ASYMMETRY DEPENDENCE OF SPECTROSCOPIC FACTORS: A STUDY OF

TRANSFER REACTIONS ON ARGON ISOTOPES AT 70 MEV/U

By

Juan José Manfredi Jr.

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Physics — Doctor of Philosophy

2018

ABSTRACT

ASYMMETRY DEPENDENCE OF SPECTROSCOPIC FACTORS: A
STUDY OF TRANSFER REACTIONS ON ARGON ISOTOPES AT 70

MEV/U

By

Juan José Manfredi Jr.

Nuclear reactions are useful tools to study the structure of the atomic nucleus. One of

the most popular reactions over the last several decades is the transfer reaction, and the ad-

vent of rare isotope beam facilities has opened up new swaths of the nuclear chart available

for exploration with this technique. In principle, different techniques should give consistent

nuclear structure information (like the spectroscopic factor which quantifies single-particle

occupancy) for a given isotope. However there is a well-established discrepancy between

spectroscopic factors extracted from transfer reaction data and those extracted from knock-

out reaction data. In particular, reduction factors (ratios of extracted spectroscopic factors

to theoretical expectation) from knockout data show a strong dependence on nuclear asym-

metry, whereas the transfer measurements show at most a weak dependence. This discrep-

ancy not only raises important questions on the influence of nucleon-nucleon correlations in

nuclear structure, but also calls into question the validity of the relevant nuclear reaction

techniques.

This dissertation describes the measurement of the 34Ar(p, d)33Ar and 46Ar(p, d)45Ar

single-neutron transfer reactions at 70 MeV/u. The motivation of this study is to measure

the same transfer reactions on argon examined in earlier work at low energy, while matching

the high beam energy of previous knockout measurements on argon. Raising the beam

energy to a regime where few reliable measurements exist could illuminate potential defects

in the transfer reaction mechanism. We performed a kinematically complete measurement

of the differential cross sections for these (p, d) reactions at the National Superconducting

Cyclotron Laboratory using several detector systems. The High Resolution Array (HiRA)

detected the outgoing deuterons, the S800 Spectrograph detected the heavy argon recoil, and

two Microchannel Plates (MCPs) tracked the incoming beam to normalize the cross section

and to better localize the transfer on the reaction target. We carried out various calibrations

on each individual detector system (including a detailed characterization of silicon detectors

in HiRA) before merging and normalizing the data to generate the cross sections of interest.

We extracted spectroscopic factors using the adiabatic distorted wave approximation

(ADWA) framework implemented in the TWOFNR code. Both the CH89 global optical

potential as well as the microscopic Jeukenne, Lejeune, and Mahaux (JLM) optical potential

produced spectroscopic factors for each reaction system. The resulting reduction factors

corroborate the low-energy results and disagree with the knockout data by showing a weak
asymmetry dependence between the neutron-rich 46Ar and the proton-rich 34Ar. Therefore,

the transfer reaction mechanism yields consistent results even at a high beam energy. We

advocate for further transfer reaction measurements at high asymmetry, as well as a deeper

theoretical understanding of both the transfer and knockout reaction mechanisms.

ACKNOWLEDGMENTS

Having come to the end of this challenging endeavor called graduate school, I find myself

in a grateful mood. Although my name is on the title page of this dissertation, the work

presented here would have been impossible without the help of many amazing people. I hope

the reader will indulge me as I do my best to express sincere appreciation to them.

First, I would like to express my utmost gratitude to Professor Betty Tsang, my advisor.

I have relied heavily on her support and wisdom, especially during the most trying times

of my graduate career. Enumerating all the lessons I learned from her would take an entire

chapter of this thesis, but I have especially admired her tenacity in the face of challenges,

her organized approach, and her willingness to search out and open doors for her students.

I will forever be grateful for her patience and guidance during my scientific education.

I would also like to thank Professor Bill Lynch, who advised me extensively over the

last several years. From him I learned the extraordinary value of asking simple questions,

as well as how to solve a problem by breaking it down into its most basic components. I

must also thank him for his patience in helping me with my research, despite my occasional

stubbornness.

In addition to Profs. Tsang and Lynch, I would like to thank Profs. Wolfgang Mittig,

Ed Brown, and Norman Birge for serving on my dissertation guidance committee. I have

sincerely appreciated their advice and their ability to keep in mind the big picture of my

graduate career trajectory to ensure the timely completion of my degree.

Of course, I am extraordinarily grateful for my funding sources. From 2013 to 2017,

I had the privilege to participate in the Department of Energy National Nuclear Security

Administration Stewardship Science Graduate Fellowship. I am indebted to this fellowship

iv

for funding that allowed me to perform this thesis work, as well as professional opportuni-

ties (like the chance to spend three months at Lawrence Livermore National Laboratory)

and a fantastic network of fellows. I would especially like to thank the Krell Institute for

administering the fellowship and providing excellent support over the years. I also thank

the National Science Foundation for their continued support of rare isotope research at the

NSCL, and the Graduate School for funding during my final year.

I wish to thank Kim

Crosslan for her help in navigating the sometimes confusing bureaucracy of Michigan State,

as well as for her encouragement.

Fortunately, I have enjoyed many exceptional classes during my time at Michigan State.

In particular I would like to thank Profs. Morten Hjorth-Jensen, Filomena Nunes, Norman

Birge, Dave Morrissey, Mark Iwen, Paul Mantica, and Wade Fisher for teaching excellent

courses. I would also like to thank Prof. Brian O’Shea for advising me in my pursuit of

a CMSE graduate certificate, as well as supervising my independent study. I thank Prof.

Saul Beceiro Novo (as well as the ISEE PDP program) for teaching me about teaching. Of

course, I also owe a debt of gratitude to the many excellent teachers I had before arriving

at MSU. Thank you to Profs. Guido Weiss and Jason Woods for guiding me through my

mathematics and physics college coursework. Thank you also to Prof. Demetrios Sarantites,

Mrs. Yacamelli, Mr. Matusiak, and Mr. Zellmer for having a particularly large impact on

my development as a student.

I have had the privilege of working with many incredible people as part of the HiRA

group. I am grateful for the mentorship of Prof. Zbigniew Chajecki early in my graduate

career, as well as for his patience in helping me learn the basics of experimental science. I

would like to thank Dr. Kyle Brown for his collaboration during my thesis experiment, his

advice on difficult analysis issues, and for many years of his friendship and office visits. I

v

have valued working with other post-docs as well, especially Drs. Rebecca Shane, Pierre

Morfouace, Giordano Cerizza, Genie Jhang, and Daniele Dell’Aquila. Working with the

other graduate students of the group has also been a privilege. I would like to thank Dr.

Jack Winkelbauer and Dr. Rachel Hodges Showalter for helping me learn the ropes of the

HiRA group as well as for their assistance running my experiment. I would also like to thank

Jon Barney and Justin Estee for playing important roles in running my thesis experiment,

and also for their valued friendship. Special thanks to Jon for giving me a cat, and to Justin

for showing me how much fun Japanese baseball games are. I am grateful for the chance

to have worked with the younger students in the group, including Sean Sweany, Zhu Kuan,

Tommy Tsang, and Chi-En Teh: as I leave it is nice to know that the HiRA group is in

good hands. I would also like to thank the many excellent undergraduates who have passed

through our group, especially Corinne Anderson for her role in the CsI crystal repair project.

It is difficult to overstate how tremendously lucky I am to have worked at the NSCL.

Nuclear physics is a team sport, and the team at the NSCL is second to none. In particular

I would like to thank Dr. Tom Ginter, Dr. Daniel Bazin, Craig Snow, Cody Norat, Dr.

Dave Sanderson, the safety office, the cyclotron operators and Dr. Jeremy Tompkins for

their help in running my thesis experiment. I am thankful to Profs. Remco Zegers, Artemis

Spyrou, and Michael Thoennessen for their sincere effort to make the lab a welcoming and

healthy place for students to do excellent science. Thank you to the lab leadership, who

have fostered an incredible scientific environment, and will certainly continue to do so in the

upcoming transition to FRIB.

The research described herein was only possible with the help of many amazing scientists

around the world. First of all, I must express my most sincere gratitude to Prof. Jenny

Lee, who originally proposed the measurement described in this thesis. Beyond serving as

vi

the PI of the experiment, her work is the foundation that this dissertation is built upon.

I am incredibly grateful for her invitation to come work with her in Hong Kong for an

entire summer. I would also like to thank Prof. Andy Rogers for all of his work during the

experimental setup, as well as for his collaboration developing the analysis code. Thank you

to all the other people who collaborated on running the experiment, especially Dr. Zhengyu

Xu, Cole Pruitt, Jon Elson, Prof. Lee Sobotka, Prof. Zbigniew Chajecki, Dr. Karl Smith,

and Prof. Zhigang Xiao. I also thank Chenyang Niu for his analysis collaboration on the

E detector characterization (as well as for his hospitality during my visit to Beijing). I am

grateful to the theorists who took the time to help me understand the details of reaction

theory, in particular Prof. Jeff Tostevin, Dr. Gregory Potel, and Prof. Filomena Nunes.

I must take a moment to especially thank Prof. Lee Sobotka at Washington University

for serving as an absolutely fantastic undergraduate advisor. I am tremendously grateful

that he responded to the email I sent him in my freshman year of college asking about

research opportunities. I certainly do not exaggerate when I say that I would not be where

I am today without his advice and support. I would also like to thank Prof. Robert Charity

for teaching me about ROOT during my undergraduate years.

One of the best features of the NSCL is the vibrant graduate student community which

has allowed me to avoid the feelings of isolation that graduate students sometimes experience.

Thank you to all the graduate students I have had the privilege to interact with over the

past six years here at Michigan State. I am in particular grateful to Dr. Chris Prokop for

years of friendship as roommates, as well as to Dr. Sam Lipschutz for ensuring that a boring

day never transpired at the NSCL. I am thankful to my wonderful officemates, especially

Dr. Wei Jia Ong, Zach Matheson, Katie Childers, and honorary officemember Becky Lewis.

Outside of the NSCL, I am thankful to Dr. Connor Glosser for his friendship and for showing

vii

me the best places to eat and drink in the Lansing area.

Despite what it feels like, I did not spend every minute of my time in graduate school

working on this thesis. I am grateful to Zach Constan for giving me the chance to engage in

science outreach both for the satisfaction of pursuing a noble mission as well as the practical

benefits of improved communication skills. I would also like to thank everyone at WaMPS for

organizing so many fantastic outreach events, and for bringing to life the SL@MS program.

My participation in organizations outside of science, especially Crossing Water, Michigan

for Bernie, and MSU Men’s Volleyball, kept me sane during the long slog of graduate school

and introduced me to incredible people and lifelong friends.

Most importantly, I would like to thank my incredible family. My father taught me

the importance and the beauty of science, and I would not have pursued this degree (or

persevered until the end) without his guidance. My mother is the kindest person I know,

and raised me to respect myself and others. My little sister always makes me laugh and

also inspires me with her perseverance through difficult circumstances. And, of course, I

must thank our dog. I would also like to thank my extended family in Spain and Argentina.

Although we are literally oceans apart, I feel the powerful influence of their love and support

in my daily life: gracias! Last but not least, my partner continues to inspire me each and

every day, and I am not sure this endeavor would have been possible without the personal

growth she has inspired within me. Her support, as well as that of our precious cat, during

the trials and tribulations of graduate school has proven inestimably important. Thank you.

viii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1

Inside the Nucleus

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Atomic Nucleus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
1.3 Understanding Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Dissertation Outline
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Reaction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
2.1.2 The Distorted Wave Born Approximation . . . . . . . . . . . . . . .
2.1.3 The Adiabatic Distorted Wave Approximation . . . . . . . . . . . . .
2.1.4 Local Energy Approximation . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Optical Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6.1 Global Optical Potentials
. . . . . . . . . . . . . . . . . . .
2.1.6.2 Microscopic Optical Potentials
. . . . . . . . . . . . . . . .
2.1.6.3 Optical Potential Uncertainty . . . . . . . . . . . . . . . . .
2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The TWOFNR Reaction Code . . . . . . . . . . . . . . . . . . . . . .

Silicon Detectors

Chapter 3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Primary Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2
Secondary Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 HiRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 CsI Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 The S800 Spectrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Cathode Readout Drift Chambers (CRDCs) . . . . . . . . . . . . . .
3.4.2
Ionization Chamber (IC) . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Plastic Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 MCPs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Position Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1
1
4
9
13
20

22
22
23
27
31
32
33
34
35
36
37
40
41

42
42
46
47
49
50
53
57
59
61
63
64
64
65
68

Chapter 4 Data Analysis I: Calibrations and Corrections . . . . . . . . . .
4.1 HiRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2
Silicon Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3
. . . . . . . . . . . . . . . . . . . . . .
Silicon Dead Layer Thickness
4.1.4 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.5 CsI Energy Calibration and Silicon Detector Thickness . . . . . . . .
4.1.6 Pixelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 S800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70
71
74
78
83
89
90
97
99
4.2.1 Beam Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.2 CRDC Position Calibration . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.3 Particle Identification and Trajectory Corrections . . . . . . . . . . . 102
4.3 MCP Beam Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3.1 Mask Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.2 Beam Spot Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 111

Chapter 5 Data Analysis II: Cross Sections and Reduction Factors . . . . 114
5.1 Reaction Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Excitation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.1 DAQ Live Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4.2 HiRA Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.3 Beam Detector Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4.3.1 OBJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.3.2 MCP Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.3.3 MCP Position . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.4 Beam Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.5 Beam Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.6 Overall Normalization Uncertainty . . . . . . . . . . . . . . . . . . . 140
. . . . . . . . . . . . . 140
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Reduction Factor Asymmetry Dependence . . . . . . . . . . . . . . . . . . . 149

5.5 Differential Cross Sections and Spectroscopic Factors

5.5.1
5.5.2

34Ar(p, d)
46Ar(p, d)

Chapter 6

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 154

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

x

LIST OF TABLES

Table 4.1: MCP position resolutions along both axes for several hole sizes in
both MCPs, as well as the intrinsic resolutions found be linearly ex-
trapolating to a 0 mm hole size.

. . . . . . . . . . . . . . . . . . . . 112

Table 5.1:

Table 5.2:

Table 5.3:

Table 5.4:

Energy levels below 5 MeV as well as known spin and parity assign-
ments for 33Ar and 45Ar.
Differential cross section for 34Ar(p, d)33Arg.s. in the COM frame with
statistical uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . 143

. . . . . . . . . . . . . . . . . . . . . . . . 117

Combined differential cross section for 46Ar(p, d)45Ar to states at 0
and 0.542 MeV in 45Ar in the COM frame with uncertainty. . . . . . 148

Extracted spectroscopic factors and reduction factors for both 34Ar
and 46Ar. Results are shown using both the CH89 global model as
well as the JLM microscopic model.

. . . . . . . . . . . . . . . . . . 149

xi

LIST OF FIGURES

Figure 1.1: Components of an atom (from [2]). . . . . . . . . . . . . . . . . . . .

Figure 1.2: Chart of the nuclides, with neutron number on the x-axis and pro-
ton number on the y-axis. The chain of calcium isotopes is magni-
fied, showing that calcium can vary from proton-rich to neutron-rich.
Taken from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3:

Single-particle orbitals generated by various mean-field potentials.
Figure taken from [12].
. . . . . . . . . . . . . . . . . . . . . . . . .

2

3

7

Figure 1.4:

Structure of 41Ca. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Figure 1.5:

Spectroscopic factors extracted from transfer (red), knockout (blue),
and electron scattering (black) data for a variety of different isotopes.
. . .
Data compiled from [26], [52], [53], [54], and references therein.

Figure 1.6:

Spectroscopic factors extracted from transfer (red) and knockout (blue)
data along the argon isotopic chain. Data taken from [55] and [52]. .

Figure 1.7:

Spectroscopic factors extracted with transfer (red), knockout (blue)
and quasi-elastic proton scattering (magenta) reactions for the oxygen
. . . . . . . . .
isotopic chain, along with corresponding trend lines.

Figure 2.1: An incoming plane wave interacts with a potential to produce outgo-
. . . . . . . . . . . . . . . . .

ing spherical waves. Figure from [77].

4.3 in [77].

Figure 2.2: Coordinates for a single-neutron transfer reaction. Based on Figure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.3: Differential cross section calculations for 46Ar(p, d)45Arg.s. with CH89
. . . . . . . . . . . . . . .

(black) and JLM (red) optical potentials.

Figure 2.4:

Spectroscopic factors extracted from transfer (red) and knockout (blue)
data along the argon isotopic chain. SFs extracted with two different
optical models are shown: the JLM (solid) and the CH89 (open).
. . . . . . . . . . . . . . . . . . . . .
Data taken from [55] and [52].

xii

14

16

19

24

29

38

39

Figure 3.1: Cartoon of the experimental setup. The incoming beam (either 34Ar
or 46Ar) was produced with the Coupled Cyclotron Facility (CCF)
at the NSCL. A polyethylene target was inserted into the beamline
to measure (p, d) reactions on the incoming beam. Upstream of the
target, two Microchannel Plates (MCPs) were placed to normalize
the cross-section and also track the beam position on target. The
outgoing heavy recoil from the transfer reaction (either 33Ar or 45Ar)
was detected and identified in the S800 Spectrograph. The outgoing
deuteron was detected and identified with the High Resolution Array
(HiRA).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.2: Kinematic curves for 46Ar(p, d)45Ar for different excitation energies
of the outgoing recoil 45Ar. This energy-angle relationship of the
emitted deuteron can be measured with HiRA.
. . . . . . . . . . . .

Figure 3.3:

Schematic of the Coupled Cyclotron Facility, showing ion sources,
the K500 cyclotron, the K1200 cyclotron, and the A1900 fragment
separator [117].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

45

47

Figure 3.4: Cartoon showing the radiation detectors inside of each HiRA telescope. 51
Figure 3.5: Kinematic curves for 46Ar(p, d)45Arg.s. (in red) and 34Ar(p, d)33Arg.s.
(in blue) at 70 MeV/u. The green dotted line corresponds to the
punch-through energy for a deuteron through the E detector.
. . . .

54

Figure 3.6: Photo of a partially assembled HiRA telescope with an E silicon de-
tector. The front surface of the E detector (EF) is visible at the top
of the figure. Wire bonds connect each individual EF strip to the
orange cable running down the left side of the telescope. The corre-
sponding cable for the back surface (EB) can be seen on the right.
The E detector is held by a G10 fiberglass frame, and this frame is
attached with screws to the metal sides of the telescope. A bundle
of four CsI scintillators can be seen underneath the silicon detector.
The DE detector (not shown) is placed above the E and secured in
place by screwing its G10 frame to the telescope. . . . . . . . . . . .

Figure 3.7:

Side-view and front-view drawings of a four-crystal CsI scintillator
bundle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.8: Photo of HiRA in the setup for experiment 09084.

. . . . . . . . . .

Figure 3.9: Geometrical efficiency versus emitted deuteron angle for HiRA. This
efficiency was calculated assuming that all detectors work perfectly.
This assumption will be revisited in detail in Chapter 5. . . . . . . .

55

59

60

61

xiii

Figure 3.10: The S800 Spectrograph and its associated analysis line.

. . . . . . .

Figure 3.11: The S800 focal plane detector box. Figure adapted from [128].

. . .

Figure 3.12: Diagram showing the operation of MCPs in the chevron configuration.
Electrons strike the conductive surface on a glass tube in the MCP,
resulting in a cascade of electrons and amplification of the signal.
After two stages of amplification, the resulting electrons are detected
on a resistive anode. Figure taken from [130]. . . . . . . . . . . . . .

Figure 3.13: On the left is a schematic of an MCP setup, and on the right is an
overhead picture of the MCP1 setup in the beamline. Two such setups
were used in this experiment. Detailed description can be found in
the text. Modified from [131].
. . . . . . . . . . . . . . . . . . . . .

Figure 3.14: A graduate student (Jonathan Barney) measures positions on the
target ladder that held the MCP0 Mylar foil and MCP0 calibration
mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.15: Angles of each pixel in all 14 HiRA telescopes calculated from ROMER
. . . . . . . . . . . . . . . . . . . . . . . . . . .

arm measurements.

Figure 4.1: Data analysis flowchart from raw data to the final differential cross
section observables. Chapter 4 describes the analysis above the gray
dotted line, and Chapter 5 describes the analysis below the gray dot-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ted line.

Figure 4.2: Flowchart for the HiRA data analysis from the raw ROOT Tree to
. . . . . . . . . . . . . . . . . . . . . . .

the pixelated ROOT Tree.

Figure 4.3: Pulser ramp for an example DE silicon strip. The top panel is a his-
togram showing the electronics output (in ADC channels) for many
clearly resolved pulser peaks. The bottom panel displays the relation-
ship between ADC channels and pulser voltage. Each peak in the top
panel corresponds to a red dot in the bottom panel. The blue line is
a linear fit to the red dots in the central ≈ 80% of the full dynamic
range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: Pulser ramps for example EF (left column) and EB (right column)
silicon strips. Both columns are set up similarly to Figure 4.3. The
blue line is a linear fit to the red dots in the central ≈ 80% of the full
dynamic range.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

63

66

67

69

69

72

73

75

76

xiv

Figure 4.5: Pulser ramp for an example CsI crystal set up similarly to Figure 4.3.
The blue line is a linear fit to the red dots up to ≈ 50% of the full
dynamic range.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.6: Decay scheme for 228Th. The green dashed box includes all decay
radiation from 212Pb, the isotope that is primarily deposited onto
each pin source. Energies for all alpha decays with branching greater
than 1% are shown in red, along with the corresponding branching
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
rates [134, 133].

Figure 4.7:

228Th calibrated alpha spectrum for an example HiRA EF detec-
tor. The DE and EB spectra look similar. Energy losses in the gold
window, the Mylar foil in the front of the HiRA telescope, and an
approximate E dead layer are taken into account. Typically, a HiRA
silicon calibration will use the 5 most prominent peaks seen in the
spectrum. The resolution in this detector is 65 keV (FWHM) [133]. .

Figure 4.8: HiRA telescope assembly. On the bottom left is a side-view photo-
graph of a HiRA telescope. On the bottom right is a cartoon of the
detectors contained within the telescope. Dotted red lines connect
the detectors in the cartoon to their approximate positions in the
telescope. On the top of the figure is the pin source frame with the
pin source installed. The pin source slot is located between the DE
and the E detector as indicated by the thick red arrow [133].
. . . .

Figure 4.9: Example pin source alpha spectrum for one pixel in one E detector
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[133].

Figure 4.10: Cartoon of pin source and dead layer. The distance h between the
pin source and the detector surface is 3.2 mm. The angle θ of the
alpha particle from the pin source determines how much of the dead
layer the particle passes through. The dead layer thickness (T) can
be extracted by studying the relationship between detected energy
and angle [133].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.11: A two-dimensional hit map for the number of detected pin source
counts for one E detector. The central pixel is shown at the intersec-
tion of the black rectangles (which correspond to the front and back
strips with the most counts) [133].
. . . . . . . . . . . . . . . . . . .

Figure 4.12: Example energy distribution with statistical error bars of the 8.785
MeV peak for different pixels across the central front strip. The cor-
. . . . . .
responding fit (using Equation 4.2) is shown in red [133].

77

79

79

81

82

84

86

87

xv

Figure 4.13: Example energy distribution with statistical error bars for the 8.785
MeV peak across all pixels in a single telescope according to different
values of 1/ cos θ, as well as the corresponding fit [133].
. . . . . . .

88

Figure 4.14: Dead layer thicknesses extracted for each E detector from both the
low (red open points) and high (blue solid points) energy peaks from
212Pb decay. All error bars are statistical, and the wide range of errors
is due to different intensities of the pin sources. The average dead
layer thickness (0.61 µm) is given by the dotted black line. Values
extracted with the higher energy peak and the lower energy peak are
consistent with each other [133].
. . . . . . . . . . . . . . . . . . . .

89

Figure 4.15: DE-E PID plot for an example telescope, with CsI thresholds applied. 90

Figure 4.16: E-CsI PID plot using raw CsI energy.

. . . . . . . . . . . . . . . . .

91

Figure 4.17:

(a) HiRA PID plot for reaction data using one CsI crystal and its
corresponding E silicon detector. The y-axis is calibrated energy in
the E detector, and the x-axis is uncalibrated CsI energy in units of
electronics channels. Protons (within the red-dashed line) can clearly
be identified, even though the CsI energy is uncalibrated. (b) The
top panel is the HiRA PID for the same CsI crystal as in (a), this
time showing the scattering data (zoomed in to the relevant region).
The red-dashed line is the same proton gate as in (a). The bottom
panel is a projection of the top panel onto the x-axis. Two peaks are
clearly visible: the higher energy peak corresponds to proton elastic
scattering off of carbon, and the lower energy peak is from inelastic
scattering (E∗(12C) = 4.439 MeV) [133].
. . . . . . . . . . . . . . .

Figure 4.18: HiRA CsI light response with a direct proton beam at low energies.
Data for one crystal is shown with a linear fit. Modified from [140].

Figure 4.19: Relationship between calculated CsI energy and raw CsI channels
for one telescope at three different E thicknesses. The blue squares
correspond to scattered protons with well known energies that deposit
a small amount of their energy in the E detector, and the open circles
are calculated via the energy-loss method as described in the text
using each of the three indicated E thicknesses. When fitting both
sets of points together, an E thickness of 1474 µm provides the points
that yield the best fit [133]. . . . . . . . . . . . . . . . . . . . . . . .

93

94

95

xvi

Figure 4.20: Summary of results from E thickness extraction in comparison with
manufacturer provided detector thicknesses. The line corresponds
to exact agreement. The measured values agree within error to the
manufacturer values. Wide variation from the nominal value of 1500
. . . . . . . . . . . . . . . . . . . . . . . . . . .
µm is evident [133].

Figure 4.21: Comparison between two adjacent strips in an example EB detector.
The diagonal lines indicate charge splitting from each alpha particle
. . . . . . . . . . . . . . . . . . . . . . . .
from the thorium source.

96

98

Figure 4.22:

Incoming beam PID for the 34Ar beam. . . . . . . . . . . . . . . . . 101

Figure 4.23:

Incoming beam PID for the 46Ar beam. . . . . . . . . . . . . . . . . 101

Figure 4.24: Example calibrated CRDC mask.

. . . . . . . . . . . . . . . . . . . 103

Figure 4.25:

(a) TOF-OBJ0 plotted against x1. (b) TOF-OBJ0 plotted against
AFP. (c) ∆E0 plotted against x1. (d) ∆E0 plotted against y1. Red
dotted lines correspond to linear fits.

. . . . . . . . . . . . . . . . . 105

Figure 4.26: S800 PID for the 34Ar beam, with the (p, d) residue 33Ar clearly

indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.27: S800 PID for the 46Ar beam, with the (p, d) residue 45Ar clearly

indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.28: MCP1 raw corner signals (left), and MCP1 pedestal-subtracted cor-

ner signals (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 4.29: MCP brass calibration masks for MCP0 (left) and MCP1 (right). The

holes in the mask pattern are 1, 2, or 3 mm in diameter. . . . . . . . 110

Figure 4.30: MCP0 xcal vs. ycal calibrated mask data (left) and MCP1 xcal vs.

ycal calibrated mask data (right).

