A STUDY OF COMPLEMENTARY REACTIONS: SINGLE-PROTON KNOCKOUT AND
SINGLE-PROTON PICKUP
by
Sean Michael McDaniel

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
PHYSICS
2011

ABSTRACT
A STUDY OF COMPLEMENTARY REACTIONS: SINGLE-PROTON KNOCKOUT AND
SINGLE-PROTON PICKUP
By
Sean Michael McDaniel

Knockout reactions using fast, exotic beams in inverse kinematics are an established tool
for probing the ground state wave function of the incident projectile and studying the
spectroscopy of the projectile-like knockout residue. In a knockout reaction, the removal
of a nucleon from the projectile populates single-hole states in the residue. Measured partial
cross sections and longitudinal momentum distributions to final states of the residue,
as tagged by the detection of γ rays emitted in-flight and correlated event-by-event to
the residue, are used to quantify the single-particle composition of the projectile wave
function.
Recently, pickup reactions using fast, exotic beams in inverse kinematics have emerged
as a complementary tool to single-nucleon knockout reactions by populating single-particle
states in the residue. Recent analyeses have shown qualitative agreement between theoretical and measured strengths and population patterns to final states in the residue.
This agreement suggests that pickup reactions can provide similar information about the
ground state wavefunction of the projectile and the spectroscopy of the residue.
Two experiments were run at the nscl that show the power and highlight the differences of these two complementary reactions. The one-proton knockout experiment
9 Be(54 Ti, 53 Sc + γ)X at 72 MeV/u was performed to probe the single-proton configurations

of the N = 32 nuclei 54 Ti and 53 Sc and test the validity of the nuclear shell model and
valance space in this interesting neutron-rich mass region. An unexpectedly large spectroscopic strength to excited states was observed. The magnitude of strength observed was

incompatible with shell model calculations using the f p model space only, and was therefore attributed to the knockout of deeply-bound sd-shell protons outside the calculation
valance space. This interpretation was substantiated through a comparison to previous
experimental work, the analogous 50 Ti(d, 3 He) 49 Sc transfer reaction and the two-proton
knockout reaction 9 Be(54 Ti, 52 Ca + γ)X, and through the comparison of the inclusive momentum distribution to a theoretical distribution that assumes sd-shell contributions. The
results highlight the need for proton cross-shell interactions for the sdpf valance space.
Three different one-proton pickup reactions on proton-rich nuclei centered around 50 Fe
were studied on two different targets, 9 Be and 12 C: 9 Be(48 Cr, 49 Mn+γ)X, 9 Be(49 Mn, 50 Fe+
γ)X, 9 Be(50 Fe, 51 Co+γ)X, 12 C(48 Cr, 49 Mn+γ)X, 12 C(49 Mn, 50 Fe+γ)X, and 12 C(50 Fe, 51 Co
+ γ)X. The investigation of these reactions, through the comparison of the experimentally
measured and theoretically predicted partial cross sections and population pattern to
final states in the pickup residue, was done with the goal of further developing pickup
reactions as a spectroscopic tool. The theoretical partial cross sections were calculated
using shell-model spectroscopic factors and single-particle cross sections calculated in
a post-form Coupled Channels Born Approximation (ccba) framework assuming twobody final states. The reaction mechanism was further probed by measured longitudinal
momentum distributions whose comparison to calculated distributions that assume the
two-body character of the final reaction product tests the direct nature of the reaction
processes, and by the use of the two different targets, through which the importance of
including only a subset of target final states can be judged. We report the results for the
first fast beam, inverse-kinematics, proton pickup reactions in the f p-shell, and the first
proton pickup reactions from a 12 C target.

Copyright by
Sean Michael McDaniel
2011

To my parents, Mike and Jan McDaniel

v

ACKNOWLEDGMENTS
This acknowledgment is an attempt to inform you, the reader, that even though my
name is on the cover, I was not alone. I received a tremendous amount of guidance, love,
and friendship from a great number of people. Without their support, this thesis would
not be possible. In their name, here is my et al..
My adviser, Alexandra Gade, guided me through my graduate career. Each stage of
this long journey—experimental proposal, data analysis, and presentation of work—has
benefited from her expertise and generous help. Her tireless editing has significantly
improved the quality of thesis. Any errors or faults that remain are mine alone.
Thank you to all current and former members of my graduate committee for your
guidance and encouragement: Alex Brown, Alexandra Gade, Thomas Glasmacher, William
Hartmann, Wayne Repko, and Vladimir Zelevinsky. And thank you to the graduate
secretary Debbie for her ability to handle all the bureaucratic frustrations I created and
dropped upon her desk.
A big thanks to the gamma group, past and present. The graduate students—Travis
Baugher, Chris Campbell, Johnathan Cook, Chris Lawrence, David “Floyd” Miller, Kyle
Siwek, Ragnar Stroberg, Andrew Ratkiewitcz, Aimee Shore, Russ Terry, Phill Voss, and
Kathy Walsh—the post-docs—Christian Diget, Geoff Grinyer, Alexander Obertelli, and
Ryan Winkler—and our excellent staff physicist Dirk Weisshaar. There is a small poster by
my desk that I have guarded for the last several years. It reads: “Mistakes: It could be that
the purpose of your life is only to serve as a warning to others.” I would like to dedicate a
signed copy of that poster in everyone’s honor.
Nuclear research is collaborative. As an experimentalist, I relied on the effort of the
a1900 group, the s800 group (especially Daniel Bazin), and the cyclotron operators to create
and deliver the appropriate rare-isotope, cocktail beam to the experimental vault. Dirk
Weisshaar and the gamma group helped set up and maintain sega. Shifts were covered

vi

by the members of the gamma group. Jeff Tostevin and Alex Brown did the theoretical
calculations. Every person in the orchestration deserves praise and credit for a superb job.
I’ve met many good friends during my time at Michigan State who have kept me sane
and reminded me of the honest importance of what lies within and beyond the walls of
the lab. In particular, Ana Beccerril, Chris Farrow, Luke Granlund, and Tony Nettelton.
They have been here throughout my graduate career, are lunch, drinking, and gaming
buddies without peer, and I will miss them greatly. I will always have an open door to
welcome them in.
I’m grateful I was able to spend the last part of my graduate career with a wonderful
woman, Sonia. I hope that we’ll continue to pickup, throw down, and push through all the
bright and wonderful or dark and sad. With you, everything’s better.
Lastly, I would like to thank my family. The nuclear crew (Dad, Mom, Devin, and
Aaron), my extended family (Grandma Rene, Grandpa Vic, Grandma Fran, Uncle Bob,
Peggy, Grandma Mary Ann, Uncle Jim, Aunt Terri, Ryan and Jamie), and the departed,
whom I hope, through grace, are acknowledged (Grandpa Stan, Grandpa Bob, and Sadie).
I love you all.
I’ve dedicated this thesis to my parents, Mike and Jan McDaniel. It’s my small remembrance and immense gratitude for their unconditional love. As an experimentalist, I can’t
prove there’s a better family. The reader will just have to accept it as an axiom of fact.
The good stuff follows.

vii

TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
CHAPTER 1 INTRODUCTION . .
1.1 Overview . . . . . . . . . . . .
1.2 The Atomic Nucleus . . . . . .
1.2.1 What is a nucleus? . . .
1.2.2 The Nuclear Force . . .
1.2.3 The Nuclear Landscape
1.2.4 Magic Numbers . . . .

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CHAPTER 2 MOTIVATION . . . . . . . . . . . . . . . . . . .
2.1 Single-particle Structure and Nuclei far from Stability
2.2 Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Summary of specific motivations . . . . . . . . . . . .

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15
15
20
21

CHAPTER 3 NUCLEAR SHELL MODEL AND REACTION THEORY . . . . .
3.1 Reactions and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Basic Shell-Model Procedure . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Nuclear Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Model Space and Truncation . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Model Space Limitations . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Shell Model Spectroscopic Factors . . . . . . . . . . . . . . . . . . .
3.3 Reaction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Knockout reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Theoretical Cross Sections . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Single Particle Cross Sections . . . . . . . . . . . . . . . . . . . . .
3.4.2.1 Spectator Core Approximation . . . . . . . . . . . . . . .
3.4.2.2 Glauber approximation . . . . . . . . . . . . . . . . . . .
3.4.2.3 Static Density Approximation to the Scattering Matrices .
3.4.3 Projectile Wave Functions . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Momentum Distributions . . . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Single Nucleon Pickup Reactions . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 General Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Early Work: The 9 Be(20 Ne, 21 Na + γ)X reaction . . . . . . . . . . .
3.5.3 Pickup Reaction Theory . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4 EXPERIMENTAL SETUP AND ANALYSIS TECHNIQUES
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Production of Exotic Nuclei . . . . . . . . . . . . . . . . . .
4.2.2 A1900 Fragment Separator . . . . . . . . . . . . . . . . . . .
4.3 S800 Magnetic Spectrograph . . . . . . . . . . . . . . . . . . . . . .
4.4 The Segmented Germanium Array (sega) . . . . . . . . . . . . . .
4.4.1 Detector Overview . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Segmentation and Doppler Reconstruction . . . . . . . . . .
4.4.3 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . .
4.4.4 SeGA Efficiency . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Analysis Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Inclusive Cross Sections . . . . . . . . . . . . . . . . . . . .
4.5.3 Longitudinal Momentum Distributions . . . . . . . . . . . .
4.5.3.1 Single-nucleon Knockout Reaction . . . . . . . . .
4.5.3.2 Single-nucleon pickup reactions . . . . . . . . . .
4.5.4 Partial Cross Sections . . . . . . . . . . . . . . . . . . . . . .

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56
56
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80

CHAPTER 5 ONE PROTON KNOCKOUT FROM 54 TI TO 53 SC . . . . . . . . . .
5.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The N = 32 shell gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Previous Studies of 53 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Expectation and Prediction of Spectroscopic Strength in the fp-Shell Model
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Secondary-Beam Production . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 SeGA Efficiency Calibrations . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Inclusive Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Observed γ Ray Transitions . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Partial Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Knockout of Deeply Bound Protons . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 Knockout From the sd Shell . . . . . . . . . . . . . . . . . . . . . . .
5.7.2 The 50 Ti(d,3 He)49 Sc and 9 Be(54 Ti,52 Ca + γ)X Reactions . . . . . . .
5.7.3 Inclusive Parallel Momentum Distribution . . . . . . . . . . . . . . .
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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117

CHAPTER 6 ONE-PROTON PICKUP REACTIONS CENTERED AROUND 50 FE
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 S800 Setup and Target Selection . . . . . . . . . . . . . . . . . . . . .

120
120
123
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124

ix

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6.3.3 SeGA Setup . . . . . . . . . . . . . .
6.3.4 Experimental Changes . . . . . . .
6.4 Proton Pickup Reactions: 9 Be target . . . .
6.4.1 Particle Identification . . . . . . . .
6.4.2 Theoretical Predictions . . . . . . .
6.4.3 Measured Inclusive Cross Sections
6.4.4 Observed Transitions . . . . . . . .
6.4.5 Partial Cross Sections . . . . . . . .
6.4.6 Momentum Distributions . . . . . .
6.5 Proton Pickup Reactions: 12 C target . . . .
6.5.1 Particle Identification . . . . . . . .
6.5.2 Theoretical Predictions . . . . . . .
6.5.3 Inclusive Cross Section . . . . . . .
6.5.4 Observed Transitions . . . . . . . .
6.5.5 Partial Cross Sections . . . . . . . .
6.5.6 Momentum Distributions . . . . . .
6.6 Summary . . . . . . . . . . . . . . . . . . .
CHAPTER 7

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179

SUMMARY AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 182

APPENDIX A DATA TABLES FOR SINGLE-PROTON PICKUP REACTIONS . 186
A.1 Shell-model Spectroscopic Factors . . . . . . . . . . . . . . . . . . . . . . . . 186
A.2 Measured and Predicted Partial Cross Sections . . . . . . . . . . . . . . . . . 186
APPENDIX B
THE BIRTH OF NUCLEAR PHYSICS . . . . . . . . . . . . . . . . 193
B.1 A Symphony of Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
B.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

x

LIST OF TABLES
1.1 Coupling strength and interaction range of the four fundamental forces relative
to the strong force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4.1 Secondary beam production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.2 Secondary and primary beam characteristics for the single-proton pickup and
knockout experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.3 Properties of the S800 Magnetic Spectrograph . . . . . . . . . . . . . . . . . . .

64

4.4 Source energies used for sega energy calibration . . . . . . . . . . . . . . . . .

70

4.5 Source properties for 152 Eu, 226 Ra, and 56 Co . . . . . . . . . . . . . . . . . . .

72

4.6 Targets used in the knockout and pickup experiments . . . . . . . . . . . . . .

77

5.1 Theoretical excitation energies and spectroscopic factors to the final states in
53 Sc following the 9 Be(54 Ti,53 Sc + γ)X reaction . . . . . . . . . . . . . . . . . .

97

5.2 Input parameters and results for the calculation of single-particle cross sections
for the 9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.3 Theoretical spectroscopic factors, single-particle cross sections, and partial
cross-sections for the 9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . . . . . . . . .

99

5.4 Secondary beam production for the 9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . 101
5.5 152 Eu and 56 Co source energies used in the SeGA efficiency calibrations for the
37◦ and 90◦ rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6 Efficiency fit results for the 152 Eu and normalized 56 Co sources . . . . . . . . . 103
5.7 Error budget for the measured transition energies in 53 Sc . . . . . . . . . . . . 109
5.8 Measured partial cross sections to final states in 53 Sc . . . . . . . . . . . . . . . 111
5.9 Comparison of theoretical predictions and experimental measurements of
summed strength to excited states in 53 Sc . . . . . . . . . . . . . . . . . . . . . 113
5.10 Summary of experimental spectroscopic results for the 50 Ti(d, 3 He) 49 Sc reaction as reported by Doll et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xi

6.1 Experimental characteristics of secondary beam production for the proton
pickup reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2 Major experimental made changes during the course of the experiment . . . . 130
6.3 Overview of the various timing measurements used in the pickup reaction
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 Additional details on the calculation of the single-particle cross sections for the
9 Be(48 Cr, 49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.5 Additional details on the calculation of single-particle cross sections for the
9 Be(49 Mn, 50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.6 Additional details on the calculation of single-particle cross sections for the
9 Be(50 Fe, 51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Theoretical predictions of the partial and inclusive cross sections for the
9 Be(48 Cr, 49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.8 Theoretical predictions of the partial and inclusive cross sections for the
9 Be(49 Mn, 50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.9 Theoretical predictions of the partial and inclusive cross sections for the
9 Be(50 Fe, 51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.10 Inclusive cross sections for the 9 Be(48 Cr, 49 Mn+γ)X, 9 Be(49 Mn, 50 Fe+γ)X, and
9 Be(50 Fe, 51 Co + γ)X reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.11 Measured energies for 49 Mn and 50 Fe with comparisons to literature . . . . . . 156
6.12 Additional details on the calculation of the single-particle cross sections for the
12 C(48 Cr, 49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.13 Additional details on the calculation of single-particle cross sections for the
12 C(49 Mn, 50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.14 Additional details on the calculation of single-particle cross sections for the
12 C(50 Fe, 51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.15 Theoretical predictions of the partial and inclusive cross sections for the
12 C(48 Cr, 49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.16 Theoretical predictions of the partial and inclusive cross sections for the
12 C(49 Mn, 50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xii

6.17 Theoretical predictions of the partial and inclusive cross sections for the
9 Be(50 Fe, 51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.18 Measured and predicted inclusive cross sections for the 12 C(48 Cr, 49 Mn + γ)X,
12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X reactions. . . . . . . . . . . . . 171
A.1 Shell-model spectroscopic factors for the 9 Be(48 Cr, 49 Mn + γ)X reaction . . . . 187
A.2 Shell-model spectroscopic factors for the 9 Be(49 Mn, 50 Fe + γ)X reaction . . . . 188
A.3 Shell-model spectroscopic factors for the 9 Be(50 Fe, 51 Co + γ)X reaction . . . . 189
A.4 Measured and predicted partial cross sections for the 9 Be(48 Cr, 49 Mn + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
A.5 Measured and predicted partial cross sections for the 9 Be(49 Mn, 50 Fe + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.6 Measured and predicted partial cross sections for the 9 Be(50 Fe, 51 Co + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.7 Measured and predicted partial cross sections for the 12 C(48 Cr, 49 Mn + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.8 Measured and predicted partial cross sections for the 12 C(49 Mn, 50 Fe + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.9 Measured and predicted partial cross sections for the 12 C(50 Fe, 51 Co + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

xiii

LIST OF FIGURES
1.1 Atomic and nuclear sizes and constituents . . . . . . . . . . . . . . . . . . . . .

4

1.2 Central projection of the nucleon-nucleon potential with labeled general features

7

1.3 Nuclear chart and characteristics of the nuclear landscape . . . . . . . . . . . .

8

1.4 Single-particle orbitals showing the emergence of “magic numbers” with the
addition of a strong spin-orbit term to the nuclear Hamiltonian . . . . . . . . .

14

3.1 Valence space for 53 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.2 Nuclear chart showing the size and location of typical valance spaces used in
shell model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.3 Knockout reaction schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.4 Pickup reaction schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.5 Population of final states in the 9 Be(54 Ti, 53 Sc + γ)X knockout reaction . . . . .

42

3.6 Theoretical momentum distributions for 53 Sc in the 9 Be(54 Ti, 53 Sc + γ)X reaction 49
3.7 Stripping and knockout comparison for 21 Na . . . . . . . . . . . . . . . . . . .

54

3.8 Post-form dwba reaction schematic . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.1 NSCL Coupled Cyclotron Facility . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.2 A1900 fragment separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.3 Side view of the S800 magnetic spectrograph . . . . . . . . . . . . . . . . . . .

63

4.4 S800 focal plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.5 Schematic view of the sega detector array . . . . . . . . . . . . . . . . . . . . .

83

4.6 Segmentation of the sega crystal . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.7 Photograph of sega from the front . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.8 SeGA side view showing features important for Doppler reconstruction . . . .

86

4.9 Contributions to the energy resolution of sega . . . . . . . . . . . . . . . . . .

87

xiv

4.10 PID for the 9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . . . . . . . . . . . . . . .

88

4.11 53 Sc γ spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.1 Nuclear landscape near the 9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . . . . . .

91

5.2 Physics of the N = 32 shell gap . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.3 Naive model for the population of states in 53 Sc following the knockout of a
proton from the f p valance space in 54 Ti . . . . . . . . . . . . . . . . . . . . . .

96

5.4 Time-of-flight spectrum for an unreacted and a reacted run with 52 Ca centered
in the focal plane of the S800 spectrograph . . . . . . . . . . . . . . . . . . . . . 100
5.5 SeGA source efficiency curves for the 37◦ and 90◦ rings. . . . . . . . . . . . . . 102
5.6 Particle identification plot for the 9 Be(54 Ti, 53 Sc + γ)X reaction, gated on the
incoming projectile 54 Ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.7 Acceptance correction for 53 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 Doppler-reconstructed γ ray spectrum for 53 Sc . . . . . . . . . . . . . . . . . . 110
5.9 Comparison of the theoretical and experimental population of 53 Sc in the
9 Be(54 Ti, 53 Sc + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.10 Overlay of the measured parallel momentum distribution with a theoretical
model assuming sd-shell knockout . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1 Nuclear chart showing the location of the full set of isotopes and reactions
studied using the single-proton pikcup reaction . . . . . . . . . . . . . . . . . . 122
6.2 Particle-identification spectra for an unreacted run and a 12 C target . . . . . . 125
6.3 Lab and Lorentz-boosted SeGA efficiencies for the 37◦ and 90◦ rings . . . . . . 129
6.4 Effect of time delays on the observation and identification of components in
the incoming cocktail beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5 Effect of the time-to-amplitude converters on time-of-flight and particle identification of components of the incoming cocktail beam . . . . . . . . . . . . . . 134
6.6 Time-of-flight splitting and the difficulties in identifying components of the
cocktail beam: 9 Be pickup data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.7 Dispersive position versus tof and dispersive position versus angle in the
focal plane showing both the contributions and method of cleanup for pid
contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xv

6.8 Particle identification plot for the 9 Be(48 Cr,49 Mn + γ)X reaction before and
after the application of the cleanup gating procedure . . . . . . . . . . . . . . . 138
6.9 Shell-model spectroscopic factors for the 9 Be(48 Cr, 49 Mn + γ)X reaction . . . . 142
6.10 Shell-model spectroscopic factors for the 9 Be(49 Mn, 50 Fe + γ)X reaction . . . . 143
6.11 Shell-model spectroscopic factors for the 9 Be(50 Fe, 51 Co + γ)X reaction . . . . 144
6.12 Cross section measurements for the 9 Be(48 Cr, 49 Mn + γ)X reaction . . . . . . . 150
6.13 Cross section measurements for the 9 Be(49 Mn, 50 Fe + γ)X reaction . . . . . . . 151
6.14 Cross section measurements for the 9 Be(50 Fe, 51 Co + γ)X reaction . . . . . . . 151
6.15 Rest-frame and laboratory-frame spectra of 49 Mn for the 9 Be(48 Cr, 49 Mn + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.16 Rest-frame and laboratory-frame spectra of 50 Fe from the 9 Be(49 Mn, 50 Fe + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.17 Measured and predicted partial cross sections for the 9 Be(48 Cr, 49 Mn + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.18 Measured and predicted partial cross sections for the 9 Be(49 Mn, 50 Fe + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.19 Measured and predicted partial cross sections for the 9 Be(50 Fe, 51 Co + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.20 Inclusive parallel momentum distribution of 49 Mn associated with the 9 Be(48 Cr,
49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.21 Inclusive parallel momentum distribution of 50 Fe associated with the 9 Be(49 Mn,
50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.22 Inclusive parallel momentum distribution of 51 Co associated with the 9 Be(50 Fe,
51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.23 Residue rest frame and laboratory frame γ-ray spectra for 49 Mn following the
12 C(48 Cr, 49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.24 Residue rest frame and laboratory frame γ-ray spectra for 50 Fe following the
12 C(49 Mn, 50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.25 Measured and predicted partial cross sections for the 12 C(48 Cr, 49 Mn + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

xvi

6.26 Measured and predicted partial cross sections for the 12 C(49 Mn, 50 Fe + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.27 Measured and predicted partial cross sections for the 12 C(50 Fe, 51 Co + γ)X
reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.28 Measured and estimated parallel momentum distributions for the 12 C(48 Cr,
49 Mn + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.29 Measured and predicted parallel momentum distributions for the 12 C(49 Mn,
50 Fe + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.30 Measured and predicted parallel momentum distributions for the 12 C(50 Fe,
51 Co + γ)X reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xvii

Chapter 1
INTRODUCTION
1.1

Overview

This thesis is concerned with the experimental observation and theoretical description
of the single-particle structure of a variety of exotic nuclei, accessed using two complementary nuclear reactions: the single-proton knockout reaction and the single-proton
pickup reaction, specifically 9 Be(54 Ti, 53 Sc + γ)X, 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe
+ γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe,
51 Co + γ)X1 .

Because this work joins a much larger narrative, the broad history of nuclear

physics, an overview of the major discoveries in the path to modern nuclear structure
physics is given in Appendix B.
The work is broken down into seven chapters. Chapter 1 provides an overview of
the atomic nucleus. This introduction proceeds a detailed motivation (Chapter 2) for the
original work contained in this thesis. Chapter 3 introduces the nuclear shell model and
the single-proton pickup and single-nucleon knockout reactions. Shell model calculations
and reaction theory (pickup or knockout) are combined to predict the experimentally
observable inclusive and partial cross sections. The technical details of the experimental
setup are covered in Chapter 4. This includes details of the production of exotic nuclei at
the National Superconducting Cyclotron Laboratory (nscl) Coupled Cyclotron Facility
(ccf), a description of the major experimental devices used, the Segmented Germanium
Array (sega) and the s800 magnetic spectrograph, and an overview of how the singleparticle observables, γ ray transition energies and cross sections, are calculated using data
1 The

nomenclature of these reactions are explained in Section 3.3

1

collected from these devices.
Chapters 5 and 6 cover the original experimental work, whose data was collected in
two separate experiments. Chapter 5 covers the single-proton knockout reaction 9 Be(54 Ti,
53 Sc + γ)X.

Chapter 6 covers the single-proton pickup reactions 9 Be(48 Cr, 49 Mn + γ)X,

9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X,

and 12 C(50 Fe, 51 Co + γ)X. A final summary is found in Chapter 7.

1.2
1.2.1

The Atomic Nucleus
What is a nucleus?

The nucleus is a femtometer scale, quantum, mesoscopic collection of protons and neutrons
bound together by the residual strong force and interacting through three of the four
fundamental forces of nature: the strong, weak, and electromagnetic forces. The nucleus
contains nearly all the mass of the nucleus (≈ 99.9%) but has negligible size in comparison
to the diameter of the atom. Atomic radii are on the order of 10−10 m; nuclear sizes are on
the order of 10−15 m.
Protons and neutrons are not fundamental (indivisible) particles. They belong to a
class of composite particles called hadrons composed of fundamental particles called
quarks. More specifically, protons and neutrons are composed of three quarks, either
two up quarks and one down quark (proton, uud) or one up quark and two down quarks
(neutron, udd). Although the term “strong force” was originally applied to the force
between protons and neutrons, so named because the force had to be stronger than the
Coulomb force to bind the positively charged protons within the nucleus (see Table 1.1 for
a comparison of the coupling constants of the four forces), the name now applies to the
more fundamental interaction between quarks. This force, similar to all other fundamental
forces, is modeled through the exchange of particles. For quarks the force carriers are
massless gluons.

2

Force

Relative strength
(ratio to strong)

Range
(m)

Strong
Electromagnetic
Weak
Gravity

1
1/137
10−6
6 × 10−39

10−15
∞
10−18
∞

Table 1.1: Coupling strength and interaction range of the four fundamental forces relative
to the strong force.

A bit of the strong force leaks out from the strongly bound triplet of quarks. This residual force, a remaining polarization analogous to the relationship between the Coulomb
and Van der Waals force, binds protons and nucleons together. This residuum is either
referred to as the “residual strong force” or the “nuclear force” (see Figure 1.1) and the
relative strength of the strong force allows the pertubative effects of the Coulomb and
the weak force2 to be ignored when searching for the basic framework that describes the
interaction and motion of nucleons within the nucleus.

1.2.2

The Nuclear Force

Despite the complexity of the nuclear force, the non-central tensor components (the
force between nucleons is not parallel to the radial position vector between them), the
dependence on the momentum and position of the nucleons, the presence of many-body
forces, and so forth, it is worth reviewing its broad phenomenological features. We
only consider only the two-body component. Because of high-quality nucleon-nucleon
scattering data, the two-body nucleon-nucleon (nn) force is well described (see Refs. [2–4]
for the modern Argonne V18, the Nijm1, Nijm2, Nijm93, and Reid93 nucleon-nucleon
2 The

mechanism of flavor changing whereby a up quark changes to a down quark or
a down quark to an up quark through the emission of an intermediate heavy W − boson.
For example, the weak interaction drives the processes called β-decay, whereby a neutron
changes to a proton through the emission of an electron and an electron antineutrino (β −
¯
decay: n → p + e− + νe ) or a proton changes to a neutron through the emission of a positron
and an electron neutrino (β + decay: p → n + νe ).
3

Figure 1.1: Approximate sizes of the constituents of a 12 C atom. At medium (d ≈ 1 – 2 fm)
and large distances ( 2 fm), the nuclear force can be modeled as the exchange of light
mesons (π, ρ, and ω), as first proposed by Hideki Yukawa in 1934 [1]. For interpretation of
the references to color in this and all other figures, the reader is referred to the electronic version
of this dissertation.

potentials, and the CD-Bonn potential). Modifications to the bare nn potential to account
for the nuclear medium and three-body forces, although important, preserve the two-body
phenomenological features reviewed below.
The complete force can be decomposed into a central, two-body spin-orbit interaction,
and a tensor component [5]. The nuclear force is both charge symmetric and charge
independent, i.e. the force that describes all nucleon-nucleon interactions is identical.
Because the nuclear Hamiltonian is invariant under a change of nucleon type, which can

4

be experimentally observed through the similar spectra of mirror nuclei, pairs of nuclei
with identical A but with switched numbers of protons Z and neutrons N 3 , a conserved
quantum number is introduced to reflect the symmetry. This quantum number is called
isospin. Isospin symmetry is broken by the electromagnetic interaction which effects
protons but not neutrons.
The dependence of the interaction on the position, spin, isospin can be written as
V (1, 2) = V (r1 , σ1 , τ1 ; r2 , σ2 , τ2 ),

(1.1)

V (1, 2) = Vc (1, 2) + VLS (1, 2) + VT (1, 2).

(1.2)

and its decomposition gives

The central interaction depends on the relative distance between the nucleons. We
have,
VC (1, 2) = V0 (r) + Vσ (r) σ1 · σ2 + Vτ (r) τ1 · τ2 + Vσ τ (r) (σ1 · σ2 ) (τ1 · τ2 ).

(1.3)

and for the two-body spin-orbit interaction,
is
iv
VLS = VLS (r) + VLS (r) τ1 · τ2 L · S

(1.4)

where L is the relative orbital momentum between the two interacting nucleons and S is
the intrinsic spin S = 1/2(σ1 + σ2 ). The labels is and iv indicate isoscaler (isospin singlet,
T = τ1 + τ2 = 0) and isovector (isospin triplet, T = τ1 + τ2 = 1) components. The tensor
component is
is
iv
ˆ
VT (1, 2) = VT (r) + VT (r)τ1 · τ2 S12 (r)

(1.5)

3
S12 (r) = 2 (σ1 · r)(σ2 · r) − σ1 · σ2
r

(1.6)

where

In all cases r = |r1 − r2 |, τ1 and τ2 are the respective isospin vectors of nucleon 1 and 2,
and σ2 and σ2 are their spin matrices.
3 Examples

include 11 B ↔ 11 C and 21 Na ↔ 21 Ne.
5

The radial part of the nuclear force can be divided into three distinct ranges. At very
short distances (d
range (0.5 fm

d

0.5 fm) the force is extremely repulsive (nearly infinite). At medium
2 fm) it is strongly attractive, and at long range (d

2 fm), only a small,

attractive exponential tail remains. Beyond 2.5 fm, the Coulomb force dominates. The
radial part of the nuclear force illustrated in Figure 1.2. At medium and long distances, the
nuclear force can be modeled by the exchange of light mesons (π, ρ and ω), as proposed by
Yukawa in 1934 [1] and experimentally seen in 1947 by Cecil Powell, Giuseppe Occhialini
and César Lattes [6, 7]. As will be described in Section 1.2.4, the single-pion exchange
is a very good approximation because on average nucleons only feel the very tails of the
nuclear force. The average nucleon-nucleon distance of d ∼ 2.4 fm is much larger than the
point (∼ 0.7 fm) at which the nucleon-nucleon binding potential is strongest.
Recent efforts have begun to derive the nuclear force from quark degrees of freedom.
Lattice quantum chromodynamics qcd calculations [8] show broad phenomenological
agreement with the central component of the nuclear force (repulsive core, attraction
at medium range, long-range tails consistent with pion exchange) but these efforts are
computationally extremely demanding and have only been done for a limited selection
of two nucleon states (e.g. 1 s0 and 3 s1 ). The derivation of nuclear properties from
fundamental qcd calculations remains a long-term goal. For calculations of multi-nucleon
systems, as studied in this work, we start with the nucleon as the basic degree of freedom.
There are several ways to label these systems by total atomic number A, nuclear charge Z,
and number of neutrons N , but in this work will adopt the conventions of only using A
(total mass) and Z (atomic number), for example 53 Sc (A = 53 and Z = 21 for Scandium)
and 48 Cr (A = 48 and Z = 24 for Chromium).

1.2.3

The Nuclear Landscape

Nuclear physicists organize the wide spectrum of nuclei according to their nuclear charge
and number of neutrons in a similar way that chemists organize the elements by their

6

V (MeV)

200
2π, ρ, ω, σ
exchange

Repulsive
core

100

π
exchange

0

-100

0.0

0.5

1.0

1.5

2.0

3.0

r (fm)
Figure 1.2: Central projection of the nucleon-nucleon (nn) potential showing general
features: the hard repulsion at short range, a minimum just below 1 fm, and a limited
(≈ 2.5 fm) nn interaction range. The average nucleon-nucleon distance of d ≈ 2.4 fm,
located well outside the minimum, is indicated by a black arrow.

nuclear charge and group. This chart, seen in Figure 1.3, is called a nuclear or Sergè chart.
A nuclear chart has a few important features:
The black squares in Figure 1.3 indicate stable elements. The entire collection of stable
elements is called the “valley of stability”. This valley ranges from the light isotopes,
starting with Hydrogen (1 H) where N = Z and N /Z = 1, all the way up through Bismuth
(209 Bi), where N > Z and N /Z = 1.5, i.e. there are 52% more neutrons than protons. Past
Bismuth at Z = 83, all isotopes are unstable. Elements heavier than Bismuth that exist
in the Earth’s crust, for example 238 U and 232 Th have half-lives on the order of the age
of our solar system (τ1/2 (238 U) = 4.468(3) × 109 y, τ1/2 (232 Th) = 1.41(1) × 1010 y) or they
lie in the decay chains of one of these very long-lived isotopes. This increasing neutronproton asymmetry with respect to proton number is caused by the interplay between
the short-range nuclear force, which favors the very strong proton-neutron combination,
and the repulsive Coulomb force which decreases the binding energy of nucleons within
the nucleus. The nuclear contribution to the binding energy increases linearly with A (a

7

120
N=Z

Proton number

100
80
60

Valley of
stability
Proton
drip line

40

Te

20
0

0

a
rr

In

co

g

n

a
it

Observed nuclei

Neutron
drip line

20

40

60

80

100

Stable nuclei

120

140

160

180

Neutron number
Figure 1.3: The nuclear chart is used for organizing the nuclear isotopes. Stable elements
(valley of stability) are shown in black. The dotted lines indicate the magic numbers at
N,Z = 2, 8,20, 50, 82 and 126. The figure is adapted from Figure 2.1 of Ref. [9]

nucleon is only affected by its nearest neighbors), and the binding energy per nucleon, once
we move beyond the very strong shell effects at low mass and reach nuclear saturation,
is an approximate B(N , Z)/A|A>12 ≈ −8.5 MeV/u (see Ref [10, page 2]). The Coulomb
contribution to the total binding energy increases quadratically with A (each proton feels
all other protons). The interplay of proton-neutron pairing, Coulomb effects, and the
contribution of the strong force to nuclear binding is shown in the semi-empirical mass
formula of Bethe and Weizäcker [10–12].
B(N , Z) = aV A + aS A2/3 + aC

8

(N − Z)2
Z2
+ ap
− δ(A),
A
A1/3

(1.7)

where the nuclear binding (volume) constant aV , surface constant aS 4 , Coulomb constant
aC , paring constant ap , and odd-even term δ(A) can be found through fits of measured
masses. The constants depend upon the fit method and included masses. Fit values and
further detail are found is Refs. [13–15]. A sample is listed below:
aV = −15.68 MeV,

as = 18.56 MeV,



34 A−3/4 ,







δA = 0







−34 A−3/4


ac = 0.717 MeV,

ap = 28.1 MeV

(1.8)

for even-even
for even-odd
for odd-odd

Nuclei off the valley of stability are not stable against β-decay, α-decay, or spontaneous
fission. Neutron-rich nuclei to the right of the valley of stability move towards stability
Z+1
Z
through β − decay (N A → N −1 A), nuclei to the left of the valley of stability (proton-rich
Z
for lighter nuclei) head towards stability through electron capture or β + -decay (N A →
Z−1
N +1 A).

β − decay, electron capture, and β + decay involve the weak process. For additional

information, see Refs. [16, 17]. Heavy elements, such as those beyond 209 Pb, typically
Z−2
Z
emit α particles (N A → N −2 (A − 4)), until they can β-decay to a stable isotope. The N , Z

boundary at which nuclei become particle-unbound, i.e Sn > 0 or Sp > 0, are called
drip-lines and are indicated in Figure 1.3.
Of the ∼ 6000 isotopes predicted to exist [18], approximately 3000 have been made
with particle accelerators or nuclear reactors. 270 are stable. The unexplored areas, labeled
“terra incognita” in Figure 1.3, contain a wealth of physics important for nuclear structure,
astrophysics, and reaction theory. Newly discovered isotopes offer a test for the predictive
power of nuclear models. Nuclei far from stability can exhibit novel properties such
as nuclear halos, shape coexistence, inversion of traditional single-particle levels, new
collective behavior, and so forth. These novel properties are often sensitive to different
aspects of the nuclear force that are broadly noticeable in the study of stable nuclei, and
4 Nucleons

near the surface of the nucleus contribute less to the total binding energy
9

fully folding these new properties with the existing theoretical framework leads to a better
understanding of the complete nuclear many-body system. Furthermore, many important
astrophysical processes, like the reaction pathways for the r and rp-processes [19, 20], lie
in areas currently unexplored and inaccessible by experiment.
The production yields of these extremely exotic nuclei are presently too low to scientifically explore, but new, second generation facilities for the production of exotic nuclei,
like Système de Production d’Ions in Ligne, Génération 25 (SPIRAL2) at Grand Accélérateur National d’Ions Lourds (ganil)6 , Facility for Antiproton and Ion Research (fair) at
¯
Gesellschaft für Schwerionenforschung7 (gsi), ribf at Rikagaku Kenkyusho (riken), and
the Facility for Rare-Isotope Beams (frib) at Michigan State University (msu), will provide
the beams of exotic nuclei necessary to start such research.

1.2.4

Magic Numbers

Magic numbers begin with the concept of nucleons as independent particles moving on
discrete orbits. Like electrons, nucleons have an intrinsic spin of 1/2 and are Fermions:
they obey the Pauli exclusion principle. Thus, protons and neutrons (separately) cannot
share the same quantum numbers. The Pauli principle and a nearly infinite repulsive core
(see Figure 1.2), lead to a diffuse nucleus where nucleons on average feel the tails of the
nuclear force8 . These two effects allow us to treat the particles individually, and it forces
nucleons, as they are added to a nucleus, to occupy consecutively higher-energy orbitals in
exactly the same manner that electrons fill the orbitals in atoms.
5 English

translation: system of ion-production online

6 English

translation: large heavy-ion accelerator

7 English

translation: Society for Heavy-Ion Research

8 The

ratio of the minimal volume possible (maximal packing fraction) of nucleons in a
nucleus to the measured volume is 1:100, Vmin /Vmeas = (c/2r0 )3 ≈ 1/100. See [10, Page 36,
Ch. 2] and [21, Section 2.5]

10

There is a crucial difference between electrons and nucleons. All electrons move in a
central potential well generated by the positively charged nucleus. Nucleons, however, are
not bound by an external potential. Because the nuclear force is short range, as inferred by
the saturation of binding energy per nucleon, the nuclear potential is a common average
over nearest neighbors. The homogeneity of the potential associated with saturation
allows us to define a nuclear mean field, an average nuclear potential. Describing a uniform
and similar potential for all nucleons allows us to decouple the many-body problem and
consider each nucleon individually. This framework of independent particle motion where
each nucleon only feels the average potential generated by all other particles is called the
Independent Particle Model (ipm), and its final formulation was done by Brueckner in
1954 [22, 23].
As in atomic physics, there are certain orbitals (the valance orbitals) that when fully
occupied lead to an especially nonreactive atom. Nuclear physics, similarly, has very stable
nuclei (experimental signatures are seen below) when certain orbits are fully filled. The
energies of the proton and neutron orbitals are not uniformly distributed. Instead they
are grouped into shells with wide energy gaps between each shell (see Figure 1.4). When
all the orbitals in a shell are full and there is a large gap in energy between the outermost
filled orbital and the next unfilled orbital, we call the nucleus magic. Nuclei where either
the protons or neutrons are in magic configurations are called semi-magic and nuclei where
both the proton and neutron shells fulfill the magicity condition are called doubly magic.
The total number of protons and neutrons necessary to fulfill this condition are called
magic numbers and they occur, for example, at N , Z = 2, 8, 20, 50, 82, and 126. Examples
of doubly magic nuclei are 4 He, 16 O, 40 Ca, 48 Ca, 56 Ni, and 208 Pb.
These magic numbers manifest experimentally in numerous ways. A few signatures:
1. High binding energies: Semi-magic and doubly magic nuclei have higher binding
energies, which can be seen in their deviation from the Weiszäcker empirical mass
formula.
11

2. Nucleon pickup cross section: The neutron or proton pickup cross section is two orders
lower in magnitude for nuclei with a magic number of neutrons or protons.
3. High excitation energy: The energy of the first excited states is higher than corresponding states in neighboring nuclei.
4. Two-nucleon removal energies: The two-neutron S2n and two-proton S2p separation
energies are higher for magic nuclei. The two-nucleon, rather than single nucleon
(Sp and Sn ) separation energies are used to eliminate the effects of the pairing force
which binds even numbers of protons and neutrons more tightly.
5. Zero electrical quadrupole moment: Indicates that magic nuclei are nearly spherical in
shape.
The location and spread of the single particle orbitals was first described in nearly
simultaneous papers in 1949 by Mayer [24] and Haxel, Jensen, and Suess [25]. Mayer and
Jensen combined the harmonic oscillator potential with a spin-orbit term9 . This spin-orbit
term breaks the degeneracy within the harmonic oscillator shells, and causes the descent
of levels with higher angular momentum into the major oscillator shell below, explaining
the origin of the magic numbers above 2010 . This split in degeneracy is seen in Figure 1.4.
Orbitals are defined by three quantum numbers: the principal quantum number N (major
oscillator shell), the orbital angular momentum , and the total angular momentum j. The
orbital angular momentum is given in spectroscopic notation where

= s, p, d, f , g, h · · ·

(sharp, principal, diffuse, and fundamental) and is equivalent to = 0, 1, 2, 3, 4, 5, · · · . Past
g the numbers increment in alphabetic order. N is usually replaced by n the number
of radial nodes in the wave function, leading to the quantum numbers n j, written as
9 The

addition of the spin-orbit term causes the agreement. Both the base harmonic
oscillator or a base realistic potential (e.g. finite square well with rounded edges) give a
similar level structure.
10 We

label the magic numbers created by the major oscillator quantum number, HO
magic numbers, and those created by the spin-orbit force, LS magic numbers.
12

n j . This number either starts at either 0 or 1 depending if the radial node at the origin
is counted. Under the convention that the first radial node is counted, the first several
orbitals are labeled 1s1/2 (s-shell), 1p3/2 , 1p1/2 (p-shell), 1d5/2 , 2s1/2 , 1d3/2 (sd-shell), and
1f7/2 .
For several decades, the location and origin of these magic numbers, seen uniformly
through the valley of stability, was a convenient dogma. New facilities like the nscl
ccf facility, however, have allowed experiments to be conducted with radioactive beams.
Starting in the 1970s, experimentalists noticed that the N = 20 shell gap did not necessarily
persist in Na, Ne, and Mg isotopes [26–28] and they concluded that the canonical shell
gaps did not hold in regions of extreme proton-neutron asymmetry. The natural question
arises: what drives these evolutions in shell structure? This question, as seen in Chapter 2
can be addressed experimentally through probing a nucleus’s single-particle structure,
and is one of the major motivations of the experimental work contained in this thesis.

