbital speed of the electrons, the electrons are moving fast enough to
be able to continuously adjust their motion to the relative position of the collision
partners. This creates a so-called “quasimolecule” composed of the ion and neutral,
whose energy levels are a function of the internuclear separation which varies slowly
over time.
The duration of the collision with respect to the characteristic electron tran-
sition time, i.e., how long the electron is bound versus being transferred between
the two collision partners, affects the adiabaticity of the collision process [Janev and
Winter, 1985]. In slow collisions, the process is adiabatic, and at higher speeds,
it becomes diabatic [Lichten, 1963]. In the CX collision, the quasimolecule moves
along the potential of the initial, inbound state (on the diabatic potential) as the
collision partners approach each other. In regions of strong curvature of two of
48
ΔV
Rc
V�
V�
2.0 2.5 3.0 3.5 4.0
-4
-3
-2
-1
0
Internuclear Distance (a.u.)
V
n
A  +H
A    +H
q+
(q-1)+
Figure 2.1 Schematic of the diabatic (solid lines) and adiabatic (dashed
lines) energy curves, with an ingoing polarization potential Vi (Equation
2.3) and an outgoing Coulomb potential Vf (Equation 2.5). The quasi-
molecule travels along the initial, inward diabatic potential curve until it
reaches a pseudocrossing Rn, where R is the internuclear distance. Here,
the quasimolecule moves to the initial adiabatic curve, then to the final
outward one. This is the electron capture process. Finally, the quasi-
molecule exits on the outgoing channel. Figure adapted from Cumbee
[2016].
these potential lines or at very small internuclear separations, electron transfer can
occur at a so-called “avoided curve crossing” or “pseudocrossing” [Eichler, 2005,
Janev, 1983]. At these regions, the quasimolecule moves along the initial, then final
adiabatic potential curves as the electron is transferred from the neutral to the ion.
The outgoing quasimolecule then moves along the final, outbound diabatic potential
curve. A schematic of this process is depicted in Figure 2.1.
Multichannel Landau-Zener theory [Landau, 1932, Zener, 1932] considers the
49
probability of electron transfer at these pseudocrossings, and can be used to derive
state-selective (nlS-resolved) cross sections for CX interactions at low energies (be-
low ∼5 keV/amu). It is most accurately applied to a system with a fully stripped
ion and neutral atomic H, because the symmetry of this system limits the density of
pseudocrossings and keeps the transition regions fairly well isolated [Janev, 1983].
The most important parameters in the MCLZ calculation are the locations of any
pseudocrossings, the magnitude of the energy separation of the two potential energy
curves that make up the pseudocrossing, and the slope of those two potential energy
curves [Cumbee, 2016].
The location of these pseudocrossings in atomic units can be approximated
with
Rn ' 2n2 q − 1
q2 − n2 (2.2)
for n < q, where n is the index of the pseudocrossing and q is the ion charge [Janev,
1983, Janev et al., 1983]. Based on this equation, we see that the n distribution of
captured electrons for n ≥ q cannot be calculated with the MCLZ method [Janev,
1983].
Alternatively, Rn can be calculated by equating the two potentials of the
inbound collision pair (one at large internuclear distances and another at small
ones) with that of the outgoing collision pair and solving for R. The MCLZ code
implemented to produce the cross sections in Section 6.5 uses this generalized form
of Rn. The potentials involved are described in the following three equations.
50
At large internuclear distances, the potential is given by the polarization po-
tential:
Vpol = −αq
2
2R4
, (2.3)
where α is the static dipole polarizability of the neutral (for H, α = 4.5) [Cumbee,
2016].
At small internuclear distances, the potential between the approaching ion and
neutral is given by the repulsive Born-Mayer potential:
VBM = Ae
BR, (2.4)
where A and B are fitting coefficients, given to be A = 25q and B = −0.8 + ξi,
where ξi is 1 for H and 1.7 for He [Butler and Dalgarno, 1980].
After electron transfer, both ion and neutral are positively charged, so the
outgoing potential can be described by Coulomb repulsion:
VCoul =
(q − 1)(qcap)
R
− E, (2.5)
where qcap is 1 for SEC, and E is the collision energy in eV/amu.
With these potentials at play, we can now show how MCLZ calculates the
probability of a transition occurring between an initial potential curve (i) and a
final curve (f) at a particular pseudocrossing Rn. This probability is given by
51
1− pn, where pn is given by the Landau-Zener formula:
pn = exp
(−2pi V 2if
∆F vrad
)
. (2.6)
∆F is the difference in the slopes of the potential energy curves at the pseudocrossing
Rn, and vrad is the radial velocity at Rn, given by
vrad = ν
√
1− b
2
R2n
− Vi
E
, (2.7)
where ν is the collision velocity, b is the impact parameter, Vi is the initial po-
tential energy, and E is the collision energy. Vif in Equation 2.6 is the potential
energy difference between the initial and final potential curves at Rn, which for the
MCLZ calculations shown in this dissertation, is given by the Olson-Salop-Taulbjerg
approximation [Salop and Olson, 1976, Olson et al., 1971, Taulbjerg, 1986]:
Vif =
(
9.13 fnl√
q
)
exp
(−1.324Rn α√
q
)
. (2.8)
α is a correction factor for the ionization potential of neutrals other than atomic H,
α =
√
2I, (2.9)
where I is the ionization potential of the neutral. fnl is a correction factor to
describe capture into nondegenerate l states for charge states lower than H-like. For
H-like ions, fnl → 1 due to degeneracies. This results in the limitation that only
n–resolved cross sections may be calculated for H-like ions [Mullen et al., 2016].
To obtain nl–resolved cross sections for this charge state, an l distribution function
52
0	
1	
p1	
V	
RR1	
Figure 2.2 One pseudocrossing p between initial potential state 1 and
final potential state 2.
must be applied. We discuss possible l distributions shortly.
This transition probability just discussed is illustrated in Figure 2.2, which
depicts one initial state (0, potential given by Equations 2.3 and 2.4), one final state
(1, potential given by Equation 2.5), and one pseudocrossing at R1 between them
with a probability of p1.
Using the probability defined in Equation 2.6, the probability of two crossings
at Rn are 2pn(1 − pn). Generalizing this to N crossings going to a final state f
requires considering the mutual interaction of all potentials. In this case, we have
53
0	
1	
2	
3	
N	
p3	
p2	
p1	
pN	
V	
R
Figure 2.3 N-1 pseudocrossings between N potential states, where each
pn represents a pseudocrossing happening between the initial state and
final state n.
[Janev et al., 1985]
Pf = p1p2 . . . pf (1− pf )[1 + (pf+1pf+2 . . . pN)2+
(pf+1pf+2 . . . pN−1)2(1− pN)2+
(pf+1pf+2 . . . pN−2)2(1− pN−1)2+
. . . p2f+1(1− pf+2)2 + (1− pf+1)2].
(2.10)
This process is depicted in Figure 2.3. In this figure, the ingoing channel is
0, and the final outgoing channel f is chosen to be one any one of the numbered
states.
54
The partial cross section for one particular transition between state i and state
f , i.e. the goal of the MCLZ calculation, is then given by integrating this probability
over all possible impact parameters:
σf = 2pi
∫ Rn
0
Pf b db . (2.11)
The total cross section for capture is given by
σ =
N∑
n=1
σf . (2.12)
As previously mentioned, MCLZ can only calculate an n distribution for bare
(going to H-like) ions; to get nl-resolved cross sections, it is thus necessary to apply
an l distribution. Janev [1983] does this by multiplying the n-resolved cross section
by an l distribution:
σnl = Wnl σn, (2.13)
where Wnl takes different forms depending on the energy regime of the collision.
Several commonly-used distributions are presented in Equations 1.7–1.10. The two
most commonly used in MCLZ calculations are repeated here for convenience. For
high collision energies of ∼1 keV/amu or higher, the l levels are found to be popu-
lated statistically [Janev and Winter, 1985]:
W statnl =
2l + 1
n2
. (2.14)
For energies below this threshold, Abramov et al. [1978] derived a low-energy
55
distribution function:
W low enl = (2l + 1)
[(n− 1)!]2
(n+ l)!(n− 1− l)! . (2.15)
Equation 2.14 is thus the same as the statistical distribution function presented in
Equation 1.7, and Equation 2.15 is the same as the “low energy I” equation used in
spex-cx from Equation 2.1.
For further reading on MCLZ theory, see, for example, Janev [1983], Bliek
et al. [1998], Salop and Olson [1976], Eichler [2005], Janev et al. [1985], Janev and
Winter [1985], and Cumbee [2016].
56
Chapter 3: The Role of Laboratory Astrophysics in Improving Charge
Exchange Models
While theoretical models are important for a nuanced understanding of CX,
they are most useful in concert with experiments that test them. Experiments
are not only important for benchmarking theoretical models, however: through ex-
periments, we learn new and often surprising things empirically that may not be
expressed in theory (e.g., Leutenegger et al. [2010]). Of course, with any experi-
ment, it is important to disentangle physical results from the processes under study
from any instrumental and experimental effects. To do this, one must have a deep
understanding of the tools used and the errors that contribute to a result. In this
vein, we will describe the operating principles of the tools used for the CX experi-
ments presented in this dissertation, namely an electron beam ion trap (EBIT) and
an X-ray microcalorimeter.
3.1 The LLNL Electron Beam Ion Trap
The EBIT-I facility at the Lawrence Livermore National Laboratory (LLNL),
where the EBIT was invented, is an ideal environment in which to perform lab-
oratory astrophysics measurements, as the EBIT can produce well-characterized
57
plasmas in specific ionization states for nearly every element through bare uranium.
The EBIT-I and EBIT-II at LLNL were the first devices utilized specifically for lab-
oratory astrophysics [Beiersdorfer, 2003]. Within the set of plasma sources available
for laboratory use (including Z-pinches, tokamaks, vacuum sparks, etc.), EBITs are
especially useful for laboratory astrophysics because the densities achieved in the
trap are low enough to be in the coronal limit, i.e. the rate of collisional processes
is much lower than that of radiative or autoionization processes. This means that
an ion, after an excitation, ionization, or recombination event, has time to de-excite
to the ground state and emit radiation that can be used for spectral diagnostics.
The main components of the EBIT are an electron gun and three sets of cylin-
drical electrodes, called drift tubes, that are oriented along a vertical axis. These
drift tubes are elevated to a tunable voltage potential by a low-noise high voltage
amplifier (which we call the drift tube rack voltage), and an additional floating
low voltage power supply superimposes additional bias to the three individual drift
tubes. The electron beam originates from the gun, and the electrons are accelerated
towards the bottom, middle, then top drift tubes, as shown in Figure 3.2. The ion
trap region is created in the 2 cm–long middle drift tube region, which is kept at a
lower bias than the bottom and top drift tubes. Adjusting the top and middle drift
tube voltages allows us to vary the depth of the ion trap. These drift tubes create
the ion trap in the axial direction; ions are radially confined due to the electrostatic
attraction to the electron beam.
The trap region is surrounded by a pair of superconducting Helmholtz coils.
The 3 T magnetic field from the coils compresses the electron beam to a diameter
58
of ∼60 µm, and also serves as the ion trapping method in the radial direction when
the beam is off (so-called magnetic trapping mode, discussed shortly). After the
electron beam passes through the drift tube assembly, it is defocused by a magnet
and directed into the collector electrode, which is cooled with liquid nitrogen to
offset the heat input from the beam. The EBIT axis is vertical, and to allow direct
line-of-sight access to the trap region, there are six radial ports, onto which are
attached various spectrometers and gas injectors. A schematic of the LLNL EBIT
is shown in Figure 3.1, and a schematic of the trap region is shown in Figure 3.2.
We inject neutral species directly into the trap, where the electron beam col-
lisionally excites and ionizes them to our desired charge state. In the experiments
presented here, we inject gases in two ways. The first is via a ballistic gas injector,
mounted on one of the radial ports. This injects a collimated stream of neutral
gas directly into the electron beam. The amount of gas injected is controlled by
the injection pressure, which is regulated by a thermal valve. We typically inject
gases from a compressed gas cylinder, and in some cases, liquids with a high vapor
pressure at room temperature. Solids used in our experiments, such as Ni, are in-
jected with a sublimation injector. The injection pressure is adjustable, and vapor
from the sublimator is collimated and aligned to intercept the electron beam. The
background EBIT chamber pressure is low (on the order of a few 10−10–10−11 torr),
and gas injection pressures are usually on the order of 10−6–10−9 torr.
The charge state of the ions is determined by the energy of the electron beam,
the amount of time the ions interact with the beam, and recombination processes
in the trap. Electrons are stripped off the ions sequentially until the next ionization
59
Wall of EBIT 
Vacuum Vessel
SuperConducting
Helmholtz Coils
Metal Vapor
Vacuum Arc
(MeVVA)
Einzel Lens
Collector
Collector 
Magnet
Liquid
Nitrogen
Liquid
Helium
Electron 
Beam
Transition
E-gun
Bucking Coil
Drift Tubes
t
m
b
X-ray Observation
Port
Gas Injector
Figure 3.1 Schematic of the LLNL EBIT-I, courtesy of Klaus Widmann.
60
Figure 3.2 Sketch of the trap region of the LLNL EBIT-I, courtesy of
https://ebit.llnl.gov/ and Natalie Hell.
potential is higher than the electron beam energy. The energy of the electron beam
is determined by
Ebeam = e(VDT + VMDT + VSC), (3.1)
where e is the electron charge, VDT is the voltage of the drift tube rack, VMDT is the
voltage of the middle drift tube, and VSC is the space charge voltage as determined
below. Negative space charge is generated by electrons in the beam, and positive
space charge is generated by trapped ions. The electron beam space charge in Volts
can be approximated by
VSC = −5.4 Ib√
Eb
, (3.2)
61
where Ib is the electron beam current in mA and Eb is the electron beam energy
in keV [Brown, 2000]. It is estimated that the positive space charge from the ions
reduces the effect of the electron beam space charge by half [Brown, 2000], so this
number is often divided by two before using the value in Equation 3.1.
Precise selection of an electron beam energy is important to produce a desired
ion charge state. The electron beam energy is nearly monoenergetic, with an energy
distribution that is roughly Gaussian in shape with a FWHM of∼30 eV [Beiersdorfer
et al., 1992]. Because of the monoenergetic nature of the electron beam in normal
operation, plasmas created in the EBIT are not thermal. The beam energy can
be tuned to as low as 100 eV and as high as 20 keV [Beiersdorfer, 2003]. For our
experiments, the energy was typically in the range of 1-12 keV.
The electron beam current is controlled by the voltage of the anode on the
electron gun. The current of the LLNL electron gun in Amperes is given by
I = pV 3/2a , (3.3)
where p is the perveance, which for the LLNL EBIT is 0.5 µperv [Gu, 2000], Va is
the anode voltage of the gun in Volts. Typical beam currents are on the order of
∼100–200 mA. The electron density in the trap, which is determined by the electron
beam current, is ∼1010–1012 cm−3 for most experiments [Beiersdorfer, 2003].
Ions are heated by elastic collisions with the electron beam. The ion temper-
62
ature in eV is approximated by
T ∼ αqV, (3.4)
where α is an efficiency factor, typically taken to be 0.2, q is the ion charge in units of
elementary charge, and V is the trap potential in Volts [Levine et al., 1989]. For the
species typically used in most laboratory astrophysics experiments, ion temperatures
are about 50 eV/amu. Ions with sufficient (thermal) kinetic energy can overcome
the potential barrier of the trap and escape in a process called evaporative cooling.
Because of the effects of evaporative cooling and the approximate nature of the
value of the efficiency factor, it is helpful to have an in-situ measurement of the ion
temperature, rather than simply extrapolating one from the trapping potential. One
can do this by taking high-resolution measurements of the doppler broadening of a
spectral line with, for example, a crystal spectrometer. This process was performed
in, for instance, Leutenegger et al. [2013].
As ions with lower charge experience a lower trapping potential and thus can
escape the trap with a lower kinetic energy, low-Z species may be intentionally
injected to carry away excess thermal energy and keep the average ion temperature
low. Conversely, high-Z elements such as Ba (Z=56) and W (Z=74), which evaporate
off the electron gun and enter the trap, can eventually dominate the ion population.
To mitigate this, the potential of the top drift tube can be lowered below that of
the middle drift tube as a function of time. In this so-called “dump,” ions are
accelerated upwards towards the collector and out of the EBIT.
63
It is possible to adjust the timing pattern of the voltage supplies to change
the EBIT parameters, charge balance, and even EBIT mode as a function of time.
The EBIT can be run in two modes: electric trapping mode and magnetic trapping
mode. In electric trapping mode, the electron beam is on and the injected gas is
collisionally ionized and trapped. In magnetic trapping mode [Beiersdorfer et al.,
1996b], the electron beam is turned off, and the ions initially created with the
electron beam are radially confined in the trap by the magnetic field instead of the
electron beam. In this mode, the EBIT is effectively a Penning trap. We study
the emission from charge exchange (CX) recombination in magnetic trapping mode.
The electron beam energy and current can also be swept as a function of time to
create a pseudo-Maxwellian thermal electron distribution [Savin et al., 2000] with a
characteristic electron temperature.
3.2 The EBIT Calorimeter Spectrometer
The EBIT Calorimeter Spectrometer (ECS) is an invaluable tool for many
of the laboratory astrophysics experiments performed at the LLNL EBIT facility,
including the CX work presented here. The ECS system is low-maintenance and
easy to use, and the non-dispersive spectrometer yields high spectral resolution and
quantum efficiency across a wide energy band. Further, the ECS data acquisition
system time-tags each photon with its arrival time, which allows us to differentiate
between photons from electric versus magnetic trapping mode for CX experiments,
or to isolate transient states such as during the charge breeding process.
64
In this section, we will provide the framework for understanding how the ECS
works, both theoretically and operationally, and how to understand its results and
uncertainties. We will first review the basics of microcalorimeter theory, then move
on to the specific ECS system, then finally give an overview of a new microcalorime-
ter spectrometer system in development at NASA’s Goddard Space Flight Center
(GSFC) that will soon be delivered to the LLNL EBIT facility.
3.2.1 X-ray Microcalorimeter Basics
X-ray microcalorimeters (referred to interchangeably as X-ray calorimeters)
are attractive for X-ray spectroscopy due to their imaging capabilities and impres-
sive spectral results across a wide bandpass (i.e., Collaboration [2016]). Here, we
will discuss general microcalorimeter theory for the ideal case, a few details to con-
sider in detector design, and finally selected non-ideal cases. For more details and
derivations, one can peruse these resources: Enss [2005], Figueroa Feliciano [2001],
and Moseley et al. [1984].
A microcalorimeter is a near-equilibrium, non-dispersive, thermal detector.
Thermal detectors measure the heat of each incoming photon that is incident upon
the detector. Non-dispersive detectors determine the energy of incident X-ray pho-
tons without diffracting the light onto a position-sensitive detector, as crystal and
grating spectrometers do. Though dispersive spectrometers exhibit extremely high
resolution (typical R∼3000), they are not suitable for observations of extended
sources, as light from different regions of the observation would be diffracted to
65
different locations on the position-sensitive detector. This confuses the spectral
information with the spatial. Using a non-dispersive spectrometer is vital for the
extended X-ray emission in the EBIT trap during magnetic trapping mode, when the
ion cloud spreads out.1. Near-equilibrium detectors measure nearly all the energy
from each photon event, with very little energy getting lost. This is in contrast to,
for example, ionization detectors such as CCDs, where some of the photon energy
goes into a detectable form of internal energy (i.e., free charge), but a significant
fraction goes into undetectable forms of internal energy, such as heat. This leads
to a fundamental constraint on the energy resolution of such detectors. In contrast,
because nearly all the energy of photons incident upon calorimeters goes into ther-
mal energy, which is exactly the quantity being detected, they can theoretically have
much higher energy resolution.2
Microcalorimeters fundamentally only consist of three parts: an X-ray ab-
sorber, a thermometer, and a heat sink kept at cryogenic temperatures (∼50 mK)
to achieve optimal detector sensitivity. Mirroring this simplicity, microcalorimeters
1It should be noted that although crystal spectrometers have limited use for extended sources,
as discussed here, and also can only operate within a narrow energy bandpass at one time, their
extremely high resolution makes them extremely useful for many EBIT experiment where extreme
precision is required and a narrow bandpass can be tolerated. These include measurements of ab-
solute cross section measurements as in Brown et al. [2006] and absolute wavelength measurements
as in Brown et al. [2002] and Gu et al. [2007a].
2The fundamental limit on the energy resolution of CCD detectors is due to the fano factor, as
discussed here, as well as counting statistics for the measured electrons. The fundamental limit for
microcalorimeters is due to energy being transferred across the weak link between the thermometer
and the heat bath. This is called phonon noise, and will be discussed shortly.
66
detect the energy of an X-ray event in just three steps. First, the X-ray is absorbed
by the absorber through the photoelectric effect and is thermalized, next, the ther-
mometer measures the temperature rise of the absorber, which corresponds to the
energy of the photon, and finally, the heat from the event bleeds out to a heat sink,
which is weakly linked to the absorber to cool the system back down to its equilib-
rium temperature to be ready for the next event. This is the picture for one detector,
which we call a pixel; by creating arrays of these absorber-thermometer pairs, we
create an imaging spectrometer. Figure 3.3 presents a schematic of a single-pixel
calorimeter.
Now including a few variables: if the energy of incident photon is E0 and ab-
sorber has some heat capacity C, assuming instant thermalization, the temperature
rise of the absorber is
∆T ' E0/C(T ). (3.5)
The absorber, which is connected to the thermal bath with a thermal conductance
G, will then cool to the bath temperature. This creates a temperature pulse profile
for each incident photon. The shape of this pulse is proportional to the energy of
the photon.3 The time-dependent equation for this profile is
T (t) =
E0
C(T )
e
−t
τ , (3.6)
3There are several metrics we can use to convert the shape of the pulse to a photon energy,
such as the pulse height or the area under the curve. In our analysis, we use the amplitude of the
pulse at all frequencies.
67
X-ray	absorber	
Thermometer	
X-ray	
Weak	thermal	link	
Heat	sink	
G(T)	
T0	
C(T)	
E0	
Figure 3.3 Schematic of an ideal microcalorimeter. An absorber with
heat capacity C(T ) is connected to a thermometer. This pair is then
connected through a weak thermal link with a thermal conductanceG(T )
to a heat bath at temperature T0, nominally 50 mK. An X-ray with
energy E0 incident upon the absorber is thermalized and the temperature
change is read out by the thermometer.
68
where the time constant of the exponential decay is
τ ' C(T )
G(T )
, (3.7)
valid only without the presence of electrothermal feedback, which is discussed shortly.
An approximation of this temperature profile is shown in Figure 3.4. This simple
equation tells us that by knowing the parameters C and G and by measuring T , we
can determine the energy of the photon. Conversely (and more realistically), with
a well-characterized energy source, we can derive the properties C and G from this
ideal case.
With this information in mind, let us now consider some qualitative design
criteria and trades for the three basic parts of a microcalorimeter. Two important
qualities for the absorber are high stopping power to measure as many photons as
possible (achieving high quantum efficiency), and quick internal thermalization (in
the crystal structure of the absorber) and a compact geometrical design, so the
energy input on the absorber is not position-dependent, which degrades the energy
resolution. In addition, the combination of a low C (of both the absorber and the
thermometer) and a high G to the heat bath allows the detector to quickly return
back to base temperature, so more photons can be collected over a short amount of
time. This avoids a phenomenon called pileup, in which a second photon is absorbed
too soon after a first one, and thus the detector has not yet returned back to its
base temperature.
See Morgan [2015] and references therein for a discussion about material
69
T	
t	
τ	=	C/G	
Figure 3.4 Schematic of an X-ray pulse from an ideal microcalorimeter.
The height of the pulse is a function of the photon energy and the heat
capacity, and the time constant of the decay is equal to the heat capacity
of the absorber divided by the thermal conductance to the heat bath.
70
choices and design trades for the absorber. The detectors fabricated at NASA/GSFC
use either a combination of mercury and tellurium (MgTe) or bismuth and gold
(BiAu).
