THE EXCITATION OF ISOVECTOR GIANT RESONANCES VIA THE 60Ni(3He, 𝑡)
REACTION AT 140 MEV/U

By

Felix Ndayisabye

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2024

ABSTRACT

Nuclear charge-exchange reactions at intermediate energies are powerful probes of the isovector

response of nuclei. They provide an opportunity to study isovector giant resonances, such as the

Gamow-Teller resonance and the isovector giant monopole and dipole resonances. The properties

of these giant resonances provide important insights into the bulk properties of nuclear matter

and have important implications for neutrino and astrophysics. In this work, the focus is on the

investigation of the properties of isovector giant resonances excited via the 60Ni(3He,𝑡) reaction

at 140 MeV/u up to excitation energies of 60 MeV to study the Gamow-Teller resonance, isobaric

analog state, isovector (spin) monopole, dipole, and quadrupole giant resonances.

The (3He,𝑡) reaction was used as an isovector probe to investigate the properties of isovector giant

resonances in 60Cu. To investigate these isovector giant resonances, the analysis was done through

a multiple decomposition analysis (MDA). The differential cross sections were fitted with a linear

combination of the distorted wave Born approximation (DWBA) angular distributions associated

with different angular momentum transfer Δ𝐿. The angular distributions were calculated in DWBA

by using the code package FOLD code. Different giant resonances were seen at different excitation

energies. The results were compared with shell-model, and normal-mode calculations.

It was found that the Gamow-Teller strengths extracted from the experiment could be repro-

duced reasonably well by the shell-model calculations. The extraction of the isovector dipole and

monopole resonances was complicated by the presence of the quasi-free continuum. A detailed

extraction of the isovector monopole resonances was not possible. For isovector dipole resonance,

a reasonable consistency was found with the normal-mode calculations, after applying a simple

estimate for the contributions from the quasifree continuum.

Overall, this work provides valuable insights into the properties of isovector giant resonances,

highlights the importance of continuum subtraction, and provides a detailed analysis of the (3He,𝑡)

reaction for probing these resonances at high excitation energies.

Copyright by
FELIX NDAYISABYE
2024

This thesis is dedicated to my parents, who have been a constant source of
love and support despite their distance, and to my uncle Pius MASUMBUKU,

whose inspiration has shaped me into the person I am today.

"Ikivi cyawe ndacyushije ukiriho"

iv

ACKNOWLEDGEMENTS

The completion of this Ph.D. journey was made possible through standing on the collective shoulder

of giants. I’ve been incredibly fortunate to have the backing of colleagues, friends, family, and

my church community throughout my graduate studies at Michigan State University, within the

Department of Physics and Astronomy, and particularly at the Facility for Rare Isotope Beams

(FRIB). The graduate school presented numerous challenges, both academic and personal, as I

balanced my roles as a student, parent, and father of two wonderful children, Abia Curie ILISA (7

years old) and Happy Austin TUZO (4 years old). I owe an immense debt of gratitude to all those

who helped me navigate this journey.

First and foremost, I would like to give a special and wholehearted appreciation to my advisor,

Remco Zegers. Your support, guidance, and unwavering encouragement have been instrumental

throughout this research endeavor. Your mentorship has been the cornerstone of my academic

journey, and I deeply value the dedication you’ve shown. I’ll always remember your daily check-ins

and your willingness to respond to emails, even during weekends and late nights.

I’ll always

remember your evening visits to my desk before you go home with words of encouragement and

were truly uplifting. Thank you for accepting me as one of your mentees, irrespective of my

background, and for allowing me to prioritize my family during challenging times. Your support

and advocacy will forever hold a special place in my heart. I am equally grateful to the members of

my thesis committee, Paul Gueye, Filomena Nunes, Danny Caballero, and Tyce DeYoung, for your

invaluable guidance, insightful discussions, questions, and feedback during our yearly meetings,

which significantly enriched the quality of this work.

Special thanks are owed to Pawel Danielewicz, Scott Pratt, and Kirsten Tollefson for your

unwavering efforts in securing my admission to the Department of Physics and Astronomy at

Michigan State University. It was not easy due to VISA issues, but you made it possible. I’ll never

forget the dedication and training I received from Pawel Danielewicz during my first semester at

MSU.

My gratitude extends to all past and current members of the FRIB charge-exchange (CE) group,

v

including R. G. T. Zegers, D. Bazin, S. Noji, J. Pereira, C. J. Zamora, S. Giraud, A. Mustak, J.

Schmitt, C. Maher, Z. M. Rahman, and J. Rebenstock. Special thanks to Shumpei, Juan, and Simon

for your invaluable contributions to graduate and undergraduate students in the CE group during

our weekly research group meetings and for patiently addressing our questions. To J. Schmitt, thank

you for your friendly assistance and guidance regarding DWBA input parameters during your time

in the CE group. The list of appreciations goes on, but a special mention to C. Maher for creating

the initial Python code to run MDA files; it proved immensely helpful in my analysis processes.

I am profoundly grateful to Mornetka Guèye, Paul Guèye, and Rachel Younger, whose unwa-

vering support has extended far beyond the academic realm. They have not only been pillars of

strength in my scholarly pursuits but have also provided invaluable emotional and social support to

both my family and myself during the extraordinary challenges we’ve confronted in dealing with

my wife’s critical health condition during my academic journey.

I’d like to express my sincere gratitude to the Professors and instructors who guided me through-

out my Ph.D. studies, including Michael Thoennessen (my first Advisor at NSCL/FRIB), Daniel

Bazin (Master’s Advisor), Vladimir Zelevinsky, Mohammad Maghrebi, Luke Robert, Andreas Von

Manteuffel, Witold Nazarewicz, Yamazaki Yoshishige, Danny Caballero, Filomena Nunes, Alex

Brown, Carlo Piermarocchi, Artemis Spyrou, David J. Morrissey, and Sean Liddick.

A special note of appreciation goes to the dedicated staff in the Department of Physics and

Astronomy, with a heartfelt thank you to Kim Crosslan and her exceptional team. Your unwavering

support, and treating students like your own, have been truly remarkable.

I’m also grateful to

the entire team at FRIB, where I’ve had the privilege to learn.

I extend my gratitude to FRIB

lab directors, Thomas Glasmacher and Brad Sherrill, and Heiko Hergert, for their exceptional

leadership in creating a safe and enriching scientific environment that operates around the clock.

I’m also appreciative of the financial assistance I received from both FRIB and the Department of

Physics and Astronomy during the challenging times I encountered. The financial support from the

U.S. National Science Foundation Grant No: PHY-2209429, Windows on the Universe: Nuclear

Astrophysics at FRIB, greatly contributed to making this work possible.

vi

I acknowledge the bridge connection between the African Institute for Mathematical Sciences

(AIMS) and MSU, specifically at FRIB, which paved the way for my Ph.D. pursuit. Lastly, I want

to express my deep appreciation to the members of the Spartan Chapter of NSBP at MSU; you’ve

been an essential part of my journey, and I’m proud to have been a member. My heartfelt thanks

also go to my fellow graduate students, officemates, and friends who provided invaluable support

along the way, including Pierre Nzabahimana, Chi-En (Fanurs) Teh, Timilehin Ogunbeku, Yani

Udiani, Jean Pierre Twagirayezu, Sylvester Agbemava, Linda H’lophe, Avik Sarkar, and anyone

reading this message! Your support has meant the world to me. To all the friends I met during this

journey, their company means a lot to me. THANK YOU!.

vii

TABLE OF CONTENTS

CHAPTER 1

1.1 Motivation .
.
1.2 Thesis Organization .

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.

.

.

.

.

.
.
.

1
1
2

CHAPTER 2

GIANT RESONANCES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
Isospin Picture .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Isovector Giant Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
2.1 Definitions and Classification of the Giant Resonances
4
2.2
7
2.3
9
2.4 Hydrodynamic Model of Giant Resonances
. . . . . . . . . . . . . . . . . . . 14
. 16
Isovector Giant Resonances in the Charge-Exchange Spectrum . . . . . . . .
2.5
2.6 Overview of the Charge-Exchange Reaction Probes . . . . . . . . . . . . . . . 18

. .

.

CHAPTER 3

Introduction into Charge-Exchange Reaction Techniques

CHARGE-EXCHANGE REACTION TECHNIQUES . . . . . . . . . . 24
3.1
. . . . . . . . . . . . 24
3.2 Calculation of Differential Cross Sections . . . . . . . . . . . . . . . . . . . . 25
3.3 Brief Description of the DWBA . . . . . . . . . . . . . . . . . . . . . . . . . 27
Inputs & Outputs for the DWBA Calculations . . . . . . . . . . . . . . . . . . 28
3.4
3.5 Eikonal Approximation and the Unit Cross Section for GT and IAS . . . . . . . 45

CHAPTER 4

DATA ANALYSIS: 60Ni(3He, 𝑡) REACTION AT RCNP . . . . . . . . . 48
4.1 Experimental Setup, Procedures & Devices
. . . . . . . . . . . . . . . . . . . 48
4.2 Grand Raiden Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Focal-Plane Detectors for Grand Raiden Spectrometer . . . . . . . . . . . . . . 50
. 53
4.4 Missing-Mass Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 56
4.5 Magnetic Fields of Grand Raiden Spectrometer
4.6 Angular Acceptance and Resolution . . . . . . . . . . . . . . . . . . . . . .
. 58
4.7 Differential Cross Sections Calculation . . . . . . . . . . . . . . . . . . . . . . 64

CHAPTER 5

RESULTS AND COMPARISON WITH THEORY . . . . . . . . . . . . 67
5.1 Calculated and Measured IAS . . . . . . . . . . . . . . . . . . . . . . . . . . 67
. 69
5.2 Multiple Decomposition Analysis Results
5.3 Gamow-Teller Strength B(GT) Extraction . . . . . . . . . . . . . . . . . . . . 75
5.4 Background or quasifree-continuum Estimation . . . . . . . . . . . . . . . .
. 87
5.5 Extraction of the Isovector Spin Giant Monopole Strength . . . . . . . . . . . . 96
5.6 Extraction of the Isovector Spin Giant Dipole Strength . . . . . . . . . . . . . 98

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

CHAPTER 6

.
6.1 Summary .
6.2 Future Outlook . .

CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . . .

. 100
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
. 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

BIBLIOGRAPHY . .

. .

.

.

.

.

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

APPENDIX A

FULL INPUT FILE IN THE WSAW CODE . . . . . . . . . . . . . .

. 118

viii

.
.

.

APPENDIX B

INPUT AND OUTPUT FILES IN THE NORMOD CODE . . . . . . . 121

APPENDIX C

FULL INPUT FILES IN THE FOLD CODE . . . . . . . . . . . . . .

. 132

APPENDIX D

FULL INPUT FILES IN THE DWHI CODE . . . . . . . . . . . . . .

. 139

APPENDIX E

PROPORTIONALITY RELATION BETWEEN 𝑑𝑝 AND 𝑑𝑘 . . . . . . 146

ix

CHAPTER 1

INTRODUCTION

1.1 Motivation

Charge-exchange (CE) reactions involve the exchange of charge between a target nucleus and a

nucleus of the incoming beam. CE reactions studies at intermediate beam energies (∼ 100 MeV/𝑢)

provide insight into isovector excitations, and isovector giant resonances in particular. These CE

reactions are associated with a change in isospin (Δ𝑇 = 1). By selecting specific types of CE

reactions, such as (n,p) or (p,n)-type CE reactions, one can define the projection of the isospin

quantum number as 𝑇𝑧 = ±1. A variety of excitations, associated with the transfer of different units

of angular momentum and spin, can be probed. Measurements of the properties of isovector giant

resonances are of interest for a variety of reasons:

• The experimental data can be used to test microscopic models of nuclei up to high excitation

energies and, therefore, test the underlying assumptions made for these models [1].

• The characteristics of isovector giant resonances are also useful to constrain macroscopic

properties of nuclei and nuclear matter. In particular, isovector giant resonances are associated

with out-of-phase oscillations of the proton and neutron "fluids" inside the nucleus. Therefore,

experimental data constrains the restoring forces when neutrons and protons are displaced

relative to each other and thus provide insight about asymmetric nuclear matter.

• The microscopic and macroscopic properties of isovector giant resonances have interesting

applications. For example, the properties of allowed and forbidden isovector transitions play

important roles in nuclear astrophysics and neutrino physics, as they determine stellar reaction

roles mediated by the weak interaction. In additions, macroscopic properties of the isovector

giant resonances reveal properties of the nuclear equation of states, with implications for

stellar phenomena such as neutron stars.

Experimentally, there are few data sets where spectra are obtained up to high excitation energies.

1

The motivation behind the present work is to investigate the 60Ni(3He,𝑡) reaction at 140 MeV/𝑢 up

to excitation energies of 60 MeV to study the:

• Gamow-Teller Resonance and Isobaric Analog State,

• Isovector (spin) monopole resonances,

• Isovector (spin) dipole resonances.

1.2 Thesis Organization

This work is divided into 6 chapters. Chapter 2 provides an overview of both microscopic and

macroscopic perspectives on giant resonances. It discusses the isospin structure of nuclei and its

significance in the isovector response. The chapter delves into the properties of isovector giant

resonances as they serve as tests of the microscopic model of nuclei at high excitation energies.

Additionally, it introduces the hydrodynamic model of giant resonances as a macroscopic model

describing a nucleus as a liquid drop of the proton and neutron fluids oscillating in different modes

around its equilibrium shape. The chapter further explores isovector giant resonances within the

charge-exchange spectrum. and concludes with an overview of the charge-exchange reaction probes.

Chapter 3 discusses a brief overview of the theoretical tools used to extract information about

charge-exchange (CE) excitations and the properties of the isovector giant resonances using the

distorted wave Born approximation in the code package FOLD. A brief overview of the Eikonal

approximation and the unit cross section for GT and IAS will be discussed in this chapter.

Chapter 4 provides an overview of the experimental setup and procedures used to investigate

isovector giant resonances in 60Ni up to excitation energy of 60 MeV by using the (3He,𝑡) reaction

at 140 MeV/𝑢. The experiment was performed with the Grand Raiden Spectrometer (GRS) at

the Research Center for Nuclear Physics (RCNP) in Osaka University. Furthermore, this chapter

describes the procedures used to extract the differential cross sections from the three rigidity settings

(low, medium, and high) of the GRS.

Chapter 5 presents the results from the multiple decomposition analysis (MDA), where the

spectra were decomposed into contributions from resonances with different angular momentum

2

transfers. It shows the location of the isovector giant resonances investigated via the 60Ni(3He, 𝑡)

reaction. This chapter details the findings for the Gamow-Teller strength, B(GT), and the study

of the B(GT) for the 𝑇0 + 1 states at 14.4 MeV and above in 60Cu, compared with known 𝑇0 + 1

states from 60Ni(𝑝, 𝑝′) and 60Ni(𝑒, 𝑒′) reactions.

It elucidates the techniques for subtracting a

quasifree-continuum from the measured data.

It underscores the significance of the continuum

subtraction. Finally, it delves into the extraction of isovector spin giant monopole and dipole

strengths. Comparisons with theoretical calculations are made

Chapter 6 marks the conclusion of the thesis with a general summary and future outlook.

3

CHAPTER 2

GIANT RESONANCES

2.1 Definitions and Classification of the Giant Resonances

2.1.1 Microscopic View of the Giant Resonances

In the microscopic picture, giant resonances are described as a coherent superposition of one-

particle one-hole (1p-1h) transitions excited from the ground state[2, 1], where the particle is a

proton (neutron), and the hole is neutron (proton). Charge-exchange reactions are associated with

the transfer of isospin (ΔT = 1). The projection of the isospin quantum number (ΔTz) can be

lowered (ΔTz = −1) or raised (ΔTz = +1 ) in the ( 𝑝, 𝑛) or (𝑛, 𝑝)- type charge-exchange reactions,

respectively. This is illustrated in Figure 2.1. The transitions to excited states from the ground state

for a certain giant resonance are characterized by the operator O

𝜇
JM [3]:

O

𝜇

JM = r𝜆 [−→𝜎 ⊗

−→
𝑌𝐿]J
M

𝜏𝜇

(2.1)

where J is the total angular momentum transfer (

−→
𝐽 =

−→
𝐿 +

−→
𝑆 ) and M is the projection of J, L is the

orbital angular momentum transfer, 𝜆 defines the radial operator and is defined by 𝜆 = Δ𝐿 + 2Δ𝑛,

where Δ𝐿 is the orbital angular momentum transfer, and 𝑛 is the change in the major oscillator for

transition. If the excitation goes across a major oscillator shell, 𝜆 will be increased by 2. This is,

for example, the case for the isovector giant monopole resonance (IVGMR) and its spin-transfer

−→
𝑌𝐿 is the spherical
partner, the isovector spin-transfer giant monopole resonance (IVSGMR).
harmonic associated with 𝐿, −→𝜎 is the spin-transfer operator, and 𝜏 is the isospin operator, with

𝜇 = ±1 for charge-exchange reactions, where 𝜇 is the isospin projection.

The total transition strength is constrained by the most coherent superposition of the particle-

hole excitations associated with a specific one-body operator as described through [1, 3, 4, 5, 6, 7].

The strength function described the response of a nucleus to an arbitrary one-body operator 𝑂

connecting the ground state of a nucleus to the excited states:

SO =

∑︁

f

(cid:12)⟨Φf |O|Φi⟩(cid:12)
(cid:12)
2𝛿(Ef − Ei) MeV
(cid:12)

(2.2)

4

Figure 2.1 Schematic representation of collective one-particle one-hole (1p-1h) excitations of the
isovector giant resonances for Δ𝑇𝑧 = −1 reactions (e.g., (p,n), (3He,𝑡)) (left) and Δ𝑇𝑧 = +1 reactions
(e.g., (n,p), (𝑡,3He) ) (right). The arrows correspond to a constructive contribution from each p-h
component. Figure taken from [3].

where (cid:12)

(cid:12)Φ𝑖⟩ and (cid:12)

(cid:12)Φ 𝑓 ⟩ stands for the ground (𝑖) and final ( 𝑓 ) states respectively. For continuum

states, the sum turns into an integral.

2.1.2 Macroscopic View of the Giant Resonances

In the macroscopic picture, giant resonances are defined as a collective motion of nucleons

involving many if not all the particles in the nucleus, causing density oscillations in the proton and

neutron nuclear fluids [8]. Such collective oscillations are categorized into two types: isoscalar

(Δ𝑇 = 0) if the proton and neutron fluids oscillate in-phase, and isovector (Δ𝑇 = 1), if they oscillate

out-of-phase.

In addition, the collective motion can be associated with spin.

If nucleons with

opposite spin oscillate in phase the resonances are referred to as electric (Δ𝑆 = 0). If nucleons

with opposite spin oscillate out-of-phase, the oscillations are referred to as magnetic (Δ𝑆 = 1).

They are depicted schematically in Figure 2.2. In this work, we will focus on the isovector giant

resonances. Table 2.1 lists the acronyms used to describe these resonances in this thesis, although

other variations exist (for example, the acronym SDR can represent the IVSGDR). The table also

includes the relevant quantum numbers for each of these giant resonances. The list of isovector giant

5

resonances in Table 2.1 can be extended to resonances of higher multipolarity, but identifying such

excitations is difficult since they are situated on top of the continuum background and their angular

distributions are not very distinctive. The IAS and GTR cannot be associated with a hydrodynamic

motion as there is no radial component to the operator (𝑟𝜆 = 1).

Figure 2.2 Schematic of the macroscopic depiction of giant resonance modes. The symbols p and
n represent proton and neutron fluids, and the small triangles indicate spin direction and refer to
either up or down. The arrows show the direction of motion of the fluid components. Single
asterisks indicate resonances for which a second contribution exists in which spin directions are
reversed. The double asterisk indicates that this resonance does not exist in first order. The figure
was modified from Ref [3].

6

Table 2.1 Overview of the isovector giant resonances and their quantum numbers. Δ𝑇 stands for
isospin-transfer, Δ𝐿 for angular momentum, Δ𝑆 for spin, Δ𝐽 for total angular momentum transfer
(Δ𝐽 = Δ𝐿 + Δ𝑆 ), and 𝜋 for the charge in parity. If the charge-exchange reaction is associated
with angular momentum and spin transfer, the excited giant resonance has several components with
different total spins. Δ𝑛 is the change in the major oscillator shell, and Δ𝜆 = Δ𝐿 + 2Δ𝑛.

Transition name
Fermi/Isobaric Analog State (Fermi/IAS)
Gamow-Teller Resonance (GTR)
Isovector Giant Monopole Resonance (IVGMR)
Isovector Spin Giant Monopole Resonance (IVSGMR)
Isovector Giant Dipole Resonance (IVGDR)
Isovector Spin Giant Dipole Resonance (IVSGDR)
Isovector Giant Quadrupole Resonance (IVGQR)
Isovector Spin Giant Quadrupole Resonance(IVSGQR)

Δ𝑇 Δ𝐿 Δ𝑆 Δ𝐽 𝜋
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1

0+
1+
0+
1+
1−
(0, 1, 2)−
2+
(1, 2, 3)+

0
0
0
0
1
1
2
2

Δ𝑛 Δ𝜆(ℏ𝑤)
0
0
0
0
2
1
2
1
1
0
1
0
2
0
2
0

2.2

Isospin Picture

The symmetry between protons and neutrons can be described by using the isospin operator t,

which is a vector quantity that behaves identically to the ordinary spin vectors. The protons are

defined as having isospin projection 𝑡𝑧 = −1/2, while neutrons have 𝑡𝑧 = +1/2. The total isospin T

of a nucleus is the vector sum of the isospins of its constituent nucleons (T =
projection 𝑇𝑧, calculated by 𝑇𝑧 = 𝑁−𝑍
2

can be positive, for 𝑁 > 𝑍, or negative, for 𝑁 < 𝑍. The

ti) [9]. The isospin

A
(cid:205)
t=i

value of 𝑇𝑧 determines the states that can be populated in the daughter nucleus. The concept of

isospin is key for describing charge-exchange reactions as isovector probes. There are two types of

charge-exchange reactions. The first one is 𝛽−-type or (p,n)-type, for which the change in isospin

projection Δ𝑇𝑧 = −1. In the Δ𝑇𝑧 = −1 direction, for nuclei with 𝑁 > 𝑍, states in residual nucleus

(with 𝑇𝑧 = 𝑇0 − 1, where 𝑇0 is the ground state isospin of another nucleus i.e. ground state of the

target nucleus) with isospin 𝑇0 − 1, 𝑇0, and 𝑇0 + 1 can be populated. The second one is 𝛽+-type

or (n,p)-type charge-exchange reaction. For nuclei with 𝑁 > 𝑍 states in the residual nucleus (with

𝑇𝑧 = 𝑇0 + 1) have isospin 𝑇0 + 1; this occurs because the minimum isospin in the residual nucleus is

equal to 𝑇𝑧. This is shown in Figure 2.3. Here we focus on (3He,𝑡) reactions, which are of 𝛽−-type

or (p,n)-type. It is possible to write the transition strength to the final states in terms of reduced

matrix elements for which the dependence on isospin quantum numbers is made explicit. These

isospin factors are

1

(2T0+1)(T0+1) for final states with isospin 𝑇0 + 1,

1
𝑇0+1 for final states with isospin

7

𝑇0 and 2𝑇0−1
nucleus is 2, and the isospin weights are 1

2𝑇0+1 for states with final isospin of 𝑇0 − 1. In the case of 60Ni, the isospin of the target
5 for final states with isospin 𝑇0 + 1, 𝑇0 and

3 and 3

15, 1

𝑇0 − 1, respectively.

Figure 2.3 Schematic depiction of the isospin scheme associated with charge-exchange reactions for
the excited states in both (p,n) and (n,p) directions. The dotted lines indicate the analog states and
the arrows show the possible transitions. For charge-exchange reactions with 𝑁 = 𝑍 ( 𝑇 = 𝑇𝑧 = 0 )
nuclei, only final states with 𝑇 = 1 can be populated in both Δ𝑇𝑧 = ±1 directions. Figure modified
from Ref [3].

