THE (P,N) CHARGE-EXCHANGE REACTION IN INVERSE
KINEMATICS AS A PROBE FOR ISOVECTOR GIANT RESONANCES
IN EXOTIC NUCLEI
By
Samuel Israel Lipschutz

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics—Doctor of Philosophy
2018

ABSTRACT
THE (P,N) CHARGE-EXCHANGE REACTION IN INVERSE KINEMATICS AS A
PROBE FOR ISOVECTOR GIANT RESONANCES IN EXOTIC NUCLEI
By
Samuel Israel Lipschutz
Charge-exchange experiments at intermediate (∟100 MeV/u) beam energies have long
been a tool to study isovector giant resonances in nuclei. Since the 1970s, many probes and
techniques have been developed to study and isolate different isovector giant resonances,
which have illuminated the spin-isospin response of nuclei. However, these experiments have
almost solely been restricted to stable nuclei. The advent of rare-isotope beam facilities has
created increased interest in studying these resonances in radioactive nuclei. Such experiments are important for better constraining models that aim to describe the properties of
nuclei and nuclear matter, with important applications in astro and neutrino physics. To
take advantage of these rare-isotope facilities, new techniques in inverse kinematics need to
be developed and validated. This thesis presents a new technique for studying isovector
giant resonances on unstable nuclei in the ∆Tz = −1 direction, through the (p,n) reaction in
inverse kinematics. The technique was validated through the measurement of the absolute
differential cross section in the neutron-rich 16 C(p,n)16 N* reaction at 100 MeV/u up to ∟20
MeV of excitation energy. From the data, the Gamow-Teller (GT) strength distribution
[B(GT)] and the spin-dipole (SD) differential cross section were extracted. The extracted
B(GT) was compared with shell-model calculations in order to test their reliability in the
neutron-rich, high-excitation energy regime. A total GT strength up to 20 MeV was extracted and compared to the model-independent sum rule, giving a quenching factor of 73.5
Âą2.8(st) Âą 15.5(sys)%. This experiment provided one of the first measurements of isovector

strength in a neutron-rich nucleus and has paved the way for future exploration of isovector
giant resonances in exotic nuclei.

This thesis is dedicated to you, the reader, so that you may feel special.

iv

ACKNOWLEDGMENTS

No PhD is completed in isolation, and certainly not this one. Without the help and
guidance of countless people, my research could not have been completed, and this thesis
could not have been written. Though I cannot thank everyone by name, know that I am
grateful for all the help you have given me and for what it has allowed me to accomplish.
I would like to give a special and wholehearted thanks to my advisor, Remco Zegers. Even
in the depths of the great “factor of four” debate of 2017, you continued to provide the same
excellent advice and mentorship that you have given me over the past six years. Though our
opinions on Dutch licorice sharply diverge, it has not impacted your tireless commitment
to see me through to the end. I would also like to thank the members of my committee
for providing guidance and assistance throughout my time at the lab: Sean Liddick, Alex
Brown, Michael Thoennessen, and Norman Birge.
Shumpei Noji requires special acknowledgment. Without your steadfast assistance much
of this thesis would not exists. Throughout the years you have gone above and beyond to
help me many times. Thank you.
By any measure, completing a PhD is a difficult task. Beyond the research and coursework
challenges, maintaining the necessary perseverance and optimism that is needed to finish is
an undertaking of its own. This was only possible with the help and friendship of my fellow
students. Alanna, Bethany, Charlie, Chris, Eric, and Juan: you have been true friends since
the beginning.
Most importantly, I would like to thank my mother and father. Naturally, without you,
I wouldn’t have gotten very far in life. It must be a distinct source of pride to have two sons
who have squandered their twenties obtaining their PhDs.
v

Finally, I want to thank Jeny for supporting and encouraging me since the day we met.
The help you have given me in just the last few days has been immeasurable. You have been
there for me throughout the toughest parts of this process. I dont know how I could have
done it without you.

vi

TABLE OF CONTENTS

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

ix

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

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . .
1.1 The Atomic Nucleus . . . . . . . . . . . . . . . . .
1.2 The Nuclear Shell Model . . . . . . . . . . . . . . .
1.3 Giant Resonances . . . . . . . . . . . . . . . . . . .
1.4 Nuclear Charge-Exchange Reactions . . . . . . . . .
1.5 Historical Development of Charge-Exchange Probes
1.6 The (p,n) Probe in Inverse Kinematics . . . . . . .
1.7 β Decay of 16 C . . . . . . . . . . . . . . . . . . . .
1.8 The Gamow-Teller Sum Rule . . . . . . . . . . . .
1.9 Quenching of Gamow-Teller Strength . . . . . . . .

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1
1
2
7
10
13
15
17
18
19

Chapter 2 Theoretical Techniques . . .
2.1 Distorted Wave Born Approximation
2.2 Effective Interaction . . . . . . . . .
2.3 Proportionality . . . . . . . . . . . .
2.4 Technical Details of Calculations . .
2.5 Shell Model Results . . . . . . . . . .
2.6 DWBA Results . . . . . . . . . . . .

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22
22
28
29
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36

Chapter 3 Experiment . . . . . . . . . . . . . . . . .
3.1 Experimental Method . . . . . . . . . . . . . . . .
3.2 Experimental Equipment . . . . . . . . . . . . . .
3.2.1 Beam Creation and Delivery . . . . . . . .
3.2.2 Ion Source . . . . . . . . . . . . . . . . . .
3.2.3 Cyclotrons . . . . . . . . . . . . . . . . . .
3.2.4 A1900 . . . . . . . . . . . . . . . . . . . .
3.2.5 S800 Spectrograph . . . . . . . . . . . . .
3.2.6 Focal Plane Detectors . . . . . . . . . . .
3.2.7 Liquid-Hydrogen Target . . . . . . . . . .
3.2.8 LENDA and VANDLE . . . . . . . . . . .
3.3 Digital Data Acquisition System . . . . . . . . . .
3.3.1 DDAS Overview . . . . . . . . . . . . . .
3.3.2 Digital Timing . . . . . . . . . . . . . . .
3.3.3 Intrinsic Timing Resolution of DDAS . . .
3.3.4 Timing Resolution of LENDA with DDAS
3.3.5 Energy Extraction Methods . . . . . . . .
3.3.6 KeVee to Tn . . . . . . . . . . . . . . . . .

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39
39
44
44
44
45
46
47
48
50
50
52
52
53
54
58
60
63

vii

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3.4

Implementation with the S800 .
3.4.1 Clock Synchronization .
3.4.2 Operation Infinity Clock
3.4.3 Trigger Logic . . . . . .

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64
64
65
67

Chapter 4 Data Analysis . . . . . . . . . . . . . .
4.1 Calibration . . . . . . . . . . . . . . . . . . . .
4.1.1 Neutron Detector Timing Calibrations .
4.1.2 LENDA and VANDLE Gain Calibrations
4.1.3 Particle Identification . . . . . . . . . . .
4.1.4 CRDC Calibration . . . . . . . . . . . .
4.2 Experimental Yield . . . . . . . . . . . . . . . .
4.3 Neutron Detector Efficiency and Acceptance . .
4.4 Background Subtraction . . . . . . . . . . . . .
4.5 Absolute Normalization Corrections . . . . . . .
4.5.1 Incident-Beam Measurement . . . . . . .
4.5.2 S800 Acceptance . . . . . . . . . . . . .
4.5.3 Target Thickness . . . . . . . . . . . . .
4.5.4 Target Density Variations . . . . . . . .
4.6 Cross Section Calculation . . . . . . . . . . . .
4.7 Systematic Error . . . . . . . . . . . . . . . . .

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70
70
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81
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96
102
102
103
108
111
112
116

Chapter 5 Results . . . . . .
5.1 Angular Distributions . .
5.2 Multipole Decomposition
5.3 B(GT) Extraction . . . .
5.4 Dipole Strength . . . . .

. . . . .
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Analysis
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Chapter 6 Discussion . . . . .
6.1 The (p,n) Probe in Inverse
6.2 Shell Model Comparison .
6.3 Quenching . . . . . . . . .
6.4 The First Two 1+ States .
Chapter 7

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121
121
122
127
130

. . . . . . .
Kinematics
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134
134
135
136
138

Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 144

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Derivation of the Gamow-Teller Sum Rule . . . .
Appendix B Energy of a Compton Edge . . . . . . . . . . . .
Appendix C The Number of Beam Particles in a Beam Bunch

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146
147
150
152

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

viii

LIST OF TABLES

Table 1.1: Summary of the results from previous β-decay measurements of 16 C and
from theoretical calculations. a taken from Ref. [73]. b taken from Ref. [71]. 18
Table 1.2: Quenching factors of GT strength for nuclei occupying the p, sd and pf
shells. The factors presented are the quenching of the GT operator. To
find the quenching of GT strength, the factor must be squared, so in the
sd shell it is 0.762 = 0.58. . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Table 2.1: Parameters for the optical potentials used in the 16 C(p,n)16 N cross-section
calculations. All parameters are given in MeV and Fermi. V refers to a
volume term and SO a spin-orbit term. The volume terms are WoodsSaxon potentials. The spin-orbit terms are derivatives of Woods-Saxons.
The p and n refer to the potentials for the proton (entrance) and neutron
(exit) channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Table 2.2: Isospin Clebsch-Gordan factors for the Ti = 2 → Tf = 1, 2, 3 transitions.
The factors give in column 3 are normalized to the T − 1 transition. . . .
Table 3.1: Table of the different decay channels accessible in the 16 C(p,n)16 N reaction for ranges of excitation energy in 16 N and their associated magnetic
rigidities. The acceptance of each rigidity setting is Âą %5. . . . . . . . . .
Table 4.1: Table of the Îł-ray calibration sources used for the neutron detector gain
calibration. See text for details. . . . . . . . . . . . . . . . . . . . . . . . .

33

43

76

Table 4.2: Summary of corrections to the cross sections. Parameters with a range in
values indicate that a run-by-run correction was performed. See text for
details on how each correction was determined. . . . . . . . . . . . . . . . 114
Table 4.3: Summary of the source and estimated size of the major systematic uncertainties in the 16 C(p,n)16 N measurement. See text for details on each
source of systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . 118
Table 6.1: Comparison of the quenching factors of GT strength for 16 C, various nearby
nuclei, and adopted values for several nuclear shells. Quenching studies
from charge-exchange reactions and β decay studies are presented. The
p-shell factor comes from the empirical relation given in Ref. [117] with
A = 16. See text for details. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

ix

Table 6.2: Charge-exchange differential cross section at zero momentum transfer and
extracted B(GT) for the first two 1+ states in 16 N. A breakdown of the
statistical and systematic error is shown. The systematic error for the
B(GT) is increased by the uncertainty in σ̂GT . . . . . . . . . . . . . . . . 139
Table 6.3: Comparison of the B(GT) values from charge-exchange, β decay, and the
shell model for the first two 1+ states in 16 N. . . . . . . . . . . . . . . . . 140

x

LIST OF FIGURES

Figure 1.1: Chart of the nuclides showing the number of neutrons on the x axis and the
number of protons on the y axis. The black squares show the stable nuclei,
the blue region shows nuclei that have been observed but are unstable,
and the red region shows nuclei that are predicted to exist but have not
been observed experimentally. The dashed lines show the nuclear magic
numbers. Figure taken from Ref. [3]. . . . . . . . . . . . . . . . . . . . .

2

Figure 1.2: Neutron single-particle states in 208 Pb with three different potential models. The left shows a harmonic oscillator potential, the center shows a
Woods-Saxon potential, and the right shows a Woods-Saxon plus spinorbit potential. The numbers in brackets give the maximum number of
neutrons that each state can hold while the adjacent numbers give the
running sum. Figure taken from Ref. [7]. . . . . . . . . . . . . . . . . . .

4

Figure 1.3: Schematic of the macroscopic depiction of various giant resonances. For
each ∆L value, the different possible spin and isospin transfers are shown.
Figure taken from Ref. [18]. . . . . . . . . . . . . . . . . . . . . . . . . .

8

Figure 1.4: Shell-model schematic showing the different 1p-1h contributions of the GT
and SD giant resonances in the 16 C(p,n)16 N reaction. . . . . . . . . . . .

17

Figure 2.1: Experimental unit σ̂GT and σ̂F as a function of mass number. The dashed
and dotted lines are the result of the parameterization of Taddeucci et al.
Figure is reproduced from Ref. [56]. . . . . . . . . . . . . . . . . . . . . .

31

Figure 2.2: The square root of the ratio of the GT and Fermi unit cross sections as a
function of incident proton energy. The data points come from reactions
on 14 C at various energies. The dashed line is a linear fit to the data
above 50 MeV. Figure taken from Ref. [56], see references therein. . . . .

32





Figure 2.3: Shell Model calculations of B(GT) for 16 C0+ ; T = 2 →16 N1+ ; T = 1, 2
transitions. The WBP (a) and WBT (b) interactions were used in the
spsdpf model space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Figure 2.4: Shell-model spectrum with the WBP interaction for the SD
16 N, presented as the peak cross-section value for each state.
into final states of 16 N with Jfπ = 0− , 1− , 2− are shown. All
T = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

xi

strength in
Transitions
states have
. . . . . . .

Figure 2.5: Scatter plot of the shell-model B(GT) for 0+ → 1+ transitions against
the theoretical σ̂ determined from DWBA. The shell-model calculation
was done with the WBT interaction in the spsdpf model space. The red
dot indicates the 2nd shell-model state. In e10003 states with strengths
below ∟ 0.05 could not be discerned. . . . . . . . . . . . . . . . . . . . .

37

Figure 2.6: The model angular distributions for ∆L = 0, 1, 2 that are used in the
MDA analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Figure 3.1: Kinematic correlations for the 16 C(p,n)16 N reaction at 100 MeV/u, in
the plane of the laboratory neutron angle (θn ) and neutron kinetic energy
(Tn ). The blue box indicates the approximate coverage of the neutron
detectors in e10003. Solid lines indicate the excitation energies in 16 N.
Dashed lines show the COM scattering angles. . . . . . . . . . . . . . . .

40

Figure 3.2: Top-view schematic of the experimental setup for the neutron detectors.

41

Figure 3.3: Picture of the complete setup for experiment e10003. In the bottom center
of the picture, the south LENDA bars are visible. The north VANDLE
bars can be seen in the top left. The liquid hydrogen target can be seen
in the center of the photograph. . . . . . . . . . . . . . . . . . . . . . . .

42

Figure 3.4: Schematic showing the coupled cyclotrons and the A1900 fragment separator. Figure taken from Ref. [95]. . . . . . . . . . . . . . . . . . . . . .

45

Figure 3.5: Schematic of the S800 analysis line and spectrometer. Figure adapted
from [94]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Figure 3.6: Schematic of the two cathode readout drift chambers, with an example
event trajectory passing through the detectors. The inset shows an example of a charge distribution detected by the pads. Figure originally taken
from Ref. [101] and modified by Ref. [102] . . . . . . . . . . . . . . . . .

49

Figure 3.7: Pressure-temperature diagram for the liquid hydrogen target showing the
phase transition from gas to liquid. The red line indicates the expected
relationship taken from [104]. . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 3.8: Example of digital trace from LENDA (filled circles) with overlaid digital CFD (filled squares). The thick line is the third-order interpolation
function calculated for this event. . . . . . . . . . . . . . . . . . . . . . .

55

Figure 3.9: Time-difference spectra from a split-signal electronics test. The time differences from the 3 different timing algorithms have been shifted to zero
for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

xii

Figure 3.10: Electronic timing resolution of DDAS as measured with a split signal from
a LENDA bar. The upside down triangles utilize a linear-interpolation
method while the filled circles use a cubic-interpolation method. See text
for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 3.11: Measured timing resolution for LENDA readout with DDAS. Upside down
triangles use a linear-interpolation method, while the filled circles use a
cubic-interpolation method. . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 3.12: A trace from a PMT of LENDA from a 22 Na source. The first integration
region is used to determine the average baseline of the signal, while the
second integrates the pulse and determines the energy of the event. . . .

60

Figure 3.13: Top panel shows a calibrated energy spectrum from 22 Na seen in a LENDA
bar. Bottom panel shows a calibrated energy spectrum from a 241 Am source. 62
Figure 3.14: Scatter plot of light output against neutron energy from a 252 Cf source
measured with LENDA and a EJ-301 liquid scintillator as the timing
reference. The cuts on maximum light output are applied in this figure. .

63

Figure 3.15: Detailed diagram of the electronics setup in experiment e10003. The
NSCL numbers refer to the NSCL E-Pool number for the module depicted. Care has been taken to draw the modules with enough detail to
make them easily recognizable. The positions of the switches depicted in
the first two modules on the left correspond to their actual setting in the
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 4.1: Two dimensional histogram showing the correlation between the neutron
TOF, and dispersive angle (a) and position (b) in the S800 focal plane.
The Îł flash is seen prominently in the figures. The neutron events are to
the right, but are cut off to clearly show the Îł flash. . . . . . . . . . . . .

72

Figure 4.2: Neutron TOF plotted against the calibrated light output from the neutron
detectors. The Îł flash can be seen centered around 0. At low energies, a
small correlation can be seen for the Îł events. . . . . . . . . . . . . . . .

73

Figure 4.3: Fully Corrected neutron time of flight spectrum for the N15 data (3.1325
Tm setting). The dashed green line indicates the TOF cut between Îł and
neutron events. It corresponds to a neutron kinetic energy of ∟67 MeV,
which is far above the energies used in the analysis. . . . . . . . . . . . .

74

Figure 4.4: Corrected neutron TOF spectra for the 14 N (2.9 Tm) (a), 16 N (3.38 Tm)
(b), 15 C (3.6565 Tm) (c), and 14 C (3.38 Tm) (d) channels. The 15 C and
14 C data from the 3.52 Tm and 3.2828 Tm settings are not shown, but
have similar TOF spectra. . . . . . . . . . . . . . . . . . . . . . . . . . .

75

xiii

Figure 4.5: Example light output calibration for south LENDA bar 2. The data points
are described in Table 4.1. The red line is a linear fit to the data. . . . .

77

Figure 4.6: Uncalibrated Particle Identification for the 3.1325 Tm setting. . . . . . .

78

Figure 4.7: Correlation between the ejectile TOF and the dispersive angle (a) and
position (b) in the S800 focal plane for 15 N events. The artifacts seen
in (b) are the result of bad events in the CRDCs, which are removed in
subsequent steps of the analysis. . . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.8: Corrected PID spectra from the 3.1325 Tm (a), 2.9 Tm (b), 3.6565 Tm (c),
and 3.38 Tm (d) settings. The main particles of interest in each setting
are indicated. The remaining two settings 3.52 Tm and 3.2828 Tm are
not shown, but are similar to the 3.6565 Tm and 3.38 Tm settings. . . .

80

Figure 4.9: The uncalibrated position (x-axis) and time (y-axis) measured by the
CRDCs during the mask runs. . . . . . . . . . . . . . . . . . . . . . . . .

82

Figure 4.10: The pad distribution for CRDC 1 before (a) and after (b) the pad-by-pad
gain matching. The data is from one experimental run of the 3.1325 Tm
rigidity setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Figure 4.11: Two dimensional histogram of lab neutron angle and kinetic energy measurements (a,c,e,g,i), and reconstructed excitation energy in 16 N and COM
scattering angle (b,d,f,h,j). In (a,c,e,g,i) the solid lines indicate excitation
energy in 16 N and the dashed lines indicate the COM scattering angle. .

87

Figure 4.12: Experimental yield for the 15 N channel (3.1325 Tm) in the Θcm = 4−6° bin. 89
Figure 4.13: A cartoon depicting the interactions of neutrons in a plastic scintillating
material. In order for the neutron to be detected it must undergo a nuclear
reaction with the hydrogen or carbon nuclei in the material. The angles of
the scattering are exaggerated for artistic purposes. The dominate elastic
scattering on hydrogen is shown at the top of the figure. Elastic (shown
at the bottom of the figure) and inelastic (not shown) scattering on 12 C
are also possible but highly quenched. See text for details. . . . . . . . .

91

Figure 4.14: Light output curves for BC-400 plastic scintillator. Each line shows the
expected light output as a function of incident particle energy. Adapted
from [114] see text for details. . . . . . . . . . . . . . . . . . . . . . . . .

92

xiv

Figure 4.15: Two dimensional histogram of light output and incident neutron energy
from the Geant4 simulation. Contributions from elastic scattering on
hydrogen and 12 C are included in the simulation. Inelastic channels for
carbon scattering are also included and create the few events above the
elastic scattering trend line. These events amount to a tiny portion of the
data (<0.05%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Figure 4.16: The black points show the measured intrinsic neutron detection efficiency
for a single LENDA bar at a light output threshold of 30 keVee. The
blue line is the result of the Geant4 simulation of the efficiency. Good
agreement is seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 4.17: Two dimensional histogram of Θcm and Ex in 16 N from the Geant4 simulation of the e10003 setup. . . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 4.18: Simulated neutron acceptance and efficiency in the Θcm =4-6 ° (a) and
Θcm =14-16° (b) bin for the e10003 setup. . . . . . . . . . . . . . . . . .

95

Figure 4.19: Two dimensional histogram of lab neutron scattering angle and neutron
kinetic energy for 15 N (a) and 11 B (b) gated events. The events from 11 B
are used to model the background seen in the 15 N channels. . . . . . . .

98

Figure 4.20: Ratio of the 15 N and 11 B gated events integrated from 40 to 60 MeV in
neutron kinetic energy as a function of the lab neutron scattering angle.
The blue points show the region of charge-exchange reactions, while the
black points indicate regions with only background. The red line is a
linear fit to the black points in the range shown. . . . . . . . . . . . . . .

99

Figure 4.21: Experimental yields with overlaid background shape from the 11 B models.
The most forward (Θcm =4-6°) and backward (Θcm =14-16°) COM angles
are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 4.22: The kinetic energy spectrum of 15 C events detected in the S800 Spectrograph. The black (blue) hashed region comes from the 3.52 (3.6565) Tm
rigidity setting. The gray line is the full distribution constructed from the
two settings. The thin blue line shows the simulated distribution from
charge-exchange reactions. The low-energy tail of the distribution is energetically inaccessible to charge-exchange reactions and so must come from
background processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 4.23: Experimental yield and background models for the 15 C and 14 C channels. The data comes from the 3.6565 Tm and 3.38 Tm rigidity settings
respectively. Yield is primarily seen at backward angles. . . . . . . . . . 101

xv

Figure 4.24: Histogram of the energy deposited in the object scintillator in the first run
of the experiment. The peaks correspond to to single- and multi-particle
hits in the scintillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 4.25: A two-dimensional histogram of the kinetic energy and dispersive angle
at the target for the 14 C channel as measured in the 3.2828 Tm rigidity
setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 4.26: Full kinetic energy distribution for 14 C ejectiles reconstructed from 4 rigidity settings. The vertical line shaded regions indicate areas of incomplete
acceptance. See text for details. . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 4.27: Experimentally determined acceptance curve for 14 C in the 3.2828 Tm
setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 4.28: Efficiency curves from the finite S800 acceptance for each rigidity setting
used in the analysis. Lines of common color show settings centered around
the same detected particle in the CE reaction. . . . . . . . . . . . . . . . 107
Figure 4.29: PID spectrum from the run where the 16 C was tuned into the focal plane
of the S800. The beam consisted of 16 C and a small containment of 14 B.

109

Figure 4.30: Kinetic energy measured by the s800 Spectrograph for the two beam particles. The peak from 16 C has been scaled down for easy comparison with
14 B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 4.31: Relationship between energy loss difference and liquid hydrogen target
thickness determined from simulation. The points are simulated values
and the red line is a linear fit. . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 4.32: The density derived from the temperature measurement of the liquid hydrogen target for each experimental run. . . . . . . . . . . . . . . . . . . 111
Figure 4.33: Differential cross section for the 16 C(p,n)16 N reaction showing the contributions from each reaction channel. Panel (a) shows the cross section
at forward angles in 250 keV bins. In (b) the same angle is shown with
a coarser binning to highlight the channels at higher excitation energy,
where the statistics is low. The most backward angle, with the poorest
resolution, is presented in (c). The shaded bans in (b) and (c) indicate
the systematic error estimation from the background subtraction for the
15 C and 14 C channels. The small vertical lines on the top of each figure
indicate the threshold energies for the 15 N, 15 C, 14 C and 14 N channels.
The color of the line matches the color of the data to indicate the channel. 115

xvi

Figure 4.34: Total differential cross section from all decay channels in the 16 C(p,n)16 N reaction. The green band shows the systematic error including the systematic error from the background subtractions in the different channels and
the systematic error in the neutron acceptance determination. . . . . . . 117
Figure 4.35: The total charge-exchange cross section from 0-20 MeV for each experimental run for the 15 N channel. The spread in the points gives an estimate
of the uncertainty in the beam normalization. . . . . . . . . . . . . . . . 119
Figure 5.1: Differential cross section for the 16 C(p,n)16 N reaction as a function of
excitation energy. Data is from Θcm =4-6° but smeared to match the
resolution at Θcm =14-16°. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 5.2: Measured COM angular distributions in 1 MeV bins from 0-20 MeV in
16 N. The black dots are the experimental data. The red, green, and blue
lines are the ∆L=0,1,2 components of each fit determined from the MDA.
The dotted line is the sum of the 3 MDA fit components. . . . . . . . . . 124
Figure 5.3: MDA results. The ∆L components of the cross section is shown as a
function of excitation energy for each angular bin. At forward angles the
spectrum is dominated by ∆L = 0 transitions, while at backward angles
∆L = 1 transitions appear at high excitation energies. . . . . . . . . . . 128
Figure 5.4: The differential cross section extrapolated to zero momentum transfer.
The error bars show statistical errors. The blue peak shows the contribution from the IAS to the spectrum. See text for details. . . . . . . . . . . 129
Figure 5.5: Extracted GT strength distribution (a) and summed strength (b) in 16 N
up to Ex =20 MeV. The green band shows the total systematic error in
both plots. The data is compared with two shell interactions WBT and
WBP, both calculated in the spsdpf model space. . . . . . . . . . . . . . 131
Figure 5.6: The black points show the dipole (∆L = 1) differential cross section determined from the MDA at 15°. The error bars give the statistical uncertainty
from the MDA fits. The black histogram gives the smeared, DWBA cross
sections from the 1,3 h̄ω shell-model calculation with the WBP interaction
in the spsdpf model space. The blue and red histograms give the results
of the calculations when pure 1 and 3 h̄ω contributions are calculated separately. Panel (a) Shows the result when the full MDA fit is used, while
(b) excludes the ∆L=2 component. The 7-8 MeV bin was excluded from
(a) since its error bar was too large to be meaningful and was consistent
with zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Figure 6.1: The amount of quenching between the experimental data and the shell
model (WBP interaction) as a function of excitation energy. . . . . . . . 139
xvii

Figure 6.2: A typical transition density with its 1p-1h components. This is the calculation of the transition to the third 1+ state in 16 N with the WBT
interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 6.3: The transition densities and their 1p-1h components for the first two 1+
states in 16 N, calculated with the WBT interaction. The details of these
states create an atypical transition density that has a significant effect on
the unit cross section for these two states. . . . . . . . . . . . . . . . . . 142
Figure C.1: The figure shows all possible ways to distribute 2 particles into 3 buckets.
Each row in the figure indicates a different arrangement of particles, giving
a total of 6 possible configurations. . . . . . . . . . . . . . . . . . . . . . 153

xviii

Chapter 1
Introduction

1.1

The Atomic Nucleus

Nuclear physics is a unique field of study. While it has had significant impact on the world
from its beginnings, a robust and a comprehensive predictive theory remains elusive. The
most basic goal of nuclear science is to understand the forces that bind the atomic nucleus
together and, with those forces, understand the many phenomena observed in nuclei and their
interactions. Though they are not fundamental particles, the nuclear problem is typically
defined in terms of the nucleon constituents—protons and neutrons—that interact via the
strong nuclear force.
Beginning with Rutherford’s discovery of the nucleus [1], the field has grown to discover
more than 3000 different isotopes out of approximately 7000 predicted to exist [2]. These
different nuclei are organized in a way similar to the periodic table of the elements, called
the chart of the nuclides. This chart is shown in Fig. 1.1, where the number of neutrons is
plotted on the horizontal axis and the number of protons is plotted on the vertical axis.
The vertical and horizontal dashed lines in Fig. 1.1 indicated the “magic numbers” (2,
8, 20, 28, 50, 82, 126). The magic numbers and their meaning were a crucial discovery for
nuclear physics. They correspond to numbers of nucleons that form and designate particularly stable or strongly bound nuclei. This is a phenomenon that is similar to the noble

1

Figure 1.1: Chart of the nuclides showing the number of neutrons on the x axis and the
number of protons on the y axis. The black squares show the stable nuclei, the blue region
shows nuclei that have been observed but are unstable, and the red region shows nuclei that
are predicted to exist but have not been observed experimentally. The dashed lines show
the nuclear magic numbers. Figure taken from Ref. [3].
gases of atomic physics, where a completely filled outer electron shell makes them inert. The
magic numbers were first recognized by Maria Mayer in 1948 [4], and later further explained
by Mayer [5], and Haxel, Jensen, and Suess in 1949 [6]. The insight made independently by
Mayer and Jensen et al. was the necessity of a strong spin-orbit interaction to reproduce
the observed magic numbers. Both Mayer and Jensen were awarded a portion of the Nobel
Prize in Physics in 1963 for creating and explaining the nuclear shell model.

1.2

The Nuclear Shell Model

The nuclear shell model organizes the protons and neutrons of a nucleus into shells that
attempt to reproduce the observed magic numbers. Strictly speaking, the nucleus is a selfbound system of A nucleons, where the attractive nuclear potential between all the particles
2

binds the nucleus. The shell model, however, is a mean-field approximation that describes
the motion of the individual nucleons within an average potential that models the total effect
of the remaining A − 1 nucleons. This marks a significant difference from the atomic case.
For an atom the negatively charged electrons are bound by the Coulomb potential well of
the positively charged nucleus, and not by the collective effect of all other electrons in the
system.
There are two traditional forms of the potential used in the nuclear shell model: a
harmonic-oscillator
1
V (r) = mω 2 r2 ,
2

(1.1)

or Woods-Saxon potential
V (r) =

V0
.
1 + exp[(r − R)/a]

(1.2)

For the harmonic-oscillator potential, m is the nucleon mass and ω is the oscillator frequency.
For the Woods-Saxon potential, V0 is the depth, R is the radius, and a is the diffuseness of
the potential well. In addition to the potential well described by V (r), a term to model the
spin-orbit interaction between the nucleons is also included. For the spin-orbit potential, it
is important to note that it must vanish at the center of the nucleus [7]. This motivates
using a potential shape that is peaked at the surface. The derivative of a Wood-Saxon is
typically used,
Vso (r) = Vso

1 dfso (r)
,
r dr

(1.3)

where
fso =

1
1 + exp[(r − Rso )/aso ]

(1.4)

These potentials are summarized in Fig. 1.2, where the neutron single-particle states of
3

10
Neutron Single-Particle Energies

0

[56] 168 N=6

[ 2] 168 3s
[10] 166 2d
[18] 156 1g
[26] 138 0i

[42] 112 N=5

single-particle energy (MeV)

-10

-20

126

[ 6] 112 2p
[14] 106 1f
[22] 92 0h

[30] 70 N=4

[ 4] 184 2d3
[16] 180 0j15
[ 8] 164 1g7
[ 2] 156 3s1
[ 6] 154 2d5
[12] 148 0i11
[10] 136 1g9

82

[ 2] 70 2s
[10] 68 1d
[18] 58 0g

50

[20] 40 N=3
[ 6] 40 1p
[14] 34 0f
[12] 20 N=2
[ 2] 20 1s

-30

28
20

[10] 50 0g9
[ 2] 40 1p1
[ 4] 38 1p3
[ 6] 34 0f5
[ 8] 28 0f7

[ 6] 14 0d5

8
[ 2] 8 0p1
[ 4] 6 0p3

[ 6] 8 0p

-40

[12] 82 0h11
[ 2] 70 2s1
[ 4] 68 1d3
[ 6] 64 1d5
[ 8] 58 0g7

[ 2] 20 1s1
[ 4] 18 0d3

[10] 18 0d
[ 6] 8 N=1

[ 2] 126 2p1
[ 6] 124 1f5
[ 4] 118 2p3
[14] 114 0i13
[10] 100 0h9
[ 8] 90 1f7

2
[ 2] 2 N=0

Harmonic
Oscillator
Potential

[ 2] 2 0s

Woods-Saxon
Potential

[ 2] 2 0s1

Woods-Saxon
Plus Spin-Orbit
Potential

-50

Figure 1.2: Neutron single-particle states in 208 Pb with three different potential models.
The left shows a harmonic oscillator potential, the center shows a Woods-Saxon potential,
and the right shows a Woods-Saxon plus spin-orbit potential. The numbers in brackets give
the maximum number of neutrons that each state can hold while the adjacent numbers give
the running sum. Figure taken from Ref. [7].