The main desired quality for a microcalorimeter thermometer is the ability to
accurately measure a very small change in temperature without adding much noise
to the system. Our choice of thermometers is fairly wide, but here we will focus on
resistive thermometers, otherwise known as thermistors, whose resistance varies as a
function of temperature. Operationally, this means that the quantity we measure af-
ter an X-ray pulse is the resistance of the thermometer, in some form. At GSFC, the
thermistors we use are either semiconductors (called “silicon-thermistor devices”)
or superconductors (called “Transition-Edge Sensors,” or TESs). Si-thermistor de-
vices were developed first, but there are practical limits on their ability to scale up to
large arrays of pixels, which we will discuss shortly. Because of this, in recent years,
the field has moved to TESs. TESs have proved to routinely yield high resolution
and can scale up to array sizes of thousands of pixels, though this technology is not
without new complications (see Section 3.3).4
The weak link to the thermal bath determines in part how quickly the detector
cools back down to its equilibrium temperature, and the optimal link must balance
competing interests. On one hand, the thermal time constant should be longer than
the time it takes for the absorber and thermometer to reach thermal equilibrium
4The fundamental limits on Si-thermistor detectors and TESs are the same, and they both can
be optimized to achieve similar energy resolution for a variety of applications. The choice between
one or another often depends on the size, mass, and power constraints of the experiment.
71
after a pulse. However, if the thermal time constant is too long, the risk of pileup
increases, which ultimately leads to lower energy resolution.
The main job of the detector heat sink is to achieve a low temperature that
is steady and well-controlled. This can be done in many ways, always involving
successive thermal stages that make discrete transitions from the outer layer at
room temperature to the inner detector cold stage. The details of the various cooling
methods are beyond the scope of this dissertation, but one such way, implemented
in the ECS system, is described in the following section.
In practice, there are countless other design considerations to take into ac-
count when choosing the materials and design of a calorimeter than what has been
briefly mentioned here. These include factors such as structural stability, robustness
against vibration or temperature changes, cost, ease of fabrication, among others.
Additionally, one must consider the thermal, mechanical, magnetic, and supercon-
ducting/semiconducting properties of the chosen materials, and realize that altering
the materials, thicknesses, geometries, temperatures, and, where applicable, super-
conducting or semiconducting properties of a chosen design will change the proper-
ties and performance of the detector. Furthermore, external factors such as noise
and magnetic fields, along with the choice of detector readout hardware and the
method of pulse analysis will add variation and complexity in detector properties
and performance.
Now that we have already begun to drift away from the ideal case, let us
consider more concretely a few more complications; mainly noise and other effects
present in a real detector system. Remember that our goal is to calculate a photon
72
energy. However, the information we measure is not just due to photons incident
upon the detector; there are various sources of noise that we must disentangle from
our true signal to accurately measure the photon energy.
There are two fundamental noise sources intrinsic to resistive X-ray microcalorime-
ters that must be considered: phonon noise and Johnson noise. Phonon noise arises
from the presence of the weak link between the detector and the cold bath: this
can be thought of as Poisson fluctuations in the number of energy carriers in the
detector that have a mean energy kBT [Enss, 2005]. These fluctuations have the
same frequency spectrum as a pulse, and contribute an average change in energy in
the detector of
〈∆E2〉 = kBT 2C(T ), (3.8)
where kB is the Boltzman constant, T is the temperature of the detector, and C(T )
is the temperature-dependent heat capacity of the device. To minimize this effect,
it is helpful to reduce T and C(T ).
Johnson noise is caused by the Brownian motion of electrons in the thermistor,
and leads to fluctuations in the temperature readout. The power spectrum from
Johnson noise is nearly white. Johnson noise is also present in other parts of the
system connected to the detector, like resistors in the circuit or from an amplifier.
Though both phonon and Johnson noise are nearly constant across all frequencies,
the natural time constant τ of the thermistor will damp out any phonon fluctuations
with frequencies greater than 2piτ .
73
Considering a microcalorimeter as part of a system with other instruments
introduces many other noise sources that are extrinsic to the detector but often
present in real calorimeter systems, and which limit their spectral performance. A
random selection of these noise sources include fluctuations in temperature of the
heat bath, RF pickup, shot noise from stray photons, noise in the amplifier chain,
1/f noise [Stahle et al., 1999], and mechanical vibrations from nearby pumps. Other
complexities arise from the fact that a detector is not made up of a single pixel, but
rather an array of them. Because of the close proximity of pixels, heat input to one
pixel may be detected by a neighboring one in a process called thermal crosstalk.
This becomes worse at high count rates, and is a major limiting factor for the
performance of large arrays of pixels.
A much more complete discussion on noise sources is presented in Enss [2005].
In general, sources of noise should be eliminated or reduced as much as possible, but
if present, they should be well-characterized in order to determine their effect on the
signal during post-processing. This can be done via a so-called “optimal filtering”
procedure, which will be described in the following section.
Besides noise sources, a commonly discussed feature of resistive thermometers
is electrothermal feedback. To understand this, we must first place the calorimeter
in a circuit. Recall that we use thermistors to determine the ∆T of a pixel after a
photon event, due to their well-known resistance versus temperature curve. We must
thus measure the resistance across the thermistor before, during, and after a pulse.
In practice, we achieve this by running current across the thermistor, measuring the
voltage drop, and calculating the resistance. For Si-thermistors, in order to give the
74
thermistor a constant supply of current, we voltage bias it through a load resistor of
high value. This causes the current through the thermistor to be mainly dependent
on the value of the load resistor. A simplified circuit diagram of this (Si-thermistor)
detector scheme is shown in Figure 3.5. One may note the presence of an amplifier:
the small voltage drop across the thermistor must be amplified near the cold detector
stage before moving the signal to subsequent warmer thermal stages. The JFET
is placed there in order to minimize capacitive vibrational noise induced in the
long cable that extends to higher temperature stages. For Si-thermistor devices
fabricated at GSFC we use junction gate field-effect transistors (JFETs, which are
voltage amplifiers), and for TESs, we use superconducting quantum interference
devices (SQUIDs, which are current amplifiers).
Electrothermal feedback occurs due to current passing through the thermistor
in order to measure the temperature. The current causes the thermistor to heat via
Joule heating, but this in turn changes the resistance of the thermistor, which then
changes the voltage drop across the resistor. Whether this is positive or negative
feedback depends on the sign of the thermistor’s resistance dependence on tempera-
ture, and whether the detector is voltage- or current-biased. Both Si-thermistor and
TES devices experience negative electrothermal feedback. In the case of a TES, the
resistance in its superconducting-to-normal transition increases with temperature.
With a constant applied voltage bias, the Joule power (PJ = V
2/RTES) will decrease
as the temperature increases. For Si-thermistor arrays, the resistance decreases as
a function of temperature, but Si-thermistor arrays are current-biased rather than
voltage-biased. Negative electrothermal feedback stabilizes the device against ther-
75
Load	resistor	
Thermistor	
Amplifier	
Voltage	bias	
Figure 3.5 A simplified Si-thermistor calorimeter bias circuit diagram
with a load resistor, a voltage-biased thermistor, and an amplifier.
76
mal runaway, and has the added effect of shortening the thermal decay constant
τ .
To wrap up our general view of microcalorimeters, noise, and optimization,
we will briefly show the dependence of the theoretical full-width half-max energy
resolution on several aforementioned parameters. A full derivation of this can be
found in Figueroa Feliciano [2001] and Enss [2005]; here, we will simply point out
that the energy resolution is limited by the various sources of noise that are present
in the measurement of a pulse. To calculate the intrinsic energy resolution of a
device, one must calculate the so-called noise-equivalent power (NEP) for each of
these sources. The NEP is the equivalent inferred power given as an input that
produces a given output noise at a particular frequency, and is related to the energy
resolution [Moseley et al., 1984] by
∆ERMS =
(∫ ∞
0
4df
NEP2(f)
)−1/2
. (3.9)
One then sums the NEP of each noise source in quadrature (assuming they
are uncorrelated), as in the following:
NEP2total = NEP
2
phonon + NEP
2
Johnson + . . . (3.10)
Inputting the relevant variables for the intrinsic sources of noise gives [Enss,
2005]
∆E ∼
√
kbT 20C
α
, (3.11)
77
where α is the dimensionless sensitivity of the thermistor, essentially, the sharpness
of the slope of its R vs T curve in the operating region:
α ≡ dlogR
dlogT
=
T
R
dR
dT
. (3.12)
Equation 3.11 tells us that we would like to run our detector as cold as possible,
choose materials that have a low heat capacity, and select a thermometer with a
very high dependence of R on T.
The most basic goal of a microcalorimeter is to determine the energy of a single
photon. Next we shall address how this is accomplished in practice, considering all
parts of the system, each with some error contribution. We will thus move from a
generalized case to a more specific one of the ECS, and follow the path of a photon
generated in the EBIT chamber, from creation as a photon to a count in a histogram
that makes up a spectrum.
3.2.2 The EBIT Calorimeter Spectrometer
The EBIT Calorimeter Spectrometer contains a Si-thermistor microcalorime-
ter array provided by GSFC that has been a permanent facility-class instrument
at the LLNL EBIT since 2007. Its predecessor, the XRS/EBIT, was based on the
engineering models of the detector systems and microcalorimeter arrays from the
Astro-E and Suzaku missions. During its seven years of operation at the EBIT, the
XRS/EBIT yielded groundbreaking laboratory astrophysics measurements, includ-
ing the first high-resolution (5.5–6.0 eV FWHM at 6 keV) observations of CX in
78
astrophysical elements [Beiersdorfer et al., 2003b]. However, the XRS/EBIT was
not meant to be a permanent instrument: it required frequent upkeep of its liquid
cryogens for operation, and its duty cycle was rather short. The ECS was developed
as an permanent upgrade to the XRS/EBIT system, building off experience operat-
ing and assembling the XRS detector array and with prototype and spare hardware
from the Astro-E and Astro-E2 programs [Porter et al., 2000, 2008b]. Many of the
upgrades made it significantly more user-friendly than the XRS/EBIT: less frequent
cryogenic maintenance, an almost completely automated cooling procedure, and a
much longer hold time at its operating temperature of 50 mK. The detector array
also exhibited improved resolution over the XRS/EBIT, achieving ∼4.5 eV FWHM
at 6 keV.
The ECS uses four steps to achieve the 50 mK operating temperature for the
detector array heat sink. The outer jacket of the ECS cryostat is cooled with liquid
nitrogen to 77 K. Next, an inner jacket is filled with liquid helium at atmospheric
pressure, which cools to 4 K. This shields a He-3/He-4 sorption cooled refrigera-
tor that cools to ∼350 mK. Finally, the sorption cooler is thermally linked via a
heat switch to an adiabatic demagnetization refrigerator (ADR) which cools the
connected detector array to its operating temperature of 50 mK.
He-3/He-4 sorption coolers work by condensing He-3 gas after bringing it in
contact with a sorption pumped He-4 reservoir at ∼1 K. The temperature of the
He-3 liquid is then reduced by pumping on it using a sorption pump.
ADRs use a paramagnetic salt crystal surrounded by a magnet to control the
direction of the magnetic moments in the salt. Bringing up the magnetic field aligns
79
the magnetic moments, which generates heat that is transferred to the He-3/He-4
cooler. Once the ADR and detector have cooled back to the He-3 temperature,
the thermal link to the He-3/He-4 cooler is broken. The magnetic field is then
reduced slowly, allowing the entropy of the salt pill to increase and thus lower its
temperature. Once the magnetic field has been reduced to zero, the cooling power
has been exhausted and the cycle must be repeated.
In the ECS system, the ADR and sorption cooler are recycled together under
software control. Temperature fluctuations of the ADR are minimal (≤ 200 nK RMS
at 50 mK), thus contributing negligibly to degradation in the detector resolution.
The ECS cryogenic package is shown in Figure 3.6.
The ECS detector is a 30-pixel array of silicon-doped thermistors divided into
mid- and high-energy pixels, which together have a dynamic range of 0.05 − > 100
keV. The 16-pixel mid-band array uses 8 µm thick HgTe absorbers each with an
area of 625 × 625 µm and 95% quantum efficiency at 6 keV, and is sensitive to
0.05–12 keV. The 14-pixel high-energy array, sensitive to 0.3–100 keV, uses 100 µm
thick HgTe absorbers with a size of 625× 500 µm and has 32% quantum efficiency
at 60 keV. Figure 3.7 shows the ECS detector chip, including both mid-band and
high-band pixels.
3.2.3 ECS Data Acquisition and Analysis
Let us now begin our task of following the path of a photon from creation to
a count in a spectrum.
80
Figure 3.6 The ECS cryogenic package. The detector assembly is on the
upper right side, the He-3/He-4 sorption cooler is on the left, and the
ADR is in the back, obscured by the other parts.
81
Figure 3.7 Image of the ECS detector chip. The pixels are in the center.
The lighter colored pixels on the left are the hard band pixels and the
darker ones on the right are the mid-band pixels.
82
3.2.3.1 ECS Filters and Windows
After the X-ray is created in the EBIT trap, if it is traveling in the direction
of the ECS radial port, it will first pass through optical and thermal blocking filters.
Four of these filters are positioned along the radiative path of the X-rays inside
the ECS, at different temperature stages of 77 K, 4 K, 300 mK, and 50 mK. The
filters are made of aluminized polyimide that block optical and infrared photons
but allow X-rays to pass through. They are present for two reasons: first, to reduce
the level of shot noise on the detector from these lower energy photons, and second,
to block the radiative thermal load from the room temperature EBIT environment
from heating the detector cold stage. The total aluminum thickness is 1470 A˚ and
the total polyimide thickness is 2386 A˚. In addition, there is a 525 A˚ polyimide
window outside the ECS dewar to isolate the lower ECS vacuum from the higher
EBIT vacuum so as to reduce the amount of background gas in EBIT. Finally, there
are two Be windows (0.5 mil and 5 mil) that can be used to reduce the soft X-ray
count rate on the ECS, allowing a higher fraction of “high-res” events (which will
be explained shortly) for some experiments. The thicknesses of all filters have been
experimentally verified to an accuracy of 10%.
While the purpose of the blocking filters is to block unwanted light but allow
X-rays, the X-ray attenuation is not negligible and must be included in all anal-
yses. The filter transmission of the aforementioned filters is shown in Figure 3.8,
with transmission constants obtained from Henke et al. [1993]. This transmission
constants are considered to be accurate to the 3% level [Saloman et al., 1988]. The
83
Figure 3.8 Filter transmission for the ECS optical and thermal blocking
filters (left), and for the optional windows along the ECS line of sight
(right). Transmission constants are from Henke et al. [1993]. The ab-
sorption edges of the elements present in the filter materials are marked.
Reproduced from Hell [2017].
dips in transmission are located at the so-called absorption edges of the elements
present in the filter materials. These energies correspond to the ionization potentials
of inner-shell electrons of those elements. Photons passing through the filter with
energies near these ionization potentials have a higher probability of ionizing one of
these inner-shell electrons, thus getting absorbed.
In addition to the photon attenuation from the filters, we must also consider
attenuation from contaminants that may freeze on the filters. We believe these
contaminants are from water ice frozen on the 77 K filter, nitrogen and oxygen gas
frozen on the 4 K filter, or both. Since the photoelectric absorption cross section
scales very nearly as E−3, each substance would produce the same transmission curve
at energies above the oxygen K threshold, assuming a given optical depth at a given
84
energy. Given this, for simplicity, we assume that the contaminant is water ice.
During each experiment, the thickness of water ice must be determined by regularly
measuring the O Ly-α to Ly-β line strength ratio in electron impact excitation at
high electron beam energy (∼ 12 keV). The measured value for this ratio is 6.25
[Beiersdorfer, 2003]. From there, one considers the photon transmission through a
material:
T (E) = e−τ(E), (3.13)
where τ is the optical depth,
τ(E) =
t
l(E)
, (3.14)
where t is the thickness of the contaminant material and l(E) is the transmission
length at a given energy.
The decrement is the measured ratio of the O Ly-α to Ly-β line fluxes, cor-
rected for filter transmission, divided by the measured value of 6.25:
Decrement =
Ly − α/Ly − β
6.25
. (3.15)
We equate this with the predicted decrement,
TLy−α
TLy−β
, (3.16)
and solve this for t to calculate the contaminant thickness:
t =
ln(Tα
Tβ
)
1
lβ
− 1
lα
. (3.17)
85
One then calculates the energy-dependent transmission curve for a layer of
water ice of the calculated thickness, and folds this in with the filter transmission
correction during spectral analysis.
3.2.3.2 Absorber
After passing through the ECS filters and frozen contaminants, the photon is
then absorbed by a pixel in the ECS detector. The quantum efficiency (QE) of the
ECS detectors is nearly unity for the energy range of astrophysical K- and L-shell
ions (∼0.1–6 keV), and does not change quickly as a function of energy in this range
(see Figure 3.9). As our CX results do not depend on flux ratios from lines that are
highly disparate in energy (as some other experiments do, e.g., absolute cross section
measurements that are normalized to the radiative recombination line), we do not
correct for quantum efficiency. In general, errors on the QE stem from uncertainties
in the absorber thickness and in the mass absorption coefficient.
3.2.3.3 Amplification, Filtering, and Digitization
After the photon is absorbed, the signal passes through a JFET that is con-
nected to that pixel. JFETs are run at a warmer temperature than the thermistor,
so they are thermally isolated from the detector cold stage. JFETs are run at this
temperature in order to minimize their noise output. At lower temperatures, there
are less electrons available in the conduction band to be able to move charge, and
shot noise increases. At higher temperatures, Johnson noise increases. 130 K is the
86
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Figure 3.9 Quantum efficiency for the 8 µm thick HgTe absorbers on
the low-energy pixels (top) and the 100 µm thick HgTe absorbers on
the high-energy pixels (bottom) of the ECS, as given by various sources.
Absorption edges are marked. Reproduced from Hell [2017].
87
point where the sum of these effects is minimized. After the signal is passed through
the JFET, it goes to the analog processing unit called the Calorimeter Analog Pro-
cessor (CAP), where the signal is further amplified. Next, the signal gets filtered,
digitized, and passed to the Software Calorimeter Digital Processor (SCDP).
3.2.3.4 The Software Calorimeter Digital Processor
The SCDP, developed for data analysis for the Astro-E/Astro-E2 missions,
takes an input data stream and converts it into photon pulse heights that correspond
to their energies. It performs this in several steps which will be explained next in
more detail: generally, it triggers the data stream, tags the photon with its arrival
time relative to the EBIT phase, applies an “optimal filter” to determine a photon
energy, and categorizes each pulse with an event grade. From there, the end user
can interact with the data using custom analysis software written in a commercial
software suite called Igor Pro.
3.2.3.5 Triggering
The SCDP first triggers the data stream. The trigger point defines (nearly)
the start of a data record of a given length, which should contain the majority of
the pulse decay. Some pre-trigger data are stored as well to check that all the event
data is contained in the data record.
88
3.2.3.6 Optimal Filtering
Next, the SCDP applies an optimal filter to the pulse. The details of optimal
filtering were first presented in Szymkowiak et al. [1993]; Adams et al. [2009] presents
the method that is currently implemented in the SCDP. The optimal filter is a
matched filter that is weighted to optimize signal-to-noise over all frequencies. It
allows us to make a best estimate of the pulse height, and therefore photon energy.
The premise is that each pulse, without noise, is a scaled version of the same shape,
i.e.
HγS(t), (3.18)
where Hγ is the amplitude of the pulse shape, and is proportional to the energy of
the photon, and S(t) is the invariant pulse shape.
The actual measurement of the pulse shape that we make, D(t) is given by
D(t) = HγS(t) +N(t), (3.19)
where N(t) is the noise. This is sometimes called a “noisy pulse.”
Our goal, then, is to find Hγ for each pulse. We obtain Hγ by minimizing,
in the least squares sense and in frequency space, the difference between the “noisy
pulse” and the model of the pulse shape, HγS(f). This can be shown by
χ2 =
∑ [D(f)−HγS(f)]2
N2(f)
. (3.20)
89
To minimize Equation 3.20, one sets the derivative of χ2 with respect to Hγ to
zero. It can be shown [Figueroa Feliciano, 2001] that after doing this, the optimal
estimate for Hγ, within a normalization constant, is given by
Hγ =
∑ D(f)S∗(f)
N2(f)
. (3.21)
This equation is now composed of elements that we can measure. To obtain
the noise power spectrum, N(f), we measure many data records without an X-ray
source to get N(t), then perform a Fourier transform. As mentioned before, D(t)
is obtained by measuring the detector response to many photons of a particular
energy. We then take a Fourier transform to obtain D(f). S(f) is found by dividing
D(f) by the energy of the incident photons.
We then transform Equation 3.21 back to the time domain:
Hγ =
∑
D(t)F (t), (3.22)
where F (t) is the inverse Fourier transform of F (f) = S(f)/N2(f).
There are two caveats to this method: optimal filtering is only truly optimal
if the detector response is linear (i.e., follows Equation 3.18), and if the noise is sta-
tionary (i.e., its power spectrum does not vary over time). These two requirements
are not always true for real detectors: saturation of the detector from a high-energy
photon will change the pulse shape, and Johnson and phonon noise both depend on
the resistance of the detector, which changes over the duration of a pulse. Other
methods of pulse analysis have been developed, with similarly good results and vary-
90
ing levels of complexity. See Peille et al. [2016] for a review and comparison of these
methods.
3.2.3.7 Event Grades and Instrumental Response
In determining the pulse height, the SCDP flags each photon with one of three
event grades: high-res, mid-res, and low-res. These correspond to the amount of
time the exponential decay from the initial event is recorded before another photon
on the same pixel causes a second pulse. High-res events only have one photon event
for the entire data record. Mid-res and low-res pulses have a secondary pulse that
rides on the tail of the primary one, the later in time after the primary pulse than
the latter. High-res events are so named because obtaining a full measurement of
the pulse shape over an entire record length leads to a better match to the optimal
filter, thus higher spectral energy resolution.5 Operationally, using the beryllium
window or decreasing the EBIT beam current can reduce the count rate on the ECS
to maximize the number of high-res events. Ishisaki et al. [2016] has more details
about event grades.
Even with a perfectly monoenergetic photon beam, the ECS instrumental
response is not a delta function with infinite resolution. The actual instrumental
response has a Gaussian core [Porter et al., 2004, Cottam et al., 2005], with its
FWHM value defining the FWHM energy resolution of each pulse. For high-res
5Optimizing the record length, thus, is a balance between minimizing pileup (where short record
lengths would be best) and maximizing the frequency space over which we can measure a pulse,
in order to maximize the energy resolution (where long record lengths would be best).
91
primary events on the ECS, this value is ∼4.5 eV at 6 keV. This finite energy
resolution is due to the fact that the detector and system noise is white [Eckart et al.,
2016]: essentially, the optimally filtered pulse height calculation for a monoenergetic
photon beam will have a normal distribution of values due to these noise effects.
3.2.3.8 Gain Scale
After calculating a pulse height, it is necessary to apply a gain scale to match
the instrumental response to an energy in eV. After we create a gain calibration file,
we apply it to our data in Igor Pro, discussed in the next section.
Creating a gain calibration file for the ECS involves injecting species with well-
known energies into the EBIT then collisionally ionizing them and recording their
spectra. The species should be chosen to map out a fairly wide range in energy space,
focusing on the energies around the lines of interest to the experiment. This must
also be done at the experimental operating temperature, since the measured pulse
height is temperature-dependent. One then inputs this spectrum into a software
program called CXRS, developed by Ming-Feng Gu [private communication], that
fits each line to find its centroid in instrumental units then maps it to its energy
in eV. This correlation is usually fit with a fourth order polynomial to make a gain
curve that extends across all relevant energies. This is illustrated in Figure 3.10.
The ECS gain is fairly linear for small energies, but large signals (and thus large
energies) reach a more non-linear region of the gain scale. This causes the pulse
template to not fit the pulse as well, leading to a degradation in energy resolution.
92
Energy (eV)
In
st
ru
m
en
ta
l U
ni
ts
Figure 3.10 Sketch of the gain calibration procedure and resulting gain
curve for the ECS. Measurements of spectral lines from K-shell elements
with well-known energy centroids are used to map ECS instrumental
units to a photon energy in eV. Figure adapted from Hell [2017].