As discussed for the IAS (section 2.3.1), because of isospin symmetry, analogs of states with the

same isospin can be found in isobaric nuclei. Besides the analog of the ground states, other states

can have analogs as well. This is illustrated in Figure 2.4. Δ𝑇𝑧 = −1 charge-exchange reactions on

a target nucleus with isospin 𝑇0 will populate states with 𝑇0 − 1, 𝑇0, and 𝑇0 + 1. States with isospin

𝑇0, and 𝑇0 + 1 have analogs in the target nucleus. These analogs can be populated through inelastic

scattering reactions. States with isospin 𝑇0 + 1 have analogs in the nucleus that can be reached

through Δ𝑇𝑧 = +1 charge-exchange reactions. Hence, in principle, it is possible to learn about the

8

𝑇0 + 1 states by performing Δ𝑇𝑧 = +1 and Δ𝑇𝑧 = −1 charge-exchange reactions. Unfortunately, it is

not easy to study the 𝑇0 + 1 states in Δ𝑇𝑧 = −1 charge-exchange reactions, in part because they are

disfavored due to the clebsch-Gordan isospin coefficients for 𝑁 > 𝑍 discussed above. In addition,

they are situated at excitation energies where there are many states that have lower isospin. Still,

the studies of analog states have been used to extract GT strengths for 𝛽+ direction from (p,n)

charge-exchange data, including for the case of 60Ni(3He,𝑡) reaction. In this work, it is attempted

to improve on that previous effort.

Figure 2.4 Schematic depiction of isospin symmetry in charge-exchange reactions. The dotted lines
indicate states of like isospin (analog states) in both Δ𝑇𝑧 = ±1 directions. In the Δ𝑇𝑧 = −1 direction,
the IAS is populated from 𝑇0 to 𝑇0 transition, but no such transition can occur in Δ𝑇𝑧 = +1 direction.

2.3

Isovector Giant Resonances

2.3.1

Isobaric Analogue State

The experimental measurements of nuclear spectra have shown that families of isobaric nuclei,

which have the same mass number (𝐴), but different neutron (𝑁) and proton (𝑍) numbers have

9

analog energy levels. The neutron and proton have similar masses and are up/down states of

a system with isospin 1/2 and have approximately identical behavior under the strong nuclear

force.

Ignoring the electromagnetic and weak interactions, nuclear spectra with identical spins

are therefore similar, which is known as isospin symmetry. The IAS excited in charge-exchange

reactions is the analog of the ground state of the mother nucleus. It has the same structure, but

with one neutron (proton) replaced by a proton (neutron). The states have the same isospin 𝑇0, but

differ by one unit in 𝑇𝑧, same quantum numbers, and the same microscopic properties.

Focusing on nuclei with 𝑁 > 𝑍, the IAS of the mother ground state can only be excited in

Δ𝑇𝑧 = −1 charge-exchange reactions: in the Δ𝑇𝑧 = +1 direction, no states with isospin identical to

that of the mother ground state are available, as shown in Figure 2.3. The excitation of the IAS

is characterized with the transfer of quantum numbers Δ𝐽 = Δ𝐿 = Δ𝑆 = 0 and Δ𝑇 = 1. In terms

of the shell-model, to excite the IAS, a target neutron is replaced with a proton that fills the same

single-particle orbit. The excitation strength of IAS exhausts the model-independent Fermi sum

rule [10] given by:

S(𝛽−) − S(𝛽+) = N − Z,

(2.3)

where 𝑆(𝛽−) and 𝑆(𝛽+) indicate the total strength of the 𝛽− and 𝛽+ decays for the Fermi transitions,

respectively. For 𝑁 − 𝑍 > 0, 𝑆(𝛽+) = 0. The simple structure of IAS can help in the understanding

of the neutron skins [11] and the nuclear equation of state [12].

For charge-exchange reactions at intermediate energies (∼ 100 MeV/𝑢 and above), it has been

shown [13] that the differential cross section at small momentum transfer is proportional to the

Fermi strength, B(F):

IAS
where ˆ𝜎𝜏 is the unit cross section. Since this excitation is due to the central isospin transfer

(cid:19)

(q = 0)

= ˆ𝜎𝜏B(F),

(2.4)

(cid:18) d𝜎
dΩ

(𝑉𝜏) component of the nucleon-nucleon interaction (see Chapter 3), the measurement of the IAS

provides direct information about this component of the interaction [13].

The IAS has an attraction property: it can be used to measure the charge-exchange cross section

at forward momentum transfers that are proportional to the 𝑁 − 𝑍 transition strength. It provides a

10

direct probe of the strength of the isospin component of the nucleon-nucleon.

2.3.2 Gamow-Teller Resonances

Over the past decades, Gamow-Teller (GT) transitions have been one of the main motivations

for performing charge-exchange experiments at intermediate beam energies. The GT transition

populated through charge-exchange reactions are associated with the 𝜎𝜏 operator and have Δ𝐿 = 0,

and Δ𝑆 = 1, and Δ𝑇 = 1. The GT transitions populates the same final states as in allowed

𝛽/electron-capture (𝛽/EC) decays. Due to the proportionality between cross section at 𝑞 = 0, and

B(GT), the Gamow-Teller transition strength can be extracted from charge-exchange reactions (see

section 5.3.1), even though CE reactions are mediated by the strong nuclear force, and 𝛽/EC decay

is mediated by the weak nuclear force. However, by using charge-exchange reactions, one can

populate states that are outside of the 𝑄-value window available for 𝛽/EC decay (more details seen

[1, 4, 14, 15, 16, 17, 18]).

The ability to extract the GT transition strength model-independently from charge-exchange

reactions allows for a stringent test of nuclear structure models. This ability has many applications

in nuclear astrophysics and neutrino physics. The well-known model-independent proportionality

relationship between GT transitions cross section and B(GT) from the charge-exchange reactions

[19, 13, 20] is given by:

ΔL=0
where ˆ𝜎GT is the Gamow-Teller unit cross sections, and on the left-hand side, the Gamow-Teller

(cid:18) d𝜎
dΩ

(cid:19)

(q = 0)

= ˆ𝜎GTB(GT)

(2.5)

cross section is extracted at zero momentum transfer (q = 0, where q = kf − ki), from theoretical

calculations by setting the 𝑄-value of the reaction used to 0 MeV and considering the cross section

at zero degrees. Experimentally, the measured cross section is the combination of all possible

charge-exchange transitions. Multiple decomposition analysis (MDA) is performed to extract the

Δ𝐿 = 0 component by fitting the experimental angular distribution with a linear combination of

various curves ( this will be discussed in more details, see Chapters 3 and 5).

The total amount of Gamow-Teller strength, including both the 𝛽+ and 𝛽− directions for a

11

particular nucleus, is constrained by the model-independent sum rule [10]:

S(𝛽−) − S(𝛽+) = 3(N − Z),

(2.6)

where N is the number of neutrons present in the target nucleus, and Z is the number of protons. In

nuclei in which N is appreciably larger than Z, Equation 2.6 reduces in a shell-model picture to :

S(𝛽−) ≈ 3(N − Z),

(2.7)

since 𝑆(𝛽+) ∼ 0 due to Paul blocking [1]. Experimentally, only 50-60% of the estimated sum-rule

strength can be accounted for in the excitation-energy range including the GTR. This is known as

the quenching of the sum rule [21, 22, 23, 24].

2.3.3

Isovector-Spin & Non-Spin-Transfer Giant Monopole Resonances

The IVGMR and IVSGMR resonances can be described as breathing modes in which the

proton and neutron fluids oscillate out-of-phase. They are of fundamental interest as collective

excitations at high excitation energies that are described microscopically by coherent 2ℏ𝜔, 1𝑝 − 1ℎ

excitations [25, 26, 27]. Their properties serve as tests for microscopic model calculation with

effective nucleon-nucleon interactions [28]. The excitation energy of the IVGMR and IVSGMR

can be used to understand the bulk properties of nuclei and nuclear matter [2, 1, 28, 29, 30, 31].

The strength distribution of the IVGMR provides a useful tool for better understanding the neutron

skin properties from which the density dependence of the symmetry energy for asymmetric nuclear

matter can be constrained [32, 33].

The isovector giant monopole resonance (IVGMR) is characterized by no change in the orbital

angular momentum (Δ𝐿 = 0), non-spin-transfer (Δ𝑆 = 0), and isospin-transfer of one unit (Δ𝑇 =

1). The properties of the IVGMR resonance provide insight into the fundamental understanding

of the isovector part of the residual nuclear interaction and isospin symmetry breaking and isospin

mixing in nuclei [31]. The IVGMR and IAS differ by the change in principal quantum number (n)

between the particles and holes. The IVGMR (2ℏ𝑤 excitation) is the overtone of the IAS (0ℏ𝑤). To

investigate the IVGMR experimentally, a probe for non-spin-transfer excitations is required, because

12

its spin-flip partner has a much larger cross section than the IVGMR at intermediate beam energies

[34, 35, 36, 37, 38, 39, 40, 41]. Therefore, the experimental evidence for the IVGMR is limited to

the results from the small number of charge-exchange experiments. Convincing evidence for the

IVGMR has been found via the spinless pion charge-exchange reaction (𝜋+, 𝜋0) [42, 43, 44, 45]. In

addition, evidence for the IVGMR has been reported in 60Ni(7Li, 7Be) charge-exchange reaction

[46]. Recently, the (10Be, 10B + 𝛾[1.74 MeV]) charge-exchange reaction at E(10Be)=100 AMeV

was reported as a new probe for isolating the isovector non-spin transfer excitations, and thus a

useful probe for studying the IVGMR, as evidenced by the successful extraction of the IVGMR and

IVGDR cross sections from 28Si(10Be, 10B + 𝛾) [29].

For probes that are not selective for spin-transfer, the spin-transfer partner of the IVGMR,

the IVSGMR is much more strongly excited than the IVGMR at intermediate beam energies

(100 < 𝐸𝑥 < 300 AMeV ) [34, 37]. This is because the spin-isospin (𝜎𝜏) component of the

nucleon-nucleon interaction is much stronger than the isospin (𝜏) component [3, 37, 41]. The

operator of the IVSGMR carries the same Δ𝐿, Δ𝑆, and Δ𝑇 as the operator of the GTR. However,

the IVSGMR is a 2ℏ𝜔 excitation, and the GT excitation is of 0ℏ𝜔 nature. So far, the progress in

the experimental studies of the IVSGMR is more advanced than for the IVGMR due to the much

higher cross section, which makes it easier to identify charge-exchange experiments with light ion

probes. [47, 48]. In general, isolating and observing the IVSGMR in Δ𝑇𝑧 = +1 charge-exchange

reactions is easier than in the Δ𝑇𝑧 = −1 charge-exchange reactions. This is because the IVSGMR in

the Δ𝑇𝑧 = +1 direction is located at lower excitation energies, where the continuum background is

lower [35, 49]. The exothermic heavy-ion charge-exchange reactions have been applied to enhance

the signature of the IVSGMR through 90Zr(12N, 12C) reaction at 175 MeV/𝑢 [50]. The exothermic

nature of this reaction reduces the momentum transfer for excitations at high excitation energy

which is beneficial for enhancing the cross section for the Δ𝐿 = 0 giant resonances.

2.3.4

Isovector-Spin & Non-spin-Transfer Giant Dipole Resonance

Significant effort to investigate the other isovector giant resonances through charge-exchange

reaction has been made, including the isovector spin giant dipole resonances (IVSGDR, Δ𝐿 =

13

1, Δ𝑆 = 1, Δ𝑇 = 1) and non-spin-transfer giant dipole resonances (IVGDR, Δ𝐿 = 1, Δ𝑆 =

0, Δ𝑇 = 1). The IVGDR was the first resonance that could be explained in a macroscopic picture

in which the neutron fluid oscillates against the proton fluid [51].

The IVSGDR consists of 3 components associated with the three possible couplings from its

spin transfer and angular momentum transfer: 𝐽 𝜋

𝑓 = 0−, 1−, 2−, assuming 𝐽 𝜋

𝑖 = 0+, as shown in Table

2.1. In the experimental data analysis, it is not easy to isolate the three spin-parity components

of the IVSGDR. All three components are associated with angular momentum transfer Δ𝐿 = 1.

Their angular distributions are therefore similar, and they peak approximately at the same scattering

angle. It is beneficial to separate the three components to perform a detailed test of the theoretical

calculations, but it requires the measurement of polarization observables, which has only been done

a few times (see e.g., [52]). The IVSGDR is more strongly excited in charge-exchange reactions

at intermediate beam energies compared to the IVGDR. This is because the spin-isospin (𝜎𝜏)

component of the nucleon-nucleon interaction is much stronger than the isospin (𝜏) component

[3, 37, 41]. However, there is no well-established proportionality between transition strength and

differential cross section for dipole resonances. There is also no model-independent sum rule for

the IVSGDR that can be used to characterize the amount of strength found in spectra [35, 36].

2.4 Hydrodynamic Model of Giant Resonances

The hydrodynamic or macroscopic model that has been applied to isovector giant resonances

describes a nucleus as a liquid drop of the proton and neutron fluids oscillating in different modes

around its equilibrium shape. The spins of the nucleons serve as an additional degree of freedom.

When describing a nucleus as an oscillation of a liquid drop, two main modes can be distinguished:

a surface-vibrational mode and a compressional mode.

It is worth noting that a vibrating drop

can have a superposition of both modes [3]. The amplitude of the oscillations is small. For

the surface-vibrational mode, harmonic oscillations about a mean spherical shape are assumed

(see more theoretical discussions of the surface-vibrational and compressional modes for giant

resonances in the References [3, 53, 6, 54, 55]).

The macroscopic description of giant resonances provides a picture that connects to the bulk

14

properties of the nucleus and nuclear matter. Therefore, the study of giant resonances can help

constrain models of these bulk properties. An example is the simple vibrational giant resonance

mode known as the ISGMR observed for the first time in 1977 by Harakeh et al.

[56, 57]

and confirmed in the same year by Youngblood et al.[58, 59]. They used inelastic 𝛼-scattering

measurements at forward scattering angles. This properties of the ISGMR yields insights into the

bulk properties of the nucleus, in particular the incompressibility of nuclear matter (Knm) [60, 55].

The location of the excitation energy of the ISGMR was estimated for the first time by Bohr and

Mottelson [6]. The energy of the isoscalar giant monopole resonance was determined as a function

of the mass of nucleus (A) [61] to be:

E1,0 =

ℏk1,0
1.2

√︂

Knm
9m

A−1/3 MeV

(2.8)

where k1,0 stands for eigenvalues of the motion for principal quantum number 𝑛 = 1 and multi-

polarity l = 0, and m is the nucleon mass. Knm was not known at the time of derivation, and was

estimated to be Knm = 135 MeV [6]. The solution for the ISGMR (E1,0) is:

EISGMR
X

= 65A−1/3 MeV

(2.9)

When the experimental data became available, the value of knm was determined to be 231 ± 5 MeV

[60, 55].

For the IVGMR, assuming that a nucleus consists of two inter-penetrating, incompressible fluids

[51, 3, 53, 6, 54, 55] of protons and neutrons, the restoring force is proportional to the surface of

the nucleus (R2). The frequency of the resulting harmonic oscillations is proportional to the square

root of a constant force over mass parameter (A). Therefore, this assumption leads to a behavior

that is linear with R−1/2 or linear with A−1/6, since R = roA1/3 [3].

In general, the vibrational frequency and excitation energy of a particular giant resonances can

be determined come from linearized Navier-Stokes equations [62]. For example, solving for the

IVGMR, the excitation energy is:

EIVGMR

x

= 170A−1/3 MeV

(2.10)

15

However, this model with volume terms only does not accurately describe the excitation energies

of the isovector giant resonances. A study by Bowman et al.

[30] indicated that besides the

volume terms, surface tension effects must be taken into account as well to describe the properties

of isovector giant resonances. Therefore, a more precise estimate of the excitation energy of the

IVGMR was found to be:

EIVGMR

x

= 88A−1/6

(cid:18)

1 +

14
3

A−1/3

(cid:19) −1/2

MeV

(2.11)

and the systematic study of experimental results for nuclei gives the excitation energy of the IVGDR

as a function of mass umber (A) [3, 63] to be:

EIVGDR

x

= 31A−1/3 + 20A−1/6 MeV

(2.12)

2.5

Isovector Giant Resonances in the Charge-Exchange Spectrum

In the microscopic picture, the excitation energies of giant resonances can be estimated from the

energy differences between the shells in which the particles and holes that participate in the coherent

one-particle one-hole (1p-1h) excitations are located. The excitation energy for giant resonances

is related to the difference in the major shell (Δ𝑁) of the particles and the holes. The energy

difference between each major shell is estimated to be ℏ𝜔 = 41A−1/3. This simple approximation

provides a basic understanding of the relative excitation energies of the isovector giant resonances.

This is illustrated in Figure 2.5(a), which displays the excitation-energy spectrum in 60Cu obtained

from the 60Ni(3He,t) reaction at 140 MeV/𝑢 for 3 ranges in scattering angle, as discussed in more

detail in Chapter 4 and 5. Different giant resonances appear at different excitation energies. The

angular distribution provides a characteristic of the angular momentum transfer associated with a

particular excitation.

16

Figure 2.5 The typical Δ𝑇𝑧 = −1 charge-exchange energy spectrum from the full range of excitation
energy (60 MeV) used for the three angular settings from the 60Ni(3He,𝑡) reaction. Various isovector
giant resonances and their corresponding spin and parity (𝐽 𝜋) are indicated. Figure (a) include
the Isobaric Analog State (IAS), Gamow-Teller Resonance (GTR), the isovector spin, and non-spin
giant dipole, monopole, and quadrupole resonances (IVSGDR/IVGDR, IVSGMR/IVGMR, and
IVSGQR/IVGQR). Their features can be identified by comparing spectra at different scattering
angles. Figure (b), the visualization of the IAS state at the lower excitation energy range between
0 to 5 MeV.

17

For example, the GTR, IAS, and monopole excitations are associated with Δ𝐿 = 0, so they peak

at 0◦ scattering angle. For the (3He,𝑡) reaction at 140 MeV/𝑢, their presence is enhanced in the

spectrum gated on scattering angles between 0 and 5 mrad. Dipole excitations peak at small but

finite angles and their features are enhanced at scattering angles between 35 and 45 mrad. The IAS

is known to reside at Ex(60Cu) = 2.55 MeV, and the main component of GTR appears at excitation

energies of ∼ 10 MeV. The IVSGDR peaks at an excitation energy of about 18 MeV. The difference

in the excitation energy between ΔN = 1ℏ𝜔 dipole resonances and the ΔN = 0ℏ𝜔 IAS is about 16

MeV, higher than estimated value of 1ℏ𝜔 = 41A−1/3 = 10.5 MeV. The ΔN = 2ℏw IVSGMR and

IVGMR are located at excitation energies of about 35 MeV and are not easy to identify due to

their large widths. The IVSGQR and IVGQR resonances are located at similar excitation energies,

but their angular distributions are relatively flat at forwarding scattering angles. Therefore, their

contributions are difficult to observe when comparing spectra at different scattering angles.

At lower excitation energies, up to 5 MeV (Figure 2.5(b)), the most dominant peak is the IAS,

and it is possible to identify individual states in 60Cu. However, the level density increases as a

function of the excitation energy and even at 4-5 MeV it becomes difficult to identify individual

states. These low-lying states are non-collective in nature.

2.6 Overview of the Charge-Exchange Reaction Probes

Over the past decades, charge-exchange reactions have been used to investigate various phenom-

ena in nuclear physics with applications in nuclear structure and astrophysics. Charge-exchange

reactions have been developed since the last half-century [64, 65], and they are used to probe the

spin-isospin response of nuclei.

As discussed above, in terms of the isospin formalism, charge-exchange reactions are charac-

terized by an isospin-transfer by one unit, Δ𝑇 = 1 (isovector), meaning that occur with Δ𝑇𝑧 = ±1

as shown in Figure 2.3. Δ𝑇𝑧 = −1 corresponds to the (p,n)-type, causing the residual nucleus

to become more proton-rich nucleus, while Δ𝑇𝑧 = +1 corresponds to the (n,p)-type, from which

the residual nucleus become more neutron-rich. Both probes, involve spin 1

2 particles and can

mediate Δ𝑆 = 0 and Δ𝑆 = 1 excitations.

In charge-exchange reactions, any amount of angular

18

momentum can be transferred, Δ𝐿 = 0 (monopole), Δ𝐿 = 1 (dipole), Δ𝐿 = 2 (quadrupole), etc.

At intermediate beam energies, the excitations with spin-transfer are strongly favored compared to

non-spin transfer.

In 𝛽-decay experiments, states may be measured in an excitation energy region from 0 MeV up

to the Q-value of the reaction, but higher-lying states will not be accessible via this reaction. In

contrast, charge-exchange reactions are not limited by the Q-value, and highly excited states can

be studied using different charge-exchange probes, such as ( 𝑝, 𝑛)/(3He,𝑡), (𝑛, 𝑝)/(𝑡,3He), (𝑑, 2He),

and other reactions (see Table 2.2). A variety of probes are used with different advantages for

the study of isovector excitations. The choice depends on the experimental considerations and the

sensitivity to different resonances (see more experimental discussions about the IVGRs probes used

in References [66, 67, 4, 1, 14]).

Table 2.2 Various charge-exchange reaction probes are classified by: 𝛽−-type or (p,n)-like reactions
(left side) and 𝛽+-type or (n,p)-like reactions (right side).

𝛽−-type
(p,n)
(3He,𝑡)
(6Li,6He)
(12C,12Be)
(10C,10B)
(10Be,10B)
𝜋+ , 𝜋0

𝛽+-type
(n,p)
(𝑡,3He)
(d,2He)
(12C,12N)
(13C,13N)
(7Li,7Be)
𝜋− , 𝜋0

The analysis of data obtained from the (n,p) or (p,n)-like reactions benefits from the reduced

complexity of the reaction mechanism compared to using composite probes such as (𝑡,3He) or

(3He,𝑡) reactions [68]. However, to achieve a better energy resolution, it is preferred that both the

projectile and ejectile are charged so that their momenta can be well constrained and analyzed. For

example, with the (3He,𝑡) reaction an excellent resolution of as low as 30 keV or less can be achieved

[69, 70, 71, 72]. Achieving a better resolution is useful for studying the reaction mechanism for

which it is preferable to isolate individual transitions, or for the study of the fine structure of giant

resonances [73].

The main difference between the (n,p) and (p,n) reactions and reactions induced by composite

19

probes such as (𝑡,3He) and (3He,𝑡) reactions is that the former probe the target nucleus relatively

deeply, while the latter predominantly probe the surface of the target nucleus. The strong absorption

near the surface of the target nucleus is a general characteristic of composite probes used for

charge-exchange reactions at intermediate beam energies. Therefore, it has been used to isolate the

excitations associated with radial nodes near the nuclear surface such as the IVSGMR and IVGMR

[74]. This specific property was used to compare the (𝑡,3He) and (n,p) reactions by measuring the

double-differential cross sections at 300 MeV/𝑢 on 208Pb and 92Zr targets at the rare isotope beam

factory of RIKEN [40].

Experiments using the (d,2He) reaction as a probe are relatively complex compared to (𝑡,3He)

and (n,p) reactions, but such experiments have been successful in tracking the isovector spin-transfer

strength for many nuclei [75, 76, 77, 78, 79, 18]. The complexity of the (d,2He) reaction is due

to the simultaneous measurement of the two emitted protons from the unbound 2He particle. The

energy and angles of both protons need to be measured accurately in order to achieve good energy

and scattering angle resolutions in the reconstructed spectra and to allow for making a precise cut on

the relative energy (𝜖 𝑝 𝑝) between the two emitted protons. [68]. However, there is a disadvantage

to using the (d,2He) reaction to study giant resonances in medium-heavy and heavy nuclei because

of the background from deuteron-breakup reactions occurring in the Coulomb fields of the target

nuclei. Consequently, the studies of the isovector giant resonances through the (d,2He) reaction

have been limited to light and medium-heavy nuclei [68].

The (6Li,6He) reaction probe has also been used as a selective filter for spin-transfer excitations

because of the transitions from the 𝐽 𝜋 = 0+ ground state in 6Li to the 𝐽 𝜋 = 1+ ground state in 6He.