4

208 Pb

are shown. Though the harmonic oscillator clearly has the wrong asymptotic behavior

(V(r→ ∞)→ ∞), it reproduces several of the observed magic numbers (2, 8, and 20). The
Woods-Saxon potential, shown in the center of Fig. 1.2, breaks the the l degeneracy by
bringing states with higher l values down in energy, but still fails to reproduce the observed
magic numbers above 20. Adding the spin-orbit term completely breaks the degeneracy of
the states and reproduces the magic numbers as Mayer and Jensen et al. first showed. The
spin-orbit term in the nuclear case is of opposite sign to the atomic case, bringing higher J
states lower in energy. This is shown as the right column in Fig. 1.2.
This basic description of the shell model is referred to as the independent-particle model
and works well for nuclei with one nucleon more or less than a magic number. In these
circumstances the nucleus can be modeled well by an inert “core” and one orbiting nucleon.
There the low-lying excited states of the nucleus can be understood as the valence nucleon
simply moving into higher orbitals of the potential created by the core. However, in the more
common case of several nucleons outside a closed core, the problem becomes more complex,
since the interaction between the valence nucleons must be taken into account.
This is done by solving the the Schrödinger equation,

ĤΨ = EΨ,

(1.5)

where Ĥ is the Hamiltonian, Ψ is the wave function, and E is the energy. Unfortunately,
we cannot write down the full expression for Ĥ, since we don’t know the true shape of the
nuclear potential. Following the notation of [7], the Hamiltonian is often modeled as,

H=

n
X



!
(Tk + Uk )

+

k=0

n
X
k<l

5


Vkl  = H 0 + W,

(1.6)

where Tk is the kinetic energy operator, Uk is the single particle potential, n is the number
of nucleons, and Vkl is a two body interaction that is chosen to reproduce the experimental
data. Using the single particle states from H 0 as a basis, the problem is cast into a matrixdiagonalization problem where the off-diagonal terms come from the interaction. This gives
a series solution of the form,
Ψ=

∞
X

= cn ÎŚ n

(1.7)

n=0

where ÎŚn is a Slater determinate for each possible arrangement of nucleons among the
different single-particle states and cn is the coefficient from the diagonalization. A clear
problem with equation 1.7 is that the sum goes to infinity, since there are an infinite number
of single-particle states at higher and higher energies. This is clearly impossible to calculate,
so the series must be truncated. This puts the problem into a specific model space where
only certain single-particle levels are used (for example only the 0p shell). This approach is
used in modern shell-model programs such as OXBASH [8], NuShell, and NuShellX [9]
The shell structure and associated magic numbers presented in Fig. 1.2 were developed
and validated with experimental information from stable nuclei. However, in unstable nuclei the energies between shells can change and the ordering of the single-particle levels can
change. The large shell gaps that create the familiar magic numbers can vanish and create
new, smaller ones at different proton and neutron numbers. This phenomenon is often referred to as shell evolution or shell inversion, and understanding its behavior is a significant
goal of nuclear physics (see Ref. [10] and references therein). Light, unstable nuclei where
large neutron-to-proton ratios can be achieved have been used as a fertile testing ground
for nuclear structure models and for understanding the role of shell evolution in nuclei.
Of particular interest to charge-exchange studies is the role that the isovector part of the

6

nucleon-nucleon (NN) interaction plays in shell evolution. This role has been investigated
previously and the interested reader is directed to Refs. [11, 12] for more information.

1.3

Giant Resonances

Giant resonances in nuclei occur at high excitation energy, where a large fraction of the
nucleons participate in a collective mode. Historically, they are considered a giant resonance
if they exhaust more than 50% of the associated sum rule (an expression for the total
amount of strength for a given operator, see Section 1.8 for more information) for that
mode. Giant resonances were first observed by Bethe and Gentner in 1937 [13], where they
excited the isovector giant dipole resonance (IVGDR) through photo absorption experiments
on a variety of targets. Systematic study of the IVGDR was enabled in 1947 with the
development of the betatron, which allowed for high-energy Îł rays to be produced through
the bremsstrahlung process [14]. This led Baldwin et al. to study the photo-fission and
photo-absorption responses of nuclei in detail from 10-100 MeV [15, 16]. In the early 1970s,
a second giant resonance was observed by Pitthan and Walcher in the (e, e0 ) reaction at
65 MeV on 208 Pb [17]. This was determined to be the isoscalar giant quadruple resonance
(ISGQR). Having found both isovector and isoscalar giant resonances, many experiments
were undertaken to study them in detail using a variety of probes. The interested reader is
referred to Ref. [14] for more information on the historical development of giant resonances.
From a macroscopic perspective, giant resonances can be thought of as resonant oscillations of a nuclear liquid drop. Some of these modes are depicted in Fig. 1.3, where they are
organized in terms of their orbital angular momentum. Shown in the figure are breathing
modes (∆L = 0), where the protons and neutrons oscillate radial outwards, dipole modes

7

(∆L = 1), where the protons and neutrons oscillate against themselves, quadrupole (∆L=2)
and octupole modes (∆L = 3). As depicted in the figure, each of these resonances can occur
with spin and isospin degrees of freedom. When a unit of spin is transferred (∆S = 1), the
spin up (down) protons and neutrons will oscillate out of phase with the spin down (up)
protons and neutrons. Similarly, a unit of isospin (∆T = 1) can be transferred, causing the
protons to oscillate out of phase with the neutrons. For example, the isovector spin giant
dipole resonance (IVSGDR or spin-dipole resonance (SDR)) can be thought of as protons
with spin up oscillating against neutrons with spin down, superimposed with the reverse.
electric mode (∆S = 0)
IS (∆T = 0)

∆L = 0

IV (∆T = 1)

IS (∆T = 0)

IV (∆T = 1)

p

p↑n↑

n

p↓n↓

p↓

pn

monopole

∆L = 1

p

dipole

magnetic mode (∆S = 1)

n

p↑
n↑

∆L = 2

p↓

n↑

p↓
n↑

p↓n↓

n

pn

p↓
n↓
p↑n↑

p
quadrupole

n↑

∆L = 3

octupole

pn

p

n

p↑n↑

p↓n↓

p↓

n↑

Figure 1.3: Schematic of the macroscopic depiction of various giant resonances. For each
∆L value, the different possible spin and isospin transfers are shown. Figure taken from Ref.
[18].

8

From a microscopic perspective, giant resonances are a coherent superposition of oneparticle, one-hole (1p-1h) excitations. A general expression for the operators that mediate
giant resonances can be written as,

Âą =
OJM

X
k

rkλ [Y∆L ⊗ σ̂k ]JM τ̂±k ,

(1.8)

where Y∆L is a spherical harmonic, σ̂ is the vector spin operator, τ̂± is the isospin ladder
operator, k runs through each nucleon in the target nucleus, λ = 2∆n + ∆L, and n is the
principle quantum number. For example, the isovector (∆T = 1) spin (∆S = 1) giant dipole
(∆L = 1) resonance and can be written as,

Âą =
SDJM

X
k

rk [Y1 ⊗ σ̂k ]JM τ̂±k

(1.9)

This picture is especially important for the Gamow-Teller (GT) and the Fermi (F) giant
resonances, which involve only spin and isospin degrees of freedom and therefore have no
macroscopic, oscillatory picture. These resonances can be excited and studied by chargeexchange reactions and will be discussed in Section 1.4.
In general, giant resonances offer powerful insight into the collective properties of nuclei
and provide a stringent testing ground for nuclear structure models at high excitation energy.
Many studies have investigated isoscalar electric resonances using the (Îą, Îą0 ) reaction to
extract bulk nuclear properties and to test theoretical models [19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29]. In particular, isoscalar monopole resonances can be used to extract the compression
modulus of nuclear matter Knm [30]. This was first done by Youngblood et al. who measured
the ISGMR strength distributions in several nuclei, from which they determined Knm =

9

231 Âą 5 MeV [31, 32]. Searches for the isovector giant monopole resonance (IVGMR) have
also been done on several stable nuclei, using nuclear [33, 34, 35] and pion [36] chargeexchange reactions. Studying the IVGMR resonance gives a fundamental understanding of
the isovector part of the residual nuclear interaction and provides information on isospinsymmetry breaking and isospin mixing in nuclei [33, 34].
The spin-dipole resonance can be used to give a model-independent measure of neutron
skin thickness through its non-energy-weighted sum rule. This was done recently for the
first time by Yako et al., where 90 Zr(p,n) and 90 Zr(n,p) were measured at 300 MeV. This
q
gave a neutron rms radius of hr2 in = 4.26 Âą 0.04 fm and a neutron skin thickness of
δnp = 0.07 ¹ 0.04 fm [37]. Neutron skin thickness has also been extracted from studies of the
pygmy dipole resonance in unstable nuclei [38]. Neutron skin thicknesses make constraints
on the neutron equation of state that are important at high density and so are useful for
neutron star properties [39, 40, 41]. Of particular interest to this thesis are the isovector giant
resonances induced by charge-exchange reactions. They have received decades of study from
a variety of probes. The following sections present details about charge-exchange reactions
and their role in the giant resonance arena.

1.4

Nuclear Charge-Exchange Reactions

Nuclear charge-exchange reactions occur when participating nuclei exchange charge. The
nuclei’s proton number changes by one (∆Z = ±1), while its mass number remains the same
(∆A = 0). In terms of isospin formalism, this occurs with ∆Tz = ±1. As an illustrative
example, consider the case of the 12 C(p,n)12 N reaction. The isospin projection for 12 C can

10

be easily calculated as,
Tz =

N −Z
6−6
=
= 0.
2
2

(1.10)

The total isospin for the ground state can be taken as the smallest value that can accommodate the projection, so 12 C must be T = 0, Tz = 0. Following similar arguments, 12 N
must be T = 1, Tz = −1. Therefore, charge-exchange reactions must change the isospin by
1, and are thus isovector transitions (∆T = 1). In general, charge-exchange reactions can
transfer any amount of orbital angular momentum (∆L) and can occur with or without a
spin transfer (∆S = 0, 1). This gives charge-exchange reactions access to a wide variety of
spin-isospin responses in nuclei [42].
The simplest reaction meditated by charge-exchange is the Fermi transition, which is
characterized by the quantum numbers ∆T = 1, ∆L = 0, ∆S = 0. Using Equation 1.8 the
operator for the Fermi transition must be,

O(F ) =

X
k

τ̂±k .

(1.11)

Nearly all the Fermi transition strength is located in the so-called isobaric analog state
(IAS). The next simplest excitation is the Gamow-Teller (GT) (∆T = 1, ∆L = 0, ∆S = 1)
transition,
O(GT ) =

X
k

σ̂k τ̂±k ,

(1.12)

where the spin-transfer operator has been added. The GT and Fermi transitions, induced
through charge-exchange reactions, have great interest since they connect the same initial
and final states as β decay. It has been shown, as outlined in Sections 1.5 and 2.3, that there
exists a proportionality between charge-exchange cross sections for Fermi and GT transitions

11

and their associated β decay matrix elements.
The GT strength is defined as the square of the matrix element between an initial state
i and final state f mediated by the GT operator,

B(GTÂą ) =

|hf ||

2
k σ̂k τ̂±k ||ii| ,

P

2Ji + 1

(1.13)

where the hf ||O||ii notation refers to the reduced matrix element and Ji is the total angular
momentum of the initial state. The GT and Fermi strength are directly related to the β
decay f t value (partial half-life times the phase space factor f ),

ft =

C
,
B(F ) + (gA /gV )2 B(GT )

(1.14)

where C is a constant, B(F ) is the Fermi strength, and gA and gV are the axial and vector
coupling constants. Throughout this thesis, the value gA /gV = 1.251 Âą 0.009 as determined
from neutron β decay is used [43]. Further, the convention for B(GT) is that the β decay of
the free neutron has B(GT)=3.
The Fermi strength is concentrated into a single transition to the isobaric analog state
(IAS), where the wave function has only been modified by a neutron becoming a proton or
vice-versa. This means that for transitions to other states, the relation simplifies to,

ft =

C
,
(gA /gV )2 B(GT )

(1.15)

connecting β decay half-life measurements directly to the GT strength. However, unlike β
decay, charge-exchange reactions are not limited in excitation energy by the β-decay Q-value
window. This gives charge-exchange reactions access to the full isovector response of nuclei,
12

up to high excitation energy, and therefore makes them a powerful tool for the investigation
of isovector excitations, including the isovector giant resonances.

1.5

Historical Development of Charge-Exchange Probes

Charge-exchange experiments began in 1958 when a proton beam was impinged on a deuterium gas target—the first (p,n) reaction [44]. In 1961, (p,n) measurements were done by
Anderson et al. [45], where the IAS was first measured. In 1962, a measurement by Bowen
et al. [46] saw the GT giant resonance, but it was incorrectly interpreted as the IAS. It was
not until 1975 that Doering et al. [47] first measured and identified the GT giant resonance
in the 90 Zr(p,n) reaction at 35 MeV. In 1980, the nature of the proportionality between (p,n)
cross section and GT strength was determined by Goodman et al. [48]. Again studying the
90 Zr(p,n)

reaction, but at 120 MeV, Goodman et al. observed that the GT giant resonance

was preferentially excited at 0° in the (p,n) charge-exchange cross section and with simple
corrections for distortion effects, proportional to GT strength. These initial experiments
began a long line of (p,n) measurements on a range of stable targets, many of which were
done at the Indiana University Cyclotron Facility (IUCF)[49, 50, 48, 51, 52, 53, 54, 55].
Throughout the 1980s, there was clear empirical evidence for the proportionality between
the (p,n) charge-exchange cross section and GT strength. However, it wasn’t until 1987 that
the theoretical framework for the proportionality was formalized by Taddeucci et al. [56].
The success of (p,n) charge-exchange reactions for understanding the collective spin-isospin
modes in nuclei motivated finding other charge-exchange probes to isolate different isovector
resonances. The (n,p) [57] and (t,3 He) [58] reactions were first used in the 1970s to explore
the ∆Tz = +1 direction. Additionally, (d,2 He) experiments have been undertaken [59, 60].

13

Heavy ion probes such as the (7 Li,7 Be) were used to add spin flip selectivity [61]. In the
mid 1980s, the resolution of the ∆Tz = −1 direction was improved over (p,n) with the
introduction of the (3 He,t) reaction [62]. For all of these probes, similar proportionality
relationships between cross section and B(GT) were established.
The aforementioned probes and techniques have all contributed to a detailed and rich
understanding of giant resonances on stable nuclei. However, with the advent of rare-isotope
beam facilities, studying isovector giant resonances on unstable nuclei has received increased
interest. Measuring the GT giant resonance in unstable, neutron-rich nuclei in the mediumheavy mass region validates nuclear-structure models needed for calculating electron capture
rates [63, 64]. These rates (particularly in the pf and sdg shells) play an important role in
type II supernovae (see Ref. [65] and reference therein). With unstable neutron-rich nuclei,
increased neutron excesses make charge-exchange studies more sensitive to neutron skin
measurements—which makes more stringent constraints than are possible with stable nuclei.
In general, studying isovector giant resonances in exotic nuclei is useful for understanding
the underlying nuclear structure since the sensitivity of the underlying degrees of freedom is
increased compared to stable nuclei [14].
To perform charge-exchange experiments with rare isotopes, new techniques need to be
developed for experiments in inverse kinematics, where the unstable nucleus of interest is
impinged on the stable probe serving as the target. In the ∆T = +1 direction this has been
done with the (7 Li,7 Be) reaction with a rare-isotope beam of 34 P [66]. For the ∆Tz = −1
direction, a relatively new technique for studying isovector giant resonances (GT and SD
giant resonance) has been developed using the (p,n) reaction in inverse kinematics. The
development and validation of this probe is the focus of this thesis.

14

1.6

The (p,n) Probe in Inverse Kinematics

In (p,n) experiments in inverse kinematics, an unstable isotope is impinged on a thick, liquidhydrogen target. The energy and angle of the low-energy recoiled neutron is measured.
Missing-mass spectroscopy is used to calculate the center-of-mass (COM) scattering angle
and excitation energy of the ejectile (see Chapter 3 for technical details). This technique has
several advantages:
ˆ The (p,n) probe is the simplest of the charge-exchange probes. It is well established

and provides model independent extractions of GT strength.
ˆ The lower beam rates in rare-isotope experiments can be offset by a thick liquid hy-

drogen target. Since the experiment is performed in inverse kinematics, the excitation
energy resolution is less sensitive to the energy loss of the beam in the target. Further,
the neutron can easily escape the target for detection.
ˆ The simplicity of the missing-mass spectroscopy allows for the extraction of the isovec-

tor response up to high excitation energy of any nucleus. The scattering angle and
excitation energy come from the measurement of the recoiled neutron.
The technique was first used to measure the 56 Ni(p,n) and 55 Co(p,n) reactions in inverse
kinematics at 110 MeV/u [63, 64]. Other measurements have also been performed, including
12 Be(p,n)12 B

[67] and an invariant-mass measurement of the 14 Be(p,n)14 B reaction at 69

MeV/u [68]. Instead of probing the isovector response of the nucleus up to high excitation
energies, Ref. [64] focused on benchmarking electron-capture rate calculations needed for
core-collapse supernovae simulations. The next round of experiments utilizing this technique
have since occurred, with 132 Sn(p,n)132 Sb at RIKEN and 16 C(p,n)16 N at NSCL (experiment
15

number e10003). The 16 C(p,n)16 N experiment is described in this thesis and seeks to validate
the probe’s ability to explore the isovector response of radioactive nuclei up to high excitation
energies.
16 C

is a good choice for this goal for the following reasons:

ˆ Since it is a light system the cross section is expected to be relatively high due to

relatively small distortion effects from the nuclear mean field.
ˆ High beam rates are achievable, ensuring that sufficient statistics will be obtained in

the measurement.
ˆ The whole GT resonance region can be covered in the experiment.
ˆ Detailed shell-model calculations are feasible for comparison.

With the full GT response measured, the quenching of the GT strength in an unstable
neutron rich-nucleus can be measured (assuming the β+ strength is zero). Alongside the
full GT giant resonance, a portion of the SD resonance will be measured. This cross section
can be compared to shell-model and reaction code predictions.
The shell-model depiction of the GT and SD resonances are given in Fig. 1.4. The
different 1p-1h contributions to the resonances are shown. 16 C is a cross-shell nucleus, with
two neutrons in the sd shell. For the GT transitions, there will be contributions from both
the p and sd shells. In the notation of the OXBASH program, configurations with two
nucleons in the sd shell are called 2h̄ω configurations. This naming is with reference to an
16 O

core, which is a 0 h̄ω configuration. Since GT transitions are associated with no change

in h̄ω, the final states in 16 N will be remain in 2h̄ω configurations. The SD resonance changes
major oscillator shells (h̄ω = 1), so it will have many contributions including transitions into
16

the pf shell. This means that there can be transitions from the 2h̄ω configuration of the
16 C

ground state to either 1 or 3 h̄ω configurations in 16 N. To include all of these different

components the shell-model calculations were done in the full spsdpf model space. The
details and results of shell-model calculations for these resonances are given in Chapter 2.
16C

16C

Gamow-Teller Resonance
protons

Spin Dipole Resonance

neutrons

protons

1p1/2

neutrons

1p1/2

0f5/2

0f5/2

1p3/2

1p3/2

0f7/2

0f7/2

0d3/2

0d3/2

1s1/2

1s1/2

0d5/2

0d5/2

0p1/2

0p1/2

0p3/2

0p3/2

0s1/2

0s1/2

Figure 1.4: Shell-model schematic showing the different 1p-1h contributions of the GT and
SD giant resonances in the 16 C(p,n)16 N reaction.

1.7

β Decay of

16

C

Several β-decay studies of 16 C have already been performed [69, 70, 71]. From the first
measurement in 1976 by Alburger et al., the low-lying structure of 16 N has generated interest
as a testing ground for shell-model interactions. Initial efforts by Snover et al. with the
Millener-Kurath (MK) interaction failed to reproduce the observed GT strength for the
two, strong GT states at 3.3 and 4.3 MeV [72]. Significant improvements were made by
Warburton et al. [73] in 1992 with the use of the WBT interaction. There, the 0p shell part
of the interaction was determined simultaneously with the cross-shell part, which improved
the results. A more recent β-decay study by Grevy et al. identified a new state at 6 MeV
17

[71]. Table 1.1 shows a summary of the 1+ states that have been observed in β decay
measurements and the corresponding shell-model results.
Experimental

Theory

Ex [MeV]

Jπ

log(ft)

B(GT)

B(GT)MK

B(GT)WBT

3.353

1+

3.551 Âą0.012

1.104 Âą 0.03

0.001a

0.84a

4.320

1+

3.83 Âą0.05

0.586 Âą 0.07

0.903a

0.21a

6.00

1+

3.86 Âą0.1

0.543 Âą 0.12

0.36b

Table 1.1: Summary of the results from previous β-decay measurements of 16 C and from
theoretical calculations. a taken from Ref. [73]. b taken from Ref. [71].
These β-decay experiments have provided constraints and insights for shell-model theories
for the low-lying states in 16 N. With the 16 C(p,n)16 N measurement, the full GT response of
16 N

will be probed. This will provide a stringent test for the shell model at higher excitation

energies, which are unavailable to β-decay studies.

1.8

The Gamow-Teller Sum Rule

Many model-independent sum rules exist for the operators that mediate the excitation of
giant resonances. They are categorized into energy-weighted (EW) and non-energy-weighted
(NEW) sum rules (SR). For general information about SRs, the reader is referred to Ref.
[14]. Of interest to this thesis is the NEWSR for the GT and the Fermi operators. The GT
sum rule is,
S− − S+ = 3(N − Z),

(1.16)

S− − S+ = (N − Z).

(1.17)

and the Fermi sum rule is,

18

The S± indicates the total strength in the β+ and β− directions, for the GT and Fermi
operators respectively. Equations 1.16 and 1.17 are simple and model independent—all you
need to know is the N and Z of the initial nucleus. A full derivation of the GT sum rule
is given in Appendix A. For the neutron-rich case of 16 C, Equation 1.16 is simplified, since
the transitions in the β+ direction will be Pauli blocked. In this case, Equation 1.16 can be
approximated as,
S− ≈ 3(N − Z).

(1.18)

This means that the total summed strength in 16 C can be probed by measuring only the β−
direction and then directly comparing the result with the sum rule. This will be done in the
16 C(p,n)16 N

reaction described in this thesis. For the Fermi transition, close to 100% of the

NEWSR is exhausted in the transition to the IAS. For 16 C, this would mean that the Fermi
strength in the β− direction would be 4.

1.9

Quenching of Gamow-Teller Strength

A reoccurring feature of charge-exchange and β -decay studies is missing GT strength. In
β-decay experiments, the strength from individual states is compared to shell model predictions where a systematic reduction is seen. With charge-exchange reactions, the whole
GT response can be probed and the summed strength can be extracted, including the GT
giant resonance. When the total strength is compared to the model-independent sum rule
for the GT operator, a systematic reduction in strength is also observed. This phenomenon
is known as quenching and has been studied for many decades [74, 75]. Consider again the

19

form of GT single particle operator,

ˆ τ̂± ,
O(GT ) = gA /gV ~σ

(1.19)

where the coupling constants are included. This equation gives the free-nucleon form of the
operator. In the β-decay context, quenching of the nuclear matrix element can be thought
of as a renormalization of the decay of the free-nucleon. Degrees of freedom not included in
the model space of the calculation require a correction to either the value of gA /gV or to the
ˆ τ̂ [76]. Different mechanisms have been proposed for quenching,
nuclear matrix element of ~σ
such as higher-order nuclear configuration mixing (2p-2h and above) and couplings to nucleon
degrees of freedom (the ∆ isobar) [77, 76, 78]. These mechanism serve to push the missing
strength to higher excitation energies. In charge-exchange experiments, a significant fraction
of the missing strength has been observed at energies above the GT giant resonance. By
measuring both the β+ and β− directions with the 90 Zr(p,n) and 90 Zr(n,p) reactions, 88±6%
of the sum-rule strength was observed. This was done by looking up to 50 MeV of excitation
energy [79, 80, 81, 82].
The amount of quenching of GT strength has been studied in the p, sd, and pf shells by
using β-decay information and comparing it to shell-model calculations. These results are
summarized in Table 1.2. As mentioned in Section 1.8, the 16 C(p,n)16 N reaction will measure
the strength in the β− direction and give a measurement of the quenching of summed GT
strength in an unstable neutron-rich nucleus. This value will be compared to the results in
table 1.2.

20

Region
p-shell

Quenching Factor
 0.35
A
1-0.19 16
[77]

sd-shell

0.76 [76]

pf-shell

0.744 [83]

Table 1.2: Quenching factors of GT strength for nuclei occupying the p, sd and pf shells.
The factors presented are the quenching of the GT operator. To find the quenching of GT
strength, the factor must be squared, so in the sd shell it is 0.762 = 0.58.

21

Chapter 2
Theoretical Techniques
I demand rigor
A foolish 1st year graduate student

This chapter gives a brief overview of the theoretical tools used in the calculation of
GT strength and charge-exchange cross section. It does not give a rigorous derivation of the
theoretical framework, but instead focuses on the necessary ideas needed for charge-exchange
studies. For a development of nuclear reaction formalism, the reader is referred to Ref. [84].

2.1

Distorted Wave Born Approximation

This section gives a brief overview of the distorted wave born approximation and its relevance to charge-exchange reactions. Here the aim is solve the time-independent Schödinger
equation for the nuclear reaction problem,

ĤΨ = EΨ.

(2.1)

In the asymptotic limit (r → ∞), the wave function should take the following form,

Ψ = Ψbeam + Ψscatt = C

22

ik r
e f
ik
z
i
e
+ f (θ, φ)
R

!
(2.2)

ik r
where eiki z represents the beam as an incoming plane wave and e f /r is the outgoing

spherical wave from the scattering. f (θ, φ) is known as the scattering amplitude, which
contains the scattering information of the reaction. The differential cross section is directly
related to the scattering amplitude by,

dσ
= |f (θ, φ)|2 .
dΩ

(2.3)

In order to calculate the charge-exchange differential cross section, we need to find an
expression for the scattering amplitude f . To do this we return to the Schrödinger equation
but write it more explicitly as,
h̄2 ∇2
+V
−
2Âľ

!
Ψ=

h̄2 k 2
Ψ,
2Âľ

(2.4)

2 2

where E has been replaced by E = h̄2µk and the kinetic part of the Hamiltonian has been
written explicitly. Rearranging this expression yields,



∇2

+ k2



2Âľ
Ψ(~r) = 2 V (~r)Ψ(~r),
h̄

(2.5)

where the dependence on the spatial coordinate ~r has been included for clarity. This equation


can be cast into the standard Green’s function problem by setting D̂ = ∇2 + k 2 and
ρ = 2¾2 V (~r)Ψ(~r) yielding,
h̄

D̂Ψ = ρ.

23

(2.6)

This equation has the solution,

2Âľ
Ψ(~r) = φ(~r) + 2
h̄

Z

G¹ (~r, ~r0 )V (~r0 )Ψ(~r0 )d~r0

(2.7)

where G± (~r, ~r0 ) is the Green’s function of the operator D̂ and φ(~r) is the solution to the
homogeneous equation D̂Ψ = 0 . The ± notation in the Green’s function refers to the post
and prior forms of the solution. Both an incoming plane wave with outgoing spherical wave
(prior form) and an outgoing plane wave with an incoming spherical wave (post form) satisfy
the boundary conditions. In the calculations of the charge-exchange cross section, the prior
form is used. It can be shown (see for example Ref. [84]) that the Green’s function takes
the form of,
GÂą

(~r, ~r0 )

0

e±ik|~r−~r |
.
=−
4π|~r − ~r0 |

(2.8)

Substituting the Green’s function into equation 2.7 and applying the approximation |~r −
q
p
2
0
~r | = r2 + r0 − 2~r · ~r0 ≈ r 1 − 2~r · ~r0 /r2 we have,
2Âľ
Ψ(~r)± = φ(~r)± −
4πh̄2

Z

e±ikr ∓i~k0~r0
e
V (~r0 )Ψ(~r0 )d~r0
r

2Âľ eÂąikr
Ψ(~r)± = φ(~r)± −
4πh̄2 r

Z

~0 0

e∓ik ~r V (~r0 )Ψ(~r0 )d~r0 .

(2.9)

(2.10)

Comparing Equation 2.10 with Equation 2.2 the scattering amplitude can be identified
as,
2Âľ
f (~k, ~k 0 ) = −
4πh̄2

Z

~0 0

e∓ik ~r V (~r0 )Ψ(~r0 )d~r0

(2.11)

So far in the derivation there has only been the well constrained approximation that r >> r0 .

24

However, the expression for the scattering amplitude in Equation 2.11 contains the total
wave function, which is the unknown we started with. To arrive at a useful result, the
Born Approximation is made where the form of the solution in Equation 2.10 is recursively
inserted into Equation 2.11. This gives a series solution, where each term contains higher
and higher orders of the potential V . Taking just the first term, we have,

f (~k, ~k 0 )

2Âľ
≈−
4πh̄2

Z

0 0

e∓i~q ~r V (~r0 )d~r0 ,

(2.12)

where ~q = ~k − ~k 0 is the momentum transferred. This gives a clear and explicit way to
calculate the scattering amplitude. This is the first order Plane Wave Born Approximation
(PWBA). This result is often written in terms of the T matrix,

T = hφ | V | Ψi

(2.13)

T ≈ hφ | V | φi ,
where φ are plane waves.
To derive the Distorted Wave Born Approximation (DWBA), it is useful to rewrite Equation 2.10 in an operator form known as the Lipmann-Schwinger equation,

|Ψ± i = |φi + G± V |Ψ± i ,

(2.14)

which, under the Born Approximation reduces to,

|Ψ± i = |φi + G± V |φ± i .