The gain calibration may vary slightly on time scales of hours to days. How-
ever, if the gain scale was created contemporaneously with the relevant measure-
ments, it is accurate to within ∼0.5 eV. This can be verified by applying a gain scale
to an experiment and comparing the Gaussian centroid of the measured line with
line energies in the literature. This accuracy is set by many parameters, including
the range of species used in the gain calibration procedure and the accuracy of the
polynomial fit to the gain curve, the temperature stability of the ECS over the mea-
surement period, and the linearity of the ECS detectors. An accurate gain scale is
important for line identification, however, for the CX measurements presented here,
our measured accuracy is sufficient for our needs.
93
3.2.3.9 Final Analysis
Our photon has finally traveled from the EBIT chamber to the ECS, and
has been transformed into an entry in an event list with an energy determination
and a set of flags. We use the commercially available software program Igor Pro to
interact with our data, as well as to adjust and apply any other post-processing tools
that may be necessary, such as the aforementioned gain calibration. Other post-
processing steps that may be necessary include excluding pulses due to crosstalk,
applying corrections for a temperature drift of the ADR, or including or excluding
pulses of a certain EBIT phase, pulse height, or time of the experiment. Finally, we
extract a spectrum.
3.3 The Transition-Edge EBIT Calorimeter Spectrometer
Together, the ECS and the EBIT have enabled fundamental laboratory mea-
surements that no other technique can achieve, demonstrated by over 40 refereed
publications to date, e.g. Leutenegger et al. [2010], Beiersdorfer et al. [2003b], Thorn
et al. [2009], Brown et al. [2006], Graf et al. [2009], Gu et al. [2007b]. This includes
benchmarking atomic theory for charge exchange, measuring absolute cross sections
of L-shell transitions for astrophysically abundant elements from Mg to Ni, and
studying K-shell emission in He-like Fe and Ni, including dielectronic recombination
satellite emission from lower charge states. ECS measurements even provided di-
agnostic laboratory data that were critical for interpreting spectra from the Hitomi
mission [Collaboration, 2016].
94
However, upgrading from a 30-pixel Si-thermistor microcalorimeter to a kilo-
pixel transition-edge sensor microcalorimeter is important for both scientific and
technological reasons. State-of-the-art silicon thermistor microcalorimeters that
have been optimized for astrophysical measurements achieve ∼3 eV resolution at
6 keV [Porter et al., 2006], and typical array sizes are 36 pixels. The energy res-
olution of these devices is hindered by a relatively small α parameter of ∼10 or
less [Enss, 2005] (see Equation 3.11). TES microcalorimeters, by contrast, use a
thin-film superconductor operated in its superconducting-to-normal transition as
the thermometer, are read out using superconducting electronics, and have values
of α close to or surpassing 100. Compared to the current-generation silicon devices,
state of the art TES microcalorimeters have higher energy resolution and can allow
higher count rates by a factor of ∼10–100 per pixel.
An important advantage of using TESs for astrophysical and laboratory ap-
plications is that they are scalable to large arrays. The current-generation silicon
thermistor arrays like the ECS appear prohibitively difficult to scale to significantly
larger pixel count primarily because each pixel requires its own wiring chain from
room temperature to the 50 mK detector cold stage. In comparison, multiplexed
readout schemes developed by collaborators at NIST/Boulder enable multiple TES
pixels to share preamplifiers and signal wiring, so that the cryogenic wiring complex-
ity of an N ×N -pixel array scales as only N , as opposed to N2 for silicon thermistor
(non-multiplexed) devices [Irwin, 2002]. In addition, TES array fabrication does not
require any hand assembly, as is required for the silicon devices, which allows for
more straightforward fabrication of large arrays through microfabrication.
95
Furthermore, in TESs, the thermal link between the detector absorber and the
cold bath is chosen to be stronger than in Si-thermistor devices, which shortens the
thermal decay time constant and thus increases the throughput. This will lead to
higher count rate capabilities, thus shorter experiment times and a higher availability
for science.
A large technological challenge is that no fully operational kilopixel TES in-
strument that meets the requirements for Athena or the proposed Lynx mission
currently exists.6 In preparation for these future missions, it is critical to develop
a proof-of-concept operated on the ground to determine the limitations on the cur-
rent baseline designs, both in hardware and software, and to address the many open
questions related to the performance and optimization of large arrays of TESs.
In response to these scientific and technological needs, the microcalorimeter
group at GSFC will soon deploy a kilopixel array of TESs called the Transition-Edge
EBIT Microcalorimeter Spectrometer (TEMS) as a user system at the LLNL EBIT.
TEMS, a result of a longstanding collaboration between GSFC, NIST/Boulder, and
LLNL, will be the first fully functional flight-like array of its kind in the world.
The goal of this new facility is both to produce scientific results that are improved
over those using the ECS, and also to be a critical stepping stone for the X-IFU on
6Our collaborators at NIST/Boulder have, however, deployed several TES arrays of up to ∼250
pixels to various light sources, accelerator facilities, and laboratory facilities. These detectors are
not kilopixel arrays of close-packed pixels as is required for Athena, but achieve excellent resolution.
The arrays that are optimized for photon energies of < 10 keV achieve resolutions of up to 2.1 eV
FWHM at 6 keV, and arrays optimized for photon energies of < 2 keV achieve resolutions as high
as 1.0 eV FWHM at 0.5 keV [Doriese et al., 2017].
96
Property ECS TEMS
Thermometer type Si themistor MoAu TES
Energy resolution at 6 keV 4.5 eV < 3 eV
Energy resolution at 60 keV 32 eV 25 eV (Planned upgrade)
Array size 30 pixels 1024 pixels
Pixels read out 30 252
Timing 10 µs < 1 µs
Count rate/array 100/s > 5000/s
Table 3.1 Comparison between the ECS and the TEMS.
Athena and the proposed detector array on Lynx. Table 3.1 shows a summary of
major design differences between the ECS and the future TEMS.
3.3.1 Technical Specifications
The TEMS detector is a 32×32 pixel array with 250 µm pixel pitch that uses a
Bi/Au X-ray absorber with a thin-film Mo/Au superconductor as the thermometer.
With the current generation of readout electronics and geometrical constraints in
the focal plane design, 252 of those pixels are wired and read out. The main detector
array is sensitive to 0.1–10 keV; a second array of 8 × 8 pixels will be added later
to cover the hard X-ray band from 0.5–100 keV, with 25 eV resolution and 70%
quantum efficiency at 60 keV [Bennett et al., 2012].
The array is contained in a 15 mm × 19 mm chip, and the pixel pitch is
250 × 250 µm. Each pixel has a 140 × 140 µm2 transition-edge sensor that is
97
Figure 3.11 The resistance vs. temperature curve of five pixels in the
TEMS array showing the superconducting-to-normal transition. The
normal-state resistance is ∼8 mΩ.
composed of 50 nm of Mo and 220 nm of Au, and has a normal-state resistance of
∼8 mΩ. A figure of a transition shape of a TEMS pixel is shown in Figure 3.11.
A T-shaped stem sits on the TES to support the 240 × 240 µm2 absorber made
of 1.5 µm of Au and 4.0 µm of Bi. Both the TES and the stem are supported by
a silicon-nitride membrane, which provides a weak thermal link to the heat bath.
There is also ∼64 µm of silicon between each pixel in order to provide a heat sink
and minimize thermal cross-talk. The TES bias leads, made of Nb microstrips, are
run along this frame. Figure 3.12 shows a photo of a detector chip much like the
one implemented in TEMS and a schematic of one pixel.
One important requirement of large multiplexed TES arrays is the uniformity
98
240	µm	
Absorber	contact	
region	
140	µm		
Nb	bias	leads	
Figure 3.12 A kilopixel array like the one currently used in the TEMS,
fabricated at NASA/GSFC. The inset on the left shows a magnification
of a few pixels, and the cartoon on the right shows the geometry of the
TES (yellow), absorber and absorber stem (tan and brown) and niobium
bias leads (green). Image courtesy of Stephen Smith.
99
of the critical temperature (Tc) of each pixel. With TDM, each column of pixels is
voltage biased by a common line, so in order for each pixel in the column to be held
at the same point on its transition (R/Rnormal), the critical temperature of the pixels
must be very similar. The TEMS detector chip has an average Tc of 91 + /− 0.12
mK across 220 currently operating pixels, exhibiting excellent uniformity.
3.3.2 Device Readout
The TEMS TESs are read out using Superconducting Quantum Interference
Devices (SQUIDs), developed at NIST/Boulder, using time-division multiplexing
(TDM) [de Korte et al., 2003]. In this scheme, each TES is coupled to a first-stage
SQUID amplifier, and the output from each SQUID per column of pixels is read out
sequentially in time. The first-stage SQUIDs from each column are coupled through
a summing coil to a second stage SQUID, whose signal gets amplified by a SQUID
series array. A schematic of a 2-row × 2-column SQUID TDM system is shown in
Figure 3.13.
Magnetic shielding for the magnetically sensitive detector and SQUIDs from
Earth’s magnetic field and the 4 T magnet in the ADR consists of a supercon-
ducting niobium shield around the detector card, and a mu-metal shield around
the circumference of the cryostat. The TEMS focal plane, including the GSFC-
fabricated detector chip shielded by a copper collimator, flexible wiring for TES
bias and readout, and NIST-supplied multiplexing and readout chips, is shown in
Figure 3.14.
100
Figure 3.13 Circuit diagram for a 2-row× 2-column time-division SQUID
multiplexer. Reproduced from Kilbourne et al. [2008]
101
Figure 3.14 The current NIST-developed TEMS focal plane assembly,
pictured here with a GSFC detector array that is obscured by a copper
collimator. Only a fraction of the pixels on the chip are wired and read
out (which trace out a rough “plus” pattern), and the collimator blocks
X-rays from being absorbed by the pixels that are not active. This
reduces the thermal crosstalk measured by the active pixels.
102
3.3.3 TEMS Cooling System
Like the ECS, the TEMS is operated at 50 mK in order to achieve optimal
spectral energy resolution. The cooling system consists of a cryogen-free 3 K pulse
tube cooler and a 300 mK closed-cycle sorption cooler that cools a single stage
ADR operating at 50 mK. The 50 mK stage currently has a hold time of 60+ hours
and recharges automatically under software control; the final system will have an
estimated hold time of ∼120 hours at 50 mK.
3.3.4 Current Status of TEMS and Outlook
We have tested several different detector chips for the TEMS system in order
to select one with the best intrinsic energy resolution for permanent installation.
During one of these tests, we achieved an average FWHM energy resolution of 3 eV
for 30 pixels in a 2×16 multiplexing configuration, for count rates of one per second
per pixel [Smith et al., 2014c]. This shows substantial progress from the results
presented in [Kilbourne et al., 2008], which demonstrated 2.9 eV FWHM resolution
at the 2 × 8 multiplexing level. Figure 3.15 shows the spectral energy resolution
of all pixels read out with this detector array during 2 × 16 multiplexing tests at
GSFC.
The detector chip that is currently installed in the TEMS system exhibits
even higher intrinsic spectral resolution and higher uniformity of the TES transition
temperature across the chip. Figure 3.16 shows the 2.35 eV spectral resolution at 6
keV for one pixel in the current detector array.
103
Figure 3.15 Spectral resolution of all pixels read out during a multiplex-
ing test of 2 columns of 16 pixels each, performed on a previous iteration
of a TEMS detector chip. The average resolution was 3 eV at 6 keV for
all pixels.
The detector system with this chip is currently undergoing tests at GSFC to
optimize the spectral resolution and performance while multiplexing all 8 columns
of 32 pixels. The team is also upgrading the SCDP software to be able to handle
the more complex TES data. The current pulse processing method requires ex-
pert knowledge of TES behavior and operation, and is done on a per-pixel basis.
This is not feasible for a large pixel array in system that will be used regularly at
a laboratory facility, sometimes for days at a time, and used by scientists of all
backgrounds. Making the TEMS a turnkey system is a major undertaking, but will
provide extremely beneficial lessons learned for future spaceflight applications.
Pulse analysis is not the only new complexity when moving from Si-thermistor
devices to TESs. The TESs themselves exhibit variations in their pulse shape or
transition shape that often depend on the operating temperature, noise conditions,
104
Figure 3.16 FWHM energy resolution of 2.35 eV at 6 keV for one pixel
in the current TEMS detector array.
105
or the magnitude of the magnetic field present. Multiplexing adds further sources of
operational complexity and noise. Even given these factors, results from TEMS are
excellent, and progress on its development and completion is promising. Once TEMS
is able to demonstrate routine resolution of <3 eV at 6 keV while multiplexing, with
minimal issues such as crosstalk and gain stability, it will be delivered to the LLNL
EBIT facility.
This has only been a quick summary of TEMS; a more detailed description of
TESs and SQUID readout electronics is out of the scope of this dissertation. For
more information about TES physics, see Enss [2005], and for further information
on SQUIDs, see Clarke and Braginski [2006].
106
Chapter 4: Observation of Highly Disparate K Shell X-ray Spectra
Produced by Charge Exchange with Bare Mid-Z ions
Now that we have discussed the theoretical and experimental tools that we can
harness to gain a better understanding of the underlying atomic physics of CX, in the
rest of this dissertation, we present recent results that stem from measurements of
both K-shell and L-shell CX spectra. The text in this chapter is taken directly from
a published paper from 2014 [Betancourt-Martinez et al., 2014c] that highlights the
complex nature of the CX collision and the need for more comprehensive quantum
mechanical theoretical calculations, even for the relatively simple case of K-shell
ions.
4.1 Introduction
Charge exchange (CX), or charge transfer, is a semi-resonant process in which
a highly charged ion captures one or more electrons from a neutral atom or molecule
during a close interaction. Charge exchange is important in setting the ionization
balance in laboratory and astrophysical plasmas, as a spectral diagnostic for fusion
plasmas, in determining the storage time in ion traps and storage rings, and in
antihydrogen production [Isler, 1977, Fonck et al., 1982, Fonck and Hulse, 1984,
107
Kallne et al., 1984, Rice et al., 1986, Sto¨hlker et al., 1998, Gabrielse et al., 2007,
Greenwood et al., 2001, Cadez et al., 2003, Mawhorter et al., 2007, Greenwood
et al., 2004, Simcic et al., 2010, Carolan et al., 1987, Beiersdorfer et al., 2005,
Isler, 1994]. Astrophysically, charge exchange occurs in planetary atmospheres and
the comae of comets interacting with the solar wind, and has been hypothesized
to occur at the rim of supernova remnants [Katsuda et al., 2011, Cumbee et al.,
2014]. Charge exchange also occurs between solar wind ions and neutrals in the
exosphere and in the heliosphere, which adds variable foreground emission for every
astrophysical observation from our solar system [Lisse et al., 1996, Cravens, 1997,
Gladstone et al., 2002, Dennerl et al., 2006, Koutroumpa et al., 2009a, Dennerl
et al., 2012, Bhardwaj et al., 2007, Slanger et al., 2008, Bodewits, 2007]. Astro-
H, a JAXA satellite observatory scheduled for launch in 2015, features an x-ray
calorimeter imaging spectrometer that will measure the first high-resolution x-ray
spectra of extended objects. Correct interpretation of these observations will require
accurate modeling of foreground CX.
In order to model CX spectra, we must know the n- and l-selective electron
capture cross section. Classical treatment predicts a sharp peak in the n capture
distribution at nc ∼ q3/4
√
IH
In
, where IH and In are the ionization potentials of
hydrogen and the neutral target, respectively, and q is the ion charge [Janev and
Winter, 1985]. The n-distribution is generally well-understood, and many papers
show good agreement between theory and experiment for n-selective capture cross
sections, as in Mawhorter et al. [2007], Otranto and Olson [2010], Igenbergs et al.
[2012] and Wu et al. [2012].
108
The l distribution, which is dependent on collision energy, is more challenging
to model correctly. One can use classical considerations to determine the electron
capture state: in high-energy collisions, in the limit of strong Stark mixing, l states
are assumed to be populated statistically, and high l states dominate [Janev and
Winter, 1985]. In low-energy collisions, the electron does not have enough angular
momentum to populate the higher l states, and statistical assumptions do not apply.
This was verified in, for example, Beiersdorfer et al. [2000a].
The Lyman series x-ray emission is a powerful diagnostic for determining the
state-selective capture cross section, σnl, in bare ions undergoing charge exchange
with a neutral gas. Electron capture into an l = 1 state is dominated by direct decay
from nc, l = 1 to the ground state [Wargelin et al., 2008], emitting a ncp → 1s Ly
series photon. If the electron is captured into a high angular momentum state, it
will decay along the Yrast chain in steps of ∆n = −1 and ∆l = −1, finally yielding a
2p→ 1s Ly-α photon. Therefore, if the electrons are captured following a statistical
cross section, the strongest line in the spectrum will be Ly-α; if the electron capture
cross section is largest for an np state, the ncp→ 1s Ly lines will dominate.
The hardness ratio, H, is the ratio of the ncp→ 1s to 2p→ 1s emission, where
n is greater than two, i.e.,
H =
∑∞
n=3 Fn→1
F2→1
(4.1)
where F represents the flux in the denoted transition lines. The hardness ratio
is expected to decrease with increasing collision energy, since as collision energy
increases, so does the cross section for capture into higher angular momentum states,
109
producing more Ly-α photons. The hardness ratio can therefore be used as a probe
of the collision velocity, allowing us to measure, for example, the velocity of the
solar wind. An added benefit of the hardness ratio is that, in principle, it can be
determined even with medium-resolution detectors that may not be able to resolve
individual high-n Lyman lines, such as CCDs or high purity germanium solid state
detectors that typically have resolutions on the order of ∼100 eV at 1.5 keV.
Besides the classical over-the-barrier (COB) method [Ryufuku et al., 1980,
Niehaus, 1986], several other more complex charge exchange models exist which
can be used to estimate total or state-selective cross sections. These include the
Landau-Zener (LZ) method [Salop and Olson, 1976], the multichannel Landau-
Zener (MCLZ) approximation [Janev et al., 1983], the atomic-orbital close-coupling
(AOCC) method, and the molecular-orbital close-coupling (MOCC) method [Fritsch
and Lin, 1991, Shimakura et al., 1993]. The classical trajectory Monte Carlo (CTMC)
method [Abrines and Percival, 1966, Olson and Salop, 1977] is the most widely used
approximation, due to its simplicity and its accuracy, compared to other models, in
predicting experimentally-measured n-state selective cross sections.
COB and CTMC models agree qualitatively with experiments that show a
decrease in hardness ratios with increasing collision velocity [Otranto et al., 2006].
Both models also agree qualitatively with experiments that show an increase in cap-
ture cross section with increasing ion charge, but have systematic uncertainties of
∼ 25− 50% when predicting absolute cross sections [Cadez et al., 2003, Mawhorter
et al., 2007, Djuric´ et al., 2008]. CTMC shows better agreement with experiments at
high collision energies (above ∼1 keV amu−1), as demonstrated in comparison with
110
experimental results from fusion plasmas excited with a very energetic hydrogen or
deuterium beam [Beiersdorfer et al., 2005]. However, discrepancies between models
and experiments arise at the edge of tokamaks, where cold ions interact with molec-
ular gas, in electron beam ion traps (EBITs), where the collision velocity is ∼10
eV amu−1, and also in space, where cometary, exospheric, and heliospheric neutrals
interact with low energy solar wind ions.
The COB and LZ methods do not make predictions for l-selective capture
cross section, and to the extent that there are measurements of l-selective capture
cross section at low collision energies, especially over a range of conditions and
interacting species other than atomic hydrogen, theoretical calculations that can
incorporate l, such as CTMC and AOCC/MOCC, show poor agreement with ex-
periments [Leutenegger et al., 2010, Beiersdorfer et al., 2000a, Stancil et al., 2002,
Wu et al., 2012, Mancˇev et al., 2013]. Furthermore, the commonly used CTMC
method cannot incorporate multi-electron capture (MEC), which becomes impor-
tant in the low-energy regime [Ali et al., 1994, Otranto et al., 2006].
Several comparisons between theory and experiment have been made, with
varying results. Otranto et al. [2006] demonstrated that while CTMC calculations
qualitatively agree with EBIT experiments where nc decreases for increasing ioniza-
tion potential, their CTMC model overestimates the flux in high-n Rydberg transi-
tions following CX onto O8+. Beiersdorfer et al. [2000a] performed experiments with
EBIT-I showing that, contrary to CTMC calculations, the hardness ratio following
CX with bare Ne, Ar, Kr and Xe was always nearly unity. Further, the disagree-
ment between calculated and measured hardness ratios worsened at higher atomic
111
numbers. Otranto et al. [2007] presented EBIT results involving O8+ demonstrating
that the hardness ratio can vary within nearly a factor of two by varying the neu-
tral gas. Leutenegger et al. [2010] presented EBIT spectra of bare Ar and P that
concurrently underwent CX with the same neutral gas, and contrary to the trend
established in Beiersdorfer et al. [2000a] and also to previous CTMC calculations,
the hardness ratios measured for Ar and P differed by a factor of two.
In this paper, we present experiments that investigate the dependence of CX
line emission on the ionization potential of the neutral gas, the number of valence
electrons in the neutral gas, and the atomic number of the ion. We probe whether
any of these characteristics are predictive of CX spectral features, and provide em-
pirical data towards more comprehensive and quantitatively accurate models.
4.2 Experimental Method
The measurements presented here were performed with the EBIT-I electron
beam ion trap at the Lawrence Livermore National Laboratory [Beiersdorfer et al.,
2003a, Beiersdorfer, 2008]. The spectra were measured using the EBIT calorimeter
spectrometer (ECS) [Porter et al., 2004, 2008a]. The ECS is a non-dispersive spec-
trometer developed at NASA/Goddard Space Flight Center with quantum efficiency
of nearly unity over a large bandwidth. The 30-pixel array of silicon-doped thermis-
tors is divided into a mid- and a high-energy array of 16 and 14 pixels, respectively,
which together have a dynamic range of 0.05–100 keV. The experiments discussed
here made use of the mid-band array, which has an energy resolution of ∼4.5 eV at
112
6 keV.
The ECS has four internal aluminized polyimide filters used to block optical
and thermal radiation at temperature stages of 77 K, 4 K, 300 mK, 50 mK, with
a total aluminum thickness of 1460 A˚ and total polyimide thickness of 2380 A˚. In
addition, we used a 500 A˚ polyimide window outside the ECS dewar to isolate the
ECS vacuum from the EBIT vacuum. These thicknesses have been experimentally
verified to an accuracy of ∼10%.
Fundamentally, the EBIT operates in one of two modes: electron trapping
mode and magnetic trapping mode. In electron trapping mode, after neutral species
are injected into the ion trap, they are collisionally ionized by the electron beam,
then confined in the trap. The ions are radially confined due to the electrostatic
attraction of the electron beam, and axially confined by a voltage potential applied
across three copper drift tubes. Typical thermal energies of trapped ions are ∼10
eV amu−1 at typical beam currents of ≥ 130 mA and trap potentials of ≥ 100 V
[Beiersdorfer et al., 1996a], and typical ion densities in the trap are ∼ 3× 109 cm−3.
For a charge exchange experiment in the EBIT-I, first, ions are created during
electron trapping mode, with the electron beam turned on for typically about 0.5
seconds to generate a sufficient number of bare ions of mid-Z elements. Next, in
magnetic trapping mode [Beiersdorfer et al., 1996b], the electron beam is turned off
and the ions are radially trapped by the 3 T magnetic field of the superconducting
Helmholtz coils and the electric field of the drift tubes, so that the EBIT is effectively
a Penning trap. The ions gain electrons and emit X-rays through CX with the
neutral species introduced into the trap. Since the beam is off, no excitations due
113
to electron impact occur. Filling the K shell of most of the trapped bare ions
takes about 0.5 seconds. After this occurs, the trap is dumped and the cycle is
repeated. It typically requires several hours to accumulate sufficient statistics for
each experiment.
We performed charge exchange experiments for the following ions and neutral
gases: Mg12++CO2, Mg
12++H2, Mg
12++He, Mg12++Ne, Cl17++C2H4Cl2, Cl
17++He,
Ar18++Ar, S16++He, and S16++SF6. In the Mg experiments, there were contami-
nant ions present in the trap, including P, S, Si and Ar, which entered from a port
on EBIT that was open in order to perform crystal spectrometer measurements.