13

N

nℓ
1i

112

nℓj
14

112
42

14

1h
30

6

2f

70

3p

22

70

82

3s

20

2

2d

40

126

12

1g

16

50

20

12

6

1f

14

8

6

2s
1d

2
10

8
2

1p
2

Isotropic harmonic
oscillator

28
20

20

2

8

6

1s

2

Realistic potential

1i13/2
3p1/2
3p3/2
2f5/2
2f7/2
1h9/2

12 1h11/2
2 3s1/2
4 2d3/2
8 2d
5/2
6

1g7/2

2
6
4

40
2p

14
2
4
6
8
10

2

1g9/2
2p1/2
1f5/2
2p3/2

8

1f7/2

4
2
6

1d3/2
2s1/2
1d5/2

2
4

1p1/2
1p3/2

2

1s1/2

10

Realistic potential
with spin-orbit

Figure 1.4: Single particle orbitals under consequentially more complicated nuclear
potentials: pure isotropic harmonic oscillator, isotropic harmonic oscillator with angular
momentum effects (labeled “releastic potential”), and the harmonic oscillator with spinorbit splitting. The harmonic oscillator with the spin-orbit terms reproduces the magic
numbers (red font text) seen in nuclei near stability. The quantum numbers (green font
text) and occupancies (blue font text) for each level are given. N for the harmonic oscillator
potential, n for the realistic potential, and n j for the realistic potential with a spin-orbit
term.

14

Chapter 2
MOTIVATION
2.1

Single-particle Structure and Nuclei far from Stability

The atomic nucleus can exhibit a wide range of phenomena, like collectivity, shape coexistence, pronounced single-particle character, and magic numbers, and as a result of the
diversity of nuclear behavior, a corresponding diversity of experimental techniques have
been developed to probe specific aspects of the nucleus. For example, we can probe the
existence of a nucleus (drip-line location), its life-time (β-decay, α-decay), nuclear mass,
nuclear shape (prolate, oblate, triaxial), proton and neutron density distributions, and
single-particle structure (the location and occupancy of orbitals). These unique nuclear
attributes, in tandem with theoretical models, allow physicists to address some of the
major and fundamental goals of nuclear science. What is the nature of the nuclear force?
What are the limits of nuclear existence? What is the origin of the heavy elements in the
Universe? And what are the basic explanations for the emergence of simple patterns in
complex nuclei like shell gaps, collective motion, deformation, and so forth? [29]
This thesis approaches the nucleus with two specific reactions, single-nucleon knockout
and single-nucleon pickup, that provide information on the single-particle degree of
freedom. Tracking the location and occupation of single-particle orbitals have several
important applications.
First, we can track the migration of the shells to aid in the quest of unraveling the
driving forces behind shell evolution. Starting in the 1970’s with with N = 20 nuclei
31 Na

[27] and 32 Mg [26, 28], as experimenters gained access to beams of unstable nuclei,

it became clear that the canonical harmonic oscillator and spin-orbit shell gaps at 8,

15

20, 28, 50, 28 and 126, softened or even disappeared for nuclei that shared similar
magic numbers but were located off the valley of β-stability. The disappearance of magic
numbers and migration of orbitals were and are viewed as critical for the wider problem
of understanding the nucleon-nucleon interaction in the nucleus (see Section 3.2 for an
overview and Equation 3.7 for the definition of the two-body matrix elements that are the
foundation of modern shell-model calculations).
Following the fall of the ephemeral N = 20, other shell gaps were found to disappear
as experimenters moved from the valley of stability to the proton or neutron drip line.
For example, N = 8 for 11 Li and 12 Be [30, 31] and N = 28 for 42,40 Si [32–35] and 44 S [36].
Shell migration can also lead to the emergence of new energy gaps, for example N = 14 for
22 O,

N = 16 for 24 O [37, 38], and N = 32 for Calcium (52 Ca) through Chromium (56 Cr)

isotones [39–46] (See Section 5.2 for additional information). For a comprehensive review
of these and other shell closures and appearances, see Refs. [5, 47] and citations therein.
Second, tracking the migration of orbitals can reveal the role and strength of specific
matrix elements of the nuclear Hamiltonian in driving shell evolution. Analyzing the
occupancies and relative energy difference of the 1d3/2 and 2s1/2 orbitals for the Z = 19
nucleus Potassium, with one hole in the π1d3/2 orbital, between N = 20 (39 K) and N = 28
19
(47 K) using (d, 3 He) and (e, e p) reactions, revealed a decrease in energy and eventual
19
crossing between the two orbitals as neutrons were added to the νf7/2 orbital. This change
in the energy spacing of the πd3/2 and π(2s1/2 )1 orbital, measured by calculating the
energy difference between the first E(3/2− ) and E(1/2− ) states, with respect to neutron
1
1
occupancy of the νf7/2 orbital, revealed the role of the monopole forces Vσ τ (d3/2 f7/2 )
and Vσ τ (s1/2 f7/2 ) in driving the proton shell evolution [48–50]. Similar analysis was
performed for the Chlorine chain, with one proton in the πd3/2 orbital in Ref. [51] through
the spectroscopy of 43 Cl and 45 Cl, and a comparable evolution of the πd3/2 and π(2s1/2 )
orbitals was seen.
1 The

radial quantum number n is only included to avoid confusion with 1s1/2 orbital.

16

The dominance of these matrix element reflects a few underlying properties of the
nuclear interaction that should be pointed out. First, the πν nn iteraction is strongest
when the proton and neutron wavefunctions strongly overlap—this occurs when the
proton and neutron share the same radial quantum number n and the same orbital
angular momentum —and second when their spins are opposite. The spin-flip ∆ = 0
and ∆ = 1 πν monopole interactions tend to drive the evolution of the single particleorbitals near shell gaps. The strongest negative interaction occurs between j< = − 1/2 and
j> = + 1/2, ∆ = 0, spin-flip partners. See Ref. [5] for a comprehensive review. Shell gaps
modified by this interaction include N = 8 (πp3/2 – νp1/2 ), N = 20 (πd5/2 – νd3/2 ), N = 28
(πf7/2 – νf5/2 ), N = 50 (πg9/2 – νg7/2 ), and N = 82 (πh1/2 – νh9/2 ). The spin-flip ∆ = 1 is
important for shell evolution in 19 K isotopes (πd3/2 – νf7/2 ), 29 Cu isotones (πf5/2 – νg9/2 ),
and 51 Sb isotones (πg7/2 – νh11/2 ) [5]. The force that drives the collapse of the N = 28
shell gap is also partially responsible for the creation of the N = 32 shell gap, a point that
will be further addressed in the discussion of the one proton knockout reaction 9 Be(54 Ti,
53 Sc + γ)X

(see Section 5.2).

Third, as we continue enumerating the advantages of probing single-particle structure,
often outwardly macroscopic properties, like deformation can be understood or explained
through single-particle behavior. Two classic examples, 32 Mg and 31 Na (N = 20) were
studied soon after rare-isotope beams became available. They belong to a cluster of nuclei
with similar deformations called the “island of inversion”.
As the neutron d3/2 and f7/2 orbitals move closer, driven by changes to the πd5/2 – νd3/2
spin-flip monopole interaction as the πd5/2 and νd3/2 orbitals are occupied, cross shell
excitations develop, i.e. neutrons are promoted from the lower energy νd3/2 orbital to one
of the higher energy fp-shell orbitals across the N = 20 shell gap, predominately the νf7/2
orbital. Because the N = 20 shell gap is a harmonic-oscillator shell gap, the orbitals above
and below the gap have opposite parity. Parity is conserved in the strong interaction, and
therefore only even-n particle-hole excitations are allowed, of which the two-particle –

17

two-hole (2p2h) are the most probable. The two promoted neutrons have the full set of f7/2
quantum states available, and they can therefore arrange themselves in the lowest energy
configuration. The system gains quadrupole correlation energy through the preferential
arrangement of the neutron orbitals that is greater than the energy of the energy difference
between the orbitals. This neutron configuration leads to pronounced deformation [52].
A single-neutron knockout experiment 9 Be(32 Mg, 31 Mg + γ)X, showing the power of the
knockout reaction process to probe the ground state wavefunction of nuclei2 , quantified
the presence of neutrons in the negative parity f p-shell. Because of this inversion between
the standard normal-order configuration, because of the observed difference in occupancy
between the d3/2 and the higher energy f7/2 orbitals, and because similar inversions are
seen in a limited selection of neighboring nuclei roughly centered on 20 ≤ N ≤ 22 and
10 ≤ Z ≤ 12 [52], this area of the nuclear landscape is called the “island of inversion.”
Forth, the description of phenomena through a single particle perspective can reveal
the limitations of the valance space used in nuclear shell model calculations. Original
efforts to study the properties of 32 Mg using the Universal sd (usd) effective interaction
[53] failed. The anomalous properties of “island of inversion” nuclei are caused by the
promotion of neutrons to the fp-shell orbits, which lie outside of the sd model space3
used in the usd effective interaction. Through the inclusion of the fp-shell in the sdpf-m
effective interaction, Refs. [54, 55] found broad agreement between the measured and
calculated spectroscopic factors for the ground-state wavefunction of 32 Mg.
The fifth and last point: experimental energies are essential in the determination
of the two-body matrix elements (tmbes), used to represent the residual interaction
between nuclei, and the single-particle energies (spes) used in shell-model effective Hamiltonians4 . The 195 tmbes and 4 spes in the gxpf1 effective interaction for the f p-shell
2 See

Section 3.4 for experimental and theoretical details on knockout reactions.

3 See

Section 3.2.4 for a definition of model spaces the associated effective interactions

4 See

Section 3.2.3 for a more through definition of tmbes and spes. Section 3.2 covers

18

(1f7/2 2p3/2 1f5/2 2p1/2 ) were calculated by fitting 70 linear combinations of tmbes to 699
energy levels in the mass range 47 – 66 [56]. Judicious experimental measurements of
the location of specific orbitals can lead to the select modifications of matrix elements
corresponding to the interaction between pairs of orbitals for specific quantum numbers.
For example, the N = 34 shell gap predicted by the gxpf1 effective interaction was not
seen in the E(2+ ) of 56 Ti [43, 44, 57]. When the experiment did not observe the predicted
1
shell gap, five T = 1 matrix elements were modified5 resulting in the gxpf1a effective
interaction [58] which has higher predictive power in the mass region for nuclei with
N ≥ 32. The gxpf1 effective interaction was used to calculate the level structure and
spectroscopic factors for the 9 Be(54 Ti, 53 Sc + γ)X reaction; the gxpf1a effective interaction
provided similar information for the six different proton-pickup proton-pickup reactions.
Theory and experiment evolve simultaneously. Often experiment reveals the limitations of shell-model theory, like the need to include the intruder states in the valance space
of 32 Mg. The observation of new phenomena, outside the traditional model space, drives
this change in scope. With this last point, we recap the applications and advantages of
tracking single-particle structure. Through tracking single-particle structure one can:
1. Work towards a microscopic explanation for the migration of orbitals and the changes
in shell spacing. This includes the disappearance of traditional shell gaps and the
emergence of new shell gaps.
2. Decompose the nucleon-nucleon, in-medium nuclear interaction into its individual
multipole components.
3. Analyze and explain macroscopic properties, like deformation, through the location
and occupation of orbitals.
the nuclear shell model.
5 Defining

V as V (2ja 2jb 2jc 2jd ; JT ), the five altered matrix elements are V (7777; 01),
V (5511; 01), V (1111; 01), V (5151; 21), and V (5151; 31). See Honma et al. [58] for additional details.
19

4. Expose limitations of the current valence spaces through the observation of new
phenomena that cannot be explained by shell-model calculations in a restricted
model space.
5. Provide data and constraints for the determination of tmbes and spes.

2.2

Techniques

It is clear that looking at the nucleus through the lens of single-particle structure can
undress hidden components of the nuclear interaction and explain a variety of observed
phenomena. The best way to extract this information is through a nuclear reaction that can
probe the single-particle degree-of-freedom. Because of the symmetry between particles
and holes in a nucleus, it is clear we can take two different routes. We can track particles
or we can track holes.
Knockout reactions, the removal of a single nucleon in a direct process from a nucleus
in the fast, projectile beam, are an established technique for tracking single-particle states
in exotic nuclei. These reactions are covered in Section 3.4 and in review articles by Hansen
and Tostevin [59] and Gade and Glasmacher [47]. Through the removal of a nucleon in the
initial projectile with mass A, hole-like states in the A − 1 residue are populated, providing
information on the ground-state wavefunction of the mass A projectile. Recently, single
nucleon pickup reactions, in inverse kinematics with fast, exotic beams have emerged
as a possible new tool for probing single-hole structure [60, 61]. Because the heavy A
projectile picks up a nucleon from the target, particle-like states are populated in the mass
A + 1 residue. Thus, the information provided by the pickup reaction is complementary:
knockout reactions populate single-hole states and pickup reactions populate singleparticle states.

20

2.3

Summary of specific motivations

This thesis approaches both the use and development of knockout and pickup reactions
for the determination of single-particle structure. The specific motivations:
1. Through the 9 Be(54 Ti, 53 Sc + γ)X reaction, probe the single-particle proton configurations of the N = 32 nuclei 54 Ti and 53 Sc, and test the validity of the shell model
and valance space in this neutron rich-region.
2. Continue the development of pickup reactions as qualitative spectroscopic tool by: (1)
Comparing measured partial and inclusive cross sections for three different protonrich reactions centered around 50 Fe to theoretical predictions that combine ccba
reaction calculations and shell model theory; (2) Varying the experimental conditions
through the use of two different targets, 9 Be and 12 C; and by (3) Comparing measured longitudinal momentum distributions to theoretical distributions that assume
a direct reaction process. The six different reactions studied, 9 Be(48 Cr, 49 Mn + γ)X,
9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe

+ γ)X, and 12 C(50 Fe, 51 Co + γ)X, are on the proton-rich side of the valley of stability,
have a conveniently limited number of excited states, and lie in a region of the
nuclear landscape with well understood shell structure.

21

Chapter 3
NUCLEAR SHELL MODEL AND
REACTION THEORY
3.1

Reactions and Structure

The size of the nucleus presents a difficult problem. How does one obtain information on
the individual constituents of a system when the total system size is femtometer scale? The
method, as will be outlined in Section 3.3, is to use a nuclear reaction. The combination of
an experimental probe and a theoretical framework in which to interpret results, allows
scientists to make powerful statements about the properties of a nucleus and its nucleons.
The choice of the nuclear reaction is critical. To probe the wavefunction of a nucleus, one
should ideally use reactions that excite only the single-particle degree of freedom. The
direct one-nucleon pickup and knockout reactions, which will be outlined in this Chapter
in Sections 3.4 and 3.5, provide such a selectivity.
For this work, nuclear reaction theory is combined with shell model calculations
to provide theoretical estimates for experimentally observable quantities like partial
and inclusive cross sections. The goal of the nuclear shell model is a fully microscopic
description of the atomic nucleus. Information on the nuclear shell model follows.

22

3.2
3.2.1

The Nuclear Shell Model
Overview

The nuclear shell model describes the nucleus from a microscopic basis under the assumptions that the nucleus is a quantum mechanical, many body system, relativistic effects can
be ignored, and the interaction between nucleons is dominantly two-body in character.
The core components of the nuclear potential were finally resolved by Mayer [24] and
Haxel, Jensen, and Suess [25] in 1949 through the addition of a strong a spin-orbit term
to the existing, central, harmonic oscillator (ho) potential. The result was an improved
agreement with the experimentally measured single-particle levels for stable nuclei including the first correct description of all experimentally observed magic numbers (see
Figure 1.4). The combination of the extremely repulsive core in the nuclear potential (see
Figure 1.2) and Pauli blocking leads to a dilute system of Fermions that in zeroth order,
can be approximated as completely independent. The many body system where each nucleon interacts with its closest neighbors is replaced by a collection of one-body equations
where each particle interacts with a mean field generated by the averaged potential of all
nucleons. This framework is called the independent particle model (ipm) [22, 23].
The ipm had immediate successes. Because nucleons usually pair to J = 0, the ipm
accurately described the spins and parities of many odd-A nuclei in terms of the last
unpaired nucleon, and as mentioned above, it also described the existence and location of
magic numbers observed in stable nuclei (see Figure 1.3). Unfortunately, there were also
clear failures that indicated the need to include effects beyond the simple ipm mean-field
model. The ipm could not describe the rotational and vibrational behavior of nuclei,
including signature rotational and vibrational bands [10, 17].
Aa. Bohr, Rainwater, Mottelson, and Nilsson tried to merge these separate phenomena
into one model in two ways: the deformed shell model where the individual nucleons
move in a deformed potential caused by the true physical deformation of the nucleus

23

and the particle plus rotor model where a few valence nucleons are coupled to a rotor
representing the internal behavior of the remaining nucleons. These “unified models” of a
particle moving in a deformed mean field are an extension of the ipm. The effect of these
deformed potentials can be seen in Nilsson diagrams [62] which plot the energies of the
single-particle levels with respect to the deformation of the nucleus.
The unified model shares one critical similarity with the ipm. The nucleons still move
in a mean field. There are no two-body interactions or higher order terms (three-body,
etc.). The desire to include these experimentally observed components of the nuclear
interaction led theorists to develop the modern Configuration-Interaction Shell Model
(ci), henceforth simply called ‘shell model’. The shell model is predicated on the idea
that calculations using a purely microscopic, two-body interaction can predict the full
range of nuclear phenomena, including collective behavior. An example of this connection
was shown in 1958 by Elliot who outlined the microscopic origin of rotational motion in
Refs. [63, 64].
Although the concept is simple, the problem is difficult in practice [65]. The shell
model has the following characteristics:
1. Many-body problem. In a nucleus composed A nucleons, each nucleon can interact
with A − 1 partners. This interaction can occur between neutrons (νν), protons (ππ),
or between a proton and a neutron (πν), in principle for all available orbitals.
2. The nuclear interaction is very complicated. See Section 1.2.2 for an overview of the
bare nucleon-nucleon force. Higher-order aspects of the force are still unknown.
3. The system is mesoscopic. The nucleon number is too small to use statistical or
thermodynamical approximations but too large to be completely computationally
tractable. Approximation methods must be used.
4. The interaction is both microscopic and effective. On microscopic: we begin with
the individual proton and neutron degrees of freedom, as opposed to a collective
24

aggregate model. On effective: each nucleon interacts within the medium of the
nuclear matter, which modifies the bare nucleon-nucleon (NN) interaction. Effective
interactions are largely derived from fits to experimental data.
The introduction of the residual (remaining two-body interaction after the subtraction
of the dominant mean-field strength) two-body interaction creates correlations between
different proton and neutron orbitals (off-diagonal interaction matrix elements) and causes
the mixing of Slater determinants associated with different possible configurations of
protons and neutrons within the model space. The result are occupation numbers for a
single nucleon that vary between 0 and 1. Through this configuration mixing, effects that
are outside of the traditional naive ipm, like deformation and other collective behavior,
are describable. CI codes include NuShell and NuShellX [66], antoine [67], mshell
[68], redstick [69], and oxbash [70]. A review of the basic steps in a shell model
calculation follow. For a more thorough coverage see one of the many reviews on the
subject [65, 71–74].

3.2.2

Basic Shell-Model Procedure

Nuclear shell model codes, through the solution of the time-independent Schrödinger
equation (tise), provide the energies and wavefunctions (used to calculate other observables) of the low-lying states in nuclei. Calculations usually involve the following choices:
1. Choice of a model space: It is currently computationally impossible to calculate the
properties of medium to large mass nuclei (A

12) from the ab initio, nucleon degree

of freedom1 . A decision must be made: which proton and neutron orbitals are
important to describe the observed experimental phenomena. The selected truncated
set of orbitals forms the valance space (the core and valance space form the complete
1 Success

up to A = 12, starting with the bare NN interaction, has been achieved with
Green’s Function Monte Carlo (gfmc) [75–77] and ab initio no core [78, 79] shell model
calculations using accurate N N potentials [2–4].
25

model space). For both the pickup and knockout experiments, covered in Chapters
5 and 6, an f p-shell (1f7/2 , 2p3/2 , 1f5/2 and 2p1/2 orbitals) valance space with 48 Ca
and 40 Ca cores was used. See the indicated chapters for specific details.
2. Choice of an effective interaction: The effective interaction, in the form of singleparticle energies and two-body matrix elements (see Section 3.2.3), define the interactions between the particles and holes within a particular model space. Matrix
elements for or of effective interactions appropriate for the same model space, e.g.
GXPF1A and KB3G, will differ, and the choice of an effective interaction may de-

pend on the particular mass region and physical phenomena one is interested in
describing.
3. Choice of a shell model code: Shell model codes execute the complex matrix algebra
(diagonalization of matrices of up to 1011 elements) and the configuration mixing
associated with the solution of the many-body Schrödinger equation. There are many
shell model codes available. Oxbash and NuShellX were used for the calculation
of energies and spectroscopic factors for the pickup and knockout experiments,
respectively. There are notable differences between codes, including the choice of a
coupling scheme. To form states with good total angular momenta, J, and projection,
M, quantum numbers, the angular momentum ji and projection mi of the active
protons and neutrons within the model space must be coupled. There are several
methods of doing so: the M-scheme [67–69], J coupled scheme, and the JT coupled
scheme [66, 70].
4. Comparison to experiment: Calculated level energies can be directly compared to
experiment, and the wavefunctions, using renormalized operators (operators renormalized to the model space), can be used to calculate other nuclear properties, like
electromagnetic transition rates and spectroscopic factors.

26

3.2.3

Nuclear Hamiltonian

The solution of the nuclear many-body problem begins with the time-independent Schrödinger equation
ˆ
ˆ ˆ
HΨ = EΨ −→ (T + V )Ψ = EΨ
where

A

ˆ
H=
i=1

p2
i
2m

(3.1)

A

Vik (|ri − rk |)

+

(3.2)

i k

Starting with Equation 3.2, we can separate the potential into a principal single-particle
term (independent motion in a mean-field) and a residual two-body term2
A

ˆ
H=
i=1


 2
  A
A

 pi
 
 ˆ

 
ˆ


+ U (|ri |) + 
Vik (|ri − rk |) −
U (|ri |) = H0 + Vresidual

 


 2m
 


(3.3)

i=1

i k

ˆ
ˆ
ˆ
Under the ipm approximation, the residual interaction Vresidual is zero Hipm ≈ H0 , and the
full wave function is the product of the wave functions of the individual nucleons φi . A
harmonic oscillator (ho) basis is chosen for the nucleon wave functions for several reasons:
H.O.
they have an analytic form (i.e. φ(rk ) = [ψn (rk ) ⊗ Y m (θk , φk )]j ); ho wave functions

are the natural framework for annihilation and creation operators and the Fock-space
basis (basis with occupational number representation); and they permit the many-body
Hamiltonian of Equation 3.3 to be be cleanly separated into intrinsic and center-of-mass
terms [65, 80].
For an A nucleon system,
sym
Ψ1,2,··· ,A

A

=

φk (rk ),

(3.4)

k=1

where the index k indicates occupancy of the nucleon in any of the N available orbitals
in the valence space and φk is appropriate the single-particle wave function. Because
nucleons are Fermions, the wave function must be properly anti-symmetrized. This is
2 Although

this method is used by B.A. Brown and Rae in their shell model codes, an
alternative method of multipole decomposition exists, see Refs. [65, 72, 73] for details.

27

represented by a single Slater determinant
φ1 (r1 ) φ1 (r2 ) · · · φ1 (rA )
asym

Ψ1,2,··· ,A =

φ2 (r1 ) φ2 (r2 ) · · · φ2 (rA )
.
.
.
...
.
.
.
.
.
.

(3.5)

φA (r1 ) φA (r2 ) · · · φA (rA )
ˆ
The presence of the residual interaction Vresid. causes mixing of the different Slater
determinants, a process called configuration mixing, and as a result, single-nucleon occupancies will vary between 0 and 1 (or for a complete orbital, occupancies will range
between 0 and 2J + 1). The final wave function with the residual interaction included is
assym

assym

Ψ1,2,··· ,A =

Cj Ψj

(1, 2, · · · , A, · · · , N ).

(3.6)

j

The sum is over all possible configurations of nucleons that preserve the conserved quantities as given by the coupling scheme. The goal of a ci code is to determine this complete
wavefunction, with coefficients Cj , and the associated energies.
The use of a harmonic oscillator basis introduces a center-of-mass (com) energy that
gives the system a non-zero energy. These states are called spurious states and must be
removed, usually by adding a center of mass Hamiltonian [81]. With the subtraction of
the com motion, the Hamiltonian of Equation 3.3, this time in the notation of second
quantization (particle creation and annihilation operators), can be written as
†
j aj aj

ˆ
H=
j

+
j1 j2 j1 j2

ˆ
j1 j2 |Vresid. |j1 j2 JT a† a† aj1 aj2 ,
j1 j2

(3.7)

where total angular momentum J and isospin T are good quantum numbers. Pairs of
nucleons are created on orbitals j1 and j2 and annihilated on orbits j1 and j2 .
The parameter j of Equation 3.7 is the single particle energy (spe) of the state j. It
represents the total energy of the proton or neutron in a mean field. spes can be experimentally estimated from the difference in binding energies between A and A ± 1 nuclei.

28

ˆ
The second term Vj j j j ≡ j1 j2 |Vresid. |j1 j2 JT is the two-body matrix element (tbme). The
1212
tbme represents the residual attraction between the nucleons. An effective interaction is
completely determined by its spes and tbmes. Popular effective interactions such as usd
[53], gxpf1 [56], and KB3G [73], have slightly different spes and tbmes depending upon
the valence space and the experimental data from which the matrix elements are derived.

3.2.4

Model Space and Truncation

The size of the matrices associated with mass A ≥ 12 nuclei becomes computationally
intractable when we consider the combination of each nucleon interacting with every
other nucleon. The size of the matrices grows exponentially with the inclusion of new
orbitals in the shell model valance space up to the modern limit of ∼ 1011 [82]. If we
are interested in low-lying states, particle or hole states near the Fermi surface may be
sufficient to describe a nucleus’ low-lying properties. This allows us to treat the other
nucleons as an inert core and fold in the core effects (e.g. the binding energy between the
active nucleons and the core) into the Hamiltonian of the valence nucleons. By ignoring
the interaction of the nucleons in the core with the valence nucleons included the valence
space, the Hamiltonian must be modified. This process of adjusting the Hamiltonian to
the valence space is called renormalization.
The choice of a model space is important and there are trade-offs to consider. With
more terms, in general, one gets more complete results, including effects that do not
manifest within a more limiting valence space. See, for example, the intruder states of
32 Mg

and neighboring nuclei [26, 28, 52, 55, 83]. Large valence spaces, unfortunately,

dramatically increase the computational complexity and the complexity of the effective
interaction. The largest basis, the M-scheme, explodes exponentially as
DT =

Dπ Dν
·
.
nπ
nν

(3.8)

Dπ,ν is the total occupancy of the orbitals in the valence space, and nπ,ν are the number

29

of nucleons filling those orbitals for protons, π, and neutrons, ν, respectively. Maximum
matrix size occurs when nπ,ν = 1 Dνπ . For the p-shell, which includes the 1p3/2 and
2
1p1/2 orbitals and has 6 holes available for both the protons and neutrons, the maximum
matrix size is 400 (e.g. 10 B). For the sd shell with the 1d5/2 , 2s1/2 , and 1d3/2 orbitals, the
maximum matrix size is approaching 1 million (853 776). For the f p shell with the 1f7/2 ,
2p3/2 , 1f5/2 and 2p1/2 orbitals, the maximum matrix size is 3.4 × 106 . Given the exposition
of matrix sizes with the inclusion of additional orbitals, it is evident why a full ab initio
calculation that includes all orbitals up through the f p space is impossible.
The large spacing between orbitals that occur at the magic numbers make N , Z = 2,
8, 20, 28, 50, 82, and 126 natural places for model space truncation. The large shell
gaps minimize the off-diagonal matrix elements. The strongest correlations occur when
the wave functions strongly overlap, i.e. when the orbitals are close in energy and share
certain quantum numbers. Other effects, as mentioned above, that originate from nucleons
outside the model space, can be folded into the valence-space nucleon-nucleon interactions.
Consider, for example 53 Sc. 53 Sc has 21 protons and 32 neutrons and thus 48 Ca with
N = 28 and Z = 20 is a natural choice for an inert core. An oxbash calculation [81] using
the gxpf1 [56] effective interaction, was used to calculate the level scheme (for 53 Sc) and
spectroscopic factors for the 9 Be(54 Ti, 53 Sc + γ)X reaction (see Section5.4). Calculations
used an f p model space and an inert 48 Ca core. For the neutron rich isotopes, 48 Cr,
49 Mn,

and 50 Fe, and their associated reactions3 ,the spectroscopic factors and energies

were calculated using the gxpf1a effective interaction [58] and a 40 Ca core. See Figure 3.1
for a cartoon representation of the 53 Sc model space and Figure 3.2 for the dimension of
several model spaces with respect to the nuclear chart.
3 9 Be(48 Cr, 49 Mn + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 12 C(49 Mn, 50 Fe
+ γ)X, 9 Be(50 Fe, 51 Co + γ)X, and 12 C(50 Fe, 51 Co + γ)X

30

nℓj
10

28

1g9/2

2
6
4

2p1/2
1f5/2
2p3/2

8

20
4
2
6

8
2
4

2

2

Protons (Z = 21)

32

10
2
6
4

28

1f7/2

8

20

1d3/2
2s1/2
1d5/2

4
2
6

8

1p1/2
1p3/2

2
4

2

1s1/2

}

pf - 109

}

sd - 105

}

p - 102

2

Neutrons (N = 32)

Figure 3.1: Valence space (yellow highlight) used in the calculation of 53 Sc excited states
and spectroscopic factors for the 9 Be(54 Ti, 53 Sc + γ)X reaction. The matrix dimensions
and names for common valence spaces are shown. Magic numbers for the neutrons and
protons (including the emergent N = 32 shell gap) are independently listed (tan circles).
The orbitals in the pf5/2 g9/2 valence space are not shown (see also Figure 3.2).

3.2.5

Model Space Limitations

A given model space can only handle phenomena associated with the configuration of
nucleons within that model space. Occasionally, behavior emerges that is caused by
processes that involve the promotion of nucleons into or out of the model space. These
states are called intruder states and they are usually occur at energies close to the excitation
energy of the first excited state in a closed shell [71]. For example in 53 Sc such intruder
states may occur near the energy of the 2+ state in 48 Ca (Ex = 3382 keV). Sometimes,
1
however, a weakening shell gap and increase in N N correlations can lead to cases where

31

50
28
20
8
2

pf5/2g9/2
fp

sd
p

2 8

82
50

20 28

Figure 3.2: Nuclear chart showing the size and applicable mass range of valance spaces
used in shell model calculations: the p, sd, f p and new pf5/2 g9/2 valance spaces are shown.
these intruder states can occur at lower energies [71]. The 1/2− state at 110 keV in 19 F or
the 1/2+ state in 11 Be are intruder states. Intruder states can only be handled by increasing
the model space and their identification can indicate the limitations of a model space and
a need for cross shell interactions. Chapter 5 covers the observation of positive-parity
intruder states in the low-lying spectrum of 53 Sc and the implications of their discovery
on the choice of valance spaces and cross shell interactions.

3.2.6

Shell Model Spectroscopic Factors

Using their calculated wavefunctions, shell model codes can provide spectroscopic factors.
For the single-nucleon pickup or knockout reactions, these spectroscopic factors are
defined as
1
| Φ A |a† |Φ A−1 |2
2Ji + 1 i k f
1
pick.
A+1 |2 ,
Sj
=
| ΦiA |ak |Φf
2Ji + 1

knock. =
Sj

32

(3.9)
(3.10)

where by convention the factor 2Ji + 1 is associated with the mass A projectile. Φi is the
internal wave function of the projectile. Φf is the internal wavefunction of the projectileˆk
like residue after the addition or removal of a nucleon, depending upon the reaction. a†
ˆ
and ak are the creation and annihilation operators for the creation or removal of a nucleon
with quantum numbers k ≡ (nj m). Spectroscopic factors provide the parentage of the
initial ground state with respect to a specific final state and thus provide information
about the occupation of orbitals in the A projectile ground state.
The cross section for the removal of a nucleon σknock. emerges from the reduced matrix
element
σknock. ∝

1
2Ji + 1

A−1 |2
| ΦiA |a† |Φf
k
Mf ,Mi ,m

=
Mi ,Mf ,m

=


 J
 f





−M


 J
 f






m Mi  −Mf
j

f

Ji



Ji 


 | Φ A |a† |Φ A−1 |2
 i k f


m Mi 
J

(3.11)

1
| Φ A |a† |Φ A−1 |2 = Sknock.
2Ji + 1 i k f

The proportionality factor between the spectroscopic factor S and the knockout cross
section σknock is the single-particle cross section, the reaction probability assuming S = 1
(see Equation 3.17). Notice the explicit sum over final states Mf and average over initial
states Mi . The cross section for the pickup reaction σpick. follows similar algebra.
A specific example may be more illustrative. Consider the 9 Be(54 Ti, 53 Sc + γ)X reaction,
where a proton is removed from the 54 Ti ground state leaving the 53 Sc heavy residue
in one of many states, including the ground state. The final states populated depend
on the quantum numbers (the orbital) of the removed proton. If a proton is removed
from the 54 Ti π(1f7/2 ) orbital we expect, in the absence of fragmentation of strength
from the f7/2 orbital, to populate the 53 Sc 7/2− ground state. The magnitude of the
g.s.
overlap scales with occupancy. If there are two protons in the proton f7/2 orbital for the
54 Ti

ground state, we expect a large spectroscopic factor of nearly 2 to the 53 Sc ground

state. If there are no protons in the π(1f7/2 ) orbital, we expect a negligible overlap with
33

the 53 Sc ground state and a πf7/2 spectroscopic factor of nearly zero. Because isospin is
a conserved quantum number, and because, as a vector, it has similar couping algebra
as angular momentum, spectroscopic factors are usually provided with the appropriate
Clebsch-Gordan isospin-coupling coefficient as C 2 S.
The number of available particle-like and hole-like states for an orbital with quantum
numbers k ≡ (n j) is clearly related to the orbital’s maximum occupancy 2j + 1. Because
knockout and pickup reactions probe particle-like and hole-like states, the relationship
between particles, holes, and occupancy for orbital k is observed in their spectroscopic
factors. This relationship is called a sum rule, and it takes the following form:
Sknock +
m−

(2Jf + 1)Spick = 2j + 1

(3.12)

m+

The sums are over all states in the A − 1 nucleus (m−) and the A + 1 nucleus (m+). The
additional 2Jf + 1 factor in front of the pickup spectroscopic factor Spick accounts for the
difference in M state averaging.
If spectroscopic factors are calculated using a harmonic oscillator basis (wave functions)
moving relative to a fixed point rather than the with respect to the center-of-mass (com)
motion of the A nucleons rather than the A − 1 (knockout) or A + 1 (pickup) nucleons, a
center-of-mass (com) correction [84] is required. These corrections Cknock. and Cpick. take
the form
Cknock. =

A N
A−1

Cstrip. =

A N
,
A+1

(3.13)

where N is the major oscillator quantum number (N = n + where n by convention, starts
at 0). This correction was necessary for both oxbash (knockout) and NuShellX (pickup)
calculations.
With configuration mixing, the wavefunction is the combination of several Slater determinants rather than one, and therefore the occupancy of orbitals is usually never a
discrete integer. So besides providing information on the occupancy of orbitals, spectroscopic factors provide a measure of the purity of the single particle state. Experimental

34

observables that relate to spectroscopic factors (e.g. partial cross sections) are some of
the most fundamental tests of shell-model wave functions. Over the last decade, single
nucleon knockout reactions have been proven as an effective method for quantifying
spectroscopic strength in exotic nuclei [85], and more recent work by Gade et al. [60, 61],
showed single-nucleon pickup reactions can also provide similar qualitative information.

3.3

Reaction Theory

In a nuclear reaction a nucleus is subjected to an external field (electromagnetic or nuclear)
created by another nucleus, and the response of the nucleus to the external field provides
information on both the nucleus’s constituent protons and neutrons (shell structure) and
the interaction between the nucleus and its interaction partner (reaction theory). Reaction
theory and shell model calculations can be combined in a straight-forward way to predict
directly observable experimental quantities: partial and inclusive cross sections.
To gain information about the single-particle structure, a small wavelength probe
must be used. Long-wavelength, low-energy probes only tell us about the strength of the
interaction. Short-wavelength, high-energy probes are needed to provide a detailed map
of the region of interaction. Ideally, the interaction should only involve the single-particle
states (nucleons) we want to probe. A class of reactions called direct reactions satisfies
these restraints. Direct reactions are characterized by short time-scales (τ ∼ 10−22 sec) and
single-step transitions from initial to final states. In inverse-kinematics these constraints
are typically satisfied by intermediate (or higher) energy projectiles and grazing reactions.
Both single nucleon knockout (Figure 3.3) and proton pickup reactions (Figure 3.4) are
direct reactions.
Pickup and knockout reactions reactions access single-particle structure either through
the population of single hole-states through the removal of a nucleon from the projectile
(knockout reactions) or through the population of single-particle states through the
addition of a nucleon to the projectile (pickup reactions). Knockout reactions allow a
35

determination of partial cross sections (which are connected to spectroscopic factors and
thus occupation numbers) [85, 86], and recent evidence suggests that single nucleon
pickup reactions may provide similar information, so far on a more qualitative basis
[60, 61].
There are two possible notations used to describe a nuclear reaction depending on the
reaction kinematics. A reaction in regular kinematics uses a target made from the isotope of
interest and the beam is used to probe the target. These reactions are written in the form
Pi (Ti , Tf )Pf

(3.14)

where Pi,f and Ti,f are the initial and final states of the target and projectile.
The first nuclear reactions were in regular kinematics (see Appendix B) and today,
established single-nucleon transfer techniques using light beams (e.g. deuteron d, tritium
t, or 3 He beams) are used for obtaining spectroscopic information on stable target nuclei.
In regular-kinematics transfer, a nucleon is transfered from the light projectile to the
target, and based on the differential cross sections, angular distribution and energy of the
light reaction residue, information on the occupancy, orbital angular momentum, and
energy of states in the target nuclei are obtained. For example, in the d(50 Ti, 49 Sc) 3 He
reaction of Ref. [87], a deuteron, d, accelerated to 52 MeV, impinged on a 50 Ti target.
The deuteron picked up a proton, leaving the target nucleus one proton lighter as 49 Sc.
Through the measurement of the cross section, energy, and angular distribution of the
3 He

nuclei, information is inferred about the occupancy of states in 50 Ti.

In contrast, in an inverse kinematics reaction, the projectile is the species of interest and
the target is used as a probe. Such reactions are written in the form
Ti (Pi , Pf )Tf ,

(3.15)

e.g. 9 Be(54 Ti, 53 Sc + γ)X and 9 Be(48 Cr, 49 Mn + γ)X. In an inverse-kinematics knockout
or pickup reaction (see Figures 3.3 and 3.4), a short-lived nucleus, moving at velocity of
about v ∼ 0.3 c interacts with a light target, typically 9 Be. The more exotic the isotope,
36

as judged by its neutron-proton asymmetry, the shorter its half-life. It is impossible to
make targets out of species with half-lives shorter than a few days, but inverse kinematics
reactions allow one to study nuclei with short half-lives. Facilities, such as the nscl, that
can produce beams of nuclei with short half-lives, have only emerged within the last few
decades. This work concentrates on the modern inverse-kinematics nucleon removal and
addition reactions with fast beams. Henceforth, unless qualified, knockout and pickup will
refer to the inverse-kinematics techniques.

Figure 3.3: Knockout reaction schematic for the example of the 9 Be(54 Ti, 53 Sc + γ)X
reaction. A nucleon from the incoming projectile of mass A (A Z) is removed (green)
creating a final heavy projectile (residue) of mass A − 1 (A−1 Z or A−1 (Z-1)). The removed
nucleon is either absorbed by the target (stripping) or, as seen in the figure, continues with
the same velocity as the incoming beam (diffraction). Both the residue and its de-excitation
γ rays are detected. The final state is a three-body system.

The knockout and pickup reactions discussed here were developed for the specific
experimental constraints of the National Superconducting Cyclotron Facility (nscl) Coupled Cyclotron Facility (ccf) [88, 89]. Nuclei are short lived: we use fast beams in inverse
kinematics. It is much easier to detect the “spectator”, the mass A − 1 residue left after the
addition or subtraction of a particle: we relate the properties of the residual nucleus to
the properties of the transfered nucleon. The high beam velocity compresses the angular
37

Figure 3.4: Pickup reaction schematic for the example of the 9 Be(48 Cr, 49 Mn+γ)X reaction.
An incoming projectile of mass A (A Z) picks up a nucleon from the target, creating a final
heavy projectile with mass A (A+1 Z or A+1 (Z+1)). The final projectile and its de-excitation
γ rays are detected. The final state is a two-body system.

distribution of the reaction products and the γ rays emitted in flight in a forward cone:
we use a magnetic spectrograph that typically accepts the full momentum width of the
residue and a γ ray spectrometer (see Section 4.4) that measures both the energy and
the angle of the emitted γ ray in the lab frame. The γ ray spectrometer, through the
measurement of the angle of γ ray emission by the excited residue, reconstructs the energy
of the γ ray energy to the projectile’s rest frame.
Despite the switch from regular to inverse kinematics, knockout and pickup reactions
can provide similar information to classic transfer reactions. Through the detection of γ
rays emitted by the residue in flight and the identification of the corresponding transitions
in the Doppler-reconstructed, residue rest-frame spectrum, final states in the residue are
identified. From the measurement of partial cross sections to the populated states in the
residue, experimental spectroscopic factors can be extracted in comparison to reaction
theory, and for knockout reactions, the shape of the parallel momentum distribution of
the residues offers information on the orbital angular momentum of the removed nucleon.

38

3.4

Knockout reactions

The kinematics of a knockout reaction can be rewritten in a slightly modified form of
Equation 3.15 to explicitly highlight the critical components of the reaction
P (= ci + x) + Ti −→ cf + Tf

(3.16)

The incoming projectile P = ci + x, written in terms of a spectator core c and participating
nucleon x, loses its nucleon x in a direct, single-step interaction with the target T (stripping
or diffraction). The core c is left in a particular final state cf . The removed nucleon x is
not detected. The incoming projectiles, assumed to be in its ground state, are identified
event-by-event through a time-of-flight difference. The orbital angular momentum of
the removed nucleon is inferred from the measured longitudinal momentum distribution
of the ensemble of projectile-like one-nucleon knockout residues. The final state of the
projectile-like residue is determined through γ ray spectroscopy. Knockout reactions are
a proven and established technique for tracking single-particle structure and probing
nuclear phenomena [59, 90]. Two representative examples follow.
Knockout reactions probed the radial extent and wave function of “halo nuclei,” nuclei
where one or more neutrons (e.g. in 11 Be, 11 Li, or 6 He) or protons (e.g. in 8 B or 17 B)
tunnel through the nuclear potential formed by a well-defined core to form an extended,
diffuse matter distribution [91, 92]. The large spatial extent of the “halo” nucleus 11 Be was
qualified by the narrow momentum distribution of the 10 Be core following the removal
of the valance nucleon from 11 Be in a knockout reaction [93]. The 11 Be ground state
wavefunction was quantified in a knockout reaction, i.e. 9 Be(11 Be, 10 Be + γ)X, through
the additional tagging and identification of final states using γ ray spectroscopy with a
position-sensitive NaI array [94].
Knockout reactions helped delineate the island of inversion. The composition of the
ground state wavefunctions of 32 Mg, 30 Mg, and 28 Ne was quantified through the knockout
reactions 9 Be(32 Mg, 31 Mg + γ)X, 9 Be(30 Mg, 29 Mg + γ)X [95], and 9 Be(28 Ne, 27 Ne + γ)X
39

[55]. A gradual onset of the intruder ground-state configurations was shown, further
expanding the early predicted boundary of 10 ≤ Z ≤ 12 and 20 ≤ N ≤ 22 [52]. For further
successes of knockout reactions, including the quantification of the effect of correlations,
both short- and long-range, and the resultant quenching of spectroscopic strength, see
Refs. [47, 59, 90, 96]

3.4.1

Theoretical Cross Sections

As first shown in Section 3.2.6 in the discussion of spectroscopic factors, one can break
down the total cross section into two separate parts: a component that describes the
reaction theory (single-particle cross section) and a component that describes the nuclear
structure (spectroscopic factor) [47, 96]. The single-particle cross section quantifies the
probability that the removal of a single nucleon (spectroscopic factor of S = 1) from a
specific orbital in the initial mass A projectile will lead to a particular final state in the
mass A − 1 residue. The total cross section, because it has to scale with the occupancy of
the orbital, includes multiplication by a spectroscopic factor. The final form is written as
[59, 90, 97, 98]
C 2 S(j, I π ) σsp j, Sν,π + Ex (I π ) .