. . . . . . . . . . . . . . . . . . . 111

Figure 4.31: Reconstructed beam spot at the target for the 46Ar beam. . . . . . . 113

Figure 5.1: Measured kinematic curves for 34Ar(p, d)33Ar shown with calcula-

tions for comparison.

. . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 5.2: Measured kinematic curves for 46Ar(p, d)45Ar shown with calcula-

tions for comparison.

. . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 5.3: Measured 34Ar(p, d)33Ar excitation energy spectrum for θCOM < 10

degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xvii

Figure 5.4: Measured 46Ar(p, d)45Ar excitation energy spectra for several dif-
ferent angular ranges. We show the excitation energies calculated
without MCP beam tracking (black) and with MCP beam tracking
(red).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 5.5: Widths of the Gaussian fits for 46Ar(p, d)45Arg.s.. The black (red)

points correspond to data without (with) MCP beam tracking.. . . . 122

Figure 5.6:

S800 PID (left) and deuteron kinematics (right) for the carbon target
runs with the 46Ar beam. The kinematics plot is gated on the 45Ar
residue indicated in the S800 PID plot.

. . . . . . . . . . . . . . . . 124

Figure 5.7:

Live time of the DAQ system plotted against run number. . . . . . . 127

Figure 5.8:

Strip-by-strip electronics live times for three example HiRA DE de-
tectors. We see variation from detector to detector, as well as within
individual detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Figure 5.9: HiRA geometrical efficiency for 46Ar(p, d) without (black) and with

(pink) dead time corrections as described in the text.

. . . . . . . . 130

Figure 5.10: RF time plots that illustrate the OBJ efficiency. The black histogram
is the ungated RF time for an example run, and the red histogram is
the RF time for the same run gated on valid counts in the OBJ.

. . 132

Figure 5.11: OBJ efficiency plotted against the experiment run number.

. . . . . 132

Figure 5.12: MCP0 (red) and MCP1 (blue) scaler efficiencies plotted against the

experiment run number. . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 5.13: Sum of the MCP1 corner signals, as well as a red line indicating the

threshold value.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 5.14: Purity of both argon secondary beams over time. . . . . . . . . . . . 136

Figure 5.15: Detected MCP cleaned-up (CU) scaler rates vs. beam attenuation
for both MCP0 (red) and MCP1 (blue). When the beam attenuation
is 1, the beam is completely removed from the experimental setup.

. 138

Figure 5.16:

(a) MCP1 CU scaler correlations with the OBJ scintillator scaler (top,
blue) and the CsI OR of OR scaler (bottom, red). We use the result-
ing linear fits (dashed green lines) to calculate the expected CU scaler
for data runs in which it was not directly recorded. (b) Percentage
difference between the expected MCP1 CU scaler calculated from the
OBJ scaler and the CsI OR of OR scaler. They agree within 10%.

. 139

xviii

Figure 5.17: Absolute differential cross sections at 7 degrees in the center-of-mass
frame for several different subsets (or groups) of data. The 34Ar
cross section is shown in blue, and the 46Ar in red. The solid line
corresponds to the cross section value for the entire data set, and the
dotted lines are 10% from the solid line. . . . . . . . . . . . . . . . . 141

Figure 5.18: Differential cross section for 34Ar(p, d)33Arg.s. in the COM frame.
The blue circles are the data from the present work. The lines cor-
respond to TWOFNR calculations normalized to the first peak using
CH89 (dotted, green) and JLM (dashed, red) optical potentials. The
calculations shown are multiplied by the best fit spectroscopic factors. 142

Figure 5.19: 45Ar excitation energy spectrum for one angular bin in which the
0 MeV peak is fit with two Gaussians. Individual Gaussian fits are
drawn in blue, and their sum in red. . . . . . . . . . . . . . . . . . . 145

Figure 5.20: ADWA calculations for 46Ar(p, d)45Ar for the ground state (red) and
first excited state (blue) of 45Ar. The ratio of the two cross sections
is given by the green dashed line. . . . . . . . . . . . . . . . . . . . . 146

Figure 5.21: Differential cross section for 46Ar(p, d)45Arg.s.,E*=0.542 MeV in the
COM frame. The blue circles are the data from the present work.
The lines correspond to TWOFNR calculations for two different final
45Ar states normalized to the SFs which were obtained as described
in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 5.22: Asymmetry dependence of reduction factors using the CH89 global
optical model. We show reduction factors from the current work
(green squares) as well as the 33 MeV/u measurement (open red cir-
cles). Each line represents the best fit to the corresponding reduction
factor data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 5.23: Asymmetry dependence of reduction factors using the JLM micro-
scopic optical model. We show reduction factors from the current
work (green squares) as well as the 33 MeV/u measurement (open
red circles). Each line represents the best fit to the corresponding
reduction factor data.

. . . . . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.24: Asymmetry dependence of reduction factors from transfer reactions
using CH89 (open green squares and open red circles) and from knock-
out reactions (blue triangles). Each line represents the best fit to the
corresponding reduction factor data. . . . . . . . . . . . . . . . . . . 152

xix

Figure 5.25: Asymmetry dependence of reduction factors from transfer reactions
using CH89 (open green squares and open red circles), from knockout
reactions (blue triangles), and from (p, 2p) reactions (pink crosses).
All data points represent measurements on oxygen isotopes except for
the open green squares, which come from the present work on argon
isotopes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xx

Chapter 1

Introduction

I, a universe of atoms, an atom in the

universe.

Richard Feynman

1.1 The Atomic Nucleus

The world around us is made of atoms. Since Democritus first theorized the existence of

the atom in ancient Greece, science has ventured within this building block of matter to

study its secrets. Each atom consists of a nucleus made up of protons (that have positive

electric charge) and neutrons (no electric charge) surrounded by a cloud of negatively-charged

electrons. Ernest Rutherford’s 1911 discovery of the atomic nucleus immediately made clear

two of its remarkable properties [1]. First, the nucleus is heavy, making up over 99.9%

of the total mass of the atom. Protons and neutrons, collectively referred to as nucleons,

are much heavier than electrons, and therefore constitute almost all the mass of ordinary

matter. Secondly, the nucleus is small. Typically the radius of the atom is more than 10,000

times bigger than the nuclear radius. This extraordinarily dense collection of interacting

neutrons and protons contains a rich and complex set of phenomena that is the focus of

nuclear physics.

Since the positively-charged protons in the nucleus repel each other via the electrical

force, there must be another force overpowering this repulsion to bind the nucleons together.

1

Figure 1.1: Components of an atom (from [2]).

This nuclear force is a residual effect of the strong interaction between the constituent quarks

that make up protons and neutrons, as shown in Figure 1.1 [2]. The aptly-named strong

force is the most powerful fundamental force in the universe, and the source of energy tapped

by nuclear energy and nuclear weapons.

Although the exact mathematical form of the nuclear force is not known, we do know

that this force acts only at very short distances. In contrast, the electrostatic force (also

called the Coulomb force) acts at long distances. The balance between short-range nuclear

attraction and long-range Coulomb repulsion limit the possible configurations of neutrons

and protons that form into allowed nuclei. Figure 1.2 shows the landscape of possible nuclei,

known as the chart of the nuclides [3]. Each square represents a single isotope, denoted by

proton number on the y-axis and neutron number on the x-axis. The black squares indicate

stable nuclei that make up most of the ordinary matter on Earth, and are collectively referred

to as the valley of stability. The green squares indicate all other observed nuclei, which are

unstable (also called exotic or rare) and will eventually decay towards the valley of stability.

2

Figure 1.2: Chart of the nuclides, with neutron number on the x-axis and proton number on
the y-axis. The chain of calcium isotopes is magnified, showing that calcium can vary from
proton-rich to neutron-rich. Taken from [3].

The largest region of the chart (in yellow) remains unexplored.

The number of protons in a nucleus (given by Z) determines its element, and each element

has a variety of isotopes determined by neutron number (given by N). An isotope is typically

represented as AX, where A = N +Z is the mass number of the nucleus and X is the chemical

symbol for its corresponding element. In Figure 1.2, the calcium isotopic chain (Z = 20) is

enlarged to show the variety of isotopes possible for a single element. Some isotopes, like
35Ca have a relatively low number of neutrons (proton-rich) and others, like 58Ca have a

relatively high number of neutrons (neutron-rich).

A central theme of modern nuclear physics research is understanding these exotic nuclei

3

on both the proton-rich and neutron-rich sides of the chart. Producing rare isotopes is

challenging, so the American nuclear physics community has coordinated the construction of

large accelerator facilities, like the National Superconducting Cyclotron Laboratory (NSCL)

and the upcoming Facility for Rare Isotope Beams (FRIB), to create and study rare isotopes.

Researchers from all over the world use these rare isotope factories to address important

questions about nuclear physics. For instance, what are the properties of exotic nuclei away
from the valley of stability? How do nuclei on one side of an isotopic chain (e.g. 35Ca)
behave differently from nuclei on the other side (e.g. 58Ca)? And what is the best way

to study these properties? This dissertation will focus on addressing such issues. First, we

must formulate these questions more precisely via a basic introduction to nuclear structure,

the study of how protons and neutrons arrange themselves in the nucleus.

1.2

Inside the Nucleus

Unlike macroscopic objects (like planets, cars, or human beings), nuclei are too small to be

accurately described by Newton’s Laws. Instead, nuclei obey the laws of quantum mechanics,

in which objects have properties of both particles and waves, and quantities like energy and

momentum are restricted to discrete values. A quantum system can exist in a superposition

of these discrete states, and so the complexity of a nucleus with A nucleons rises very quickly

with A. As a result, treating an arbitrary nucleus exactly as a quantum many-body problem

is an extraordinarily challenging, and often impossible, task from both the analytical and

the computational point of view.

Given the difficulty of solving this problem exactly, nuclear physicists employ different

approximations in order to further theoretical understanding of the nucleus and to make

4

sense of experimental observations. An early example of such an approach is the Bethe-

Weizsäcker formula, first published in the 1930s, based on the liquid drop model (LDM) of

the nucleus [4, 5]. The LDM considers the nucleus as a drop of incompressible fluid made

up of neutrons and protons, similar to a drop of water made up of H2O molecules [6, 7].
The Bethe-Weizsäcker formula puts the LDM in quantitative form with only five terms to
calculate the binding energy for a particular nucleus1; these terms relate to the volume,

surface area, Coulomb repulsion of constituent protons, neutron-proton asymmetry, and

nuclear pairing energy contributions [4, 5, 9]. Despite its simplicity, this formula successfully

describes broad trends in nuclear binding energy, putting early mass measurements into

useful theoretical context.

The LDM’s intuitive appeal outweighs its predictive power. For instance, experimental

evidence showed that nuclei with certain numbers of neutrons or protons (2, 8, 20, 28, 50, 82,

and 126) have excess binding energy compared to values predicted by the Bethe-Weizsäcker

formula [10, 11, 12, 13]. Eugene Wigner, intrigued by the discrepancy, referred to these points

of extra stability as magic numbers, which is how they have subsequently been referred to

since then [13].

In 1949, Marie Goeppert-Mayer and Hans Jensen left behind the LDM to develop the

first shell model of the nucleus, shedding light on the mystery of these magic numbers [14, 15,

16]. The shell model had a transformative impact on nuclear physics, and to this day is the

most successful and robust nuclear structure framework. Mayer and Jensen were awarded

the Nobel Prize in 1963 for their work.

The simplest shell model is called the Independent Particle Model (IPM). The central idea
1The binding energy of a nucleus is the minimum energy required to dismantle that nucleus. By Einstein’s
principle of mass-energy equivalence, the binding energy can also be expressed as a mass with the relation
E = mc2 [8]. In the present discussion, binding energy and mass will be used interchangeably.

5

of the IPM is the assumption that each nucleon moves in a mean-field potential independently

from all the other nucleons. This potential is meant to represent the average influence on

one particular nucleon from all the others, and is typically defined in a harmonic oscillator

or Woods-Saxon form [12, 16]. In the mean field, each nucleon occupies a single-particle

eigenstate orbital characterized by its discrete energy and quantum numbers n, l, and j,

where n is the number of nodes in the radial wave function, and l and j are the orbital and

total angular momenta. Figure 1.3 shows these orbitals for different choices of the potential.

Due to the Pauli exclusion principle no two nucleons can occupy the same quantum state,

and the lowest energy configuration comes from filling in orbitals from the bottom up [17].

Protons and neutrons fill orbitals independently. Groups of orbitals with similar quantum

numbers and energies are called shells.

In Figure 1.3 we see that the IPM predicts energy gaps between separate shells. Therefore

exciting a nucleon to a higher energy orbital in a nucleus with a fully occupied, or closed,

shell bears a large energy cost. This is exactly what results in extra stability for nuclei with

a magic number of neutrons or protons (or both), and each magic number corresponds to a

fully-occupied shell of single-particle orbitals. Using a simple mean-field potential with an

added spin-orbit component (see the rightmost column in Fig. 1.3), the IPM can reproduce

all the magic numbers listed above [12, 16].

The IPM essentially substitutes a very difficult A-body problem with A more tractable

single-body problems. In fact, we typically solve far less than A single-body problems, as

we also assume that only the highest energy (or valence) orbitals dictate nuclear structure

properties due to the inert nature of the filled inner shells. Although these assumptions

substantially reduce the number of degrees of freedom, the IPM performs quite well.

In

addition to reproducing the magic numbers, the IPM also provides the correct ordering of

6

Figure 1.3: Single-particle orbitals generated by various mean-field potentials. Figure taken
from [12].

7

single-particle states for the vast majority of stable nuclei, and a basic description of their

static and dynamic properties [18].

This mean field approach provides a useful basic framework for nuclear structure, but

the IPM does not generalize well beyond closed-shell nuclei. One reason is the influence

of nucleon-nucleon correlations. Short-range correlations, due to strong repulsion between

nucleons at very short distances, push nucleons to higher-momentum orbitals [19, 20]. Long-

range correlations between valence nucleons result in collective behavior like deformation

and giant resonances [21]. Both short and long range correlations disrupt the IPM mean

field picture, reducing the occupancy of single-particle orbitals [22, 23]. More sophisticated

models, like the large-basis shell model (LBSM), can account for some of these correlation

effects (mostly of the long range variety) by expanding to a larger single-particle model space

to describe configuration mixing in the valence states, or by using specialy designed effective

interactions [22, 24]. Still, they are based on the foundation set by the mean-field theory of

Mayer and Jensen.

We are interested in evaluating how well state-of-the-art shell models actually describe

reality using experimental data. Which orbitals are occupied in a given nucleus, and to what

extent? How does this orbital structure change away from the valley of stability? And for

that matter, what type of data can we use to compare to theory? In the next section, we

will discuss one potential point of contact between experimental data and nuclear structure

theory.

8

1.3 Understanding Nuclear Structure

The occupancy of a single-particle orbital in a particular state of a given nucleus is quantified

by its spectroscopic factor (SF) [25]. This quantity is defined as follows:

Z

(cid:12)(cid:12)(cid:12)(cid:12)DΨA−1(cid:12)(cid:12)(cid:12)a~p

(cid:12)(cid:12)(cid:12)ΨAE(cid:12)(cid:12)(cid:12)(cid:12)2

d~p

SF =

(1.1)

where ~p is the set of quantum numbers for a particular single-particle orbital and ΨX is

a wave function for an X-nucleon system. The annihilation operator a~p acts on a wave
function to remove a nucleon in that single-particle state. SFs can be calculated directly
from shell model wave functions, and range from 0 (no overlap between ΨA−1 and a~pΨA)
to 2j + 1 (perfect overlap) where j is the total angular momentum of ~p. Normalizing to

a maximal value of 1, the SF can be interpreted as the probability that given an A-body
wave function ΨA, removing a nucleon in state ~p will yield a particular A − 1-body core
configuration ΨA−1. In this work, we will normalize SFs to 2j + 1.

Figure 1.4 illustrates this concept with an example. For the ground state of 41Ca, the

spectroscopic factor of the f7/2 orbital is approximately 1 [26]. This means that we can
consider 41Ca as consisting of a doubly-magic (Z=20, N=20) core with an extra single

neutron fully occupying the f7/2 orbital. In this instance, the single-particle orbital is fully
occupied due to the stability of the inert core. For many other cases SFs will have sub-

maximal values, indicating the presence of nucleon-nucleon correlations disrupting the simple

picture of a single-particle in a mean field.

Spectroscopic factor calculation depends on details of the chosen Hamiltonian and basis

states used to define the wave functions [27, 28]. Therefore, the SF is not a true observable

and cannot be directly measured in the same way as an energy or cross section. Instead, the

9

Figure 1.4: Structure of 41Ca.

standard procedure is to first induce a nuclear reaction on the nucleus of interest and measure

an observable quantity (e.g. reaction cross section). Experimental spectroscopic factors must

then be extracted from data via comparison of this observable to nuclear reaction theory [27].

The specific extraction procedure depends on the chosen reaction probe, but the general idea

is to use reaction theory to calculate the expected observable assuming the orbital of interest

is fully occupied. Then the ratio of the experimental observable to the calculated observable

gives a measure of the actual orbital occupancy. This will be discussed further in Chapter

2. The ratio of the resulting experimental SF to the SF from nuclear structure theory is

called the reduction factor (typically denoted R or Rs), and quantifies how well the theory

describes the actual nuclear structure.

In the literature, some discuss whether the non-observability of the SF prevents it from

being a valid measure of nuclear structure [28, 29, 30]. Ultimately, many studies have

shown that meaningful information can be gleaned from SFs within a systematic extraction

10

approach [23, 31, 32, 33, 34, 35], including the work of Tsang, Lee, and Lynch in which they

re-analyzed transfer reaction data from 80 nuclei within a consistent theoretical framework

to extract SFs [36, 37, 38]. A single SF in isolation suffers from high theoretical uncertainty,

but comparing SFs from several nuclear states extracted in a similar way from similar data
can yield real insight about the structure within. Consider the previous example of 41Ca:

the SF ranges from 0.75 to 1.06 depending on the optical potential used for the reaction

model in the extraction. However, the SF trend across many calcium isotopes is consistent

for different optical potentials as well as IPM calculations [39].

Several types of nuclear reactions can be used to access SFs. One possibility is the (e, e0p)
electron scattering reaction, in which a high-energy electron scatters off a stationary target
nucleus and knocks loose a proton in the process. The major advantages of (e, e0p) are that
it penetrates the nuclear interior and that it occurs via the well-understood electromagnetic

interaction. In principle, using an electromagnetic probe rather than a hadronic one reduces

theoretical uncertainty related to our less developed understanding of the nuclear force com-
pared to the electromagnetic force. The results from (e, e0p) data show substantial reduction
(R ≈ 0.6 − 0.7) compared to the IPM for stable nuclei across a wide mass range [40]. This

indicates significant influence from nucleon-nucleon correlations. When compared to other

nuclear structure models, the reduction fluctuates [39].

Unfortunately, the (e, e0p) technique does not easily apply to the study of rare isotopes.
First of all, electron scattering cannot access particle states (for which nucleons must be

added rather than removed from the nucleus) or hole states involving neutron removal.

More importantly, most rare isotopes are too difficult to produce and too short-lived to

serve as targets in an electron scattering experiment. Instead, we need reaction mechanisms

that allow for the study of isotopes in an accelerated beam.

11

Transfer reactions, in which a nucleon is transferred from one nucleus to another, have

been a popular choice for extracting SFs since the 1950s [41, 42, 43]. Transfer reactions

experiments can study both hole and particle states, selectively, in either an accelerated beam

or stationary target nucleus. With the advent of fast radioactive beams, transfer reactions

have again come to prominence as an important tool for exploring the exotic regions of the
nuclear chart [43, 44, 45]. Single-nucleon transfer reactions (like (p, d), (d, p), and (d,3 He))

are particularly useful for probing single-particle states, and have been called “the perfect

tool for shell structure studies” [46]. The typical procedure is to measure the differential cross

section for a specific transfer reaction, and then divide by the calculated differential cross

section from reaction theory. The reaction theory (which we discuss in detail in Chapter

2) dictates that the shape of the differential cross section strongly depends on the angular

momentum, so the shapes of the measured and calculated cross sections usually match [41].

The SF then comes from the ratio of the magnitude between experiment and theory in the

differential cross section.

A more recently developed technique for extracting SFs is the single-nucleon knockout

reaction, in which a nucleon is removed from an intermediate-energy beam nucleus [44, 47,

48, 49, 50]. Knockout experiments can run with low beam intensity, and therefore can

reach further away from the valley of stability than other techniques [51, 44]. This reaction

probe is particularly well suited for studying the structure of weakly bound states [47, 48].

In a knockout experiment, the isotope in the beam impinges on a light reaction target.
When knockout occurs, the mass A − 1 recoil is measured and identified in a spectrometer.

Gamma ray detectors can then tag on individual final states in the recoil, while longitudinal

momentum distributions allow for angular momentum determination [44, 52]. The ratio of

the experimental total cross section to the theoretical value then gives the SF.

12

1.4 Motivation

It stands to reason that if the structure of a given nucleus is invariant, then using differ-

ent reaction probes to study the same nucleus should yield consistent results. Surprisingly,

different techniques do in fact disagree significantly on extracted spectroscopic factors for

well-bound nucleons in asymmetric nuclei. Figure 1.5 illustrates this inconsistency by sum-

marizing spectroscopic factor studies performed with several different reaction mechanisms.

The y-axis is the reduction factor defined above (relative to LBSM calculations), and the
x-axis is defined as ∆S = Sn − Sp (for neutron removal) or ∆S = Sp − Sn (for proton re-
moval) where Sn is the neutron separation energy (i.e. the energy cost to remove a neutron

from the nucleus) and Sp is the proton separation energy. For neutron-removal reactions,

proton-rich nuclei have a very positive ∆S since the neutrons, as the deficient species, are

more bound relative to the protons. Removing a neutron from a neutron-rich nucleus has a

negative ∆S since the neutrons are relatively less bound. The opposite applies to proton-

removal reactions. The plot shows reduction factors extracted via single-nucleon transfer

(red), single-nucleon knockout (blue), and electron-induced proton knockout (black). The

red, blue, and black points mostly agree for stable nuclei with ∆S close to 0. For removal

of weakly bound nucleons, the transfer and knockout both show that reduction factors are

at least as high as for stable nuclei (although the trend is more clear from the knockout

results). However, in the case of tightly bound nucleon removal, the transfer and knockout

show drastically different results. Although both data sets show decreased reduction factors

(which indicate the increased influence of correlations), the knockout results show a much

stronger decrease than the transfer results. The knockout also shows further reduction for

very asymmetric nuclei not currently accessible by transfer reaction studies.

13

Figure 1.5: Spectroscopic factors extracted from transfer (red), knockout (blue), and electron
scattering (black) data for a variety of different isotopes. Data compiled from [26], [52], [53],
[54], and references therein.

14

S (MeV)∆(cid:173)20020Reduction Factor00.20.40.60.811.2n(cid:173)transfern(cid:173)knockoutp(cid:173)transferp(cid:173)knockout(e,e’p)}}p (cid:173) SnS = S∆n (cid:173) SpS = S∆To more carefully examine this strange phenomenon, we focus on a single isotopic chain.

Figure 1.6 shows the transfer-knockout discrepancy clearly for neutron removal from argon

isotopes [55, 56]. The reduction factors for single-neutron transfer in red come from [55,

56], and those extracted from single-neutron knockout in blue are from [57, 58, 59]. Again,
the red and blue points agree in the case of neutron-rich 46Ar, in which the neutrons being
removed are loosely bound. For the proton-rich 34Ar, in which neutrons are tightly bound,

the knockout data show a much stronger decrease in reduction factor than the transfer
results, and even further reduction for 32Ar (which was not measured with transfer). Note
that the ∆S value for 34Ar differs between the transfer and knockout measurements. The

transfer measurement was exclusive, so Fig. 1.6 shows the ∆S value calculated with the

ground state of the final argon recoil. The knockout measurement, on the other hand, was
inclusive, so its ∆S value was calculated with a weighted combination of final states in 33Ar.
The knockout reduction factor shown for 34Ar therefore represents an upper limit on the

ground state to ground state reduction factor measured by the transfer. This potentially

widens the disagreement between transfer and knockout even further. We note that the

transfer reactions were measured at a lower beam energy (33 MeV/u) than the knockout

reactions (70 MeV/u).

In summary, transfer reactions and knockout reactions produce substantially different

results regarding the neutron-proton asymmetry dependence of single-particle structure in

argon isotopes. The transfer data tells us that the shell model calculations do a reasonably
good job of calculating, for instance, the 34Ar single-particle structure (compared to the 46Ar

case). On the other hand, the knockout data shows that the same shell model calculations

are in fact woefully inadequate, and correlations are playing a stronger role than expected.

This discrepancy has also been observed in asymmetric oxygen isotopes [53]. Tostevin and

15

Figure 1.6: Spectroscopic factors extracted from transfer (red) and knockout (blue) data
along the argon isotopic chain. Data taken from [55] and [52].

16

S (MeV)∆(cid:173)20020Reduction Factor00.20.40.60.811.2n(cid:173)transfern(cid:173)knockoutAr46Ar36Ar34Ar3233 MeV/u70 MeV/uGade present a summary of knockout results for many different systems that consistently

show the same quenching of the reduction factor [52]. Some recent knockout studies use the

systematic results from Tostevin and Gade to re-normalize SFs in order to compare them

to shell model calculations in the sum-rule limit without worrying about suppression due to

correlations [60, 61].

This disagreement between transfer and knockout has important consequences. Firstly,
the single-particle structure of the neutrons in 34Ar (and other such cases of the deficient

species in asymmetric nuclei) is still unresolved. Furthermore, the transfer-knockout discrep-

ancy reveals that at least one of the two reaction mechanisms is not completely understood.

This situation is analogous to two astronomers, each with a different type of telescope (opti-

cal and x-ray, for instance), looking upon the same star: if they describe the star differently,

then perhaps the telescopes themselves are not well understood.

The nuclear physics community has attempted to address this inconsistency between

transfer and knockout probes. From a reaction theory perspective, Nunes, et al. suggested

that Lee’s analysis of the argon transfer measurements was not definitive, and that an alter-

native treatment of the reaction theory resolves the discrepancy in asymmetry dependence

[62]. However other results have since provided further evidence of inconsistency between

transfer and knockout SFs [53, 63, 64]. Timofeyuk and others controversially advocate that

unresolved problems with nonlocality in the single-nucleon transfer reaction theory make it

an unreliable probe for SFs [65, 66]. In general, the large spread in the transfer data (e.g. in

Figure 1.5), as opposed to the more clearly defined knockout trend, hints at shortcomings

in the reaction theory. There are also theoretical challenges regarding the knockout mecha-

nism. Flavigny et al. have shown that for well-bound nucleons the approximations made in

knockout reaction theory break down, resulting in altered cross sections [67]. However, the

17

tell-tale distorted momentum distributions of the non-sudden effects described by Flavigny

are not seen by Gade [57, 58, 59]. Another possibility is that core excitations, typically ne-

glected in knockout analysis, become important when knocking out a deeply bound nucleon

[68, 69].

