CONSTRAINING ELECTRON-CAPTURE RATES USING (𝑑,2He) REACTION IN INVERSE
KINEMATICS

By

Zarif Mubassir Rahman

A DISSERTATION

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

Physics—Doctor of Philosophy

2025

ABSTRACT

This thesis describes the use of the (𝑑,2 He) reaction in inverse kinematics with rare-isotope beams

to constrain electron-capture rates, which are important for accurately simulating the heating and

cooling processes in neutron star crusts. Electron-capture rates are calculated from Gamow-Teller

(GT) strengths, which can be extracted experimentally in a model-independent way using charge-

exchange reactions at intermediate energies. Following a successful pilot study with the 14O(𝑑,2 He)

and 13N(𝑑,2 He) reactions in inverse kinematics using the Active-Target Time Projection Chamber

with the S800 spectrometer at the Facility of Rare Isotope Beam, a second experiment was performed

to investigate the 33Al(𝑑,2 He) and 32Mg(𝑑,2 He) reactions using an identical setup.

The nuclei 33Al and 32Mg are located in or near the 𝑁 = 20 "island of inversion," a region where

shell evolution leads to the inversion of the 𝑠𝑑 and 𝑝 𝑓 shells. These nuclei play an important role

in understanding the Urca mechanism in neutron star crusts, where rapid cycles between electron

capture and 𝛽− decay result in significant cooling due to strong neutrino emission. This thesis

presents the experimental setup, data analysis, and results for the extracted GT strengths from these

reactions. GT strengths were extracted for one state in the 33Al(𝑑,2 He) reaction and two states

in the 32Mg(𝑑,2 He) reaction. The results provide evidence that the assumption of a closed-shell

structure is invalid for 33Al and 32Mg. Although the uncertainties in the extracted GT strengths are

large, there is reasonable agreement with shell-model calculations that include contributions from

both the 𝑠𝑑- and 𝑝 𝑓 -shell configurations.

In addition to the experimental work, weak interaction rates and their impact on neutron

star crusts were examined for nuclei in the 𝑠𝑑 and 𝑝 𝑓 shells, for which precise shell-model

calculations are already established. Neutron star crust simulations utilized three different sets of

weak interaction rates: (1) rates derived solely from Quasiparticle Random Phase Approximation

(QRPA) calculations, (2) shell-model rates for 𝑠𝑑 and 𝑝 𝑓 shell nuclei, combined with QRPA rates

for all other nuclei, and (3) experimental data for 𝑠𝑑 and 𝑝 𝑓 shells when available, supplemented by

shell-model calculations, with QRPA rates used for all other nuclei. The inclusion of experimental

data for weak interaction rates leads to a suppression of both heating and cooling effects, and the

results align more closely with the rates from shell-model calculations than with those from QRPA

calculations.

The results from this work underscore the importance of precise weak interaction rates for

realistic neutron star modeling and demonstrate the effectiveness of the (𝑑,2 He) reaction in inverse

kinematics as a tool for extracting essential nuclear data.

Copyright by
ZARIF MUBASSIR RAHMAN
2025

Thank you, Remco and Simon.

ACKNOWLEDGEMENTS

v

TABLE OF CONTENTS

CHAPTER 1

BACKGROUND AND INTRODUCTION . . . . . . . . . . . . . . . .
1
1.1 Neutron Stars .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
6
1.2 The Island of Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Electron-Capture Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Charge-Exchange Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

.

.

.

CHAPTER 2

THEORETICAL CALCULATION . . . . . . . . . . . . . . . . . . . . 20
2.1 Shell-Model .
. 21
2.2 Normal-Mode calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 CCBA calculations

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

.

.

.

.

.

.

CHAPTER 3

EXPERIMENTAL SETUP AND PROPERTIES . . . . . . . . . . . . . 30
. 30
3.1 Beam Production .
3.2 S800 Magnetic Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . 35
3.3 Active-Target Time Projection Chamber
. 40
3.4 Magnetic Rigidity Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

.

.

CHAPTER 4

.

.

.

. .

DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
. 42
4.1 S800 CRDC Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Beam Gate . .
. 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Event Reconstruction in the AT-TPC . . . . . . . . . . . . . . . . . . . . . . . 49
. 58
4.5 Drift Velocity Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Simulation .
4.7 Kinematic Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8 Event Selection . .
. 75
4.9 Excitation Energy Spectra and Differential Cross-Sections
. . . . . . . . . .
. 80
4.10 Multipole Decomposition Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.11 B(GT) Extraction .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.12 B(GT) Comparison with Theory . . . . . . . . . . . . . . . . . . . . . . . . . 89

.

.

.

.

.

.

.

.

CHAPTER 5

NEUTRON STAR CRUST SIMULATION . . . . . . . . . . . . . . . . 91
5.1 Using the Neutron Star Crust Simulation Code . . . . . . . . . . . . . . . . . . 94
5.2 Calculation at 𝑇 = 0.5 GK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Urca Pairs .

.

.

.

.

.

.

CHAPTER 6

CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . 111

BIBLIOGRAPHY . .

. .

.

.

.

.

.

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

. 114

vi

CHAPTER 1

BACKGROUND AND INTRODUCTION

1.1 Neutron Stars

Neutron stars stand out as the smallest and densest known stellar entities in our universe [1, 2].

Baade and Zwicky first proposed in 1934 that a neutron star could arise from the aftermath of a

massive star (≥ 8𝑀⊙) explosion [3]. As a massive star exhausts its fuel generated through fusion

in its core, it collapses under its own gravitational pressure and explodes into a supernova [4]. The

cataclysmic eruption leaves behind a sphere of extremely dense matter. These remnants form what

we call neutron stars, which typically have radii on the order of 10 km and masses roughly between

1 and 2 solar masses [5, 6]. The neutron star is so dense that the atomic electrons collapse from

their quantum orbits into the depths of the atomic nuclei fusing with protons to form neutrons. The

neutrons are tightly packed in these conditions resulting in a high neutron degeneracy pressure due

to the Pauli exclusion principle, along with the strong nuclear force, together counteracting gravity

and supporting the neutron star against collapsing further [7].

Neutron stars can be divided into five major regions: atmosphere, ocean, crust, mantle and

core. Fig. 1.1 shows different layers of a neutron star with approximate depth of each layer. The

atmosphere is composed of normal atoms, but the temperature is not high enough for nuclear

reactions to take place. The ocean, consisting of free electrons and ionized nuclei, is situated

beneath the atmosphere and plays a role in transportation and release of thermal energy from the

surface of the star. The atmosphere and the ocean are collectively referred to as the envelope.

The crust forms the next region of the neutron star and is made of primarily ionized nuclei and

electrons. The crust is sometimes divided into inner crust and outer crust based on the density. The

composition of the core is not well established, but it is speculated that the outer core is formed

of nucleons, electrons, and muons whereas the inner core possibly contains exotic particles such

as hyperons, pions, or kaons [8]. Presence of a quark condensate has also been suggested to be

possibly found in the center of a neutron star [1].

1

Figure 1.1 Regions of a neutron star with their approximate depths. Note that the depths of different
layers are not drawn to scale.

Neutron stars have been observed both in isolation and in binary systems. The companion star

can be any star, a black hole, or another neutron star. When the neutron star is in a binary system

with a white dwarf or an evolved star- such as a giant - it may accrete mass. The mass transfer results

in highly variable X-ray emission [2]. Continual accretion can result in recurrent X-ray bursts that

are roughly periodic with rapidly sharp increases in luminosity, peaking in the X-ray regime of the

electromagnetic spectrum. There are also soft X-ray transients which emit thermal X-rays for some

time but then turn off for extended period of times [9]. Instabilities in the accretion process are

suggested as a reason for such emission. On the extreme end, there can also be superbursts which

are long and more energetic X-ray bursts [10]. The surface temperature of these accreting neutron

stars can be deduced from the observation of these X-ray photons.

2

AtmosphereOceanCrustCore~10 m~1 m~1 km~10 kmEnvelopeContinuous accretion of matter onto a neutron star fuels thermonuclear burning on its surface.

The ashes from these surface reactions are gradually compressed into the crust. At shallow depths

in the crust, the Fermi energy (𝐸𝐹) is smaller than the absolute value of the electron-capture (EC)

𝑄−value, |𝑄𝐸𝐶|, as illustrated by Fig. 1.2(a). In this region, the electron capture is energetically

suppressed [11].