Since 6He has no particle-stable excited states, effects from ejectile excitation can be avoided [80].

Experiments with the (6Li,6He) reaction were performed at ∼ 100 MeV/𝑢 on 12C, 13C, 58Ni, and

90Zr target nuclei [81]. While it was not possible to measure the scattering angle, this reaction was

successful in terms of isolating the spin-transfer of giant resonances [80].

The (7Li,7Be) and (7Li,7Be+𝛾) reactions have been used at beam energy of 𝐸/𝐴 = 50 MeV on

6Li, 12C, 90Zr, 120Sn, 208Pb and polystyrene targets [82]. The gamma rays detected from excited

20

states in 7Be via (7Li,7Be+𝛾) were successfully used to isolate excitations associated with and

without spin-transfers. With this probe, either the 3/2− ground state or the 1/2− excited state at 429

keV in 7Be can be populated making it possible to separate excitations associated with Δ𝑆 = 0 and

Δ𝑆 = 1. This technique has been applied to investigate the isovector giant resonances in forwarding

kinematics at beam energies of 50 − 70 AMeV [82, 83, 84, 46].

The (12C,12B) and (12C,12N) reactions also have selective spin-transfer properties and they can

be used to investigate Δ𝑇𝑧 = −1 and Δ𝑇𝑧 = +1 transitions, respectively. The (12C,12B) reaction has

been used to measure GT transitions on 12C, 26Mg, 54Fe, 58Ni, and 90Zr targets, and the (12C,12N)

reaction have been used on 56Fe targets at a beam energy of 𝐸/𝐴 = 70 MeV [85], 135 AMeV [86],

and others [50, 87, 88]. However, with large negative reaction Q-values, the significant momentum

transfers associated with these reactions make them less useful to study the isovector transitions,

specifically the monopole excitations.

The charge-exchange probe (13C,13N) has been studied to locate isovector (Δ𝑇 = 1) non-spin-

transfer (Δ𝑆 = 0) giant resonances via 60Ni(13C,13N)60Co reaction at 𝐸/𝐴 = 100 MeV [89].

Through a distorted wave Born approximation analysis, the isovector dipole resonance was found at

𝐸𝑥 = 8.7 MeV and quadrupole resonance was found at 𝐸𝑥 = 20 MeV. The proton separation energy

in 13N is less than the excitation energy of the first excited state, the only transition contributed was

the one from the 13C( 1
2

−

) ground state to the 13N( 1
2

−

) [89, 90]. This transition is dominated by the

large Fermi matrix element, which is why this reaction provides some selectivity for ΔS = 0.

The pion (𝜋±, 𝜋0) charge-exchange reactions are another probe used to investigate the isovector

excitations. Since pions are spinless particles, they are used as an ideal tool for studying spinless-

transfer excitations [91]. The (𝜋+, 𝜋0) and (𝜋−, 𝜋0) reactions were used to investigate the properties

(energies, widths, and cross sections) of the isovector monopole, and the other isovector giant

resonances in nuclei between 14C and 208Pb [43]. The production of pion beams is complicated

and it is difficult to achieve a high resolution in the 𝜋0 exit channel through the analysis of the

𝜋0 → 𝛾𝛾 decay [3]. However, at Los Alamos Meson Physics Facility (LAMPF) successful the pion

charge-exchange reactions have been performed with aim of extracting information on the IVGMR

21

[43, 44, 42].

The availability of rare-isotope beams has resulted in the use of new unstable charge-exchange

reaction probes in forward kinematics. Rare-isotope beams provide new possibilities to selec-

tively excite specific giant resonances, for example, the (10C,10B(0+,IAS)) reaction [92, 93] and the

(10Be,10B(0+,IAS)) reaction [29] have been used to selectively excite the non-spin-transfer excita-

tions. In addition, by using rare isotope beams it is possible to create exothermic reactions. The

released energy can offset the energy needed for the excitation of the target nucleus. Therefore, this

reduces the linear momentum transferred between the projectile and the target nucleus and results

in creating more favorable conditions to study monopole excitations. For example, the (12N,12C)

charge-exchange reaction at 175 MeV/𝑢 was successfully developed and applied to 90Zr as a novel

probe for studying the excitation of the IVSGMR [50]. This probe comes with an additional

advantage of spin-transfer transitions that are selectively excited.

As charge-exchange reactions probe the spin-isospin response of nuclei, they are very attractive

for studying the excitations from nuclei with asymmetric neutron-to-proton ratios. By impinging the

rare-isotope beam on a hydrogen target, the (p,n) reaction in inverse kinematics can be used to study

unstable nuclei. In the first experiments, the neutron from the (p,n) reaction was not detected. This

method relies on measuring the residual nuclei from the projectile only and avoided the detection

of the recoil neutron from the proton target in the process [94, 95, 96, 97, 98, 99, 100, 101].

More recently, inverse-kinematics (p,n) experiments using the missing-mass method in which

the neutrons are detected have been developed at NSCL [102] and at RIBF group [92]. The

NSCL group has used the setup called Low Energy Neutron Detection Array (LENDA) designed

to facilitate the study of (p,n) charge-exchange reactions in inverse kinematics at intermediate

energies using unstable beams. The LENDA detector was successfully developed and used to study

the 56Ni(p,n) and 55Co(p,n) reactions at 110 MeV/𝑢 in inverse kinematics in order to extract GT

strengths for transitions to 56Cu and 55Ni, respectively [103], which are of astrophysical importance.

Other experiments where this technique has been utilized are 132Sn(p,n) and 12Be(p,n) reactions at

RIBF [104] and 16C(p,n) at NSCL [105].

22

The development of a new technique to perform charge-exchange experiments in inverse kine-

matics using the (n,p) reaction has been more complicated. This is because a neutron target is not

available. Therefore, one of the other probes in Δ𝑇𝑧 = +1 direction must be used [68]. The first

successful attempt to study (n,p) reactions in iverse kinematics used the 34P(7Li,7Be + 𝛾(429 keV))

charge-exchange reaction in inverse kinematics at 100A MeV to measure GT transition strengths in

the 𝛽+ direction from 34P, populating states in 34Si [106]. Meharchand et al. (2010) used (7Li,7Be)

reaction in inverse kinematics to study the spectroscopy of 12Be [107]. This technique relies on

tagging the charge-exchange reactions with 𝛾-rays emitted from the decay of the 1/2− excited state

at 429 keV from 7Be and use the measurement of the projectile-like residual in a spectrometer to

learn about reaction kinematics and be able to extract scattering angles and the excitation energies.

As a consequence, its use is limited to relatively light nuclei and to excitation energies below the

particle-decay threshold.

The most recent (n,p)-type charge-exchange reaction developed by Giraud et al. (2021) [108]

uses the (d,2He) reaction in inverse kinematics to study exotic nuclei. This was done by using

an active-target time-projection chamber (AT-TPC). The inner volume of the AT-TPC was filled

with deuterium gas, which serves as the target and the detector medium for the tracking of the two

protons emitted from the unbound 2He nucleus. The charge-exchange residual nuclei were detected

in the 𝑆800 spectrometer, and served as a trigger for the time-projection chamber data acquisition

system [109, 110].

23

CHAPTER 3

CHARGE-EXCHANGE REACTION TECHNIQUES

Charge-exchange (CE) reactions involve the exchange of a proton and neutron between a target

nucleus and an incoming beam nucleus.

In these reactions, the isospin changes by Δ𝑇 = 1

(isovector), with the potential for Δ𝑆 = 1 (spin transfer) or Δ𝑆 = 0 (non spin transfer) and the ability

to transfer various units of angular momentum: Δ𝐿 = 0 (monopole), Δ𝐿 = 1 (dipole), Δ𝐿 = 2

(quadrupole), Δ𝐿 = 3 (octupole), and so forth. This chapter gives a brief overview of CE reactions

and associated techniques.

The theoretical cross sections calculated in this thesis were performed using the distorted

wave Born approximation (DWBA) method discussed in section 3.3 and 3.4 with the FOLD code

[111]. The DWBA calculations were used to perform a multiple decomposition analysis for

extracting the isovector response associated with different units of angular momentum transfer. To

convert extracted cross sections into Gamow-Teller strength, the Eikonal approximation is applied

as discussed in section 3.5.

3.1

Introduction into Charge-Exchange Reaction Techniques

To extract information about charge-exchange excitations and giant resonances from measured

excitation-energy spectra, such as the one shown in Figure 2.5, several steps are required. The

experimental differential cross sections are determined from the data with:

(cid:19) exp

(cid:18) d𝜎
dΩ

=

Y
NbNt𝜖1𝜖2dΩ

mb/sr,

(3.1)

where Y represents the total number of counts in a specific angular bin, Nb is the total number of

nuclei that struck the target foil, Nt is the number of nuclei in the target, dΩ is the opening angle,

𝜖1 is the correction for the lifetime of the data acquisition system (DAQ) and 𝜖2 is the correction for

the target purity.

To extract the contribution from transitions associated with the transfer of different units of

angular momentum transfers (ΔL) a multiple decomposition analysis (MDA) [112, 113] is per-

formed. The measured differential cross section for each peak or energy bin is fitted with a linear

24

combination of theoretical angular distributions associated with these different angular momentum

transfers:

(cid:19)

(cid:18) d𝜎
dΩ

exp

= a0

(cid:18) d𝜎
dΩ

(cid:19) DWBA

ΔL=0

+ a1

(cid:18) d𝜎
dΩ

(cid:19) DWBA

ΔL=1

+ a2

(cid:18) d𝜎
dΩ

(cid:19) DWBA

ΔL=2

+ ...,

(3.2)

where

(cid:16) d𝜎
dΩ

(cid:17) DWBA

ΔL=i

are the theoretical differential cross sections for ΔL = i and ai are their corre-

sponding fit parameters.

In this work, the theoretical differential cross sections are calculated in Distorted Wave Born

Approximation (DWBA), which will be discussed in more details in the following sections, focusing

on the case of the 60Ni(3He,𝑡) reaction at 140 MeV/𝑢.

3.2 Calculation of Differential Cross Sections

Theoretical calculations for the differential reaction cross section in the (3He,𝑡) experiment were

performed using the FOLD code package [111, 114]. This code uses the Distorted Wave Born

Approximation (DWBA), discussed in section 3.3, where both incoming and outgoing waves are

distorted by the mean field of the target nucleus. Details regarding the inputs and outputs for the

DWBA calculations are provided in section 3.4.

The FOLD code was used to calculate the differential cross sections for charge-exchange

(CE) reactions involving composite particles composed of three component codes, including the

WSAW, FOLD, and DWHI. The WSAW code was used for computing radial wave functions for

relevant shell-model orbitals, as discussed in section 3.4. The second component, FOLD, was

used to calculate the form factor based on a double folding of the one-body transition densities,

providing structural information for the target and projectile system based on the nucleon-nucleon

interaction. The third component, DWHI, was employed to calculate the differential cross section

while considering the distortion of incoming and outgoing particles due to the mean field of the

target nucleus, incorporating optical potentials. Additionally, more details about the DWHI code

were discussed in section 3.4.3

For the 60Ni(3He,𝑡) reaction, one-body transition densities (OBTDs) were obtained within the

normal-modes (NM) formalism using the NORMOD code [115], as detailed in section 3.4.2. The

25

folding procedure was executed using the Love-Franey effective nucleon-nucleon (NN) interaction

at 140 MeV/𝑢 [41, 37].

Figure 3.1 shows a schematic description of the inputs and outputs for the DWBA calculations

in FOLD code to calculate differential cross sections for CE reactions with composite particles that

consist of three parts: WSAW, FOLD, and DWHI.

Figure 3.1 Hierarchy image describing the steps in the distorted wave Born approximation (DWBA)
used to compute theoretical differential cross sections through FOLD code.

26

3.3 Brief Description of the DWBA

Above beam energies of about ∼ 100 MeV/𝑢, the charge-exchange reaction predominantly

proceeds in a single step. Therefore, to good approximation, two-step (or more-step) processes do

not have to be taken into account. Here, we assume that the nucleons of the projectile and target

system are non-identical. Participating nucleons of the target and the projectile systems are treated

equally and the initial and final states have definite isospins. For the reaction:

projectile(p) + target(t) = ejectile(e) + residual(R),

the Hamiltonian in the prior form can be written as:

H = Hp + Ht + Tpt + Vpt,

(3.3)

(3.4)

where Hp and Ht are the internal Hamiltonians for the projectile and target, Tpt is the kinetic

energy of the relative motion of projectile and target, and Vpt is the interaction potential. The

scattering potential Vpt is divided into two parts: one with known solution, which contains the

elastic scattering potential (Uelastic) including the Coulomb potential, and one that contains any

residual potential, which mediate the charge-exchange (Wcharge−exchange):

Vpt = Uelastic + Wcharge−exchange.

The transition between the initial and final states is described by a T-matrix [3, 116]:

Tfi = ⟨Φ|Uelastic| 𝜒+⟩ + ⟨𝜒−|Wcharge−exchange|Ψ+⟩,

(3.5)

(3.6)

where 𝜒+ and 𝜒− represent incoming and outgoing waves distorted by the mean field of the target

(Uelastic). Ψ+ is the solution of the Schr (cid:165)𝑜dinger equation in prior form, with an incoming plane

wave and outgoing spherical wave. Φ is the incoming plane wave. The first term in equation 3.6,

which is of isoscalar type, drops out because it does not connect the initial and final states of a

charge-exchange reaction. Potential U is still used to determine the distortion of the incoming and

outgoing waves. The remaining term describes the charge-exchange reaction.

27

Tfi is used to calculate the theoretical differential cross sections of the nuclear reaction.

d𝜎
dΩ

=

(cid:16) 𝜇
2𝜋ℏ2

(cid:17) 2 kf
ki

|Tfi|2.

(3.7)

where ki and kf are momenta of incoming and outgoing channels, respectively, 𝜇 is the reduced

energy. For the charge-exchange reactions, Tfi is usually calculated by using:

Tfi = ⟨𝜒+

𝑓 ( (cid:174)𝑘 𝑓 , (cid:174)𝑅′)|𝐹 ( (cid:174)𝑅′)| 𝜒−

𝑖 ( (cid:174)𝑘𝑖, (cid:174)𝑅′).

(3.8)

Equation 3.8 consists of the incoming and outgoing distorted waves 𝜒+

𝑓 and 𝜒−
𝑖

, respectively, and

a form factor, 𝐹 ( (cid:174)𝑅′), that describes the interaction between the projectile and the nucleons in the

target.

The form factor is a double folding of the nucleon-nucleon (𝑁 𝑁) interaction over the projectile

and target-system transition densities. The transition densities for projectile and target systems

contain the overlap between the initial and final states of these separate systems and directly relate

to the transition strengths, such as the Gamow-Teller transition strengths. The transition densities

and strengths are calculated in a nuclear structure model.

The DWBA calculations for the charge-exchange reactions used in this work are done by using

the package known as FOLD [111], which consists of three modules: WSAW, FOLD, and DWHI.

The module FOLD calculates the form factor 𝐹 ( (cid:174)𝑅′). Besides the nuclear-structure information

obtained from a structure model, it requires a single-particle radial wave function of the orbitals

of the nucleus involved in the calculation. These single-particle wave functions are calculated in

WSAW. In addition, FOLD requires a 𝑁 𝑁-potential. In this work, we use the 𝑁 𝑁-interaction of

Love and Franey [41, 37]. The form factor calculated in FOLD serves as input for DWHI, which is

used for the distorted-wave calculation resulting in the transition matrix 𝑇 𝑓 𝑖 and the cross sections.

3.4

Inputs & Outputs for the DWBA Calculations

3.4.1 Single-Particle Wave Functions (WSAW)

As mentioned in the above section 3.3, one of the important ingredients needed to calculate

form factor 𝐹 ( (cid:174)𝑅′) include single-particle wave functions calculated through the first module from

28

the FOLD package known as WSAW [111]. WSAW is used to calculate the radial part of the

single-particle wave functions for the one-particle and one-hole states connected by the relevant

transition operator. A Woods-Saxon potential is assumed, and for protons, the Coulomb potential

is also included.

In addition, a spin-orbit potential is included for both protons and neutrons.

The necessary nuclear physics inputs to the WSAW calculations are the nuclear charge of the

target nucleus, core mass, and the binding energies (BEs) of every single proton and neutron in

orbitals that participate in the excitation. These BEs were obtained through the shell-model code

NUSHELLX@MSU [115, 117] with the DENS function, employing the SK20 [118] interaction in

60Ni and 60Cu.

In WSAW, the depth of the Woods-Saxon potentials are varied to match binding energies for

each of the single-particle orbitals separately. The Woods-Saxon potential has the general form:

f(R; rn, an) =

V0
1 + exp(cid:0)R − rnA1/3(cid:1) /an

,

(3.9)

where V0 is the maximum depth of the volume potential, rn is the radius of the potential, A is the

mass number of the target nucleus, and an is diffuseness of the potential [119]. In this work, the

starting value of the Woods-Saxon potential is set to 60 MeV. The radius parameter rn is fixed to

1.25 fm, and the diffuseness is fixed to 0.6 fm. The Coulomb potential has a radius parameter of

1.25 fm. The spin-orbit potential has a radial form that goes with 1

r times the derivative of the

Woods-Saxon potential. The spin-orbit potential strength is set to 7 MeV. Note that, the program

assumes a spin-operator of the form (cid:174)𝐿 · (cid:174)𝜎 and that a factor of

(cid:17) 2

(cid:16) ℏ
m 𝜋 c

≈ 2 is already included.

Table 3.1 describes input parameters used in the WSAW program for target (60Ni) and residual

(60Cu) system . The full input files are included in the Appendix A, Table A.1. The outputs are

the radial wave functions for each of the orbitals involved in the calculation. The plots of these

relevant single-particle wave functions are shown in Figure 3.2. Note that all wave functions are
positive near r = 0 (by design) and are normalized such that ∫ ∞

𝜓(𝑟)𝑟 2 𝑑𝑟 = 1. The densities for

0

3He-projectile and 3H-ejectile system are calculated from Variational Monte-Carlo results [120].

29

Figure 3.2 The Woods-Saxon radial single-particle wave functions calculated from the WSAW
program for n=0 (top), n=1 (middle) and n=2 (bottom). The left panels show single-particle wave
functions for protons and on right panels, for neutrons.

30

Table 3.1 Description of the input parameters in the WSAW file for calculating single-particle wave
functions as shown in the Appendix A, Table A.1- A.3.

Particle

Input parameters Meaning of the input values

line number 1

0.1 fm
20 fm
1
150

Step size for radius
Maximum value of radius
Step size (first point)
Total number of points needed to plot radius

line number 2

Ni60Cu60

File name containing radial wave functions

line number 3

For each orbital a line with

59
60
.65 fm
1.25 fm
1.25 fm
7.0 MeV

x MeV
1.
l
n
1.
j
s

Core mass of target nucleus
Starting value of the volume potential depth
Diffuseness of the potential
Radius parameter of the potential
Coulomb radius parameter
Spin-orbit potential strength

Binding energy of paricle (proton or neutron)
Mass number of particle
Orbital angular momentum of proton
Number of interior nodes for this particular particle
Charge of particle (either 0 or 1)
Total angular momentum of the particle orbit
Spin of the particle

3.4.2 Calculation of the Formfactor (FOLD)

The FOLD program [111] is used to calculate the form factor 𝐹 ( (cid:174)𝑅′):

𝐹 ( (cid:174)𝑅′) = ⟨ΦeΦr|Veff |ΦtΦp⟩

(3.10)

where Φe, Φr, Φt, and Φp are the wave functions of the ejectile, residual, target, and projectile,

respectively, and Veff is the effective nucleon-nucleon (𝑁 𝑁) interaction between nucleons in the

target and projectile nuclei. The ingredients for the calculation of the formfactor are the single

particle wave functions from WSAW, the effective 𝑁 𝑁 interaction, and one-body transition densities

(OBTDs) calculated in the NORMOD code [121], which is discussed later in this section. The

formfactors are calculated from these ingredients by a double-folding of the effective 𝑁 𝑁 interaction

over the transition densities of the projectile-ejectile and target-residual systems. The double-folding

over the transition densities of the participant nuclei in the reaction is necessary to account for the

composite nature of the nuclei involved.

31

The phenomenological nucleon-nucleon interaction used to model charge-exchange reactions

were described by Love and Franey in 1981 through a phaseshift analysis of 𝑁 𝑁 scattering data [37].

Later on, with an updated data set, this interaction was improved by Franey and Love in 1985 [41].

The Love and Franey interaction is particularly useful in the charge-exchange studies since Veff is

parameterized in terms of the central (VC), spin-orbit (VLS), and tensor (VT) terms contributing to

the interactions that are of interest for charge-exchange reactions. It takes the following form:

𝑉𝑖 𝑗 = 𝑉𝐶 (𝑟𝑖 𝑗 ) + 𝑉𝐿𝑆 (𝑟𝑖 𝑗 ) (cid:174)𝐿 · (cid:174)𝑆 + 𝑉𝑇 (𝑟𝑖 𝑗 )𝑆𝑖 𝑗 ,

where 𝑆𝑖 𝑗 is the tensor operator, defined as

𝑆𝑖 𝑗 = 3

( (cid:174)𝜎𝑖 · (cid:174)𝑟𝑖 𝑗 ) ( (cid:174)𝜎𝑗 · (cid:174)𝑟𝑖 𝑗 )
𝑟 2
𝑖 𝑗

− (cid:174)𝜎𝑖 · (cid:174)𝜎𝑗

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

𝑉 (𝑟) =

∑︁

𝑖

𝑉𝑖𝑌 (𝑟/𝑅𝑖),

(3.11)

(3.12)

(3.13)

where 𝑌 (𝑥) = 𝑒−𝑥

𝑥 . The parameters 𝑉𝑖 and the ranges 𝑅𝑖 are fit to nucleon-nucleon scattering data.
The result of the work by Love and Franey provides effective 𝑁 𝑁 T-matrix interaction strengths

applicable at various incident beam energies. For the case of the 60Ni(3He,𝑡) reaction at the Ex =

420 MeV, the Love and Franey interaction at 140 MeV/𝑢 was used.

Information about the transitions for which differential cross sections are calculated come in

the form of one-body transition densities (OBTDs), which are amplitudes for one-particle one-

hole (1p-1h) excitations that are connected by the specific operators of Eq. 2.1 for each of the

transitions. The OBTDs for all giant resonances being investigated in the 60Ni(3He,𝑡) reaction were

calculated in a normal-modes formalism and the calculations were performed with the NORMOD

code [121]. Unlike the shell-model calculations with Nushellx (see section 3.4 ), in the normal-

modes formalism, the OBTDs are calculated by producing the most coherent superposition of the

1p-1h excitations, thereby maximizing the transition strength.

For the normal-modes calculations, it was assumed that the 28 protons fill all single-particle

shells up to 0 𝑓7/2. The 32 neutrons are assumed to also fill the 0 𝑓7/2 shell and the 1𝑝3/2 shell,

32

as shown in Figure 3.3(a). Figure 3.3(b) and (c) show examples for the exciting of two isovector

transitions in the 60Ni(3He,𝑡) reaction.

Table 3.2 describes the input parameters of the NORMOD input file. The full NORMOD input

file and the output tables of the OBTDs for each giant resonance being investigated in this work are

included in the appendix B, Table B.1-B.2 and Tables B.3-B.15 respectively. The plots of the radial

transition densities, which represent the overlaps between the initial and final states in the target

nucleus, for all giant resonances being investigated in this work as listed in Table 2.1 are shown in

Figure 3.4. These are calculated in FOLD code, and serve as input for the formfactor calculation.