σ (I π ) =

(3.17)

j

Whenever the spectroscopic factors are calculated using Harmonic oscillator wave functions about a fixed center, as was true for both the oxbash and NuShellX codes used to
calculate the spectroscopic factors for the knockout and pickup analyses of Chapters 5 and
6, the com correction of Equation 3.13 must be applied. Equation 3.17, with the addition
of the com correction, is written as
σ (I π ) =
j

A N 2
C S(j, I π ) σsp j, Sν,π + Ex (I π )
A−1

(3.18)

The sum in Equations 3.17 and 3.18 is over all angular momenta j of the removed nucleon
that can couple the spin and parity I π of the residue to the spin and parity of the incident
projectile J π . Ex is the excitation energy of the residue and Sπ and Sν are the proton and
40

neutron separation energies of the incident projectile4 . If a proton is removed Sπ,ν ≡ Sπ
and if a neutron is removed Sπ,ν ≡ Sν . N is the major harmonic oscillator quantum
number.
For an even-even incident projectile with a 0+ ground state, the angular momentum
coupling is especially simple. The total angular momentum and parity I π of the residue has
the same total angular momentum and parity j π of the removed nucleon. The relationship
between the population of specific states with quantum numbers I π and the removal
of a nucleon with quantum numbers j π is the fundamental spectroscopic background
of knockout reactions (see Figure 3.5). If the complete strength (summed partial cross
sections) from the ground state in the projectile is accounted for—particularly for cases
where the spectroscopic strength is spread over several final states in the residue, a
phenomena called fragmentation—experimental spectroscopic factors can be extracted
with the aid of reaction theory (single-particle cross sections)5 (see Equation 3.18).

3.4.2

Single Particle Cross Sections

Single-particle cross sections are broken down into three components
σsp = σstrip + σdiff + σCoul .

(3.19)

The Coulomb component is negligible for light, low-Z targets like 9 Be6 and the reaction
cross section can be reliably calculated from just the stripping and diffractive components.
In stripping, the removed nucleon interacts at most inelastically with the target, exciting
4 The

separation energies are written as Sπ and Sν instead of the more traditional Sn
and Sp to avoid confusion with the nucleon and core scattering matrices Sn and Sc
5 Spectroscopic factors cannot be directly extracted from experiment; only inclusive and

partial cross sections can.
6 Using

a light, 9 Be target has several advantages: the coulomb component is negligible,
and the highly-absorptive residue-target interaction (9 Be has no bound excited states),
combined with the experimental observation of the residue (the residue must survive),
ensures a peripheral collision [59].
41

2p3/2
1f7/2
1d3/2
2s1/2
1d5/2

Z = 28

3/2-

Z = 20

7/2-

Ground State
54Ti

53Sc

Figure 3.5: Population of final states in the example 9 Be(54 Ti, 53 Sc+γ)X knockout reaction.
For even-even nuclei with zero total angular momentum I = 0 (the ground state) the
removal of a nucleon with quantum numbers j π populates final states with the same total
angular momentum I π . For example, the removal of a π(f7/2 ) proton from the ground
state in 54 Ti populates the 7/2− ground and possibly higher-lying f7/2 fragments in 53 Sc.
g.s.
The removal of a p3/2 proton may populate one or many 3/2− states in 52 Sc. If the strength
to the 7/2− and 3/2− states can be quantified through the measurement of their partial
cross sections, information about the wavefunction of 54 Ti can be extracted. In the case
above, comparatively large partial cross section to the ground state of 53 Sc in comparison
to the partial cross section to the population of the first 3/2− state (assuming minimal
fragmentation), reveals that the two valence protons are predominately in the π(f7/2 )
orbital. See the analysis of the 9 Be(54 Ti, 53 Sc + γ)X reaction in Chapter 5.

the target, and is absorbed. Its contribution to σsp is [99]
σ (strip) =

1
2I + 1

c
c
φIM |(1 − |SxT |2 )|ScT |2 |φIM .

db

(3.20)

M

The total stripping reaction probability is the product of the probability of elastically
scattering the core |ScT |2 and the probability of inelastic absorption of the nucleon by
the target (1 − |SxT |2 ) integrated over all impact parameters b and averaged over all M
c
substates. The quantum numbers I and M refer to final states of the core. φIM is the

single-particle, relative motion wave function of the removed nucleon and the projectilelike core (residue) in the final state (see Section 3.4.3). Note that the norm of the scattering

42

matrices for the nucleon-target and core-target systems, SxT and ScT , are the transmission
probabilities, e.g. TcT = |ScT |2 is the probability the core survives (emerges) in the elastic
channel and TxT = |SxT |2 is the probability that the nucleon emerges in the elastic channel.
1 − |SxT |2 is the probability that a nucleon is removed.
The contribution of the diffractive or elastic component, where the removed nucleon
interacts elastically with the target and is dissociated from the residue, is more complicated.
It includes a summation over continuum relative-motion states [90]
σ (diff) =

1
2I + 1

dk

db | φkσ |(1 − SxT SxT )|φIM |2 ,

(3.21)

σ ,M

but because the spectator-core approximation allows us to assume that φkσ is an eigenstate
of the same bound Hamiltonian whose eigenvector gives φjm 7 , we can rewrite Equation 3.21 without the integration over the continuum as
σ (diff) =

1
2I + 1

c
c
φIM ||1 − ScT SxT |2 |φIM −

db
M

c
c
| φIM |(1 − ScT SxT )|φIM |2 . (3.22)
MM

This is the form used in the calculation of the single-particle cross sections. A similar
integration over all impact parameters and average over all M substates occurs. The
removed nucleon is either absorbed or diffracted according to the processes (stripping or
diffraction).
The effects of the interaction of the nucleon and the core with the target including
absorption are seen through phase shifts included in SxT and ScT . The approximations
used in the derivation of Equations 3.20 and 3.22—spectator-core, Glauber (eikonal), and
static density approximations—are covered in the subsequent sections.
7 We have assumed that the two-body Hamiltonian has only one excited state.

Additional
bound states, assumed to have a small contribution at intermediate energies, will slightly
decrease the diffractive contribution.

43

3.4.2.1

Spectator Core Approximation

The momentum distribution of the projectile-like fragment, measured at small grazing angles (given the absorptive core-target interaction, the reaction is assumed to be peripheral),
peaks near the momentum of the projectile. Given this observation, a spectator-core approximation is used: the interaction between the projectile and the target can be approximated
by the interaction between the participating nucleon and the target. The core is treated as
a spectator, i.e. it is only allowed to elastically scatter. The influence of the target on the
incoming projectile and outgoing residue is only reflected in their optical wavefunctions.
These wavefunctions are calculated using the eikonal Glauber approximation.

3.4.2.2

Glauber approximation

The Glauber approximation is rooted in physical optics. It assumes the incident projectile
will continue in a straight path (eikonal approximation) as it interacts with the target; it
ignores the negligibly small contribution to the integral in the calculation of the wavefunction that come from components that deviate from the incoming projectile wave vector k.
Thus the multi-nucleon projectile can be broken down into a series of one-dimensional
problems at frozen impact parameters. The entire interaction is reflected in a phase [100]
whose imaginary component represents the strong absorption of the projectile, core, or
nucleon by the target. This formalism greatly decreases the dependence on the entrance
and exit channel optical model potentials used as in dwba calculations for classic transfer
reactions [47, 101].
There are two conditions that must be satisfied: the energy, E, of the incident particle
must be greater than the magnitude of the potential, V , and the wavelength of the particle
must be much shorter than the size of the potential. The first statement, written as V /E

1

is equivalent to the statement that V (x) changes slowly within a particle wavelength. This
places kinetic energy restrictions on the incident projectile, and the Glauber approximation
is only used for incident projectile energies around 50 MeV/u or greater. The projectile
44

beam energy for the knockout data analysis contained in Chapter 5 was 72 MeV/u.
The Glauber phase ξi for the projectile-target, core-target, and nucleon-target interaction can be written as
zb
1
∆k dz.
ν −∞ i

ξi (bi ) =

(3.23)

and represents the amount of Born-approximation phase-shift accumulated along the
straight trajectory defined by the beam axis. ∆k is defined as
∆ki ≡

k
k
Vi (z, b) = i (Ui (r) + iWi (r)),
2E
2Ei

(3.24)

where W is the imaginary component of the strongly-absorptive nuclear potential and E is
the energy of the projectile.
The Glauber S-matrices for elastic scattering SpT and SxT naturally emerge as
SxT = e2iδxT (b)

(3.25)

2iδpT (b)

(3.26)

SpT = e

where the total phase shift accumulated δi (b) is
z

δi (b) =

3.4.2.3

dz
−∞

ik
W (z , b).
2E

(3.27)

Static Density Approximation to the Scattering Matrices

The strong nuclear absorption W contained in the S-matrices phase shifts is represented
by double-folding the projectile nuclear density and the target nuclear density with the
NN nuclear interaction. The general result for Eqns. 3.25 and 3.26 is
f
Si (b) = exp − N N
2

(z)

(z)

d 2 x ρi (|x|) ρT (|bj − x|)

(3.28)

where i = cT , xT (core-target and nucleon-target), bj is the impact parameter, ρP and ρT
are the ground state densities of the projectile and the target, and fN N is the empirical,
free, isospin-weighted, nucleon-nucleon, forward scattering amplitude evaluated at the
45

incident projectile energy [102], as parametrized by L. Ray [103], and includes the ratio of
the real to imaginary parts of the forward scattering amplitude to account for incomplete
absorption [103–106]). Sj (bj ) is integrated over the plane (d 2 x) perpendicular to b. The
ˆ
superscript (z) indicates the projection of the density distribution onto the unit vector z
(parallel to b) that defines the beam axis, e.g.
∞

(z)

ρi (b) =

−∞

dz ρi (|b + z|)

(3.29)

Gaussian matter distributions have been assumed for the core ρc and target ρT density
distributions before 2003, but for more recent calculations, as done in this thesis, matter
distributions were taken from Hartree-Fock calculations using the SkX Skyrme force [81].
For the core-target integration, the S-matrix (Equation 3.28) becomes
f
ScT (bj ) = exp − N N
2

(z)

(z)

d 2 xρP (|x|)ρT (|b − x|)

(3.30)

and for the core-nucleon S-matrix, where density distribution for a single nucleon reduces
to a delta function, we have
f
(z)
SxT (bj ) = exp − N N ρT (bj ) .
2

(3.31)

These scattering matrices are calculated for a specific impact parameter (Glauber approximation). Equations 3.30 and 3.31 are the final forms used in Equations 3.22 and
3.20.

3.4.3

Projectile Wave Functions

c
The projectile wave functions φIM of the nucleon-core system, as seen in Equation 3.20

and Equation 3.22, are constructed from the bound states (eigenvalues) of a two-body
interaction approximated by a Woods-Saxon potential
V =

V0
1 + exp r−R
a
46

.

(3.32)

The potential strength V0 is fixed to the core / valence-nucleon separation energy in the
final state, i.e. the addition of the proton or neutron separation energy (Sπ or Sν ) and
the excitation energy of the A − 1 residual, E = Sπ,ν + Ex (I π ). Originally, Woods-Saxon
radii R = r0 A1/3 were calculated using r0 = 1.25 fm and the mass of the core [47, 59].
For current calculations, the root-mean-square (rms) separation R was taken from a selfconsistent Hartree-Fock (hf) calculation using the SkX Skyrme interaction [47, 81] at a
fixed diffuseness a = 0.7 fm. Gade et al. [107] tested the sensitivity of the single-particle
cross sections, for the one-neutron knockout reaction from three N = 16 isotones (32 S,
33 Cl,

and 34 Ar), to the radius R and diffuseness parameters a through finite difference

calculations. They found that the single-particle cross sections were insensitive to the
choice of diffuseness parameter a8 , but a 0.1 fm error of the rms radius R translated to a
13 % relative error. Ref. [107] also noted the accuracy of the hf calculations: calculated
charge radii were accurate to within 0.02 fm of existing experimental data. Thus, through
the hf calculation of the relative core-nucleon rms radius R, the uncertainty of the singleparticle cross section on input parameters that depend on nuclear sizes was reduced.

3.4.4

Momentum Distributions

As defined in the introduction, the longitudinal momentum distributions of the heavy
residues were first used to determine the radial extent of the nuclear halos [91]. Further
theoretical and experimental development allowed the extraction of the orbital angular
momentum

of the removed nucleon from the experimental longitudinal momentum

distribution. The longitudinal momentum distribution is less sensitive to Coulomb effects
and is easier to calculate than the transverse parallel momentum distribution.
Longitudinal momentum distributions were calculated using the code momdis [106].
momdis uses the same eikonal framework (e.g. elastic scattering matrices Sc and Sn ) used
8 The full partial derivative evaluated by Ref. [107] through finite difference calculations
was δσsp /σsp = (1.286 fm−1 ) δR + (0.181 fm−1 ) δa.

47

to calculate the single-particle cross sections whose stripping and diffractive components
were shown in Equations 3.20 and 3.22. The shape of the momentum distributions
of the diffractive dissociation and stripping components, as shown by Ref. [108], have
similar shapes, and thus momdis only calculates the stripping component. The full form
of the longitudinal momentum distribution in cylindrical coordinates (where kz is the
longitudinal component of the momentum of the core kc , and r ≡ (ρ, z, φ) = rx − rc ), as
taken from Ref. [106], is
dσstr.
1
1
=
2 2 +1
dkz
(2π)
m

∞
0

d 2 bx 1 − |SxT (bx )|2 ×
∞

∞

d 2 ρ |S
0

cT (bc

)|2
−∞

c
dz exp[−ikz z]φIM (r)

2

(3.33)

where SxT and ScT are the nucleon-target and core-target eikonal S-matrices of Equations
c
3.31 and 3.30, bx,c = |ρ − bx,c |, and φIM is the single-particle, bound-state wave function

of the core-nucleon (c + x) system.
Sample theoretical momentum distributions, calculated in the center of mass of the
projectile, are shown in Figure 3.6.

3.4.5

Accuracy

The single particle cross sections and momentum distributions have been verified against
other approximations, including solution of the full time-dependent Schrödinger equation
[109], semi-classical transfer to continuum including isolated an overlapping resonances
[110], and continuum-discretized coupled channels calculations [111, 112]. The current
formalism incorporating the Glauber approximation, gives adequate single-particle cross
sections, and thus allows an accurate comparison to experimental spectroscopic strength.

48

Calculated Parallel Momentum Distributions

Counts [arb. units]

1d3/2
1d5/2
1d5/2
1f7/2
2p1/2
2p3/2
2s1/2

-600

-400

-200

0

∆p (MeV/c)

200

400

600

Figure 3.6: Theoretical momentum distributions of 53 Sc in the 9 Be(54 Ti, 53 Sc+γ)X reaction
for the removal of a proton from the indicated orbital from 54 Ti. There are clear shape
differences between the removal of a nucleon with s, d, p, and f ( = 0,1,2 and 3) orbital
angular momentum. Additional steps are necessary before comparing these distributions
to theory. These steps are covered in Section 4.5.3.

3.5
3.5.1

Single Nucleon Pickup Reactions
General Features

In knockout reactions, a particle is removed from the mass A projectile leaving a mass
A − 1 residue. By removing a nucleon, knockout reactions selectively populate hole-state
configurations relative to the projectile. In pickup reactions, by contrast, a particle is
added to the projectile, selectively populating particle-state configurations in the residue.
This suggests that knockout and pickup reactions will populate a different set of final
states and therefore serve as complementary spectroscopic tools.
In pickup reactions, momentum matching is important [113–115]. The maximum
Fermi momentum pF can be calculated from the number density of infinite nuclear matter
49

ρ = 0.17 fm−3 through pF = (2πρi )1/3 . This gives an approximate Fermi momentum
of pF ≈ 250 MeV/c. The Fermi momentum is related to the mean-square momentum
2
through pF = 5/3 P 2 . Measurements of the Fermi momentum for the light target nuclei
9 Be

and 12 C used in this thesis have found results consistent with this estimation (e.g.

for 12 C, Ref. [116] measured a Fermi momentum of 221 MeV/c). The measured Fermi
momentum, pF , however, is still much smaller than the translational momenta of nucleons
in a nucleus accelerated to intermediate energy (≥ 50 MeV/u), such as the beams delivered
by the nscl ccf facility. For example, 48 Cr, at a projectile kinetic energy of 70 MeV/u,
has a translational momentum of 17.6 GeV/c or a momentum per nucleon of 367 MeV/c.
Fortunately, the high binding energy of nucleons in light nuclei, like 9 Be (Sπ = 16.9 MeV)
and 12 C (Sπ = 16.0 MeV), and the nucleons’ small spatial confinement with the tightly
bound nucleus, lead to wide momentum distributions with high momentum components
that overlap with the momentum of the incoming projectile. These advantages lead to
experimentally reasonable cross sections of ∼ 1 mb when a deeply bound proton from 9 Be
or 12 C is picked up [60]. Momentum matching also implies a sensitivity to the higher j
single-particle orbitals with fast beams are used [117]. Such sensitivity may be useful for
mapping the descent of higher j orbitals into a lower oscillator shell, for example mapping
high momentum intruder states, like the

= 3 neutron intruder states in the island of

inversion around N = 20 [60].
There is an important difference between knockout and pickup reactions. In pickup
reactions, no information on the orbital angular momentum of the picked-up nucleon
can be extracted from the parallel momentum distribution of the residue. The theoretical
width is equal to the momentum distribution of the incoming beam convoluted with the
differential momentum loss in the target. There is little broadening from the reaction
process. This insensitivity comes from the two-body final state (target, mass A + 1 residue)
in contrast to the three-body final state (target, mass A − 1 residue, removed nucleon) in
knockout reactions, where the removed nucleon is related through momentum coupling

50

to the A − 1 residue. This can be seen mathematically. The differential momentum
distribution of the reaction (dσreac /dp ) can be related to the differential momentum seen
in the laboratory frame (dσ /dΩL ) through [60]
dσreac 2π
mR
=
dp
pR mR + mT

pP
dσ
cos θL − 1
.
pR
dΩL

(3.34)

pP and pR are the momenta of the incoming projectile and outgoing residue and θL is
the angle of the heavy residue as seen in the laboratory frame. With high (≥ 50 MeV/u)
energies and a peripheral reaction process, the lab angles are small and the momentum
transfer is negligible, e.g. pR ≈ PP (A − 1)/A. The broadening due to the reaction process
is overwhelmed by the experimental contributions to the momentum width (e.g. target
thickness).
Even though the parallel momentum distribution cannot provide spectroscopic information about the projectile wave function, deviation from the predicted distribution
can reveal the presence of reaction channels (coupled-channels effects, continuum, etc.)
beyond the enumerated two-body states.

3.5.2

Early Work: The 9 Be(20 Ne, 21 Na + γ)X reaction

The investigation of fast-beam pickup reactions in inverse kinematics began with experiments at riken by Shimoura [118] and Michimasa et al. [119] on 23 F. Because of the
high momentum components associated with tightly bound nuclei, Shimoura and Michimasa et al. used 4 He as a target. 4 He has a proton separation energy of nearly 20 MeV
(Sπ = 19.8 MeV). The cocktail beam on target offered four reaction paths to 23 F: the proton
transfer (pickup) α(22 O, 23 F + γ), inelastic scattering α(23 F, 23 F + γ), two-nucleon removal
α(25 Ne, 23 F + γ), and one neutron knockout α(24 F, 23 F + γ). The pickup reaction from 22 O
to 23 F populated states in 23 F with a cross section of approximately 1 mb.
Based on the success of the riken experiments, Gade et al. [60] considered the pickup
reaction 9 Be(20 Ne, 21 Na + γ)X off a much less experimentally complicated 9 Be target.

51

Doppler reconstructed γ rays were observed in coincidence with 21 Na at 330(3) keV,
1832(4) keV, and 2420 keV and attributed to the decay of the 5/2+ state to the 3/2+
g.s
1
ground state, the decay of the first 7/2+ to the 5/2+ state at 330 keV, and the decay of the
first 1/2+ state to the ground state. 86(6)% of the total reaction strength populated the 3/2+
state. A residue rest-frame γ ray transition at 981 keV was observed and was attributed to
the decay of the 1+ state to the ground state in the target residue 9 Be − 1p → 8 Li(1+ ).
The states and population strengths of 21 Na were compared to the direct (d, n) and
(3 Ne, d) reactions, that is 20 He(d, n) 21 Na and 20 Ne(3 He, d) 21 Na [120]. These transfer
reactions showed quantitative information on agreement with the inverse kinematics
results, as seen in Figure 3.7. Because transfer reactions are a well known spectroscopic
tool for providing quantitative single-particle structure, the similar qualitative population
of excited states for the proton pickup reaction implied similar potential as a spectroscopic
tool.

3.5.3

Pickup Reaction Theory

Pickup reactions are treated as direct, two-body reactions. The couplings to intermediate states are small at high beam energies and are assumed to be zero for the present
case. Therefore, for the 9 Be(48 Cr, 49 Mn+γ)X, 12 C(48 Cr, 49 Mn+γ)X, 9 Be(49 Mn, 50 Fe+γ)X,
12 C(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, and 12 C(50 Fe, 51 Co + γ)X reactions, the single-

particle cross sections were calculated in post-form Coupled Channels Born Approximation (ccba) (see Figure 3.8) using the reaction code fresco [121]. The initial state is
three-body and the final state is two-body. The final state includes treatment of both the
residue and the target. For example, in the 9 Be(48 Cr, 49 Mn + γ)X reaction, rewritten as
9 Be(48 Cr, 49 Mn(j π )) 8 Li(I + ),

the I π = 2+ (g.s.), 1+ (981 keV), and 3+ states in 8 Li, formed

from 9 Be after the removal of a proton, are explicitly included. For the 12 C target, the
I π = 3/2− (g.s.), 1/2+ (2125 keV), 5/2− (4445 keV) states in 11 B are included. Breakup
effects, associated with the slow relative motion compared to the center-of-mass motion of

52

the participating nucleon target core, are ignored because of the high binding energy of
the nucleons in 9 Be and 12 C.
The dwba reaction process is given by the form
b + A → B + a.

(3.35)

Using the post-form formalism, the 9 Be(48 Cr, 49 Mn + γ)X reaction components b (target,
final state), A (projectile, final state), B (target, initial state), and a (projectile, initial state)
are combined as A(49 Mn) = a(48 Cr) + x and B(9 Be) = b(8 Li) + x where x is the transfered
nucleon. The exit channel potential is
VaB = Vax + Uab ,

(3.36)

and the transition matrix, in bracket form, is
T = χβ φa φB |(Vaν + Uab − UaB )|χα φA φb

(3.37)

The potential includes the binding potential Vaν of the removed nucleon8 Li(I + ) to
the core, and the optical potentials of 48 Cr on 8 Li (Uab ) and 49 Mn on 8 Li (UaB ). The
absorptive part of the 49 Mn + 8 Li component in Equation 3.37 can be calculated, as in
knockout reactions by double folding the neutron and proton densities of 49 Mn (from
Hartree-Fock calculations) with the density of 8 Li and an effective NN interaction. See
Equations 3.28, 3.30, and 3.31. χβ and χα are the entrance and exit channel distorted
wave functions.
The nucleon-core relative motion wavefunctions φB (9 Be) and φA (49 Mn) are created
through the coupling of an excited core to the removed nucleon. In the most basic singleparticle form, these are written as φB (9 Be) = [8 Li ⊗ φj ]3/2− or φB (49 Mn) = [48 Cr ⊗ φj ]j .
The 49 Mn states were calculated using a Woods-Saxon potential with a spin orbit term of
6 MeV using the standard radius and diffuseness parameters of r0 = 1.25 fm and a = 0.7
fm. The separation energies are adjusted to the binding energy of the proton in the target
and the excitation energy of the pickup residue. See Section 6.4.2 for further theoretical
details and the full reaction calculations.
53

Individual cross section (%)

80
70
60
50
40
30
20
10

9Be(20Ne, 21Na)X

63.4 MeV/nucleon

1/2+

7/2+

5/2+

3/2+

Figure 3.7: Comparison of the energies and population pattern of excited states in the
transfer reaction 20 Ne(3 He, d) 21 Na and the pickup reaction 9 Be(20 Ne, 21 Na + γ)X. The top
half of the figure was extracted from Mukhamedzhanov et al. [120].

54

A
a

Target
8
+
(e.g. Li(I ))

A + 1 residue
b

(e.g. 49Mn(jπ))

Figure 3.8: Post-form dwba reaction schematic.

55

Chapter 4
EXPERIMENTAL SETUP AND
ANALYSIS TECHNIQUES
4.1

Overview

The spectrum of experimental requirements for the study of exotic nuclei using singlenucleon pickup and knockout reactions include the production of a rare-isotope beam,
particle identification of the mass A projectile, particle identification and momentum
measurement of the mass A − 1 or mass A +1 residue, and γ ray spectroscopy of the residue
to tag the final state. The National Superconducting Cyclotron Laboratory (nscl) Coupled
Cyclotron Facility (ccf) [88], Segmented Germanium Array (sega) [122, 123], and the
s800 magnetic spectrograph [124] provide the means. A schematic of the nscl ccf facility
is shown in Figure 4.1. Further information on the s800 and sega are found in Sections
4.3 and 4.4. Chapter 4 also covers the analysis techniques used in in the extraction of
partial and inclusive cross sections and longitudinal momentum distributions common to
the analysis of the 9 Be(54 Ti, 53 Sc + γ)X knockout reaction of Chapter 5 and the six pickup
reactions1 of Chapter 6.
19 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn +
γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X

56

ECR K500

K1200

N1

A1900

N2/N3

S1

N4

S2

N5

S3

Reaccelerator

0

10 m

Figure 4.1: NSCL Coupled Cyclotron Facility showing both the location of beam production (ecr, k500, k1200, and a1900) and the experimental vaults that receive and study the
nuclei in the secondary beam using a wide array of equipment. Both the knockout and
pickup experiments were run in the S3 vault.

4.2
4.2.1

Beam Production
Production of Exotic Nuclei

Because of their short decay half-lives, the radioactive ions studied at the nscl do not
exist in nature. They must be produced as beams of short-lived nuclei. Beams at the
nscl are produced by projectile fragmentation at 80 – 150 MeV/u. A stable, primary beam,
accelerated to velocities nearly half the speed of light, impinges on a thick production
target of 9 Be (target thickness are several 100 mg/cm2 ). Nuclei in the stable beam, as
they pass through the production target, may fragment into residues of smaller mass and
different ratios of N /Z with respect to the initial projectile. Because of the high beam
energy, the fragmentation products exit the back of the target with nearly beam velocity.
Typically the farther the N /Z ratio is from the isobaric ratio defined by the valley of
stability, that is the more proton or neutron asymmetric the fragmentation products are,
the smaller the production cross section and the more exotic the fragmentation product is.
From the remainder of the primary beam and the vast number of fragmentation
products, one or a few specific isotopes must be selected using a highly selective filter
called a fragment separator. Fragment separators offer purification of up to 1012 , e.g. an

57

exotic nuclear species, produced at a rate of 1 pps in the fragmentation processes, can be
selected from a total primary beam rate of 1012 pps (see Figure 4.2 for a schematic of the
a1900 fragment separator). In summary, in-flight separated, rare-isotope beam production
can be broken down into three main parts: primary beam production, fragmentation, and
selection of the isotope of intent.
To accelerate an originally charge-neutral atom in the intense electric fields of the
cyclotron, it must be stripped of its electrons. The acceleration process begins in one of
two Electron Cyclotron Resonance (ecr) ion sources, the Advanced TEMperatures Ion
Source (artemis) [125] or the Superconducting Source for Ions (susi) [126]. Artemis and
susi ionize the stable ions, confined by a strong magnetic field within a small volume of
plasma called a magnetic bottle, through repeated collisions with thermal electrons. A
single charge state of the now ionized plasma of atoms is extracted from the ion source
and injected into the k500 cyclotron, the first of the two coupled cyclotrons at the nscl.
In the cyclotrons, the charged nuclei are confined to a plane by a magnetic field and are
accelerated in an outward spiral by an oscillating electric field. The frequency is tuned
to exert a consistent force on the bunches of ions. In the k500 and k1200, the electric
field is produced by three fan-shaped poles, called dees, charged to high voltages at radio
frequencies (∼ 100 kV and ∼ 24 MHz for the k1200). The high magnetic fields that confine
the ion bunches to a plane are created by high-current, superconducting magnets. The
ions are injected into the center of the k500, and, as they are accelerated, spiral out to the
edge of the accelerating plane, where they are extracted by electrostatic deflectors.
The k500 accelerates the injected ions to about v ∼ 0.15 c (15 % the speed of light). The
maximum possible energy provided by either device is given by the name. The K in k500
or k1200 come from the relation,
q 2
E
=k
A
A

(4.1)

where E/A is in MeV/u2 .
2 In

practice, the energies of the nucleons leaving the k1200 are under one-half of this
58

The nuclei extracted from the k500 are injected into a second, even more powerful
cyclotron, the k1200. A carbon stripper foil located 30 cm from the center of the k1200
removes additional electrons. The stripper foil leaves medium mass, primary beam nuclei,
such as those used in this thesis, fully stripped. The maximally stripped nuclei are
accelerated to an energy between 45 MeV/u for 238 U and 170 MeV/u for 24 Mg, depending
on the mass and charge of the nuclei accelerated. These energies correspond to a highly
relativistic velocity of up to v ∼ 0.5 c. Average beam currents are around 50 pnA where
1 pnA is 6.25 × 109 pps (particles per second). The stable, high intensity beam is extracted
from the k1200 and is delivered onto on a thick Beryllium production target located in
front of the a1900 fragment separator.

4.2.2

A1900 Fragment Separator

To select a nuclear species from the fragmentation products created from the interaction of
the primary beam with the production target, the high-velocity nuclei are filtered through
the high momentum (∆p/p = 5 %) and angular (∆Ω = 5 msr) acceptance a1900 fragment
separator. The entire 35 m long device is composed of 24 quadrupole magnets (3 per
cryostat) used to focus the beam, and four 45◦ dipole magnets used to disperse the beam.
The a1900 selects fragments based on their magnetic rigidity Bρ, the product of a dipole
magnet’s field strength and the trajectory curvature of the fragment through the dipole,
and based upon their atomic charge Z. Bρ is related to the properties of the fragment
through the following relation,

Bρ =

p
q

(4.2)

where p is the relativistic momentum p = γ mv, and q is the observed charge (equal to the
atomic number Z for fully stripped nuclei).
value

59

Experiment

Primary
Beam

Prod. Target
Thickness
(mg/cm2 )

Beam
Al Wedge Momentum
Energy Thickness Acceptance
( MeV/u) (mg/cm2 ) (%)

Knockout
Pickup

76 Ge

423
893

130
160

58 Ni

300
300

1
1

Table 4.1: The primary beam, 9 Be production target thickness, beam energy, aluminum
wedge thickness, and momentum acceptance of the a1900 used in the production of the
secondary beam.

The optics of the a1900 can be divided into two parts. The first half, up to the Image
2 position seen in Figure 4.2, selects particles based upon their magnetic rigidity. The
fragments, dispersed in momentum by the first dipole magnet, pass through a collimating
slit that selects a narrow range of magnetic rigidities. A single cut based on rigidity,
however, does not uniquely define a nuclear specifies. Several fragmentation products may
have similar ratios of p/q. Therefore, an achromatic wedge, placed at the Image 2 position
of the a1900, is typically inserted to slow nuclei according to their atomic number Z.
Nuclei with the same rigidity but different atomic charge experience a different differential
momentum loss in the wedge [127], and thus nuclei with a common magnetic rigidity
before the wedge, have different rigidities after the wedge. The second half of the magnetic
separator, composed of a matching set of dipoles, quadrupoles, further purifies the beam
through momentum dispersion and collimation.
Although the a1900 is highly selective, the final secondary beam often consists of
several species and, in such cases, is called a cocktail beam. The primary beam, production
target thickness, beam energy, aluminum wedge thickness, and momentum acceptance of
the a1900 spectrograph for the knockout and pickup experiments are shown in Table 4.1.
The secondary and primary beam characteristics of the cocktail beams used in this thesis—
the primary beam, secondary beam component (the primary three constituents of the
cocktail beam are listed), approximate rate at the s800 target position, and the energy of
the beam before the reaction target located at the s800 pivot point, the reaction target

60

position, of the s800—are listed in Table 4.2.
primar y
beam

isotope
selection

A1900

production
target

focal plane
momentum
selection

image 2

momentum
selection

wedge-shaped
degrader

Figure 4.2: Schematic of the a1900 fragment separator. The location of the production
target and wedge are shown. The bream direction is from left to right. The red, blue, and
green lines show, for a fragment the a1900 is fully tuned to accept, the dispersion and
focus of the beam in the dipole and quadrupole elements.

4.3

S800 Magnetic Spectrograph

From the focal plane of the a1900, the exotic cocktail beam is delivered to a reaction
target at the pivot point of the s800 spectrograph. In the reactions of interest in this
thesis, a nucleon is removed (knockout) or added (pickup) to the incoming projectile A Z,
A−1
A+1
leaving either a residue N −1 Z or A−1 (Z − 1) (knockout reaction) or N +1 Z or A+1 (Z + 1)
N
N

(pickup reaction) that continues with approximately the same translational momentum
per nucleon as the incoming beam (spectator-core and eikonal approximations). The target
is surrounded by sega, a highly-segmented, high-purity germanium detector (hpge) array,
which detects and measures the de-excitation γ rays emitted in-flight by the residue.
The s800 is divided into two parts, the analysis beam line and the spectrograph (see
Figure 4.3). The analysis beam line, leading up to the target position, is used for beam
diagnostics and tuning the optics mode of the spectrograph. The constituents of the
cocktail beam are identified through their difference in velocity, which appears as a

61

Primary
beam

Secondary
beam

Approximate
rate (pps)

Purity
(%)

Energy
(MeV)

76 Ge

54 Ti

103
85
77

37
30
27

84
89
90

2.5 × 103
2.0 × 103
0.5 × 103

39
40
11

61
63
66

55 V
57 Cr
58 Ni

48 Cr
49 Mn
50 Fe

Table 4.2: Secondary and primary beam characteristics of the cocktail beams used in this
thesis: the primary beam, secondary beam component (the primary three constituents of
the cocktail beam are listed), approximate rate of the cocktail component on target, and
the energy of the beam before the target. The target refers to the reaction target positioned
at the pivot point of the s800 spectrograph and not the production target located before
the a1900 fragment separator. All rates are quoted at 1 % a1900 momentum acceptance
and include transmission losses in the beam lines.

difference in tof between two thin plastic scintillators (127 µm thick) located at the end
of the a1900 fragment separator (xfp scintillator) and in the object box of the s800 analysis
line (obj scintillator). The spatial and angular profile of the beam can be detected by
two Parallel Plate Avalanche Counters (ppacs) inserted at the intermediate image of the
analysis line. Although the ppacs were inserted for both experiments, they were not used
in the analysis.
The analysis line is operated in one of two modes: focused mode or dispersion-matched
mode. In dispersion-matched mode, the beam is dispersed at the target and focused in
the s800 focal plane, i.e. the focal plane is achromatic3 . The dispersion at the target is
10 cm/% momentum. With a 2 in (5 cm) target, the dispersion matched mode limits the
momentum profile of the projectile beam to an effective ±0.25 % or 0.5 % total momentum
width. The increased momentum resolution at the s800 focal plane is gained at the expense
of a momentum cut and thus the rate on target.
3 From

optics, an chromatic lens separates light into its constituent wavelengths, and an
achromatic lens transmits light without separating it into its constituent wavelengths. In
beam optics, achromatic and chromatic refer to the dispersion of momentum.

62

Focal Plane
(focal plane detectors)

Object Box
(plastic scintillator)

Intermediate
image

Target + SeGA

Analysis Line

Spectrograph

Figure 4.3: Side view of the s800 magnetic spectrograph. The secondary cocktail beam
enters from the left and is guided to the target by the analysis line. The reaction residue of
interest is selected by the magnetic rigidity of the s800 spectrograph and is identified in
the s800 focal plane. A plastic scintillator (obj scintillator) located in the object box is used
to provide a tof difference relative to a timing scintillator located at the end of the a1900
(xfp scintillator) and a 3 mm scintillator located in the s800 focal plane (e1 scintillator)
.
Because knockout and pickup reactions are often run at very low beam rates down to
a few particles per second, a consequence of the low yield of the exotic nuclei studied,
the analysis line was tuned to focused mode. The beam was focused at the target, but
dispersed at the focal plane (achromatic at the target and chromatic at the focal plane).
The momentum width of the beam is directly folded into the momentum width observed
in the focal plane of the s800. In focused mode, the full acceptance of the analysis line
can be used to deliver a full ±2 % momentum spread to the spectrograph. The advantage
of running in focused mode is increased rate that can be accepted on target. The target
size does not restrict the momentum since the beam is focused on the target. A list of the
acceptances and measured resolutions for the two optics mode is shown in Table 4.3.
The s800 spectrograph is located downstream of the target and consists of two dipoles,
a quadrupole triplet, and a focal plane containing several detector systems (see Figure 4.4).
The spectrograph is used for particle identification, and momentum and angle measure63

Property

Value

Spectrograph momen- Both modes: ±3.0 % with greatly reduced acceptance
tum acceptance
near the edges [128]. In dispersion matched mode,
a small target will restrict the momentum spread
of the beam, much like the collimating slits in the
a1900, e.g. a typical 2 in target restricts the beam
momentum to ±0.25 %. In focused mode, the target
does not restrict the momentum width.
Angular acceptance

Ω = 20 msr. The angular acceptance is elliptical:
±3.5◦ in the dispersive plane and ±5◦ in the nondispersive plane.

Energy resolution

Intrinsic resolution: ∆E/E = 1/10 000. The measured momentum resolution depends on the beam
tune and the s800 spectrograph mode. In dispersion
matched mode, a typical resolution is ∆E/E ≈ 1/2000.
In focused mode, the energy resolution is dominated
by the intrinsic differential energy width of the beam.

Momentum resolution

Intrinsic resolution: ∆p/p = 1/20 000. The measured
resolution, like the energy resolution, depends upon
the beam tune and the optics mode. A typical value
for dispersion matched mode is ∆p/p = 1/4000. In focused mode, the measured momentum widths reflect
the momentum width of the beam (∆E/E ≈ 1/100 for
this work).

Angular resolution

∆Ω = 2 mrad.

Position resolution

0.5 mm in the dispersive and non-dispersive directions (position resolution of CRDC1 and CRDC2).

Dispersion
plane)

9.5 cm/% at CRDC1.

(focal

Table 4.3: Properties of the s800 Magnetic Spectrograph.

64

ments of the reaction residues. These measurements occur in the compact array of devices
in the focal plane: two Cathode Readout Drift Chambers (crdcs) for positions x, y and
angles θdisp. , θnon-disp. (dispersive and non-dispersive), a segmented ion chamber (ic) for
energy loss ∆E, and a series of thick plastic scintillators tof information.

Scintillators

Ion chamber
CRDCs

Figure 4.4: S800 focal plane showing the crdcs (red), ion chamber (green), and scintillators (yellow). In the present work, the crdcs were used for position and angle information,
the ion chamber was used for energy loss, and the signal from the first (3 mm) scintillator
was is used by the trigger logic to issue the master trigger.

The two crdcs, separated by 1 m inside the focal plane, are position sensitive and
are therefore responsible for the position and angle information of the heavy residue.
Because of the momentum dispersion of the s800, the x-position is connected to the
longitudinal momentum of the residue after the target. The active area of the crdcs is
±28 cm (dispersive) and ±13 cm (non-dispersive). In the dispersive x direction, the position
is determined by a fit of induced charges on 224 cathode pads. The non-dispersive
y direction is determined via drift time of the ionization electrons (see Ref. [129] for
65

additional details). Both the dispersive and non-dispersive directions yield a position
resolution of r ∼ 0.5 mm. The drift time depends on the gas mixture, and it may may shift
over the course of the experiment as the properties of the gas mixture change (80 % CF4
and 20 % isobutane C4 H4 at a pressure of 50 Torr). To ensure proper calibration, masks
with a known pattern were placed in front of both crdcs. Based on the known position
of the mask holes and the measured position in the crdcs, linear calibration parameters
were applied to the raw values. The crdcs limit the rate in the focal plane to 6 × 103 pps.
At higher rates, rapid aging of the anode wire and performance degradation is typically
observed.
The ion chamber (ic), collects ionization charge produced when an ion passes through
its gas (P10 mixture of 90 % Ar and 10 % methane CH4 at 140 Torr), and gives a ∆E
measurement proportional to Z 2 . Because any medium Z projectile will create ionization
charge in the ic, the ic is assumed to be 100 % efficient. Therefore the ion chamber
serves as the reference to which the efficiency of all other particle detectors upstream are
measured.
Three scintillators are located behind the ion chamber of thickness 3 mm, 5 cm, and
10 cm. The signal from the first 3 mm scintillator was used by the trigger module, for the
s800 singles or particle-γ (p-γ) coincidence settings, to issue the master trigger (for pγ
coincidences, a secondary trigger signal from the sega array was also required).

4.4
4.4.1

The Segmented Germanium Array (sega)
Detector Overview

The energy of an excited state of the residue is determined by the detection of its deexcitation γ rays emitted in flight. For the present work, γ rays were detected with a
high-resolution germanium array position around the reaction target, the Segmented
Germanium Array (sega) [122, 123], as seen in Figure 4.5 and Figure 4.7. Sega, in its

66

so-called “classic” configuration, consists of 17 mounted detectors in two rings located
at 37◦ and 90◦ relative to the beam axis. The detection volume is a n-type, High Purity
germanium (hpge) (single) crystal, 7 cm in diameter, and 8 cm long, and in a closed-axial
configuration. Each single crystal is electrically segmented into 32 segments, formed
from 8 slices and 4 quadrants (see Figure 4.6). Details on the role and necessity of this
segmentation is found in Section 4.4.2. The central core, seen in Figure 4.6, provides the
signal for timing and energy. A set of preamplifiers (33 total; one for the central contact
signal and 32 for the segment signals) are included close to the crystal, and the entire
crystal assembly is cooled down to liquid nitrogen temperatures in a cryostat (T ∼ 100 K)
that includes a large liquid nitrogen Dewar.

4.4.2

Segmentation and Doppler Reconstruction

The γ rays of energy Erest , emitted in-flight by the reaction residues traveling at v ∼ 0.3 c,
γ

are subject to a Doppler shift into the laboratory frame to an energy Elab 4
γ

γ
γ
Elab (Erest ) =

Erest
,
γ(1 − β cos θlab )

γ

γ

(4.3)

or with rearrangement
γ

Erest (Elab ) = Elab γ(1 − β cos θlab ),

(4.4)

where θlab is the angle of the emitted γ ray relative to the direction of the beam in the
laboratory frame, and β = v/c is the average residue velocity at the point of γ ray emission.
The Doppler shift has important experimental consequences. Because we are interested in
measuring the energies emitted by the nucleus in its rest frame, according to Eqn. 4.3 we
need the angle of emitted γ ray and the velocity of the moving nucleus β to reconstruct
Erest . Any uncertainties of the emission angle of the γ ray or of the beam velocity will
4A

notation distinction is made for the energy of a γ ray emitted by the residue in
γ
its rest frame (Erest ), the energy of the emitted γ ray boosted into the laboratory frame
γ
through Eqn. 4.3 (Erest ), and any arbitrary γ ray detected by sega (Eγ ) (e.g. γ rays emitted
by a source or the environment).
67

γ

γ

directly affect the energy resolution of the observed transition (∆Erest /Erest ). To greatly
improve the energy resolution of the residue rest-frame spectrum, reconstructed from
the lab-frame spectra through Eqn. 4.3, each crystal is electronically segmented into 32
segments and the energy deposited within each crystal segment was recorded for each
event. In the case of multiple interaction points, as seen in Figure 4.8, the first interaction
point is taken from the segmented with the highest energy deposition. With segmentation,
the angular resolution of sega is approximately ∆θlab ≈ 50 mrad (fwhm).
Because of the thickness of the target and the unknown or variable reaction point
within the target, the pickup or knockout reaction occurs over a spread of energy and
momentum. The spread in velocity, ∆β, associated with the momentum width of the
beam, together with the uncertainty in opening angle, ∆θlab , taken from the 1 cm crystal
segmentation and the distance from the target position to the crystal (≈ 20 cm to the crystal
edge and 24 cm to the crystal center), contribute to the total energy resolution through
 γ 2
2
2
 ∆Erest 
β sin θlab
cos θlab − β



(∆β)2 +
(∆θlab )2 .