Figure 1.2 (a) Electron capture is suppressed at shallow depths because the reaction Q-value is
(b) At greater depths, the higher Fermi energy allows
higher than the available Fermi energy.
electron capture, but the reverse 𝛽− decay is not possible. If the Fermi energy is high enough,
electron capture can also occur to excited states, which then deexcite by emitting 𝛾-rays that locally
heat the crust.

At greater depths, where 𝐸𝐹 is significantly higher than |𝑄𝐸𝐶 | (as shown in Fig. 1.2(b)), EC has

occurred, but the phase space for electron emission via 𝛽− decay is blocked by degenerate electrons.

If 𝐸𝐹 > |𝑄𝐸𝐶 | + 𝐸𝑥, the first EC transition will occur for an excited state in the daughter nucleus.

𝐸𝑥 is the excitation energy of the lowest state in the daughter nucleus for which EC is allowed. The

subsequent de-excitation of this nucleus via 𝛾-ray emission deposits energy locally, contributing to

3

ExEx(Z,A)|QEC|EF(Z-1,A)Energy𝛾(Z,A)|QEC|EFEC(Z-1,A)Energy(a)(b)heating. Due to odd-even staggering in nuclear binding energy, odd-odd nuclei have lower |𝑄𝐸𝐶 |

values than even-even nuclei. Consequently, EC often proceeds in pairs for even-mass nuclei, as

illustrated in Fig. 1.3. When 𝐸𝐹 greatly exceeds the 𝑄𝐸𝐶 of the reaction for the odd-odd nucleus

in the second step, the energy difference is split between the escaping neutrino and heat deposition

into the crust [12]. These two-step EC processes are significant sources of heat in the neutron star

crust.

Figure 1.3 Two-step electron captures occur due to odd-even staggering in nuclear binding energy.
Both reactions can proceed through excited states in the daughter nuclei, which deexcite by de-
positing heat locally. This makes two-step electron captures a significant heat source in neutron
star crusts.

When 𝐸𝐹 ≈ |𝑄𝐸𝐶 |, 𝛽− decay is not completely blocked at finite temperature. Therefore, both

electron-capture and 𝛽− decay can cycle back-and-forth via:

𝐴
𝑍 𝑋 + 𝑒−− > 𝐴

𝑍−1 𝑌 + 𝜈𝑒

𝐴

𝑍−1𝑌 − > 𝐴

𝑍 𝑋 + 𝑒− + 𝜈𝑒

(1.1)

(1.2)

Both of these reactions emit neutrinos that carry away heat energy from the crust. The rapid cycling

between EC and 𝛽− decay, known as the Urca process [13], serves as a cooling mechanism for

the star. The two nuclei involved in this process are known as the Urca pair. Fig. 1.4 illustrates

4

ExEx𝛾EC𝛾EC(Z,A)(Z-1,A)(Z-2,A)EFEF|QEC||QEC|Energyschematic of the Urca process, where EC reaction occurs to the ground state of the daughter nucleus.

However, if the 𝐸𝐹 is sufficiently high, EC can also occur to a low-lying excited state.

Figure 1.4 The Urca process occurs through cyclic electron capture and 𝛽− decay at finite temper-
ature, where the Fermi energy is close to the reaction Q-value. Both processes emit neutrinos, and
their strong emission significantly contributes to neutron star crust cooling.

As EC in even-mass nuclei generally proceed in steps of two due to the odd-even staggering in

the nuclear binding energy, the Urca process is inhibited. As a result, Urca cooling pairs are mostly

formed by odd-mass nuclei as odd-even and even-odd nuclei allow for EC and 𝛽− decay cycles at

each step.

Heating and cooling in the neutron stars are governed by neutron-rich rare isotopes.

It is

important to have accurate EC rates for these isotopes in the neutron star crust simulation to

determine the sources that cause change in heat energy. Nuclei in the 𝑁 = 20 region, commonly

known as the "island of inversion" are of particular interest for these simulations and calculations.

Schatz et al.

[11] used a full reaction network that included both EC and 𝛽− decay rates to

determine the Urca pairs that may play a significant role in the cooling of neutron stars. The

reaction network incorporated weak interaction rates based on the Quasiparticle Random Phase

Approximation (QRPA) formalism of Ref. [14]. However, Cole et al. [15] have shown that these

QRPA calculations are not well-suited to accurately predict transitions to individual low-lying final

states. In contrast, shell-model calculations can provide more reliable results.