25

(2.15)

In nuclear reactions, the potential is often divided into two parts, V = U + W , where U is
the part of the potential that describes elastic scattering for the system while W contains
the residual interaction that is not included in U . U is chosen such that,

|χi = |φi + GU
± U |φi ,

(2.16)

where |χi is referred to as a distorted wave and GU
Âą is the Greens function for the problem
containing only U . For charge-exchange calculations, W contains the interaction responsible
for exciting the spin-isospin degrees of freedom in the target nucleus. The purpose of this
separation becomes clear when you consider the total T matrix with these two potentials,

T = hφ | W + U | Ψi .

(2.17)

For clarity, the full wave function is used in the above relation. It can be shown that,


    
   
T = φ−  U  χ+ + χ−  W  Ψ

(2.18)

Since U only models the average potential of elastic scattering, it cannot connect the initial
to the final states we are interested in for charge-exchange reactions. This leaves only the
second term,

   
T = χ−  W  Ψ ,

(2.19)

where the effect of the U potential is incorporated into the determination of the distorted
wave χ− . Reapplying the Born Approximation, we arrive at the T matrix for first order

26

DWBA,

   
T DW BA = χ−  W  χ+

(2.20)

So far the internal state of target and residual nucleus has been omitted from the derivation to focus on the details of the reaction theory. Equation 2.20 is rewritten in terms of a
~
form factor F (R),
T DW BA

=

D







 ~  +
χ− F (R)
χ

E

,

(2.21)

~ is the spatial distance between the center of the target nucleus and the position of
where R
the incident probe. The form factor is,

~ =
F (R)

Z
drT drp ρp Vef f ρT ,

(2.22)

where the residual interaction has been relabeled as Vef f . ρ are the transition densities of
the target and projectile,

ρ=

X
Îą

OBT Dα hφfα ||σ̂τ̂± ||φiα i .

(2.23)

The one-body transition densities (OBTD) are calculated in the shell model between the
initial and final states. The sum over α runs over the contributions from the different 1p1h combinations for single particle level transitions φiα to φfα . The single-particle wave
functions are modeled with Woods-Saxon potentials. With the formalism above, the only
remaining unknown is the form of the effective NN interaction that can meditate chargeexchange reactions.

27

2.2

Effective Interaction

In this section, the phenomenological nucleon-nucleon interaction that is used to model
charge-exchange reactions is discussed (Vef f ). The long-standing choice for interaction in
the charge-exchange community has been the Love and Franey (LF) N-N interaction [85, 86].
The LF interaction is attractive for charge-exchange studies since it is directly parameterized
in terms of the spin and isospin operators that are of interest for charge-exchange reactions.
The potential between nucleons i and j is structured into central, spin-orbit, and tensor
terms,
~ ¡S
~ + V T (rij )Sij ,
Vij = V C (rij ) + V LS (rij )L

(2.24)

where Sij is the tensor operator

Sij = 3

(σ̂i · rij )(σ̂j · rij )
2
rij

− σ̂i · σ̂j

(2.25)

The radial dependence of each of these terms is expanded as a sum of Yukawa potentials,

V (r) =

X

Vi Y (r/Ri ),

(2.26)

i

where Y (x) = e−x /x. The parameters Vi and the ranges Ri are fit to nucleon-nucleon
scattering. The result of the work by Love and Franey gave effective nucleon-nucleon T
matrices at a variety of incident beam energies. For the 16 C(p,n)16 N at 100 MeV/u, the
LF interaction at 140 MeV was used. The 140 MeV interaction was chosen over the 100
MeV interaction because the parameters were better determined from the nucleon-nucleon
scattering data [85, 86]. The GT component of the interaction does not change significantly

28

between these energies. However, the magnitude of the τ component of the NN interaction
reduces with increasing energy. This is corrected for by using Equation 2.31. The interaction
is folded over the transition densities of the target and projectile to arrive at the form factor
given in Equation 2.23.

2.3

Proportionality

This section provides a brief overview of proportionality between GT strength and chargeexchange cross section. The interested reader is referred to [56] for the complete derivation
and a detailed discussion. Taddeucci et al. showed that the form of this relationship is a
simple proportionality,



dσ
= σ̂B(GT ),
dΩ q→0

(2.27)

where the q → 0 indicates that the cross section needs to be extrapolated to zero linear
momentum transfer. The σ̂ is the proportionality constant of the expression and is called
the unit cross section. Within DWBA and under the limits of high beam energy (>∟100
MeV/u) and the Eikonal approximation, Taddeucci et al. showed that the unit cross section
factors into three components,
σ̂ = KND |Jστ |2 .

(2.28)

K is the kinematic factor,
Ei Ef

kf
.
(h̄2 c2 π 2 )2 ki

(2.29)

ND is a distortion factor and Jστ is a volume integral of the relevant interaction (in this case,
the Love Franey interaction). ND can be calculated as the ratio of distorted- and plane-wave
cross sections (see Equation 2.23 of Ref. [56]).
29

While the factorization of the proportionality gives insight, the unit cross section is often
determined empirically. When β-decay information for a low-lying state is available, σ̂ can
be found by using the known B(GT) value and the measured charge-exchange cross section.
When such information is not available, the unit cross section can sometimes be estimated
by looking at compiled dependencies between σ̂ and mass number. Fig. 2.1 shows the mass
dependence of the unit cross section over a range of nuclei for the (p,n) reaction at 120 MeV.
The parameterization was determined by Taddeucci et al. [56].
The ratio of GT and Fermi unit cross sections was also investigated by Taddeucci et al.
They defined,
σ̂ (Ep , A)
R2 = GT
σ̂F (Ep , A)

(2.30)

The dependence of R2 on incident energy and mass number was investigated experimentally
by measuring R2 for a set of targets at a variety of beam energies. A linear dependence on
Ep was found,
R(Ep ) =

Ep
.
55.0 Âą 0.4 MeV

(2.31)

Fig. 2.2 shows this dependence for 14 C. The deviations below 50 MeV are below the incident
energy region where the proportionality is applicable.

2.4

Technical Details of Calculations

The cross-section calculations for the 16 C(p,n)16 N reaction at 100 MeV/u were completed
using the DW81 code [87] (see Ref. [88] for a description of the original formalism). The
theoretical GT strengths and one-body transition densities were calculated using OXBASH
[8] in the spsdpf model space using either the WBT or WBP interactions. The optical

30

Figure 2.1: Experimental unit σ̂GT and σ̂F as a function of mass number. The dashed and
dotted lines are the result of the parameterization of Taddeucci et al. Figure is reproduced
from Ref. [56].

31

Figure 2.2: The square root of the ratio of the GT and Fermi unit cross sections as a function
of incident proton energy. The data points come from reactions on 14 C at various energies.
The dashed line is a linear fit to the data above 50 MeV. Figure taken from Ref. [56], see
references therein.
potentials for the proton-16 C entrance channel and the neutron-16 N exit channel come from
the parameterizations given in Refs. [89, 56]. The parameters used for the 16 C(p,n)16 N calculation are given in Table 2.1 for convenience. The LF interaction at 140 MeV was used for
the N-N potential. The binding energies of the single-particle wave functions in the initial
and final state were calculated using a Skyrme interaction.

32

Potential

V

r

a

W

r

a

p-V

30.68

1.23

0.71

7.12

1.41

0.57

p-SO

3.54

1.00

0.65

-1.07

0.97

0.62

n-V

25.16

1.23

0.71

7.12

1.41

0.57

n-SO

4.01

1.00

0.65

-1.07

0.97

0.62

Table 2.1: Parameters for the optical potentials used in the 16 C(p,n)16 N cross-section calculations. All parameters are given in MeV and Fermi. V refers to a volume term and SO
a spin-orbit term. The volume terms are Woods-Saxon potentials. The spin-orbit terms are
derivatives of Woods-Saxons. The p and n refer to the potentials for the proton (entrance)
and neutron (exit) channels.

2.5

Shell Model Results

This section presents the results from the shell-model calculations performed with OXBASH.


We start in the J π = 0+ ; T = 2 16 C ground state. In charge-exchange reactions (∆T = 1),
the final state in 16 N can be Ti = 2 ⊗ ∆T = 1 → Tf = 1, 2, 3. If Ti is high, transitions to
higher T states are suppressed by their Clebsch-Gordan coefficients. For the 16 C(p,n)16 N reaction, these factors are given in Table 2.2.
Tf

Coefficient

Factor

T+1

1
(T +1)(2T +1)

0.11

T

1
T +1

0.33

T-1

2T −1
2T +1

1

Table 2.2: Isospin Clebsch-Gordan factors for the Ti = 2 → Tf = 1, 2, 3 transitions. The
factors give in column 3 are normalized to the T − 1 transition.
In terms of angular momentum, the important transitions are Jfπ = 1+ (GT transitions),
and Jfπ = 0− , 1− , 2− (dipole transitions). Fig. 2.3 shows the GT strength distributions
33

for 0+ → 1+ transitions calculated with OXBASH in the spsdpf model space, and the
WBP and WBT interactions. Configurations in both the p and sd shells are included in
the calculation. Transitions to T = 1 and T = 2 finals states are shown. Within this
model space, the full strength of the sum rule is exhausted in the ∆Tz = −1 direction (the
∆Tz = +1 is Pauli blocked), so the sum of all calculated states is ∼121 . The T = 3 states
contribute < 0.002 to the total strength and are neglected. Both interactions predict the
majority of the GT strength to be below approximately 12 MeV, with strong states near 10
MeV. The Tf = 2 strength is a small component appearing at higher energies. The shell
model does not give intrinsic widths to these states, which since they occur above particle
separation thresholds are expected to be several MeV wide.
Fig. 2.4 shows the more complicated case of spin-dipole transitions, calculated with the
WBP interaction. Unlike the GT transitions, no simple proportionality between spin-dipole
strength (B(SD)) and charge-exchange cross section has been established. The typical approach is to directly compare measured cross sections with theoretical cross section calculated
in DWBA. Fig. 2.4 therefore shows the cross section value at the angle where it is maximized
for each shell model state with Jfπ = 0− , 1− , 2− . The excitation energy of the states are
aligned such that the first 2− state is at 0 MeV (the ground state of 16 N is 2− ). The shell
model predicts a 0− ground state with slightly lower energy (∼300 keV), and so appears
negative in Fig. 2.4. This difference is within the error of the calculation and below the
resolution of the experiment. The shell model predicts several strong states at ∟5-10 MeV.
Above ∟12 MeV, substantial strength comes form many states. The resonances is expected
to extend above 20 MeV.
1 For 16 C, the GT sum rule is: 3(N − Z) = 3(10 − 6) = 12

34

3

WBP
T=1
T=2

2.5

B(GT)

2
1.5
1
0.5
0
0

5

10

15

Ex [MeV]

20

25

30

25

30

(a)

3

WBT
T=1
T=2

2.5

B(GT)

2
1.5
1
0.5
0
0

5

10

15

Ex [MeV]

20

(b)





Figure 2.3: Shell Model calculations of B(GT) for 16 C0+ ; T = 2 →16 N1+ ; T = 1, 2 transitions. The WBP (a) and WBT (b) interactions were used in the spsdpf model space.

35

2

Jπ=0
Jπ=1
Jπ=2
-

1.8

[mb/sr]

1.6
1.4
max

1.2
1

dσ
dΩ

0.8
0.6

0.4
0.2
0

0

5

10

15

20

25

Ex [MeV]
Figure 2.4: Shell-model spectrum with the WBP interaction for the SD strength in 16 N,
presented as the peak cross-section value for each state. Transitions into final states of 16 N
with Jfπ = 0− , 1− , 2− are shown. All states have T = 1.

2.6

DWBA Results

Within the shell-model and DWBA theory the proportionality between the charge-exchange
cross section at vanishing momentum transfer and B(GT) can be explored. This is done
by taking many shell-model states and calculating their differential cross sections at q = 0.
Then, a theoretical proportionality constant (σ̂) is found for each state by dividing the cross
section by the B(GT) value from the shell model. This analysis is presented in Fig. 2.5
A consistent proportionality is seen for the stronger states, with a gradual decrease in
quality for the weakest transitions. The spreading in the weaker states comes from the interplay between the στ and the tensor-τ portion of the effective NN interaction. The tensor-τ
portion mediates ∆L=2 ∆S=1 transitions that interfere with the desired GT transitions
(see Chapter 6 for more details). Since these proportionality breaking effects occur in weak
transitions, they don’t contribute much to the final GT spectrum. The scatter in the strong

36

20
18

σ [mb/sr]

16
14
12
10
8
6
4
2
0
10−5

10−4

10−3

10−2

B(GT)

10−1

1

Figure 2.5: Scatter plot of the shell-model B(GT) for 0+ → 1+ transitions against the
theoretical σ̂ determined from DWBA. The shell-model calculation was done with the WBT
interaction in the spsdpf model space. The red dot indicates the 2nd shell-model state. In
e10003 states with strengths below ∟ 0.05 could not be discerned.
states tends to be evenly distributed giving the average σ̂GT a small error. The red dot indicates the transition to the 2nd 1+ shell model state and is a clear outlier with a significantly
higher predicted σ̂. The detailed properties of this state will be discussed in Chapter 6.
In order to isolate the different orbital angular momentum transfers in the data, a multipole decomposition analysis (MDA) will have to be performed. Details on the procedure for
the MDA is given in Section 5.2. Here the calculations of the characteristic angular distributions needed for the MDA are shown. Angular distributions with pure ∆L = 0 ∆S = 1 components from the 0+ → 1+ transitions were used as a model for the characteristic ∆L = 0
angular distribution. The calculations to 1− final states were used for the characteristic
dipole angular distribution. Transitions to 2+ final states were used for the quadrupole component. For each multipole, the first 100 shell-model states were examined with the Q value
in the reaction calculation set to zero. From these angular distributions, a representative

37

strong state was chosen as the model for each ∆L transfer. The deviation between these
states was small compared to the overall systematic error of the measurement. These model
angular distributions are shown in Fig. 2.6.

3

∆L=0
∆L=1
∆L=2

dσ [arb]
dΩ

2.5
2

1.5
1

0.5
0
0

5

10

15

Θcm [deg]

20

25

Figure 2.6: The model angular distributions for ∆L = 0, 1, 2 that are used in the MDA
analysis.

38

Chapter 3
Experiment
Nothing works! Everything is broken!

Ancient Proverb

The (p,n) charge-exchange experiment in inverse kinematics involves the detection of lowenergy neutrons in coincidence with fast residues in a magnetic spectrometer. The neutron
laboratory scattering angle and neutron kinetic energy provide event-by-event kinematic
information necessary to reconstruct the excitation energy in 16 N and the center-of-mass
(COM) scattering angle. For experiment e10003, the neutrons were detected using two lowenergy neutron-detector arrays: LENDA [90, 91] and VANDLE [92, 93]. The fast residues of
16 N

were detected by the S800 Spectrograph [94]. The experiment ran in March of 2015 with

the goal of extracting the Gamow-Teller strength distribution up to 20 MeV of excitation
energy in 16 N. Details on the experimental equipment and method will be given in the
sections below.

3.1

Experimental Method

Fig. 3.1 shows kinematic correlations between the laboratory neutron kinetic energy and
scattering angle, and the excitation energy in 16 N and COM scattering angle for the 16 C(p,n)16 N
reaction at 100 MeV/u. As Fig. 3.1 shows, to cover the excitation energy regime of interest
(0-20 MeV) and the COM scattering angles of interest (0-15 degrees), a large range of labo39

ratory neutron scattering angles and low neutron energy must be measured.

30 15

20

EX= 0 MeV

C(p,n)16N
at100 MeV/u

16

Tn Lab [MeV]

18
16
14

200

12
10

150

8
6

100

4
2
0
0

θCM=50
20

40

60

80

100

Θ n Lab [Deg]

120

140

160

180

Figure 3.1: Kinematic correlations for the 16 C(p,n)16 N reaction at 100 MeV/u, in the
plane of the laboratory neutron angle (θn ) and neutron kinetic energy (Tn ). The blue box
indicates the approximate coverage of the neutron detectors in e10003. Solid lines indicate
the excitation energies in 16 N. Dashed lines show the COM scattering angles.

To this end, the centers of the LENDA and VANDLE bars were placed 1 m from the
center of the target so as to cover laboratory scattering angles between approximately 40
and 140 degrees. A schematic of the setup can seen in Fig 3.2. Each array was split in
half and arranged such that one section of LENDA was placed on the opposite side of the
beam line from the corresponding section of VANDLE. The angles of the bars were further
arranged such that the gap in angular coverage between two LENDA bars was filled by the
VANDLE bars on the opposite side of the beam line. The arrangements of different sections
of LENDA and VANDLE allowed both arrays to span the full laboratory angular range being
measured, while also preventing any systematic bias between the two arrays from differences
in the neutron background on either side of the beam line. The choice to arrange the angles
40

of the bars to fill the angular gaps in the detectors on the opposite side of the beam line
allowed for continuous laboratory angular coverage and reduced systematic uncertainties in
the estimation of geometrical acceptance. This is because regions of near zero acceptance,
where systematic uncertainties due to slight deviations in detector placement are large, were
avoided. Fig. 3.3 shows a picture of the complete experimental setup.

North VANDLE

North LENDA

Beam

South LENDA

South VANDLE

Figure 3.2: Top-view schematic of the experimental setup for the neutron detectors.

41

Figure 3.3: Picture of the complete setup for experiment e10003. In the bottom center of
the picture, the south LENDA bars are visible. The north VANDLE bars can be seen in the
top left. The liquid hydrogen target can be seen in the center of the photograph.

42

To isolate events that could be due to charge-exchange reactions, the ejectile was analyzed
in the S800 Spectrograph. Since the experiment aimed to measure up to excitation energies of
20 MeV in 16 N, the 1n, 1p, 2n, 1p1n emission thresholds were exceeded. This caused several
different nucleon decay channels to open (16 N, 15 N, 15 C, 14 C, 14 N), each of which needed
to be measured to find the total charge-exchange cross section. The momentum spread of
these residual ejectiles after the momentum kick from the decayed nucleons can be greater
than the momentum acceptance of the S800 Spectrograph ( dp
p = 5% [94]) . To account for
this, seven different magnetic-rigidity settings were used to capture a large enough fraction
of the momentum spectrum for each decay channel. The different decay channels and their
associated rigidity settings used during the experiment are shown in Table 3.1.
Ex in 16 N

Reaction

Detected Particle

Set Centroid Rigidities

0 → 2.4 MeV

16 C(p,n)16 N

16 N

3.2828, 3.338 Tm

>2.489 MeV

16 C(p,n)16 N

→ 15 N + n

15 N

2.98, 3.1325 Tm

>11.478 MeV

16 C(p,n)16 N

→ 15 C + p

15 C

3.52, 3.6565 Tm

>12.696 MeV

16 C(p,n)16 N

→ 14 C + p + n

14 C

3.2828, 3.38 Tm

>13.322 MeV

16 C(p,n)16 N

→ 14 N + n + n

14 N

2.98, 2.9 Tm

Table 3.1: Table of the different decay channels accessible in the 16 C(p,n)16 N reaction for
ranges of excitation energy in 16 N and their associated magnetic rigidities. The acceptance
of each rigidity setting is Âą %5.

With the kinematic information provided by the low-energy recoiled neutron and the momentum information of the fast ejectile, the differential cross section could be reconstructed
up to 20 MeV of excitation energy in 16 N.

43

3.2

Experimental Equipment

3.2.1

Beam Creation and Delivery

The 16 C(p,n)16 N reaction was performed with a beam of radioactive 16 C nuclei. Radioactive
ion beams are created at the NSCL through the process of projectile fragmentation, where
an accelerated stable ion beam is impinged on a thick 9 Be production target. The impinging
ions are fragmented in flight by the target, creating a wide array of different residual isotopes,
some of which are the unstable, exotic isotopes that are of interest for study. The specific
isotopes of interest are then selected in flight by their magnetic rigidity using the A1900
fragment separator [95], possibly resulting in a nearly pure beam of the isotope of interest.
This beam can be directed to one of several different experimental areas for study.

3.2.2

Ion Source

At the NSCL, there are two options for ion sources: the Superconducting Source for Ions
(SuSI) [96, 97] and the Advanced Room TEMperature Ion Source (ARTEMIS) [98]. Both
take a stable element and ionize it using the Electron Cyclotron Resonance (ECR) method.
In the ECR method, a plasma is confined by a magnetic field while microwave power is
applied at the electron cyclotron frequency,

ωc =

eB
,
me

(3.1)

where e is the electron charge, me is the electron mass, and B is the applied magnetic field
strength. The particles are ionized by collisions with the moving electrons. The ions are
then extracted and injected into the K500 cyclotron for acceleration. For e10003, the SuSI

44

source was used to produce 18 O3+ ions.

3.2.3

Cyclotrons

The coupled cyclotron facility at the NSCL [99, 100] houses two super-conducting cyclotrons,
the K500 and the K1200, where the numeric designations refer to the maximum extraction
energy achievable for protons. The cyclotrons accelerate charged ions by applying a strong
radio-frequency (RF) electric field, while a magnetic field constrains the path of the particles
to a circular orbit. Since the acceleration is more effective with higher charge states, a
stripper foil designed to remove electrons from the ions is placed at the entrance to K1200.
For light and medium-heavy nuclei, this allows for the creation of completely ionized beams.
Fig. 3.4 shows a schematic of the two cyclotrons.

Figure 3.4: Schematic showing the coupled cyclotrons and the A1900 fragment separator.
Figure taken from Ref. [95].
For e10003, the K500 cyclotron accelerated a beam of 18 O3+ ions to an energy of 10.91
MeV/u. These ions were extracted and sent through the stripper foil at the entrance to the
K1200 cyclotron, where the completely ionized 18 O nuclei were further accelerated to 120
MeV/u. The beam was then impinged on a 846 mg/cm2 target of 9 Be for fragmentation.
45

The resulting fragments then entered the A1900 fragment separator.

3.2.4

A1900

The A1900 fragment separator [95] consists of four dipole bending magnets and eight quadrupole
triplet focusing magnets. The magnetic field of the dipole magnets serves to disperse the
incoming particles according to their magnetic rigidity,

Bρ = p/q,

(3.2)

where B is the magnetic field, ρ is the bending radius, q is the charge and p is the relativistic
momentum. Together, Bρ is called the magnetic rigidity. Further selection is accomplished
by placing a wedge at the intermediate image of the A1900 (see “image 2” in Fig. 3.4).
The wedge causes species of different atomic numbers to lose different amounts of energy,
allowing the particles to be separated by their new magnetic rigidities in the following two
dipole magnets.
For e10003, the magnets were tuned to select 16 C. The slits at the first image of the
separator were set such that the momentum acceptance was dp
p = 0.54%. An aluminum
wedge of 240 mg/cm2 was placed at the intermediate image of the A1900 to further remove
contaminants. This resulted in a 97.8% pure beam of 16 C, with an intensity of 3.0 ¡ 104
pps
pnA 18 0

as measured at the focal plane of the A1900. The largest remaining contaminant in

the beam was 14 B.

46

3.2.5

S800 Spectrograph

Following the selection of 16 C in the A1900, the beam was transported through the transfer
line to the S3 experimental vault. The S3 vault consists of the S3 analysis line and the S800
Spectrograph as can be seen in Fig. 3.5. The spectrograph has a momentum acceptance of
5% and covers a solid angle of 20 msr. The analysis line can operate at maximum rigidity
of 4.9 Tm and is limited to 4 Tm for the spectrograph itself. Since the reconstruction of
the excitation energy and COM scattering angle for the (p,n) reaction in inverse kinematics
comes entirely from the measurement of the recoiled neutron, high resolution of the ejectile
was not needed. Therefore, S800 beam line was operated in achromatic beam-transport
mode.
The fast ejectile particles from the 16 C(p,n)16 N reaction were bent through the dipoles
of the spectrograph into a series of focal-plane detectors. The purpose of the focal-plane
detectors were to determine the hit position, angle, energy loss, and time-of-flight of the
ions. From these parameters, the momentum, scattering angle, and particle identity (PID)
can be determined. These focal-plane detectors are briefly described below.

Figure 3.5: Schematic of the S800 analysis line and spectrometer. Figure adapted from [94].
47

3.2.6

Focal Plane Detectors

The first particle detectors at the S800 focal plane [101] are the Cathode Readout Drift
Chambers (CRDCs). There are two CRDCs separated by 1073 mm allowing for two horizontal and vertical position measurements. Each CRDC is filled with a mixture of 20%
isobutane and 80% carbon tetrafluoride. As the particle passes through the CRDCs, it ionizes the gas mixture, causing the freed electrons to drift towards an array of 224 cathode
pads which lie along the dispersive (x) axis. As is illustrated in Fig. 3.6, the dispersive
position can be determined by analyzing the image charge distribution on the pads. The
non-dispersive (y) position is inferred from the drift time of the electrons in the detector.
The drift time is determined from the difference between the S800 DAQ trigger time and
the CRDC anode signal. Calibration details for the CRDCs are given in Chapter 4. From
these two position measurements the dispersive and non-dispersive angles of the particle can
be determined. This yields a total of four values: the dispersive position and angle (xfp and
afp) and the non-dispersive position and angle (yfp and bfp).
After the CRDCs, the particle passes through an ion chamber filled with a gas mixture
(90% argon and 10% methane), where the energy loss of the beam is measured. As in the
CRDCs the particle ionizes the gas mixture, which causes electrons to drift to one of 16
anodes, where the charge is read out. The amount of energy lost in the ion chamber is
proportional to the square of the charge of the particle as given in the Bethe-Bloch formula
[103],


4πe4 Z 2
2m0 v 2
dE
=
n z
ln
− ln 1 −
−
dx
I
m0 v 2 abs abs

v2
c2



v2
− 2
c


,

(3.3)

where Z is the charge of the incident particle, nabs and zabs are the number density and atomic
number of the absorbing material, m0 is the electron’s rest mass, I is the average ionization
48

Figure 3.6: Schematic of the two cathode readout drift chambers, with an example event
trajectory passing through the detectors. The inset shows an example of a charge distribution
detected by the pads. Figure originally taken from Ref. [101] and modified by Ref. [102]
potential of the absorbing material respectively, and v is the velocity of the incident particle.
Hence, by measuring the energy loss, the square of the atomic number can be determined
after correcting for the difference in velocity across the focal plane.
The final detector at the S800 focal plane was a 5-mm thick plastic scintillator, referred
to as the E1 scintillator. It serves as the trigger for the S800 data acquisition and gives time
and energy readouts from the PMTs attached at each end of the detector. The E1 scintillator
measures the TOF of the outgoing ejectiles in the 16 C(p,n)16 N reaction with reference to
the object scintillator described below. The TOF from the object scintillator to the focal
plane is proportional to the mass number of the ejectile and is used with the energy-loss
measurement to determine the particle identification at the S800 focal plane.
The object scintillator is a thin plastic scintillator that is placed in the path of the beam
in the object box of the S800 analysis line (see “object” in Fig. 3.5). For e10003, a 1127 µm

49

thick object scintillator was used. To ensure high-precision timing measurements could be
made with these detectors and the neutron detectors, the signals from the object and focal
plane scintillators were split and sent into both the S800 DAQ and the LENDA/VANDLE
digital data acquisition system.

3.2.7

Liquid-Hydrogen Target

The reaction target in e10003 was the Ursinus liquid-hydrogen target. It had an average
thickness of approximately 66 mg/cm2 and a radius of 35 mm. The temperature was kept
at approximately 19 K, at a pressure of just above 1 atm. Two 125 Âľm Kapton foils contain
the liquid in the target. Fig. 3.7 shows the measured phase transition from gas to liquid,
when the target was filled and cooled at the beginning of the experiment. The temperature
and pressure were monitored throughout the experiment and any fluctuations of the density
were corrected for on a run-by-run basis. Details on the target thickness determination are
given in Section 4.5.3.

3.2.8

LENDA and VANDLE

The Low Energy Neutron Detector Array (LENDA) [90, 91] is an array of 24 BC-408 plastic
scintillator bars built by the charge-exchange group at the NSCL. Each bar has dimensions
of 4.5 x 2.5 x 30 cm. It is designed to measure the kinetic energy of neutrons via the timeof-flight method with kinetic energies from 100 keV-10 MeV and with a neutron kineticenergy resolution of approximately 5%. A Hamamastu H6410 photomultiplier tube (PMT)
is attached directly to each end of the LENDA bars to read out the scintillation light. At 1-m
distance, the laboratory scattering-angular coverage of a LENDA bar is 2.58 degrees, and

50

1140
1120

P [Torr]

1100
1080
1060
1040
1020
1000
980
960
19

19.5

20

20.5

21

21.5

22

22.5

23

T [Kelvin]
Figure 3.7: Pressure-temperature diagram for the liquid hydrogen target showing the phase
transition from gas to liquid. The red line indicates the expected relationship taken from
[104].
its vertical position resolution, determined from the time difference of the top and bottom
PMTs, is 6 cm.
VANDLE, the Versatile Array of Neutron Detectors at Low Energy, is a similar set of
neutron TOF detectors from the University of Tennessee [92, 93]. VANDLE is comprised of
three different sized detectors, 3 x 3 x 60 cm (small) 3 x 6 x 120 cm (medium) and 5 x 5 x 200
cm (large), allowing different experimental layouts to be optimized to the energy resolution
and range needed. For e10003, 24 of the small sized plastic scintillator bars were used.
Similar to LENDA, VANDLE is able to measure low-energy neutrons with comparable energy
resolution and detection efficiency. Both sets of detectors had the signals from each PMT
sent into the Pixie DDAS System. Details about the digital system and the performance of
LENDA are given in Section 3.3.

51

3.3

Digital Data Acquisition System

The data acquisition system (DAQ) for LENDA was upgraded prior to the running of this
experiment in order to utilize a digital data acquisition system (DDAS). This upgrade effort
was a significant portion of the work that makes up this thesis. Using a digital system
allowed the VANDLE detectors to be easily integrated with the new DAQ, since adding
more detectors only meant using more DDAS modules. The following sections give a brief
overview of DDAS and describe the capabilities achieved for LENDA with DDAS.

3.3.1

DDAS Overview

The fundamental function of DDAS is to produce a digitized waveform, which is referred to
as the trace. Each signal from the detector system is sent through a Nyquist filter and into
a high-frequency analog to digital converter (ADC), where the analogue voltage is sampled
to produce a digital waveform [105].
LENDA and VANDLE were instrumented with XIA’s Digital Gamma Finder Pixie-16
boards [105]. For each of the 16 channels on the board, 250 megasamples per second (MSPS),
14-bit flash ADCs were used to perform the digitization of the incoming detector signals.
After capturing the waveforms, 4 field-programmable gate arrays (FPGAs) implement onboard trace-processing routines to extract both time and energy information for each pulse.
The Pixie-16 modules were placed in a PXI/PCI crate, which was controlled by a National
Instruments MXI-4 crate controller connected with fiber-optic cable to a dedicated computer.

52

3.3.2

Digital Timing

Unlike an analog system, where timing and energy measurements are performed by two independent circuits, DDAS extracts this information simultaneously from the trace. This is
accomplished for both time and energy through digital filtering algorithms, either on the
board or in off-line analysis. For the 250-MSPS digitizers (a period of 4 ns between clock
ticks) used in e10003, some form of interpolation scheme is needed to achieve timing resolution below the sampling frequency. This can be accomplished by selecting an appropriate
function for the detector signal of interest and fitting that function to the waveform to extract a sub-clock tick time for each pulse. Prior to e10003, this method was successfully
implemented for VANDLE where a detector-limited timing resolution of 0.6 ns was achieved
over the full energy range using the fit function,

4
f (t) = αe−(t−φ)/β (1 − e−(t−φ) /γ ).