 γ  =


1 − β cos θlab
(1 − β 2 )(1 − β cos θlab )
Erest
γ

(4.5)

γ

Typical values for the fwhm energy uncertainty ∆Erest /Erest at beam velocities of β = 0.3
range between 2 % to 4 %. The relative contribution of ∆θ and ∆β depend on the angle
γ

γ

of emission of the γ ray. This dependency of (∆Erest /Erest ) with respect to θlab is shown
in Figure 4.9. The two rings of detectors, located at 37◦ and 90◦ are, positioned near the
minimum of ∆Er /Er . The contributions to the resolution from ∆β and ∆θlab dominate the
intrinsic resolution of sega, as determined, for example, from the measurements using
standard sources. The intrinsic resolution is approximately 2.5 keV for a 1 MeV γ ray.
The angle of Eqn. 4.3 is defined with respect to an assumed target position located in
the center of the sega array at the pivot point of the s800 spectrograph. The exact target
position, however, is not precisely known. Typically uncertainties are of the order of a few
millimeters. The uncertainty in target position affects the angle θlab used to reconstruct
the residue rest-frame γ spectrum. To compensate, one can adjust the ztar position in
68

software so that known energies (only transitions with short half-lives of order τ1/2 ∼ 1 ps
are considered) are reconstructed satisfactory. The β value was chosen to align the energy
of a γ ray transition in both the 37◦ and 90◦ rings. The β value is sensitive to the lifetime of
the state due to the location of the decaying nucleus and should be determined separately
for each observed transition (the optimal value of β = 0.371 for the 2.1 MeV transition
in 53 Sc is close to the mid-target β value calculated using the magnetic rigidity of the
analysis line). β and ztar are independent for small changes about their optimal values and
therefore they can both be independently optimized without worrying about simultaneous
minimization on a grid of possible β and ztar values. Details for the knockout and pickup
reactions are found in Sections 5.6.2, 6.4.4 and 6.5.4.

4.4.3

Energy Calibration

Neither the energy response of sega nor the electronics is perfectly linear. Calibration
parameters, mapping raw to true energies, must be calculated per detector for both the energies of the central contact (cce) and the energies of the segments. For the central contact,
a 2nd order polynomial fit between the measured transitions and known transitions of a
calibration source (226 Ra or 152 Eu) provides the relationship between channel and energy.
The fitting procedure for the 32 segments was automated in the analysis code SpecTcl
[130] and is based upon Ref. [131]. The energies used for the 226 Ra or 152 Eu calibration
procedure are listed in Table 4.4.

4.4.4

SeGA Efficiency

For sega we are interested in the full-energy-peak efficiency, the probability that a photon
will both be detected and the photon will deposit its full energy in a sega crystal. There
are three main ways that a γ ray interacts with matter: photoelectric absorption, Compton
scattering, and pair production. Photoelectric absorption is the full absorption of a photon
by an atom. The photon’s energy is completely converted into a free electron whose energy
69

226 Ra

152 Eu

(keV)

(keV)

186.2
240.2
295.90
609.3
1120.3
1764.5
2447.8

244.70
344.30
778.90
964.00
1408.00

Table 4.4: Source energies used for sega energy calibration.

is equal to the photon energy minus the binding energy of the electron. Compton scattering
is the scattering of a γ ray off a bound electron. The electron carries away part of the
photon’s energy, leaving the photon to Compton scatter again or deposit its energy through
one of the other two processes. Frequently, a γ ray will deposit a portion of its energy
in a sega crystal through a Compton scatter event, and then leave the crystal without
depositing its full energy. Lastly, pair production is the creation of an electron-positron
pair in the intense field of the proton. The positron immediately annihilates, creating two
511 keV γ rays which may or may not be detected within the crystal volume.
These three processes dominate within unique energy ranges. For germanium, complete absorption through the photoelectric effect dominates for energies lower than
140 keV. Between 140 keV and 10 MeV (approximate range), Compton scattering dominates. Pair production only becomes possible if the γ ray energy is equal to or exceeds
the rest mass of the created electron and positron, i.e. above 1.022 MeV. Pair production
becomes increasingly important as the energy of the incident γ ray increases and begins
to approach the probability of Compton scattering around 10 MeV. These processes are
not exclusive. The most likely scenario for the energies observed in this work is multiple
Compton scattering followed by a single photoelectric event. Further details of these
interactions can be found in Knoll [132], Chapters 10 and 12.
The efficiency of sega was measured using standard calibration sources, e.g. 226 Ra,

70

152 Eu,

or 56 Co. The efficiencies for the measured transitions were fit by a curve, extrap-

olating the efficiency for any incident energy. The fit curve is empirical and is of the
form
R
sega (Eγ ) =

a
(Eγ − b + e

The efficiency is calculated per ring R (e.g.

−0.269Eγ c
)

37◦
sega

(4.6)

.

90◦
and sega ), instead of for the entire

sega array, because the angle-dependent Lorentz transformations necessary to adjust
the efficiency curve for a source at rest to a source moving at a typical beam velocity of
v ∼ 0.3 c depends upon the angle of the detector. We have, for each ring, the following
γ

i
experimental definition of efficiency sega at energy Eγ :
R
i
sega (Eγ ) =

d Id (Eγ )
.
i
b(Eγ )R t

(4.7)

The summed number of counts in the photopeak for the ring, measured in time t, is the
summed number of counts for all properly-calibrated detectors

i
d Id (Eγ ).

is the live-

i
time fraction of the data acquisition, and b(Eγ ) is the source branching ratio for the given

transition i, i.e. the fractional number of γ rays expected per source decay (a given excited
state may decay through more than one “branch”, and each decay route has an associated
probability). The source intensity, R, is calculated from the initial source activity R0 and
the time from source creation t> (standard radioactive decay law) as R = R0 e−t> /τ . See
Section 5.5.2 and Section 6.3.3 for the experimental calibration curves for the knockout
and pickup experiments.
A γ ray emitted in the residue rest frame will be seen as a distribution of energies in
γ,low

each detector, from Elab

γ,high

to Elab

, where the prime indicates the lab boosted energies,

because each detector covers an angular range and the Lorentz boost is angle dependent
(see Eqn. 4.3). The efficiency sega , averaged over the angle covered by each crystal, for a
residue rest-frame energy Eγ and a ring R is
γ
R
¯sega (Erest ) =

1
γ,high γ
γ,low γ
Elab (Erest ) − Elab (Erest )

71

γ,high γ
E
(Erest )
lab
dE
γ,low γ
E
(Erest )
lab

R
sega (E

),

(4.8)

152

226

Eu

56

Ra

Co

Eγ
(keV)

γ per Bq

d(g)
(%)

Eγ
(keV)

γ per Bq

d(g)
(%)

Eγ
(keV)

γ per Bq

d(g)
(%)

121.8
244.7
344.3
367.8
411.1
444.0
488.7
678.6
688.7
778.9
867.4
964.0
1005.1
1085.8
1089.7
1112.1
1212.9
1299.1
1408.0

0.28370
0.07530
0.26570
0.00874
0.02238
0.03125
0.00414
0.00479
0.00859
0.12970
0.04214
0.14630
0.00744
0.10130
0.01731
0.13540
0.01412
0.01626
0.20850

0.46
0.53
0.41
1.05
0.47
0.45
1.24
1.29
1.06
0.46
0.59
0.41
2.01
0.49
0.52
0.44
0.51
0.68
0.43

186.2
242.0
295.2
351.9
609.3
768.4
934.1
1120.3
1238.1
1377.7
1509.2
1729.6
1764.5
1847.4
2118.5
2204.1
2447.7

0.03560
0.07240
0.18370
0.35590
0.45570
0.04870
0.03093
0.14920
0.05860
0.03972
0.02113
0.02839
0.15290
0.02031
0.01156
0.04900
0.01553

0.60
0.40
0.40
0.40
0.40
0.50
0.50
0.40
0.50
0.50
1.30
0.60
0.40
0.80
0.80
0.60
0.70

846.8
977.5
1037.9
1175.1
1238.3
1360.2
1771.4
1810.8
1963.9
2015.2
2034.8
2113.0
2214.5
2598.5
3010.1
3202.0
3253.8
3273.0
3451.2
3548.3

0.99933
0.01619
0.14130
0.02239
0.66070
0.04256
0.15490
0.01299
0.01099
0.03029
0.07771
0.00400
0.00400
0.16960
0.00799
0.03130
0.07620
0.01780
0.00930
0.00178

0.01
6.27
0.35
0.49
0.29
0.35
0.32
46.17
18.22
0.43
0.35
25.02
25.02
0.35
37.52
2.90
3.15
3.37
4.30
5.06

Table 4.5: Energies Eγ , number of emitted γ rays per decay, and the uncertainty on the
number of emitted γ rays per decay for the three sources used in energy calibration. Data
was taken from Refs.[133, 134]. See the appropriate experimental sections, Section 5.5.2
or Section 6.3.3, for additional details.
R
rest

is the fit efficiency curve (Eqn. 4.6 with defined coefficients) for ring R. The Lorentzγ,low

boosted laboratory frame energies, Elab

γ,high

and Elab

, are evaluated at a residue rest frame

γ

R
R
energy Erest through Eqn. 4.3, and at the polar angles θlow and θhigh . These angles are the

lower and upper bounds of the polar angle covered by each germanium crystal, and thus
◦

37
are naturally ring dependent. Per ring, the angles are defined as θlow,high = 37.0◦ ± 9.5◦
◦

90
and θlow,high = 90.0◦ ± 9.5◦ .

One additional Lorentz transformation must be included. Although we make the
assumption that the emitted γ rays are emitted in an isotropic distribution in the residue
rest frame, they are not observed as an isotropic distribution in the lab frame. The
72

angular distribution is boosted in the forward direction. To take the boost into account the
efficiencies are modified in the following way:
γ
R
εsega (Erest ) =

θhigh
θlow dθlab

π

sin θlab Wlab (θlab )/ 0 dθlab sin θlab Wlab (θlab )
θhigh

2π θ
dθlab sin θlab /4π
low

γ
R
· ¯sega (Erest )

(4.9)

where the use of εsega rather than ¯sega indicates the energy-averaged and boosted efficiency. Wlab is defined as
Wlab (θrest ) =

sin2 θrest + γ(cos θrest + β)2

3/2

γ(1 + β cos θrest )

(4.10)

where
θrest = tan−1

sin θlab
.
γ(cos θlab − β)

(4.11)

The array efficiency, for the detection of a residue rest frame energy E, is the summation
of the fully averaged and boosted efficiencies for both rings, e.g.
γ

◦

γ

◦

γ

90
total
37
εsega (Erest ) = εsega (Erest ) + εsega (Erest )

4.5

Analysis Techniques

4.5.1

(4.12)

Particle Identification

Nuclei in the focal plane are identified through tof and energy loss (∆t – ∆E). For example,
see the particle identification (pid) spectrum for the 54 Ti reaction products, including
53 Sc,

in Figure 4.10. Energy loss and tof, as mentioned earlier, are determined by the ion

chamber (ic) and the tof between the scintillator in the object box and the first plastic
scintillator in the focal plane (e1). The tof depends on the nuclei’s mass to charge ratio
A/q, and the energy loss depends on the nuclei’s atomic charge Z 2 .
Additional tof corrections had to be made for the variations in the nuclei’s flight
path through the spectrograph. These corrections depend on the scattering angle and the
73

momentum (approximated, because of the dispersion of the beam in the focal plane, by
the position in the first crdc, xcrdc1 ) as
t = tobj + a · θafp + b · xcrdc1 .

(4.13)

Larger angles about the central trajectory lead to a longer flight path as do lower momenta
projectiles. The parameters a and b were experimentally adjusted by optimizing a matrix
of the angle in the focal plane (afp) versus the tof to provide the best resolution for
particle identification.
Spectra, such as the pid of Figure 4.10, combine the signals from several detectors.
The total particle detection efficiency folds in the efficiencies of all included detectors.
In general, these efficiencies depend on the beam rate, beam energy, atomic charge, and
mass of the nuclei identified. The product of the efficiencies, the efficiency of particle
identification in the focal plane, is
det

= (obj · xfp) · (crdc1 · crdc2)

(4.14)

To provide accurate cross sections, the total efficiency must be measured. Efficiencies
are determined relative to the ic which is assumed to be 100 % efficient. A software gate
on the ic is applied to a spectrum that folds in the efficiencies of all detectors, a twodimensional matrix of the time of flight difference between the xfp and obj scintillators
versus the angle in the focal plane (the angle requires a valid crdc1 and crdc2 position).
A ratio of the counts in an ion chamber gate to the total number of counts in the twodimensional matrix with the IC gate applied gives the efficiency. The combined efficiencies
for the detectors during the knockout data runs was near 0.99, and for the knockout data
runs the detector efficiencies ranged between 0.97 and 0.99.

4.5.2

Inclusive Cross Sections

The inclusive cross section includes a sum over all reaction channels leading to bound
states and can be measured solely by counting the number of knockout or pickup residues
74

detected in the s800 focal plane. The basic formula for the determination of the inclusive
cross section from experiment is
σinc =

1 Nr
.
NT Np

(4.15)

Nr is the total number of reaction products (residues) of the specific reaction channel
and Np is the total number of incident A projectiles. The measured numbers of incident
projectiles and reaction products are tempered by the angular and momentum acceptance
limitations and efficiencies of the detectors and these effects must be folded into Eqn. 4.15.
The cross section is the probability of the reaction occurring Nf /Nb , per total number of
scattering centers seen by the projectile per unit area, NT , in the target. NT is defined in
terms of the target thickness, given in the traditional units of mg/cm2 and the mass density,
the number of atoms per unit mass, as NT = nx. x is the target thickness in mg/cm2 and n
is the target number density, the total number of atoms per cm2 . The value of n can be
calculated from the atomic mass/mol and Avagadro’s constant of 6.0221415 × 1023 mol−1 ,
the total number of atoms per mol.
Usually, the spectrograph cannot accept both the projectile beam and all reaction
components created through the interaction of the incident beam with the target. This
limitation has two causes. First, the momentum acceptance of the s800 is limited. Only a
portion of the beam can be seen for a chosen Bρ and usually it is not possible to simultaneously observe the unreacted beam and the desired fragmentation product. Second,
the detectors in the focal plane are rate limited. The crdcs can only handle a total rate
of 6 kHz. Because of small reaction probabilities, the rate limitation does not affect the
reaction products, but it does affect the measurement of the beam rate incident on target.
These limitations force separate measurements of the incoming rate and the rate of the
reaction products. To measure the incoming rate, the Bρ of the spectrograph is set to center
the momentum distribution of the unreacted projectile passing through the target into the
focal plane. The spectrograph, when tuned in this manner, is said to be in an unreacted
setting. Because the reaction probabilities are so small, the presence of the target does

75

not affect the number of detected unreacted projectiles in a statistically significant way.
For example, for a 9 Be 188 mg/cm2 target, a 1 mb cross section corresponds to a reaction
from one out of every ∼ 80 thousand incident nuclei. All summed reaction channels, of a
maximum order of a few hundred mb, affect the cross section at a statistically insignificant
level.
The number of detected nuclei includes experimental detector efficiencies (see Eqn.
4.14). The total number of detected fragments (Nr or Np ), for either the unreacted or
reacted run is
Ni =

i
Nobs Di
i
i
Nnorm i det Ai

(4.16)

.

where i = r, p for the reaction residue (r) or incident projectile (p). Nobs is the number of
observed (detected) fragments of interest, Nnorm is the normalization of Nobs to the beam
rate, and D is the downscaler (ds). Because separate runs were used to measure the total
number of reaction residues and incoming projectiles, Nf and Nb (see Eqn. 4.15) were
normalized to the beam rate through the average of the obj and xfp scalers.
To minimize deadtime in unreacted settings or conversely maximize the livetime of the
data acquisition system, a downscaling factor Di can be chosen, where 1 out of every N
particles is used to trigger the system. The ds factor was 1 for both experiments. Deadtime
is the fractional time that the daq is not able or ready to accept events.

is the percentage

livetime of the system and is the probabilistic reciprocal of the deadtime.

= 1 − ddt . The

livetime can either be taken from the number of total triggers (events) to the number of
processed triggers (live.trigger/raw.trigger), or from a free running clock (oscillator)
and clock that is gated by the daq not-busy (clock.live/clock.raw), where the total
number of clock output by both clocks are counted by separate scalars which are not part
of the daq system and thus are not impacted by the dead time. For both experiments, the
two methods gave statistically similar results.
The last unknown component in Eqn. 4.15 is the acceptance A. The spectrograph is
physically limited to a given momentum and angle acceptance, and for many reasons the

76

Knockout
9 Be
Target type
Thickness (mg/cm2 )
376(4)
3 ) 1.85
Target density (g/cm

Pickup
9 Be

12 C

188(4)
1.85

73(1)
1.87(4)

Table 4.6: Targets used in the knockout and pickup reactions

properties of the beam exceed these limitations: the momentum width of the reaction
residues caused by the reaction, the location of the momentum distribution relative to the
center momentum of the spectrograph, the tune of the analysis line, the momentum width
of the incoming beam, and so forth. Usually these losses occur in the dispersive direction.
The direct nature of the pickup and knockout reactions are reflected in a narrow reaction
residue momentum distribution, and therefore only minimal acceptance corrections of
order ∼ 5 % were necessary. These losses were estimated by simple parametric fits to the
momentum and angle distributions reconstructed at the target from position and angle
measurements at the focal plane.
With the full assortment of experimental adjustments, the inclusive cross section is


 
p
r

 Dr
Nobs
Nobs

  Dp
1 


 
 (4.17)
/ 
 r

σinc =

 

  p p
 r
1 (N (obj) + N (xfp))
1 N (obj) + N (xfp) 

nx
Ar
Ap
r
r
det
p
p
2

det

2

The xfp and obj scalars and the number of observed residues or projectiles in the focal
i
plane are given by the labels Ni (obj), Ni (xfp), and Nobs for i ≡ r, p. Di is the ds factor,

is the live time, det is the combined detector efficiency, and A is the acceptance of
the spectrograph. Uncertainties from the target thickness, statistical uncertainties, and
systematic uncertainties all contribute to the final uncertainty budget.
Three different targets were used in the two experiments. Their properties are seen in
Table. 4.6. For the beryllium target, the target thickness and uncertainty was provided by
the manufacturer. For the carbon target, the thickness and density were calculated using a
measured target area of 46.5(5) cm2 , thickness of 390(5) µm, and a weight of 3.38 g.

77

4.5.3

Longitudinal Momentum Distributions

The crdcs measure the position and angle of the fragments entering the focal plane of
the spectrograph. As in the a1900 fragment separator, the momentum dispersion of the
spectrograph maps projectiles with different magnetic rigidities (Bρ = p/q) to different
positions. In the a1900, the dispersion was used to select the secondary beam from the
multitude of different reaction products. In the s800, the dispersion allows a position
measurement to substitute for a momentum measurement. Using an optics map for s800
spectrograph, the position and angles of the nuclei in the focal plane, measured with
the two crdcs, can be mapped back to angle, position, and momentum of the nuclei at
the target. The inverse matrix map used to relate vectors in the focal plane to vectors at
the target was calculated using the optics code cosy infinity [124, 135]. Details on the
comparison of the measured and theoretical longitudinal momentum distributions for the
knockout and pickup reaction follow.

4.5.3.1

Single-nucleon Knockout Reaction

Additional concerns, like the focused mode of the s800 analysis line, which folds in the
momentum spread of the incoming beam to the resolution in the focal plane require
additional treatment. Rather than try to subtract these effects through de-convolutions
of the experimental longitudinal momentum distribution, it is far easier to apply to
the effects to the theoretical longitudinal momentum distributions (for details on the
calculation of the theoretical distributions for single-nucleon knockout reactions, see
Section 3.4.4). After the theoretical distribution includes similar momentum-broadening
effects via folding, it can be compared to the experimental distribution. The similarities
between the distributions, an example of which is shown in Figure 5.10 are used to make
orbital angular momentum assignments to the removed nucleon.
A list of steps in this process follow:

78

1. Use the inverse map to map the measured positions and angles in the focal plane to
a differential kinetic energy at the target position.
2. Convert the lab differential energy at the target dt to differential momentum. The
ratio between the two quantities is
γ
∆p
∆E
=
p
γ +1 E

(4.18)

which can be found from the differential of relativistic kinetic energy E = (γ −1) mA c2
to relativistic momentum p = γ ma v with respect to beam velocity β = v/c.
3. Convert the theoretical distribution, typically calculated in the residue rest frame, to
the lab frame. The lab frame and rest frame parallel momentum are related through
plab = γprest

(4.19)

4. Because the spectrograph was operated in focused mode, the theoretical parallel
momentum distribution, now converted from the rest frame to the lab frame (see
Figure 3.6 for sample rest frame distributions), must be folded with the experimental
resolution of the incoming beam. The experimental distribution intrinsically includes
this folding. An analytical function5 was fit to the unreacted beam and the result
of the fit was convoluted with the theoretical momentum distribution. A typical
width for the incoming momentum spread of the unreacted beam is 1 %. This value
is determined by the momentum acceptance of the a1900 and is broadened by energy
losses in the detectors and the target.
5. Finally, the theoretical distribution, converted to the lab frame and folded with
the momentum resolution of the incoming beam, must account for the differential
momentum loss in the target caused by the difference in reaction location in the
target. Consider the case of the one proton knockout 9 Be(54 Ti, 53 Sc + γ)X. Because
5 See

the experimental Sections 5.7.3, 6.4.6, and 6.5.6 for the reaction specific details.
79

energy loss in a target is roughly proportional to Z 2 , 54 Ti will lose more energy as
it moves through the target than will 53 Sc. Thus, if the reaction takes place at the
front of the target, the projectile will have more momentum relative to a reaction
occurring at the center of the target (the average), and if the reaction takes place
at the back of the target the projectile will have less momentum relative to the
average. Because energy loss is roughly linear over the target thickness, the effect
on momentum resolution due to the thickness of the target can be approximated by
a top-hat function. This top-hat function is convoluted with modified theoretical
distribution of step 4. The differential momentum losses were calculated using the
computer program Lise++ [136].
After the theoretical distributions are converted to the lab frame, and folded with
all the experimental resolutions, they can be directly overlaid. The inclusive parallel
momentum distribution for 53 Sc is shown in Figure 5.10 (page 118).

4.5.3.2

Single-nucleon pickup reactions

For single-nucleon pickup reactions, because the final state is two-body (as addressed in
Section 3.5.1), there should be no broadening from the reaction processes. Thus, experimentally measured longitudinal momentum distributions are overlaid with a calculated
distribution that convolutes the momentum broadening in the target with the momentum
width of the incident beam.

4.5.4

Partial Cross Sections

Through the total number of detected γ rays emitted at the transition energy (see Figure 4.11), one can extract the partial cross section for the reaction channel. The experimental partial cross sections, in combination with the theoretical single-particle cross sections,
are used to calculate the experimental spectroscopic factors for knockout reactions (see

80

Eqn. 3.18). Although the single-particle cross sections for single-nucleon pickup reactions
have not yet been shown to provide quantitatively accurate spectroscopic factors, the
extraction of partial cross sections can be used to both test reaction theory and provide
qualitative estimates of relative orbital occupation. See Chapter 6 for experimental details
and discussion on the set of proton pickup reactions 9 Be(48 Cr, 49 Mn+γ)X, 12 C(48 Cr, 49 Mn
+ γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 12 C(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, and 12 C(50 Fe,
51 Co + γ)X.

The partial cross section for the population of a final state in the residue are calculated
from the total number of γ rays in the photopeak corresponding to the decay of the
populated state. The math is similar to the calculation of the inclusive cross section in
Eqn. 4.15, i.e.
σpart. (E, J π ) =

1 Nγ
,
Nt Nb bγ

(4.20)

with appropriate corrections for feeding and the branching ratio bγ of the populated
state. E and J π are the energy and associated quantum numbers of the final state, Eγ is
the measured energy of the observed transition. The decay of a state with energy E may
populate several lower energy states, the fractional number of γ rays emitted from the
i
decay of the initial state to possibly one of many final states i is the branching ratio bγ .

The state E may also be fed through the decay of higher lying transitions. This feeding
strength must be subtracted. Thus,
j

σpart. = σpop. −

σfeed. .

(4.21)

j

where σpop is the total population cross section of Eqn. 4.21 (i.e. no feeding subtraction).
The effects of feeding and branching can be visualized through the creation of a level
diagram which places the transitions in the proper hierarchical ordering. Level ordering
and assignment are assisted by intensity balancing, angular momentum assignments
extracted from the parallel momentum distributions, and if needed, comparisons to shell
model.
81

Although we can proceed in a similar way as Eqn. 4.16 through defining all efficiencies
and quantities in Nγ and Np , e.g.
γ

γ
Nγ (Erest ) =

Nobs Dγ

(4.22)

γ
total γ
r
i
Nnorm εsega (Erest ) det γ Aγ bγ (Erest )
p

Nb =

p

Nobs. Dp

(4.23)

p

Nnorm εdet p Ap ,

γ

b
Nobs and Nobs are the number of observed γ rays and beam projectiles. These quantities

must measured in separate runs because of the limitations of the spectrograph. As in
γ

Section 4.5.2, this forces separate normalizations relative to the xfp and obj scalars (Nnorm
γ

p

p

and Nnorm ), measurements of particle identification detector efficiency ( det and det ),
live time ( γ and p ), and spectrograph acceptance (Aγ and Ap ).
Fortunately, we can make use of the s800-sega coincidence condition to simplify the
math. When sega and the s800 spectrograph are run in coincidence mode, γ rays are only
recorded when there is a detected residue in the focal plane. Under this condition, we can
calculate a partial cross section as
γ

σpartial = σinc

Nobs.

p

total
Nobs. bγ (Eγ )εsega (Eγ )

− σfeed

(4.24)

The quantities det , and A cancel in the ratio and Nnorm are not needed. SeGA (Eγ ) is the
Lorentz-boosted sega efficiency for energy Eγ , evaluated at the residue rest-frame energy
Eγ of the transition. The partial cross section to the ground state is
σg.s. =

i
σpart

−
inc

i

assuming all feeding has been accounted for in the sum

82

(4.25)
i
i σpart .

Detector
volume
Beam pipe

Target

37o

Beam direction

90o

To S800

t
ta

os
ry
C

90o Ring

37o

Figure 4.5: Schematic view schematic of the sega detector array with a cut along the
mid-plane and through the beam axis. The two rings of detectors are arranged 37◦ and
90◦ with respect to the target position. The active detector volume is in yellow. The target
is position equidistant from the crystals, at a distance of d ≈ 24 cm. The remainder of
the detector hardware includes the cryostat (composed of the Dewar and crystal housing
assembly), electronics and the appropriate housing for both.

83

H G F E D C B A

4 1
3 2

(a) side view

H1

(b) front view

H4

H3

A1
4
A
A2
3
A

(c) isometric view

Figure 4.6: Segmentation of the sega crystal. The side view (a) shows the eight slices and
the location of the central core. The signal from the central core is used for timing and
total energy deposition. The front view (b) shows the four quadrants. And the isometric
view (c) shows the complete detector volume.

84

Figure 4.7: Photograph of the sega array in its “classic” configuration from a viewpoint
upstream of the target position. Key components are labeled: the dewar and crystal
housing for a single sega detector, the mounting ring, the 37◦ (7 detectors) and 90◦
rings (10 detectors), target location, 6 in diameter beam pipe, and the yellow entrance
quadrupole to the s800 spectrograph. In the photograph one detector is missing from the
forward ring (the only aqua circle that does not tag a crystal housing). This detector was
present for both experiments run in this thesis.

85

Target
θ

90o Ring

37o Ring

Figure 4.8: Side view of sega showing features important for the Doppler reconstruction
of the γ ray emitted by a relativistic projectile. The two rings of detectors are arranged at
angles of 37◦ and 90◦ with respect to the beam axis. The γ ray emitted in-flight by the
residue following a nuclear reaction in the target is detected in one of the sega germanium
crystals. The interaction point is determined by the segment with the highest energy
deposition. The measured angles were not corrected for the different trajectories (angles)
of the residues.

86

Resolution (FWHM) [%]

37º ± 9.5º

90º ± 9.5º

5
Velocity (∆β)
Angle (∆θ)
Total

4
3
2
1
30

60

90

120

150

180

Angle (degree)
Figure 4.9: Contributions of the angular and velocity uncertainty to the total energy
resolution of sega for β = 0.371, angular uncertainty of ∆θlab = 0.04 (from crystal segmentation size) and velocity uncertainty of ∆β = 0.05 (from target energy loss). The polar
angular coverage of each ring of detectors (37◦ and 90◦ ) is shown in yellow.

87

Figure 4.10: PID for the 9 Be(54 Ti, 53 Sc + γ)X reaction. 53 Sc, the one proton knockout
residue from the incoming 54 Ti beam, is labeled, along with other reaction products. Isotopic lines are indicated for scandium and calcium to show how pid spectra are interpreted.
Each reaction product is separated and has a definite loci. The colors, from blue to red,
indicate intensity on a logarithmic scale. The high intensity loci to the upper left of 53 Sc is
the q = 21 charge state of 54 Ti.

88

Counts / 16 keV

400
1111(2) keV

300

2110(3) keV
1273(2) keV

200

1539(4) keV
2459(5) keV

100
0

0

500

1000

1500

2000

2500

3000

3500

4000

Energy (keV)
Figure 4.11: Doppler-reconstructed γ spectra in coincidence with the 53 Sc residues in
the 9 Be(54 Ti, 53 Sc + γ)X reaction. A few transitions are labeled, including several newly
reported transitions at 1111(2), 1273(2), 1539(4), and 2459(5) keV. For interpretation and
further experimental details, see Section 5.6.2 (page 108).

89

Chapter 5
ONE PROTON KNOCKOUT FROM 54TI
TO 53SC
5.1

Introduction and Motivation

The first experiment, whose analysis, results, and summary are contained in this chapter,
focused on probing the single-proton structure of the N = 32 nuclei 54 Ti and 53 Sc with
the goal of testing the validity of the nuclear shell model and valance space in this neutron
rich region. The region around N = 32 from 52 Ca to 56 Cr has attracted considerable
attention in recent years [40, 41, 43–46, 57, 137–143] as theorists and experimentalists
(using a variety of experimental probes), have delineated the extent of the sub-shell gap
and explicated its emergence in terms of fundamental components of the nuclear force.
Given the stated goals, knockout reactions provided the appropriate means. Knockout
reactions, through the measurement of partial cross sections and longitudinal momentum
distributions to final states in the knockout residue, can disentangle the ground-state
nuclear wavefunction of the projectile and provide spectroscopic details of excited states
in the residue (for example, the orbital angular momentum). For the 9 Be(54 Ti, 53 Sc
+ γ)X reaction, the experimentally extracted partial cross sections were starkly different
than those calculated using a combination of shell model calculations (gxpf1 effective
interaction and f p-shell model space) and single-particle cross sections, calculated in
the eikonal framework reviewed in Section 3.4. Far more strength to excited states was
measured (10.9(17) mb) than predicted (0.7 mb) (see Table 5.3). The investigation and
resolution of this difference in strength, attributed to the knockout of deeply bound sd-

90

shell protons that lie outside the f p-shell model space, is addressed in this chapter. Before
continuing with the details of the observed and predicted single-particle strength, it is
worth reviewing the physics and background of the nuclear landscape around 54 Ti and
53 Sc,

i.e. the N = 32 sub-shell gap. See Figure 5.1 for the location of the 9 Be(54 Ti, 53 Sc

+ γ)X reaction within the nuclear landscape.

Fe

Cr
V
Ti

54Ti

Sc

Neutron drip line

Mn

53Sc

Ca

Z = 20
N = 28

N = 32
Neutron drip line

Figure 5.1: Nuclear landscape near the 9 Be(54 Ti, 53 Sc + γ)X reaction, showing the stable
isotopes in the valley of stability (filled black squares), the canonical N = 28 and Z = 20
shell gaps (red rectangles), and the emergent N = 32 sub-shell gap (blue rectangle),
predicted and observed to occur between 52 Ca and 56 Cr. The gap disappears in 58 Fe
because of the strength of the Vσ τ (πf7/2 , νf5/2 ) monopole force has sufficiently lowered
the νf5/2 orbital. The proton knockout reaction reaction from 54 Ti, in yellow, leads to the
even more neutron-rich isotope 53 Sc.

91

5.2

The N = 32 shell gap

The N = 32 shell gap was first proposed by Tondeur [144] in 1981 from a self-consistent
energy-density calculation. Following his prediction, a trove of experimental evidence has
emerged confirming and delimiting the shell gap between the isotones 52 Ca and 56 Cr. The
evidence includes mass measurements [39, 145], high E(2+ ) energies [40, 44, 138, 140],
1
low electromagnetic transition probabilities B(E2; 0+ → 2+ ), and spacing of higher-spin
1
structures [140, 143]. Two complementary mechanisms driving the change in shell structure were proposed [142]: a large gap between the neutron spin-orbit partners ν(2p3/2 )
and ν(2p1/2 ), and the reduction in strength of the strong Vσ τ (πf7/2 , νf5/2 ) monopole interaction between the π(f7/2 ) and ν(f5/2 ) orbitals as protons are removed from the f7/2
orbital. The reduction of the attractive force pushes the ν(f5/2 ) orbital to higher energy.
As mentioned in Section 2.1, the spin-flip, πν interactions, especially between J< = − 1/2
π ν
π ν
and J> = + 1/2 partners, are specially strong. V (j< , j> ) or V (j> , j< ) are proposed as the

driving force behind the erosion of several major shells. The effect of the large spin-orbit
splitting and reduction in the Vσ τ (f7/2 , f5/2 ) monopole interaction on the emergence of
the N = 32 sub-shell gap for 53 Sc is shown in Figure 5.2.
A brief summary of the work on the N = 32 shell gap, per constituent isotope 52 Ca,
53 Sc, 54 Ti, 55 V,

and 56 Cr, follows. Early β-decay work by Huck et al. [39] first identified

the high 2+ energy of 52 Ca at 2.56 MeV. They attributed the high energy of the 2+ state to
1
1
a subshell closure at N = 32. The energy of the 2+ state was later verified through γ ray
1
spectroscopy following the 9 Be(54 Ti, 52 Ca+γ)X reaction by Gade et al. [40]. The measured
longitudinal momentum distributions of the 52 Ca residues were consistent with removal
of two f7/2 protons outside a 52 Ca core.
53 Sc,

with one proton in the 1f7/2 orbital, had the least amount of spectroscopic

information available. The first 3/2− state was reported in Ref. [138] and its high excitation
1
energy of 2109 keV was consistent with the existence of a N = 32 sub-shell gap. For
additional details on 53 Sc, see Section 5.3.
92

nℓj
1g9/2
1g9/2

}

Large
spin-orbit
splitting

nℓj

2p1/2
1f5/2

1g9/2
N = 32

2p3/2
N = 28

V(f
5

/2 f
7/2 )

1f7/2
1d3/2
2s1/2
1d5/2

Z = 28
Z = 20

N = 20

Neutrons (N = 32)

2p1/2
1f5/2
2p3/2
1f7/2
1d3/2
2s1/2
1d5/2

}

Reduction
in V(f5/2,f7/2)
strength
through the
removal of
1f7/2 protons

Protons (Z = 21)

Figure 5.2: Physics of the N = 32 shell gap. Two effects are responsible: the large splitting
between the spin-orbit partners ν(2p3/2 ) and ν(2p1/2 ), and the decrease in the strength
of the strong, attractive Vσ τ (f7/2 , f5/2 ) monopole force as protons are removed from the
π1f7/2 orbital. There are six protons shown in the πf7/2 orbital. The first proton, shown as
a red circle ( ), represents 53 Sc. The addition of five more protons, shown as pink circles
( ), to 53 Sc, creates 56 Cr, the upper limit of the N = 32 shell gap.
The spectroscopy of 54 Ti, by Janssens et al. [140], whom combined the spectroscopic
results of two reactions, β-decay (54 Sc → 54 Ti) and deep-inelastic (48 Ca + 208 Pb), provided
evidence for the existence of a N = 32 sub-shell gap in 54 Ti through the measurement of
the energy of the E(2+ ) state and though the measurement of the spacing and energies
1
of the higher-spin states (I ≥ 8). First, the measurement of the E(2+ ) energy for 54 Ti
1
extended the systematics of the E(2+ ) states for even-even Titanium isotopes to 50,52,54 Ti,
1
and the energy staggering, ending in a high E(2+ ) energy in 54 Ti of 1495 keV, mirrored
1
similar staggering in 48,50,52 Ca and 52,54,56 Cr [142], nuclei previously shown to have a
closed-shell character at N = 32.
Second, Ref. [140] noted the higher-energy spin states in 54 Ti, J π = 7+ – 10+ , involve
neutron particle-hole excitations out of the closed-shell 52 Ca core. The 7+ and 8+ states are
dominated by a [π(f7/2 )2 ]Jp =6 ⊗ [ν(f7/2 )8 (p3/2 )3 (p1/2 )1 ]Jn =2 configuration, and the 9+ and
93

10+ states are dominated by a [π(f7/2 )2 ]Jp =6 ⊗ [ν(f7/2 )8 (p3/2 )3 (f5/2 )1 ]Jn =3,4 configuration.
The large spacing between the J π = 0+ , 2+ , 4+ , 6+ orbitals, formed through the coupling
g.s 1 1
of two πf (f7/2 ) protons to the magic, N = 32 core, e.g. [π(f7/2 )2 ]Jp ⊗ [ν(f7/2 )8 (p3/2 )4 ]Jn ,
and the higher-spin 8+ , 9+ , and 10+ orbitals, created by the promotion of neutrons above
1 1
1
the N = 32 sub-shell gap, reflects the high energy necessary to overcome the sub-shell gap.
The spacing among the higher spin-states also reflects the relative spacing of the νp1/2 and
νf5/2 orbitals and suggests, in reference to shell model calculations, the possible presence
of a N = 34 shell gap.
This conjecture was later experimentally tested through the spectroscopy of 56 Ti by
Fornal et al. [44] following heavy-ion, deep-inelastic collisions, and by Liddick et al. [57]
in the β-decay of 56 Sc → 56 Ti. Both experiments observed a E(2+ ) energy of 1128 keV that
1
was not consistent with the presence of a N = 34 shell gap. Lastly, the work by Dinca et al.
[43], confirmed the existence of the N = 32 shell gap through a simultaneous measurement
of the energy of the E(2+ ) state and the reduced matrix element B(E2; 0+ → 2+ ) by the
1
1
Coulomb excitation of 56 Ti. They did not find evidence for the N = 34 gap in 56 Ti.
The excitation spectra of 55 V and 55 Ti show similar evidence for a N = 32 shell gap
[143]. The lower-spin 7/2− , 11/2− , and 15/2− states of 55 V (ground state, 1433 keV and
g.s.
1
1
2508 keV respectively) are dominated by excitations in the proton πf7/2 orbital, i.e. they
have a dominate [π(f7/2 )3 ]Jp ⊗ [ν(f7/2 )8 (p3/2 )4 ]Jn configuration. The higher-lying spin
states are dominated by neutron particle-hole excitations—for example, the 19/2− state
1
at 4696 keV has a dominate [π(f7/2 )3 ] ⊗ [ν[(f7/2 )8 (p3/2 )3 (f5/2 )1 ] configuration—and the
high excitation necessary to break the neutron configuration is evidence of the shell gap.
Analysis of the spectrum of 55 Ti, in comparison to shell model calculations, showed similar
jumps in energy in the excitation spectrum caused by a break in the neutron core. Thus, as
reported in Ref. [143], 55 V and 55 Ti can be treated as a single proton and neutron outside
of a semi-magic 54 Ti core.
As seen in Figure 5.1, 56 Cr marks the upper limit for the observed N = 32 shell gap.

94

The first work was done by Prisciandaro et al. [142], who completed the systematics of the
even-even N = 32 isotones by measuring the energy of the first 2+ state in 58 Cr through
1
the β-decay of 58 K. A peak in excitation energy of the 2+ state of 56 Cr in the chain of
1
even-even Cr isotones suggested the presence of a shell gap. Additional spectroscopy was
performed on the high-spin states following the heavy-ion, fusion-evaporation reaction
48 Ca(11 B, pxn) 58−x Cr

[46]. Both the energy of the E(2+ ) state and the ratio E(4+ )/E(2+ ),
1
1
1

which approaches the vibrational limit, were good indicators of the N = 32 shell gap in
56 Cr.

The complementary B(E2; 0+ → 2+ ) values for 54,56,58 Cr were measured by Bürger
1

et al. [137], who observed a low B(E2; ↑). The combination of high excitation energy of
the 2+ state and low electromagnetic transition probability are characteristic of shell gaps
1
(see Section 1.2.4). The minimal B(E2; ↑) values in the observed Chromium staggering
occurred at N = 28 and N = 32. Finally, mass measurements of 56,57 Cr confirmed the
increase of binding for 56 Cr through the difference between the measured two-neutron
separation energy S2n (56 Cr) and a linear extrapolation based on S2n (54 Cr) and S2n (52 Cr).

5.3

Previous Studies of 53 Sc

Compared to its even mass neighbors 52 Ca and 54 Ti, less information was known about
53 Sc.

A lower limit on the β-decay half-life of τ1/2 (53 Sc) > 3 s was established by Sorlin

et al. [146] during studies of the neutron-rich nuclei 53−55 Sc and 56−69 V. No decays
were observed in the γ-ray spectrum following β decay within the 3.3 s observation
window. Additional information was provided by Crawford et al. [138] through the β
decay 53 Ca33 → 53 Sc32 . A high energy of the 3/2− state of 2110 keV was measured. The
20
21
1
3/2− state was placed within a quintet of states from 3/2− to 11/2− formed by the coupling
1
of the π(1f7/2 ) orbital to the excited 52 Ca core: [52 Ca(2+ )] ⊗ [π(f7/2 )]Jp =7/2 . Additional
states in this quintet were measured at ganil using deep-inelastic, multi-nucleon transfer
reactions [147]. Of the measured transitions of 345(7), 2283(18), and 2617(20) keV, the
two highest energy transitions were ascribed to the 9/2− and 11/2− yrast states.
1
1
95

5.4

Expectation and Prediction of Spectroscopic Strength in the fp-Shell
Model Space

We can make a naive prediction for the strength of population of final states in 53 Sc,
following the proton knockout reaction 9 Be(54 Ti, 53 Sc + γ)X, based upon a model of 54 Ti
with two protons outside a doubly magic 52 Ca core. The removal of a proton from the
π(1f7/2 ) orbital in 54 Ti should strongly populate the 7/2− ground state of 53 Sc (see
g.s.
Figure 3.5), i.e. knockout to the ground state of 53 Sc should nearly exhaust the complete
spectroscopic strength of 2 (πf7/2 occupation number).

nℓj
2p1/2
1f5/2
2p3/2

Z = 28

1f7/2
1d3/2
2s1/2
1d5/2

5/21
3/21

Z = 20

7/2g.s.