We did not observe any significant effect from these contaminants in the spectra,
other than the presence of their K shell emission lines. The charge exchange cross
section for ion-ion interactions is negligible. We verified that the x-rays recorded
were nearly all from interactions of trapped ions with the chosen neutral gas, and
not with background gases in the trap, by reducing the neutral CX partner gas
injection pressure to zero while holding all other experimental parameters constant.
From contemporaneous measurements, we estimate that the background CX rate
for the Mg experiments was about 10% of the experimental CX rate; for all other
experiments the background rate is on the order of 1%.
4.3 Analysis and Spectra
We fit Lyman series lines from H-like ions, assuming a Gaussian instrumental
function [Kelley et al., 2007]. We used reference energies calculated with the Flex-
114
ible Atomic Code (FAC) [Gu, 2008]. We accounted for partial line blending where
two transitions from different ions could be disentangled due to differing energy cen-
troids. In a few cases, however, ion line flux could not be separated from other lines
due to contaminants with transitions that overlapped with the lines of interest: for
example, Mg11+ Ly-γ is nearly coincident with the strong Si12+ 1s2s 3S1 → 1s2 1S0
transition. In order to account for this in our hardness ratios, we determined a lower
limit assuming zero flux in Ly-γ, which appears in the text of table 4.3 and Figures
4.4 and 4.4, and an upper limit by including the entire blended line, mentioned in
the caption of table 4.3. In the absence of independent measurements of the Si lines,
we assert that the lower limit is the more realistic one; the Si forbidden line in CX
is more likely to be stronger than the Mg11+ Ly-γ line.
We corrected the observed flux for attenuation from the optical and thermal
blocking filters as well as frozen contaminants on the filters, which we believe is
either or both of water ice or nitrogen gas frozen on the 77 K filter, or nitrogen
gas frozen on the 4 K filter. Since the photoelectric absorption cross section scales
very nearly as E−3, either substance would produce the same transmission curve at
energies above the oxygen K threshold, assuming a given optical depth at a given
energy. Therefore, we make a fiducial assumption that the contaminant is water
ice. We determine the thickness of water ice using the decrement in the ratio of the
O Ly-α to Ly-β line strengths in electron impact excitation experiments compared
to the measured ratio of 6.25 in the limit of high incident electron beam energy
[Beiersdorfer, 2003]. The oxygen measurements were taken regularly during the
experiments. The inferred fiducial ice layer thickness was typically ∼1 µm, which
115
3400 3600 3800 4000 4200 4400
Energy (eV)
0
25
50
75
C
o
u
n
ts
Lyα Kβ Kγ KδLyβ
Kǫ
Kζ
Kη
Kθ
Kι
Lyγ
Lyδ
Lyǫ
Lyζ
Lyη
Lyθ
Lyι
Figure 4.1 Spectrum of bare argon undergoing CX with neutral argon.
corresponds to a line flux correction of ∼10–20% in the Mg Ly band from 1300–1900
eV, and ∼1% for the S, Cl, and Ar experiments with line energies above ∼2400 eV.
This corresponds to a maximum effect on the ratio of Ly-n:Ly-α of ∼10% for the
Mg experiments and ∼1% for the S, Cl, and Ar experiments.
Selected spectra are presented in Figures 4.3–4.3. Lyman lines refer to the
hydrogen-like ion emission; “K” lines refer to He-like ion emission. K lines can be
used for various diagnostics, but in this paper we focus on the Lyman emission.
Hardness ratios and other diagnostic ratios for all experiments are summarized in
tables 4.3 and 4.3. Individual line fluxes are also presented in these tables.
4.4 Discussion
The hardness ratios determined for the experiments presented here span a
larger range than previous experiments have shown, and can deviate widely from
CTMC calculations, as can be seen in Figure 4.4. In earlier CX experiments using
116
1500 1600 1700 1800 1900 2000
Energy (eV)
0
40
80
 
Mg12+  + CO2 Si12+
Lyα
Kβ Lyβ Lyγ Lyδ
Lyǫ
Lyζ
Lyη
Si13+
0
20
40
 
Mg12+  + He
Si12+
Lyα
Kβ Kγ Kδ Lyβ
Lyγ Lyδ Lyǫ Si13+
0
15
30
 
Mg12+  + Ne
Si12+
Lyα
Kγ Lyβ
Lyδ
Lyǫ
Lyζ
Lyη
0
20
40
 
Mg12+  + H2
Si12+Lyα
Kβ Kγ Kδ Lyβ
Lyγ
Lyδ Lyǫ Lyζ
C
o
u
n
ts
Figure 4.2 Spectra from bare Mg undergoing CX with four different
neutral gases. Relevant transitions, along with background ions, are
labeled.
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0
75
150
C
o
u
n
ts Lyα Kβ Kγ Lyβ
Kδ Kǫ Kζ Lyγ
Lyδ
Lyǫ
Lyζ
Lyη
Figure 4.3 S16+ and He charge exchange. Note the extremely strong
8→ 1 transition (Ly-η), identified in red.
117
T
ab
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4.
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0
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6
2
.2
2
±
0
.4
5
1
.1
6
±
0
.2
7
1
.2
6
±
0
.1
9
1
.4
1
±
0
.1
9
H
0
.7
3
8
±
0
.0
5
4
2
.5
8
1
±
0
.3
7
6
0
.5
0
9
±
0
.0
5
9
0
.4
9
0
±
0
.0
6
5
1
.8
6
8
±
0
.3
7
4
2
.6
6
0
±
0
.2
6
5
0
.7
1
4
±
0
.1
1
6
2
.3
8
3
±
0
.3
0
7
0
.4
6
0
±
0
.0
4
7
0
.6
1
4
±
0
.0
4
1
118
T
ab
le
4.
2
T
h
is
ta
b
le
p
re
se
n
ts
th
e
n
or
m
al
iz
ed
H
e-
li
ke
se
ri
es
li
n
e
st
re
n
gt
h
s
fo
r
th
e
ex
p
er
im
en
ts
p
re
se
n
te
d
in
th
e
te
x
t
ex
ce
p
t
fo
r
M
g1
1
+
+
N
e,
w
h
ic
h
h
ad
si
gn
ifi
ca
n
t
li
n
e
b
le
n
d
in
g
in
th
e
M
g
fo
rb
id
d
en
(z
),
in
te
rc
om
b
in
at
io
n
(y
an
d
x
),
an
d
re
so
n
an
ce
(w
)
li
n
es
w
it
h
N
e9
+
.
F
il
te
r
tr
an
sm
is
si
on
-
an
d
ic
e-
co
rr
ec
te
d
li
n
e
st
re
n
gt
h
s
ar
e
n
or
m
al
iz
ed
to
w
.
U
n
ce
rt
ai
n
ti
es
ar
e
on
e-
si
gm
a
st
at
is
ti
ca
l
er
ro
rs
.
A
ls
o
p
re
se
n
te
d
ar
e
th
e
H
e-
li
ke
li
n
e
ra
ti
os
d
efi
n
ed
in
G
ab
ri
el
an
d
J
or
d
an
[1
96
9]
:
R
=
F
z
/(
F
x
+
F
y
),
G
=
F
x
+
F
y
+
F
z
/F
w
,
an
al
te
rn
at
e
q
u
an
ti
ty
gi
v
in
g
th
e
tr
ip
le
t-
to
-s
in
gl
et
ca
p
tu
re
ra
ti
o,
G′
=
(F
x
+
F
y
+
F
z
)/
(F
w
+
F
3
+
),
th
e
H
e-
li
ke
h
ar
d
n
es
s
ra
ti
o
H
=
F
3
+
/(
F
2
),
an
d
a
ra
ti
o
in
tr
o
d
u
ce
d
in
L
eu
te
n
eg
ge
r
et
al
.
[2
01
0]
,
H′
=
F
3
+
/F
w
.
F
re
p
re
se
n
ts
th
e
fl
u
x
in
th
e
d
en
ot
ed
tr
an
si
ti
on
li
n
es
.
T
ra
n
si
ti
o
n
M
g
1
1
+
+
C
O
2
M
g
1
1
+
+
H
e
M
g
1
1
+
+
H
2
S
1
5
+
+
H
e
S
1
5
+
+
S
F
6
C
l1
6
+
+
H
e
C
l1
6
+
+
C
2
H
4
C
l 2
A
r1
7
+
+
A
r
z
1
.5
3
±
0
.1
6
3
.7
1
±
0
.5
9
2
.9
0
±
0
.2
4
5
.1
2
±
0
.2
5
1
.6
0
±
0
.1
2
5
.8
2
±
0
.3
8
1
.5
2
0
±
0
.0
9
6
1
.8
8
9
±
0
.0
8
4
x
/
y
1
.2
3
±
0
.1
4
1
.2
8
±
0
.2
4
0
.8
6
1
±
0
.0
9
2
1
.7
7
7
±
0
.0
9
8
1
.6
1
±
0
.1
2
2
.2
3
±
0
.2
1
1
.6
5
±
0
.1
4
1
.3
9
±
0
.1
8
1
s
−
3
p
0
.2
1
9
±
0
.0
4
3
0
.7
1
±
0
.1
6
0
.3
4
7
±
0
.0
5
1
0
.3
4
9
±
0
.0
3
0
0
.2
3
1
±
0
.0
3
1
0
.4
5
0
±
0
.0
5
1
0
.1
8
5
±
0
.0
2
4
0
.2
9
6
±
0
.0
2
2
1
s
−
4
p
0
.1
8
4
±
0
.0
4
0
0
.5
6
±
0
.1
4
0
.2
0
9
±
0
.0
3
6
0
.2
1
6
±
0
.0
2
3
0
.1
1
7
±
0
.0
2
1
0
.3
0
7
±
0
.0
3
8
0
.1
0
7
±
0
.0
1
8
0
.1
3
5
±
0
.0
1
4
1
s
−
5
p
0
.1
0
8
±
0
.0
2
4
0
.4
7
±
0
.1
1
0
.1
2
5
±
0
.0
2
6
0
.1
7
6
±
0
.0
2
6
0
.0
6
5
±
0
.0
2
3
0
.1
4
1
±
0
.0
2
3
0
.0
6
2
±
0
.0
1
3
0
.0
7
6
±
0
.0
1
0
1
s
−
6
p
0
.0
9
0
±
0
.0
2
4
0
.1
4
5
±
0
.0
8
6
0
.0
3
0
±
0
.0
1
1
0
.1
4
1
±
0
.0
1
8
0
.0
4
1
1
±
0
.0
0
5
9
0
.1
1
2
±
0
.0
2
2
0
.0
4
4
0
±
0
.0
0
8
9
0
.0
6
1
6
±
0
.0
0
9
2
1
s
−
7
p
0
.1
1
4
±
0
.0
2
9
0
.4
2
±
0
.1
2
0
.0
2
9
±
0
.0
1
2
0
.1
6
0
±
0
.0
1
9
0
.0
1
5
2
±
0
.0
0
6
1
0
.2
2
0
±
0
.0
3
3
0
.0
2
1
2
±
0
.0
0
4
7
0
.0
2
9
0
±
0
.0
0
5
7
1
s
−
8
p
0
.3
1
5
±
0
.0
9
8
0
.0
1
8
±
0
.0
1
8
0
.0
0
5
0
±
0
.0
0
5
3
0
.0
2
4
4
±
0
.0
0
7
8
0
.0
6
6
9
±
0
.0
0
9
5
0
.0
1
3
7
±
0
.0
0
4
7
0
.0
2
8
7
±
0
.0
0
6
0
1
s
−
9
p
0
.0
0
5
5
±
0
.0
0
4
2
0
.0
2
9
3
±
0
.0
0
7
7
0
.0
0
0
2
±
0
.0
0
3
3
0
.0
0
2
9
±
0
.0
0
3
7
0
.0
1
9
5
±
0
.0
0
4
9
1
s
−
1
0
p
0
.0
0
2
6
±
0
.0
0
1
5
0
.0
0
1
2
±
0
.0
0
4
1
0
.0
0
9
7
±
0
.0
0
2
8
0
.0
1
4
0
±
0
.0
0
4
6
0
.0
3
2
7
±
0
.0
0
6
1
R
1
.2
4
8
±
0
.1
2
6
2
.9
0
4
±
0
.4
2
0
3
.3
6
5
±
0
.3
0
2
2
.8
7
9
±
0
.1
1
0
0
.9
9
4
±
0
.0
6
4
2
.6
0
4
±
0
.2
0
0
0
.9
2
2
±
0
.0
7
5
1
.3
6
2
±
0
.1
7
9
G
2
.7
5
5
±
0
.2
6
9
4
.9
9
1
±
0
.7
7
7
3
.7
5
8
±
0
.3
0
6
6
.8
9
3
±
0
.3
2
5
3
.2
1
4
±
0
.2
1
5
8
.0
5
3
±
0
.5
3
2
3
.1
6
9
±
0
.2
0
4
3
.2
7
6
±
0
.2
1
8
G′
1
.6
0
6
±
0
.1
2
9
1
.3
7
8
±
0
.1
3
9
2
.1
3
7
±
0
.1
4
3
3
.3
5
4
±
0
.1
2
1
2
.1
0
8
±
0
.1
2
2
3
.4
9
1
±
0
.1
6
8
2
.1
8
7
±
0
.1
2
7
1
.8
5
7
±
0
.0
8
2
H
0
.1
9
1
±
0
.0
2
0
0
.4
3
8
±
0
.0
4
8
0
.1
5
9
±
0
.0
1
5
0
.1
3
4
±
0
.0
1
0
0
.1
2
4
±
0
.0
1
7
0
.1
4
4
±
0
.0
0
8
0
.1
0
8
±
0
.0
0
9
0
.1
5
9
±
0
.0
1
3
H
′
0
.7
1
5
±
0
.0
9
1
2
.6
2
2
±
0
.4
4
8
0
.7
5
9
±
0
.0
8
5
1
.0
5
5
±
0
.0
9
0
0
.5
2
5
±
0
.0
7
8
1
.3
0
7
±
0
.1
0
5
0
.4
4
9
±
0
.0
4
0
0
.6
7
8
±
0
.0
5
3
119
trapped ions, the hardness ratio measured at low collision velocity was often ∼1
[Wargelin et al., 2005, Beiersdorfer et al., 2000a, Otranto et al., 2006, Leutenegger
et al., 2013, Allen et al., 2008, Beiersdorfer et al., 2001]. All experiments in this
work were performed at similar collision energies of . 25 eV amu−1 [Beiersdorfer,
2003], but the hardness ratios vary between ∼0.5 to ∼2.6. This demonstrates that
the contribution from the np capture cross section, as normalized to the total cap-
ture cross section, occupies a broader range than previously supposed; the initially
surprising variation in H between bare Ar and P as shown in Leutenegger et al.
[2010] seems to be less of an anomaly than formerly thought.
One might expect to see H scale with the atomic number of the ion (as CTMC
calculations in Beiersdorfer et al. [2000a], Wargelin et al. [2008] show), the number
of valence electrons in the neutral, or the ionization potential of the neutral. For
example, Ali et al. [2005] notes that SEC (single electron capture) from a neutral
with a large ionization potential would require a smaller impact parameter, leading
to a low l state of the captured electron. However, our results also show that there is
no clear scaling between the l distribution of captures and any of the aforementioned
parameters, as can be seen in Figures 4.4 and 4.4.
The fact that we do not see a scaling of H with the ionization potential in
particular may stem from the relative dominance of SEC versus MEC. As pointed
out by Ali et al. [2005], MEC produces multiply charged ions that autoionize until
reaching a lower n, l state that radiatively decays. This leads to fewer high-n Lyman
lines and thus a smaller H. In our experiments, we find that charge exchange with
helium as the neutral partner shows an especially highH. This is likely due to a high
120
0 10 20 30 40 50 60
Atomic Number
0.0
0.5
1.0
1.5
2.0
2.5
3.0
H
a
rd
n
e
ss
 R
a
ti
o
CTMC calc., Wargelin et al. [57]
This work
Beiersdorfer et al. [31]
Allen et al. [59]
Otranto et al. [41]
Wargelin et al. [57]
Beiersdorfer et al. [60]
Leutenegger et al. [43]
Figure 4.4 Hardness ratio as a function of the atomic number of the ion.
All experiments were performed at less than ∼25 eV amu−1.
121
10 12 14 16 18 20 22 24 26
IP (eV)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
H
a
rd
n
e
ss
 R
a
ti
o
C2H4Cl2
CO2
H2
Ar
Ne
He
SF6
0 10 20 30 40 50
Nvalence
0.0
0.5
1.0
1.5
2.0
2.5
3.0
He
H2
Ar and Ne
CO2
C2H4Cl2
SF6
Figure 4.5 Hardness ratio as a function of the ionization potential (left)
and the number of valence electrons (right) of the neutral gas. Colors
and symbols represent different bare ions: Mg in black circles, Cl in red
squares, Ar in green stars, S in blue upward-pointing triangles, and P in
cyan downward-pointing triangles. The neutral species is indicated on
the plot.
122
percentage of single electron capture, as Ali et al. [2005] demonstrated, in addition
to the high ionization potential of He. However, we measure a low H (∼0.5) for
experiments with molecular H, even though one would expect SEC to be dominant
for H2.
We suggest that in the case of SEC, the differences we measure in hardness
ratio from CX with otherwise similar neutrals stem from inherent differences in the
np cross section of those species. These differences can likely only be understood in
the context of a rigorous quantum mechanical treatment of the interaction, which
must be guided by further experimental benchmarks.
The large variation in hardness ratio we have shown from charge exchange
experiments performed at low collision energies, in comparison with both previous
work in the literature and theoretical calculations using models such as CTMC,
demonstrates that open questions still exist in determining the l-selective CX cap-
ture cross section distribution, and therefore the resulting x-ray spectrum and its
accompanying diagnostics. This is an issue that is imperative to address now, so
as to properly interpret the high-resolution spectra from the Astro-H x-ray satel-
lite observatory and high-resolution x-ray microcalorimeters to be implemented on
future space missions.
4.5 Acknowledgments
The authors would like to thank Ed Magee and David Layne at LLNL for
their technical support. We also thank the anonymous referees for their helpful
123
comments. Work at the Lawrence Livermore National Laboratory was performed
under the auspices of the U.S. Department of Energy under Contract DE-AC52-
07NA27344 and supported by NASA APRA grants to LLNL and NASA/GSFC.
124
Chapter 5: High-Resolution Spectra of L-Shell Charge Exchange with
Ne-like Ni
5.1 Introduction
Most observational and modeling efforts for CX have concentrated on K-shell
ions, and we have just seen in the previous chapter that there is still much work
to be done to properly understand CX interactions involving these ions. However,
L-shell ions make up a non-negligible fraction of X-ray emission from a wide variety
of X-ray sources. L-shell sulphur may contribute to CX in Jupiter’s polar regions
[Cravens et al., 2003, Gladstone et al., 2002, Elsner et al., 2005], and L-shell Si, S,
Mg, Ne, Fe, and O ions, found in the solar wind [Schwadron and Cravens, 2000],
may also lead to CX lines in the solar system. L-shell Ni ions, though less cosmically
abundant than Fe, have been identified in spectra of stellar coronae [Behar et al.,
2001, Peretz et al., 2015, Gu¨del and Naze´, 2009] as well as in high-resolution spectra
of the sun [Phillips et al., 1982]. As high-resolution spectral measurements become
more routine, it will become increasingly important to understand the behavior of
Ni L-shell lines compared to those resulting from neighboring L-shell Fe ions in order
to properly interpret spectral line diagnostics.
125
Despite its potential significance in astrophysics (see Section 1.5), the study of
CX lacks a deep understanding of the subtleties in the atomic physics governing the
process. Experimental data are often in conflict with models, even just for K-shell
[Beiersdorfer et al., 2003a, Wargelin et al., 2005, Beiersdorfer et al., 2003b, 2000a,
Otranto et al., 2007, Leutenegger et al., 2010]. Some of this disagreement may
stem from the fact that multi-electron capture (MEC), which may be important to
consider in CX [Ali et al., 1994, 2005, Hasan et al., 2001], is not modeled. Indeed, ex-
periments performed with atomic H, ensuring single electron capture (SEC), agreed
well with models [Beiersdorfer et al., 2005]; though this could also be due to the
fact that the collision energy in the experiment was high (40 keV/amu). In general,
experiments at high collision energy (above 0.1–1 keV/amu), where the detailed
pseudo-molecular structure of the electron capture interaction is not as important
(see Section 2.3), agree better with theoretical calculations [Cariatore and Otranto,
2013, Igenbergs et al., 2012, Otranto and Olson, 2010]. However, it is imprudent
to simply ignore the cases of low collision energy or MEC. For example, solar wind
ions typically have initial energies of tens of keV/amu, but this may drop to tens of
eV/amu after passing through the bow shock of a comet [Wegmann et al., 1998]. As
gas in the neutral medium undergoes CX with the surface layers of shock-heated gas
in supernova remnants, the collision energies are dominated by the thermal motion
of the hot gas, which are tens of eV/amu [Wise and Sarazin, 1989]. Ions in the
ISM may also interact with nearby neutrals at these thermal velocities [Steigman,
1975, Christensen et al., 1977]. In addition, MEC likely plays an important role in
SWCX, as the multi-electron neutrals CO2, H2O, and H2 are abundant in cometary
126
comae and planetary atmospheres.
While K-shell spectra are challenging to fully understand, the situation is
even worse for L-shell due to its more complex atomic structure. Compounding the
problem, few experimental data of L-shell CX exist, especially at high resolution.
The only data in the literature are for the low-Z elements C, N, and O in the EUV
range [Bliek et al., 1998, Crandall et al., 1979, Dijkkamp et al., 1985, Folkerts et al.,
1995, Lubinski et al., 2000, 2001, Soejima et al., 1992], and X-ray measurements
of Kr, U [Beiersdorfer et al., 2000b, Tawara et al., 2002, 2003], Fe [Frankel et al.,
2009b, Beiersdorfer et al., 2008], and S [Frankel et al., 2009b], the latter being
the only previous L-shell X-ray measurement at high spectral resolution. Results
from comparisons of these experiments to theoretical models, ranging from classical
and semi-classical to quantum mechanical, are mixed, but tend towards agreeing at
high collision energies and showing significant discrepancies at low collision energies
[Soejima et al., 1992, Lubinski et al., 2000, 2001].
Performing experimental measurements of CX between astrophysically abun-
dant L-shell ions and neutrals at high spectral resolution is a crucial step towards
understanding the detailed atomic physics of CX, and benchmarking and improving
CX theory. In this chapter, we present recent measurements of CX with Ni19+ and
He and H2
1 with an Electron Beam Ion Trap (EBIT) and an X-ray microcalorime-
1One may note that we do not present measurements with atomic H, even though this is the
most astrophysically relevant target. This harkens back to Section 1.7: we did in fact use an atomic
H source for some experiments, but the results were unsatisfactory. Our efforts to characterize
the fraction of atomic vs. molecular hydrogen from our atomic H source were limited by poor
127
ter. In the following chapter, we will compare our experimental results to theory.
5.2 Experimental Method
For our experiments, we used the EBIT-I electron beam ion trap at the
Lawrence Livermore National Laboratory (LLNL) [Beiersdorfer et al., 2003a, Beiers-
dorfer, 2008] and measured the spectra with the EBIT calorimeter spectrometer
(ECS) [Porter et al., 2004, 2008a]. For a more detailed description of the LLNL
EBIT and the ECS, see Chapter 3. A brief description of the tools used follows.
The EBIT is a plasma source with with six radial ports, onto which we attach
various spectrometers or gas injectors. The EBIT operates in one of two modes:
electron trapping mode and magnetic trapping mode. For our CX experiments,
we used both in sequence, beginning with electron trapping mode. In this mode,
neutral species that are injected into the EBIT trap region are collisionally ionized
by a tunable electron beam, then confined in the trap. These newly-created ions
are radially confined due to the electrostatic attraction of the electron beam, and
axially confined by a voltage potential applied across three copper drift tubes. We
adjust the electron beam energy and the length of time for which the electron beam
is on to achieve our desired charge state. Next, we turn off the beam and enter
into magnetic trapping mode [Beiersdorfer et al., 1996b], where the ions are radially
confined by the 3 T magnetic field of a pair of superconducting Helmholtz coils, and
signal to noise, so it was difficult to know this fraction with sufficient accuracy. We are currently
working towards developing a more satisfactory and well-characterized method of performing CX
experiments with atomic H.