In this thesis, we build on the work of Cole et al., by using shell-model weak interaction

5

|QEC|EFEC(Z,A)(Z-1,A)Energyβ¯rates for 𝑠𝑑- and 𝑝 𝑓 - shell nuclei in a neutron star crust simulation and compare results with the

previous model. In addition, we extend our study to the so-called 𝑁 = 20 "island of inversion" by

experimentally studying two reactions: 33Al(𝑑,2He)33Mg and 32Mg(𝑑,2He)32Na and constraining

EC rates in this region. The results are used to provide information for the development of new

theoretical models and for understanding the uncertainties in state-of-the-art shell-model calculation

models.

1.2 The Island of Inversion

The well-established nuclear shell model is a theoretical framework that aims to describe

the nuclear structure in terms of shells much like the electron shells in atomic physics [16]. One

fundamental difference between the atomic and nuclear shell model is that in an atom, the negatively

charged electrons are bound by the Coulomb potential of the positively charged nucleus. However, in

the nuclear shell model, the mean-field approximation describes the motion of individual nucleons

as if they move independently within an average potential, which accounts for the collective

influence of all other nucleons in the nucleus.

The foundation of the shell model comes from the observation of quantum magic numbers of

nucleons, a phenomenon where certain numbers of nucleons are more tightly bound [17]. Relatively

large energy gaps appear after the orbitals corresponding to the conventional magic numbers - 2,

8, 20, 28, 50, 82, and 126, as shown in Fig. 1.5 - for stable nuclei. However, far from the valley of

stability, the shell structure evolves as neutrons are added to isotopic chains, sometimes leading to

a reduction in the shell gap or inversion of shells.

6

Figure 1.5 Low-lying energy levels shown for a single-particle shell model.

It has been well established that in the 𝑁 = 20 region for 𝑍 = 10 − 12, as a result of the

shell evolution, shell model orbitals can switch order for certain nuclei. The first observation of

this phenomenon came from the mass measurement of 31Na [18], which suggested the possibility

of a breakdown in the magic number structure at 𝑁 = 20. Theoretical calculations using both

Hartree-Fock [19] and the shell model [20, 21, 22] reproduced these results by allowing neutron

excitations from the 𝑠𝑑 shell to the 𝑝 𝑓 shell, leading to a deformed ground-state configuration.

Mass measurements in the 𝑁 = 28 region have shown a similar pattern [23], where deformation

of the ground state has also been predicted. Regions in the Segrè chart, a graphical representation

of isotopes in a two-dimensional grid, where orbital inversions occur are referred to as islands of

7

N = 0s (l = 0)N = 1p (l = 1)N = 2d (l = 2)s (l = 0)N = 3f (l = 3)p (l = 1)N = 4g (l = 4)d (l = 2)s (l = 0)1s 1/2  [2]1p 3/2  [4]1p 1/2  [2]1d 5/2  [6]1d 3/2  [4]2s 1/2  [2]1f 7/2   [8]2p 3/2  [4]1f 5/2   [8]2p 1/2  [2]1g 9/2  [10]1g 7/2  [8]2d 5/2  [6]2d 3/2  [4]3s 1/2  [2]{50}{28}{20}{8}{2}MajorOscillatorShelll = N, N-2, …Spin-OrbitCoupling[No. of nucleonsin shell]MagicNumberinversion. Fig. 1.6 shows the island of inversion in the 𝑁 = 20 region in the Segrè chart. Note that

the shores of the island of inversion are not firm; some of the surrounding nuclei are also impacted

by the breaking of the 𝑁 = 20 shell.

Figure 1.6 Island of inversion in the 𝑁 = 20 region represented by the orange boxes in the Segrè
chart.

The inversion occurs due to the properties of the nucleon-nucleon (NN) interaction, which

can be further decomposed into three main terms: monopole, pairing, and all other components

[24]. The largest contribution to the inversion of shells comes from the monopole component of

the NN interaction induced by the tensor force. The monopole interaction induces shifts in the

orbital energies, thereby triggering alterations in shell configurations. The energy shifts create

varying energy gaps as 𝑍 and 𝑁 change, with large energy gaps leading to the emergence of new

magic numbers while potentially causing the disappearance of some traditional magic numbers.