Finally, the formfactors for each giant resonance are calculated. Given the total angular mo-

mentum transfer in the projectile (3He,𝑡) system, ΔJp, and the total angular momentum transfer in

the target (60Ni,60Cu) system, ΔJT, the relative angular momentum transfer is calculated with:

Δ𝐽𝑅 = Δ𝐽𝑝 ⊕ Δ𝐽𝑇 ,

(3.14)

where ΔJp = ΔLp + ΔSp equals either 0 (Sp = 0) or 1 (Sp = 1), since ΔLp = 0. ΔJT is defined by

the type of giant resonance excited. Since ΔLp = 0, the orbital angular momentum transfer in the

target system ΔLT, defines the charge in parity:

(−1)Δ𝐿𝑇 = (𝜋𝑖 · 𝜋 𝑓 )Δ𝑇 ,

(3.15)

where 𝜋𝑖 is the parity of the 60Ni ground state and 𝜋 𝑓 is the parity of the final state in 60Cu. Since

the 60Ni ground state has a spin-parity of 0+, 𝜋i = 1. 𝜋f is negative (positive) if ΔLT = odd (even).

Note that for certain excitations, two sets of formfactors contribute. For example for 0+ → 1+

excitations, ΔJp = 1 and ΔJT = 1, allowing for both ΔJR = ΔLT = 0 and ΔJR = ΔLT = 2. Because

of the tensor interaction in the nucleon-nucleon interaction, the two components interfere with each

other. However, for strong excitations such as the giant resonances, the impact of such interferences

is small [20]. For the MDA, only formfactors with pure ΔJR = ΔLT are used, as its purpose is to

decompose contributions with different ΔLT. Each type of transition as listed in Table 2.1 requires

its own FOLD input file. Table 3.3 describes the input parameters of the FOLD input file. Example

of FOLD input files for several transitions are included in the Appendix C, Tables C.1 - C.7. The

33

Figure 3.3 Figure (a) shows the configuration of 60Ni ground state assumed in the NORMOD
calculations. Protons are in red color, they fill single particle shells up to 0 𝑓7/2, while neutrons are
in blue color and fills up to the 1𝑝3/2 shell. Figures(b) and (c) are examples of the exciting GT and
IVSGMR transitions respectively. The GT transition can only populate states in 60Cu with pf-shell
contributions (60Ni[g.s] → 60Cu[0ℏ𝑤]) while IVGMR transition can populate states in 60Cu with
pf, sdg and pfh-shell contributions (60Ni[g.s] → 60Cu[2ℏ𝑤])).

outputs of each FOLD input file are the formfactors as discussed above, which are shown in Figures

3.5 and 3.6. Since the 𝑁 𝑁 interaction has real and imaginary terms, the formfactors also have

real and imaginary components. Note that excitations involving the transfer of spin (ΔS = 1) have

strong real formfactors, while those without the transfer of spin (ΔS = 0) have strong imaginary

formfactors.

34

Table 3.2 Description of the input parameters in the NORMOD input file for calculating OBTDs
useful for FOLD input files for each giant resonance being investigated as shown in the Appendix
C, Tables C.1 - C.7.

Line number
line number 1

Input parameters Meaning of the input values
1

Wave functions are defined to be
positive near origin
Number of proton shells
Describing parameters of every single proton orbit
Number of nodes in particular wave Function
Angular momentum of proton
Twice total angular momentum of
single particle wave function
Fullness of proton-shell
1 for proton, 0 for neutron
n is major shell number of proton
x is an arbitrary number for each shells
Number of neutron shells
Same description as line 3 to 16
Parameters describing IAS
Parameters describing GT transition
Parameters describing Dipoles transition
Parameters describing IVGQR transition
Parameters describing IVSGQR transition
Parameters describing IVGMR transition
Parameters describing IVSGMR transition
Parameters describing Octupole transition
Discussed in operator of Eq. 2.1
Total transferred angular momentum
Product of parities
Mass number of target
Isospin transfer
reaction type (T=1) for CE reactions
Minimum and Maximum difference
between major shells number

line number 2
line number 3 to 16: Protons
Column number 1
Column number 2
Column number 3

14

n-value
L-value
2J-value

1.0
1 or 0
nℏ𝜔
x-value
11

Column number 4
Column number 5
Column number 6
Column number 7
line number 17
line number 18 to 28: Neutrons Same as 3 to 16
line number 29
line number 30
line number 31-33
line number 34
line number 35-37
line number 38
line number 39
line number 40
Column 1
Column 2
Column 3
Column 4
Column 5
Column 6
Column 7 and 8

IAS
GT
Dipoles
IVGQR
IVSGQR
IVGMR
IVSGMR
Octupole
M-value
ΔJ
J𝜋
m
T
1
Δnℏ𝜔

35

Table 3.3 Description of the input parameters in the FOLD files for calculating transition densities
and formfactors as shown in the Appendix C, Tables C.1 - C.7.

Line number

Input parameters Meaning of the input values

line number 1

line number 2 to 7
parameters of (3He,𝑡)

line number 3

line number 4

line number 5
line number 6

line number 7
line number 8
line 9 to 10 same as
3 to 4 but for (60Ni,60Cu)
line number 10

1
1FOLDNI
600
0.03
420
3
1
1
1
0.5+
0.5+
0.5
+0.5
0.5
-0.5

3

0.000

3
1
1
ΔJp
0.707
-1
-1
HE3H3
ΔJ𝜋
J𝜋 = 0+ for 60Ni
1 (2 for IAS)
1
2
2
3

Selects CE reaction type
File name to save formfactors
Total number of integration steps
Step size (fm)
Beam energy (MeV)
Mass of projectile (3He)
set to be 1 for printout for r-space densities
set to be 1 for printout for q-space densities
set to be 1 for printout for formfactors
Spin and parity (J𝜋) of ejectile (3H)
Spin and parity (J𝜋) of projectile (3He)
Isospin (T) of ejectile (3H)
Isospin projection (Tz) of ejectile (3H)
Isospin of projectile particle (3He)
Isospin projection of projectile particle (3He)

Selection of format of OBTDs particle
Single-particle wave function (ex: 0𝑠1/2)
hole-particle wave function (ex: 0𝑠1/2)
1 for spin transfer and 0 for no spin transfer
OBTD for (3He,𝑡) system
fixed end line
File name of projectile and ejectile

Initial (ground) spin and parity of target
Isospin (T) of residual (60Cu)
Isospin projection (Tz) of residual (60Cu)
Isospin (T) of target (60Ni)
Isospin projection of target (60Ni)
Selection of format of OBTDs

R = 1+ for GT Relative final spin and parity of each transition

3

0.000

line 11 same as 5
line number 13 to x-line OBTDs (1p-1h) Values calculated from NORMOD code
Column 1
Column 2
Column 3
Column 4
Column 5
line number 14
line number 15

Single-particle wave function
hole-particle wave function
Depends on type of giant resonance excited
ignored
OBTD
fixed end line
File name containing radial wave functions

1p
1h
ΔJT
0.0
value
-1
-1
Ni60Cu60

36

Table 3.4 Description of the input parameters in the FOLD file for calculating transition densities
and formfactors, a continuation of Table 3.3.

line number 16

0.939

2.650

line number 17
line number 18

Scaling parameter for transformation
of interaction strengths from NN system
to nucleon-nucleon system
Momentum parameter used to calculate
exchange contribution.
Not used.

1.000
Love and Franey Filename contains parameters at 140 MeV/𝑢.
NN interaction
1 or 2
ΔJR
JP
JT

Number of formfactor(s).
Relative spin transfer
Total angular momentum for each formfactor
Total angular momentum transfer in
the target system
Select components of NN interaction
(the -1 is all components)
Scaling factors for formfactor components.
(Usually set to 1)

−1

line number 19-20 or 19-22 Fixed values

37

Figure 3.4 Transition densities calculated from FOLD code for a target nucleus (60Ni) of the IAS,
GTR, monopoles, dipoles, quadrupoles, and octupoles transitions being investigated in this work
as all listed in Table 2.1.

38

Figure 3.5 The real (left panels) and imaginary (right panels) part of the formfactors for transitions
listed in the panels. They were calculated in the FOLD code. The excitations with spin transfer
(ΔS = 1) have strong real formfactors, while these without spin transfer (ΔS = 0) have strong
imaginary formfactors. Also note that the 2ℏ𝑤 resonances have a node in the formfactior.

39

Figure 3.6 The real (left panels) and imaginary (right panels) part of the formfactors for transitions
listed in the panels. They were calculated in the FOLD code. The excitations with spin transfer
(ΔS = 1) have strong real formfactors, while these without spin transfer (ΔS = 0) have strong
imaginary formfactors.

40

3.4.3 Calculation of Differential Cross Section (DWHI)

The final part of the FOLD package, called DWHI, calculates the transition matrix elements,

𝑇 𝑓 𝑖 in Eq. 3.7 and the differential cross sections. The DWHI code uses the previously calculated

formfactor (see Eq. 3.10) together with the optical model potential (Eq. 3.9) as the ingredients.

The optical potential distorts the incoming and outgoing waves. The real and imaginary optical

potential parameters (depths of the potential, radius and diffuseness) used in this work for the

60Ni-3He and 60Cu-3H systems, were taken from measurement of 3He elastic scattering on 58Ni at

443 MeV [119]. The potential depths of 60Cu-3H system were scaled to 85% of the one of 60Ni

following the procedure in Ref. [122], while the other parameters remain the same. The potential

parameters are shown in Table 3.5.

Table 3.5 Optical potential parameters used in DWHI calculations for 60Ni(3He,𝑡)60Cu reaction at
140 MeV/𝑢. These values were taken from Ref. [119]. and defined for 3H following Ref. [122].

Nuclides

V
[MeV]
60Ni + 3He
35.16
60Cu + 3H 29.89

rv
(fm)
1.32
1.32

av
[fm]
0.84
0.84

W rw

[MeV]
44.43
37.77

[fm]
1.021
1.021

aw
[fm]
1.018
1.018

As discussed in section 3.4.2, each giant resonance listed in Table 2.1 requires its own DWHI

input file. Table 3.6 describes the input parameters of the DWHI input files. Examples of DWHI

input files for several transitions are included in the Appendix D, Tables D.1 - D.14. The output of

each DWHI calculation file is shown in Figure 3.7. The differential cross sections will be used in

the data analysis and serve as inputs for the MDA (Eq. 3.2. The results from MDA calculation are

discussed in Chapter 5.

41

Table 3.6 Description of the input parameters in the DWHI files for calculating the angular distri-
butions for each giant resonance as shown in the Appendix D, Tables D.1 - D.14.

Line number

Input parameters

Meaning of the input values

line number 1

1210000041000000 Options for calculation and plotting

line number 2
line number 3

line number 4

line number 5

line number 6
(For incoming channel)

line number 7
(Real part of optical
potential parameters
of 60Ni-target)

(Imaginary part of
optical potential
parameters of 60Ni-target)

line number 8

FOLDNI
40
0
0.2

value-1=160
value-2=1 or 2

value-3=1
value-4=1
value-5=0
value-6

value-1=0.03
value-2=600
value1= 420
value-2 = 3
value-3 = 2
value-4 = 60
value-5 = 28
value-6 = 1.25
value-7=1
value-8=0
value-1=1
value-2=-35.16
value-3=1.32
value-4=0.84
value-5=0

value-6=-44.43
value-7=1.021
value-8=1.018
value-9=0
value-10=0
value=0

cross sections for each form factor
File name containing form factor
Total number of angle used
Initial angle
Angle step size

Number of partial waves for elastic
Number of form factors to expect
must match number produced by FOLD
2 × ΔJ-projectile spin in initial channel (3He)
2 × ΔJ-ejectile spin in final channel (𝑡)
2 × ΔJ-target spin initial channel (60Ni)
2 × ΔJ-residual transfer (60Cu)

Step size
Total number of integration steps
Beam/lab energy
Projectile (3He) mass number
Projectile proton number
Target (60Ni) mass number
Target proton number
Coulomb radius
Twice the spin value of the incident projectile
Not used in entrance channel
Woods-Saxon potential (WS)
Real well depths of WS potential
Radius
Diffuseness
Indicate that beam energy used (420 MeV)
is the lab energy or reaction Q-value

Imaginary well depths of WS potential
Radius
Diffuseness
Imaginary spin-orbit factor (not used)
Computing factor (not used)
Fixed end line

42

Table 3.7 Description of the input parameters in the DWHI file for calculating the angular distribu-
tions. Continuation of Table 3.6.

line number 9
(For outgoing channel)

line number 10
(Real part of optical
potential parameters
of 60Cu-residual )

(Imaginary part of
optical potential
parameters of 60Cu)

line number 11
line number 12

line number 13

Q-value
Ejectile (𝑡) mass number
Ejectile proton number
Residual (60Cu) mass number
Residual proton number
Coulomb radius
Twice the spin value of the ejectile
Not used in entrance channel
Wood-Saxon potential

value-1= −6.2
value-2 = 3
value-3= 1
value-4= 60
value-5= 29
value-6= 1.25
value-7=1
value-8=0
Value-1=1
value-2=-37.77 Real depth of WS potential
value-3=1.021
Radius
value-4=1.018 Diffuseness
value-5=0

Indicate that beam energy used (420 MeV)
is the lab energy or reaction Q-value

Imaginary depth of WS potential
Radius

value-6=-44.43
value-7=1.021
value-8=1.018 Diffuseness
value-9=0
value-10=0
value=0
ΔJR
2 × JP
2 × ΔJT
0
0

0 1

Imaginary spin-orbit factor
Computing factor
Fixed end line
Relative spin transfer
Same as JP in FOLD
Same as JT in FOLD
Fixed line ending form factor,
where 0. terminate further form factor
and 1. enter data for single form factor
Filename of output

line number 14

Plot filename

43

Figure 3.7 Differential cross sections calculated for the following giant resonances excited via
the 60Ni(3He,𝑡) reaction: the isobaric analog state (top-left), Gamow-Teller resonance (top-right),
isovector(spin) monopole resonances (middle-left), isovector (spin) dipole resonances (middle-
right), isovector (spin) quadrupole resonances (bottom-left) and isovetor (spin) octupole giant
resonances (bottom-right).

44

02468cm [deg]012345d/d[mb/sr]IAS/Fermi transition(J,L,S)=(0+,0,0)02468cm [deg]050100150200250d/d[mb/sr]GTR transition(J,L,S)=(1+,0,1)02468cm [deg]050100150d/d(mb/sr)IV(S)GMR transitions(J,L,S)=(0+,0,0)(J,L,S)=(1+,0,1)02468cm [deg]0102030d/d(mb/sr)IV(S)GDR transitions(J,L,S)=(0,1,1)(J,L,S)=(1,1,1)(J,L,S)=(2,1,1)(J,L,S)=(1,1,0)02468cm [deg]0246810d/d(mb/sr)IV(S)GQR transitions(J,L,S)=(1+,2,1)(J,L,S)=(2+,2,1)(J,L,S)=(3+,2,1)(J,L,S)=(2+,2,0)02468cm [deg]0.0000.0050.0100.0150.0200.025d/d(mb/sr)IV(S)GOR transitions(J,L,S)=(3,3,1)(J,L,S)=(3,3,0)3.5 Eikonal Approximation and the Unit Cross Section for GT and IAS

The extraction of Gamow-Teller strength, 𝐵(GT), can be performed directly via a 𝛽/EC decay

half-life measurement. However, 𝛽-decay can only populate states in the daughter nucleus that are

energetically accessible given the 𝑄-value of the transition. As shown in Figure 3.8, CE reactions

are not limited by a 𝑄-value, and transitions to high-lying states can be studied. CE reactions are

mediated by the strong force, and the weak force mediates 𝛽-decay. However, the operators involved

in each process are both of the same 𝜎𝜏𝜎𝜏𝜎𝜏 type and the initial and final states that are connected by

this operator are the same. It turns out that, at intermediate beam energies, the differential cross

sections,

(cid:17)

(cid:16) d𝜎
dΩ

ΔL=0

, at small momentum transfer (q ≈ 0) measured via charge-exchange reactions

is proportional to the Gamow-Teller strength of the transition [123]. This proportionality was

experimentally shown by Taddeucci 𝑒𝑡 𝑎𝑙 [123, 19, 20, 124], and is given by:

(cid:18) 𝑑𝜎
𝑑Ω

(cid:19)

(q = 0)

Δ𝐿=0

= ˆ𝜎𝐵(GT),

(3.16)

where the ˆ𝜎 is the proportionality constant, called the unit cross section. For this proportionality to

hold, the ΔL = 0 component of the differential cross section must be used, which can be obtained

through the multipole decomposition analysis described in section 3.1 and 5.2.

In the Eikonal

approximation, the unit cross section can be decomposed:

ˆ𝜎GT = 𝐾 𝑁 |𝐽𝜎𝜏 |2,

(3.17)

where ˆ𝜎GT is the Gamow-Teller unit cross sections, K is a kinematical factor, carrying information

about masses and energies of particles, and is expressed in terms of incoming and outgoing wave

momenta ( ki and kf) and their corresponding initial and final reduced energies (Ei and Ef):

K =

EiEf
(𝜋ℏ2c2)2

kf
ki

,

(3.18)

N is a distortion factor defined by the ratio of the DWBA to the PWBA (plane-wave Born approxi-

mation) cross sections [123, 19]:

N =

(cid:18) 𝜎DWBA
𝜎PWBA

(cid:19)

,

45

(3.19)

Figure 3.8 Schematic representation of both 𝛽-decay (red colors) and CE reactions (blue colors)
from parent nucleus Y to the daughter nucleus X. 𝛽-decay can only populate states in the daughter
nucleus that are energetically accessible with the Q-value of the decaying nucleus. However, CE
reactions are not limited by Q-value, and highly excited states (nuclei that do not 𝛽-decay) can be
studied via CE reactions.

and J𝜎𝜏 (or J𝜏) is the volume integral of the corresponding effective 𝑁 𝑁 interaction (see Eq.

3.11) between the projectile and target nucleons. In practice, 𝜎GT is conveniently calibrated by

using transitions for which the transition strengths are known from 𝛽/EC decay and one does not

have to rely on a calculation. This makes the extraction of GT strength model independent, which

is very important for testing theoretical models. Of course, not for all nuclei are measured GT

strength from 𝛽/EC decay available. However, the unit cross sections for Gamow-Teller transitions

in such cases can be calculated by using expressions that only depend only on the mass number (𝐴)

of the nucleus which have been established experimentally. For (3He,𝑡) reactions at 𝐸 ≈ 420 MeV,
this relationship is ˆ𝜎GT = 109/A0.65 as shown in Figure 3.9. The proportionality described above
for GT transitions also hold for the Fermi transition to the IAS and the mass dependent ˆ𝜎𝐹 is also

included in Figure 3.9.

46

Figure 3.9 Figures (a) and (b) are measured of Fermi and Gamow-Teller unit cross sections respec-
tively, as a function of mass number (𝐴) for (3He,𝑡) reaction at 420 MeV. Both figures are taken
and modified from Ref. [19].

The proportionality between strength and differential cross sections holds for beam energies

of 𝐸 ≳ 100 MeV/𝑢. This energy is sufficiently high to strongly reduce the contribution of

multistep processes to the CE reaction. The proportionality has the least uncertainty for strong

transitions. For weak transition, interference between the 𝑉𝜎𝜏 and 𝑉𝑇 𝜏 components of the 𝑁 𝑁

interaction become significant [19]. The uncertainty as a function of B(GT) was estimated to be

𝜎𝑟𝑒𝑙.𝑠𝑦𝑠𝑡.𝑒𝑟𝑟𝑜𝑟 ≈ 0.03 − 0.035𝑙𝑛[𝐵(GT)]. For example, states at 𝐸𝑥 = 2.07[𝐵(GT) = 0.091] and

𝐸𝑥 = 2.74[𝐵(GT) = 0.113], standard deviations of 11.4% and 10.6% were expected, respectively

[20].

47

CHAPTER 4

DATA ANALYSIS: 60Ni(3He, 𝑡) REACTION AT RCNP

This dissertation focuses on probing isovector giant resonances in 60Ni up to high excitation energies

(60 MeV) using the (3He,𝑡) reaction at 140 MeV/𝑢. The experiment was performed at the Grand

Raiden Spectrometer (GRS) at Osaka University’s Research Center for Nuclear Physics (RCNP).

The method involved directing a 3He beam at 420 MeV produced by the RCNP Ring Cyclotron with

an intensity of ∼ 4 pnA on a 60Ni foil target of 2 mg/cm2. Section 4.1 describes the experimental

setup, procedures, and devices. An overview of the GRS and its focal plane detectors are discussed

in section 4.2 and 4.3, respectively. The method used to extract the excitation energy spectrum

from 60Ni(3He,𝑡) reaction was the missing-mass method, discussed in section 4.4. The magnetic

fields of GRS and the angular acceptance are discussed in section 4.6 and 4.5, respectively. The

measured differential cross sections from 60Ni(3He,𝑡) data are discussed in section 4.7.

4.1 Experimental Setup, Procedures & Devices

The 60Ni(3He,𝑡) experiment was performed at the Research Center for Nuclear Physics (RCNP)

in Osaka University, Japan by using a primary 3He beam of 140 MeV/𝑢 and the high-resolution

QQDD-type Grand Raiden Spectrometer described in section 4.2. The spectrometer was set at
−0.5◦ relative to the beam axis. The intermediate-energy 3He2+

beam at the RCNP facility has been

used extensively in various experiments for studying the Gamow-Teller strength distribution and

other giant resonances via (3He,𝑡) reactions at intermediate beam energies [19, 119, 125, 126, 127].

The Azimuthally Varying Field (AVF) and Ring cyclotrons shown on Figure 4.1 were coupled to

accelerate a beam of 3He nuclei to 420 MeV and transported to the target through the WS beam

line [128] connecting the separated-sector Ring Cyclotron and the GRS. Beam intensities of up to

4 pnA were impinged on a 60Ni-target foil of 2 mg/cm2 installed in the scattering chamber. The

60Ni target was 98% pure.

48

Figure 4.1 Schematic layout of the RCNP Ring Cyclotron facility.

4.2 Grand Raiden Spectrometer

The Grand Raiden Spectrometer (GRS) shown in Figure 4.2, is used to identify and analyze

the momentum of tritons produced in (3He,𝑡) reactions [129]. The GRS was designed for high-

resolution measurements. The GRS contains three dipoles magnets (D1, D2 and DSR), two

quadrupoles (Q1 and Q2), one sextupole (SX), and one multipole (MP) magnet as shown in Figure

4.2. The multipole magnet can produce fields that are dipole, quadrupole, octupole, sextupole,

49

and decapole. It is used to correct for aberrations in the ion optics. The Dipole Magnet for Spin

Rotation (DSR), is meant for polarized beam experiments and not used in this experiment. The

GRS can achieve a momentum resolution of Δp/p = 2.7 × 10−5 and operate at magnetic rigidity

of up to 5.4 Tm allowing the measurement of tritons at 140 MeV/𝑢 (𝐵𝜌 = 5.31 Tm ) [130, 128].

Specification parameters of the GRS are shown in Table 4.1.

Figure 4.2 Schematic representation of the Grand Raiden Spectrometer set at 0◦. The position and
angle are measured in the focal plane detectors. The momentum vector of the ejectile is deduced,
from which the excitation energy in the residual nucleus and scattering angles are determined.
Figure modified from Ref. [131].

4.3 Focal-Plane Detectors for Grand Raiden Spectrometer

The Focal-plane (FP) detectors for the GRS contain two sets of Multi-Wire Drift Chambers

(MWDCs, see Figure 4.3 and Table 4.2) placed in the focal plane of the spectrometer. They are

used to collect information about the positions and angles of the particles. Each MWDC is filled

with a gas mixture typically composed of argon (71.4%), and isobutane (28.6%). They have two

anode-wire planes known as X and U anode-wire planes. The X layer has wires perpendicular to

50

Table 4.1 Specification parameters of Grand Raiden Spectrometer.