(3.4)

With the highest energy signals, a timing resolution of 0.5 ns was achieved [92]. The fitting
method requires significant computational post processing and the creation of a library
of appropriate functions for each detector signal. An alternate choice—and the one that
was utilized in e10003 for both LENDA and VANDLE—is to use a digital constant fraction
discriminator (CFD) algorithm similar to the one implemented on the Pixie boards [105, 106].
Higher-order interpolations of the CFD are done off line in software that are unavailable on
the board. The implementation for this work is as follows. The trace is processed through

53

a symmetric trapezoidal filter,

F F [i] =

i
X
j=i−(Tr −1)

T r[j] −

i−(Tr +Tg )
X

T r[j]

(3.5)

j=i−(2Tr +Tg −1)

where Tr is referred to as the rise time, Tg the gap time and T r the trace points. The filtered
trace is used to form a digital CFD,

CF D[i + D] = F F [i + D] − w F F [i]

(3.6)

where D is a user specified delay and w is a scaling factor [105, 107]. The units of Tr , Tg , and
D are in clock ticks and w is a unitless scaling factor. The sub clock-tick timing information
is then determined from a linear interpolation of the CFD’s zero crossing (this technique will
be referred to as the Linear Algorithm). In Fig. 3.8, an example of this procedure is shown.
The filled circles are a trace for a gamma ray from a 22 Na source generated by one photo
multiplier tube (PMT) of a LENDA bar. The filled squares are its corresponding CFD signal
with filter parameters Tr = 6, Tg = 2, D = 4, w = 5/8.

3.3.3

Intrinsic Timing Resolution of DDAS

One of the significant challenges for high-precision timing with DDAS is the occurrence of
non-linearities in the system’s response to fast scintillator pulses [92, 106]. This behavior
makes it difficult to rely on a simple linear interpolation of the CFD’s zero crossing. This
non-linearity is maximized when the arrival time of a signal places its zero crossing in between
two clock ticks, which for the 250 MHz digitizers used here is a time difference of 2 ns. To
explore this behavior for LENDA, a 22 Na source was placed on a LENDA bar where the

54

ADC Units

20000

Trace
Interpolation

10000

CFD

0

-10000
-20000
-30000
40

50

60

70

80

90

100

Clock Ticks
Figure 3.8: Example of digital trace from LENDA (filled circles) with overlaid digital CFD
(filled squares). The thick line is the third-order interpolation function calculated for this
event.
signal from one PMT of the bar was split into DDAS with a 2 ns delay added to one copy
of the signal. Signals from both the 511 and 1275 keV gamma rays from 22 Na were used in
the following analysis. The CFD algorithm described above was performed on each signal
with the filter set Tr = 2, Tg = 0, D = 4, and w = 1/8. This parameter set corresponds
to the optimum choice for the Cubic Algorithm (described below) and is one example of
the linear-interpolation algorithm failing in the presence of a non-linear CFD signal. Figure
3.9 shows the time difference between these two signals after correcting for the dependence
of the time difference on signal amplitude (a phenomenon known as “walk”). The walk
correction was done by plotting the pulse height against the time difference and correcting
for any observed correlation of the time difference on the pulse height. The pulse heights in
this data set ranged from approximately 4% to 58% of maximum ADC value. The linear
interpolation with the above parameter set (labeled in the plot as “Unoptimized Linear”)

55

introduces systematic biases and fails to achieve good timing resolution.
However, the shape of the digital CFD signal in the neighborhood of the zero crossing is
a function of the CFD filter parameters. By performing an optimization of those parameters, significant improvement can be achieved in the performance of the linear-interpolation
method. As the parameters for the CFD algorithm are discrete, a brute-force approach was
used for the optimization. Each trace in the data was processed with every possible filter
combination in the ranges Tr = 1 − 8, Tg = 0 − 3, D = 1 − 8, w = 1/8 − 7/8 (Tr , Tg , D
were limited to integer values, while w was limited to steps of 1/8). The parameter set that
achieved the best resolution was taken as the optimum set. The result of that optimization
(Tr = 6, Tg = 2, D = 4, and w = 5/8) is shown in Fig. 3.9, where it is labeled as “Optimized
Linear.” Significant improvement is made in the linear method after the optimization, but

Counts

as Fig. 3.9 shows, there is still some systematic bias in the distribution.

25000

Unoptimized Linear Algorithm
Optimized Linear Algorithm

20000

Optimized Cubic Algorithm

15000
10000
5000
0
-0.5 -0.4 -0.3 -0.2 -0.1

0

0.1 0.2 0.3 0.4 0.5

Time Difference [ns]
Figure 3.9: Time-difference spectra from a split-signal electronics test. The time differences
from the 3 different timing algorithms have been shifted to zero for comparison.

Further improvements can be made through the use of a higher-order interpolation of the
56

CFD signal’s zero crossing. Here the two points before the zero crossing and the two points
after are used to calculate a third-order polynomial interpolating function. An example of
this is shown in Fig. 3.8 as a thick red line. Given the clock ticks and corresponding ADC
values, the coefficient for the interpolating function are found by inverting the following
equation,

3
(x1 )


(x2 )3



(x3 )3


(x4 )3

   
(x1
(x1
(x1  c3  y1 
   
   
2
1
0
   
(x2 ) (x2 ) (x2 ) 
 c2  y2 
  =  
   
   
(x3 )2 (x3 )1 (x3 )0 
 c1  y3 
   
c0
(x4 )2 (x4 )1 (x4 )0
y4
)2

)1

)0

(3.7)

where x1,2 and y1,2 are the two points before the zero crossing and x3,4 and y3,4 are the two
points after. The same optimization procedure as described above was used for this method
(referred to as the Cubic Algorithm). The result of this optimization (Tr = 2, Tg = 0, D = 4,
and w = 1/8) applied to the same data set as above is also shown in Fig. 3.9, labeled as
“Optimized Cubic.” It is clear that the resolution has improved substantially using the cubic
interpolation scheme and any systematic biases in the time difference spectrum have been
removed.
The intrinsic resolution results using the four points and a third-order polynomial were
much better than the expected resolution of the detector (∟420 ps) Therefore, higher-order
or more complicated interpolating schemes were not pursued.
Fig. 3.10 shows the full-width at half-maximum (FWHM) electronic timing resolution of
DDAS from the same data shown in Fig. 3.9. The width of the time difference distribution
is plotted as a function of pulse height for both the optimized cubic- and linear-interpolation
algorithms. Figure 3.10 shows that the Cubic Algorithm performs significantly better at low
pulse amplitudes, while both techniques provide electronic resolution well below the expected
57

FWHM [ps]

performance of a LENDA bar at an appreciable fraction of the ADCs range.

1400

Linear Algorithm

1200

Cubic Algorithm

1000

2ns Delay

800
600
400
200
0
0

0.1

0.2

0.3

0.4

0.5

0.6

ADC Fraction

Figure 3.10: Electronic timing resolution of DDAS as measured with a split signal from a
LENDA bar. The upside down triangles utilize a linear-interpolation method while the filled
circles use a cubic-interpolation method. See text for details.

3.3.4

Timing Resolution of LENDA with DDAS

The timing resolution of LENDA with DDAS was measured using the two correlated 511
keV gamma rays generated after the positron decay of 22 Na. Two LENDA bars were placed
on opposing sides of the 22 Na source at a distance of 1 m. Events were recorded in a 4-way
coincidence between the PMTs of the two bars. The time difference between the two bars
was calculated as:

1
1
Tdiff = (T1,top + T1,bottom ) − (T2,top + T2,bottom )
2
2

(3.8)

where the numerical subscripts refer to bar 1 and bar 2. The width of the Tdiff distribution,
√
corrected for walk and divided by 2, is taken as the timing resolution of a single LENDA
58

bar. In the same way described above, the parameters for this data set were optimized for
the best resolution, yielding Tr = 2, Tg = 2, D = 4, and w = 1/8 for the linear algorithm
and Tr = 2, Tg = 0, D = 2, and w = 1/8 for the cubic algorithm. Fig. 3.11 shows the
measured timing resolution in FWHM of one LENDA bar as a function of pulse amplitude
and energy in keVee for the linear and cubic timing algorithms discussed above. Unlike
the data presented in Section 3.3.3, the signals in this measurement arrive randomly across
all time differences, instead of always at a time difference of 2 ns. Because of the intrinsic
resolution of the detector, the difference in achieved resolution between the cubic and linear
algorithms is less than above. However, an improvement when using the cubic algorithm is
still achieved, reaching about 400 ps at the highest energies. The saturation of the resolution
at approximately 400 ps is a result of the intrinsic properties of the detector, and is not from

FWHM [ps]

the electronics.

0
1400

Energy [keVee]

50 100 150 200 250 300 350 400 450

Linear Algorithm

1200

Cubic Algorithm

1000
800
600
400
200
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ADC Fraction

Figure 3.11: Measured timing resolution for LENDA readout with DDAS. Upside down
triangles use a linear-interpolation method, while the filled circles use a cubic-interpolation
method.

59

3.3.5

Energy Extraction Methods

There are many algorithms for extracting energy from digitized data, especially for highprecision spectroscopy (for example, see Ref. [108]). In the case of a plastic scintillator
array, where signal-amplitude resolution is expected to be limited, relatively straightforward
methods can be utilized. With the waveform saved, the energy of the event can be easily
determined from the maximum trace value or, as is done for the e10003 analysis, by integrating the digitized pulse. In Fig. 3.12, a trace from one PMT in LENDA is shown, taken with
a 22 Na source. The beginning 24 clock ticks of the trace is used to determine the average
baseline of the signal, while a window around the maximum point in the trace is used to find

ADC Units

the pulse’s amplitude.

1600
1400
1200
1000
800
0

20

Baseline Window

40

60

Event Window

80

100

Clock Ticks

Figure 3.12: A trace from a PMT of LENDA from a 22 Na source. The first integration
region is used to determine the average baseline of the signal, while the second integrates
the pulse and determines the energy of the event.
To determine the resolution, an energy spectrum for 22 Na was measured using a LENDA
bar. The source was placed in the center of the bar, and the energy of the event was taken
60

as the geometric average of the top and bottom PMT’s pulse integration values,

EBar =

p
ET EB .

(3.9)

The top panel in Fig. 3.13 shows the calibrated energy spectrum in keVee obtained from
this measurement. The two Compton edges from the 511 keV and 1274.5 keV gamma rays
are seen at 341 and 1062 keVee, respectively. The Klein-Nishina formula convolved with a
Gaussian [109] was used to fit the two Compton edges in order to extract a FWHM resolution
of 23.5% at 341 keVee and 16% at 1062 keVee. Further, it was found that the position of the
Compton edge was at 63.2 Âą 1.9 % of the maximum yield of the Compton spectrum, where
the error was determined from the fitting of the Compton edges.
The bottom panel in Fig. 3.13 shows an energy spectrum from a similar measurement
using a 241 Am source. Here, the photo peaks from the low-energy x-rays at 26.3 and 59.5
keVee can be seen. The photo peaks were fit with Gaussians to obtain a FWHM energy
resolution of approximately 42.3%.

61

Counts

Na Source

22

103
102
10
1

Counts

0

200 400 600 800 1000 1200 1400 16001800 2000

Light Output [keVee ]

Am Source

241

1400
1200
1000
800
600
400
200
0
0

20

40

60

80

100

120

140

Light Output [keVee ]

Figure 3.13: Top panel shows a calibrated energy spectrum from 22 Na seen in a LENDA
bar. Bottom panel shows a calibrated energy spectrum from a 241 Am source.

62

3.3.6

KeVee to Tn

The correspondence between keVee and neutron kinetic energy in keV can be measured using
a 252 Cf fission source. This measurement was performed with an EJ-301 liquid scintillator
placed next to the 252 CF source, which was positioned 1 m from a LENDA bar. The liquid
scintillator served as the start signal for a neutron time-of-flight (TOF) measurement with
the LENDA bar, where a 3-way coincidence between the two PMTs of the LENDA bar and
the PMT of the liquid scintillator was required. Fig. 3.14 shows a result of this measurement,
where light output in keVee from neutrons in LENDA versus their kinetic energy determined

Light Output [keVee]

from time-of-flight is presented.

4000
3500
3000
2500
2000
1500
1000
500
0
0

1

2

3

4

5

6

7

8

9

10

Tn [MeV]
Figure 3.14: Scatter plot of light output against neutron energy from a 252 Cf source measured
with LENDA and a EJ-301 liquid scintillator as the timing reference. The cuts on maximum
light output are applied in this figure.

For high neutron energies (>3 MeV), the relationship between maximum light output
(full energy of the neutron deposited in the bar) and kinetic energy is linear. The relationship

63

approximately follows,

Lmax (E) = 518.1 Tn − 499.5, Tn > 3 MeV

(3.10a)

where Lmax (E) is the maximum light output in keVee and Tn is the kinetic energy of the
neutron in MeV. This relationship is non-linear at low neutron energies where it follows,

Lmax (E) = 18.53 + 95.08 Tn + 81.58 Tn2 , Tn < 3 MeV.

(3.10b)

To remove background events, cuts on the light output of the neutron detectors following
equations 3.10a and 3.10b were made in the analysis of e10003.

3.4

Implementation with the S800

Confident with the internal performance of DDAS, both LENDA and VANDLE were instrumented with the Pixie-16 250 MSPS modules for e10003. This section will discuss how the
clock synchronization between the two systems was achieved and provide some details of the
coincident trigger logic.

3.4.1

Clock Synchronization

A 50 MHz clock was taken from the backplane of the DDAS crate, downscaled to 12.5 MHz
and sent into a JTEC XLM72 module in the VME crate of the S800’s DAQ. Since this
clock was from the backplane of the crate, it was a down-scaled version of the internal fast
ADC clocks that perform the digitization of the detector signals and assign the events’ time
stamps. To achieve the synchronization two specialized DDAS modules were used. The first
64

was inserted in the front of the crate and served as a breakout module for the clock. It is
shown in the bottom of Fig. 3.15, labeled as clock breakout module. The second was a clock
extraction module connected to the backplane of the crate behind slot 2. Slot 2 contained the
master Pixie module, which distributed the clock to other Pixie modules over the backplane.
The clock extraction module connected to the clock breakout module with a Cat5 cable, as
indicated in Fig. 3.15.
For each event, the scaler from the clock signal was latched and readout into the S800
data stream, allowing for a common time stamp between the two systems at a granularity of
12.5 MHz (80 ns intervals). This level of synchronization is only used for event correlation
between the two systems, so it does not need the precision timing resolution achieved for the
neutron TOF. Prior to the experiment, the setup was tested for stability and was found to
remain synchronized for greater than 12 hours.
Under the firmware available at the time of the e10003 (March 2015), the internal clocks
of each channel would pick up a random 1 clock tick phase each time they were reset (the
default of the readout system at the beginning of each run). Further, reseting the clocks
in DDAS would mean that the clock in the S800 would have to be cleared and the two
systems synchronized again. These issues motivated finding a way to reset these clocks as
infrequently as possible. As will be discussed in Section 3.4.2, this complicated the setup.

3.4.2

Operation Infinity Clock

The goal of Operation Infinity Clock was to allow the internal clocks of each DDAS channel
to count continuously during the whole length of the experiment, without a reset at the
beginning of each run. To achieve this, the DDAS readout program was edited such that the
firmware call to reset the modules was only called once when the first run was started. The
65

S800 DAQ software was also modified to allow the scaler which keeps track of the DDAS
clock, to remain uncleared between runs. Therefore, the clocks had to be reset only once
at the beginning of the experiment and any time the DAQ crashed (which unfortunately
occurred several times, requiring new timing offsets to be calculated for every channel).
When the DDAS system was reset, it would take approximately 10 seconds before the
clocks were cleared and the system entered a running state. During this time, it was necessary
to ensure that the S800 system did not receive any triggers or clock cycles, so that only when
the DDAS system reset would the S800 begin processing timestamps and events. This was
accomplished using a specialized module that output a constant signal when DDAS was in
a running state. This module is labeled “DDAS State Module” in the diagram show in Fig.
3.15. This module was connected to the master DDAS module in slot 2 of the crate by a
cat5 cable. This signal was used as a veto for:
1. The clock signal from DDAS (which is output continuously)
2. The LENDA/VANDLE master trigger to the S800
3. The S800 DAQ
When starting the first run, the veto of the clock was necessary since the S800 could not start
counting the clock signals until DDAS reset its internal clocks and switched to a running
state. The veto of the LENDA triggers and the S800 DAQ were necessary to ensure that
no events happened in the S800 DAQ before the clock signal was ready. Reseting the two
systems in this way synchronized the clocks and ensured that no events were triggered until
both systems were ready.
However, since the clocks needed to be free running after they were synchronized in the
first run, the veto for the clock signal needed to be removed during the first run. This way,
66

the veto is not reapplied when the DDAS system stops taking data at the end of the run,
and the two systems remain synchronized. To accomplish this, the veto cable for the clock
was patched to the data-U and then back down to the experimental setup. This way, during
the first run, the cable could be unplugged, thereby sending the clock signal to the S800
DAQ regardless of what state the DDAS crate was in. Experimental shift takers were given
detailed instructions about the role of this “magic veto cable” to ensure the clock remained
synchronized during the experiment.

3.4.3

Trigger Logic

In order to select charge-exchange events from the 16 C(p,n)16 N reaction, both a neutron and
a fast ejectile needed to be detected. To achieve this, the DAQ trigger logic was constructed
to only record coincidence events between the two systems. For LENDA and VANDLE,
the definition of a good trigger occurred whenever any bar had a signal in both the top
and bottom PMT, within a 24-ns coincidence window. This initial check was conveniently
performed internally by the FPGAs on the Pixie boards. The 24-ns coincidence window
was chosen because it was the smallest value where the behavior of internal coincidence was
reliable (at 8 and 16 ns, some coincidences would miss validation). The OR of all of the
top-bottom coincidence in a given Pixie module was output from the Pixie breakout modules
(this happend on the channel labeled o7 ). The breakout modules have lemo connectors for
the available trigger inputs and outputs of the Pixie boards and are labeled as PIXIE 16
Breakout modules in Fig. 3.15. The outputs (o7 ) of all 8 Pixie modules were or-ed together
to form the LENDA and VANDLE master trigger.
This master trigger was sent up to the S800 DAQ where it was input as the secondary
trigger to the S800 trigger box. If a coincidence was found, the S800 DAQ would send a
67

live trigger signal back down to LENDA and VANDLE. During the time that the initial
neutron master trigger went up to the S800 DAQ and back down, the Pixie boards buffered
the detector signals (the delay was approximately 1.2 Âľs). The live trigger was sent back
into the boards external validation input (i4 ) on the breakout modules, which triggered
an internal validation window of 700 ns. The size of this window was chosen to be long
enough to validate low-energy neutron events, while short enough to keep the rate of random
coincidence from overwhelming the DAQ system. Under these conditions, a raw coincident
event rate of approximately 500 per second was written to disk. The S800 singles rate was
approximately 10k.

68

DDAS State
Module
NSCL#3316

3 ns

TTL or NIM

Clock to S800
fast patch 1
on 20 ns cable

A
C

X
Y

2 ns

A
C

X
Y

DDAS Trig
to S800
fast Patch 2 on
20 ns cable

A
C

X
Y

A
C

X
Y

A
B
C
D
X
X

A
B
C
D
+
+

S

L

A
B
C
D
X
X

A
B
C
D
+
+

S

L

A
B
C
D
X
X

A
B
C
D
+
+

S

L

A
B
C
D
X
X

A
B
C
D
+
+

S

L

A

B

Patch
#108
DATA-U6

D

X

1 ns

Y

B

X

Patch
#107
DATA-U6

Y

1 ns

D

C

X
Y

A
C

X
Y

B

A

D

C

X

X

1 ns

Y

Y

A

D

C

X

X

1 ns

A
B
C
D
+
+

S

L

A
B
C
D
X
X

A
B
C
D
+
+

S

L

A
B
C
D
X
X

A
B
C
D
+
+

S

L

B

1 ns

D

1 ns

o7

B

1 ns

D

1 ns

Y

A
B
C
D
X
X

A
B
C
D
+
+

o7

S

L

OUT

2ns

o7

Out

B

2 ns

D

2 ns

o7
o7

~30 ns

Y

2 ns

D

2 ns

2 ns

Obj->DDAS

Obj->DDAS

~30 ns
Comp

Phillips 726
NSCL#2965

69

PIXIE-16 250MHz

Clock breakout
module

To DDAS Clock
mod on 5 ns
cable

5 ns

1

2

3

4

5

6

7

x2 for FP signal
DDAS breakout
module
i4: Validate DDAS

5 ns

i4

To NSCL#2965

Mod
0

From
NSCL#3261

Obj from
patch 71

Norm

1 ns

S800 Sync
fast patch3
on 20 ns cable

Out

2 ns

Patch #110
DATA-U6
Magic VETO
Patch #109

X

B

2 ns

~8 ns

PIXIE 16
Breakout Module

Obj->DDAS

Comp

Y

Y

4-Way Splitter
NSCL#1166
Obj->DDAS

o7

X

X

QUAD FAST AMP
(Green face)
NSCL#0679

IN

S800 live
trigger
from fast
patch 4
on 20 ns
cable

X
Y

FAN IN / FAN OUT
MODULE (1X16)
NSCL#3261
LVTTL

o7
o7

4x4 Logic
21X6421 P-1
NSCL#0416

4x4 Logic
21X6421 P-1
NSCL#1341

TTL NIM NIM

2 ns

B

Y

A
B
C
D
X
X

LeCroy 688AL
NSCL#0439

o7
A
B

A
B
C

Cat5 cable to DDAS
Back plane
(to 2nd port from top)

x8

A
B
C
S/N #

1047 | 1039 | 1043 | 1035 | 1040 | 1042 | 1036 | 1044

Figure 3.15: Detailed diagram of the electronics setup in experiment e10003. The NSCL numbers refer to the NSCL E-Pool
number for the module depicted. Care has been taken to draw the modules with enough detail to make them easily recognizable.
The positions of the switches depicted in the first two modules on the left correspond to their actual setting in the experiment.

69

Chapter 4
Data Analysis
Have you tried looking at the traces?

Remco Zegers

The purpose of this chapter is to discuss how the raw data from e10003 was processed and
dσ ). To accomplish this, a ROOT based analyfiltered to arrive at differential cross sections ( dΩ

sis package was developed to unpack, build, and analyze the experimental data. Throughout
the data, analysis the different “reaction channels” are discussed. This language refers to
the different final ejectile particles that are possible in the 16 C(p,n)16 N reaction (see Table
3.1). In some of the reaction channels, there is more than one rigidity setting of the S800
spectrograph used in the analysis of that channel. These parts of the data are referred to as
“settings” or “rigidity settings”.

4.1

Calibration

This section describes the details of the calibrations procedures used in e10003 for the neutron
detectors and the focal plane detectors of the S800 spectrograph.

4.1.1

Neutron Detector Timing Calibrations

Since the basis of the (p,n) measurement is the TOF of the neutrons, the calibration of
the timing signals from the neutron detectors is crucial. The first step in the calibration
70

procedure is to choose a start time for the neutron TOF. With the digital system, each
signal in the setup has an absolute timestamp, which allows any signal to be chosen as the
reference time for the TOF after the data taking. Originally, the in-beam scintillator located
in the S800 object box was going to be used for this purpose. However, as the analysis
progressed it became clear that this detector suffered significant radiation damage due to
the high incident beam rate. This deteriorated the detector’s timing properties beyond what
was acceptable to achieve the physics goals of the measurement.
Instead, the timing of the beam-like ion measured by the E1 scintillator at the S800
focal plane was used as the reference time for the neutron TOF. This has the advantage
of significantly lowering the incident particle rate (thousands instead of millions per second), preventing any radiation damage. However, using the S800 E1 scintillator introduces
dependence of the neutron TOF on the dispersive angle and position in the focal plane.
Figures 4.1 shows the correlation between neutron TOF, and the dispersive angle and
the dispersive position. These events come from the experimental data, where 15 N has been
selected in the S800 focal plane (see Section 4.1.3 for details about the particle identification).
The dependencies can be corrected by looking at the events from the Îł flash seen prominently
in both figures. Since the Îł events must all occur at the same time, the correlation can be
fit and removed. After these corrections are performed the position of the Îł flash is used
to calibrate this time difference to reflect the actual TOF from the target to the neutron
detectors. Since the distance from the center of the target to the bars is known to be 1 m,
the time that the Îł flash should appear can be easily calculated (1m/c=3.33-ns).

71

3
1000

afp [Deg]

2

800

1
0

600

−1

400

−2

200

−3

−4

−2

0

2

0

4

TOF [ns]
(a)

300

250

250
200

xfp [mm]

200
150

150

100
100

50
0

50

−50
−100
−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

0

TOF [ns]
(b)

Figure 4.1: Two dimensional histogram showing the correlation between the neutron TOF,
and dispersive angle (a) and position (b) in the S800 focal plane. The Îł flash is seen
prominently in the figures. The neutron events are to the right, but are cut off to clearly
show the Îł flash.

72

In addition to the corrections for the focal-plane parameters, the neutron TOF must
also be corrected for any walk (dependance of the TOF on the signal height in LENDA or
VANDLE). Fig. 4.2 shows this dependence for events where 15 N has been selected in the
S800 focal plane.

500

E [keVee]

400
300
200
100
0
−5

−4

−3

−2

−1

0

1

2

3

4

5

TOF [ns]
Figure 4.2: Neutron TOF plotted against the calibrated light output from the neutron
detectors. The Îł flash can be seen centered around 0. At low energies, a small correlation
can be seen for the Îł events.

Combining all of these effects provides the following expression for the neutron TOF,

T OF = Tn − TE1 − a(af p) − b(xf p) −

c
− Tof f set ,
Lo

(4.1)

where Tn is the timestamp from LENDA or VANDLE, TE1 is the timestamp from the top
PMT of the S800 focal-plane scintillator, af p is the dispersive angle in the focal plane, xf p is
the dispersive position in the focal plane, Lo is the light output seen in LENDA or VANDLE
in keVee, and a, b and c are correction coefficients.
Fig. 4.3 shows the fully corrected TOF spectrum for the 15 N data. The Îł events are seen

73

cleanly separated from the neutron events by TOF. A FWHM timing resolution of ∟700-ps
is achieved in the experimental data. Similar calibration procedures were used for the other
channels of interest 14 N, 15 C, 14 C, 16 N. Their TOF spectra are shown in Fig. 4.4.
50000
45000

Îł

40000

N

15

35000
30000
25000
20000
15000

n

10000
5000
0

0

20

40

TOF [ns]

60

80

100

Figure 4.3: Fully Corrected neutron time of flight spectrum for the N15 data (3.1325 Tm
setting). The dashed green line indicates the TOF cut between Îł and neutron events. It
corresponds to a neutron kinetic energy of ∟67 MeV, which is far above the energies used
in the analysis.
As can be seen in the Fig. 4.4, the Îł flash for the 15 C channel has a long tail and what
appears to be poor resolution. Since the target material is hydrogen, the Îł rays must come
from some excited state in the ejectile. However, below the neutron emission threshold, 15 C
has only 1 known state at 740-keV. Since this state has a long lifetime of 2.61 ns, the Îł flash
appears smeared out. The 15 C can emit the Îł ray later along its flight path, increasing the
total distance it must travel to the neutron detectors. This effect was simulated and included
in the timing calibration of the 15 C data.
Fig. 4.4 also shows a sharp peak to the right of the Îł flash in the 15 C and 14 C TOF
spectra. These peaks come from a strong neutron-knockout background that dominates at

74

forward lab angles. Details of this background and its subtraction are given in Section 4.4.

4500
4000
3500
3000
2500
2000
1500
1000
500
0

0

20

40

60

TOF [ns]

80

200
180
160
140
120
100
80
60
40
20
0

100

0

(a) 14 N Events

14000
12000
10000
8000
6000
4000
2000
0

0

20

40

60

TOF [ns]

20

40

60

TOF [ns]

80

100

80

100

(b) 16 N Events

80

35000
30000
25000
20000
15000
10000
5000
0

100

(c) 15 C Events

0

20

40

60

TOF [ns]

(d) 14 C Events

Figure 4.4: Corrected neutron TOF spectra for the 14 N (2.9 Tm) (a), 16 N (3.38 Tm) (b),
15 C (3.6565 Tm) (c), and 14 C (3.38 Tm) (d) channels. The 15 C and 14 C data from the 3.52
Tm and 3.2828 Tm settings are not shown, but have similar TOF spectra.

4.1.2

LENDA and VANDLE Gain Calibrations

Though the light output from the neutron detectors is not directly used in the determination
of the excitation energy and COM scattering angle, it is necessary for setting the neutron
detection threshold. The threshold affects the neutron detection efficiency and therefore the
differential cross section. To accomplish this the light output for each bar is calibrated into

75

keVee using Îł sources 241 Am, 22 Na and 137 Cs. The specific gamma rays used and their
specifications are summarized in the Table 4.1.

Source

EÎł

Type

EÎłReduced

22 Na

1274.5

CE

1061.7

22 Na

511

CE

340.7

137 Cs

661.7

CE

477.4

241 Am

59.5

Ph

59.5

241 Am

26.3

Ph

26.3

Table 4.1: Table of the Îł-ray calibration sources used for the neutron detector gain calibration. See text for details.
EÎł is the true energy of the emitted gamma ray, CE refers to compton edge, Ph refers
to photo-absorption and EÎłReduced is the expected energy of the gamma ray in the neutron
detectors. For photo-absorption the expected energy is unchanged, but for compton edges
the energy is reduced (see Appendix B for more details).
The gain for the top and bottom PMT of each bar were matched in software using the
compton edge of 137 Cs. The total energy of the bar was then defined as,

E=

p
ET EB ,

(4.2)

where ET,B are the gain matched energies for the top and bottom PMTs. The energy for the
bar was then calibrated into keVee using the compton edges from 22 Na and the photo-peaks
from 241 Am,
E Cal = g ¡ E + T h,

76

(4.3)

where g is the gain and T h is the threshold from the fit.

1400

E [KeVee]

1200
1000
800
600
400
200
0
0

200

400

600

800

1000

1200

E [arb]
Figure 4.5: Example light output calibration for south LENDA bar 2. The data points are
described in Table 4.1. The red line is a linear fit to the data.

Fig. 4.5 shows an example of the calibration curve from a LENDA bar (south LENDA
2) using the 4 points from 241 Am and 22 Na. The neutron detection threshold is set using
this calibrated energy for each bar. The uncertainty in the calibrated energy from fitting
uncertainties and drifts in the gain during the experiment are included in the final systematic
error for the differential cross section.

4.1.3

Particle Identification

Though the experimental conditions are tuned to explore charge-exchange reactions, many
other processes can trigger the DAQ by producing a neutron and an ejectile in the 16 C
+ p reaction. To filter away the events from these processes the timing and energy loss
information from the E1 scintillator and the ion chamber in the S800 are used to make a
particle identification plot. This allows only the ejectile of interest (16 N, 15 N, 14 N, 15 C, 14 C)

77

to be included in the data analysis. Fig. 4.6 shows an uncalibrated particle identification
plot for the 3.1325 Tm rigidity setting (the main setting for 15 N), where the TOF of the ion
to the S800 focal has been shifted to near zero for convenience.
1600
1400

∆ E [arb]

1200
1000
800

N
13
C
15

600
400
200
0
−30

−20

−10

0

TOF [ns]

10

20

30

Figure 4.6: Uncalibrated Particle Identification for the 3.1325 Tm setting.