54Ti (ground state)

53Sc

Figure 5.3: Naive model for the population of states in 53 Sc following the knockout of
a proton from 54 Ti in the f p model space. With Z = 22, 54 Ti has two protons in the
f p-model space that reside nearly exclusively, because of the canonical Z = 28 shell gap,
in the π(1f7/2 ) orbital. The removal of one of the π(f7/2 ) protons will populate the 53 Sc
7/2− ground state (see Table 5.1). This strength is indicated by the large black arrow.
g.s.
There is a small probability of one or more protons occupying a higher orbital, e.g. the
π(2p3/2 ) orbital. The low occupancy is marked by dashed circles. The removal of a proton
from the π(2p3/2 ) orbital will predominately populate the 3/2− state in 53 Sc, which will
1
decay through γ ray emission. Note that by confining shell model calculations to the f p
model space, strength from more deeply bound protons (e.g. sd shell) is not accounted for.

This basic prediction can be compared to shell-model calculations using the gxpf1 or

96

Jπ

Ex
(MeV)

C2 S

7/2−
3/2−
1/2−
7/2−
5/2−
5/2−
5/2−
7/2−
7/2−
3/2−
7/2−

0 (g.s.)
2.194
3.231
2.246
3.361
5.294
5.513
3.486
5.154
5.178
5.582

1.85
0.062
0.012
0.008
0.005
0.005
0.004
0.003
0.002
0.002
0.001

Table 5.1: Theoretical energies and spectroscopic factors to final states in 53 Sc following
the 9 Be(54 Ti, 53 Sc + γ)X reaction, listed in order of decreasing spectroscopic strength. Only
values with C 2 S ≥ 0.001 are listed. Calculations were performed in oxbash using the
gxpf1 effective interaction and a f p model space. The spectroscopic strength to the ground
state dominates, i.e. the removal of a proton from 54 Ti should almost exclusively populate
the 7/2− ground state of 53 Sc.

gxpf1a effective interaction [56, 58]. The gxpf1 effective interaction has been effective
in predicting the spectroscopy of several isotopic chains through N = 32. That predictive
power was experimentally shown to break down at N = 34. The gxpf1 effective interaction
predicted a N = 34 shell gap in titanium that was not observed by Refs. [43, 44, 57],
leading to the modification of five matrix elements mostly involving the T = 1 f5/2 and
p1/2 orbitals. Shell model calculations using the gxpf1 and gxpf1a effective interactions
gave similar excitation spectra (for 53 Sc) and spectroscopic factors for the 9 Be(54 Ti, 53 Sc
+γ)X reaction. The results for the gxpf1 interaction are shown in Table 5.1. Note that there
are no positive-parity states. These states, resulting from 1 ω particle-hole excitations
from the sd-shell, are outside the model space. The fp model space used, π(f7/2 p3/2 f5/2 p1/2 )
and ν(p3/2 f5/2 p1/2 ) (48 Ca core), excluded lower harmonic-oscillator shells.
The shell-model spectroscopic factors can be combined with calculated single-particle
cross sections to provide theoretical partial cross sections that can be directly compared
to experimental data (see Eqn. 3.18). Single-particle cross sections were calculated using
97

Ex
(MeV)

n j

Sp + Ex
(MeV)

V0
(MeV)

r0
(fm)

a
(fm)

σstrip
(mb)

σdiff σtotal
(mb) (mb)

0.00
2.11
3.22
3.38

1f7/2
2p3/2
1f5/2
2p1/2

15.26
17.37
18.48
18.64

64.2
81.0
72.0
82.1

1.29
1.19
1.31
1.22

0.7
0.7
0.7
0.7

6.57
6.17
5.44
6.13

1.68
1.67
1.32
1.64

8.25
7.85
6.76
7.77

Table 5.2: Input parameters and results for the calculation of single-particle cross sections
for the 9 Be(54 Ti, 53 Sc + γ)X reaction for fp-orbital knockout. The input parameters include
the energy of the final state in 53 Sc (Ex ), the orbital of the knockout nucleon in 54 Ti (n j ),
and the Woods-Saxon parameters used to calculate the wavefunction of the nucleon-core
system (V0 , r0 , and a). The results of the calculation are broken down into stripping (σstrip )
and diffractive components (σdiff ). The total single-particle cross section is the sum of
both.

the method outlined in Section 3.4.2. A Hartree-Fock calculation using the SkX Skyrme
interaction [81] provided the density profile of the core. The density profile of the target
was approximated as a Gaussian with a width σ of 2.36 fm. The core-nucleon wavefunction
was calculated from the bound state of a Woods-Saxon Hamiltonian with a set diffuseness
of a = 0.7 fm, and a orbital rms separation radius of of Rsp = r0 A1/3 . r0 was adjusted so
that Rsp matched the separation radius Rhf predicted by the core Skyrme Hartree-Fock
sp
calculation. The potential depth V0 was adjusted to reflect the physical separation of the
nucleon from the core, i.e. Esep = Sp (54 Ti) + Ex (53 Sc). The proton separation energy of
54 Ti

is Sp = 15.26 MeV (systematic estimation, see Ref. [148]).

Combining the theoretical spectroscopic factors in Tables 5.1 and 5.2 with the center-ofmass correction of Section 3.2.6 and the single-particle cross sections, gives the theoretical
partial cross sections for the population of final states in 53 Sc. These results are seen in
Table. 5.3. The theoretical predictions, reflect our naive assumption. Nearly all the strength
of the 9 Be(54 Ti, 53 Sc + γ)X reaction should populate the ground state of 53 Sc. The 53 Sc γ
ray spectrum should be principally background, reflecting the negligible population of
excited states of 53 Sc in the 9 Be(54 Ti, 53 Sc + γ)X reaction.

98

Ex
(keV)

n j

C2S

σsp
(mb)

σth
(mb)

0
2110
2331
3383

1f7/2
2p3/2
1f5/2
2p1/2

1.85
0.06
0.01
0.01

8.25
7.85
6.76
7.77

16.2
0.5
0.1
0.1

Table 5.3: Theoretical spectroscopic factors (C 2 S), single-particle cross sections (σsp ), and
theoretical partial-cross sections to final states in 53 Sc (σth ). Only states with spectroscopic
factors C 2 S ≥ 0.01 are shown (see Table 5.1).

5.5
5.5.1

Experimental Setup
Secondary-Beam Production

A 76 Ge primary beam, accelerated to 130 MeV/u and directed onto a 423 mg/cm2 production target, produced a wide range of fragmentation products. The a1900 fragment
separator, with a 300 mg/cm2 Al wedge reduced the fragmentation products to a cocktail
beam with three primary constituents: 54 Ti, 55 V and 57 Cr (see Figure 5.4 and Table 5.4).
The cocktail beam was delivered to a 376 mg/cm2 9 Be target located at the pivot point
of the s800 magnetic spectrograph. This secondary target was surrounded by sega in its
classic configuration (see Section 4.4) consisting of 17 detectors arranged in two rings 37◦
and 90◦ with respect to the beam direction. The reaction residues produced in the reaction
of the reaction beam with the secondary target were detected by the s800 spectrograph.
The s800 detector systems were operated in coincidence with sega and therefore the heavy
residuals detected in the spectrograph’s focal plane could be correlated event-by-event to
their de-excitation γ rays.

5.5.2

SeGA Efficiency Calibrations

As outlined in Section 4.4.3, the efficiency of SeGA is energy and threshold dependent
and must be calculated on a per-experiment basis. Two sources were used to calibrate

99

Counts / chn.

Unreacted
setting

55V

2000

54Ti

1500
1000
500

57Cr

0

Counts / chn.

One-p
knockout
setting

55V

2000

54Ti

1500
1000
500
0
300

350

400

450

500

550

600

time-of-flight (arb. units)
Figure 5.4: Time-of-flight spectrum for a unreacted run (top panel; projectiles centered in
the s800 focal plane) and a reacted run (bottom panel; desired reaction channel centered in
the focal plane) with 52 Ca centered in the focal plane. See text for more details. The tof
for the unreacted beam clearly shows the three cocktail components, 54 Ti, 55 V, and 57 Cr,
listed in order of A/Q and therefore time-of-flight. The magnetic rigidity of the reacted
run, set to center 52 Ca in the focal plane, also allows fragments created in the a1900 wedge
with similar rigidity to enter the s800 focal plane. The fragmentation products produced
in the wedge appear as broad hump at higher channels. Fortunately, a clean TOF gate can
be placed around 54 Ti (red dotted line). The fragmentation products coincident with only
incoming 54 Ti nuclei are shown in the pid plot of Figure 5.6.

100

Primary beam

76 Ge

Wedge

300 mg/cm2 Al (achromatic)

at 130 MeV/u and 25 pnA

A1900 momentum Restricted 1 % momentum for the 53 Sc-centered and
54 Ti-centered runs used for the calculation of the inacceptance
clusive and partial cross sections. No momentum restriction for γ-ray spectroscopy data collection runs
(two-proton knockout residue 52 Ca centered in the s800
focal plane).
Typical intensity

54 Ti

at 103 pps, 55 V at 85 pps, 57 V at 77 pps (1 % momentum acceptance)

S800 optics mode

Focus mode

Table 5.4: Secondary beam production for the 9 Be(54 Ti, 53 Sc + γ)X reaction.
the stationary-source efficiency of SeGA over a wide energy range, 152 Eu and 56 Co. The
calibrations were done consecutively in two separate runs. The photopeak areas of nine
peaks in the 152 Eu spectrum were extracted for energies between 121.8 keV to 1408.0 keV
and the photopeak areas of eleven peaks from the 56 Co spectrum were extracted for
energies between 846.8 keV to 3548.2 keV. The source calibration energies, listed in
Table. 5.5, are a subset of the transitions listed in Table 4.5. The range of energies covered
by the two calibration sources, 121.8 keV to 3451.2 keV, covered the full range of Dopplerboosted energies emitted by the fast-moving residue (see Eqn. 4.3). The highest energy
transition seen in the residue’s rest frame for 53 Sc, 2.4 MeV, was boosted to ≈ 3.0 MeV in
the 37° ring.
The photopeak areas were then used to calculate the efficiency of sega for the specific
source energies according to Eqn. 4.7. See Section 4.4.4 for further details. Given the
high source activity, e.g. 313 100 Bq for 152 Eu, the daq livetime was low, below 20 %.
The half-life of 152 Eu is τ1/2 (152 Eu) = 13.543 y; that of 56 Co, τ1/2 (56 Co) = 77.31 d. The
56 Co

source was not absolutely calibrated and therefore the fit efficiency curve was scaled

relative to the calibrated 152 Eu efficiency. Despite the lack of an absolute calibration, the
56 Co

source was essential for providing calibration data for high-energy γ rays. The final

101

90o ring

5
4

152Eu
56Co
Fit

Absolute Efficiency (%)

3
2
1
4

37o ring

3
2
1
0

500

1000 1500 2000 2500 3000 3500

Energy (keV)
Figure 5.5: SeGA source efficiency curves for the 37° and 90° rings. The 56 Co source
calibration was normalized to the calibrated 152 Eu source. The number of detectors in
the 90◦ ring is greater than the number of detectors in the 37◦ ring (10 versus 7). This
difference accounts for the increased efficiency of the 90° ring.

102

152 Eu

56 Co

(keV)

(keV)

121.8
244.7
344.3
444.0
778.9
867.4
964.0
1112.1
1408.0

846.8
1037.9
1238.3
1360.2
2105.2
2034.8
2598.5
3202.0
3253.4
3273.0
3451.2

Table 5.5: 152 Eu and 56 Co energies used in sega efficiency calibrations for the 37◦ and
90◦ rings.
a
37◦
90◦

b

c

2.943
3.259

0.792
0.759

-106.5
-99.51

Table 5.6: Efficiency fit results for the 152 Eu and normalized 56 Co sources. The fit function
is a slight rewrite of Eqn. 4.6: f (Eγ ) = a exp(−b log(Eγ − c + exp(0.269Eγ ))).
efficiency curves for the 37◦ and 90◦ ring are shown in Figure 5.5, and the fit parameters
a, b, and c, are listed in Table 5.6.
The efficiency for a source at rest must be transformed to the efficiency for a residue
rest-frame transition boosted into the lab frame. The method of calculating the efficiency
for a measured residue rest-frame energy is covered in Section 4.4.4. The boosted efficiency
for the two highest-intensity transitions observed in the γ ray spectrum of 53 Sc, 2.11 MeV
and 1.76 MeV, are 2.8(1) % and 1.76(5) %.

5.5.3

Particle Identification

The complete reaction process requires the identification of three components: the incoming projectile species, the projectile-like residue, and the γ rays coincident with the

103

reaction residue. As covered in Section 4.3, the incoming projectile is identified through
its time-of-flight between two thin plastic scintillators located at the end of the a1900
fragment separator and at the object position of the s800 analysis beam line (xfp and obj
scintillators). The associated spectra are shown in Figure 5.4 for a reacted run (upper
panel) where the magnetic rigidity of the s800 spectrograph was changed (decreased
relative to 54 Ti) to center 52 Ca in the focal plane, and an unreacted run (lower panel)
where the magnetic rigidity of the s800 was tuned to center the incoming cocktail beam
in the focal plane. The detection of a residue by the e1 scintillator is used by the s800
trigger module, along with input from a secondary detector depending on the trigger logic
(s800 singles or s800-sega coincidences) to issue a master trigger. Because a projectile
must survive to the focal plane to issue a trigger, the observed spectra for the obj or xfp
scintillators depends on the rigidity of the spectrograph. Thus the difference in the two
tof spectra of Figure 5.4.
By applying the slice gate seen in the lower panel of Figure 5.4 to the reaction products
detected in the s800 focal plane, only the reaction products of 54 Ti are selected. A particleidentification (pid) matrix of tof versus energy loss in the ion chamber, gated on incoming
54 Ti

projectiles, provides an identification of the reaction residues event-by-event. The

pid for 54 Ti, corrected in the manner shown in Section 4.5.1, is shown in Figure 5.6.

5.6
5.6.1

Results
Inclusive Cross Section

A multitude of different fragmentation products, including the proton knockout reaction
channel 9 Be(54 Ti, 53 Sc + γ)X, were detected in the focal plane1 . Through the application
of a tof gate on the incident projectile 54 Ti (see Figure 5.6) and a contour gate on the
1 The 9 Be(54 Ti, 53 Sc + γ)X

knockout data was taken from a larger data set devoted to
the two-proton knockout reaction 9 Be(54 Ti, 52 Ca + γ)X [40].

104

Figure 5.6: Particle identification plot for the 9 Be(54 Ti, 53 Sc + γ)X reaction, gated on the
incoming 54 Ti projectile shown in the lower panel of Figure 5.4. The tof is corrected for
the ion’s flight path through the magnetic spectrograph. See Section 4.5.1.
knockout reaction residue 53 Sc (see Figure 5.4), events from the 9 Be(54 Ti, 53 Sc + γ)X reaction were selected on an event-by-event basis. Although the momentum acceptance of the
s800 is large (∆p/p = ±2.5 %), it could not accommodate the full momentum distributions
of both 53 Sc and 52 Ca in the s800 focal plane. As a result, for most runs, 53 Sc was not
centered in the focal plane, and instead entered at the edge of the s800 acceptance. With
a big and uncertain acceptance correction, these runs could not be used for the determination of the inclusive cross section. To extract an accurate inclusive cross section, 53 Sc
was briefly centered in the s800 focal plane. From the integration of 53 Sc in the pid for the
53 Sc-centered

run, relative to the number of incoming 54 Ti and the 9 Be target density, the

105

inclusive cross section was calculated. See Eqn. 4.15 for the full list of required quantities.
The final inclusive cross section for the one-proton knockout reaction from 54 Ti to 53 Sc is
σinc = 17.6(6) mb.
Acceptance corrections have a complex dependency on the dispersive and non-dispersive angles and momenta of the reaction residues. These quantaties can be calculated using
the position and angles of the residues measured in the focal plane, and then transformed
to angles and momenta at the target using an inverse map (see Section 4.5.3). The four
reconstructed measurements used in acceptance calculations are labeled: ata (dispersive
angle at the target), dta (dispersive differential energy at the target), bta (non-dispersive
angle at the target), and yta (non-dispersive position at the target).
Fortunately, because of the high fragment velocity of v ∼ 0.3 c and heavy mass of the
residue, the reaction products were very forward focused. With the magnetic rigidity of
the spectrograph set to center 53 Sc in the focal plane, acceptance losses were minimal and
only evident in the dispersive direction (see Figure 5.7). To account for these losses, the
dta distribution was fit by several different curves. Examples include a single Gaussian,
a double Gaussian (two floating Gaussians), and a skew Gaussian (single Gaussian with
different widths σ− and σ+ ). The missing area in the tail was approximated by the area
under these curves. The ratio the uncorrected area, to of the total area, including the
estimated area under the tail, is the acceptance correction A of Eqn. 4.15. The difference
in tail area resulting from the different fits contributed to the systematic error.
The error of 0.6 mb includes both statistical and systematic contributions. The statistical Poisson uncertainties (see Eqn. 4.17) are dominated by the number of detected
projectiles (54 Ti) and the number of detected residues (53 Sc)2 .
2 Errors

are combined through the standard propagation of errors, e.g.
a
2
σf

σi2

=
i=1

f
xi

a

2

+2

σi σj
i j

f
xi

f
+ ··· ,
xj

(5.1)

where f is the full inclusive cross section σinc. of Eqn. 4.17. Because the different errors
106

500

Data
Fit range
Tail

Counts

400
300
200
100
0
-6

-4

-2

0

2

4

6

DTA (%)
Figure 5.7: Acceptance correction for 53 Sc. A skew Gaussian fit is shown. The 53 Sc
residues lost to acceptance cuts are estimated by the area under the left tail. Several
different parametrizations were used. The differences in area due to these different
parametrizations contributed to the total uncertainty for the inclusive cross section.

There were several additional sources of systematic error beyond the uncertainty in the
acceptance correction: target thickness and stability of the cocktail beam. The uncertainty
on the target thickness was provided by the manufacturer. Details on the stability of the
cocktail beam follow. By taking both 53 Sc and 54 Ti of Eqn. 4.17 relative to the integrated
beam rate in the obj and xfp scalars, we assume that the composition of the cocktail
beam does not change. While this approximation is reasonable, the local fluctuation of
the beam composition should be folded into the systematic uncertainty. An estimation
of the fluctuation came by measuring the proportion of 54 Ti relative to the total rate as
taken from the obj and xfp scalers for several runs on either side of the run used for the
determination of the cross section. The fluctuation on the ratio, 3 %, provided a reasonable
estimate for the stability of the beam composition. The extracted uncertainty was added
in quadrature to the other contributions listed above.
were uncorrelated, the second term, and other higher order terms were zero.
107

5.6.2

Observed γ Ray Transitions

The Doppler-reconstructed γ ray spectrum, in coincidence with the 53 Sc residues of is
shown in Figure 5.8. As outlined in Section 4.4.2 the γ rays were boosted from the lab frame
to the residue rest frame using the angle of γ ray emission calculated from the position of
the detector Ge crystal segment with the highest energy deposition relative to the position
of the target. The fragment velocity β = v/c at the moment of γ decay was adjusted to
align the Doppler-reconstructed spectra in the forward (37◦ ) and backwards (90◦ ) ring. β
values were determined transition-by-transition because of possible differences in lifetime
effects. γ rays are emitted in flight by the projectile-like residue, and the velocity of
the residue when it decays depends on the amount of target material transversed. For
very short lifetimes, e.g. of order τ1/2 < 1 ps, the average decay position is approximately
mid-target (at a incident velocity of v ∼ 0.3 c, the projectiles are traveling at approximately
1 cm per ns). For longer lifetimes of τ1/2 > 1 ps the average decay position will be further
downstream. Significant lifetimes of a state are noticed in the β value that is near or equal
to the end-target value βend and in the shape of the γ ray peak in the spectrum.
The target position ztar effects all angles in the Doppler reconstruction. To fix the target
position, a transition of known energy and short half-life, ensuring an average mid-target
decay, was used as a reference. For this data, the reference was the Eγ (1494.8 keV; 2+ → 0+ )
0.9 ps transition in 54 Ti.
Both β and ztar were optimized over a grid of possible ztar and β values. ztar was
optimized to minimized the absolute difference of the true peak position of 1495 keV and
the measured peak position in the Doppler reconstructed spectrum. The optimal β value,
as mentioned, was chosen to minimize the absolute difference in energy between the 37°
and 90° rings. A target position of ztar = 0.15 cm was extracted, and the uncertainty of
transition energies in 53 Sc was directly connected to the uncertainty in this choice of
ztar through finite difference calculations. ztar was varied at ztar ± σz and the effect on
Eγ (β, ztar ) was measured.
108

Energy
(keV)
2110
2459
1111
1273
1539

δE(fit) δE(sys.) δE(∆β) δE(∆z)
(keV)
(keV)
(keV)
(keV)
0.80
3.12
0.80
1.31
3.32

0.36
0.72
0.08
0.18
1.19

0.42
2.13
0.17
0.39
1.30

2.30
3.03
0.98
1.08
1.86

δE(tot.)
(keV)
3
5
2
2
4

Table 5.7: Error budget for the measured transition energies in 53 Sc.
With the ztar position fixed to the 1495 keV transition in 54 Ti, the energy of the first
Eγ (3/2− → 7/2− ) = 2110(3) keV transition was in agreement with the previous reported
g.s
1
value of 2109 keV by Ref. [138]. In addition to confirming this transition, four new
transitions at 1111(2), 1273(2), 1539(4) and 2459(5) keV were reported [42]. The knockout
reaction is not expected to populate the 9/2− and 11/2− high-spin states of 2283 keV and
2617 keV reported by Ref. [147] because of the connection between angular momentum j π
of the removed nucleons and the angular momentum of the residue I π . For single-nucleon
knockout from an even-even nuclei like 54 Ti, and assuming no coupled-channels effects,
these values are identical.
The energy resolution was dominated by the target thickness. See Section 4.4.2 and
Figure 4.9. The peak position uncertainties combine the statistical uncertainty from fitting,
as done by a fitting code written in the root Data Analysis Framework [149], and the
systematic uncertainty that includes both the uncertainty in target position (δztar ) and the
uncertainty in velocity (δβ) (see Table. 5.7).

5.6.3

Partial Cross Sections

Only the 53 Sc-centered run could be used for the extraction of the partial cross section.
Both the stripping and diffractive components of the single-particle cross section involve
an average over all momentum projections m of the removed nucleon with total angular
momentum j (see Eqn. 3.20 and Eqn. 3.22). Each projection m has a different differential

109

Energy (keV)

Counts / 16 keV

0

1000

3000

4000

400
300
200
100
0

1111(2) keV

250

Counts / 15 keV

2000

2110(3) keV

1273(2) keV

200
150

1539(4) keV

100

2459(5) keV

50
0
1000

1500

1500

2000

Energy (keV)
Figure 5.8: Doppler-reconstructed γ ray spectrum of 53 Sc. The 2110(3) keV transition,
first reported in Ref. [138], was confirmed, and four new transitions at 1111(2), 1273(2),
1539(4), and 2459(5) keV were seen. The lower panel is an expansion of the red rectangle
insert of the upper spectrum.

110

Energy
(keV)

δA(stat.)
(counts)

SeGA (boost.)

(%)

(mb)

δ ( i σi )
(mb)

2110
2459
1111
1273
1539

99
–
39
30
–

1.760
1.570
2.814
2.551
2.222

10.9
–
2.7
2.3
–

1.7
–
0.4
0.8
–

i σi

Table 5.8: Measured partial cross sections to final states in 53 Sc. The summation over
index i indicates the feeding has not been subtracted. Because of the low statistics in the
53 Sc-centered run, the areas for two transitions could not be extracted. These are indicated
by an en dash (–).

momentum distribution and carries a different fraction of the total, angle-integrated cross
section. By selecting a portion of the momentum distribution, the contributions of the
different projections to the cross section changes, affecting the final averaged value. Partial
cross sections can only be reliably extracted for runs with small acceptance corrections.
The large reduction in statistics forced by the use of the single 53 Sc-centered run, increased
the statistical uncertainty of the partial cross sections and reduced the number of peaks
from which a peak area could be confidently extracted.
Partial cross sections must be corrected for feeding, the contribution to the photopeak
area from transitions decaying into the state of interest. Unfortunately, because of low
statistics associated with the only 53 Sc-centered run, the orbital angular momentum
assignments associated with the exclusive parallel momentum distributions could not
be used to construct a level diagram. Level assignments were only based upon intensity
balances and comparisons with shell model.

111

5.7
5.7.1

Knockout of Deeply Bound Protons
Knockout From the sd Shell

With a large overlap between the ground-state wavefunctions of 54 Ti and 53 Sc (C 2 S = 1.85),
theory, restricted to the proton f p-shell, predicts most (95 %) of the cross section strength
will directly populate the 53 Sc ground state. From Table 5.3, the summed strength to
excited states is only 0.7 mb. In contrast, as inferred from Table 5.8, experiment observes a
large percentage, a clear majority, of the total knockout strength feeds excited states in
53 Sc.

There is an obvious discrepancy between the measured and expected population of

excited states. To provide an upper limit on the population of the 53 Sc ground state, we
can assume the first 3/2− state at 2.1 MeV acts like a funnel. The cross section to of all
1
states higher in energy, both observed and unobserved, funnel through this state. Making
this approximation, we have
σg.s. = σinc − σ2.1 MeV = 6.7(18) mb.

(5.2)

In this extreme condition, 6.7 mb or 60 % of all cross section goes to excited states (see
Table 5.9). The factor of 0.4 between the theoretical partial cross section to the ground
state of 16.2 mb and the experimental upper bound of σg.s ≤ 6.7 mb is in agreement with
the reduction factor systematics reported in Ref. [150].
Experimentally, measured spectroscopic strengths are lower than both the simple ipm
predictions (single Slater determinant and full occupancy), and shell model calculations
that account for some of the reduction in single-particle strength through configuration
mixing. The additional reduction in strength beyond ipm and ci calculations are caused
by correlations, for example short-range and long-range. The short-range correlations
come from both the extreme repulsion at small distances and the tensor nn interaction.
The long-range correlations include coupling of the single-particle strength to low-lying
collective excitations (e.g. 2p1h and 1p2h excitations and giant resonances). The result

112

Energy
(keV)

σth.
(mb)

σexp.
(mb)

0
2110
2331
3383

16.2
0.5
0.1
0.1

6.7(18)
–
–
–

0.7

10.9(17)

excited
i σi

Table 5.9: Comparison of theoretical predictions and experimental measurements of the
summed strength to states in 53 Sc. The measured partial cross sections are divided between
the ground and excited states assuming all excited states decay through the 2.1 MeV 3/2−
1
state (see Eqn. 5.2). The factor of 0.4 between the theoretical and measured partial cross
section to the ground state is commensurate with the reduction factor systematics of
Ref. [150]. The large discrepancy between the measured and predicted strength of excitedstate population can be resolved if knockout of deeply-bound sd-shell protons (outside
the f p-shell model space) is considered.

of this coupling is a fragmentation of the single-particle strength to high energies, well
beyond the limited model space of most shell-model calculations.
The observed experimental strength to excited states must lie outside the model space
of the gxpf1 shell-model calculations. That is, we must consider knockout from more
deeply bound sd-shell protons. In this new picture, the removal of protons from the
π(1f7/2 ) orbital directly populates the 53 Sc ground state, in accordance with the theoretical
predictions of Table 5.1, and the strength to excited states is attributed nearly exclusively
to sd-shell knockout of protons. This strength populates higher-energy states above the
first 3/2− state, which then feed through the 3/2− state to the ground state. The schematic
1
1
of this model is seen in Figure 5.9.
Although the strength of population to excited states strongly suggests a picture where
spectroscopic strength originates from the knockout of deeply-bound sd-shell protons, we
can look for further support in the population of excited states for the analogous regularkinematics reaction 50 Ti(d, 3 He) 49 Sc and in the inclusive parallel momentum distribution
of 53 Sc.

113

S n = 5.66 MeV
.
.
.

sd-shell states

fp-shell states
σex = 0.4 mb

Feeding
(3/2-)

2.110(3)

3/2-

2.194

σex = 0.9 mb

σex = 10.9(17) mb
g.s.

σ gs ≤ 6.7(18) mb

(7/2-)

Experiment

g.s.

σ gs = 16.1 mb

7/2-

Theory

Figure 5.9: Comparison of the theoretical and experimental population of 53 Sc in the
9 Be(54 Ti, 53 Sc + γ)X reaction. The energy in MeV, the assumed spin-parity, and summed
partial cross sections (sum of feeding and direct population) in mb are displayed. Theory,
restricted to the pf -shell, predicts most of the knockout strength will directly populate
the ground state (95 %); experimentally a large population of excited states (≥ 60 %) is
observed.

5.7.2

The 50 Ti(d,3 He)49 Sc and 9 Be(54 Ti,52 Ca + γ)X Reactions

Support for the interpretation of the excited state strength from particle-hole excitations
in the sd-shell comes from sd-shell strength observed in the 9 Be(54 Ti, 52 Ca + γ)X [40] and
50 Ti(d, 3 He) 49 Sc

reactions [87]. In the two-proton knockout from 54 Ti [40], appreciable

strength was observed to feed the 3990 keV 3− state in 52 Ca, bypassing the first 2+ state
1
at 2563 keV. Of the total inclusive strength of σinc. = 0.32(4) mb, 67(7) % populated the
ground state. Given the selectivity of the knockout to the occupancies and orbitals of
the removed proton in the initial 54 Ti nucleus, the population of the ground state was
attributed to the removal of two correlated protons in the π(1f7/2 ) orbital and the population of the 3− state was attributed to the simultaneous removal of two correlated protons
1
from a π(1f7/2 ) orbital and an sd-shell orbital. This initial conjecture was confirmed by

114

shell model calculations using the modified wbmb Hamiltonian [151]. Although effective
interactions for the excitation of protons into the fp-shell are poor—the calculation placed
the 3− state ∼ 3 MeV above the experimental value—based on the negative-parity 52 Ca
spectrum and the theoretical maximum possible cross section to the 3− state that was
several times the experimental value, the shell model calculations provided qualitative
support of the strong presence of sd-shell particle-hole excitations in the 9 Be(54 Ti, 52 Ca
+ γ)X reaction.
50 Ti

and 54 Ti share the same number of protons (Z = 22) and are semi-magic in the

number of neutrons (the canonical N = 28 shell gap for 54 Ti; the emergent N = 32 shell
gap for 54 Ti). Thus, we can look to the similar, regular-kinematics transfer reaction 50 Ti(d,
3 He) 49 Sc,

as performed by Doll et al. [87], to gauge the importance of sd-shell excitations.

They observed significant spectroscopic strength to excited states, and based on the shape
of the angular distributions of the 3 He residues for different final state of 49 Sc, from
which angular momentum assignments of the removed nucleon in 50 Ti could be extracted,
most of the strength was attributed to the removal of sd-shell protons. In particular, this
includes the population of the 2.23 MeV and 3.36 MeV states that exhaust nearly the full
spectroscopic strength of the 2s1/2 and 1d3/2 orbitals [C 2 S(1/2+ , 2s1/2 ) = 1.40(10) and
C 2 S(3/2+ , 1d3/2 ) = 3.62(20), respectively].
The results of both reactions, 9 Be(54 Ti, 52 Ca + γ)X and 50 Ti(d, 3 He) 49 Sc strongly support that in the case of the 9 Be(54 Ti, 53 Sc + γ)X reaction, a sizeable fraction if not all of the
strength to excited states in 53 Sc can be attributed to the removal of sd-shell protons.

5.7.3

Inclusive Parallel Momentum Distribution

As discussed in Section 3.4.4, the parallel momentum distributions of the knockout
residues provide information on the orbital momentum distribution of the removed nucleon. Although statistics prevented any meaningful extraction of parallel momentum
distributions for individual states, there was enough statistics to extract the experimental

115

Energy
(MeV)

n j

C2S

0.00
2.23
2.36
3.55
3.77
4.01
4.86
5.28
5.60
6.32
6.32
6.43
7.37
7.60
8.59
9.20

1f7/2
2s1/2
1d3/2
1d5/2
1d5/2
2s1/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2
1d5/2

1.91(20)
1.40(10)
3.62(20)
0.30
0.30
0.22
0.43
0.20
0.20
≤ 0.29
≤ 0.50
≤ 0.50
0.42
0.29
0.29
0.19

Table 5.10: Summary of experimental spectroscopic results for the 50 Ti(d, 3 He) 49 Sc reaction as reported by Doll et al. [87]. Strong transfer involving the sd-shell protons is seen in
the large spectroscopic factors for the positive parity 1/2+ and 3/2+ 2.3 MeV and 2.36 MeV
levels.
inclusive parallel momentum distribution for the 53 Sc-centered run. The inclusive momentum distribution includes the summation of all bound final states in the 9 Be(54 Ti,
53 Sc + γ)X

reaction. As per the conjecture of the previous two sections, the dominance of

sd-shell strength to excited states in 53 Sc, the inclusive momentum distribution is likely
a superposition of distributions associated with the removal of protons from the f7/2
and sd-shell orbitals. To check the experimental inclusive cross section for consistency, a
theoretical distribution consisting of f7/2 and sd components with a relative ratio extracted
from experiment—6.7(18) mb for f7/2 knockout (38 %) and 10.9(17) mb for sd-shell knockout (62 %)—was fit to the experimental longitudinal momentum distribution. Additional
details will follow.
The experimental parallel momentum distribution was extracted, as explained in
Section 4.5.3, from the position and angles of the knockout residues measured by the

116

position-sensitive crdc detectors, and then mapped from the focal plane to the target
through trajectory reconstruction. This differential energy distribution at the target was
then transformed to differential momentum through the use of a relativistic scale factor
(see Eqn. 4.18).
The theoretical distributions were calculated according to Section 3.4.4 using the same
eikonal Glauber model and S-matrices used in the calculation of the single-particle cross
sections. The distribution for sd-shell knockout was constructed from an admixture of
π(1d3/2 ) and π(2d1/2 ) orbitals. These contributions were weighted by their full spectroscopic strength: C 2 S(d3/2 )/C 2 S(s1/2 ) = 2. After combining the sd-distribution and the
f7/2 distributions, weighted according to their partial cross sections, the final admixture
was boosted to the lab frame and then folded with the profile of the incoming beam (see
Figure 5.10). With the theoretical and experimental distributions on the same footing, the
theoretical distribution was scaled to fit to the experimental distribution.
The resultant fit is seen in Figure 5.10, and the overlay is consistent with the removal
of protons from both the f7/2 and sd-shell (1d5/2 and 2s1/2 ) orbitals with the fraction
proposed before. The shape of the high-momentum component of the distribution places
the greatest constraint on the orbital angular momentum assignment of the removed
nucleon. The low momentum side is affected by a commonly observed artifact: a low
momentum tail formed from a shift of cross section from higher to lower momentum. This
effect is likely caused by events where there is significant energy exchange with the target
nucleus (stripping component) and it is not described by the simple Glauber model used
in the single-particle cross section and parallel momentum distribution calculations. This
effect is discussed further in Ref. [152].

5.8

Summary

In the 9 Be(54 Ti, 53 Sc+γ)X reaction, the 2110(3) keV, 3/2− → 7/2− ground-state transition
g.s
1
was identified and four more, previously unidentified γ rays were measured at 1111(2) keV,
117

1f 7/2 + sd

Events/bin

800
600
400
200
0
17.4

17.6

17.8

18.0

p (GeV/c)

18.2

18.4

18.6

Figure 5.10: Momentum profile of the incoming 54 Ti beam (triangles), a fit to the 54 Ti
momentum profile (used in the creation of the theoretical distribution; see Section 4.5.3)
and the measured parallel momentum distribution of 53 Sc (circles) compared to theoretical
calculations. The theoretical distribution is composed from the addition of f7/2 and
sd-shell contributions, as weighted by their measured partial cross sections: 38 % f7/2
(σ = 6.7 mb) and 62 % sd-shell (σ = 10.9 mb). The sd-shell distribution was created from
the addition of d3/2 and s1/2 contributions as weighted by their full spectroscopic strength,
C 2 S(d3/2 )/C 2 S(s1/2 ) = 2.
1273(2) keV, 1539(4) keV, and 2459(5) keV. Based on the overwhelming population of
excited states (≈ 60 %), in comparison to the negligible feeding predicted by shell model
calculations within the fp model space (see Table 5.3), these excited states were identified
as positive-parity states populated through the removal of deeply-bound sd-shell protons.
This picture is consistent with the results of the two-proton knockout reaction 9 Be(54 Ti,
52 Ca+γ)X and the regular-kinematics transfer reaction 50 Ti(d, 3 He) 49 Sc, and with the fit of

the experimental inclusive parallel momentum distribution with a calculated distribution
that includes an admixture of both f7/2 and sd shell components mixed according to their
extracted partial cross sections.
These results highlight the need for improved cross-shell interactions in the shell

118

model in this region, and the equivalent experimental need for quantitative spectroscopic
information on sd-shell configurations at low excitation energy.

119

Chapter 6
ONE-PROTON PICKUP REACTIONS
CENTERED AROUND 50FE
6.1

Motivation

9 Be-induced,

proton [60, 61, 118, 119] and neutron [153] pickup reactions, with fast

beams and inverse kinematics, have recently emerged as a spectroscopic tool for probing
particle-like states in exotic nuclei whose short half-lives and low production rates make
them inaccessible by traditional, low-energy transfer reactions using light beams (e.g.
(d, p), (3 He, d), etc.) in normal kinematics and conventional, stable targets. Because of
momentum matching considerations, the single-nucleon pickup reaction with fast beams
in inverse kinematics preferentially populates single-particle states in the projectile with
higher orbital angular momentum. As suggested in Refs. [60, 61, 153] and covered in
Section 3.5.1, this preferential population might be used to track the descent of higher orbital angular momentum intruder states, for example the ν(1f7/2 ) orbital whose migration
enables the hallmark 2p2h configurations in the ground state wavefunction of “island of
inversion” nuclei. Lastly, the developments with light, non-cryogenic, non-gaseous, targets
(e.g. 9 Be) offer experimental advantages over the earlier, more-complex, liquid 4 He targets
[118, 119] (see Section 3.5.2).
The motivation for this work is two-fold: first, we seek to further develop single-proton
pickup reactions as a spectroscopic tool by comparing the magnitude and population pattern of final states following proton-pickup from light targets in three f p shell nuclei. Proton and neutron pickup reactions using a beryllium target and neutron pickup reactions

120

using both a beryllium and carbon target have been performed along and near the N = 10
isotonic chain: 9 Be(20 Ne, 21 Na+γ)X [60], 9 Be(22 Mg, 23 Al+γ)X[61], 9 Be(22 Mg, 23 Mg+γ)X
[153], and 12 C(22 Mg, 23 Mg+γ)X [153]. These measurements have shown consistent agreement between the magnitudes and population of excited states as predicted by shell model
and reaction theory. In Ref. [60] (see Section 3.5.2) the population patterns observed in
the 9 Be(20 Ne, 21 Na + γ)X reaction were consistent with the traditional 20 Ne(3 He, d) 21 Na
reaction [119]. We seek to perform the first proton pickup reactions on f p shell nuclei:
9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X,
12 C(49 Mn, 50 Fe + γ)X,

and 12 C(50 Fe, 51 Co + γ)X. See Figure 6.1 for the location of these

reactions within the nuclear landscape. Further consistency between measured and predicted yields will show the applicability of single-nucleon pickup reactions across a wide
range of the nuclear chart.
Second, we seek to test the reaction theory mechanism through the use of two different
targets—a 188 mg/cm2 9 Be target offering well-bound (Sp = 16.9 MeV) protons, and a
72.8 mg/cm2 , 12 C target, offering well-bound (Sp = 16.0 MeV) protons—and through the
analysis of the measured, lab-frame momentum distributions of 49 Mn, 50 Fe, and 51 Co,
where differences from the expected momentum distributions, created from the convolution of the momentum width of the incoming beam with the differential momentum loss
in the target, can reveal higher order, possibly multi-step processes. The target structure
enters the calculation of the single-particle cross sections through the overlaps of the form
ˆ
φ(r) = A Z|a† | A−1 (Z-1)(I π ) and the associated spectroscopic factors, e.g. S = dr |φ(r)|2 .
See Section 6.4.2 and Section 6.5.2 for more details. Using two different targets, the effects
of the magnitude and inclusiveness over final target residue states can be roughly gauged.
Altogether, six different reactions were analyzed: 9 Be(48 Cr, 49 Mn+γ)X, 9 Be(49 Mn, 50 Fe
+ γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe,
51 Co + γ)X.

The work contained herein is broken down into two main sections, proton

pickup reactions induced by a beryllium target (Section 6.4) and proton pickup reactions

121

induced by a carbon target (Section 6.5). Within each section, the experimental results
are compared with theoretical predictions. The complete work is reviewed in the final
summary, Section 6.6.

Z = 28
Ni

Proton drip line

Co

51Co

Fe

50Fe

Mn

49Mn

Cr

48Cr

V
Ti
Figure 6.1: Nuclear chart showing the inclusive set of isotopes and reactions studied in this
work (yellow squares): T(48 Cr, 49 Mn + γ)X, T(49 Mn, 50 Fe + γ)X, and T(50 Fe, 51 Co + γ)X
for target choices T ≡ 9 Be and T ≡ 12 C. Stable nuclei are indicated by black squares. The
unbound nucleus 49 Co, revealing the close proximity of the proton drip line, is indicated
by a diagonal hash pattern, and the nearby Z = 28 magic number, corresponding to the
close of the πf7/2 orbital, is highlighted by a red rectangle. The direction of the pickup
reaction from 48 Cr to 49 Mn, from 49 Mn to 50 Fe, and from 50 Fe to 51 Co is indicated by
black arrows. Note that the final A + 1 pickup residues are more proton-rich than the
initial A projectile.

122

6.2

Background

A limited amount of spectroscopic information is available for the A + 1 proton-pickup
reaction products 49 Mn, 50 Fe, and 51 Co. For 49 Mn and 50 Fe, the available information includes the location and tentative spins and parities of the low-lying states as
probed through a variety of different reactions. A full evaluation for 49 Mn is found
in Ref. [154] and includes spectroscopy from the β + -decay of 49 Fe (following the β + delayed proton emission of 50 Co) [155] and γ ray spectroscopy following deep inelastic
12 C(40 Ca, 2npγ) 49 Mn

[156] and 24 Mg(28 Si, p2nγ) 49 Mn [157] reactions. The full collec-

tion of identified states span from the 5/2− ground state to a high-spin (31/2− ) state at
g.s
10.7 MeV.
Full evaluations for 50 Fe and 51 Co can be found in Ref. [158] and Ref. [159] respectively.
50 Fe has been studied through 50 Co β +

decay [155, 160], 51 Ni β + -delayed proton emission

[160], and intermediate energy Coulomb excitation [161]. Nine states up to 8.5 MeV have
been identified, including the first 2+ , 4+ , 6+ , and 8+ states. No spectroscopic information
1 1 1
1
was available for the most proton-rich of the isotopes 51 Co, potentially because of both
its low-production cross section and its low proton separation energy of only Sp = 88 keV
(Ref. [148]; from systematic trends).

6.3
6.3.1

Experimental Setup
Beam Production

The final cocktail beam, optimized for the production of 50 Fe, used a primary beam of
58 Ni

accelerated to 160 MeV/u. The primary beam was fragmented on a 893 mg/cm2

Be production target and the desired reaction products were selected by rigidity and
charge with the a1900 fragment separator, operating at 1 % momentum acceptance for all
measurements (reacted and unreacted settings). A 300 mg/cm2 nat Al wedge, inserted at

123

Primary beam 58 Ni at 160 MeV/u
Wedge 300 mg/cm2 27 Al (achromatic)
A1900 momentum acceptance 1 % for all measurements.
Cocktail composition and intensity 48 Cr at 2.5 × 103 pps;
2.0 × 103 pps; and 50 Fe at 0.5 × 103 pps

49 Mn

at

S800 optics mode Focus mode
Table 6.1: Experimental characteristics of secondary beam production for the proton pickup reactions 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X,
12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X.

the Image 2 position of the a1900 (see Figure 4.2) separated nuclei with similar rigidity
according to their nuclear charge via energy loss. See Section 4.2 for additional details.
A summary of the beam production settings and the final composition and component
rates of the cocktail beam used in this analysis is found in Table. 6.1. A two-dimensional
pid plot of xfp time-of-flight versus obj time-of-flight1 showing the composition of the
incoming cocktail beam is presented in Figure 6.2. The data shown is taken from a subset
of data collection runs using the 12 C target. The 12 C target data was taken later in the
experiment, after changes to the tof electronics. Further details will be covered in later
sections.