Parameters
Intrinsic momentum resolution (ΔP/P)
Intrinsic energy resolution (ΔE/E)
Position resolution
Maximum B𝜌
Maximum B (D1, D2 )
Maximum magnetic gradient (Q1)
Maximum magnetic gradient (Q2)
Momentum range
Focal plane tilt
Mean orbit radius
Total deflection angle
Angular range
Horizontal magnification (x|x)
Vertical magnification (y|y)
Maximum momentum dispersion
Horizontal acceptance angle
Vertical acceptance angle
Solid angle
Weight
Flight path for the central ray

Value
2.7 × 10−5
4.5 × 10−5
300 𝜇𝑚 (both horizontal and vertical)
5.4 Tm
1.8T
0.13 T/cm
0.033 T/cm
5%
45%
3 m
160◦
−5◦ to 90◦
−0.417
5.98
15.45 m
±20 mr
±40 mr (in over-focus mode)
5.6 msr ( 3.2 in over-focus mode)
600 tons
20 m

the medium plane of the spectrometer, and the U layer has wires at an angle of 48.19◦ [129] with

respect to the X plane. The potential wires are charged and serve to generate a uniform electrical

field between the cathode and anode planes [131]. Charged particles ionize the gas atoms in the

trajectory and the ionized electrons drift perpendicularly to the anode plane and are detected by the

grounded sense wires. Drift times from the four sets of anode wires were measured and particle

trajectories were determined with a position resolution of around 100 𝜇m in each plane.

Event rates were such that the corrections for the lifetime of the data acquisition system (DAQ)

were ≈ 95%.

The energy loss and time-of-flight information for each hit were measured by using a set of

10-mm thick plastic scintillators (PS1 and PS2), which are mounted behind the drift chambers.

The first scintillator triggers the data acquisition system, serves as the start of the time-of-flight

measurement, and benefits particle identification. The stop signal is provided by the cyclotron radio

frequency (RF) signal. A 1-mm thick aluminum plate placed between the scintillators improves the

particle identification (PID) by increasing the energy loss in the second scintillator. At the highest

51

Figure 4.3 Schematic structure of an X-plane of the MWDCs for Grand Raiden Spectrometer.
Anode wires and Cathode planes are represented with a typically charged particle track.

rigidity setting, beside the tritons, singly-charged 3He

+

the same mass-to-charge ratio as the tritons. The 3He

+

ions enter the focal plane, as they have
ions are produced when the 3He2+

beam

particles pick up an electron in the reaction target. The particle-identification plot for this situation

is shown in Figure 4.4. The gates used for selecting the 3H

+

and 3He

+

events are also indicated.

The vertical bands associated with each species are due to pile-up. The 3He

+

particles produced

via the atomic charge-exchange in target material [132] can be used for the calibration of scattering

angles. Since the 3He

+

particles have the same rigidity as the tritons but have negligible scattering

angles, they are useful for determining the central beam axis, corresponding to 0◦ scattering angle.

The 3He

+

measurement is also helpful to obtain reliable energy calibration.

52

Figure 4.4 Particle identification from the plastic scintillator signals at Grand Raiden focal plane.
+
Gates used to select tritons (3H
) and 3He (3He
) charge-state particles are indicated. Particles
near channel 0 in both axes are due to cosmic rays and noise in the scintillators.

+

The 3He2+

beam transport to the GRS target location was achromatic. More details about the

ion-optical modes for experiments at the GRS can be found in Ref. [126].

4.4 Missing-Mass Calculation

From the measured positions and angles in the MWDC detectors at the focal plane, the momen-

tum vector of the triton is deduced, from which the excitation energies in the residual nucleus and

scattering angle are determined from the missing-mass calculation [133]. The ray-trace matrix that

is used to reconstruct the momenta and scattering angles from the positions and angles measured

in the focal plane is determined empirically by using a sieve slit measurements [134]. The sieve

slit is a block of a distinctive hole pattern (see Figure 4.5). It is installed 60 mm downstream from

53

020040060080010001200140016001800020040060080010001200140016001800102103104105ΔE1 (a.u.)ΔE2 (a.u.)3H+3He1+Table 4.2 Specification parameters of the MWDC for Grand Raiden Spectrometer.

Parameters
Anode:

Wire configuration
Sense wire
Potential wires
Number of sense wires
Anode wire spacing
Sense wire spacing

Cathode:

Material
Supplied voltage
Cathode-anode gap
Active area
Gas mixture:

Distance between two MWDCs

Value

X(0◦ = vertical ), U(48.2◦)
𝜙 20 𝜇𝑚 gold-plated tungsten wire
𝜙 50 𝜇𝑚 gold-plated beryllium-copper wire
192 (X) and 208 (U)
2 mm
6 mm (X) and 4 mm (U)

10 𝜇𝑚-thick carbon-aramid film
−5.6 keV
10 mm
1200𝐻 mm × 120𝑊 mm
Argon (71.4%) + Isobutane (28.6%) +
Isopropyl alcohol (vapor pressure at 2◦C)
250 𝜇𝑚

the target and runs with this sieve slit are taken for every setting of the spectrometer for calibration

purposes. To reconstruct horizontal and vertical scattering angles based on the hole pattern in the
sieve slit, 6𝑡ℎ order polynomials relying on XFP, YFP, Θhorizontal

are used. Then, the

and Θvertical
FP

FP

sieve slit is removed from the beam-line and the polynomials are used to reconstruct the target

angles for the rest of the data. For the momentum calibrations, excitation energy spectra of nuclei

with states for which the excitation energies are well known are used. For this experiment the

26Mg(3He,t) reaction was the primary calibration reaction.

54

Figure 4.5 An example of the structure of sieve slit of the GRS for calibration of scattering angles.
Figure taken from [135].

The missing-mass is calculated using the missing energy and momentum after reconstruction:

√︃

𝑚𝑚𝑖𝑠𝑠 =

𝑚𝑖𝑠𝑠 − 𝑃2
𝐸 2

𝑚𝑖𝑠𝑠

The excitation energy is determined based on the mass of the residual nucleus;

𝐸𝑥 (60Cu) = 𝑚𝑚𝑖𝑠𝑠 − 𝑚(60Cu),

(4.1)

(4.2)

where m(60Cu) is the ground-state mass of 60Cu. The missing energy (𝐸𝑚𝑖𝑠𝑠) is defined as the

excess energy in the 60Ni(3He,𝑡) reaction:

𝐸𝑚𝑖𝑠𝑠 = 𝐸𝑖 − 𝐸 𝑓

= 𝐸𝑘 (3He) + 𝑚(3He) + 𝑚(60Ni) − 𝐸𝑘 (3H) − 𝑚(3H)

(4.3)

where 𝐸𝑘 is the kinetic energy. 𝐸𝑘 (3H) is determined from the reconstructed momentum of the

triton.

√︃

𝐸𝑘 (3H) =

𝑝2(3H) − 𝑚2(3H) − 𝑚(3H)

(4.4)

55

The missing momentum is calculated from:

√︃

𝑝𝑚𝑖𝑠𝑠 =

𝑚𝑖𝑠𝑠,𝑥 + 𝑝2
𝑝2

𝑚𝑖𝑠𝑠,𝑦 + 𝑝2

𝑚𝑖𝑠𝑠,𝑧

where,

𝑝𝑚𝑖𝑠𝑠,𝑖 = 𝑝𝑖 (3He) − 𝑝𝑖 (3H)

(4.5)

(4.6)

In this calculation 𝑝𝑥(3He) and 𝑝𝑦(3He) are assumed to be 0. The x,y, and z components of the 3H

momentum are calculated from the measured momenta and scattering angle of the 3H on the 𝐺 𝑅𝑆:

𝑝𝑥 (3H) = 𝑝𝑧 (3H) · 𝑡𝑎𝑛(Θ𝑥)

𝑝𝑦 (3H) = 𝑝𝑧 (3H) · 𝑡𝑎𝑛(Θ𝑦)

𝑝𝑧 (3H) = 𝑃𝑥 (3H) · 𝑐𝑜𝑠(Θ)






Θ = 𝑎𝑡𝑎𝑛

(cid:18)√︃

𝑡𝑎𝑛2(Θ𝑥) + 𝑡𝑎𝑛2(Θ𝑦)

(cid:19)

(4.7)

(4.8)

where,

and Θ𝑥 and Θ𝑦 are the reconstructed horizontal and vertical components of the scattering angle.

4.5 Magnetic Fields of Grand Raiden Spectrometer

As discussed in section 2.1, the strength distribution of the isovector-spin and non-spin-transfer

giant monopole and dipole resonances are expected to extend up to high excitation energies. These

features require experimental measurements with an energy range that covers the whole width of

the resonances, approximately up to 60 MeV.

The momentum acceptance of the GRS is ≈ 5%. Therefore, the energy acceptance for a setting

of the magnetic field is ≃ 10% (this is shown in Appendix E, since

2𝑘 ). This implies that
the energy range for a 140 MeV/𝑢 (420 MeV) triton covered in a single setting is about 40 MeV.

𝑑𝑝

𝑝 ≈ 𝑑𝑘

In order to cover the range of excitation energies above 40 MeV, three overlapping settings of the

magnetic field were used. Examples of the measured spectra at these three settings are shown in

Figure 4.6, for scattering angles between 25 and 30 mrad.

56

Figure 4.6 Singles spectra obtained for the 60Ni(3He,𝑡) reaction for scattering angles between 25
and 30 mrad. Top: high magnetic field for the low excitation energy setting; Middle: medium
magnetic field for the medium excitation energy setting; Bottom: low magnetic field for the high
excitation energy setting.

57

4.6 Angular Acceptance and Resolution

The 3He

+

charge state provides a convenient way to determine the angular resolution. In Figure

4.7(a) and (b), the Θ𝑥 and Θ𝑦 distribution for the charge state are shown, respectively. The width

(FWHM) of these distributions are 2.4 mrad (x) and 3.3 mrad (y), which constitute the angular

resolutions in each direction.

The angular acceptance of the GRS has an irregular shape as shown in Figure 4.8, which displays

Θ𝑥 versus Θ𝑦 for the low excitation energy runs. Near the edges of the acceptance, the angular

acceptance is uncertain and depends on the momentum of the tritons. For Θℎ ≤ −6 mrad, the ray

tracing of Θ𝑣 is uncertain, especially at low triton momentum (high excitation energy), resulting in

too many events at Θ ≈ 0. To ensure that only events with well-reconstructed angles are utilized

and the acceptance is well understood, only the data with −6 ≤ Θ𝑥 ≤ 21 mrad and −36 ≤ Θ𝑦 ≤ 36

mrad were used, as indicated by the black box in Figure 4.8. Within the angular range defined by

this box, 8 angular slices, each of 5 mrad wide, were used to create angular distributions as shown

in Figure 4.9.

The solid angle of each of these 8 angular bins was determined in a simple Monte-Carlo

integration. The result is shown in Figure 4.10. Due to the finite angular resolution, the effective

solid angle of each angular bin is slightly distorted. The effect was included in the Monte-Carlo

integration.

The estimated excitation energy resolution in 60Cu via the 60Ni(3He,𝑡) was 0.11 MeV, see Figure

4.11.

58

Figure 4.7 Figures(a) and (b) are the distribution of angular widths [FWHM] in horizontal (Θ𝑥 = 2.4
mrad) and vertical (Θ𝑦 = 3.3 mrad) projection, respectively. The angular resolution [FWHM] is
∼ 3 mrad.

59

Figure 4.8 Reconstructed angles from the highest magnetic field setting. The black box indicates
the angular ranges used in further analysis.

60

-50-40-30-20-1001020304050-20-100102030050100150200250300350400Θx (mrad)Θy (mrad)Figure 4.9 Figure illustrating vertical and horizontal angular acceptance of the GRS for the three
angular settings used for 60Ni(3He,𝑡) reaction. The 8 angular slices, each of 5 mrad wide are shown
in circles.

61

Figure 4.10 Calculated effective opening angles (𝑑Ω) for the angular bins used in the analysis of
the 60Ni(3He,𝑡) data.

62

Figure 4.11 The estimated energy resolution [FWHM] from the analysis of the 60Ni(3He,𝑡) data
was 0.11 MeV.

63

4.7 Differential Cross Sections Calculation

As discussed in section 4.5, data from the three angular settings (low, medium, and high

excitation energies) of the GRS were merged for every scattering angle bin of 5 mrad in order to

have one spectrum from the full range of excitation energies up to 60 MeV. The differential cross-

sections of the 60Ni(3He,𝑡) reaction at 140 MeV were calculated using the following equation:

𝑑𝜎
𝑑Ω

=

𝑌
𝑁𝑏𝑒𝑎𝑚 𝑁𝑡𝑔𝑡𝜖1𝜖2𝑑Ω

,

(4.9)

where 𝑌 is the total number of counts in an angular bin, 𝑁𝑏𝑒𝑎𝑚 is the number of beam particles
(3He2+) on target, 𝑁𝑡𝑔𝑡 is the number of nuclei in the target 60Ni foil of 2 mg/c𝑚2, 𝜖1 is the correction
for the lifetime of the data acquisition system (95%), 𝜖2 is the correction for the target purity (98%),
and 𝑑Ω = 2𝜋 ∫ 𝜃 𝑓
𝜃𝑖

sin 𝜃𝑑𝜃 is the solid angle for angular bin as shown in Figure 4.10, where 𝜃𝑖 and

𝜃 𝑓 are the lower and upper angular bin limits.

Figure 4.12 and 4.13 shows results of the measured differential cross sections for different

angular bins of 5 mrad from 60Ni(3He,𝑡) data. The dominant observed peaks in the spectra are the

GTR, please note that since the angular distributions of the IAS and GTR are similar and they both

peak at forward angles. The IAS can be seen at an excitation energy of 2.55 MeV. The GTR peaks

around 10 MeV. At higher excitation energies, the flat structure is a combination of states related to

different angular momentum transfers. The statistical uncertainties are represented in the spectra.

64

Figure 4.12 Extracted differential cross sections for each 5-mrad wide angular bin for the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢. Angles are in the laboratory frame. The uncertainties presented in the data
are statistical.

65

010203040500.00.5t,lab<5 mrad010203040500.00.55t,lab<10 mrad010203040500.00.510t,lab<15 mrad010203040500.00.515t,lab<20 mradd/ddE [mb/(sr 20 keV)]Ex(60Cu) [MeV]Figure 4.13 Extracted differential cross sections for each 5-mrad wide angular bin for the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢. Angles are in the laboratory frame. The uncertainties presented in the data
are statistical.

66

010203040500.00.520t,lab<25 mrad010203040500.00.525t,lab<30 mrad010203040500.00.530t,lab<35 mrad010203040500.00.535t,lab<40 mradd/ddE [mb/(sr 20 keV)]Ex(60Cu) [MeV]CHAPTER 5

RESULTS AND COMPARISON WITH THEORY

5.1 Calculated and Measured IAS

The next step in the analysis of the experimental data is to investigate the contribution from

different types of transitions and to identify the location of the giant resonances. As discussed

in Chapter 2, resonances associated with different units of angular momentum transfer peak at

different scattering angles, and we can use a multipole decomposition analysis (MDA) to separate

these contributions. In the following, we first look to the IAS, before performing a MDA of the

whole excitation energy spectrum.

In 60Cu, it is possible to identify individual states at low excitation energies.

In Figure 5.1

(a) the excitation energy spectrum for 𝐸𝑥 < 5 MeV is shown, at scattering angles of Θ𝑡,𝑙𝑎𝑏 < 5

mrad. Figure 5.1 (b) shows the differential cross section for the transition to the IAS of the 60Ni:

ground state, located at an excitation energy of 2.55 MeV as indicated in Figure 5.1 (a). The error

bars show statistical errors. The measured differential cross section is compared with the theory,

based on the distorted wave Born approximation (DWBA) see section 3.4.3. The theoretical cross

section was higher than the measured cross-section and scaled down by a factor of 1.35 to match

the measured cross sections. The likely reason for this overestimation by the theory is the use of

an approximation to the exchange contribution in the 𝑁 𝑁 interaction used in the calculation [37].

Other than this scaling factor, the theoretical calculation matches well with the experimental data,

indicating the ability of the DWBA calculation to accurately describe the experimental angular

distributions.

67

Figure 5.1 Figure (a) shows the excitation-energy spectrum between 0 and 5 MeV for the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢. Figure (b) shows the comparison of the measured differential cross section
for the IAS with theory, based on the DWBA calculation as shown with the black solid line. The
DWBA calculation was scared down by a factor of 1.35.

68

5.2 Multiple Decomposition Analysis Results

As discussed in chapter 3, the MDA is a method used to extract contributions to the measured

spectra that are associated with the transfer of different units of angular momentum (Δ𝐿). The

MDA was performed by fitting the measured differential cross section shown in Figures 4.12 and

4.13 for each energy bin with a linear combination of theoretical angular distributions (see Eq.

3.2), each associated with a different value of Δ𝐿, as discussed in section 3.4.3.

Figures 5.2 and 5.3 illustrates the MDA fits for a few selected excitation energy bins. A bin

size of 40 keV was used in order to ensure adequate statistics for the MDA. In DWBA, the Δ𝐿=0,

1, and 2 shapes were computed with matching excitation energy for each bin. Note that systematic

uncertainties related to the choice in which monopole, dipole, or quadrupole formfactor is used, is

relatively small as the angular distributions for different choices are very similar.

The top four figures in Figure 5.2 depict a fit at excitation energies below 14 MeV, where the

monopole (Δ𝐿 = 0) contribution dominates over other Δ𝐿 values. However, as the scattering angles

increase, the dipole (Δ𝐿 = 1) contribution becomes significant. The bottom two panels in Figure

5.2 show fits around 20 MeV, where Δ𝐿 = 1 and 2 contributions are relatively strong. Below 1◦,

the Δ𝐿 = 2 component dominates, while above 1◦, the dipole contributions dominate.

Figure 5.3 (top two figures) shows fits around 30 MeV, where all Δ𝐿 values contribute to the

total strength. Below 1.25◦, the monopole contribution dominates, while above 1.25◦, the dipole

contribution dominates. For the fit around 47 and 48 MeV in Figure 5.3 (middle two figures), the

Δ𝐿 = 2 component dominates at all scattering angles, but other Δ𝐿 = 0 and 1 values also contribute

to the total strength. Finally, the bottom two panels of Figure 5.3 show the fit around 54 and 57

MeV, where all Δ𝐿 values contribute to the total strength. Below 1.25◦, a significant contribution

of the Δ𝐿 = 0 component was observed, while above 1.25◦, the Δ𝐿 = 1 component dominates.

Please note that the components Δ𝐿 = 2 likely also include contributions from excitations with

higher Δ𝐿, but given the relatively narrow angular range covered in the experiment, these cannot

be separated.

The results from the MDA for each individual bin can now be combined to obtain a picture of

69

the multipole response as a function of excitation energy. The combined plots are shown in Figure

5.4 and 5.5 for each of the angular bins. Figure 5.4 shows the results for angular bins below 20

mrad. Each figure contains the contributions from each Δ𝐿 component, the sum of these three

components, and the experimental data.

The Δ𝐿 = 0 contributions are enhanced at low scattering angles and below 𝐸𝑥 = 12 MeV it is the

dominant contribution to the spectrum. This strength can be attributed to Gamow-Teller transitions,

aside from the IAS at 2.55 MeV. Note that the 0+ → 1+ GT transitions have a Δ𝐿 = 0 and Δ𝐿 = 2

component. Therefore, in regions where GT transitions are strong, a Δ𝐿 = 2 contribution must

be present as well. Of course, Δ𝐿 = 2 contributions can also come from transitions to 2+ and 3+

final states as well. At higher excitation energies, contributions to the Δ𝐿 = 0 response from the

IVGMR and IVSGMR are expected. Indeed, Δ𝐿 = 0 strength is observed above 20 MeV, but the

interpretation is complicated by contributions from quasifree reactions, as discussed below.

Dipole strength (with Δ𝐿 = 1) peaks at a larger scattering angle. Indeed a broad resonance

that peaks at about 19 MeV is visible at the largest scattering angles. This broad structure is due

to a combination of the IVSGDR (𝐽 𝜋 = 0−, 1−, 2−) and IVGDR 𝐽 𝜋 = 1−. Because the extracted

Δ𝐿 = 2 distribution also contains contributions from transitions with Δ𝐿 > 2, the interpretation of

this distribution is ambiguous. It doesn’t display a clear resonance like structure even though the

IVGQR and IVSGQR should contribute to the spectrum at high excitation energies. This is due to

the fact that the quasifree reactions contribute significantly to the Δ𝐿 = 2 response, as discussed

below. In the next sections, the responses will be examined in more detail.

70

Figure 5.2 Figures show the MDA fit for a few selected measured center-of-mass angular distribu-
tions in 0.04 MeV bins between 0 and 60 MeV in 60Cu. The experimental data are represented by
black dots, while the dashed red, blue, and yellow lines correspond to the Δ𝐿=0,1,2 components
respectively of each fit determined from the MDA. The green solid line is the sum of the three
MDA fit components. For more details, see the text.

71

Figure 5.3 𝐶𝑜𝑛𝑡 Figures show the MDA fit for a few selected measured center-of-mass angular
distributions in 0.04 MeV bins between 0 and 60 MeV in 60Cu. The experimental data are
represented by black dots, while the dashed red, blue, and yellow lines correspond to the Δ𝐿=0,1,2
components respectively of each fit determined from the MDA. The green solid line is the sum of
the three MDA fit components. For more details, see the text.

72

Figure 5.4 MDA results presented as the differential cross sections of 60Ni(3He,𝑡) reaction at 140
MeV/𝑢 for scattering angle bin from 0 up to 20 mrad angular distribution. Each 0.04 MeV energy
bin’s associated angular distribution was fit in the MDA with Δ𝐿 =0, 1 and 2 theoretical angular
distribution.

73

010203040500.00.51.0t,lab<5 mradL = 2L = 1L = 0data010203040500.00.51.05t,lab<10 mradL = 2L = 1L = 0data010203040500.00.51.010t,lab<15 mradL = 2L = 1L = 0data010203040500.00.51.015t,lab<20 mradL = 2L = 1L = 0datad/ddE [mb/(sr 40 keV)]Ex(60Cu) [MeV]Figure 5.5 MDA results presented as the differential cross sections of 60Ni(3He,𝑡) reaction at 140
MeV/𝑢 for scattering angle bin from 25 up to 40 mrad angular distribution. Each 0.04 MeV energy
bin’s associated angular distribution was fit in the MDA with Δ𝐿 =0, 1 and 2 theoretical angular
distribution.

74

010203040500.00.20.40.620t,lab<25 mradL = 2L = 1L = 0data010203040500.00.20.40.625<t,lab<30 mradL = 2L = 1L = 0data010203040500.00.20.430<t,lab<35 mradL = 2L = 1L = 0data010203040500.00.10.20.335<t,lab<40 mradL = 2L = 1L = 0datad/ddE [mb/(sr 40 keV)]Ex(60Cu) [MeV]5.3 Gamow-Teller Strength B(GT) Extraction

5.3.1 Extrapolation to q=0

Section 3.5 introduced a relationship between charge-exchange cross sections extracted to zero

momentum transfer and Gamow-Teller strength, B(GT), as shown in Eq. 3.16. The top four figures

in Figure 5.2 and Figure 5.4 indicate that the cross-section at forward angles is dominated by Δ𝐿 = 0

transitions, which enables the extraction of B(GT) as a function of excitation energy. Firstly, the

cross section needs to be extrapolated to zero linear momentum transfer (𝑞 → 0). For this purpose

the cross section at 0◦ for the Δ𝐿 = 0 component from the MDA was extracted from finite physical

Q value to 𝑄 = 0, using the DWBA calculations:

d𝜎
dΩ

(q = 0)

(cid:12)
(cid:12)
(cid:12)
(cid:12)ΔL=0

=

(cid:34) d𝜎
dΩ (Q = 0, 𝜃 = 0◦)
d𝜎
dΩ (Q = Q, 𝜃 = 0◦)

(cid:35)

×

(cid:20) d𝜎
dΩ

DWBA

(cid:21) ΔL=0

(𝜃 = 0◦, Q)

experiment

(5.1)

The DWBA scaling factor is displayed in Figure 5.6. Finally, after extracting the experimental

cross section at 𝑞 = 0 and by using the Gamow-Teller unit cross section ( ˆ𝜎𝐺𝑇 = 7.614, see Figure

3.9): the B(GT) was calculated by dividing the differential cross section at 𝑞 = 0 by the unit cross

section.