While even at this level separation between different particles can be seen, the resolution
of both the TOF and energy loss signals can be improved by correcting for the dependence
on dispersive position and angle in the focal plane. Fig. 4.7 shows these dependencies for the
TOF when the 15 N events are selected in the uncalibrated PID spectrum shown in Fig. 4.6.
Similar correlations and corrections exist for the energy loss measured in the ion chamber.
After applying these corrections a particle identification plot is constructed for each
rigidity setting. Fig. 4.8 shows a corrected PID spectrum from 4 of the rigidity settings
measured in the experiment, each of which highlights an exit channel of interest. Elliptical
gates are made on each particle extending out to 2 sigma in TOF and ∆E. The good
events cut away in this process are included in the final normalization of the differential
cross section. These gates are shown as black ellipses in Fig. 4.8.
78

3

afp [Deg]

2
1
0
−1
−2
−3
−3

−2

−1

0

1

2

3

TOF [ns]
(a)

300
250

xfp [mm]

200
150
100
50
0
−50
−100
−150
−200

−4

−2

0

2

4

TOF [ns]
(b)

Figure 4.7: Correlation between the ejectile TOF and the dispersive angle (a) and position
(b) in the S800 focal plane for 15 N events. The artifacts seen in (b) are the result of bad
events in the CRDCs, which are removed in subsequent steps of the analysis.

79

(a)

(b)

(c)

(d)

Figure 4.8: Corrected PID spectra from the 3.1325 Tm (a), 2.9 Tm (b), 3.6565 Tm (c),
and 3.38 Tm (d) settings. The main particles of interest in each setting are indicated. The
remaining two settings 3.52 Tm and 3.2828 Tm are not shown, but are similar to the 3.6565
Tm and 3.38 Tm settings.

80

4.1.4

CRDC Calibration

In order to find the particle’s position and angle from the pad distribution and electron drift
time measured by the CRDCs, the detectors must be calibrated. To accomplish this a metal
plate (a “mask”) with a series of known holes and strips is placed 8 cm upstream of each
CRDC for a calibration measurement. The result of these calibration runs are shown in Fig.
4.9, where the mask hole and strip pattern is clearly visible. Taking the measured drift time
and pad position of several of these holes a linear calibration of the form,

X1,2 (mm) = sx1,2 (mm/pad) ¡ X(pad) + cx1,2 (mm)
y

y

Y1,2 (mm) = s1,2 (mm/ns) ¡ Y (ns) + c1,2 (mm),

(4.4)

(4.5)

x,y

can be fit to the data. The s1,2 coefficients refer to the slope of the calibration for x and
x,y

y of CRDCs 1 and 2. Similarly C1,2 are the offsets from these calibrations. Since the x
measurement comes from discrete PADs, the slope for that calibration is just the width of
the pads, 2.54 mm/pad. The constant offset can be found by using any of the holes seen in
Fig. 4.9. For the Y direction, 3 vertical points in the center of the mask pattern were used
in the fit.
To accurately determine the x position from the PAD distribution, the gains of the PADs
must be matched and the resulting distribution fit on an event-by-event basis. If the gains
of some PADs are substantially larger than the others, the fits can be biased towards those
PADs, creating artifacts in the spectrum. Fig. 4.10 shows the effect of this calibration step
for CRDC 1.
Once the positions for both CRDCs are fully calibrated, the dispersive and non-dispersive

81

3000

40

2500

35
30

2000

TAC

25

1500

20

1000

15

500

10
5

0
0

50

100

150

200

250

300

0

Pad Number
(a) CRDC 1

35

2500

30

2000

25

TAC

3000

20

1500

15

1000

10
500
5
0
0

50

100

150

PAD Number

200

250

300

0

(b) CRDC 2

Figure 4.9: The uncalibrated position (x-axis) and time (y-axis) measured by the CRDCs
during the mask runs.

82

5000
4500

50

4000

E [arb]

3500

40

3000
30

2500
2000

20

1500
1000

10

500
0
0

20

40

60

80

100

120

140

160

180

200

220

0

PAD Number
(a)

5000
4500

50

4000

E [arb]

3500

40

3000
30

2500
2000

20

1500
1000

10

500
0
0

20

40

60

80

100

120

140

160

180

200

220

0

PAD Number
(b)

Figure 4.10: The pad distribution for CRDC 1 before (a) and after (b) the pad-by-pad gain
matching. The data is from one experimental run of the 3.1325 Tm rigidity setting.

83

angles can be calculated,
af p = atan((X2 − X1 )/1073.0)

(4.6)

bf p = atan((Y2 − Y1 )/1073.0),

(4.7)

where 1073.0 is the distance between the CRDCs in mm. This defines 4 parameters for the
particle’s track in the focal plane.
However, determining the particle’s properties at the target position is often a more
critical piece of information. Specifically for this measurement, the ejectiles momentum distribution is needed so that the cross section can be corrected for the momentum acceptance
of the spectrometer, and the different rigidity settings can be stitched together. To accomplish this the transfer map of the spectrograph is calculated using the ion optics code COSY
infinity [110]. With the inverse map (M −1 ) calculated by COSY the target parameters can
be derived,


 
xf p
ata


 


 
af p
yta


 
.
  = M −1 


 
yf p
 bta 


 
 


dta
bf p

(4.8)

The “ta” (as opposed to “fp”) designation is a common convention to indicate that these
are the values at the target position. Similar to the focal plane parameters ata (bta) is the
dispersive (non-dispersive) angle at the target, yta is the non-dispersive position, and dta
is the deviation of the particles’ energy relative to the energy of the center track through
the spectrometer (E0 ). The kinetic energy of the ejectile can be determined from dta easily
through,
84

E = E0 (1 + dta).

(4.9)

From the energy and angles, the momentum vector p~ can be determined. The ejectiles kinetic
energy is used to investigate the acceptance of the spectrometer (see Section 4.5.2 for more
information).
With the above calibrations completed, we have the ingredients necessary to calculate
the COM scattering angle and excitation energy for each event. The following sections go
through the major analysis steps needed to extract absolute differential cross sections from
the data.

4.2

Experimental Yield

In order to calculate the COM differential cross section as a function of excitation energy
in 16 N, the lab measurements must be transformed into the correct variables. For the
(p,n) reaction in inverse kinematics this is done with missing-mass spectroscopy, where the
excitation energy of 16 N can be found with,

Ex =

q

2 + P 2 − 2P P
(Mp + M16 C + T16 C − Mn − Tn )2 − (P16
n 16 C Cos(θn )) − M16 N .
n
C

(4.10)
Mx , Tx and Px refer to the rest mass, kinetic energy and momentum of particle x in the
lab frame respectively. θn is the lab scattering angle of the neutron. The momentum of the
neutron and 16 C are given by,

Px =

q
2Mx Tx + Tx2 .
85

(4.11)

To find the center of mass scattering angle the following relation is used,

Θcm = Sin−1

!
Pnlab
Sin(θn ) ,
Pncm

(4.12)

where the Pncm is found by Lorentz boosting the neutron lab momentum vector into the
COM frame.
The above relations were used to find the excitation energy and COM scattering angle
for each event. The yields in the lab and COM are presented in Fig. 4.11. The plots on the
left show a two dimensional histogram of lab neutron kinetic energy (y-axis) and neutron
scattering angle (x-axis) for the different decay channels. The figures on the right show the
corresponding two dimensional histogram of COM scattering angle (y-axis) and excitation
energy in 16 N (x-axis). On the left, the expected kinematic relationships for the 16 N, 15 N,
and 14 N channels are clearly visible. However, for the 15 C and 14 C channels there is little
apparent structure. Instead the spectra are overwhelmed with forward peaking background
events. These events come from neutron-knockout reactions that have the same outgoing
particles as true charge-exchange reactions. The neutrons from these events scatter off the
downstream quadrapole magnet and hit the neutron detectors, creating false coincidences.
Section 4.4 will discuss this background in more detail.
The data were separated into two-degree bins in COM scattering angle and the excitation
energy histogramed to form the experimental yield. As an illustrative example, Fig. 4.12
shows the experimental yield in the Θcm = 4 − 6° bin for the 15 N channel. The acceptance
of the neutron detectors is clearly visible in the spectrum, where it creates artificial peak-like
structures. Also evident in the spectrum are background events that need to be subtracted.
The following sections will discuss the neutron acceptance corrections and the background

86

Θcm

Tn [MeV]

30 20 10 0 MeV
20
18
16
103
14
20o
12
10
102
8
15o
6
10o
4
2
5o
10
0
40 50 60 70 80 90 100110120130140

θn

20
18
16
14
12
10
8
6
4
2
0
−20

102
10
1
−10

10

20

30

Ex [MeV]
(b) 15 N

30 20 10 0 MeV
20
103
18
16
14
20o
12
102
10
8
15o
6
10o
4
10
2
5o
0
40 50 60 70 80 90 100110120130140

Θcm

(a) 15 N

Tn [MeV]

0

θn

20
18
16
14
12
10
8
6
4
2
0
−20

102
10
1
−10

0

10

20

30

Ex [MeV]

(c) 15 C

(d) 15 C

Figure 4.11: Two dimensional histogram of lab neutron angle and kinetic energy measurements (a,c,e,g,i), and reconstructed excitation energy in 16 N and COM scattering angle
(b,d,f,h,j). In (a,c,e,g,i) the solid lines indicate excitation energy in 16 N and the dashed lines
indicate the COM scattering angle.

87

Figure 4.11: cont’d
(f) 14 C

30 20 10 0 MeV
20
18
16
14
103
20o
12
10
8
15o
102
6
10o
4
5o
2
10
0
40 50 60 70 80 90 100110120130140

Θcm

Tn [MeV]

(e) 14 C

θn

20
18
16
14
12
10
8
6
4
2
0
−20

102
10

−10

10

20

30

(h) 16 N

30 20 10 0 MeV
20
18
16
10
14
o
20
12
10
1
8
15o
6
10o
4
10−1
5o
2
0
40 50 60 70 80 90100110120130140

Θcm

Tn [MeV]

0

Ex [MeV]

(g) 16 N

θn

20
18
16
14
12
10
8
6
4
2
0
−20

10
1
10−1
−10

0

10

Ex [MeV]

20

30

(j) 14 N

30 20 10 0 MeV
20
18
16
102
14
20o
12
10
8
15o
10
6
10o
4
5o
2
1
0
40 50 60 70 80 90 100110120130140

Θcm

(i) 14 N

Tn [MeV]

1

θn

88

20
18
16
14
12
10
8
6
4
2
0
−20

10
1

−10

0

10

Ex [MeV]

20

30

10−1

subtraction methods used for the different channels.

6000
5000

Yield

4000
3000
2000
1000
0
−5

0

5

10

15

20

25

30

Ex [MeV]
Figure 4.12: Experimental yield for the 15 N channel (3.1325 Tm) in the Θcm = 4 − 6° bin.

4.3

Neutron Detector Efficiency and Acceptance

Of significant importance to determining the absolute cross section for the 16 C(p,n)16 N reaction is the intrinsic neutron detector efficiency and geometrical acceptance of the LENDA
and VANDLE bars. As was done in previous work with LENDA, the efficiency for each
bar is determined through a Monte Carlo simulation that has been validated against experimental measurement [63, 64]. For e10003 this was done using the GEANT4 simulation
toolkit [111] and its associated high-precision neutron physics library [112]. This is the same
approach used by the VANDLE collaboration [93] and the European Low Energy Neutron
Spectrometer (ELENS) [113].
In order to be detected by LENDA or VANDLE the neutron must first interact with
the hydrogen or the carbon nuclei in the plastic scintillator. A cartoon of this process is

89

show in Fig. 4.13. In principle the neutron can interact with any combination of hydrogen
and carbon nuclei, however the light output from interactions with the carbon is highly
quenched [113, 93]. These carbon reactions can therefore contribute only a small amount
to the detector’s response in the neutron energy region of interest (< 20 MeV). The elastic
scattering on hydrogen dominates the spectrum.
Fig. 4.14 shows the light output for plastic scintillator in keVee for protons, 12 C, and 4 He
as a function of incident particle energy (adapted from [114]). The quenching of the carbon’s
light output can be clearly seen–a 1 MeV proton will result in a 177.7 keVee of light output
but a 1 MeV 12 C will make only 6.6 keVee. This effect pushes the carbon interactions below
the light-output threshold for all but the most energetic carbon scatterings. The low-energy
tail of the light-output function for protons was modified to better reproduce the measured
neutron response from the 252 Cf data presented in Fig. 3.14. This is shown as the dotted
blue line in Fig. 4.14. These light-output curves were used in the simulation of the neutron
detection efficiency.
Fig. 4.15 shows the result of the Geant4 simulation, where neutrons between 0 and 20
MeV were impinged on a single LENDA bar from a point like source. The light output was
determined in the simulation by taking the energy loss from each simulation step inside the
LENDA bar and converting it to a light output using the functions shown in Fig. 4.14.
The light output of all the steps are added together to arrive at a total light output for the
simulated event.
In the previous work with LENDA [63, 64], MCNP was successfully used to simulate the
response of the neutron detectors. Using Geant4 has allowed all aspects of the simulation to
be done in one framework. Within the simulation the beam profile, the bulging of the target
windows, the energy loss of the beam in the target, and the neutron detector response are
90

p
H(n,n)

n

12C(n,n)

n

12C

Figure 4.13: A cartoon depicting the interactions of neutrons in a plastic scintillating material. In order for the neutron to be detected it must undergo a nuclear reaction with the
hydrogen or carbon nuclei in the material. The angles of the scattering are exaggerated for
artistic purposes. The dominate elastic scattering on hydrogen is shown at the top of the
figure. Elastic (shown at the bottom of the figure) and inelastic (not shown) scattering on
12 C are also possible but highly quenched. See text for details.

91

105
104

Modified Proton

Light [keVee]

Proton
3

10

2

10

12

C

Alpha

10
1
10−1
10−2 −2
10

10−1

1

E [MeV]

10

Figure 4.14: Light output curves for BC-400 plastic scintillator. Each line shows the expected
light output as a function of incident particle energy. Adapted from [114] see text for details.

Figure 4.15: Two dimensional histogram of light output and incident neutron energy from the
Geant4 simulation. Contributions from elastic scattering on hydrogen and 12 C are included
in the simulation. Inelastic channels for carbon scattering are also included and create the
few events above the elastic scattering trend line. These events amount to a tiny portion of
the data (<0.05%).

92

all included at the same time.
To obtain at a simulated efficiency for the LENDA bar, a light output threshold must be
applied in the simulation to match the data. The threshold for the data was set at 30 keVee
as determined for each bar following the calibration procedure outlined in Section 4.1.2. An
effective threshold of 25 keVee and a scaling factor of 0.92 was applied to the simulation
since it better reproduced the measured efficiency at low neutron energies. Fig. 4.16 shows
the result of the Geant4 simulation with the measured efficiency overlaid at a threshold of
30 keVee. This efficiency was measured using a 252 Cf fission source as described in Ref. [90].

0.5
0.45

Efficiency

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0

0.5

1

1.5

2

2.5

3

3.5

4

Tn [MeV]
Figure 4.16: The black points show the measured intrinsic neutron detection efficiency for a
single LENDA bar at a light output threshold of 30 keVee. The blue line is the result of the
Geant4 simulation of the efficiency. Good agreement is seen.

Confident with the simulations ability to reproduce the intrinsic neutron detector efficiency, the simulation was used to calculate the total neutron acceptance as a function of
COM scattering angle and excitation energy for the full experimental setup. The beam was
simulated through the bulged liquid hydrogen target, where it underwent a charge-exchange
reaction. When the scattered neutron hit one of the detectors, the event was recorded. The
93

result of this simulation is shown in Fig. 4.17, where the neutron energy and angle have
smeared to match the experimental resolution and transformed into the COM frame.
40
5000

35

Θ cm [Deg]

30

4000

25
20

3000

15

2000

10

1000

5
0
−10

−5

0

5

10

Ex in

15

16

20

N [MeV]

25

30

35

40

0

Figure 4.17: Two dimensional histogram of Θcm and Ex in 16 N from the Geant4 simulation
of the e10003 setup.

The total detection efficiency as a function of COM scattering angle and excitation energy
is obtained by taking projections of Fig. 4.17 onto the x-axis in the same COM scattering
angle binning used in the analysis and dividing by the total number of events simulated
in that bin. Figs. 4.18 shows the simulated efficiency for the most forward (Θcm =4-6°)
and most backward (Θcm =14-16°) angles used in the analysis. The sharp, periodic features
seen in both figures are the result of measuring the lab scattering angle with discrete bars
of differing geometrical acceptance–the VANDLE bars cover approximately twice the solid
angle as the LENDA bars. These simulated curves were used to correct the experimental
yields for the acceptance in each COM scattering angle bin used in the analysis.
The simulation was also used to determine the experimental excitation-energy resolution
of the full setup. This was done by simulating the response of the neutron detectors to

94

0.05
0.045

Efficiency

0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−5

0

5

10

15

Ex in

16

20

25

30

35

40

25

30

35

40

N [MeV]

(a)

0.05
0.045

Efficiency

0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−5

0

5

10

15

20

Ex in 16N [MeV]
(b)

Figure 4.18: Simulated neutron acceptance and efficiency in the Θcm =4-6 ° (a) and Θcm =1416° (b) bin for the e10003 setup.

95

events with discrete excitation energies. As was seen in Ref. [64], the resolution worsened
at large COM scattering angles. The resolution varied from σ(5°)FWHM ≈ 750 keV to
σ(15°)FWHM ≈ 2 MeV. The following section describes the background subtraction used in
the analysis.

4.4

Background Subtraction

There are various possible background sources that can contribute to the charge-exchange
spectrum. They can be categorized as:
ˆ Foil background. There may be reactions off of the non-hydrogen nuclei in the Kapton

foils that surround the liquid-hydrogen target.
ˆ Random background. The TOF spectrum may have random coincidences between

neutrons and uncorrelated beam bunches.
ˆ Beam-induced background. Reactions on the hydrogen other than charge exchange

that produce the correct ejectile of interest and a neutron.
The foil induced background was studied by taking data with the empty target at the
conclusion of the experiment. This was done in all of the rigidity settings used in the
experiments. The contribution from these events was found to be vanishingly small (< %0.5)
and could not be determined accurately with the statistic of the background runs. Instead
the thickness of the target was increased to include the hydrogen nuclei in the Kapton foils
and a systematic uncertainty of 0.5% was included. The random background was inspected
by looking at events that occurred at negative TOFs. The rate of random coincidences was
also found to be very small (< 1%). Similar to the foil background, the random background
96

shape could not be accurately determined and was included as a systematic error. The
choice to use the S800 E1 scintillator for the neutron TOF made the random rate far less
than in previous experiments when an in beam scintillator was used [63]. See Section 4.7 for
a breakdown of the systematic errors.
The vast majority of the background events came from knockout or fragmentation reactions on the hydrogen in the target that produced the correct ejectile and a neutron. These
reaction types are associated with fast neutron emission at forward scattering angles. These
neutrons can scatter off the surrounding materials and hit a LENDA or VANDLE bar, sometimes creating a false coincidence. In this circumstance the TOF of the neutron is not related
to its energy and creates a complex background. Since modeling the different background
processes and the scattering of the neutrons from different materials in the experimental
vault was a difficult task, a background model built off the experimental data was used.
Different techniques were used for the different reaction channels since the background
shapes and intensity were different. For the 15 N and 14 N channels, a background model was
based on events that created 11 B in the spectrometer. Events that create 11 B cannot be
from charge-exchange reactions in the excitation energy range of interest of 0-20 MeV (the
11 B

channel would open at 42 MeV). Therefore they should give a good representation of

the neutrons from the possible background processes. A similar technique was used in the
first LENDA (p,n) experiment [63, 64]. Fig. 4.19 shows the two dimensional histogram of
neutron energy and scattering angle for 15 N gated events and 11 B gated events. Even at
the preliminary level the shape of the 11 B spectrum seems to reproduce the shape of the
background seen in the 15 N data.
A more detailed look at the 11 B model for the background reveals that some rescaling
is needed to characterize the background at higher excitation energy. Fig. 4.20 shows the
97

60
3

50

10

40

102

30
20

10

10

Tn [MeV]

Tn [MeV]

60

103

50
40

102

30
20

10

10

0
1
40 50 60 70 80 90 100110120130140

0
1
40 50 60 70 80 90 100110120130140

θn

θn

(a) 15 N

(b) 11 B

Figure 4.19: Two dimensional histogram of lab neutron scattering angle and neutron kinetic
energy for 15 N (a) and 11 B (b) gated events. The events from 11 B are used to model the
background seen in the 15 N channels.
ratio of the 15 N and 11 B data integrated between neutron kinetic energy of 40 and 60 MeV
as a function of lab scattering angle. The blue points show the region of θn that contains
charge-exchange events, while the black points contain only background. A linear function
was fit to the black points to yield a rescaling function that models the background seen in
the 15 N channel.
Using this model yields the background shape in Fig. 4.21, where the forward (Θcm =
4−6°) and backward (Θcm = 14−16°) most angles are shown. The background is normalized
to the data in the unphysical region below the channel’s threshold (2.5 MeV for 15 N). The
data presented in Fig. 4.21 still contains the jagged acceptance of the neutron detectors
and so contains periodic structures. Building the background model from the experimental
data reduces systematic error because mistakes in the acceptance of the bars are the same
between the data and the background model. A similar procedure was performed for the 14 N
data. The 11 B data was again used as the background model, but with a different rescaling
function.

98

3.5
3

Ratio

2.5
2
1.5
1
0.5
0

40

60

80

100

θn [Deg]

120

140

7
6
5
4
3
2
1
0
−5 0

Data
Background
arb

arb

Figure 4.20: Ratio of the 15 N and 11 B gated events integrated from 40 to 60 MeV in neutron
kinetic energy as a function of the lab neutron scattering angle. The blue points show
the region of charge-exchange reactions, while the black points indicate regions with only
background. The red line is a linear fit to the black points in the range shown.

5 10 15 20 25 30 35 40

Ex [MeV]

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−5 0

5 10 15 20 25 30 35 40

Ex [MeV]

(a) 15 N at Θcm =4-6°

(b) 15 N at Θcm =14-16°

Figure 4.21: Experimental yields with overlaid background shape from the 11 B models. The
most forward (Θcm =4-6°) and backward (Θcm =14-16°) COM angles are shown.

99

Characterizing the background for the 15 C and 14 C channels proved to be difficult due to
the large amount of neutron-knockout and fragmentation reactions. After many iterations,
events from the low-energy tail of the ejectiles kinetic energy distribution was used to estimate
the shape of the background. Fig. 4.22 shows the kinetic energy distribution for the 15 C
channel. The full distribution is reconstructed from two rigidity settings: 3.52 and 3.6565
Tm. The gray line represent the full distribution after combining the two settings with
corrections for incomplete acceptance. The spectrum contains both charge-exchange and
background events. However, charge-exchange events up to 20 MeV of excitation energy
cannot create the events seen in the low-energy tail of the distribution. This makes these
events a good candidate for a background model.
−6

25 ×10
20

arb

15
10
5
0
1250

1300

1350

1400

1450

1500

1550

1600

T15C [MeV]
Figure 4.22: The kinetic energy spectrum of 15 C events detected in the S800 Spectrograph.
The black (blue) hashed region comes from the 3.52 (3.6565) Tm rigidity setting. The
gray line is the full distribution constructed from the two settings. The thin blue line
shows the simulated distribution from charge-exchange reactions. The low-energy tail of the
distribution is energetically inaccessible to charge-exchange reactions and so must come from
background processes.

Fig. 4.23 shows the data and the above background model for the forward and backward

100

most scattering angles gated on the 15 C and 14 C channels. In both cases, the signal to
background is clearly much lower than the 15 N case. Both channels have limited yield for
the Θcm = 4 − 6° bin with increased strength appearing at backwards angles. The statistics
for the 15 C background model are clearly limited. However, this technique was the best
compromise between statistical and systematic error for this channel.

6

Data
Background

4

arb

arb

5
3
2
1
0
−5 0

5 10 15 20 25 30 35 40

0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−5 0

Ex [MeV]

Ex [MeV]
(b) 15 C Θcm = 14 − 16°

Data
Background
arb

arb

(a) 15 C Θcm = 4 − 6°

18
16
14
12
10
8
6
4
2
0
−5 0

5 10 15 20 25 30 35 40

5 10 15 20 25 30 35 40

Ex [MeV]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−5 0

5 10 15 20 25 30 35 40

Ex [MeV]

(c) 14 C Θcm = 4 − 6°

(d) 14 C Θcm = 14 − 16°

Figure 4.23: Experimental yield and background models for the 15 C and 14 C channels. The
data comes from the 3.6565 Tm and 3.38 Tm rigidity settings respectively. Yield is primarily
seen at backward angles.
The 16 N channel does not suffer from a complex background because the PID cut on 16 N
ensures that no breakup or fragmentation reaction occurred. A flat background subtraction

101

was done based on a linear fit to the data at higher excitation energies where there is no
strength in the 16 N channel.

4.5

Absolute Normalization Corrections

The goal of e10003 was to measure absolute cross sections in the 16 C(p,n)16 N reaction. This
meant measuring the beam rate incident on the liquid-hydrogen target and determining the
efficiencies and acceptances of each detector and analysis step. The neutron acceptance
and efficiency calculation has already been discussed in its own section (see Section 4.3).
This section discusses some of the remaining corrections that were used in final cross-section
normalization.

4.5.1

Incident-Beam Measurement

The scaler readout of the object scintillator was used to monitor in the incoming beam. The
measured rate varied between 2 and 3 MHz throughout the experiment. To determine the
total number of particles incident on the scintillator in each run, the averaged measured
rates were averaged together and multiplied by the length of the run. This gave a run-byrun tracking of the number of incident beam particles during the experiment. To check the
reliability of this measure, the correlation between the object scintillator rate and primary
beam rate monitors like the A1900 Faraday bar were confirmed.
However, at rates that are an appreciable fraction of the cyclotron’s RF frequency, the
measured rate will diverge from the true incident rate. This occurs because the probability
of finding more than 1 particle in a beam bunch becomes significant. If there are 2 or more
beam particles in a beam bunch the object scintillator’s scaler will only count once and the

102

measurement will be incorrect. The measured rate by the object scintillator needs to be
corrected for this effect. Appendix C gives a detailed derivation of this correction, which can
be given as a linear function,

Rcor = 5.6 · 10−8 Robject + 0.99,

(4.13)

where Robject is the raw scaler measurement. This effect was corrected on a run-by-run basis
and increased the rate by approximately 10-15%.
The energy loss spectrum (∆E) in the object scintillator was also measured. This is
shown in Fig. 4.24, where events with 1 and 2 particles in the beam bunch can be seen.
With the clear separation between these two cases, a separate determination of the incident
rate can be made. As Appendix C shows the average incident beam rate (r) is related to
the ratio of single hits (S1 ) and multi hits (S>1 ), and the RF rate (n),

r
S>1
= .
S1
n

(4.14)

This technique confirmed the scaler measurement with the above correction to within 3%.

4.5.2

S800 Acceptance

The S800 Spectrograph has a finite acceptance in both momentum and angle. To extract
a properly normalized cross section this acceptance must be taken into account. Further,
since the measurement of the cross section was broken across different, partially overlapping
rigidity settings these acceptances are key to understanding how to merge the settings together. Here, the case of 14 C, where the largest correction is required, is discussed. The full

103

104

1 Hit

Counts

103

2 Hits

102

3 Hits

10

1
0

5000

10000

15000

∆ E [arb]

20000

25000

Figure 4.24: Histogram of the energy deposited in the object scintillator in the first run of
the experiment. The peaks correspond to to single- and multi-particle hits in the scintillator.
kinetic energy distribution of 14 C is spread across 4 rigidity settings: 3.1325 Tm, 3.2828 Tm,
3.28 Tm, and 3.52 Tm. Fig. 4.25 shows a two-dimensional histogram of the kinetic energy
of the 14 C ejectiles and the dispersive angle at the target from the 3.2828 Tm setting. The
acceptance cut of the S800 is clearly visible. There is complete acceptance in the center of
the distribution, but the reduced acceptance in the dispersive angle removes events on both
ends of the distribution. No acceptance cuts were seen in the non-dispersive direction. To
correct for these acceptance effects, the full distribution for 14 C was reconstructed.
Fig. 4.26 shows the full kinetic energy distribution of 14 C. The distribution from the 3.52
Tm setting was extended into the area of partial acceptance (< 1420 MeV) by correcting
for the evens cut off due to the finite angular acceptance. The same procedure was applied
to the low energy side of the 3.2828 Tm setting. These regions are shown in Fig. 4.26 with
vertical line shading. Between the central two rigidity settings (3.2828 and 3.38 Tm), no
correction for partial acceptance was needed because the areas of full acceptance in the two

104

5

104

4

ATA [Deg]

3
103

2
1
0

102

−1
−2

10

−3
−4
−5
1100

1150

1200

1250

1300

1350

1400

1450

1

T14C [MeV]
Figure 4.25: A two-dimensional histogram of the kinetic energy and dispersive angle at the
target for the 14 C channel as measured in the 3.2828 Tm rigidity setting.
settings overlapped.
−6

50 ×10
45

3.38 Tm
3.28 Tm
3.13 Tm
3.52 Tm

40
35

arb

30
25
20
15
10
5

0
1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600

T14C [MeV]
Figure 4.26: Full kinetic energy distribution for 14 C ejectiles reconstructed from 4 rigidity
settings. The vertical line shaded regions indicate areas of incomplete acceptance. See text
for details.

With the full distribution reconstructed, acceptance curves can be constructed from the
data by dividing the uncorrected distribution in each setting by the total. This gives the
105

acceptance of the S800 as a function of kinetic energy of the ejectile. For the 14 C channel
only the two central settings (3.2828 and 3.38 Tm) were used to extract charge-exchange
information. Fig. 4.27 shows the acceptance curve for the 3.2828 Tm setting obtained in
this way. The acceptance on the low-energy side of the distribution has significant statistical
fluctuations since it comes from an area of incomplete acceptance.

1.2

Acceptance

1
0.8
0.6
0.4
0.2
0
1100

1150

1200

1250

1300

1350

1400

T14C [MeV]
Figure 4.27: Experimentally determined acceptance curve for 14 C in the 3.2828 Tm setting.

The width of the ejectile’s kinetic energy distribution from charge-exchange reactions is
a function of excitation energy in 16 N. The higher the excitation energy, the more energy is
available for the decaying nucleon(s) to carry away—widening the ejectile’s kinetic energy
distribution. To study this effect, the distribution from charge-exchange reactions needs to
be known, but Fig. 4.26 gives the the distribution for background plus charge-exchange
reactions. For the 15 N and 14 N channels, the charge-exchange distribution was estimated by
gating around the kinematic relationships seen in Fig. 4.11. From the measured distribution
a model for the neutron decay scheme was built into the Geant4 simulation to calculate
excitation energy dependent acceptances for each channel. For the 15 C and 14 C channels this
106

was not possible since there were no clear kinematic relationships seen in the two channels.
For these cases a decay scheme to the evenly spaced levels in the final nucleus was assumed.
The systematic error associated with this procedure is discussed below.
1.2

Efficiency

1
0.8

3.1325 Tm (15N)
2.9 Tm (14N)
3.38 Tm (14C)
3.2828 Tm (14C)
3.6565 Tm (15C)
3.52 Tm (15C)

0.6
0.4
0.2
0
0

2

4

6

8

10

12

Ex [MeV]

14

16

18

20

Figure 4.28: Efficiency curves from the finite S800 acceptance for each rigidity setting used
in the analysis. Lines of common color show settings centered around the same detected
particle in the CE reaction.

