INDIRECT REACTION METHODS FOR NUCLEAR ASTROPHYSICS:

EXPLORING CHARGE-EXCHANGE AND TRANSFER REACTION

MODELS

By

Terri Elizabeth Poxon-Pearson

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics — Doctor of Philosophy

2020

ABSTRACT

INDIRECT REACTION METHODS FOR NUCLEAR ASTROPHYSICS: EXPLORING

CHARGE-EXCHANGE AND TRANSFER REACTION MODELS

By

Terri Elizabeth Poxon-Pearson

Indirect reaction methods play an important role in probing many astrophysical nucle-

osynthetic processes in cases where a direct measurement in the laboratory is technically

challenging or not possible. However, the resulting astrophysical data is only as good as

the quality of the reaction theories used to extract it from experimental measurements. In

this thesis we explore two indirect reaction methods, transfer and charge-exchange reactions,

with an emphasis on the reaction theory models used to interpret the measurement.

Deuteron induced transfer reactions are a useful tool for probing single particle capture

reactions. We discuss a methodology that has been developed to extract spectroscopic factors

from transfer to low lying resonances. Spectroscopic factors are used to experimentally

constrain the astrophysical reaction rate of interest via the resonance strength. Here, we

present results of three transfer reaction studies: 30P(d,n) to extract the 30P(p,γ) reaction

in classical novae, 23Al(d,n) to extract the 23Al(p,γ) reaction in type-I x-ray bursts, and

56Ni(d,n) to extract the 56Ni(p,γ) reaction, also important in x-ray bursts. In all of these

cases, the transfer data was able to reduce the uncertainty in the astrophysical reaction rate

and this marks the first experimental constraints on the 30P(p,γ) reaction rate.

Charge-exchange reactions have diverse applications to astrophysical processes, ranging

from constraining electron capture rates in core collapse supernovae, to probing the nuclear

symmetry energy, important to understanding neutron stars and their mergers. However, the

reactions models which describe charge-exchange reactions remain relatively underdeveloped

compared to those used to describe other reactions. In this thesis we present an initial study

exploring several aspects of charge-exchange reaction models.

We conduct a systematic study of charge-exchange transitions to 0+ isobaric analog states

over a range of targets and beam energies using the distorted wave Born approximation. We

use a two-body framework, which is characterized by a nucleon-target Lane potential, and a

three-body framework, which uses an NN interaction to describe charge-exchange between

a scattering nucleon and a valence nucleon bound to an inert core. We explore the impact

of different interactions, varying both the potential which mediates the charge-exchange and

the interaction which describes the incoming and outgoing distorted waves.

We find that the two-body formalism was better able to describe both the shape and mag-

nitude of charge-exchange data, capturing 31% of the data within the error band created by

normalized calculations using two different optical potentials. This is opposed to describing

less that 15% of the data in the three-body model. Although there was a 50% difference, on

average, between the charge-exchange cross sections produced using Koning-Delaroche [1]

and Chapel-Hill [2] parameter sets, neither parameter set is preferred by the data.

The shape of the angular distributions produced by the three-body framework differ

significantly from their two-body counterparts and from experimental data. We determined

that this difference arises from a selection of different partial waves between the formalisms.

The Lane interaction in the two-body framework selects lower partial waves, indicating a

more central interaction, while the Gogny and AV8’ interactions select higher partial waves,

resulting in a reaction located near the surface of the target where the active nucleons are

in close proximity. Overall, the charge-exchange cross section is very sensitive to the choice

of interaction, indicating that charge-exchange could be a useful tool to further constrain

nuclear interactions.

ACKNOWLEDGMENTS

First I want to thank my advisor, Filomena Nunes, for all of her support and guidance

over the past six years. While I have at times questioned my abilities and my goals, I have

never questioned my choice of advisor. Filomena has helped me grow professionally, listening

intently as I discerned my career goals, and connecting me with people and opportunities

that support that growth. She has helped me foster resilience and grit, lending me confidence

and reassurance when I was in doubt. At the same time, Filomena has supported me as a

whole person, encouraging my involvement in science outreach and WaMPS and affording

me the flexibility to nurture a long distance relationship and marriage. Thank you.

I want to thank my parents, without whom, I would not have been possible. Their

support throughout my life was been unwavering, regardless of what endeavor I was pursing,

because they have never really cared what I chose to do as long as I was happy and healthy. I

am grateful to have been unburdened by their expectations and uplifted by their pride in my

accomplishments. Of course, I need to thank my husband, Michael Swift, who has been my

loudest cheerleader and strongest support since I began this process. Thank you for calming

me down when I was overwhelmed, celebrating every small victory and milestone, moving

from sunny California to less sunny Michigan while I finished my degree, and, most of all,

for loving me.

I am grateful for the support offered by the NSCL, from the camaraderie of lab graduate

students, all the way up to the highest levels of leadership which has intentionally supported

graduate student initiatives, such as WaMPS, both with their words and financially. I am

grateful for the opportunity to work as a lab tour guide and for Zach Constan’s support of

graduate student guides. Showing off this facility and the amazing research performed here

iv

has reinvigorated my passion for my work on more than one occasion.

I would like to thank my guidance committee members, Remco Zegers, Alex Brown, Pawel

Danielewicz, and Laura Chomiuk for their advice and input. In particular, I am grateful for

the time Remco has taken to share some of his deep knowledge of charge-exchange reactions,

offering particularly helpful guidance, and for Pawel whose help was indispensable in getting

my reaction code running. I am grateful for the insights gleaned from conversations with

several colleagues, particularly Bob Wiringa. Finally, I am grateful for the support of Jolie

Cizewski who is a fierce advocate for young scientists and has been a great support in moving

into the next stage of my career.

While graduate school was not always easy for me, I never lacked good friends and a

strong support system of fellow graduate students. Specifically, I want to thank Brandon

Elman (and honorary graduate student, Alice Mills), Chris Izzo, Andrew LaJoie, Dennis

Foren, Justin Lietz, Rachel Sandler, Amy Lovell, and Christa Haverly for their friendship

and support. I am grateful for every meal, drink, game, and conversation we shared.

I wanted to thank the members of the few-body reactions group, past and present, for

all of their feedback on practice talks, company during travel, and opportunities to pass

on some of my knowledge: Amy Lovell, Gregory Potel, Weichuan Li, Linda H’lophe, Luke

Titus, Alaina Ross, Nick Cariello, Manuel Catacora-Rios, Jimmy Rotureau, Garrett King,

Paul Wright, and Simon Sundberg.

Finally, this work would not have been possible without the funding sources that have

supported my research, including the National Science Foundation, the National Nuclear

Security Administration Stewardship Science Academic Alliance program through the De-

partment of Energy, Michigan State University, and the National Superconducting Cyclotron

Laboratory.

v

TABLE OF CONTENTS

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

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

Chapter 1

Indirect Reactions for Nuclear Astrophysics

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
1.1
. . . . . . .
1.2 Transfer Reactions as a Probe of Astrophysical Proton Capture
1.3 Charge-Exchange Reactions as an Astrophysical Probe . . . . . . . . . . . .
1.3.1 Electron Capture in Supernovae . . . . . . . . . . . . . . . . . . . . .
1.3.2 Neutron Stars and the Nuclear Equation of State . . . . . . . . . . .
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Reaction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Optical Model Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Transfer Reaction Formalism and DWBA . . . . . . . . . . . . . . . . . . . .
2.3.1 Adiabatic Distorted Wave Approximation . . . . . . . . . . . . . . .
. . . . . . . . . . . .
2.3.2
2.4 Charge-Exchange Framework . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Two-Body Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Three-Body Formalism . . . . . . . . . . . . . . . . . . . . . . . . . .

Spectroscopic Factors and Resonance Strengths

Chapter 3 Calculations and Inputs . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Transfer Reactions with FRESCO . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Bound State Approximation . . . . . . . . . . . . . . . . . . . . . . .
3.2 Charge-Exchange Reactions with CHEX . . . . . . . . . . . . . . . . . . . .
3.2.1 Two-Body Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Three-Body Calculations . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Details of NN Potentials . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Selected Transfer Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Motivation for Studying 30P(p,γ)31S via 30P(d,n)31S . . . . . . . . .
4.1.2 Experimental and Theoretical Methods . . . . . . . . . . . . . . . . .
4.1.3 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Charge-Exchange Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Two-Body Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Three-Body Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Comparing Reaction Formalisms
. . . . . . . . . . . . . . . . . . . .

23Al(d,n)24Si

vi

x

1
1
2
4
5
8
12
15

17
17
20
23
26
27
30
31
34

42
42
43
47
49
50
51

56
56
56
57
58
61
65
67
75
87

4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Limitations of Current Work . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . .
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90
93

96
96
99

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
APPENDIX A Summary of Additional Transfer Results
. . . . . . . . . . . . . . 103
A.1 56Ni(d,n)57Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2 Using the ATTPC to Validate the Combined Method . . . . . . . . . . . . . 106
A.2.1 Asymptotic Normalization Coefficients and the Combined Method . . 106
A.2.2 Proposed Measurement and Preliminary Calculations . . . . . . . . . 110
APPENDIX B Derivation of Charge-Exchange Two-Body T-Matrix . . . . . . . . 113
APPENDIX C Derivation of Charge-Exchange Three-Body T-Matrix . . . . . . . 119
C.1 Central T-Matrix Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.2 Tensor T-Matrix Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
C.3 Spin-Spin T-matrix Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 151
C.4 Spin-Orbit T-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
APPENDIX D Benchmarking and Testing CHEX . . . . . . . . . . . . . . . . . . 177

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

vii

LIST OF TABLES

Table 3.1: Comparison of angle integrated transfer cross sections to states in 31S cal-
culated using the bound state approximation and the continuum binning
method. As the resonance energy increases, and the value of l decreases,
the approximation breaks down.
. . . . . . . . . . . . . . . . . . . . . . .

47

Table 4.1: Summary of results from [3] for the study of 30P(p,γ)31S via 30P(d,n)31S.
Transitions which were not observed in the experiment provide upper limits
for the experimental cross sections and spectroscopic factors. Theoretical
spectroscopic factors are shown for observed transitions and were produced
via the shell model using the USDA Hamiltonian [4] for positive parity
states and the WBP Hamiltonian [5] for negative parity states. Resonance
strengths are shown for states that will contribute significantly to the proton
capture reaction rate, noting that states with high l value will be suppressed. 59

Table 4.2: Summary of results from [6] for the study of 23Al(p,γ)24Si via 23Al(d,n)24Si.
Tentative spin allocations for states in 24Si are shown in parenthesis and a
final spin assignment could not be determined for the 3471 keV state. The
ground state cross section represents an upper limit. Theoretical spectro-
scopic factors were produced in the shell model framework using the USDB
interaction [4] and theoretical spectroscopic factors were determined using
the relation given in Equation (4.1).
. . . . . . . . . . . . . . . . . . . . .

Table 4.3: Percent difference evaluated at the first peak for cross sections produced
using different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.4: Numerical comparison of two-body charge-exchange angular distributions
to experimental data. The final column shows the percentage of the data,
including experimental errors, which falls between the theoretical curves
produced by two-body calculations with the KD and CH89 OMP. . . . . .

Table 4.5: Bound state parameters used in the three-body calculations with CHEX. .

Table 4.6: Numerical comparison of three-body charge-exchange angular distributions
to experimental data. The final column shows the percentage of the data,
including experimental errors, which falls between the theoretical curves
produced by three-body calculations with the AV8’ NN interaction and the
KD and CH89 OMP used for distorted waves. . . . . . . . . . . . . . . . .

63

73

74

76

84

viii

Table 4.7: Numerical comparison of three-body charge-exchange angular distributions
to experimental data. The final column shows the percentage of the data,
including experimental errors, which falls between the theoretical curves
produced by three-body calculations with the Gogny NN interaction and
. . . . . . . . . . . . .
the KD and CH89 OMP used for distorted waves.

Table 4.8: Original and refit parameters for the central component of the Gogny NN
interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

85

Table A.1: Summary of results from [7] for the study of 56Ni(p,γ)57Cu via 56Ni(d,n)57Cu
and the mirror reaction of 56Ni(d,p)57Ni. States which were not observed
in this study are listed with upper limits in the experimental cross section.
A large proton branching ratio in inferred for decay of the 7/2− resonance
at 1.835 MeV, so no experimental spectroscopic factor is listed for that
state here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Table D.1: Analytic result for the first few partial waves for the two-body charge-
exchange cross sections calculated using the T-matrix from Equations (2.42)
and (D.1). In the expressions above, the radial integrals are set to 1.

. . . 182

ix

LIST OF FIGURES

Figure 1.1: The chart of the nuclides with various nucleosynthetic processes shown
schematically. The rp-process is shown in red along the proton dripline,
the s-process is shown in pink close to the valley of stability, and the r-
process is shown in purple, reaching out towards the neutron drip line.
Figure from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.2: a) (n,p) charge-exchange reactions populate the same initial and final
states as electron-capture and β+ decay, although charge-exchange pro-
ceeds through the strong nuclear force and electron-capture and β decay
are mediated by the weak force. b) Charge-exchange is a versatile probe
of B(GT) because it is able to populated final states in the Z-1 isobar up
to high excitation energies, while β+ decay is limited to final states in
the Z-1 daughter with excitation energies smaller than the β+ Q-value.
Figure is from [9]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3: Constraints from different theoretical and experimental sources on the
symmetry energy at saturation density (aV
a ) and the slope of the symme-
try energy at the saturation density (L). The constraints from the isobaric
analog state (IAS) study by Danielewicz et al. are shown in yellow [10]
and predictions from neutron matter calculations within chiral effective
field theory (χEFT) in N 3LO are shown in brown [11]. Constraints from
observables include neutron skin (n-skins) thickness [12] shown in green,
neutron-star observations [13] shown in purple, nuclear masses [14] shown
in blue, and heavy-ion collisions [15] shown in pink. Figure is an adapta-
tion from [10], first presented in [9].
. . . . . . . . . . . . . . . . . . . .

Figure 2.1: Coordinates for A(a,b)B transfer reactions in the a) initial channel (prior
form) and b) final channel (post form) where a=b+v and B=A+v . . . .

Figure 2.2: Coordinates for A(p,n)B charge-exchange reactions where the residual nu-
cleus, B, is an isobar of the target, A. . . . . . . . . . . . . . . . . . . . .

Figure 2.3: Coordinates for A(p,n)B charge-exchange reactions within the three-body
reaction formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.1: Comparing the single-particle wave functions created using the bound
state approximation (red) and the exact calculation using continuum bin-
ning (black) for a proton resonance in 31S at Ex=6833 keV. The green
curve shows the difference between the two wave functions. . . . . . . . .

3

5

9

24

31

35

45

x

Figure 3.2: Flowchart showing the general organization of CHEX, the charge-exchange
reaction code created for this thesis. The primary observables calculated
by CHEX are charge-exchange cross sections.
. . . . . . . . . . . . . . .

Figure 3.3: Plot of the radial form of the Lane potential calculated using parameters
. . . . . . . .

from KD and CH89. The case shown is for 48Ca and L=0.

Figure 3.4: Plot of the radial form factors of the AV8’ and Gogny potential operators
. . . . . . . . . . . . . . . . . . . . . . . . . . .

considered in this work.

Figure 4.1: The proton capture reaction rate for various final states of 30P(p,γ)31S in
the astrophysical temperature range relevant for novae. Solid lines indicate
the use of spectroscopic factors extracted from this work. In cases where
a final state was not observed, the spectroscopic factor is provided by
theoretical shell model calculations. At low temperatures the 3/2− state
at 196 keV dominates the reaction rate, but at high temperatures, the
3/2+ state at 260 keV might become dominant, although this state was
not observed in [3]. Figure adapted from [3]. . . . . . . . . . . . . . . . .

Figure 4.2: Differential cross section for 23Al(d,n)24Si in the center of mass system us-
ing the LENDA detector (blue dots) compared with the sum of theoretical
distributions calculated using ADWA, weighted by experimental spectro-
scopic factors (solid pink). The error bands (dashed lines) are due to
the uncertainty in the experimental spectroscopic factors. Figure adapted
from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.3: The 1-σ uncertainty band of the reaction rate determined by [6] is shown
in grey, compared with the previous 1-σ uncertainty outlined in black.
The green dashed line shows the recommendations from the REACLIB
database which contain reaction rates to be used in astrophysical model
calculations. The blue line shows the REACLIB value decreased by a
factor of 30, which fell within previous uncertainties, but would be able to
remove the by pass of material from the 22Mg waiting point [16]. Figure
adapted from [6].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: Charge-exchange cross sections for the 14C(p,n)14NIAS transition at Elab =
25 (a), 35 (b), and 45 (c) MeV. The left column shows the results for cal-
culations using a two-body formalism with Lane potentials using OMP
parameters from KD (solid blue) and CH89 (dashed pink). The right col-
umn shows the results for calculations using a three-body formalism. The
solid/dashed blue lines utilized the AV8’ NN interaction and distorted
waves derived from KD/CH89. The solid/dashed pink lines utilized the
Gogny NN interaction and distorted waves derived from KD/CH89. In-
sets show the same results in a log scale. Experimental data from [17] is
shown in black.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

50

54

60

64

65

70

xi

Figure 4.5: Charge-exchange cross sections for the 48Ca(p,n)48ScIAS transition at
Elab = 25 (a), 35 (b), and 45 (c) MeV. The left column shows the results
for calculations using a two-body formalism with Lane potentials using
OMP parameters from KD (solid blue) and CH89 (dashed pink). The right
column shows the results for calculations using a three-body formalism.
The solid/dashed blue lines utilized the AV8’ NN interaction and distorted
waves derived from KD/CH89. The solid/dashed pink lines utilized the
Gogny NN interaction and distorted waves derived from KD/CH89. Insets
show the same results in a log scale. Experimental data from [18] is shown
in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.6: Charge-exchange cross sections for the 90Zr(p,n)90NbIAS transition at
Elab = 25 (a), 35 (b), and 45 (c) MeV. The left column shows the results
for calculations using a two-body formalism with Lane potentials using
OMP parameters from KD (solid blue) and CH89 (dashed pink). The right
column shows the results for calculations using a three-body formalism.
The solid/dashed blue lines utilized the AV8’ NN interaction and distorted
waves derived from KD/CH89. The solid/dashed pink lines utilized the
Gogny NN interaction and distorted waves derived from KD/CH89. Insets
show the same results in a log scale. Experimental data from [18]is shown
in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.7: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS,
and 90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calcu-
lated using the AV8’ potential. The contribution to the cross section from
various potential operators are shown in dashed black for central, dotted
green for spin-spin, dash-dotted red for tensor, and solid blue for the total
cross section. Calculations shown here use the KD OMP for to calculate
distorted waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: Charge-exchange cross sections for the 58Fe(p,n)58CoIAS transitions at
Elab = 23 MeV calculated using the AV8’ potential. The contribution
to the cross section from various potential operators are shown in dashed
black for central, dotted green for spin-spin, dash-dotted red for tensor,
and solid blue for the total cross section. Calculations shown here use the
KD OMP for to calculate distorted waves.
. . . . . . . . . . . . . . . . .

Figure 4.9: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS,
and 90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calcu-
lated using the Gogny potential. The contribution to the cross section
from various potential operators are shown in dashed black for central,
dotted green for spin-spin, and solid blue for the total cross section. Cal-
culations shown here use the KD OMP for to calculate distorted waves. .

71

72

78

79

81

xii

Figure 4.10: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS,
and 90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calculated
using a refit Gogny potential with only a central contribution (back). The
red curve shows the original calculation using the isospin central Gogny
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
interaction.

Figure 4.11: Angle integrated charge-exchange cross sections and a function of scatter-
ing angular momentum for 90Zr(p,n)90Nb at 25 MeV for the three-body
formalism with the Gogny and AV8’ interactions, and the two-body for-
malism with the Lane potential parameterized by KD and CH89.
. . . .

Figure 4.12: Dependence of radial integral on the range of various radial parameters.
. . . .

The x-axis shows the internal radial cut in each radial parameter.

86

88

89

Figure A.1: The 1-σ uncertainty band of the reaction rate determined in [7] is shown
in red, compared with the previous calculations from refs.
[19, 20, 21].
Results match previous results from Rehm et al. at lower temperatures,
but begin to diverge at higher temperatures because of the constraints
imposed on higher lying states by this work. Figure from [7]. . . . . . . . 106

Figure A.2: Single-particle wave functions ψsp(rAN ), produced by changing the radius
of the Woods-Saxon potential from r0 = 1.2 to r0 = 1.3. These values
are both within a reasonable range for the single-particle geometry but
create significant differences in the wave functions which, in turn, directly
impact the cross section. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure A.3: Left: A cartoon of the spectroscopic factor as a function of single-particle
ANC, blj, at two different beam energies. Right: A cartoon of the many-
body ANC, Clj, as a function of single-particle ANC blj, at two different
beam energies.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure A.4: Left: Preliminary calculations showing the many-body ANC, Clj, as a
function of single-particle ANC blj, for the ground state of 87Kr. Left:
Preliminary calculations showing the many-body ANC, Clj, as a function
of single-particle ANC blj, for the 3/2+ excited state of 87Kr at 1.47
MeV excitation energy. The orange points correspond to calculations at 5
MeV/u and the blue points correspond to calculations at 8 MeV/u. These
energies represent a conservative energy range attainable with the ATTPC.112

Figure C.1: Dependence of radial integral to the range of various radial parameters
for the case of 48Ca(p,n)48Sc. The x-axis shows the internal radial cut in
each parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

xiii

Figure D.1: Real and Imaginary parts of the scattering wave function as a function of
scattering radius for the L=0, J=0.5 partial wave for 48Ca(p, p)48Ca at Ep
= 30.0 MeV. The solid black (blue) lines show the real (imaginary) part
of scattering wave function calculated in CHEX, the dashed red (green)
lines show the same result produced by FRESCO. . . . . . . . . . . . . . 178

Figure D.2: Differential elastic scattering relative to Rutherford as a function of scat-
tering angle for 48Ca(p, p)48Ca at Ep = 30.0 MeV. The solid black line
shows the elastic scattering cross section calculated in CHEX, the dashed
red line shows the same result produced by FRESCO. . . . . . . . . . . . 179

Figure D.3: Bound state wave functions as a function of radius. The left plot shows
n+47Ca bound state wave function and the right plot shows the p+47Ca
bound state wave function, which correlate to the in the initial and fi-
nal states of the 48Ca(p,n)48Sc reaction. The solid black line shows re-
sults from CHEX, the dashed red line shows the same result produced by
FRESCO.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure D.4: Comparing the charge-exchange cross sections for 48Ca(p,n)48Sc at 35
MeV, produced by [10] and by CHEX. (a): there is a small difference
resulting from the way spin-orbit was included. (b): once this spin-orbit
contribution is neglected, the two calculations match exactly.

. . . . . . 183

xiv

Chapter 1

Introduction

1.1 Indirect Reactions for Nuclear Astrophysics

Nuclear astrophysics is a unique field which links the fundamental interactions of protons and

neutrons across length and energy scales with the evolution of elements in stars throughout

the universe. Although the field has been very successful in creating a general picture of

how heavy elements have built up since the time of the big bang, there remain many open

lines of investigation, including understanding the complex dynamics of supernovae, verify-

ing astrophysical sites for different nucleosynthetic processes, and gaining an understanding

of the most exotic nuclear matter through the study of neutron stars and their mergers.

Exploring all of these research areas requires large amounts of nuclear physics data, from

reaction rates to masses, and requires extensive reaction networks, models, and simulations

to link astrophysical observations with underlying physics.

This process is further complicated by the fact that many nucleosynthetic processes of

interest involve incredibly short lived nuclei and reactions at sub-Coulomb energies, making

them difficult, or even impossible, to recreate in the lab. Without the ability to directly

measure these nuclear properties, the field must rely on indirect reaction methods which

measure a different, more accessible, reaction process that can probe some aspect of the

reaction of astrophysical interest. Indirect reaction methods require reaction and structure

1

theory to link measured quantities to the reaction rates required for astrophysical calculations

and simulations. Because of this, the extracted value of astrophysical interest is only as good

as the theory tools used to link it to the experimentally measured observable. In this thesis

we will discuss how charge-exchange has been used as an indirect probe for electron capture

and the nuclear equation of state, and how transfer reactions can be used to probe proton

capture reactions. In both cases, we will be focusing on the reaction theory used to describe

these processes.

1.2 Transfer Reactions as a Probe of Astrophysical Pro-

ton Capture

Single-particle capture reactions are ubiquitous in a wide range of astrophysical environ-

ments. Many of these processes are illustrated in Figure 1.1. Along the proton dripline,

nucleosynthesis of light to medium mass nuclei can proceed through the rapid proton cap-

ture process (rp-process) in environments such as Oxygen Neon (ONe) novae explosions [22]

and x-ray bursters [23]. In these hydrogen rich environments, the reaction network consists

of proton captures followed by β+ decays back towards stability.

On the neutron rich side of the chart of the nuclei, the slow neutron capture process

(s-process) builds up elements heavier than lead in Asymptotic Giant Branch (AGB) and

red giant stars through a chain of neutron captures, followed by beta decay back towards

stability [24]. In these environments, the neutron density is low enough that beta decay rates

can compete with the neutron capture rate, creating a reaction chain that clings closely to

the valley of stability and stalls out with the production of lead and bismuth.

The recent observation of a neutron-star merger by LIGO and subsequent multi-messenger

2

Figure 1.1: The chart of the nuclides with various nucleosynthetic processes shown schemati-
cally. The rp-process is shown in red along the proton dripline, the s-process is shown in pink
close to the valley of stability, and the r-process is shown in purple, reaching out towards the
neutron drip line. Figure from [8].

observations of the remnant kilonova have confirmed neutrons stars as a site of the rapid

neutron capture process (r-process) [25].

In this scenario, there is a very high density of

excess neutrons, allowing for many subsequent neutron captures, resulting in highly exotic

nuclei which eventually beta decay back towards stability, forming the heaviest elements.

These astrophysical capture processes involve thousands of reaction and decay rates,

many of which are essential to understanding the observed isotopic abundances. However, in

all but a few cases, direct measurement in the laboratory is impossible. Astrophysical proton

capture reactions proceed below the Coulomb barrier, leading to plummeting cross sections.

As an additional complication, many of these captures are on unstable nuclei, limiting the

3

experiments to relatively low beam rates accessible through current rare isotope beams. The

situation is even more dire when studying neutron capture onto rare r-process isotopes. In

this scenario, both the target isotope and the neutron are unstable, meaning that neither can

be made into a target for an experiment. Instead, these processes must be explored through

an indirect probe and, in both of these cases, (p,γ) and (n,γ), transfer reactions provide a

helpful tool.

A(d,p)B and C(d,n)D transfer reactions have the same initial and final state as their

corresponding A(n,γ)B and C(p,γ)D astrophysical reactions, but are experimentally much

more feasible. For reactions involving an unstable target, the experiment runs in inverse

kinematics, with a deuterated target and a rare isotope beam. Unlike direct proton capture

experiments, these experiments can run well above the Coulomb barrier (> 30 MeV/u) and

still occupy the low lying final state resonances relevant to nuclear astrophysics. Once transfer

to these final states is measured, a variety of theoretical techniques can be used to connect

the transfer data to the astrophysical capture process of interest, including extraction of

the asymptotic normalization coefficient (ANC) or spectroscopic factor (S), which will be

discussed in greater detail throughout this work.

1.3 Charge-Exchange Reactions as an Astrophysical Probe

Charge-exchange reactions are isobaric transitions where a neutron in the target is exchanged

with a proton in the projectile, or vice-versa. These reactions can be performed using single-

nucleon probes, such as (n,p) or (p,n) reactions but, experimentally, it is often advantageous

to use composite probes such as (t,3He) or (d,2He), or even heavy-ion probes such as (12C,

12N) and (7Li, 7Be). These reactions are mediated by the strong nuclear force via meson-

4

exchange but populate the same initial and final states as processes mediated by the weak

force and, therefore, can be used as a probe in regions where β decay or beta delayed neutron

emission (β-n) data is unavailable or energetically forbidden (see Figure 1.2). In general,

charge-exchange reactions provide insight into two aspects of nuclear astrophysics: they

serve as an indirect probe for stellar electron-capture processes and as a tool for exploring

bulk properties of nuclear matter, such as the nuclear equation of state, which is central to

understanding neutron stars and their mergers. First, we will introduce charge-exchange in

the context of electron-capture.

1.3.1 Electron Capture in Supernovae

Figure 1.2: a) (n,p) charge-exchange reactions populate the same initial and final states
as electron-capture and β+ decay, although charge-exchange proceeds through the strong
nuclear force and electron-capture and β decay are mediated by the weak force. b) Charge-
exchange is a versatile probe of B(GT) because it is able to populated final states in the
Z-1 isobar up to high excitation energies, while β+ decay is limited to final states in the Z-1
daughter with excitation energies smaller than the β+ Q-value. Figure is from [9]

Supernovae are an important site for nucleosynthesis and produce significant amounts of

elements heavier than Iron. In both core-collapse and type Ia supernovae, electron-capture
reactions on nuclei in the pf shell (Z ∼ 21-40), neutronize the nuclear material, affecting

the dynamics of the nuclear explosion [26]. Understanding these electron-capture reactions

are a key component for interpreting the observed isotopic abundances produced in these

5

stellar explosions. In most cases, relevant electron-capture rates cannot be measured directly,

but can be estimated with knowledge of the Gamow-Teller transition strengths in the β+

direction. Gamow-Teller transitions are mediated by the στ operator and change the total

spin (S) and isospin (T) of the nucleus, but not the orbital angular momentum (L) (∆L=0,

∆S=1, ∆T=1). Charge-exchange reactions have become an important tool to probe Gamow-

Teller transition strengths (B(GT)) because they can be used to excite transitions that are

energetically blocked to β decay (see Figure 1.2).

Extracting B(GT) from charge-exchange reaction cross sections relies upon an approx-

imate proportionality relation between these two quantities, first established by [27] and

expressed as

(cid:105)

(q = 0)

(cid:104) dσ

dΩ

= ˆσB(ST )

ST

(1.1)

where dσ

dΩ is the charge-exchange cross section, ST specifies the transition of interest (Gamow-

Teller, Fermi, etc), and q is the momentum transfer. A key ingredient is the unit cross section

(ˆσ) which can be determined by direct comparison to β decay data, when available, or by

extraction from a well defined correlation between mass number and unit cross section (e.g.

[28, 29, 30]). This proportionality in Eq. 1.1 is only valid at energies where a single-step
process can be assumed. These studies are commonly performed at ∼ 100 MeV/u, where

the single-step approximation is warranted, although its region of validity could extend down
to ∼ 50 MeV. At lower energies, multistep reactions become increasingly important.

It

has also been extended to, and experimentally verified for, a wide range of charge-exchange

probes including (p,n) [28], (n,p) [31], (d,2He) [32, 33], (t,3He)/(3He,t) [34, 30], and (7Li,7Be)

[35, 36].

6

Undoubtedly, there has been great success using charge-exchange reactions to constrain

astrophysical models. Although core-collapse supernovae models incorporate a large network

of electron-capture reactions, sensitivity studies such as [37] guide experimental efforts by

determining which reactions have the greatest impact on observables such as peak neutrino

luminosity. The study in Sullivan et al.

[37] highlighted the ”high-sensitivity region” near

the N=50 shell gap closure. Guided by this and other sensitivity studies [38, 39], B(GT)

was extracted from the 86Kr(t,3He)86Br charge-exchange reaction and introduced in the

calculation of stellar electron-capture rates [40]. The extracted electron-capture rates were

significantly smaller than those often derived from a simple single-state approximation of-

ten used in regions without experimental results or high quality structure inputs. When

input into core-collapse supernovae simulations, this difference leads to a reduction in the

deleptonization in the core-collapse, which has effects on observables such as peak neutrino

luminosity and the frequency of gravitational waves emitted from the collapsing star [41].

Both of these are potentially important signals for understanding core-collapse supernovae

as we move into the multimessanger era of astronomy.

In this work, we are only considering transitions between 0+ isobaric analog states and,

therefore, do not explore these Gamow-Teller transitions. Still, Fermi transitions can be im-

portant for electron capture in supernovae. High stellar temperatures allow for the population

of excited states and Fermi transitions from these states become possible [42]. Extraction

of electron capture rates are one of the most important applications of charge-exchange cal-

culations and future extension of the framework presented in this thesis should be made to

additionally include Gamow-Teller transitions.

7

1.3.2 Neutron Stars and the Nuclear Equation of State

Charge-exchange reactions are also a versatile tool for exploring several aspects of bulk nu-

clear matter. These constraints are vital to modeling neutron stars and their mergers, which

were recently confirmed as a central site for the production of r-process elements. One way

charge-exchange reactions constrain bulk nuclear matter is by placing limits on the nuclear

symmetry energy [43]. The symmetry energy encompasses the energy penalty for an im-

balance of neutrons and protons within nuclear matter and is directly linked to the nuclear

equation of state, a key component for modeling the behavior of neutrons stars. Understand-

ing the evolution of the symmetry energy with changes to the neutron-proton asymmetry is

essential for extrapolating from experimental observations of nuclei, with relatively low levels

of asymmetry, to the extreme of asymmetric nuclear matter in neutron stars. In uniform

nuclear matter, with neutron density ρn, proton density ρp, and total density ρ = ρn + ρp,

the energy per nucleon can be expressed as

(cid:16)ρn − ρp

(cid:17)2

ρ

.

E
A

=

E0
A

(ρ) + S(ρ)

(1.2)

E0 represents the energy of symmetric nuclear matter and S(ρ) is the density dependent

symmetry energy. S(ρ) can then be expanded around nuclear saturation density, ρ0 as

S(ρ) = aV

a +

ρ − ρ0
ρ0

L
3

+ ....

(1.3)

where aV

a is the symmetry energy at normal nuclear density and L is the slope of the symme-

try energy. These values directly impact quantities, such as the pressure of nuclear matter,

which inform the nuclear equation of state [44].