(cid:12)
(cid:12)
(cid:12)
(cid:12)ΔL=0

d𝜎
dΩ (q = 0)
ˆ𝜎GT

B(GT) =

(5.2)

A 10% uncertainty that scales all GT strengths attributed to the uncertainty in ˆ𝜎𝐺𝑇 was taken into

consideration in the further analysis [136, 19, 48]. Figure 5.7(a) presents the B(GT) distribution

extracted as a function of excitation energy, while Figure 5.7(b) shows the cumulative B(GT)

strength up to 20 MeV. In the subsequent section, the comparison with the shell-model calculation

for the same excitation energy range is discussed.

For excitation energies up to 20 MeV, the MDA analysis yielded a total GT strength of 10.6 ±

1.45 (stat) ± 1.19 (sys). At higher excitation energy additional strength is found that is associated

with Δ𝐿 = 0 and likely contains some additional GT strength. However, at high excitation energies

contributions from the isovector giant monopole resonances are expected to dominate the Δ𝐿 = 0

response.

75

Figure 5.6 The ratio of the differential cross sections at 𝜃 = 0◦ and at zero Q-value (𝑄 = 0) to that
of 𝜃 = 0 and 𝑄 = 𝑄𝑒𝑥 𝑝 as calculated in DWBA.

76

0510152025Q [MeV]1.001.251.501.752.002.252.50dd(Q=0,=0)dd(Q=Q,=0)Figure 5.7 Figure (a) shows the extracted B(GT) in 60Cu up to 20 MeV excitation energy, while
Figure (b) displays the cumulative B(GT) strengths after removing IAS. The uncertainties shown
are statistical. An additional uncertainty of 10% that scales all GT strengths is due to the uncertainty
in ˆ𝜎 [136, 19, 48].

77

05101520Ex[60Cu] [MeV]0.000.050.100.150.20B(GT)(a)05101520 Ex(60Cu) (MeV) 0.02.55.07.510.012.515.0B(GT)(b)5.3.2 Comparison with Theory, Shell-Model Calculations

The B(GT) distributions as a function of the excitation energy extracted from 60Ni(3He,𝑡) reac-

tion were compared to the strengths distributions calculated in shell-model calculations performed

in pf-shell-model space using the GXPF1A interaction [137]. Truncations in the shell-model

space were necessary to calculate the strength up to high excitation energy. The calculations were

performed with the code NushellX [117]. Three calculations were performed, one in which the

pf-shell-model space was not truncated, one in which at least 7 neutrons and protons were in 𝑓7/2

orbit (strong truncation), and one in which at least 6 neutrons and protons were in the 𝑓7/2 orbit

(mild truncation). Calculations with no truncation could be performed up to an excitation energy

of 6.5 MeV. Calculations with a mild truncation could be performed up to ∼ 15 MeV. Calculations

with a strong truncation were performed up to 20 MeV. All shell-model calculations were scaled by

a factor of 0.56 (see section 2.3.2) to account for a well-known quenching of the GT strength [138].

The comparisons of the experimental and theoretical GT strengths are shown in Figure 5.8 and

5.9. The theoretical strength distributions shown in Figure 5.8 match the overall structure of the

experimental spectra quite well. There are several relatively strength transitions at low excitation

energies and a broad distribution of strength between 5 and 15 MeV. On the other hand, it is not

possible to make a one-to-one correspondence between experimental and theoretical transitions

even at low excitation energies, and reducing the truncation does not provide a significantly better

result, even though the theoretical spectra change significantly with different levels of truncation.

Above 17 MeV more strength is found than in theory. Some of this strength might be attributed

to the quenched strength at low excitation energy. However, some of this strength could also be

due to the excitation of the isovector giant monopole resonances and to the quasifree-continuum as

discussed in later sections.

To compare the total strength, the cumulative strengths are plotted in Figure 5.9. The level of

truncation modifies the summed strength, and the total strength is lower with reduced truncation.

Slightly more strength is found in the experiment compared to the theoretical calculations. The

shell-model calculations plateau above 15 MeV, whereas some GT strength is observed. As

78

discussed above, this could be due to quenched strength at low excitation energies shifting to higher

excitation energies or due to misinterpreted isovector monopole strength or contributions from the

quasifree-continuum.

79

Figure 5.8 The comparison of the measured B(GT) strength and these from shell-model calculations.
Figure (a) shows the measured B(GT) strengths up to 20 MeV. Figure (b) to Figure (d) shows
theoretical shell-model calculations in black, green, and red for no truncation, mild truncation, and
strong truncation, respectively. Note that the IAS is visible in the extracted strengths from the
experiment, but does not appear in the theoretical calculations (Gamow-Teller only).

80

0.02.55.07.510.012.515.017.520.00.00.10.2IAS                   (a)Data0.02.55.07.510.012.515.017.520.00.00.10.2(b)No Truncation0.02.55.07.510.012.515.017.520.00.00.10.2(c)Mild Truncation 0.02.55.07.510.012.515.017.520.00.00.10.2(d)Strong TruncationB(GT)Ex(60Cu) [MeV]Figure 5.9 The extracted cumulative B(GT) distribution after removing IAS was compared to theo-
retical shell-model calculations depicted in blue, green, and red for no truncation, mild truncation,
and strong truncation, respectively. All shell-model calculations were scaled by a factor of 0.55
[138] to account for a well-known quenching of the GT strength.

81

05101520 Ex[60Cu] [MeV] 05101520B(GT)Strong truncationMild truncationNo truncationData B(GT)5.3.3 Extracted B(GT) at 14.4 MeV for 𝑇> State

Studying 𝑇0 + 1 excitations is of interest as these states are the analogs of states populated in the

60Ni(n,p) reaction, see Figure 2.3. These transitions can be used to estimate the electron-capture

rate in core-collapse supernovae [139] for which GT strengths in the 𝛽+ direction are important.

The extraction of the first 𝑇0 + 1 state, located at 𝐸𝑥 (60Cu) = 14.4 MeV was previously analyzed

through the 60Ni(p,n) reaction [139], see Figure 5.10. The extracted B(GT) for the 60Ni(n,p)

reaction after taking into consideration the isospin scaling factor as discussed in section 2.2 to

scale the strength from the 60Ni(p,n) reaction to 60Ni(n,p) reaction was 0.95 ± 0.15. The scaling

factor was 15 calculated by using the equation in section 2.2 [139]. A similar analysis but with the

(3He, 𝑡) reaction was performed in this thesis.

The process of extracting the B(GT) for the transition to the 𝑇0 + 1 state around 14.4 MeV in the

60Ni(3He,𝑡) reaction at 140 MeV/𝑢 is shown in Figure 5.11. Figure 5.11(a) illustrates the differential

cross section between 13.5 and 16 MeV at forward scattering angles emphasizing the prominent

𝑇0 + 1 state at an excitation energy of 𝐸𝑥 (60Cu) = 14.4 MeV. Figure 5.11(b) displays the fitted

data using a first-order polynomial background and a Gaussian function. Following background

subtraction, Figure 5.11(c) visualizes the resulting peak centered around 14.4 MeV. Subsequently,

a MDA was performed to extract the cross section at 0◦ and the B(GT) following the same process

as in section 5.3.1.

In order to convert the cross-section from data to B(GT), the cross sections were extrapolated

to 𝑞 = 0 momentum transfer and divided by unit cross-section ( ˆ𝜎𝐺𝑇 = 7.614).

In this work,

the extracted B(GT) of 𝑇0 + 1 state at 14.4 MeV is 0.86 ± 0.09 (stat.)±0.08 (syst.). The value is

consistent with the result from the 60Ni(p,n) experiment [139].

82

Figure 5.10 Spectra for 58,60,62,64Ni(p,n) reactions at 134.4 MeV, the 𝑇0 + 1 state in 60Ni nucleus is
shown at 14.4 MeV. Figure taken from [139].

83

Figure 5.11 Extracting B(GT) for the transition to the 𝑇0 + 1 state at 14.4 MeV in the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢 . Figure (a) depicts the differential cross section between 13.5 and 16 MeV.
Figure (b) shows the fitted peak, while Figure (c) displays the background subtracted peak centered
at 𝐸𝑥 = 14.4 MeV. Figure (d) presents the angular distribution and the MDA.

84

13.514.014.515.015.516.0Ex(60Cu) [MeV]0.1250.1500.1750.2000.2250.250d/d [mb/sr](a)13.514.014.515.015.516.0Ex(60Cu) [MeV]0.1250.1500.1750.2000.2250.250d/d [mb/sr](b)FitData14.3514.4014.4514.5014.55Ex(60Cu) [MeV]0.000.020.040.060.080.10d/d [mb/sr](c)10203040Center-of-Mass Angle [mrad]0.000.050.100.150.200.250.30d/d [mb/sr]L=0L=2SumData5.3.4 Comparison with Theory, (𝑝, 𝑝′) and (𝑒, 𝑒′)

In addition to the transition to the 𝑇0 + 1 state at 14.4 MeV, it is likely that are other GT

transitions to 𝑇0 + 1 states above 14.4 MeV. In this work, it was attempted to extract the strength

of these transitions by using a simple background subtraction, as shown in Figure 5.12. Figure

5.12(a) shows the differential cross section between 𝐸𝑥 = 13 MeV and 𝐸𝑥 = 20 MeV for scattering

angles below 5 mrad, where GT transitions are dominant. Besides the peak at 14.4 MeV, several

other weaker peaks are visible. A simple linear background was subtracted, as shown in Figure

5.12(b), where the red dashed lines indicates the estimated uncertainty in this background model.

The background-subtracted differential cross sections are shown in Figure 5.12(c). The extracted

B(GT) strengths are shown in Figure 5.12(d). To verify that these states are 𝑇0 + 1 states, the

excitation energies of the states were compared with known corresponding 𝑇0 + 1 states measured

in the 60Ni(𝑝, 𝑝′) and 60Ni(𝑒, 𝑒′) reactions [140, 141], these excitation energies are indicated in red

and green arrows, respectively, in Figure 5.12(d). Up to about 16.5 MeV, it appears that several

𝑇0 + 1 states can be identified. At higher excitation energies, no (𝑝, 𝑝′) or (𝑒, 𝑒′) data are available

and it cannot be proven that the peaks found are transitions to 𝑇0 + 1 states. The results were

compared with shell-model calculations with the GXPF1A interaction in the pf-shell model space

(no truncation) in the (n,p) direction. They are shown as the black dots in Figure 5.12(d).

Aside from the transition to the first 𝑇0 + 1 states, it is not possible to find a one-to-one

correspondence between states found in the experiment and states calculated in the shell-model. In

addition, more strength is found in the theoretical calculations than observed in the experiment.

85

Figure 5.12 Extraction of the B(GT) for the transitions to possible 𝑇0 + 1 states. Figure (a) illustrates
the differential cross sections between 13 and 20 MeV. Figure (b) shows the background estimation,
while Figure (c) shows the differential cross sections of the same spectrum after the background
has been subtracted. Figure (d) presents the extracted B(GT) from the measured in the 60Ni(3He,𝑡)
data in blue dots and shell-model calculations in the (n,p) direction in black dots. The location of
the states are compared with available data for the location of the 𝑇0 + 1 states in (𝑝, 𝑝′) in red and
(𝑒, 𝑒′) in green data [140, 141].

86

14161820Ex(60Cu) [MeV]0.140.160.180.200.22d/d [mb/sr](a)14161820Ex(60Cu) [MeV]0.140.160.180.200.22d/d [mb/sr](b)back groundData14161820Ex(60Cu) [MeV]0.000.020.040.060.08d/d [mb/sr](c)back groundData1415161718Ex [MeV] 0.000.250.500.751.00B(GT) (d)pp0ee0No truncation(3He,t)5.4 Background or quasifree-continuum Estimation

Extracting the Isovector Spin Giant Monopole Resonance (IVSGMR), Isovector Spin Giant

Dipole Resonance (IVSGDR), and other isovector giant resonances with higher multipolarities at

higher excitation energies is complicated by the presence of the quasifree-continuum in charge-

exchange reactions. As the excitation energy increases, the quasifree-continuum becomes more

pronounced, making it difficult to observe and analyze the resonant structures of interest. The

quasifree-continuum arises from non-resonant processes. At energies of an excess of 100 MeV/𝑢

the dominant component of the continuum are quasifree reactions in which the projectile interacts

with a single nucleon and the other nucleons in the target are spectators.

In quasifree reactions in the (3He,𝑡) reaction, the 3He projectile interacts with a single neutron

in the target, where the rest of the nucleus acts as a spectator, and the neutron behaves as a free

particle, except for its binding energy. This CE process involves the transformation of this neutron

into a proton, leading to its "knockout" from the nucleus. The process must occur above the

proton separation energy to facilitate this knockout. As the transferred energy increases, removing

neutrons from deeper shells with higher binding energies becomes possible. More details about

the quasifree-continuum origin can be found in Refs. [142, 143, 144, 145, 146, 147, 42, 145, 148].

The contribution of quasifree reactions is uncertain.

If quasifree reactions do not peak at

forward scattering angles, the MDA conveniently includes contributions from the continuum into

components associated Δ𝐿 ≥ 2, making the extraction of Δ𝐿 = 0 and Δ𝐿 = 1 strength straight-

forward. However, if the continuum contribution peaks at forward scattering angles, it will impact

the strength extraction of Δ𝐿 = 0 and Δ𝐿 = 1 resonances and it is better to remove the continuum

contribution before the MDA. In this work, we will compare the two approaches. The quasifree-

continuum has been estimated using a phenomenological approach.

Initially, Erell et al.

[42]

introduced this description for 𝜋 charge-exchange reactions, and later, it was also applied to (3He,𝑡)

reactions [145, 146, 147, 148]. For the case of (3He,𝑡) reaction, it has the following form:

𝑑2𝜎
𝑑Ω𝑑𝐸

= 𝑁 1 − 𝑒𝑥 𝑝 [(𝐸 − 𝐸0)/𝑇]
1 + [(𝐸 − 𝐸𝑄𝐹)/𝑊]2

,

(5.3)

where 𝐸𝑄𝐹 is the centroid energy of a Lorentzian distribution that undergoes a shift relative to

87

the energy of the free process, 𝐸𝑡 ( 𝑓 𝑟𝑒𝑒), due to various contributing factors. These factors include

the proton binding energy 𝑆 𝑝, the excitation energy 𝐸𝑥 of the neutron-hole state, and the Coulomb

barrier 𝐵𝐶𝑜𝑢𝑙 experienced by the proton:

𝐸𝑄𝐹 = 𝐸𝑡 ( 𝑓 𝑟𝑒𝑒) − (𝑆 𝑝 + 𝐸𝑥 + 𝐵𝐶𝑜𝑢𝑙),

(5.4)

The energy 𝐸𝑥 becomes zero only when the neutron is removed from the orbit closest to the Fermi

level. The width 𝑊 is attributed to the Fermi motion inside the nucleus, while the exponential term

arises from Pauli blocking effects. The cut-off energy 𝐸0 is defined by:

and 𝐸 is derived from 𝐸𝑥 by:

𝐸0 = 𝐸𝑡 (𝑔𝑠) − 𝑆 𝑝,

𝐸 = 𝐸 𝑝𝑟𝑜 𝑗 + 𝑄𝑔𝑠 − 𝐸𝑥

(5.5)

(5.6)

The parameter T is a temperature parameter. The normalization (N) is typically selected to

match the full cross section at high excitation energies, which likely leads to an overestimation

of the quasifree-continuum due to the contribution from non-quasifree reactions at high excitation

energies.

In this work, equation 5.3 was used to estimate the quasifree-continuum in spectra from

60Ni(3He,𝑡) reaction as shown in Figure 5.13 and Figure 5.14. The parameters in Eq. 5.3 used in

calculating the quasifree curves are described in Table 5.1. The value of the normalization factor

N as a function of the scattering angle is shown in Figure 5.15.

It was determined by scaling

the calculated quasifree curve to the experimental differential cross sections for excitation energies

above 55 MeV. N peaks at forward scattering angles, although the angular dependence is weak.

In order to extract monopole and dipole strengths from 60Ni(3He,𝑡) data, the estimated quasifree-

continuum was removed from data as shown in Figures 5.16 to 5.19. A comparison was made

between the MDA results for 1 MeV bins up through 60 MeV in excitation energy before and

after removing the quasifree-continuum in the 60Ni(3He,𝑡) data for a scattering angle bin ranging

from 0 to 40 mrad as shown in Figure 5.16 to 5.17. The top figure displays the MDA results

88

before removing the quasifree-continuum, whereas the bottom figure displays the MDA after the

quasifree-continuum removal within the same angular range. It is observed that, after the subtraction

of the quasifree-continuum, tails that extend to high excitation energies for Δ𝐿 = 0 and Δ𝐿 = 1

components are strongly reduced.

Figure 5.13 Estimating quasifree-continuum for each 5 mrad wide angular bin via the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢 for scattering angle bin from 5 to 20 mrad angular distribution. The black
dots in the figure represents the experimental data, while the blue dots illustrate the quasi-free curve
used for comparison.

89

02040 Ex (MeV) 02468d2/ddE (mb/sr MeV) 5 mradQF-continuumData02040 Ex (MeV) 02468d2/ddE (mb/sr MeV) 10 mradQF-continuumData02040 Ex (MeV) 02468d2/ddE (mb/sr MeV) 15 mradQF-continuumData02040 Ex (MeV) 0246d2/ddE (mb/sr MeV) 20 mradQF-continuumDataFigure 5.14 Estimating quasifree-continuum for each 5 mrad wide angular bin via the 60Ni(3He,𝑡)
reaction at 140 MeV/𝑢 for scattering angle bin from 25 to 40 mrad angular distribution. The black
dots in the figure represents the experimental data, while the blue dots illustrate the quasi-free curve
used for comparison.

90

02040 Ex (MeV) 0123456d2/ddE (mb/sr MeV) 25 mradQF-continuumData02040 Ex (MeV) 0123456d2/ddE (mb/sr MeV) 30 mradQF-continuumData02040 Ex (MeV) 0123456d2/ddE (mb/sr MeV) 35 mradQF-continuumData02040 Ex (MeV) 0123456d2/ddE (mb/sr MeV) 40 mradQF-continuumDataFigure 5.15 Normalizing parameters (N) from fit was used to match the full cross-section at high
excitation energies for each 5 mrad wide angular bin in (3He,𝑡) data.

Table 5.1 Parameters used in calculating the quasi-free curve/quasifree-continuum via the
60Ni(3He,𝑡) reaction at 140 MeV/𝑢.

Normalization, fit
Q value for 60Ni(3He,𝑡)
Q value for n(3He,𝑡)p
Beam energy (3He)

N
𝑄 (𝑔𝑠)
nQ
𝐸 𝑝𝑟𝑜 𝑗
𝐸𝑡 ( 𝑓 𝑟𝑒𝑒) Energy of the free triton, (cid:2)𝐸𝑡 ( 𝑓 𝑟𝑒𝑒) = 𝐸 𝑝𝑟𝑜 𝑗 + 𝑛𝑄(cid:3)
𝐸𝑡 (𝑔𝑠)
𝑆 𝑝
𝐵𝐶𝑜𝑢𝑙
𝐸𝑥
𝐸𝑄𝐹
𝐸0
𝑊
𝑇

Ground state energy of the triton, (cid:2)𝐸𝑡 (𝑔𝑠) = 𝐸 𝑝𝑟𝑜 𝑗 − 𝑄(cid:3)
Proton separation energy for 60Ni
Coulomb barrier for the proton
Excitation energy of the neutron hole state
Quasi-free energy, 𝐸𝑄𝐹 = 𝐸𝑡 ( 𝑓 𝑟𝑒𝑒) − (𝑆 𝑝 + 𝐸𝑥 + 𝐵𝐶𝑜𝑢𝑙)
𝐸𝑡 (𝑔𝑠) − 𝑆 𝑝
Width of resonance, fit
Temperature parameter

fit
-6.147
0.762 MeV
420 MeV
420.762 MeV
413.852
9.922 MeV
7.3 MeV
2 MeV
401.932 MeV
404.3196 MeV
fit
100

91

010203040 Scattering Angle () in mrad 051015202530 Scaling parameter Figure 5.16 The MDA results at 1 MeV bin from 60Ni(3He,𝑡) data for a scattering angle bin ranging
from 0 to 10 mrad. The top figure illustrates the MDA results before removing the quasifree-
continuum, while the bottom figure shows the same results after the quasifree-continuum removal
within the same angular range.

92

10203040500.02.55.07.5t,lab<5 mradL = 2L = 1L = 0data010203040500.02.55.07.5t,lab<5 mrad   QF continuum  subtractedL = 2L = 1L = 0data10203040500.02.55.07.55t,lab<10 mradL = 2L = 1L = 0data010203040500.02.55.07.55t,lab<10 mrad   QF continuum  subtractedL = 2L = 1L = 0datad/ddE [mb/(sr 1 MeV)]Ex(60Cu) [MeV]Figure 5.17 The MDA results at 1 MeV bin from 60Ni(3He,𝑡) data for a scattering angle bin ranging
from 10 to 20 mrad. The top figure illustrates the MDA results before removing the quasifree-
continuum, while the bottom figure shows the same results after the quasifree-continuum removal
within the same angular range.

93

10203040500.02.55.07.510t,lab<15 mradL = 2L = 1L = 0data01020304050024610t,lab<15 mrad   QF continuum  subtractedL = 2L = 1L = 0data1020304050024615t,lab<20 mradL = 2L = 1L = 0data01020304050024615t,lab<20 mrad   QF continuum  subtractedL = 2L = 1L = 0datad/ddE [mb/(sr 1 MeV)]Ex(60Cu) [MeV]d/ddE [mb/(sr 1 MeV)]Ex(60Cu) [MeV]Figure 5.18 The MDA results at 1 MeV bin from 60Ni(3He,𝑡) data for a scattering angle bin ranging
from 20 to 20 mrad. The top figure illustrates the MDA results before removing the quasifree-
continuum, while the bottom figure shows the same results after the quasifree-continuum removal
within the same angular range.

94

1020304050024620t,lab<25 mradL = 2L = 1L = 0data01020304050024620t,lab<25 mrad   QF continuum  subtractedL = 2L = 1L = 0data1020304050024625t,lab<30 mradL = 2L = 1L = 0data0102030405002425t,lab<30 mrad   QF continuum  subtractedL = 2L = 1L = 0datad/ddE [mb/(sr 1 MeV)]Ex(60Cu) [MeV]Figure 5.19 The MDA results at 1 MeV bin from 60Ni(3He,𝑡) data for a scattering angle bin ranging
from 30 to 40 mrad. The top figure illustrates the MDA results before removing the quasifree-
continuum, while the bottom figure shows the same results after the quasifree-continuum removal
within the same angular range.

95

1020304050024630<t,lab<35 mradL = 2L = 1L = 0data0102030405002430t,lab<35 mrad   QF continuum  subtractedL = 2L = 1L = 0data1020304050024635t,lab<40 mradL = 2L = 1L = 0data0102030405002435t,lab<40 mrad   QF continuum  subtractedL = 2L = 1L = 0datad/ddE [mb/(sr 1 MeV)]Ex(60Cu) [MeV]5.5 Extraction of the Isovector Spin Giant Monopole Strength

Due to the common transition of Δ𝐿 = 0 for the GT, IVGMR, and IVSGMR excitations, their

strength contributions cannot be distinguished solely through the MDA. Although some GT strength

can be situated at excitation energies above 20 MeV, it is of a non-resonant nature and is expected

to be weakly excited compared to the IVGMR and IVSGMR. In this work, separating the strength

contribution of the IVSGMR/IVGMR from that of GT transition was done by a simple assumption

that the cross sections associated with Δ𝐿 = 0 at low (≤ 20 MeV) and high (> 20 MeV) excitation

energies correspond to the excitations of GT states and the IVSGMR/IVGMR, respectively. The

extraction of the B(GT) strength was discussed in section 5.3.