The results of these simulations are shown in Fig. 4.28, where the efficiency from the
finite S800 acceptance is shown for each rigidity setting used in the analysis. The start of
the lines for the higher lying channels indicates the threshold energy. Only for the 15 C and
14 C

channels were there two different settings used in the analysis. For these channels the

cross section was determined for each setting separately, then averaged together to form a
final cross section.
As Fig. 4.28 shows, the acceptance for most of the rigidity settings was quite high in the
excitation-energy regime of interest. Therefore uncertainties in the procedure to determine
these corrections will have a small effect on the final cross section. The exceptions to this
were the corrections for the 3.2828 and 3.52 Tm settings for 14 C and 15 C respectively. Both
107

of these settings corresponded to the low-energy tail of their respective channels where it was
clear that there was less charge-exchange reactions below 20 MeV of excitation energy. Since
this approach corrects each setting into a total cross section, the two results can be checked
against one another for consistency as a measure of the systematic error of the procedure.
In both cases the results were statistically equivalent, and the systematic error was deemed
smaller than the statistical error.

4.5.3

Target Thickness

When filled, the windows of the liquid-hydrogen target bulged beyond the nominal 7 cm
thickness of the frame. This bulging needs to be taken into account in the determination of
target thickness so that the cross section can be properly determined. Originally the thickness
was going to be estimated by looking at the energy loss of the 16 C beam going through the
full and empty target into the S800 focal plane. This requires changing the rigidity of the
spectrometer. However, during the analysis it became clear that the high rigidity of the 100
MeV/u 16 C beam put the spectrometer too close to its maximum operational rigidity and
saturated the magnets (Bρ=3.94 (empty) and 3.9 (filled) Tm out of a maximum of 4 Tm).
It became impossible to very accurately determine the momenta from the runs with different
Bρ. To alleviate this issue the target thickness was determined by looking at the energy loss
difference between the 16 C and the containment 14 B. With this approach the systematic
error between the two settings from the saturation of the magnets was reduced.
Fig. 4.29 shows the raw PID spectrum from the run where the beam was tunned through
the spectrometer to the focal plane. Selecting events for 16 C and 14 B and reconstructing
their energy yields Fig. 4.30, where the measured kinetic energy of the two particles after
traveling through the target is shown. This gives an energy loss difference of 311.7Âą 0.3
108

2000
1800

∆ E [arb]

1600
1400
1200
1000
800

B

C

14

600

16

400
200
0
940

945

950

955

960

965

970

TOF [arb]

975

980

985

990

Figure 4.29: PID spectrum from the run where the 16 C was tuned into the focal plane of
the S800. The beam consisted of 16 C and a small containment of 14 B.
MeV.
120
100

B

C

14

80

16

60
40
20
0
1000

1100

1200

1300

1400

1500

Kinetic Energy [MeV]

1600

1700

Figure 4.30: Kinetic energy measured by the s800 Spectrograph for the two beam particles.
The peak from 16 C has been scaled down for easy comparison with 14 B.

With the measured energy loss difference through the target between 16 C and 14 B, the
Geant4 simulation was used to determine a relationship between the target thickness and
109

energy loss difference. This was done with the full simulation including the bulged surface
of the target and the realistic beam profile determined from the data. A different simulation
was run for both particles for a range of target bulge values. The result of these simulations
is shown in Fig. 4.31, where a clear linear relationship is seen. From this relationship a half
bulge amount of ∟1.0 mm was found, giving a total target thickness of 9.1 ¹ 0.3 mm.
313

∆E [MeV]

312.5
312

311.5
311

310.5
310

0.6

0.8

1

1.2

1.4

1.6

Half Bulge [mm]
Figure 4.31: Relationship between energy loss difference and liquid hydrogen target thickness
determined from simulation. The points are simulated values and the red line is a linear fit.

The curved surface of the target and the finite extent of the beam spot must also be taken
into account. An average thickness over the width of the beam spot (∟3.9 mm FWHM) was
calculated from the amount of bulging and the size of the beam. This reduced the effective
target thickness by an amount substantially less than the 0.3 mm error bar. This corresponds
to an aerial density of 65.8 Âą 2.2 mg/cm2 .

110

4.5.4

Target Density Variations

The temperature and pressure of the liquid hydrogen target were monitored throughout the
experiment so that any changes in the density could be taken into account in the final cross
section normalization. The density was determined from the temperature according to the
following expression,

4
3
2
ρH
f orLH = −0.0008T + 0.0672T − 2.1467T + 29.219T − 65.934,
2

(4.15)

where T is the temperature in Kelvin [104]. Fig. 4.32 shows the density measurements
averaged over each experimental run. The fluctuations are small (< 0.5%) and are corrected
on a run-by-rub basis.

73.2
73.1

ρ [mg/cm3]

73
72.9
72.8
72.7
72.6
72.5
72.4
72.3
560

580

600

620

640

660

680

700

720

740

760

780

Run Number
Figure 4.32: The density derived from the temperature measurement of the liquid hydrogen
target for each experimental run.

111

4.6

Cross Section Calculation

In addition to the corrections discussed in the above sections, there are other run-by-run
corrections for various detectors and other overall normalization corrections. These factors
are shown in Equation 4.16 and summarized in Table 4.2. With these many effects, the
absolute differential cross section can be extracted. For e10003 the full definition for the
differential cross section, containing all efficiency factors and corrections is,

dσ
Ns (1 + Îą)
=
[b/sr],
dΩ
∆Ω10−24 ρ∗ daq ndectors tof pid crdc IC nhit s800 Trun trans Rcor NT

(4.16)

where Ns is the number of scattered particles from charge-exchange reactions in a given COM
scattering angle bin, ∆Ω (sr) is the solid angle coverage of that bin, Trun (s) is the length
of the run, ndetectors is the neutron detection efficiency, and Rcor (1/s) is the corrected
incident beam rate described in Section 4.5.1. NT (1/cm2 ) is the number of target nuclei
from the liquid hydrogen and the small contribution from the hydrogen in the cell’s Kapton
foils given by,

NT = 

ρH
LH tcell NA
2

M MH

+

ρH
kapton tf oil NA
M MH


,

(4.17)

where tcell is the target thickness, tf oil is the thickness of the Kapton foils (250 Âľm), NA
is Avogadro’s number, M MH is the molar mass of hydrogen, and ρH
x is the density of
hydrogen in either liquid hydrogen or kapton. The remaining factors, their descriptions, and
their approximate values are summarized in Table 4.2. The following paragraph gives a brief
explanation of how each of these corrections was obtained.
The DAQ live time (daq ) was determined from the ratio of the S800’s live and raw
clocks. Though the singles rate in the S800 focal plane was high (∟10k), the coincidence

112

condition with the neutron detectors brought live trigger rate down substantially—decreasing
the overall dead time of the system. The PID TOF efficiency (tof ) adjusted for the number
of events when the correct beam bucket could not be found in the digitized waveform of the
object scintillator signal. This timing signal was needed for the PID, so events lacking the
signal were discarded. The PID gate efficiency (pid ) was the 2σ radius of the PID gates.
The CRDC efficiency (crdc ) was determined by comparing the number of hits in the IC to
hits in the CRDCs for a given run. The IC pile up efficiency (IC ) was found by inspecting
the number of events in the high-energy tail of the IC distribution for a given particle. It
was found to be constant across different runs and detected particles. The single neutron hit
efficiency (nhit ) was found by examining the neutron multiplicity spectrum. Only events
in which 1 neutron was detected could be used in the analysis so higher multiplicities were
discarded. The correction was found to be constant across the different rigidity settings. The
beam transmission (trans ) from the object scintillator to the target position was measured
at the beginning (77.5%) and end (82.3%) of the experiment by tuning the beam into the
S800 focal plane. The increased transmission seen in the second measurement was applied
to runs 631 and above. A jump in the cross section measurement was observed at run 631,
motivating the use of the second transmission measurement. The object pile-up correction
(Îą) was found by counting the number of multi-hit events in the object scintillator. Only
events that deposited energy consistent with 1 particle in the object scintillator were used
in the analysis. The timing resolution for the multi-particle hits was poor and so were
discarded.

113

Factor

Description

Value

daq

DAQ live time

90-96%

tof

PID TOF efficiency

90-95%

pid

PID gate efficiency

95%

crdc

CRDC efficiency

50-70%

IC

IC pileup efficiency

98%

nhit

Single neutron hit efficiency

93%

trans

Beam transmission to target

77.5,82.3%

Îą

Object pileup correction

8-40%

ρ∗

Target Density Correction

ÂĄ0.5%

Table 4.2: Summary of corrections to the cross sections. Parameters with a range in values
indicate that a run-by-run correction was performed. See text for details on how each
correction was determined.
Fig. 4.33 shows charge-exchange differential cross section calculated from Equation 4.16.
The cross section is shown for the Θcm = 4−6° and Θcm = 14−16° bins, with the contribution
from each decay channel shown. From the higher-resolution spectrum at forward angles, it
is clear that the (p,n) reaction in inverse kinematics can give detailed structure information.
The two known states from β decay at 3.3 and 4.3 MeV are resolved and can be identified.
Alongside the low-lying structure information the response up to high excitation energy is
mapped out. The error in the total cross section for excitation energies above ∟12 MeV
increases with the opening of the other decay channels, where the statistical and systematic
uncertainties are increased.

114

3

15

N
N
14
N
14
C
15
C

dσ/dΩ [mb sr-1]

2.5

16

2
1.5
1
0.5
0

0

5

10

15

20

Ex [MeV]
(a) Θcm =4-6°

8

15

N
N
14
N
14
C
15
C

dσ/dΩ [mb sr-1]

7

16

6
5
4
3
2
1
0

0

5

10

15

20

Ex [MeV]
(b) Θcm =4-6°

Figure 4.33: Differential cross section for the 16 C(p,n)16 N reaction showing the contributions
from each reaction channel. Panel (a) shows the cross section at forward angles in 250 keV
bins. In (b) the same angle is shown with a coarser binning to highlight the channels at higher
excitation energy, where the statistics is low. The most backward angle, with the poorest
resolution, is presented in (c). The shaded bans in (b) and (c) indicate the systematic error
estimation from the background subtraction for the 15 C and 14 C channels. The small vertical
lines on the top of each figure indicate the threshold energies for the 15 N, 15 C, 14 C and 14 N
channels. The color of the line matches the color of the data to indicate the channel.

115

Figure 4.33: cont’d
(c) Θcm =14-16°

dσ/dΩ [mb sr-1]

3

15

N
N
14
N
14
C
15
C
16

2.5
2
1.5
1
0.5
0

0

5

10

15

20

Ex [MeV]
Fig. 4.34 presents the total cross section for the Θcm = 4 − 6° and Θcm = 14 − 16° bins.
The systematic error from the background subtraction and uncertainty in the neutron acceptance is shown as a green band. Section 4.7 gives more details about the systematic error
estimation.

4.7

Systematic Error

No absolute cross-section measurement is complete without an estimate of the systematic
errors. Table 4.3 gives a breakdown of the major systematic errors in experiment e10003.
Systematic error in the extracted differential cross section can either give an overall uncertainty in the normalization, or create a more complex uncertainty, like in the case of the
background subtraction. The systematic error from the background subtraction is not included in Table 4.3 since it is treated as a function of COM scattering angle and excitation

116

3

dσ/dΩ [mb sr-1]

2.5
2
1.5
1
0.5
0

0

5

10

15

20

15

20

Ex [MeV]
(a) Θcm =4-6°

2

dσ/dΩ [mb sr-1]

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

5

10

Ex [MeV]
(b) Θcm =14-16°

Figure 4.34: Total differential cross section from all decay channels in the 16 C(p,n)16 N reaction. The green band shows the systematic error including the systematic error from the
background subtractions in the different channels and the systematic error in the neutron
acceptance determination.

117

energy. Details of the estimation technique for each of the systematic errors are given in the
following paragraphs.
Source

Estimate

Incident rate determination

15.6%

Background subtraction

See text

Neutron Acceptance

5%

Target Thickness

3.3%

Neutron Energy Threshold

1.8%

Random Background

< 1%

Empty cell induced background

< 0.5%

Total

16.8%

Table 4.3: Summary of the source and estimated size of the major systematic uncertainties in
the 16 C(p,n)16 N measurement. See text for details on each source of systematic uncertainty
The largest systematic error was in the incident beam rate normalization procedure.
This error was estimated by looking at the total charge-exchange cross section from 0-20
MeV for each 15 N experimental run. If the normalization was perfect, each run should give
statistically equivalent values. Therefore, the spread seen between the runs will give an
estimate of the uncertainty in the incident beam rate measurement. These integrated cross
sections are shown in Fig. 4.35. The rigidity setting appropriate for detecting 15 N (3.1325
Tm) was measured three times during the experiment, so the large gaps between the points
correspond to the runs when the other rigidity settings were measured. From this plot a
FWHM systematic error of 15.6% for the normalization procedure was estimated.
Unlike the beam normalization, the background subtraction affects the shape of the
extracted cross section. To estimate the uncertainty in the subtraction, the background
118

40

dσ
ÎŁ20
0 dΩ [mb/sr]

35
30
25
20
15
10
5
0
560

580

600

620

640

660

680

700

720

740

760

780

Run Number
Figure 4.35: The total charge-exchange cross section from 0-20 MeV for each experimental
run for the 15 N channel. The spread in the points gives an estimate of the uncertainty in
the beam normalization.
model was varied for each rigidity setting—giving a bin-by-bin estimate of the uncertainty.
For the neutron-decay channels (15 N and 14 N), this involved varying the integration ranges
used to find the scaling functions shown in Fig. 4.20. This indicated that the error from the
background subtraction in the regions of interest was small (∟ 4%). For the proton-decay
channels (15 C and 14 C), the uncertainty was higher. A similar procedure was followed to
estimate the uncertainty where the different background models were compared. For 15 C
the portion of the low-energy tail of ejectile kinetic energy distribution that was used to
model the background was varied. For 14 C a separate background model using events that
created 12 C was compared with the technique used above. This gave an uncertainty that
varied from 10-30% depending on angle and excitation energy.
The error in neutron acceptance comes from uncertainty in the exact positions of the
neutron detectors. As Fig. 4.18 showed, the neutron acceptance is not a smooth function, so
small changes to the angles of the neutron detectors can change the acceptance significantly.

119

During the analysis process small adjustments were made to the positions of the neutron bars
to correct for any unphysical discontinuities seen in the excitation energy spectrum. The
size of these adjustments gave a scale to vary the angles in the estimation of this systematic
error. The Geant4 simulation was used to propagate the variations in detector angle to the
simulated yield in each COM bin. This gave a relatively flat change of 5%, except for the bins
in excitation energy from 9-11 MeV at forward angles. There the spectrum is particularly
sensitive to the position of the bars because of the steepness of the kinematic lines. An
additional 8% systematic error was added to these bins.
The error in the target thickness determination was already discussed in Section 4.5.3.
The error comes from the uncertainty in the energy loss difference between 16 C and 14 B.
This error was estimated by comparing the shape of the measured distribution for the two
nuclei.
The error due to uncertainties in the neutron detection threshold were estimated in a
similar way using the Geant4 simulation. The error in the threshold was determined by
comparing the neutron detector gain calibrations before and after the experiment. The gain
at the end of the experiment had decreased by approximately 9%, lowering the yield. This
error was combined with any fitting uncertainties in the gain calibration and propagated
through the full Geant4 simulation of the setup. This showed a 1.8% change to the neutron
yield.
The systematic error associated with the random and cell backgrounds were already
discussed in Section 4.4 and are placed in Table 4.3 for convenience. As indicated in Table
4.3 these systematic errors combined to give a total systematic error of 16.8%. The final
systematic error bands presented in chapter 5 are a combination of the systematic error from
the background subtraction and the 16.8% error in the overall normalization.
120

Chapter 5
Results
The trick is to just look at the question
and write down the answer.
Works every time

This chapter gives the results of the 16 C(p,n)16 N reaction at 100 MeV/u. The differential
cross section is used to form angular distributions, needed to perform a multiple decomposition analysis (MDA). The MDA isolates the different ∆L components of charge-exchange
cross section so that GT strength and dipole cross sections can be extracted.

5.1

Angular Distributions

dσ ) as a function of COM scattering angle)
Angular distributions (differential cross section ( dΩ

are formed from the experimental cross section data. To make these distributions for each
excitation energy bin, the cross-section data was separated into 2 degree angular bins from
4 to 16 degrees in the COM. The bin from 0-2 degrees was below the neutron detection
threshold. The bin from 2-4 degrees was excluded due to the large amount of low energy
neutron background, which could not be subtracted reliably. Angles beyond 16 degrees in
the COM were not used in the analysis.
As discussed in Section 4.3, the excitation-energy resolution worsens at larger scattering
angles. To correctly form angular distributions, the data at forward scattering angles must
121

be smeared to match the excitation-energy resolution at the most backward scattering angle
used in the analysis. Fig. 5.1 shows an example of this smearing procedure. The figure
shows the same data at Θcm = 4 − 6 ° as Fig. 4.34, but with additional smearing to match
the excitation energy resolution for the Θcm =14-16° bin.

2

dσ/dΩ [mb sr-1]

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

5

10

15

20

Ex [MeV]
Figure 5.1: Differential cross section for the 16 C(p,n)16 N reaction as a function of excitation
energy. Data is from Θcm =4-6° but smeared to match the resolution at Θcm =14-16°.

5.2

Multipole Decomposition Analysis

A multipole decomposition analysis is used to separate the different angular momentum
transfer components of the cross section. The smeared angular distributions are fit with a
linear combination of theoretical cross sections calculated in DWBA,




 
 
 
dσ
dσ
dσ
dσ
=a
+b
+c
,
dΩ total
dΩ ∆L=0
dΩ ∆L=1
dΩ ∆L=2

122

(5.1)

where a, b, and c are the fitting parameters. The fitting was done through χ2 minimization,
where the error bars in the angular points came from a combination of the statical error and
the systematic uncertainty from the neutron detector acceptance. For each ∆L component,
the shape of the angular distribution is represented by a characteristic DWBA calculation.
Section 2.4 gives details on the cross-section calculations and presents the characteristic
angular distributions used in the MDA. The result of the MDA is shown in Fig. 5.2. Angular
distributions in 1 MeV bins of excitation energy were prepared from the smeared excitation
energy distributions and fit according to Equation 5.1. Good agreement between the fits
and the data is seen. Of specific interest is the success of the fits in first bin (0-1 MeV),
where a reduced χ2 of 1.09 is achieved. Between 0 and 1 MeV there are only expected to be
4 negative parity states at 0.0, 120, 298, and 397 keV. Starting from a 16 C ground state of
0+ , transitions to these states must all be ∆L=1. This is confirmed by the MDA. It is clear
from the remaining fits that much of the data is strongly ∆L = 0 in nature.
It is important to note that the ability of the MDA to distinguish between ∆L=1 and
∆L=2 is limited. Over the angular range used in the analysis the angular distributions for
∆L=1,2 transitions are similar. To explore this sensitivity, decompositions including only
∆L=0,1, and only ∆L=0,2 were compared with the full fit. In both cases, the portion of the
distribution coming from GT type transitions did not change significantly. The fits without
∆L=1 were of a similar quality to the fits without ∆L = 2, yet the full fit strongly prefers
the ∆L=1 component. Given this, all three components were used in the final analysis, but
with the understanding that differentiating ∆L=1 and 2 is difficult.
Fig. 5.3 shows the ∆L components of the cross section in each angular bin as determined
by the MDA. As expected from the theoretical calculations, the dipole component peaks
at the backward angles and appears at high excitation energy. In the 14-16° bin, there is
123

0.5
0.4
0.3
0.2

Data
∆ L=0
∆ L=1
∆ L=2
Total

0<Ex<1 MeV

0.1
0
4

6

8

10

12

Θcm [Deg]

14

0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
4

dσ/dΩ [mb sr-1]

dσ /dΩ [mb sr-1]

0.6

16

1<Ex<2 MeV

6

8

8

10

12

Θcm [Deg]

14

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

2.5

3<Ex<4 MeV

2
1.5
1
0.5
0
4

16

6

8

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

4<Ex<5 MeV

2
1.5
1
0.5
6

8

10

10

12

Θcm [Deg]

14

16

(d)

3

0
4

16

3

(c)

2.5

14

(b)

2<Ex<3 MeV

6

12

Θcm [Deg]

(a)

1.4
1.2
1
0.8
0.6
0.4
0.2
0
4

10

12

Θcm [Deg]

14

3
2.5
2
1.5
1
0.5
0
4

16

(e)

5<Ex<6 MeV

6

8

10

12

Θcm [Deg]

14

16

(f)

Figure 5.2: Measured COM angular distributions in 1 MeV bins from 0-20 MeV in 16 N.
The black dots are the experimental data. The red, green, and blue lines are the ∆L=0,1,2
components of each fit determined from the MDA. The dotted line is the sum of the 3 MDA
fit components.

124

Figure 5.2: cont’d

3.5
3
2.5
2
1.5
1
0.5
0
4

(h)

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

(g)

6<Ex<7 MeV

6

8

10

12

Θcm [Deg]

14

3
2.5
2
1.5
1
0.5
0
4

16

7<Ex<8 MeV

6

8

8

10

12

Θcm [Deg]

14

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]
16

7
6
5
4
3
2
1
0
4

6

8

8

10

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

10

12

Θcm [Deg]

14

16

(l)

10<Ex<11 MeV

6

16

9<Ex<10 MeV

(k)

7
6
5
4
3
2
1
0
4

14

(j)

8<Ex<9 MeV

6

12

Θcm [Deg]

(i)

4
3.5
3
2.5
2
1.5
1
0.5
0
4

10

12

Θcm [Deg]

14

16

125

5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
4

11<Ex<12 MeV

6

8

10

12

Θcm [Deg]

14

16

Figure 5.2: cont’d

4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
4

(n)

dσ /dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

(m)

12<Ex<13 MeV

6

8

10

12

Θcm [Deg]

14

16

3.5
3
2.5
2
1.5
1
0.5
0
4

13<Ex<14 MeV

6

8

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

14<Ex<15 MeV

2
1.5
1
0.5
6

8

10

12

Θcm [Deg]

14

15<Ex<16 MeV

2
1.5
1
0.5
0
4

16

6

8

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

16<Ex<17 MeV

1.5
1
0.5
6

8

10

10

12

Θcm [Deg]

14

16

(r)

2.5

0
4

16

2.5

(q)

2

14

(p)

3

0
4

12

Θcm [Deg]

(o)

2.5

10

12

Θcm [Deg]

14

3
2.5
2
1.5
1
0.5
0
4

16

126

17<Ex<18 MeV

6

8

10

12

Θcm [Deg]

14

16

Figure 5.2: cont’d
(t)

2.5

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

(s)

18<Ex<19 MeV

2
1.5
1
0.5
0
4

6

8

10

12

Θcm [Deg]

14

2.5

1.5
1
0.5
0
4

16

19<Ex<20 MeV

2

6

8

10

12

Θcm [Deg]

14

16

significant dipole strength above ∼12 MeV. Conversely, the 4-6° bin is dominated by ∆L=0
strength. This motivates directly using the cross section in the 4-6° bin, where the best
excitation energy resolution is available to extract the B(GT) spectrum.

5.3

B(GT) Extraction

Knowing that the cross section is dominated by ∆L=0 transitions in the 4-6° bin, the
B(GT) can be readily extracted. First the cross section must be extrapolated to zero linear
momentum transfer (q → 0). The MDA results are used to extrapolate the cross section to
0 degrees. Then DWBA calculations are used to extrapolate the Q value to 0 for each bin in
excitation energy. With both Θcm → 0 and Q → 0, the cross section is extrapolated to zero
linear momentum transfer. Fig. 5.4 shows the ∆L=0 cross section after this extrapolation.
The blue histogram seen in Fig. 5.4 is a simulation of the IAS assuming full exhaustion
of the Fermi sum rule. Within the 16 N exit channel a peak corresponding to the Îł decay
of the IAS was observed with a branching ratio of 0.6 Âą 0.2(stat)%. The peak location
was determined from a calculation of the difference in coulomb energy yielding a value of
127

4
3

Θcm=4-6
∆L=0
∆L=1
∆L=2

2
1
0
0 2 4 6 8 10 12 14 16 18 20

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

6
5

6
5
4
3
2
1
0
0 2 4 6 8 10 12 14 16 18 20

Ex [MeV]

Ex [MeV]
(b)

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

(a)

5
4.5
4
Θcm=8-10
3.5
3
2.5
2
1.5
1
0.5
0
0 2 4 6 8 10 12 14 16 18 20

4
3.5
3
Θcm=10-12
2.5
2
1.5
1
0.5
0
0 2 4 6 8 10 12 14 16 18 20

Ex [MeV]

Ex [MeV]
(d)

dσ/dΩ [mb sr-1]

dσ/dΩ [mb sr-1]

(c)

4
3.5
3
Θcm=12-14
2.5
2
1.5
1
0.5
0
0 2 4 6 8 10 12 14 16 18 20

Θcm=6-8

3
2.5
2

Θcm=14-16

1.5
1
0.5
0
0 2 4 6 8 10 12 14 16 18 20

Ex [MeV]

Ex [MeV]

(e)

(f)

Figure 5.3: MDA results. The ∆L components of the cross section is shown as a function
of excitation energy for each angular bin. At forward angles the spectrum is dominated by
∆L = 0 transitions, while at backward angles ∆L = 1 transitions appear at high excitation
energies.

128

10.025 MeV. This value was consistent with the peak position seen in the 16 N exit channel.
The detectors response to the IAS was determined in the full Geant4 simulation and was
subtracted from the measured cross section.

dσ/dΩ(q=0,∆L=0) [mb/sr]

4
3.5
3
2.5
2
1.5
1
0.5
0

0

5

10

15

20

Ex [MeV]
Figure 5.4: The differential cross section extrapolated to zero momentum transfer. The
error bars show statistical errors. The blue peak shows the contribution from the IAS to the
spectrum. See text for details.

In order to transform this into a B(GT) spectrum, a suitable σ̂GT must be determined.
This can be done by looking at a low-lying state that is accessible by β decay and using
the known log(f t) value and measured charge-exchange cross section to find σ̂GT . If β
decay data is not available, σ̂GT can be estimated from well-established mass-dependent
systematics [56]. The β decay of 16 C has been measured previously [69, 70, 71], but as will
be discussed in Chapter 6, the low-lying states have peculiar structural effects that break
the proportionality. To avoid the uncertainty associated with the two low-lying states, the
unit cross section was determined from nearby (p,n) measurements on 14 C [115, 56] and 18 O
[116].
For the 14 C data, the σ̂F was taken from two measurements at 80 and 120 MeV and
129

averaged to arrive to find σ̂F at 100 MeV. It is well known that the σ̂F varies with incident
beam energy, while the σ̂GT remains nearly constant [56]. From this, σ̂GT was found using
the empirical relationship between σ̂GT and σ̂F ,
Ep
σ̂GT
=
,
σ̂F
55.0

(5.2)

where Ep is the incident proton energy [56]. This yielded σ̂GT =5.4 mb/sr. The 18 O data
gave σ̂GT =4.2 mb/sr [116]. The average of these two measurements with a systematic error
of 13% was used in the final analysis of 16 C: σ̂GT =4.82 ± 0.61 mb/sr.
Fig. 5.5 (a) shows the extracted Gamow-Teller strength distribution in 16 N compared
with two shell model calculations using the WBT and WBP interactions in the spsdpf
model space. The shell-model states have been smeared with the experimental resolution.
States with T=1 and 2 are included in the figure. Fig. 5.5 (b) shows the corresponding
sum spectrum for the data and shell model calculations. Chapter 6 will discuss this result
in detail.

5.4

Dipole Strength

The data can also give some insight into dipole (∆L = 1) transitions. Dipole strength seen
at this excitation energy and at this beam energy will predominately come from spin-flip
transitions, so it should correspond to excitations of the isovector spin dipole giant resonance
(SDR). Looking at the most backward angle (where the dipole strength is highest), the
∆L = 1 component of the cross section was extracted from the MDA fits. These results are
presented in Fig. 5.6. Unlike the GT strength, the spin-dipole (SD) strength does not have

130

1.2

Data
WBP
WBT

1

B(GT)

0.8
0.6
0.4
0.2
0

0

5

10

15

20

15

20

Ex [MeV]
(a)

12

Data
WBP
WBT

ÎŁB(GT)

10
8
6
4
2
0

0

5

10

Ex [MeV]
(b)

Figure 5.5: Extracted GT strength distribution (a) and summed strength (b) in 16 N up to
Ex =20 MeV. The green band shows the total systematic error in both plots. The data is
compared with two shell interactions WBT and WBP, both calculated in the spsdpf model
space.

131

a well-established proportionality. Instead direct comparisons of measured cross section and
DWBA based calculations are typically used. The black histogram in Fig. 5.6 shows the
calculated cross section for all the dipole shell-model states discussed in Section 2.5. For each
shell-model transition, the angular distribution is calculated with the program DW81 and
the value at 15° is taken. These states are then smeared with the experimental resolution
and histogramed.
As mentioned above, the sensitivity between ∆L = 1, 2 of the MDA is limited. The
significant error bars seen in the top panel of Fig. 5.6 are a result of this. The bottom panel
shows the result of the MDA if ∆L=2 is excluded from the fits.

132

1.4

Data
1,3 h ω
1 hω
3 hω

dσ/dΩ [mb/sr]

1.2
1
0.8
0.6
0.4
0.2
0
0

5

10

15

20

25

30

Ex [MeV]
(a) Full fit

1.4

Data
1,3 h ω
1 hω
3 hω

dσ/dΩ [mb/sr]

1.2
1
0.8
0.6
0.4
0.2
0
0

5

10

15

20

25

30

Ex [MeV]
(b) ∆L=2 excluded.

Figure 5.6: The black points show the dipole (∆L = 1) differential cross section determined
from the MDA at 15°. The error bars give the statistical uncertainty from the MDA fits. The
black histogram gives the smeared, DWBA cross sections from the 1,3 h̄ω shell-model calculation with the WBP interaction in the spsdpf model space. The blue and red histograms
give the results of the calculations when pure 1 and 3 h̄ω contributions are calculated separately. Panel (a) Shows the result when the full MDA fit is used, while (b) excludes the
∆L=2 component. The 7-8 MeV bin was excluded from (a) since its error bar was too large
to be meaningful and was consistent with zero.