128
axially by the electric field of the drift tubes. In this mode, CX may occur between
ions and neutrals that are simultaneously injected into the trap via another radial
port. These ions lower their charge state with each successive CX interaction, until
most CX involves ions of a low enough charge state that the emitted photons are
no longer in the X-ray band. At this point, the trap is emptied and the cycle is
repeated. One cycle that includes electric trapping mode, followed by magnetic
trapping mode, then finally emptying the trap, is called an EBIT cycle. This is
repeated until the resulting CX spectrum has sufficient counts in each individual
line.
For the experiments presented here, nickel was supplied by sublimation of
nickelocene (C10H10Ni) which flowed directly into the trap region via one of the six
radial ports around the EBIT. In the charge-breeding electric trapping phase, called
direct excitation (DE), we tuned the electron beam energy to breed mostly F-like
Ni (Ni19+), which leads to Ne-like (Ni18+) following single electron capture (SEC)
in CX. Ne-like Ni was also present during charge breeding. This was necessary in
order to avoid creating O-like, thus F-like following CX, which has several spectral
lines within the Ne-like spectrum. We injected neutral He and H2 via a ballistic gas
injector connected to another radial EBIT port.
The EBIT cycle consisted of 0.35 seconds of charge breeding followed by 0.35
seconds of CX, after which the trap was dumped and the cycle repeats. Figure 5.1
shows individual X-ray counts as a function of the EBIT cycle, where the significant
reduction in count rate delineates where DE ceases and CX begins. Figures 5.4–5.7
show comparisons of the DE and CX spectra.
129
Time	(s)	
En
er
gy
	(e
V)
	
0.1	 0.2	 0.3	 0.5	 0.6	 0.7	 0.8	
500	
1000	
1500	
DE	 CX	
Trap	is	dumped	
Electron	beam	is	turned	off	
Figure 5.1 Measured X-ray energy as a function of the EBIT cycle, show-
ing ∼35 ms of DE followed by ∼35 ms of CX.
130
Neutral Ni was also present during CX, along with background neutrals present
in the trap such as C, N, and O, and trace amounts of Ar, which are from residual air
in the EBIT vacuum space. We provide an analysis and estimate of our background
CX rate in the next section. X-rays may also be produced during the CX phase
from metastable states, but we exclude this from our spectra by commencing data
analysis a few microseconds after the electron beam is turned off, which is long
enough for these states to relax to the ground state.
Typical thermal energies of trapped ions in the EBIT are ∼10 eV amu−1 (∼50
km s−1). This is the approximate collision energy/velocity at which CX occurs in
our experiments. CX experiments performed with EBITs are thus an ideal way to
probe the important low collision energy regime, which, as mentioned in Section 5.1,
is not well described by theory.
We measured our spectra with the EBIT Calorimeter Spectrometer, which
is connected to the EBIT via a radial port. The ECS is a silicon-thermistor X-
ray microcalorimeter developed at NASA/GSFC as part of the Astro-E/Suzaku
program. It has a 30-pixel array of silicon-doped thermistors which are divided
into a mid- and a high-energy array of 16 and 14 pixels, respectively. With these
two arrays, the ECS has a total dynamic range of 0.05–100 keV. The experiments
discussed here made use of the mid-band array, which has an energy resolution of
∼4.5 eV at 6 keV and an absorber quantum efficiency of nearly unity. Each X-ray
measured by the ECS is time-tagged with its arrival time relative to the EBIT cycle
during acquisition, which allows us to temporally distinguish the X-rays from the
DE and CX phases during analysis.
131
The ECS has four internal aluminized polyimide filters used to block optical
and thermal radiation at temperature stages of 77 K, 4 K, 300 mK, and 50 mK,
with a total aluminum thickness of 1460 A˚ and total polyimide thickness of 2380
A˚. We also used a 500 A˚ polyimide window outside the ECS dewar to isolate the
ECS vacuum from the EBIT vacuum. These thicknesses have been experimentally
verified to an accuracy of ∼10%.
5.3 Analysis
We created a gain scale to correlate the pulse height of each event measured on
the ECS to a photon energy in eV (see Section 3.2.1 for an overview of the physics
of microcalorimeters and Section 3.2.2 for the function of a gain calibration). We
accomplished this by injecting into the EBIT several different species that have K-
shell lines at well-known energies within and beyond the Ni L-shell energy band,
measuring the pulse height of these lines with the ECS, and fitting a 4th order
polynomial to these points to map out a well-sampled gain scale across our energy
band of interest.
We removed excess counts in our spectra due to cosmic ray events in the
detector frame using a software algorithm that searches for nearly simultaneous
events measured on multiple pixels on the ECS. We then made time cuts in our
data separating the DE and CX portions of each EBIT cycle. We combined spectra
from several measurements with the same experimental conditions, namely neutral
gas species, Ni and neutral gas injection pressures, and background conditions, to
132
create one dataset for each ion and neutral combination.
We took into account X-ray attenuation from the ECS blocking filters, as well
as from contaminants present in EBIT that froze on one or more filters. We believe
these contaminants are water ice frozen on the 77 K filter, N2 or O2 gas frozen on the
4 K filter, or both. The shapes of the N2 and O2 transmission curves are very similar
above the oxygen K edge, and our Ni L-shell lines of interest all occur above this edge.
We therefore made a fiducial assumption that the contaminant is water ice, and
determined the thickness of the ice by performing contemporaneous measurements
of the decrement in the ratio of the O Ly-α to Ly-β line strengths in electron
impact excitation experiments. The measured ratio of these lines at high electron
beam energy (∼ 12 keV) is 6.25 [Beiersdorfer, 2003]. From our measurements of the
decrement, we calculated an ice thickness of ∼0.3 µm for the time period during
which our experiments were performed. See Section 3.2.2 for a discussion on this
calculation. Combining the effects from the ECS filters and the ice corresponds to
a maximum flux correction of 42% for the strongest line in the CX spectra at ∼880
eV, but closer to 9% for the higher energy lines towards ∼1500 eV. The spectra
shown in the following section are not corrected for this attenuation, but the models
used in the cross section calculations described in Section 6.3 take this into account.
The total effective transmission accounting for attenuation from both the filters and
the ice frozen on the filters is shown in Figure 5.2.
We performed contemporaneous measurements of the background CX rate
present in the trap by injecting only Ni with no other neutral gas. The only neutrals
present were neutral Ni due to the continuous nature of the injection method, as
133
800 900 1000 1100 1200 1300 1400 1500 1600
Energy (eV)
0.6
0.7
0.8
0.9
1.0
T
ra
n
sm
is
si
o
n
Figure 5.2 X-ray transmission in our energy range of interest of the
infrared blocking filters and 0.3µm of water ice frozen on one or more of
the filters.
well as C, N, and O that are often present at a low level in the EBIT chamber.
While we do identify H- and He-like C, N, and O lines in our DE spectra, CX with
bare, H-like, and He-like C, N, and O, do not lead to spectral lines in our region
of interest. For example, even electron capture into n = 15, l = 1 onto O8+ would
yield an emission line at ∼868 eV, whereas the lowest energy line in the L-shell
band of Ne-like Ni occurs around ∼880 eV. Another common background gas is Ar.
The electron beam energy for the experiments presented here was sufficient to breed
primarily He-like Ar, which would lead to Li-like Ar following SEC in CX. Li-like
Ar would yield spectral lines from its most likely high-n capture state according to
Equation 1.2 (n = 8 − 11 → n = 2), which do fall in our energy range of interest.
However, we measure a negligible number of photons resulting from this species in
DE, so do not expect Li-like Ar to be significant in our CX spectra.
We do perform a correction for background CX between F-like Ni (Ni19+) and
neutral nickel. We estimate the strength of the background contribution in our
134
experiments by considering the relative count rate of a pure background spectrum
compared to a “signal” spectrum. We expect a background CX spectrum to have
a fairly low and constant count rate over the CX phase, since the neutral gas load
injected during this phase is relatively low and the ions created during DE would not
have many neutral partners with which to undergo CX quickly. Indeed, we measured
this in our background spectra (see Figure 5.3). Conversely, our experiments with
He and H2 injection begin with the same number of ions as in our background
spectra at the beginning of the CX phase, but have a higher gas load of neutrals
available for CX, so these ions quickly undergo CX, emitting X-rays and reducing
their charge state with each interaction. The number of ions in the trap decreases
exponentially as:
Ni(t) = Ni,0e
−t/τ , (5.1)
where Ni is the time-dependent number of ions, Ni,0 is the initial number of ions at
the end of the DE phase and start of the CX phase, t is the time, and τ is the time
constant. Note that Ni,0 is the same for both signal and background spectra. The
average number of ions throughout the CX phase is thus
Ni =
1
φCX
∫ φCX
0
Ni(t)dt = Ni,0
τ
φCX
(
1− e−φCXτ
)
, (5.2)
where φCX is the total amount of time of the CX portion of the EBIT cycle. An
example of the signal and background gas loads as a function of time is shown in
Figure 5.3. This shows the aforementioned exponential decay of the count rate in
135
the CX experiment (top) and the lower, more constant count rate of the background
measurement (bottom).
Before we subtract our background spectrum from our signal spectrum, we
must weight the background spectrum by the relative number of background ions
present in the signal spectrum ( Ni
Ni,0
), as well as the relative integration time of each
experiment (Ts
Tb
, where Ts and Tb are the total integration times of the signal and
background experiments, respectively). This corresponds to a spectral correction of
the form
Background-subtracted spectrum = Signal – Background ·
(
Ni
Ni,0
)(
Ts
Tb
)
. (5.3)
We measured τ by creating a histogram of all the counts in the CX phase in
each signal spectrum and fitting the curve with an exponential. We then calculated
an average τ for each neutral species, weighted by the integration time for each
experiment that contributed to the total spectrum. We chose φCX at the onset of our
experimental run to be ∼0.35 seconds. This background subtraction corresponded
to a maximum reduction in line flux of ∼10.7% in the H2 experiment and ∼6.6%
for the He experiment.
The following sections show the background-subtracted spectra, and in Chap-
ter 6, we use this background-subtracted data as the input for our cross section
models.
136
Co
un
ts
/b
in
	
Time	in	CX	phase	(s)	
Ni(t)	=	Ni,0e(-t/τ))	
	
From	fit:	
Ni,0	=	90.322	
τ	=	0.1212	s	
Ni19+	+	H2	CX	
Co
un
ts
/b
in
	
Time	in	CX	phase	(s)	
Ni19+	+	Ni	(background)	CX	
Figure 5.3 Measurements of the X-ray count rate as a function of time
for our signal measurement (CX with desired ions and neutrals, top) and
our background measurement (no injection of neutral partners in exper-
iment, bottom). In the signal measurement, one can see the exponential
decay of the count rate over time, where the best fit parameters for an
exponential decay are shown. The background measurement shows a
lower and fairly constant count rate, as expected.
137
5.4 Spectra
Upon measurement of the L-shell DE and CX spectra, the striking spectral
differences between the DE and CX line strengths are immediately obvious. These
spectra are presented in Figures 5.4–5.7. Each spectrum in these figures is approx-
imately normalized to the peak height of the DE line near 890 eV (M2/3G) and
is split into two energy bands with different y-axis scales in order to highlight the
stark differences in individual line strengths.
The two largest differences between DE and CX, present in both the Ni19++H2
and Ni19++He spectra, are at opposite ends of the L-shell spectra. At the lower
energy end, the 3F, 3C, and 3G lines are greatly suppressed in comparison with
the DE spectrum. In contrast, the M2/3G line is fairly strong in both CX and
DE spectra, but it dominates the CX spectrum. On the high energy end, the DE
spectral lines decrease in strength with increasing energy. However, in CX, the
second strongest line is from a high-n → n = 2 transition. This can be seen as
an analog to strongly enhanced high-n Lyman lines seen in K-shell CX spectra (see
Chapter 4)
The count rate is also significantly higher in DE than it is in CX. This is
because in DE, an ion can be excited and radiatively de-excite many times within
one EBIT cycle. However, in CX, the count rate depends on the density of the
neutral species present. Typically, neutral densities are low [Beiersdorfer et al.,
1996b]—we estimate this to be ∼ 109−10 cm3 in the experiments presented here—
which leads to lower total counts in the CX spectra.
138
850 900 950 1000 1050 1100 1150
Energy (eV)
0
5000
10000
15000
20000
C
o
u
n
ts
a
b
c d e f g
h
i
j 1 2 k l
CX∗67, Ni+H2
DE, Ni+H2
Figure 5.4 Spectra of Ni19+ (F-like) created in direct excitation (blue)
and CX of Ni19++H2 (red), from 850–1175 eV. Note the comparatively
weaker 3F (line ID b), 3D (line ID h), and 3C (line ID i) lines in the
CX spectrum. Line identifications correspond to the entries in Tables
5.1 and 5.2. The CX spectrum is normalized to the M2/3G line in DE
(line ID a).
139
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0
500
1000
1500
2000
2500
3000
3500
C
o
u
n
ts
m n
3
o
4
5 6 7 p
q
r s t u v w x y z aa
CX∗67, Ni+H2
DE, Ni+H2
Figure 5.5 Spectra of Ni19+ (F-like) created in direct excitation (blue)
and CX of Ni19++H2 (red), from 1175–1550 eV. Note that while the
lines in the DE spectrum decrease in strength with increasing energy, in
CX there is a strong line near ∼1500 eV (line ID aa) resulting from a
high-n → n = 2 (n ∼11) transition. Line identifications correspond to
the entries in Tables 5.1 and 5.2. The CX spectrum is normalized to the
M2/3G line in DE (line ID a).
140
850 900 950 1000 1050 1100 1150
Energy (eV)
0
5000
10000
15000
20000
C
o
u
n
ts
a
b
c d e f g
h
i
j 1 2 k l
CX∗213, Ni+He
DE, Ni+He
Figure 5.6 Spectra of Ni19+ (F-like) created in direct excitation (blue)
and CX of Ni19++He (red), from 850–1175 eV. Like in the Ni19++H2
spectra (Figures 5.4–5.5), one can note the suppressed 3F, 3D, and 3C
lines (line IDs b, h, and i, respectively) in CX. Line identifications cor-
respond to the entries in Tables 5.1 and 5.2. The CX spectrum is nor-
malized to the M2/3G line in DE (line ID a).
141
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0
500
1000
1500
2000
2500
3000
3500
C
o
u
n
ts
m n
3
o
4
5 6 7 p
q
r s t u v w x
y
z aa
CX∗213, Ni+He
DE, Ni+He
Figure 5.7 Spectra of Ni19+ (F-like) created in direct excitation (blue)
and CX of Ni19++He (red), from 1175–1550 eV. As in the Ni19++He
spectra, note the strongly enhanced high-n→ n = 2 line (n ∼8–9) near
1475 eV (line ID y). Line identifications correspond to the entries in
Tables 5.1 and 5.2. The CX spectrum is normalized to the M2/3G line
in DE (line ID a).
142
Line identifications in the CX spectra (letters, a–aa) are presented in Figure
5.8 and Table 5.1. We identified the strongest lines present in our CX spectra
by aligning our DE and CX spectra in energy, selecting lines that had more than
∼2 counts in CX, and estimating the energy centroid of those lines in the DE
spectrum. We then compared these line energies to calculations with the Flexible
Atomic Code (fac) and measurements of L-shell Ni lines with a high-resolution
crystal spectrometer from Gu et al. [2007a]. See Section 2.1 for a description of fac.
We corrected the ionization energies to make Ne-like Ni according to Scofield [private
communication], and the 3 → 2 transition energies to match those experimentally
verified in Gu et al. [2007a]. In our identifications, we only considered lines with a
branching ratio greater than 5%, except in a few cases indicated in the table. In
some cases, one line may be a combination of several transitions presented in the
table. Some F-like Ni lines are also included in the table; F-like lines would be
present following SEC in CX with an O-like ion. Although we include them in our
table, they should only be a minor contribution to the spectrum due to the fact
that the O-like 2p1/22p
2
3/23d5/2(J = 3) → 2p23/2(J = 2) transition, which should be
the strongest one in the O-like series in DE [Gu et al., 2007a], is not present in our
spectra.
143
F
ig
u
re
5.
8
C
X
sp
ec
tr
a
of
N
i1
9
+
+
H
2
(b
lu
e)
an
d
N
i1
9
+
+
H
e
(r
ed
),
n
or
m
al
iz
ed
to
th
e
3G
/M
2
li
n
e
in
th
e
N
i1
9
+
+
H
2
sp
ec
tr
u
m
.
L
in
es
w
it
h
m
or
e
th
an
∼2
co
u
n
ts
ar
e
id
en
ti
fi
ed
b
y
le
tt
er
s
w
h
ic
h
co
rr
es
p
on
d
to
an
en
tr
y
in
T
ab
le
5.
1.
N
ot
e
th
e
d
iff
er
en
ce
in
th
e
st
ro
n
g
h
ig
h
-n
→
n
=
2
tr
an
si
ti
on
s
fr
om
14
50
eV
–1
50
0
eV
b
et
w
ee
n
th
e
H
2
an
d
th
e
H
e
sp
ec
tr
a.
144
As described in Section 1.4, CX typically leads to electron capture into a high
n state. The l capture state, while harder to predict, has been shown to vary with
the collision energy. At high collision energies, the states are populated statistically,
favoring higher angular momentum states [Janev and Winter, 1985]. After capture
into such a state, the electron will de-excite via Yrast transitions, changing its l state
by one with each step and eventually landing in n = 3, 4, 5, 6 which can then decay
to n = 2 (see Figure 1.3). However, at the low collision energies produced with
the EBIT, lower angular momentum states are favored [Ryufuku and Watanabe,
1979, Beiersdorfer et al., 2000a]. Given this, we would expect significant capture
into l = s, p, and d in our experiments. We also naively assume that most parent
F-like ions would have an electron hole in 2p, so capture into l = s, d could directly
decay into l = p, yielding a high energy line.
The strongest line in both Ni19++H2 and Ni
19++He CX spectra is a blend
of the M2 (2p33/23s1/2(J = 2) → 2p43/2(J = 0)) and 3G (2p33/23s1/2(J = 1) →
2p43/2(J = 0)) lines, which are not individually resolved with the ∼4.5 eV resolution
of the ECS. This is a stark difference from Ne-like Ni spectra following collisional
excitation, where the M2/3G line is nearly as strong as the 3C line (2p11/23d
3
3/2(J =
1) → 2p43/2(J = 0)), and the 3F (2p11/23s11/2(J = 1) → 2p43/2(J = 0)) and 3D
(2p33/23d
1
5/2(J = 1) → 2p43/2(J = 0)) lines are also fairly strong. This can be seen
in Figures 5.4 and 5.6. This result indicates that capture into a previously F-like
ion with an electron hole in 2p3/2 and a radiative cascade whose penultimate state
is 3s1/2 is strongly preferred. This fits with our naive view of CX at low collision
energies. The relative enhancement of the M2/3G line and suppression of the 3F,
145
Table 5.1 CX lines identified in this work. fac energy calculations for
Ne-like and F-like ions are described in Section 2.1. All lower states for
Ni19+ are 2p43/2(J = 0), and lower states for the Ni
20+ transitions here are
2p33/2(J = 3/2). An asterisk indicates that the branching ratio for this
transition is less than 5%, so this line is unlikely but may contribute at
a low level. Such lines are only included when the measured line energy
is different from the closest Ni19+ theoretical line energy by more than 5
eV. Names for F-like ions follow those from Gu et al. [2007a].
Line ID Energy in DE (eV) fac Energy (eV) Energy from Gu et al. [2007a] (eV) Upper State Name
a 882 880.753 880.827 2p3
3/2
3s1
1/2
(J = 2) M2
882.888 882.96 2p3
3/2
3s1
1/2
(J = 1) 3G
b 900 899.807 899.877 2p1
1/2
3s1
1/2
(J = 1) 3F
c 919 920.037 918.542 2p3
3/2
3p1
1/2
(J = 2) E2L
d 925 926.041 2p3
3/2
3p3
3/2
(J = 2)
e 932 930.711 931.586 2p2
3/2
3s1
1/2
(J = 5/2) F4
f 941 942.002 940.346 2p1
1/2
3p1
3/2
(J = 2) E2U
g 967 969.255 967.947 2p3
3/2
3d1
3/2
(J = 1) 3E
h 980 979.487 979.571 2p3
3/2
3d1
5/2
(J = 1) 3D
i 997 997.139 997.218 2p1
1/2
3d1
3/2
(J = 1) 3C
j 1023 1023.523 1023.475 2p2
3/2
3d1
5/2
(J = 5/2) F14
1021.840 1023.475 2p2
3/2
3d1
5/2
(J = 3/2) F15
k 1070 1073.451 2s1
1/2
3p1
1/2
(J = 1) 3B
l 1074 1079.017 1074.844 2s1
1/2
3p1
3/2
(J = 1) 3A
1076.882 2s1
1/2
3p1
1/2
(J = 2) *
m 1188 1190.006 2p3
3/2
4s1
1/2
(J = 1) 4G
n 1207 1207.353 2p1
1/2
4s1
1/2
(J = 1) 4F
o 1226 1228.023 2p3
3/2
4d1
5/2
(J = 1) 4D
p 1322 1322.537 2p3
3/2
5s1
1/2
(J = 1) 5G
q 1340 1338.760 2p3
3/2
5d1
3/2
(J = 1) 5E
1340.036 2p1
1/2
5s1
1/2
(J = 1) 5F
1341.376 2p3
3/2
5d1
5/2
(J = 1) 5D
r 1356 1356.458 2s1
1/2
4p1
1/2
(J = 1) 4B
1358.142 2p1
1/2
5d1
3/2
(J = 1) 5C
1358.583 2s1
1/2
4p1
3/2
(J = 1) 4A
s 1390 1391.931 2p3
3/2
6s1
1/2
(J = 1) 6G
t 1401 1402.566 2p3
3/2
6d1
5/2
(J = 1) 6D
u 1423 1419.787 2p1
1/2
6d1
3/2
(J = 1) 6C
v 1432 1432.827 2p3
3/2
7s1
1/2
(J = 1) 7G
w 1438 1439.425 2p3
3/2
7d1
5/2
(J = 1) 7D
x 1462 1462.704 2p3
3/2
8d1
3/2
(J = 1) 8E
1463.288 2p3
3/2
8d1
5/2
(J = 1) 8D
y 1479 1476.579 2p3
3/2
9s1
1/2
(J = 1) 9G
1476.658 2p1
1/2
8s1
1/2
(J = 1) 8F
1479.618 2p3
3/2
9d1
5/2
(J = 1) 9D
1480.824 2p1
1/2
8d1
3/2
(J = 1) 8C
1481.395 2s1
1/2
5p1
1/2
(J = 1) 5B
1482.437 2s1
1/2
5p1
3/2
(J = 1) 5A
z 1490 1489.084 2p3
3/2
10s1
1/2
(J = 1) 10G
1491.282 2p3
3/2
10d1
5/2
(J = 1) 10D
aa 1499 1497.226 2p1
1/2
9d1
3/2
(J = 1) 9C
1498.262 2p3
3/2
11s1
1/2
(J = 1) 11G
1499.902 2p3
3/2
11d1
5/2
(J = 1) 11D
146
T
ab
le
5.
2
L
in
es
on
ly
p
re
se
n
t
in
th
e
D
E
sp
ec
tr
a
or
th
e
a
c
x
an
d
sp
e
x
-c
x
m
o
d
el
s
d
es
cr
ib
ed
in
C
h
ap
te
r
6,
n
ot
in
th
e
ex
p
er
im
en
ta
l
C
X
sp
ec
tr
a.
fa
c
en
er
gy
ca
lc
u
la
ti
on
s
fo
r
N
e-
li
ke
an
d
F
-l
ik
e
io
n
s
ar
e
d
es
cr
ib
ed
in
S
ec
ti
on
2.
1.