To make an assessment of how much of the expected resonance strength for the IVSGMR/IVGMR

is found in the analysis, the extracted cross sections associated with Δ𝐿 = 0 in the MDA at 0◦

were compared with the calculated cross sections in DWBA with transition densities calculated
in normal modes (see section 3.4.2). Figure 5.20 shows the extracted ratio of 𝜎(0◦)(cid:12)
(cid:12)
𝜎(0◦)(cid:12)
(cid:12)

𝐷𝑊 𝐵𝐴
𝐼𝑉 𝑆𝐺 𝑀 𝑅/𝐼𝑉𝐺 𝑀 𝑅 for the MDA analysis performed before and after the removal of the quasifree-

𝑒𝑥 𝑝
𝑑𝐿=0 to

continuum contributions. Figure 5.20 also includes the summed fractions as a function of excitation

energy, which provides a measure of how much of the expected normal-modes strength is observed

in the analysis. Figure 5.20(a) and (b) show the extracted ratios after quasifree-continuum subtrac-

tion, while Figure 5.20(c) and (d) shows the same analysis before quasifree-continuum subtraction.

In the former case, monopole contribution at excitation energy between 20 to 50 MeV was found

to be 8% of the expected strength from the normal-modes calculation. It is therefore likely the

background subtraction removed a fraction of the isovector spin giant monopole strength.

In the latter case, the strength distribution shows no sign of resonance structure, and three

times the expected strength is extracted. This indicates that the removal of the quasifree-continuum

is important. We conclude that in order to extract strengths for the IVGMR and IVSGMR, an

experimental method for removing the quasifree-continuum is required. In the past, this has been

achieved by requiring a coincidence between the (3He,𝑡) and protons at backward angles [2, 34, 149].

The reactions that contribute to the quasifree-continuum are associated with a proton emitted at the

96

forward scattering angles. Protons emitted follow the excitation of the IVGMR/IVSGMR and are

distributed isotropically. Hence the coincidence with protons at backward angles provides a good

filter for reactions in which the giant resonances are excited.

𝐷𝑊 𝐵𝐴
Figure 5.20 Figure (a) and (b) shows the extracted ratios of 𝜎(0◦)(cid:12)
𝐼𝑉 𝑆𝐺 𝑀 𝑅,𝐼𝑉𝐺 𝑀 𝑅 from
(cid:12)
60Ni(3He,𝑡) reaction at 140 MeV/𝑢 after the quasifree-continuum subtraction. Conversely, Figures
(c) and (d) illustrate the same ratios derived from the original data without the quasifree-continuum
subtraction. In Figure (b) we observe that the anticipated resonance strength for IVSGMR/IVGMR
is 8% of the expected normal-modes strength, while in Figure (d) more than 300% of expected
normal-modes strength is observed.

𝑑𝐿=0 to 𝜎(0◦)(cid:12)
(cid:12)

𝑒𝑥 𝑝

97

20304050Ex(60Cu) (MeV)0.0000.0050.0100.015(0)|expL=0/(0)|DWBAIV(S)GMR(a)20304050 Ex(60Cu) (MeV) 0.000.020.040.060.080.100.12(0)|expL=0/(0)|DWBAIV(S)GMR(b)20304050Ex(60Cu) (MeV)0.000.050.100.150.200.250.30(0)expL=0/(0)DWBAIV(S)GMR(c)20304050 Ex(60Cu) (MeV) 0123(0)expL=0/(0)DWBAIV(S)GMR(d)5.6 Extraction of the Isovector Spin Giant Dipole Strength

As previously discussed in section 2.5, the IV(S)GDR consists of three components (0−, 1−,

and 2−). However, in the data, these components are not distinguishable from one another due to

their overlap and equal Δ𝐿 value and angular distribution. The summed strength peaks at around

18 MeV.

Here, we compared the differential cross sections extracted for Δ𝐿 = 1 transitions with the

DWBA calculations using the normal-mode framework. The comparison is performed for 35≤

Θ𝑡,𝑙𝑎𝑏 < 40 mrad, where the dipole transitions peak. As was done in the previous section 5.5 for

the IVGMR/IVSGMR, the results from the analysis with and without subtraction of the quasifree-

continuum were compared. The results are shown in Figure 5.21. With the removal of the

quasifree-continuum, a strong resonance is observed that peaks at 𝐸𝑥 = 18 MeV. The ratio between

the extracted and theoretical cross sections is about 1.25, suggesting that the quasifree-continuum

subtraction works quite well. Since, the Δ𝐿 = 1 excitations peak at backward angles, unlike

the quasifree-continuum, systematic uncertainties induced by the subtraction of the quasifree-

continuum appear less severe than for the monopole transitions. Without the subtraction of the

quasifree-continuum, a large amount of excess cross section above 𝐸𝑥 = 30 MeV is seen, indicating

that the subtraction of the quasifree-continuum is necessary.

98

Figure 5.21 The isovector spin giant dipole strength was extracted from the 60Ni(3He,𝑡) reaction at
140 MeV/𝑢. In Figure (a), we depict the ratios of 𝑑𝐿 = 1 components at 0◦ derived from both the
measured differential cross sections (MDA) and the theoretical calculation using DWBA. Figure
(b) showcases the cumulative ratio of the results presented in Figure (a). In contrast, Figures (c)
and (d) depict the identical ratios calculated from the original data without quasifree-continuum
subtraction.

99

01020304050Ex(60Cu) (MeV)0.000.020.040.060.08|expL=1/|DWBAIV(S)GDR(a)01020304050 Ex(60Cu) (MeV) 0.000.250.500.751.001.25|expL=1/|DWBAIV(S)GDR(b)01020304050Ex(60Cu) (MeV)0.000.020.040.060.080.10|expL=1/|DWBAIV(S)GDR(c)01020304050 Ex(60Cu) (MeV) 0123|expL=1/|DWBAIV(S)GDR(d)CHAPTER 6

CONCLUSION AND OUTLOOK

6.1 Summary

This dissertation focused on probing isovector giant resonances in 60Cu up to high excitation

energies (60 MeV) using the (3He,𝑡) reaction at 140 MeV/𝑢. The experiment was performed at the

Grand Raiden Spectrometer at Osaka University’s Research Center for Nuclear Physics (RCNP).

The method involved directing a 3He beam of 420 MeV with an intensity of 4 pnA onto a 60Ni foil

target of 2 mg/cm2. The tritons were identified and analyzed in the focal plane. From the measured

position and angles in the focal plane, the momentum vector of the triton was deduced, from which

the excitation energy in 60Cu and scattering angle were determined.

To investigate the Gamow-Teller strengths, the isobaric analog state, the isovector (spin)

monopole, and dipole giant resonances, a multiple decomposition analysis (MDA) was performed.

The differential cross sections were fitted with a linear combination of the angular distributions

associated with different angular momentum transfer Δ𝐿 (Δ𝐿 = 0, 1, 2, ...) calculated in distorted

wave Born approximation.

Different giant resonances were observed at different excitation energies. For the (3He,𝑡)

reaction at 140 MeV/𝑢, the GTR, IAS, and monopole excitations are associated with Δ𝐿 = 0, so

they peak at 0◦ scattering angle and their presence is strongly enhanced in the spectrum gated on

scattering angles between 0 and 5 mrad. Dipole excitations peak at small but finite angles and their

features are enhanced at scattering angles between 35 and 45 mrad. The IAS was found to reside at

Ex(60Cu) = 2.55 MeV, and the main component of the GTR appears at excitation energies of ∼ 10

MeV. The IVSGDR peaks at an excitation energy of about 18 MeV. The IVSGMR and IVGMR

were seen at the excitation energies of about 35 MeV and are not easy to identify due to their large

widths.

For excitation energies up to 20 MeV, the MDA analysis resulted in a total GT strength of 10.6 ±

1.45 (stat.) ± 1.19 (syst.). At higher excitation energy additional strength is found that is associated

with Δ𝐿 = 0 and likely contains additional GT strength. However, at high excitation energies

100

contributions from IVSGMR are also expected. The study compared the extracted Gamow-Teller

Strength B(GT) running sum with shell-model calculations using the GXPF1A interaction. Various

truncation levels were applied to calculate strength distributions, and all calculations were scaled

by a factor of 0.56 to address quenching effects. The comparison of the GT strengths revealed

reasonable agreement between experiment and shell-model calculations.

Gamow-Teller strengths for 𝑇0 + 1 states were extracted. The extracted B(GT) for the 𝑇0 + 1

state at 14.4 MeV was found to be 0.86 ± 0.09 (stat.) ± 0.08 (syst.), consistent with results from

the 60Ni(p,n) experiment. Additional GT transitions to 𝑇0 + 1 states above 14.4 MeV in 60Cu were

investigated. Several peaks, apart from the 14.4 MeV peak, were observed. To confirm that these

states are 𝑇0 + 1 states, their excitation energies were compared with known 𝑇0 + 1 states from

60Ni(𝑝, 𝑝′) and 60Ni(𝑒, 𝑒′) reactions. Up to about 16.5 MeV, multiple 𝑇0 + 1 states were identified.

However, beyond this energy, there is no available data to conclusively determine if the observed

peaks via 60Ni(3He,𝑡) data and shell-model calculations correspond to 𝑇0 + 1 states.

The contributions from the GT and IVGMR/IVSGMR excitations were extracted simultaneously

as all are characterized by Δ𝐿 = 0 transitions. While some GT strength likely appears above 20 MeV,

it lacks a resonant nature. To separate the IVSGMR/IVGMR from GT transitions, an assumption

was made that at low excitation energy (≤ 20 MeV) Δ𝐿 = 0 cross sections correspond to GT

states, Δ𝐿 = 0 strength at high excitation energies (> 20 MeV) is due to the excitation of the

IVSGMR/IVGMR. After quasifree-continuum subtraction, monopole contributions at 20-50 MeV

were found to be 8% of the normal-mode strength, indicating that the continuum removal also

removed a portion of the isovector spin giant monopole strength. Without the quasifree-continuum

subtraction, the strength distribution lacks resonance structure and is three times the expected

strength highlighting the importance of continuum removal. To obtain a better result for the

IVSGMR, an experimental technique to remove the continuum must be used. This is possible by

requiring coincidences with protons from the decay of the IVSGMR, as pursued in the experiments.

The contributions from the dipole resonances were extracted, and the differential cross sections

for Δ𝐿 = 1 transitions were compared with DWBA calculations using the normal-mode framework.

101

Similar to the analysis for the IVGMR/IVSGMR, results with and without quasifree-continuum

subtraction were compared. With continuum subtraction, a strong resonance around 𝐸𝑥 = 18 MeV

was observed, with a ratio of extracted to theoretical cross sections at about 1.25, suggesting that

the continuum subtraction is reasonable. Systematic uncertainties from continuum subtraction

are less severe for Δ𝐿 = 1 excitations due to their backward-angle peaking behavior, in contrast

to monopole transitions. Without continuum subtraction, significant excess cross section above

𝐸𝑥 = 30 MeV emphasizes the necessity of the subtraction.

102

6.2 Future Outlook

In this work, it has been shown that the (3He,𝑡) reaction at 140 MeV/𝑢 is a valuable tool for

extracting information about isovector giant resonances. However, the analysis is complicated by

the presence of the quasifree-continuum, and future studies with this probe would benefit from

the measurement of protons emitted from the excited nucleus. While the quasifree-continuum

is associated with protons directed at forward scattering angles, the decay after the excitation of

isovector monopole giant resonances is associated with protons emitted isotropically. Therefore

a coincidence measurement with protons at backward angles removes the contribution from the

continuum. Such experiments have been performed successfully in the past, but require thick

silicon detectors that are difficult to produce. It is well-kown that the extraction of the Gamow-

Teller strengths via the (3He,𝑡) reaction provides high-quality tests of theoretical models. In the

experiment studied here, the excitation energy resolution was modest (0.11 MeV) [FWHM]. By

applying the dispersion matching techniques [130], superior resolutions can be achieved (∼ 30 keV),

which would be helpful for the study of 𝑇0 + 1 Gamow-Teller states observed at high excitation

energies.

The CE group at FRIB is preparing for the next (3He,𝑡) experiment at RCNP. The target will be

92Zr and the goals will be to study transitions that are important for understanding the production

site of 92Nb in the universe. Due to its long lifetime (92Nb) is a potential cosmochronometer, but

the production site is still unknown, which makes its use as a cosmochronometer difficult. Many

of the techniques used in this thesis will also be used in the next experiment.

103

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Stephan, M. Boivin, and P. Radvanyi. The (3He,𝑡) reaction at intermediate energies: Spin-
isospin transitions to states in 12N and 13N. Nuc. Phys. A, 469(4):648–668, 1987.

[147] D. L. Prout, C. Zafiratos, T. N. Taddeucci, J. Ullmann, R. C. Byrd, T. A. Carey, P. Lisowski,
J. B. McClelland, L. J. Rybarcyk, W. Sailor, W. Amian, M. Braunstein, D. Lind, D. J. Mercer,
D. Cooper, S. DeLucia, B. Luther, D. G. Marchlenski, E. Sugarbaker, J. Rapaport, B. K.
Park, E. Gülmez, C. A. Whitten, C. D. Goodman, W. Huang, D. Ciskowski, and W. P. Alford.
Cross sections and analyzing powers for quasielastic scattering at 795 and 495 MeV using
the (p→,n) reaction. Phys. Rev. C, 52:228–242, 1995.

[148] K. Pham, J. Jänecke, D. A. Roberts, M. N. Harakeh, G. P. A. Berg, S. Chang, J. Liu, E. J.
Stephenson, B. F. Davis, H. Akimune, and M. Fujiwara. Fragmentation and splitting of
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C, 51:526–540, 1995.

[149] H. Akimune, I. Daito, Y. Fujita, M. Fujiwara, M. B. Greenfield, M. N. Harakeh, T. Inomata,
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proton decay from the gamow-teller resonance in 208Bi. Phys. Rev. C, 52:604–615, 1995.

117

APPENDIX A

FULL INPUT FILE IN THE WSAW CODE

Table A.1 The file with full input parameters in WSAW input program for 60Ni(3He,𝑡) reaction as
described in Table 3.1.

0.1 20.
Ni60Cu60
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
2.9
59.
4.8
59.
1.9

28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.

1

150

0

60.
1.
60.
1.
60.
3.
60.
3.
60.
5.
60.
5.
60.
0.
60.
2.
60.
2.
60.
4.
60.
4.
60.
1.
60.
1.
60.
3.

.65
2.
.65
2.
.65
1.
.65
1.
.65
0.
.65
0.
.65
2.
.65
1.
.65
1.
.65
0.
.65
0.
.65
1.
.65
1.
.65
0.

1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.

7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5

1.25
1.5
1.25
0.5
1.25
3.5
1.25
2.5
1.25
4.5
1.25
5.5
1.25
0.5
1.25
1.5
1.25
2.5
1.25
3.5
1.25
4.5
1.25
0.5
1.25
1.5
1.25
2.5

118

Table A.2 The file with full input parameters in WSAW input program continuation.

28.

28.

28.

28.

28.

28.
1.
28.

59.
8.6
59.
14.7 1.
59.
13.4 1.
59.
17.9 1.
59.
24.7 1.
59.
26.8 1.
59.
35.4 1.
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
3.0
59.
3.0
59.
1.0
59.
1.0
59.
1.0
59.
1.0
59.
5.4

28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.
28.
1.

60.
3.
60.
0.
60.
2.
60.
2.
60.
1.
60.
1.
60.
0.
60.
1.
60.
1.
60.
3.
60.
3.
60.
5.
60.
5.
60.
0.
60.
2.
60.
2.
60.
4.
60.
4.

.65
0.
.65
1.
.65
0.
.65
0.
.65
0.
.65
0.
.65
0.
.65
2.
.65
2.
.65
1.
.65
1.
.65
0.
.65
0.
.65
2.
.65
1.
.65
1.
.65
0.
.65
0.

1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.

7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5

1.25
3.5
1.25
0.5
1.25
1.5
1.25
2.5
1.25
0.5
1.25
1.5
1.25
0.5
1.25
1.5
1.25
0.5
1.25
3.5
1.25
2.5
1.25
4.5
1.25
5.5
1.25
0.5
1.25
1.5
1.25
2.5
1.25
3.5
1.25
4.5

119

Table A.3 The file with full input parameters in WSAW input program continuation.

28.
1.
28.

28.
1.
28.

28.
1.
28.

59.
8.3
59.
10.2 1.
59.
8.7
59.
15.4 1.
59.
20.
59.
20.1 1.
59.
24.6 1.
59.
30.7 1.
59.
32.9 1.
59.
40.

28.
1.

28.

28.

28.

60.
1.
60.
1.
60.
3.
60.
3.
60.
0.
60.
2.
60.
2.
60.
1.
60.
1.
60.
0.

.65
1.
.65
1.
.65
0.
.65
0.
.65
1.
.65
0.
.65
0.
.65
0.
.65
0.
.65
0.

1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.

7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5

1.25
0.5
1.25
1.5
1.25
2.5
1.25
3.5
1.25
0.5
1.25
1.5
1.25
2.5
1.25
0.5
1.25
1.5
1.25
0.5

120

APPENDIX B

INPUT AND OUTPUT FILES IN THE NORMOD CODE

Table B.1 The file with full input parameters in NORMOD input program for 60Ni(3He, 𝑡) reaction
as described in Table 3.2. The outputs are One-body transition densities (OBTDs).

1
14
1 1 3 1.0 1 3 10
0 3 5 1.0 1 3 11
1 1 1 1.0 1 3 12
0 4 9 1.0 1 4 13
0 4 7 1.0 1 4 14
1 2 5 1.0 1 4 15
1 2 3 1.0 1 4 16
2 0 1 1.0 1 4 17
0 5 11 1.0 1 5 18
0 5 9 1.0 1 5 19
1 3 7 1.0 1 5 20
1 3 5 1.0 1 5 21
2 1 3 1.0 1 5 22
2 1 1 1.0 1 5 23
11
0 0 1 1.0 0 0 1
0 1 3 1.0 0 1 2
0 1 1 1.0 0 1 3
0 2 5 1.0 0 2 4
1 0 1 1.0 0 2 5
0 2 3 1.0 0 2 6
0 3 7 1.0 0 3 7
1 1 3 1.0 0 3 8
0 3 5 0.0 0 3 9
1 1 1 0.0 0 3 10
0 4 9 0.0 0 4 11
0 0 1 60 1

1 0 0

121

Table B.2 The file with full input parameters in NORMOD input program.

0 1 1 60 1
0 0 -1 60 1
0 1 -1 60 1
0 2 -1 60 1
0 2 1 60 1
0 1 1 60 1
0 2 1 60 1
0 3 1 60 1
2 0 1 60 1
2 1 1 60 1
0 3 -1 60 1

1 0 0
1 1 1
1 1 1
1 1 1
1 0 2
1 0 2
1 0 2
1 0 2
1 2 2
1 2 2
1 1 1

Table B.3 The OBTDs for IAS , ΔJ = 0+. P stand for proton and H is hole (neutron). The NP stand
for principle quantum (n) number of a single-particle (proton), LP is the orbital quantum number
of particle, 2JP is twice total angular momentum of particle and LP, 2JH have the same meaning
but for hole (neutron). TYPE =1, for proton and 0 for neutron.

NP LP 2JP

TYPE

NH

LH

2JH

TYPE

1
0
1
0

1
3
1
4

3
5
1
9

1
1
1
1

1
0
1
0

1
3
1
4

3
5
1
9

0
0
0
0

S=0
L=0

-1.0000
-0.0000
-0.0000
-0.0000

S= 1
L=-1
0.0000
0.0000
0.0000
0.0000

S=1
L=0
0.0000
0.0000
0.0000
0.0000

S=1
L=1
0.0000
0.0000
0.0000
0.0000

Table B.4 The OBTDs for Gamow-Teller transition, ΔJ = 1+.

NP LP 2JP TYPE

NH

LH

2JH

TYPE

1
1
1
0
0
0
1
1
0
0

1
1
1
3
3
3
1
1
4
4

3
3
3
5
5
5
1
1
9
7

1
1
1
1
1
1
1
1
1
1

1
0
1
0
1
0
1
1
0
0

1
3
1
3
1
3
1
1
4
4

3
5
1
7
3
5
3
1
9
9

0
0
0
0
0
0
0
0
0
0

S=1
L=0

S=0
L=1
0.0000 -0.5092
0.0000 -0.0000
0.0000
0.0000
0.0000 -0.7303
0.0000 -0.0000
0.0000
0.0000
0.0000 -0.4554
0.0000
0.0000
0.0000 -0.0000
0.0000 -0.0000

S=1
L=2

S=1
L=1
0.0000 -0.2267
0.0000 -0.0000
0.0000 -0.0000
0.4064
0.0000
0.8481
0.0000
0.0000
0.0000
0.0000
0.2534
0.0000
0.0000
0.0000 -0.0000
0.0000
0.0000

122

Table B.5 The OBTDs for dipole transition, ΔJ = 0−.

NP LP 2JP

TYPE

NH

LH

2JH

TYPE

1
0
1
0
1
1
2
0

1
3
1
4
2
2
0
5

3
5
1
7
5
3
1
9

1
1
1
1
1
1
1
1

0
0
1
0
0
1
1
0

2
2
0
3
3
1
1
4

3
5
1
7
5
3
1
9

0
0
0
0
0
0
0
0

S=0
L=0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=-1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
S=1
L=1
L=0
0.0000
0.2236
0.0000 -0.5123
0.0000
0.2500
0.0000 -0.6708
0.0000
0.0000
0.0000 -0.4183
0.0000
0.0000
0.0000 -0.0000

Table B.6 The OBTDs for dipole transition, ΔJ = 1−, where values of states with ΔS = 1, ΔL = 1
and ΔS = 0, ΔL = 1 were used.

NP LP 2JP TYPE

NH

LH

2JH

TYPE

1
1
1
0
0
1
1
0
0
0
1
1
1
1
1
1
2
2
0
0
1

3
1
3
1
3
1
5
3
5
3
1
1
1
1
9
4
7
4
7
4
5
2
5
2
5
2
3
2
3
2
3
2
1
0
0
1
5 11
9
5
7
3

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
1
0
0
0
1
0
0
0
0
0
1
0
1
0
1
1
1
0
0
0

2
0
2
2
2
0
2
3
3
3
3
1
3
1
3
1
1
1
4
4
4

5
1
3
5
3
1
3
7
7
5
7
3
5
3
5
1
3
1
9
9
9

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=0
L=1
-0.2108
-0.2485
-0.0703
0.1054
-0.3944
0.1757
-0.1571
-0.6086
0.1029
-0.0000
-0.2520
-0.3944
-0.0000
0.1315
-0.0000
-0.0000
-0.2222
-0.0000
-0.0000
0.0000
-0.0000

S=1
L=1

S=1
L=0
0.0000 -0.1338
0.1577
0.0000
0.1784
0.0000
0.0000
0.4015
0.0000 -0.2504
0.2230
0.0000
0.0998
0.0000
0.0000
0.3863
0.5224
0.0000
0.0000 -0.0000
0.0000 -0.1600
0.2504
0.0000
0.0000
0.0000
0.3338
0.0000
0.0000
0.0000
0.0000 -0.0000
0.0000 -0.1411
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000 -0.0000

S=1
L=2
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

123

Table B.7 The OBTDs for dipole transition, ΔJ = 2− (ΔL = 1).

NP LP 2JP TYPE

NH

LH

2JH

TYPE

1
1
1
0
0
0
0
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
0
0
1
1

3
1
3
1
3
1
5
3
5
3
5
3
5
3
1
1
1
1
9
4
9
4
7
4
7
4
7
4
5
2
5
2
5
2
5
2
3
2
3
2
3
2
3
2
1
0
0
1
5 11
9
5
7
3
5
3

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
1
0
1
0
1
0
1
1
0
0
0
0
0

2
0
2
2
0
2
4
2
2
3
3
3
1
3
3
1
3
1
3
1
3
1
1
3
4
4
4
4

5
1
3
5
1
3
9
5
3
7
5
7
3
5
7
3
5
1
7
3
5
1
3
5
9
9
9
9

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=0
L=2

S=1
L=1

0.0000 -0.2024
0.0000 -0.3492
0.0000
0.0883
0.0000 -0.2164
0.0000 -0.0000
0.1623
0.0000
0.0000
0.0000
0.0000 -0.2164
0.0000
0.0442
0.0000 -0.4795
0.0000
0.0000
0.0000 -0.2892
0.0000
0.0000
0.0000
0.0000
0.0000 -0.2125
0.0000 -0.3787
0.0000
0.0000
0.0000
0.0000
0.0000 -0.2833
0.0000 -0.1653
0.0000
0.0000
0.0000
0.0000
0.0000 -0.3123
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.0000

S=1
L=2

S=1
L=3

0.0000 -0.0597
0.0000
0.0000
0.0000 -0.1367
0.2152
0.0000
0.3537
0.0000
0.0000
0.2870
0.0000 -0.0000
0.0558
0.0000
0.0000
0.2734
0.0000 -0.2120
0.0000 -0.0000
0.3515
0.0000
0.6213
0.0000
0.0000
0.0000
0.0000 -0.1252
0.0000 -0.1036
0.0000 -0.0000
0.0000 -0.0000
0.0939
0.0000
0.2374
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000 -0.0000
0.0000 -0.0000
0.0000
0.0000
0.0000 -0.0000
0.0000
0.0000

124

Table B.8 The OBTDs for the IVGQR and IVSGQR transitions with Δ𝐽 = 2+.