6.3.2

S800 Setup and Target Selection

The full set of s800 detectors and tof scintillators were individually calibrated in the
standard way. See Chapter 5 for the calibration procedures associated with the analysis
of the 9 Be(54 Ti, 53 Sc + γ)X knockout reaction. The crdc pads, critical for position and
momentum measurements, were adjusted for similar gain, bad pads were removed from
the center-of-gravity calculations of dispersive position, and position measurements were
calibrated using the aid of masks with known hole positions placed in front of each
1 See

Table 6.3 for an overview of the physical tof measurements corresponding to the
labels obj and xfp.
124

Figure 6.2: Particle identification spectra for an unreacted run (components of the incident
cocktail projectile beam are centered in the focal plane) on the 12 C target showing the
composition of the incoming cocktail beam. The upper pane shows a standard s800 pid
plot of the tof between the object scintillator (obj) and the back of the s800 focal plane
(E1 scintillator) versus energy loss in the focal plane ion chamber. See Table 6.3 for a
description of the obj, xfp, and OBJE1 labels given to the different tof measurements.
The lower pane is a two-dimensional tof spectrum of xfp versus obj time-of-flight that
can be used, like the one-dimensional difference spectrum shown in the bottom panel of
Figure 6.4, to select a single component of the incoming cocktail beam.

125

crdc. The tof of projectiles through the spectrograph were adjusted using the position
and angles of the projectiles in the focal plane to account for differing trajectories (see
Section 4.5.1).
Run-by-run calibrations were performed for the crdc non-dispersive position (to
account for changes in the drift gas mixture), ion chamber energy loss, and various timeof-flights (obj and xfp). With a large number of experimental changes throughout the
experiment (see Section 6.3.4), careful run calibrations ensured that the collection of gates
and calibrations done for one condition (e.g. xfp-obj tof before the addition of delays)
would still apply after changes to the experimental setup (e.g. after the addition of delays).
Three different targets were used in the experiment: 9 Be, 12 C and 93 Nb. Experimental
difficulties associated with the heavy 93 Nb target, including the increased production
of charge states, prevented its inclusion in the present analysis. The 9 Be target thickness was 188(4) mg/cm2 (as provided by the manufacturer). The 12 C target thickness
of 72.8(13) mg/cm2 was calculated from the weight, area, and thickness of the carbon
foil. The error associated with the thickness of the carbon target was calculated from
the associated errors in the geometric parameters as combined according to the standard
propagation of errors equation.

6.3.3

SeGA Setup

The standard sega “classic” configuration, as used for the 9 Be(54 Ti, 53 Sc + γ)X experimental setup, with two rings of detectors arranged at 37◦ and 90◦ with respect to the beam
axis (see Figures 4.7 and 4.5), was also used for the present experiment. Although 10 total
detectors can be mounted on the 90◦ ring, only 9 were available for this experiment. The
full set of 7 detectors were mounted on the forward ring.
Two different sources were used for the energy and efficiency calibration, 226 Ra and
152 Eu.

The energy and efficiency calibration procedure was covered before (Sections 4.4.3

and 4.4.4) and only details specific to or different in the present analysis will be covered

126

here. Separate energy calibrations were performed using both the 152 Eu and 226 Ra sources
and the results were tested against each other and absolute literature values. Both sources
provided similar energy residuals (sega calibrated energies minus true source energies) of
∼ 0.5 keV between 121.8 keV and 2447.7 keV, the maximum range covered by either 152 Eu
or 226 Ra sources. The Lorentz-boosted energies seen in the laboratory-frame following the
γ-decay from the in-flight 49 Mn, 50 Fe and 51 Co proton-pickup reaction products all lay
between this range. Energy calibrations were performed at the beginning, middle, and end
of the experiment to check for detector drifts and other changes over time. All consistency
checks passed.
The 152 Eu source activity was absolutely calibrated at date of issue; the 226 Ra sources
was not. To get accurate efficiencies across a wide energy range, the uncalibrated 226 Ra
source had to be included. The minimum to maximum energies emitted by 152 Eu that
were suitable for efficiency calibration were 121.8 keV to 1408.0 keV. For 226 Ra that range
was 768.4 keV to 2447.7 keV. Although 226 Ra and its decay products emit γ rays down to
186.2 keV, the mounting ring used for the 226 Ra source absorbed the lower energy γ rays.
This effect was particularly prominent for the 90◦ ring. Using the individual efficiencies of
each observed transition as calculated from the source full-energy peak areas, decay rate,
and expected number of emitted γ rays per decay, the efficiency curve of Equation 4.6
was fit. A customized fitter was written for root [149] that kept the parameters that
described the curve shape identical for the 152 Eu and 226 Ra sources (parameters b and c
of Equation 4.6), but allowed for two separate amplitude parameters, one for each source
(a226Ra and a152Eu ).
The 226 Ra amplitude a226Ra was allowed to vary in relation to the absolutely determined 152 Eu amplitude a152Eu . The 226 Ra fit was then rescaled according to the difference
in amplitude a152Eu /a226Ra . The final efficiency curves for both the 37◦ and 90◦ rings
are shown in Figure 6.3. Both the laboratory-frame and projectile-rest frame efficiency
curves (Lorentz-boosted efficiency for a sample velocity of β = 0.327) are displayed. The

127

Lorentz-boosted efficiency, as detailed in Section 4.4.4, includes the effects of the boost
of energy from the residue rest-frame to the laboratory frame and the forward bias of
angular emission as observed in the laboratory-frame. The angular contraction accounts
for the increased efficiency of the 37◦ ring in relation to the 90◦ ring. The final residuals
and function errors are small, showing the magnitude of the efficiency calibration across a
wide energy range.

6.3.4

Experimental Changes

This experiment was designed to probe the pickup reaction mechanism under the use of
different reaction targets, and the experimental differences associated with this change,
including the change of spectrograph rigidity to center the reaction products using the
different thickness targets, were expected. Additional unexpected changes, however,
complicated analysis. A chronographic list of the different experimental changes, with
the duration of data collection for each setting, the spectrograph rigidity, reaction setting,
and description of the change is shown in Table 6.2. The neutron pickup reactions lie
outside the scope of the original intent of the experiment and this thesis. The reaction
channel may form the basis of future work, but the smaller difference in rigidity between
the unreacted beam and the pickup reaction products for the neutron pickup reaction
(∆Z = 0) in comparison to the proton pickup reaction (∆Z = 1) will complicate the analysis
for heavier nuclei. The smaller difference of rigidity permits a greater proportion of the
unreacted beam to enter the focal plane.
Problems with timing drove many of the changes made during the course of the
experiment. There were two major problems. The first is shown in Figure 6.4. As
shown, the faster tof components of the cocktail beam were not detected, resulting in the
reduction in intensity and even non-observation of several cocktail-beam components. The
initial delay on the signal generated by the obj and xfp scintillators—see Table 6.3 for the
physical time-of-flights corresponding to the labels xfp, obj, and xfp-obj—was too short,

128

500

1500

2000

226Ra

scaled data
152Eu data
Efficiency fit to all data
226Ra residuals
152Eu residuals
Fit function error
Lorentz-boosted efficiency

3

Efficiency [%]

1000

2.5
2
1.5

37 degree ring

2.5
2
1.5

0.5

0.5

0

0

3.5

3.5

3

3

Efficiency [%]

3

1

1

2.5

2.5

2

2

1.5

1.5
1

1

90 degree ring

0.5

0.5
0

2500

Residuals

0

Residuals

3.5

0
0

500

1000

1500

2000

2500

Energy(keV)
Figure 6.3: Sega efficiencies for the 37◦ and 90◦ rings, as calculated from measured fullenergy peak yields extracted from a calibrated 152 Eu source and an uncalibrated 226 Ra
source. Both the lab efficiency curve and the Lorentz-boosted efficiency curve, for a γ ray
emitted by a projectile moving at β = 0.324, are shown. The assumed isotropic angular
distribution of γ rays emitted in the residue rest frame is biased to forward angles in the
lab frame because of the Lorentz boost, resulting in the perceived increase of efficiency for
the 37◦ ring.

129

Time Bρ
(hr) (Tm)

Setting

Change
Initial setup. This data was not used in the
current analysis.
Insertion of parallel-plate avalanche counters
(ppacs) at the intermediate image of the s800
analysis line.
Changed to the rigidity of the s800 spectrograph to center the neutron pickup products.
Returned the rigidity of the spectrograph to the
+1p reaction setting.
Increased the delay on the obj scintillator by
200 ns and the xfp scintillator by 50 ns.
Added tacs in parallel with the existing tdc
branch to processes the obj and xfp time-offlights. Changed the rigidity of the s800 to center +1n reaction products.
Change of target to 93 Nb. Corresponding
change of spectrograph rigidity.
Change of target to 12 C. Corresponding change
of spectrograph rigidity.
Change of spectrograph rigidity to center 1n
pickup reaction products.

1. 12.3

1.745

9 Be,

+1p

2. 16.7

1.746

9 Be,

+1p

3. 2.0

1.886

9 Be,

+1n

4. 2.9

1.745

9 Be,

+1p

5. 14.5

1.745

9 Be,

+1p

6. 13.9

1.886

9 Be,

+1n

7. 14.4

1.834

93 Nb,

8. 24.4

2.050

+1p
12 C, +1p

9. 2.4

2.050

12 C,

+1n

Table 6.2: Major experimental changes during the course of the experiment including
changes of spectrograph rigidity to center the new reaction channels and electronics
modifications. Only the one-proton pickup data on the 9 Be and 12 C targets are presented
in this thesis. The analysis of other reaction channels may form the basis of future work
but are outside the scope of this thesis. See Table 6.3 for a description of the tof labels.

130

Label

Description

XFP

Elapsed time between the master trigger (start) and a thin plastic
scintillator (xfp scintillator) located at the end of the A1900
fragment separator (stop).

OBJ

Elapsed time between the master trigger (start) and a thin plastic
scintillator (obj scintillator) physically located at the start of the
s800 analysis line (stop).

OBJE1

Identical to the obj time-of-flight with addition corrections for the
flight path of nuclei through the spectrograph. These corrections
depend on the measured angle and momentum of fragments
detected in the focal plane.

XFP-OBJ

tof difference between the obj and xfp scintillators. Calculated
in the analysis software.

DIA1

Elapsed time between the obj scintillator and the master trigger
(stop). The tof difference is measured by the tac branch and
reverses the regular trigger start and stop conditions.

DIA2

Elapsed time between the master trigger (start) and the xfp scintillator (stop). This tac time-of-flight uses the tac branch to
measure the tof but is otherwise identical to the xfp tof.

Table 6.3: Overview of the various timing measurements used in the pickup reaction
analysis. Time-to-amplitude converters (tacs), added in parallel to the existing tdc
branch, mapped in the data acquisition software to the DIA1 and DIA2 channels, were
added in mid-experiment, before the pickup reactions on the 12 C target. See Figure 6.4
for sample xfp, obj, and xfp-obj spectra.

and the signal generated by the scintillator photomultiplier tubes (pmts) lay outside the
400 ns coincidence gate opened by the master trigger. To present the entire cocktail beam
to the electronics, delays were added, 50 ns to the xfp signal and 200 ns to the obj signal,
pushing the full spread of tofs of the incoming beam within the coincidence window. The
juxtaposition of the tofs before and after the addition of the delays is shown in Figure 6.4.
Black lines show the tof before delays (summation of settings 2 and 4 of Table 6.3); red
lines show the tof after the delays (setting 5 of Table 6.3). The spectrograph was set to
observe the 1p pickup reaction products.
A splitting of the tof into two components was observed in most tof spectra (obj,

131

× 103

(1) OBJ

50
40
30

20

20

10
0

Counts

30

40

Counts

40

× 103

× 103

× 103

(2) XFP

50
40

700

900

30

20

20

10
500

30

10

10

0

0

0

Time-of-flight (arb. unit)

600

Before delays
After delays

× 103

× 103

(3) XFP - OBJ

Before delays
After delays

100

80

80

60

60

40

40

20

Counts

1400

Time-of-flight (arb. unit)

Before delays
After delays

100

1000

20

0

200

300

400

Time-of-flight (arb. unit)

0
500

Figure 6.4: Effect of the addition of delays to the xfp and obj scintillators. The new delays
shifted the measured time of cocktail beam components that previously lay outside the
time coincidence window opened by the master trigger within the coincidence window
permitting their detection. All three panes reflect the full sum of data taken on the 9 Be
target with the spectrograph set to measure the pickup reaction products. See Table 6.2
for the location of this experimental setting within the complete experiment timeline. The
xfp-obj spectrum was used for identification of the cocktail components for the proton
pickup reactions using the 9 Be target.

132

xfp, and xfp-obj) associated with the 9 Be target runs and was attributed to problems
in the Time-to-Digital Converter (tdc) branch receiving the signal from the obj and
xfp scintillators. This splitting widened the time resolution of each cocktail component
and led to significant overlap of similar A/Q nuclei in the one-dimensional xfp-obj and
two-dimensional xfp versus obj time spectra used to identify the incoming projectiles.
The tacs were added in parallel to the existing tdc branch, and the signals generated by
the obj and xfp scintillators were split and simultaneously processed by both branches.
The tac channels were mapped in software to the labels DIA1 and DIA2 (see Table 6.3). As
seen in Figure 6.5, the tacs significantly eliminated the broadening of the tof signal.

6.4
6.4.1

Proton Pickup Reactions: 9 Be target
Particle Identification

There are two separate particle identifications necessary to label the entire direct, protonpickup reaction channel: the identification of the incoming A Z projectile and the identification of the outgoing A+1 (Z+1) residue. The experimental difficulties outlined in
Section 6.3.4 complicated both identifications.
For the 9 Be pickup data, before the addition of the tacs, wide incoming time gates
were used to compensate for the effects of the tof broadening. See for example Figure 6.6
for the two-dimensional histograms of xfp time-of-flight versus obj time-of-flight (upper
panel) and the one-dimensional histogram of xfp-obj time-of-flight. The data shown was
taken from setting 5 of Table 6.2, i.e. after the addition of the delays to the obj and xfp
scintillators but before the addition of the tacs. These plots show unavoidable overlap
in the tof of the different cocktail components and therefore the difficulty of placing
gates that minimize contamination but that do not restrict the observed rate of the desired
reaction channel.
The width of the time gates was set by checking the gate placement against two-

133

Figure 6.5: Effect the time-to-amplitude (tac) converters on the tof and pid for components of the incoming cocktail beam. The upper panel shows a two-dimensional tof using
the tdcs installed before the start of the experiment. The lower panel shows the effect of
the tac branch added in parallel. The physical time-of-flight measured in both panes are
identical. Notice the cleaner separation of isotopes in the lower pane. The more diffuse
loci in the upper pane is caused by the tof broadening covered in Section 6.3.4.

134

component Gaussian fits to the xfp-obj spectra, and by looking for any positive correlation
between the width of the time gate and the accuracy of the inclusive cross section. The
final gate choice allowed cocktail beam components adjacent in tof to contaminate the
identification of the final heavy residue, see the top pane of Figure 6.8, but it ensured that
the final inclusive cross section would not have to be adjusted for artificial software cuts
to the observed rate of the incoming projectile. The addition of the tacs repaired these
problems, and clean tof gates could be used for identification of the cocktail components
impinging on the 12 C target.
There were several sources of contamination in the final pid of OBJE1 versus energy loss
in the ion chamber: contamination from one or more components of the unreacted beam
(e.g. for the 9 Be(48 Cr, 49 Mn + γ)X reaction, the 49 Mn pid is affected by contamination
from both the unreacted 48 Cr and 49 Mn isotopes), possible scattering of beam off the
beam blocker located immediately in front of the s800 focal plane (see the lower panel of
Figure 6.7), and tails of one-neutron knockout residues. These issues are shown in the
standard pid plot of Figure 6.6. The upper panel shows the plain pid before cleanup. The
lower panel shows the pid after cleanup. Following cleanup, unambiguous software gating
contours, free of most contamination, could be easily placed around the pickup reaction
product. Note that the same gating scheme was used for the data analysis on both the 9 Be
and 12 C targets (details follow).
To eliminate the contamination, all combinations of identifying spectra were tested.
These spectra included dispersive and non-dispersive position and angle in the focal plane,
tof (see Table 6.3 for a full list), and energy loss. The combination of xfp time-of-flight and
dispersive position (CRDC1 x-position), shown in the upper panel of Figure 6.7 eliminated
all sources of contamination. The pid before and after the application of this gate is
seen in the upper and lower panels of Figure 6.8. Any artificial cuts to the momentum
distribution caused by software cuts to the CRDC1 spectrum were handled during the
acceptance corrections covered in Section 6.4.3.

135

Figure 6.6: Time-of-flight broadening observed in in the 9 Be pickup data. The runs
summed above occurred after the addition of the delays but before the addition of the
tacs. The top pane shows the two-dimensional histogram of xfp versus obj time-of-flight;
the bottom plane is their difference. See Table 6.2 for the list of tof definitions. The
addition of the new electronics resolved the tof broadening. See Figure 6.5.

136

Figure 6.7: Contamination and cleanup of various components present in the pid histogram used to identify the pickup reaction products. The upper histogram of dispersive
position in the focal plane (CRDC1 x-position) versus xfp time-of-flight provided the best
method of discriminating between true A + 1 pickup events and contamination. The lower
panel of CRDC1 x-position versus dispersive angle shows contamination consistent with
scattering of beam in the focal plane. Assumed sources of contamination are labeled.

137

Figure 6.8: Particle identification plot for the 9 Be(48 Cr,49 Mn + γ)X reaction before and
after the application of the cleanup gating procedure shown in Figure 6.7. In the upper
pane, the pid is only gated upon the incoming tof and particle singles events. The
49 Mn residues are overwhelmed by contamination from components of the cocktail beam
(unreacted 48 Cr and 49 Mn isotopes that enter the focal plane at the edge of the acceptance)
and possible scattering off the beam blocker (see Figure 4.3). The lower pane shows the
pid after the application of the contour cleanup gate shown in Figure 6.7.

138

6.4.2

Theoretical Predictions

Theoretical predictions for the strength and pattern of population of states in the pickup
residues were taken from the multiplicative combination of single-particle cross sections
[162], spectroscopic factors [163], and a center-of-mass correction2 . The same approach
was used for the calculation of the 9 Be(54 Ti, 53 Sc + γ)X partial and inclusive cross sections
(see Equation 3.18). There are several assumptions guiding the pickup reaction calculation.
First, the pickup reaction was assumed to be direct. The direct nature is reflected in
the longitudinal momentum distribution whose broadening from the reaction processes,
because of the two-body the initial and final states (see Section 3.5.3), is assumed to be a
delta function. A increased width of the measured distribution relative to the calculated
distribution outlined above indicates possible deviations from the direct reaction process.
Second, it is assumed that the proton is transfered in a single step, from an initial
bound state in the 9 Be target, to a bound, single-particle state in the incident projectile
(48 Cr, 49 Mn and 50 Fe), creating a A+1 (Z + 1) pickup residue (49 Mn, 50 Fe, and 51 Co respectively). The 9 Be target, now missing a proton, is assumed to be left in one of three
states of 8 Li, the 2+ ground state, the 1+ excited state at 981 keV, and the 3+ state at
g.s.
1
1
2255(3) keV. With Sn (8 Li) = 2024 keV, the 3+ 8 Li state is unbound. These three states
exhaust most of the overlap between the 9 Be ground state and all final 8 Li states (spectroscopic factor sum rule). Using 9 Be(48 Cr, 49 Mn + γ)X as an example to illustrate the
reaction processes for all three pickup reactions using the 9 Be target, we have the final
ccba form: 48 Cr(9 Be, 8 Li(J π )∗ ) 49 Mn(I π )∗ .
The spectroscopic factors were calculated by B.A. Brown [163] using the NuShellX
2 The

com correction for a single-nucleon pickup reaction, which arises from shell
model calculations that use harmonic oscillator wave functions relative to the A rather
than A + 1 com are of the form
A N
,
A+1
where N = n + (n is defined to start at 0, rather than the convention of defining n to start
at 1 as adopted in this thesis). For further details, see Section 3.2.6 and Equation 3.13.
139

code [66] with the gxpf1a [58] effective interaction. The f p shell calculations assume a
40 Ca

core and a proton/neutron 1f7/2 , 2p3/2 , 1f5/2 , and 2p1/2 valence space. The result

of these calculations are shown in Figures 6.9, 6.10, and 6.11. These figures show the
spectroscopic strength, energies, and quantum numbers associated with final states in
49 Mn, 50 Fe, and 51 Co (respectively).

The data was drawn from Tables A.1, A.2, and A.3. In

all three cases, the spectroscopic strength is spread thinly over a multitude of states above
and below the proton separation energy. For the 9 Be(48 Cr, 49 Mn + γ)X reaction, most of
the overlap, with a spectroscopic factor of C 2 S[48 Cr(0+ ) ⊗ 1f7/2 ]7/2− = 0.425 occurs with
the 7/2− state calculated to lie at 0.219 MeV and experimentally observed at 261 keV3 .
1
Negligible overlap occurs with the ground state (shown in bold in the aforementioned
tables) and therefore the expected yield (partial cross section) to this state is also expected
to be near zero. Most of the summed overlap proceeds to states above the 49 Mn proton
separation energy of Sp = 2.1 MeV (the neutron separation energy is much higher, i.e.
Sn = 16.3 MeV).
For the 9 Be(49 Mn, 50 Fe + γ)X reaction, because of the higher proton separation enˆ
ergy (Sp = 4.2 MeV) of the even-even nucleus, most of the 49 Mn|aj π | 50 Fe(I π ) overlap4
occurs with bound states in 50 Fe. The largest overlaps occurs with the first 2+ state,
1
C 2 S[49 Mn(5/2− ) ⊗ 1f7/2 ]2+ = 1.145 and the 1+ C 2 S[5/2− ⊗ 1f7/2 ]1+ = 0.530, calculated to
1
1

1

lie at 3.349 MeV (the state was not observed experimentally; for additional details, see
Section 6.4.4). The spectroscopic strength slowly declines thereafter at higher energies.
3 Pickup

and knockout spectroscopic factors are related through the sum rule of Equation 3.12. The number of hole-like states for orbital n j in the initial A nucleus includes
a multiplication by 2j + 1, e.g. 8 for proton transfer to the 1f7/2 orbital in the 9 Be(48 Cr,
49 Mn + γ)X and 12 C(48 Cr, 49 Mn + γ)X reactions. The multiplication of 8 and 0.425 gives
a value of 3.40, close to the naive expectation of 4 hole-like states in the 1f7/2 orbital for
48 Cr.
4 The

ˆ
ˆj
overlaps 49 Mn|aj π | 50 Fe(I π ) or 50 Fe(I π )|a†π | 49 Mn 49 Mn are identical within

a multiplicative phase (in general φa |A|φb ∗ ≡ φb |A† |φa where ∗ indicates complex
conjugation). This phase disappears in the square of the matrix element. The definition
does not affect the calculation of the spectroscopic factor.
140

Note, there are several different ways to couple the angular momentum of the initial
49 Mn

ground state spin of 5/2− and the total angular momentum of the added nucleon
g.s

(proton 1f7/2 , 2p3/2 , 1f5/2 , and 2p1/2 orbitals) to the final 50 Fe(I π ) angular momentum.
Each of these different couplings has a separate associated spectroscopic factor. For
the lowest states included in the reaction calculations the total spectroscopic factor is
dominated by one configuration. This configuration is listed (nlj column; Tables A.1, A.2,
and A.3). For example, the first 2+ state, predicted to be at 0.788 MeV (experimentally
1
observed to lie at 0.765 MeV) is dominated by the [5/2− ⊗ 1f7/2 ]2+ configuration with a
g.s
1

spectroscopic factor of C 2 S = 1.145. The other configurations, involving the 2p3/2 , 1f5/2 ,
and 2p1/2 orbitals have associated spectroscopic factors of 0.005, 0.000, and 0.007.
Lastly, for 51 Co, only one state is known to be proton-bound, with C 2 S[50 Fe(0+ ) ⊗
1f7/2 ]7/2− = 0.246. The proton separation energy for this very proton-rich nucleus is
estimated from mass systematics to be Sp = 88 keV.
The single-particle cross sections were calculated by Tostevin [162], as detailed in
Section 3.5.3 in the Coupled Channels Born Approximation (ccba) framework using the
reaction code fresco. The calculations were post-form and fully finite range, and were
evaluated at the expected 9 Be 188 mg/cm2 mid-target energies of 50.7 MeV/u (9 Be(48 Cr,
49 Mn + γ)X),

52.8 MeV/u (9 Be(49 Mn, 50 Fe + γ)X), and 54.8 MeV/u (9 Be(50 Fe, 51 Co + γ)X).

As mentioned, it is assumed that the projectile and target are bound systems and the
transfer occurs from one bound state to another.
The calculation of the single-particle cross sections requires the overlaps and spectroscopic factors of the target and projectile, the proton-core, relative motion wave functions
of the target and projectile, and the absorptive optical nuclear interactions (calculations
details follow).
ˆ
The target overlaps φ(r) = 9 Be|a† | 8 Li(I π ) and spectroscopic factors S = dr |φ(r)|2 ,
for the three included 8 Li final states, the 2+ ground state, the 1+ state at 980.80(10 keV)
1
1
(literature value), and the 3+ state at 2255(3) keV, were calculated using Variational
1

141

Sp = 2085 keV

1f7/2
1f5/2
2p3/2
2p1/2

Figure 6.9: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(48 Cr, 49 Mn + γ)X reaction. All values, i.e. energies, spectroscopic factors, spins, and parities, are taken from
the shell model calculations of Brown [163]. Only levels with spectroscopic strength of
C 2 S ≥ 0.04 are displayed. The spin and parity of the final state 49 Mn(I π ) is indicated
by the color of the bars. See the figure label. The orbital n j of the transfered proton
(in 49 Mn) is illustrated through the shapes of the bar caps: circle (1f7/2 ), square (1f5/2 ),
diamond (2p3/2 ), and pentagon (2p1/2 ). The proton separation energy of Sp = 2085(5) keV
is indicated by a dotted line. The states used in the calculation of the theoretical partial
and inclusive cross sections (see Table 6.7) are noted by yellow-filled arrows. The cross
section to the ground state is nearly zero. The figure data is taken from Table A.1.

Monte Carlo (vmc) wave functions, as provided by Wiringa [164]. These calculations
used a combination of the two-body Argonne v18 Hamiltonian [3] and the three-body
Urbana IX [165] Hamiltonian. The final vmc overlaps were then fit by π(1p3/2 ) and
π(1p1/2 ) single-particle wave wave functions calculated in a Woods-Saxon potential. The
final spectroscopic factors, involving the integrated norm of the overlap functions are
listed in Table 6.7. The summed spectroscopic strength for the 1p3/2 and 1p1/2 orbital
is 0.97 and 0.36. The total sum of 1.33 is smaller than the inclusive value of 2. The
remaining overlap of 0.67 is spread to higher unbound states that were not included in
the reaction calculations. Contrast the 9 Be target with the 12 C target, where nearly all of
the spectroscopic overlap of 4 (see Section 6.5.2) is accounted for by the ground state and

142

Sp = 4150 keV

1f7/2
1f5/2
2p3/2
2p1/2

Figure 6.10: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(49 Mn, 50 Fe + γ)X reaction. All values, i.e. energies, spectroscopic factors, spins, and parities, are taken from
the shell model calculations of Brown [163]. Only levels with spectroscopic strength of
C 2 S ≥ 0.04 are listed. The spin and parity of the final state 50 Fe(I π ) is indicated by the
color of the bars. See the figure label. The orbital n j of the transfered proton (in 50 Fe) is illustrated through the shapes of the bar caps: circle (1f7/2 ), square (1f5/2 ), diamond (2p3/2 ),
and pentagon (2p1/2 ). The proton separation energy of Sp = 4150(60) keV is denoted by
a dotted line. The states used in the calculation of the theoretical partial and inclusive
cross sections (see Table 6.8) are noted by yellow-filled arrows. Two possible fragmented
6+ states are considered based upon the proximity of their energies of 3.257 MeV and
2.959 MeV to the literature value of 3.159 MeV. Note the high degree of fragmentation to
states above and below the proton separation energy (compare this figure to Figures 6.9
and 6.11). The figure data is taken from Table A.2.

143

Sp = 88 keV

1f7/2
1f5/2
2p3/2
2p1/2

Figure 6.11: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(50 Fe, 51 Co + γ)X. All
values, i.e. energies, spectroscopic factors, spins, and parities, are taken from the shell
model calculations of Brown [163]. Only levels with spectroscopic strength of C 2 S ≥ 0.04
are listed. The spin and parity of the final state 51 Co(I π ) is indicated by color. See the
figure label. The orbital n j of the transfered proton (in 51 Co) is illustrated through the
shapes of the bar caps: circle (1f7/2 ), square (1f5/2 ), diamond (2p3/2 ), and pentagon (2p1/2 ).
The low proton separation energy of Sp = 88 keV is denoted by a dotted line. The proton
separation energy of Sp = 2085(5) keV is indicated by a dotted line. Only the ground state,
as indicated by a yellow-filled arrow, was used in the calculation of the theoretical partial
and inclusive cross sections (see Table 6.9). The figure data is taken from Table A.3.
two excited states of 11 B.
ˆ
Projectile overlaps, e.g. φ(r) = 48 Cr|a| 49 Mn(I π ) and spectroscopic factors for the
9 Be(48 Cr, 49 Mn + γ)X reaction,

were calculated within the shell model using the NuShellX

code and gxpf1a effective interaction. The proton-core p + A Z relative-motion wave functions were calculated in the same manner as the p + 8 Li∗ relative-motion wave functions: in
a Woods-Saxon potential with a spin-orbit term. The depth of the Woods-Saxon potential
was adjusted to the physical separation energy, e.g. Sp (9 Be) + Ex (49 Mn∗ ). Standard values
S.O. = 6.0 MeV, r S.O. = 1.25 fm, and aS.O. = 0.7 fm were used
of r0 = 1.25 fm, a = 0.7 fm, V0
0

for all three 9 Be cases.
The final component, the absorptive optical nuclear interactions, were calculated,

144

Target (9 Be → 8 Li∗ )

49 Mn

σs.p
s.p.
i σi

π j

res.
Sp (eff.)
(MeV)

Jπ
8 Li)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

σ s.p.
(mb)

0.262

1f7/2

1.813

2+
1+
3+

16.888
17.870
19.143

-14.803
-15.785
-17.058

2.212
0.882
0.811

3.904

1/2−

1.703

1f7/2

0.382

2+
1+
3+

16.888
17.870
19.143

-14.803
-15.785
-17.058

0.021
0.014
0.004

0.039

3/2−

1.741

1f7/2

0.344

2+
1+
1
3+
1

16.888
17.870
19.143

-14.803
-15.785
-17.058

0.049
0.020
0.014

0.082

Iπ

Ex
(MeV)

7/2−

(mb)

Table 6.4: Additional details on the calculation of the single-particle cross sections for
the 9 Be(48 Cr, 49 Mn + γ)X reaction evaluated at a mid-target energy of 50.7 MeV/u. The
complete single-particle cross section for the transfer of a proton from the target to the
orbital n j in 48 Cr, forming the pickup residue 49 Mn, includes a summation over target
final states. The pickup residue 49 Mn is labeled by its quantum numbers I π , excitation
final
res.
res.
energy Ex , and its effective separation energy of Sp (eff) = Sp (49 Mn)g.s − Ex . The
res.
ground state separation energy for 49 Mn is Sp (49 Mn)g.s = 2.085 MeV. Each target final
state—for the target residual 8 Li this includes the 2+ , 1+ , and 2+ states—is labeled by
g.s. 1
2
π , effective separation energy S tar. (eff) = S (9 Be)
8
its quantum numbers J
p
g.s. + Ex ( Li), and
p
res.
tar.
reaction Q value Q = Sp (49 Mn)g.s. − Sp (eff). The single-particle cross section to the

ground state of 49 Mn was not calculated because of the extremely small spectroscopic
factor C 2 S = 0.002 seen in Figure 6.9 and Table A.1. The 7/2− energy of 262 keV was taken
1
from experiment. The 1/2− and 3/2− energies were not observed in this experiment and
1
1
literature values do not exist. Their energies were taken from shell-model calculations, see
Table A.2, and are italicized to note this origin.
as done for the analysis of the 9 Be(54 Ti, 53 Sc + γ)X reactions (see Section 3.4.2.3), using
eikonal S-matrices that include the double folding of the proton and neutron densities of
the projectile (taken from Hartree-Fock calculations using the SkX Skyrme Hamiltonian)
and the target (estimated by a Gaussian) with an effective NN interaction. Folding all
these components up within the ccba calculations gives the single-particle cross sections
shown in Tables 6.4, 6.5, and 6.6.
Combining the spectroscopic factors of Figures 6.9, 6.10, and 6.11 with the single-

145

Target (9 Be → 8 Li∗ )

50 Fe

σs.p
s.p.
i σi

π j

res.
Sp (eff.)
(MeV)

Jπ
8 Li)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

σ s.p.
(mb)

0

1f7/2

4.15

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.020
0.027
0.007

0.054

2+

1.703

1f7/2

3.385

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.210
0.083
0.078

0.371

4+

1.851

1f7/2

2.299

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.370
0.147
0.138

0.655

6+

3.159

1f7/2

0.991

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.516
0.204
0.189

0.909

2+

2.672

1f7/2

1.478

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.201
0.080
0.074

0.355

2+

3.044

2p3/2

1.106

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.011
0.004
0.003

0.018

1+

3.349

1f7/2

0.801

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.118
0.047
0.043

0.208

3+

3.398

2p3/2

0.752

2+
1+
1+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.014
0.006
0.004

0.023

3+

3.544

2p1/2

0.606

2+
1+
3+

16.888
17.870
19.143

-12.738
-13.720
-14.993

0.011
0.008
0.002

0.021

Iπ

final
Ex
(MeV)

0+

(mb)

Table 6.5: Additional details on the calculation of single-particle cross sections for the
9 Be(49 Mn, 50 Fe + γ)X reaction evaluated at a mid-target energy of 52.8 MeV/u. For additional information on the column labels, please refer to the caption of Table 6.4. The
ground-state proton separation energy of 50 Fe is 4.150 MeV. Listed energies are from
literature when available. Exceptions, e.g. the 2+ state at 2.672 MeV, the 2+ state at
2
3
3.044 MeV, and the 1+ and 3+ states, were taken from shell model calculations and are
italicized (see Table A.2). The two fragmented 6+ states of Table A.2 are calculated at the
experimental energy of 3.544 MeV.

146

Target (9 Be → 8 Li∗ )

51 Co

Iπ

Ex
(MeV)

0+

0

σs.p

π j

res.
Sp (eff.)
(MeV)

Jπ
8 Li)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

1f7/2

0.164

2+
1+
3+

16.888
17.870
19.143

-16.724
-17.706
-18.979

1.632
0.642
0.595

s.p.
i σi

σ s.p.
(mb)

(mb)
2.869

Table 6.6: Additional details on the calculation of the single-particle cross sections for
the 9 Be(50 Fe, 51 Co + γ)X reaction evaluated at a mid-target energy of 54.8 MeV/u. For
additional information on the column labels, please refer to the caption of Table 6.4. The
ground-state separation energy of 51 Co is 88 keV. No excited states were observed and
therefore only the partial cross section to the ground state was calculated.
Efinal
Shell model
(MeV) configuration

7/2−
1/2−
3/2−

0.272
1.703
1.741

COM

corr.

[0+ ⊗ 1f7/2 ]
[0+ ⊗ 2p1/2 ]
[0+ ⊗ 2p3/2 ]

0.940
0.940
0.940

C 2 Ss.m.

σs.p.
(mb)

σtheory
(mb)

0.425
0.039
0.373

3.904
0.245
0.082

1.56
0.01
0.03

Inclusive sum:

Jπ

1.60

Table 6.7: Theoretical predictions of the partial and inclusive cross sections for the
9 Be(48 Cr, 49 Mn + γ)X reaction. Transitions corresponding to the decays of the 1/2− and
1
3/2− states were not observed in the current experiment and literature values values do
1
not exist. Italics indicate that shell model energies were used. A statistically insignificant
population of the 49 Mn ground state is predicted because of the calculated, near zero
C 2 S(48 Cr → 49 Mng.s. ) = 0.002 spectroscopic factor of Table A.1.

particle cross sections of Tables 6.4, 6.5, and 6.6 yields the theoretical predictions for
the partial and inclusive cross sections for the 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe +
γ)X, and 9 Be(50 Fe, 51 Co + γ)X reactions. The detailed comparison of the theoretical and
experimental results are shown in the analysis of the inclusive (Section 6.4.3) and partial
cross sections (Section 6.4.5).

6.4.3

Measured Inclusive Cross Sections

Following the cleanup procedure of Section 6.4.1, the number of pickup reaction products
could be accurately counted. From the number of pickup residues with respect to the
147

Jπ

Efinal
(MeV)

Shell model
configuration

COM

0+
2+
4+
6+
6+

0
0.765
1.851
3.159
3.159

[5/2− ⊗ 1f5/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]

2+
2+

2.672
3.044

1+
3+
3+

3.349
3.398
3.544

C 2 Ss.m.

σs.p.
(mb)

σtheory
(mb)

0.941
0.941
0.941
0.941
0.941

0.033
1.145
0.208
0.023
0.371

0.054
0.371
0.655
0.909
0.909

0.00
0.40
0.13
0.02
0.32

[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 2p3/2 ]

0.941
0.941

0.194
0.232

0.355
0.018

0.06
0.00

[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 2p3/2 ]
[5/2− ⊗ 2p1/2 ]

0.941
0.941
0.941

0.529
0.309
0.214

0.208
0.023
0.021

0.10
0.01
0.00

Inclusive sum:

1.05

corr.

Table 6.8: Theoretical predictions of the partial and inclusive cross sections for the
9 Be(49 Mn, 50 Fe+γ)X reaction. The 0+ ground state, 2+ , and 4+ states were observed in the
g.s.
1
1
50 Fe residue rest-frame γ-ray spectrum of Figure 6.16. For the higher
experiment. See the
π
spin states, i.e. J π = 2+ , 4+ , 6+ , several different couplings of the form [5/2− (49 Mng.s. ) ⊗ jp ]
are possible depending on the spin and parity of the orbital in 49 Mn accepting the transfered proton (the f p model space allows for 1f7/2 , 2p3/2 , 1f5/2 , and 2p1/2 ; see text for
further details). For all cases, however, one configuration dominates the spectroscopic
strength. This configuration is listed. There are two 6+ fragments predicted by shell model
about 500 keV apart. The reaction calculation was done for each at the experimental
energy.

Efinal
Shell model
(MeV) configuration

7/2−

0

[0+ ⊗ 1f7/2 ]

COM

corr.
0.942

C 2 Ss.m.

σs.p.
(mb)

σtheory
(mb)

0.246

2.869

0.66

Inclusive sum:

Jπ

0.66

Table 6.9: Theoretical predictions of the partial and inclusive cross sections for the
9 Be(50 Fe, 51 Co + γ)X reaction.

148

σtheory
(mb)

Reaction
9 Be(48 Cr, 49 Mn + γ)X
9 Be(49 Mn, 50 Fe + γ)X
9 Be(50 Fe, 51 Co + γ)X

σexp.
(mb)

1.60
1.05
0.66

1.63(8)
1.38(7)
0.57(8)

Table 6.10: Inclusive cross sections for the 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X,
and 9 Be(50 Fe, 51 Co + γ)X reactions. The predictions for the 9 Be(48 Cr, 49 Mn + γ)X and
9 Be(50 Fe, 51 Co + γ)X reactions agree with calculated values. The 9 Be(49 Mn, 50 Fe + γ)X
theoretical result under-predicts the measured value. This discrepancy may be associated
with the difficulty of including the wide spread of spectroscopic strength observed below
Sp (50 Fe) in the calculations.
number of incoming projectiles, as measured in one of four unreacted runs5 on the 9 Be
target, and with respect to the target number density (see Equation 4.17), inclusive cross
sections were calculated for the 9 Be(48 Cr, 49 Mn+γ)X, 9 Be(49 Mn, 50 Fe+γ)X, and 9 Be(50 Fe,
51 Co + γ)X

reactions. The results are summarized in Table 6.10 which also include the

theoretical results of Section 6.4.2. The magnitude of the inclusive cross sections for the
9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X,

and 9 Be(50 Fe, 51 Co + γ)X reactions, of order

σinc ∼ 1 mb, and the manner in which the inclusive cross sections decrease with respect
to increasing proton asymmetry along the N = 24 isotonic chain, are similar in behavior
to the proton pickup reactions measured by Gade, et al. across the N = 10 isotonic chain:
9 Be(20 Ne, 21 Na + γ)X

[60] with σinc = 1.85(12) mb and 9 Be(22 Mg, 23 Al + γ)X [61] with

σinc = 0.54(5) mb.
To ensure the consistency of experimental conditions between the reacted and unreacted measurements, only data runs immediately adjacent in time were used. The
only exception was for the 9 Be(50 Fe, 51 Co + γ)X reaction. Because of the low production
rate of 51 Co, two, rather than one, neighboring reacted runs were summed. Acceptance
losses were only seen in the dispersive direction and were estimated by fits to the dta
5 Recall

that for an unreacted run the spectrograph’s field is set to center the incoming
beam passing through the target in the focal plane. For a reacted run, the spectrograph is
set to center the pickup reaction product.
149

distribution. See Section 5.6.1 for similar analysis for the 9 Be(54 Ti, 53 Sc + γ)X reaction.
The contributions to the final, weighted runs are shown in Figures 6.12, 6.13, and 6.14.
The consistency of the separate cross section measurements, whose weighted sums form
the final reported values, with each other show that the differences associated with the

Cross section (mb)

different experimental settings have been accurately accounted for.

Cross section measurement
Weighted average
Error on weighted average

2.3
1.9
1.5
1.1
0

1

2

3

4

5

Data set
Figure 6.12: Cross section measurements for the 9 Be(48 Cr, 49 Mn + γ)X reaction. Each
cross section measurement is associated with a separate unreacted run. The consistency
of the measurements with each other show the consistency of the separate calibrations
applied to each different experimental setting.

The comparison of experiment and theory, contrasted in Table 6.10, shows good
agreement between the theoretical predictions and experimental measurements of the
inclusive cross section for the even-even pickup reactions, i.e. 9 Be(48 Cr, 49 Mn + γ)X and
9 Be(50 Fe, 51 Co+γ)X. The theoretical result for the 9 Be(49 Mn, 50 Fe+γ)X reaction, although

similar in magnitude, under predicts the observed inclusive cross section. This difference
is likely caused by the limited number of fragmented states with significant spectroscopic
strength included in the final theoretical calculations.

150

Cross section (mb)

Cross section measurement
Weighted average
Error on weighted average

2

1.6

1.2

0.8
1

2

Data set

3

Cross section (mb)

Figure 6.13: Cross section measurements for the 9 Be(49 Mn, 50 Fe+γ)X reaction. Each cross
section measurement is associated with a separate unreacted run. The consistency of the
measurements with each other show the consistency of the separate calibrations applied
to each different experimental setting. Only three runs were summed, in comparison
to the 9 Be(48 Cr, 49 Mn + γ)X and 9 Be(50 Fe, 51 Co + γ)X reactions, because of pid problems
associated with Setting 5 of Table 6.2.