The pairing component involves interactions between protons and neutrons with total angular

8

MgAlSiPNaNeF10121410121416182022Proton Number (Z)Neutron Number (N)Stable NucleiUnstable NucleiNuclei confirmed to be in the N=20 Island of Inversion Regionmomentum 𝐽 𝜋 = 0+. All other terms account for remaining contributions e.g., the quadrupole

interaction. Fig. 1.7 shows the contribution to ground-state energy of Mg isotopes with even

number 𝑁. Mg istopes with 𝑁 = 8 − 20 were calculated using shell model in the 𝑠𝑑 model space

and for 32Mg, the 𝑠𝑑 − 𝑝 𝑓 shell model space was also used. The USD Hamiltonian interaction is

used for up to 𝑁 = 18 and SDPF-M interaction was used for 𝑁 = 20. Two calculations are shown

for the latter: a full calculation and one in which excitations from the 𝑠𝑑 to 𝑝 𝑓 shell are switched

off. It can be clearly seen that the monopole contribution is larger for 𝑁 = 20 compared to other

neutron numbers. Fig. 1.7 shows the ground-state expectation values of these components for the

Mg isotopes with even 𝑁 values ranging from 8 to 20, calculated using a shell model in the 𝑠𝑑 shell

for 20−30Mg, and in the 𝑠𝑑 − 𝑝 𝑓 shell for 32Mg.

Figure 1.7 Contribution from NN interaction and single-particle energy (SPE) shown for the ground-
state energy of Mg isotopes with even N. The NN interaction includes monopole, pairing, and other
effects. The other effects are dominated by quadrupole interaction. Two calculations are shown for
𝑁 = 20 - a full calculation with excitations across the gap to the 𝑝 𝑓 shell and calculation without
any such excitations. The calculation without across-gap excitations has a positive value for all
other terms because of a repulsive contribution. Figure is taken from Ref. [25].

Several studies [26, 27, 28, 29, 30, 31, 32, 33] have reported presence of shape deformation

9

around the 𝑁 = 20 region for Z ≤ 12. Ref. [34] proposed 0p-0h and 2p-2h mixed configuration for

33Al ground state based on shell-model calculations. Fig. 1.8(a) shows the “normal” configuration

for 33Al, which is a 𝑁 = 20 nucleus. The normal configuration would mean that the 𝑠𝑑 shell

is full. However, in an “intruder" 2p-2h configuration, a pair of neutrons in the 𝑁 = 20 shell

closure is located in the pf shell, as illustrated in Fig. 1.8(b). Shell-model calculation shows that

for many nuclei in this region, the intruder 2p-2h configuration has lower energies than normal

configuration for nuclei with 20 ≤ N ≤ 22 and 10 ≤ Z ≤ 12. In fact, for 32Mg, a sudden inversion

has experimentally been observed between the states where the 2p-2h configuration is the ground

state [35]. The inclusion of the 𝑝 𝑓 shell extends the model beyond the closed shell structure,

allowing the consideration of higher energy orbitals. This permits the possibility of higher-order

excitations and interactions, which can enable Gamow-Teller (GT) transitions that were previously

blocked in the closed-shell assumption.

Figure 1.8 (a) The 0p-0h configuration for 33Al, where no nucleons are excited across the energy gap
(represented by the dashed line) between the 𝑠𝑑 and 𝑝 𝑓 shells. In this configuration, all nucleons
remain in the 𝑠𝑑-shell orbitals. (b) The 2p-2h configuration, where two nucleons from the 𝑠𝑑-shell
are excited into the 𝑝 𝑓 -shell, creating two particle-hole excitations.
In this case, the shell gap
has been reduced, making the 2p-2h configuration more energetically favorable. Although such
configurations are typically associated with excited states, for certain nuclei in the 𝑁 = 20 region,
the 2p-2h configuration has been observed as the ground state.