8

Knowledge of the neutron skin thickness, defined as the difference between the root mean

square radii of proton and neutron distributions inside nuclei, constrains the symmetry en-

ergy [45, 46]. Because of this, precise measurements of the neutron skin thickness have

become a goal for many types of reaction probes. However, while neutron stars contain a

vast imbalance of excess neutrons, ordinary nuclear matter, even the unstable nuclei acces-

sible with rare isotope beams, have relatively small proton-neutron asymmetry. This small

asymmetry shrinks the neutron skin thickness, making its precise determination difficult.

Charge-exchange reactions allow access points to this difference of proton and neutron den-

sities, referred to as the isovector density.

Figure 1.3: Constraints from different theoretical and experimental sources on the symmetry
energy at saturation density (aV
a ) and the slope of the symmetry energy at the saturation
density (L). The constraints from the isobaric analog state (IAS) study by Danielewicz et
al. are shown in yellow [10] and predictions from neutron matter calculations within chiral
effective field theory (χEFT) in N 3LO are shown in brown [11]. Constraints from observables
include neutron skin (n-skins) thickness [12] shown in green, neutron-star observations [13]
shown in purple, nuclear masses [14] shown in blue, and heavy-ion collisions [15] shown in
pink. Figure is an adaptation from [10], first presented in [9].

Fermi transitions (∆L=0, ∆S=0, ∆T=1) between isobaric analog states (IAS) provide a

9

unique tool for exploring isovector densities. In (p,n)-type reactions, the IAS maintains the

same structure as the target nucleus, except replacing one neutron with a proton. Isospin

symmetry holds that the excitation energy of the IAS will be approximately equal to the

Coulomb energy of the incoming proton. This energy matching means that Fermi transitions

to the IAS are often considered to be ”elastic” in nature, except that the isospin projection

of the projectile is flipped by the isovector term of the interaction potential, transforming a

proton to a neutron or vice-versa. There have been several theoretical efforts, informed by

measurements of IAS reactions, to explore the isovector properties of nuclei.

These isospin flipping transitions to the IAS can be described using the Lane optical

potential [47]

U (r) = U0(r) +

τ T
4A

U1(r),

(1.4)

where τ is the isospin operator which acts on the scattering particle and T is the isospin

operator which acts on the target nucleus. Additionally,

U1(r) ∝ Un(r) − Up(r)

(1.5)

is the isovector term which drives the IAS transition. Phenomenological potentials fit to

proton/neutron elastic scattering data on a wide variety of targets and scattering energies

are often used for Up/n(r). They take the form of Woods-Saxon potentials with real and

imaginary terms, as well as terms to describe absorption at the surface of the target and

a spin-orbit interaction. Adjusting the radius and diffuseness of these potentials will affect

the shape of both the elastic and charge-exchange reaction cross sections they produce.

Recent work by Danielewicz et al.

[10] allowed the radius and diffuseness parameters for

10

a particular parameterization [1] of Up/n(r) to vary in order to simultaneously fit data for

proton elastic scattering, neutron elastic scattering, and charge-exchange to the IAS. The

modified potential parameters from this procedure are then compared to the values fit only

to elastic scattering data. These values were then related to various aspects of the symmetry

energy, notably, the symmetry energy at normal nuclear density (aV

a ) and the slope of the

symmetry energy as a function of density, evaluated at saturation density (L). Figure 1.3

demonstrates the constraints the work by Danielewicz et al. [10] put on these properties of the

symmetry energy (labeled IAS) as well as other theoretical and experimental efforts, including

calculations using ab-initio nuclear interactions [11], measurements of nuclear masses [14],

heavy-ion collisions [15], neutron skin thicknesses [12], and neutron star observations [13].

Similar theoretical efforts use a more microscopic approach. In this vein, charge-exchange

transitions to the IAS are studied within the folding model where effective NN interactions

are integrated over the proton and neutron densities of target nuclei. In the case of (3He,t)

reactions, this potential is also folded over the projectile nucleus in the so-called double-

folding model [48, 49, 50, 51, 52]. These calculations require neutron and proton densities

for the target which can be taken from experiment or calculated using a realistic nuclear

interaction. The radius parameter of the proton and neutron densities can then be adjusted

to best produce IAS charge-exchange data. From these adjusted potentials, a neutron skin

thickness can be extracted [51].

In particular, (3He,t) reactions are of interest because some argue that the spatial overlap

between the probe and target could create nuclear densities close to or above the nuclear

saturation density, allowing a unique probe of nuclear symmetry energy. By varying the

sensitivity of the effective interaction to isovector density and comparing these results with

observational constraints from x-ray bursters, Ref.

[50] concluded that equations of state

11

with a ”soft” density dependence on isovector density are unrealistic and data favored a

stiffer equation of state.

One challenge in this field is that current IAS data comes from measurements on stable

targets. Then, results based on these measurements must be extrapolated to the limits

of nuclear asymmetry inside neutron stars. In order to more effectively probe the nuclear

symmetry energy, there is a need for high quality measurements of IAS transitions on neutron

rich nuclei, such as those that will be produced by the Facility for Rare Isotopes Beams. It

is notable, however, that such measurements will create new experimental challenges, as

these reactions must be run in inverse kinematics. The IAS has been measured successfully

measured in unstable, neutron rich isotopes, such as 16C [53]. Because neutrons are also

unstable, (n,p) type reactions will require the use of composite probes such as (d,2He),

although this is limited to measuring Gamow-Teller transitions.

1.4 Motivation

Although there have been significant efforts to improve precision and validate new charge-

exchange probes in the experimental regime, there are many opportunities to explore ad-

ditional complications in the reaction mechanics. For example, almost exclusively, charge-

exchange calculations are performed assuming a single-step process in the framework of the

distorted wave Born or impulse approximation (see Section 2.3), regardless of the beam
energy of the reaction. When comparing to experiments at intermediate energies (∼100

MeV/u), a single-step process is likely a good approximation. This is evidenced by a rel-

atively good description of the g shape (although, for the case of composite probes, not

necessarily the magnitude) of experimental angular distributions. However, charge-exchange

12

reaction experiments which investigate the population of the have been performed at ∼25-50

Mev/u (see [17, 18]). While DWBA is within its region of validity for the electron-capture
studies run at ∼100 Mev/u, it will become increasingly less accurate with decreased beam en-

ergy and dedicated study is necessary to understand where this approximation breaks down.

As with all indirect reaction methods, an incomplete understanding of charge-exchange re-

action theory will impact the quality of extractions made for astrophysical applications.

Investigations of reaction dynamics in charge-exchange has not yet implemented recent

developments in the realm of other reactions such as the incorporation of non-local interac-

tions (see [54, 55, 56, 57, 58]). In this work we hope to begin to broaden our understanding

of charge-exchanges reactions, including the impact of using two-body versus three-body

reaction formalisms, and the effect of nuclear interactions on charge-exchange observables.

There are several opportunities to explore the impact of interactions which mediate

isospin transitions in charge-exchange. Most commonly, for experimental charge-exchange

studies, the effective nucleon-nucleon (NN) interaction parameterized by Love and Franey

[59, 60] is used to describe the isospin transition. This phenomenological potential has an

operator form, and uses a sum of real and imaginary Yukawa potentials. Of course, the

choice of the effective NN interaction will directly affect the shape and magnitude of the

calculated cross section. The Love and Franey potential is best constrained above about

100 MeV. It would be informative to explore the effects of different effective interactions,

including microscopic and non-local potentials. In particular, in this initial study we will

be looking at reactions in the 25-45 MeV energy range, where LF interaction has not been

parameterized. In this thesis, we take this as an opportunity to investigate the effect of other

NN interactions on charge-exchange observables. Having this goal in mind, the reaction code

developed here is not restricted to interactions with Yukawa form factors. In future reac-

13

tion studies, however, it would be informative to implement the LF interaction for ease of

comparison with calculations using different formalisms. Although studying the effects of

non-local potentials is outside of the scope of this thesis, the code developed for this work,

CHEX, can readily be extended to include non-local interactions.

In many IAS studies which have been used to study properties of nuclear matter, optical

model potentials are used to construct, not only the incoming and outgoing scattering states,

but also the Lane potential which describes the transition. It has been shown in the realm

of transfer reactions that the phenomenological optical model potentials used to produce

initial and final distorted wave functions are not very well constrained and produce large

uncertainties [61]. In this work we begin to investigate the uncertainty derived from these

phenomenological interactions, although similar, more rigorous, studies are needed in the

realm of charge-exchange before uncertainties can be properly quantified.

Finally, we begin to explore the ways in which the reaction formalism effects charge-

exchange observables. The two-body formalism which utilizes Lane potentials offers a simple

way to explore charge-exchange transitions, but a three-body formalism can better describe

microscopic transitions between single-particle states in the target. Instead of using a dif-

ference of proton and neutron optical potentials that adjusts the isospin projection of the

target, we employ an NN potential that operates directly on the valence nucleon. In this

work, we present a systematic study, over a variety of target nuclei and beam energies, to

highlight the impact of various reaction formalisms in charge-exchange.

14

1.5 Outline

This thesis is organized in the following way: In Chapter 2, we begin by presenting the

necessary reaction theory to build up to our description of transfer and charge-exchange

reactions, including an introduction to optical model potentials, which are used throughout

this work, and the fundamentals of elastic scattering theory. After this initial discussion,

this thesis proceeds on a dual track, with the first part of each chapter discussing transfer

reactions and the second half discussing how the topic of that chapter pertains to charge-

exchange reactions. For example, after the general introductions, Chapter 2 goes on to

introduce the reactions framework used for our transfer reaction studies, and the theoretical

tools necessary to connect these reactions to astrophysical capture processes. Then, in the

final section of Chapter 2, we introduce the reactions frameworks used to study charge-

exchange in this thesis.

Chapter 3 discusses details of implementing the theoretical methods discussed in the

previous chapter in reaction calculations. First, in Section 3.1, we discuss transfer calcu-

lations, including the bound state approximation which we have used to calculate transfer

to unbound resonances. Then, in Section 3.2, we discuss the details of the charge-exchange

reactions code, CHEX, developed for this thesis and the implementation of our two-body and

three-body formalisms in that code, including a discussion of the various potentials explored

in this work.

In Chapter 4 we present results for transfer and charge-exchange reactions. This thesis

contains three separate transfer reaction studies, all utilizing a similar methodology. Section

4.1 highlights one of these transfer studies, the case of 30P(d,n), in detail and discusses

how this methodology was expanded in a study of 23Al(d,n)24Si. The remainder of the

15

transfer reaction results can be found in Appendix A, including a preliminary study for a

proposed new method for constraining spectroscopic factors in transfer measurements using

the Active Target Time Projection Chamber (AT-TPC). Section 4.2 discusses the results

of our charge-exchange reaction study, for both the two-body and three-body frameworks

presented here. Finally, Chapter 5 presents conclusions and outlooks for both the transfer

and charge-exchange studies discussed in this thesis.

Some of the more technical developments made for this thesis have been included here

in the appendices. In addition to the aforementioned transfer results in Appendix A, Ap-

pendix B contains the full derivation of the two-body charge-exchange T-matrix presented

in this work. Similarly, Appendix C contains T-matrix derivations for each of the operators

considered in the three-body charge-exchange framework:

isospin central, isospin tensor,

isospin spin-spin, and isospin spin-orbit. Finally, Appendix D presents the tests performed

to benchmark and check the charge-exchange code, CHEX, developed for this thesis.

16

Chapter 2

Reaction Theory

Regardless of which astrophysical process we are probing, reliable results can only be ex-

tracted if there is an adequate and valid reaction theory used in the extraction.

In this

chapter we will build up to more complicated reaction descriptions by, first, introducing the

basics of scattering theory, starting with the introduction of optical potentials and elastic

scattering, and then expand on these topics to describe the transfer and charge-exchange

formalism used in this thesis.

2.1 Optical Model Potentials

The effective nuclear interaction between a projectile nucleon and an A-body target results

from a complicated, many-body problem and, although there are current efforts to develop

scattering potentials from a many-body framework (e.g.

[62]), it is not currently feasible

to describe the full dynamics of complex reactions using an ab-initio framework. Therefore,

throughout this work, in both the realms of transfer and charge-exchange reactions, inter-

actions between the projectile/ejectile and the target/residual nucleus are described using

optical model potentials (OMP). These are phenomenological potentials fit to reproduce elas-

tic scattering data. They have real components, as well as imaginary terms which account

for flux loss to non-elastic processes. The optical model effectively imposes a two-body ap-

proximation, freezing the A-body target into an inert object in its ground state. This implies

17

that many-body effects such as NN-correlations and anti-symmeterization are incorporated

into the phenomenological fit via a strong mass and beam energy dependence [63, 64, 65].

When data is available, these fits can be developed for a particular projectile/target pair

at the energy of interest. However, for many cases, this data is unavailable and global optical

potentials are used instead. Global potentials are created by simultaneously fitting elastic

scattering for a wide range of targets over a large energy range. These fits produce potentials

which vary smoothly over mass and energy, allowing for interpolation or extrapolation to

targets and energies without dedicated fits (Examples include [66, 1, 2]). Although these

potentials are able to capture general trends over different mass and energy regions, it is

worth noting that they are fit to stable nuclei and their validity for unstable nuclei has not

been demonstrated. Despite this, they are currently used to describe reactions on unstable

targets, because there are no better alternatives available.

Typically optical potentials follow a common form which can be expressed as

U (R) = V (R) + i(W (R) + Ws(R)) + VSO(R) + VC (R),

(2.1)

where V (R)/W (R) are real/imaginary central terms, Ws(R) is the imaginary surface term,

VSO(R) is the real spin-orbit term, and VC (R) is the Coulomb term. Most terms are param-

eterized by a Woods-Saxon potential or its derivative. The central real and imaginary terms

have the form

V (R) = −

Vr

1 + e(R−Rr)/ar

,

(2.2)

where Vr (Wr for imaginary central) is the depth parameter which specifies the strength of

the potential, Rr is the radius term defined by Rr = rrA1/3, and ar is the diffuseness.

18

The surface term has only an imaginary contribution and accounts for reactions that

occur at the surface of the target (usually strongest at low energies). It takes the form of

the derivative of a Woods-Saxson such that

Ws(R) = −4as

d
dR

Ws

1 + e(R−Rs)/as

,

with Rs = rsA1/3.

The spin-orbit term is typically real and also has a derivative form expressed as

(cid:16) ¯h

(cid:17)2 2L · s

mπc

R

VSO(R) =

d
dR

VSO

exp[(R − RSO)/aSO]

,

(2.3)

(2.4)

where mπ is the pion mass [67]. Finally, for potentials where both the projectile and target

are charged, there is an additional Coulomb term. Outside of the Coulomb radius defined

as RC = rcA1/3, a simple point-Coulomb potential is valid, defined by

VC (R) =

ZpZte2

R

,

(2.5)

where Zp is the projectile charge and Zt is the target charge. Inside Rc, the point approx-

imation is no longer valid and a homogeneous charge distribution is taken for the target,

resulting in:

VC (R) =

ZpZte2

RC

(cid:16)3

2

(cid:17)

.

− R2
2R2
C

(2.6)

Here, we will simply note that optical potential fits for each target and energy contain
a large number of fit parameters (∼ 12) and these fits are not unique, even when these

parameters have limited ranges of applicability. This introduces large uncertainties that

19

propagate through to the observables calculated with these potentials as inputs (e.g. transfer

cross-sections). Recent efforts to quantify this uncertainty using Bayesian methods have

found that the uncertainty is larger than previously assumed, at times larger that 100%

[68]. This insight is important to note when comparing calculations using these potentials

to experimental results.

2.2 Elastic Scattering

We will begin with a description of two-body, elastic scattering of projectile, p, with target,

t, under the influence of a spherically symmetric potential. The reduced mass of the system

is µ and the center of mass scattering energy is E. The dynamics of the system are dependent

on the wave function that describes the scattering particle, so we must therefore solve the

Schr¨odinger equation

[ ˆT + U (R) − E]ψ(R, θ) = 0,

(2.7)

where U (R) is the nuclear potential, and T is the kinetic energy operator. In general, we

can fix the incoming particle momentum along the z-axis. In this picture, the incoming wave

function is modeled as a plane wave, characterized by its momentum. After interacting with

U (R), the outgoing wave function is a superposition of the incoming plane wave and an

outgoing spherical wave according to the relation

ψasym(R, θ) = eikz + f (θ)

eikR
R

,

(2.8)

where f (θ) is the scattering amplitude which characterizes the strength of the interaction

20

where, for example, U (r) = 0 would lead to a scattering amplitude f (θ)=0. When both

short-range nuclear interactions and Coulomb potentials are present, the total scattering

amplitude is equal to the sum of both the Coulomb scattering amplitude, fc(θ) and the

nuclear scattering amplitude, under the influence of Coulomb, f

(c)
n (θ). Because the Coulomb

interaction is long range, but the nuclear potential goes to 0 at far distances, we use different

methods to calculate these scattering amplitudes. The effect of the nuclear interaction can

be found by expanding in Legendre polynomials, PL(cos θ), which is known as a partial wave

decomposition. The expansion takes the form

∞(cid:88)

ψ(R, θ) =

(2L + 1)iLPL(cos θ)

L=0

1
kR

χL(R),

(2.9)

where k is the projectile momentum. The Legendre polynomials (PL) are a convenient

choice for this expansion because they are eigenfunctions of both the ˆL2 and ˆLz operators

with eigenvalues L(L + 1) and mL = 0 for the case of a central potential, respectively. Thus,

we are able to solve the Schr¨odinger equation for each partial wave separately. This means

that equation 2.7 can be decomposed to give

(cid:104) − ¯h2

2µ

(cid:16) d2
dR2 − L(L + 1)

R2

(cid:17)

(cid:105)

+ U (R) − E

χL(R) = 0.

(2.10)

Solving this expression is then straightforward. Inside the range of the interaction, Equa-

tion (2.10) can be solved using a trial wave function uL(R) according to a chosen numerical

method utilizing the constraints that the wave function must go to 0 at R = 0 and have a

non-zero derivative at R = 0. Once this solution, uL(R), is found, it can be matched to the

the behavior at large distances (R>a), outside of the range of the nuclear potential. The

true wave function is χL(R) = BLuL(R), where we impose a normalization consistent with

21

the plane wave for the non-interaction process. Outside the range of the nuclear force, there

are still effects from the long ranged Coulomb interaction, so the external form of the wave

function, as given in [67], is

χL(R > a) =

i
2

[H−

L (η, kR) − SLH+

L (η, kR)].

(2.11)

In this expression, H±

L (η, kR) are the Hankel functions as defined in [67], η is the dimen-

sionless Sommerfeld parameter defined as

(cid:16) µ

(cid:17)1/2

ZpZte2

¯h

2E

,

η =

(2.12)

and SL is the S-matrix for each individual partial wave. The S-matrix can be calculated

through the use of the R-matrix which matches the logarithmic derivatives of the true and

trial wave functions at the matching radius a such that

RL =

1
a

χL(a)
χ(cid:48)
L(a)

=

1
a

uL(a)
u(cid:48)
L(a)

(2.13)

and

SL =

H−
L − aRLH(cid:48)−
L − aRLH(cid:48)+
H+

L

L

.

(2.14)

Finally, the nuclear scattering amplitude (under the influence of a Coulomb potential)

can be written as

fn(θ) =

1
2ik

∞(cid:88)

(2L + 1)PL(cos θ)e2iσL(η)(SL − 1),

(2.15)

L=0

22

where σL(η) is the Coulomb phase shift defined as

σL(η) = argΓ(1 + L + iη).

(2.16)

the point Coulomb scattering amplitude is given in [67] as

fc(θ) =

−η

2ksin2(θ/2)

exp[−iηln(sin2(θ/2)) + 2iσ0(η)]

(2.17)

and the total differential cross section is given by

= |fc(θ) + fn(θ)|2.

dσ
dΩ

(2.18)

2.3 Transfer Reaction Formalism and DWBA

For many reactions, a two-body description is insufficient to describe the complexities of the

reaction process. This is particularly true in cases where mass partitions are rearranged,

such as in (d,N) transfer reactions. For both transfer and charge-exchange reactions, we will

therefore employ a three-body formalism which freezes an inert core within the target, but

allows for the rearrangement of the other two particles during the reaction. We will begin by

introducing the three-body formalism used to describe transfer reactions, such as A(d,n)B,

where a proton is transferred from the scattering deuteron to the inert target. For these

reactions, it is useful to use the T-matrix formalism, which is based on integral relations, as

opposed to the S-matrix used in the previous section, which is based on asymptotic matching.

The T-matrix can be described equivalently using either the initial (prior form) or final

(post form) coordinates of the system (see Figure 2.1) which correspond to the mass partition

23

rearrangement during this process.

Figure 2.1: Coordinates for A(a,b)B transfer reactions in the a) initial channel (prior form)
and b) final channel (post form) where a=b+v and B=A+v

Using these coordinates, the prior T-matrix can be expressed as [67]

prior = (cid:104)Ψ(−)exact(rvA, Rf )|VvA + VbA − Ui|ΦIb:Ia(rvb)χi(Ri)(cid:105)
T exact

(2.19)

and the post T-matrix can be written as

post = (cid:104)ΦIA:IB
T exact

(rvA)χf (Rf )|Vvb + VbA − Uf|Ψexact(rvB, Ri)(cid:105)

(2.20)

where ΦIi:If

is the overlap between the i and f wave functions, Vif is the interaction between

the ith and f th bodies, U is an optical potential between the scattering body and the target

in either the initial or final channels, χ is a distorted scattering wave produced using U , and
Ψexact is the exact three-body wave function. The last two terms of the interaction, VbA− Ui
in the prior formalism and VbA − Uf in the post formalism, are referred to as the remnant

term. In the case of deuteron induced transfer reactions, the two potentials in the post-form

remnant term have similar magnitudes, and it is often a good approximation to neglect it.

Therefore, we will continue using the post T-matrix form.

Ψexact

i

is the solution to the three-body Shr¨odinger equation such that

24

+ Trvb + Vbv + VbA + VvA − E]Ψexact = 0,

[TRi

(2.21)

where TRi

is the kinetic energy operator of the scattering vb system, and Trvb is the kinetic

energy of the internal motion of particles v and b.

Similarly to Equation (2.11), we can formulate an expression for Ψexact at far distances

for each set of quantum numbers in the initial and final channels (αi/α) with the form

ψ(Rf ) = δααiFLi

(ηL, kf Rf ) + T ααiH+

L (ηL, kf Rf )

(2.22)

where FL is the regular Coulomb function, H+

L is the Hankel function, and T ααi is the T-

matrix. It is often challenging to solve for this exact wave function, so there are a number of

approximations that can be made to simplify the T-matrix expression. Most commonly, the

Distorted Wave Born Approximation (DWBA) is applied. In DWBA, Ψexact is simplified to

the elastic channel, a product of the scattering distorted wave and the corresponding bound

state wave function. This simplification models the reaction as a single-interaction process

and will generally break down at low scattering energy. Applying the DWBA and neglecting

the remnant term, transforms Equation (2.20) into

T DW BA
post

= (cid:104)ΦIA:IB

(rvA)χf (Rf )|Vvb|ΦIb:Iv (rvb)χi(Ri)(cid:105).

(2.23)

Once the T-matrix is calculated, it can be easily related to the scattering amplitude by

µpi

µti
µtµp

f

(ki; kf ) =

µα
2π¯h2 T

µpi

µti
µtµp (ki; kf ),

(2.24)

where µp/t is the spin projection of the projectile and target and µα is the reduced mass of

25

the system. Finally, averaging over initial state spin projections and summing over final spin

projections, gives us the cross section

(cid:88)

µpiµti

µpµt

σ(θ) =

1
ˆI2
ˆI2
pi
ti

where ˆI =(cid:112)(2I + 1).

|f

µpi

µti
µtµp

(ki; kf )|2,

(2.25)

2.3.1 Adiabatic Distorted Wave Approximation

Often, DWBA is not sufficient to describe deuteron induced reactions accurately. The

deuteron is weakly bound (2.2 MeV) and, therefore, has a large probability for breakup

in the potential field of the target nucleus. Theoretical studies have demonstrated that tak-

ing this breakup into account is critical for reproducing experimental observations [69]. In

DWBA, the exact incoming wave function is simplified as the product of the deuteron ground

state wave function and the deuteron distorted wave, which means that this breakup is not

included, apart for its influence on the optical potential which describes the deuteron elastic

scattering. The adiabatic distorted wave approximation (ADWA) allows for the considera-

tion of deuteron breakup, but offers a less computationally expensive alternative to including

the full three-body wave function.

ADWA is based on a separation of scales between the deuteron beam energy and its

internal binding energy (E >> 
i, where 
i is the excitation energy of the n/p system).

With this separation of scales, the excited states of the deuteron can be taken as essentially

degenerate with the ground state. To ensure accuracy, the deuteron beam energy should

be above about 20 MeV to ensure the separation of scales between the beam and internal

energies. One consequence exploited by [70, 71] is that the three-body wave function is only

26

necessary within the relatively short range of the deuteron binding potential, Vnp. Within

that limited range, the exact wave function can be expanded in some choice of basis functions.

In this case Weinberg states [72], φi(r), are used to describe the relative motion inside the

deuteron and are complete within the range of the deuteron interaction. χi(R) are used to

describe the relative target-deuteron motion. This expansion can be expresses as

Ψexact ≈ ΨADW A =

∞(cid:88)

i=0

φi(r)χi(R).

(2.26)

ADWA retains only the first term of this expansion, ΨADW A ≈ φ0(r)χ0(R), where φ0(r)

is an eigenfunction of the internal deuteron Hamiltonian with the deuteron binding energy as

its eigenenergy and χ0(r) is referred to as the adiabatic scattering wave function. Retaining

just the first term has been shown to be a good approximation for our purposes [73]. χ0(r)

is calculated using the adiabatic potential which is defined using neutron-target and proton-

target optical potentials. It is worth noting that this adiabatic potential is no longer useful

to describe elastic scattering. The final ADWA T-matrix for (d,N) transfer is written as

T ADW A
post

= (cid:104)ΦIA:IB

(rvA)χf (Rf )|Vvb|ΨADW A(cid:105).

(2.27)

2.3.2 Spectroscopic Factors and Resonance Strengths

Once transfer cross sections have been calculated, a theoretical framework must be used

to connect this value to the astrophysical capture process. Commonly, and for the cases

discussed throughout this thesis, this is achieved through the extraction of spectroscopic

factors. From a theory perspective, the single-nucleon spectroscopic factor is defined as the

norm of the single-nucleon overlap function:

27

S = |(cid:104)φf|φi(cid:105)|2,

(2.28)

where φi and φf are the radial parts of ΦIi:If

from equations (2.19) and (2.20).

In an intuitive sense, the spectroscopic factor can be thought of as a measure of how

well a given state can be well described in a ”single-particle” picture where the transferred

nucleon occupies a single orbital in the mean field of the target. For states that are highly

single-particle in nature, S will be close to one, and a small S can be interpreted as a more

fractured structure that involves the participation of nucleons in several orbitals. Therefore,

the full wave function overlap for the composite A+1 nucleus in both the single-particle

transfer reaction or the particle capture reaction that it correlates to, can be related to the

overlap in a single-particle picture ψsp by the relation

φ(rAN ) = S1/2ψsp(rAN ),

(2.29)

where rAN is the coordinate that connects the target nucleus with the transferred (captured)

nucleon. As shown in Equation (2.27) the transfer cross section is proportional to this over-

lap squared. Because the experimentally measured transferred cross section will contain

information about the full wave function overlap, and our calculations were performed by

modeling the final bound state with a single-particle, Woods-Saxon potential, the spectro-

scopic factor can easily be obtained by normalizing the experimental cross section to the

theoretical value as shown below:

S =

dσ/dΩ|exp
dσ/dΩ|th

.

28

(2.30)

Practically, this normalization is often performed at the first peak of the differential cross

section. In many cases, in (d,n) reactions where the outgoing neutron is difficult to measure,

angular information is unavailable and the comparison must be performed by simply taking

the ratio of the full angle-integrated cross sections.

Once the spectroscopic factor is extracted, it can be used to place constraints on the

astrophysical capture rate of interest. For particle capture cases where the final state reso-

nances are narrow and isolated (often the case for proton captures into low lying resonances)

the reaction cross section to a isolated state is described by a Breit-Wigner resonance. It

can be shown that the astrophysical reaction rate is [74]

Na(cid:104)σν(cid:105) ∝ (T )−3/2(cid:88)

i

− Ei
kT

(ωγ)ie

(2.31)

where Na(cid:104)σν(cid:105) is the reaction rate, T is the temperature of the astrophysical environment,

Ei is the excitation energy of the final resonance states, and ωγi is the resonance strength.

The negative exponent in Equation (2.31) dictates that only low lying resonances will con-

tribute significantly to the reaction rate. For most cases of interest, these excitation energies

have been measured experimentally, however, ωγi remains relatively unconstrained. The

resonance strength ωγ is described by the relationship

ωγif =

(2Jf + 1)

ΓpΓγ

(2jp + 1)(2Ji + 1)

Γp + Γγ

,

(2.32)

where Jf is the spin of the final state, Ji is the spin of the initial state, jp is the spin of the

captured proton, Γγ is the gamma decay width for the final state and Γp is the proton decay

width for the final state. However, for many of these low lying resonances, the Coulomb

barrier reduces the proton decay width by orders of magnitude relative to the gamma decay

29

width, such that the resonance strength can, to very good approximation, be expressed as

ωγif ≈

(2Jf + 1)

(2jp + 1)(2Ji + 1)

Γp.

(2.33)

Finally, the proton decay width can be related to the single-particle decay width through

the spectroscopic factor such that

Γp = C2SΓsp.

(2.34)

In this expression, C is a Clebsch-Gordan coefficient and Γsp is the single-particle decay

width. Γsp is obtained by calculating elastic scattering from a central Woods-Saxon potential

where the depth is adjusted to reproduce the energy of the resonance of interest and is easily

calculated. By these means, the spectroscopic factor extracted from the transfer reaction

can be used to constrain the calculation of the proton decay width and, therefore, directly

constrain the proton capture reaction rate.

2.4 Charge-Exchange Framework

Similar to the reaction framework presented here for transfer reactions, charge-exchange

reactions can be described using a T-matrix formalism. Within this work we have em-

ployed a two-body and a three-body framework, both using single step DWBA, to analyze

charge-exchange reactions to the isobaric analog state. Here, we will introduce both of these

methods, and more in depth derivations of the relevant T-matrix expressions can be found

in Appendices B and C.

30

2.4.1 Two-Body Formalism

The first description of charge-exchange reactions involves a two-body formalism where the

charge-exchange transition is caused by changes in a bulk optical potential which describes

the interaction between the projectile nucleon and the target. The coordinates for two-body

charge-exchange are show in Figure 2.2. Because charge-exchange is an isobaric transition,

the mass difference between the target and residual nucleus is neglected and we utilize the

same scattering coordinate, R1A, in the initial and final channel. For (p,n) reactions, the

projectile, 1, is a proton in the incoming channel, and a neutron in the outgoing channel,

representing a isospin flip interaction.

Figure 2.2: Coordinates for A(p,n)B charge-exchange reactions where the residual nucleus,
B, is an isobar of the target, A.

In this formalism, the transition potential is defined using the Lane potential [47] which

is cast in the form

U (R1A) = U0(R1A) +

τ · T
4A

U1(R1A),

(2.35)

where U0(R1A) is the isoscalar potential which drives elastic scattering, U1(R1A) is the

isovector potential which drives the charge-exchange transition, and τ and T are the isospin

operators which act on the projectile nucleon and target nucleus, respectively. The dot

product of the isospin operators can be expanded as

31

τ · T = τzTz + τ+T− + τ−T+.

(2.36)

where τ+/− are the isospin raising/lowering operators.

Throughout this work, we use the convention that neutrons have isospin projection 1
2

and protons have isospin projection − 1

2. The first term in Equation (2.36) can be neglected

in charge-exchange reactions, because it will not cause an isospin-flip. The third term will
produce 0 because τ−| 1

2(cid:105) = 0. Therefore, only the second term, which increases the

2 − 1

isospin projection of the proton and lowers the isospin projection of the target, can contribute.

The factors in Equation (2.35) are chosen such that it can be rearranged to express elastic

scattering of neutrons and protons:

Un,p = U0(R1A) ± N − Z

A

U1(R1A)

(2.37)

where the upper sign corresponds to neutron scattering potentials and the lower sign corre-

sponds to protons. Neutron number, N, charge number, Z, and mass number, A, pertain to

the target nucleus. This makes it clear that the isovector part of the Lane potential can be

written in terms of proton and neutron elastic scattering potentials giving us

U1(R1A) =

A

2(N − Z)

[Un(R1A) − Up(R1A)].

(2.38)

Un and Up are typically taken to be optical potentials with a form similar to that introduced

in Section 2.1. The choice of optical potential, however, is not unique and we will discuss

the potentials used in this study more thoroughly in the next chapter.

The transition matrix element for charge-exchange can been expressed simply as

32

2B = (cid:104)χf (R1A)|2
T CE

(cid:112)|N − Z|

A

U1(R1A)|χi(R1A)(cid:105).

(2.39)

χi(R1A) and χf (R1A) represents the p+A/n+B distorted waves, which are calculated us-

ing an optical potential. As discussed earlier, the incoming and outgoing scattering wave

functions can be expanded in partial waves, giving us a T-matrix expression of

2B = (cid:104)χf (R1A)|2
T CE

(cid:112)|N − Z|

A

U1(R1A)|χi(R1A)(cid:105) =

(cid:90) (cid:88)

Lf Jf Mf LiJi

i

(cid:2)Y

Li−Lf e

Li−(Mf−µ(cid:48)

iσLi (−1)

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf−Mf
√|N−Z|

(cid:16)

kf ki

1)+Jf−Mf (4π)3/2 ˆLi

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji
(cid:17)

µ1

χLiJi

(R1A)

2

A

U1(R1A)

χ∗
Lf Jf

(R1A)

C

Jiµ1
Li0I1µ1

C

Y

Lf
−(Mf−µ(cid:48)
1)

Jf Mf
Lf (Mf−µ(cid:48)
(ˆkf )

1)I1µ(cid:48)

1

(2.40)

R2

1A

dΩ1Adξ1dR1A.

where primed variables have undergone an isospin transition.