133

Chapter 6
Discussion
Everything works, Nothing is Broken

Samuel Lipschutz

6.1

The (p,n) Probe in Inverse Kinematics

Figs. 5.5 and 5.6 readily show the achievement of the main goal of e10003 and this thesis—to
validate the (p,n) probe in inverse kinematics for the study of isovector giant resonances in
radioactive nuclei. The full GT response and a portion of the spin-dipole resonance have
been measured in a neutron-rich isotope. This technique will allow future experiments to
explore the isovector response of unstable nuclei up to high excitation energy. As Figs. 5.5
and 5.6 shows, it is clearly a powerful probe, providing enough resolution to see the low-lying
structure of 16 N, while at the same time capturing a large range of excitation energy.
Higher excitation energies and the remaining portion of the spin-dipole resonance would
have been available if more particle decay channels were measured (13 C opens at ∟21 MeV).
Given the finite time of the experiment and the wide momentum distributions of the multinucleon decay channels, a cutoff had to be made. However, for heavier systems this will be
become easier as the change in rigidity from few nucleon decays becomes less severe. This
will simplify the experimental procedure and analysis, since many decay channels can be
fully measured in a single spectrometer setting.
134

6.2

Shell Model Comparison

The spectrum in Fig. 5.5 allows us to see the quality of shell-model calculations for the
GT response in neutron-rich nuclei up to high excitation energies for the first time. It is
clear that the details of the spectrum are not well reproduced by either of the interactions.
This is a departure from previous charge-exchange data where shell-model calculations have
achieved better agreement (for example, Refs. [78, 64]). In Fig. 5.5, the positions of many of
the strong states seen in the spectrum are shifted from the data by several MeV. The WBP
interaction does a better job at spreading the strength out in the spectrum, while the WBT
interaction concentrates a large amount in the state at ∟11 MeV. However, it is important
to note that the higher-lying states most likely have large intrinsic widths since the particledecay thresholds are exceeded. Such effects are not included in the shell model—only the
experimental resolution is used to smear the shell-model states. In light of this, an exact
match in the shape of the spectrum is not expected. Further, the shell-model spectrum is
unquenched and is expected to over predict what is seen in the data. These limitations
often make the summed strength spectrum a more reliable indicator of the quality of the
calculation.
The spin-dipole spectrum also gives guidance to the validity of the shell model. Presented
in Fig. 5.6 is the measured ∆L=1 differential cross section in the 14-16° bin, overlaid with
the cross section calculated in DWBA. The theoretical cross section is broken down into the
components coming from pure 1 and 3 h̄ω configurations in the final state. The calculations
shows that the ground state comes from the 1 h̄ω component. There is additional 1 h̄ω
strength predicted at ∟6 MeV, where an enhancement in the data is seen, but the strength
is underestimated by the calculation. The 3 h̄ω component opens at ∼4 MeV and vastly over

135

predicts the strength in the region. At higher excitation energy (∟12MeV and above) the
calculation does a reasonable job at predicting the location and strength of the resonance.
The full calculation includes both of these components where they mix with each other.
The mixing typically reduces the strength of the states, but in this case does not alleviate
the issues seen in the 3 h̄ω calculation. The full calculation still vastly over predicts the low
lying strength compared to the data. Further investigation and improvements in the model
are needed.

6.3

Quenching

Fig. 5.5 shows the measured, summed GT strength in 16 N. As discussed in Section 1.9
this experiment has given one of the first measurement of the quenching of summed GT
strength in a neutron-rich, unstable nucleus. Under the assumption that the strength in β+
direction is small due to fermi blocking, the data shows 73.5% Âą2.8(st) Âą 15.5(sys)% of the
model-independent sum rule up to 20 MeV. Table 6.1 shows a comparison of this result with
the known quenching values in the region from charge-exchange and β decay studies. The
charge-exchange studies shown in Table 6.1 give their quenching values relative to the GT
sum rule, while the β-decay numbers are from comparisons to shell-model calculations.

136

Relative to Sum Rule

Relative to Shell Model

System

Quenching Factor

System

Quenching Factor

18 O

67% [116]

p-shell

65.6% [117]

14 C

60% [49]

18 Ne

62.2Âą0.2% [76]

16 C
Ex 0→20

73.5 Âą2.8(st) Âą 15.5(sys)%

sd-shell

57.8% [76]

16 C
Ex 0→12

50.7 Âą1.2(st) Âą 10.7(sys)%

pf-shell

55.4% [83]

16 C
Ex 0→12

61.1 Âą1.5(st) Âą 12.8(sys)%

16 C
Ex 0→20

79.0 Âą3.0(st) Âą 16.7(sys)%

Table 6.1: Comparison of the quenching factors of GT strength for 16 C, various nearby
nuclei, and adopted values for several nuclear shells. Quenching studies from charge-exchange
reactions and β decay studies are presented. The p-shell factor comes from the empirical
relation given in Ref. [117] with A = 16. See text for details.
The β-decay quenching results come from measurements of individual states. Since they
are restricted by the β-decay Q-value window, they can only probe the properties of the lowlying strength. To make a fair comparison with the 16 C charge-exchange data, the summed
strength up to the end of the main part of the resonance (∟12 MeV) was compared with the
shell model. This yields a quenching factor of 61.1 Âą1.5(st)Âą12.8(sys)%, which is consistent
with the β-decay factors seen in Table 6.1. Though the systematic error is large, the 61.1%
value follows the trend of decreasing quenching between the p and sd shells. As discussed in
Section 1.9, the strength is thought to be pushed to higher excitation energies through several
proposed mechanisms, such as higher-order configuration mixing. The charge-exchange data
confirms this for 16 C, where 79.0 Âą3.0(st) Âą 16.7(sys)% is found up to 20 MeV of excitation
energy in 16 N. This shows that about a third more strength is found at excitation energies
above the GT giant resonance.

137

More insight into this behavior is given in Fig. 6.1, where the quenching factor relative
to the shell model as a function of excitation energy is shown. After the end of the main
resonance at ∟12 MeV the quenching factor steadily increases. This shows that the measured
distribution is “catching up” to the calculation, which has predicted too much strength in
the main portion of the resonance. The calculation does predict strength after the resonance,
going all the way up to ∟ 30 MeV of excitation energy before the sum rule is exhausted.
However, the data increases faster than the calculation in the measured region recovering
a portion of the missing strength. With the (p,n) reaction in inverse kinematics a similar
recovery of strength is seen in the neutron-rich 16 C as was seen in the 90 Zr experiment
discussed in Section 1.9.
The first half of Table 6.1 compares the quenching in this measurement relative to the
model-independent sum rule with 18 O and 14 C—the closest measurements where the full
GT spectrum was measured through charge exchange. Here a quenching factor of 73.5
Âą2.8(st) Âą 15.5(sys)% up to 20 MeV was found. This is consistent with the measurements
on 18 O and 14 C. The number is naturally lower than the quenching factor relative to the
shell model since the full sum rule is not exhausted in the shell model at 20 MeV.

6.4

The First Two 1+ States

As alluded to earlier, there are GT proportionality breaking effects in the two low-lying
states of 16 N. With the differential cross section and extracted B(GT) spectra shown in
Figs. 5.4 and 5.5 respectively, the cross section and B(GT) values for the first two states can
be determined. These values along with a breakdown of the statistcal and systematic errors
are shown in Table 6.2. Table 6.3 shows a comparison of the B(GT) values from charge-

138

1
0.9

Quenching

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

11

12

13

14

15

16

17

18

19

20

Ex [MeV]
Figure 6.1: The amount of quenching between the experimental data and the shell model
(WBP interaction) as a function of excitation energy.
exchange, β decay and the shell model. It is clear that the strengths of the first two states
from charge-exchange and β decay are significantly different. The summed strength, however,
is a near match. The predictions from the WBT interaction also do a good job at predicting
the total strength, but does worse than WBP at properly recreating the individual strengths
seen in β decay. Further, the separation of the two states in the shell-model calculations are
underestimated: 375 keV for the WBP interaction and 625 keV for the WBT interaction
versus 967 keV from the β decay data. In the following analysis the WBT interaction is used
to investigate the two states.
Ex

(dσ/dΩ)q→0 [mb/sr]

B(GT)CE

3.353 MeV

3.39Âą 0.19(st)Âą0.56(sys)

0.70 Âą 0.04 (st) Âą 0.15(sys)

4.320 MeV

4.62Âą 0.20(st)Âą0.76(sys)

0.96 Âą 0.04 (st) Âą 0.20(sys)

Table 6.2: Charge-exchange differential cross section at zero momentum transfer and extracted B(GT) for the first two 1+ states in 16 N. A breakdown of the statistical and systematic error is shown. The systematic error for the B(GT) is increased by the uncertainty in
σ̂GT .
139

Ex

B(GT)W BT

B(GT)W BP

B(GT )CE

B(GT)β

3.353 MeV

1.34

0.97

0.70

1.104

4.320 MeV

0.27

0.39

0.96

0.586

Sum

1.61

1.36

1.66

1.69

Table 6.3: Comparison of the B(GT) values from charge-exchange, β decay, and the shell
model for the first two 1+ states in 16 N.
Proportionality breaking effects in charge-exchange reactions have been seen before [78,
118]. These are often explained by interference from the ∆L = 2 ∆S = 1 transitions
that are mediated by the tensor-τ portion of the NN interaction, which interferes with the
∆L = 0 ∆S = 1 στ portion. This effect is typically seen in states with small B(GT) values.
The calculation of σ̂GT presented in Fig. 2.5 shows this feature in 16 C. The spread in the
calculated σ̂GT values increase significantly below B(GT) values of ∼0.1. This spreading can
be largely removed by turning off the tensor portion of the effective NN interaction.
However, the first two states are not weak. Further, the DWBA calculations with and
without the tensor force included in the interaction do not show a large enough sensitivity
to explain the inconsistency in Table 6.3. However, the calculation presented in Fig. 2.5
does show that the expected σ̂GT for the second state is ∼ 52% higher than the average
of the other strong states. If the unit cross section for that state is abnormally high, then
the measured charge-exchange cross section should be enhanced, which is seen in the data.
Since shell-model calculations are available, the details of these two states can be explored.
To do this the transition densities for these two states are examined (see Equation 2.23).
Fig. 6.2 shows a typical transition density (shown as r2 ρ), broken down into the significant 1p-1h contributions. This transition density comes from the shell-model calculation of
the 4th 1+ state in 16 N. As the figure shows the sum of the different 1p-1h components of
140

the transition gives a smooth function. This is not the case when looking at the 1st and 2nd
1+ states in 16 N.
0.06

State3
0p1/2
0p1/2
0p3/2
1s1/2
0d5/2
0d5/2
0d3/2
Total

0.05

r2ρ [arb]

0.04
0.03
0.02
0.01

0p1/2
0p3/2
0p1/2
1s1/2
0d5/2
0d3/2
0d5/2

0

−0.01
−0.02
−0.03
0

2

4

r [fm]

6

8

10

Figure 6.2: A typical transition density with its 1p-1h components. This is the calculation
of the transition to the third 1+ state in 16 N with the WBT interaction.

Fig. 6.3 shows the transition densities for the first two states. In both cases there is
a node, and in the second state, the transition density switches sign in the interior. In β
decay, the whole transition density is probed and its integral squared proportional to B(GT).
In the case of the second state, the portion of the transition density that is negative will
bring down the strength. However, in the (p,n) charge-exchange case, the proton does not
fully penetrate the target and so only probes a portion of the transition density. Further,
distortion effects bias the integration of the transition density—it enhances the role of the
surface and decreases the effect of the interior. For the 2nd state, this means that the chargeexchange cross section will increase because the negative portion of the transition density is
suppressed. This increased cross section will give the state a increased unit cross section as
seen in Fig. 2.5.

141

0.12

State0
0p1/2
0p1/2
0p3/2
1s1/2
0d5/2
0d5/2
0d3/2
Total

0.1

r2ρ [arb]

0.08
0.06
0.04
0.02

0p1/2
0p3/2
0p1/2
1s1/2
0d5/2
0d3/2
0d5/2

0

−0.02
−0.04
−0.06
0

2

4

r [fm]

6

8

10

(a) State 0

State1
0p1/2
0p1/2
0p3/2
1s1/2
0d5/2
0d5/2
0d3/2
Total

r2ρ [arb]

0.1
0.05
0

0p1/2
0p3/2
0p1/2
1s1/2
0d5/2
0d3/2
0d5/2

−0.05
−0.1
0

2

4

r [fm]

6

8

10

(b) State 1

Figure 6.3: The transition densities and their 1p-1h components for the first two 1+ states
in 16 N, calculated with the WBT interaction. The details of these states create an atypical
transition density that has a significant effect on the unit cross section for these two states.

142

The distortion can have this effect only because of the unique and subtle details of the
state. There is not one, or even two, dominate transitions that create this behavior for the
second state. Instead, many things must go “wrong” at the same time. The 1s1/2 → 1s1/2
transition must be strong enough to create the abnormal shape, while the p and d transitions
must have the right amplitudes to keep the shape and bring a portion of the transition density
below 0. While the shell-model calculation predicts this behavior for only the second state,
the B(GT) extracted from the CE data for the first state also is inconsistent with the β
decay value. Given the closeness of the two states in the shell model, some mixing of the
two must be closer to the true states.
Since both of the states likely suffer from these effects, it motivated using the systematics
of the unit cross section to find σ̂GT for 16 C. It is interesting to note that the unit cross
section determined from the sum of the two states is consistent with the σ̂GT = 4.83 ±
0.61 mb/sr used in the analysis. The sum is σ1 + σ2 = 8.01 mb/sr, which gives σ̂GT =
8.01/1.69 = 4.74 mb/sr. This consistency gives confidence in the normalization procedure
and the determination of the unit cross section.

143

Chapter 7
Conclusion and Outlook
Soon we will all be graduated

Unknown origin

The (p,n) charge-exchange reaction in inverse kinematics has been successfully validated
as a tool for studying isovector giant resonances in neutron-rich nuclei up to high excitation
energy. This was done through the measurement of the absolute, differential cross section
of the 16 C(p,n)16 N reaction at 100 MeV/u and the subsequent extraction of GT and SD
strength distributions up to 20 MeV of excitation energy. This was one of the first measurements of isovector giant resonances in an unstable, neutron-rich nucleus. Future experiments
utilizing this technique will allow the isovector response of exotic nuclei to be probed up to
high excitation energy.
Key to the success of the measurement was the development of a new digital data acquisition system for LENDA and its integration with the S800 DAQ. E10003 was the first
experiment at the NSCL to combine the Pixie digital system with the analog S800 DAQ—
laying the foundation for future NSCL experiments. The results of the experiment were
compared with detailed shell-model calculations, which gave a measure of their reliability in
the neutron-rich, high-excitation energy regime. With the full GT resonance measured, a
summed GT spectrum was produced, allowing for the quenching of GT strength in a neutronrich nucleus to be determined and discussed. Alongside this broad information, there was

144

sufficient resolution to analyze the details of the first two 1+ states in 16 N, where peculiar
structure effects were found.
There is an exciting future for the (p,n) probe inverse kinematics. The first measurement
of GT strength in a heavy, neutron-rich system (132 Sn(p,n)132 Sb) has already been performed
at RIKEN and is in its final stages of analysis. The next NSCL experiment has already been
approved, in which the (p,n) probe will be validated on the proton-rich side through the
11 C(p,n)11 N

and 12 N(p,n)12 O reactions at 100 MeV/u. These initial, proof-of-principle

experiments prepare the charge-exchange group for future (p,n) experiments in the FRIB
era. There increased beam rates and optimized detector systems will allow for ambitious
measurements of isovector strength in highly exotic nuclei.

145

APPENDICES

146

Appendix A
Derivation of the Gamow-Teller Sum
Rule
This appendix presents a derivation of the model independent Gamow-Teller (GT) sum rule,
first derived by Ikeda in 1964 [119]. Sum rules exists for the other giant resonances, but
are not disccused here. The interested reader is directed to Ref. [14] for more information.
Beginning with the definition of the total GT strength,

2


X

SÂą = 
hf |β̂µ± |ii ,
 Âľf


(A.1)

and
A

1X
β̂µ± =
σ̂µ τ̂±k ,
2

(A.2)

k

where the sum over µ goes through the components of σ̂µ (−1, 0, +1), and sum over f goes
through all possible final states—summing up the total strength. Each σ̂µ is a Pauli matrix
and is related to the spin operator by ŝ = 2σ̂. The expression above can be rewritten as,

SÂą =

X
Âľf

∗

hf |β̂µ± |ii hf |β̂µ± |ii ,

147

(A.3)

SÂą =

X
Âľf

SÂą =

†

hi|β̂µ± |f i hf |β̂µ± |ii ,

X
Âľ

†

hi|β̂µ± β̂µ± |ii .

(A.4)

(A.5)
†

Considering the difference S− − S+ and noting that τ̂± = τ̂∓ ,
S− − S+ =

Recognizing that

†
µ σ̂µ σ̂µ

P

1X
†
†
hi|σ̂µk σ̂µk τ̂+k τ̂−k − σ̂µk σ̂µk τ̂−k τ̂+k |ii .
4

(A.6)

Âľk

= 3, the expression reduces to,

S− − S+ =

3X
hi|τ̂+k τ̂−k − τ̂−k τ̂+k |ii .
4

(A.7)

k

Using the following values for the τ operators, the above expression can be further simplified: τ̂+ |ni = τ̂− |pi = 0, τ̂+ τ̂− |ni = 4 |ni, and τ̂− τ̂+ |pi = 4 |pi. The first term in A.7 will
count 4 times the number of neutrons, while the second term will count 4 times the number
of protons in the initial state. This gives the resulting sum rule,

S− − S+ = 3(N − Z).

(A.8)

For the case of neutron-rich nuclei the strength in the ∆Tz = +1 direction can be small
due to Pauli blocking, resulting in the approximation,

S− ≈ 3(N − Z).

148

(A.9)

This gives a simple and powerful statement on the total GT strength one can expect to
see for a given initial nucleus.

149

Appendix B
Energy of a Compton Edge
This appendix presents a short derivation of the expected energy of a Compton edge in a
detector given a known Îł decay energy. It is written down so that future students will not
make the same foolish mistake that I have made. This mistake is indicated below in bold
font. We being with the Compton scattering formula,

λf − λi =

h
(1 − cos(θ)) ,
me c

(B.1)

where Ν denotes the wave length, h the plank constant, me the mass of the electron, and θ
the scattering angle. Rewriting this in terms of energy,

1
1
1
(1 − cos(θ))
γ − γ =
me c2
Ef
Ei

(B.2)

where it is important to remember that this is the energy of the recoiling Îł not
the electron that will deposit energy into the detector. Rearranging,
Îł

Îł

Ef =

Ei me c2
Îł

(1 − cos(θ))Ei + me c2

150

.

(B.3)

The Compton edge will occur at the energy where the recoiled electron is maximized, meaning
the final energy of the γ is minimized. Setting cos(θ) = −1 gives,
Îł

Îł
Ef

=

Ei me c2
Îł

2Ei + me c2

.

(B.4)

Îł
As an example consider the 661 keV Îł ray from 137 Cs. Equation B.4 gives Ef = 184 keV.

This makes the energy of the scattered electron 661 keV - 184 keV=477 keV. Hence we
expect to see the Compton edge at 477 keV.

151

Appendix C
The Number of Beam Particles in a
Beam Bunch
Beam-based nuclear physics experiments performed at incident particle rates that are an
appreciable fraction of the underlying RF frequency of the accelerator require special considerations when determining the true incident beam rate from a direct measurement. This
appendix will discuss some of the difficulties and derive some important relations.
Suppose we have an incident beam rate given by R and a RF frequency given by RF .
This means that in a 1-second period we have a total of RF buckets to place randomly R
particles. The total number of ways that the particles can be arranged where any number
of particles are allowed in a bucket is given by,

n Hr

=n+r−1 Cr =

(n + r − 1)!
,
r!(n − 1)!

(C.1)

where n is number of buckets, r is the number of particles, and C is a binomial coefficient.
Figure C.1 shows a simple example of the case of 2 particles distributed in 3 possible buckets.
From inspection it is clear that there are 6 possible configurations. For this case we can

152

readily confirm the validity of Equation C.1 as

3 H2

=

(4)!
(4)!
(3 + 2 − 1)!
=
=
= 6.
2!(3 − 1)!
2!(2)!
4

(C.2)

Arragement 1
Arragement 2
Arragement 3
Arragement 4
Arragement 5
Arragement 6
Bucket 1

Bucket 2

Bucket 3

Figure C.1: The figure shows all possible ways to distribute 2 particles into 3 buckets. Each
row in the figure indicates a different arrangement of particles, giving a total of 6 possible
configurations.

The number of instances when a given bucket contains zero particles is given by,

n−1 Hr

n Hr

=

(n + r − 2)!
.
r!(n − 2)!

(C.3)

gives the total number of possible arrangements for n buckets and r particles. The

case where there are n + 1 buckets but that extra bucket contains no particles is the same
situation as the number of arrangements where a particular bucket contains zero particles in
the case of n buckets and r particles. This gives an intuitive reason for the n − 1 in Equation
153

C.3. Utilizing Equation C.3 and comparing it to Fig. C.1, we can confirm that the total
number of arrangements with a given empty bucket should be 3,

3−1 H2

=

(3 + 2 − 2)!
(3)!
=
= 3.
2!(3 − 2)!
2!

(C.4)

Generalizing this relation to the number of configurations where a bunch has k particles
in it,
n−1 Hr−k

=

(n + r − k − 2)!
.
(r − k)!(n − 2)!

(C.5)

To complete the example, we can calculate the number of configurations where a bucket
contains 1 and 2 particles given 3 possible buckets and 2 particles,

3−1 H2−1

=

(3 + 2 − 1 − 2)!
2!
=
= 2,
(2 − 1)!(3 − 2)!
1!

(C.6)

3−1 H2−2

=

(3 + 2 − 2 − 2)!
1!
=
= 1.
(2 − 2)!(3 − 2)!
1!

(C.7)

and

Knowing the total number configurations for each number of particles, the probability to
find a certain number of particles in a bucket can be calculated as,

Pk (n, r) =

n−1 Hr−k
n Hr

.

(C.8)

Using Equations C.1 and C.3, we can calculate the ratio of the number of buckets with

154

greater than 1 particle S>1 to the number with just 1 particle S1 ,
H
+
H
+ . . .n−1 H0
S>1
= n−1 r−2 n−1 r−3
S1
n−1 Hr−1
=
=

−n−1 Hr +n−1 Hr −n−1 Hr−1 +n−1 Hr−1 +n−1 Hr−2 +n−1 Hr−3 + . . .n−1 H0
n−1 Hr−1
n Hr

−n−1 Hr −n−1 Hr−1
,
n−1 Hr−1
(C.9)

where in the last step, the following was used,

n Hr

=n−1 Hr−1 +n−1 Hr−2 . . . +n−1 H0 .

(C.10)

Reducing the expression further,

=

n Hr

−n−1 Hr −n−1 Hr−1
n−1 Hr−1

Hr
H
− n−1
− n−1 r−1
n−1 Hr−1
n−1 Hr−1
n−1 Hr−1
Hr
n Hr
=
− n−1
−1
n−1 Hr−1
n−1 Hr−1
=

n Hr

(n + r − 1)!(r − 1)!(n − 2)! (n + r − 2)!(r − 1)!(n − 2)!
=
−
−1
r!(n − 1)!(n + r − 3)!
r!(n − 2)!(n + r − 3)!
=
=

(C.11)

(n + r − 1)(n + r − 2) (n + r − 2)
−
−1
r(n − 1)
r
r−1
.
n−1

This yields a useful relation for incident beam rate,

S>1
r−1
=
,
S1
n−1

155

(C.12)

which for r >> 1 and n >> 1 reduces further to,

S>1
r
= .
S1
n

(C.13)

Given the known number of buckets in a 1 second period (the RF rate) and the ratio of single
particle to multi-particle buckets, the average incident particle rate can be determined.

Correcting an In-Beam Scaler Measurement
If the incoming beam rate is directly measured with an in-beam detector (a plastic scintillator
for example), corrections will have to be made to the measured rate because of multi-particle
buckets. The scaler will count only once if there is a bucket containing more than one beam
particle, therefore the probabilities for each number of particles needs to be calculated. This
can be done using Equation C.8,
H
P0 (n, r) = n−1 r−0
n Hr
(n − 2 + r)!r!(n − 1)!
=
r!(n − 2)!(n + r − 1)!
=

(n − 1)
,
(n + r − 1)

H
P1 (n, r) = n−1 r−1
n Hr
(n − 3 + r)!r!(n − 1)!
=
(r − 1)!(n − 2)!(n + r − 1)!
=

(C.14)

r(n − 1)
,
(n + r − 1)(n + r − 2)

156

(C.15)

and
H
P2 (n, r) = n−1 r−2
n Hr
(n − 4 + r)!r!(n − 1)!
=
(r − 2)!(n − 2)!(n + r − 1)!
=

(C.16)

r(r − 1)(n − 1)
.
(n + r − 1)(n + r − 2)(n + r − 3)

Continuing the clear pattern, the following are presented for completeness:

H
P3 (n, r) = n−1 r−3
n Hr
r(r − 1)(r − 2)(n − 1)
=
,
(n + r − 1)(n + r − 2)(n + r − 3)(n + r − 4)

H
P4 (n, r) = n−1 r−4
n Hr
r(r − 1)(r − 2)(r − 3)(n − 1)
=
,
(n + r − 1)(n + r − 2)(n + r − 3)(n + r − 4)(n + r − 5)

H
P5 (n, r) = n−1 r−5
n Hr
r(r − 1)(r − 2)(r − 3)(r − 4)(n − 1)
=
,
(n + r − 1)(n + r − 2)(n + r − 3)(n + r − 4)(n + r − 5)(n + r − 6)

(C.17)

(C.18)

(C.19)

and
H
P6 (n, r) = n−1 r−6
n Hr
=

r(r − 1)(r − 2)(r − 3)(r − 4)(r − 5)(n − 1)
.
(n + r − 1)(n + r − 2)(n + r − 3)(n + r − 4)(n + r − 5)(n + r − 6)(n + r − 7)
(C.20)
157

Taking the RF rate from e10003 of 21.8259 MHz and considering a one second period,
the following table can be generated using Equations C.14 through C.20,
RT

N0

N1

N2

...

N6

Rm

RT /Rm

5.00E+5

2.13E+07

4.78E+05

1.07E+04

...

2.69E-03

4.89E+05

1.023

1.00E+6

2.09E+07

9.14E+05

4.01E+04

...

1.48E-01

9.56E+05

1.046

1.50E+6

2.04E+07

1.31E+06

8.45E+04

...

1.44E+00

1.40E+06

1.069

2.00E+6

2.00E+07

1.68E+06

1.41E+05

...

6.99E+00

1.83E+06

1.092

2.50E+6

1.96E+07

2.01E+06

2.07E+05

...

2.31E+01

2.24E+06

1.115

3.00E+6

1.92E+07

2.32E+06

2.80E+05

...

5.97E+01

2.64E+06

1.137

3.50E+6

1.88E+07

2.60E+06

3.59E+05

...

1.31E+02

3.02E+06

1.160

4.00E+6

1.84E+07

2.86E+06

4.42E+05

...

2.55E+02

3.38E+06

1.183

4.50E+6

1.81E+07

3.09E+06

5.29E+05

...

4.51E+02

3.73E+06

1.206

5.00E+6

1.78E+07

3.31E+06

6.17E+05

...

7.45E+02

4.07E+06

1.229

5.50E+6

1.74E+07

3.51E+06

7.06E+05

...

1.16E+03

4.39E+06

1.252

where RT is the true number of particles, Nx is the number of buckets containing x particles
and Rm is the sum

P6

x=1 Nx .

An in beam detector will measure Rm since the scalar will

only count once when there is one or more particles in the bucket. To correct for this the
measured rate only needs to be multiplied by the correction factor seen in the last column
of the table. As the true rate increases, the size of this effect becomes more significant.
It is convenient to express this correction as a linear fit to the points given in the last
column as a function of the measured rate in the object scintillator (Robject ),

Rcor = 5.6 · 10−8 Robject + 0.99.
158

(C.21)

This expression is used in the analysis of the experimental data.

159

BIBLIOGRAPHY

160

BIBLIOGRAPHY

[1] E. Rutherford, “LXXIX. The scattering of α and β particles by matter and the structure of the atom,” The London, Edinburgh, and Dublin Philosophical Magazine and
Journal of Science, vol. 21, p. 669, 1911.
[2] J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac, and
M. Stoitsov, “The limits of the nuclear landscape,” Nature, vol. 486, p. 509, 2012.
[3] K. Gelbke et al., Isotope Science Facility at Michigan State University. 2006.
[4] M. G. Mayer, “On Closed Shells in Nuclei,” Physical Review, vol. 74, p. 235, 1948.
[5] E. Feenberg, K. C. Hammack, and M. G. Mayer, “On Closed Shells in Nuclei. II,”
Physical Review, vol. 75, p. 1877, 1949.
[6] O. Haxel, J. H. D. Jensen, and H. E. Suess, “On the “Magic numberNOTs” in Nuclear
Structure,” Physical Review, vol. 75, p. 1766, 1949.
[7] B. A. Brown, “Lecture Notes in Nuclear Structure Physics,” November 2005.
[8] B. A. Brown, “Oxbash for Windows,” 2004.
[9] B. A. Brown and W. D. M. Rae, “The Shell-Model Code NuShellX@MSU,” Nuclear
Data Sheets, vol. 120, p. 115, 2014.
[10] A. Gade and T. Glasmacher, “In-beam nuclear spectroscopy of bound states with fast
exotic ion beams,” Progress in Particle and Nuclear Physics, vol. 60, p. 161, 2008.
[11] T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki, “Magic
numberNOTs in exotic nuclei and spin-isospin properties of the NN interaction,” Physical Review Letters, vol. 87, p. 082502, 2001.
[12] T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, and Y. Akaishi, “Evolution of nuclear
shells due to the tensor force,” Physical Review Letters, vol. 95, p. 232502, 2005.
[13] W. Bethe and W. Gentner, “Weitere Atomumwandlunger, durch γ-Struhlen,”
Zeitschrift für Physik, vol. 19, p. 191, 1937.
[14] M. Harakeh and A. V. D. Woude, Giant Resonances Fundemental High-Frequency
Modes of Nuclear Excitation. Oxford University Press, 2001.