A
ll
lo
w
er
st
at
es
fo
r
N
i1
9
+
ar
e
2p
4 3
/
2
(J
=
0)
,
an
d
lo
w
er
st
at
es
fo
r
th
e
N
i2
0
+
tr
an
si
ti
on
s
h
er
e
ar
e
2p
3 3
/
2
(J
=
3/
2)
,
ex
ce
p
t
fo
r
li
n
e
ID
6,
w
h
ic
h
is
2p
1 1
/
2
(J
=
1/
2)
.
L
in
e
ID
s
9–
12
re
fe
r
to
th
e
h
ig
h
-n
tr
an
si
ti
on
s
in
th
e
a
c
x
m
o
d
el
w
h
ic
h
w
e
h
av
e
d
et
er
m
in
ed
to
h
av
e
en
er
gi
es
d
iff
er
en
t
fr
om
th
os
e
in
fa
c
b
y
as
m
u
ch
as
∼
10
eV
(s
ee
S
ec
ti
on
6.
2)
.
T
h
e
A
C
X
en
er
gi
es
w
er
e
d
et
er
m
in
ed
b
y
A
d
am
F
os
te
r
[p
ri
va
te
co
m
m
u
n
ic
at
io
n
].
A
ll
lo
w
er
st
at
es
fo
r
N
i1
9
+
ar
e
2p
4 3
/
2
(J
=
0)
,
an
d
lo
w
er
st
at
es
fo
r
th
e
N
i2
0
+
tr
an
si
ti
on
s
h
er
e
ar
e
2p
3 3
/
2
(J
=
3/
2)
,
ex
ce
p
t
fo
r
li
n
e
ID
6,
w
h
ic
h
is
2p
1 1
/
2
(J
=
1/
2)
.
N
am
es
fo
r
F
-l
ik
e
io
n
s
fo
ll
ow
th
os
e
fr
om
G
u
et
al
.
[2
00
7a
].
L
in
e
ID
P
ea
k
E
n
er
g
y
(e
V
)
fa
c
E
n
er
g
y
(e
V
)
E
n
er
g
y
fr
o
m
G
u
et
a
l.
[2
0
0
7
a
]
(e
V
)
E
n
er
g
y
fr
o
m
a
c
x
U
p
p
er
S
ta
te
N
a
m
e
1
1
0
3
6
1
0
3
6
.3
7
3
1
0
3
6
.7
4
0
2
p
1 1
/
2
2
p
3 3
/
2
3
d
1 3
/
2
(J
=
5
/
2
)
F
1
7
2
1
0
4
8
1
0
4
7
.1
8
8
1
0
4
8
.0
4
4
2
p
1 1
/
2
2
p
3 3
/
2
3
d
1 3
/
2
(J
=
3
/
2
)
F
2
0
1
0
4
8
.2
0
9
1
0
4
8
.0
4
4
2
p
1 1
/
2
2
p
3 3
/
2
3
d
1 3
/
2
(J
=
5
/
2
)
F
2
0
3
1
2
1
7
1
2
2
2
.9
4
6
2
p
3 3
/
2
4
d
1 3
/
2
(J
=
1
)
4
E
4
1
2
4
3
1
2
4
4
.3
5
9
2
p
1 1
/
2
4
d
1 3
/
2
(J
=
1
)
4
C
5
1
2
9
8
1
2
9
7
.1
9
4
2
p
2 3
/
2
4
d
1 5
/
2
(J
=
5
/
2
)
6
1
3
0
6
1
3
0
5
.5
8
4
2
p
1 1
/
2
2
p
3 3
/
2
4
d
1 5
/
2
(J
=
1
/
2
)
7
1
3
1
2
1
3
1
1
.0
2
4
2
p
1 1
/
2
2
p
3 3
/
2
4
d
1 3
/
2
(J
=
5
/
2
)
8
1
4
1
7
1
4
1
9
.7
8
7
2
p
1 1
/
2
6
d
1 3
/
2
(J
=
1
)
9
1
4
8
8
∼1
4
7
7
–
1
4
8
1
∼1
4
8
6
–
1
4
8
8
2
p
1 1
/
2
8
l,
2
p
3 3
/
2
1
0
l
1
0
1
4
9
9
∼1
4
8
9
–
1
4
9
2
∼1
4
9
8
–
1
5
0
1
2
p
3 3
/
2
1
0
l
1
1
1
5
0
6
∼1
4
9
4
–
1
4
9
8
∼1
5
0
4
–
1
5
0
8
2
p
1 1
/
2
9
l
1
2
1
5
1
8
∼1
5
0
7
–
1
5
0
9
∼1
5
1
6
–
1
5
1
9
2
p
1 1
/
2
1
0
l
147
3D, and 3C lines may thus be seen as strongly diagnostic of the presence of CX.
A key feature of CX in K-shell H-like spectra is a strong np→ 1s Lyman line,
where n is large [Betancourt-Martinez et al., 2014c, Leutenegger et al., 2010]. An
analog to this for L-shell spectra would be a strong ns → 2p or nd → 2p line. We
do in fact observe this in our spectra, with the strong line most likely being the 11s
or 11d → 2p transition in the Ni19++H2 spectrum and 8d or 9d → 2p transition in
the Ni19++He spectrum.
The canonical equation to estimate the primary n capture state of the trans-
ferred electron is Equation 1.2, which is repeated here for convenience:
nmax ∼ q
√
IH
In
(
1 +
q − 1√
2q
)−1/2
∼ q3/4, (5.4)
where IH and In are the ionization potentials of hydrogen and the neutral target,
respectively, and q is the ion charge [Janev and Winter, 1985].
Following this equation leads to an estimate of nmax ∼ 9 for Ni19++H2 and
nmax ∼ 7 for Ni19++He. Though the trend of nmax being inversely proportional to
the ionization potential of the neutral gas is observed in our experiment (with an
nmax of ∼ 11 for Ni19++H2 and nmax ∼ 9 for Ni19++He), this canonical estimate
differs from our measurement by several n levels. This highlights the approximate
nature of this equation.
We also observe emission from intermediate states such as 3d, 4s, 4d, 5s, 5d, 6s,
and 6d → 2p. Emission from transitions between n < nmax, l = s, d and 2p may
result from one of three options.
148
First, if a high angular momentum state is populated after capture, the electron
can decay via the Yrast chain as described in Section 1.4, ending in a 2p state.
As capture into low angular momentum states is preferred in EBIT experiments,
however, this is not likely to be the dominant reason for these intermediate-energy
lines.
Second, electron capture into n ∼ nmax, l = p may decay in one step to
a n < nmax, l = s, d state, then subsequently to 2p. We have determined via fac
calculations that the branching ratios required for this to occur are not insignificant,
so this may play a role in creating these lines.
Third, these intermediate-energy transitions may result from double-electron
capture into a doubly excited state, followed by autoionization of one electron while
the second drops to a lower n state. This second electron would then radiatively
de-excite from that lower state to n = 2. It has been shown experimentally that
double electron capture can account for nearly half of all CX interactions [Ali et al.,
2005]. Further, the relative strength of these intermediate-energy lines is higher
in the Ni19++H2 spectrum than it is in the Ni
19++He spectrum. The ionization
potential of H2 is ∼15.4 eV, while the first and second ionization potentials of He
are ∼24.6 eV and ∼54.4 eV, respectively. It thus follows that multi-electron capture
is more likely to occur with neutral H2 than with He, which would suggest that the
dominant process creating these intermediate-energy lines in our spectra is in fact
from the double-electron capture scenario.
Although these empirical observations are extremely valuable, in order to fully
disentangle the relative dominance of all relevant processes, detailed radiative cas-
149
cade models are required. The following chapter begins to address this need.
A particularly fascinating feature of our spectra is the unexpectedly strong
presence of the 3F and 3A/3B lines. These lines result from an electron hole in
2s1/2 being filled by either a 3p3/2 or 3p1/2 electron (to create the 3A and 3B lines,
respectively), or a hole in 2p1/2 being filled by a 3s1/2 electron (to create the 3F
line). The presence of these lines is surprising because as previously described, our
naive assumption of the F-like parent ion core structure was the lowest energy con-
figuration with a hole in 2p3/2, and we do not expect CX to influence this core
configuration. We do not observe O-like Ni in our DE spectrum, which could po-
tentially populate these levels after capturing two electrons. One possible reason
for the presence of these lines could be from the effects of mixing: the energy state
of a given core configuration with one excited electron may be similar to a different
core configuration with an excited electron. More detailed theoretical calculations
are necessary to obtain an accurate estimate of the strength of this effect.
5.5 Summary
In this chapter, we have described experiments performed at the LLNL EBIT
of CX between F-like Ni and neutral H2 and He to create Ne-like Ni. These are the
first high-resolution L-shell data of Ni L-shell CX, adding to the very short list of
high resolution L-shell CX data. This was also performed in the low collision energy
regime, where most discrepancies between experiment and theory occur.
The spectra resulting from collisional excitation and CX are dramatically dif-
150
ferent. We found that a significantly enhanced M2/3G line is likely highly diagnostic
of CX, and that the ratio of the higher energy lines to M2/3G may be a good indi-
cator for the capture state, similar to the hardness ratio for K-shell spectra.
There are obvious differences between the Ni19++H2 spectrum and the Ni
19++He
spectrum. The strong high energy line is shifted to a higher energy for the Ni19++H2
spectrum, as is qualitatively predicted by the canonical equation for nmax (Equation
1.2), and the intermediate energy lines are stronger for the Ni19++He spectrum.
It remains to be determined whether these differences are due to the difference in
ionization potential of the neutral partner, the relative significance of MEC, or due
to innate variations in the cross sections. With these experimental results in hand,
we can now begin comparing them to theory.
151
Chapter 6: L-shell Ni CX: Comparisons to Models and Relative Cap-
ture Cross Section Calculations
6.1 Introduction
In the previous chapter, we presented the first high-resolution spectra of CX
between L-shell Ni and two astrophysically relevant neutral species, H2 and He. We
now compare our experimental spectra with those created using the commonly-used
CX models spex-cx [Gu et al., 2016] and acx [Smith et al., 2014a], and show that
significant differences exist. See Section 2.2 for details on these models. We also
describe a pipeline we created that utilizes radiative cascade models with atomic
data from fac in order to fit our data to a model that assumes single electron
capture (SEC) into various quantum states. See Section 2.1 for a brief discussion on
fac and how we use the code. We then use this model fit to extract state-selective
relative capture cross sections from our experimental data.
We first demonstrate the functionality of our model with the relatively simpler
case of K-shell S undergoing CX with neutral He, and compare our results to recent
Multi-Channel Landau-Zener (MCLZ) state-selective cross section calculations. An
overview of MCLZ theory is given in Section 2.3. We then move back to L-shell Ni
152
CX, and present our model fit to the data in Chapter 5, along with the state-selective
cross sections we derive, and discuss our next steps.
6.2 Comparisons to Spectral Models
We used the CX spectral synthesis codes spex-cx and acx to simulate spectra
for interactions between Ni19++H (spex-cx, acx) and Ni19++He (acx only) to then
compare to our experimental spectra.
We gave the spex-cx model as an input the lowest allowable collision velocity
of 50 km/s, the approximate ionization temperature of 500 eV, and only considered
Ni ions (zeroing out the abundance of all the other ions in the model). While this
does not exactly describe our system—in particular, the monoenergetic electron
beam in the EBIT does not have a thermal Maxwellian distribution in the mode
in which we operated it—spex-cx requires a temperature, and 500 eV yielded the
closest match to our spectra upon visual inspection. We then convolved the result
of this model with a Gaussian line profile to match the instrumental response of the
ECS (see Section 3.2.3.7).
To compare our data to the acx model, we used the acxion model, and again
convolved the resulting spectrum with a Gaussian line profile. We set the ion and
charge state to be Ni19+, and considered CX with both H and He by adjusting
the fracHe0 parameter. We used two different l distributions by varying the model
parameter: model “8,” which is the default model used in acx and which assumes a
separable l distribution (Equation 1.8), and model “15,” which uses a Landau-Zener
153
weighting function for the total L distribution (Equation 1.7). Model 15 was chosen
because Landau-Zener methods are most applicable to low-energy collisions such as
those in EBITs [Janev et al., 1985], and because model 15 best matched previously
measured K-shell EBIT CX spectra [Betancourt-Martinez et al., 2016].
Figures 6.1–6.17 show the results of the comparison between our experimental
data and these models. These figures show background-subtracted data (see Section
5.3) and model-predicted counts (i.e., the models have been adjusted to account for
the X-ray attenuation in the data due to the ECS filters; see Section 3.2.3.1). Figures
6.1, 6.8, and 6.13 show the entire spectrum for each of the comparisons we perform
in order to obtain a global view. Each spectrum is then split into four plots to
more easily identify the differences in line strength. spex-cx cannot incorporate
neutral He, so it is not compared to our Ni19++He data. The line identifications on
the plots correspond to the entries in Tables 5.1 and 5.2. Table 5.1 presents lines
that we identified in our experimental CX spectra. The lines presented in Table 5.2
correspond to lines that are present in the DE spectra or in the models, but not in
our experimental CX spectra. In the figures, all model line strengths are normalized
to the total number of counts in the M2/3G line in the data, so that line is not
shown in the zoomed-in plots.
154
1
0
0
2
0
0
3
0
0
4
0
0
S
P
E
X
-C
X
A
C
X
, 
m
o
d
e
l 
8
A
C
X
, 
m
o
d
e
l 
1
5
D
a
ta
, 
N
i+
H
2
9
0
0
1
0
0
0
1
1
0
0
1
2
0
0
1
3
0
0
1
4
0
0
1
5
0
0
E
n
e
rg
y
 (
e
V
)
0
1
0
2
0
3
0
4
0
Counts
F
ig
u
re
6.
1
sp
e
x
-c
x
m
o
d
el
(m
ag
en
ta
),
a
c
x
m
o
d
el
8
(c
ya
n
),
an
d
a
c
x
m
o
d
el
15
(y
el
lo
w
),
as
su
m
in
g
C
X
b
et
w
ee
n
N
i1
9
+
an
d
H
,
p
lo
tt
ed
ag
ai
n
st
N
i1
9
+
+
H
2
E
B
IT
C
X
d
at
a
(b
la
ck
).
N
ot
e
th
e
b
ro
ke
n
y
-a
x
is
.
T
h
e
d
at
a
sp
ec
tr
u
m
h
as
b
ee
n
b
ac
k
gr
ou
n
d
-s
u
b
tr
ac
te
d
,
an
d
th
e
m
o
d
el
sp
ec
tr
a
sh
ow
m
o
d
el
-p
re
d
ic
te
d
co
u
n
ts
,
i.
e.
,
th
ey
h
av
e
b
ee
n
ad
ju
st
ed
to
ac
co
u
n
t
fo
r
th
e
at
te
n
u
at
io
n
in
th
e
d
at
a
d
u
e
to
th
e
E
C
S
fi
lt
er
s.
A
ll
sp
ec
tr
al
li
n
es
ar
e
n
or
m
al
iz
ed
to
th
e
fl
u
x
in
th
e
M
2/
3G
li
n
e
in
th
e
d
at
a.
In
d
iv
id
u
al
li
n
es
ar
e
la
b
el
ed
in
su
b
se
q
u
en
t
p
lo
ts
.
155
We first compare our Ni19++H2 data to the models assuming capture from
atomic H. It is relevant to compare CX between H2 and H because of their similar
ionization potentials (15.4 eV and 13.6 eV, respectively) and number of electrons
available to be captured (2 and 1), though MEC does have to be considered in the
H2 case, and intrinsic differences in the state-selective cross section may be present.
Figures 6.2–6.5 show the spex-cx and acx models along with the Ni19++H2 EBIT
data. Although some lines are well approximated by the models, both in strength
and energy centroid, (namely c, d, g; k and l with the acx model; and q with the
spex-cx model), most lines are dramatically over- or under-predicted. There are
strong lines in the models that did not have enough counts in the data to be initially
identified according to our ∼2-count minimum threshold, and the acx model even
predicts flux at energies that do not have any Ni19+ transitions, according to fac
structure calculations, such as the line between 1140–1160 eV and the line to the
low energy side of line ID p.
An important energy regime that the models fail to correctly reproduce is
near the strong high-n → n = 2 transition(s) around 1500 eV, which, in the EBIT
Ni19++H2 spectrum, is likely dominated by 11s, 11d→ 2p transitions (see Chapter
5). Recall that Equation 1.2 predicts the primary n level of electron capture from
neutral H2 to be 9.0. With atomic H, which is the assumption in the models, this
number changes to n = 9.6. Models in acx that weight this distribution should
put 60% of electron capture into n = 10 and 40% into n = 9. If we compare
the lines in the acx spectra to our FAC energy calculations, it would appear that,
first, acx actually populates n = 8 − 12, and second, models 8 and 15 differ in
156
900 920 940 960 980 1000 1020 1040
Energy (eV)
0
50
100
150
200
C
o
u
n
ts
a b c d e f g h i j
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.2 spex-cx model (magenta), acx model 8 (cyan), and acx
model 15 (yellow) plotted against Ni19++H2 EBIT CX data (black), from
888 eV–1048 eV. The data spectrum has been background-subtracted,
and the model spectra show model-predicted counts, i.e., they have been
adjusted to account for the attenuation in the data due to the ECS filters.
All spectral lines are normalized to the flux in the M2/3G line in the
data. Labels correspond to line IDs in Tables 5.1 and 5.2.
157
1060 1080 1100 1120 1140 1160 1180 1200
Energy (eV)
0
2
4
6
8
10
12
C
o
u
n
ts
k l
m n
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.3 Same as Figure 6.2, but for 1048 eV–1210 eV.
158
1220 1240 1260 1280 1300 1320 1340 1360
Energy (eV)
0
2
4
6
8
10
12
C
o
u
n
ts
o 4
p q r
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.4 Same as Figure 6.2, but for 1210 eV–1368 eV.
159
1380 1400 1420 1440 1460 1480 1500 1520
Energy (eV)
0
2
4
6
8
10
12
C
o
u
n
ts
s t 8 u v w x y z/9
aa/10
11
12
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.5 Same as Figure 6.2, but for 1368 eV–1528 eV.
160
their treatment of the n distribution. Figure 6.6 shows line identifications of these
high-n → n = 2 transitions assuming energies calculated from fac, showing this
unexpected behavior. However, we found that this behavior can be explained with
further information from the authors of acx: the line energies used in the acx
spectra differ from those of the fac calculation by up to ∼10 eV [Adam Foster,
private communication]. In addition to this energy discrepancy, acx calculations
do not include transitions that involve n > 10, meaning that even with line energies
closer to those of our calculations, the model will be unable to reproduce certain
high-n lines in our spectra. Figure 6.7 shows the same high-n line region of the
spectrum, with transitions determined by the line energies used in acx. We can see
from Figure 6.6 that the spex-cx model behaves as we expect (see Section 2.2),
i.e., it appears to only calculate capture into n = 10 states.
Next, we investigated whether the spectral models using atomic H as the neu-
tral partner would better reproduce the Ni19++He data. This may be the case, as
He is more likely than H2 to undergo SEC rather than MEC. A greater contribution
of SEC vs. MEC would alter the relative line strengths, especially at intermediate
energies. Figures 6.8–6.12 show this comparison between models and data. This
time, only lines c and d are fairly well reproduced. The models again fail to repro-
duce the high-n lines. This is in line with what we expect, however: besides the
discrepancy in line energy as discussed above, the higher ionization potential of He
should lower nmax relative to that of CX with atomic H, shifting the high-n lines to
lower energies. We observe this in Figure 6.12.
161
1480 1485 1490 1495 1500 1505 1510 1515 1520 1525
Energy (eV)
0
5
10
15
20
C
o
u
n
ts
10l− 2p3/2
9l− 2p1/2
11l− 2p3/2
10l− 2p1/2
12l− 2p3/2 11l− 2p1/2
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.6 Comparison of high-n lines in the Ni19++H2 data, the spex-
cx model, and the two acx models. Here, possible transitions contribut-
ing to each line are identified using line energies as calculated by fac.
From this analysis, it appears that the acx model does not behave in
the way we expect. This is due to the fact that the line energies differ
between methods by ∼10 eV. The data spectrum has been background-
subtracted, and the model spectra show model-predicted counts, i.e.,
they have been adjusted to account for the attenuation in the data due
to the ECS filters. Lines are normalized to the flux in the M2/3G line
in the data.
162
1480 1485 1490 1495 1500 1505 1510 1515 1520 1525
Energy (eV)
0
5
10
15
20
C
o
u
n
ts
8l− 2p1/2
9l− 2p3/2 10l− 2p3/2 9l− 2p1/2 10l− 2p1/2
ACX, model 8
ACX, model 15
Data, Ni+H2
Figure 6.7 Comparison of high-n lines in the Ni19++H2 data and the two
acx models, with transitions determined by the energies used in acx.
This shows an n- and l-distribution we would expect from ACX, unlike
in Figure 6.6. The data spectrum has been background-subtracted, and
the model spectra show model-predicted counts, i.e., they have been
adjusted to account for the attenuation in the data due to the ECS
filters. Lines are normalized to the flux in the M2/3G line in the data.
163
5
0
1
0
0
1
5
0
S
P
E
X
-C
X
A
C
X
, 
m
o
d
e
l 
8
A
C
X
, 
m
o
d
e
l 
1
5
D
a
ta
, 
N
i+
H
e
9
0
0
1
0
0
0
1
1
0
0
1
2
0
0
1
3
0
0
1
4
0
0
1
5
0
0
E
n
e
rg
y
 (
e
V
)
05
1
0
Counts
F
ig
u
re
6.
8
sp
e
x
-c
x
m
o
d
el
(m
ag
en
ta
),
a
c
x
m
o
d
el
8
(c
ya
n
),
an
d
a
c
x
m
o
d
el
15
(y
el
lo
w
)
as
su
m
in
g
C
X
b
et
w
ee
n
N
i1
9
+
an
d
H
,
p
lo
tt
ed
ag
ai
n
st
N
i1
9
+
+
H
e
E
B
IT
C
X
d
at
a
(b
la
ck
).
N
ot
e
th
e
b
ro
ke
n
y
-a
x
is
.
T
h
e
d
at
a
sp
ec
tr
u
m
h
as
b
ee
n
b
ac
k
gr
ou
n
d
-s
u
b
tr
ac
te
d
,
an
d
th
e
m
o
d
el
sp
ec
tr
a
sh
ow
m
o
d
el
-p
re
d
ic
te
d
co
u
n
ts
,
i.
e.
,
th
ey
h
av
e
b
ee
n
ad
ju
st
ed
to
ac
co
u
n
t
fo
r
th
e
at
te
n
u
at
io
n
in
th
e
d
at
a
d
u
e
to
th
e
E
C
S
fi
lt
er
s.
A
ll
sp
ec
tr
al
li
n
es
ar
e
n
or
m
al
iz
ed
to
th
e
fl
u
x
in
th
e
M
2/
3G
li
n
e
in
th
e
d
at
a.
In
d
iv
id
u
al
li
n
es
ar
e
la
b
el
ed
in
su
b
se
q
u
en
t
p
lo
ts
.
164
Finally, we compared the acx models, which, unlike spex-cx, can be altered
to use He as a neutral partner, to our Ni19++He data. As can be seen in Figures
6.13–6.17, however, the models and data are just as inconsistent as with the pre-
vious comparisons. Most notably, the primary state for electron capture for the
experimental data appears to be 8s or 9s, and neither acx model predicts flux in
this line. Instead, they predict a dominant 7d→ 2p transition, which is in fact what
Equation 1.2 would predict.
165
900 920 940 960 980 1000 1020 1040
Energy (eV)
0
10
20
30
40
50
60
70
C
o
u
n
ts
b c d e f g h i
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+He
Figure 6.9 spex-cx model (magenta), acx model 8 (cyan), and acx
model 15 (yellow) assuming CX with atomic H plotted against Ni19++He
EBIT CX data (black), from 888 eV–1048 eV. The data spectrum
has been background-subtracted, and the model spectra show model-
predicted counts, i.e., they have been adjusted to account for the attenu-
ation in the data due to the ECS filters. All spectral lines are normalized
to the flux in the M2/3G line in the data. Labels correspond to line IDs
in Tables 5.1 and 5.2.