This expression can be simplified using a number of properties, including the fact that

the Lane potential is a scalar operator, so the total angular momentum in the initial and

final channels must couple to 0. Once the expression is fully simplified, we are left with

(cid:112)|N − Z|

A

(cid:104)χf (R1A)|2

(cid:90) (cid:88)

iσLi

e

(4π)3/2

kf ki

ˆLi

LiJi

C

Jiµ1
Li0I1µ1

C

Jiµ1
Li(µ1−µ(cid:48)

1)I1µ(cid:48)

1

U1(R1A)|χi(R1A)(cid:105) =
√|N−Z|

(cid:16)

χLiJi

(R1A)

2

A

(cid:17)

U1(R1A)

χ∗
Lf Jf

(R1A)

(2.41)

R2

1A

Y

Li
µ1−µ(cid:48)

1

(ˆkf )dR1A.

33

To get the cross section, we average over initial state projections, and sum over the final

projections to give us

(cid:88)
µ1µ(cid:48)

1µAµB

dσ
dΩ

=

kf
ki

µ1µf

1

4π2(¯hc)4

ˆJA ˆµ1

|2

|T
µ1
µ(cid:48)
1

(2.42)

where JA and JB are the initial and final spin of the target and equal 0 in the case of 0+

transitions.

2.4.2 Three-Body Formalism

The two-body formalism discussed previously offers a simple way to explore charge-exchange

transitions but freezing the target into an inert, single body, necessarily erases interesting

phenomenon introduced by the internal structure of the target. Although in most cases of

interest computing reactions using the full, A+1 body system remains intractable, we will

introduce an additional complication into the formalism by now expressing the target as

an inert core with a valence nucleon. This three-body formalism can describe transitions

between single-particle states in the target.

Instead of using a difference of proton and

neutron optical potentials that adjusts the isospin projection of the target, we will now

employ an NN potential that operates directly on the valence nucleon.

The coordinates for the three-body charge-exchange formalism are show in Figure 2.3.

The valence particle, 2, occupies a well defined single-particle state with a given value of l

and j, which we assume can be described by a Woods-Saxon potential.

In this description, the formalism is built on two coordinates, R1A which runs from the

projectile/ejectile to the target/residual, and r2c which runs from the valence nucleon to the

inert core. The NN interaction discussed above will, therefore, run along the coordinate r12,

34

which connects the projectile and valence nucleons. Because this coordinate is different than

the basis coordinates which describe our wave functions, it poses an additional complication.

In our work, the NN potential is expanded in a multipole expansion.

Figure 2.3: Coordinates for A(p,n)B charge-exchange reactions within the three-body reac-
tion formalism.

NN potentials utilized in this three-body framework must be cast in an operator form

in coordinate space. Because we are interested in charge-exchange reactions, we will only
include terms of the potential which contain the τ · τ , Fermi operator where, in the three-

body framework, the operator acts on the projectile and valence nucleons. These potentials

contain a radial dependence that can be phenomenologically fit to charge-exchange data or

derived from a more ab-initio approach. They can be wholly real or contain a mixture of real

and imaginary components. Some interactions are energy dependent while others are not. In

addition to a central isospin term, these potentials can contain other isospin operators such
as the spin-spin, (σ · σ)(τ · τ ), operator or the spin-orbit operator, (L · S)(τ · τ ), among

others. The choice of potential will be discussed in more detail in the next chapter.

The charge-exchange three-body T-matrix can be written in a similar form as the two-

body T-matrix:

35

3B = (cid:104)χf (R1A)ΦIc:I2(cid:48) (ξc, ξ2, r2c)|tN N|ΦIc:I2
T CE

(ξc, ξ2, r2c)χi(R1A)(cid:105),

(2.43)

where now the target/residual wave functions have a dependence on the core spin compo-

nents, the valence spin components, and the internal coordinate r2c. The Lane potential is

replaced by an NN potential. As previously mentioned, the NN potential is directed along

the r12 coordinate which runs from the projectile proton to the valence neutron. Various NN
potentials are parameterized using different terms, but only terms that contain the τ1 · τ2

operators will contribute to charge-exchange. For example, the AV8’ potential considered in

this work ([75]) has 8 terms, 4 of which contain isospin operators. Because all of the opera-

tors I discuss for charge-exchange contain the isospin operators, I will often drop this from

the description of the operators and refer to them, instead, as the central, tensor, spin-spin,

and spin-orbit terms. In this case, tN N is written as

(cid:114)24π

(cid:104)
(τ1 · τ2)

tN N = VC (r12)τ1 · τ2 + VT (r12)
+ VSS(r12)(σ1 · σ2)(τ1 · τ2) + VLS(r12)(L · S)(τ1 · τ2)

5

Y2(ˆr12) · [σ1 ⊗ σ2]2

(cid:105)

(2.44)

where VC (r12), VT (r12), VSS(r12), and VLS(r12) give the radial dependence of the cen-

tral, tensor, spin-spin, and spin-orbit terms, respectively. An expression for the three-body

charge-exchange T-matrix with an isospin central interaction is given here, along with a brief

commentary on the expressions involving the isospin-tensor, isospin-spin-spin, and isospin-

spin-orbit operators. For a full derivation of these expressions, see Appendix C. For the case

of only an isospin central term, Equation (2.44) simplifies to

36

tN N = VC (r12)τ1 · τ2

(2.45)

In addition to making the isospin flip, the isospin operators will give a factor of 1

2 and

we will also add a (2j + 1) factor to account for all available neutrons in the valence shell.

Now, we must express this quantity using an expansion in our current coordinates using the

method from [67]. The relationship between the two coordinate vectors and r12 is

r12 = R1A − mc
mA

r2c,

(2.46)

where mc is the mass of the core and mA is the mass of the target. The potential only

depends on the magnitude of r12, so we can calculate r2
12

r2
12 = R2

1A +

m2
c
m2
A

2c − 2mc
r2
mA

R1Ar2cz,

(2.47)

where z = cos θ12 and θ12 is the angle between R1A and r2c. Then we can build a multipole

function, Fλ, using the potential so that:

Fλ(R1A, r2c) =

(cid:90) 1

−1

1
2

VN N (r12)Pλ(z)dz,

(2.48)

where Pλ are Legendre Polynomials. Then, our final potential is

tN N = V (r12)

(2j + 1)

2

=

(cid:88)

λ

(2λ + 1)Fλ(R, r)

4π

2λ + 1

(cid:88)

mλ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

(2j + 1)

2

.

(2.49)

As before, we can expand our incoming and outgoing wave functions in partial waves.

37

In addition to the scattering wave functions previously used in the two-body description,

there will also be an expression for the internal valence-core wave function. Including all the

appropriate factors, the T-matrix can be written as

(cid:104)χf (R1A)ΦIc:I2(cid:48) (ξc, ξ2, r2c)|tN N|ΦIc:I2

(ξc, ξ2, r2c)χi(R1A)(cid:105) =

Lf
Mf

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

C

(ˆkf )C

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

1

Y

(cid:2)Y

C

C

Jiµ1
Li0I1µ1

JAµA
jimji

Icµc

(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

JB µB
jf mjf

c

Icµ(cid:48)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

µ1

(cid:90) (cid:88)
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf

mjf

Lf Jf Mf LiJi
µcµ(cid:48)
mji
c

(cid:2)Y

k1

−(Mf +µ(cid:48)
1)

∗Ic
µ(cid:48)
Ξ
c

(ξc)ΞIc

µc(ξc)

mji

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2dξc,

where

Lf−Li(−1)

k1 = i

Li+Lf +Mf +µ(cid:48)

1+mjf e

iσLi

3
2

.

ˆLi(4π)
kf ki

(2.50)

(2.51)

Ideally, we simplify all angular components of this integral in order to ease its implemen-

tation in our calculations. We can accomplish this by using a number of angular momentum

identities and, again, taking advantage of the scalar nature of the central operators (see

Appendix C). After all of these manipulations, we are left with a final expression that only

contains a radial integral:

38

(cid:90) (cid:88)
(cid:104)χf (R1A)ΦIc:I2(cid:48) (ξc, ξ2, rrc)|tN N|ΦIc:I2
Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

(ˆkf )C

Lf
Mf

k5 Y

1

(ξc, ξ2, r2c)χi(R1A)(cid:105) =

C

Jiµ1
Li0I1µ1

C

λMJ
Jf−(Mf +µ(cid:48)

C

1)Jiµ1

Lf Jf Mf LiJi
MJ λmji
mjf
µc

Cλ0
Li0Lf 0Cλ0

lf 0li0C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

λMJ
jf−mjf

jimji

(2.52)

1

Li

Ji

I1

Jf Lf λ



 li

jf



ji

I2

lf λ

φjf

(r2c)FLf Jf

(R1A)Fλ(R, r)φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

where

Lf +Li(−1)

k5 = i

Mf +µ(cid:48)

1+mjf

−MJ +Ji+I1+ji+lf +I2

iσLi (2ji + 1)

e

ˆLi

2 ˆLf

3
(4π)
2
2kf ki

ˆji ˆjf

ˆli

ˆlf

.

ˆJf
ˆJi
ˆλ2

Now the total cross section expression is

dσ
dΩ

=

kf
ki

µ1µf

(cid:88)
µ1µ(cid:48)
×(cid:12)(cid:12)(cid:104)χf (R1A)ΦIc:I2(cid:48) (ξc, ξ2, r2c)|tN N|ΦIc:I2
1
µAµB

1
ˆJ 2
A ˆµ2
1

4π2(¯hc)4

(2.53)

(2.54)

(ξc, ξ2, r2c)χi(R1A)(cid:105)|2,

where JA and JB are the initial and final spins of the target. A similar expression can also

be worked out for additional terms in the interaction. Here, we just remark on a few unique

features of each term considered in this work.

The spin-spin operator, which has the form

39

VSpin−Spin = VSS(r12)(σ1 · σ2)(τ1 · τ2),

(2.55)

is relatively straightforward to consider because the additional operator, σ1 · σ2, acts only

in spin space. The operator therefore factorizes and much of the T-matrix expression re-

mains unchanged. This operator, often referred to as the Gamow-Teller operator or spin-flip

operator, is capable of switching the spin projection of the projectile and valence nucleus,
causing a net change of ∆S = 1. However, like the τ · τ operator, it can be broken down
into three terms: σ1 · σ2 = σ1zσ2z + σ1+σ2− + σ1−σ2+. Because we are only considering

Fermi transitions in this work, we only consider contributions from σ1zσ2z (∆S=0).

The tensor operator offers an additional challenge, however, because it mixes spin and

position space operators. The tensor operator has the form

VT ensor = VT (r12)

(cid:104)
(τ1 · τ2)

Y2(ˆr12) · [σ1 ⊗ σ2]2

(cid:105)

,

(2.56)

(cid:114)24π

5

where Y2(r12) is a rank two spherical harmonic directed along the r12 coordinate. Because

our initial and final wave functions are written in the R1A and r2c coordinates, the tensor

operator’s spherical harmonic must be expanded in terms of the other two coordinates. This

greatly complicates the expression. Details for this process can be found in Appendix C.

Finally, we also considered the spin-orbit operator. This operator has a similar compli-

cation to the tensor operator, where it operates both on spin space and angular momentum

space (which contains an angular dependence). Unlike the spherical harmonic from the ten-

sor operator, however, there is not a simple spherical harmonic expansion that can be used

to express the angular momentum operator, L, in terms of the coordinates we have chosen to

express the problem. Although it is possible to express this term exactly, it would introduce

40

the need for a numerical angular integral in the T-matrix expression and require a signifi-

cant alteration to the framework used for our calculations code. Therefore, we experimented

with several approximate way of including the spin-orbit interaction which is described in

detail in Appendix C. After these investigations, we felt we were unable to reliably extract

meaningful information regarding the magnitude or shape of the spin-orbit contribution, so

we have instead, chosen to not include this operator in our calculation and leave this as an

open area for future development and investigation.

Once we have developed our formalism for charge-exchange reactions, the challenge is

to operationalize these expressions in a reaction code. The next chapter discusses the code,

CHEX, that has been developed to explore these reaction formalisms.

41

Chapter 3

Calculations and Inputs

3.1 Transfer Reactions with FRESCO

All transfer cross sections were calculated using the reaction code FRESCO [76]. Each of

the different transfer reactions considered in this thesis followed the same basic prescription

for calculating cross sections to various final states of interest. Any differences, if present,

are noted in the discussion of the results. For all cases considered, (d,n) and (d,p) theo-

retical cross sections were calculated using the finite-range adiabatic approximation [70, 71]

described in Section 2.3.1 which explicitly incorporates deuteron breakup in the field of the

target. To implement the ADWA formalism, the adiabatic optical potential for the incom-

ing channel must be constructed from proton and neutron optical potentials and then fed

into FRESCO. The effective adiabatic potentials for (d,p) and (d,n) were computed with

TWOFNR [77]. The CH89 [2] optical potential was used for the nucleon-target interactions

in the initial and final channels.

The deuteron bound state in the initial channel is parameterized using the Reid soft

core NN interaction [78] (built into FRESCO). For consistency, the Vvb interaction in the

transition amplitude from Equation (2.27) is also from [78]. In these calculations, the target

nucleus is taken as inert and the final state of the target-like nucleus is modeled by placing

the transferred nucleon into a single-particle bound state described by a real Woods-Saxon

42

potential with central and spin-orbit terms. The radius and diffuseness of the Woods-Saxon

are chosen to be standard values of r=1.25 fm and a=0.65, respectively, and the depth of the

spin-orbit term is chosen to be 6 MeV. For further discussion of the shape of this potential,

see Section A.2. In the case of bound final states, FRESCO automatically adjusts the depth

of the central potential to reproduce the experimental binding energy. In the case of unbound

resonances in the final channel, we implemented either continuum binning or a bound state

approximation which we explore in further detail now.

3.1.1 Bound State Approximation

For all of the cases considered in this work, the final states of astrophysical interest are low-

lying, positive energy, resonances where the transferred particle sits in an excited state above

the one proton separation energy. Still, these low-lying resonances exhibit many aspects of

a bound excited state, including a confined wave function and well defined excitation energy

with a relatively small width.

In these cases, there are a couple of different choices for

representing the final states.

Resonance states can be represented with continuum bins [67]. Using this method, one

can place the resonance at the exact experimental excitation energy and calculate a wave

packet (the integral of the corresponding wave function over energy or momentum) within

a chosen energy window around the resonance. In this configuration, the potential depth of

the single-particle Woods-Saxon is not automatically adjusted by FRESCO. Therefore, we

manually tuned the depth of the Woods-Saxon until the phase shift goes through π/2 at the

appropriate resonance energy. For narrow proton resonances at low excitation energies, the

complete transition from 0 to π in phase can happen over the span of less than 1 eV (as

compared to the natural energy scale of MeV) and locating this resonance is often difficult.

43

Once the resonance is located, the width can be calculated by taking the derivative of

the phase shift and determining the full width at half max. Although using FRESCO to

determine widths is an accurate and reliable method, it is inefficient and, for the narrowest

resonances, infeasibly time consuming.

For most results presented in this thesis, we have sidestepped this process by using a

bound state approximation. In this approximation, a low lying resonance is artificially bound

by about 0.001 MeV. This is based on the idea that a resonance which is just slightly unbound

will have a similar spacial distribution as a very shallow bound state. This is particularly

valid in the case of proton transfer where the Coulomb barrier contributes to an even more

localized resonance state. Of course, at some point, this approximation will break down as

the resonance energy increases. Although this approximation has been widely used prior to

the applications presented in this thesis, its accuracy had not been rigorously tested. Below,

we present the effect of the bound state approximation on reaction cross section results for

the particular case of resonant states in 31S, and note that these results generalize to other

nuclei studied in this thesis.

A reasonable place to start is to check that the wave functions produced by the artificially

bound states behave similarly to the wave functions calculated using continuum binning.

Figure 3.1 shows this comparison for a l = 1 proton resonance in 31S at Ex = 6.833 MeV

(Eres = 702 keV). It is clear that the unbound resonance behaves very similarly to a shallowly

bound state and, therefore, it is reasonable to expect similar cross sections produced by the

two configurations.

Next, we examined the impact of the bound state approximation on the angle integrated

transfer cross section which will be used to constrain the astrophysical capture rate (see

Section 2.3.2). To evaluate whether or not the bound state approximation is accurate, it

44

Figure 3.1: Comparing the single-particle wave functions created using the bound state
approximation (red) and the exact calculation using continuum binning (black) for a proton
resonance in 31S at Ex=6833 keV. The green curve shows the difference between the two
wave functions.

is important to compare the error introduced through this method with other errors and

uncertainties present in our model. The main source of uncertainty in our calculations are

the OMPs used to describe our incoming and outgoing distorted scattering waves. These

optical potentials are obtained from fits to elastic scattering data between the nucleons and

the initial (final) state nuclei at the initial (final) energies. However, for target and energy

combinations for which this experimental information do not exist, one uses global optical

potentials which are created by simultaneously fitting large data sets over a wide variety of

energies and masses. These parameterizations describe trends across the nuclear chart, but

are not necessarily valid for a given target at a particular energy, especially if it is unstable.

Additionally, the parameterization for each potential contain around 15 different parameters

and the choice of parameterization is not unique, leading to a large uncertainty, typically

around 30%, but which has in some cases been shown to be over 100% [61].

45

Another, less rigorous, way to gauge the scale of the uncertainty introduced by the optical

potential is to repeat the calculation using different global parameterizations and observe the

effect on the angle integrated cross section. For example, all transfer results presented in

this thesis use the parameterization by Varner et al. [2] (CH89), but these calculations can

be repeated using another choice of global optical potential, such as the Becchetti-Greenlees

parameterization [66]. When such a substitution is made, the propagated difference to the

angle integrated cross section is at least on the order of about 10-15%, depending on the

case. Errors from the bound state approximation will be independent of errors introduced by

the optical potential and, thus, will be added in quadrature to those errors in our theoretical

calculations. This means that the errors introduced through this bound state approximation

should be significantly less than the errors introduced by the optical potentials to avoid a

large proliferation of errors.

Table 3.1 shows the result of comparing the angle integrated transfer cross sections ob-

tained using continuum bins to describe the unbound resonances and those obtained via the

bound state approximation. As we would expect, the errors associated with this method

increase as the resonance energy increases. Another clear trend is that the value of orbital

angular momentum has a large impact on the effect of the approximation. For l=1 states

the error is consistently higher than the error associated with l=2 or 3 states. This is not

surprising as the potential barrier will be increased for larger values of angular momentum.

This increased barrier will better mimic the potential of a bound state in an attractive po-

tential well. This effect is evidenced in the single-particle widths calculated for these states.

As the angular momentum increases, the resonances become narrower and behave more like

bound states.

For 31S, the error associated with the bound state approximation can be tolerated for

46

l=1 states including and below the resonance located at 226 keV, as we would expect the

error to continue decreasing for lower energies. For l=1 states above that threshold, the

bound state approximation introduces too large an error into our calculated cross sections,

and continuum binning should be performed. Continuum binning is, however, much more

feasible for these higher energy resonances with wider widths. It appears that the bound

state approximation is safe to use for resonances with l >1 up to resonance energies of at least

450 keV. The error associated with the l=2 resonance with Eres=452 keV is only 2.4%, so

it is reasonable to assume that the region of validity extends to significantly higher energies.

When applying this approximation to other systems, it will be imperative to locate the

energy at which the error associated with the bound state approximation become too large.

For l=0 and l=1 systems, this point seems to be much lower than was previously thought.

For states with large orbital angular momentum, the approximation is very likely to hold for

most resonances of interest.

Final State

Comparison

Eres (keV) Ex (keV)

n,l,j

Bound State Approximation xs (mb) Continuum Binning xs (mb) xs % Difference Width (eV)

226.0
246.0
262.0
411.0
452.0
702.0

6357.0
6377.0
6393.0
6542.0
6583.0
6833.0

2,1,1.5
1,3,3.5
1,2,2.5
2,1,1.5
1,2,1.5
2,1,1.5

1.52
6.68
1.14
1.04
0.82
1.55

1.43
6.75
1.16
0.92
0.80
1.23

6.13
-1.05
-1.39
12.05
2.36
22.51

3.67E-04
1.02E-06
2.64E-04
1.02E00
6.51E-02
2.02E+02

Table 3.1: Comparison of angle integrated transfer cross sections to states in 31S calculated
using the bound state approximation and the continuum binning method. As the resonance
energy increases, and the value of l decreases, the approximation breaks down.

3.2 Charge-Exchange Reactions with CHEX

All charge-exchange calculations presented here were produced using the reaction code,

CHEX, developed for this thesis. Of course, there are a number of charge-exchange reaction

47

codes on the market, such as DW81 [79] for (p,n)/(n,p) reactions and FOLD [80] which can

be used for the analysis of composite probes, such as (3He,t). However, older codes, such as

DW81, were not written with the expressed goal of exploring reaction dynamics and prove

difficult for outside users to explore and extend. More modern codes such as FOLD are more

accessible, but we decided it was necessary to build our code with components that we could

fully understand and control, allowing for a stronger base for future extensions in the reaction

mechanism. The goal of CHEX is for a flexible and user friendly tool that can be used for

exploring and expanding the description of reaction dynamics for charge-exchange reactions.

To this end, CHEX is built on a modular design where well described tasks occupy different

subroutines. Additionally, CHEX utilizes a front end interface to solicit input parameters

from the user. The general flow of CHEX calculations are shown in Figure 3.2.

Figure 3.2: Flowchart showing the general organization of CHEX, the charge-exchange reac-
tion code created for this thesis. The primary observables calculated by CHEX are charge-
exchange cross sections.

The wave functions in CHEX are created using subroutines adapted from the NLAT

reaction code [81]. NLAT (nonlocal adiabatic transfer) is a transfer reaction code which

allows for the inclusion of non-local nucleon-target and bound state interactions. Within

this context, the Schr¨odinger equation is solved using an iterative method. Although the

effects of non-local potentials have not yet been explored in charge-exchange reactions, CHEX

utilizes the NLAT Schr¨odinger equation solver to allow for an easy extension to including

48

effects from non-locality in future studies. Elastic scattering cross sections for the entrance

and exit channels are also calculated. The remainder of this section will discuss some of the

key inputs and methods used in CHEX charge-exchange calculations. For an outline of the

benchmarks and checks performed for CHEX, see Appendix D.

3.2.1 Two-Body Calculations

As previously discussed, charge-exchange reactions described in a two-body framework utilize

a Lane potential, defined in the equations in Section 2.4.1, which is proportional to the

difference of neutron and proton optical potentials. The user is free to choose the form of these

optical potentials. CHEX has incorporated Lane potentials based on the global potentials

developed by Koning-Delaroche (KD) [1] and Varner et al.
(CH89) [2]. Lane potentials
derived from these OMPs have a non-zero range ∼7-8 fm, with the exact range depending

on the case, and have real and imaginary components. As demonstrated in Figure 3.3 , Lane

potentials derived from different optical potentials differ from one another significantly. In

addition to the Lane interaction which mediates the charge-exchange, an optical potential

must be chosen to describe the incoming and outgoing distorted waves. CHEX has built in

options to use the KD or CH89 global potentials, or the user can input parameters for a

specific OMP which is cast in the form discussed in 2.1.

In the two-body picture, the target has no internal structure and, therefore, bound-state

wave functions do not enter into the cross-section calculations. Additionally, there is only

an isospin central component. This greatly simplifies the two-body calculations and allows

for short computation times.

49

Figure 3.3: Plot of the radial form of the Lane potential calculated using parameters from
KD and CH89. The case shown is for 48Ca and L=0.

3.2.2 Three-Body Calculations

CHEX also includes the implementation of charge-exchange using the three-body formalism

presented in this work. Similar to the two-body calculations described above, the user has the

ability to specify the optical potential used to describe the incoming and outgoing distorted

waves, and to specify the interaction that will mediate charge-exchange. CHEX has built in

options to implement the isospin components of the NN interactions developed by Decharg´e-

Gogny (Gogny) [82] and Pudliner et al. (AV8’), which is based on the Argonne v18 potential

[75, 83]. These potentials will be discussed in more detail in the following section. The user

can also decide which operators to include in the calculation.

In the three-body picture, the target is comprised of an inert core with a valence particle

in a well defined single-particle state. In this work we study nuclei that undergo transitions

to their IAS. The IAS is the lowest energy excited state in the final nucleus with the same

50

isobaric spin, T, as the target nucleus. In a basic picture, the wave function of the IAS can

be thought of as identical to the initial state, with a neutron replaced by a proton for the case

of isospin lowering reactions or vice versa for isospin raising reactions. Unlike other reaction

codes, such as DW81 and FOLD, the current version of CHEX does not input transition

densities to many possible final states created by shell-model or other structure calculations.

Instead, the user selects a specific initial and final single particle state with a value of l and

j. This state is modeled using a Woods-Saxon potential with a radius of r0=1.25 fm and

a diffuseness of a=0.65. The user must adjust the depth of the potential to reproduce the

experimental binding energies of these states. Because of this set up, CHEX is currently best

suited to calculate specific microscopic transitions with well defined initial and final states.

3.2.3 Details of NN Potentials

One of the most important inputs in the charge-exchange calculations is the choice of NN

interaction which mediates the charge-exchange reaction. One of the goals of this work

is to explore the sensitivity of the charge-exchange observable to the choice of interaction.

As a general note, CHEX is capable of incorporating any NN interaction which can be

written in an operator form in position space. Most commonly, the effective NN interaction

parameterized by Love and Franey (LF) [59, 60] is used to describe the isospin transition.

LF is a phenomenological, energy dependent interaction that is cast in an operator form with

isospin central, tensor, and spin-orbit terms. Each term uses a sum of real and imaginary

Yukawa potentials with different ranges, correlating to π, ρ, and 2π meson exchange. The LF

potential was fit to reproduce NN scattering data and, its updated form, includes in-medium

effects which have been demonstrated to alter the bare NN potential [84]. LF has been

tabulated for energies ranging from 50 MeV to 1000 MeV. The LF interaction has proven

51

to be a useful interaction for describing charge-exchange reaction data, although it is best

constrained above about 100 MeV.

In the LF potential, the parameterization is chosen to explicitly and separately account

for the knock-on exchange contributions to the transition amplitude. Exchange contributions

arise from interchange of position of the projectile and valence nucleons [85]. The exchange

contribution is estimated to be about 35% in [18]. At this time, CHEX does not implement

an exchange term, either approximately or exactly. For this reason, the LF potential was

not used in this analysis. Based on this, we impose the additional constraint that all NN

interactions currently implemented in CHEX should not contain explicit parameterizations

for knock-on exchange.

With this in mind, we selected two other NN interactions to implement in CHEX and

explore the effect on charge-exchange cross sections. The first interaction chosen was AV8’

[75]. AV8’ is a reprojection of the Argonne v18 potential [83] which reduces the 18 terms to

just 8, while still reproducing key features including the deuteron binding energy. v18 is a

high-quality, bare, NN potential with 40 adjustable parameters that were fit to a database

of thousands of nn, np, and pp scattering data, as well as the deuteron binding energy.

Within this framework, v18 obtains a χ2 per datum of 1.09 over an energy range of 0-350

MeV. Unlike the Lane and LF interactions, v18 is fully real and is not energy dependent. It

can be written in an operator form with 18 distinct operators, half of which contain isospin

operators.

A full implementation of the 9 relevant isospin terms for v18 is a complicated task and

lies outside the scope of this thesis. Additionally, choosing to implement only some of those

9 operators is risky because v18 is not fit on an operator by operator basis and, therefore,

it can be difficult to predict the effect of excluding operators. This effect is amplified by

52

the fact that reaction amplitudes from various components are able to add destructively.

For this reason we, instead, implemented AV8’. AV8’ has only 8 operator terms and can be

expressed as

(cid:88)

p=1,8

v(cid:48)
ij =

v(cid:48)
p(rij)Op
ij

(3.1)

where the various operators are: central, isospin central, spin-spin, isospin spin-spin, tensor,

isospin tensor, spin-orbit, and isospin spin-orbit. These operators are expressed as

Op=1,8

ij

= 1, (τi · τj), (σi · σj), (σi · σj)(τi · τf ), Sij, Sij(τi · τf ), (L · S), (L · S)(τi · τf ),

respectively, and v(cid:48)

p(rij) gives the radial form of each term. The strength of the radial forms

are derived from a recombination of the first 14, charge-independence components of v18.

For the purpose of charge-exchange reactions, we only need to consider contributions from

isospin dependent forms, reducing the potential which must be implemented to just four

(3.2)

terms:

v(cid:48)
ij(rij) = v(cid:48)
+ v(cid:48)

2(rij)(τi · τj) + v(cid:48)
6(rij)Sij(τi · τf ) + v(cid:48)

4(rij)(σi · σj)(τi · τf )
8(rij)(L · S)(τi · τf ).

(3.3)

The T-matrix for each operator is calculated in a separate subroutine in CHEX, and then

combined. The radial form factors of the operators are implemented using a subroutine from

[86] and are shown in Figure 3.4.

53

Figure 3.4: Plot of the radial form factors of the AV8’ and Gogny potential operators con-
sidered in this work.

AV8’ is a bare NN interaction, like v18. We know that nucleons undergoing charge-

exchange reactions will feel the effects of other nucleons in the target. However, one could

imagine a scenario where, as the valence nucleon undergoing charge-exchange in our model

becomes less and less bound, these effects might begin to decrease. For this reason, we would

like to study the charge-exchange response to bare NN interactions. Additionally, for this

work, we have chosen the Gogny interaction [82], which does include in-medium effects.

The Gogny interaction is a real, effective interaction. Although it also contains a density

dependent and spin-orbit term, for our purposes studying charge-exchange reactions, it takes

the simple form of the sum of Gaussians,

54

(cid:88)

V (r) =

(HPτ − M PσPτ )ie

−r2/µ2
i

(3.4)

i=1,2

where r is the distance between the nucleons, M , H, and µ are parameters which specify the

strength of the interaction, and Pτ = 1

2(1 + ˆτ1 · ˆτ2). Because we are only concerned with

terms which can cause an isospin transition, this expression can be stated as

(cid:88)

i=1,2

1
2

V (r) =

(H(ˆτ1 · ˆτ2) − M (ˆσ1 · ˆσ2)(ˆτ1 · ˆτ2))ie

−r2/µ2
i .

(3.5)

The Gaussians used simulate a short range and intermediate range interaction. Because

the Gogny interaction seeks to describe in-medium effects, it was fit using a different, and

much smaller set of nuclear data from AV8’, including empirical data in nuclear matter, such

as nuclear symmetry energy. The parameters used in these calculation are: µ1 = 0.7fm,
µ2 = 1.2 fm, H1 = −496.2 MeV, H2 = 37.27 MeV, M1 = −32.56 MeV, M2 = −68.81 MeV

and produce the forms shown in Figure 3.4.

55

Chapter 4

Results

4.1 Selected Transfer Results

This section contains a detailed look at the methods and results for a study of the 30P(d,n)31S

reaction with applications to the astrophysical 30P(p,γ)31S capture reaction. Additionally,

Section 4.1.4 discusses an extension of this method to study 23Al(d,n)24Si for applications

to the 23Al(p,γ)24Si astrophysical reaction rate. The analysis of the 56Ni(d,n)57Cu and

56Ni(d,p)57Ni reactions are briefly summarized in Appendix A.1.

4.1.1 Motivation for Studying 30P(p,γ)31S via 30P(d,n)31S

The astrophysical proton capture reaction rate for 30P(p,γ)31S remains one of the largest

uncertainties in models of oxygen neon (ONe) novae explosions. In these systems, an ONe

white dwarf star is in a binary system with a hydrogen-rich companion star which accretes

mass onto the surface of the white dwarf until the temperature and pressure build up enough

to ignite an explosive thermonuclear runaway [87]. Nucleosynthesis in this environment

follows the path of the rp-process discussed in Section 1.2, terminating with the production

of elements near Ca. Although the rp-process involves a complex network of capture reactions

and beta decays, it has been shown that uncertainty in the 30P(p,γ)31S reaction rate plays

a key role in determining the synthesis of heavier elements in the rp-process [88].

56

Additionally, the structure of the final state nucleus, 31S, remains largely unknown. Past

experimental studies that examined this nucleus yielded conflicting results for the final state

spins and parities [89, 90, 91] and updated shell model calculations which include the full

sd-pf model space indicated that contributions from negative parity resonances are likely to

dominate the proton capture reaction rate in the Gamow window [92, 93]. Determining the

exact location and spins of these negative parity states is crucial to minimizing the large

uncertainties remaining in the proton capture reaction rate.

However, direct measurements of 30P(p,γ)31S are not currently feasible. 30P has a half

life of 2.5 minutes, so experiments must be run in inverse kinematics with rare isotope beams.

Beam energies must be well below the Coulomb barrier to directly populate the final states

of interest (Ep < 500 keV), greatly suppressing the reaction cross section and rendering the

radioactive isotope beam intensity insufficient for a direct measurement. For these reasons,

the 30P(d,n)31S transfer reaction has been used to probe the astrophysical capture rate

through the extraction of spectroscopic factors, as described in Section 2.3.2.

4.1.2 Experimental and Theoretical Methods

The experimental 30P(d,n)31S transfer cross sections were measured by our collaborators

at the National Superconducting Cyclotron Laboratory at Michigan State University [94].

A 30 MeV/u beam of 30P was created via in-flight fragmentation and impinged on a solid

deuterated target, populating the excited states of interest in 31S. These excited states

quickly decay to bound states, releasing γ rays that were detected by GRETINA (Gamma-

Ray Energy Tracking In-beam Array) which surrounded the target. GRETINA is a high-

resolution, high-efficiency, γ ray detector capable of an energy resolution around 2-3 keV in

the energy region of interest [95]. The heavy 31S reaction products have a large forward

57

momentum and continue moving in the direction of the beam until detected in the S800

Spectrograph [96]. This method can be used to determine the transfer cross section to

each of the excited states of interest through 31S-γ coincidence. The cross section to the

ground state, however, cannot be measured directly because it will not decay via γ emission.