From a structure point of view, the idea that nucleon-nucleon correlations play a more

important role for well-bound nucleons in asymmetric systems has some support. Calcula-

tions done with the dispersive optical model [70, 71], the self-consistent Green’s functions

method [72], the inhomogeneous equation [73], and the microscopic coupled-cluster method

[74] all indicate an asymmetry-dependence in SF reduction, but (in most cases) weaker than

the one observed in the knockout data.

There is also experimental evidence for the slightly higher influence of nucleon-nucleon

correlations away from stability. For instance, some transfer reaction measurements already

discussed ([55, 56]) could indicate a weak asymmetry dependence. In addition, recent results

from (p, 2p) quasi-elastic proton scattering experiments on oxygen isotopes show a similar

asymmetry trend as the transfer reaction results [75, 76]. These results are plotted in Figure

1.7 in comparison with relevant transfer and knockout data. Quasi-elastic proton scattering is

an entirely different reaction mechanism from single-nucleon transfer: the fact that these two

probes agree with each other provides strong support for this weaker asymmetry dependence.

Finally, electron scattering results in stable nuclei suggest that short-range correlations more

strongly affect the deficient species in asymmetric nuclei, but how much this effect manifests

in exotic nuclei is unclear [19]. To be clear, however, there is so far no experimental evidence

for strong asymmetry dependence outside of the single-nucleon knockout results.

Despite productive steps forward in understanding this problem, it remains unresolved.

Nuclear structure theorists are working to further understand the role correlations play in

18

Figure 1.7: Spectroscopic factors extracted with transfer (red), knockout (blue) and quasi-
elastic proton scattering (magenta) reactions for the oxygen isotopic chain, along with cor-
responding trend lines.

19

S (MeV)∆(cid:173)20020Reduction Factor00.20.40.60.811.2n(cid:173)transfern(cid:173)knockoutp(cid:173)transferp(cid:173)knockout(p,2p)their models, while nuclear reaction theorists continue to study sources of uncertainty in SF

extraction. On the experimental side, we must gather more data while varying asymmetry,

mass number, and other variables to test the validity of both the transfer and knockout

reaction mechanisms. One possible variable to explore is the beam energy.

Single-nucleon knockout measurements are performed with intermediate-to-high beam

energy (70 MeV/u for the argon knockout data) to ensure the applicability of the eikonal

approximation, whereas transfer reactions are usually measured at lower energy (33 MeV/u

for the argon transfer data) where the cross sections are higher. Comparing these two

mechanisms at the same energy could provide important evidence either for or against the

validity of the transfer reaction probe. In this dissertation we present (p, d) transfer reaction
measurements (on 46Ar and 34Ar) similar to those performed by Lee except with a beam

energy of 70 MeV/u to match Gade’s knockout measurement. If this higher energy transfer

measurement agrees with Lee’s results at low energy, then it bolsters confidence in the

consistency of the transfer reaction method at high energies for well-bound nucleons.

If

the two transfer measurements disagree, then there could be a problem with the current

understanding of the single-nucleon transfer mechanism.

1.5 Dissertation Outline

In this dissertation we present (p, d) transfer reaction measurements on two different argon

isotopes, as well as the subsequent SF extraction. Chapter 2 introduces the relevant transfer

reaction theory and the methodology for extracting SFs from measured differential cross

sections. Chapter 3 details the experimental setup at the NSCL, with particular emphasis

on the three main radiation detector systems: the High Resolution Array (HiRA), the S800

20

Spectrograph, and the Microchannel Plates (MCPs). In Chapter 4, we discuss calibration

and analysis procedures for each individual detector system. Then, in Chapter 5, we combine

data from these separate detectors, and further analyze them to produce differential cross

sections for the transfer reactions of interest. We then present the SF extraction from the

measured cross sections using transfer reaction calculations, in addition to the corresponding

reduction factors and asymmetry trend. We summarize this work and provide concluding

thoughts in Chapter 6.

21

Chapter 2

Reaction Theory

I do not like it and I’m sorry I had

anything to do with it.

Erwin Schrödinger, on quantum

mechanics

Spectroscopic factors are not observable, and therefore must be extracted from exper-

imental data via comparison to reaction theory. This chapter motivates the basic mathe-

matical framework for calculating transfer reaction differential cross sections, along with the

relevant assumptions and commonly used potentials. The methodology used to extract SFs

presented in [26] and [36] will be followed.

2.1 Theoretical Framework

In the initial system of a nuclear reaction, an incoming nucleus (referred to as the projectile,

or the beam) moves towards a stationary target nucleus. Some sort of interaction between

the beam and the target occurs, resulting in a new system of reaction products. In a transfer

reaction, a nucleon or group of nucleons is transferred between the projectile and the target.

The primary concern of this thesis is a type of transfer reaction called neutron-pickup,
in which a neutron is transferred from the heavy target to the light proton beam1. For
1The distinction between beam and target is entirely dependent on reference frame. The reaction model
is motivated here assuming a proton beam, but in the experimental setup the proton acted as the target (see
Chapter 3).

22

a neutron-pickup reaction on an argon isotope with mass number A, the reaction can be

expressed as

AAr + p → A−1Ar + d

and can also be represented in the more convenient notation AAr(p, d)A−1Ar.

The observable of interest is a differential cross section, given by dσ

dΩ(θ, φ), which quan-
tifies the likelihood that a particle will be detected coming out of the reaction system at

the solid angle element dΩ with angles θ and φ relative to the incoming beam axis. Due

to azimuthal symmetry, only θ dependence is considered here. The general idea is to mea-

sure a differential cross section experimentally, calculate the differential cross section for the

same transfer reaction using a theoretical reaction model, and compare the experimental and

theoretical differential cross sections to extract the SF.

Calculating the differential cross section requires the appropriate mathematical infras-

tructure. We will first develop this framework in the context of two-body scattering: although

this is a simpler problem than the transfer reaction of interest, understanding it will be useful

in the subsequent description of transfer reaction formalism.

2.1.1 Scattering Theory

As previously mentioned, a fundamental principle of quantum mechanics is that particles

have wave-like properties, and in particular that a physical system can be described using a

wave function. This wave function is a complex-valued probability amplitude for properties

of the system like position and momentum. The Schrödinger equation describes the behavior

23

Figure 2.1: An incoming plane wave interacts with a potential to produce outgoing spherical
waves. Figure from [77].

of a wave function:

(cid:21)
(cid:20) ˆT + V
ˆT = −¯h2
2µ

Ψ = EΨ

∇2

where ˆT is the kinetic energy operator, V is the potential energy, and Ψ is the wave function.
In the definition of ˆT, ¯h is the reduced Planck constant, µ is the reduced mass of the system,
and the Laplacian operator is ∇2. E is the discrete energy value corresponding to the state

given by Ψ.

A schematic of the scattering problem is shown in Figure 2.1: the incoming projectile

plane wave packet approaches the target along the z-axis, and after the interaction, an

outgoing spherical wave emerges (along the radius R). Asymptotically, the wave function

can be written as

Ψasym(R, θ) = eikz + f(θ)eikR
R

(2.1)

24

where k is the wavenumber and f(θ) is the scattering amplitude, which is the probability

amplitude of the scattered wave at angle θ relative to the incident beam. Via consideration

of the incoming and outgoing flux (e.g. in [78]), it can be shown that the differential cross

section is the square of this amplitude, meaning that

dσ

dΩ(θ) = |f(θ)|2

(2.2)

The boundary conditions of scattering are such that at infinite distance, the beam and the
target do not interact (V ≈ 0). So, the wave-function Ψ is given by the free wave function
φ where [E − ˆT]φ = 0. As the beam approaches the target, the wave function changes due

to the influence of V . The naive solution to the Schrödinger equation that satisfies these

boundary conditions is given by the Lippmann-Schwinger equation:

Ψ = φ + ˆG+

0 V Ψ

where ˆG+

0 is the Green’s function operator given by

ˆG+
0 =

1

E − T + i


(2.3)

(2.4)

We use the fact that the free solution φ can be given by a plane wave (which we choose to

be in the z direction), and then take the position representation of the Lippmann-Schwinger

Equation to get

Ψ( ~R) =D ~R

(cid:12)(cid:12)(cid:12)ΨE = eikz +D ~R
Z

= eikz +

(cid:12)(cid:12)(cid:12) ˆG+
0 V ΨE
0 ( ~R, ~R0)V ( ~R0)Ψ( ~R0)

d ~R0G+

(2.5)

25

Using contour integration (e.g. in [79]), we can find the following expression for the Green’s

function:

Now we plug in Equation 2.6 into Equation 2.5 to get

0 ( ~R, ~R0) = − µ
G+
2π¯h2

Z

Ψ( ~R) = eikz − µ
2π¯h2

d ~R0 e

ik
e

(cid:12)(cid:12)(cid:12) ~R− ~R0(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ~R − ~R0(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ~R− ~R0(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ~R − ~R0(cid:12)(cid:12)(cid:12) V ( ~R0)Ψ( ~R0)

ik

(2.6)

(2.7)

(cid:12)(cid:12)(cid:12)
We are concerned with the asymptotic behavior of this wave function, so we let R = (cid:12)(cid:12)(cid:12) ~R
go to infinity to evaluate (cid:12)(cid:12)(cid:12) ~R − ~R0(cid:12)(cid:12)(cid:12) by ignoring R02 terms and then applying the binomial

approximation:

(cid:12)(cid:12)(cid:12) ~R − ~R0(cid:12)(cid:12)(cid:12) =

lim
R→∞

q

q
( ~R − ~R0)2 =
vuut1 − 2

ˆR · ~R0

R

≈ R

R2 + R02 − 2 ~R · ~R0 = R
(cid:18)

(cid:19)

1 − ˆR · ~R0

≈ R

= R − ˆR · ~R0

R

vuut1 − (2 ~R − ~R0)

R2

(2.8)

Similarly, we can also neglect ~R0 in the denominator of the Green’s function expression.
Plugging into Equation 2.7 yields

R→∞ Ψ( ~R) = eikz − µ
lim
2π¯h2

eikR
R

Z

d ~R0e−ik ˆR· ~R0

V ( ~R0)Ψ( ~R0)

(2.9)

By comparing Equation 2.9 to Equation 2.1, we can see that the scattering amplitude

f(θ) can be related to the potential V via

Z

f(θ) ∝

d ~R0e−ik ˆR· ~R0

V ( ~R0)Ψ( ~R0) = hφ∗|V |Ψi

(2.10)

26

For convenience, we can define the quantity on the right of Equation 2.10 as the transition

matrix (T-matrix)

T = − 2µ
¯h2k

hφ∗|V |Ψi

(2.11)

where φ∗ denotes the complex conjugate of the plane wave φ. Using the T-matrix and
scattering amplitude, we now have a way to directly calculate the differential cross-section

for a given potential V .

2.1.2 The Distorted Wave Born Approximation

Let V be given by two separate components so that V = U1 + U2, where U1 is larger. Then
consider separately the full case with V = U1 + U2, the free field case with no potential at
all, and the distorted case with the potential given by only U1. The associated Schrödinger
equations and wave functions are given by

[E − ˆT]φ = 0
[E − ˆT − U1]χ = 0
[E − ˆT − U1 − U2]Ψ = 0

As discussed above, with no potential the wave function is simply given by φ. The

Lippmann-Schwinger equations for the distorted and full case, respectively, are

χ = φ + ˆG+

0 U1χ

Ψ = φ + ˆG+

0 (U1 + U2)Ψ

(2.12)

We would like to derive a T-matrix expression to understand the difference between U1
scattering and U1 + U2 scattering. Plugging in U1 + U2 to Equation 2.11 gives T(1+2) =

27

R [φ(U1 + U2)Ψ]dR, from which follows

− 2µ
¯h2k

−¯h2k
2µ

T(1+2) =

Z (cid:20)
χ(U1 + U2)Ψ − ( ˆG+
Z (cid:20)

(cid:21)
φU1χ + χU2Ψ

=
= hφ∗|U1|χi + hχ∗|U2|Ψi

dR

(cid:21)
0 U1χ)(U1 + U2)Ψ

dR

(2.13)

which we can equivalently write as the two-potential formula:

T(1+2) = T(1) + T2(1)

(2.14)

where T2(1) = − 2µ
¯h2k

hχ∗|U2|Ψi refers to the T-matrix contribution from U2 coupling with
U1. Equation 2.14 shows that to calculate scattering due to U1 + U2, one can calculate the
effect of the dominant potential U1 and then the effect of U2 on top of U1 (which shows
up via the distorted wave-function χ in T2(1)). We can rewrite the Lippmann-Schwinger

equation for the full U1 + U2 case to reflect this:

Ψ = χ + ˆG+

1 U2Ψ

(2.15)

where ˆG+

1 = [E − T − U1 + i
]−1.

Since Ψ is on both sides of Equation 2.15, we can iterate this implicit equation to yield

a Born series that converges for a weak enough U2:

Ψ = χ + ˆG+

1 U2[χ + ˆG+

1 U2[χ + ˆG+

1 U2[. . .]]]

(2.16)

Now, we can plug Eq. 2.16 into Eq. 2.14 and truncate after the first term, resulting in the

28

Figure 2.2: Coordinates for a single-neutron transfer reaction. Based on Figure 4.3 in [77].

distorted-wave Born approximation (DWBA) given by

TDWBA = T(1) − 2µ
¯h2k

hχ∗|U2|χi

(2.17)

The DWBA treats U2 as a perturbation on U1. In the case of elastic scattering, U1 is the
long-range Coulomb interaction and U2 is a short-range nuclear interaction. Equation 2.17
is linear in U2, which can be interpreted as corresponding to a single reaction “step,” and is
therefore called the first-order DWBA. Some reactions call for two or more steps, in which

case more terms from Equation 2.16 are included in the truncation.

The above considerations on elastic scattering can be generalized to the three-body trans-

fer reaction system. Figure 2.2 shows the three-body system of interest for the reaction

A(p, d)B: a proton (p), a neutron (n), and a nucleus B, treated as a structure-less core.

A consists of the neutron in a bound state with B. We can write the Hamiltonian using

coordinates (defined in Fig. 2.2) chosen to represent the three-body system either before

29

(the prior form) or after (post) the transfer occurs:

H = Hprior = TRpA
= Hpost = TRdB

+ Ui(RpA) + HA(rnB) + Vi
+ Uf(RdB) + Hd(rnp) + Vf

(2.18)

where TRpA,RdB
are kinetic energies, Ui,f are the entrance and exit channel potentials, HA,d
are the internal Hamiltonians of A and the deuteron, and Vi,f are the interaction terms given
by

Vi = Vnp(rnp) + UpB(RpB) − Ui(RpA)
Vf = VA(rnB) + UpB(RpB) − Uf(RdB)

(2.19)

The post and prior forms provide equivalent descriptions of the three-body system, so for

convenience we choose to work in the prior form. This has the advantage that in the in-
teraction term Vi, the optical potential between p and A is most likely very similar to the
optical potential between p and B, and therefore UpB(RpB) − Ui(RpA) ≈ 0 (for all but
light targets). Therefore the interaction term is simply given by the relatively well-known

neutron-proton interaction term Vnp(rnp).

Equation 2.17 can apply to transfer reactions with a few minor modifications. First,

the distorting potential U1 is typically chosen to be a spherical potential that describes the
elastic scattering between projectile and target. Since no angular momentum is exchanged,

this potential cannot induce a nucleon transfer. This means that the T-matrix contribution
due to U1 (T(1) in Equation 2.17) is 0, simplifying the DWBA expression. Secondly, the
outgoing reaction products of a transfer reaction are by definition different from the incoming

reactants. The entrance and exit channels are characterized by particular combinations of

30

mass and quantum numbers. We express the T-matrix from an entrance channel α to an

exit channel β by

TDWBA
αβ

= − 2µ
¯h2k

D

χ∗
αφpφA

(cid:12)(cid:12)(cid:12)Vnp

(cid:12)(cid:12)(cid:12)φdφBχβ

E

(2.20)

where φ are the internal wave functions of the incoming and outgoing particles for the

A(p, d)B reaction.

2.1.3 The Adiabatic Distorted Wave Approximation

The DWBA framework as described above assumes that the outgoing deuteron from a (p, d)

reaction remains intact as it travels away from the target. In reality, the breakup energy

of the deuteron is quite low (2.224 MeV), so it will break apart easily in the field of the

target nucleus. This clearly has a substantial effect on the measured reaction cross-section.

Johnson and Soper modified the DWBA in order to take deuteron breakup into account

[80] by making the Adiabatic Distorted Wave Approximation (ADWA): the internal motion

of the neutron and proton in the deuteron is slow compared to the motion of the deuteron

center of mass. Therefore, the full three-body wavefunction for the n + p + target system

is only needed within the small range of the neutron-proton interaction Vnp. In the limit of

a zero-range potential, the interaction between the deuteron and the target can be simply

modeled as a combination of target-nucleon interactions, where the nucleons equally share

the deuteron energy between them. The deuteron adiabatic potential is defined by

Z (cid:26)

(cid:27)
2~r) + Up( ~R − 1
2~r)

Un( ~R + 1

Ud( ~R) = 1
D0

Vnp(~r)φd(~r)d~r

(2.21)

31

where Un and Up are the neutron and proton optical potentials at half the deuteron energy
[81], Vnp is the neutron-proton interaction potential, φd is the deuteron wave function, ~R
is the coordinate of the deuteron center of mass, and ~r is the relative coordinate between

the neutron and the proton [80]. D0 is the strength of the neutron-proton interaction. In
addition to the explicit treatment of deuteron breakup, another advantage to ADWA is that

it uses nucleon optical potentials, which are typically better constrained experimentally than

deuteron optical potentials (see Section 2.1.6 for further discussion of optical potentials).

This approximation is valid as long as the deuteron energy is much larger than the

binding energy of the deuteron. The finite-range ADWA has been benchmarked against the

exact (and more difficult to solve) three-body Faddeev calculations for simple systems, and

is within 10% agreement [82].

2.1.4 Local Energy Approximation

We have assumed, so far, that the neutron-proton interaction is zero-range. Given that

the real neutron-proton interaction has a finite range, using the zero-range potential risks

overestimating the transfer contribution from the interior of the nucleus. For a small finite-

range, we can apply a first-order correction to the zero-range strength called the local energy

approximation (LEA) [83]. The LEA instructs us to simply replace the zero-range strength
D0 with an effective zero-range strength D = (1 + k2
b β2)D0, where kb is the deuteron wave
number and β is a parameter that defines the finite-range effective radius.

32

2.1.5 Nonlocality

All potentials discussed so far have depended only on the distance in between the two inter-

acting particles. Reality is far messier. A potential is considered nonlocal if it depends on

the value of the potential and scattering wave function at all points in space. Nonlocality

can play a significant role in transfer reactions, due to momentum dependence and reaction

channels coupling to other degrees of freedom. Exact three-body calculations for light sys-

tems show that nonlocality sometimes (but not always) significantly affects the differential

cross section [84].

There are two angles by which to approach the problem of nonlocality. One can either

explicitly calculate the nonlocal potential (which is difficult), or adjust a local potential in

order to effectively approximate the behavior of the fully nonlocal one (which is much easier).

Historically, the latter approach has been more common.

For example, Perey and Buck [85] modeled nonlocality with a simple Gaussian adjustment

to the local potential, where the Gaussian parameters are fit to experimental data. Since the

Perey-Buck approach is easy to implement, it is a common method of including nonlocality

in transfer calculations. Of particular importance to this work is that Lee et al. used

Perey-Buck in calculating ADWA (p, d) cross sections on argon isotopes at 33 MeV/u [26,

55, 56]. More recent ADWA studies by the University of Surrey group suggest calculating

the local nucleon potentials with large and positive energy shifts (around 40 MeV) from the

standard prescription of using half the deuteron energy [65, 66]. This shift is meant to take

into account the large relative kinetic energy between the neutron and proton induced by

their short-range interaction in the deuteron while still using well-known local potentials. A

follow-up study suggested also including the deuteron D-wave, increasing the average energy

33

shift to 70 MeV [86, 87].

In light of the popularity of these simplified methods for treating nonlocality, some have

taken a more rigorous approach to identify potential problems. Several studies have pointed

out that cross sections calculated with Perey-Buck can differ substantially from those calcu-

lated with more explicitly nonlocal theoretical models by 20% or more [88, 89, 90, 91, 92].

Another recent paper shows that Perey-Buck requires nontrivial energy dependence to ac-

curately describe elastic scattering across a wide energy range [93]. The more contemporary

energy shift method of Timofeyuk and Johnson has also faced criticism. When compared to

the explicit nonlocal formalism, the energy shift method does not reliably improve (quanti-

tatively or qualitatively) the pure local calculation [90].

This is a fascinating and active area of research. Since in reality these many-body systems

are inherently nonlocal, much of the theoretical work thus far has justifiably focused on

probing the details of the underlying physics rather than developing a widely-applicable

experimental tool. At this point, there is no settled upon solution for how to include nonlocal

effects into single-nucleon transfer reactions, and the importance of nonlocality (as well as

how to properly account for it) varies from system to system. Deeper investigation into the

effects of nonlocality on the specific argon isotopes that we focus on here is beyond the scope

of this thesis. So, for the purpose of comparison to the low-energy transfer work of [55, 56],

we will use the standard Perey-Buck nonlocality.

2.1.6 Optical Potentials

Above, we outlined how to calculate the differential cross-section using quantities that de-

pend primarily on the potentials. As such, the choice for which potentials to use can have

a significant impact on the calculated cross section. The real, many-body potentials are ex-

34

tremely complicated and beyond the reach of current theoretical tools: the nucleon-nucleus

optical potential reduces this difficulty to scattering of a single particle on a complex one-

body potential. Optical potentials use effective imaginary components to remove flux due to

reactions not included in the model, analogous to the imaginary potentials used to account

for absorption as light passes through a cloudy medium.

Ideally, an optical potential for a given nucleus-nucleus interaction can be fit directly to

elastic scattering data for that particular system and beam energy. Unfortunately such data

does not typically exist for rare isotope beams. Below, we describe global optical models and

microscopic optical models, the two most common approaches in the face of this obstacle.

2.1.6.1 Global Optical Potentials

A global optical model is a parameterized, simultaneous fit across many angular distributions

for a wide range of target nuclei with a given projectile. This global model can then be used

to interpolate (or even extrapolate) to the case of interest. The nucleon-nucleus global optical

potential is given by

U(r) = −Vrfws(r, RV , aV ) − iWrfws(r, RW , aW ) + 4iWsaw

fws(r, RW , aW )

d
dr

+2(Vso + iWso)

fws(r, Rso, aso)L · σ

(cid:18)1

d
dr

r

(cid:19)

(2.22)

+ VC
(cid:21)−1

. V and W

(cid:20)

1 + e

r−R
a

where fws is the Woods-Saxon potential given by fws(r, R, a) =

refer to real and imaginary potentials, respectively. Potentials due to the interior volume

(given by subscript r) as well as the exterior surface (s) are both included, as well as a

spin-orbit potential (so) and a Coulomb term (c) that treats the nucleus as a homogenous

sphere of charge. R is the nuclear radius parameter and a indicates the diffuseness of the

35

nuclear surface. L is the orbital angular momentum of the projectile relative to the target,

and σ is the spin operator.

The global potential used in this work is the Chapel-Hill 89 (CH89) potential developed

by Varner, et al. [94]. CH89 is fit to nearly 300 proton and neutron differential cross sections
with A = 40 − 209 and E = 10 − 65 MeV, and has been shown to describe single-nucleon

transfer data better than other global models on the market [95, 26].

2.1.6.2 Microscopic Optical Potentials

In a microscopic optical model, nucleon density distributions (known from electron scattering

experiments or nuclear structure calculations) are folded together via a convolution integral

to generate the optical potential. As opposed to the global optical model approach in which

data from a wide range of nuclei is fit, a microscopic optical model leverages nuclear structure

theory to generate a potential specific to a single nucleus given a particular nucleon-nucleon

interaction. Suppose a structureless projectile (e.g. a proton) is approaching a target with

many nucleons. If we know the interaction potential between the projectile and individual

nucleons in the target, we can estimate the overall potential via the following convolution

integral:

Z

U(r) =

drtVpN(r − rt)ρt(rt)

(2.23)

where r is the position of the projectile relative to the target, rt is the internal position
coordinate over the target, VpN is the potential between the projectile and each nucleon of
the target, and ρt is the target density.

The microscopic potential used in this work was developed by Jeukenne, Lejeune, and

Mahaux (JLM) [96, 97]. The JLM model uses Reid’s hard core nucleon-nucleon interaction

36

[98] in the many-body framework of the Brueckner-Hartree-Fock approximation. Reasonable

agreement between theoretical and empirical cross-sections [97, 99, 100, 101, 102] shows that
the JLM is valid for A = 12 − 208 and energies up to 160 MeV.

As shown in Equation 2.23, nucleon densities are required as input to calculate the JLM

potential. In this case, these densities are taken from Hartree-Fock calculations with the SkX

parameter set [103] that accounts for a wide range of experimental observables [104, 105,

106]. When using the JLM potential, we adjust the radius parameter r0 to reproduce the
mean squared radius of the transferred neutron orbital obtained via Hartree-Fock calculation

[39].

2.1.6.3 Optical Potential Uncertainty

Earlier in this chapter, the theoretical differential cross section was connected directly to

the interactions in the three-body system. Different potentials, therefore, yield different

cross sections. This is problematic given that in most cases the relevant potentials are not

directly constrained to data from the system of interest, and are instead obtained from

a phenomenological fit (CH89) or from a microscopic calculation using nuclear densities

(JLM). In other words, there is ample room for theoretical error. As an example, Figure

2.3 shows the differential cross section for a transfer reaction calculated with CH89 (black)

and JLM (red). Although the shapes of the calculations match, the magnitude of the JLM

cross section is slightly higher due to larger neutron bound-state wave functions [26, 95,

39]. More recent work on uncertainty quantification suggests that theoretical uncertainties

due to the optical potential are often underestimated [107]. The validity of the ADWA for

transfer reactions at high energies has even been called into question in the past [108, 109].

How can meaningful physics information be extracted using transfer reactions with so much

37

Figure 2.3: Differential cross section calculations for 46Ar(p, d)45Arg.s. with CH89 (black)
and JLM (red) optical potentials.

theoretical ambiguity?

Lee, Tsang, and Lynch directly addressed this problem by studying a wide range of differ-

ential cross sections and extracting spectroscopic factors using a single systematic approach

[36, 26, 38, 39]. For a given individual cross section or spectroscopic factor, one can expect

the theoretical error to be quite large. However, within a consistent framework, the rela-

tive trends across a range of isotopes can provide insight that is often independent of the

choice in optical potential. This point is exemplified in Figure 2.4, which shows (in red)

the reduction factors of the previous argon transfer study calculated with both the CH89

and JLM potentials. The reduction factors differ in magnitude, but the trend shown by the

CH89 reduction factors is the same as for the JLM: both choices of optical potential show

substantial disagreement with the knockout reaction results (in blue), and so in this sense

the large error bars for a specific SF are not of crucial importance.