161

[15] G. C. Baldwin and G. S. Klaiber, “Photo-fission in heavy elements,” Physical Review,
vol. 71, p. 3, 1947.
[16] G. C. Baldwin and G. S. Klaiber, “X-ray yield curves for γ-n reactions,” Physical
Review, vol. 73, p. 1156, 1948.
[17] F. Buskirk, H.-D. Gräf, R. Pitthan, H. Theissen, O. Titze, and T. Walcher, “Evidence
for E2 resonances at high excitation energies in 208 Pb,” Physics Letters B, vol. 42,
p. 194, 1972.
[18] S. Noji, A new spectroscopic tool by the radioactive-isotope-beam induced exothermic
charge-exchange reaction. PhD thesis, University of Tokyo, 2012.
[19] Y.-W. Lui, D. H. Youngblood, S. Shlomo, X. Chen, Y. Tokimoto, Krishichayan, M. Anders, and J. Button, “Isoscalar giant resonances in 48 Ca,” Physical Review C, vol. 83,
p. 044327, 2011.
[20] D. Youngblood, Y.-W. Lui, and H. Clark, “Giant resonances in 24 Mg,” Physical Review
C, vol. 60, p. 014304, 1999.
[21] Y. Tokimoto, Y. W. Lui, H. L. Clark, B. John, X. Chen, and D. H. Youngblood, “Giant
resonances in 46,49 Ti,” Physical Review C, vol. 74, p. 044308, 2006.
[22] Y. W. Lui, H. L. Clark, and D. H. Youngblood, “Giant resonances in 16 O,” Physical
Review C, vol. 64, p. 064308, 2001.
[23] Y.-W. Lui, P. Bogucki, J. D. Bronson, D. H. Youngblood, and U. Garg, “Giant resonaces in 112 Sn,” Physical Review C, vol. 30, p. 51, 1984.
[24] M. Uchida, H. Sakaguchi, M. Itoh, M. Yosoi, T. Kawabata, Y. Yasuda, H. Takeda,
T. Murakami, S. Terashima, S. Kishi, U. Garg, P. Boutachkov, M. Hedden, B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, M. Fujiwara, H. Fujimura, H. P. Yoshida, K. Hara,
H. Akimune, and M. N. Harakeh, “Systematics of the bimodal isoscalar giant dipole
resonance,” Physical Review C, vol. 69, p. 051301, 2004.
[25] T. Li, U. Garg, Y. Liu, R. Marks, B. K. Nayak, P. V. Madhusudhana Rao, M. Fujiwara,
H. Hashimoto, K. Nakanishi, S. Okumura, M. Yosoi, M. Ichikawa, M. Itoh, R. Matsuo, T. Terazono, M. Uchida, Y. Iwao, T. Kawabata, T. Murakami, H. Sakaguchi,
S. Terashima, Y. Yasuda, J. Zenihiro, H. Akimune, K. Kawase, and M. N. Harakeh,
“Isoscalar giant resonances in the Sn nuclei and implications for the asymmetry term
in the nuclear-matter incompressibility,” Physical Review C, vol. 81, p. 034309, 2010.
[26] M. Itoh, H. Sakaguchi, M. Uchida, T. Ishikawa, T. Kawabata, T. Murakami, H. Takeda,
T. Taki, S. Terashima, N. Tsukahara, Y. Yasuda, M. Yosoi, U. Garg, M. Hedden,
B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, H. Fujimura, M. Fujiwara, K. Hara,
162

H. P. Yoshida, H. Akimune, M. N. Harakeh, and M. Volkerts, “Systematic study of
L ≤ 3 giant resonances in Sm isotopes via multipole decomposition analysis,” Physical
Review C, vol. 68, p. 064602, 2003.
[27] M. Uchida, H. Sakaguchi, M. Itoh, M. Yosoi, T. Kawabata, H. Takeda, Y. Yasuda,
T. Murakami, T. Ishikawa, T. Taki, N. Tsukahara, S. Terashima, U. Garg, M. Hedden,
B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, M. Fujiwara, H. Fujimura, K. Hara,
E. Obayashi, H. P. Yoshida, H. Akimune, M. N. Harakeh, and M. Volkerts, “Isoscalar
giant dipole resonance in 208 Pb via inelastic Îą scattering at 400 MeV and nuclear
incompressibility,” Physics Letters B, vol. 557, p. 12, 2003.
[28] B. K. Nayak, U. Garg, M. Hedden, M. Koss, T. Li, Y. Liu, P. V. Madhusudhana
Rao, S. Zhu, M. Itoh, H. Sakaguchi, H. Takeda, M. Uchida, Y. Yasuda, M. Yosoi,
H. Fujimura, M. Fujiwara, K. Hara, T. Kawabata, H. Akimune, and M. N. Harakeh,
““Bi-modal” isoscalar giant dipole strength in 58 Ni,” Physics Letters B, vol. 637, p. 43,
2006.
[29] M. Itoh, H. Sakaguchi, M. Uchida, T. Ishikawa, T. Kawabata, T. Murakami, H. Takeda,
T. Taki, S. Terashima, N. Tsukahara, Y. Yasuda, M. Yosoi, U. Garg, M. Hedden,
B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, H. Fujimura, M. Fujiwara, K. Hara, H. P.
Yoshida, H. Akimune, M. N. Harakeh, and M. Volkerts, “Compressional-mode giant
resonances in deformed nuclei,” Physics Letters B, vol. 549, p. 58, 2002.
[30] J. P. Blaizot, “Nuclear compressibilities,” Physics Reports, vol. 64, p. 171, 1980.
[31] D. H. Youngblood, H. L. Clark, and Y.-W. Lui, “Incompressibility of Nuclear Matter
from the Giant Monopole Resonance,” Physical Review Letters, vol. 82, p. 691, 1999.
[32] D. H. Youngblood, H. L. Clark, and Y.-W. Lui, “Compressibility of Nuclear Matter
from the Giant Monopole Resonance,” Nuclear Physics A, vol. 649, p. 49, 1999.
[33] S. Nakayama, H. Akimune, Y. Arimoto, I. Daito, H. Fujimura, Y. Fujita, M. Fujiwara,
K. Fushimi, H. Kohri, N. Koori, K. Takahisa, T. Takeuchi, A. Tamii, M. Tanaka, T. Yamagata, Y. Yamamoto, K. Yonehara, and H. Yoshida, “Isovector Electric Monopole
Resonance in 60 Ni,” Physical Review Letters, vol. 83, p. 690, 1999.
[34] R. G. T. Zegers, A. Van den Berg, S. Brandenburg, F. Fleurot, M. Fujiwara, J. Guillot,
V. Hannen, M. Harakeh, H. Laurent, K. Van der Schaaf, S. Van der Werf, A. Willis,
and H. Wilschut, “Search for Isovector Giant Monopole Resonances via the Pb(3 He,t p)
Reaction,” Physical Review Letters, vol. 84, p. 3779, 2000.
[35] M. Scott, R. G. T. Zegers, R. Almus, S. M. Austin, D. Bazin, B. A. Brown, C. Campbell, A. Gade, M. Bowry, S. Galès, U. Garg, M. N. Harakeh, E. Kwan, C. Langer,
C. Loelius, S. Lipschutz, E. Litvinova, E. Lunderberg, C. Morse, S. Noji, G. Perdikakis,
T. Redpath, C. Robin, H. Sakai, Y. Sasamoto, M. Sasano, C. Sullivan, J. A. Tostevin,
163

T. Uesaka, and D. Weisshaar, “Observation of the Isovector Giant Monopole Resonance via the 28 Si(10 Be, 10 B [1.74 MeV]) Reaction at 100 AMeV,” Physical Review
Letters, vol. 118, p. 172501, 2017.
[36] A. Erell, J. Alster, J. Lichtenstadt, M. A. Moinester, J. Bowman, M. Cooper, F. Irom,
H. Matis, E. Piasetzky, U. Sennhause, and Q. Ingram, “Properties of the Isovector
Monopole and Other Giant Resonances in Pion Charge Exchange,” Physical Review
Letters, vol. 52, p. 2134, 1984.
[37] K. Yako, H. Sagawa, and H. Sakai, “Neutron skin thickness of 90 Zr determined by
charge exchange reactions,” Physical Review C, vol. 74, p. 051303(R), 2006.
[38] A. Klimkiewicz, N. Paar, P. Adrich, M. Fallot, K. Boretzky, T. Aumann, D. CortinaGil, U. D. Pramanik, T. W. Elze, H. Emling, H. Geissel, M. Hellström, K. L. Jones,
J. V. Kratz, R. Kulessa, C. Nociforo, R. Palit, H. Simon, G. Surówka, K. Sümmerer,
D. Vretenar, and W. Waluś, “Nuclear symmetry energy and neutron skins derived from
pygmy dipole resonances,” Physical Review C, vol. 76, p. 051603(R), 2007.
[39] B. A. Brown, “Neutron radii in nuclei and the neutron equation of state,” Physical
Review Letters, vol. 85, p. 5296, 2000.
[40] S. Typel and B. A. Brown, “Neutron radii and the neutron equation of state in relativistic models,” Physical Review C, vol. 64, p. 027302, 2001.
[41] S. Yoshida and H. Sagawa, “Neutron skin thickness and equation of state in asymmetric
nuclear matter,” Physical Review C, vol. 69, p. 024318, 2004.
[42] Y. Fujita, B. Rubio, and W. Gelletly, “Spin-isospin excitations probed by strong, weak
and electro-magnetic interactions,” Progress in Particle and Nuclear Physics, vol. 66,
p. 549, 2011.
[43] D. H. Wilkinson, “Renormalization of the Axial-Vector Coupling Constant in Nuclear
β Decay,” Physical Review C, vol. 7, p. 930, 1973.
[44] M. P. Nakada, J. Anderson, C. Gardner, J. McClure, and C. Wong, “Neutron Spectrum
from p+d Reaction,” Physical Review, vol. 110, p. 594, 1958.
[45] J. Anderson and C. Wong, “Evidence For Charge Independence in Medium Weight
Nuclei,” Physical Review Letters, vol. 7, p. 250, 1961.
[46] P. H. Bowen, G. C. Cox, G. B. Huxtable, J. J. Thresher, and A. Langsford, “Neutrons
Emitted at 0° From Nuclei Bombarded by 143 MeV Protons,” Nuclear Physics, vol. 30,
p. 475, 1962.

164

[47] R. R. Doering and A. Galonsky, “Observation of Giant Gamow-Teller Strength in (p,n)
Reactions,” Physical Review Letters, vol. 35, p. 1691, 1975.
[48] C. Goodman, C. Goulding, M. Greenfield, J. Rapaport, D. Bainum, C. Foster, W. Love,
and F. Petrovich, “Gamow-Teller Matrix Elements from 0° (p, n) Cross Sections,”
Physical Review Letters, vol. 44, p. 1755, 1980.
[49] J. Rapaport and D. Wang, “(~p,n) reaction on carbon isotopes at Ep =160 MeV,” Physical Review C, vol. 36, p. 500, 1987.
[50] T. N. Taddeucci, J. Rapaport, and C. C. Fostster, “Spin excitations in 40 Ca(p,n),”
Physical Review C, vol. 28, p. 2511, 1983.
[51] B. D. Anderson, R. J. Mccarthy, M. Ahmad, A. Fazely, A. M. Kalenda, J. N. Knudson, J. Watson, R. Madey, and C. C. Foster, “Comparison of 0°(p,n) cross sections
and B(M1) values for seperating current and spin contributions to isovector M1 transitions,” Physical Review C, vol. 26, p. 8, 1982.
[52] X. Yang, L. Wang, J. Rapaport, C. D. Goodman, C. Foster, Y. Wang, W. Unkelbach,
E. Sugarbaker, D. Marchlenski, S. de Lucia, B. Luther, J. L. Ullmann, A. G. Ling, B. K.
Park, D. S. Sorenson, L. Rybarcyk, T. N. Taddeucci, C. R. Howell, and W. Tornow,
“Dipole and spin-dipole resonances in charge-exchange reactions on 12 C,” Physical
Review C, vol. 48, p. 1158, 1993.
[53] L. Wang, X. Yang, J. Rapaport, C. D. Goodman, C. C. Foster, Y. Wang, R. A.
Lindgren, E. Sugarbaker, D. Marchlenski, S. De Lucia, B. Luther, L. Rybarcyk, T. N.
Taddeucci, and B. K. Park, “10 B(p,n)10 C reaction at 186 MeV,” Physical Review C,
vol. 47, p. 2123, 1993.
[54] J. Rapaport, T. N. Taddeucci, C. Gaarde, C. D. Goodman, C. C. Foster, C. A. Goulding, D. Horen, E. Sugarbaker, T. Masterson, and D. Lind, “12 C(p,n)12 N reaction at
120, 160, and 200 MeV,” Physical Review C, vol. 24, p. 335, 1981.
[55] M. Bhattacharya, C. D. Goodman, R. S. Raghavan, M. Palarczyk, J. Rapaport, I. J. V.
Heerden, and P. Zupranski, “Measurement of Gamow-Teller Strength for 176 Yb→176 Lu
and the Efficiency of a Solar Neutrino Detector,” Physical Review Letters, vol. 85,
p. 4446, 2000.
[56] T. N. Taddeucci, C. A. Goulding, T. A. Carey, R. C. Byrd, C. D. Goodman, C. Gaarde,
J. Larsen, D. Horen, J. Rapaport, and E. Sugarbaker, “The (p,n) reaction as a probe
of beta decay strength,” Nuclear Physics A, vol. 469, p. 125, 1987.
[57] K. P. Jackson, A. Celler, W. P. Alford, K. Raywood, R. Abegg, R. E. Azuma, C. K.
Campbell, S. El-Kateb, D. Frekers, P. W. Green, O. Häusser, R. L. Helmer, R. S.
Henderson, K. H. Hicks, R. Jeppesen, P. Lewis, C. A. Miller, A. Moalem, M. A.
165

Moinester, R. B. Schubank, G. G. Shute, B. M. Spicer, M. C. Vetterli, A. I. Yavin, and
S. Yen, “The (n,p) reaction as a probe of Gamow-Teller strength,” Physics Letters B,
vol. 201, p. 25, 1988.
[58] E. R. Flynn and J. D. Garrett, “Excitation of the Parent Analogs of the 58 Ni Giant M1
Resonance by the Reaction 58 Ni(t,h)58 Co,” Physical Review Letters, vol. 29, p. 1748,
1972.
[59] S. Rakers, F. Ellinghaus, R. Bassini, C. Bäumer, A. M. Van den Berg, D. Frekers,
D. De Frenne, M. Hagemann, V. M. Hannen, M. N. Harakeh, M. Hartig, R. Henderson,
J. Heyse, M. A. De Huu, E. Jacobs, M. Mielke, J. M. Schippers, R. Schmidt, S. Y. Van
der Werf, and H. J. Wörtche, “Measuring the (d,2 He) reaction with the focal-plane
detection system of the BBS magnetic spectrometer at AGOR,” Nuclear Instruments
and Methods in Physics Research A, vol. 481, p. 253, 2002.
[60] H. Okamura, S. Fujita, Y. Hara, K. Hatanaka, T. Ichihara, S. Ishida, K. Katoh, T. Niizeki, H. Ohnuma, H. Otsu, H. Sakai, N. Sakamoto, Y. Satou, T. Uesaka, T. Wakasa,
and T. Yamashita, “Tensor analyzing power of the (d,2 He) reaction at 270 MeV,”
Physics Letters B, vol. 345, p. 1, 1995.
[61] T. Annakkage, J. Jgnecke, J. S. Winfield, G. R. A. Berg, J. A. Brown, G. Crawley,
S. Danczyk, M. Fujiwara, D. J. Mercer, K. Pham, D. A. Roberts, and G. H. Yoo,
“Isovector giant resonances in 6 He, 12 B, 90 Y, 120 In and 208 Tl observed in the (7 Li,7 Be)
charge-exchange reaction,” Nuclear Physics A, vol. 648, p. 3, 1999.
[62] M. Fujiwara, H. Akimune, I. Daito, and H. Ejiri, “Spin-Isospin Resonances in Nuclei,”
Nuclear Physics A, vol. 599, p. 223, 1996.
[63] M. Sasano, G. Perdikakis, R. G. T. Zegers, S. M. Austin, D. Bazin, B. A. Brown,
C. Caesar, A. L. Cole, J. M. Deaven, N. Ferrante, C. J. Guess, G. W. Hitt, R. Meharchand, F. Montes, J. Palardy, A. Prinke, L. A. Riley, H. Sakai, M. Scott, A. Stolz,
L. Valdez, and K. Yako, “Gamow-Teller Transition Strengths from 56 Ni,” Physical
Review Letters, vol. 107, p. 202501, 2011.
[64] M. Sasano, G. Perdikakis, R. G. T. Zegers, S. M. Austin, D. Bazin, B. A. Brown,
C. Caesar, A. L. Cole, J. M. Deaven, N. Ferrante, C. J. Guess, G. W. Hitt, M. Honma,
R. Meharchand, F. Montes, J. Palardy, A. Prinke, L. A. Riley, H. Sakai, M. Scott,
A. Stolz, T. Suzuki, L. Valdez, and K. Yako, “Extraction of Gamow-Teller strength
distributions from 56 Ni and 55 Co via the (p,n) reaction in inverse kinematics,” Physical
Review C, vol. 86, p. 034324, 2012.
[65] K. Langanke and G. Martı́nez-Pinedo, “Nuclear weak-interaction processes in stars,”
Reviews of Modern Physics, vol. 75, p. 819, 2003.

166

[66] R. G. T. Zegers, R. Meharchand, Y. Shimbara, S. M. Austin, D. Bazin, B. A. Brown,
C. A. Diget, A. Gade, C. J. Guess, M. Hausmann, G. W. Hitt, M. E. Howard, M. King,
D. Miller, S. Noji, a. Signoracci, K. Starosta, C. Tur, C. Vaman, P. Voss, D. Weisshaar, and J. Yurkon, “34 P(7 Li,7 Be+γ) reaction at 100A MeV in inverse kinematics.,”
Physical Review Letters, vol. 104, p. 212504, 2010.
[67] K. Yako Unpublished.
[68] Y. Satou, T. Nakamura, Y. Kondo, N. Matsui, Y. Hashimoto, T. Nakabayashi, T. Okumura, M. Shinohara, N. Fukuda, T. Sugimoto, H. Otsu, Y. Togano, T. Motobayashi,
H. Sakurai, Y. Yanagisawa, N. Aoi, S. Takeuchi, T. Gomi, M. Ishihara, S. Kawai, H. J.
Ong, T. K. Onishi, S. Shimoura, M. Tamaki, T. Kobayashi, Y. Matsuda, N. Endo,
and M. Kitayama, “14 Be(p,n)14 B reaction at 69 MeV in inverse kinematics,” Physics
Letters B, vol. 697, p. 459, 2011.
[69] D. Alburger and D. Wilkinson, “Beta decay of 16 C and 17 N ,” Physical Review C,
vol. 13, p. 835, 1976.
[70] C. A. Gagliardi, G. Garvey, N. Jarmie, and R. Roberston, “0+ →0− beta decay of
16 C,” Physical Review C, vol. 27, p. 1353(R), 1983.
[71] S. Grévy, N. L. Achouri, J. C. Angélique, C. Borcea, A. Buta, F. De Oliveira, M. Lewitowicz, E. Liénard, T. Martin, F. Negoita, N. A. Orr, J. Peter, S. Pietri, and
C. Timis, “Observation of a new transition in the β-delayed neutron decay of 16 C,”
Physical Review C, vol. 63, p. 037302, 2001.
[72] K. Snover, E. Adelberger, P. G. Ikossi, and B. A. Brown, “Proton capture to excited
states of 16 O M1, E1, and Gamow-Teller transitions and shell model calculations,”
Physical Review C, vol. 27, p. 1837, 1983.
[73] E. K. Warburton and B. A. Brown, “Effective interactions for the 0p1s0d nuclear
shell-model space,” Physical Review C, vol. 46, p. 923, 1992.
[74] G. F. Bertsch and H. Esbensen, “The (p,n) reaction and the nucleon-nucleon force,”
Reports on Progress in Physics, vol. 50, p. 607, 1987.
[75] A. Arima, “History of giant resonances and quenching,” Nuclear Physics A, vol. 649,
p. 260, 1999.
[76] B. A. Brown and B. H. Wildenthal, “Experimental and theoretical Gamow-Teller
beta-decay observables for the sd-shell nuclei,” Atomic Data and Nuclear Data Tables, vol. 33, p. 347, 1985.
[77] B. H. Wildenthal, M. S. Curtin, and B. A. Brown, “Predicted features of the beta
decay of neutron-rich sd-shell nuclei,” Physical Review C, vol. 28, p. 1343, 1983.
167

[78] R. G. Zegers, H. Akimune, S. M. Austin, D. Bazin, A. M. D. Berg, G. P. Berg,
B. A. Brown, J. Brown, A. L. Cole, I. Daito, Y. Fujita, M. Fujiwara, S. Galès, M. N.
Harakeh, H. Hashimoto, R. Hayami, G. W. Hitt, M. E. Howard, M. Itoh, J. Jänecke,
T. Kawabata, K. Kawase, M. Kinoshita, T. Nakamura, K. Nakanishi, S. Nakayama,
S. Okumura, W. A. Richter, D. A. Roberts, B. M. Sherrill, Y. Shimbara, M. Steiner,
M. Uchida, H. Ueno, T. Yamagata, and M. Yosoi, “The (t,3 He) and (3 He, t) reactions
as probes of Gamow-Teller strength,” Physical Review C, vol. 74, p. 024309, 2006.
[79] K. Yako, H. Sakai, M. B. Greenfield, K. Hatanaka, M. Hatano, J. Kamiya, H. Kato,
Y. Kitamura, Y. Maeda, C. L. Morris, H. Okamura, J. Rapaport, T. Saito, Y. Sakemi,
K. Sekiguchi, Y. Shimizu, K. Suda, A. Tamii, N. Uchigashima, and T. Wakasa, “Determination of the Gamow-Teller quenching factor from charge exchange reactions on
90 Zr,” Physics Letters B, vol. 615, p. 193, 2005.
[80] K. J. Raywood, B. M. Spicer, S. Yen, S. A. Long, M. A. Moinester, R. Abegg, W. P.
Alford, A. Celler, T. E. Drake, D. Frekers, P. E. Green, O. Häusser, R. L. Helmer,
R. S. Henderson, K. H. Hicks, K. P. Jackson, R. G. Jeppesen, J. D. King, N. S. P.
King, C. A. Miller, V. C. Officer, R. Schubank, G. G. Shute, M. Vetterli, J. Watson,
and A. I. Yavin, “Spin-flip isovector giant resonances from the 90 Zr(n,p)90 Y reaction
at 198 MeV,” Physical Review C, vol. 41, p. 2836, 1990.
[81] T. Wakasa, H. Sakai, H. Okamura, H. Otsu, S. Fujita, S. Ishida, N. Sakamoto, T. Uesaka, Y. Satou, M. Greenfield, and K. Hatanaka, “Gamow-Teller strength of 90 Nb in
the continuum studied via multipole decomposition analysis of the 90 Zr(p,n) reaction
at 295 MeV,” Physical Review C, vol. 55, p. 2909, 1997.
[82] H. Sakai and K. Yako, “Experimental determination of Gamow-Teller quenching value,
Landau-Migdal parameter gN ∆ and pion condensation,” Nuclear Physics A, vol. 731,
p. 94, 2004.
[83] G. Martı́nez-Pinedo, A. Poves, E. Caurier, and A. P. Zuker, “Effective gA in the pf
shell,” Physical Review C, vol. 53, p. R2602, 1996.
[84] F. M. Nunes and I. J. Thompson, Nuclear Reactions for Astrophysics. Cambridge
University Press, 2009.
[85] W. G. Love and M. A. Franey, “Effective nucleon-nucleon interaction for scattering at
intermediate energies,” Physical Review C, vol. 24, p. 1073, 1981.
[86] M. A. Franey and W. G. Love, “Nucleon-nucleon t-matrix interaction for scattering at
intermediate energies,” Physical Review C, vol. 31, p. 488, 1985.
[87] R. Schaeffer, J. Raynal, and J. Comfort, “Program DW81,” unpublished, 1981.

168

[88] J. Raynal, “Multipole Expansion of a Two-Body Interaction in Helicity Formalism
and its Applications to Nuclear structure and Nuclear Reaction Calculations,” Nuclear
Physics A, vol. 97, p. 572, 1967.
[89] P. Schwandt and M. D. Kaitchuck, “Analyzing power of proton-nucleus elastic scattering between 80 and 180 MeV,” Physical Review C, vol. 26, p. 55, 1982.
[90] G. Perdikakis, M. Sasano, S. M. Austin, D. Bazin, C. Caesar, S. Cannon, J. M. Deaven,
H. J. Doster, C. J. Guess, G. W. Hitt, J. Marks, R. Meharchand, D. T. Nguyen,
D. Peterman, A. Prinke, M. Scott, Y. Shimbara, K. Thorne, L. Valdez, and R. G. T.
Zegers, “LENDA: A low energy neutron detector array for experiments with radioactive
beams in inverse kinematics,” Nuclear Instruments and Methods in Physics Research
A, vol. 686, p. 117, 2012.
[91] S. Lipschutz, R. Zegers, J. Hill, S. Liddick, S. Noji, C. Prokop, M. Scott, M. Solt,
C. Sullivan, and J. Tompkins, “Digital data acquisition for the Low Energy Neutron
Detector Array (LENDA),” Nuclear Instruments and Methods in Physics Research A,
vol. 815, p. 1, 2016.
[92] S. V. Paulauskas, M. Madurga, R. Grzywacz, D. Miller, S. Padgett, and H. Tan, “A
digital data acquisition framework for the Versatile Array of Neutron Detectors at
Low Energy (VANDLE),” Nuclear Instruments and Methods in Physics Research A,
vol. 737, p. 22, 2014.
[93] W. A. Peters, S. Ilyushkin, M. Madurga, C. Matei, S. V. Paulauskas, R. K. Grzywacz,
D. W. Bardayan, C. R. Brune, J. Allen, J. M. Allen, Z. Bergstrom, J. Blackmon,
N. T. Brewer, J. A. Cizewski, P. Copp, M. E. Howard, R. Ikeyama, R. L. Kozub,
B. Manning, T. N. Massey, M. Matos, E. Merino, P. D. O’Malley, F. Raiola, C. S.
Reingold, F. Sarazin, I. Spassova, S. Taylor, and D. Walter, “Performance of the
Versatile Array of Neutron Detectors at Low Energy (VANDLE),” Nuclear Instruments
and Methods in Physics Research A, vol. 836, p. 122, 2016.
[94] D. Bazin, J. A. Caggiano, B. M. Sherrill, J. Yurkon, and A. Zeller, “The S800 spectrograph,” Nuclear Instruments and Methods in Physics Research B, vol. 204, p. 629,
2003.
[95] D. J. Morrissey, B. M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhoever, “Commissioning the A1900 projectile fragment separator,” Nuclear Instruments and Methods
in Physics Research B, vol. 204, p. 90, 2003.
[96] P. A. Zavodszky, B. Arend, D. Cole, J. DeKamp, G. Machicoane, F. Marti, P. Miller,
J. Moskalik, J. Ottarson, J. Vincent, and A. Zeller, “Design of SuSI - superconducting
source for ions at NSCL/MSU - II. The conventional parts,” AIP Conference Proceedings, vol. 749, p. 131, 2005.

169

[97] P. A. Zavodszky, B. Arend, D. Cole, J. Dekamp, G. MacHicoane, F. Marti, P. Miller,
J. Moskalik, J. Ottarson, J. Vincent, A. Zeller, and N. Y. Kazarinov, “Status report on
the design and construction of the Superconducting Source for Ions at the National Superconducting Cyclotron Laboratory/Michigan State University,” Review of Scientific
Instruments, vol. 77, p. 03A334, 2006.
[98] G. Machicoane, D. Cole, J. Ottarson, J. Stetson, and P. Zavodszky, “ARTEMIS-B: A
room-temperature test electron cyclotron resonance ion source for the National Superconducting Cyclotron Laboratory at Michigan State University,” Review of Scientific
Instruments, vol. 77, p. 03A322, 2006.
[99] X. Wu, H. Blosser, D. Johnson, F. Marti, and R. York, “The K500-to-K1200 coupling
line for the coupled cyclotron facility at the NSCL,” Proceedings of the 1999 Particle
Accelerator Conference, vol. 2, p. 1318 vol.2, 1999.
[100] D. Morrissey, “The coupled cyclotron project at the NSCL,” Nuclear Physics A,
vol. 616, p. 45, 1997.
[101] J. Yurkon, D. Bazin, W. Benenson, D. J. Morrissey, B. M. Sherrill, D. Swan, and
R. Swanson, “Focal plane detector for the S800 high-resolution spectrometer,” Nuclear
Instruments and Methods in Physics Research A, vol. 422, p. 291, 1999.
[102] G. W. Hitt, The 64 Zn(t,3 He) Charge-Exchange Reaction at 115 MeV Per Nucleon and
Application to 64 Zn Stellar Electron-Capture. PhD thesis, Michigan State University,
2008.
[103] G. F. Knoll, Radiation Detection and Measurement, vol. 3. John Wiley and Sons, 2010.
[104] R. Zegers. private communication.
[105] XIA, “Digital Gamma Finder (DGF) PIXIE-16 User’s Manual,” 2005.
[106] C. Prokop, S. Liddick, N. Larson, S. Suchyta, and J. Tompkins, “Optimization of
the National Superconducting Cyclotron Laboratory Digital Data Acquisition System
for use with fast scintillator detectors,” Nuclear Instruments and Methods in Physics
Research A, vol. 792, p. 81, 2015.
[107] C. J. Prokop, S. N. Liddick, B. L. Abromeit, A. T. Chemey, N. R. Larson, S. Suchyta,
and J. R. Tompkins, “Digital data acquisition system implementation at the National
Superconducting Cyclotron Laboratory,” Nuclear Instruments and Methods in Physics
Research A, vol. 741, p. 163, 2014.
[108] H. Tan, M. Momayezi, A. Fallu-Labruyere, Y. X. Chu, and W. K. Warburton, “A fast
digital filter algorithm for gamma-ray spectroscopy with double-exponential decaying
scintillators,” IEEE Transactions on Nuclear Science, vol. 51, p. 1541, 2004.
170

[109] N. Kudomi, “Energy calibration of plastic scintillators for low energy electrons by using
Compton scatterings of γ rays,” Nuclear Instruments and Methods in Physics Research
A, vol. 430, p. 96, 1999.
[110] M. Berz, K. Joh, J. A. Nolen, B. M. Sherrill, and A. F. Zeller, “Reconstructive correction of aberrations in nuclear particle spectrographs,” Physical Review C, vol. 47,
p. 537, 1993.
[111] S. Agostinelli et al., “GEANT4 - A simulation toolkit,” Nuclear Instruments and Methods in Physics Research A, vol. 506, p. 250, 2003.
[112] E. Mendoza, D. Cano-Ott, T. Koi, and C. Guerrero, “New standard evaluated neutron
cross section libraries for the GEANT4 code and first verification,” IEEE Transactions
on Nuclear Science, vol. 61, p. 2357, 2014.
[113] L. Stuhl, A. Krasznahorkay, M. Csatlós, A. Algora, J. Gulyás, G. Kalinka, J. Timár,
N. Kalantar-Nayestanaki, C. Rigollet, S. Bagchi, and M. A. Najafi, “A neutron spectrometer for studying giant resonances with (p,n) reactions in inverse kinematics,”
Nuclear Instruments and Methods in Physics Research A, vol. 736, p. 1, 2014.
[114] Saint-Gobain, “Organic Scintillation Materials and Assemblies,” Catalogue, p. 16,
2015.
[115] T. N. Taddeucci, C. D. Goodman, R. C. Byrd, T. A. Carey, D. J. Horen, J. Rapaport,
and E. Sugarbaker, “A neutron polarimeter for (p,n) measurements at intermediate
energies,” Nuclear Instruments and Methods in Physics Research A, vol. 241, p. 448,
1985.
[116] B. D. Anderson, A. Fazely, R. J. McCarthy, P. C. Tandy, J. W. Watson, R. Madey,
W. Bertozzi, T. N. Buti, J. M. Finn, J. Kelly, M. A. Kovash, B. Pugh, B. H. Wildenthal,
and C. C. Foster, “Gamow-teller strength in the 18 O(p,n)18 F reaction at 135 MeV,”
Physical Review C, vol. 27, p. 1387, 1983.
[117] W. T. Chou, E. K. Warburton, and B. A. Brown, “Gamow-Teller beta-decay rates for
A≤18 nuclei,” Physical Review C, vol. 47, p. 163, 1993.
[118] S. M. Austin, N. Anantaraman, and W. G. Love, “Charge Exchange Reactions and the
Efficiency of Solar Neutrino Detectors,” Physical Review Letters, vol. 73, p. 30, 1994.
[119] K. Ikeda, “Collective Excitation of Unlike Pair States in Heavier Nuceli,” Progress of
Theoretical Physics, vol. 31, p. 434, 1964.

171