Additionally, the outgoing neutron is not detected and the γ ray statistics are insufficient to

extract angular distribution information, so only the angle integrated cross section to each

final state can be compared to theoretical calculations. This method was first validated for

the case of 26Al(d,n)27Si, successfully reproducing known spectroscopic factors and resonance

strengths [94].

We then calculated theoretical transfer cross sections for each observed state, according

to the procedure described in Section 3.1. For each of the excited states of astrophysical in-

terest, the bound state approximation provided sufficient accuracy. Spectroscopic factors for

each transition were then extracted by taking the ratio of the experimental angle integrated

cross section to the theoretical cross sections. In several cases where a transition was not

observed in GRETINA, there was an upper bound placed on the cross section and, therefore,

the spectroscopic factor. Once S is obtained, resonance strengths for each state that will

significantly contribute to the astrophysical capture rate were calculated according to the

procedure presented in Section 2.3.2.

4.1.3 Results and Conclusions

A summary of the results for the cross sections, spectroscopic factors, and relevant resonance

strengths are summarized in Table 4.1. Full results for this work are available in [3].

A key result from this work is that the three strongest states measured in this experiment

are all negative parity states and they correlate with the three states predicted to have the

58

Ex (keV) Eres (KeV)
6138.6(6)
6158.5(5)

8.0(6)
27.9(6)

6255.3(5)
6279.0(6)
6327.0(5)
6357.3(2)
6376.9(4)

6390.2(7)
6392.5(2)
6394.2(2)
6541.9(4)
6583.1(20)

124.7(6)
148.4(6)
196.4(6)
226.7(3)
246.3(5)

259.6(7)
261.9(3)
263.6(3)
411.3(5)
452.5(20)

i → J π
J π

f

(3/2+, 7/2+) → 7/2+ ≤ 0.030
0.177(33)

σexp (mb) σth (mb)

C2Sexp
≤ 0.16(7)
0.036(13)

C2SSM l
0
3

0.26

ωγ

≤ 0.19
≤ 0.16
0.023(12)
≤ 0.011
0.051(17)
≤ 0.22
0.007(3)
≤ 0.002
≤ 0.007

≤ 5.9 × 10−3

0.29

0.39

0.0032

9.5 × 10−12
0
0
3.5(19) × 10−7
1
1 ≤ 1.4 × 10−6
3

2.4 × 10−5
0
4.8(21) × 10−7
2
4
2 ≤ 1.7 × 10−4
3

0.188
4.94

.0938
0.188
1.07
1.60
6.21

0.189
4.66
1.20
6.21
3.72

7/2(−) → 7/2−
7/2(−) → 5/2+
1/2+ → 1/2+
3/2− → 1/2+
9/2− → 7/2−
9/2− → 7/2+
5/2(+) → 3/2+

3/2+
5/2−

3/2+

11/2+
7/2+
(7/2)

≤ 0.019
≤ 0.029
0.025(10)
≤ 0.017
0.32(5)
≤ 0.042
0.034(9)
≤ 0.018
≤ 0.037
≤ 0.027

Table 4.1: Summary of results from [3] for the study of 30P(p,γ)31S via 30P(d,n)31S. Tran-
sitions which were not observed in the experiment provide upper limits for the experimental
cross sections and spectroscopic factors. Theoretical spectroscopic factors are shown for ob-
served transitions and were produced via the shell model using the USDA Hamiltonian [4]
for positive parity states and the WBP Hamiltonian [5] for negative parity states. Reso-
nance strengths are shown for states that will contribute significantly to the proton capture
reaction rate, noting that states with high l value will be suppressed.

largest spectroscopic factors by shell model calculations.

It is notable however, that all

of the measured spectroscopic factors corresponding to negative parity states are about an

order of magnitude smaller than those predicted in the shell model. This indicates that

the single-particle strength is, in fact, highly fragmented. This is not unreasonable given

the complicated structure of the odd-odd nucleus, 30P, which has many low-lying excited

states. This result also indicates that the remaining single-particle strength must lie higher in

excitation energy than about 6.7 MeV (about 0.5 MeV above the proton emission threshold).

Above this energy, proton decays begin to compete with and dominate γ decay as a means to

depopulate excited states in 31S and, once this occurs, it is no longer a good approximation
to purport that Γγ (cid:29) Γp. In this case, the resonance strength becomes sensitive to both the

proton decay width and gamma decay width, invalidating the approximation in Equation

(2.33) and the experimental method.

59

Figure 4.1: The proton capture reaction rate for various final states of 30P(p,γ)31S in the
astrophysical temperature range relevant for novae. Solid lines indicate the use of spectro-
scopic factors extracted from this work. In cases where a final state was not observed, the
spectroscopic factor is provided by theoretical shell model calculations. At low temperatures
the 3/2− state at 196 keV dominates the reaction rate, but at high temperatures, the 3/2+
state at 260 keV might become dominant, although this state was not observed in [3]. Figure
adapted from [3].

The resonance strengths determined or constrained by upper limits in this study can

then be used to calculate proton capture rates over the temperature range relevant for novae

events. Those reaction rates are shown in Figure 4.1. States that were experimentally con-

strained by (d,n) in this study are shown in the full lines, while results from fully theoretical

(p,γ) predictions are indicated by dashed lines. In particular, the resonance strength of the

3/2+ resonance at 260 keV comes from shell model calculations using the USDE interaction
calculated in [89]. For temperatures between about 0.10 and 0.17 GK, the 3/2− state at

196 keV which was first constrained by this work is expected to dominate the reaction rate.

It is notable that, because the spectroscopic factor extracted here was much smaller than

shell model predictions, this marks a significant decrease in the predicted reaction rate at

low temperatures. Above about 0.17 GK, the 3/2+ resonance at 260 keV is predicted to

60

dominate the reaction rate. However, this state was not measured in the work presented

here and, therefore, the reaction rate shown in Figure 4.1 is based solely on shell model

predictions, which is likely to be significantly higher than the true reaction rate. Measuring

this state will be crucial to further reducing the uncertainty in the 30P(p,γ)31S reaction rate.

Nevertheless, this study marks the first experimental constraints of the 30P(p,γ)31S reaction

rate and has reduced the overall uncertainty, particularly at temperatures between about

0.10 and 0.17 GK, in this key reaction for novae nucleosynthesis.

While this work was able to successfully constrain the spectroscopic factors for states in

31S, it is oftentimes difficult to determine the correct l and j values for the single particle,

final state, resonances. This information is encoded in the shape of the angular distribution,

whose first peak is pushed towards higher angles for higher values of angular momentum

transfer (l). This probably is particularly pronounced for targets with non-zero spin, allowing

for more angular momentum couplings. Additionally, spectroscopic factors extracted at the

first peak of the distribution, as opposed to from the total angle integrated cross section,

are more exact. For this reason, an experimental advance that allowed for a measurement of

angular distributions would be useful. This is difficult in the case of (d,n) transfer reactions

where beam rates for radioactive isotopes are relatively low and, of course, detection of the

emitted neutrons in the final state is much more difficult that charged particle detection.

4.1.4 23Al(d,n)24Si

In type-I x-ray bursts (XRBs), a neutron star is in a binary system with a companion, low

mass main sequence or red giant star, which is accreting hydrogen rich material onto the

surface of the neutron stars. The transferred mass reaches high temperatures and densities

until it eventually ignites thermonuclear runaway powered by hydrogen and helium burning

61

[23]. Recent advances in modeling XRBs could allow us to constrain neutron star properties,

such as the mass and radius, but these models depend on nuclear reaction network inputs.

Therefore, it is essential to reduce the large uncertainties in some key hydrogen and helium

burning reaction rates in order to obtain meaningful neutron star properties from these

models.

The 23Al(p,γ)24Si reaction rate has been shown by systematic studies to have a significant

effect on the XRB light curve [97, 98]. These curves, which show luminosity over time, are

the primary observables of XRBs. Similarly to the 30P(p,γ) case, direct measurement of the

proton capture reaction on 23Al is not possible because of Coulomb suppression of the cross

section to final excited states of interest. Additionally, 23Al is unstable, with a half-life of

about 0.47 seconds, so the experiment must be performed in inverse kinematics. Therefore,

the 23Al(d,n)24Si transfer reaction is used as a probe for the astrophysical capture rate.

A similar procedure to that described in Section 4.1 was used to measure the 23Al(d,n)24Si,

this time using a 48 MeV/u beam of 23Al impinging on the deuterated target. As before,

GRETINA was used to measure γ rays from the de-exciting final state in 24Si in coincidence

with heavy ion detection of 24Si in the S800. Unique to this experiment was the addition

of the low-energy neutron detector array (LENDA) to also measure the outgoing low en-

ergy neutrons from the (d,n) transfer. This setup allows for a complete measurement of the

transfer reaction and, if enough neutrons are detected, could provide angular distribution

information for the cross sections of various excited states, simplifying the comparison to

theoretical calculations.

Theoretical transfer cross sections were produced in the ADWA framework using FRESCO

and TWOFNR using the same potentials and procedures discussed in Section 3.1. As before,

the theoretical spectroscopic factors were calculated in the framework of the shell model, this

62

time using the USDB interaction [4]. Because the ground state of 23Al is 5/2+, there are

many possible values of l transfer that couple to the appropriate initial and final spin of the

target nucleus. Therefore, these theoretical cross sections for each l value were combined to

obtain experimental spectroscopic factors according to the relation

C2Si

exp =

(cid:80)

C2Si
i(cid:48)(C2Si(cid:48)

theo × σi
theo × σi(cid:48)

theo

theo)

× σexp

σi
theo

(4.1)

where the sum over i(cid:48) runs over all of the possible lj quantum number combinations for each

of the final excited states. A summary of these results for the observed populations of states

in 24Si are shown in Table 4.1. The partial cross section listed for the ground state was

obtained by subtracting the sum of all excited state cross sections measured in GRETINA

from the total cross section measured in the S800. The state at 3471 keV has two possible

spin assignments, 0+ or 4+, and no conclusion was made regarding this state, so both results

are given. Resonance strengths are shown for the resonance states measured in this work.

Ex (keV) Eres (KeV)

0

1874(3)

f

i → J π
J π
1 → 0+
2+

g.s.

3449(5)

156

2 ) → 2+
(2+

1

3471(6)

178

1 ) → 2+
(4+
(0+) → 2+

1

1

σexp (µb) σth (µb) C2Sexp C2SSM l
≤ 271
2
263(83)
0
2
2
0
2
2
2
2
2

≤2.8
0.6(2)
0.07(2)
0.4(1)
0.7(4)
0.002(1)
0.3(2)
0.07(4)
0.004(3)
0.8(4)

3.44
0.27
0.03
0.17
0.45
0.001
0.176
0.016
0.001
0.24

98
139
473
411
86
402
349
722
629
69

78(41)

54(30)

54(30)

j

5/2
1/2
3/2
5/2
1/2
3/2
5/2
3/2
5/2
5/2

ωγ

4.2 × 10−5

5.2 × 10−7
5.0 × 10−6

Table 4.2: Summary of results from [6] for the study of 23Al(p,γ)24Si via 23Al(d,n)24Si. Ten-
tative spin allocations for states in 24Si are shown in parenthesis and a final spin assignment
could not be determined for the 3471 keV state. The ground state cross section represents an
upper limit. Theoretical spectroscopic factors were produced in the shell model framework
using the USDB interaction [4] and theoretical spectroscopic factors were determined using
the relation given in Equation (4.1).

Unfortunately, due to low reaction yield, there were only about 100 counts registered in

63

LENDA between 6 and 16 MeV at the measured angles. With these low statistics, it was not

possible to distinguish individual neutron distributions for the various final states measured

in 24Si. However, the total differential cross section could still be used to independently verify

the results obtained from the GRETINA/S800 measurement. Each of the single l transfer

theoretical cross sections could be weighted by their respective experimental spectroscopic

factors and summed to create a total differential cross section and compared to the total cross

section measured in LENDA. This comparison is shown in Figure 4.2 where the dashed lines

indicate the uncertainty in the experimental spectroscopic factors. The agreement between

the two measurements is remarkable and demonstrates the potential of using this method to

extract all of the required information for astrophysical reaction rates with one experiment.

Figure 4.2: Differential cross section for 23Al(d,n)24Si in the center of mass system using the
LENDA detector (blue dots) compared with the sum of theoretical distributions calculated
using ADWA, weighted by experimental spectroscopic factors (solid pink). The error bands
(dashed lines) are due to the uncertainty in the experimental spectroscopic factors. Figure
adapted from [6].

When the results of this work are applied to the astrophysical reaction rate for 23Al(p,γ)24Si,

the uncertainty is reduced by as much as 3-4 orders of magnitude in the temperature region

64

relevant for XRBs. This is shown in Figure 4.3. This reduction in uncertainties will allow

for improved constrains on neutron-star compactness from XRB observations.

Figure 4.3: The 1-σ uncertainty band of the reaction rate determined by [6] is shown in grey,
compared with the previous 1-σ uncertainty outlined in black. The green dashed line shows
the recommendations from the REACLIB database which contain reaction rates to be used
in astrophysical model calculations. The blue line shows the REACLIB value decreased by
a factor of 30, which fell within previous uncertainties, but would be able to remove the by
pass of material from the 22Mg waiting point [16]. Figure adapted from [6].

4.2 Charge-Exchange Results

Investigation of charge-exchange reaction dynamics has not yet incorporated some of the

recent developments developed for other reaction channels, such as transfer reactions and

Coulomb dissociation (see [99, 55, 56, 57, 58]). In this work, we have begun some of these

explorations and, in this chapter, we present the results of this initial study. In this work, we

examined many facets of charge-exchange through a reaction theory lens, including the sensi-

65

tivity to the interaction which mediates charge-exchange and what effect the charge-exchange

formalism has on the resulting cross sections (i.e. two-body vs. three-body framework).

We focused on (p,n) charge-exchange reactions to IAS in 14C, 48Ca, and 90Zr. These

targets were chosen because they span a large range of nuclear masses, their valence nucleons

occupy shells with a large range of orbital angular momentum values (l), their final IAS are

bound and, therefore, amenable to being modeled by our three-body formalism, and there is

experimental charge-exchange cross section data available for each target’s IAS transition.

Each of these reactions are studied at three different lab energies: E=25, 35, and 45 MeV. It is

worth noting that this energy range is much lower than charge-exchange reaction experiments

typically used to extract transition strengths (i.e. B(GT)) using relations such as Equation

(1.1). We chose to focus on this energy regime for a couple different reasons. First, in our

current study we chose to study transitions between 0+ IAS. Experimental data for these

types of transitions are most common in this 20-50 MeV energy region. Additionally, a large

motivation for studying charge-exchange reactions is to probe properties of the isovector

density, which is most efficient with lower energy projectiles which probe the surface region

of the target.

Experimental cross sections for 14C(p,n) are taken from Taddeucci et al., where the

reaction to the IAS was measured at Elab = 25.7, 35 and 45 MeV [17]. Experimental error

bars were estimated to be around 10% for all data points. Experimental charge-exchange

cross section for 48Ca(p,n) and 90Zr(p,n) come from Doering et al., in which transitions were

measured at Elab = 25, 35 and 45 MeV for both targets [18]. For most data points in these

sets, a 7% error bar was reported, although some data points, usually at large angles, have

a larger percent error.

The charge-exchange results are organized in the following way: Section 4.2.1 presents

66

the results from our study of the two-body formalism introduced in Section 2.4.1. Calcu-

lations were performed using CHEX. This work is similar to the (p,n) reactions calculated

in Danielewicz et al.[10], but here we will extend our calculations to explore the effect of

utilizing different OMPs in the Lane potential which mediates the charge-exchange. We

present results utilizing KD and CH89 optical potentials [1, 2]. Additionally we compare

these results to experimental data.

Next, in section 4.2.2 we will present the results for charge-exchange reactions in the

three-body framework introduced in Section 2.4.2. First, we comment on overall features of

the cross sections in the framework, including trends with beam energy, target mass, and

angular momentum of the valence nucleon. In this work we also explore the effect of various

NN potentials, specifically the Gogny and AV8’ interactions [82, 75, 83] and the effect of

using different OMPs for the scattering wave functions. Finally, we compare these results to

data and discuss the quality of that description.

In Section 4.2.3 we will explore the differences between results produced in the two-

body and three-body calculations. Specifically, we explore the cause of significant differences

in the magnitude and shapes of the angular distributions from two-body and three-body

calculations. Finally, Section 4.2.5 discusses the limitations of this study, and the models

used herein. We explore the possible effects of these constraints and present key areas for

future exploration and extension.

4.2.1 Two-Body Results

In the first part of our charge-exchange study we calculated IAS charge-exchange transi-

tions using a two-body formalism and a Lane-type potential to mediate the charge-exchange

transition, as introduced in Section 2.4.1. For all cases presented here, we calculated the

67

charge-exchange cross section using two choices for the OMPs (KD and CH89). For con-

sistency, the given parameter set was used to calculate both the distorted waves for the

projectile-like nucleon and the Lane potential. Calculations for each of the cases studied

here required less than 10 partial waves to converge and each calculation takes only a few

seconds to run. In addition to the charge-exchange presented here, each run calculates the

elastic scattering cross section for the incoming and outgoing nucleon. Although it does not

effect the charge-exchange results presented here, we note that, as a general rule, the CH89

parameter set requires more partial waves than KD to converge the elastic scattering cross

sections. This could be relevant to future work which makes use of these elastic scattering

cross sections, particularly in applications to uncertainty quantification.

The two-body charge-exchange cross sections for each of the targets considered in this

work are shown in left hand panels of Figures 4.4 (14C), 4.5 (48Ca), and 4.6 (90Zr). Panels

(a), (b), and (c) show the results for Elab = 25, 35, and 45 MeV, respectively. Insets show the

same information in a log scale, which is particularly helpful for clarifying behavior at large

angles where cross sections are typically small. First we will remark on general properties of

the results. Notably, all calculations are peaked at forward angles, which is compatible with

the data that indicates a direct reaction process.

When comparing cross sections produced with KD versus CH89, it is clear that the choice

of OMP has a large impact on the angular distributions that are produced. Although the

general features of the cross sections for all targets and energy are preserved, the magnitude

can change drastically between KD and CH89. Additionally, the diffraction patterns in

the angular distributions are often shifted relative to one another, which is to be expected

because the diffraction pattern is closely related to the radius parameter of the interaction

(r0) which varies between OMPs. One way to quantify the spread caused by the choice of

68

OMP is to look at the percent difference in the value of the cross section at the first peak.

This percent difference ranges from 21% for 48Ca(p,n) at 25 MeV, to 140% for 90Zr(p,n) at

25 MeV. There is no clear trend with either target mass or projectile energy. The average

percent difference over all of the cases presented here is 50%. As mentioned earlier, the

variation produced through various choices of OMP gives a rough idea of the uncertainty in

the calculation. The 50% value mentioned here is similar magnitude to the standard 30%

uncertainties that are often cited in transfer reactions [100]. Of course, this is a very rough

estimate and the 50% value here is the result of a given choice of two OMPs. This value

would surely be different if we compared two different parameter sets. Still, this variation is

sizable and a more rigorous uncertainty quantification effort, like those by Lovell et al. [68]

which apply Bayesian methods to uncertainty quantification in transfer reactions, should

be applied to charge-exchange. The full results for the percent difference are shown in the

column labeled ’2-Body ∆OM P ’ in Table 4.3.

69

Figure 4.4: Charge-exchange cross sections for the 14C(p,n)14NIAS transition at Elab =
25 (a), 35 (b), and 45 (c) MeV. The left column shows the results for calculations using a
two-body formalism with Lane potentials using OMP parameters from KD (solid blue) and
CH89 (dashed pink). The right column shows the results for calculations using a three-body
formalism. The solid/dashed blue lines utilized the AV8’ NN interaction and distorted waves
derived from KD/CH89. The solid/dashed pink lines utilized the Gogny NN interaction
and distorted waves derived from KD/CH89. Insets show the same results in a log scale.
Experimental data from [17] is shown in black.

70

Figure 4.5: Charge-exchange cross sections for the 48Ca(p,n)48ScIAS transition at Elab =
25 (a), 35 (b), and 45 (c) MeV. The left column shows the results for calculations using a
two-body formalism with Lane potentials using OMP parameters from KD (solid blue) and
CH89 (dashed pink). The right column shows the results for calculations using a three-body
formalism. The solid/dashed blue lines utilized the AV8’ NN interaction and distorted waves
derived from KD/CH89. The solid/dashed pink lines utilized the Gogny NN interaction
and distorted waves derived from KD/CH89. Insets show the same results in a log scale.
Experimental data from [18] is shown in black.

71

Figure 4.6: Charge-exchange cross sections for the 90Zr(p,n)90NbIAS transition at Elab =
25 (a), 35 (b), and 45 (c) MeV. The left column shows the results for calculations using a
two-body formalism with Lane potentials using OMP parameters from KD (solid blue) and
CH89 (dashed pink). The right column shows the results for calculations using a three-body
formalism. The solid/dashed blue lines utilized the AV8’ NN interaction and distorted waves
derived from KD/CH89. The solid/dashed pink lines utilized the Gogny NN interaction
and distorted waves derived from KD/CH89. Insets show the same results in a log scale.
Experimental data from [18]is shown in black.

Our two-body calculations can also be compared to experimental charge-exchange cross

section data. As can be seen in panels (a)-(c) of Figures 4.4, 4.5, and 4.6, it is clear that

the two-body charge-exchange calculations are able to capture the overall features of the

charge-exchange data at small and large angles. Quantitatively, one option to explore the

”goodness of fit” is through the traditional χ2/N , defined as [67]

72

Reaction

Target Elab (MeV)
14C
14C
14C
48Ca
48Ca
48Ca
90Zr
90Zr
90Zr

25
35
45
25
35
45
25
35
45

Percent Difference Evaluated at θpeak

2-Body ∆OM P Gogny ∆OM P AV8’ ∆OM P

3-Body ∆N N

21
39
40
67
12
35
141
51
53

12
20
21
54
37
30
143
104
66

15
17
18
51
146
45
145
101
83

190
191
192
161
159
168
144
149
146

Table 4.3: Percent difference evaluated at the first peak for cross sections produced using
different models.

N(cid:88)

i=1

χ2 =

(σth(i) − σexp(i))2

∆σ(i)2

,

(4.2)

where i sums over all of the data points, and ∆σ(i) is the error in data point i and N

is the number of data points. In the cases studied here, the typical experimental percent

error was 10% for the reactions on 14C and 7% for reactions on 48Ca and 90Zr. The second

metric we will use is the percent of data, including experimental error bars, which falls

between the normalized curves produced for each target/energy pair with differing choices of

OMP. The rational is that two curves resulting from varied OMPs give a rough idea of the

theoretical error, so this number represents that percent of the data which can be reproduced,

within experimental and theoretical error bars. A summary of these values for the two-body

calculations are given in Table 4.4.

The average χ2/N and normalization values for two-body calculations produced with KD

and CH89 OMP are very similar, implying that neither potential is preferable to describe

the data. This notion is backed up by visual inspection of the cross section calculations

in comparison to data. Although a particular potential may describe any specific data set

73

2-Body KD 2-Body CH89
Target Elab (MeV) χ2/N Norm χ2/N Norm
14C
0.80
14C
0.60
14C
0.60
48Ca
2.30
48Ca
1.40
48Ca
1.40
90Zr
7.10
90Zr
2.90
90Zr
3.70

1.67
5.73
0.51
16.87
5.94
5.63
21.40
22.87
5.28

25
35
45
25
35
45
25
35
45

5.20
8.33
0.86
16.02
22.05
3.22
13.87
15.10
8.34

0.60
0.60
0.50
1.00
0.50
1.40
1.10
1.10
1.00

Reaction

Average

2-Body ∆OM P

% Data

33.33
33.33
70.00
13.89
30.30
24.00
24.24
16.67
34.38
31.13

Table 4.4: Numerical comparison of two-body charge-exchange angular distributions to ex-
perimental data. The final column shows the percentage of the data, including experimental
errors, which falls between the theoretical curves produced by two-body calculations with
the KD and CH89 OMP.

better than the other, the opposite may be true for a different target/energy combination.

This is unsurprising because OMPs are fit to large data sets and, while their fit might favor

particular nuclei, they are optimized to best describe trends in elastic scattering data over a

wide range of energies and target masses.

Finally, the percent of data captured by two-body calculations with various OMPs are

shown in the final column of Table 4.4. The two-body calculations do a reasonable job of

capturing trends in the data considering that these calculations included no free parameters,

describing at least some data points in all cases. The percentage of captured data ranges

from 13.89% for the case of 48Ca(p,n) at 25 MeV, and 70.00% for the case of 14C(p,n) at 45

MeV. The average percent of data captured among all data sets for our two-body calculations

is 31.13%.

74

4.2.2 Three-Body Results

Next we present the results of our charge-exchange calculations utilizing the three-body

formalism introduced in Section 2.4.2. For these calculations, we study the same IAS transi-

tions in 14C, 48Ca, and 90Zr at Elab = 25, 35, and 45 MeV. For these calculations, the Lane

potential operator is replaced by a NN operator which acts directly between the projectile

and valence nucleons. For our study here, we considered the bare AV8’ and in medium

Gogny interactions (see Section 3.2.3) [75, 83, 82] . For the case of AV8’, we included isospin

contributions from the central, tensor, and spin-spin operators. For the Gogny interactions,

we included isospin contributions from the central and spin-spin operators. Additionally,

incoming and outgoing distorted waves were calculated using the KD and CH89 OMPs for

all cases.

In the three-body formalism, we incorporate a bound state wave function for the valence

nucleon in the target which will undergo charge-exchange. Each bound state is calculated

using a real Woods-Saxson potential with quantum numbers, l and j. The angular momen-

tum of each state, j, should match the spin of the core because we only consider transitions

between 0+ IAS in this work. The excitation energy of the bound state is an input to CHEX

and the user then adjusts the depth of the Woods-Saxson potential (Vv) to reproduce the

experimental binding energy. The other Woods-Saxon parameters, r0 and a, are fixed at

1.25 fm and 0.65, respectively. There is also a spin-orbit term included in the bound state

potential with a depth of 6.0 MeV. The parameters used to define the bound states used in

these calculations are given in Table 4.5.

Charge-exchange reactions can involve a complicated admixture of transitions between

initial and final states.

In the work presented here, we only include a single transition

75

Nucleus Ex(M eV )

14C
14N
48Ca
48Sc
90Zr
90Nb

g.s.
2.31
g.s.
6.67
g.s.
5.01

l
1
1
3
3
4
4

j
0.5
0.5
3.5
3.5
4.5
4.5

Core Spin Binding Energy (MeV) Vv (MeV)

0.5
0.5
3.5
3.5
4.5
4.5

8.17
5.24
9.95
2.67
11.97
0.07

48.6
48.7
51.7
51.2
46.6
58

Table 4.5: Bound state parameters used in the three-body calculations with CHEX.

between single particle states. While this is not, in general, a good approximation, here we

study a limited set of cases involving transitions between 0+ isobaric analog states. In each

of the target nuclei, 14C, 48Ca, and 90Zr, there is a shell closure or subshell closure for both

proton and neutrons, indicating a simpler nuclear structure. To further test the validity of

this assumption, we explicitly calculated the one body transition densities between 48Ca and

48Sc [101]. The one body transition density is defined as

OBT D =

(cid:104)f||[a+

kα,q(cid:48) ⊗ ˜akβ ,q]λ||i(cid:105)
√

2λ + 1

,

(4.3)

where λ = 1 and a+
kα,q(cid:48) is the creation operator which creates a particle with isospin projec-
tion q = −1/2 (neutron) with single particle quantum numbers, kα. ˜akβ ,q is the annihilation

operator which destroys a particle with isospin projection q = 1/2 (proton) with single par-

ticle quantum numbers kβ. The calculation yielded a one body transition density very close
to 1 (0.98648) for the f7/2 → f7/2 transition and gave negligible contributions for each other

transition included in the calculation. Similar calculations were performed indicating the the

14C transition was similarly dominated by a single p1/2 transition. For the IAS in 90Zr, the
g9/2 → g9/2 transition is the most important configuration with a OBTD of 0.85, although
there is a secondary p1/2 → p1/2 transition with a OBTD of 0.14 which is not included in

76

this work. However, as discussed later, we find that the cross section for a p1/2 → p1/2
transition is negligible compared to the cross section from the g9/2 → g9/2 configuration.

This gives us confidence that the single-state approximation made throughout this work is

valid these these specific transitions.

First we examine the three-body charge-exchange cross sections produced using the AV8’

interaction, shown in Figure 4.7. Calculations for three-body cross sections with AV8’ require

about 10 partial waves to converge for all cases studied here, and typical run times range

from about 1 min for the isospin central term to about 5-10 minutes for the isospin spin-

spin term, although these calculations could be further optimized for speed. This quick

convergence indicates that, even though our NN interactions have large radial form factors

at r12 = 0, once the interaction is folded with our wave functions, the resulting T-matrix

is well behaved. As with the two-body case, we observe that cross sections are peaked at

relatively forward angles, consistent with the data which indicates a direct reaction process

in all cases. In this figure, the cross section resulting from the isospin central, tensor, and

spin-spin operators are shown separately, as well as the total cross section. We observe that,

depending on the case, contributions from various components of the interaction can add

constructively or destructively.

77

Figure 4.7: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS, and
90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calculated using the AV8’
potential. The contribution to the cross section from various potential operators are shown
in dashed black for central, dotted green for spin-spin, dash-dotted red for tensor, and solid
blue for the total cross section. Calculations shown here use the KD OMP for to calculate
distorted waves.

As the beam energy increases, both the individual components of the cross sections

and the total cross sections increase in magnitude and their first peaks shift to more forward

angles. The location of the first peak of the total cross section increases with target mass, but

in the cases chosen here, this also correlates to an increase in the angular momentum of the

valence neutron, l. In order to try to disentangle these two effects, we ran a test calculation

78

of charge-exchange on 58Fe which is heavier than 48Ca, but has a valence nucleon with l=1,

opposed to l=3 in 48Ca or l=4 in 90Zr. The results of this calculation are shown in Figure

4.8. The 58Fe(p,n)58Co cross section peaks around 30 degrees, compared to 52 degrees for

48Ca(p,n)48Sc and 25 degrees for 14C(p,n)14N. This indicates that the location of the first

peak is a convolution of target mass and valence nucleon angular momentum.

Figure 4.8: Charge-exchange cross sections for the 58Fe(p,n)58CoIAS transitions at Elab =
23 MeV calculated using the AV8’ potential. The contribution to the cross section from
various potential operators are shown in dashed black for central, dotted green for spin-spin,
dash-dotted red for tensor, and solid blue for the total cross section. Calculations shown
here use the KD OMP for to calculate distorted waves.

Contributions from the central term dominate the calculations at all energies in 14C(p,n)14N,

but, expectedly, contributions from the tensor term grow with increased values of l, com-

peting with the central term in 48Ca (l=3) and dominating in 90Zr (l=4). Calculations

of 58Fe(p,n)58Co show a dominant contribution from the central term, indicating that the

growth in the tensor term is, in fact, related to larger values of l, as opposed to target mass.

It is worth noting that while the spin-spin term is negligible in 48Ca and 90Zr, it repre-

sents a sizable contribution in 14C l=1 . This can also be observed in 58Fe. One possible

explanation is related to the spin-spin operator’s radial form factor which can be seen in

79

Figure 3.4. The spin-spin radial form is the only component of the NN interactions consid-

ered in this work which contains both an attractive and repulsive region, crossing through

0 MeV at around 0.5 fm. The degree to which various regions of the potential form factor

will contribute to the cross section is related to the two dimensional radial integral of the

potential along with the incoming and outgoing scattering and bound state wave functions.

Because 58Fe and 14C have l=1, their bound state wave functions peak at lower values of

the internal radial coordinate, r2c. This could result in capturing a larger portion of the

attractive spin-spin interaction, resulting in a larger contribution to the overall cross section.

Next we examine the three-body charge-exchange cross sections produced using the

Gogny interaction, shown in Figure 4.9. Calculation time and convergence properties for

calculations with the Gogny interaction are very similar to their counterparts in AV8’ calcu-

lations. Again, we observe forward peaked cross sections, consistent with a direct reaction

process in all cases. In this figure, the cross section resulting from the isospin central and

spin-spin operators are shown separately, as well as the total cross section.

80

Figure 4.9: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS, and
90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calculated using the Gogny
potential. The contribution to the cross section from various potential operators are shown
in dashed black for central, dotted green for spin-spin, and solid blue for the total cross
section. Calculations shown here use the KD OMP for to calculate distorted waves.

As with the AV8’ calculations, the cross sections produced using Gogny grow in mag-

nitude and become more forward peaked with increased beam energy. Unlike the results

from AV8’, the central term of the Gogny interaction dominates for all target and energy

combinations. This is unsurprising given that, in Figure 3.4, we see that the isospin central

and spin-spin terms are both repulsive potentials with similar form factors, except that the

central term is about 5 times larger in magnitude.

81

We next consider the effect of the choice of KD versus CH89 OMP for the distorted waves

on the charge-exchange cross section in the three-body framework. These results, including

contributions from all potential operator terms, can be seen in panels (d)-(f) of Figures 4.4,

4.5, and 4.6. Note that in these figures, the cross sections utilizing the AV8’ interaction have

been enhanced by a factor of 10 so they could be viewed on the same scale as the calculations

using the Gogny interaction. The results for the calculations using AV8’ are shown in solid

lines, while the results utilizing Gogny are shown in dashed lines. It is clear that, while the

choice of OMP can have significant effects on the magnitude of the cross section, the angular

distributions are relatively similar, regardless of the choice of OMP. This is in contrast to

results from the two-body formalism where changing the OMP could significantly change the

shape of the cross sections, although in the two-body case, using a different OMP for distorted

waves also corresponds to changing the Lane interaction. Additionally, in the two body case,

we saw that in some cases using the KD parameters led to a larger magnitude in the cross

sections, while sometimes the opposite was true and CH89 lead to the larger cross sections.

However, in the three-body formalism, the KD OMP consistently produces the larger cross

section. This is likely related to the fact that the CH89 parameter set is more absorptive,

particularly at the surface, decreasing the contribution to the charge-exchange cross section.