38

Figure 2.4: Spectroscopic factors extracted from transfer (red) and knockout (blue) data
along the argon isotopic chain. SFs extracted with two different optical models are shown:
the JLM (solid) and the CH89 (open). Data taken from [55] and [52].

39

2.2 Methodology

To extract spectroscopic factors, we will measure the differential cross section of (p, d) transfer

reactions in inverse kinematics, and then compare this experimental data with the theoreti-

cally calculated cross section. The ratio of the experimental cross section to the theoretical

cross section gives the spectroscopic factor.

The T-matrix amplitude expressed in Equation 2.20 can be expressed in terms of spec-

troscopic factors. First, the matrix element D

φα

(cid:12)(cid:12)(cid:12)Vnp

(cid:12)(cid:12)(cid:12)φβ

E can be expanded using the nuclear

overlap integral that defines the SF:

hφA|φBi = X

hJAjMAm|JBMBiq

SnljΦnljm

nljm

(2.24)

where hJAjMAm|JBMBi is a Clebsch-Gordan coefficient, Snlj is the spectroscopic factor
(SF), and Φnljm is the single-particle bound-state wave function for the A = B + n system.
Rearranging the integration variables and plugging Equation 2.24 into Equation 2.20 yields

∝ X

q

nljm

TDWBA
αβ

D
χ∗
αΦnljm

(cid:12)(cid:12)(cid:12)D

φp

(cid:12)(cid:12)(cid:12)Vnp

(cid:12)(cid:12)(cid:12)φn

E(cid:12)(cid:12)(cid:12)φβχβ

E

Snlj

(2.25)

Since the square of transition amplitude given by the T-matrix is proportional to the differ-

ential cross section (as shown in Equations 2.10 and 2.11), Equation 2.25 gives us a direct

connection between SF and differential cross section. For a theoretical cross section, the

SF is typically assigned to its maximum value (1 or 2j + 1 depending on normalization).

Therefore, the ratio of the experimentally measured cross section to the theoretical cross

section will equal the ratio of the experimental SF to the maximal SF. The shape of the

cross section is dominated by the angular momentum exchanged (which is known), so we

40

can have confidence that the shape of the theoretical cross section will match the shape of

the data.

2.2.1 The TWOFNR Reaction Code

Our approach follows the one prescribed by Lee, which is described in detail in [26]. The

(p, d) reaction calculations in this work were performed with the University of Surrey version

of TWOFNR, a direct reaction code for calculating DWBA with finite-range [110].

All calculations were performed using the finite-range ADWA described by Johnson and

Soper [80] using as input experimentally determined angular momenta and reaction Q-values.

Calculations were carried out over 30 fm integration ranges with 0.1 fm steps and 70 partial

waves. The bound-state wave function potential was a Woods-Saxon shape, in which the

depths of the central potential wells were adjusted to reproduce experimentally determined

Q-values. For the CH89 calculations, we used typical values for the radius (1.25 fm) and

diffuseness (0.65 fm) parameters. For the JLM calculations, we followed the procedure

described in [39] to tune the radius parameter in order to reproduce the mean-squared

radius of the neutron orbital. In every case, we use the local energy approximation (with

β = 0.7457 fm) to account for finite-range effects in the neutron-proton interaction. The
Reid soft-core 3S1−3 D1 neutron-proton interaction was chosen with the zero-range strength
0 = 15006.25 MeV2 fm3 and the range β = 0.7457 fm. Nonlocality corrections for both the
D2
proton and deuteron channels were included, with ranges of 0.85 fm and 0.54 fm, respectively.

In this chapter we developed the theoretical machinery needed to extract spectroscopic

factors from transfer reaction data. Next, we will discuss the transfer reaction experiment

itself.

41

Chapter 3

Experimental Methods

One must imagine Sisyphus happy.

Albert Camus

Measurements of the transfer reactions 46Ar(p, d)45Ar and 34Ar(p, d)33Ar were performed

with a beam energy of 70 MeV/u at the National Superconducting Cyclotron Laboratory

(NSCL) at Michigan State University. This chapter discusses the experimental methods used

in this measurement. Section 3.1 describes the method of measurement and motivation for

the experimental design. Section 3.2 discusses the production of the rare isotope beams used

for this experiment. Sections 3.3, 3.4, and 3.5 include descriptions of the High Resolution

Array (HiRA), the S800 Spectrograph, and the Micro-Channel Plate Detectors (MCPs),

respectively. In Section 3.6, we discuss the precise position characterization of the setup.

3.1 Experimental Design

Transfer reactions have been a crucial tool to study nuclear structure for many decades. In

the 1950s, a typical transfer measurement consisted of an accelerated beam of light particles

(usually an isotope of hydrogen) impinging upon a fixed “target” composed of stable, heavy

isotopes [42]. These normal kinematic measurements require that the target isotope have a

relatively long half-life and that it can be easily produced in large quantities. Exotic isotopes

satisfy neither of these two criteria, and so we cannot study them using normal kinematics.

42

Therefore, contemporary transfer reaction measurements are done in inverse kinematics in

which the roles are reversed: an accelerated beam of heavy (usually radioactive) nuclei

impinges upon a much lighter target.

In the center-of-mass frame of the system, inverse

kinematics is equivalent to the traditional normal kinematics. Inverse kinematics not only

opens up large swaths of the nuclear chart for study with transfer reactions, but also ensures

that all fragments originating from the incoming beam escape the target, due to their high

momenta.

Since the argon isotopes of interest in this work are unstable, we measured the trans-
fer reactions 46Ar(p, d)45Ar and 34Ar(p, d)33Ar in inverse kinematics. The radioactive, 70

MeV/u argon beams were produced at the Coupled Cyclotron Facility (CCF) at the NSCL

as described in Section 3.2. This beam energy was chosen to match that of the corresponding

knockout measurement (see [59] and Chapter 1). For each reaction, a (CH2)n polyethylene
target provided the proton. Although this plastic target is not purely composed of protons,

the background from reactions on carbon is low and in fact can be measured (and then

subtracted) using a pure carbon target. Furthermore, polyethylene can be obtained easily at

a wide range of thicknesses and with good uniformity, in stark contrast to the considerable

technical challenges involved in using a liquid or gaseous pure hydrogen target.

Figure 3.1 shows a schematic of the experimental setup. The observable of interest in this

experiment is the differential cross section of the transfer reaction from which a spectroscopic

factor can be extracted, as discussed in Chapter 2. Measuring a cross section corresponding

to a particular reaction requires knowledge of both the outgoing flux from that reaction (i.e.

the number of times that reaction occurs) as well as the incoming flux into the target (i.e.

the number of incoming beam particles). The former quantity we measure using the High

Resolution Array (HiRA) for detecting the outgoing deuterons and the S800 Spectrograph for

43

Figure 3.1: Cartoon of the experimental setup. The incoming beam (either 34Ar or 46Ar) was
produced with the Coupled Cyclotron Facility (CCF) at the NSCL. A polyethylene target
was inserted into the beamline to measure (p, d) reactions on the incoming beam. Upstream
of the target, two Microchannel Plates (MCPs) were placed to normalize the cross-section
and also track the beam position on target. The outgoing heavy recoil from the transfer
reaction (either 33Ar or 45Ar) was detected and identified in the S800 Spectrograph. The
outgoing deuteron was detected and identified with the High Resolution Array (HiRA).

detecting the outgoing heavy argon recoils in coincidence. The latter quantity we determine

using Micro-Channel Plate detectors (MCPs), which measure the incoming beam particles

before they impinge upon the target. The MCPs also enable event-by-event beam tracking,

which provides a valuable boost to the angular resolution of the emitted particles detected

in HiRA. The MCPs and HiRA sat with the reaction target in a vacuum chamber at the

entrance of the S800. A gap was left in HiRA at very forward angles to allow the heavy

recoil and unreacted beam to pass through the array into the S800. Since both the deuteron

as well as the appropriate heavy recoil must be detected to reconstruct the transfer reaction

of interest, the key trigger condition of the data acquisition was a coincidence between HiRA

and the S800.

44

Figure 3.2: Kinematic curves for 46Ar(p, d)45Ar for different excitation energies of the out-
going recoil 45Ar. This energy-angle relationship of the emitted deuteron can be measured
with HiRA.

Due to the straightforward two-body kinematics of the outgoing transfer reaction prod-
ucts, the energy and angle of the deuteron uniquely determine the final state in the 45Ar

recoil. For example, Figure 3.2 shows the calculated relationship between deuteron energy
and angle for 46Ar(p, d) transfer into different states of 45Ar. Clearly, different final states of

the heavy recoil can be distinguished by measuring the emitted deuteron. By transforming

the deuteron energy from the laboratory frame to the center-of-mass frame, we can calculate

the excitation energy spectrum for the Ar recoil. We then use the number of counts in each

peak for a given angular range to calculate a differential cross section for a transfer reaction

into the corresponding state in the argon recoil.

Nuclear physics experiments like this one often feature multiple detectors and a high

quantity of data being generated from each of those detectors every second. Therefore, a

well-designed data acquisition system (DAQ) is critical in order to properly coordinate and

record the data. The vast majority of this data is uninteresting, so ideally the DAQ will

45

Deuteron Lab Angle (Degrees)010203040Deuteron Lab Energy (MeV)141618202224262830 g.s.Ar45Ar(p,d) 46  1.7Ar45Ar(p,d) 46  2.4Ar45Ar(p,d) 46  3.95Ar45Ar(p,d) 46 recognize in real-time what data is important, and what data is not worth recording. The

trigger condition determines when the DAQ does or does not record data. In this experiment,

the most crucial ingredient of the trigger was a coincidence between HiRA and the S800, as

discussed above, while the MCPs ran as slaves (i.e. signals from the MCPs would be recorded

when the trigger condition was satisfied). An electronic pulser was included in the trigger

to properly track the dead time of the silicon detectors in HiRA (more details are provided

in Chapter 5). The DAQ also recorded downscaled counts from the HiRA and MCPs. A

CAEN V830 scaler continuously counted logic signals for a variety of electronics circuits.

Each of the detector electronics systems fed into the DAQ (written with the NSCLDAQ

software suite [111]) that managed and merged the data flows. NSCLDAQ employs a ring

buffer structure that allows for fast and flexible data transfer. The SpecTcl C++ framework

interfaced with the DAQ allowed for online monitoring of the data in real-time [112].

As described in Chapter 1, the vital information this work seeks to study is the comparison
between two transfer reactions: 46Ar(p, d) and 34Ar(p, d). Maximizing the validity of this

comparison requires making systematic uncertainties for these measurements as similar as

possible. Therefore, the most important principle of the experiment design is that it allows

each of these two reactions to be studied, one immediately after the other, with an identical

setup. The experimental systems detailed below have the necessary flexibility to achieve this

goal.

3.2 Beam Production

The NSCL’s Coupled Cyclotron Facility (CCF) consists of two superconducting cyclotrons

that can accelerate a wide range of stable primary beams [113, 114, 115] to semi-relativistic

46

Figure 3.3: Schematic of the Coupled Cyclotron Facility, showing ion sources, the K500
cyclotron, the K1200 cyclotron, and the A1900 fragment separator [117].

speeds up to 0.5c. The accelerated primary beam then impinges on a beryllium target to

produce a variety of stable and unstable isotopes via fragmentation. These isotopes then

proceed to the A1900 Fragment Separator, where they can be filtered by mass and charge

[116]. The resulting secondary beam is then delivered to one of many experimental areas for

further study. A schematic of the CCF is shown in Figure 3.3.

3.2.1 Primary Beams

Every beam accelerated at the NSCL begins in an ion source that removes electrons from

stable atoms to provide charged stable ions to the cyclotrons. Ion production relies on the

electron cyclotron resonance (ECR) phenomenon, in which electrons rotating at cyclotron

frequency in a stable magnetic field are heated by co-circulating electric fields. The electrons

heat into a plasma, and then collide with atoms to knock out other electrons from their

atomic orbitals. Eventually, cycles of collisions produce highly charged ions. These ions can

47

then be extracted electrostatically. The NSCL has two ion sources: the ARTEMIS ECR

source and the SUSI ECR source.

A cyclotron uses an oscillating electric field and a static magnetic field to accelerate

charged particles. As the name suggests, the CCF features two cyclotrons: the K500 and the

K1200. The number in the name of each accelerator simply refers to the theoretical maximum

proton energy (in MeV) that each cyclotron can achieve. Particles are first accelerated in

the lower-energy K500 cyclotron and then further accelerated in the K1200.

The ions extracted from the ion source are injected into the K500 cyclotron at the center,

and begin moving in a circular trajectory due to the magnetic field. A negative electric

potential is alternated on the three metal electrodes (referred to as “dees”) to increase the

kinetic energy of the particles. As the energy increases, the radius of the circular path traced

out by the ion increases. Eventually, the accelerated ion reaches the edge of the cyclotron

and exits. The ions, now accelerated to an intermediate energy, then pass through a thin

carbon foil to strip off more electrons and become more highly charged. To achieve optimum

acceleration, the charge-to-mass ratio of the ion must be maximized, and therefore high

charge states (full ionization if possible) are ideal. After going through the foil, the ions

enter the K1200 cyclotron where they are further accelerated. The K1200 is more powerful

than the K500 but operates via the same principles that are described above.

The ions, now referred to collectively as the primary beam, then impinge on a beryllium

production target. Although many ions in the beam pass through the target, some collide

with beryllium nuclei and undergo fragmentation. A wide range of isotopes are created

in this fragmentation process. Most of the isotopes produced then proceed to the A1900

fragment separator for purification.

In this experiment, we used two primary beams. For the first beam, 36Ar ions (with a

48

7+ charge state) were extracted from the SUSI ion source and then accelerated in the K500

to 13.06 MeV per nucleon. The ions were then stripped fully (to an 18+ charge state) and

injected into the K1200 where they were accelerated to 150 MeV per nucleon. For the second
beam, 48Ca ions (with an 8+ charge state) were extracted from the ARTEMIS ion source

and then accelerated in the K500 to 12.28 MeV per nucleon. As in the case of the first beam,

the ions were stripped fully (this time to a 20+ charge state). Then, they were accelerated

to 140 MeV per nucleon in the K1200.

3.2.2 Secondary Beams

The fragmentation process produces many different isotopes. However, most experiments

require a secondary beam made up of a single isotope. The role of the A1900 fragment

separator is to filter away the unreacted primary beam and any unwanted fragments, leaving

behind only the nuclei of interest. This isotope selection process is quite powerful, and serves

as a vital component to virtually every single experiment at the CCF.

The A1900 (shown in Fig. 3.3) consists of four 45◦ superconducting dipole magnets and
24 superconducting quadrupole magnets in eight cryostats. Filtration occurs in two stages:

in the initial stage, the first two dipole magnets after the production target select particles

by dispersing them according to their magnetic rigidity Bρ:

Bρ = p
q

∝ A
Z

v

(3.1)

where p refers to the relativistic momentum of the particle, v is the velocity, q is the charge,

B is the magnitude of the magnetic field, and ρ is the bending radius of the dipole. A and Z

refer to the mass and atomic numbers of the particle, respectively. Only particles within a

49

given range of magnetic rigidity (i.e. within a given range of A

Z ) pass through both dipoles.
In the second stage, an aluminum wedge is placed at the Intermediate Image position (see

Figure 3.3). The amount of energy loss in a medium by a given particle depends strongly

on its charge, so the wedge induces a velocity shift that depends on the Z of the fragment.

Following the wedge are another two dipole magnets that again disperse the beam according

to Bρ. At various points along the separator, slits are used to block out fragments that do

not match the set rigidity. The resulting isotope beam (known as the secondary beam) is

then transported to an experimental hall.

We used two secondary beams in this experiment. First, the 36Ar primary beam gen-
erated a 70 MeV/u 34Ar secondary beam (94% pure). Immediately afterwards, the 48Ca
primary beam generated a 70 MeV/u 46Ar secondary beam (more than 99% pure). In both

cases, the beam was transported to the reaction target and detector setup in the S3 vault at

the NSCL.

3.3 HiRA

The High Resolution Array (HiRA) is a modular array of charged particle detectors developed

at Michigan State University in collaboration with Washington University, Indiana Univer-

sity, Southern Illinois University Edwardsville, and the Istituto Nazionale di Fisica Nucleare

[118]. HiRA is composed of individual “telescopes,” each containing a 65-um, single-sided,

32-strip silicon detector (referred to as the “DE” detector), a 1500-um, double-sided, 32-

strip silicon detector (referred to as the “E” detector), and an array of four 3.9-cm-thick CsI

scintillator crystals, with each crystal spanning roughly a quadrant of the preceding silicon

detectors. Figure 3.4 shows a cartoon with each of these detectors. A thin Mylar foil in front

50

Figure 3.4: Cartoon showing the radiation detectors inside of each HiRA telescope.

of the DE on each telescope completes a Faraday cage around the detectors within, helping

to reduce noise. Depending on the needs of a given measurement, HiRA can be arranged in

different geometrical configurations. This flexibility enables HiRA to address a wide variety

of topics within the realm of nuclear physics, including direct reactions for nuclear structure,

nuclear astrophysics, exotic two-proton decay, and the nuclear symmetry energy [119, 120,

121, 122].

HiRA has two key characteristics crucial for measuring the differential cross sections of

interest in this experiment: excellent position resolution and unambiguous particle identifi-

cation.

An accurate and precise measurement of the angle of the emitted deuteron is crucial

for properly probing the energy-angle kinematic relationship discussed in Section 3.1. To

51

achieve the necessary precision, HiRA features the double-sided silicon detector referred to

as the E detector. The 32 strips on the front of the E (“EF”) and the 32 strips on the back

(“EB”) are perpendicular to each other. The coincidence between one strip on the EF and

one strip on the EB corresponds to a particle going through a 1.95 mm by 1.95 mm “pixel” at
the intersection of the two strips. Each pixel corresponds to an angular resolution of ±0.16

degrees at 35 cm from the target.

To study (p, d) transfer, we need to unambiguously identify light charged-particles. HiRA’s

crucially important particle identification (PID) feature emerges from the comparison of en-

ergies measured in different detectors within a telescope. To show why, we consider the

Bethe-Bloch equation which relates the energy deposited by a particle with atomic number

Z and mass number A in a given medium to the total energy of the particle E:

− dE
dx

∝ Z2A

E

(3.2)

Therefore, the ratio of the energy that a charged particle deposits in a “thin” detector to the

energy measured in a corresponding “thick” detector (where the charged particle deposits

the rest of its energy) allows for unique determination of both Z and A. HiRA provides two

separate stages of PID. In one case the DE acts as the “thin” detector which is compared to

the “thick” E detector. In the other case, when a particle is energetic enough to pass through

the E, the E acts as the “thin” detector with one of the CsI crystals as the corresponding

“thick” detector. These two PID stages allow for a relatively large dynamic range of particle

identification. For instance, HiRA can properly identify deuterons with energies from ~3

MeV to ~150 MeV.

While some HiRA experiments only require one stage of PID, this transfer reaction

52

measurement needs both. For the reactions of interest, the kinematic relationship between

deuteron energy and emitted angle can be calculated unambiguously via straightforward ap-
plication of conservation laws. Figure 3.5 shows this relationship for both 46Ar(p, d)45Arg.s.
and 34Ar(p, d)33Arg.s. at 70 MeV/u, as well as the punch-through energy1 for a deuteron in
the E detector. Below the green line, deuterons will stop in the E detector and therefore we

identify them via the E-DE PID stage, whereas above the green line we use the CsI-E stage.

In order to probe a reasonably large angular range for both reactions of interest with the

same experimental setup (as discussed in Section 3.1), the kinematic curves show that both

stages of PID are necessary.

3.3.1 Silicon Detectors

Each HiRA telescope holds two separate semiconductor detectors made of silicon. The silicon

detectors were manufactured by Micron Semiconductor using the design BB7. The active
surface area is 6.4 cm × 6.4 cm, and the thickness is either 1500 µm (E) or 65 µm (DE). The

DE detector is subdivided into 32 vertical strips, while the E detector has 32 vertical strips

on its front side (EF), and 32 horizontal strips on its back (EB). The pitch of each strip is

1.95 mm for all detectors, with an inter-strip gap of 25 µm for the EF and the DE, and 40

µm for the EB. Figure 3.6 shows an example of an E detector sitting in a partially assembled

HiRA telescope. Each telescope features a slot between the DE and the E detectors to allow

for insertion of a pin source, a small radioactive metal pin used to calibrate the E detector

without having to remove the DE (see Chapter 4 for more detail).

The periodic lattice of a crystalline material generates energy bands along which electrons
1The punch-through energy of a given detector for a given particle refers to the minimum amount of

energy needed for that particle to pass completely through the detector.

53

Figure 3.5: Kinematic curves for 46Ar(p, d)45Arg.s. (in red) and 34Ar(p, d)33Arg.s. (in blue)
at 70 MeV/u. The green dotted line corresponds to the punch-through energy for a deuteron
through the E detector.

54

Lab Angle (degrees)010203040Lab Energy (MeV)5101520253035404550g.s.Ar45Ar(p,d) 46 g.s.Ar33Ar(p,d) 34 d Punchthrough (1.5 mm)Figure 3.6: Photo of a partially assembled HiRA telescope with an E silicon detector. The
front surface of the E detector (EF) is visible at the top of the figure. Wire bonds connect
each individual EF strip to the orange cable running down the left side of the telescope. The
corresponding cable for the back surface (EB) can be seen on the right. The E detector is
held by a G10 fiberglass frame, and this frame is attached with screws to the metal sides of
the telescope. A bundle of four CsI scintillators can be seen underneath the silicon detector.
The DE detector (not shown) is placed above the E and secured in place by screwing its G10
frame to the telescope.

55

in the material can travel. The valence band corresponds to outer-shell electrons in the

lattice atoms, and the higher energy conduction band corresponds to electrons that can

travel freely throughout the lattice. The energy separating these two bands is called the

band gap: electrical insulators typically have band gaps upwards of 5 eV, while electrical

conductors often have no band gap at all (the valence band and conduction band overlap).

A semiconductor, which has electrical conductivity between that of an insulator and that

of a conductor, usually has a band gap of ~1 eV. Using a simple linear relation, the band

gap is related to the ionization energy, or the amount of energy needed to generate a single

electron-hole pair [123]. In the case of silicon at room temperature the band gap is 1.12 eV

and the ionization energy is 3.62 eV [124]. So, if a particle deposits 1 MeV of energy into the
detector, then (1 × 106 eV/3.62 eV per pair) ≈ 2.8 × 105 pairs are generated. By applying

an electric field across the detector, these pairs can be collected to measure the deposited

energy.

The conductivity of a semiconductor can be modified via the introduction of impurities,

or doping. Doping silicon with impurities from group V of the periodic table results in an

n-type material that has an excess of electrons. On the other hand, doping with impurities

from group III results in a p-type material with an excess of holes. Silicon radiation detectors

like the ones used in HiRA typically consist of a p-n junction in which an n-type silicon and

a p-type silicon are grown face-to-face. At this junction, free electrons drift from the n-type

side to the p-type side, creating a region of space charge where charge carriers are depleted.

This depletion region can be expanded by applying bias voltage across the junction. When a

charged particle deposits energy in a fully depleted p-n junction, the generated electron-hole

pairs are immediately swept towards the surfaces of the detector and collected on metal

contacts. Since the number of collected pairs is linear with deposited energy, the detected

56

current provides a measure of the energy.

In HiRA, the electrical signals from the silicon detectors are processed using the HINP16C

application-specific integrated circuit (ASIC) electronics designed at Washington University

and Southern Illinois University Edwardsville [125]. Each HINP16C channel, which corre-

sponds to a single strip of silicon, contains charge-sensitive amplifiers, a pseudo constant

fraction discriminator, a shaping amplifier, a time-to-voltage converter, and associated digi-

tal logic. A single ASIC chip contains 16 individual channels, so one 32-channel detector (EF,

EB, or DE) requires two ASIC chips (usually placed together on a single chipboard). These

chipboards are then placed onto motherboards (each of which can fit at most 16 chipboards)

that communicate with the data acquisition system.

In this experiment, we used 14 telescopes, each with three 32-channel silicon detectors.

We placed three motherboards inside the chamber behind the towers of the HiRA array for

the EF and EB silicon. Signals from the DEs were pre-amplified externally before being sent

to the HINP16C electronics due to high noise associated with the low capacitance of the

thin silicon detector. The DE pre-amplifiers sat on top of the vacuum chamber along with

another HINP16C motherboard.

3.3.2 CsI Detectors

Each HiRA telescope contains an array of four cesium iodide (CsI) thalium-doped scintillators

to detect particles that have enough energy to punch through both layers of silicon. These

3.9-cm-thick CsI crystals can detect deuterons up to 150 MeV in energy. Each crystal has

been cut into a trapezoidal shape, with the front surface being 3.5 cm by 3.5 cm and the

back 3.9 cm by 3.9 cm [118]. In order to assemble a full array of four packed crystals, the

two inner edges of each crystal are cut straight while the two outer edges are cut with a

57

5.3-degree taper. Each crystal is wrapped in cellulose nitrate membrane filter paper and

aluminized Mylar, and also is optically coupled (using BC600 optical cement) to a 1.3-cm

light guide. The light guide is painted with BC600 reflective paint. A 1.8-cm-by-1.8-cm

photodiode sensitive to the wavelength of light produced in the crystal is attached to the

back of each light guide using RTV615 silicon rubber. Figure 3.7 shows drawings of a CsI

bundle.

CsI is a scintillator material, meaning that it emits scintillation light when excited by

ionizing radiation. When a charged particle hits a CsI crystal, electron-hole pairs are created.

The return of an electron to the valence band results in emission of a photon [124]. However,

this process is inefficient for the pure CsI crystal, and the photon is too high in energy

to fall in the visible range.

Impurities (also known as activators) like thalium are added

to the lattice to modify the band structure of the crystal by adding in new energy states

by which electrons can de-excite sequentially to the valence band. This recombination has

higher efficiency than for the pure crystal, and yields visible photons [124]. These photons

then impinge on the photodiode, where they induce electron-hole pairs that are measured
as current. About 6.5 × 104 photons are created per MeV of deposited energy, which is

much less than the number of electron-hole pairs created per MeV in silicon. Having fewer

information carriers corresponds to a higher statistical uncertainty for the CsI compared to

the silicon, and therefore a worse energy resolution.

Signals from the CsI crystals are amplified via a charge sensitive preamplifier board

located at the back of each HiRA telescope. From there, the signals exit the vacuum chamber

and are input into CAMAC shaper/discriminator modules (made by Pico Systems). Output

from these modules is fed into a CAEN V785 ADC and a CAEN 1190 TDC.

58

Figure 3.7: Side-view and front-view drawings of a four-crystal CsI scintillator bundle.