Cross section measurement
Weighted average
Error on weighted average

1.2

0.8

0.4

0

1

2

3

Data set

4

Figure 6.14: Cross section measurements for the 9 Be(50 Fe, 51 Co + γ)X reaction. Each cross
section measurement is associated with a separate unreacted run. The consistency of the
measurements with each other show the consistency of the separate calibrations applied
to each different experimental setting.

151

6.4.4

Observed Transitions

The residue rest-frame and laboratory spectra for 49 Mn and 50 Fe are shown in Figures
6.15 and 6.16. No transitions were observed for 51 Co and its spectrum is not shown.
The Doppler-reconstructed (residue rest-frame) spectra, shown in the upper frames of
these figures, are used, as in the case of the 9 Be(54 Ti, 53 Sc + γ)X reaction, to extract the
transition energies and partial cross sections of the projectile-like reaction products. The
laboratory frame spectra, shown in the bottom panes, are used to extract the partial cross
sections to the final states of the target, stripped of a proton by the heavy projectile. For
the 9 Be target, the proton pickup reaction leads to one of several 8 Li(J π ) states. The target
final states are included in the dwba analysis outlined in Section 3.5. Only three excited
states are considered in the sum: the 2+ ground state, the first 1+ state at 980.80(10) keV
g.s
1
(leading to a Eγ = 980.80(10) keV transition), and the first 3+ state at 2255(3) keV (leading
1
to a Eγ = 2255(3) keV transition) [166]. The 1+ state was observed by Ref. [60] in the
1
laboratory-frame spectra coincident with the 21 Na pickup residue. These transitions were
not observed in the current, lower-statistics experiment.
Because the Lorentz boost (see Equation 4.3) depends upon the angle of emission, restframe energies are boosted to different energies in the sega 37◦ and 90◦ rings. The differences in the Lorentz boost accounts for the split of the Doppler-reconstructed 260.8(11) keV
transition, shown in the upper pane of Figure 6.15 into the two peaks in the lower pane.
The measured transition energies, which agree with literature (see Table 6.11), were extracted from spectra with optimized target position ztar and beam velocity β parameters.
See Section 5.6.2 for an overview of the method. 48 Cr, with a fast 4+ → 2+ transition of
1
1
1106.30(13) keV and 1.23(12) ps [167] provided the best reference to fix the target position
for both the 9 Be(48 Cr, 49 Mn + γ)X and 9 Be(49 Mn, 50 Fe + γ)X reactions. For a beam velocity
of β ∼ 0.3 or 0.1 mm/ps, a 1.23 ps lifetime corresponds to an average path length of an
excited nucleus before decay of 0.12 mm. Thus, for the 188 mg/cm2 , 1.1 mm target, 48 Cr
decays, on average, roughly in the middle of the target.

152

The spectra of each of the pickup reaction products—49 Mn, 50 Fe, and 51 Co—will be
addressed individually.
In the 49 Mn rest-frame spectrum, shown in the upper panel of Figure 6.15, only one
(−)

transition is seen, the 7/21 → 5/2− decay. The measured energy of 260.8(11) keV agrees
g.s
with the compiled literature value of 261.38(13) keV [154]. Other higher-energy and higher(−)

spin states, like the 9/21 state at 1059.18(24) keV (Eγ = 798.28 keV) and 1541.31(25) keV
(−)

(Eγ = 482.19, 1278.8 keV) [154] were not observed. The strong population of the 7/21

state is quantified with a measured partial cross section in Section 6.4.5. No 8 Li transitions
were observed in the laboratory-frame spectrum (lower pane) suggesting a low cross
section for the population (statistics were to low to assert a conclusion). Both displayed
spectra were created using a tight contour gate on the pid spectrum (with additional gates
selecting particle-singles, selecting incoming 48 Cr, and eliminating contamination). The
tight gate decreases contamination from neighboring 48 Cr (the 752.19(11) keV 2+ → 0+
g.s.
1
is barely visible in the upper pane of Figure 6.15) and 47 Cr but decreases the total peak
area in the 260.8(11) keV transition.
Two transitions were seen for 50 Fe at 765.6(20) keV and 1086.1(51) keV. Both transitions agree with the literature values of 764.9(3) keV and 1086.6(3) keV [158] (see Table 6.11). These transitions correspond to the 2+ → 0+ and 4+ → 2+ decays respectively.
g.s.
1
1
1
Similarly to the 9 Be(48 Cr, 49 Mn + γ)X reaction, a tight contour gate was used. For 50 Fe
this was used to eliminate contamination from 49 Mn and 48 Mn. No 8 Li transitions were
observable in the rest frame spectrum.
No γ rays were observed for 51 Co. The statistics for 51 Co are much lower than either
50 Fe

or 49 Mn because of both the smaller incident rate of incoming parent 50 Fe (as

compared to the rates of 48 Cr and 49 Mn; see Table 6.1), because of the comparatively lower
inclusive cross section for the production of this proton-rich nucleus—carrying through
with the simple arithmetic shows that the total production of 51 Co in pps is 14 times less
than the production of 49 Mn and 10 times less than the production of 50 Fe—and because

153

Counts / 8 keV

β = 0.273

120
100
80
60
40

48Cr contamination

20
00

Counts / 8 keV

49Mn: rest frame

260.8(11) keV

140

200

400

600

800

1000

Energy (keV)

40

1400

49Mn: lab frame

Lorentz-boosted
transitions for the
37o and 90o rings

50

1200

β=0

30
20

Expected 8Li transition

10
00

200

400

600

800

Energy (keV)

1000

1200

1400

Figure 6.15: Rest-frame and laboratory frame spectra for 49 Mn for the 9 Be(48 Cr, 49 Mn +
γ)X reaction. The observed 260.8(11) keV transition, corresponding to the decay of the
(−)

first 7/21 state to the ground state is shown. The energy of the 260.8(11) keV γ rays
emitted is boosted from the residue rest frame to the laboratory frame according to
Equation 4.3. Because the Lorentz-boost depends upon the emission angle, the transition is
observed as two different energies in the laboratory frame. This effect is seen in the bottom
panel. Of additional note, the excited states of the 9 Be target after proton removal, e.g.
Eγ (9 Be → 8 Li∗ ) = 981 keV (indicated by a black arrow), are not observed. Contamination
from 48 Cr is observed in the 49 Mn rest-frame spectrum (at Eγ ≈ 750 keV).

154

Counts / 16 keV

40

765.6(20) keV

50Fe: Rest frame

β = 0.284

35
30
25
20

1086.1(51) keV

15
10
5
00

500

25

1000

Energy (keV)

20

2000

50Fe: Lab frame

B

β=0

{

Counts / 16 keV

{

A

1500

Expected 8Li transition

15
10
5
00

500

1000

Energy (keV)

1500

2000

Figure 6.16: Rest-frame and laboratory frame spectra for 50 Fe for the 9 Be(49 Mn, 50 Fe
+ γ)X reaction. The observed 765.6(20) keV and 1087.1(51) keV transitions (upper panel),
corresponding to the 2+ → 0+ and 4+ → 2+ decays are indicated. The 765.6(20) keV
g.s.
1
1
1
residue rest-frame transition is Doppler-boosted to energies indicated by the labels A (90◦
ring) and B (37◦ ring) in the laboratory spectrum (lower panel). The 37◦ ring, Dopplerboosted transition (B) overlaps the expected 8 Li 981 keV transition. As in the 9 Be(48 Cr,
49 Mn + γ)X case (see Figure 6.15), the decays of the 8 Li 1+ and 3+ states at E = 981 keV
γ
1
1
and 2255 keV are not observed.

155

Isotope

Transition

49 Mn

7/21 → 5/2−
g.s.
2+ → 0+
g.s
1
4+ → 2+
1
1
–

50 Fe
51 Co

(−)

Measured
Energy (keV)

Literature
Energy (keV)

260.8(11)
765.6(20)
1086.1(51)
–

261.38(13)
764.9(3)
1086.6(3)
–

Table 6.11: Measured energies for 49 Mn and 50 Fe from the 9 Be(48 Cr, 49 Mn + γ)X and
9 Be(49 Mn, 50 Fe + γ)X reactions with comparison to theory. No γ rays were observed for
51 Co, populated in the 9 Be(50 Fe, 51 Co + γ)X reaction.
51 Co

is barely bound (Sp = 88 keV as evaluated by Ref. [148]). Although the presence of a

high Coulomb barrier at the proton drip line can dramatically increase the lifetimes of
unbound states, it is possible that the population of 51 Co leads to unbound states with
lifetimes smaller than the time necessary to detect 51 Co in the s800 focal plane.
For all spectra, energy errors were calculated, as in the analysis of the 9 Be(54 Ti, 53 Sc
+ γ)X reaction (see Section 5.6.2) from the uncertainty on the energy calibration, the
velocity of projectile (β), the target position, and the fitting procedure. These errors were
considered to be independent and were added in quadrature. Separate β values were
chosen for each transition because of possible differences in lifetimes of the states.

6.4.5

Partial Cross Sections

Partial cross sections were calculated from the full-energy peak yield relative to the number
of detected heavy residues. Experimental details are similar to the 9 Be(54 Ti, 53 Sc + γ)X
reaction detailed in Section 5.6.3. See Section 4.5.4 for an introduction to the method.
The tight gates used to produce the γ spectra of Section 6.4.4 were enlarged. The larger
gates included contamination from the inelastic excitation and neutron knockout reaction
products, but ensured all γ rays from the pickup reaction products were included. The
uncertainty on the size of this gate was included as a systematic contribution to the final
uncertainty. The partial cross section to the excited states in the pickup residue accounted

156

for any feeding, e.g. the 4+ → 2+ transition in 49 Mn. The inclusive cross section was
1
1
subtracted from the yield to excited states to extract the partial cross section to the ground
state: σg.s. = σinc. − σex. . For 49 Mn, the 1/2− and 3/2− states, with calculated energies of
1
1
1.703 MeV and 1.741 MeV, if they are populated in the reaction (they were not observed
in the γ ray spectrum; see Figure 6.15), should decay to the ground state. Therefore the
measured cross section to the ground state can only be considered an upper bound. For
50 Fe,

it is expected that the 4+ and 6+ states will funnel through the first 2+

Special care had to be taken for the 9 Be(49 Mn, 50 Fe + γ)X reaction. The pid for the
reaction runs after the addition of the delays (setting 5 of Table 6.2) was very diffuse and
smeared, possibly because of non-linearities in the tdc associated with the new 9 Be(49 Mn,
50 Fe + γ)X

time-of-flight. The partial cross sections extracted from the good and poor pids

were statistically inconsistent with each other. The poor quality pid had much lower yield
relative to the number of incoming 49 Mn nuclei. A decision was made to exclude the data
associated with setting 5 and the uncertainties associated with the drop in statistics, see
Table A.5, are reflected in the larger uncertainties for the 9 Be(49 Mn, 50 Fe + γ)X partial
cross sections.
The comparison of theoretical and experimental partial cross sections are found in
Figures 6.17, 6.18, and 6.19. The data for these figures was taken from Tables A.4, A.5,
and A.6. The highest statistics case, the 9 Be(48 Cr, 49 Mn + γ)X reaction, whose results are
shown in Figure 6.17, shows good agreement between the theoretical and experimental
yields to the ground and excited states. Theory predicts and experiment observes a zero
cross section to the ground state and a high (nearly the full inclusive strength) partial cross
section to the 7/2− state at 260.8(11) keV.
1
The predictions associated with the 9 Be(49 Mn, 50 Fe + γ)X reaction are shown in Figure 6.18 (the figure data was taken from Table A.5). The experimental partial cross
sections to the ground state and first 2+ state match the prediction. The experimental
1
partial cross section to the 4+ state should be considered an upper limit. It includes
1

157

{

7/2-

1.5

1.0

5/2-

{

0.5

3/2-

X

X

1703

1741

{
{

1/2-

0.0

0 (g.s.)

260.8(11)

Energy (keV)
Figure 6.17: Measured and predicted partial cross sections for the 9 Be(48 Cr, 49 Mn + γ)X
reaction. The 1/2− and 3/2− states were not observed (indicated by an X) in the current
1
1
experiment or previous spectroscopic studies and their assigned energies (in italics) were
taken from shell model calculations (see Figure 6.9). If the 1/2− and 3/2− states are fed
1
1
experimentally, they should directly decay to the ground state and bypass the 7/2− state
1
at 0.272 MeV. Because the feeding to the ground state could not be quantified, an upper
bound for the population of the ground state, i.e. σg.s. ≤ 0.20(17) mb, is given (the upper
limit is indicated by a grey triangle). The population pattern observed, with a population
of the ground state consistent with 0 mb (≤ 0.20(17) mb) and a partial cross section to the
7/2− state of 1.43(15) mb (nearly all of the inclusive strength), is consistent with theoretical
1
predictions of 0 mb and 1.56 mb. See Table A.4 for the data plotted in this figure.

158

the effects of feeding from the 6+ and other higher spin states that were not observed
1
because of statistics. The high statistical uncertainty associated with the 4+ state—only
1
27 counts were observed—impacts the partial cross section to the 2+ state because of
1
feeding subtraction. Although Figure 6.18 includes the 4+ state, the results in Table A.5
1
are presented with and without the inclusion of the state. In both cases, the partial cross
section to the highest reported state, the 4+ state of column 4 and the 6+ state of column 6,
1
1
should be considered an upper limit because of the inclusion of observed and unobserved
feeding from higher-lying states predicted by shell-model calculations and the partial
cross section predictions of Figure 6.18.
No transitions were observed in the 51 Co spectrum. The theoretical inclusive cross
section is equal to the partial cross section to the 7/2− ground state (Figure 6.19). All other
states are above the proton separation energy. The theoretical prediction of 0.71 mb is
close to the observed value of 0.57(8) mb.
In all three reactions, 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, and 9 Be(50 Fe, 51 Co
+ γ)X, the pattern and magnitude of population to states in the pickup reaction residues
are well-matched by theory, with the exception of the higher lying 4+ state of 50 Fe which
1
is likely fed by higher-lying states whose individual partial cross sections are too low to be
experimentally observed.

6.4.6

Momentum Distributions

The longitudinal momentum distributions, unlike the distributions associated with the
knockout reactions, do not provide direct spectroscopic information. This critical difference is a result of the two-body final state of the pickup reaction (target residue and
pickup reaction product) in comparison to the three-body final state of the knockout
reaction (target residue, knockout reaction product, and removed nucleon). Further details
are shown in Section 3.5.1. The momentum broadening due to the reaction processes is
negligible (delta function) and the final distribution widths include the simple convolution

159

2+

{

A.

4+

B.

{

1.0

C.

6+

{

0.5

0+

{

2+

3-

1-

1-

X

X

X

X

X

{
{
{
{
{

2+

X

0.0

0

)

.s

(g

X

4
9
9
9
8
4
2
6)
2)
2( 315 315 267 304 334 339 354
6(
5
6
7
18

Figure 6.18: Measured and predicted partial cross sections for the 9 Be(49 Mn, 50 Fe + γ)X
reaction. States whose energy labels include uncertainties (0+ , 2+ , and 4+ states) were
g.s 1
1
observed in the experiment. Experimental states labeled with an X were not observed: the
6+ state (partition A), the 2+ and 2+ states (partition B), and the odd spin 3+ , and 1+ states
1
2
3
1
(partition C). Their energies were taken from the shell model calculations of Figure 6.10
and are italicized to note their origin (also, Table A.2). The feeding to both the 4+ state and
1
the ground state could not be accurately estimated, and therefore the values reported are
upper bounds (indicated by grey rectangles). The two 6+ states represent two fragments
1
predicted by shell model calculations with different spectroscopic factors. See Table A.5
for the data plotted in this figure.

160

0.5

0.0

Ground State

Figure 6.19: Measured and predicted partial cross sections for the 9 Be(50 Fe, 51 Co + γ)X
reaction. Because no transitions were observed, the cross section to the ground state is
equal to the inclusive cross section. ccba and shell model calculations (theory) reproduce
the magnitude of the observed strength. See Table A.6 for the data plotted in this figure.

of the momentum profile of the beam with the differential momentum loss in the target.
Any deviation from this simple model, e.g. an increase in measured width, suggests
reaction processes beyond the simple, direct transfer of a proton from a target bound state
to a projectile ground state. The increase in differential momentum width is especially
significant for reactions that add or remove a proton because of the high Z-dependence of
the energy (momentum) loss of a charged particle through matter.
The final, lab-frame momentum distributions for the 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn,
50 Fe + γ)X,

and 9 Be(50 Fe, 51 Co + γ)X reactions are presented in Figures 6.20, 6.21, and

6.22. These figures include the measured longitudinal momentum distribution of the
pickup reaction product, the longitudinal momentum distribution of the incoming beam
as measured during a separate run with the unreacted incoming cocktail beam centered
in the focal plane, and the theoretical distribution composed from the convolution of
the unreacted beam with the differential momentum loss in the target. The central
161

momentum for the observed reacted (unreacted) parallel momentum distributions are
13.1 GeV/c (13.0 GeV/c) for the 9 Be(48 Cr, 49 Mn + γ)X reaction, 13.7 GeV/c (13.6 GeV/c)
for the 9 Be(49 Mn, 50 Fe + γ)X reaction, and 14.3 GeV/c (14.2 GeV/c) for the 9 Be(50 Fe, 51 Co
+ γ)X reaction.

49
500 p(48Mn) = 13.1 GeV/c

49

Mn

Counts / 31.3 MeV/c

p( Cr) = 13.0 GeV/c

48

Cr

400

Estimate

300
200
100
0
-400

-200

0

∆p (MeV/c)

200

400

Figure 6.20: Measured lab-frame parallel momentum distribution of 49 Mn (9 Be(48 Cr,
49 Mn+γ)X reaction). The unreacted momentum distribution of 48 Cr (shown; red triangles)
and its fit by Equation 6.1 (red dotted line) are used, along with the differential momentum
width associated with the position of the reaction process within the target, to create the
theoretical distribution (solid black curve). The observed width of the 49 Mn distribution is
broadened, as hypothesized in the text, by scattering off the beam blocker. This scattering
moves cross section from higher to lower momentum and creates the prominent lowmomentum hump observed at about ∆p ≈ −200 MeV in the spectrum.

Two artifacts of Figures 6.20, 6.21, and 6.22 must be addressed: the shape of the
unreacted beam and the significant difference in width between the measured lab-frame
distribution and the theoretical prediction.
Because the profile of the unreacted beam is folded into the final theoretical distribu162

Counts / 62.6 MeV/c

250

p(50Fe) = 13.7 GeV/c
p(49Mn) = 13.6 GeV/c

50

Fe

49 Mn

200

Estimate

150
100
50
0

-400

-200

0

200

400

∆p (MeV/c)
Figure 6.21: Measured, lab-frame, inclusive, parallel momentum distribution of 50 Fe
(9 Be(49 Mn, 50 Fe + γ)X reaction; blue circles). As detailed in Figure 6.20, the unreacted
beam (solid, red triangles), its fit (dashed, red line), showing more skew than the 9 Be(48 Cr,
49 Mn + γ)X reaction, and the theoretical distribution (solid, black line) created from the
convolution of the fit to the unreacted 49 Mn momentum distribution with the differential
momentum loss in the target, are displayed. The broadening of the momentum distribution
and the low momentum hump are hypothesized to occur as artifacts from scattering off
the beam blocker located in front of the focal plane.

163

Counts / 62.6 MeV/c

60

p(51Co) = 14.3 GeV/c
p(50Fe) = 14.2 GeV/c

51

Co

50

Fe
Estimate

50
40
30
20
10
0
-400

-200

0

∆p (MeV/c)

200

400

Figure 6.22: Measured, lab-frame, inclusive parallel momentum distribution of 51 Co
(9 Be(50 Fe, 51 Co + γ)X reaction; blue circles). The unreacted 50 Fe and theoretical momentum distributions are also shown. The unreacted beam shows significant skew and
asymmetry caused by the a1900 slits. The low-momentum broadening, observed in Figures
6.20 and 6.21, was significantly reduced in the 9 Be(50 Fe, 51 Co + γ)X reaction because of
decreased scattering by the beam blocker (see text for additional details). The final overlay
of the measured distribution and the theoretical prediction is consistent with the direct
reaction assumption of the ccba calculations.

164

tion, is important to correctly parametrize the shape. The beam profile depends upon the
beam tune through the a1900 fragment separator and analysis line. Typically, the settings
of the a1900 are optimized for one fragment and other components of the cocktail beam
may be at the edge of the acceptance and have asymmetric beam profiles. For the current
experiment, deviations are minor for the 9 Be(48 Cr, 49 Mn + γ)X reaction, stronger for the
9 Be(49 Mn, 50 Fe + γ)X

reaction, and strongest for the 9 Be(50 Fe, 51 Co + γ)X reaction where

clear asymmetry of the unreacted beam is observed.
The beam profile was successfully parametrized by an equation of the form [168]
D(ξ, ) = A exp

1
1
1
1
1
ξ−
k − sδ − (2 + k)δ2 + sδ3 + kδ4 where δ =
8
2
4
6
24
σ

(6.1)

A is the amplitude, is the centroid, σ is the Gaussian width, k is the shape parameter (from
Gaussian to top-hat), and s is the skew-parameter. The shape is completely determined by
five parameters and achieves an excellent fit. See especially Figure 6.22.
The shape of the low-momentum side of the momentum distribution, particularly
Figure 6.20 with the low momentum hump, suggests that the increase in width is caused
by an experimental rather than reaction artifact. One possibility is the scattering of the
reaction residues off the beam blockers. The beam blockers are plates, located before the
focal plane detectors, that can be inserted in the path of the beam. Usually this is done to
protect the detectors against the unreacted beam or high-intensity reaction products that
enter the focal plane at the edge of the acceptance. Scattering would move the differential
cross section in a manner characteristic of the shapes observed. The effect appears to
disappear for 51 Co and is smaller for 50 Fe than 49 Mn.
Because of the artificial broadening of the 9 Be(48 Cr, 49 Mn + γ)X and 9 Be(49 Mn, 50 Fe
+ γ)X parallel momentum distributions, the behavior of the reaction—are there processes beyond the single-step transfer?—cannot be unambiguously determined. For 51 Co,
because of the smaller affect of the beam blocker on the final lab-frame momentum distribution, the estimated and experimental distributions are consistent with each other and

165

therefore consistent with 9 Be(50 Fe, 51 Co + γ)X proceeding as a direct reaction, a critical
assumption of the ccba calculations.

6.5
6.5.1

Proton Pickup Reactions: 12 C target
Particle Identification

The identification of the incoming projectiles and outgoing pickup residues was easier
for the 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X reactions in
comparison to the equivalent pickup reaction using the 9 Be target because of the addition
of the tacs, added in parallel to the existing electronics, that removed the time-of-flight
splitting. The new tacs finally permitted clean identification of components of the
cocktail beam using the two-dimensional spectrum of DIA1 time-of-flight versus DIA2
time-of-flight6 . There was minimal contamination from neighboring tof components in
the cocktail beam.
A nearly identical suite of gates was taken from Section 6.4.1 and used for the cleanup
and identification of pickup reaction residues off the 12 C target: the two-dimensional
histogram of xfp time-of-flight versus dispersive position in the focal plane (see Figure 6.7
for an example), and the standard pid of OBJE1 time-of-flight versus energy loss in the ion
chamber. As with the 9 Be analysis, the full set of gates eliminated contamination from the
final pid used to count the reaction residues.

6.5.2

Theoretical Predictions

Calculation of the single-partial cross sections follow the earlier 9 Be pickup analysis
presented in Section 6.4.2 with a few critical differences associated with the treatment
of the target. The same assumptions used in the ccba calculations of the 9 Be(48 Cr,
6 DIA1 and DIA2 time-of-flights are physically identical to the obj and xfp time-of-fights.

The only difference lies in the electronics used to processes the signal See Table 6.3.

166

49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X,

and 9 Be(50 Fe, 51 Co + γ)X reactions apply: a proton

is transfered in a single-step from a bound state of a 12 C target to a bound state of the
projectile. The 12 C target is left in one of three bound states of 11 B, the 3/2− ground state,
g.s.
the 1/2− excited state at 2125 keV, and the 3/2− excited state at 5020 keV. Because a proton
1
1
is removed from the carbon p-shell (1p1/2 and 1p3/2 orbitals), the 5/2− state, located at
1
4445 keV and below the energy of the 3/2− state, is not expected to be populated. All three
11 B

states included in the analysis are bound. The proton threshold is Sp (11 B) = 11.2 MeV.

ˆ
The p-shell proton overlaps 11 B|a| 12 C are taken directly from a Woods-Saxon potential and the parameters of Ref. [85]: the radius parameter of r0 = 1.310 fm and the
diffuseness parameter of a = 0.55 fm were fixed to the second and fourth moments of the
12 C

charge radius and the depth of the Woods-Saxon well was adjusted to the separa-

tion energy of the proton relative to the 48 Cr core. The spectroscopic factors for target
ˆ
initial and final state overlaps, e.g. S = | 11 B(j π )∗ |a| 12 C |2 , were calculated in the shell
model using the wbp interaction [169]. The inclusion of the three 11 B final states nearly
exhausted the spectroscopic strength with values of 3.16, 0.58, and 0.19 for overlaps of the
12 C

ground state with the 3/2− , 1/2− and 3/2− states of 11 B respectively. The remaining
g.s.
1
2

strength of 0.07 is thinly spread up to the proton separation energy of Sp = 11.2 MeV.
The remaining details associated with the calculation of the 12 C(48 Cr, 49 Mn + γ)X,
12 C(49 Mn, 50 Fe + γ)X,

and 12 C(50 Fe, 51 Co + γ)X single-particle cross sections follow the

earlier 9 Be analysis of Section 6.4.2. The final results are listed in Tables 6.12, 6.13, 6.14 as
provided by Tostevin [162].
The single-particle cross sections, combined with the reaction-independent spectroscopic factors listed in Section 6.4.2, yield the theoretical estimates for the inclusive and
partial cross sections. See Tables 6.15, 6.16, and 6.17. The calculated population pattern
and magnitudes are very similar to the 9 Be results of Tables 6.7, 6.8, and 6.9. The partial
and inclusive cross sections are slightly higher which may be attributed to the nearly
complete inclusion of all target spectroscopic strength by the 3/2− , 1/2− , and 3/2− states.
g.s.
1
2

167

Target (12 C → 11 B∗ )

49 Mn

σs.p
s.p.
i σi

π j

res.
Sp (eff.)
(MeV)

Jπ
11 B)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

σ s.p.
(mb)

0.272

1f7/2

1.813

3/2−
1/2−
3/2−

15.957
18.082
20.977

-13.872
-15.997
-18.892

3.723
0.237
0.179

4.139

1/2−

1.703

1f7/2

0.382

3/2−
1/2−
3/2−

15.957
18.082
20.977

-13.872
-15.997
-18.892

0.029
0.008
0.003

0.040

3/2−

1.741

1f7/2

0.344

3/2−
1/2−
3/2−

15.957
18.082
20.977

-13.872
-15.997
-18.892

0.103
0.008
0.003

0.114

Iπ

Ex
(MeV)

7/2−

(mb)

Table 6.12: Additional details on the calculation of the single-particle cross sections
for the 12 C(48 Cr, 49 Mn + γ)X reaction evaluated at a mid-target energy of 56.7 MeV/u
[162]. For additional information on the column labels, please refer to the caption of
Table 6.4. The ground-state separation energy of 49 Mn is 2.085 MeV. The single-particle
cross section to the ground state of 49 Mn was not calculated because of the extremely small
spectroscopic factor C 2 S = 0.002 listed in Table A.1. The 7/2− energy of 272 keV was taken
from experiment; see the caption of Table 6.5 for additional details. The 1/2− and 3/2−
1
1
energies were not observed in this experiment and literature values do not exist. These
energies were taken from shell-model calculations, see Figure 6.10, and are italicized to
note this origin.

Comparison with experimental data is found in the experimental analysis of the inclusive
and partial cross sections, Sections 6.5.3 and 6.5.5.

6.5.3

Inclusive Cross Section

Details on the calculation of the inclusive cross sections from experiment can be found in
Section 6.5.3. The inclusive cross sections for the 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe +
γ)X, and 12 C(50 Fe, 51 Co+γ)X reactions combined two separate cross section measurements
using unreacted and reacted runs located at the beginning and end of the reaction setting.
The two results for all three reactions were consistent (within 1σ ) of each other. Unlike the
9 Be

analysis, the 12 C pickup data was all taken in one experimental setting; the problems

associated with the faulty electronics were eliminated. The cross sections extracted from
168

Target (12 C → 11 B∗ )

50 Fe

σs.p
s.p.
i σi

Iπ

Ex
(MeV)

π j

res.
Sp (eff.)
(MeV)

Jπ
11 B)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

σ s.p.
(mb)

0+

0

1f7/2

4.15

3/2−
1/2−
3/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.031
0.011
0.011

0.054

2+

1.703

1f7/2

3.385

3/2−
1/2−
3/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.360
0.022
0.018

0.400

4+

1.851

1f7/2

2.299

3.159

1f7/2

0.991

15.957
18.082
20.977
15.957
18.082
20.977

-11.807
-13.932
-16.827
-11.807
-13.932
-16.827

0.628
0.039
0.031
0.863
0.054
0.041

0.698

6+

3/2−
1/2−
3/2−
3/2−
1/2−
3/2−

2+

2.672

1f7/2

1.478

3/2−
1/2−
3/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.339
0.021
0.016

0.376

2+

3.044

2p3/2

1.106

3/2−
1/2−
3/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.023
0.002
0.001

0.025

1+

3.349

1f7/2

0.801

3/2−
3/2−
1/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.198
0.012
0.009

0.219

3+

3.398

2p3/2

0.752

3/2−
3/2−
1/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.029
0.002
0.001

0.032

3+

3.544

2p1/2

0.606

3/2−
3/2−
1/2−

15.957
18.082
20.977

-11.807
-13.932
-16.827

0.015
0.005
0.001

0.021

(mb)

0.959

Table 6.13: Additional details on the calculation of single-particle cross sections (spcss)
for the 12 C(49 Mn, 50 Fe + γ)X reaction evaluated at a mid-target energy of 59.0 MeV/u
[162]. For additional information on the column labels, please refer to the caption of
Table 6.4. The ground-state separation energy of 50 Fe is 4.150 MeV. Listed energies are
from literature when available. Exceptions, e.g. the 2+ state at 2.672 MeV, the 2+ state
2
3
at 3.044 MeV, and the 1+ and 3+ states, were taken from shell model calculations and
are italicized (see Figure 6.10, Table A.2). The two fragmented 6+ states of Table A.2 are
calculated at the experimental energy of 3.544 MeV.

169

Target (12 C → 11 B∗ )

51 Co

σs.p

Iπ

Ex
(MeV)

π j

res.
Sp (eff.)
(MeV)

Jπ
11 B)
(

tar.
Sp (eff.)
(MeV)

Q
(MeV)

σ s.p.
(mb)

0+

0

1f7/2

0.164

3/2−
1/2−
3/2−

15.957
18.082
20.977

-15.793
-17.918
-20.813

2.718
0.166
0.129

s.p.
i σi

(mb)
3.013

Table 6.14: Additional details on the calculation of the single-particle cross sections for the
12 C(50 Fe, 51 Co + γ)X reaction evaluated at a mid-target energy of 61.2 MeV/u [162]. For
additional information on the column labels, please refer to the caption of Table 6.4. The
ground-state separation energy of 51 Co is 88 keV [148]. No excited states were observed
and therefore only the partial cross section to the ground state was calculated.
Efinal
Shell model
(MeV) configuration

7/2−
g.s.
1/2−
1
3/2−
1

0.272
1.703
1.741

[0+ ⊗ 1f7/2 ]
[0+ ⊗ 2p1/2 ]
[0+ ⊗ 2p3/2 ]

COM

corr.
0.940
0.940
0.940

C 2 Ss.m.

σs.p.
(mb)

σtheory
(mb)

4.139
0.040
0.114

0.425
0.245
0.373

1.65
0.01
0.04

Inclusive sum:

Jπ

1.70

Table 6.15: Theoretical predictions of the partial and inclusive cross sections for the
12 C(48 Cr, 49 Mn + γ)X reaction.
Efinal
Shell model
(MeV) configuration

COM

0+
g.s.
2+
1
4+
1
2+
2
2+
3
6+
1
6+
1
1+
1
3+
1
3+
2

[5/2− ⊗ 1f5/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 2p3/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 1f7/2 ]
[5/2− ⊗ 2p3/2
[5/2− ⊗ 2p1/2 ]

0.941
0.941
0.941
0.941
0.941
0.941
0.941
0.941
0.941
0.941

0
0.765
1.851
2.672
3.044
3.159
3.159
3.349
3.398
3.544

corr.

C 2 Ss.m.

σs.p.
(mb)

σtheory
(mb)

0.054
0.400
0.698
0.376
0.025
0.959
0.959
0.530
0.309
0.213

0.033
1.145
0.208
0.194
0.232
0.023
0.371
0.219
0.032
0.021

0.00
0.43
0.14
0.07
0.01
0.02
0.34
0.11
0.01
0.00

Inclusive sum:

Jπ

1.12

Table 6.16: Theoretical predictions of the partial and inclusive cross sections for the
12 C(49 Mn, 50 Fe + γ)X reaction. The 0+ ground state, 2+ , and 4+ states were observed in
1
1
the experiment. Further details are found in the caption of Figure 6.8.

170

Jπ

Efinal
(MeV)

Shell model
configuration

COM

7/2−
g.s

0

[0+ ⊗ 1f7/2 ]

0.942

σs.p.
(mb)

σtheory
(mb)

3.013

0.246

0.70

Inclusive sum:

corr.

C 2 Ss.m.

0.70

Table 6.17: Theoretical predictions of the partial and inclusive cross sections for the
9 Be(50 Fe, 51 Co + γ)X reaction.
σtheory
(mb)

Reaction

σexp.
(mb)

1.70
1.12
0.70

2.00(13)
1.72(24)
0.53(13)

12 C(48 Cr, 49 Mn + γ)X
12 C(49 Mn, 50 Fe + γ)X
12 C(50 Fe, 51 Co + γ)X

Table 6.18: Measure and predicted inclusive cross sections for the 12 C(48 Cr, 49 Mn + γ)X,
12 C(49 Mn, 50 Fe+γ)X, and 12 C(50 Fe, 51 Co+γ)X reactions. The predictions for the 12 C(48 Cr,
49 Mn+γ)X and 12 C(50 Fe, 51 Co+γ)X reactions agree with calculated values. The 12 C(49 Mn,
50 Fe + γ)X theoretical results under predict the measured value. This discrepancy may
be associated with the difficulty of including the wide spread of spectroscopic strength
observed below Sp (50 Fe) = 4150(60) keV.
the yields of 49 Mn, 50 Fe, and 51 Co are found in Table. 6.18. There is good agreement
between experiment and theory for the 12 C(48 Cr, 49 Mn + γ)X and 12 C(50 Fe, 51 Co + γ)X
reactions, but not for the 12 C(49 Mn, 50 Fe + γ)X reaction, which theory under-predicts.
This behavior, also seen in the analysis of 9 Be(50 Fe, 51 Co + γ)X, was attributed to the high
fragmentation of strength to states under the proton separation in 50 Fe, for example the
2+ and 6+ states, and the difficulty of including the multitude of states in the theoretical
calculations of the single-particle cross sections.

6.5.4

Observed Transitions

The analysis of 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X
follows the earlier analysis of Section 6.4.4. The effective target position (common for
all reactions on the 12 C target) and effective β values (uniquely determined for each

171

transition) were chosen in an identical way, with the exception of the 50 Fe 4+ → 2+
1
1
transition. Additional details will follow. The γ-ray spectra observed in coincidence
with 49 Mn and 50 Fe for the 12 C(48 Cr, 49 Mn + γ)X and 12 C(49 Mn, 50 Fe + γ)X reactions
are shown in Figures 6.23 and 6.24. The same transitions are observed in both the 9 Be
proton pickup analysis and the 12 C proton pickup analysis with the exception of the 50 Fe
6+ → 4+ transition, which was likely observed because of the additional statistics on the
1
1
12 C target.

Thus, for 49 Mn, following the 12 C(48 Cr, 49 Mn+γ)X reaction, the 7/2− → 5/2−
g.s.
1

at 260.7(12) keV was observed; for 50 Fe, following the 12 C(49 Mn, 50 Fe + γ)X reaction, the
6+ → 4+ transition at ∼ 1300 keV, the 4+ → 2+ transition at 1088(7) keV, and the 2+ → 0+
g.s
1
1
1
1
1
transition at 764.2(28) keV were observed; and as for the 9 Be target, for 51 Co, following
the 12 C(50 Fe, 51 Co + γ)X reaction, no residue rest-frame or laboratory-frame γ rays were
observed.
The expected cross section for the population of target excited states, following the
removal of a proton from 12 C leading to one of three considered states in 11 B, is low—
see the calculated single-particle cross sections of Tables 6.12 6.13, and 6.14—and the
three possible decays of 11 Be∗ , 2125 keV (1/2− → 3/2− ), 2895 keV (3/2− → 1/2− ), and
g.s.
g.s.
1
1
5019 keV (3/2− → 3/2− ) were not observed.
g.s.
1
Special care was taken for the 50 Fe 4+ → 2+ decay. There were insufficient statistics to
1
1
align both rings. The transition was assumed to occur in the target. Although a half-life
for the 4+ state is not available for 50 Fe, the mid-target decay assumption was based on
1
the short half-life of the 2+ state of 1.5(3) ps and the larger 4+ → 2+ transition energy of
1
1
1
2λ+1 where λ is the multipolarity; λ = 2 here). The
1087 keV (lifetimes scale as τ1/2 ∝ 1/Eγ

energy uncertainty of the 4+ state was dominated by the choice of β which was assumed
1
to lie between the βmid and βend values of β = 0.340 and β = 0.327.

172

Counts / 8 keV

80

49Mn: Rest frame

260.7(12) keV

70
60
50
40
30

48Cr contamination

20
10

35

200

49Mn:

400

600

800

Energy (keV)

Lab frame

30
25
20

Counts / 16 keV

Counts / 16 keV

00

1000

1200

4
2

2000 2500 3000
Energy (keV)
11B transitions
Expected

15
10

1400

1500

3500

5
00

1000

2000

3000

4000

Energy (keV)

5000

6000

Figure 6.23: Residue rest frame (upper panel) and laboratory frame (lower panel) γ-ray
spectra following the 12 C(48 Cr, 49 Mn+γ)X reaction. Only the 7/2− → 5/2− 260.7(12) keV
g.s.
1
transition is observed. The 752 keV transition observed in the rest-frame spectra comes
from unwanted 48 Cr contamination. The cross sections for the population of the 11 B∗ 1/2−
1
and 3/2− states were too low to be experimentally observed. See Table 6.12. Transitions
associated with the decay of the excited 11 B target occur at 2125 keV, 2895 keV, and
5019 keV.

173

Counts / 16 keV

30

764.2(28) keV

25
20
15

1088(7) keV

10

~1300 keV

5
500

1000

10
8
6

1500

Energy (keV)
Counts / 16 keV

Counts / 16 keV

00

2000

4
2

0
1500 2000 2500 3000 3500
Expected 11B transitions Energy (keV)

4
2
00

1000

2000

3000

4000

Energy (keV)

5000

6000

Figure 6.24: Residue rest frame and laboratory frame γ-ray spectra for 50 Mn following the
12 C(49 Mn, 50 Fe+γ)X reaction. Both the 4+ → 2+ transition at 764.2(28) keV, the 2+ → 0+
g.s
1
1
1
+ → 4+ transition at ∼ 1300 keV were observed. The
transition at 1088(7) keV, and the 61
1
cross sections for the population of the 11 B∗ 1/2− and 3/2− states were too low to be
1
experimentally observed based on the lack of observed transitions in the rest-frame γ-ray
spectrum at the expected energies of 2125 keV, 2895 keV and 5019 keV. The suggested
peak at 5019 keV is consistent with background.

174

6.5.5

Partial Cross Sections

The comparison of experimental and theoretical partial cross sections for the 12 C(48 Cr,
49 Mn+γ)X, 12 C(49 Mn, 50 Fe+γ)X, and 12 C(50 Fe, 51 Co+γ)X reactions are shown in Figures

6.25, 6.26, and 6.27. The plotted data was taken from Tables A.7, A.8, and A.9. A similar
pattern of excited-state population is experimentally observed for both 9 Be and 12 C targets
and the same conclusions reached in Section 6.4.5 with respect to the 9 Be target also apply
to the 12 C target pickup results: the experimentally observed magnitude and population
pattern of states tracks theoretical predictions, with an understood deviation for the
12 C(49 Mn, 50 Fe + γ)X

reaction.

There is one difference that should be pointed out, although its investigation is beyond
the current scope of this thesis (current theory is only aimed at qualitative agreement).
Theory under predicts the increase of cross section observed when moving from the
9 Be

to 12 C target. Taking the two highest statistics cases, the proton pickup reactions

from 48 Cr to 49 C, i.e. 9 Be(48 Cr, 49 Mn + γ)X and 9 Be(49 Mn, 50 Fe + γ)X, and the proton
pickup reactions from 49 Mn to 50 Fe, i.e. 9 Be(49 Mn, 50 Fe + γ)X and 12 C(49 Mn, 50 Fe + γ)X,
we find that theory predicts an increase of 6 % and 7 % respectively, while experimental
increases of 23(2) % and 25(4) % were observed. This difference could come from the nearly
complete treatment of the 12 C and 11 Be∗ target overlaps through the inclusion of just three
states in 11 B (3/2− , 1/2− , and 3/2− states) in comparison to the 9 Be case where much the
spectroscopic overlap goes to unbound states in 8 Li which are difficult to include in the
calculation of the single-particle cross sections. The decreased complexity in calculations
for both proton and neutron (see Ref. [153]) pickup reactions using a carbon target may
make 12 C a better choice for future spectroscopic studies using pickup reactions.

6.5.6

Momentum Distributions

The expected shapes of the parallel momentum distributions are close in width to the
measured distributions. See Figures 6.28, 6.29, and 6.30. The similarity in width between
175

2.0

{

7/2-

1.5

1.0

{

5/2-

0.5

3/2-

{
{

1/2-

0.0

X
0 (g.s.)

260.7(12)

1703

X
1741

Energy (keV)
Figure 6.25: Measured and predicted partial cross sections for the 12 C(48 Cr, 49 Mn + γ)X
reaction. The 1/2− and 3/2− states were not observed experimentally, which is indicated by
an X above the experimental state. The energies assigned to these two states are taken from
shell-model calculations (see Figure 6.9). This is indicated by an italic font and the absence
of experimental error. The experimental population of the ground state is considered
an upper bound, as indicated by a grey triangle, because the feeding to the ground state
could not be quantified; see also Figure 6.17 for the 9 Be(48 Cr, 49 Mn + γ)X reaction. The
population pattern observed, with a low (consistent with zero) partial cross section to the
ground state and a high cross section of 1.60(22) mb (nearly all of the inclusive strength) to
the 7/2− state is consistent with theoretical predictions. See Table A.7 for the data plotted
1
in this figure.

176

1.5

2+

{

A.

4+

{

B.

1.0

6+

{

0.5

0+

C.

{
2+

3-

1-

1-

X

X

X

X

X

{
{
{
{
{

2+

0.0

0

X

)

.s
(g

X

2) 2(6)
55 155 672 044 349 398 544
6(
5
3
3
3
31
3
2
3
76 18

Figure 6.26: Measured and predicted partial cross sections for the 12 C(49 Mn, 50 Fe + γ)X
reaction as taken from Table A.8. Results are presented with inclusion of the 4+ state. See
1
the caption of Table A.8 and Figure 6.18 (from the similar 9 Be(49 Mn, 50 Fe + γ)X reaction)
for further details. The transitions marked by an X (all levels above the 4+ ) were not
1
observed experimentally and the energies listed (in italics) are taken from the shell model
calculations of Figure 6.10. The partial cross section to the 4+ state should be considered
1
upper bound because of unobserved feeding (indicated by a grey triangle). The two
6+ states represent two shell-model fragments with different spectroscopic factors. The
experimental partial cross section to the 6+ state, observed in the γ-ray spectrum of
1
Fig. 6.24 was not extracted and plotted because of the very low statistics and the high
resultant uncertainty. The data plotted in this figure was taken from Table A.8.