The percent difference between the cross sections produced with KD versus CH89, evaluated

at the first peak, are given in the columns labeled ”AV8’ ∆OM P ” and ”Gogny ∆OM P ” of

Table 4.3. We note that the percent difference seems to be loosely correlated with the mass

of the target, and has no observable trend with regards to the beam energies studied here.

The average percent difference over all targets and energies is 58% for AV8’ and 55% for

Gogny, which is similar to the 50% demonstrated for the two-body calculations.

Comparing the results from AV8’ to those obtained with the Gogny interaction, we note

82

that the cross sections produced with the Gogny interactions are consistently about an order

of magnitude larger than those produced with AV8’. The percent difference between cross

sections produced using these two NN interactions are given in the ”3-Body ∆N N ” column of

table 4.3. The average percent difference between calculations produced with the AV8’ and

Gogny interactions, across all cases studied here, is 167%, which represents a significantly

larger variation than is introduced through the choice of OMP. This is easily understandable

given the relative magnitudes of the Gogny and AV8’ central terms in Figure 3.4. This

mismatch in the radial form factor could be a result of Gogny being fit to observables which

include in medium effects, as opposed to the bare AV8’ NN interaction.

Finally, we can compare results from our three-body calculations to experimental data.

As we can see in Figures 4.4, 4.5, and 4.6, calculations using both the AV8’ and Gogny

interaction do a poor job of describing the experimental data, with severe mismatches in

both the magnitude and the shape of the distributions. As in the two-body case, we can

normalize our calculations to the data and calculate the percent of data, within errors,

captured by our normalized calculations. These results are shown in Tables 4.6 and 4.7 for

AV8’ and Gogny, respectively. For the case of 14C(p,n), the cross sections utilizing the AV8’

interactions needed to be enhanced by a factor of about 50 to match the magnitude of the

data, while the magnitude of the cross section was similar to data in all other cases. The

Gogny interaction produced results that is similar to the magnitude to the data in all cases,

although it consistently overshoots experimental results.

Additionally, we can asses the quality of the calculation through the percentage of data

captured inside the rough uncertainty band defined by the two curves corresponding to

calculations with different OMPs. Visual inspection asserts that both AV8’ and Gogny do

a poor job reproducing experimental data and this is backed up with this metric. Both the

83

AV8’ and Gogny interaction have one case where they capture none of the data within error.

Additionally, the maximum percent of data reproduced by the AV8’ calculation is 38% for

the case of 14C(p,n) at 35 MeV, with an average across all cases of just 13%. The picture is

similar for the Gogny interaction, which has a maximum of 34% for the case of 90Zr(p,n) at

45 MeV and an average across all data sets of 14%.

Reaction

3-Body KD 3-Body CH89
Target Elab (MeV) χ2/N Norm χ2/N Norm
14C
14C
14C
48Ca
48Ca
48Ca
90Zr
90Zr
90Zr

5.51
6.78
1.71
12.66
7.24
5.84
18.89
9.69
6.62

63
45
32
3.90
2.70
2.50
6.20
2.00
1.20

25
35
45
25
35
45
25
35
45

4.84
7.35
1.95
12.17
6.85
6.26
13.81
9.39
6.91

58
34
25
2.10
1.80
1.60
1.00
0.23
0.50

Average

3-Body ∆OM P

% Data

0.00
37.50
3.13
0.00
15.63
12.50
9.38
9.38
25.00
12.50

Table 4.6: Numerical comparison of three-body charge-exchange angular distributions to ex-
perimental data. The final column shows the percentage of the data, including experimental
errors, which falls between the theoretical curves produced by three-body calculations with
the AV8’ NN interaction and the KD and CH89 OMP used for distorted waves.

The large variation in the magnitude and shape of charge-exchange cross sections that

results from the implementation of different interactions implies that the charge-exchange

cross section is very sensitive to the choice of interaction. With this in mind, we wished to

explore whether or not charge-exchange across a range of target masses and beam energies,

such as those explored in the work, could be well described by a potential with a simple

form, such as the Gogny interaction. For this brief exploration, we tuned the 4 parameters

which define the isospin central term of the Gogny interaction to best fit experimental data

84

3-Body KD 3-Body CH89
Target Elab (MeV) χ2/N Norm χ2/N Norm
14C
0.80
14C
0.40
14C
0.40
48Ca
0.40
48Ca
0.10
48Ca
0.30
90Zr
1.20
90Zr
0.40
90Zr
0.20

6.83
11.25
4.22
8.88
18.29
2.01
14.09
7.20
4.54

25
35
45
25
35
45
25
35
45

6.80
11.47
6.42
9.93
13.72
2.38
13.95
7.07
4.00

0.70
0.30
0.20
0.20
0.10
0.20
0.20
0.10
0.10

Reaction

Average

3-Body ∆OM P

% Data

9.38
28.13
6.25
3.13
28.13
9.38
0.00
9.38
34.38
14.24

Table 4.7: Numerical comparison of three-body charge-exchange angular distributions to ex-
perimental data. The final column shows the percentage of the data, including experimental
errors, which falls between the theoretical curves produced by three-body calculations with
the Gogny NN interaction and the KD and CH89 OMP used for distorted waves.

Parameter Set
Original Gogny

Refit Gogny

µ1 (fm) µ2 (fm) H1(MeV) H2(MeV)

0.7
0.71

1.2
1.2

-469.2
-760

37.37
135

Table 4.8: Original and refit parameters for the central component of the Gogny NN inter-
action.

for the case of 48Ca(p,n)48Sc at 35 MeV. This was done qualitatively, just fitting the data

by eye. We then fixed these values and calculated the charge-exchange cross sections for all

other cases. The original Gogny parameters, and the refit parameters are shown in Table 4.8

and the results of these calculations are shown in Figure 4.10. We note that this fitting is

not unique. Our refitting involved increasing the strength parameters, but it is likely that an

equally good fit could be achieved through another combination parameters with decreased

strength.

85

Figure 4.10: Charge-exchange cross sections for the 14C(p,n)14NIAS, 48Ca(p,n)48ScIAS,
and 90Zr(p,n)90NbIAS transitions at Elab = 25, 35, and 45 MeV calculated using a refit
Gogny potential with only a central contribution (back). The red curve shows the original
calculation using the isospin central Gogny interaction.

Qualitatively, the refit Gogny interaction has better success describing the experimental

data. The cross sections produced for all cases are similar in magnitude to the data and,

in some cases, the description of the angular distribution is much better than for the unfit

Gogny interaction. Additionally, the quality of the description does seem to improve with

beam energy. Still, cross sections utilizing the refit interaction struggle to describe cross

sections that peak at 0 degrees, suggesting that the assumed gaussian shape for the NN

interaction is not physically adequate.

86

4.2.3 Comparing Reaction Formalisms

Now that we have examined results from both formalisms separately, we will comment on

how results calculated in the two-body versus three-body formalisms compare. In particular,

there are significant differences in the angular distributions between the two frameworks.

First, in the three-body calculations, the angular diffraction at large angles are washed out

for both NN interactions considered here relative to the two-body calculations.

Second, the angular distributions for the three-body calculations display a much smaller

variation between the various target/beams energies explored in this work than their two-

body counterparts. For almost every case, the three-body cross section is dominated by a

single peak around 30-50 degrees, while the two-body calculations peak between 0 and 40

degrees and exhibit more complicated structures with multiple peaks. In order to understand

this difference, we examined which partial waves were contributing most to the charge-

exchange cross section for the two-body and three-body calculations.

Generally speaking, the shape of the angular distributions is created by combining the

spherical harmonics used to describe the partial wave expansion of the cross section, weighted

by factors that include angular momentum conservation and the radial behavior of the reac-

tions. Lower values of scattering angular momentum, L, correspond to more central inter-

actions with small impact parameters, while cross sections that more heavily weight larger

L contributions are more peripheral. Figure 4.11 shows the angle integrated cross section

as a function of scattering angular momentum, L, for the case of 90Zr(p,n)90Nb at 25 MeV,

where the angular distributions produced by the two-body and three-body frameworks are

severely mismatched. In this figure the three-body cross section with the Gogny interaction

is scaled so that all contributions can be shown on the same plot. The cross sections for

87

the three-body formalisms peak at larger values of angular momentum, implying that the

reaction mechanism mediated by the two-body Lane interactions is more central, and the

charge-exchange reactions mediated by NN interaction in the three-body formalism are more

peripheral.

Figure 4.11: Angle integrated charge-exchange cross sections and a function of scattering an-
gular momentum for 90Zr(p,n)90Nb at 25 MeV for the three-body formalism with the Gogny
and AV8’ interactions, and the two-body formalism with the Lane potential parameterized
by KD and CH89.

To further explain this result, we can explore the radial parameter space to demonstrate

which regions significantly contribute to the charge-exchange cross section. To do this, we

calculate the radial integral of the interaction with the initial and final wave functions,

including bound states for the three-body formalism, in different portions of the radial pa-

rameter space. For the two-body formalism, there is only one radial variable, R1A, but for

the three-body formalism, there are three radial variables, R1A, r2c, and r12. Figure 4.12

shows the value of the radial integral as a function of an interior radial cut in each of the

these variables for the AV8’, Gogny, and Lane interactions. The specific case shown here is

88

for charge-exchange on 48Ca at Elab=35 MeV, although these results are consistent across

different targets and beam energies.

Figure 4.12: Dependence of radial integral on the range of various radial parameters. The
x-axis shows the internal radial cut in each radial parameter.

As expected, for both of the short range NN interactions, the radial integral falls off

quickly as a function of r12, but when we make cuts in the r2c and R1A parameters, the

radial integral does not significantly decrease until about 3 fm. The 47Ca + n bound state

89

that describes the target has a radius of R = r0A

1/3
T where, in this case, r0 = 1.25 and

R = 4.5. This indicates that, in the three-body formalism, the reaction is occurring near

the surface of the target, where the proton and neutron are close together, resulting in small

values of r12.

In the two-body formalism, however, there is only one scattering variable

and the radial integral begins to drop off significantly around 1.5 fm, indicating a reaction

that is involving much more of the nuclear interior. This reaffirms our prior assertion that

the three-body framework results in a reaction concentrated closer to the surface region,

modifying the resulting angular distribution compared to the two-body approach.

4.2.4 Discussion

Here, we comment on our study in the context of other methods which have been used to

study charge-exchange reactions. First, we examine the theoretical calculations presented

alongside the data sets used in the analysis. Doering et al. includes DWBA calculations for

the 48Ca(p,n) and 90Zr(p,n) reactions considered here [18]. For their calculations, they used

the OMP parameterized by Bechetti and Greenlees to calculate distorted waves [66]. In this

work they considered a number of effective interactions, including two parameterizations of

simple Yukawa potentials with a 1 fm range, and a realistic NN potential derived from the

Reid soft core potential [78, 102]. Doering et al. included the direct and knock-on exchange

contributions exactly. As mentioned before, we do not address knock-on exchange in our

calculations. These calculations yielded a reasonably good description of the data, similar in

quality to our two-body calculations. The methods used by Doering et al. are similar to our

three-body formalism in the sense that they can describe microscopic transitions between

single particle states in the initial and final nucleus and they describe their incoming and out-

going distorted waves using a separate scattering coordinate. However, it is unclear exactly

90

what approximations and methods they implement in their calculations, so it is difficult to

draw more detailed comparison. Additionally, their choice of interaction is such that they

do not see large contributions from non-central isospin terms, such as the isospin tensor and

spin-orbit interactions.

In contrast, we see increasing contributions from the tensor term

as the value of l increases in our target. Finally, our current calculations only include one

initial and final single particle state configuration. For the case of 48Ca, Doering et al. also

models the initial and final states as a pure f7/2 to f7/2 transition, so differences between the

results presented here and their calculations arise solely from the different interactions and

frameworks implemented in the two cases. However, for 90Zr, the shell model configurations

from Doering et al. are 0.8(1g9/2)10 + 0.6(1gg/2)8(2p1/2)2. In our case, we used a pure g9/2

configuration. We performed a preliminary calculation showing that the cross section for

the p1/2 transition is negligibly small compared to the g9/2 transition, so our single orbital

calculation for 90Zr should be slightly larger than the calculation that uses the configuration

from Doering et al.

Next, we can examine the results presented alongside the experimental data for 14C(p,n)

in Taddeucci et al.

[17]. They, again, use an charge-exchange reactions framework similar

to that presented in [27]. For the case of 14C(p,n), Taddeucci et al. does not use a global

parameterization to describe the incoming and scattering waves but, instead, uses an optical

model parameterization fit specifically for that work. The parameterization was chosen by

fitting a Lane-type interaction to their charge-exchange data to the isobaric analog state in

14N. For the effective NN interaction, Taddeucci et al. employs the BBML interaction [103],

which is based on the M3Y interaction [104]. Again, they calculated direct and knock-on

exchange terms explicitly. In this work they found they were unable to describe the charge-

exchange data accurately unless they modified the BBML interaction, demonstrating a high

91

sensitivity in the NN interactions as we observed in the work presented here.

Additionally, our two-body study employs a very similar methodology to that employed

by Danielewicz et al.

[10]. In that work, Danielewicz used an equivalent charge-exchange

framework to calculate transition between 0+ IAS using a Lane potential. Danielewicz et

al. considered a wide range of targets, including 48Ca(p,n) and 90Zr(p,n), and used the KD

optical model parameterization for their distorted waves and Lane interaction. Because of

this, our two-body results using KD are almost identical to those presented in Danielewicz

et al., although there is a slight variation owning to the fact that in Ref [10], the authors

modified the KD to use the same geometry for proton and neutron scattering potentials.

Danielewicz et al. simultaneously fit proton elastic scattering, neutron elastic scattering, and

charge-exchange data in order to uncover differences between isovector and isoscalar potential

geometries, which were then used to extract a neutron skin thickness. They determined

that there was a significant difference between the geometries. However, their study only

utilized one optical model parameterization, but our results indicate that using the CH98

parameterization can have a significant effect on both the magnitude and shape of angular

distributions. We know that there are large parametric uncertainties arising from OMPs.

It would be informative to explore how propagating these parametric uncertainties through

this simultaneous fitting procedure effects the uncertainties associated with the resulting

isovector skin predictions.

In our work, we chose a position space representation. However, the formalisms can

equivalently be cast in a momentum space representation. Lenske et al. has developed

a microscopic DWBA formalism which, in momentum space, separates into projectile and

target transition form factors, with a distortion coefficient [105]. This model has been used

to describe single charge-exchange using heavy ions (i.e. 18O+40Ca), as opposed to the (p,n)

92

reactions discussed here. In this formalisms, the ground states of the target and projectile

nuclear densities are derived from Hartree-Fock-Bogolubov with excited states described by

the Quasiparticle Random Phase Approximation. Lenske et al. [105] suggests that a similar

factorization between structure and reaction dynamics as is used in Equation(1.1) for (p,n)

Gamow-Teller transitions is appropriate. This factorizability could be possible for beam

energies of a few 10s of MeV per particle in a heavy ion probe. This model was not applied

to systems with light probes and, therefore, cannot be compared to our work.

Finally, the 48Ca(p,n) and 90Zr(p,n) data considered here has also been analyzed using

a microscopic model by Khoa et al. [50]. In that work they looked at both (p,n) and (3He,t)

reactions using the folding model (double-folding model in the case of (3He,t) reactions)

which integrates over the nuclear density of the target and probe. This is in contrast to

our work which does not incorporate nuclear densities and, instead, describes the projectile-

target interaction with a global optical potential. In their work, they use a coupled channel

formalism to describe elastic and charge-exchange reactions. The effective NN interaction

is, in this case, density dependent (the CDM3Y6 interaction based on the M3Y interaction

[106]). The goal of their work is to fit data by adjusting the isospin dependence of the NN

interaction. This can be used to make realistic predictions about the density dependence of

the symmetry energy. Their calculations were able to produce a good description of the data,

which were improved by tuning the complex components of the OMPs to each individual

target. Similar methods are used to analyze charge-exchange reactions in [52, 51, 48, 49].

4.2.5 Limitations of Current Work

Finally, this work represents a preliminary effort to study the reaction dynamics of charge-

exchange reactions and there are a number of limitations in this current study. Perhaps

93

most importantly, all cross sections calculated in this work use a single step DWBA, how-

ever, at beam energies as low as 25 MeV, there are likely contributions to the charge-exchange

cross section from higher order processes which are not included in our current model. Al-

though there have been efforts to quantify possible effects from multistep processes in charge-

exchange (i.e.

[107]), here we will just note that these effects are likely significant and the

reaction formalism should be extended to better understand the role that multistep process

play in low energy charge-exchange reactions.

Additionally, as discussed in Section 2.4.2, this work does not include contributions from

the isospin, spin-orbit term of the AV8’ interaction. We did not include this component

because of complications of expanding the angular momentum operator, ˆL, in the coordinate

system used to describe our framework. Although the contribution from this term is possibly

small, it should be included explicitly and a derivation explaining how to do this can be found

in Appendix C.4. This is particularly important because AV8’ is a reprojection of the more

complicated v18 interaction that reduces 14 operator terms to just 8 terms. This means that

it is difficult to predict the effect of leaving out any particular term of the interaction.

Finally, the theoretical uncertainty quantification discussed in this work is very basic.

In the future, a more robust uncertainty quantification study should be conducted in the

realm of charge-exchange reactions using the Bayesian methods recently developed by other

members of the reaction theory group at Michigan State University. These analysis tools

for studying uncertainty from the OMPs are already in place for transfer reactions and

extending this framework to charge-exchange reactions should be fairly straightforward, as

CHEX already calculates the elastic scattering cross sections which are necessary for the

analysis. Additionally, it would be illuminating to study the role that charge-exchange cross

sections play in constraining the uncertainties in OMPs, given the demonstrated sensitivity

94

of charge-exchange to the choice of interaction.

95

Chapter 5

Conclusions and Outlook

5.1 Conclusions

Indirect reaction methods are an important tool for exploring the evolution of the elements

in our universe, but they are only useful if we have reliable theories to link indirect reaction

mechanisms with the astrophysical processes of interest.

In this thesis we explored two

indirect reaction methods, transfer and charge-exchange reactions, with an emphasis on the

reaction theory models used to understand them.

We discussed several examples of using deuteron induced transfer reactions to probe the

astrophysical proton capture reaction rate. We introduced a methodology that allows us to

extract spectroscopic factors of low lying resonances of interest. These values then directly

inform the astrophysical reaction rate via the resonance strength, ωγ. These spectroscopic

factors were extracted from angle integrated cross section measurements using transfer cal-

culations which utilize the adiabatic distorted wave approximation. In many cases, a bound

state approximation was used to simplify transfer calculations to low lying resonances with

non-zero orbital angular momentum, l. We presented results of three transfer reaction stud-

ies. The study of 30P(d,n) for applications to the 30P(p,γ) reaction in classical novae marks

the first experimental constraints on the 30P(p,γ) reaction rate. In the study of 23Al(d,n)

for applications to the 23Al(p,γ) reaction in type-I x-ray bursts, the uncertainty in the as-

96

trophysical 23Al(p,γ) reaction rate was reduced by 3-4 orders of magnitude in the relevant

temperature range. Finally, the work examining 56Ni(d,n) for applications to the 56Ni(p,γ)

reaction, also important in x-ray bursts, constrained the 56Ni(p,γ) reaction rate via the first
measurement of the 7/2− state in the 57Ni mirror nucleus.

In the realm of charge-exchange reactions we conducted a systematic study of transitions

to 0+ isobaric analog states, over a range of target nuclei and beam energies. All charge-

exchange calculations presented in this study were produced using the charge-exchange re-

action code, CHEX, developed for this thesis. This study marks an initial investigation

into how various aspects of the reaction model effect charge-exchange observables. We ex-

plored a two-body framework where the charge-exchange transition is mediated by a Lane

potential. These types of interactions are described by a difference in proton+target and

neutron+target optical potentials. We explored the impact of using two different optical

model parameterizations (KD and CH89) on charge-exchange cross sections. This simple

two-body model was reasonably successful in describing the shape and magnitude angular

distribution of data. On average, there was a 50% difference in the cross sections produced

by the two parameterizations, which is similar to the spread observed in other reactions.

Finally, we examined the extent to which the data could be described, within error, by these

calculations. We defined rudimentary theoretical error bands with the normalized cross sec-

tions produced by the KD and CH89 OMPs and calculated the percent of data captured by

these bands. On average, 31% of data in our study was captured by our normalized two-body

calculations.

Next we extended our description of charge-exchange to a three-body framework, which

uses an NN interaction to describe charge-exchange between a scattering nucleon and a

valence nucleon bound to an inert core. This model allows a valence nucleon to transition

97

between two well defined single-particle states. We explored the effect of using two different

NN interactions: the AV8’ [75, 83] and Gogny [82] interactions. We found that the three-

body formalism, regardless of the choice interaction, was unsuccessful in reproducing either

the shape or magnitude of the charge-exchange angular distribution data used in this study.

Additionally, the normalized error bands for the AV8’ and Gogny interactions only captured

13% and 14% of the data, respectively.

We noted that the three-body charge-exchange cross section is highly dependent on the

choice of NN-interaction. The cross sections produced with the Gogny interaction are typi-

cally about an order of magnitude larger than those produced by the AV8’ interaction and,

on average, the choice of interaction results in a 167% difference between the cross sections

at the first peak. This is significantly larger than deviations caused by the choice of OMP

used to calculate the incoming and outgoing distorted waves, which lead to a 55% difference,

on average.

Finally, we noted that the shape of the angular distributions in the three-body model, re-

gardless of the choice of interaction, differ significantly from their two-body counterparts and

from experimental data. We determined that the two-body formalism more heavily weighted

lower partial waves in the cross section calculation, indicating a more central reaction that

involves a significant portion of the nuclear interior. In contrast, the three-body formalism

using the Gogny and AV8’ interactions select higher partial waves, resulting in a reaction

located near the surface of the target.

98

5.2 Outlook

This thesis presents initial findings regarding charge-exchange reactions in a two-body and

three-body formalism, but there are a number of important limitations in this investigation,

as well as a broad outlook regarding further studies. Because the two-body formalism pre-

sented here has a relatively straightforward implementation and short calculation time, it is

a particularly good candidate for a more rigorous uncertainty quantification study. Our pre-

liminary analysis shows that, on average, the charge-exchange cross section varies by 50%,

depending on the choice of OMPs. In order to better understand how parametric uncer-

tainties affect the precision of charge-exchange calculations, the Bayesian methods recently

developed to quantify parametric uncertainties in transfer reactions should be extended to

this reaction channel [61]. Additionally, all interactions considered for this work are purely

local, but it has been demonstrated that including the effects of non-local interactions can

have a large impact on the transfer cross section. The CHEX charge-exchange reaction

code incorporates the non-local Schr¨odinger equation solver from the NLAT reaction code

[81]. Although this effect was not investigated in our current study, CHEX allows for easy

extension to examine the effect of non-local potentials in future studies.

There are also several opportunities to improve and expand on our study of the three-

body formalism. As discussed in Section 2.4.2, we have implemented three of the four isospin

operators in the AV8’ interaction, but the isospin spin-orbit interaction is not included due to

complications expressing the ˆL operator in the coordinate system used to describe scattering

and bound states. Although the full implementation is outside of the scope of this thesis, a

derivation outlining how to explicitly include this term can be found in Appendix C.4.

In this work, we implemented the AV8’ and Gogny interaction, but CHEX is readily able

99

to include any position space nuclear interaction which is cast in an operator form. Most

notably, the effective NN interaction developed by Love and Franey [59, 60] is commonly

used to analyze charge-exchange reactions and has a parameterization for scattering energies

ranging from 50-1000 MeV. The Love-Franey interaction was not included here because it

is not parameterized for energies below 50 MeV and is considered to be less accurate at

low energies. Additionally, there are complications involving the knock-on exchange term.

Future developments to include this interaction with a knock-on exchange term would allow

for a clearer comparison between charge-exchange calculations presented in this work and

other models, such as in [17].

This thesis limited its study to charge-exchange transitions to 0+ isobaric analog states.

However, there are other important transitions that should be considered. Most notably, the

Gamow-Teller transition (∆L=0, ∆S=1, ∆T=1) is of particular physical interest as it is used

to constrain electron-capture rates in supernovae. CHEX was designed to be generalizable,

so this implementation should only require an extension of the spin-spin T-matrix expression

to include ∆S=1 transitions. Additionally, we know that the structure of many nuclei are

not well described by one single-particle state and, instead, are highly fragmented. In order

to describe these nuclei accurately, the CHEX front end program would need to be modified

to input transitions between multiple orbitals.

Finally, both the two and three-body frameworks presented in this thesis use single step

DWBA. While the single-step approximation is a valid reaction model at medium to high

beam energies, this study includes beam energies as low as 25 MeV, where there are likely

contributions to the charge-exchange cross section from higher order processes. There have

been efforts to quantify effects from multistep processes in charge-exchange, including work

by Madsen et al. which proposes an energy dependent correction factor [107]. We note

100

that these effects are likely significant and it is crucial that future investigations of charge-

exchange reaction models explicitly implement a formalism which accounts for multi-step

processes.

101

APPENDICES

102

APPENDIX A

Summary of Additional Transfer

Results

This appendix contains a summary of the methods and results used to analyze the 56Ni(d,n)57Cu

and 56Ni(d,p)57Ni reactions for applications to astrophysical capture, as well as a brief intro-

duction to a new method that has been proposed to utilize the Active Target Time Projection

Chamber (AT-TPC) to more precisely extract the spectroscopic factor from transfer reac-

tions.

A.1 56Ni(d,n)57Cu

The 56Ni(p,γ)57Cu reaction plays a key role in the rp-process in XRBs, acting as a key

waiting point that impedes flow along the proton drip line [108]. This characteristic is due

to the unique structure of 56Ni, which is doubly magic. This N=Z=28 isotope lies right on

the edge of stability with a 6.08 day half-life. In order to reduce the uncertainty in this key

reaction, the 56Ni(d,n)57Cu transfer reaction was measured using the same GRETINA/S800

setup described in Section 4.1, this time using 33.6 MeV/u beam. However, this experiment

was unique in that it also measured the transfer reaction to the mirror nucleus of 57Cu via

56Ni(d,p)57Ni. This measurement allows for a test of the isospin symmetry between these

two nuclei. Additionally, information on excited states of the mirror nucleus, 57Ni, can allow

103

for constraints on the astrophysical proton capture reaction rate to 57Cu in cases where these

states are not directly measured.

The theoretical cross sections for 56Ni(d,n)57Cu and 56Ni(d,p)57Ni were calculated ac-

cording to the same process described in Section 3.1. In this case, the final states in 57Ni

are bound, so the final bound states can be modeled with a real Woods-Saxon potential

where the depth is adjusted to reproduce the experimental binding energy. In the case of

57Cu final states, we employ the bound state approximation to model the final resonance

states. The shell model calculations which produced theoretical spectroscopic factors used

the GPFX1A Hamiltonian [109] to describe the isospin-conserving strong interaction, along

with the Coulomb and charge-dependent Hamiltonians from [110]. A brief summary of these

results are given in Table A.1.

56Ni(d,n)57Cu

Ex (MeV) Eres (KeV)

1.028
1.109
2.398
2.525

338
418
1708
1835

J π
5/2−
1/2−
5/2−
7/2−

σexp (mb) σth (mb)
2.00(40)
0.28(6)
<0.2
<0.2

2.62
0.45
2.61
14.5

C2Sexp
0.76(28)
0.62(22)
< 8 × 10−2

C2SSM
0.75
0.71

1.8 × 10−3
3.9 × 10−2

l

3
1
3
3

ωγ

1.7 × 10−11
1.9 × 10−7
1.0 × 10−2
2.7 × 10−2

56Ni(d,p)57Ni

Ex (MeV)

0.768
1.122
2.443
2.579

J π
5/2−
1/2−
5/2−
7/2−

σexp (mb) σth (mb)
2.10(60)
0.50(15)

2.77
0.68
2.61
14.9

<0.4

1.24(36)

C2Sexp
0.77(31)
0.73(31)
8(3) × 10−2

<0.1

C2SSM
0.74
0.69

3 × 10−4
4.1 × 10−2

l

3
1
3
3

Table A.1: Summary of results from [7] for the study of 56Ni(p,γ)57Cu via 56Ni(d,n)57Cu
and the mirror reaction of 56Ni(d,p)57Ni. States which were not observed in this study are
listed with upper limits in the experimental cross section. A large proton branching ratio
in inferred for decay of the 7/2− resonance at 1.835 MeV, so no experimental spectroscopic
factor is listed for that state here.

By comparing experimental and theoretical spectroscopic factors, it is clear that the low

104

lying states in both 57Cu and 57Ni exhibit a strong single-particle nature, and agree well with

the results of shell model calculations. Most importantly, charge-symmetry is preserved. For

higher lying states, spectroscopic factors, either extracted from measurement or constrained

by upper limits, are much smaller, consistent with the shell model predictions. Shell model
predictions for spectroscopic factors for the 5/2− state around 2.4 MeV in both final nuclei

are small, consistent with their non observation in this case. Finally, the observation of the
7/2− state in 57Ni constitutes the first measurement of this state.

These results, including constraining the 7/2− state in 57Cu through the measurement

in the 57Ni mirror, can be used to calculate the reaction rate in the temperature range

relevant for XRBs, as shown in Figure A.1. For temperatures below about 1 GK, the rate

is dominated by the l = 2 state at 418 keV. However, at higher temperatures, the higher

lying l = 3 states both contribute substantially and come to dominate the reaction rate. The

rate calculated in [7] which incorporates the these newly constrained spectroscopic factors is

significantly higher than the previous result from Zhou et al. [21] which used experimental

excitation energies, but only theoretical widths. For lower temperatures, the result from [7]

agrees well with the central value from Rehm et al.

[20] which incorporated experimental

spectroscopic factors for the lower-lying resonance strengths states derived from their study of

the 56Ni(d,p)57Ni reaction. However, at higher temperatures the rate predicted by [7] begins

to diverge from Rehm et al. because of the inclusion of the newly constrained spectroscopic

factors for the higher-lying l = 3 states.

105

Figure A.1: The 1-σ uncertainty band of the reaction rate determined in [7] is shown in
red, compared with the previous calculations from refs. [19, 20, 21]. Results match previous
results from Rehm et al. at lower temperatures, but begin to diverge at higher temperatures
because of the constraints imposed on higher lying states by this work. Figure from [7].

A.2 Using the ATTPC to Validate the Combined Method

Although it is common to directly extract spectroscopic factors by normalizing experimental

cross sections to theoretical calculations, this practice produces a large uncertainty because of

ambiguity in the single-particle potential used to create the final bound state wave function.

The combined method, first put forth by Mukhamedzhanov et al.

[111], proposes that

measuring transfer reactions at both a high and low energy offers a way to constrain single-

particle geometry and, therefore, the spectroscopic factor. Time projection chambers offer a

unique opportunity to apply the combined method in one measurement.

A.2.1 Asymptotic Normalization Coefficients and the Combined

Method

As mentioned before, the final bound state in transfer calculations is modeled as a nucleon

in a mean-field, Woods-Saxon potential, where the depth of the potential is adjusted to

reproduce the experimental binding energy for the final, A + N nucleus. The radius and

106

diffuseness of the potential well, r0 and a, are chosen to be realistic, but usually arbitrary

values. However, if these values are adjusted, even within reasonable limits, they will directly

impact the single-particle wave function produced in the calculation and can have a large

impact on the cross section produced in DWBA (see Figure A.2). This will, in turn, introduce

an uncertainty in the spectroscopic factor.

Figure A.2: Single-particle wave functions ψsp(rAN ), produced by changing the radius of the
Woods-Saxon potential from r0 = 1.2 to r0 = 1.3. These values are both within a reasonable
range for the single-particle geometry but create significant differences in the wave functions
which, in turn, directly impact the cross section.

One way to improve this description is through the use of asymptotic normalization co-

efficients (ANCs). The ANC describes the strength of the tail of the exponentially decaying

wave function. The many-body ANC (Clj) describes the behavior of the full, many-body
overlap function, φ(rAN ) = (cid:104)ΨA+1|ΨA(cid:105), of the A+1 nucleons in the final state, which is usu-

107

ally not known, and the single-particle ANC (blj) describes the tail of the single-particle wave

function, ψsp(rAN ) which is easy to calculate, but affected by the shape of the Woods-Saxon

geometry (see Figure A.2). As mentioned in Section 2.3.2, in standard DWBA calculations

φ(rAN ) is often replaced by S1/2ψsp(rAN ), an approximation of unknown validity. Although

the many-body and single-particle wave functions can be very different in the interior region,

they exhibit similar behavior at large distances (i.e. Whittaker functions), differing by only

a normalization factor (S1/2). Therefore outside the range of the nuclear interaction, the

overlap function defined in Equation (2.29) can be written as:

φ(rAN )

r>Rn= CljW−η,l+1/2(2kr)/r = S1/2bljW−η,l+1/2(2kr)/r

(A.1)

where W is the Whittaker function. It is important to stress that this expression is only

exact at large distances, where both the single-particle and many-body wave functions are

described by the Whittaker form. Therefore, the spectroscopic factor can be defined as

S =

C2
lj
b2
lj

.

(A.2)

With this in mind, Mukhamedzhanov and Nunes [111] have proposed the combined

method, which involves two measurements of the same transfer reaction, one at a low beam

energy and one at a higher beam energy, to more efficiently constrain the spectroscopic fac-

tor. At low beam energies (<10MeV/u), the projectile can only probe the tail of the nuclear

wave function, resulting in a peripheral reaction. Because the cross section is proportional to

the square modulus of the overlap function, and the form factor of the overlap is well known

at peripheral distances, the overall normalization Cij can be extracted without sensitivity to

the single-particle geometry. The second measurement, at higher energies, will probe deeper

108

into the nuclear wave function, and is sensitive to both the nuclear interior and exterior.

However, the many-body ANC is a property of the structure of the target, not the beam

energy, so it can be used to constrain the spectroscopic factor extracted from the high energy

measurement. This will, in turn, constrain the single-particle geometry through the ANC.