3.3.3 Geometry

The geometry for this experiment was based on the one used in experiment 05133. Figure 3.8

shows a picture of HiRA as it was setup for this experiment. Calculations with TWOFNR

indicate that detecting deuterons at relatively forward angles, roughly in the range of 8

degrees to 40 degrees in the laboratory frame, is sufficient to properly identify states in the

outgoing recoil nucleus. At very forward angles, there is no coverage in order to allow the

recoil nucleus ample space to pass through gap between HiRA telescopes to then proceed

into the S800.

Understanding the geometrical coverage of HiRA is a critical ingredient in extracting the

angular-dependence of the cross-section. In other words, we must know the probability that

a deuteron emitted at a given angle will be detected in HiRA. The geometric efficiency of

HiRA can be studied using Monte Carlo simulations in order to quantify exactly the fraction

of particles emitted at a given angle that hit HiRA.

59

Figure 3.8: Photo of HiRA in the setup for experiment 09084.

We used the program NPTool to calculate the geometric efficiency of the HiRA array.

NPTool is a framework that combines the Geant4 simulation code with the ROOT analysis

package to simulate arbitrary experimental geometries with a wide array of different detector

types [126]. The precise locations of the HiRA telescopes (see Section 3.6 for a description of

the measurement) were input into the NPTool calculation, as well as a flat cross section for

the (p, d) reactions in the center-of-mass frame in order to focus purely on the experimental

geometry. Figure 3.9 shows the calculated geometrical efficiency as it depends on the emitted

deuteron angle assuming that all detectors in the array work optimally. However, this

assumption is unrealistic. The calculation in Figure 3.9 is shown here to provide the reader

with a rough approximation of the angular coverage of HiRA. Chapter 5 will detail a more

careful treatment of the efficiency and its dependence on angle.

60

Figure 3.9: Geometrical efficiency versus emitted deuteron angle for HiRA. This efficiency
was calculated assuming that all detectors work perfectly. This assumption will be revisited
in detail in Chapter 5.

3.4 The S800 Spectrograph

The S800 Spectrograph (or simply, the S800) is a magnetic spectrograph with large accep-

tance in both solid angle and momentum [127]. In this experiment, we used the S800 to

detect and identify heavy recoils from transfer reactions. The S800 (shown in Figure 3.10)

consists of a large bore magnetic quadrupole doublet followed by two dipole magnets, which

are set at a certain magnetic rigidity to select particles of interest from reactions at the
target. Downstream from the dipoles is the focal plane2 detection system used to measure

position, energy loss, and timing information (see Fig. 3.11) [128]. The S800 is preceded by

an analysis line composed of four superconducting dipole magnets (outlined in blue in Fig.

3.10) and 5 superconducting magnetic quadrupole triplets (outlined in green). Immediately
2The focal plane is the plane perpendicular to the S800 optical axis that passes through the focal point
of the S800. The focal point is the point of convergence for particles with ideal rigidity that enter the S800
parallel to the optical axis.

61

Deuteron lab angle (degrees)510152025303540Efficiency00.050.10.150.20.250.30.350.40.45Figure 3.10: The S800 Spectrograph and its associated analysis line.

upstream of the analysis line is a scintillator (known as the object scintillator) that can be

used as a start signal in time-of-flight (TOF) measurements.

The S800 can be tuned in either of two possible optical modes. In focused mode the sec-

ondary beam is focused on the target position and then dispersed in momentum at the focal

plane. In dispersion-matched mode, the S800 and analysis line are tuned to an achromatical

focus at the focal plane so that the beam is dispersed in momentum at the target. Focused
mode allows for larger momentum acceptance (±2%) but worse energy resolution (1 part
in 1000) than dispersion-matched mode (±0.5% and 1 part in 2000, respectively). For this

experiment, we ran the S800 in focused mode in order to maximize momentum acceptance,

which allows for a higher momentum spread in the beam and therefore a higher beam inten-

sity. Furthermore, the primary purpose of the S800 was to identify reaction products, and

this does not require the excellent momentum resolution of dispersion-matched mode.

Beam-like fragments enter the S800 from the reaction target and are dispersed by the

magnetic dipoles. The dipoles are tuned to a particular magnetic rigidity in order to max-
imize acceptance for a particular isotope (in this case, usually either 45Ar or 33Ar). After

passing through the dipoles, the fragments enter the focal plane detector box as shown

62

Figure 3.11: The S800 focal plane detector box. Figure adapted from [128].

in Figure 3.11. The detectors in this box include two Cathode Readout Drift Chambers

(CRDCs) which track the trajectory of the fragment, one Ionization Chamber (IC) which

measures energy loss (∆E), and a plastic scintillator (known as the E1) used to measure

the time-of-flight (TOF) and also to trigger the S800 data acquisition. Scintillators located

upstream from the S800 provide timing information to identiy the beam-like fragment before

and after a reaction occurs. These scintillators are included in the S800 data acquisition out

of convenience. Comparison of TOF and ∆E yields particle identification, provided that the

data is sufficiently corrected using focal plane coordinate information from the CRDCs. See

Chapter 4 for details of these corrections.

3.4.1 Cathode Readout Drift Chambers (CRDCs)

The CRDCs are two identical gas-filled, single-wire drift detectors, separated by approx-

imately 1 m, that provide position information of the fragment’s trajectory as it travels

63

through the spectrograph towards the focal plane. Each CRDC has an active area of 30 cm
× 59 cm (with a thickness of 1.5 cm) and is filled with a mixture of 80% CF4 and 20% C4H10
gas. There is a constant electric field in the vertical (non-dispersive) direction across the

CRDC, so as a fragment passes through the CRDC and ionizes the gas within, the ionized

electrons drift to the anode wire for detection. The drift time of these electrons provides the

non-dispersive position of the fragment in the CRDC. The dispersive position comes from

224 cathode pads, each with a pitch of 2.54 mm, that collect image charge induced by the

anode current. Position measurements in both of the CRDCs allow for the calculation of

angles relative to the central trajectory in both the dispersive and non-dispersive directions.

3.4.2 Ionization Chamber (IC)

The IC measures the energy deposited by a fragment as it travels through the S800 to the

focal plane. Located downstream from the CRDCs, the IC is filled with P-10 gas (90% argon

and 10% methane) and segmented into 16 1-inch sections. As the fragment passes through

the IC, it deposits energy in the gas by producing ion-electron pairs, and the number of such

pairs created is proportional to the energy deposited. Therefore, detection of these pairs

allows for a direct measure of the energy lost by the fragment.

3.4.3 Plastic Scintillators

Thin plastic scintillators measure the time-of-flight (TOF) of the fragment, which is critical

for particle identification. When a particle hits the plastic, scintillation light is emitted and

then collected in photomultiplier tubes at either ends of the scintillators. Although the S800

focal plane has several plastic scintillators, the only one used in this analysis is the most

64

upstream one (known as the E1 scintillator). This 5-mm thick scintillator doubles as the

exit window of the IC to minimize energy straggling, and it stops essentially all particles of

interest. The E1 scintillator also serves as the trigger signal for the S800 data acquisition.

We also use another plastic scintillator called the object (OBJ) scintillator. Located

at the object position of the S800 analysis line upstream of the reaction target, the OBJ

scintillator is included in the S800 data acquisition. The time difference between the OBJ

and E1 scintillators provides a measure of the TOF of the recoil particle emerging from the

transfer reaction of interest, and the time difference between the OBJ and RF signal from

the K1200 cyclotron identifies the incoming beam.

3.5 MCPs

Micro-channel plates (MCPs) are compact arrays of single-channel electron multipliers that

can be used in a wide range of scientific applications [129, 130]. Each MCP used in this

work is a 0.5-mm thick array of tiny glass tubes (approximately 10 µm in diameter and

12 µm apart, center to center) painted with a conductive surface that allows for a uniform

bias voltage to be applied across the MCP and therefore across each tube [130]. Each MCP

has a circular active area with a diameter of 40 mm. When an electron strikes the interior

surface of a tube, it starts a cascade of electrons that are accelerated further down the tube

and subsequently produce more electrons upon each collision with the interior surface. The

resulting avalanche of electrons then exits on the other side. Figure 3.12 illustrates this

amplification process for two MCPs in the “chevron” stack configuration that was used for

this experiment. The change in direction of the glass tubes reduces positive ion feedback by

preventing ions generated at the outgoing surface to re-enter the tubes and generate false

65

Figure 3.12: Diagram showing the operation of MCPs in the chevron configuration. Electrons
strike the conductive surface on a glass tube in the MCP, resulting in a cascade of electrons
and amplification of the signal. After two stages of amplification, the resulting electrons are
detected on a resistive anode. Figure taken from [130].

signals. For convenience of nomenclature, in general the term “MCP” will be used to refer

to a stack of MCPs (like the one shown in Figure 3.12), rather than a single wafer.

In this experiment, two MCPs tracked the secondary beam upstream from the target

position. The upstream MCP (referred to as MCP0) sat 1 m upstream from the downstream

MCP (referred to as MCP1), which was located about 10 cm upstream from the target. Each

MCP was set up as shown in Figure 3.13 with a bias voltage of approximately 2000 V. In this

figure the beam, traveling left to right, passes through a layer of aluminized Mylar foil. The

beam striking the foil excites electrons, and those electrons excite more electrons. These

secondary electrons are accelerated towards the MCP by a 1000 volt potential difference

between the foil and the surface of the MCP. A permanent magnet (manufactured by Magnet

Sales & Manufacturing Inc with a surface magnetic field of 14 kG) confines the electrons

within a tight helical orbit as they travel towards the MCP. After amplification, the electrons

emitted from the MCP are collected on a square resistive anode. We measure the signal

amplitude at each of the four corners of the anode, and we can use this information to

reconstruct the position of the electrons on the anode (and subsequently the position of

66

Figure 3.13: On the left is a schematic of an MCP setup, and on the right is an overhead
picture of the MCP1 setup in the beamline. Two such setups were used in this experiment.
Detailed description can be found in the text. Modified from [131].

the beam hitting the Mylar foil). This entire tracking-detector setup is rotated to allow for

the beam to continue downstream. We installed a magnetic shield between the downstream

MCP and the target position in order to keep the magnetic field from influencing the reaction

products.

We implemented two gain stages for each MCP corner signal in order to extend the

dynamic range. From the MCP, each corner signal goes to a fast amplifier, after which it is

split. One signal proceeds directly to a CAEN V792 QDC for digitization (referred to as the

low-gain channel), while the other passes through an additional stage of amplification first

(the high-gain channel) before proceeding to the QDC. The timing signal from each MCP

is amplified, sent to a constant-fraction discriminator, and digitized with a CAEN V1190A

TDC.

67

3.6 Position Measurements

Measuring a differential cross-section requires precise and accurate knowledge of the tra-

jectories of particles going into and out of the reaction target. And knowledge of these

trajectories requires precisely knowing the positions of all experimental equipment in the

reaction chamber. These positions were determined with a portable coordinate measuring

machine (CMM) arm after the conclusion of the experiment. This commercially available

CMM arm, called the ROMER Arm Infinite 5024, is capable of measuring exact positions in

three dimensions within 100 µm. Furthermore, by tracking the ROMER Arm probe along

a surface, higher dimensional features (like planes) can be measured. The ROMER Arm

was used to determine points and planes on each of the two MCP setups, the target ladder,

each HiRA telescope, and several calibration pucks glued to the walls of the chamber. We

then measured these pucks with a laser tracker system in order to relate the local ROMER

Arm coordinate system to the global laboratory coordinate system. Figure 3.14 shows the

ROMER Arm being used during the position measurements.

The ROMER Arm requires physical contact between the probe and the point or surface of

interest. Clearly, this means that we could not measure the delicate silicon detectors directly.

Instead, we measured the aluminum frame that holds the Mylar foil on each telescope, and

extrapolated the positions of each silicon pixel using the HiRA design drawings. We then

transformed these values from the ROMER Arm coordinate system to the global coordinate

system of the laboratory frame. Figure 3.15 shows the determined laboratory angles of each

pixel of HiRA.

68

Figure 3.14: A graduate student (Jonathan Barney) measures positions on the target ladder
that held the MCP0 Mylar foil and MCP0 calibration mask.

Figure 3.15: Angles of each pixel in all 14 HiRA telescopes calculated from ROMER arm
measurements.

69

Theta (degrees)51015202530354045Phi (degrees)200−150−100−50−050100150200Chapter 4

Data Analysis I: Calibrations and

Corrections

Although the information was not

perfect, it had the merit of existing.

Thomas Piketty

Data gathered by the experimental systems described in Chapter 3 does not immediately

provide scientific insight. We must analyze the data to turn the raw values encoded by the

electronics into physically meaningful quantities for interpretation. This chapter describes

in detail the analysis procedures applied to the raw data from the HiRA, S800, and MCP

experimental systems.

We performed the analysis using the ROOT object-oriented analysis framework written

in C++ [132]. For each detector system, a C++ object defined characteristics of the detector

in a hierarchical structure known as a ROOT Tree. Consider as an example our ROOT

HiRA data structure: each HiRA telescope object contains three 32-strip silicon detector

objects (one each for the DE, EF, and EB) and one four-crystal CsI array object. Each

object has associated data members and methods specifically designed to store and analyze

data from the particular detector element that the object represents. These objects are

stored in a ROOT file format for each event. This organization makes it easy to loop over all

events when doing analysis tasks like making histograms or applying cuts. ROOT enables

70

both advanced, large-scale processing as well as simple, exploratory analysis via the ROOT

interpreter.

Figures 4.1 and 4.2 show flow charts describing the overall data analysis scheme. During

the experiment, the data acquisition system recorded the data into a binary file format. The

first step in the analysis pipeline is to unpack this raw binary file into the ROOT file format,

in which the electronics signals in the binary file are mapped to the appropriate detector

objects. Then, detector-specific processing methods calibrate and correct each individual

detector system. The resulting calibrated detector files correspond to real, physical quantities

(like energy or time). Next, we combine the calibrated detector files so that the separate

detector systems can be associated with each other on an event-by-event basis. Here we can

gate on transfer reactions and calculate more observables like excitation energy spectra and

angular distributions. After the final steps of background subtraction and normalization, we

have in hand the desired absolute differential cross sections.

This chapter will discuss the calibrations and corrections for each individual detector

system.

In other words, we describe here all steps in Figure 4.1 above the gray dotted

line, including the entirety of Figure 4.2. The event generation stage and all the analysis

downstream will be presented in Chapter 5.

4.1 HiRA

We measured outgoing deuterons from the (p, d) transfer reactions with the High Resolution

Array (HiRA). As described in Chapter 3, each HiRA telescope contains silicon strip de-

tectors (the single-sided, 65-µm, DE detector and the double-sided, 1500-µm E detector) as

well as four CsI scintillator crystals. The intersection of front and back strips in the E pro-

71

Figure 4.1: Data analysis flowchart from raw data to the final differential cross section
observables. Chapter 4 describes the analysis above the gray dotted line, and Chapter 5
describes the analysis below the gray dotted line.

72

Figure 4.2: Flowchart for the HiRA data analysis from the raw ROOT Tree to the pixelated
ROOT Tree.

73

vide excellent position resolution. We can identify charged particles by comparing detected

energies in different detector layers.

For each run, the unpacker creates a ROOT file filled with raw data (e.g. energy units

of ADC channels) that needs analysis in order to become physically meaningful (e.g. energy

units of MeV). HiRA provides the energy and angle information for reconstructing the ex-

citation energy spectra (and therefore the differential cross sections), so calibrating HiRA

carefully is a worthwhile investment of effort. This section describes the stages of analysis

of the HiRA data, including pulser linearity checks, energy calibrations, detector characteri-

zations, particle identification, and pixelation. Some of this analysis was recently published

[133], and portions of that article are included below in accordance with the Elsevier per-

mission guidelines.

4.1.1 Linearity

An increase in the input signal going into an electronics circuit should cause a corresponding

increase in the output signal. When this relationship can be fit well with a first-order

polynomial, the electronics response is called linear. We checked the linearity of the detector

electronics for both the silicon and the CsI using a BNC PB-5 pulser. The pulser introduced

a known amount of charge into the electronics circuit in order to simulate the detector signal.

By “ramping” the pulser across a range of voltages, the output of the electronics circuit can

be compared to the incoming pulser signal in order to determine if the electronics output

scales linearly with the pulser voltage.

If so, then we can be confident that the detector

electronics will respond linearly to the charge coming from the real detector, and therefore

to the energy of the detected particle.

This linearity check can be performed directly on the raw HiRA pulser data. Figures

74

Figure 4.3: Pulser ramp for an example DE silicon strip. The top panel is a histogram
showing the electronics output (in ADC channels) for many clearly resolved pulser peaks.
The bottom panel displays the relationship between ADC channels and pulser voltage. Each
peak in the top panel corresponds to a red dot in the bottom panel. The blue line is a linear
fit to the red dots in the central ≈ 80% of the full dynamic range.

4.3, 4.4, and 4.5 show example pulser linearity checks for an example strip from each silicon

detector and an example CsI crystal, respectively. The silicon strip electronics is linear in
the central ≈ 80% of its dynamic range and the CsI crystal electronics is linear up to ≈ 50%.

Both of these electronics systems provide enough dynamic range to measure the deuterons

of interest.

75

Counts per channel50100Raw DE Energy (ADC channels)50001000015000Pulser Voltage (V)024Figure 4.4: Pulser ramps for example EF (left column) and EB (right column) silicon strips.
Both columns are set up similarly to Figure 4.3. The blue line is a linear fit to the red dots
in the central ≈ 80% of the full dynamic range.

76

Counts per channel5010015020025030050100150200250300Raw EF energy (ADC channels)50001000015000Pulser voltage (V)01234Raw EB energy (ADC channels)5000100001500001234Figure 4.5: Pulser ramp for an example CsI crystal set up similarly to Figure 4.3. The blue
line is a linear fit to the red dots up to ≈ 50% of the full dynamic range.

77

Counts per channel200400600800Raw CsI Energy (ADC channels)01000200030004000Pulser Voltage (V)0244.1.2 Silicon Energy Calibration

We now associate the raw silicon electronics output (in ADC channels) to detected particle

energy (in MeV). To do this mapping, we placed a radioactive source with well-known decay

energies in front of the silicon detectors. Each strip of each silicon detector (1344 in total)

requires a unique calibration. Since we have already checked that the electronics response is

linear in the previous section, the energy calibration can be a simple linear function of the

raw channels.

A 228Th alpha source is commonly used to calibrate silicon detectors and is the calibration
source of choice for the HiRA E and DE detectors. Using a 228Th source has three key

advantages. First, there are six clearly separated peaks with energies from about 5 MeV to
about 9 MeV. Second, 228Th sources of various strengths are commercially available. Lastly,
228Th has a relatively long half-life (1.9 years). The source used in the current work was

electroplated onto a platinum surface, and then fixed in an aluminum holder (12.7 millimeters
in diameter and 6.35 millimeters tall) with a 100 µg/cm2 gold window. Figure 4.6 shows the
decay radiation of 228Th and its daughters [134], and Figure 4.7 shows an example E energy1

spectrum in which the peaks from these decays can be clearly seen. In HiRA calibrations,

typically the five largest peaks are used.

Because the DE detector blocks alpha particles from passing through to the E, calibrating
the E with a 228Th source requires removing the DEs from all telescopes. This can only be

done by disassembling the entire array, removing the DE detectors, and then reassembling

the array. Since energy calibrations can be sensitive to minor changes in electronics and cable

configurations, we must confirm that the performance of the E detector after reassembling
1This phrase can be somewhat ambiguous since the E detector detects each particle twice: in the front
(EF) and the back (EB). In this work, “E energy” always refers to the energy from the EF due to its higher
resolution and the EB having a higher probability of charge splitting (discussed in Section 4.1.6).

78

Figure 4.6: Decay scheme for 228Th. The green dashed box includes all decay radiation
from 212Pb, the isotope that is primarily deposited onto each pin source. Energies for all
alpha decays with branching greater than 1% are shown in red, along with the corresponding
branching rates [134, 133].

Figure 4.7: 228Th calibrated alpha spectrum for an example HiRA EF detector. The DE
and EB spectra look similar. Energy losses in the gold window, the Mylar foil in the front
of the HiRA telescope, and an approximate E dead layer are taken into account. Typically,
a HiRA silicon calibration will use the 5 most prominent peaks seen in the spectrum. The
resolution in this detector is 65 keV (FWHM) [133].

79

Energy (MeV)456789Counts/17 keV 050100150200250300350the array is consistent with the performance of the E prior to disassembly. Ideally, if an alpha

source can be inserted between the DE and E detectors, the E detector can be calibrated

without disassembling the HiRA telescope.

To resolve this issue, each HiRA telescope can features a slot between the DE and the

E detectors for insertion of a pin source, which is made from a small metal pin [133]. One
end of the pin is covered with isotopes from the decay chain of 228Th (primarily 212Pb).

To make these pin sources, we applied a negative bias to a bundle of pins placed in front
of a 13 µCi 228Th powder source. When 228Th decays to 220Rn, which has a relatively
long half-life (56 seconds), some of the gaseuos 220Rn ions drift to the negatively-charged
pin electrodes. The 220Rn then decays into the short-lived 216Po and the much longer-lived
212Pb (0.1 seconds and 10.6 hours, respectively), which are metals and generally stick to the

head of the pin. After about 24 hours of irradiation, we mounted each of the pins on a frame

and then inserted each frame into the aligned slot between the DE and E detectors (as seen

in Figure 4.8) so that alpha particles are emitted directly onto the E detector without having

to disassemble the array. The HiRA telescope was designed so that the pin mounted in the

frame sits 3.2 mm above the surface of the E detector when the frame is inserted into the
slot. Since the 212Pb on the pin source is far down the decay chain of 228Th, only two peaks

feature prominently in the observed energy spectrum (as shown in the green box in Figure

4.6) [134]. The collected data can then be used as a standalone calibration or to validate the
228Th calibrations at the end of an experiment. The latter option is in general preferable
since the 228Th data have more peaks to use as calibration points.

The relevant decays for the pin source inserted into HiRA are shown in the green dashed

box in Figure 4.6. An example spectrum in Figure 4.9 illustrates that there are only two

peaks in the pin source data: one that corresponds to an 8.785 MeV decay and a lower

80

Figure 4.8: HiRA telescope assembly. On the bottom left is a side-view photograph of a HiRA
telescope. On the bottom right is a cartoon of the detectors contained within the telescope.
Dotted red lines connect the detectors in the cartoon to their approximate positions in the
telescope. On the top of the figure is the pin source frame with the pin source installed. The
pin source slot is located between the DE and the E detector as indicated by the thick red
arrow [133].

81

Figure 4.9: Example pin source alpha spectrum for one pixel in one E detector [133].

energy peak that actually consists of two unresolved alpha decay peaks (one at 6.051 MeV

and another at 6.090 MeV). The relative probabilities of these two unresolved decays are

well known, so these two energies can be combined via a weighted sum to yield a single peak

energy of 6.062 MeV. These are not the peak energies that are seen in the detector: alpha

particles will lose energy in the dead layer so the measured energies will be below the decay

values. Fortunately, these deviations allow for measurement of the dead layer thickness (as

shown in the next section).

When the preliminary 228Th calibration is applied, the pin source peaks look reasonably

close to the expected energies. This validates that the detector performance after the exper-

iment is consistent with that of the experiment itself. Now, the calibration can be further

improved by measuring the thickness of the dead layer on the E detector.

82

Energy (MeV)56789Counts/15 keV 0200400600800100012004.1.3 Silicon Dead Layer Thickness

The pin source also provides another ingredient to a proper E energy calibration: determina-

tion of the thickness of the dead layer. During fabrication of a silicon detector, a “dead layer”

is typically formed on the surface to protect the active silicon wafer of the detector [124].

Energy deposited by charged particles in this dead layer is not detected. Therefore the

dead layer affects the accuracy of charged-particle energy measurements. Silicon-detector

dead layers have previously been measured using low energy electrons [135] and proton

bremsstrahlung [136].

The pin source is nestled between the DE and the E, so alpha particles from the pin do

not need to pass through a gold window or Mylar foil to reach the detector (as is the case

with alpha particles from the thorium source). Therefore the dead layer is the only potential

cause of energy loss between the source and the active detector volume, as shown in the

cartoon in Figure 4.10. We perform this calibration under vacuum, and the thickness of the
212Pb deposition can be neglected. Furthermore, the pin itself is only 3.2 millimeters above

the detector surface, so alpha particles will travel through the dead layer at a wide range of

incident angles. Although the dead layer can contain a variety of materials, we assume an

effective dead layer of pure silicon since our only concern is the resulting effect on charged

particles. Assuming a uniform dead layer and a constant value for the energy loss, simple

geometrical considerations (illustrated in Figure 4.10) dictate that the energy of the alpha

particles detected at different pixels can be used to extract the dead layer thickness via the

following relation:

E = E0 − dE
dx

∗ T
cos θ

(4.1)

where E is the detected energy, E0 is the initial energy of the alpha particle, θ is the emission

83

Figure 4.10: Cartoon of pin source and dead layer. The distance h between the pin source
and the detector surface is 3.2 mm. The angle θ of the alpha particle from the pin source
determines how much of the dead layer the particle passes through. The dead layer thickness
(T) can be extracted by studying the relationship between detected energy and angle [133].

angle, dE

dx is the stopping power of an alpha particle in silicon at energy E0 (since the dead
layer is so thin, the stopping power is assumed to be constant), and T is the dead layer

thickness [133]. Since the amount of dead layer that the alpha particle traverses depends

on the incident angle of the particle, the final energy of that alpha particle will also depend

on the incident angle. By measuring the relationship between incident angle and detected

energy, we can extract T.

The first step of measuring the dead layer has already been described in the previous

section: an initial calibration using the thorium source data with a reasonable guess for the

dead layer thickness (on the order of 1.0 µm based on previous studies). Since the dead

layer thickness is not yet precisely known, this calibration is inexact. In the fits described

below, the initial energy of the alpha particle is a free parameter in order to account for this

84

potential imperfection.

The next step in determining the dead layer thickness is to find the central pixel, which

is the pixel that the pin source is closest to. In principle the pin source should be exactly

at the center of the detector (which falls in between pixels), but in practice the pin source

could be slightly misaligned and is slightly closer to one pixel than the rest. The central

pixel provides a good first approximation to the precise location of the pin source. Because

of its relatively large solid angle coverage, the central pixel should have more counts than

any other pixel. Figure 4.11 shows a two-dimensional hit map in which the back strip axis

is along the y-direction and the front strip axis is along the x-direction. The central pixel

(which by definition is at the intersection of the central front strip and the central back strip)

has the most counts, and the counts decrease as distance from the central pixel increases.

The hit map is not perfectly concentric due to asymmetries in the deposition on the pin

head, but this is a negligible effect.