177

0.5

0.0

Ground State

Figure 6.27: Measured and predicted partial cross sections for the 12 C(50 Fe, 51 Co + γ)X
reaction . Because no transitions were observed, the cross section to the ground state is
identically equal to the inclusive cross section. The data plotted in this figure was taken
from Table A.9.

the predicted and measured distributions, as mentioned earlier in the analysis of proton
pickup of the 9 Be target (see Section 6.5.6), is consistent with a direct reaction. Unlike the
momentum distributions measured after the interaction of the projectiles with the 9 Be
target, the 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X, and 12 C(50 Fe, 51 Co + γ)X parallel
momentum distributions are not strongly distorted by the presence of the beam blocker,
possibly because the decreased target thickness of the 72.8 mg/cm2 12 C target, relative
to the 188 mg/cm2 9 Be target, narrowed the observed momentum distributions of the
12 C

pickup reaction products (less differential momentum broadening) and increased the

separation from the beam. A possible affect, however, may be seen in the low-momentum
tails. See especially Figure 6.29. Low-momentum tails are indication of dissipative
mechanisms beyond the one-step, direct, two-body reaction mechanism.

178

Counts / 31.3 MeV/c

250
200

p(48Cr) =
15.2 GeV/c
p(49Mn) =
15.5 GeV/c

49Mn
48

Cr
Estimate

150
100
50
0
-400

-200

0

∆p (MeV/c)

200

400

Figure 6.28: Measured and estimated parallel momentum distributions for the 12 C(48 Cr,
49 Mn + γ)X reaction. The width of the estimated distribution is close to the width of the
measured parallel momentum distribution showing the direct nature of the reaction. A
low momentum tail is observed suggesting the presence of a more complicated, dissipative
reaction channels that were not included in the theoretical calculations. A similar tail was
observed in the one-proton pickup reaction 9 Be(22 Mg, 23 Al + γ)X [61].

6.6

Summary

The general intent of the current research was to show the general applicability of the
proton pickup reaction as a qualitative spectroscopic tool by comparing experimental
results and theoretical predictions for the magnitude and pattern of population to states
in the excited states yields under different experimental conditions, specifically through
the used of two different targets, a 188 mg/cm2 beryllium target and a 72.8 mg/cm2
carbon target. The results, taken from the comparison between measured and predicted
partial and inclusive cross section shown in Figures 6.17, 6.18, 6.19, 6.25, 6.26, and 6.27
show a general broad agreement between the magnitude of values and the population
179

Counts / 31.3 MeV/c

120
100

50

p(49Mn) =
15.8 GeV/c
50Fe) =
p(
16.1 GeV/c

Fe

49 Mn

Estimate

80
60
40
20
0

-400

-200

0

∆p (MeV/c)

200

400

Figure 6.29: Measured and predicted parallel momentum distributions for the 12 C(49 Mn,
50 Fe + γ)X reaction. The width of the measured distribution is close to the width of the
measured parallel momentum distribution showing the direct nature of the reaction. A
low momentum tail is observed, indicating the presence of additional channels beyond
the direct processes included in the ccba framework.
pattern. Furthermore, the measured parallel momentum distributions for the 9 Be(50 Fe,
51 Co+γ)X, 12 C(48 Cr, 49 Mn+γ)X, 12 C(49 Mn, 50 Fe+γ)X, and 12 C(50 Fe, 51 Co+γ)X reactions,

are consistent with a direction reaction processes (a similar deduction for the 9 Be(48 Cr,
49 Mn+γ)X, and 9 Be(49 Mn, 50 Fe+γ)X reactions could not be made because of experimental

complications; see Section 6.4.6), a critical assumption of the ccba calculations.
This new body of work, the first 9 Be, fast-beam, inverse-kinematics proton pickup reaction study of f p-shell nuclei, and the first proton pickup from a 12 C target, will provide
future experimentalists additional confidence in fast-beam proton pickup reactions, which
enables experimentalists, in comparison to the well-established single-nucleon knockout
reactions, to populate single-proton states in the heavy residue and offer an increased
180

Counts / 62.6 MeV/c

20

p(50Fe) =
16.5 GeV/c
p(51Co) =
16.8 GeV/c

51

Co

50

Fe
Estimate

15
10
5
0

-400

-200

0

∆p (MeV/c)

200

Figure 6.30: Measured and predicted parallel momentum distributions for the 12 C(50 Fe,
51 Co + γ)X reaction. The low momentum bump observed around -300 MeV/c can be
attributed to the presence of the beam blocker. For a similar affect see Figures 6.20 and
6.21

sensitivity to states with higher orbital angular momentum.

181

Chapter 7
SUMMARY AND FUTURE WORK
Chapters 5 and 6 present the analysis and summary of the single-proton knockout and
pickup reactions. Detailed conclusions for the two experiments are found in Sections
5.8 and 6.6. In brief, for the 9 Be(54 Ti, 53 Sc + γ)X knockout reaction, the observation of
strong strength to excited states in 53 Sc was incompatible with shell-model predictions
using the f p valance space. The discrepancy was attributed to the knockout of sd-shell
protons outside the valance space and suggested the need for improved cross-shell proton
interactions. For the 9 Be(48 Cr, 49 Mn + γ)X, 9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X,
12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X,

and 12 C(50 Fe, 51 Co + γ)X pickup reactions,

the measured magnitude and distribution of strength to final states in the pickup residues
qualitatively matched theoretical predictions, the momentum distributions were consistent
with a direct reaction processes assumed in the ccba reaction calculations, and the
qualitative conclusions were consistent under the choice either the 9 Be or 12 C targets.
These results provided further confidence in inverse-kinematics, fast-beam single-nucleon
pickup reactions as a spectroscopic tool.
Experimental constraints prevented a more detailed conclusion for the 9 Be(54 Ti, 53 Sc
+ γ)X knockout reaction. As addressed in Section 5.6.1, the spectrograph was set to center
to the two-proton knockout residue 52 Ca for most reacted runs. These runs, with an
unknown acceptance correction, could not be used for the measurement of partial cross
sections or longitudinal momentum distributions. A re-designed experiment, with a
specific focus on the 9 Be(54 Ti, 53 Sc + γ)X reaction and perhaps a thinner target to reduce
the momentum broadening caused by the differential energy loss in the target, would
permit the extraction of partial cross sections to final states in 53 Sc and, through the

182

measurement of the longitudinal momentum distributions, the corresponding assignment
of quantum numbers associated with the orbital of the removed proton. This additional
information can be used to constrain the tbme of proton cross-shell interactions for
developed sdpf valance-space effective interactions. The new germanium γ-ray tracking
array gretina [170], with high segmentation and γ-ray tracking and available for use
at the nscl and other labs on a rotating basis starting in 2012, can be used, in addition
to its high resolving power and efficiency, through the measurement of γ-γ angular
correlation and polarization measurements, to directly measure the spin and parities of γ
rays associated with a given transition. In general, knockout reactions could be used to
probe other proton (neutron) cross-shell interactions, e.g. sd-fp or fp-g9/2 , for nuclei with
one or two protons (neutrons) above a closed shell.
Reaction theory is confirmed through agreement with experiment. The successes of
ccba reaction theory and shell-model results at qualitatively describing experimentally
observed partial cross sections for fast-beam pickup reactions are still confined to the
transfer of a proton or neutron to a limited selection of orbitals in the pickup residue:
the proton and neutron sd-shell and the proton f p shell. Due to momentum-matching
considerations, neutron pickup reactions in regions of shell evolution, where highintruder orbitals descend and change the nuclear structure significantly, would selectively
populate these intruder states and allow quantification of their strength.
Additional target tests are necessary from both an experimental and theoretical viewpoint. Experimentally, for very exotic isotopes produced at rates of a few particles per
second, choosing a target to maximize the pickup reaction cross section is critical. The
choice is not always obvious. In the neutron pickup reactions of Ref. [153], the surprisinglylarge measured inclusive cross section for the 9 Be(22 Mg, 23 Mg + γ)X reaction suggested
that the weakly-bound 9 Be valance neutron (α + α + n) was not sufficiently matched in
momentum, and the cross section was likely dominated by pickup from deeply-bound
9 Be

neutrons. The neutron pickup cross sections were slightly higher for the 9 Be target

183

than the 12 C target. In the results of Chapter 6, for proton pickup from the f p-shell, the
situation is reversed. Higher cross sections were observed for the 12 C target.
A higher statistics experiment would enable a more accurate probe of target final
states, a critical test of reaction theory. There were insufficient statistics in the pickup
reactions of Chapter 6 to accurately probe target final states in the current experiment.
Laboratory-frame γ-ray spectra for any of the six studied pickup reactions did not reveal
transitions associated with the decay of excited 8 Li or 11 B target residuals.
The spectroscopic information obtained through single-nucleon pickup reactions, as
addressed in this thesis, are still qualitative rather than quantitative. Additional work is
needed to place pickup reactions on a similar quantitative basis as knockout reactions. A
series of complementary reactions, for example 9 Be(54 Ti, 53 Sc+γ)X and 9 Be(54 Ti, 55 V+γ)X,
related through spectroscopic factor shell-model sum rules (see Equation 3.12), could use
the improved model accuracy of the knockout Eikonal single-particle cross sections to
constrain current pickup single-particle cross sections.
There are many open questions that require the fertile collaboration of theorists and
experimentalists, and whose answers will take us forward into an exciting future offered
by next generation experimental facilities, improved models, powerful computers, and
new insights into the atomic nucleus.

184

APPENDICES

185

Appendix A
DATA TABLES FOR SINGLE-PROTON
PICKUP REACTIONS
A.1

Shell-model Spectroscopic Factors

Shell-model spectroscopic factors for the (48 Cr,49 Mn), (49 Mn,50 Fe), and (50 Fe,51 Co) overlaps, as calculated by Brown [163] and plotted in Figures 6.9, 6.10, and 6.11, are listed in
Tables A.1, A.2, and A.3.

A.2

Measured and Predicted Partial Cross Sections

The comparison of measured and predicted partial cross sections for 9 Be(48 Cr, 49 Mn + γ)X,
9 Be(49 Mn, 50 Fe + γ)X, 9 Be(50 Fe, 51 Co + γ)X, 12 C(48 Cr, 49 Mn + γ)X, 12 C(49 Mn, 50 Fe + γ)X,

and 12 C(50 Fe, 51 Co + γ)X reactions are given in Tables A.4, A.5, A.6, A.7, A.8, and A.9.
This data is plotted visually in Figures 6.17, 6.18, 6.19, 6.25, 6.26, and 6.27 (respectively).

186

Iπ
(49 Mn)

n j

Ex
C 2 Ss.m.
(MeV)

Below Sp = 2085(25) keV
7/2−
1f7/2 0.219
−
3/2
2p3/2 1.675
−
1/2
2p1/2 1.696
5/2−
1f5/2
0

0.425
0.373
0.246
0.002

Above Sp = 2085(25) keV
5/2−
1f5/2 2.857
−
1/2
2p1/2 4.291
1/2−
2p1/2 6.061
−
5/2
1f5/2 2.079
3/2−
2p3/2 3.267
1/2−
2p1/2 5.663
−
3/2
2p3/2 4.244
3/2−
2p3/2 3.924
−
7/2
1f7/2 2.447
−
7/2
1f7/2 2.318
3/2−
2p3/2 2.250
1/2−
2p1/2 5.363
3/2−
2p3/2 4.749

0.232
0.225
0.195
0.142
0.102
0.089
0.076
0.058
0.049
0.047
0.045
0.043
0.041

Table A.1: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(48 Cr, 49 Mn + γ)X reaction arranged in order of decreasing spectroscopic strength and separated into states
located above and below the proton separation energy of Sp = 2085(5) keV [163]. The
final-state, 49 Mn energies (Ex ) are taken from the shell model predictions. Only levels with
spectroscopic strength of C 2 S ≥ 0.04 are listed. The spectroscopic factor to the ground
state is nearly zero and is included separately (bold text). All states listed below Sp , i.e.
Ex < Sp , are used in the theoretical calculation of the inclusive and partial cross sections
(see Table 6.4). The quantum numbers n j refer to the 50 Fe orbital of the transfered proton.
The table data is plotted in Figure 6.9.

187

Iπ
(50 Fe)

n j

Ex
C 2 Ss.m.
(MeV)

Iπ
(50 Fe)

Below Sp = 4150(60) keV
2+
1f7/2 0.788
1.145
1+
1f7/2 3.349
0.530
+
6
1f7/2 3.527
0.371
+
3
2p3/2 3.398
0.309
2+
2p3/2 3.044
0.232
3+
2p1/2 3.544
0.214
+
4
1f7/2 1.776
0.208
2+
1f7/2 2.672
0.194
+
4
2p5/2 3.956
0.169
3+
1f5/2 3.544
0.123
+
2
2p1/2 3.044
0.112
+
3
1f7/2 3.634
0.094
2+
2p3/2 3.948
0.076
4+
2p3/2 3.956
0.074
3+
2p3/2 3.544
0.069
4+
2p3/2 3.733
0.061
+
2
2p3/2 3.464
0.040
0+
1f5/2
0
0.033
4+
1f7/2 3.320
0.031
4+
2p5/2 3.080
0.027
4+
1f7/2 3.733
0.023
Continued on right column. . .

n j

Ex
C 2 Ss.m.
(MeV)

From left column. . .
2+
2p3/2 2.672
6+
1f7/2 2.959
+
4
1f7/2 3.956
+
5
1f5/2 3.577

0.023
0.023
0.022
0.026

Above Sp = 4150(60) keV
1+
2p3/2 5.499
+
3
1f5/2 4.679
+
4
2p5/2 5.007
2+
1f5/2 4.411
+
4
2p5/2 4.836
+
1
2p3/2 4.410
4+
2p3/2 5.075
+
3
1f5/2 4.529
5+
1f5/2 4.313
+
5
1f5/2 5.773
+
3
2p3/2 5.554
4+
2p3/2 4.666
+
5
1f5/2 5.285
5+
1f7/2 4.295
+
2
2p1/2 4.411
+
4
2p3/2 4.836

0.481
0.221
0.155
0.147
0.120
0.117
0.098
0.091
0.075
0.064
0.058
0.049
0.048
0.045
0.040
0.040

Table A.2: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(49 Mn, 50 Fe + γ)X reaction arranged by descending spectroscopic strength and separated into states below and
above the proton separation energy of Sp = 4150(60) keV [163]. Only values for which
C 2 S ≥ 0.04 are reported. The final-state, 50 Fe energies (Ex ) are taken from the shell-model
predictions. The ground state is highlighted and the data continues from the bottom
of the second column to the top of the first. The states used in the calculation of the
theoretical partial and inclusive cross sections (see Table 6.8) are noted by blue text. Two
possible fragmented 6+ states are considered based upon the proximity of their energies
of 3.257 MeV and 2.959 MeV to the literature value of 3.159 MeV. Note the high degree
of fragmentation to states above and below the proton separation energy. The quantum
numbers n j refer to the 50 Fe orbital of the transfered proton. The table data is plotted in
Figure 6.10.

188

Iπ
(51 Co)

Ex
C 2 Ss.m.
(MeV)

n j

Below Sp = 2085(25) keV
7/2−
1f7/2
0
0.246
Above Sp = 2085(25) keV
3/2− 2p3/2 0.709
1/2− 2p1/2 0.808
5/2−
1f5/2 1.264
1/2− 2p1/2 3.834
3/2− 2p3/2 1.857
5/2−
1f5/2 2.325
−
1/2
2p1/2 4.655
−
3/2
2p3/2 2.891
5/2−
1f5/2 3.171
−
3/2
2p3/2 3.490
−
1/2
2p1/2 4.343
1/2− 2p1/2 5.095
1/2− 2p1/2 2.866
3/2− 2p3/2 2.351

0.366
0.237
0.228
0.187
0.146
0.127
0.112
0.086
0.071
0.065
0.063
0.061
0.061
0.051

Table A.3: Shell-model spectroscopic factors (C 2 Ss.m. ) for the 9 Be(50 Fe, 51 Co+γ)X reaction
arranged in order of decreasing spectroscopic strength and separated by states above
and below the calculated separation energy of Sp = 88 keV [163]. Only levels with a
spectroscopic strength of C 2 S ≥ 0.04 are included. The final-state, 51 Co energies (Ex )
are taken from the shell model predictions. Only the ground state was used in the
calculation of the theoretical partial and inclusive cross sections (see Table 6.9). The
quantum numbers n j refer to the 50 Fe orbital of the transfered proton. The table data is
plotted in Figure 6.11.
Iπ
(49 Mn)

Ex
(keV)

σtheory
(mb)

σexp.
(mb)

5/2−
g.s.
7/2−
1
1/2−
1
3/2−
1

0
260.8(11)
1703
1741

0
1.56
0.01
0.03

≤ 0.20(17)
1.43(15)
–
–

Inclusive sum:

1.60

1.63(8)

Table A.4: Measured and predicted partial cross sections for the 9 Be(48 Cr, 49 Mn + γ)X
reaction as plotted in Figure 6.17. See the figure caption for additional details. Energies
taken from shell model calculations, the 1/2− and 3/2− states, are indicated by an italic
font.

189

4+ state included

4+ state excluded

Iπ
(50 Fe)

Ex
(keV)

σtheory
(mb)

σexp.
(mb)

σtheory
(mb)

σexp.
(mb)

0+
1
2+
1
4+
1
6+
6+

0
765.6(20)
1851.7(55)
3159
3159

0.00
0.40
0.13
0.02
0.32

≤ 0.21
0.81(41)
0.59(34)
–
–

0.0
0.87
–
–
–

≤ 0.21
1.41(22)
–
–
–

2+
2
2+
3

2672
3044

0.07
0.00

–
–

–
–

–
–

1+
1
3+
1
3+
2

3349
3398
3544

0.10
0.01
0.00

–
–
–

–
–
–

–
–
–

1.05

1.38(7)

0.87

1.38(7)

Inclusive sum:

Table A.5: Measured and predicted partial cross sections for the 9 Be(49 Mn, 50 Fe + γ)X
reaction as plotted in Figure 6.18. The high uncertainty on the yield of the 4+ state affects
1
the uncertainty on the yield of the 2+ state because of feeding subtraction. Thus the results
1
are presented with and without the 4+ (and higher lying) states included. The 6+ , 1+ and
1
3+ states were not observed experimentally. The energies were taken from shell model
calculations and are italicized to note their origin. The highest energy experimental cross
sections listed in column 4 (4+ state included) and column 6 (4+ state excluded) should be
considered upper bounds because of unobserved feeding. The theoretical prediction of
column five includes feeding contributions from the 4+ and 6+ states. The two 6+ states
1
1
1
represent two fragments predicted by shell model calculations with different spectroscopic
factors.

Jπ
(51 Co)
7/2−

Ex
σtheory
(keV) (mb)

σexp.
(mb)

0

0.66

0.57(8)

Sum:

0.66

0.57(8)

Table A.6: Measured and predicted partial cross sections for the 9 Be(50 Fe, 51 Co + γ)X
reaction as plotted in Figure 6.19. No excited states were observed, and therefore the the
cross section to the ground state was taken from the inclusive cross section.

190

Jπ
(49 Mn)

Ex
(keV)

σtheory
(mb)

σexp.
(mb)

5/2−
7/2−
1/2−
3/2−

0
260.8(11)
1703
1741

0
1.65
0.01
0.04

≤ 0.40(25)
1.60(22)
–
–

Sum:

1.70

2.00(13)

Table A.7: Measured and predicted partial cross sections for the 12 C(48 Cr, 49 Mn + γ)X
reaction as plotted in Figure 6.25. See the figure caption and the caption of Table A.7 for
additional details. Energies taken from shell model calculations, i.e. the 1/2− and 3/2−
states, are indicated by an italic font.

4+ state included

4+ state excluded

Jπ
(50 Fe)

Ex
(keV)

σtheory
(mb)

σexp.
(mb)

σtheory
(mb)

σexp.
(mb)

0+
g.s.
2+
1
4+
1
6+
1
6+
2

0
765.6(20)
1851.7(55)
3159
3159

0.00
0.43
0.14
0.02
0.34

≤ 0.45
0.88(45)
0.82(24)
–
–

0.02
1.04
–
–
–

≤ 0.45
1.70(38)
–
–
–

2+
2
2+
3

2672
3044

0.07
0.01

–
–

–
–

–
–

1+
1
3+
1
3+
2

3349
3398
3544

0.11
0.01
0.00

–
–
–

–
–
–

–
–
–

1.12

1.72(24)

1.12

1.72(24)

Inclusive sum:

Table A.8: Measured and predicted partial cross sections for the 12 C(49 Mn, 50 Fe + γ)X
reaction as plotted in Figure 6.26. Results are presented with and without the inclusion of
the 4+ state. The transitions in italics (all levels higher in energy than the 4+ state) were not
1
1
observed experimentally and the energies listed are taken from shell model calculations.
The highest energy experimental cross sections listed in column 4 (4+ state included) and
column 6 (4+ state excluded) should be considered upper bounds because of unobserved
feeding. The theoretical prediction of column five includes feeding contributions from the
4+ and 6+ states. The two 6+ states represent two shell-model fragments with different
1
1
spectroscopic factors. The experimental partial cross section to the 6+ state was not
1
extracted because of the very low statistics and the high resultant uncertainty.

191

Jπ
(51 Co)
7/2−
g.s.

Ex
σtheory
(keV) (mb)
0

Inclusive sum:

σexp.
(mb)

0.70

0.53(13)

0.70

0.53(13)

Table A.9: Measured and predicted partial cross sections for the 12 C(50 Fe, 51 Co + γ)X
reaction as plotted in Figure 6.27. Because no transitions were observed, the cross section
to the ground state is identically equal to the inclusive cross section.

192

Appendix B
THE BIRTH OF NUCLEAR PHYSICS
B.1

A Symphony of Effort

The grand mall of nuclear physics, like all science, was built through the effort of thousands of people. Most of the people who laid down their bricks are forgotten, but their
contribution was essential in creating the path that other intrepid explorers used and
lengthened. When creating any history, people only mention the exceptionals, the intellectual forces that alter the landscape. In sake of brevity, this must be done and we do the
same below. But one should always keep in mind that these brilliant men and women do
not work in isolation.
Only topics that are important to the central themes of nuclear experimentation and
shell structure are covered below. This viewpoint is biased and gives artificial gravity to
certain topics, but in pursuit of relevance and conciseness, these concessions were made.

B.2

History

The modern concept of the nucleus, as the tiny but extremely dense aggregate of protons
and neutrons occupying the center of all atoms, emerged within the last one hundred
years. At the turn of the 19th century there were still open questions about the existence of
the nucleus. Certainly there was a growing body of evidence. The chemists Robert Boyle
and John Dalton (considered the father of modern atomic theory), showed atoms were a
convenient concept that explained such phenomena as pressure, why chemicals combined
in integer ratios, and why elements could not be further broken down. But atoms as
isolated entities could not be studied. This lack of a direct experimental probe allowed
193

legitimate objections to exist. Max Planck, invoking entropy and the irreversibility of
physical processes, stated “the consistent implementation of the second law is incompatible
with the assumption of finite atoms. One may anticipate then in the course of further
development of the theory a battle will develop between the two hypothesis which will
cost one of them its life” [171].
It was Joseph John “J.J.” Thompson who defined the atom as a composite object through
the discovery of the electron in 1897 [172]. The great connection was made: given a
particle that is lighter than any known matter, and given that it comes from matter, it
must be a component of matter. And if a component of an atom exists, an atom must exist
as well. J.J. Thompson was an experimentalist, and he discovered the electron from his
studies of cathode ray tubes. Thompson found that the glowing beam in a cathode-ray
tube was not made of light waves or atoms as previous experimenters had thought, but
was instead composed of charged negatively charged particles. Through clever application
of magnetic and electric fields, he discovered that these particles had a mass far smaller
than a hydrogen atom. Thompson would later embed these electrons in a sea of charged
mass. This “plum pudding” model (1904) [173], named because electrons were distributed
throughout the atom like raisins in plum pudding, was the first theoretical model of
nuclear physics.
Radioactivity was discovered by Henri Becquerel, an expert in fluorescence and phosphorescence, in 1896, as he sought to continue Wilhelm Conrad Röntgen’s work on x-rays.
In 1895, Röntgen (English spelling William Roentgen) found x-rays emanating from the
fluorescing glass wall of a covered cathode-ray tube as inferred from the nearby fluorescence of barium platinocyanide [174]. Röntgen would earn the first Nobel prize in Physics
for this work. Becquerel, recalling a public lecture by Röntgen, knew that the x-rays were
emanating from the fluorescent material in the cathode-ray tube, and decided to try uranic
salts (potassium uranium sulfate) to see if they fluoresced x-rays. Becquerel sprinkled
the salts on a photographic plate covered by black paper with the intent of exposing

194

the crystals to direct sunlight. Cloudy weather postponed his experiment for two days.
Bequerel decided to develop his plates without exposure to the sun expecting only a faint
image. Instead, the photographic plate showed dramatic blackening. Subsequent probes
of this effect led Becquerel to conclude that the crystals themselves emitting radiation
[175, 176]. Radioactivity had been discovered, and we mark this point as the beginning of
nuclear physics.
These initial sparks beget a torrent of discoveries. Marie Curie, née Skołodowska, using
an electrometer invented by Pierre and Jacques Curie (brother of Pierre), examined the
radiation emitted by Uranium. This radiation was constant despite different methods of
preparing the samples (solid, powder, etc.). The intensity of the radiation was only affected
by the amount of Uranium in the sample. M. Curie had a critical insight: this radiation
must be an intrinsic property of Uranium and not an effect of chemistry. She continued
her work, now with her husband P. Currie. Together they discovered and isolated the new
radioactive elements Polonium and Radium [177–179]. M. Curie provided the name for
this new material property: radioactivity.
Max Planck, in 1900, introduced the quantization of photons to explain the spectrum
of black-body radiation: how do the intensity and frequency of radiation relate in a cavity
that perfectly absorbs and emits incident radiation [180, 181]. His solution contained the
fuel for a revolution, but Planck at that time did not grasp the full importance of his work.
The fire was largely provided by Albert Einstein 1905 during his miracle year (Annus
Mirabilis) in his paper describing the photoelectric effect, and later in 1907 and 1909 in
papers on the quantum theory of the specific heats of solids and on energy fluctuations
[182–184].
Starting his work in the Cavendish Laboratory, and later at McGill Laboratory and
Manchester University, Ernest Rutherford began systematically dissecting the atom.
Rutherford’s work would span several decades, and the stream of discoveries that emerged
from his labs were so critical in the early development of nuclear physics, Rutherford

195

today is called the father of nuclear physics. Based on the absorption or stopping power
of radiation in matter, Rutherford defines α-rays (a 4 He nucleus1 ) and β-rays (electrons)
as distinct types of radiation. In 1900, at McGill Laboratory, he discovers, along with
his current partner the radio chemist Fredrick Soddy, that radiation is emitted by the
transmutation of atoms. The original discovery came from the observation of the decay
from Thorium into Radium (228 Th −→ 228 Ra + α) [185–188]. Later they discover that
Thorium and Radium decays into the noble gas 220 Ra (Radon)2 . Rutherford and Soddy
note that these different radioactive materials have different characteristic times it takes
them to decay to half their previous value. A half-life is defined as a unique property of
these radioactive materials [189]. They start categorizing these values. Rutherford would
name γ radiation (photons) as a distinct type in 1903, based upon Paul Ulrich Villard’s
discovery in 1900 of a new highly penetrating type of radiation.
The Geiger-Marsden experiment ran in 1909. Hans Geiger and Ernest Marsden, a young
undergraduate, directed a collimated beam of α particles emitted by Radium decay upon
a thin foil gold target. The target was surrounded by a sheet of fluorescing zinc sulfide.
Earlier Rutherford had noticed that α particles when projected through a small slit of
mica produce an image blurred near the edges. The electrical forces necessary to produce
the deviation in flight path of the alpha particles was enormous. Rutherford hoped that
the gold scattering experiments would solve the puzzle of the mica slit experiment and
provide insight into the structure of the atom.
Rutherford suggested that Geiger and Marsden check for backscattering for alpha
particles off the gold foil. To their great surprise they found scattering greater than 90
degrees. Rutherford was astonished. In his words, “It was quite the most incredible event
1 Rutherford,

continuing work in Manchester with Hans Geiger on alpha radiation,
notes that alpha particles have a mass to charge ratio of two. With Thomas Roylds, in
1907, he lets alpha particles penetrate through a thin glass wall of an evacuated discharge
tube. The spectrum of the gas inside the tube was identified as Helium and the connection
was made: alpha particles are the nuclei of Helium atoms.
2 The

endpoint of the 228 Th decay chain is stable 208 Pb
196

that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch
shell at a piece of tissue paper and it came back and hit you” [190]. Rutherford realized
that the only way to get large deflections was to concentrated most of the positive mass
into the center of the atom. It took Rutherford two years to publish [191, 192]. Rutherford
had found the nucleus.
Although core concept of Rutherford’s model, that every atom has a tiny nucleus where
all of its positive charge is concentrated, is true, there were major flaws with Rutherford’s
theory, even despite the experimental evidence. First, despite each electron’s attraction
to the positively charged core, the repulsive force betwen electrons leads to instabilities
which rip the atom apart. A stable solution does not exist. Second, an accelerating
charged electron should radiate energy. The gradual loss of electron energy would lead
the electrons to spiral into the core of the atom. For these reasons, Rutherford’s theory
of the atomic nucleus was not taken seriously by leading contemporaries. It would take
further effort by a unique mind to resolve these difficulties and further connect the internal
majesty of the atom to experiment. This person was the great theoretical physicist Niels
Bohr.
Bohr, in 1912, provided several great insights into the nature of the atom. First, the
radioactive properties come from the atomic nucleus; the chemical properties come from
the electron stucture. Second, it is the nuclear charge that determines a nuclei’s placement
in the period table. And third, during radioactive decay, the position of an element on the
periodic chart shifts two places to the left if the atom emits an α particle and one space to
the right if the atom emits a β particle (nuclear displacement law).
Bohr hoped to resolve the instabilities associated with Rutherford’s model and he did
so by bringing quantization to the atom, and in a similar way this radical concept had
resolved the problems with black-box radiation (Planck) and the energy of photoelectrons
(Einstein), quantization solved the problems associated with the instability of the atom.
Bohr’s atom placed electrons in orbits around the atom, like planets around the sun. The

197

quantized energies of photons emitted by atoms were caused by the transition between
these different levels. Bohr’s revelation came from a last minute connection atomic physics
and transitional spectroscopy. A contemporary, J.W. Nicholoson, had published a paper
proposing a Saturian system to explain the unusual properties of the Sun’s corona. It
was immediately clear to Bohr that spectral lines were caused by the different transitions
between the different orbitals of the atom [193].
Bohr’s paper, “On the constitution of atoms and molecules”, was sent in three parts to
Rutherford in 1913 [194–196]. He proposed his new model of the atom and calculated
2 me4
Ryberg’s constant to great precision with the formula R = 2π 3 . This work would
h

win Bohr the 1922 Nobel Prize in Physics. The young experimentalist Henry “Harry”
Gwyn Jefferys Moseley began his tireless work looking for shifts in the x-ray spectra of
different elements (the spectra were generated by passing x-rays through different crystals
of material). Moseley’s experimental work [197, 198], which in 1914 firmly established
the connection between nuclear charge and atomic number, convinced contemporaries
that the Bohr-Rutherford model of the atom was correct. Moseley, one of the most talented
experimentalists of his time, would die at age 27 in World War I.
Scientific progress stalled during World War I (1914 - 1918), but the small resumption
of normalcy that the end of the war provided further advances. Francis William Aston
invented the modern mass spectrograph in 1919 [199] based on work by J.J. Thompson.
His work was motivated by the separate discovery of isotopes in 1913 by Soddy and
Thompson [200–203]. Isotopes (from the Greek “at the same place”, as named by a distant
relative of Soddy, the Scottish Physician Margret Todd) are atoms with the same number of
protons but a different number of neutrons. Soddy noticed from his work on radioactive
decay that positions on the periodic table appear to be occupied by multiple entities,
different elements with the same chemical properties. J.J. Thompson, on his work on
anode rays, put a beam of neon atoms through an electric and magnetic field and noticed
that there were two separate marks for neon on his photographic plate. This original

198

basic setup of Thompson and others was refined into the precision instrument of Aston’s.
Aston would go on to identify 212 naturally occurring isotopes. Aston’s basic design has
evolved into a variety of different devices. The modern S800 magnetic spectrograph at the
National Superconducting Cyclotron Laboratory (NSCL) allows experimenters to easily
identify different isotopes that enter the focal plane of the detector [124, 204]. The S800
spectrograph is covered in greater detail in Sec. 4.3, page 61.
Ernest Rutherford created the first artificial nuclear reaction α(14 N, 17 N + p) by bombarding nitrogen gas with α particles [205]. In 1921, Bohr connects the periodic table with
the Bohr-Rutherford model of the atom [206]. Bohr’s insight: as electrons are added to
an nucleus they fill up the orbitals in order of lowest to highest energy; the lower energy
shells must be full before electrons are placed in higher energy orbitals3 . Each orbital,
also called a sub-shell, are arranged in collections that share the same principle quantum
number n. The number of allowed electrons per shell depends only on the principle
quantum number n as 2n2 and the number of allowed electrons per sub-shell (orbital)
depends only upon the azimuthal quantum number
several subshells whose

as 2(2 + 1). Each shell can have

value must be less than n (see Sec. 1.2.4 for the relationship

between spectroscopic notation and the associated integer value of angular momentum).
The occupancy of an orbital n and are used to describe the configuration of an atom. For
example, consider Argon and its shell configuration labeled 1s2 2s2 2p6 . Here the subscript
is used to indicate the filled occupancy of the orbital. With Argon, we make one final
note. Each shell is completely full. This leads to an especially stable, nonreactive atom.
Other atoms have similar shell closure and the collection of these elements forms the noble
gasses. In Sec. 1.2.4, we will see how a similar concepts applies to the configuration of
nucleons in an atom.
Wolfgang Pauli, in 1924, explained the behavior of why electrons built up to certain
3 This maxim works well with lower mass elements.

occur with increasing electron number

199

An increasing number of exceptions

closed shells by noting that each electron had to have a unique set of labeling attributes
[207]. For Pauli’s formulation to work, he had to introduce a new fourth quantum number,
of degeneracy two, that was indescribable classically. George Uhlenbeck and Samuel
Goudsmit proposed that this quantum number was associated with intrinsic angular
momentum of the electron, a unique, inherent property just like mass and charge, and
was not caused by an electrons movement though the atom [208]. Pauli would provide the
complete mathematical description of intrinsic spin in 1927 [209] and a year later the full
relativistic treatment was done by Paul Dirac [210].
Beginning with Louis de Broglie, in 1924, the old system of quantum mechanics where
action is quantized but physics is elsewhere classical, was overturned. de Broglie, in his
doctoral dissertation [211], postulated that all moving matter, not just light, could be
described as a wave where λ = p/h. From this kernel grew the wave formulation and matrix
formulation of quantum mechanics, ascribed to Erwin Schrödinger and Werner Heisenberg
respectively. Heisenberg’s work was published first [212], and showed how quantum jumps
could occur between the stationary orbits of an atom. To do so, Heisenberg’s paper made
a bold claim: certain quantities, like position, could not be exactly known. Two years
later, Heisenberg would publish the generalized version of this principle, Heisenberg’s
uncertainty relationship [213]. Together with the mathematical physicsts Pascual Jordan,
Max Born, and the mathematician David Hilbert, Heisenberg would put this framework
within the context of matrix algebra [214–216].
Erwin Schröndinger published his work after Heisenberg [217]. Schrödinger’s wave
equation begins with de Broglie’s breakthrough, that particles can be described as waves,
and provides the full description of how the particle wavefunction evolves in both time
and space total energy associated with the wavefunction. Schrödinger paper did allowed
for certain classical concepts to persist, and so his paper was more widely accepted than
Heisenberg’s work, but today both the matrix and wave formulations are considered
equivalent and responsible for ushering in the modern era of quantum mechanics.

200

In 1929, Ernest Orlando Lawrence invented the cyclotron [218], a compact method
of accelerating charged ions. Experimentalists faced their own barrier when trying to
untangle nucleus: the long-range Coulomb repulsion between the incoming projectile and
the nucleus of interest. The Coulomb force prevents nuclei from getting close enough
for the projectile and target to interact through the very short-range strong force4 . The
kinetic energies of the projectiles should be significantly above that barrier, and this
kinetic energy is greater than the decay Q value (kinetic energy of a particle emitted
during radioactive decay) for the commonly available radioisotopes. The maximum energy
of α particles produced during decay of common radioisotopes is at most 5 MeV (e.g. the
Q-values for the alpha particles emitted during 238 U and 226 Ra decay are 4.2 MeV and
4.8 MeV). The calculated barriers for α particles on 56 Fe is 4.5 MeV, and for α particles
on 197 Au, the barrier is 15.7 MeV. To study the full range of nuclei, the problem lay with
the experimentalists to accelerate a stream of highly energetic particles that could probe
the very center of the atom. Lawrence, provided the best and most efficient method of his
time. Today, the core of this technology is found in the Coupled Cyclotron Facility (CCF)
at the NSCL [88, 89].
Other less efficient methods of acceleration had also been developed and were producing good science. John Cockcroft and Ernest Walton used a complicated transformer and
voltage multiplying system to accelerate a beam of protons (Cockcroft-Walton accelerator)
[219]. This work would lead to the first nuclear reaction using an accelerator (1932) [220],
the disintegration of 7 Li through the reaction p(7 Li, 8 Be), where the unbound 8 Be disintegrates into two α particles. Other accelerating machines were introduced at a similar
time: The linear accelerator, also overseen Lawrence and constructed by Ernest Sloan, was
constructed simultaneously as the cyclotron at Berkeley, and the new Robert Van de Graaff
generator was used to create high accelerating voltages for Merle Tuele’s work at Princeton
4 The

processes of tunneling, as described by George Gamov, means the probability of a
charged particle at sub-Coulomb kinetic energies reaching the core is extremely small but
not zero.
201

(Van de Graff accelerators).
James Chadwick found the neutron in 1932 [221]. The existence of the neutron had
been theorized by Rutherford twelve years earlier in 1920 [222]. Chadwick built upon the
work Walter Bothe and Herbert Becker who noticed a highly penetrating type of radiation
of was emitted by alpha particles incident upon 9 Be and other light metals, and upon
the work of Irène Joliot-Curie and Frédéric Joliot-Curie who noticed that this new type of
radiation, incident upon paraffin wax, knocked out energetic protons. James Chadwick
realized in his experiments that these protons could not come from γ radiation. Such a
conclusion would violate energy and momentum conservation. The particle that interacted
with the proton had to be massive: the neutron.
Heisenberg quickly incorporated the neutron into a model of the atom (proton-neutron
model) later that year and quickly published the new model of the atom [223–225].
The addition finished the major modern constituents of the atom: electron, proton, and
neutron. Much later (in 1964) it was proposed that protons and neutrons were themselves
not elementary particles, and were composed of quarks, but modern nuclear physics still
treats protons and neutrons as the fundamental building blocks of the atom. Neutrons
and protons are bound together in the nucleus, and whatever force that bound them had
to be much stronger than the Coulomb force (to overcome the Coulomb repulsion between
the protons) and be short range (to explain nuclear saturation, i.e. the proportionality of
binding energy to atomic mass A).
Hideki Yukawa published his theory of meson exchange in 1935 [1]. The meson (from
the Greek word for intermediate, “mesos”) had a heavy mass postulated by Yukawa to be
between that of an electron and proton. The mass would explain the short range force
between nucleons. The pi meson, or pion π, was discovered in 1947 by Occhialini and
Powell [6] and Lattes et al. [7] and was found to participate in the nuclear force (then
called the strong force). Today, although meson exchange is still an applicable and useful
concept in nuclear physics, the strong force is understood as the force between quarks as

202

mediated by gluons. Only a small amount of that force sneaks out beyond the confines of
a nucleon. It is this small, fluctuating residuum, much stronger than the electromagnetic
force, that binds protons and neutrons together.
Slowly models were proposed that helped explain how the nucleus as a unified whole
behaved. George Gamov proposed the liquid drop model, the first theory of nuclear energy
and mass [226, 227], in 1928, the same year he published his theory on α decay [228]
and before the discovery of the neutron. With the discovery of the neutron by Chadwick,
Gamov’s theory was further developed by Niels Bohr, John Archibald Wheeler, and Carl
Friedrich von Weizsäcker and this revised model was used by Lise Meitner and Otto
Robert Frisch to explain the spontaneous splitting of a heavy atom, a process which they
called “fission” [229]. These experimental work communicated to Meitner and Frisch was
performed by Fredrich Wilhelm “Fritz” Straßmann and Otto Hahn, who first observed
fission by detecting Barium after bombarding Uranium with neutrons.
Two papers appeared in 1949 that finally completed the major ideas behind the nuclear
shell model, the theory of how protons and neutrons are arranged in orbitals in the
atomic nucleus. The concept is similar to the early model of the electron orbits N. Bohr
proposed in 1913. Protons and neutrons independently fill orbitals of specific maximum
occupancy, starting with the lowest energy orbitals first (see Fig. 1.4, page 14. Both Maria
Goeppert Mayer at Argonne National Laboratory and the team of Johannes Hans Daniel
Jensen (Hans Jensen), Otto Haxel, and Hans Eduard Suess realized the importance of the
spin-orbit interaction, the interaction between a nucleons orbital angular momentum
and intrinsic spin, in the mean potential that each nucleon feels. They published nearly
simultaneously [24, 25] and both Mayer and Jensen won Nobel Prizes for their work.
The addition of the spin-orbit potential to the potential of a three-dimensional harmonic
oscillator finally described the level structure of a nucleus, in particular the large stability
of nuclei with specific number of protons or neutrons: 2, 8, 20, 28, 50, 82, and 126. This
final configuration, explained many of the current nuclear phenomena including spins

203

and magnetic moments.
The shell model, also called the independent shell model (IPM), worked well for certain
properties of the atom, as enumerated above, but not others, e.g. the existence of rotational
bands and large quadrupole moments and quadrupole transition probabilities, properties
that could be described well by rotational or vibrational models. Aage Bohr (son of Niels
Bohr), Ben Mottelson, and James Rainwater combined the single-particle and collective
behaviors into a comprehensive collective mode, the deformed shell model and the particle
plus rotor model.
The shell model assumes as spherical potential (isotropic harmonic potential), but
for many nuclei, it is experimentally apparent that the nucleus has a stable and static
deformation. Rainwater introduced the concept of deformed shell model where the
individual nucleons move in a deformed mean potential [230, 231]. The deformations to
the harmonic oscillator potential necessary to shift the spherical shell model results to
the experimental observations was given to Sven Gösta Nilsson. Both the model and the
diagrams showing the shift of single-particle energies with respect to nuclear deformation
are named after Nilsson in appreciation of his original paper, published in 1955 [62].
Bohr and Mottelson tried similar methods of combining collective and single-particle
behavior. The particle plus rotor model took the valence particles and coupled these
valence particles to a collective rotor, where the collective rotor accounts for the behavior
of the core [232]. The work of Bohr, Mottelson and Rainwater was rewarded with the
Nobel prize in 1975.
Nuclear physics has evolved far beyond these beginnings. Current vanguard, experimental nuclear facilities, like the NSCL, involve over a hundred people and use complicated and powerful machinery to accelerate radioactive beams. And current theoretical
calculations involve computers that can perform calculations, like the multiplication of
massive matrices, in a few seconds, calculations that would take the whole assortment
of the early brilliant physicists years to do. The scope and size of the field has increased

204

dramatically. Still the basics remain the same. The three original models, although embellished, remain as important as ever. Calculations using the single-particle shell model
are cruicial to this work, and work testing the validity of “magic numbers” remains a
contemporary topic.

205

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