In practice, the single-particle ANC is calculated directly from theory, (in our case, us-

ing the reaction code FRESCO) and for each blj, the spectroscopic factor is extracted by

normalizing the corresponding theoretical cross section to experimental results. Because of

the relation in Equation (A.2), this uniquely determines the many-body ANC. A cartoon

demonstrating how this would work is shown in Figure A.3. Calculations can be repeated

using a range of geometries, resulting in a range of single-particle ANCs and, therefore, spec-

troscopic factors. In the left plot, it is clear that depending on the single-particle geometry

chosen (which translates to different values of blj), a wide range of spectroscopic factors can

be extracted. However, the right plot demonstrates that at low energies, the reaction is pe-

ripheral and the many-body ANC (Clj) is flat, meaning it is insensitive to the single-particle

mean field geometry and, therefore, Clj can be reliably obtained. Then at high energies, we

use the previously established Clj to fix the range for blj and are able to obtain a constrained

spectroscopic factor.

This method was first validated by Walter et al. with an analysis of 86Kr(d,p)87Kr from

two different experiments, one at 5.5 MeV/u [112] and one at 33 MeV/u [113]. This allowed

for the extraction of a spectroscopic factor that was dominated by experimental errors, as

opposed to theoretical uncertainties associated with the bound state geometry.

109

Figure A.3: Left: A cartoon of the spectroscopic factor as a function of single-particle ANC,
blj, at two different beam energies. Right: A cartoon of the many-body ANC, Clj, as a
function of single-particle ANC blj, at two different beam energies.

A.2.2 Proposed Measurement and Preliminary Calculations

One practical drawback of the combined method is that it requires the combination of results

from two different experiments, which introduces its own uncertainties and complications.

Often different facilities and setups are necessary to make measurements at high and low

energies, introducing different systematic errors across the measurements. In addition, ob-

taining the beam time necessary for two experiments can be a challenge. Thus, a new

technique has been proposed to measure both a high and low energy transfer reaction in a

single experiment. This would be accomplished through the use of a time projection cham-

ber (TPC) with an active target configuration. The main benefit of this setup is that the

detector gas which fills the TPC volume, for our purposes deuterium gas, also serves as the

reaction target [114]. As a beam passes through the gas in the TPC, it will slow down.

The beam can react anywhere within the TPC volume, allowing for a continuous reaction

measurement at different energies, approaching 0 Mev/u at the far end of the chamber. Ad-

ditionally, this method allows for a nearly 4π solid angle coverage which is important for

110

measuring reactions utilizing low-intensity rare isotope beams.

It has been proposed to perform this type of measurement using the Active Target Time

Projection Chamber (AT-TPC) [115] at the National Superconducting Cyclotron Laboratory

[116]. The initial experiment to validate this new method would measure 86Kr(d,p)87Kr,

which is relevant to neutron capture in the weak r-process and has the advantage of a well

know structure at low excitation energies. A beam would be introduced to the AT-TPC

from the ReA6 facility at around 18 MeV (9 MeV/u). As the beam moves through the gas

chamber, the energy is attenuated, allowing for the collection of transfer reaction data at a

wide range of energies. Preliminary calculations for [116] indicate that, in the energy range

currently available at the NSCL, it might be possible to implement the combined method in

a single measurement.

The results of these calculations are shown in Figure A.4. The single particle ANC, blj

is calculated using FRESCO for a range of mean-field geometries. The overall normalization

is arbitrary: a transfer calculation with an reasonable choice of single-particle geometry, as

described in Section 3.1, is chosen to act as the ”data” that is used in the extraction of spec-

troscopic factors for the rest of the calculations. Once the spectroscopic factor is extracted,

the many-body ANC can be determined using relation A.2. This process is repeated at

a lower reaction energy (in this case, 5 MeV) and a higher energy (in this case, 8 MeV).

These energies represent relatively conservative estimates of the energy window that could

be attained with the AT-TPC at the NSCL. Ideally the lower energy many-body ANC will

show little to no dependence on the mean field geometry, and the constrained mean field pa-

rameters can be determined by the crossing point of the calculations from different energies.

The hope is to constrain, not only the ground state geometry as indicated in the left plot of

Figure A.4, but also a number of excited states in 87Kr, for example the 3/2+ state at 1.47

111

MeV excitation energy shown in the right plot of Figure A.4. If successful, this could be a

powerful new tool in the realm of transfer spectroscopy.

Figure A.4: Left: Preliminary calculations showing the many-body ANC, Clj, as a function of
single-particle ANC blj, for the ground state of 87Kr. Left: Preliminary calculations showing
the many-body ANC, Clj, as a function of single-particle ANC blj, for the 3/2+ excited state
of 87Kr at 1.47 MeV excitation energy. The orange points correspond to calculations at 5
MeV/u and the blue points correspond to calculations at 8 MeV/u. These energies represent
a conservative energy range attainable with the ATTPC.

112

APPENDIX B

Derivation of Charge-Exchange

Two-Body T-Matrix

This derivation is for the simplified case of (p,n) charge-exchange with a two-body formalism.

The target has mass A, N neutrons, and Z protons. For this study we are only looking at

isobaric analog state transition on 0+ targets. The scattering coordinate, R1A, runs from the

center of mass of the target to the scattering proton/neutron. The interaction is moderated

through a bulk potential, in this case, the Lane Potential [47]. It is defined as

(cid:112)|N − Z|

A

A

2(N − Z)

(cid:104)

Un(r) − Up(r)

(cid:105)

1(cid:112)|N − Z|

=

(cid:104)

Un(r) − Up(r)

(cid:105)

,

(B.1)

VLane = 2

where Un and Up are proton and neutron optical potentials.

In our study we consider

OMPs parameterized by Koning-Delaroche [1] and Varner et al. [2]. All angular momentum

identities used here come from [117]. For this derivation we use the time reversed phase

convention such that

Y l∗
m (ˆr) = (−1)l−mY l−m(ˆr).

(B.2)

In the incoming channel, the initial scattering state is

113

(cid:88)

Li

iLi(−1)Lie

iσLi χLiJi

(R1A)Y

Li
0 ( ˆR1A)ΞI1,µ1

(ξ1)

√
4π ˆLi
kiR1A

,

(B.3)

χ+
i =

2Li + 1 and we have set Mi = 0, which correspods to alligning the beam with

√

where ˆLi =

the z axis. We can couple the spinor to the incoming spherical harmonic to get

(cid:88)

LiJi

χ+
i =

iLi(−1)Lie

iσLi

√
4π ˆLi
ki

χLiJi

(R1A)

R1A

C

Jiµ1
Li0I1µ1

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji

µ1

.

(B.4)

There is an equivalent expression for the outgoing channel, written as:

−Lf (−1)

(cid:88)
Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf−Mf

Lf Jf Mf

i

χ∗−
f =

(cid:2)Y

Y

Lf
−(Mf−µ(cid:48)
1)

(ˆkf ),

−(Mf−µ(cid:48)

1)+Jf−Mf 4π
kf

χ∗
Lf Jf

(R1A)

R1A

C

Jf Mf
Lf (Mf−µ(cid:48)

1)I1µ(cid:48)

1

(B.5)

Where we have applied the phase convention from Equation (B.2). Combing the incoming

and outgoing channels with the interaction, we the our T-matrix expression, written as

(cid:104)χ∗−
f (R1A)|V |χ+

i (R1A)(cid:105) =

Li−Lf e

i

iσLi (−1)

Li−(Mf−µ(cid:48)

1)+Jf−Mf (4π)3/2 ˆLi

(cid:90) (cid:88)

Lf Jf Mf LiJi

(R1A)VL(R1A)χ∗

χLiJi

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji

R2

1A

(R1A)

Lf Jf

Y

Lf
−(Mf−µ(cid:48)
1)

µ1

C

C

Jiµ1
Li0I1µ1

Jf Mf
Lf (Mf−µ(cid:48)
(ˆkf )dΩ1Adξ1dR1A.

1)I1µ(cid:48)

1

kf ki

(cid:2)Y

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf−Mf

(B.6)

114

Next we can couple together the angular momentum in the initial and final channels:

(cid:2)Y

=

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf−Mf
(cid:110)(cid:2)Y
(cid:88)

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji
Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji(cid:111)J

µ1

CJM
Jf−Mf Jiµ1

M

JM

(B.7)

.

But we know that the Lane potential is a scalar operator, so only terms that couple to total

angular momentum 0 can contribute. This gives us

(cid:110)(cid:2)Y

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji(cid:111)0

(cid:110)(cid:2)Y

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji(cid:111)0

0

δJiJf

δMf µ1

(B.8)

.

0

= C00
Jf−Mf Jiµ1
(−1)
Jf +Mf
ˆJf

=

Next, we couple the spinors and spherical harmonics together by rearranging the couplings

using [117] pg. 70, Equation (11)



a b

c

d e f

g h k

{{P a⊗ Rd}g ⊗{Qb⊗ Se}h}k, (B.9)

(cid:88)

gh

ˆc ˆf ˆgˆh

{{P a⊗ Qb}c⊗{Rd⊗ Se}f}k =

giving us

115

(cid:110)(cid:2)Y
Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Ji(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji(cid:111)0
(cid:88)

I1 Ji

Lf

0

=

ˆgˆh ˆJ 2
i

Li

I1 Jf

(cid:110)(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h(cid:111)0

0



(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)g

mg

(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h

mh

(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)g

mg

(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)g−mg

.






Lf

I1 Ji

Li

I1 Jf

g

h

0



Lf

I1 Ji

Li

I1 Jf

g

g

0



gh

(cid:88)

mgmhgh

g

h

0

ˆgˆh ˆJ 2

i C00

gmghmh

(cid:88)

mgg

(−1)g−mg

ˆg

ˆg2 ˆJ 2
i

=

=

The spinors must couple to 0 because the interaction is scalar. This leaves us with





Lf

I1 Ji

Li

I1 Jf

ˆ02 ˆJ 2
i

1
ˆ0

0
δJiJf

0
0
δI1I1
δLiLf
ˆI1 ˆJi
ˆLf

= ˆJ 2
i

(cid:2)Y
(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)0
Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)0

(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0
(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0

0

0

0

0.

(B.10)

(B.11)

Integrating over the coupled spinors gives ˆI1 and we can simplify the spherical harmonics

with

116

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)0
(cid:88)

(−1)Li−mi

0 =

(cid:2)Y

=

mi

ˆLi

Y

Li−mi

( ˆR1A)Y

Li
mi ( ˆR1A).

(cid:88)

mi

(−1)Li−mi

ˆLi

δLiLf

δmi−mf Y

Lf−mi

( ˆR1A)Y

Li
mi ( ˆR1A)

(B.12)

Then we integrate over the ˆR1A variable using [117] pg. 148 Equation (3) to get

(cid:90) 2π

(cid:90) π

0

0

Y

Li−mi

( ˆR1A)Y

Li
mi ( ˆR1A)dΩ1A = δLiLi

δmi−mi(−1)mi.

(B.13)

Putting that all together, we have

(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)0

0 =

(cid:88)

mi

(−1)Li

ˆLi

= ˆLi(−1)Li.

(B.14)

Combining all of these expressions, we get

(cid:110)(cid:2)Y

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Ji(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji(cid:111)0

= ˆJ 2
i

δJiJi

δI1I1
δLiLf
ˆI1 ˆJi
ˆLf

(cid:2)Y

Lf ( ˆR1A)Y Li( ˆR1A)(cid:3)0

0

0

(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0

0. = (−1)Li ˆJi.

(B.15)

Inserting this back into our T-matrix and simplifying all phases and prefactors, our final

result is

117

(cid:104)χ∗−
f (R1A)|V |χ+

i (R1A)(cid:105) =

(cid:90) (cid:88)

LiJi

iσLi

e

(4π)3/2

kf ki

ˆLi

χLiJi

(R1A)VL(R1A)χ∗

Lf Jf

(R1A)

R2

1A

C

Jiµ1
Li0I1µ1

C

Jiµ1
Li(µ1−µ(cid:48)

1)I1µ(cid:48)

1

Y

Li
µ1−µ(cid:48)

1

(ˆkf )dR1A.

To get the cross section, we note that

(cid:88)
µ1µ(cid:48)

1µAµB

dσ
dΩ

=

kf
ki

µ1µf

1

4π2(¯hc)4

ˆJA ˆµ1

|2

|T
µ1
µ(cid:48)
1

where JA and JB are the initial and final spin of the target.

(B.16)

(B.17)

118

APPENDIX C

Derivation of Charge-Exchange

Three-Body T-Matrix

This appendix contains the derivations for various terms of the three-body, (p,n) charge-

exchange T-matrix. In this framework, the interaction is mediated by an NN-potential with

various isospin operators. For this study, we consider the isospin central, tensor, spin-spin,

and spin-orbit operators. All derivation use the coordinate system described in Figure 2.3.

C.1 Central T-Matrix Derivation

This derivation is for (p,n) charge-exchange with a three-body formalism and a central NN

potential. The potential only acts between the projectile proton and the target’s valence

neutron. This derivation uses the Condon-Shortly phase convention which can be expressed

as

m(θ)∗ = (−1)mY l−m(θ).
Y l

(C.1)

The isospin central NN potential is directed along the r12 coordinate which runs from the

projectile proton to the valence neutron and has the form

119

tN N = V (r12)τ1 · τ2.

(C.2)

The isospin operators will give a factor of 1

2 and we will also add a factor of (2j + 1) to

account for all available neutrons in the valence shell. This leaves us with

tN N = V (r12)

(2j + 1)

2

.

(C.3)

Additionally the central potential, V (r12), is not directed along either of the coordinates

used to describe our systems, so we express the radial form factor using an expansion in our

current coordinates using the method from [67]. The relationship between the two coordinate

vectors and r12 is expressed as

r12 = R1A − mc
mA

r2c.

The potential only depends on the magnitude of r12, so we can calculate r2
12

r2
12 = R2

1A +

m2
c
m2
A

2c − 2mc
r2
mA

R1Ar2cz,

(C.4)

(C.5)

where z = cos θ12 and θ12 is the angle between R1A and r2c. Then we can build a multipole

function, Fλ, using the potential so that:

Fλ(R1A, r2c) =

(cid:90) 1

−1

1
2

V (r12)Pλ(z)dz,

(C.6)

where Pλ are Legendre Polynomials. Inserting this into a full expression for the potential,

leaves us with

120

(cid:88)

λ

V (r12) =

(2λ + 1)Fλ(R, r)

4π

2λ + 1

(cid:88)

mλ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A).

(C.7)

Then, our final potential is

tN N = V (r12)

(2j + 1)

2

=

(cid:88)

λ

(2λ + 1)Fλ(R, r)

4π

2λ + 1

(cid:88)

mλ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

(2j + 1)

2

.

The initial scattering wave function can be expressed, in this phase convention, as

(cid:88)

Li

χi =

i−Li(−1)Li

√
4π
kiR1A

ˆLie

iσLi FLi

(R1A)Y

Li
0 ( ˆR1A)

and including spin-orbit effects we have

(C.8)

(C.9)

(cid:88)

LiJi

χi =

i−Li(−1)Li

√

4π
kiR1A

ˆLie

iσLi FLiJi

(R1A)C

Jiµ1
Li0I1µ1

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji

µ1

(C.10)

and our entire incoming channel wave function can be expressed as

|ψ

µ1
µA

(cid:105) =

i−Li(−1)Li

(cid:88)
(cid:110)(cid:2)Y li(ˆr2c)ΞI2(ξ2)(cid:3)jiΞIc(ξc)

√
4π
kiR1A

LiJi

ˆLie

iσLi φji
(cid:111)JA

,

µA

(r2c)FLiJi

(R1A)C

Jiµ1
Li0I1µ1

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji

µ1

(C.11)

where ψ

µ1
µA

= χi(R1A)ΦIc:I2

(ξc, ξ2, r2c). We can create a similar expression for the outgoing

121

scattering wave function

χf =

−Lf (−1)

(cid:88)
Lf ( ˆR1A)ΞI1(ξ1)(cid:3)Jf

i

Lf Jf Mf

(cid:2)Y

Lf−Mf

Mf +µ(cid:48)

1

.

4π

kf R1A

F∗
Lf Jf

(R1A)Y

Lf−Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

(C.12)

So the entire outgoing wave function can be expressed as

−Lf (−1)

(cid:88)
Lf ( ˆR1A)ΞI1(ξ1)(cid:3)Jf

Lf Jf Mf

i

µ(cid:48)
1
µB

|ψ

(cid:105) =

(cid:2)Y

Lf−Mf

Mf +µ(cid:48)

1

kf R1A

4π

φjf

(r2c)F∗

(R1A)Y

Lf−Mf
(cid:111)JB
lf (ˆr2c)ΞI2(ξ2)(cid:3)jf ΞIc(ξc)

Lf Jf

,

(cid:110)(cid:2)Y

µB

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

where ψ

µ(cid:48)
1
µB

= χf (R1A)ΦIc:I

2(cid:48) (ξc, ξ2, r2c) But we need (cid:104)ψ

µ(cid:48)
1
µB

complex conjugates, we are left with

(C.13)

|. After taking the appropriate

Lf (−1)
i

(cid:88)
Lf ( ˆR1A)ΞI1(ξ1)(cid:3)∗Jf

Lf Jf Mf

Lf

(cid:104)ψ

µ(cid:48)
1
µB

| =

(cid:2)Y

4π

kf R1A

Lf
Mf

(cid:110)(cid:2)Y

φjf

(R1A)Y

(r2c)FLf Jf

lf (ˆr2c)ΞI2(ξ2)(cid:3)jf ΞIc(ξc)

(ˆkf )C

(cid:111)∗JB

µB

Mf +µ(cid:48)

1

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

.

(C.14)

The first step is to free the core from its coupling because it will be unchanged by the

interaction and can be immediately integrated out:

122

(cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

| =

(cid:2)Y

µ(cid:48)

c

Lf Jf Mf mjf

Lf ( ˆR1A)ΞI1(ξ1)(cid:3)∗Jf

Lf (−1)
i

Lf

4π

kf R1A

Mf +µ(cid:48)

1

C

JB µB
jf mjf

Icµ(cid:48)

c

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

Lf
Mf

φjf

(r2c)FLf Jf

(R1A)Y

(cid:2)Y

lf (ˆr2c)ΞI2(ξ2)(cid:3)∗jf

∗Ic
µ(cid:48)
Ξ
c

(ξc).

mjf

(C.15)

(C.16)

(C.17)

(C.18)

|ψ

µ1
µA

(cid:105) =

(cid:88)

i−Li(−1)Li

√
4π
kiR1A

µc

LiJimji

(cid:2)Y Li( ˆR1A)ΞI1(ξ1)(cid:3)Ji

C

JAµA
jimji

Icµc

µ1

ˆLie

iσLi φji

(r2c)FLiJi

(cid:2)Y li(ˆr2c)ΞI2(ξ2)(cid:3)ji

(R1A)C

Jiµ1
Li0I1µ1

ΞIc
µc(ξc).

mji

Now we can note that, in our phase convention,

Lf ( ˆR1A)ΞI1(ξ1)(cid:3)∗Jf

Mf +µ(cid:48)

1

(cid:2)Y

and

= (−1)

Mf +µ(cid:48)

1(cid:2)Y

Lf ( ˆR1A)ΞI1(−ξ1)(cid:3)Jf

−(Mf +µ(cid:48)
1)

(cid:2)Y

lf (ˆr2c)ΞI2(ξ2)(cid:3)∗jf

mjf

mjf(cid:2)Y

lf (ˆr2c)ΞI2(−ξ2)(cid:3)jf−mjf

.

= (−1)

This leaves us with a total T-matrix expression of

123

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

Lf Jf Mf LiJi
µcµ(cid:48)
mji
c

mjf

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

mji

−(Mf +µ(cid:48)
1)
∗Ic
µ(cid:48)
Ξ
c

(ξc)ΞIc

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµ(cid:48)

c

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µc(ξc)

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

µ1
(R1A)

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2dξc,

where

Lf−Li(−1)

k1 = i

Li+Lf +Mf +µ(cid:48)

1+mjf e

iσLi

3
2

.

ˆLi(4π)
kf ki

Immediately, we see that

(cid:90)

∗Ic
µ(cid:48)
Ξ
c

leaving us with

(ξc)ΞIc

µc(ξc)dξc = δµcµ(cid:48)

c

(C.19)

(C.20)

(C.21)

124

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

mji

Lf Jf Mf LiJi
mji

mjf

µc

−(Mf +µ(cid:48)
1)
φjf

(r2c)FLf Jf

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

C

Icµc

(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

JB µB
jf mjf

Icµc

(C.22)

The central isospin operator is a scalar, so all of the angular momentum must couple to 0 to

contribute. To achieve this, we will couple all the angular momentum operators together.

(cid:90)

(cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

µc

JMJ jmj mji

Lf Jf Mf LiJi
mjf

(cid:110)(cid:2)Y
lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)j

JB µB
jf mjf

JAµA
jimji

jimji

Icµc

Icµc

C

C

jmj
jf−mjf

C

(cid:110)(cid:2)Y

mj

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:111)J

MJ

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2.

(C.23)

Now we can rearrange the spin and angular momentum components using Equation (B.9).

Applying this identity gives

125

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:111)J

MJ

(cid:88)
(cid:110)(cid:2)Y

JMJ
jmj

=

(cid:110)(cid:2)Y
(cid:110)(cid:2)Y

C

C

jimji

1)Jiµ1

jmj
jf−mjf

JMJ
Jf−(Mf +µ(cid:48)

(cid:110)(cid:2)Y
lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)j
(cid:88)
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h(cid:111)J
lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)j

 C



I1 Jf

ˆJf ˆgˆh

I1 Ji

Lf

ˆJi

mj

Li

g

h

J

JMJ jmj gh

MJ

.

mj

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

jmj
jf−mjf

jimji

The spinors must couple to 0 because the interaction is scalar in spin space. This leaves us

(C.24)

with

=



 C

JMJ jmj g

Li

ˆJi

Lf

ˆJf ˆg

I1 Ji

I1 Jf

JMJ
Jf−(Mf +µ(cid:48)

(cid:88)
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:111)J
lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)j

J

0

g

.

(cid:110)(cid:2)Y
(cid:110)(cid:2)Y

MJ

mj

1)Jiµ1

C

jmj
jf mjf

jimji

(C.25)

Now we can perform the exact analogy for the internal angular momentum variables, l and

j, leaving us with

126







 C

jf

ji

j

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

jmj
jf−mjf

jimji

JMJ jmj gh

lf

Li

ˆJi

Lf

ˆJf

I1 Ji

I1 Jf

ˆji ˆjf ˆgˆh

(cid:88)
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:111)J
lf (r2c)Y li(r2c)(cid:3)h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)0(cid:111)j

li

h

J

0

g

.

I2

I2

0

MJ

=

(cid:110)(cid:2)Y
(cid:110)(cid:2)Y

mj

Next, we couple everything together, allowing us to ultimately couple all of the angular

momentum operators together.

(C.26)







lf

li

 C

I2

I2

jf

ji

=

JMJ jmj ghT MT

Li

ˆJi

Lf

ˆJf

I1 Ji

I1 Jf

ˆji ˆjf ˆgˆh

(cid:88)
(cid:34)(cid:110)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)0(cid:111)j

J

0

T MT
JMJ jmj

C

(cid:110)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:111)J

h

0

g

j

(cid:35)T

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

jmj
jf−mjf

jimji

.

MT

(C.27)

Again, rearranging using Equation(B.9) gives us

127











g 0 J

h 0 j



(cid:88)

=

ˆJˆj ˆxˆy ˆJi

ˆJf

ˆji ˆjf ˆgˆh

Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

JMJ jmj ghT MT xy

(cid:34)(cid:110)(cid:2)Y
(cid:35)T
(cid:110)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)0(cid:111)y

JMJ
Jf−(Mf +µ(cid:48)

jmj
jf−mjf

T MT
JMJ jmj

1)Jiµ1

jimji

C

C

C

g

0

.

j

0

J

h

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

x y T

lf (r2c)Y li(r2c)(cid:3)h(cid:111)x

MT

(C.28)

It is immediately clear that y must equal 0. If we uncouple the spin operators and apply

this simplification we get

128



(cid:88)

=

ˆJˆj ˆx ˆJi

ˆJf

ˆji ˆjf ˆgˆh

JMJ jmj ghT MT xmx

Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

g 0 J

h 0 j

g

0

J

(cid:110)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:110)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)0(cid:111)0

jmj
jf−mjf

jimji

C

h

0

C

C

JMJ
Jf−(Mf +µ(cid:48)
T MT
JMJ jmj

C

1)Jiµ1

T MT
xmx00

0

j

x 0 T

lf (r2c)Y li(r2c)(cid:3)h(cid:111)x







mx













Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

(cid:88)

=

ˆJi

ˆJf

ˆji ˆjf

ˆJˆj

J

JMJ jmj T MT

(cid:110)(cid:2)Y
(cid:110)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)0(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)0(cid:111)0

JMJ
Jf−(Mf +µ(cid:48)

jmj
jf−mjf

T MT
JMJ jmj

1)Jiµ1

jimji

C

C

C

J

0

0

j

j

0

Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y

lf (r2c)Y li(r2c)(cid:3)j(cid:111)T

MT

(C.29)

We can now integrate over the ξ1 and ξ2 variables, giving us a factor of ˆI1 ˆI2. As discussed

before, we need the total angular momentum, T, to be equal to 0 because of our scalar

operator. This leaves us with

129

=

ˆI1 ˆI2 ˆJi

ˆJf

ˆji ˆjf

ˆJˆj

Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y

J

0

j

J

lf (r2c)Y li(r2c)(cid:3)j(cid:111)T

0

j

=

ˆI1 ˆI2 ˆJi

ˆJf

ˆji ˆjf

ˆJˆj

Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

jmj
jf−mjf

jimji

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

jmj
jf−mjf

jimji










 C
 C

MT

j

0

J

lf (r2c)Y li(r2c)(cid:3)j(cid:111)0

j

0

0

J

(cid:110)(cid:2)Y
Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y
(−1)J−MJ δJjδMJ−mj
(cid:110)(cid:2)Y

ˆJ

C

jmj
jf−mjf

jimji

1)Jiµ1





0

ˆI1 ˆI2 ˆJi

ˆJf

ˆji ˆjf

ˆJˆj

Lf

I1 Jf

Li

I1 Ji

lf

li

I2

I2

jf

ji

Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y

J

0

j

0

J

lf (r2c)Y li(r2c)(cid:3)j(cid:111)0

j




li

J

lf









(−1)J−MJ ˆI1 ˆI2 ˆJi

ˆJf

ˆji ˆjf

ˆJ

JMJ

C

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

JMJ
jf−mjf

jimji

(cid:110)(cid:2)Y

Lf

I1 Jf

Li

I1 Ji

I2

I2

jf

ji

J

0

J

Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y

0

J

lf (r2c)Y li(r2c)(cid:3)J(cid:111)0

JMJ jmj

T MT

(cid:88)
(cid:110)(cid:2)Y
(cid:88)

JMJ jmj

C

T MT
JMJ
jmj

C00

JMJ jmj

(cid:88)

=

JMJ jmj

C

JMJ
Jf−(Mf +µ(cid:48)
(cid:88)

=

0

.

(C.30)

We can use [117], pg. 357 Equation (2) to simplify the 9j symbols

130



d c e

a c

b

g 0 g

 =

(−1)b+d+c+g

ˆcˆg

leaving us with

a b

e d g

 ,

c

(cid:88)

JMJ

=

(−1)

J−MJ +Ji+Lf +I1+ji+lf +I2 ˆJi

Li
Lf (R1A)Y Li(R1A)(cid:3)J(cid:2)Y

Jf Lf

ˆji ˆjf

1
ˆJ

ˆJf

Ji

I1


 li

lf (r2c)Y li(r2c)(cid:3)J(cid:111)0

I2

jf

lf

ji

J

J

.

0

C

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

JMJ
jf−mjf

jimji

(cid:110)(cid:2)Y

(C.31)

(C.32)

We can now take advantage of the rotational invariance of the integrand. The value of the

interaction is unchanged by a rotation of the reference frame and we can align R1A with

the z axis. In this configuration, r2c forms the angle θ2c with the z axis. This simplifies

Y

(R1A) to

Lf /Li
Mi/Mf
φ2c. This results in an overall factor of 8π2. So we have

ˆL√
4π

and results in a factor of 8π2 from the integration over θ1A, φ1A, and

131

(−1)

J−MJ +Ji+Lf +I1+ji+lf +I2C00

J0J0CJ0

Li0Lf 0CJ0

lf mf li−mf

2π ˆLi

ˆLf

ˆJi

ˆJf

ˆji ˆjf

1
ˆJ

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

JMJ
jf−mjf

jimji

lf
mf

Y

(ˆr2c)Y

li−mf

(ˆr2c)

(−1)

−MJ +Ji+Lf +I1+ji+lf +I2CJ0

Li0Lf 0CJ0

lf mf li−mf

2π ˆLi

ˆLf

ˆJi

ˆJf

ˆji ˆjf

1
ˆJ 2

(cid:88)

JMJ mf

=

Li
Li

=

Ji

I1

Jf Lf

J

(cid:88)

JMJ mf

Ji

I1

Jf Lf

J




jf

 li
 li

jf

 C
 C

ji

lf

I2

J

ji

lf

I2

J

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

C

JMJ
jf−mjf

jimji

lf
mf

Y

(ˆr2c)Y

li−mf

(ˆr2c).

(C.33)

Combining this result with our full T-matrix expression, we have

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:90)

(cid:88)

Lf Jf Mf LiJi
mjf

JMJ mf mji

µc

k2 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

JMJ
jf−mjf

jimji

CJ0
Li0Lf 0CJ0

lf mf li−mf

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc



 li

jf

 Y

ji

lf

I2

J

C

Li

Ji

I1

J

Jf Lf
(r2c)FLf Jf

φjf

lf
mf

(ˆr2c)Y

li−mf

(ˆr2c)

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c sin θ2cdθ2c,

(C.34)

where, noting that (−1)

2Lf = 1 and i−Li(−1)Li = iLi,

132

Lf +Li(−1)

k2 = i

Mf +µ(cid:48)

1+mjf

−MJ +Ji+I1+ji+lf +I2

iσLi

e

2π(4π)
kf ki

3
2

ˆLi

2 ˆLf

ˆji ˆjf

ˆJf

ˆJi
ˆJ 2

.

(C.35)

Now we can look more closely at the form of our potential:

tN N = V (r12)

(cid:88)

(2λ + 1)Fλ(R, r)

λ

(cid:88)

mλ

4π

2λ + 1

(2j + 1)

2

=

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

(2j + 1)

2

(cid:88)

λmλ

=

Fλ(R, r)2π(2ji + 1)Y ∗

λ,mλ

(ˆr2c)Yλ,mλ

( ˆR1A).

(C.36)

Because we aligned the R1A coordinate with the z axis, Equation (C.36) simplifies to

(cid:88)

λ

tN N =

Fλ(R, r)

ˆλ2π(2ji + 1)

√
4π

Y λ
0 (ˆr2c).

(C.37)

We can now complete the simplification of the angular variables. First, we use [117] pg. 144

Equation (9):

Yl1m1

(θ)Yl2m2

(θ) =

(cid:88)

LM

ˆl1
ˆl2√
4π ˆL

Applying this gives

l10l20CLM
CL0

l1m1l2m2

YLM .(θ)

(C.38)

lf
mf

Y

(ˆr2c)Y

li−mf

(ˆr2c)Y λ

0 (ˆr2c) =

(cid:88)

L

ˆlf
ˆli
√
ˆL
4π

CL0
lf 0li0CL0

lf mf li−mf

Y L
0 (ˆr2c)Y λ

0 (ˆr2c).

(C.39)

133

Next, we can use [117] pg. 149 Equation (10):

(cid:90)

Y ∗
lm(θ)Yl(cid:48)m(θ) sin θdθ =

δll(cid:48)
2π

,

(C.40)

giving us the T-matrix expression:

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:90)

(cid:88)

Lf Jf Mf LiJi
mjf

JMJ mf λmji

µc

k3 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

JMJ
jf−mjf

jimji

CJ0
Li0Lf 0CJ0

lf mf li−mf

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

C

Li



 li

jf

 Cλ0

ji

lf

I2

J

Ji

I1

J

Jf Lf
(r2c)FLf Jf

φjf

lf 0li0Cλ0

lf mf li−mf

(R1A)Fλ(R, r)φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

(C.41)

where

Lf +Li(−1)

k3 = i

Mf +µ(cid:48)

1+mjf

−MJ +Ji+I1+ji+lf +I2

iσLi (2ji + 1)

e

ˆLi

2 ˆLf

3
(4π)
2
2kf ki

ˆji ˆjf

ˆli

ˆlf

.

ˆJf
ˆJi
ˆJ 2

Finally, we can simplify

giving a final result of

(cid:88)

mf

CJ0
lf mf li−mf

Cλ0
lf mf li−mf

= δJλδ00,

134

(C.42)

(C.43)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:90) (cid:88)

Lf Jf Mf LiJi
MJ λmji
µc
mjf

k4 Y

Lf
Mf

(ˆkf )C

C

λMJ
jf−mjf

jimji

Cλ0
Li0Lf 0Cλ0

lf 0li0C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

 li

jf

 φjf

ji

I2

lf λ

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

λMJ
Jf−(Mf +µ(cid:48)

1)Jiµ1

Li

Ji

I1

Jf Lf λ



(r2c)FLf Jf

(R1A)Fλ(R, r)φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

(C.44)

where

Lf +Li(−1)

k4 = i

Mf +µ(cid:48)

1+mjf

−MJ +Ji+I1+ji+lf +I2

iσLi (2ji + 1)

e

ˆLi

2 ˆLf

3
(4π)
2
2kf ki

ˆji ˆjf

ˆli

ˆlf

.

ˆJf
ˆJi
ˆλ2

(C.45)

The total cross section expression is

(cid:88)
µ1µ(cid:48)

1µAµB

dσ
dΩ

=

(cid:12)(cid:12)(cid:104)ψ

µ(cid:48)
1
µB

kf
ki

µ1µf

4π2(¯hc)4

1
ˆJ 2
A ˆµ2
1

|tN N|ψ

µ1
µA

(cid:105)|2,

(C.46)

and JA and JB are the initial and final spin of the target.