Next, we performed fits for each peak of each pixel across the central front strip to

determine precisely the measured energies for each pixel made by the intersection of the

central front strip with a back strip. According to Equation 4.1, the central pixel should

measure the highest detected energy, since the incident angle of the alpha particle is closer

to 0 than for any other pixel. Figure 4.12 shows the energies for several pixels across the

central front strip for an example detector, as well as a fit performed using a modified version

of Equation 4.1:

E(sb) = E0 − dE
dx

∗ T ∗

s

1 +

(cid:18) d0

h

(cid:19)2 ∗ (sb0 − sb)2

(4.2)

where d0 is the width of each strip (fixed to 1.95 mm), h is the distance from pin source
to detector surface (fixed to 3.2 mm), and s is the back strip number for each pixel. The

85

Figure 4.11: A two-dimensional hit map for the number of detected pin source counts for
one E detector. The central pixel is shown at the intersection of the black rectangles (which
correspond to the front and back strips with the most counts) [133].

86

 Front strip number101520 Back strip number1012141618202202000400060008000100001200014000Figure 4.12: Example energy distribution with statistical error bars of the 8.785 MeV peak
for different pixels across the central front strip. The corresponding fit (using Equation 4.2)
is shown in red [133].

back pin source position sb0 refers to the exact coordinate of the pin source along the back
strip axis. This position can take fractional values, e.g. sb0 = 15.5 would indicate that the
pin source is located above the space in between back strips 15 and 16. The free parameters

are T, sb0, and E0: E0 is treated as a free parameter since the dead layer thickness is
not known in the initial 228Th calibration, so E0 may be slightly off. A similar fit can be
performed across the central back strip in order to find the front pin source position sf0,
defined similarly to sb0.

We can extend Equation 4.2 straightforwardly to two dimensions resulting in Equation

4.3, in which case the number of data points increases since the fit is no longer limited to a

single central strip:

E(sf , sb) = E0 − dE
dx

∗ T ∗

s

1 +

(cid:18) d0

h

(cid:19)2 ∗ [(sf0 − sf)2 + (sb0 − sb)2]

(4.3)

87

Back strip number101214161820Energy (MeV)8.58.68.7Figure 4.13: Example energy distribution with statistical error bars for the 8.785 MeV peak
across all pixels in a single telescope according to different values of 1/ cos θ, as well as the
corresponding fit [133].

where sf and sb are the front and back strip numbers of the pixel, and sf0 and sb0 come
from the one-dimensional fits described above. The free parameters in this fit are only T and

E0. Results of an example fit are shown in Figure 4.13. The quality of the fit shows that
our assumption of dead layer uniformity is valid, at least for the central area of the detector

defined by the central 10 to 12 front strips and the central 10 to 12 back strips. We show the

extracted dead layer thicknesses for 14 telescopes in Figure 4.14, with results for both the

8.785 MeV peak (blue square symbols) as well as the 6.062 MeV peak (red open circles). The

error bars are statistical uncertainties that mainly depend on the intensity of the pin source

for a given telescope. Since many pins were bundled together during the source irradiation,
the 212Pb isotopes were distributed unevenly across the pins. The mean dead layer value
across all telescopes of 0.61 ± 0.07 µm is indicated by the dotted line [133]. Within error,

this average matches the value provided by the manufacturer of 0.5 µm [137].

We can then incorporate this more accurate dead layer thickness into the thorium cali-

88

θ 1/cos 12345 Energy (MeV)8.48.58.68.7Figure 4.14: Dead layer thicknesses extracted for each E detector from both the low (red
open points) and high (blue solid points) energy peaks from 212Pb decay. All error bars are
statistical, and the wide range of errors is due to different intensities of the pin sources. The
average dead layer thickness (0.61 µm) is given by the dotted black line. Values extracted
with the higher energy peak and the lower energy peak are consistent with each other [133].

bration for both the EF and the EB.

4.1.4 Particle Identification

We now have a good calibration for both the E and the DE detectors. As discussed in

Section 3.3, we can use HiRA for particle identification (PID) by comparing energies in

different detectors within a telescope.

Once all strips in a given detector have been calibrated, their energy signals are matched

to the same energy scale and can be combined. Figure 4.15 shows an example DE-E PID

plot for a single telescope in which proton, deuteron, and triton lines are clearly separated.

Since this plot is only meant to identify particles that stop in the E detector, Figure 4.15 is

gated on thresholds for the CsI crystals behind the E. In other words, to generate the DE-E

PID, we only consider events where there are counts in the E and DE and no above-threshold

89

Detector number024681012m)µDead layer thickness (0.30.40.50.60.70.80.91.01.11.28.785 MeV6.062 MeVAverage thicknessFigure 4.15: DE-E PID plot for an example telescope, with CsI thresholds applied.

counts in any of the four CsI scintillators.

For particles energetic enough to punch through the E detector, we use the E-CsI PID

stage. Figure 4.16 shows an example for a single CsI crystal. Because each CsI crystal is only

a single detector element (as opposed to the silicon detectors, which are made up of many

strips that require many individual calibrations), we can get PID without a CsI calibration.

This is rather fortuitous since CsI detector response depends strongly on Z and therefore

PID is required to do a proper CsI calibration (as discussed in Section 4.1.5).

4.1.5 CsI Energy Calibration and Silicon Detector Thickness

Calibrating the CsI is not as straightforward as calibrating the silicon since the CsI is hidden

behind two layers of silicon, has a relatively high dynamic range, and responds differently to

particles with different Z. Below we present our CsI calibration method. An important part

of this method consists of determining the thickness of each E detector, which influences the

energetics of particles that punch through the E to the CsI.

90

Calibrated E Energy (MeV)5101520Calibrated DE Energy (MeV)0.511.522.533.540510152025Figure 4.16: E-CsI PID plot using raw CsI energy.

One way to calibrate a CsI crystal is to accelerate and elastically scatter light-charged-

particles at known energies into the crystal. For this work, we used a beam of hydrogen

isotopes with a magnetic rigidity of 1.10 Tm (corresponding to a proton energy of 56 MeV).

Only protons were considered in this analysis due to low intensities for the deuteron and

triton beams. A CH2 reaction target (75 µm thick) scattered the incoming light charged
particles. For the most part, these particles scatter off the carbon in the target. From

two-body kinematics, the energies of the elastically scattered particles at a given angle are

known, and range from approximately 54 to 56 MeV. These energies are high enough to punch

through both the DE and the E detectors easily (the proton punch-through energies for the

DE and E detectors are 2.45 MeV and 15.6 MeV, respectively), while also stopping within

the CsI, which has a proton punch-through energy upwards of 110 MeV. Since the kinematic

relationship between scattering angle and energy is relatively flat (in other words, the energy

is only weakly dependent on scattering angle), this results in one elastically scattered proton
calibration point per crystal, as shown in Fig. 4.17. 12C has an excited state at 4.439

91

Uncalibrated CsI Energy (ADC channels)1000200030004000Calibrated E Energy (MeV)2468101214051015202530MeV, so there is an inelastically scattered proton calibration point as well. Together, these

elastically and inelastically scattered protons constrain the calibration at high energy for a

given crystal. There is some scattering off the hydrogen in the plastic target, but due to the

low statistics and a steep kinematic relationship (compared to proton-carbon scattering) we

did not use this data.

The scattering data constrain the calibration only at high energy. The calibration of

the CsI at low energies utilizes the Bethe-Bloch formula by comparing the energy lost by a

charged particle punching through the E detector to the energy deposited in the CsI where

the particle stops [138]. If we know the E detector thickness and have a reliable E energy

calibration, we can calculate the energy deposited by a proton in the E detector (ESi) for a
range of incoming-proton energies (Ep) using energy-loss tables [139]. The energy deposited
in the CsI crystal is therefore ECsI = Ep − ESi. We then use ECsI to calibrate the raw CsI
ADC channels corresponding to the calibrated E energy as seen in Fig. 4.17.

This procedure (which will be referred to as the energy loss method) allows for a calibra-

tion that extends well into the low-end of the dynamic range of the CsI. However, proper

implementation requires precise knowledge of the thickness of the E detector since it is a crit-

ical ingredient in determining the incoming-proton energy that corresponds to the energy

deposited in the E (and therefore the calculated CsI energy used to perform the calibra-

tion). Although the nominal thickness of each E detector is 1500 µm, the true value for the

thickness can differ from this value by up to 100 µm [133].

We make two important notes concerning the validity of this approach. First, the energy

loss method relies on the assumption that the CsI detector response is linear at low energies.

To confirm this, HiRA crystals were tested via direct proton beam at Western Michigan

University using several different beam energies (see Figure 4.18). The crystals were found

92

Figure 4.17: (a) HiRA PID plot for reaction data using one CsI crystal and its corresponding
E silicon detector. The y-axis is calibrated energy in the E detector, and the x-axis is
uncalibrated CsI energy in units of electronics channels. Protons (within the red-dashed
line) can clearly be identified, even though the CsI energy is uncalibrated. (b) The top panel
is the HiRA PID for the same CsI crystal as in (a), this time showing the scattering data
(zoomed in to the relevant region). The red-dashed line is the same proton gate as in (a).
The bottom panel is a projection of the top panel onto the x-axis. Two peaks are clearly
visible: the higher energy peak corresponds to proton elastic scattering off of carbon, and
the lower energy peak is from inelastic scattering (E∗(12C) = 4.439 MeV) [133].

to be linear down to approximately 1 MeV [140]. Secondly, since the CsI light output

depends on the detected particle species, the energy loss method requires that only data

from protons hitting the detector be used. Fortunately, by comparing single CsI crystals to

their corresponding calibrated E detector, we can unambiguously identify protons in the CsI

crystal even without a calibration (see Figure 4.17 and Figure 4.16).

We combined two separate CsI calibration methods in two different energy regions in

order to determine the thickness of the E silicon detector within each telescope. The first

calibration method is simply to use protons scattered from the CH2 reaction target. The
kinematics are well understood, so for a given angle we know the proton energy. Since these

protons are at relatively high energies, they do not deposit a large amount of energy in the

E, and they do not have high sensitivity to the E detector thickness; i.e. a large change

in the detector thickness will only slightly change the energy lost by a high energy proton.

93

Uncalibrated CsI energy (ADC channels)1000200030004000Calibrated E energy (MeV)5101502468101214161820(a)Cal. E energy (MeV)24680246810121416(b)Uncalibrated CsI energy (ADC channels)1000110012001300Counts010203040Figure 4.18: HiRA CsI light response with a direct proton beam at low energies. Data for
one crystal is shown with a linear fit. Modified from [140].

However, the proton energy is so high that there is a long “lever arm” when extrapolating

down to low energy. Small deviations in the high energy points will have a major impact on

the low end of the dynamic range.

The second calibration method is the energy-loss method described above, which allows

for a series of calibration points at low energy to be generated for a given detector thickness.

These points are highly sensitive to the E thickness. The correct detector thickness should

result in consistency between the low energy points calculated with the energy-loss method

and the high energy scattering points. In this energy range the detector (Fig. 4.18) and

electronics (Fig. 4.5) responses are linear, so therefore the energy-loss and scattering points

should be collinear. To check this, the energy-loss and scattering points were calculated

using detector thicknesses from 1400 µm to 1600 µm [133].

94

Figure 4.19: Relationship between calculated CsI energy and raw CsI channels for one tele-
scope at three different E thicknesses. The blue squares correspond to scattered protons with
well known energies that deposit a small amount of their energy in the E detector, and the
open circles are calculated via the energy-loss method as described in the text using each of
the three indicated E thicknesses. When fitting both sets of points together, an E thickness
of 1474 µm provides the points that yield the best fit [133].

Figure 4.19 shows example plots of CsI energy vs raw CsI channels for one crystal, with

CsI energies calculated using three different values for the E detector thickness: 1400 µm,

1474 µm, and 1600 µm. The blue squares are the scattering data, and the open circles are

the energy-loss calculation data. All points were calculated using the indicated thickness. At

the correct thickness value, these points should be collinear. As the assumed thickness value

diverges from the correct value, the fit quality drops. Thus, we can extract the thickness

by finding the fit with minimum Chi-square. The resulting thicknesses for 10 telescopes are

directly compared to the thicknesses provided by the manufacturer in Figure 4.20 [133].

Now we can use the appropriate E thicknesses to generate deuteron calibrations for each

crystal that are reliable up to ≈ 56 MeV.

95

5001000Calculated CsI energy (MeV)1020304050mµ1400 50010001020304050mµ1474 50010001020304050mµ1600 Raw CsI energy (ADC channels)Figure 4.20: Summary of results from E thickness extraction in comparison with manufac-
turer provided detector thicknesses. The line corresponds to exact agreement. The measured
values agree within error to the manufacturer values. Wide variation from the nominal value
of 1500 µm is evident [133].

96

4.1.6 Pixelation

The analysis described in the previous sections results in calibrated EF, EB, and DE silicon

strips, as well as calibrated CsI crystals. Next, we need to put together the calibrated

information from different detectors in a given event (e.g. EDE MeV deposited into strip
iDE of the DE detector, EEF MeV deposited into strip iEF of the EF detector, etc.) into
a cohesive description of a single particle (e.g. a deuteron with total energy E at angle θ).

We have already shown that we can identify particles (including deuterons) in two separate

stages of PID: we now proceed to the rest of the analysis needed to achieve the stated goal.

Our approach is to first categorize all events in a given telescope depending on which

detectors in that telescope contain data. By doing so, we leverage the fact that the data we

seek will necessarily feature coincidences and correlations between detectors in a telescope.

For example, any events that only contain data in the CsI and DE and none in the E clearly

do not correspond to deuterons from (p, d) reactions, and can therefore be discarded. Most

likely, the CsI+DE counts and other nonsensical categories are either noise or due to detector

inefficiencies. The only categories with relevant data are DE+EF+EB, DE+EF+EB+CsI,
and EF+EB+CsI2. Using this logic, we can significantly narrow down the amount of data

we need to process.

We now focus on the E detector for events with potentially relevant data. When a particle

deposits energy in the E, the resulting cloud of charge carriers can diffuse across the 1.5-mm

thickness of the detector and be collected on more than one strip. This problem is more

severe for the strips at the back of the detector (EB) than for those at the front (EF). For

example, Figure 4.21 shows thorium-source data for two adjacent strips in an example EB
2EF+EB+CsI can contain good data despite the lack of a DE hit. For particles energetic enough to
punch through the E into the CsI, the amount of energy deposited in the DE can be quite small and not
always detected. We account for this easily by simulating the expected energy loss in the DE.

97

Figure 4.21: Comparison between two adjacent strips in an example EB detector. The
diagonal lines indicate charge splitting from each alpha particle from the thorium source.

detector. For about 10% of the events shown, charge from a single alpha particle is split

across the two strips resulting in the diagonal lines in Figure 4.21. We see a smaller charge

splitting effect (1%) in the EF, as well.

Charge splitting can be accounted for by “gluing” together energies in adjacent strips

that likely correspond to the same particle. For each detector in each event, we search for

adjacent pairs of strips with positive energies. If the strip with a smaller energy is above

a particular threshold (i.e. not noise), then we combine the energies from both strips and

assign the combined energy to the strip that originally had the larger energy.

Our next step is to associate a strip in the EF with a strip in the EB to form a pixel. In the

case of this experiment, HiRA detects a single deuteron whenever the (p, d) reaction occurs.

This simplifies the pixelation algorithm as we need not be concerned with disentangling

98

Telescope 5 EB, strip 11 energy (MeV)02468Telescope 5 EB, strip 10 energy (MeV)024680102030405060multiple particles passing through a single E detector. We check that the location of the

E pixel is consistent with information from other detectors. For events with data in a CsI

crystal, the E pixel must be in front of that particular crystal. Otherwise, the event is thrown

out. For events that stop in the E, we use the fact that DE strips and EF strips are parallel

to each other to ensure that the DE and EF signals are consistent. If the DE strip is not

within 2 strips of the EF strip, the event is discarded.

After performing these checks, we apply the PID gates and throw out all non-deuteron

particles. Using our pixel determination, we can assign a particular detection angle to each

deuteron. We know what polar angle θ each pixel corresponds to from the ROMER Arm

measurements described in Section 3.6.

Finally, we calculate the correct total energy of the deuteron by combining energies from

all detectors in the telescope. We include energy loss effects from various materials including

the aluminized Mylar foil around the CsI crystal, the EB dead layer, the EF dead layer, the

DE dead layers (on both sides of the detector), and the aluminized Mylar foil on the front of

the telescope. Furthermore, we account for the energy lost by the deuteron in the reaction

target, which depends on the deuteron angle. For now we assume that the reaction occurs at

the center of the target. With the MCP, we can improve our determination of the deuteron

angle as discussed in Chapter 5.

4.2 S800

The S800 Spectrograph (which consists mainly of two dipole magnets and a downstream set

of detectors as described in Chapter 3) measured outgoing recoils from the (p, d) transfer

reactions. The dipole magnets select particles (according to a set rigidity) that enter the focal

99

plane detector box. There, two Cathode Readout Drift Chambers (CRDCs) track particle

trajectory, an Ion Chamber (IC) measures energy loss, and a plastic scinatillator (known as

the E1) serves as both a time-of-flight detector and a trigger for the data acquisition system.

The two primary goals for the S800 analysis are to identify heavy recoils from (p, d)

reactions, and to identify the incoming beam. Unlike in the case of HiRA, we are not

concerned with precise calibrations and measurements using the S800: our aim is simply to

identify particles. To do so, we use the energy loss (∆E) measured in the ion chamber as well

as various times-of-flight (TOFs) relative to the E1 scintillator. Here we describe the S800

analysis, including beam identification, the CRDC position calibration, and the trajectory

corrections necessary for the S800 PID.

4.2.1 Beam Identification

Identifying incoming beam particles is critical in order to ensure that we are studying the

correct transfer reactions. To perform this identification, we take advantage of the fact that

at a given rigidity, particles with different mass and/or charge will have different velocities.

Therefore, we can use the measured TOF through a given segment of the beam line to

distinguish beam particles. The spread in the momentum of the beam makes this distinction

challenging if only using one TOF, so we instead look at the correlation between two TOFs.

The E1 scintillator at the focal plane of the S800 served as the S800 data acquisition

trigger as well as the start signal for all TOF measurements. The TOF-OBJ is between the

OBJ scintillator in the analysis line of the S800 and the E1. The TOF-RF is between the RF

signal coming from the K1200 cyclotron and the E1. PID plots comparing the TOF-OBJ
and the TOF-RF are shown in Figures 4.22 and 4.23 for the 34Ar and the 46Ar beams,

respectively. We can clearly identify separate beam species.

100

Figure 4.22: Incoming beam PID for the 34Ar beam.

Figure 4.23: Incoming beam PID for the 46Ar beam.

101

TOF(cid:173)OBJ (ns)(cid:173)60(cid:173)40(cid:173)200TOF(cid:173)RF (ns)180200220240110210Ar34TOF(cid:173)OBJ (ns)(cid:173)80(cid:173)60(cid:173)40(cid:173)20TOF(cid:173)RF (ns)160180200220110210Ar46Using these PID plots, we can gate on the events with the appropriate incoming beam
particle. We can also calculate the beam purity for each system: the 34Ar beam is 94% pure,
and the 46Ar beam is 99.5% pure.

4.2.2 CRDC Position Calibration

Each CRDC measures the position at which the fragment passed through the detector.

With the CRDC1 and CRDC2 positions, we can calculate the angle of the fragment as it

travels towards the focal plane. To do this, we need to calibrate both CRDCs (from the raw

electronics channels to physical distance) so that they are on the same scale.

We performed this calibration by inserting a mask with a well-defined pattern of holes

in front of each CRDC. We then fit this mask spectrum with first-order polynomials in both

the x (dispersive) and y (non-dispersive) directions. The slope in the x direction is known,

since it depends only on the fixed geometry of the 2.54-mm cathode pads. Figure 4.24 shows

example data for CRDC2. Mask data was taken periodically throughout the experiment to

ensure that the CRDC detector response was stable.

Using the calibrated CRDC data, we can now calculate the angles (in both the dispersive

and non-dispersive directions) of the fragment in the S800.

4.2.3 Particle Identification and Trajectory Corrections

We must identify recoil fragments in the S800 in order to properly tag on the transfer reaction

channels we would like to study. We use the ∆E-TOF method to obtain the PID.

The first ingredient of this method is ∆E, the energy lost by the fragment in the ion
chamber. According to Equation 3.2, this quantity is proportional to Z2A. The second is

102

Figure 4.24: Example calibrated CRDC mask.

TOF-OBJ, the time-of-flight between the OBJ scintillator upstream of the reaction target

and the E1 scintillator. As seen in Equation 3.1, the mass-to-charge ratio A/Z is related

to the velocity at fixed Bρ, and therefore related to the TOF. We choose to use the object

scintillator as the TOF stop signal, but in principle other signals could be used. Both

∆E and TOF-OBJ depend on the particle species in different ways, so we can look at

the correlation between the two quantities to unambiguously identify the isotopes passing

through the spectrograph.

So far we have assumed that all particles travel the exact same path through the S800.

In fact, the S800 acceptance allows for substantial variation in the particle trajectory, which

affects both ∆E and TOF-OBJ. To illustrate this, consider two particles identical in Z, A,

and Bρ that exit the target at slightly different angles. They will travel different trajectories

of different lengths and will arrive at the E1 scintillator at different times (resulting in

different times-of-flight). These two particles will also have different values for ∆E. The

situation is further complicated by other path-dependent effects, like the propagation time

103

CRDC2 X (mm)(cid:173)1000100CRDC2 Y (mm)(cid:173)40(cid:173)30(cid:173)20(cid:173)100102030405060051015202530of the light in the E1 scintillator traveling to the photomultiplier tubes on either end.

Rather than microscopically modeling these effects, we follow the standard procedure of

empirically correcting both time-of-flight and ∆E for dependence on focal plane coordinates

(i.e. on trajectory) in an iterative process [141]. The first iteration is described in the

following two equations:

TOF-OBJ1 = TOF-OBJ0 −

(cid:19)

(cid:18)d(TOF-OBJ0)
(cid:19)
(cid:18)d(∆E0)
x1 −

d(x1)

(cid:18)d(TOF-OBJ0)
x1 −
(cid:18)d(∆E0)

d(AFP)

(cid:19)

y1

(cid:19)

AFP

(4.4)

(4.5)

∆E1 = ∆E0 −

d(x1)

d(y1)

where x1 is the x-coordinate in CRDC1, y1 is the y-coordinate in CRDC1, and AFP is the
dispersive focal-plane angle. TOF-OBJn and ∆En are the time-of-flight and energy loss,

respectively, after n correction iterations (so n = 0 corresponds to no corrections at all).

Equations 4.4 and 4.5 portray the subtraction of linear focal plane coordinate dependencies

from both TOF-OBJ and ∆E. These dependencies are correlated with each other, so fur-

ther iterations are necessary to converge on the goal of having all derivatives in the above

equations equal to 0.

To demonstrate these corrections further, we consider the 34Ar beam. Figure 4.25 depicts

TOF-OBJ0 (a,b) and ∆E0 (c,d) plotted against x1 (a,c), AFP (b), and y1 (d). In each figure,
we see several bands (each of which correspond to a different Z). In all four plots we see a

clear linear dependence on the focal plane coordinate. We can fit these linear correlations in
order to get the parameters d(TOF-OBJ0)

, d(TOF-OBJ0)

for Equations

d(x1) , and d(∆E0)
, d(∆E0)
d(y1)

d(AFP)

d(x1)

4.4 and 4.5. The results from this first correction TOF-OBJ1 and ∆E1 are still slightly
dependent on focal plane coordinates, so we apply a second iteration of corrections. In this

work, two stages of corrections for x1 and AFP resolved the isotopes of interest. From this

104

Figure 4.25: (a) TOF-OBJ0 plotted against x1. (b) TOF-OBJ0 plotted against AFP. (c)
∆E0 plotted against x1. (d) ∆E0 plotted against y1. Red dotted lines correspond to linear
fits.

point forward, we refer to TOF-OBJ2 and ∆E2 as the corrected TOF and ∆E, respectively.
We performed this correction procedure for each final recoil isotope of interest. Figures

4.26 and 4.27 show the ∆E-TOF PID for the uncorrected and corrected quantities. Different

isotopes, each one represented by a “blob”, can be clearly resolved in both corrected PID

plots. The isotopic resolution will be further improved later in the analysis by gating on

emitted deuterons in HiRA. We then look at data in which the S800 blocker was moved

slightly in order to let the unreacted beam into the focal plane. Using this data we can

unambiguously identify which of the many “blobs” in the PID plot corresponds to the beam

species. We can now easily determine the identities of all other blobs, including the heavy
recoils 33Ar and 45Ar (indicated in Figures 4.26 and 4.27 by red circles). Gates around these

heavy recoils allow us to narrow our analysis to the (p, d) reactions of interest.

105

 (mm)1x(cid:173)150(cid:173)100(cid:173)500 (ns)0TOF(cid:173)OBJ(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)505AFP (mrad)(cid:173)0.06(cid:173)0.04(cid:173)0.020 (ns)0TOF(cid:173)OBJ(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)505 (mm)1x(cid:173)1000100200 (arb. units)0E∆190020002100220023002400250026002700 (mm)1y(cid:173)100(cid:173)50050 (arb. units)0E∆10001200140016001800200022002400260028003000Figure 4.26: S800 PID for the 34Ar beam, with the (p, d) residue 33Ar clearly indicated.

Figure 4.27: S800 PID for the 46Ar beam, with the (p, d) residue 45Ar clearly indicated.

106

TOF(cid:173)OBJ (ns)(cid:173)20(cid:173)100E (arb. units)∆140016001800200022002400260028003000110210Ar33TOF(cid:173)OBJ (ns)(cid:173)25(cid:173)20(cid:173)15(cid:173)10E (arb. units)∆20002200240026002800110Ar454.3 MCP Beam Tracking

Two Microchannel Plate setups (MCPs) detect the beam upstream of the reaction target.

Each MCP setup has two individual Microchannel Plate detectors, each of which is a 0.5-

mm-thick array of tiny glass tubes coated with a conductive surface. Each tube acts as an

electron amplifier. The two detectors, arranged in the chevron stack configuration shown in

Figure 3.12, sit near the beamline and detect secondary electrons from the beam impinging

on a layer of aluminized Mylar. One MCP sits about one meter upstream from the target

position, and the other is about 10 cm upstream. Section 3.5 features a more detailed

technical description of the MCPs.

We used MCPs in this experiment for two reasons. First, the MCPs count the number

of incoming beam particles, therefore giving us the incident flux necessary to calculate the

transfer reaction differential cross sections. Secondly, the position sensitivity of the MCPs

allows for tracking of the beam and subsequently the localization of the reaction on target.

This provides a valuable boost in angular and energy resolution. We describe analysis

relevant to the latter objective below, and leave the discussion of beam normalization to

Chapter 5.

Each anode has four corner signals, and the amplitude of each signal is inversely pro-

portional to the distance between the corner and the location on the anode where electrons

are deposited. The signal amplitude will be higher if the deposition is closer to that corner,

and conversely the amplitude will be lower if the deposition is farther away. We then can

reconstruct the deposition location (and therefore the beam position on the aluminized My-

lar foil) using these corner signals. Once we have the beam position at each MCP setup, we

can trivially project to the target position.