166
1060 1080 1100 1120 1140 1160 1180 1200
Energy (eV)
0
1
2
3
4
5
6
C
o
u
n
ts
k l
m n
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+He
Figure 6.10 Same as Figure 6.9, but for 1048 eV–1210 eV.
167
1220 1240 1260 1280 1300 1320 1340 1360
Energy (eV)
0
1
2
3
4
5
6
C
o
u
n
ts
o
4
p q r
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+He
Figure 6.11 Same as Figure 6.9, but for 1210 eV–1368 eV.
168
1380 1400 1420 1440 1460 1480 1500 1520
Energy (eV)
0
2
4
6
8
10
12
14
C
o
u
n
ts s t u v w x y 9 10
11 12
SPEX-CX
ACX, model 8
ACX, model 15
Data, Ni+He
Figure 6.12 Same as Figure 6.9, but for 1368 eV–1528 eV.
169
5
0
7
5
1
0
0
1
2
5
A
C
X
, 
H
e
, 
m
o
d
e
l 
8
A
C
X
, 
H
e
, 
m
o
d
e
l 
1
5
D
a
ta
, 
N
i+
H
e
9
0
0
1
0
0
0
1
1
0
0
1
2
0
0
1
3
0
0
1
4
0
0
1
5
0
0
E
n
e
rg
y
 (
e
V
)
05
1
0
1
5
Counts
F
ig
u
re
6.
13
a
c
x
m
o
d
el
8
(c
ya
n
)
an
d
a
c
x
m
o
d
el
15
(y
el
lo
w
)
as
su
m
in
g
C
X
b
et
w
ee
n
N
i1
9
+
an
d
H
e,
p
lo
tt
ed
ag
ai
n
st
N
i1
9
+
+
H
e
E
B
IT
C
X
d
at
a
(b
la
ck
).
N
ot
e
th
e
b
ro
ke
n
y
-a
x
is
.
T
h
e
d
at
a
sp
ec
tr
u
m
h
as
b
ee
n
b
ac
k
gr
ou
n
d
-s
u
b
tr
ac
te
d
,
an
d
th
e
m
o
d
el
sp
ec
tr
a
sh
ow
m
o
d
el
-p
re
d
ic
te
d
co
u
n
ts
,
i.
e.
,
th
ey
h
av
e
b
ee
n
ad
ju
st
ed
to
ac
co
u
n
t
fo
r
th
e
at
te
n
u
at
io
n
in
th
e
d
at
a
d
u
e
to
th
e
E
C
S
fi
lt
er
s.
A
ll
sp
ec
tr
al
li
n
es
ar
e
n
or
m
al
iz
ed
to
th
e
fl
u
x
in
th
e
M
2/
3G
li
n
e
in
th
e
d
at
a.
In
d
iv
id
u
al
li
n
es
ar
e
la
b
el
ed
in
su
b
se
q
u
en
t
p
lo
ts
.
170
While the availability and relative ease of use of models such as acx and
spex-cx are beneficial for increasing the amount of non-CX experts incorporating
CX into their spectral analyses, these comparisons of models to data show that these
models must be used with extreme caution. These discrepancies again highlight the
need for careful energy calculations and more detailed state-selective cross section
calculations, especially at low collision energies. Continuing to benchmark models
such as these will aid us in spurring on this effort.
6.3 A Pipeline to Extract State-Selective Relative Cross Sections
from Spectra: Method
In the absence of state-selective cross section calculations for our ion and
collision energy of interest for the experiments presented here, we have created a
procedure to extract these values from our data. This procedure involves four basic
steps, which will be described in more detail presently. First, we calculate the atomic
structure of various configurations of Ne-like Ni, beginning with a base F-like ion
plus the addition of one excited electron in many different quantum states. Second,
we use the atomic data to perform a radiative cascade for each of these states,
creating a spectrum for each one. Third, from this set of spectra, we select a subset
(which we dub a spectral basis set) that we will add together into a model to fit
our data. Finally, we run our model, which uses this spectral basis set in a least-
squares minimization procedure to calculate the relative weight to be applied to each
spectrum in the set to create a best fit to the data. Recall that each spectrum in
171
900 920 940 960 980 1000 1020 1040
Energy (eV)
0
10
20
30
40
50
C
o
u
n
ts
b c d e f g h i
ACX, He, model 8
ACX, He, model 15
Data, Ni+He
Figure 6.14 acx model 8 with neutral He (cyan) and acx model 15
with neutral He (yellow) plotted against Ni19++He EBIT CX data
(black), from 888 eV–1048 eV. The data spectrum has been background-
subtracted, and the model spectra show model-predicted counts, i.e.,
they have been adjusted to account for the attenuation in the data due
to the ECS filters. All spectral lines are normalized to the flux in the
M2/3G line in the data. Labels correspond to line IDs in Tables 5.1 and
5.2.
172
1060 1080 1100 1120 1140 1160 1180 1200
Energy (eV)
0
1
2
3
4
5
6
C
o
u
n
ts
k l m n
ACX, He, model 8
ACX, He, model 15
Data, Ni+He
Figure 6.15 Same as Figure 6.14, but for 1048 eV–1210 eV.
173
1220 1240 1260 1280 1300 1320 1340 1360
Energy (eV)
0
1
2
3
4
5
6
C
o
u
n
ts
o
4
p q r
ACX, He, model 8
ACX, He, model 15
Data, Ni+He
Figure 6.16 Same as Figure 6.14, but for 1210 eV–1368 eV.
174
1380 1400 1420 1440 1460 1480 1500 1520
Energy (eV)
0
2
4
6
8
10
12
14
C
o
u
n
ts s t u v w x y 9
ACX, He, model 8
ACX, He, model 15
Data, Ni+He
Figure 6.17 Same as Figure 6.14, but for 1368 eV–1528 eV.
175
the basis set is representative of electron capture into one particular quantum state;
thus the weight assigned to each spectrum in the basis set after the fit corresponds
to the relative cross section of electron capture into that quantum state.
For the initial step, we first consider a base F-like ion in its ground configura-
tion. We initially assumed that the F-like ion would be in its ground state (with an
electron hole in 2p3/2), but as described in Section 6.6, we later found it necessary
to include core excited states (electron hole in 2p1/2 and 2s1/2). We then add one
electron in an excited state to the configuration, making the F-like ion a Ne-like one.
This represents the electron capture process during CX. We consider many quan-
tum states for the excited electron, going as high in energy as n = 15 and including
all possible l, j, and coupled J values. For each of these Ne-like configurations,
we calculate the atomic structure, including level energies and transition energies
between all levels, along with the oscillator strength and Einstein A coefficient for
each transition. This is done with fac, assuming jj-coupling (see Section 2.1).
Next, we calculate the radiative cascade paths for each of these excited Ne-like
states in order to generate spectra. We do this by using the previously calculated
atomic data, we create matrices of transition energies, Einstein A coefficients, and
branching ratios between all calculated energy levels. At this stage, we choose an
initial subset of excited states to consider by choosing a range of n values that
might be most relevant to our experimental spectra. We choose this by examining
the experimental spectra to determine the highest energy spectral line with signif-
icant flux. We then conservatively choose an nmin and nmax range for our cascade
calculations that bounds this primary capture channel, where nmax–nmin> 5. For
176
example, our Ni19++H2 spectrum shows a strong high-n line that corresponds to the
n = 11→ n = 2 transition, so for our cascade calculation, we consider all configura-
tions with the excited electron in n = 5− 13. We then generate a list of all possible
l, j, and J values for each unpaired electron within this range of n values. Finally,
we input each of these configurations into the cascade matrix to generate a list of
spectral line energies and strengths. From this linelist, we generate a spectrum of
that cascade following capture. An example spectrum is shown in Figure 6.18.
At the end of this second stage, we thus have a set of spectra that corresponds
to a given F-like core configuration with one excited electron that radiatively cas-
cades to the ground state of Ne-like Ni. This is a large number of spectra—on the
order of hundreds—which is certainly more than should be used to fit a spectrum
with only about 15 lines. It is thus necessary to reduce the number of spectra that
will make up a basis set for our fitting code. The spectra in the basis sets used in
this work were chosen by visual inspection, based on their ability to produce lines
present in our experimental spectra, generally limited to low values of l to reflect
the low collision velocities that occur in EBIT experiments, and are biased towards
the observed nmax from the data. In some cases, an element of a basis set may be a
statistical J-weighted average of several J states. We also created several different
basis sets to determine how this changed both the model spectrum generated and
the resulting cross sections.
Choosing the spectra to make up a basis set may be accomplished in various
other ways, for example: determining which spectra are most distinct and least
degenerate; using statistical weighting to make cuts according to J value; only
177
2p1/29s1/2	 (J=0)	
2p1/29s1/2	 (J=1)	
Figure 6.18 Simulated L-shell spectra for capture into Ne-like Ni configu-
rations 2p1/29s1/2(J = 0) (blue) and 2p1/29s1/2(J = 1) (green), assuming
a Gaussian response with 1.5 eV FWHM resolution.
178
including spectra with low values of l and a value of nmax as empirically determined
by the data, along with other options or combinations of the above. While limiting
the number of spectra in the basis set is necessary, any of these methods will lead to
some error in our cross section calculation. This is because in the true experiment, a
large number of capture configurations contribute to our measured spectra at some
level. An added level of complication is that a set of given electron configurations
that we deem to be significant might lead to very similar spectra. Thus each element
in our chosen basis set may not be representative of a single n, l, j, J-resolved state,
but a sum of several.
With a chosen spectral basis set, the fourth and final stage of our pipeline is
then adding each spectrum in the set into our model. Our model takes as an input
each element in our spectral basis set. Within each of these basis set elements, the
possible free parameters are the capture state itself (i.e., the base F-like configuration
and the quantum state of the additional excited electron), the Gaussian sigma of
the spectral lines, and the overall weight of that particular spectrum versus all the
other spectra in the basis set. In our fit, we freeze the capture state (after we have
selected the ones we want to consider), tie together the sigma for all spectral lines
across all the models, and only allow the overall weight of each particular spectrum
in the basis set to vary.
Our fit is done through least-squares minimization, which we perform through
a Python script using the package lmfit. During the fit, each simulated spectrum
is adjusted to match the ECS filter transmission (see Section 3.2.2). Each spectral
bin is weighted using the Gehrels method, which is better suited for low counts
179
[Gehrels, 1986]. After the fit, the weight assigned to each spectrum in the basis set
corresponds to the relative cross section of the n, l, j, J-resolved state that lead to
that spectrum. For an example basis set with three spectra A, B, and C, this can
be represented as
Best fit to data=(Spectrum A*W1) + (Spectrum B*W2) + (Spectrum C*W3),
(6.1)
where W1, W2, and W3 correspond to the weights given to spectra A, B, and C,
respectively, after the fit.
The absolute value of each weight (or relative cross section) is arbitrary, as it
corresponds to the necessary strength of that element in the spectral basis set to
match a spectrum with an arbitrary number of counts. The physical result is in
comparing the relative strengths of each weight. For the spectra in our model basis
set that are degenerate (i.e., that result from more than one capture state), we treat
the cross section as derived from the fit as that of the sum of all states included in
this degeneracy. Plots of the relative cross sections extracted from our data via the
method just described are shown in Figures 6.20, 6.21, and 6.22 for S+He CX, and
Figures 6.28, 6.29, and 6.30 for Ni+H2 CX.
6.4 Cross Section Calculations: Result for S+He
As an initial test of our pipeline and fitting procedure, we began with the
simpler case of K-shell spectra. We fit experimental data of CX between H- and
180
He-like S and neutral He, first presented in Chapter 4, with the process described
in the previous section. Because the gain calibration for the experiment from which
the data are taken led to line energies that were shifted with respect to those in the
models by ∼0.6 eV near the high-n Ly lines (possibly due to gain drift or a non-ideal
polynomial fit to the gain curve; see Section 3.2.3.8), we shifted the data for these
energies by hand to match the model. We created three different model basis sets
to fit to the data. The spectra that formed the basis set for the best fit to the data
are presented in Appendix A.
The spectrum generated with the best fit to the data is shown in Figure 6.19.
In general, the fit is very good: though the He-like forbidden line near 2400 eV
is slightly over-predicted and the H-like Ly-η line near 3450 eV is slightly under-
predicted, most other lines are well-described by the model.
The weights for each model allow us to determine the relative cross section for
each capture state described in our basis set. This is plotted in Figures 6.20 and
6.21.
Our results suggest that for H-like S in our experiments, the primary capture
channel is into 8p1/2 and 8p3/2, which have nearly indistinguishable spectra. Also
significant is capture into 8s1/2. This is to be expected, since capture into low l
states should be preferred for low collision velocity EBIT CX experiments.
For the He-like ion, capture into low l states is also preferred, though in this
case, the relative strength of capture into s is higher than that of p. Although the
spectra resulting from capture into n = 6 − 9, l = s are fairly indistinguishable,
as capture into n = 7 is preferred for l = p and f , we assume that the point
181
2400 2500 2600 2700 2800 2900
Energy (eV)
0
100
200
300
400
500
In
te
n
si
ty
z, y/x, w
Lyα Kβ
Data
Model
3000 3100 3200 3300 3400
Energy (eV)
0
20
40
60
80
100
120
140
160
In
te
n
si
ty Kγ Kδ Lyβ K² Kζ Lyγ Lyδ Ly² Lyζ
Lyη
Figure 6.19 Model fit (red) to the data (blue) for H- and He-like S+He
CX.
182
s p d f
l value
0
200
400
600
800
1000
R
e
la
ti
v
e
 C
ro
ss
 S
e
ct
io
n
n=6
n=7
n=8
n=6-9
J=1/2
J=3/2
J=5/2
J=7/2
J=1/2, 3/2
Figure 6.20 Relative n, l, J-resolved capture cross sections for H-like
S+He CX. Each point reflects one quantum state of electron capture,
or in the case of degenerate states, a sum of several. The color, symbol,
and x-value of each point describes the quantum state presented. The
absolute value of the calculated cross section is arbitrary; the relevant
physics is in the relative values. This shows that the primary capture
channel is into 8p, followed in strength by 8s.
183
s p1/2 p3/2 d3/2 d5/2 f5/2 f7/2
l value
0
500
1000
1500
2000
2500
R
e
la
ti
v
e
 C
ro
ss
 S
e
ct
io
n
n=4
n=5
n=6
n=7
n=6-9
J=0
J=1
J=2
J=3
J=4
Figure 6.21 Relative n, l, j, J-resolved capture cross sections for He-like
S+He CX. Each point reflects one quantum state of electron capture, or
in the case of degenerate states, a sum of several. The color, symbol,
and x-value of each point describes the quantum state presented. Again,
the absolute value of the calculated cross section is arbitrary. Though
the spectra for n = 6 − 9, l = s are fairly indistinguishable, we deduce
from the relative importance of n = 7 into higher angular momentum
states that electrons are primarily captured into 7s (J=1).
184
corresponding to the largest cross section (at l = s) is also dominated by n = 7.
A fairly surprising result is the significance of capture into l = f for both ion
species. It is possible that this results from double electron capture with at least
one electron in l = p, followed by autoionization and a simultaneous transition of
the second electron into nf .
Now that we have a working procedure to go from an experimental spectrum
to state-selective relative cross sections, which yields a good model fit to the data
and cross sections, we will compare our results to calculations performed by Renata
Cumbee using the Multichannel Landau-Zener (MCLZ) method.
6.5 K-shell S+He Cross Sections: Pipeline Model Comparison to
MCLZ Calculations
Dr. Cumbee used the Kronos CX database [Mullen et al., 2016] to calculate
MCLZ cross sections for CX between H-like S and neutral He. She applied a low-
energy l distribution (Equation 1.7) to obtain nl resolved cross sections at a collision
energy of 10 eV/amu. This method was performed assuming LS-coupling, so the
levels were then transcribed to their equivalent jj-coupled states using statistical
J-weighting in order to compare to our model. This process yielded a list of relative
cross sections for each significant state, normalized so that the sum of all the cross
sections is 1.
We then matched up the states reflected in our model basis set to those in
the MCLZ calculations. The result of this comparison is shown in Figure 6.22.
185
We normalized each relative cross section point in both data and theory figures to
its respective maximum value. In the case presented here, each cross section was
normalized to capture into 8p(J = 1/2)+8p(J = 3/2). In the cases where one of the
models in our basis set served as a template for more than one state, we adjusted
the corresponding MCLZ point on the figure to reflect the sum of all states included.
For completeness, we wished to ensure that the states with the largest cross
sections were presented. We thus sorted the MCLZ cross sections by strength and
added four more states to Figure 6.22 that were not included in our initial model
fit. With the addition of these four points, surrounded by light grey parentheses in
the right-hand plot of Figure 6.22, the 16 most significant capture states are shown.
The normalized cross sections obtained from these two methods show fair
agreement. Both methods find that most capture is into 8p(J = 1/2) + 8p(J =
3/2), with significant capture into both 8s(J = 1/2) and nf(J = 5/2), (n=6–9).
Considering only s angular momentum states, both methods predict that there is
more capture into 8s1/2 than into 7s1/2.
One interesting difference between the two methods is the relative importance
of nf5/2 versus nf7/2: the two methods show opposing results. As the MCLZ cross
sections were performed in LS coupling, these two states were only separated out
according to their J-weights after the cross section for the general nf state was
found. It is thus to be expected that the state with the larger J value would
dominate, and interesting that our model predicts the opposite to be true.
The MCLZ calculations predicted a relatively high cross section for several
states that we did not initially include in our model basis set. We did not include
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MCLZ Calculations
Figure 6.22 Normalized relative cross sections as obtained with the
pipeline and fitting procedure described in this chapter (left) and MCLZ
calculations (right) for CX between H-like S and neutral He. Each point
on these plots reflects one quantum state of electron capture, or in the
case of degenerate states, a sum of several. The color, symbol, and x-
value of each point describes each quantum state presented. The y-value
indicates the relative cross section of that state. To interpret this figure,
one should compare the y-position of points of the same x-value, color,
and symbol shape across the left-hand and right-hand plots. Points in
parentheses refer to states that were not included in the fit presented in
this chapter, but that have a high relative cross section according to the
MCLZ calculations. The two methods show fair agreement; they both
assign a relatively high cross section to the 8s and 8p states, but the
MCLZ calculation emphasizes capture into higher l states (d, f, g) than
our pipeline does.
187
these either because the spectra only displayed subtle differences in line strengths
from spectra resulting from capture states already included in the basis set (for the
cases of 7d5/2 and 8d5/2), or because we limited the maximum angular momentum
state to l = f (for the cases of 8g9/2 and 8g7/2). However, because of the high
relative cross section of these states in the MCLZ calculation, we performed another
fit to our data that included these four capture states to determine if our pipeline
would identify them as important. After the fit, each of these states was assigned a
weight of approximately 0, meaning they should not contribute significantly to our
experimental spectrum.
We then worked backward from the MCLZ calculations to simulate the spec-
trum predicted by the MCLZ results in order to compare it to our data. We allowed
the He-like lines to be fit according to the elements in our spectral basis set, and
fixed the relative contributions of the H-like spectral basis set elements according to
their MCLZ-calculated cross section. This fit/simulated spectrum and comparison
to both the experimental data and our pipeline model fit is shown in Figure 6.23.
It can be seen that the match between experiment and theory for the H-like lines is
fairly poor; in particular, the strong Lyman-η line is greatly under-predicted. This
shows that although the MCLZ calculations correctly assign the highest cross section
to capture states that lead to this strong line (8p), it over-emphasizes the relative
importance of capture states with other l values, in particular d, f , and g states.
The Lyman-ζ and Lyman-α lines, as simulated in the spectrum according to the
MCLZ calculations, also show poorer matches to the data than with our pipeline.
There are several next steps for this K-shell comparison, both involving the
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Figure 6.23 Data (cyan), model fit to the data through our pipeline
(black), and simulated spectrum assuming MCLZ cross sections to set
the strength of the H-like lines (red). In this simulated spectrum (red),
the He-like lines are fit with our pipeline. The strength of the Lyman-α,
Lyman-ζ, and Lyman-η lines, as simulated with the MCLZ cross sections,
show poor fits to the data. Our pipeline performs better in all cases.
189
pipeline and the MCLZ calculations. With the pipeline, we hope to break the
degeneracies that exist in our spectral basis set in order to distinguish as many
individual quantum states as possible. This can be done by utilizing L-shell S lines,
which currently are not present in our experimental spectra due to the experimental
conditions at the time. On the MCLZ side, we hope to obtain results for He-like S
shortly to compare to our results. Additionally, recall that MCLZ calculations for
bare ions undergoing CX cannot predict an l-distribution for the captured electron;
one must be applied afterwards. We thus would like to investigate how different
l-distributions affect the resulting cross section calculations and simulated spectra.
Mullen et al. [2016] showed that systematically shifting the l-distribution to lower
values led to better agreement with EBIT data. Given our results, we expect a
similar technique would also work well here. Finally, we wish to extend this pipeline
and comparison with MCLZ calculations to many more K-shell ions for which we
have experimental data.
6.6 Cross Section Calculations: Result for Ni19++H2
We now return to our Ni19++H2 CX spectrum discussed earlier in this chapter
and apply our pipeline to calculate relative cross sections.
To create a basis set of spectra to model our Ni+H2 spectrum, we first started
with the assumption that the parent F-like ion would have the lowest possible energy
configuration, 1s21/22s
2
1/22p
2
1/22p
3
3/2, and only considered states with a 2p3/2 core hole.
We then created spectra for n = 8 − 11, l = s − g, and all possible J values. As
190
described in Section 6.3, although each spectrum results from a different initial
electron configuration, many of the resulting spectra appear very similar to one
another. These states would thus not be able to be disentangled in the result.
Thus, to compose our spectral basis set, we used spectra that were visually distinct,
and noted the degeneracies for each spectrum used.
The model and residuals that resulted from this initial basis set are shown in
Figures 6.24 and 6.25. The line identifications refer to those in Table 5.1. Though
the fit to the M2/3G line is good, the fit to many higher energy lines is problematic.
In particular, flux in lines k and l is not predicted at all, and lines b, c, d, and v are
severely underpredicted. By examining the spectral basis set, we determined that
this is because these lines are either weak or nonexistent in this core configuration;
a better fit would require contributions from core-excited states. It is interesting to
note that for lines b, k, and l, which all result from core-excited states, the spex-cx
and acx model predictions are either well-matched to the data or overpredict flux
compared to the experiment. This suggests that these models include and perhaps
even over-emphasize these core-excited states.1
1Through communication with the creators of the ACX model, we have learned that these
lines do not result from direct capture into a core-excited state (e.g. 2s12p5ns1). Instead, the
excited states are formed during the radiative cascade. By comparing details of various transitions
side-by-side, it was determined that some line strengths might differ between the two methods due
to drastic differences in the atomic structure calculations between fac and hullac, an atomic
structure code that is implemented for certain transitions in AtomDB [Duane Liedahl, private
communication]. In certain cases, the Einstein A coefficient differs by an order of magnitude. We
will continue to investigate this discrepancy.
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Figure 6.24 Data (blue), model fit to the data (red), and fit residuals
(green) for CX between Ni19+ and H2, for the M2 and 3G lines, using a
model that solely includes capture states with a 2p3/2 core hole. These
lines are well-predicted with this first iteration of our cross section model.
We thus created a new basis set that included spectra that were most heavily
weighted in the first basis set along with spectra resulting from the 2p1/2 and 2s1/2
excited core configurations. Several states with a 2p1/2 core configuration lead to a
strong b line, along with some flux in c, d, and v, and can also explain the large
observed flux in the aa line. The 2s1/2 core configuration was necessary to populate
the k and l lines. We performed five iterations of various combinations of these
states, removing states that were repeatedly de-emphasized in the fit and replacing
them with others that might contribute to flux in underpredicted lines. We adjusted
the initial guess of fitting weight for certain states that we believed needed to be
moved out of a local minimum, and experimented with pegging certain weights at
a high or low value.