C.2 Tensor T-Matrix Derivation

This derivation is for the isospin tensor interaction for Fermi, (p,n) charge-exchange reactions.

The interaction is defined as:

135

(cid:114)24π

5

(cid:104)
(τ1 · τ1)Vt(r12)

VT EN =

(cid:105)

Y(2) · [σ1 ⊗ σ2](2)

(C.47)

where Y(2) is a tensor whose components are the spherical harmonics Y2m, τ1 · τ2 is the

isospin operator, Vt(r12) is a form factor for the radial dependence of the isospin tensor
interaction, and [σ1 ⊗ σ2](2) is the tensor represented by the cross product of the proton

and neutron spin functions. Just as in the central case, the isospin operator will give a

factor of 1

2 and there will be a factor of (2j + 1) to account for the number of nucleons in

the valence shell. This can be inserted in the same T-matrix expression used for the isospin

central interaction, giving us

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµ(cid:48)

c

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

Lf Jf Mf LiJi
µcµ(cid:48)
mji
c

mjf

−(Mf +µ(cid:48)
1)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

(ξc)ΞIc

µc(ξc)

∗Ic
µ(cid:48)
Ξ
mji
c
(R1A)

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2dξc

(C.48)

24π

.

(C.49)

ˆLi(4π)

√

3
2
kf ki

where

and

Lf−Li(−1)

k1 = i

Li+Lf +Mf +µ(cid:48)

1+mjf e

iσLi

136

(cid:104)

V (r12)

Y(2)[σ1 ⊗ σ2](2)

(cid:105)0

0

tN N =

(2ji + 1)

2

,

where we have used [117] pg. 65 Equation (35):

(MJ · NJ ) = (−1)−J ˆJ{MJ ⊗ NJ}00.

Immediately, we see that

(cid:90)

∗Ic
µ(cid:48)
Ξ
c

(ξc)ΞIc

µc(ξc)dξc, = δµcµ(cid:48)

c

(C.50)

(C.51)

(C.52)

leaving us with

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:90) (cid:88)

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y
(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

Lf Jf Mf LiJi
mji

mjf

µc

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

−(Mf +µ(cid:48)
1)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

mji
(R1A)

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2.

(C.53)

The central isopin operator is a scalar, so the total angular momentum must couple to 0 to

contribute. To achieve this, we will couple all the angular momentum operators together:

137

(cid:110)(cid:2)Y

JT −(Mf +µ(cid:48)
Jf−(Mf +µ(cid:48)

1)+µ1
1)Jiµ1

C

(cid:88)

JT

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:111)JT

−(Mf +µ(cid:48)

1)+µ1

.

(C.54)

Now we can rearrange the spin and angular momentum components using Equation (B.9),

giving us

ˆJi

JT −(Mf +µ(cid:48)
Jf−(Mf +µ(cid:48)

ˆJf ˆgˆhC

(cid:88)
(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h(cid:111)JT

JT ,g,h

1)+µ1
1)Jiµ1

−(Mf +µ(cid:48)

1)+µ1



Lf

I1 Jf

Li

I1 Ji

h JT

g

.



(cid:110)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g

(C.55)

In this case, the spinors must couple to 1 in order to couple to a rank two tensor. Applying

this, and also making the replacement that MJf

= Mf + µ(cid:48)

1, we are left with



Lf

I1 Jf

Li

I1 Ji

g

1 JT



(cid:110)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g

(C.56)

ˆJi

ˆJf ˆgˆ1C

JT −MJf
Jf−MJf

(cid:88)
(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)1(cid:111)JT

JT ,g

+µ1
Jiµ1

−MJf

.

+µ1

We can perform the exact analogy for the internal angular momentum variables, leaving us

with

138

(cid:88)

jt,h

ˆji ˆjf

ˆhˆ1C

jtmji
jf−mjf

−mjf
jimji



lf

li

h



jf

ji

jt

(cid:110)(cid:2)Y

I2

I2

1

lf (r2c)Y li(r2c)(cid:3)h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)1(cid:111)jt

mji

.

−mjf

(C.57)

To deal with the operator, we can break up the tensor operator into its spin and isospin

parts:

(cid:105)0

0

(cid:104)

Y(2)[σ1 ⊗ σ2](2)

(cid:88)
m2m(cid:48)

2

=

C00
2m22m(cid:48)

2

Y 2
m2

(ˆr12)S2
m(cid:48)

2

(ξ1ξ2) =

(cid:88)

m2

(−1)−m2

ˆ2

Y 2
m2

(ˆr12)S2−m2

(ξ1ξ2),

(C.58)

and couple all angular momentum operators:

(cid:34)(cid:110)(cid:2)Y
(cid:88)
lf (r2c)Y li(r2c)(cid:3)h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)1(cid:111)jt
(cid:110)(cid:2)Y

+µ1jt−mjf

+mji

C

T MT
JT −MJf

=

T MT

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)1(cid:111)JT
(cid:35)T

.

MT

(C.59)

139

Rearranging using Equation (B.9) gives us



 C

ˆJT

ˆjt ˆxˆy

g 1 JT

(cid:34)(cid:110)(cid:2)Y
(cid:88)
lf (r2c)Y li(r2c)(cid:3)h(cid:111)x(cid:110)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)1(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)1(cid:111)y

+µ1jt−mjf

T MT
JT −MJf

x y T

T MT xy

+mji

h 1

jt

=

(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g
(cid:35)T

.

MT

(C.60)

As mentioned before, because the operator is a scalar, we need the total spin components to

couple to 2, forcing y = 2. If we uncouple the spinors operators and apply this simplification

we get



 C

=

ˆJT

h 1

ˆjt ˆxˆ2

g 1 JT

(cid:88)
T MT xmxm(cid:48)
(cid:104)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)1(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)1(cid:105)2

T MT
xmx2m(cid:48)

x 2 T

jt

2

2

.

mx

m(cid:48)

2

C

T MT
JT −MJf

+µ1jt−mjf

+mji

(C.61)

Finally we rearrange the spinor components one more time using Equation (B.9) to have a

projectile and target nucleon coupled together in each channel giving us

140



(cid:88)
T MT xmxm(cid:48)

=

ˆJT

ˆjt ˆxˆ2ˆ1ˆ1ˆα ˆβ

g 1 JT

h 1

jt

2αβ

x 2 T

(cid:104)(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)1(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)1(cid:105)2

+µ1jt−mjf

T MT
JT −MJf

T MT
xmx2m(cid:48)

+mji

C

C

2

α β 2

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y



1
2 1

1
2 1





1
2

1
2

.

m(cid:48)

2

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x

mx

(C.62)

The 9j symbol simplifies to to δαβδα1

9. This will can cancel out the ˆα ˆβˆ1ˆ1 factor, leaving us
1

with



 C

=

ˆJT

h 1

ˆjt ˆxˆ2

g 1 JT

(cid:88)
T MT xmxm(cid:48)
(cid:104)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)1(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)1(cid:105)2

T MT
xmx2m(cid:48)

x 2 T

jt

2

2

C

mx

.

m(cid:48)

2

T MT
JT −MJf

+µ1jt−mjf

+mji

(C.63)

Because we have the angular momentum operators and spin operators coupled separately,

it is clear how to impose conservation of the projection of the angular momentum in both

spaces separately. Including the projections from the tensor operator, we are left with the

relations

141

m(cid:48)
2 − m2 = 0

m(cid:48)
2 = m2

mx + m2 = 0

mx = −m2.

We see from the Clebsch-Gordan coefficient that MT = mx + m(cid:48)

2. Using the substitutions

from above, we notice that MT = 0. Incorporating all of these changes into the T-matrix

expression, we are left with

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf

(cid:90) (cid:88)

Lf Jf MJf
LiJi
mji
mjf
µcgh
JT jtm2xT
jt−mjf
jf−mjf

+µ1
Jiµ1

C



+mji
jimji





Y 2
m2

(ˆr12)S2−m2

(ξ1ξ2)

lf

li

I2

I2

jf

ji

Lf

I1 Jf

Li

I1 Ji

1

h

jt

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
(cid:104)(cid:2)Y
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)1(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)1(cid:105)2

g

1 JT

−m2

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

m2
(R1A)

k2 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

CT 0
x−m22m2

CT 0
JT −MJf

+µ1jt−mjf

+mji





g 1 JT

h 1

jt

x 2 T



R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2.

(C.64)

where

142

Li−Lf (−1)

Li+Lf +MJf

k2 = i

−m2

+mjf

iσLi ˆLi
e

ˆJi

ˆJf ˆgˆhˆ12 ˆji

3 ˆjf

ˆJT

ˆjt ˆx

√

24π
2

3
(4π)
2
kikf

(C.65)

and

(cid:88)

λmλ

tN N =

Fλ(R1A, r2c)4πY λ
mλ

(R1A)Y λ
mλ

(r2c)∗,

(C.66)

where Fλ contains the interaction over the r12 coordinate expressed as a function of the

other two coordinates. We can then apply the spin operator:

(cid:90) (cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)1

mq

(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)1

mQ

(cid:88)

mqmQ

C

2m2
1mq1mQ

S2−m2

(ξ1ξ2).

(C.67)

This is equivalent to

(cid:88)

mqmQ

C

2m2
1mq1mQ

(cid:104)1
2

1
2

1mq|{σ1 ⊗ σ2}2−m2

|1
2

1
2

1mQ(cid:105),

(C.68)

which can be simplified using [117] pg. 479 Equation (28):

143

2j(cid:48)m(cid:48)|{ ˆPa(1) ⊗ ˆQb(2)}cγ|n1j1n2j2jm(cid:105)
2j(cid:48)
1n(cid:48)
1j(cid:48)
(cid:104)n(cid:48)

(cid:104)n(cid:48)

2j(cid:48)
1|| ˆPa(1)||n1j1(cid:105)(cid:104)n(cid:48)
1j(cid:48)
2|| ˆQb(2)||n2j2(cid:105).

(C.69)

(C.70)

||σp||1
2

(cid:105)(cid:104)1
2

||σn||1
2

(cid:105).



a
j(cid:48)

1

j1

b
j(cid:48)

2

j2

c
j(cid:48)

j

= (−1)2cˆcˆjCj(cid:48)m(cid:48)

jmcγ

In our case, this gives us

(cid:88)

mqmQ

C

2m2
1mq1mQ

|1
2

1
2

1mQ(cid:105) =

(cid:104)1
2

1
2

1mq|{σ1 ⊗ σ2}2−m2
(cid:88)

C

2m2
1mq1mQ

ˆ2ˆ1C

1mq
1MQ2−m2

mqmQ



1
2

1
2

1 1 2

1
2 1

1
2 1

(cid:104)1

2

The reduced matrix elements can be evaluated using [117] pg. 495 Equation (95):

(cid:104)s(cid:48)|| ˆS1||s(cid:105) = δss(cid:48)(cid:112)s(s + 1)(2s + 1),

(C.71)

keeping in mind that 2 ˆS = ˆσ. The sum over Clebsch-Gordan coefficients can also be evaluated

explicitly, giving us

144

(cid:88)

mqmQ

=

C

2m2
1mq1mQ

ˆ2ˆ1C

1mq
1MQ2−m2

(cid:104)1

2



1 1 2

1
2 1

1
2 1

1
2

1
2

(

1
2

(cid:114)1

2

4

+ 1)(2

1
2

+ 1)

√
√
5√
3
15

1
9

2
3

=

=

||σp||1
2

(cid:105)(cid:104)1
2

||σn||1
2

(cid:105)

(C.72)

Substituting this into the T-matrix expression gives

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

k3 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

Lf Jf MJf
LiJi
mjf
mji
µcgh
JT jtm2xT
jt−mjf
jf−mjf

+µ1
Jiµ1

C

+mji
jimji





g 1 JT

h 1

jt

x 2 T





(cid:2)Y

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf





lf

li

I2

I2

jf

ji

Lf

I1 Jf

Li

I1 Ji

g

1

h

jt

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x

1 JT
φjf

−m2

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1A,

where

CT 0
x−m22m2

CT 0
JT −MJf

+µ1jt−mjf

+mji

Y 2
m2

(ˆr12)

(cid:104)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

(C.73)

145

Li−Lf (−1)

Li+Lf +MJf

k3 = i

−m2

+mjf

iσLi ˆLi
e

ˆJi

ˆJf ˆgˆh ˆji

3 ˆjf

ˆJT

ˆjt ˆx

√

24π

3
(4π)
2
kikf

,

(C.74)

and

(cid:88)

λmλ

tN N =

Fλ(R1A, r2c)4πY λ
mλ

(R1A)Y λ
mλ

(r2c)∗.

(C.75)

Next we can simplify the angular momentum operators. Because the tensor operator is a

scalar operator, all of the angular momentum operators must couple to 0.

(cid:40)(cid:104)(cid:2)Y

C00
x−m22m2
(−1)2+m2

δx2δm2m2

ˆ2

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

Y 2(ˆr12)

lf (r2c)Y li(r2c)(cid:3)h(cid:105)2

(cid:41)0

=

0

(C.76)

Y 2
m2

(ˆr12).

−m2

As in the central case, we have the freedom to choose the ˆR1A direction to be along the ˆz

axis, meaning that Mi and Mf are 0 and the initial and final scattering spherical harmonics

reduce to

ˆLi
ˆLf
4π . We can also perform the integration over sin θ1Adθ1Adφ1Adφ2c, giving a

factor of 8π2. Incorporating these simplifications, Equation (C.76) becomes

(cid:104)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

lf (r2c)Y li(r2c)(cid:3)h(cid:105)2

C00
2−m22m2

C

2−m2
gmghmh

C

hmh
lf mf limi

C

gmg
Lf 0Li0

−m2
ˆLf
ˆLi
4π

Y 2
m2

(ˆr12)

(C.77)

(−1)2+m2

ˆ2

(cid:88)

δx2δm2m2
(−1)2+m2

=

m2mhmg

mimf

ˆ2

lf
mf

Y

(ˆr2c)Y

li
mi(ˆr2c)Y 2
m2

(ˆr12)8π2.

146

It is immediately clear that mg = 0 and, therefore, mh = −m2. Also, the Clebsch-Gordan

coefficient coupled to 0 can again be simplified. Finally, we can make the substitution at
mi = −m2 − mf . Combining these simplifications, we are left with

(cid:88)

m2mf

(−1)2(2+m2)

ˆ22

2−m2
g0h−m2

C

h−m2
lf mf li−m2−mf

C

Cg0

Lf 0Li0

ˆLi

ˆLf

lf
mf

Y

(ˆr2c)Y

li−m2−mf

(ˆr2c)Y 2
m2

(ˆr12)2π.

(C.78)

The T-matrix expression is now given by

k4 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf

(cid:90) (cid:88)

LiJi
Lf Jf MJf
mjf
mji
µcgh
JT jtm2mf T
jt−mjf
+µ1
jf−mjf
Jiµ1

C

2−m2
g0h−m2

C

h−m2
lf mf li−m2−mf

C

Cg0

Lf 0Li0

CT 0
x−m22m2

CT 0
JT −MJf

+µ1jt−mjf

+mji

+mji
jimji







lf

li

h

I2

I2

1

jf

ji

jt

Lf

I1 Jf

Li

I1 Ji

g

1 JT





g 1 JT

h 1

jt

2 2 T



Y

lf
mf
φjf

(r2c)Y
(r2c)FLf Jf

li−m2−mf

(r2c)Y 2
m2
(R1A)tN N φji

(ˆr12)
(r2c)FLiJi

R2

1A

where

(R1A)

R2
1Ar2

2cdR1Adr2c sin θ2cdθ2c,

(C.79)

147

Li−Lf (−1)

k4 = i

Li+Lf +MJf

+mjf

−m2

iσLi ˆLi

2 ˆLf

e

ˆJi

ˆJf ˆgˆh ˆji

3 ˆjf

ˆJT

ˆjt

√

2π

24π

ˆ22

3
(4π)
2
kikf

. (C.80)

Additionally, aligning R1A with the z-axis will mean that mλ = 0 in the tN N operator, and

the spherical harmonic over the scattering angle simplifies to

ˆλ√
4π

:

(cid:88)

λ

tN N =

Fλ(R1A, r2c)4πY λ

0 (r2c)

λ√
4π

.

(C.81)

Explicitly, Fλ is

(cid:90) 1

−1

1
2

V (r12)Pλ(z)dz.

(C.82)

Y 2
m2

(ˆr12) needs to be expanded in terms of the coordinate used within the problem using

[117] pg. 167 Equation (35):

rLYLM (θ, φ) =(cid:112)4π(2L + 1)!

L(cid:88)

l1,l2=0
l2−l1=L

Applying this to our system gives us

(−1)l2

(cid:112)(2l1 + 1)!(2l2 + 1)!

l2
l1
r
1 r
2

{Yl1

(Ω1) ⊗ Yl2

(Ω2)}LM .

(C.83)

(ˆr2c).

(C.84)

(cid:32)

ˆn

2(cid:88)

n=0

Y 2
m2

(ˆr12) =

(cid:33) 1
2(cid:16) − mc

mA

!

(cid:17)n

r2c

R2−n

1A

1
r2
12

C

2m2

(2−n)m2n0Y 2−n

m2

120
ˆ(2 − n)

2

ˆn2!

Now the four spherical harmonics over the internal angle can be simplified using Equation(C.38

148

twice, giving us

Y

(ˆr2c)Y

lf
mf
CL(cid:48)0
2−n0λ0C

(ˆr2c)Y λ

li−m2−mf
L(cid:48)m2
(2−n)m2λ0Y L−m2

(cid:88)

L(cid:48)L

(ˆr2c) =

ˆli

ˆ(2 − n)ˆλ
ˆlf
ˆL ˆL(cid:48)4π

CL0
lf 0li0C

L−m2
lf mf li−m2−mf

0 (ˆr2c)Y 2−n
m2
(ˆr2c)Y L(cid:48)

m2

(ˆr2c).

(C.85)

Next, we can use Equation (C.40), giving us

(cid:88)

L

=

ˆli

ˆ(2 − n)ˆλ
ˆlf
ˆL28π2

CL0
lf 0li0C

L−m2
lf mf li−m2−mf

CL0
2−n0λ0C

Lm2
(2−n)m2λ0.

(C.86)

Putting it all together, along with the simplification that

(cid:88)

mf

h−m2
lf mf li−m2−mf

L−m2
lf mf li−m2−mf

C

C

= δLhδm2m2,

(C.87)

leaves us with the final T-matrix expression:

149

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf

2−m2
g0h−m2

C

Cg0

Lf 0Li0

k5 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

LiJi
Lf Jf MJf
mjf
mji
µcgh
JT jtm2mf T λ
jt−mjf
+µ1
jf−mjf
Jiµ1

C

+mji
jimji





lf

li

h

I2

I2

1

jf

ji

jt

Lf

I1 Jf

Li

I1 Ji

g

1 JT



CT 0
x−m22m2

CT 0
JT −MJf

+µ1jt−mjf

+mji





g 1 JT

h 1

jt

2 2 T

 Ch0

lf 0li0

Ch0
2−n0λ0C

hm2
(2−n)m2λ0

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

(C.88)

where

Li−Lf (−1)

Li+Lf +MJf

k5 = i

+mjf e

iσLi

ˆLi

2 ˆLf

ˆJi

ˆJf ˆg ˆji

3 ˆjf
ˆJT
ˆh2

ˆjt ˆli

ˆlf

ˆ(2 − n)ˆλ

√
8

6

5

3
(π)
2
kikf

(C.89)

and

(cid:90) 1

−1

1
2

tN N =

(cid:32)

ˆn

2(cid:88)

n=0

V (r12)

r2
12

Pλ(z)dz

The total cross section expression is

(cid:33) 1
2(cid:0)− mc

mA

!

(cid:1)nR2−n

1A C

r2c

2m2
(2−n)m2n0. (C.90)

120
ˆ(2 − n)

2

ˆn2!

150

(cid:88)
µ1µ(cid:48)

1µAµB

dσ
dΩ

=

(cid:12)(cid:12)(cid:104)ψ

µ(cid:48)
1
µB

kf
ki

µ1µf

4π2(¯hc)4

1
ˆJ 2
A ˆµ2
1

|tN N|ψ

µ1
µA

(cid:105)|2,

(C.91)

where JA and JB are the initial and final spin of the target.

C.3 Spin-Spin T-matrix Derivation

This derivation is for (p,n) charge-exchange with a three-body formalism and an NN poten-

tial. The potential only acts between the projectile proton and the target’s valence neutron.

This derivation uses the Condon-Shortly phase convention from Equation (C.1). The isospin

spin-spin NN potential is directed along the r12 coordinate which runs from the projectile

proton to the valence neutron and has the form

tN N = V (r12)(τ1 · τ2)(σ1 · σ2).

(C.92)

The isospin operators will give a factor of 1

2 and we will also add a (2ji + 1) factor to account

for all available neutrons in the valence shell. This leaves us with

tN N = V (r12)σ1 · σ2

(2ji + 1)

2

.

(C.93)

As with the central and tensor cases, V (r12) is expanded in terms of R1A and r2c, giving

the full expression for the interaction:

151

tN N = V (r12)σ1 · σ2
(cid:88)

(2λ + 1)Fλ(R, r)

=

(2ji + 1)

(cid:88)

2
4π

2λ + 1

mλ

λ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

σ1 · σ2(2ji + 1)

2

(C.94)

.

Similarly to the central and tensor cases, we can create a T-matrix expressing using the

incoming and outgoing wave functions:

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

Lf Jf Mf LiJi
mji

mjf

µc

−(Mf +µ(cid:48)
1)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

mji
(R1A)

1Ar2
R2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2,

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

where

Lf−Li(−1)

Li+Lf e

iσLi

k1 = i

3
2

ˆLi(4π)
kf ki

2ji + 1

2

.

We couple the initial and final angular momentum and spin operators together to get

152

(C.95)

(C.96)

k1Y

Lf
Mf

(ˆkf )

µc

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

(C.97)

(cid:90)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:88)

Lf Jf Mf LiJi
J(cid:48)
mji
T M

J(cid:48)

T

mjf

JT MJT
J(cid:48)
J(cid:48)
T M
Jf−(Mf +µ(cid:48)

T

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

JT MJT
Jiµ1jimji

C

C

C

C

Jiµ1
Li0I1µ1

1)jf−mjf

lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:111)J(cid:48)
(cid:110)(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y
(cid:110)(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)JT

T
M

J(cid:48)

T

MJT

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2.

Now we can rearrange the spin and angular momentum components using Equation (B.9)

for both the scattering and bound states, giving us

153

J(cid:48)

J(cid:48)
T M
Jf−(Mf +µ(cid:48)

T

C

1)jf−mjf

(cid:110)(cid:2)Y

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf(cid:111)J(cid:48)

T
M

J(cid:48)

T

(cid:88)

(cid:88)

J(cid:48)
T M

JT MJT
JT MJT
Jiµ1jimji

C

T

J(cid:48)

(cid:110)(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji(cid:111)JT

MJT

ˆJf

ˆjf

ˆJiˆjiˆgˆhˆg(cid:48)ˆh(cid:48)C

J(cid:48)

J(cid:48)
T M
Jf−(Mf +µ(cid:48)

T

1)jf−mjf

C

JT MJT
Jiµ1jimji

=

J(cid:48)
ghg(cid:48)h(cid:48)

T

JT MJT
J(cid:48)
M

T
Lf (R1A)Y

lf (r2c)(cid:3)g(cid:48)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)h(cid:48)(cid:111)J(cid:48)

(cid:110)(cid:2)Y
(cid:110)(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g(cid:2)ΞI1(ξ1)ΞI2(ξ2)(cid:3)h(cid:111)JT

T
M

.

J(cid:48)

T

MJT



Li

I1 Ji

li

g

I2

ji

h JT





Lf

I1 Jf

I2

lf
jf
g(cid:48) h(cid:48) J(cid:48)

T



Then we break the coupling to JT so we can apply the spin-spin operator:

(cid:88)

ˆJf

ˆjf

ˆJiˆjiˆgˆhˆg(cid:48)ˆh(cid:48)C

J(cid:48)

J(cid:48)
T M
Jf−(Mf +µ(cid:48)

T

1)jf−mjf

C

JT MJT
Jiµ1jimji

C

J(cid:48)
J(cid:48)
T M
g(cid:48)m(cid:48)
gh(cid:48)m(cid:48)

T

h

(C.98)

C

JT MJT
gmghmh

=



J(cid:48)
J(cid:48)
T M
JT MJT
gm(cid:48)
g(cid:48)h(cid:48)mgmhm(cid:48)

T

h

gh

Li

I1 Ji

li

I2

ji





Lf

I1 Jf

I2

lf
jf
g(cid:48) h(cid:48) J(cid:48)



g

h JT

(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g

T

(cid:2)ΞI1(ξ1)ΞI2(ξ2)(cid:3)h

mg

(cid:2)Y

Lf (R1A)Y

lf (r2c)(cid:3)g(cid:48)

m(cid:48)

g

(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)h(cid:48)

m(cid:48)

h

mh

.

(C.99)

154

The spin and position operators from tN N act on independent operators, so we can evaluate

the spin space first. Collecting the relevant operators gives us



J(cid:48)
J(cid:48)
T M
g(cid:48)m(cid:48)
gh(cid:48)m(cid:48)

T

h

ˆhˆh(cid:48)C

(cid:90) (cid:88)
(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)h(cid:48)

hh(cid:48)mhm(cid:48)

h

σp · σn

m(cid:48)

h

C

JT MJT
gmghmh

Li

I1 Ji

li

I2

ji

g

h JT

(cid:2)ΞI1(ξ1)ΞI2(ξ2)(cid:3)h





Lf

I1 Jf

I2

lf
jf
g(cid:48) h(cid:48) J(cid:48)

T



dξ1dξ2,

mh

which is equivalent to

(cid:88)
hh(cid:48)mhm(cid:48)
hm1
m2m(cid:48)
1m(cid:48)
2

Li

I1 Ji

li

g

I2

ji

h JT

=







ˆhˆh(cid:48)C

h(cid:48)m(cid:48)
I1m(cid:48)

h

1I1m(cid:48)

2

C

hmh
I1m1I2m2

C

J(cid:48)
J(cid:48)
T M
gh(cid:48)m(cid:48)
g(cid:48)m(cid:48)

T

h

C

JT MJT
gmghmh

(cid:104)I1m1I2|σp · σn|I1m(cid:48)

1I2m(cid:48)
2(cid:105).

Lf

I1 Jf

I2

lf
jf
g(cid:48) h(cid:48) J(cid:48)

T

The matrix element can be simplified using [117]pg. 479 Equation (26):

1n(cid:48)
2j(cid:48)
1m(cid:48)
1j(cid:48)
(cid:104)n(cid:48)
2m(cid:48)
(cid:88)
(−1)−αC

=

1
ˆj(cid:48)
ˆj(cid:48)

1

2

α

2|( ˆPa(1) · ˆQa(2))|n1j1m1n2j2m2(cid:105)

j(cid:48)
1m(cid:48)
j1m1aαC

1

j(cid:48)
2m(cid:48)
2j(cid:48)
1|| ˆPa(1)||n1j1(cid:105)(cid:104)n(cid:48)
1j(cid:48)
j2m2a−α(cid:104)n(cid:48)
2|| ˆQa(2)||n2j2(cid:105).

2

(C.100)

(C.101)

(C.102)

In this work, we are only considering the case of Fermi transitions, where there is no spin

flip between the projectile and target nucleon. This restricts us to α = 0 in the expression

155

above and insists that m1 = m(cid:48)

1 and m2 = m(cid:48)

2. Applying this, we get

ˆhˆh(cid:48)C

h(cid:48)m(cid:48)
I1m1I2m2

h

C

hmh
I1m1I2m2

C

J(cid:48)
J(cid:48)
T M
g(cid:48)m(cid:48)
gh(cid:48)m(cid:48)

T

h

C

JT MJT
gmghmh



Li

li

g



I1 Ji

I2

ji

h JT

(C.103)

1
ˆI1 ˆI2

C

I1m1
I1m110C

I2m2

I2m210(cid:104)1

2

||σp||1
2

(cid:105)(cid:104)1
2

||σn||1
2

(cid:105).

(cid:88)
hh(cid:48)mhm(cid:48)

h

Lf

I1 Jf

I2

lf
jf
g(cid:48) h(cid:48) J(cid:48)

T

=



m1m2



Summing over the projections of the spins and the Clebsch-Gordon coefficients gives
(−1)1+mh

. This reduces the expression to

3

δhh(cid:48)δmhm(cid:48)

h

(cid:88)

hmh

=

ˆh2
6

C

J(cid:48)
T M
g(cid:48)m(cid:48)

J(cid:48)
ghmh

T



Li

I1 Ji

li

g

I2

ji

h JT





Lf

I1 Jf

lf
g(cid:48)

I2
jf
h J(cid:48)

T

(cid:104)1

2

C

JT MJT
gmghmh

||σp||1
2

(cid:105)(cid:104)1
2

||σn||1
2

(cid:105).

These matrix elements can be simplified using [117] pg. 495 Equation (95):

(cid:104)s(cid:48)|| ˆS1||s(cid:105) = δss(cid:48)(cid:112)s(s + 1)(2s + 1).

Applying this, the final expression for the spin operators is

(C.104)

(C.105)

156

(cid:88)

hmh

=

ˆh2
4

C

J(cid:48)
T M
g(cid:48)m(cid:48)

J(cid:48)
ghmh

T



Li

I1 Ji

li

g

I2

ji

h JT





Lf

I1 Jf

lf
g(cid:48)

I2
jf
h J(cid:48)

T

 .

C

JT MJT
gmghmh

(C.106)

Recombining this in the T-matrix expression gives

(cid:90)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:88)

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

Lf
Mf

Lf Jf Mf LiJiJT MJT
mjf

gmji

gg(cid:48)mgm(cid:48)
J(cid:48)
J(cid:48)
T M
g(cid:48)m(cid:48)
ghmh

T

C

JT MJT
gmghmh

k2 Y

T

J(cid:48)
J(cid:48)
T M
µchmh
J(cid:48)
T M
Jf−(Mf +µ(cid:48)

J(cid:48)

C

T

JAµA
jimji

Icµc

C

JB µB
jf mjf

C

Icµc





C



Li

I1 Ji

li

I2

ji

g

h JT
(r2c)FLf Jf

φjf



(cid:2)Y

Lf

I1 Jf

lf
g(cid:48)

I2
jf
h J(cid:48)
T
(R1A)tN N φji
(r2c)FLiJi

R2

1A

1)jf−mjf

C

JT MJT
Jiµ1jimji

lf (r2c)(cid:3)g(cid:48)

m(cid:48)

g

(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g

mg

(C.107)

Lf (R1A)Y

(R1A)

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1A,

where

and

Lf−Li(−1)

k2 = i

Li+Lf +Mf +µ(cid:48)

1+mjf e

iσLi

1
4

ˆJf

ˆjf

ˆJiˆjiˆgˆh2ˆg(cid:48) ˆLi(4π)
kf ki

3
2

,

(C.108)

157

(cid:88)

λ

tN N =

(2λ + 1)Fλ(R, r)

4π

2λ + 1

(cid:88)

mλ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

(2ji + 1)

2

.

(C.109)

Now we can deal with the angular momentum operators. Because the interaction is scalar,

we want all of the angular momentum variables to couple to total angular momentum 0.

Applying this produces

(cid:110)(cid:2)Y
(cid:88)
(cid:88)
(−1)g−mg δgg(cid:48)δmg−m(cid:48)

T MT
gmgg(cid:48)m(cid:48)

gT MT

C

g

g

gg(cid:48)mgm(cid:48)

=

gmg

ˆg

lf (r2c)(cid:3)g(cid:48)(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g(cid:111)T
lf (r2c)(cid:3)g(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g(cid:111)0

MT

Lf (R1A)Y

Lf (R1A)Y

(cid:110)(cid:2)Y

(C.110)

.

0

We can now take advantage of the rotational invariance of the integrand, as in the central and

tensor interactions. The value of the interaction is unchanged by a rotation of the reference

frame and we can align R1A with the z axis. Once we integrate over Ω1Aφ2c, we gain an

overall factor of 8π2. So we have

(cid:110)(cid:2)Y

Lf (R1A)Y

lf (r2c)(cid:3)g(cid:2)Y Li(R1A)Y li(r2c)(cid:3)g(cid:111)0

0

=

(cid:88)
m(cid:48)
gm(cid:48)(cid:48)

g mf mi

C00
gm(cid:48)

ggm(cid:48)(cid:48)

g

gm(cid:48)
Lf 0lf mf

g

C

(C.111)

g

gm(cid:48)(cid:48)
Li0limi

(cid:88)

ˆLi
ˆLf
8π2
4π
(−1)g−mi

C

=

ˆg

mi

li
mi(ˆr2c)Y
Y
g−mi
Lf 0lf−mi

C

lf
mf

(ˆr2c)

C

gmi
Li0limi

2π ˆLi

ˆLf Y

li
mi(ˆr2c)Y

lf−mi

(ˆr2c).

Combining this result with our full T-matrix expression, we have

158

(cid:90)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

g−mi
Lf 0lf−mi

C

C

gmi
Li0limi



li

g

R2

1A

φjf

(r2c)FLf Jf

(R1A)tN N φji

(cid:88)

C





k3 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

gmgmji

Lf Jf Mf LiJiJT MJT
miJ(cid:48)
J(cid:48)
T M
J(cid:48)
J(cid:48)
T M
g−mghmh

JT MJT
gmghmh

mjf

C

T

T

µchmh
J(cid:48)
T M
Jf−(Mf +µ(cid:48)

J(cid:48)

C

Li

I1 Ji

Lf

I1 Jf

lf

g

I2
jf
h J(cid:48)

T

I2

ji

h JT
(r2c)FLiJi

1)jf−mjf

C

JT MJT
Jiµ1jimji

li
mi(ˆr2c)Y

lf−mi

(ˆr2c)

T

 Y

(R1A)

R2
1Ar2

2cdR1Adr2csin(θ2c)dθ2c,

(C.112)

where

k3 = i

Lf−Li(−1)

Li+Lf +Mf +µ(cid:48)

1+mjf

−mi−mg

iσLi

e

1
4

ˆJf

ˆjf

ˆJiˆji

ˆh2

ˆL2
i

ˆLf (4π)
kf ki

3
2

.