107

Using the charge-division method [130], we can calculate raw x and y positions using the

upper left (U L), lower left (LL), upper right (U R), and lower right (LR) corner signals:

xraw = (U L + LL) − (U R − LR)
U L + LL + U R + LR
yraw = (U L − LL) + (U R − LR)
U L + LL + U R + LR

(4.6)

With a proper calibration, we can associate xraw and yraw with real physical coordinates for

tracking the beam as it travels towards the target.

As discussed in Section 3.5, we recorded a low-gain and a high-gain version of each

corner signal. Due to complications with the high-gain signals as well as the reasonably

good efficiency of the low-gain signals, we only use the low-gain signals in this analysis. All

subsequent discussion of the corner signals refer to the low-gain signals.

Before we can sensibly combine the corner signals as described in Equation 4.6, we must

ensure that these signals are on the same (albeit uncalibrated) scale. Although all corners

undergo the same amount of amplification, they do not have the same pedestal values. The

pedestal is the mean value of the signal when no beam is present. We determined this pedestal

by taking data with no beam in the vault. Then, we can subtract from each corner signal the

corresponding pedestal so that they all have the same zero offset, as shown in Figure 4.28.

Now that all MCP corner signals have value equal to 0 when there is no incoming signal, we

can combine the corners to produce meaningful position information.

Beam tracking is not available for the 34Ar data due to a misconfigured electronics fast

clear circuit. We discovered this problem during the experiment and fixed it in time for the
46Ar beam, so below we discuss the beam spot reconstruction for this system. During the
34Ar beam tuning, we estimated the size of the beam spot to be roughly 50 mm2, which is

108

Figure 4.28: MCP1 raw corner signals (left), and MCP1 pedestal-subtracted corner signals
(right).

much smaller than the size of the 46Ar beam spot. So, fortunately, beam tracking is not as
critical for the 34Ar beam.

4.3.1 Mask Calibration

In order to measure the mapping from raw position calculated with charge division to cali-

brated position corresponding to a particular location relative to the beam axis, we inserted

brass masks with well-defined hole patterns (shown in Figure 4.29) in front of the aluminized

Mylar foils for several data runs. These masks feature asymmetric hole patterns to unam-

biguously tell their orientation relative to the beam axis. Although the brass masks do not

stop the beam from going through, they do change the rigidity of the beam. Therefore, the

S800 deflects any beam particles that passed through either mask such that they do not pass

through the focal-plane box, and do not trigger the DAQ. We measured the exact positions

109

Raw signal (ADC channels)120140160180200Counts020406080100120140160180200220Sub. signal (ADC channels)(cid:173)40(cid:173)2002040Counts020406080100120140160180200Figure 4.29: MCP brass calibration masks for MCP0 (left) and MCP1 (right). The holes in
the mask pattern are 1, 2, or 3 mm in diameter.

of the masks in the laboratory frame using the ROMER arm described in Section 3.6. We

can reconstruct the mask pattern with the MCP data to make a calibration. Once we have

this mapping, we can use it to calculate the position of the beam in the transfer reaction

data.

Since we know the exact locations of the holes in the mask, we can match the raw x and

y values to their appropriate positions with a two-dimensional, third-order polynomial:

xcal = a0 + a1xraw + a2yraw + a3x2
raw + a7xrawyraw + a8x2
ycal = b0 + b1yraw + b2xraw + b3y2
raw + b7yrawxraw + b8y2

raw + a5x3
raw + a4y2
raw
rawyraw + a9xrawy2
raw
raw + b5y3
raw + b4x2
raw
rawxraw + b9yrawx2
raw

+a6y3

+b6x3

(4.7)

The calibrated masks are shown in Figure 4.30. All of the holes are within 1 millimeter

110

Figure 4.30: MCP0 xcal vs. ycal calibrated mask data (left) and MCP1 xcal vs. ycal calibrated
mask data (right).

of the expected mask position. To determine the position resolution of each MCP, we

select holes of different sizes near the center of the detector (where the beam is likely to be

concentrated). Then we perform a Gaussian fit for the vertical and horizontal projections

for each hole. Table 4.1 lists the widths of a sample hole peak in both the vertical (y) and

horizontal (x) directions for holes of the three different sizes, as well as the intrinsic position

resolutions for both MCPs found via extrapolation to a hole size of 0 mm. We assume that

the resolution is uniform across the entire MCP.

4.3.2 Beam Spot Reconstruction

We now have a mapping from the MCP corner signals to real position on the beamline.

Next, we can apply this mapping to the real data for each MCP and then use the resulting

MCP0 position and MCP1 position to calculate the position of the beam at the reaction

target using simple geometrical considerations. As in [140], we assume that the deflection of

111

 (mm)calx(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)5051015202530 (mm)caly(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)5051015202530 (mm)calx(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)5051015202530 (mm)caly(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10(cid:173)5051015202530MCP
0
0
0
0
1
1
1
1

Hole size (mm)

σx (mm) σy (mm)

3
2
1

→0 Intrinsic

3
2
1

→0 Intrinsic

1.5
1.3
1.2
1.0
1.5
1.3
1.0
0.73

1.6
1.5
1.3
1.2
1.5
1.2
1.0
0.77

Table 4.1: MCP position resolutions along both axes for several hole sizes in both MCPs, as
well as the intrinsic resolutions found be linearly extrapolating to a 0 mm hole size.

the beam due to the field from the permanent magnets at each MCP is negligible. Figure
4.31 shows the resulting beam spot for the 46Ar beam, which has FWHM of 16 mm in the

horizontal direction and 26 mm in the vertical direction.

This beam spot reconstruction gives us an event-by-event determination of the transfer
reaction location in the plane of the target for the 46Ar beam. As mentioned above, the
MCPs did not function properly for the 34Ar beam, and so we will assume the reaction locus

is at the center of the target plane for every event.

112

Figure 4.31: Reconstructed beam spot at the target for the 46Ar beam.

113

X (m)(cid:173)0.04(cid:173)0.0200.020.04Y (m)(cid:173)0.04(cid:173)0.03(cid:173)0.02(cid:173)0.0100.010.020.030.04051015202530354045Chapter 5

Data Analysis II: Cross Sections and

Reduction Factors

In God we trust, others must provide

data.

Edwin R. Fisher

The calibrated and corrected detector data do not by themselves tell us about the transfer

reactions of interest: we must perform more analysis to reach our goal. This chapter describes

the process of producing differential cross sections from calibrated HiRA, S800, and MCP

data.

We first merge the data via an event generation procedure in which data from separate

detectors are assembled into a single transfer reaction event. For each beam, we isolate the

(p, d) transfer reactions by gating on the incoming beam PID, on the appropriate recoil in

the S800 PID, and on deuterons in HiRA. We also gate on the appropriate pixel codes (see
Section 4.1.6) in HiRA, and we discard HiRA strips (or, in a few cases, entire detectors1) that
are not working properly. In the case of the 46Ar beam, we gather beam tracking information

from the MCPs. This merged event data is then written to a ROOT Tree, allowing for direct

study of the (p, d) transfer reactions.

Once we have gated on the (p, d) reaction channel, we calculate the reaction kinematics
1For instance, two telescopes had EB detectors that did not function correctly. This made pixelation

impossible, so the data from these telescopes was discarded.

114

as well as the recoil excitation energy. For the 46Ar beam, we can also include MCP beam

tracking to boost the deuteron energy resolution. We then generate angular distributions by

fitting peaks in the excitation energy spectra. Next, we normalize these angular distributions

with the incoming number of beam particles as well as corrections for detector performance

and geometry. This process yields the differential cross sections of interest. We use these

cross sections in combination with ADWA calculations to extract spectroscopic factors for

(p, d) reactions to different final states in the heavy argon recoil. Finally, we compare the

extracted SFs to shell model calculations to calculate reduction factors and study their

asymmetry dependence.

We note that the term “normalization” is used in two separate ways in this chapter.

First, we discuss normalization of angular distributions to produce our observable of interest,

differential cross sections. Later in the chapter, normalization refers to the comparison of

differential cross sections to ADWA reaction calculations in order to extract spectroscopic

factors. To avoid confusion, we advise the reader to consider the context in which this term

is used throughout the chapter.

5.1 Reaction Kinematics

After event generation, we examine the two-body kinematic relationship between the emitted

deuteron angle and its energy as detected in HiRA. This relationship does not depend on

any advanced underlying physics, and we can calculate it via simple conservation laws.

Furthermore, as depicted in Figure 3.2, the kinematics are sensitive to the final state of the

heavy recoil. Figures 5.1 and 5.2 show the kinematics data for each reaction system along

with several calculated kinematic lines corresponding to different final states in the argon

115

recoil. Table 5.1 shows the energy levels for both 33Ar and 45Ar below 5 MeV. The states

listed in bold are represented with kinematic lines in Figures 5.1 and 5.2. Although in most

cases the measured kinematics agreed with the calculated lines, for some telescopes (despite

our best efforts in calibration) we had to shift the measured energy in order to better match

the expected kinematics. These shifts were on the order of 100 keV for particles in the silicon

and 1 MeV for particles in the CsI. Because the goal of this experiment is measuring cross

sections involving known states rather than characterizing new states, these energy shifts

are inconsequential to our measurement.

In both reaction systems, resolution worsens going from forward to backward angles. This

is due to the higher energy particles at backward angles punching into the CsI crystals, which

have inherently worse resolution than the E silicon detectors. Kinematic broadening further

exacerbates this effect. This issue can at least partially be mitigated using the improved

angular resolution via the MCP beam tracking system. As mentioned in Chapter 4, the
MCPs did not function properly during the 34Ar beam run. In that case, we are largely
concerned with the forward-angle peak corresponding to the 33Ar ground state. Fortunately

this peak is almost entirely detected in the silicon detectors at forward angles, so the inferior

resolution at backward angles is not a major obstacle. Furthermore, the beam spot for the
34Ar was rather small (50 mm2), as pointed out in Chapter 4.

5.2 Excitation Energy

So far, this analysis has taken place entirely in the reference frame of the laboratory. By

transforming to the center-of-mass (COM) frame, we can reconstruct the excitation energy

of the heavy argon recoil left behind after the (p, d) transfer reaction. We use the fact that

116

33Ar

Energy (MeV)

0

1.359
1.798
2.439
3.154
3.361
3.456
3.819

J π
1/2+
3/2+
5/2+
3/2+
3/2+
5/2+
7/2+
5/2+

J π

5/2−, 7/2−
1/2−, 3/2−
1/2−, 3/2−

1/2−, 3/2−

1/2−, 3/2−

45Ar

Energy (MeV)

0
0.542
1.340
1.416
1.660
1.734
1.770
1.876
1.911
2.420
2.510
2.757
3.230
3.295
3.718
3.950
4.280
4.326
4.800

Table 5.1: Energy levels below 5 MeV as well as known spin and parity assignments for 33Ar
and 45Ar.

117

Figure 5.1: Measured kinematic curves for 34Ar(p, d)33Ar shown with calculations for com-
parison.

the kinematics of the deuteron detected in HiRA tells us the final state of the argon recoil,

as discussed in Section 5.1. So, we can use HiRA to identify the final state of the heavy Ar

recoil, and generate differential cross sections to individual final states.

We first must calculate the reaction Q-value, or the amount of energy released in the

reaction. Conceptually, the Q-value is simply the difference between the initial and final

kinetic energies of the system. Via mass-energy equivalence, we define the Q-value for the

reaction A(p, d)B as:

Q = mA + mp − md − m∗

B

(5.1)

where mA, mp, md, and m∗
B are the rest masses of the beam particle, the proton, the
deuteron, and the heavy recoil. The superscript on m∗
B denotes that B can have non-zero
excitation energy, and that we include this in the rest mass. Both the proton and deuteron

118

Lab Angle (Degrees)101520253035Lab Energy (MeV)1820222426283032343638Figure 5.2: Measured kinematic curves for 46Ar(p, d)45Ar shown with calculations for com-
parison.

have no particle-bound excited states, and we assume that the incoming projectile is in the

ground state. All of the ground-state rest masses of particles measured in this experiment
are known, so m∗
immediately have the Q-value.

B is the only variable quantity in Equation 5.1: once we have m∗

B, we

We determine m∗

mentum) to determine the deuteron kinetic energy in the COM frame (tCOM

B by first performing a Lorentz transformation (using the beam mo-
). Because

d

this transformation has no effect on momentum vectors perpendicular to the beam momen-
tum, the relationship between tCOM

and the kinetic energy in the lab depends on both the

d

deuteron energy as well as its angle. Applying conservation of energy and momentum, we

119

Lab Angle (Degrees)10152025303540Lab Energy (MeV)1416182022242628303234derive a relation between tCOM

d

and m∗
B:

(m∗

B)2 = (M − md)2 − 2M tCOM

d

(5.2)

where M is the invariant mass of the system which we calculate using the beam energy,

the beam rest mass, and the target rest mass. Now, we have a complete prescription for

calculating the Q-value event-by-event starting from the deuteron energy and angle in the

laboratory frame.

We can easily calculate the expected Q-value for the ground state using masses from

literature tabulations like [134]. Therefore, we can compute the excitation energy of B for

a given event by simply subtracting the Q-value for the event from the known ground-state

Q-value. Once we have the event-by-event excitation energy, we can generate excitation

energy spectra to unambiguously identify final states in the heavy Ar recoil.

Figures 5.3 and 5.4 show excitation energy spectra for 33Ar and 45Ar (respectively). Fig.
5.3 depicts excitation energies for 33Ar at forward angles (the approximate angular range

of the main peak of interest) in the center-of-mass frame. We see a prominent peak at 0

MeV with a standard deviation of about 240 keV, which agrees with the expected resolution
from simulation. As shown in Table 5.1, the lowest lying excited state of 33Ar is at 1.359

MeV, so we are confident that the 0 MeV peak is well-separated from any excited state
contributions. For the 46Ar(p, d)45Ar case, each panel of Figure 5.4 shows excitation energy

spectra calculated both with (red) and without (black) MCP beam tracking for a given

angular range. On the left panel (forward angles), we see a peak at 0 MeV, but cannot
separate the ground state of 45Ar from the first excited state at 0.542 MeV. In any case, the

peak at 0 MeV is distinguishable from higher-lying peaks. The right panel, corresponding to

120

Figure 5.3: Measured 34Ar(p, d)33Ar excitation energy spectrum for θCOM < 10 degrees.

more backward angles and worse experimental resolution, shows the 0 MeV peak with a long

tail at higher excitation. We note that the spectra calculated without MCP beam tracking

(in black) are scaled down for comparison to spectra that did use the MCP beam tracking.

This is necessary because of the imperfect efficiency of the MCP beam tracking system.

Figure 5.4 also depicts the excitation energy both with and without MCP beam tracking

in the angle determination. At forward angles (left panel) the excitation energy resolution
for the ground state of 45Ar is similar with and without beam tracking, but for backwards

angles (right panel) the MCP beam tracking provides a boost in resolution. Fig 5.5 shows

explicitly the improvement of MCP beam tracking for large angles.

To generate angular distributions, we gate the excitation energy on several angular slices

in the center of mass. We then fit these spectra with a sum of several Gaussian functions,

each one corresponding to a separate peak in a given reaction system. All peaks are fit

simultaneously to account for overlap between different peaks. Integrating these Gaussian

121

Ar excitation energy (MeV)33024Counts per 100 keV050100150200250300350400Ar33Ar(p,d)34Figure 5.4: Measured 46Ar(p, d)45Ar excitation energy spectra for several different angular
ranges. We show the excitation energies calculated without MCP beam tracking (black) and
with MCP beam tracking (red).

Figure 5.5: Widths of the Gaussian fits for 46Ar(p, d)45Arg.s.. The black (red) points corre-
spond to data without (with) MCP beam tracking..

122

Ar excitation energy (MeV)45(cid:173)2024Counts per 100 keV020406080100120140160180200 < 8COMθAr45Ar(p,d)46With MCPWithout MCPAr excitation energy (MeV)45(cid:173)2024Counts per 280 keV20406080100120140160180 > 20COMθLab angle (degrees)102030Exc. energy resolution (MeV)0.40.60.811.2No MCPWith MCPfunctions yields the number of counts for each peak, which corresponds to the number of

deuterons emitted from transfer reactions where the argon recoil occupied one (or more)

particular state(s).

Because the ground state peak in the 33Ar excitation spectra is relatively well separated,
the fitting is straightforward. The 45Ar case is slightly more complicated. At backward
angles, the 45Ar 0 MeV peak blends into a smooth spectrum without distinguishable peaks

at higher excitation (as in the right panel of Figure 5.4). To address this problem, we fit

the uncontaminated lower-energy half of the 0 MeV peak to get the Gaussian function. At

forward angles, we expect to see some contribution in the 0-MeV peak from the l = 1 first
excited state in 45Ar at 542 keV. Unfortunately we are unable to resolve these two peaks

cleanly. However we can see a distortion in the 0-MeV peak which indicates the presence

of the unresolved first-excited-state peak, as shown in the left panel of Figure 5.4. We deal

with this issue by fitting the 0 MeV peak with two Gaussian functions in order to constrain

the excited state contribution to the cross section. We also use the angular distribution at

more backward angles (where the cross section for the excited state drops significantly) to

check these results. Section 5.5 features more details about the excitation energy fits.

Now, our goal is to produce differential cross sections by normalizing the angular distri-

butions to account for detector performance and beam purity. Before taking this step, we

consider the possibility of non-(p, d) contamination of our spectra from background.

5.3 Background Subtraction

Because the reaction target we used in this measurement is polyethylene, it is possible that

some of our deuteron and heavy recoil coincidences are due to non-transfer reactions on

123

Figure 5.6: S800 PID (left) and deuteron kinematics (right) for the carbon target runs with
the 46Ar beam. The kinematics plot is gated on the 45Ar residue indicated in the S800 PID
plot.

carbon nuclei. To address contributions from reactions on carbon, we used a 17 mg/cm2

C target to evaluate the background. This target is substantially thicker than our reaction

targets, so any carbon-induced reactions potentially coming from the reaction target should

be plainly visible using the thick C target.

Figure 5.6 shows the S800 PID spectrum for the C target data runs, as well as the
deuteron kinematics for these runs gated on the 45Ar residue. The number of background

counts in the kinematics plot is negligible compared to the statistics measured with the
polyethylene reaction target. The same is true for the case of the 34Ar beam on C target

runs. Therefore, a background subtraction is not necessary.

124

TOF(cid:173)OBJ (ns)(cid:173)30(cid:173)25(cid:173)20(cid:173)15(cid:173)10E (arb. units)∆140016001800200022002400260028003000Ar45 (degrees)labθ10203040Lab energy (MeV)14161820222426283032345.4 Normalization

The results of our analysis to this point consist of detected deuterons with known angle and

energy that have been emitted from (p, d) transfer reactions for two separate beams. In order

to produce our desired differential cross sections, we must account for the efficiency of our

experimental setup and also the number of argon-hydrogen beam-target interactions. This

process is called normalization, and we discuss it in detail for both beams in the following

section. First, we quickly introduce the general plan for normalization, and then delve into

the various necessary corrections.

For the (p, d) transfer reactions measured here, the differential cross section is given by:

(cid:18) dσ

dΩ

(cid:19)

θi

=

Nd(θi)

NtargetNbeam

(5.3)

where θi is the ith angular bin, Ntarget is number of hydrogen nuclei per square centimeter in
the polyethylene reaction target, Nbeam is the total number of beam particles that impacted
the target, and Nd(θi) is the number of deuterons emitted from the reaction in θi.

Ntarget is fixed throughout the experiment for a given target thickness. We calculate

Nbeam with the following relation:

Nbeam = η

NDet
beam
εMCP

(5.4)

where NDet

beam is the number of beam particles counted by the MCP1 scaler, η is the purity

of the beam, and εMCP is the MCP1 scaler efficiency.

To get Nd(θi), we count the number of deuterons detected within the given angular bin

125

θi and then adjust for detector efficiencies in the following way:

Nd(θi) = NDet

(θi)/εHiRA(θi)
εDAQεOBJεMCPTrack

d

(5.5)

d

where NDet

(θi) is the number of deuterons detected in our experimental setup at the angular
bin θi, and εHiRA(θi) is the angle-dependant efficiency of HiRA detecting a deuteron in θi
(and includes the solid angle subtended by θi). The terms in the denominator are overall
efficiency corrections: εDAQ is the live time of the data acquisition (DAQ) system, εOBJ
is the efficiency of the OBJ scintillator used for beam and S800 particle identification, and
εMCPTrack is the overall efficiency of the MCP beam tracking system2.

We first will calculate the assorted efficiency corrections needed to determine Nd(θi) and

Nbeam. Then we will use Equation 5.3 to compute differential cross sections.

5.4.1 DAQ Live Time

Recording an event with the DAQ takes a nonzero amount of time. If another event arrives

while the previous event is being processed, this new event is lost completely. The fraction

of time that the DAQ is available to record data is called the live time (the fraction of time

the DAQ is not available for data taking is fittingly called the dead time). Because transfer

reaction events can occur during the DAQ dead time, we use the live time (εDAQ) as a
correction factor in Equation 5.5.

During the experiment a scaler module counted logic signals for a variety of electronics

circuits, which allows us to track how much those circuits are firing. For example, we recorded

the scaler rates of the OBJ and HiRA detectors. We can split a logic signal coming from
2This quantity is distinct from εMCP, the scaler efficiency of the MCP which comes from the MCP time

signal.

126

Figure 5.7: Live time of the DAQ system plotted against run number.

the detector, sending one side directly into the scaler module as the raw scaler. We block

the other side with the overall DAQ trigger before sending it to the scaler module as the

live scaler. In other words, the live scaler only fires when (a) there is a logic pulse in the

corresponding raw scaler and (b) the DAQ is available for recording data. The ratio of the

live scaler to the raw scaler is a measure of the total DAQ live time.

In this analysis, we use the HiRA scaler to compute the live time. Figure 5.7 shows the

live-to-raw scaler ratio for the HiRA trigger plotted against the run number. On average,
the live time is 98.3% for the 46Ar beam, and 88.1% for the 34Ar beam.

127

Run number300400500600Live time0.50.60.70.80.91DAQ live timeAr34Ar465.4.2 HiRA Efficiency

HiRA does not have perfect angular coverage, and therefore deuterons emitted at an angle

not covered by HiRA are not detected. To correct for this, we simulated the geometrical

acceptance of HiRA using the NPTool Monte Carlo simulation package (as described in Sec-

tion 3.3.3). The outcome of this simulation is εHiRA(θi), the fraction of deuterons emitted
within the θi angular bin that are detected in HiRA. NPTool allows us to easily remove in-
dividual faulty detector elements (e.g. missing silicon strips or poorly biased detectors) from

the Monte Carlo calculations to fold these effects into the resulting efficiency. Accounting

for the variable electronics dead time across the array requires a more subtle approach.

Individual noisy strips in the silicon can clog up the fast clear circuits in the HiRA ASIC
electronics, resulting in a high electronics dead time3. This problem is notably pronounced

in the thin, high-capacitance DE detectors. We measured this electronics dead time during

the experiment by pulsing the silicon detectors at a fixed frequency and an energy far above

what was relevant for the transfer reaction deuterons. By comparing the number of measured

pulses in each silicon strip to the total number of expected pulses (found by multiplying the

known pulser frequency by the total run-time), we can extract the electronics dead time

for individual strips. Figure 5.8 shows that the electronics dead time varied not only from

detector to detector but also within individual detectors. This is due to each detector having

two ASIC chips (with each chip responsible for 16 channels) that can have different amounts

of noise.

We include the effects of individual-strip electronics dead times into the Monte Carlo

simulation via an extra step of random sampling for each event. We see a pronounced effect
3Note that here we mean the dead time of just the HiRA electronics system, not the overall DAQ dead

time discussed in Section 5.4.1.

128

Figure 5.8: Strip-by-strip electronics live times for three example HiRA DE detectors. We
see variation from detector to detector, as well as within individual detectors.

129

DE strip number0102030Live time00.10.20.30.40.50.60.70.80.91Telescope 0Telescope 6Telescope 9Figure 5.9: HiRA geometrical efficiency for 46Ar(p, d) without (black) and with (pink) dead
time corrections as described in the text.

on the final geometrical efficiency, in particular at forward angles below the E punch through

energy. Figure 5.9 shows the efficiency dependence on laboratory angle from simulations with

and without the dead time corrections.

5.4.3 Beam Detector Efficiencies

Several important detectors lay directly in the path of the incoming beam: the OBJ scintil-

lator upstream from the S800 (which identifies the incoming beam and the heavy recoil), and

the MCPs (which count the number of incoming beam particles, and also track the beam

trajectory). These detectors are not perfectly efficient due to the sustained high beam rate.

Clearly, we require these detector efficiencies in order to properly normalize the data.

130

Deuteron lab angle (degrees)10203040Efficiency00.050.10.150.20.250.30.350.40.45UncorrectedCorrected for dead timeWe first assume that both the RF signal the E1 scintillator in the S800 are 100% efficient.

The RF signal is trivially 100% efficient, and the E1 assumption is grounded in the fact that

due to the presence of the blocker in the S800, the rate of particles hitting the E1 scintillator

is multiple orders of magnitude below the full beam rate. Then, we gate either the RF

or E1 time on the corresponding beam detector signal (OBJ, MCP time, or MCP position

signals). The relative presence of any of these detectors when compared to the RF or E1

gives a measure of the corresponding detector efficiency.

5.4.3.1 OBJ

We monitored the OBJ scintillator efficiency during the course of the experiment to track

performance deterioration related to radiation damage. If necessary, we raised the scintillator

voltage or adjusted the scintillator position so that the beam was focused on a different spot.

Figure 5.10 shows the ungated and gated RF time spectra for the OBJ.

Figure 5.11 depicts the OBJ efficiency over the course of the experiment. For the total

normalization, we use the efficiency values that result from combining data from all runs.
For the 34Ar beam, this efficiency is 90.1%, and for the 46Ar beam it is 78.7%.

5.4.3.2 MCP Time

The MCP scalers (triggered by the MCP TDC signals) count the number of particles that

impinge upon each MCP. The TDC signal from each MCP triggers the MCP scaler circuits.

Therefore, the efficiency of the MCP scalers are given by the respective efficiencies of the

MCP timing signals. We are especially concerned with the MCP1 scaler because of its

proximity to the target, but we also measure MCP0 as a consistency check (as discussed

further in Section 5.4.5).

131

Figure 5.10: RF time plots that illustrate the OBJ efficiency. The black histogram is the
ungated RF time for an example run, and the red histogram is the RF time for the same
run gated on valid counts in the OBJ.

Figure 5.11: OBJ efficiency plotted against the experiment run number.

132

RF Time (ns)0100200300400500Counts0200400600800100012001400160018002000RF UngatedRF Gated on OBJRun number300400500600Efficiency0.50.60.70.80.91OBJ EfficiencyAr34Ar46Figure 5.12: MCP0 (red) and MCP1 (blue) scaler efficiencies plotted against the experiment
run number.