S=1
L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=2
0.0000
0.0844
0.2136
-0.0000
-0.0000
-0.0000
0.1724
-0.1290
-0.2997
-0.2180
-0.0000
0.0000
0.0844
-0.2403
-0.0000
0.2670
-0.0000
0.2643
-0.2266
-0.0000
-0.0000
0.1290
0.1090
0.0000
0.1090
0.1580
-0.0000
-0.0617
0.0755
0.3776
0.3708
-0.0000
-0.0000
0.2266
0.0000

S=1
L=3
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

NP LP 2JP TYPE NH

LH

2JH

TYPE

1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
2
2
0
0
0
1
1
1

3
1
3
1
3
1
3
1
3
1
3
1
5
3
5
3
5
3
5
3
5
3
5
3
1
1
1
1
1
1
9
4
9
4
7
4
7
4
7
4
5
2
5
2
5
2
5
2
3
2
3
2
3
2
1
0
1
0
5 11
9
5
9
5
7
3
7
3
7
3

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
0
0
1
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0

1
1
3
1
3
1
1
1
3
1
3
1
1
1
3
2
4
2
2
4
2
0
2
4
2
0
2
2
2
3
3
3
3
1
3

3
1
7
3
5
1
3
1
7
3
5
1
3
3
5
5
9
5
3
9
5
1
3
9
5
1
3
5
3
7
7
5
7
3
5

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=0
L=2
-0.0746
-0.0746
0.2830
0.2122
0.0000
0.0000
0.0913
-0.1709
-0.1134
-0.1155
0.0000
0.0000
0.0746
-0.2122
0.0000
-0.3537
0.0000
0.1001
-0.3002
-0.0000
-0.1155
-0.1709
-0.0578
0.0000
0.0578
0.1395
-0.0882
-0.0817
-0.0667
-0.5003
0.1092
-0.0000
-0.1544
-0.3002
-0.0000

125

Table B.9 The OBTDs for the IVGQR and IVSGQR transitions with Δ𝐽 = 2+ continuation.

1
1
1
1
2
2
2
2
2
2

3
3
3
3
1
1
1
1
1
1

5
5
5
5
3
3
3
3
1
1

1
1
1
1
1
1
1
1
1
1

0
1
0
1
0
1
0
1
1
0

3
1
3
1
3
1
3
1
1
3

7
3
5
1
7
3
5
1
3
5

0
0
0
0
0
0
0
0
0
0

0.0535
0.1225
-0.0000
-0.0000
-0.1070
-0.1248
-0.0000
-0.0000
0.1248
-0.0000

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

0.1413
0.2312
0.0000
-0.0000
-0.0807
0.0000
0.0000
0.0000
0.1413
0.0000

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

126

Table B.10 The OBTDs for the IVSGQR transition, ΔJ = 1+.

NP LP 2JP TYPE NH

LH 2JH

TYPE

1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
1
1
1
1
1
2
2
0
1
1
1
1
1
2
2
2
2
2

1
1
1
1
1
3
3
3
3
1
1
1
1
4
4
4
2
2
2
2
2
0
0
5
3
3
3
3
3
1
1
1
1
1

3
3
3
3
3
5
5
5
5
1
1
1
1
9
7
7
5
5
3
3
3
1
1
9
7
7
5
5
5
3
3
3
1
1

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
0
1
0
1
0
0
1
0
0
0
1
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
1
1
1

1
1
1
3
1
1
3
1
3
1
1
1
1
4
2
4
2
2
2
0
2
0
2
3
3
3
3
1
3
1
3
1
1
1

3
1
3
5
1
3
7
3
5
3
1
3
1
9
5
9
5
3
5
1
3
1
3
7
7
5
7
3
5
3
5
1
3
1

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=1
L=0
-0.0000
0.0000
-0.5092
-0.0000
0.0000
0.0000
-0.7303
-0.0000
0.0000
-0.0000
0.0000
-0.4554
0.0000
-0.0000
-0.0000
-0.0000
0.0000
-0.0000
0.0000
-0.0000
-0.0000
0.0000
-0.0000
0.0000
0.0000
-0.0000
0.0000
0.0000
-0.0000
-0.0000
0.0000
0.0000
-0.0000
0.0000

S=1
L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=2
0.0314
0.0351
-0.0894
-0.0000
-0.0000
-0.2644
0.1603
0.3345
0.0000
-0.0351
-0.0993
0.0999
0.0000
-0.0000
-0.4240
0.0000
0.0596
0.0557
-0.0557
-0.1971
-0.1115
0.0000
0.0942
-0.6051
0.0872
0.0000
-0.0755
-0.3547
-0.0000
0.0526
0.0000
0.0000
-0.0588
-0.0000

S=0
L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

127

Table B.11 The OBTDs for IVSGQR transition, ΔJ = 3+ (ΔL = 2).

NP LP 2JP TYPE NH

LH

2JH TYPE

1
1
1
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
1
1
1
1
1
2
0
0
0
0
0
1
1
1

3
1
3
1
3
1
3
1
5
3
5
3
5
3
5
3
5
3
5
3
1
1
1
1
9
4
9
4
9
4
7
4
7
4
7
4
7
4
5
2
5
2
5
2
5
2
3
2
3
2
3
2
0
1
5 11
5 11
9
5
9
5
9
5
7
3
7
3
7
3

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
0
1
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0

1
3
1
3
1
1
3
1
3
1
3
3
2
2
4
2
0
2
4
2
0
2
4
2
2
4
2
3
3
3
1
3
3
1
3

3
7
3
5
3
1
7
3
5
1
7
5
5
3
9
5
1
3
9
5
1
3
9
5
3
9
5
7
5
7
3
5
7
3
5

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=1
L=2

-0.1321
0.2445
0.3758
-0.0000
-0.0576
0.0322
0.2264
0.0729
-0.0000
-0.0000
0.2823
-0.0000
-0.2617
0.3866
0.0000
-0.0998
0.0000
0.0865
0.0000
-0.1339
-0.2255
0.0729
0.0000
-0.0729
0.0223
0.0000
-0.1078
-0.3401
0.0000
-0.1461
-0.0000
0.0000
-0.1533
-0.2594
0.0000

S=1
L=4

S=1
L=3
0.0000 -0.0000
0.0541
0.0000
0.0000
0.0000
0.0000
0.0000
0.0956
0.0000
0.0000
0.1710
0.0000 -0.1377
0.0000 -0.1814
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.0468
0.0000 -0.0000
0.0000 -0.0772
0.0000 -0.0522
0.0000
0.0000
0.2023
0.0000
0.3258
0.0000
0.0000
0.2337
0.0000 -0.0000
0.0000 -0.0395
0.0000 -0.0000
0.0000 -0.0725
0.0000
0.0000
0.0725
0.0000
0.2369
0.0000
0.0000 -0.0000
0.0000
0.0000
0.0000 -0.1504
0.0000 -0.0000
0.3501
0.0000
0.6238
0.0000
0.0000
0.0000
0.0000 -0.0904
0.0000 -0.0722
0.0000 -0.0000

S=0
L=3
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

128

Table B.12 The OBTDs for IVSGQR transition, ΔJ = 3+ (ΔL = 2) Continuation.

1
1
2
2
2

3
3
1
1
1

5
5
3
3
1

1
1
1
1
1

0
1
0
1
0

3
1
3
1
3

7
3
7
3
7

0
0
0
0
0

0.0000
0.0000
0.0000
0.0000
0.0000

-0.1067
-0.0773
-0.0924
-0.2210
-0.1067

0.1298
0.0000
0.0000
0.2422
0.0000 -0.0409
0.0000 -0.0000
0.0354
0.0000

Table B.13 The OBTDs for IVGMR transition, ΔJ = 0+, where only values of state with ΔS = 0
and ΔL = 0 was used.

NP LP 2JP TYPE NH

LH

2JH TYPE

1
1
1
1
2
1
1
2
2

1
1
2
2
0
3
3
1
1

3
1
5
3
1
7
5
3
1

1
1
1
1
1
1
1
1
1

0
0
0
0
1
0
0
1
1

1
1
2
2
0
3
3
1
1

3
1
5
3
1
7
5
3
1

0
0
0
0
0
0
0
0
0

S=0
L=0
0.2840
0.2008
0.4115
0.3360
0.2840
0.5388
0.0000
0.4752
0.0000

S=1
L=-1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

129

Table B.14 The OBTDs for IVSGMR transition, ΔJ = 1+.

NP LP 2JP TYPE NH

LH

2JH

TYPE

1
1
0
1
1
0
1
1
1
1
1
2
2
0
1
1
1
1
1
2
2
2
2
2

1
1
3
1
1
4
2
2
2
2
2
0
0
5
3
3
3
3
3
1
1
1
1
1

3
3
5
1
1
7
5
5
3
3
3
1
1
9
7
7
5
5
5
3
3
3
1
1

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
1
1
1

1
1
1
1
1
2
2
2
2
0
2
0
2
3
3
3
3
1
3
1
3
1
1
1

3
1
3
3
1
5
5
3
5
1
3
1
3
7
7
5
7
3
5
3
5
1
3
1

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=1
S=0
L=0
L=1
0.0000
0.2117
0.0000 -0.1893
0.0000
0.0000
0.0000
0.1893
0.0000 -0.0669
0.0000 -0.0000
0.0000
0.2811
0.0000 -0.3005
0.0000
0.3005
0.0000 -0.0000
0.0000 -0.1503
0.0000
0.2840
0.0000 -0.0000
0.0000
0.0000
0.0000
0.3527
0.0000 -0.0000
0.4073
0.0000
0.0000
0.0000
0.0000 -0.0000
0.3542
0.0000
0.0000
0.0000
0.0000 -0.0000
0.3168
0.0000
0.0000 -0.0000

S=1
L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

S=1
L=2
0.0350
0.0392
-0.1896
-0.0392
-0.1108
-0.3715
0.0854
0.0799
-0.0799
-0.2355
-0.1599
0.0000
0.1576
-0.6267
0.1529
0.0000
-0.1324
-0.4804
-0.0000
0.0921
0.0000
0.0000
-0.1030
-0.0000

130

Table B.15 The OBTDs for the IVSGOR and IVGOR with ΔJ = 3−.

NP LP 2JP TYPE

NH

LH

2JH

TYPE

1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
1
1
1
2
2
0
0
1
1
2

3
1
3
1
3
1
5
3
5
3
5
3
5
3
1
1
9
4
9
4
9
4
7
4
7
4
7
4
7
4
5
2
5
2
5
2
5
2
3
2
3
2
3
2
1
0
0
1
5 11
9
5
7
3
5
3
3
1

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

0
0
0
0
1
0
0
0
0
1
0
0
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0

2
2
4
2
0
2
4
2
3
1
3
3
1
3
1
3
1
3
1
3
1
3
3
3
4
4
4
4
4

5
3
9
5
1
3
9
5
7
3
5
7
3
5
1
7
3
5
1
7
3
5
7
5
9
9
9
9
9

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

S=0
L=3
0.1606
0.1967
-0.0000
-0.1577
-0.2277
0.1932
0.0000
-0.1796
0.4037
0.4554
0.0000
-0.1805
-0.2037
0.0000
0.0000
0.2666
0.2791
0.0000
0.0000
-0.1539
-0.3418
0.0000
0.2352
0.0000
0.0000
-0.0000
0.0000
-0.0000
0.0000

S=1
S=1
L=3
L=2
0.0000
0.0466
0.0000 -0.2281
0.0000 -0.0000
0.0000 -0.2744
0.0000 -0.2640
0.0560
0.0000
0.0000
0.0000
0.0000 -0.2083
0.0000 -0.1170
0.0000 -0.3960
0.0000 -0.0000
0.0000 -0.4186
0.0000 -0.3542
0.0000
0.0000
0.0000
0.0000
0.0773
0.0000
0.0000 -0.0809
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.2678
0.0000 -0.3963
0.0000 -0.0000
0.0000
0.2045
0.0000 -0.0000
0.0000 -0.0000
0.0000 -0.0000
0.0000
0.0000
0.0000 -0.0000
0.0000
0.0000

S=1
L=4
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

131

APPENDIX C

FULL INPUT FILES IN THE FOLD CODE

Table C.1 The file with full input parameters in FOLD input program for GT transition as described
in Table 3.3.

1

1FOLDNIB

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

+0.5

0.5

-0.5

0.000
1

0.0

3
1
-1

1.0+

0.0+

0.707

1.0
3
8
9
7

-1

3
8
10
8

-1

Ni60Cu60
0.939

1.0

2.0

2.0

0.000
1
1
1

0.0
0.0
0.0

-0.5092
-0.7303
-0.4554

2.650

1.000

love_140

2
0
1.00
1.00
2
1.00
1.00

1

1

1
1.00
1.00
1
1.00
1.00

-1

-1

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

132

Table C.2 The file with full input parameters in FOLD input program for IAS as described in Table
3.3.

1

1FOLDNIA

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

2.0
3
8

-1

+0.5

0.5

-0.5

0.000
0

0.0

3
1
-1

0.707

0.0+

0.0+

1.0

2.0

2.0

3
8

0.000
0

0.0

-1

-1.0

Ni60Cu60
0.939

1
0
1.00
1.00

2.650

1.000

love_140

-1

0

0
1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

133

Table C.3 The file with full input parameters in the FOLD input program for IVSGDR with
ΔJ = 1, ΔS = 1 as described in Table 3.3.

1

1FOLDNIE

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

1.0
3
8
8
8
9
9
7
7
15
14
13
13
12
11

-1

+0.5

0.5

-0.5

0.000
1

0.0

3
1
-1

0.707

1.0-

0.0+

1.0

2.0

2.0

0.000
1
1
1
1
1
1
1
1
1
1
1
1
1

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

3
6
4
5
6
5
4
5
10
10
10
8
8
8

-1

-0.1338
0.1577
0.1784
0.4015
-0.2504
0.2230
0.0998
0.3863
0.5224
-0.1600
0.2504
0.3338
-0.1411

Ni60Cu60
0.939

1
1
1.00
1.00

2.650

1.000

love_140

-1

1

1
1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

134

Table C.4 The file with full input parameters in FOLD input program for IVSGQR with ΔJ = 3 as
described in Table 3.3.

1

1FOLDNIM

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

1.0
3
8
8
8
9
9
9
9
7
15
15
14
14
13
13
13
12
.
.
-1

+0.5

0.5

-0.5

0.000
1

0.0

3
1
-1

0.707

3.0+

0.0+

1.0

2.0

2.0

0.000
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
.
.

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
.
.

3
3
10
8
3
2
10
8
10
6
5
6
5
6
4
5
6
.
.

-1

-0.1321
0.2445
0.3758
-0.0576
0.0322
0.2264
0.0729
0.2823
-0.2617
0.3866
-0.0998
0.0865
-0.1339
-0.2255
0.0729
-0.0729
.
.

Ni60Cu60
0.939

1
2
1.00
1.00

2.650

1.000

love_140

-1

1

3
1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

135

Table C.5 The file with full input parameters in FOLD input program for IVGMR as described in
Table 3.3.

1

1FOLDNIN

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

1.0
3
8
7
13
12
11
19
17

-1

+0.5

0.5

-0.5

0.000
0

0.0

3
1
-1

0.707

0.0+

0.0+

1.0

2.0

2.0

0.000
0
0
0
0
0
0
0

0.0
0.0
0.0
0.0
0.0
0.0
0.0

3
3
2
6
5
4
10
8

-1

0.2840
0.2008
0.4115
0.3360
0.2840
0.5388
0.4752

Ni60Cu60
0.939

1
0
1.00
1.00

2.650

1.000

love_140

-1

0

0
1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

136

Table C.6 The file with full input parameters in FOLD input program for IVSGMR as described in
Table 3.3.

1

1FOLDNIO

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

1.0
3
8
8
7
7
13
13
12
12
11
19
18
17
16

-1

+0.5

0.5

-0.5

0.000
1

0.0

3
1
-1

0.707

1.0+

0.0+

1.0

2.0

2.0

0.000
1
1
1
1
1
1
1
1
1
1
1
1
1

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

3
3
2
3
2
6
5
6
5
4
10
10
8
8

-1

0.2117
-0.1893
0.1893
-0.0669
0.2811
-0.3005
0.3005
-0.1503
0.2840
0.3527
0.4073
0.3542
0.3168

Ni60Cu60
0.939

2
0
1.00
1.00
2
1.00
1.00

2.650

1.000

love_140

1

1

1
1.00
1.00
1
1.00
1.00

-1

-1

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

137

Table C.7 The file with full input parameters in FOLD input program for IVSGOR with
ΔJ = 3, ΔS = 1 transition as described in Table 3.3.

1

1FOLDNIP

600 0.03

0.5+

420.

0.5+

3.

1

1

1

0.5
3
1
-1
HE3H3

1.0
3
8
8
9
9
9
7
15
15
14
14
13
13
12
12
11

-1

+0.5

0.5

-0.5

0.000
1

0.0

3
1
-1

0.707

3.0-

0.0+

1.0

2.0

2.0

0.000
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

3
6
5
6
4
5
6
10
8
10
8
10
8
10
8
10

-1

0.0466
-0.2281
-0.2744
-0.2640
0.0560
-0.2083
-0.1170
-0.3960
-0.4186
-0.3542
0.0773
-0.0809
-0.2678
-0.3963
0.2045

Ni60Cu60
0.939

1
3
1.00
1.00

2.650

1.000

love_140

-1

1

3
1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

1.00
1.00

138

APPENDIX D

FULL INPUT FILES IN THE DWHI CODE

Table D.1 The file with full input parameters in DWHI input program for IAS transition as described
in Table 3.6.

0

1

0.

1
600

1210000041000000
FOLDNIA
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

2.

3.
-35.16 1.32

1.

3.
-29.89 1.32

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
0

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25

1.

0.

-44.43 1.021

1.018

0.

0.

1.25

1.

0.

-37.77 1.021

1.018

0.

0.

0 0

0.

0
0.

IAS.plot

0.

1.

Table D.2 The file with full input parameters in DWHI input program for GT transition as described
in Table 3.3.

0

1

0.

1
600

210000041000000
FOLDNIB
40.
160 2
0.03
420.
1.
0.
-6.2
1.
0.

2.

3.
-35.16 1.32

1.

3.
-29.89 1.32

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
2

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25

1.

0.

-44.43 1.021

1.018

0.

0.

1.25

1.

0.

-37.77 1.021

1.018

0.

0.

0 2

0.

2 2

0.

2
0.
2
0.

GTA.plot

0.

0.

1.

1.

139

Table D.3 The file with full
ΔJ = 0, ΔS = 0 transition as described in Table 3.3.

input parameters in DWHI input program for IVGMR with

0

1

0.

1
600

1210000041000000
FOLDNIN
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

2.

3.

1.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
0

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

0 0

0
0.
IVGMR.plot

0.

0.

1.

Table D.4 The file with full input parameters in DWHI input program for IVSGMR with ΔJ = 1,
ΔS = 1 as described in Table 3.3.

1

0

0.

1
600

1210000041000000
FOLDNIO
40.
160 2
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

3.

1.

2.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
2

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

0 2

0.

2 2

0.

2
0.
2
0.

IVSGMRa.plot

0.

0.

1.

1.

140

Table D.5 The file with full input parameters in DWHI input program for IVSGDR with ΔJ = 0,
ΔS = 1 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNID
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

3.

3.

1.

2.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
0

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

1 2

0.

0
0.

DT0.plot

0.

1.

Table D.6 The file with full input parameters in DWHI input program for IVSGDR with ΔJ = 1,
ΔS = 1 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNIE
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

1.

3.

3.

2.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
2

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

1 2

0.

2
0.

DT1A.plot

0.

1.

141

Table D.7 The file with full input parameters in DWHI input program for IVSGDR with ΔJ = 1,
ΔS = 0 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNIF
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

2.

3.

1.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
2

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

1 0

0.

2
0.

DT1B.plot

0.

1.

Table D.8 The file with full input parameters in DWHI input program for IVSGDR with ΔJ = 2,
ΔS = 1 as described in Table 3.6.

0

1

0.

1
600

1210000041000000
FOLDNIZ
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

3.

2.

3.

1.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
4

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

1 2

0.

4
0.

DT2.plot

0.

1.

142

Table D.9 The file with full input parameters in DWHI input program for IVSGQR with ΔJ = 1,
ΔS = 1 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNIJ
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

2.

3.

3.

1.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
2

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

2 2

0.

2
0.

IVSGQR1.plot

0.

1.

Table D.10 The file with full input parameters in DWHI input program for IVSGQR with ΔJ = 2,
ΔS = 1 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNIK
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

1.

2.

3.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
4

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

2 2

0.

4
0.

IVSGQR2a.plot

0.

1.

143

Table D.11 The file with full input parameters in DWHI input program for IVSGQR with ΔJ = 2,
ΔS = 0 as described in Table 3.6.

0

1

0.

1
600

1210000041000000
FOLDNIG
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

2.

3.

1.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
4

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

2 0

0.

4
0.

IVGQRa.plot

0.

1.

Table D.12 The file with full input parameters in DWHI input program for IVSGQR with ΔJ = 3,
ΔS = 1 as described in Table 3.6.

1

0

0.

1
600

1210000041000000
FOLDNIM
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-35.16

-29.89

1.

3.

3.

2.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
6

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

2 2

0.

6
0.

IVSGQR3.plot

0.

1.

144

Table D.13 The file with full input parameters in DWHI input program for IVGOR with ΔJ = 3,
ΔS = 0 as described in Table 3.6.

1

0

0.

1
600

1220000041000000
FOLDNIL
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

2.

3.

1.

3.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
6

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

3 0

0.

6
0.

0.

1.

Octupole3b.plot

Table D.14 The file with full input parameters in DWHI input program for IVSGOR with ΔJ = 3,
ΔS = 1 as described in Table 3.6.

0

1

0.

1
600

1220000041000000
FOLDNIP
40.
160 1
0.03
420.
1.
0.
-6.2
1.
0.

-29.89

-35.16

3.

3.

2.

1.

MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTL

0.2
6

60.

0.84

60.

0.84

28.
0.

29.
0.

1.25
-44.43 1.021

1.

0.
1.018

0.

0.

1.25
-37.77 1.021

1.

0.
1.018

0.

0.

1.32

1.32

3 2

0.

6
0.

0.

1.

Octupole3a.plot

145

(E.1)

(E.2)

(E.3)

APPENDIX E

PROPORTIONALITY RELATION BETWEEN 𝑑𝑝 AND 𝑑𝑘

We can show that

𝑑𝑝

𝑝 ≈ 𝑑𝑘

2𝑘 : where 𝑝 is the momentum and 𝑘 is kinetic energy.

√

For 𝑝 =

𝐸 2 − 𝑚2 and 𝐸 = 𝑘 + 𝑚

𝑝 =

√︁(𝑘 + 𝑚)2 − 𝑚2 =

√︁𝑘 2 + 2𝑘𝑚

If m >> k, then: 𝑝 ≈

√

2𝑘𝑚 =⇒ 𝑚 =

𝑝2
2𝑘

𝑑𝑝
𝑑𝑘

=

2𝑚
√
2𝑘𝑚

2

=

√

𝑚
2𝑘𝑚

=

𝑝2
2𝑘 𝑝

=

𝑝
2𝑘

𝑑𝑝
𝑝

=

𝑑𝑘
2𝑘

146