(C.113)

Now we can look more closely at the form of our potential:

(cid:88)

λ

tN N =

(2λ + 1)Fλ(R, r)

4π

2λ + 1

(cid:88)

mλ

(ˆr2c)∗Yλ,mλ

Yλ,mλ

( ˆR1A)

(2ji + 1)

2

.

(C.114)

After aligning R1A with the z axis, tN N simplifies to

(cid:88)

λ

tN N =

Fλ(R, r)

ˆλ2π(2ji + 1)

√
4π

Y λ
0 (ˆr2c).

(C.115)

159

We can now complete the simplification of the angular variables using Equations (C.38) and

(C.40):

Y

li
mi(ˆr2c)Y

lf−mi

(ˆr2c)Y λ

0 (ˆr2c) =

(cid:88)

L

ˆli
ˆlf
√
ˆL
4π

lf 0li0CL0
CL0

lf−milimi

0 (ˆr2c)Y λ
Y L

0 (ˆr2c)

(C.116)

and

(cid:90)

Y L
0 (ˆr2c)Y λ

0 (ˆr2c) sin θ2cdθ2c =

δLλ
2π

.

(C.117)

Combining everything gives us

(cid:90)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:88)

k4 Y

Lf
Mf

(ˆkf )C

λ

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

gmgmji

Lf Jf Mf LiJiJT MJT
miJ(cid:48)
J(cid:48)
T M
J(cid:48)
J(cid:48)
T M
g−mghmh

JT MJT
gmghmh

mjf

C

C

T

T

µchmh
J(cid:48)
T M
Jf−(Mf +µ(cid:48)

J(cid:48)

C

T

1)jf−mjf

C

JT MJT
Jiµ1jimji

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc



Li

I1 Ji

li

g

I2

ji

h JT







Lf

I1 Jf

lf

g

I2
jf
h J(cid:48)

T

(C.118)

g−mi
Lf 0lf−mi

C

C

gmi
Li0limi

C

g−mi
Lf 0lf−mi

C

gmi
Li0limi

Cλ0
lf 0li0Cλ0

lf−milimi

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

where

160

Lf−Li(−1)

Li+Lf +Mf +µ(cid:48)

1+mjf

k4 = i

−mi−mg

iσLi

e

1
4

ˆJf

ˆjf

ˆJiˆj3
i

ˆh2ˆli

ˆlf

3
2

ˆL2
i

ˆLf (4π)
2kf ki

.

(C.119)

We can simplify some of the Clebsch-Gordan coefficients by implementing equation 12 on

pg. 260 of [117]:

(cid:88)

αβδ

Ccγ
aαbβCe


dδbβCdδ

aαf φ = (−1)b+c+d+f ˆc ˆdCe


cγf φ

a b

e f d

 .

c

(C.120)

First, however, we have to rearrange some Clebsch-Gordan coefficients using expressions

from equation 10 on page 245:

aαbβ = (−1)a+b−cCcγ
Ccγ

bβaα = (−1)b+β ˆc

ˆa

Ca−α
c−γbβ.

(C.121)

Applying this we get

(cid:88)

mi

g−mi
Lf 0lf−mi

C

C

=

gmi
Cλ0
lf−milimi
Li0limi
li+mi+Lf +lf−g ˆg
ˆLi

(−)

= (−)

li+mi+2Lf +lf−g+g+li+λ ˆg
ˆLi

C

Li0
λ0Lf 0

Cλ0
lf−milimi

C

Li0
g−milimi

g−mi
lf−miLf 0

(C.122)

C

lf

li

Li Lf

 .

λ

g

Taking into account that (−1)

2Lf = 1 and (−1)2li = 1 we obtain, as our final T-matrix

expression,

161

(cid:90)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:88)

k5 Y

Lf
Mf

(ˆkf )C

λ

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

1)jf−mjf

C

JT MJT
Jiµ1jimji

(C.123)

Lf Jf Mf LiJiJT MJT
J(cid:48)
J(cid:48)
T M
T
J(cid:48)
T M
g−mghmh

gmgmji
J(cid:48)

JT MJT
gmghmh

mjf

C

C

µchmh
J(cid:48)
T M
Jf−(Mf +µ(cid:48)

J(cid:48)

C

T





Li

I1 Ji

li

g

I2

ji

h JT

Lf

I1 Jf

lf

g

I2
jf
h J(cid:48)

T



C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

Cλ0
lf 0li0C

Li0
λ0Lf 0

lf

li

Li Lf



λ

g

T



φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2c,

where

Lf−Li(−1)

k5 = i

Li+Lf +Mf +µ(cid:48)

1+mjf

−mg+lf +λ

iσLi
e

1
4

ˆJf

ˆjf

ˆJiˆj3
i

ˆh2ˆgˆli

ˆlf

ˆLi

ˆLf (4π)
2kf ki

3
2

.

(C.124)

The total cross section expression is

(cid:88)
µ1µ(cid:48)

1µAµB

dσ
dΩ

=

(cid:12)(cid:12)(cid:104)ψ

µ(cid:48)
1
µB

kf
ki

µ1µf

4π2(¯hc)4

1
ˆJ 2
A ˆµ2
1

|tN N|ψ

µ1
µA

,(cid:105)|2

(C.125)

where JA and JB are the initial and final spin of the target.

162

C.4 Spin-Orbit T-Matrix

This derivation is for (p,n) charge-exchange with a three-body formalism and an NN poten-

tial. The potential only acts between the projectile proton and the target’s valence neutron.

This derivation uses the Condon-Shortly phase convention from Equation (C.1). The isospin

spin-orbit NN potential is directed along the r12 coordinate which runs from the projectile

proton to the valence neutron and has the form

VLS = (τ1 · τ2)VLS(r12)L · S.

(C.126)

Here, we note that the L and S operators act between the two active nucleons on our

system. This presents a challenge because our coordinate system is defined with by Li/f and

li/f , so L cannot act directly on these angular momentum operators, making this operator

particularly difficult to implement. In this section we will first present a derivation for a full

implementation of the spin-orbit interaction in our system, and then we will discuss some

simplifications which we considered for this work.

Just as in the central case, the isospin operator will give a factor of 1

2 and there will be

a factor of (2j + 1) to account for the number of nucleons in the valence shell. This can be

inserted in the same T-matrix expression used for the isospin central interaction, giving us

163

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

Lf Jf Mf LiJi
µcµ(cid:48)
mji
c

mjf

−(Mf +µ(cid:48)
1)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

(ξc)ΞIc

µc(ξc)

∗Ic
µ(cid:48)
Ξ
mji
c
(R1A)

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµ(cid:48)

c

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2dξc,

where

and

Lf−Li(−1)

k1 = i

Li+Lf +Mf +µ(cid:48)

1+mjf e

iσLi

3
2

,

ˆLi(4π)
kf ki

tN N =

(2ji + 1)

2

V (r12)L · S.

(C.127)

(C.128)

(C.129)

Immediately, we see that

(cid:90)

∗Ic
µ(cid:48)
Ξ
c

leaving us with

(ξc)ΞIc

µc(ξc)dξc = δµcµ(cid:48)

c

,

(C.130)

164

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

C

JB µB
jf mjf

Icµc

1

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

(cid:2)Y
Lf (R1A)ΞI1(−ξ1)(cid:3)Jf
(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji

µ1

k1 Y

Lf
Mf

(ˆkf )C

Jf Mf +µ(cid:48)
Lf Mf I1µ(cid:48)

1

(cid:2)Y

lf (r2c)ΞI2(−ξ2)(cid:3)jf−mjf

Lf Jf Mf LiJi
mji

mjf

µc

−(Mf +µ(cid:48)
1)

(cid:2)Y li(r2c)ΞI2(ξ2)(cid:3)ji

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

mji
(R1A)

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2.

(C.131)

The spin-orbit operator is a scalar, so all of the angular momentum must couple to 0 to

contribute. To achieve this, we will couple together all the angular momentum operators:

(cid:110)(cid:2)Y

JT −(Mf +µ(cid:48)
Jf−(Mf +µ(cid:48)

1)+µ1
1)Jiµ1

C

(cid:88)

JT

Lf (R1A)ΞI1(−ξ1)(cid:3)Jf(cid:2)Y Li(R1A)ΞI1(ξ1)(cid:3)Ji(cid:111)JT

−(Mf +µ(cid:48)

1)+µ1

.

(C.132)

Now we can rearrange the spin and angular momentum components using Equation (B.9)

and make the replacement that MJf

= Mf + µ(cid:48)

1, giving us giving us

(cid:88)

JT ,g,h

ˆJi

ˆJf ˆgˆhC

JT −MJf
Jf−MJf

+µ1
Jiµ1

Lf

I1 Jf

Li

I1 Ji





165

(cid:110)(cid:2)Y

g

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h(cid:111)JT

h JT

(C.133)

.

+µ1

−MJf

We can perform the analogous couplings and rearrangements for the internal angular mo-

mentum variables, leaving us with





(cid:88)

jt,˜g,˜h

ˆji ˆjf

ˆ˜hˆ˜gC

jt−mjf
jf−mjf

+mji
jimji

lf

li

I2

I2

jf

ji

(cid:110)(cid:2)Y

˜h

lf (r2c)Y li(r2c)(cid:3)˜h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)˜g(cid:111)jt

jt

˜g

(C.134)

.

+mji

−mjf

Also, we can break up the spin-orbit operator into its spin and angular momentum parts:

0

L · S = (−1)1ˆ1{L1S1}0

(−1)1ˆ1C00

(cid:88)
(cid:88)
(−1)−mLL1mL

mLmS

=

=

mL

S1−mL

.

1mL1mS

L1mL

S1mL

(C.135)

Next, we couple all of the operators together so we can ultimately apply conservation of spin

and angular momentum:

(cid:88)

T MT

(cid:34)(cid:110)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)h(cid:111)JT

T MT
JT −MJf

C

(cid:110)(cid:2)Y

+µ1jt−mjf

+mji

lf (r2c)Y li(r2c)(cid:3)˜h(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)˜g(cid:111)jt

(cid:35)T

.

MT

(C.136)

We, again, rearrange these operators using Equation (B.9), giving

166



 C

ˆJT

ˆjt ˆxˆy

g h JT

(cid:34)(cid:110)(cid:2)Y
(cid:88)
lf (r2c)Y li(r2c)(cid:3)h(cid:111)x(cid:110)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)˜g(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)˜h(cid:111)y

+µ1jt−mjf

T MT
JT −MJf

x y T

T MT xy

+mji

˜h ˜g

jt

=

(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g
(cid:35)T

.

MT

(C.137)

As mentioned before, because the operator is a scalar, we need the spin components to couple

to 1, so they can couple with the S operator to give 0. This constraint implies that y=1. If

we uncouple the spin operators and apply this simplification we get



 C

=

ˆJT

ˆjt ˆxˆ1

˜h ˜g

g h JT

(cid:88)
T MT xmxm(cid:48)
(cid:104)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI1(ξ1)(cid:3)˜g(cid:2)ΞI2(−ξ2)ΞI2(ξ2)(cid:3)˜h(cid:105)1

T MT
xmx1m(cid:48)

x 1 T

jt

1

1

.

mx

m(cid:48)

1

C

T MT
JT −MJf

+µ1jt−mjf

+mji

(C.138)

Finally we rearrange the spinor components using Equation(B.9 once again more time to

have a projectile and target nucleon coupled together in each channel giving us

167



 C

˜g

˜h

1
2

1
2

α β 1

=

ˆJT

g 1 JT

ˆjt ˆxˆ1ˆ˜gˆ˜hˆα ˆβ

(cid:88)
T MT xmxm(cid:48)
(cid:104)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)α(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)β(cid:105)1

x 2 T

h 1

1αβ

jt





1
2

1
2

mx

.

m(cid:48)

1

T MT
xmx1m(cid:48)

1

C

T MT
JT −MJf

+µ1jt−mjf

+mji

(C.139)

Because we have the angular momentum operators and spin operators coupled separately,

we can impose conservation of the projection of the angular momentum in both spaces

separately.

Including the projections from the spin-orbit operator, we are left with the

relations

m(cid:48)
1 − mL = 0

m(cid:48)
1 = mL

mx + mL = 0

mx = −mL.

Also, we see from the Clebsch-Gordan coefficient in Equation (C.139) that MT = mx + m(cid:48)
1.

Using the substitutions from above, we notice that MT = 0.

Incorporating all of these

changes into the T-matrix expression, we are left with

168

(cid:90) (cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

LiJi
Lf Jf MJf
µcgh˜g˜h
mjf
mji
JT jtmLxT αβ
jt−mjf
+µ1
jf−mjf
Jiµ1

C

+mji
jimji

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf

k2 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc







CT 0
x−mL1m1

CT 0
JT −MJf

+µ1jt−mjf

+mji





1
2

1
2

1
2

1
2



˜g

˜h

x 1 T

α β 1

L1mL

S1−mL

lf

li

I2

I2

jf

ji

Lf

I1 Jf

Li

I1 Ji

g h JT

˜h ˜g

jt

g

˜g

˜h

jt

(cid:104)(cid:2)Y
lf (r2c)Y li(r2c)(cid:3)h(cid:105)x
Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y
(cid:104)(cid:2)ΞI1(−ξ1)ΞI2(−ξ2)(cid:3)α(cid:2)ΞI2(ξ2)ΞI1(ξ1)(cid:3)β(cid:105)1

h JT

−mL





mL
(R1A)

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

R2

1A

where

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1Adξ1dξ2,

(C.140)

Li−Lf (−1)

Li+Lf +MJf

k2 = i

−mL

+mjf

iσLi ˆLi
e

ˆJi

ˆJf ˆgˆhˆ1 ˆji

3 ˆjf

ˆJT

ˆjt ˆxˆα ˆβˆ˜gˆ˜h

3
(4π)
2
2kikf

,

(C.141)

and

(cid:88)

λmλ

tN N =

Fλ(R1A, r2c)4πY λ
mλ

(R1A)Y λ
mλ

(r2c)∗.

(C.142)

Fλ contains the interaction over the r12 coordinate expressed as a function of the other two

169

coordinates. We can then apply the spin operator. This amounts to calculating

(cid:88)

mαmβ

qmqrmr(cid:104)I1I2αmα|S1,−mL
1mL

C

|I1I2βmβ(cid:105),

(C.143)

which gives a factor of − 1√

2

δαβδ1α. Substituting this in gives us

k3 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf



(cid:90) (cid:88)

Lf Jf MJf
mji

LiJi
µcgh˜g˜h

mjf
JT jtmLxT
jt−mjf
jf−mjf

C

+µ1
Jiµ1





CT 0
x−mL1m1

CT 0
JT −MJf

+µ1jt−mjf

+mji

+mji
jimji











˜g

˜h

1
2

1
2

1
2

1
2

1 1 1

lf

li

I2

I2

jf

ji

Lf

I1 Jf

Li

I1 Ji

g h JT

˜h ˜g

jt

L1mL

(cid:104)(cid:2)Y

˜g

˜h

jt

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

g

x 1 T

h JT

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x

−mL

φjf

(r2c)FLf Jf

(R1A)tN N φji

(r2c)FLiJi

(R1A)

R2

1A

R2
1Ar2

2cdR1Adr2cdΩ2cdΩ1A,

where

Li−Lf (−1)

Li+Lf +MJf

k3 = i

−mL

+mjf

iσLi ˆLi
e

ˆJi

ˆJf ˆgˆh ˆji

3 ˆjf

ˆJT

ˆjt ˆxˆ˜gˆ˜h

3

(cid:113) 3

2(4π)
kikf

(C.144)

3
2

.

(C.145)

Next we can simplify the angular momentum operators by coupling to total angular momen-

170

tum 0:

C00
x−mL1mL
(−1)1+mL

=

ˆ1

δx1δmLmL

(cid:40)(cid:104)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

lf (r2c)Y li(r2c)(cid:3)h(cid:105)x

(cid:41)0
lf (r2c)Y li(r2c)(cid:3)h(cid:105)1

L1

0

(cid:104)(cid:2)Y

Lf (R1A)Y Li(R1A)(cid:3)g(cid:2)Y

(C.146)

L1

mL

.

−mL

If we now align the R1A coordinate along the ˆz axis, the spherical harmonics over ˆR1A

reduce to

ˆLi
ˆLf
4π . This is additionally helpful, because in this configuration, θ2c = θ12 and

φ12 = π + φ2c. Finally, we can perform the integration over sin θ1Adθ1Adφ1A gives a factor

of 4π. Applying this and uncoupling the angular momentum operators, we are left with

(cid:88)

=

mLmhmgmimf

(−1)1+mL

ˆ1

C00
1−mL1mL

C

1−mL
gmghmh

C

hmh
lf mf limi

C

gmg
Lf 0Li0

ˆLf
ˆLi
4π

(C.147)

lf
mf

Y

(ˆr2c)Y

li
mi(ˆr2c)L1

mL

4π.

It is immediately clear that mg = 0 and, therefore, mh = −mL. Also, the Clebsch-Gordan

coefficient coupled to 0 can again be simplified. Finally, we can make the substitution at
mi = −mL − mf . Combining these simplifications, we are left with

(cid:88)

mLmf

(−1)2(1+mL)

ˆ12

1−mL
g0h−mL

C

h−mL
lf mf li−mL−mf

C

Cg0

Lf 0Li0

ˆLi

ˆLf Y

lf
mf

(ˆr2c)Y

li−m2−mf

(ˆr2c)L1

mL

(C.148)

The T-matrix expression is now

171

k4 Y

Lf
MJf

−µ(cid:48)

1

(ˆkf )C

Jf MJf
Lf MJf

−µ(cid:48)

1I1µ(cid:48)

1

C

Jiµ1
Li0I1µ1

C

JAµA
jimji

Icµc

(cid:90)

(cid:88)

(cid:104)ψ

µ(cid:48)
1
µB

|tN N|ψ

µ1
µA

(cid:105) =

mji

Lf Jf MJf

LiJi
µcgh˜g˜hmf
mjf
JT jtmLxT
jt−mjf
jf−mjf

+mji
jimji

C

JB µB
jf mjf

C

Icµc

JT −MJf
Jf−MJf

+µ1
Jiµ1

lf

li

I2

I2

jf

ji

Lf

I1 Jf

Li

I1 Ji

g h JT

˜h ˜g

jt

C







˜h
˜g
1−mL
g0h−mL
φjf

C

h JT

g

jt
h−mL
lf mf li−mL−mf

C

(r2c)FLf Jf

(R1A)tN N φji

R2

1A

where

CT 0
x−mL1m1

CT 0
JT −MJf

+µ1jt−mjf

+mji









1
2

1
2

1
2

1
2



˜g

˜h

x 1 T

1 1 1

lf
Cg0
Lf 0Li0Y
mf
(r2c)FLiJi

(R1A)

(ˆr2c)Y

li−mL−mf

(ˆr2c)L1

mL

R2
1Ar2

2cdR1Adr2c sin θ2cdθ2cdφ2c,

(C.149)

(cid:113) 3

2(4π)
kikf

3
2

(C.150)

Li−Lf (−1)

Li+Lf +MJf

k4 = i

−mL

+mjf

iσLi ˆLi
e

2 ˆLf

ˆJi

ˆJf ˆgˆh ˆji

3 ˆjf

ˆJT

ˆjt ˆxˆ˜gˆ˜h

and, in the tN N operator, mλ = 0 and the spherical harmonic over ˆR1A simplifies to

ˆλ√
4π

.

Incorporating this, the expression becomes

(cid:88)

λ

tN N =

Fλ(R1A, r2c)4πY λ

0 (r2c)

ˆλ√
4π

.

(C.151)

Explicitly, Fλ is

172

(cid:90) 1

−1

1
2

V (r12)Pλ(z)dz.

(C.152)

To implement this expression, one could apply the angular momentum operator, L1

mL

, the

the spherical harmonics in the initial channel and then perform the remaining angular inte-

gral over the this result and all other remaining spherical harmonics. Explicitly, L can be

represented by Equation (18) from pg. 41 of [117]:

∂θ

eiφ(cid:110) ∂
e−iφ(cid:110) ∂

L1,+1 = − 1√
2

L1,0 = −i
∂
∂φ
L1,−1 = − 1√
2

(cid:111)

∂
∂φ

+ icotθ

(cid:111)

.

− icotθ

∂
∂φ

(C.153)

∂θ

However, CHEX is not currently set up to handle these angular integrals, so we hoped to

find an approximation that could give us an idea of the shape and magnitude of the spin-

orbit contribution without this explicit integration. The first simplification we considered

was a zero-range approximation for the valence nucleon and core. In this approximation, the

valence particle would sit directly on top of the core, collapsing the r2c coordinate. In this
configuration, r12 = R1A and applying L · S is straightforward.

While this would simplify the T-matrix expression considerably, Figure C.1 demonstrates

the dependence of the radial integral over the NN potential and both scattering and bound

state wave functions on the various radial variables. The case shown is for the central term

of the AV8’ potential for the 48Ca(p,n)48Sc reaction, but the behavior is similar across all

terms of both NN potentials studied here. The x-axis shows the cutoff on the lower bound

of the radial integral for each radial variable. As expected, we see that the radial integral

173

quickly goes to zero if we cut out contributions from even r12 < 1 fm. This is reflected in

the geometry of the NN potentials which are all peaked in magnitude at 0 fm and approach

0 in the range of 2-3 fm. However, we can see that a similar radial cut in the bound state

coordinate, r2c or the scattering coordinate, R1A does not cause a large decrease to the

radial integral until about 2 fm and the fall-off is less steep. This indicates that the reaction

is occurring near or just below the surface of the target nucleon when the scattering and

valence nucleons are in close proximity, as opposed to in the internal region of the core.

Figure C.1: Dependence of radial integral to the range of various radial parameters for the
case of 48Ca(p,n)48Sc. The x-axis shows the internal radial cut in each parameter.

Based on this, we experimented with applying an approximation that is a hybrid of a

zero-range approximation in the r12 coordinate. The zero-range approximation would model
the projectile/valence potential as a delta function with a weighing factor, VN N (r12)φ(r12) ∼

D0δ(r12). The strength of D0 can be found by taking the integral of the proton-neutron

wave function and the NN potential.

Additionally, we need to adjust our integrals over the bound state and scattering state

variables to reflect the fact that we are only considering contributions in a limited region of

the nuclear volume. Instead of integrating over the entire length of r12 and R1A, we weight

174

the interaction by only including an integral over these wave functions in the region near the

nuclear surface. The strength of the radial interaction will depend on the cutoffs chosen for

this approximation and the choice should depend on the size of the target considered in the

reaction. After studying the response in targets of various size, we chose a weighting factor
r0A1/3

which included contributions from the bound and scattering wave functions between

2.5

to r0A1/3, where r0 is the radius parameter of the Woods-Saxon potential that describes the

bound state.

The biggest challenge to this approximation is that our coordinates do not contain a nat-

ural wave-function to describe the relative motion between proton and neutron. The form of

this wave function will directly impact the magnitude of D0 in the zero range approximation,

and, therefore, the magnitude of the cross section. We considered several different forms of

this wave function, including a simple deuteron wave function parameterized by the sum

of exponentials from [118]. This wave function has both an s-wave and d-wave component.

However, the spin-orbit contribution from from relative angular momentum L = 0 is 0, so

we only consider the d-wave contribution and, thus, will only consider L=2 contributions to

the spin-orbit potential. Strictly speaking, the zero-range approximation would only include

L=0, but we allow for this deviation in order to explore the possible effect of including a

spin-orbit term from the NN interaction.

We also considered simpler forms, like rL or a gaussian form factor. Because D0 was so

sensitive to the choice of wave function, and we had no consistent way of choosing an appro-

priate factor given our representation, we felt we could neither conclude that the spin-orbit

contribution is small enough to disregard nor be convinced that any of our approximations

would give an accurate representation of the magnitude or shape. Thus, we decided it was

best to refrain from implementing an approximate spin-orbit contribution and leave this

175

problem as an area of future investigation.

176

APPENDIX D

Benchmarking and Testing CHEX

The charge-exchange code, CHEX, was developed to calculate charge-exchange cross sec-

tions in both the two and three-body formalisms presented in this thesis. Throughout its

development, CHEX was benchmarked and tested against other reaction codes for several

cases in order to ensure it was working properly. Additionally, we compared analytic forms

with known solutions, for specific simplified cases. Some of those tests are presented and

discussed here. Wave functions and elastic scattering cross sections were compare to results

from the reaction code FRESCO [76]. Charge-exchange cross sections in the two-body for-

malism are compared to results from the unpublished charge-exchange code developed for

the IAS analysis work in [10].

Scattering Wave Functions

First, we verify that the scattering wave functions produced by CHEX are correct. This is

demonstrated in Fig. D.1 for the case of the reaction 48Ca(p,p)48Ca at Ep=30 MeV. The

solid black (blue) lines show the real (imaginary) part of the scattering wave function for the

L=0, J=0.5 partial wave calculated in CHEX. These calculations are compared to results

from FRESCO shown by the dashed red (green) lines. For these calculations, we used a step

size of 0.05 fm, a matching radius of 30 fm, and the OMP parameterized by Koning-Delaroche

[1]. The calculation is fully converged using these parameters. Scattering wave functions

177

calculated by CHEX agree with Fresco within 1%, with an average percent difference of

0.33% for the real scattering wave function and 0.46% for the imaginary scattering wave

function in the region shown here.

Figure D.1: Real and Imaginary parts of the scattering wave function as a function of
scattering radius for the L=0, J=0.5 partial wave for 48Ca(p, p)48Ca at Ep = 30.0 MeV. The
solid black (blue) lines show the real (imaginary) part of scattering wave function calculated
in CHEX, the dashed red (green) lines show the same result produced by FRESCO.

Elastic Scattering

Next, we demonstrated that the elastic scattering cross sections calculated by CHEX re-

produce results from FRESCO. This can be seen in Fig. D.2, again for the case of the

reaction 48Ca(p,p)48Ca at Ep=30 MeV. The solid black line shows the differential elastic

scattering cross section relative to Rutherford calculated in CHEX. This is compared to the

178

result from FRESCO shown by the dashed red line. For this calculation, we used the same

input parameters as for the scattering wave function benchmarking tests (stepsize of 0.05 fm,

matching radius of 30 fm, and optical model parameterized by Koning-Delaroche [1]) and

the calculation is fully converged using these parameters. Elastic scattering cross sections

produced by CHEX agree with Fresco within 1%, for small angles, with a slight divergence

at large scattering angles. The average percent different over all angles shown in Figure D.2

is 1.84%.

Figure D.2: Differential elastic scattering relative to Rutherford as a function of scattering
angle for 48Ca(p, p)48Ca at Ep = 30.0 MeV. The solid black line shows the elastic scattering
cross section calculated in CHEX, the dashed red line shows the same result produced by
FRESCO.

179

Bound States

Calculations performed in the three-body formalism require the calculation of a bound state

wave function. These wave functions have been benchmarked with bound state calculations in

FRESCO. Fig. D.3 shows the n+47Ca (left) and p+47Ca (right) bound state wave functions

which correlate to the target and residual of the 48Ca(p,n)48Sc. The cross section calculated

using CHEX is shown in solid black and the result from FRESCO is shown in dashed red.

In this case, the n+47Ca bound state corresponds to the f7/2 ground state in 48Ca, and the

p+47Ca bound state corresponds to the the IAS in 48Sc. The calculation used a step size

of 0.5 fm, a matching radius of 2.5 fm and a maximum radius of 30 fm and the calculation

is fully converged with these parameters. Bound state wave functions produced by CHEX

agree with results from FRESCO well within 1%, with an average percent difference of 0.03%

within the region shown if Figure D.3.

Figure D.3: Bound state wave functions as a function of radius. The left plot shows n+47Ca
bound state wave function and the right plot shows the p+47Ca bound state wave function,
which correlate to the in the initial and final states of the 48Ca(p,n)48Sc reaction. The solid
black line shows results from CHEX, the dashed red line shows the same result produced by
FRESCO.

180

Two-Body Charge-Exchange

Two-body calculations performed using CHEX can be directly compared to results from

[10]. Before checking the implementation in CHEX, we first verified that the expressions we

developed for the charge-exchange T-matrix and cross sections, shown in Equations (2.41)

and (2.42), give the same result as the expression presented in [10]. This task was not trivial,

as the expressions were developed using sums over different parameters. The cross section

expression from [10] gives

dσ(p,n)

dΩ

=

1
k2
p

1

2s + 1

(cid:88)

L

(2L + 1)A

(p,n)
L

PL(cos θ),

(D.1)

A

(p,n)
L

where

and

=4µpµnkpkn

(2J(cid:48) + 1)(2l(cid:48) + 1)

(cid:88)

Jl

(2J + 1)(2l + 1)



l(cid:48) L
J(cid:48) J s

2

Re[I∗

J(cid:48)l(cid:48)IlJ ],

l

J(cid:48)l(cid:48)

l(cid:48) L

(cid:88)
2 l
(cid:112)|N − Z|
(cid:90) ∞

A

0

0 0 0

(D.2)

(D.3)

Ilj = 2

drr2u+

nJl(r)U Jl

1 (r)u+

pJl(r).

In this formulation, u(r) are the radial scattering wave functions, U1 is the charge-exchange
term from the Lane potential shown in Equation (2.35). In these expressions, J/J(cid:48) and l/l(cid:48)

do not correlate to the initial and final angular momentum values from our definition but,

instead, correspond to the different indexes used while taking the modulus squared of the

181

reaction amplitude. There is no direct correlation between L in the expression above, and

our Equation (2.41). To prove equivalence of these two expressions, we can demonstrate

that they give the same results for particular partial waves, as well as an arbitrary sum of

partial waves. We do this by setting the values of the integrals to 1 (Ilj in the expression

above) and checking the analytical forms of the expressions using Mathematica. It is worth

noting that the wave functions in [10] include a factor of

1
ki/f

and, for the case of proton

distorted waves, include the Coulomb phase shift. Excluding these factors, the first three

partial waves, in both expressions, have the analytical forms shown in Table (D.1).

L value

Analytic Expression

0

1

2

0+1+2

40kf µf µi
ki(¯hc)4

ki(¯hc)4

360kf µf µiCos(θ)2

250kf µf µi

ki(¯hc)4 − 1500kf µf µiCos(θ)2
− 540kf µf µiCos(θ)2

ki(¯hc)4

+

2250kf µf µiCos(θ)4

+
1800kf µf µiCos(θ)3

ki(¯hc)4

90kf µf µi

ki(¯hc)4 − 360kf µf µiCos(θ)

ki(¯hc)4

ki(¯hc)4

ki(¯hc)4

+

2250kf µf µiCos(θ)4

ki(¯hc)4

Table D.1: Analytic result for the first few partial waves for the two-body charge-exchange
cross sections calculated using the T-matrix from Equations (2.42) and (D.1). In the expres-
sions above, the radial integrals are set to 1.

Once we determined that our expressions for the two-body T-matrix and cross sections are

sound, we could check that the implementation in CHEX was correct by directly comparing

charge-exchange cross sections with those produced by Danielewicz et al. In making this

comparison, we found there was a slight difference in the way that the spin-orbit term is

included in the Lane interaction (see Figure D.4(a)).

In the work by Danielewicz et al.,

they varied parameters to create best fits to charge-exchange data and, as a result, their

code initialized the spin-orbit parameters at slightly different values than the standard KD

values. Figure D.4(b) shows that, if this small spin-orbit term is neglected, the cross sections

agree well within 1% difference.

182

Figure D.4: Comparing the charge-exchange cross sections for 48Ca(p,n)48Sc at 35 MeV,
produced by [10] and by CHEX. (a): there is a small difference resulting from the way spin-
orbit was included. (b): once this spin-orbit contribution is neglected, the two calculations
match exactly.

Three-Body Charge-Exchange

Benchmarking the three-body formalism presented in this thesis, as well as its implementa-

tion in CHEX, presents a unique challenge because there is no option for direct comparisons

with another charge-exchange code (this is the first implementation of its kind). Still, we

took many steps to ensure that the resulting calculations in CHEX were correct. These

tests generally fell into one of two categories: tests which ensured that components were

implemented accurately in CHEX, and limiting cases that simplify the T-matrix expressions

and allow us to verify our results.

While implementing the formalism for each operator’s T-matrix expression, sums in

CHEX were checked numerically against the same sums coded in Mathematica. This ensures

that errors were not introduced to the code through FORTRAN syntax errors or through

issues calling subroutines that compute algebraic factors, such as 6 and 9-j factors. Addi-

tionally, we could use Mathematica to ensure that our numerical integrals, including those

183

that utilize a multipole expansion, were working properly. We did this by importing the

wave functions and potentials produced by CHEX into Mathematica and using its interpo-

lation and numerical integration functions to calculate integrals. When we compared these

integrals to those produced by CHEX, they were in agreement.

We noticed that in cases where the magnitude of the potential integral was very small,

the value of the integral could be sensitive to the radial cut-off and number of integration

points used in the calculation. This was resulting from adding many small contributions

from regions where the potential should be, effectively, zero. This problem was solved by

adding a routine which sets the value of the integrant to 0 if its magnitude drops below a

certain tolerance value, which depends on the strength of the potential.

Finally, we were able to check components of the CHEX implementation, as well simpli-

fied cases with the reaction code DW81 [79]. These cases included replacing incoming and

outgoing distorted waves and our NN interaction with simple Yukawa interaction. However,

a detailed comparison of charge-exchange cross sections was hindered by differences in the

reaction formalism.

We also performed a number of analytical checks to ensure that our T-matrix expressions

behave as we expect in limiting cases. These tests included using spin=0 for the projectile

and the valence nucleon, which results in Li/f = Ji/f and li/f = ji/f , greatly reducing the

number of sums and reducing 6 and 9-j symbols to simpler expressions. We also checked, for

each T-matrix, the results when the partial wave expansion was reduced to just L=0 or L=1.

Finally, we checked our T-matrix expression for the limiting case where the T-matrix operator

is assumed to be unity, VN N =1.

In this case, you would expect δLi,Lf δli,lf

δJi,Jf

δji,jf

.

CHEX passed each of these tests.

184

BIBLIOGRAPHY

185

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