We compare the ungated S800 (E1 scintillator) data with data gated on each MCP TDC

signal to compute the efficiency. The resulting efficiencies are shown in Figure 5.12. Increases

in MCP efficiency correspond to increases in MCP bias voltage.

5.4.3.3 MCP Position

For the 46Ar beam, we need to know the overall efficiency of the MCP beam tracking system

(εMCPTrack). This quantity expresses what fraction of the data has valid beam positions
in both MCP0 and MCP1. In the case of the 34Ar beam, we simply set εMCPTrack to 1 in
Equation 5.5.

Each MCP has four corner signals that together give us the position of the beam particle

in the plane of the aluminized Mylar foil sitting in the beam path. A naive approach to

133

Run number300400500600Efficiency0.50.60.70.80.91MCP0MCP1Ar34Ar46Figure 5.13: Sum of the MCP1 corner signals, as well as a red line indicating the threshold
value.

calculating the MCP tracking efficiency would be to set a threshold on each individual

corner signal for a given MCP and count how many events have all four corners above the

threshold (compared to counts in the S800). However, this prescription results in throwing

out legitimate data. Some counts of interest might have above-threshold signal in only 2

or 3 MCP corners, depending on where the charge deposition occurs. So, we instead set a

threshold on the sum of all the corner signals in an MCP, as shown in Figure 5.13. The

prominent peak below the threshold corresponds to events with pedestal data in all four

corners, which are exactly the events we wish to discard.

We also cut on the calculated MCP position to remove non-physical data that results

from negative values in the pedestal-subtracted corner signals (due to the non-zero width of

the pedestal peak) that are used to calculate the raw x and y positions.

134

MCP1 corner signal sum (arb. units)0.511.5310×Counts05101520310×Now we can calculate the overall beam tracking efficiency using the same approach as

with the MCP scalers. We calculate the ratio of counts in the S800 gated on both MCP

corner-sum cuts and both MCP position cuts to the ungated S800 counts. The resulting

efficiency εMCPTrack is 81.1%.

5.4.4 Beam Purity

To correct for the incoming beam purity, we simply generate beam identification plots (see

Figs. 4.22 and 4.23) for each data run and calculate the relative presence of the desired

beam species. The resulting beam purity for each run is shown in Figure 5.14. The average
beam purities were 94% for the 34Ar beam and 99.6% for the 46Ar beam. The beam purity

for each species remained consistent throughout the experiment.

5.4.5 Beam Normalization

We used the MCP scalers to count the incoming beam particles and determine the NDet
beam
term needed for the absolute differential cross sections (see Equations 5.3 and 5.4). Unfor-

tunately, the experiment suffered from high noise in the MCP electronics, which adversely

affected the scalers. Much of the problem was due to multiple-firing of the circuit for a

single beam particle. In order to address this issue, we set up another scaler circuit, which

we will refer to as the cleaned-up (CU) scaler, that was blocked off for 200 ns after the arrival

of a signal from the MCP TDC. The CU scaler essentially vetoed the multiple-firing, and

significantly reduced the noise.

However, the CU scaler is not completely free of noise. To further understand the effect

of noise on the MCP scalers we performed beam “ramps” in which the beam rate was

135

Figure 5.14: Purity of both argon secondary beams over time.

136

Run number300400500600Beam purity0.80.850.90.951Ar34Ar46changed incrementally to determine how much of the scaler data was real and how much

was electronics noise. Figure 5.15 shows plots of the beam ramp scaler data for both MCP0

and MCP1. Clearly, there is still a positive scaler rate for both MCPs even when the beam

is completely attenuated (i.e. when there is no beam present at all). We use this rate as a

measure for the noise in each MCP.

Once we have the rate of noise for each MCP, we then subtract it from the MCP CU

scaler rates in each run of the data to yield a noise-corrected MCP scaler. We use this noise-

corrected scaler as well as the MCP scaler efficiencies (found in Section 5.4.3.2) to calculate

the number of beam particles on target for each run, which we then sum together to get
NDet
beam. In principle, the number of beam particles detected at MCP0 and MCP1 should be
consistent within 5 to 10 %. A small number of runs had large deviations (20%) between

the number of beam particles on MCP0 and on MCP1, and so these runs were discarded.
For the 46Ar beam, the result is 8.04 × 1010 total beam particles (the associated uncertainty

will be discussed in the next chapter).

Unfortunately, the CU scalers were only implemented towards the end of the 34Ar beam.
So while there are a few runs of 34Ar data that have these scalers, most of the runs for this
beam do not. Therefore we need a different approach to normalize the 34Ar data. We must
also keep in mind that the ultimate goal of this work is a comparison between the 34Ar(p, d)
and 46Ar(p, d), so the normalization procedures must be consistent between the two reaction

systems.

Our approach is to use the 34Ar data runs (many of which were for calibration) that do

have CU scalers and find other scalers (which exist in all of the data) that correlate well.

Then, we can fit the relationship and calculate expected values for the CU scaler. Figure

5.16 shows two such scaler correlations. The left plot shows correlations for MCP1 relative

137

Figure 5.15: Detected MCP cleaned-up (CU) scaler rates vs. beam attenuation for both
MCP0 (red) and MCP1 (blue). When the beam attenuation is 1, the beam is completely
removed from the experimental setup.

138

Beam attenuation00.51MCP CU scaler rate (kHz)0200400600MCP0MCP1MCP0 Noise RateMCP1 Noise RateFigure 5.16: (a) MCP1 CU scaler correlations with the OBJ scintillator scaler (top, blue)
and the CsI OR of OR scaler (bottom, red). We use the resulting linear fits (dashed green
lines) to calculate the expected CU scaler for data runs in which it was not directly recorded.
(b) Percentage difference between the expected MCP1 CU scaler calculated from the OBJ
scaler and the CsI OR of OR scaler. They agree within 10%.

to the CsI OR of OR scaler as well as the OBJ scaler. We then use the resulting fits to

calculate expected CU scaler rates. The right plot shows the percentage difference between

the CU scaler calculated from the CsI scaler and the CU scaler calculated from the OBJ
scaler. Each point corresponds to an 34Ar data run that lacks the CU scaler. Clearly, using

either the CsI scaler or the OBJ scaler yields similar results within less than 10%.

Using the calculated CU scalers in combination with the run-by-run MCP scaler efficiency
and beam purity, we calculate 6.28 × 1010 total 34Ar beam particles from the OBJ scaler,
and 6.09 × 1010 from the CsI scaler. These values agree within 3%, so we simply use the
average of the two (6.18 × 1010) as our measure of the total number of beam particles on

target.

We applied a similar approach to the 46Ar data to demonstrate that this method produces
consistent results for both beams. Using this correlation method, we get 8.13 × 1010 46Ar

139

OBJ scaler rate (kHz)0200400600800MCP1 CU scaler rate (kHz)200400CsI scaler rate (kHz)0200400MCP1 CU scaler rate (kHz)200400Run number350400450Percentage difference051015beam particles from the OBJ scaler and 7.99 × 1010 from the CsI scaler. These values are
within 1% of the 8.04 × 1010 value calculated above with the measured CU scalers. Therefore,

we are confident that our normalization technique yields consistent results for both reaction

systems.

5.4.6 Overall Normalization Uncertainty

In order to assess the uncertainty of the normalization procedure, we divide the data up into

five groups and calculate cross sections for each. The fluctuation of the cross section across

separate data groups gives us an estimate of the overall normalization uncertainty.

Figure 5.17 presents the absolute differential cross section at 7 degrees in the center-of-
mass frame for both the 46Ar (to 45Arg.s. and 45ArE*=0.542 MeV, shown in red) and 34Ar
(to 33Arg.s., shown in blue) reaction systems. The solid lines indicate the cross section value
taken from the complete data set. Although there is some fluctuation from group to group,

the total normalization uncertainty is on the order of 10% (indicated by the dotted lines).

5.5 Differential Cross Sections and Spectroscopic Fac-

tors

After normalization, we have a set of differential cross sections that we can compare to

theory to elicit nuclear structure information. Here, we present the final differential center-
of-mass cross sections for 34Ar(p, d) and 46Ar(p, d) reactions. We also extract SFs from the

differential cross sections by normalizing the data to (p, d) reaction calculations generated

in the ADWA framework using the TWOFNR code as described in Chapter 2. For the

140

Figure 5.17: Absolute differential cross sections at 7 degrees in the center-of-mass frame for
several different subsets (or groups) of data. The 34Ar cross section is shown in blue, and
the 46Ar in red. The solid line corresponds to the cross section value for the entire data set,
and the dotted lines are 10% from the solid line.

ground state transitions, we use two different optical models in the ADWA calculations: the

CH89 global optical potential, and the JLM microscopic optical potential constrained by

Hartree-Fock density calculations. Each of these two optical models yields different SFs, so

we label them SF(CH89) and SF(JLM), accordingly. The JLM calculations are not available

for excited states because the nucleon density calculations are more complicated, so for the
45Ar excited-state calculations we only use the CH89 potential.

The depicted uncertainty for each angular bin in the differential cross sections includes

statistical error as well correlated parameter uncertainty from the Gaussian fit. The cross

section also has an overall normalization uncertainty (discussed in Section 5.4.6) which is

only taken into account during the spectroscopic factor extraction.

141

Group number12345 (mb/sr)Ω/dσd0510152025Ar46Ar34Figure 5.18: Differential cross section for 34Ar(p, d)33Arg.s. in the COM frame. The blue
circles are the data from the present work. The lines correspond to TWOFNR calculations
normalized to the first peak using CH89 (dotted, green) and JLM (dashed, red) optical
potentials. The calculations shown are multiplied by the best fit spectroscopic factors.

5.5.1 34Ar(p, d)

Figure 5.18 shows the measured differential cross section in the COM frame for 34Ar(p, d)33Arg.s.,
along with SF-normalized single-nucleon transfer reaction calculations. The l = 0 shape of

the data matches the calculations well, as expected. We observe a large peak at forward
angles, and another (smaller) peak at ≈ 18 degrees. The relative uncertainty on lower cross

section points between 10 and 15 degrees is high compared to larger cross section points.

This is due to decreased statistics, as well as increased parameter correlations (i.e. higher

sensitivity to small parameter shifts). Table 5.2 lists the cross section values and associated

uncertainties.

142

COMθ5101520 (mb/sr)Ω/dσd(cid:173)110110210g.s.Ar33Ar(p,d)34CH89JLM34Ar(p, d)33Ar, E* = 0 MeV

θCOM (degrees) dσ/dΩ (mb/sr) Uncertainty (mb/sr)

4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
16.5
17.5
18.5
19.5
20.5
21.5

8.11
8.98
6.38
5.03
3.15
1.75
0.82
0.71
0.51
0.69
0.65
0.66
0.86
0.94
1.29
1.23
0.84
0.90

0.77
0.49
0.29
0.21
0.16
0.12
0.29
0.08
0.07
0.18
0.07
0.07
0.09
0.17
0.10
0.13
0.17
0.25

Table 5.2: Differential cross section for 34Ar(p, d)33Arg.s. in the COM frame with statistical
uncertainty.

We follow the methodology outlined in [26] and extract the spectroscopic factor by com-

paring the TWOFNR calculations to the most forward peak. The normalization procedure
is a χ2-minimization using the six data points located in the first peak. We get a 10% uncer-

tainty from the minimization and another 10% uncertainty from the overall normalization

described in the previous section. Combining these two sources of uncertainty yields 14%

relative error. Figure 5.18 shows normalized calculations for both the CH89 global optical

model (dotted, green) and the JLM-HF microscopic optical model (dashed, red). We extract
SF(CH89) = 1.00 ± 0.14 and SF(JLM) = 0.73 ± 0.10. Comparing to SF(LBSM) = 1.31
yields Rs(CH89) = 0.76 ± 0.11 and Rs(JLM) = 0.56 ± 0.08.

143

5.5.2 46Ar(p, d)

To extract the spectroscopic factor corresponding to the 45Ar ground state, we must first

disentangle the cross section contributions from the ground state and first excited state.

Although we are unable to cleanly resolve each peak, at forward angles we see a “shoulder”

at slightly positive excitation energy on the 0-MeV peak that most likely represents minor

population of the first excited state. Our approach is to fit the 0-MeV peak with two

Gaussians (one for each state). We generate excitation energy spectra for different angular

slices and fit these two Gaussians (as well as higher lying peaks) simultaneously so that the

sum of all Gaussian functions reproduces the measured histogram. We also fix the widths

of the two Gaussian functions to be identical, since the experimental response for each state

should be the same. Figure 5.19 depicts an example excitation spectrum. The individual

Gaussian fits are drawn in blue, and their sum in red. The best-fit means for the ground

state and first excited state are indicated in green. Although we do not exactly reproduce

the correct values of 0 and 0.542 MeV, the values from the fit are reasonably close and there
are no other nearby states in 45Ar. As described above, we integrate each Gaussian peak

to generate angular distributions at forward angles for both the ground state and the first
excited state, and then perform a χ2-minimization to get the SFs. The resulting spectroscopic
factors are SF(CH89) = 4.77 ± 0.67 and SF(JLM) = 3.59 ± 0.50 for the ground state, and
SF(CH89) = 0.38 ± 0.12 for the first excited state. Comparing to the LBSM values of 5.16

for the ground state and 0.78 for the first excited state, we get reduction factors of Rs(CH89)
= 0.92 ± 0.13 and Rs(JLM) = 0.70 ± 0.10 for the ground state, and Rs(CH89) = 0.49 ±
0.15 for the first excited state. As in the 34Ar case, we combine 10% statistical uncertainty

with 10% overall normalization uncertainty for 14% total uncertainty.

144

Figure 5.19: 45Ar excitation energy spectrum for one angular bin in which the 0 MeV peak
is fit with two Gaussians. Individual Gaussian fits are drawn in blue, and their sum in red.

Although we extract the SFs from only the most forward angles, we can still use other

data in the differential cross section to check our results. The expected differential cross

section for the p3/2 first excited state drops with increasing COM angle when compared to
ground state cross section. In Figure 5.20, we show ADWA calculations for transfer into the
ground state (red) and first excited state (blue) of 45Ar. For illustration, each calculation

is scaled by the LBSM predicted SF (5.16 for the ground state, 0.78 for the excited state).

The green dashed line shows the ratio between the two cross sections. Between 11 and 18

degrees, the ground state to first excited state ratio reaches a relative maximum of about an

order of magnitude. Therefore, we assume that in this angular range the 0-MeV peak purely
(within 10%) consists of the 45Ar ground state. Normalizing the ground state calculations to

the data in this angular range yields SF(CH89) = 4.67 and SF(JLM) = 3.43. These results

145

Ar excitation energy (MeV)45(cid:173)1012Counts per 200 keV0102030405060 < 6COMθ5 < (cid:173)15 keV580 keVFigure 5.20: ADWA calculations for 46Ar(p, d)45Ar for the ground state (red) and first
excited state (blue) of 45Ar. The ratio of the two cross sections is given by the green dashed
line.

are well within the stated 14% uncertainty of the SFs extracted from the forward-angle

double-Gaussian fits.

Fig. 5.21 and Table 5.3 show the final differential cross section for the combined (p, d)

transfer to both the ground state and the 542 keV excited state. The cross section drops

steadily with increasing center-of-mass angle without any significant peaks. We note that

the shape of the ground state calculation matches the data quite well with the exception of a

small enhancement at the most forward angles. This is exactly the region at which the first

excited state calculation has a maximum. This, in combination with the distorted shape of

the 0-MeV peak described above, gives us confidence that we are in fact seeing contribution

from the first excited state.

146

 (degrees)COMθ510152025 (mb/sr)Ω/dσd1107/2f3/2pRatioFigure 5.21: Differential cross section for 46Ar(p, d)45Arg.s.,E*=0.542 MeV in the COM frame.
The blue circles are the data from the present work. The lines correspond to TWOFNR
calculations for two different final 45Ar states normalized to the SFs which were obtained as
described in the text.

147

COMθ510152025 (mb/sr)Ω/dσd(cid:173)110110210g.s., E* = 0.542 MeVAr45Ar(p,d)46 (0 MeV)7/2f (0.542 MeV)3/2pTotal46Ar(p, d)45Ar, E* = 0, 0.542 MeV

θCOM (deg.) dσ/dΩ (mb/sr) ∆dσ/dΩ (mb/sr)

3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
16.5
17.5
18.5
19.5
20.5
21.5
22.5
23.5
24.5
25.5

15.36
14.34
14.21
12.75
12.68
12.02
10.79
10.85
11.31
8.94
8.34
9.47
6.25
5.32
8.29
6.61
5.33
4.42
5.55
3.64
2.60
2.39
2.41

2.92
2.56
1.46
1.73
1.68
1.12
1.42
1.42
0.48
0.78
0.80
1.05
0.58
0.68
1.27
0.91
0.75
0.58
0.95
0.60
0.42
0.41
0.56

Table 5.3: Combined differential cross section for 46Ar(p, d)45Ar to states at 0 and 0.542
MeV in 45Ar in the COM frame with uncertainty.

148

Isotope ∆S (MeV) SF(LBSM)
34Ar
46Ar

Rs(JLM)
0.56 ± 0.08
0.70 ± 0.10
Table 5.4: Extracted spectroscopic factors and reduction factors for both 34Ar and 46Ar.
Results are shown using both the CH89 global model as well as the JLM microscopic model.

SF(CH89)
1.00 ± 0.14
4.77 ± 0.67

Rs(CH89)
0.76 ± 0.11
0.92 ± 0.13

SF(JLM)
0.73 ± 0.10
3.59 ± 0.50

12.40
-10.63

1.31
5.16

5.6 Reduction Factor Asymmetry Dependence

Table 5.4 shows a summary of the extracted SFs and reduction factors. We compare the

current work to the previous 33 MeV/u transfer measurement [55] using both the CH89 global

potential and the JLM microscopic potential in Figures 5.22 and 5.23, respectively. The green

squares represent reduction factors from this work, and the open red circles are from [55]. For

both optical potentials, the reduction factor magnitudes are comparable between the current

work and the low-energy measurement. More importantly, the asymmetry trend (indicated

by the colored lines) from the current work agrees with the previous measurement with either

potential. We conclude that with a consistent analysis methodology, single-neutron pickup

reactions give consistent results for the same reactions at different energies.

In Figure 5.25, we compare both the low and high energy transfer reduction factors (red

circles and green squares, respectively) to those extracted from knockout reaction data (blue

triangles). We see a clear discrepancy between the asymmetry trend in the transfer data

(red and green lines) to the one in the knockout data (blue line) [52]. We conclude that

our 70 MeV/u single-neutron pickup reactions do not reproduce the systematic asymmetry

dependence seen in the single-nucleon knockout results. Although we plot the CH89 results

to illustrate the disagreement, it persists when using the JLM as well (even though the

positive-∆S reduction factors are closer in magnitude to the knockout).

Finally, we compare our results to recent quasi-elastic proton scattering work in Fig-

149

Figure 5.22: Asymmetry dependence of reduction factors using the CH89 global optical
model. We show reduction factors from the current work (green squares) as well as the 33
MeV/u measurement (open red circles). Each line represents the best fit to the corresponding
reduction factor data.

150

S (MeV)∆(cid:173)20(cid:173)1001020Reduction Factor00.20.40.60.811.233 MeV, CH8970 MeV, CH89Ar46Ar36Ar34Figure 5.23: Asymmetry dependence of reduction factors using the JLM microscopic optical
model. We show reduction factors from the current work (green squares) as well as the 33
MeV/u measurement (open red circles). Each line represents the best fit to the corresponding
reduction factor data.

151

S (MeV)∆(cid:173)20(cid:173)1001020Reduction Factor00.20.40.60.811.233 MeV, JLM70 MeV, JLMAr46Ar36Ar34Figure 5.24: Asymmetry dependence of reduction factors from transfer reactions using CH89
(open green squares and open red circles) and from knockout reactions (blue triangles). Each
line represents the best fit to the corresponding reduction factor data.

152

S (MeV)∆(cid:173)20(cid:173)1001020Reduction Factor00.20.40.60.811.233 MeV, CH8970 MeV, CH89n(cid:173)knockoutAr46Ar36Ar34Ar32Figure 5.25: Asymmetry dependence of reduction factors from transfer reactions using CH89
(open green squares and open red circles), from knockout reactions (blue triangles), and from
(p, 2p) reactions (pink crosses). All data points represent measurements on oxygen isotopes
except for the open green squares, which come from the present work on argon isotopes.

ure 5.24. The red open square (circle) points show reduction factors from single-neutron

(proton) transfer measurements [53], while the blue triangles that face downward (upward)

correspond to neutron (proton) knockout measurements [52]. The pink crosses indicate re-

duction factors from (p, 2p) quasi-elastic scattering data from [75] and [76]. Lastly, the green

squares are from the current work using the JLM microscopic model. All points in this figure

represent reduction factors from oxygen isotopes except the green squares, and each colored

line represents a best fit to the data of the corresponding color. We see that our results show

an asymmetry dependence consistent with the one seen in the oxygen transfer data in [53]

as well as the one seen in the (p, 2p) data from [75] and [76].

153

S (MeV)∆(cid:173)20020Reduction Factor00.20.40.60.811.2This work (JLM)n(cid:173)transferp(cid:173)transfern(cid:173)knockoutp(cid:173)knockout(p,2p)Chapter 6

Summary and Conclusions

All of physics is either impossible or

trivial. It is impossible until you

understand it, and then it becomes

trivial.

Ernest Rutherford

In this work, we explored the asymmetry dependence of single-particle structure across
the argon isotopic chain by studying two argon isotopes: 34Ar and 46Ar. Our goal was to

extract spectroscopic factors (SFs), which quantify the occupancy of a single-particle orbital,

in order to constrain the influence of nucleon-nucleon correlations along the argon isotopic

chain. Previous transfer measurements of single-particle structure in the argon isotopes indi-

cate weak asymmetry dependence in the reduction factor, whereas knockout measurements

of these isotopes show a strong dependence. This discrepancy between transfer and knockout

techniques is an unresolved problem that has consequences on both our understanding of

nuclear structure as well as the theoretical reaction models that describe these experimental

techniques. To address this inconsistency, we performed (p, d) measurements on argon iso-

topes at 70 MeV/u, a much higher beam energy than the previously used 33 MeV/u. This

is the first high energy transfer reaction measurement for nuclei with this much asymmetry.

If the 70 MeV/u results disagree with the 33 MeV/u trend, this would indicate deficiency

in the current understanding of the transfer mechanism. To ensure a consistent comparison

154

with the lower energy measurement, we used the adiabatic distorted wave approximation

(ADWA) as a theoretical framework to extract SFs.

To obtain the SFs, we measured (p, d) single-neutron transfer reaction differential cross

sections on each isotope. We determined these differential cross section in a kinematically

complete measurement using the High Resolution Array (HiRA) to detect the outgoing
deuterons, the S800 Spectrograph to detect the heavy recoil (either 33Ar or 45Ar), and Mi-

crochannel Plates (MCPs) to normalize the cross section and also localize the single-neutron

transfer on the reaction target. Each individual detector system required separate calibration

techniques. As part of this calibration process, we characterized the dead layer and detector

thicknesses for the HiRA “E” silicon detectors. We then merged the calibrated data to study

events which featured the coincidence of a deuteron in HiRA and the appropriate heavy ar-

gon recoil in the S800. From these coincidence events we generated angular distributions,

which we then normalized to produce the absolute differential cross sections.

We then extracted experimental spectroscopic factors (SFs) taking the ratio of the experi-

mental cross sections to the ADWA transfer reaction cross section calculations. The resulting

ground-state spectroscopic factors were then compared to the large-basis shell model (LBSM)
calculations to produce reduction factors. From 46Ar to 34Ar we see a relatively weak re-

duction trend. This corroborates the previous transfer data and is not consistent with the

quenching observed in the knockout results. This trend holds true for both of our choices

of optical model: the CH89 global optical model and the JLM microscopic model. There-

fore, we conclude that the transfer reaction mechanism is consistent in this energy range.

Although this conclusion does not resolve the transfer-knockout discrepancy, these results

do support the reliability of the transfer reaction as a probe of nuclear structure.

At this point we wish to emphasize that despite the discrepancy in absolute spectroscopic

155

factors, both knockout and transfer are crucial probes for studying nuclear structure. Each

of these probes has unique advantages, and together they offer a complementary approach to

studying single-particle structure rather than a competing one. Furthermore, each technique

has consistently reproduced the asymmetry trends discussed here across different systems

and different experimental groups. We assert, therefore, that the difference in asymmetry

dependence between transfer and knockout originates from an incomplete understanding of

one or both of these reaction mechanisms. Clearly, more work is needed before this disparity

is settled.

A better theoretical understanding of the knockout model is important, especially re-

garding the validity of the eikonal approximation for deeply bound nucleons as well as the

role of core excitations in the knockout mechanism [67, 68]. On the transfer side, theoretical

work is underway to address open questions regarding the proper treatment of nonlocality in

optical potentials, and also to expand the applicability of the Faddeev equations in transfer

reactions [142]. More generally, the nuclear theory community has begun making concerted

efforts towards proper uncertainty quantification in reaction models (see [143] for an ex-

ample of this work in the transfer reaction context). Aside from increasing transparency,

putting explicit error bars on theoretical calculations will enable better diagnosis of critical

weaknesses in nuclear reaction models.

More transfer reaction measurements for asymmetric systems are necessary, as well as

further investigation of new techniques like quasi-elastic proton scattering and electron scat-

tering on unstable isotopes [144]. Electron scattering is a particularly promising avenue

of investigation, as its electromagnetic nature provides a different pathway to access nu-

clear structure than hadronic probes like transfer and knockout. More systematic studies in

the spirit of [26] would be useful in order to better understand the wide spread of transfer

156

reaction reduction factors at a given asymmetry. Future rare isotope facilities like FRIB

will significantly expand the boundaries of reachable asymmetry for any of these reaction

mechanisms.

Ultimately, the specific question of reduction-factor asymmetry dependence illuminates

a larger issue in nuclear physics: the separation between nuclear structure and nuclear re-

actions. Transfer, knockout, and other reaction probes are commonly used to study nuclear

structure, despite the fact that the underlying theoretical scheme of the reaction technique is

segregated from the one used to understand the structure. This holds true in our work: the

LBSM spectroscopic factors were calculated with an entirely different Hamiltonian than the

one used by the ADWA model (which assumes that the A(p, d)B reaction is a three-body

system). This gap between the two disciplines is an inherent weak link in our ability to distill

insight into the properties of nuclei from experimental data. In order to make progress, the

field of nuclear physics faces the challenging task of uniting nuclear structure and nuclear

reactions within a single theoretical framework based in first principles. There are already

encouraging results on this front, albeit limited to simple systems like alpha-alpha scattering

or single-neutron transfer on light nuclei [145, 146, 147, 148]. Next-generation experimen-

tal facilities, innovations in reaction theory, and advanced leadership-class supercomputing

facilities all promise a deeper understanding of nuclei and their interactions.

157

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