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Figure 6.25 Data (blue), model fit to the data (red), and fit residuals
(green) for CX between Ni19+ and H2, for lines with energies between
890–1530 eV, for a model that solely includes capture states with a 2p3/2
core hole. Though the model performs better than the acx and spex-
cx models, it significantly underpredicts lines b, c, d, s, t, v, and w, and
does not predict any flux in lines k and l. These lines are primarily a
result of capture into states with 2p1/2 and 2s1/2 core hole configurations.
193
The result from the best fit is shown in Figures 6.26 and 6.27. The fit is gen-
erally excellent: though it still underpredicts flux in some lines, it is much improved
over the initial fit that only included 2p3/2 core hole configurations, and is vastly
better than the spex-cx and acx models shown earlier in this chapter. Line g is
the most severely underpredicted, with d, m, and v also lacking flux. The model
also predicts flux in a line near 1510 eV, which is not clearly present in the data.
From the shape of the residuals of the underpredicted lines, it appears that the dis-
crepancies may be due to an energy offset between the model predicted line and the
data. However, it is also possible that this underprediction is due to not including
a capture state that is important, or due to errors in the fac-calculated transition
probability that generated the line strengths in the spectral basis sets.
As we saw from the MCLZ cross section calculations for CX between S16+ and
He, higher l states like f and g may be more important than we initially assumed for
low-energy EBIT experiments. However, for the fit presented here, our spectral basis
set includes states with l as high as g for the 2p3/2 and 2p1/2 core configurations.
Therefore, errors in the fit that can be attributed to omitting important capture
states are likely not due to preferentially selecting low l states, but rather because
there are many more important capture states than we can incorporate into one
basis set.
In general, however, this particular potential source of error, along with any
errors in the fac transition probabilities, are likely not dominant in our current fit.
These potential errors would solely alter the predicted flux in a given line, and as
previously mentioned, the errors in the fit appear to primarily result from differences
194
in line energy between the model and data. Besides the shape of the residuals at
these lines, this is also evidenced by the fact that fixing certain models with higher
weights in an attempt to better match the flux in the experimental spectrum led
to a higher chi-squared. In some cases, the energy mismatch is evident via visual
inspection of the spectra. This is especially obvious in line g.
The main peak of line g in the data is close to the model line centroid, but
there is also a low-energy shoulder that the model does not include. Interestingly,
the energy centroid for that line in the data, as determined both by our analysis
of the DE spectrum (see Section 5.4) and Gu et al. [2007a], is ∼967–968 eV. fac
calculates the energy of this line to be ∼969. This means that in DE, the fac
transition energy appears to be shifted from the data by almost 2 eV, however,
in CX, the experimental result matches better with fac. This difference merits
additional investigation. It is possible that the low-energy shoulder on the CX line
is a satellite line that results from MEC followed by autoionization and inner-shell
excitation. If this is the case, however, it remains to be determined why the main
line centroid would shift to a higher energy in CX. One way of testing whether the
main error is in the model energy centroids is by manually correcting them to match
experimental results such as in Gu et al. [2007a], and by calculating the energy of
any potential satellite lines and including them in the spectra.
We do not believe the energy mismatch is due to errors in the gain calibration,
since the lines with this error are scattered across the L-shell band.
From these model fits to the data, we then extracted relative cross sections for
each considered capture state. Figures 6.28, 6.29, and 6.30 show these cross sections
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Figure 6.26 Data (blue), model fit to the data (red), and fit residuals
(green) for CX between Ni19+ and H2, for the energy region that includes
the M2 and 3G lines. This model includes states with all three core hole
configurations. Though the model appears to somewhat underpredict
the flux in 3G, it is generally a good fit.
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Figure 6.27 Data (blue), model fit to the data (red), and fit residuals
(green) for CX between Ni19+ and H2, for lines with energies between
890–1530 eV. This model includes states with all three core hole configu-
rations. Most lines are fit very well, with the exception of lines g, h, s, t,
and v. The model also predicts a line past 1500 eV where the data does
not show much flux. These mismatches could be caused by not includ-
ing a capture state that is relevant, by slight offsets in energy centroid
of experimental lines compared to the model-predicted lines causing the
fit to not weight a relevant capture state more heavily, or errors in the
fac-calculated transition probabilities that went into the model spectra.
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for the 2p3/2, 2p1/2, and 2s1/2 core hole configurations, respectively. Degeneracies
across electron configurations are indicated with solid black lines across points.
As can be seen in Figure 6.28, the most prominent capture channel is into
2p3/2np1/2(J = 1), where n is likely to be between 8–11. Also important is capture
into 2p3/2nf5/2(J = 4) or 2p3/2ng7/2(J = 5) (n=8–11), which produce nearly iden-
tical spectra, and following that, capture into 2p3/2np3/2(J = 1) (n=8–11). Like
in our S+He results, the significant capture into l = f or l = g that we find is
surprising, given our naive assumption that capture into low angular momentum
states are preferred. We note that capture into these high-l values with a 2p1/2
core hole configuration is not preferred overall. However, within the subset of 2p1/2
core hole configurations, these states have the highest cross section. This highlights
the possibility of mixing between the 2p1/2 and 2p3/2 core excited configurations,
as discussed in Section 5.4. Comparison to theoretical state-selective relative cross
sections may help confirm whether this result is physical. We hope this work spurs
on such theoretical calculations so that we can compare them to our results.
The fourth strongest capture channel is into 2p3/211s1/2(J = 1). This spectrum
is unique compared to other similar configurations with only differing values of n.
The fact that 11s is favored over states with lower n values indicates that the primary
n value of capture for the states with higher cross sections may also be 11. This
highlights the fact that we must use caution applying Equation 1.2, which predicts
nmax to be 9 for this experiment.
Another significant capture channel is into a 2s1/2 core hole configuration:
2s1/211d5/2(J = 2). The importance of this capture channel underscores the possi-
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s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2 g7/2 g9/2
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Figure 6.28 State-selective cross sections extracted from the model fit
to Ni19+ and H2 data, for states with a 2p3/2 core hole. Lines that
span multiple points reflect degeneracies between those states. The most
significant capture channel is into np1/2(J = 1), with n = 8 − 11. This
ambiguity in n is due to the degenerate nature of the resulting spectra
across that range of states. The next highest capture cross section is into
nf5/2(J = 4) or ng7/2(J = 5), with n = 8− 11. This is surprising due to
the high angular momentum of those states. Comparing this figure to
Figures 6.29 and 6.30, we see that the highest overall cross sections are
in the 2p3/2 core hole configuration.
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s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2 g7/2 g9/2
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Figure 6.29 State-selective cross sections extracted from the model fit to
Ni19+ and H2 data, for states with a 2p1/2 core hole. Lines that span
multiple points reflect degeneracies between those states. Most states
with this core hole configuration do not have a very high cross section
compared to the other configurations, but the primary capture channel
in this configuration is for a combination of the states nd5/2(J = 3),
nf7/2(J = 4), and ng9/2(J = 5), with n = 8− 11.
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Figure 6.30 State-selective cross sections extracted from the model fit to
Ni19+ and H2 data, for states with a 2s1/2 core hole. The only capture
channel with significant cross section in this configuration is 11d5/2(J =
2), though it is surprising that this state is significant at all. We would
naively expect the base F-like Ni ion prior to CX to have a 2p3/2 core
hole, or, less likely (due to a higher energy), a 2p12 core hole. A 2s1/2
core hole has the highest energy configuration of the three. It is possible
that this configuration results from MEC followed by autoionization and
inner-shell excitation.
201
bility of n = 11 being the primary n value of the other degenerate states. As dis-
cussed in Section 5.4, this core configuration is unexpected due to its high energy,
but perhaps this state results from multi-electron capture followed by inner-shell
ionization. We will be able to better determine the likelihood of this possibility
from a theoretical standpoint with total cross section calculations of SEC versus
MEC for our collision energy regime and neutral species. Experimentally, by per-
forming CX experiments with atomic H, we can ensure SEC and determine whether
these core excited states are present in the resulting spectra.
6.7 Summary
In this chapter, we have compared our experimental L-shell CX data to com-
monly used spectral synthesis codes. The spex-cx and acx models do not accu-
rately reproduce our CX spectra, with disconcerting differences across the L-shell
energy band. In some cases, energy centroids are offset from fac energy calcula-
tions and our observed spectrum, and both models significantly overpredict flux in
several lines that arise from core excited states. With further communication with
the creators of these spectral synthesis codes, we hope to determine the quantitative
differences between our spectra and begin to work towards a better match between
experiment and theory.
We have also presented a powerful pipeline to extract relative cross sections
from experimental (and if desired, simulated) spectra. We can directly compare
state-selective relative cross sections calculated through this pipeline to MCLZ cal-
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culations, where they are available. MCLZ calculations for S16++He CX only show
fair agreement with our method, and we have shown that a simulated spectrum
resulting from MCLZ calculations does not match our experimental spectrum well.
This underscores the fact that more progress must be made on the theory side.
However, it is important that we have a promising analysis tool that allows us to
directly and quantitatively compare the results of these two methods. This pipeline
can thus be used both in conjunction with and in the absence of detailed theoretical
calculations.
When applying the pipeline to our Ni19++H2 CX data, we find that a core-
excited state (2s1/2) is required to produce the k and l lines, which may result from
MEC followed by autoionization and inner-shell excitation. Emission from the 2p1/2
core excited configuration may result from configuration mixing. This highlights
the need for more comprehensive theoretical calculations that can incorporate these
complex processes to compare to our results and guide our analysis.
We will continue to improve on this modeling process. In particular, we will
focus on the errors present in our cross section calculations. Currently, our fitting
procedure uses chi-squared minimization (with Gehrels weighting), which is not ideal
for our case, as our spectra have many bins with low counts. In the future, we wish
to experiment with different methods of calculating a fit and confidence intervals,
such as the C-statistic [Cash, 1979]. Our top priority is adding confidence intervals
to our calculations of cross sections in order to have a more scientifically robust
result.
In the future, we can apply this method to other experimental data, including
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those from techniques that achieve higher collision energies such as merged-beam
experiments, in order to determine how the relative capture cross sections change.
We can even apply this same procedure to spectra generated from MCLZ or other
theoretical cross section calculations as verification of self-consistency. Although we
can still work to make the selection of the model basis set and the determination of
configuration degeneracies more robust (see Chapter 7), this pipeline is a powerful
new tool that allows us to directly probe the physical processes at work during a
CX experiment, and quickly visualize the spectral lines that result from any electron
configuration in any H, He-, and Ne-like ion.
Our modeling pipeline has already helped us identify the most important states
for electron capture, and can aid in determining more detailed diagnostics for the
occurrence of CX in astrophysical observations. This combination of experiment
and theory will bring us closer to understanding the detailed atomic physics of
CX, properly identifying it in our astrophysical observations, and harnessing its
diagnostic power.
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Chapter 7: Outlook
Improving our understanding of the complex and subtle nature of CX requires
incremental progress. In this dissertation, we have pushed the boundaries of our
knowledge a bit further by using powerful experimental and theoretical tools where
they existed already, and by developing new ones where they did not. We have shown
that CX is an important process to consider in many astrophysical environments
(Section 1.5), and that with high enough resolution and a large enough signal, if we
properly understand the underlying atomic processes that affect CX spectra, it can
be diagnostic of the velocities, elemental abundances, and densities of the emitting
region (Section 1.6). Our work is far from over, however: much remains to be done
in order to reach our goal of verifying and improving the accuracy of CX spectral
models to within the resolution of XARM, Athena, and Lynx in order to utilize
these diagnostics in the future.
There are many natural continuations of the work presented here. We will
continue to develop the cross section modeling pipeline, working to make it more
robust with more automated selections of basis spectral states, determinations of
state degeneracy, and a quantitative error analysis. In order to cross-check our re-
sults, we will also apply our pipeline to model-predicted spectra, including those
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from SPEX-CX, ACX, and future MCLZ results. MCLZ calculations for the other
spectra presented in this thesis are forthcoming [Renata Cumbee, private commu-
nication], and we are looking forward to a detailed comparison of our results to this
theory.
There is also much to be done on the experimental side. It is imperative to
perform more high-resolution spectral measurements, especially of L-shell CX, and
over a wide range of collision energies. We believe that MEC is likely a contributor
to our experimental CX spectra, but disentangling its spectral effect from SEC
is challenging. We can experimentally approach this challenge in two ways. The
first is with time-of-flight measurements (e.g. Ali et al. [2005]) that measure the
relative contribution of MEC versus SEC by identifying the charge state of the
neutral target after CX. The second approach is through high-spectral resolution
measurements of CX with atomic H. Using H ensures SEC, and in addition, simulates
most astrophysical CX. We currently have an H source at LLNL for this purpose
which operates by thermally cracking H2 molecules into H, but this method has
proved to be difficult to properly characterize. Instead, we have begun discussions
with colleagues at GSFC, LLNL, and collaborators at Pacific Union College to design
an H beam that results from neutralizing a proton beam from a duoplasmatron
source with an alkali jet.
CX is finally starting to gain more attention in the astronomical field as we
begin to see hints of it in various astrophyisical environments and learn more about
the complexities of the process. To gain a better understanding of the atomic
physics at work during CX, we must merge high-resolution experimental data with
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comprehensive theoretical models. Observers, theorists, and experimentalists alike
must thus work together to continue to learn about this fascinating process.
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Appendix A: Spectral Basis Set Elements for Model Fit to S+He CX
Data
This appendix shows simulated spectra for H-, then He-like S that result from
electron capture into a given electron configuration in jj-coupling, then the sub-
sequent cascade to ground. These spectra formed the basis set for a least-squares
minimization fit to data presented in Chapter 4. The procedures for this fit and the
generation of these spectra are described in Chapter 5.
208
0 500 1000 1500 2000 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0800000001
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.1 Cascade spectrum of H-like S with initial electron configura-
tion 8s1/2.
0 500 1000 1500 2000 2500
Energy (eV)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0700000001
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.2 Cascade spectrum of H-like S with initial electron configura-
tion 7s1/2.
209
0 500 1000 1500 2000 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0801000003
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.3 Cascade spectrum of H-like S with initial electron configura-
tion 8p3/2.
0 500 1000 1500 2000 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0701000003
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.4 Cascade spectrum of H-like S with initial electron configura-
tion 7p3/2.
210
0 500 1000 1500 2000 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0802000003
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.5 Cascade spectrum of H-like S with initial electron configura-
tion 8d3/2.
0 500 1000 1500 2000 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0803000005
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.6 Cascade spectrum of H-like S with initial electron configura-
tion 8f5/2.
211
0 500 1000 1500 2000 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0803000007
2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.7 Cascade spectrum of H-like S with initial electron configura-
tion 8f7/2.
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0600010100
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.8 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/26s1/2(J = 0).
212
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0700010100
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.9 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27s1/2(J = 0).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0600010102
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.10 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/26s1/2(J = 1).
213
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0701010002
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.11 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27p1/2(J = 1).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0601010102
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.12 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/26p3/2(J = 1).
214
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0701010102
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.13 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27p3/2(J = 1).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0702010002
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.14 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27d3/2(J = 1).
215
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0702010004
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.15 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27d3/2(J = 2).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0702010104
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.16 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27d5/2(J = 2).
216
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0702010106
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.17 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27d5/2(J = 3).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0401010102
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.18 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/24p3/2(J = 1).
217
2400 2420 2440 2460 2480 2500
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0501010102
3000 3050 3100 3150 3200
Energy (eV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.19 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/25p3/2(J = 1).
2400 2420 2440 2460 2480 2500
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0703010106
3000 3050 3100 3150 3200
Energy (eV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure A.20 Cascade spectrum of He-like S with initial electron configu-
ration 1s1/27f7/2(J = 3).
218
Appendix B: Spectral Basis Set Elements for Model Fit to Ni+H2,
Ni+He CX Data
This appendix shows simulated spectra for Ne-like Ni that result from electron
capture into a given electron configuration in jj-coupling, then the subsequent cas-
cade to ground. These spectra formed the basis set for a least-squares minimization
fit to data presented in Chapter 5. The procedures for this fit and the generation of
these spectra are described in Chapter 6.
219
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0800010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.1 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/28s1/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0900010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.2 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/29s1/2(J = 1).
220
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1000010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.3 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/210s1/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1100010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.4 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/211s1/2(J = 1).
221
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1101010002
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.5 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/211p1/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1101010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.6 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/211p3/2(J = 1).
222
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0902010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.7 Cascade spectrum of Ne-like Ni with initial electron configu-
ration 2p3/29d5/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.05
0.10
0.15
0.20
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0803010004
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.05
0.10
0.15
0.20
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.8 Cascade spectrum of Ne-like Ni with initial electron config-
uration as one or a combination of 2p3/28f5/2(J = 2), 2p3/29f5/2(J =
2), 2p3/210f7/2(J = 2), 2p3/211f7/2(J = 2), 2p3/28g7/2(J = 3),
2p3/29g7/2(J = 3), 2p3/210g7/2(J = 3), or/and 2p3/211g7/2(J = 3).
223
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0803010104
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.9 Cascade spectrum of Ne-like Ni with initial electron config-
uration as one or a combination of 2p3/28f7/2(J = 2), 2p3/29f7/2(J =
2), 2p3/210f5/2(J = 2), 2p3/211f5/2(J = 2), 2p3/28g9/2(J = 3),
2p3/29g9/2(J = 3), 2p3/210g9/2(J = 3), or/and 2p3/211g9/2(J = 3).
224
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1103010008
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.10 Cascade spectrum of Ne-like Ni with initial electron config-
uration as one or a combination of 2p3/28f5/2(J = 4), 2p3/29f5/2(J =
4), 2p3/210f5/2(J = 4), 2p3/211f5/2(J = 4), 2p3/28g7/2(J = 5),
2p3/29g7/2(J = 5), 2p3/210g7/2(J = 5), or/and 2p3/211g7/2(J = 5).
225
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.2
0.4
0.6
0.8
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 1103010110
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.2
0.4
0.6
0.8
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.11 Cascade spectrum of Ne-like Ni with initial electron config-
uration as one or a combination of 2p3/28f5/2(J = 4), 2p3/29f5/2(J =
4), 2p3/210f5/2(J = 4), 2p3/211f5/2(J = 4), 2p3/28g7/2(J = 5),
2p3/29g7/2(J = 5), 2p3/210g7/2(J = 5), or/and 2p3/211g7/2(J = 5).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 0700010102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.12 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p3/27s1/2(J = 1).
226
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 080100000102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.13 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p1/28s1/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 090100000102
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.14 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p1/28s1/2(J = 1).
227
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 110102000106
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.15 Cascade spectrum of Ne-like Ni with initial electron con-
figuration 2p1/28d5/2(J = 3), 2p1/29d5/2(J = 3), 2p1/210d5/2(J = 3),
2p1/211d5/2(J = 3).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 090103000004
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.16 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p1/29f5/2(J = 2).
228
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 110103000108
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.17 Cascade spectrum of Ne-like Ni with initial electron con-
figuration one or a combination of 2p1/28d5/2(J = 3), 2p1/29d5/2(J =
3), 2p1/210d5/2(J = 3), 2p1/211d5/2(J = 3), 2p1/28f7/2(J = 4),
2p1/29f7/2(J = 4), 2p1/210f7/2(J = 4), 2p1/211f7/2(J = 4),
2p1/28g9/2(J = 5), 2p1/29g9/2(J = 5), 2p1/210g9/2(J = 5), or/and
2p1/211g9/2(J = 5).
229
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 060102000002
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.18 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p1/26d3/2(J = 1).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 090102000002
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.1
0.2
0.3
0.4
0.5
0.6
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.19 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2p1/29d3/2(J = 1).
230
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 110000010100
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.20 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2s1/211s1/2(J = 0).
850 900 950 1000 1050 1100 1150 1200
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s) 110002010004
1200 1250 1300 1350 1400 1450 1500 1550
Energy (eV)
0.05
0.10
0.15
0.20
0.25
0.30
In
te
n
si
ty
 (
a
rb
it
ra
ry
 u
n
it
s)
Figure B.21 Cascade spectrum of Ne-like Ni with initial electron config-
uration 2s1/211d5/2(J = 2).
231
Appendix C: X-Ray Emission Mechanisms
X-rays are emitted from a wide variety of astrophysical objects, and one can
either measure those X-rays via imaging, timing, polarization, or spectral studies.
In this dissertation, I have focused on X-ray spectroscopy, which can yield a wealth
of physical information, especially if at high resolution: with sufficient data from
an astrophysical target or laboratory plasma, along with a correct interpretation
of the underlying X-ray emission mechanisms, one can derive information about
composition, temperature and density distributions, structure, turbulent and bulk
velocities, and energy balance.
The first step in interpreting the spectra from the objects and regions we ob-
serve is to understand the emission mechanism. Many atomic processes involving
highly charged ions can lead to the emission of X-rays in astrophysical and labo-
ratory environments, either in the form of an X-ray continuum or via spectral line
formation. Though the focus for this dissertation has been CX, in the regions where
it occurs, is often just one of many processes happening simultaneously. It is thus
important understand other mechanisms that can generate competing X-ray emis-
sion. For a more detailed theoretical discussion of these mechanisms, see Mewe
[1999].
232
Figure C.1 Radiative cascade.
Figure C.2 Scattering of X-ray light by a neutral.
The most basic of these processes is a radiative cascade (Figure C.1): elec-
trons in an excited state cascade down to a lower energy level, following quantum
mechanical selection rules, and emitting photons at each step with an energy equal
to the difference in energy between the initial and final states.
X-ray photons can also be elastically scattered by electrons in an atom. (Figure
C.2).
Bremmstrahlung radiation (Figure C.3) is produced when a free electron is
233
Figure C.3 Bremmstrahlung.
decelerated as it passes near an ion. To satisfy the law of conservation of energy, the
electron’s loss of kinetic energy from the deceleration is released back into the system
as an X-ray photon. Because this loss in energy depends on several variable factors,
such as the initial velocity of the electron, the charge of the ion, and the impact
parameter, the resulting X-rays have a distribution of energies. Bremmstrahlung is
thus a continuum process.
Electrons in an ion can be collisionally excited from one subshell to another
by other electrons (collisional excitation, Figure C.4) or a photon (photoexcita-
tion, Figure C.5). After collisional excitation, the colliding electron is left with a
lower kinetic energy and the excited electron subsequently radiatively decays back
to ground. Photoexcitation is a resonant process, meaning the energy of the in-
coming photon must exactly match the difference in energy states of the electron
to be excited. If the energy of the electron collision or photon is sufficiently large,
inner-shell excitation or ionization may occur.
234
Figure C.4 Collisional excitation by electron impact.
Figure C.5 Photoexcitation.
235
Figure C.6 Collisional ionization.
Much like excitation, electrons may be ejected from their initial subshell by a
collision with an electron (Figure C.6) or stimulation by a photon (Figure C.7) in
a process called ionization. The electron or photon must have energy higher than
the binding energy of the electron that it ejects, and its post-collision energy is
lowered by the interaction. An excited ion may also undergo autoionization (Figure
C.8): the spontaneous emission of an outer-shell electron past the bound state.
Auger ionization (Figure C.9) is a two-step process that involves a free electron
that collisionally ionizes an inner-shell electron in the ion. A bound electron then
fills the electron hole. In Auger ionization, the change in energy is transferred into
another electron which gets ejected from the ion. If the change in energy instead
gets released as a photon, the process is called fluorescence (Figure C.10).
Recombination is an inverse process to ionization: a free electron is captured
into a bound level of an ion. In radiative recombination (Figure C.11), after electron
capture, a photon is released with an energy equal to the sum of the free electron’s
236
Figure C.7 Photoionization.
Figure C.8 Autoionization.
237
Figure C.9 Auger ionization.
Figure C.10 X-ray fluorescence.
238
Figure C.11 Radiative recombination.
initial kinetic energy and the binding energy of the state into which it is captured.
In a plasma with a distribution of free electron energies, this leads to a contin-
uum spectrum. Dielectronic recombination (Figure C.12) is a resonant process in
which the energy released during electron capture goes into simultaneously exciting
a bound electron to an excited state, leading to a doubly excited ion. From here,
the ion may autoionize, converting the process into an electron scattering event, or
one excited electron may radiatively decay, leading to a satellite line. This satellite
line will have a slightly lower energy than from a normal transition from that level
due to the nearby recombined and excited electron.
239
Figure C.12 Dielectronic recombination.
240
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