The ground state of 33Al is suggested to have a spin-parity of (5/2)+ [36]. An allowed

GT transition to 33Mg would indicate that the 33Al ground state does not exhibit a closed-shell

10

pn0d5/21s1/20d3/2(a)(b)0f7/2pn0f7/20d3/21s1/20d5/233Al 0p0h configuration33Al 2p2h configurationstructure and shows evidence of being in the 𝑁 = 20 island of inversion. The first study of 33Mg

was conducted in 1984 through the 𝛽− decay of 33Na [37], but a detailed level scheme was not

established until a 2001 investigation [38]. Despite multiple efforts to determine the spin-parity of

the 33Mg ground state, it remains a subject of debate. A positive-parity state in 33Mg could allow a

GT transition from the ground state of 33Al. Current evidence from hyperfine and nuclear magnetic

resonances along with knockout reactions suggests that the spin-parity of the 33Mg ground state is

most likely 3/2− [39, 40, 41]. However, the low log 𝑓 𝑡 value of 5.6 established based on the study

of 𝛽− decay of 33Mg suggested the possibility of a GT transition to a positive-parity (5/2)+ ground

state [34]. A state at 484 keV had been assigned a spin-parity of (5/2)− based on band assignments

and Coulomb excitation data [42], but has been suggested to be (3/2, 5/2)+ based on knockout

reactions [41]. Theoretical calculations in Refs. [38] and [43] predict a spin-parity of (3/2, 5/2)+

for a state at 705 keV, making it a candidate for a GT transition from the 33Al ground state. In

addition, Ref. [41] proposed spin-parities of (1/2, 3/2)− for the states at 1.24 MeV and 1.85 MeV.

A 2p-2h configuration has been experimentally observed for the 32Mg ground state [35], and

its spin-parity has been established as 0+ [44]. While earlier studies suggested a spin-parity of

(3,4)− for the 32Na ground state [44], this assignment remains uncertain. Additionally, the excited

states in 32Na remain experimentally unconfirmed. The present study aims to provide new insights

into these excited states, contributing to a deeper understanding of nuclear structure in the 𝑁 = 20

region.

1.3 Electron-Capture Rates

Electron capture (EC) is a fundamental nuclear process mediated by the weak nuclear force. In

an EC reaction, a nucleus captures an electron converting a proton into a neutron and produces an

electron neutrino in the process. Through the process, the atomic number of the nucleus is reduced

by 1, but the mass number is conserved. ECs play a vital role in various astrophysical processes,

including the heating and cooling within neutron star crusts, as discussed earlier.

It also plays

an important role in the core-collapse phase of massive stars prior to a supernovae [45, 46, 47],

final evolution of ONeMg cores in intermediate-mass stars [48, 49], nucleosynthesis of Type 1a

11

supernovae, and many more [50, 51]. Several nuclei relevant to these astrophysical phenomena

are neutron-rich and have negative Q-values, making direct study of these nuclei through EC

experiments impossible. In addition, stellar ECs are different from laboratory EC as a result of

the high-temperature and/or high-density environment and, therefore, cannot usually be accessed

through experiments directly. Additionally, in high-temperature stellar environments, transitions

from excited states are possible. GT distributions obtained through experiments in the laboratory

are typically limited to transitions from the ground state of the mother nucleus only. At relatively

low temperatures and densities, EC rates are dominated by allowed GT transitions. At higher

densities and temperatures, forbidden transitions also contribute [45, 46, 52, 53].

The following equation can be used to calculate EC rates [45]:

𝜆𝐸𝐶 = 𝑙𝑛(2)

∑︁

𝑖 𝑗

𝑓𝑖 𝑗 (𝑇, 𝜌, 𝑈𝐹)𝐵(𝐺𝑇)𝑖 𝑗 ,

(1.3)

where 𝑓 (𝑇, 𝜌, 𝑈𝐹) is the phase-space factor, which depends on the density (𝜌), temperature (𝑇),

and chemical potential (𝑈𝐹) and B(GT) is the GT strength. The rate is obtained by summing over

the B(GT) distribution and is weighted by the phase-space factor. ( 𝑓 𝑡)𝑖 𝑗 is related to B(GT) through

the following relationship:

𝐾
𝑔2
𝐴
where 𝑡 is half-life of the transition 𝑖 → 𝑗, K is the weak interactions constant, 𝑔𝐴 is the axial-vector

1
B(GT)𝑖 𝑗

( 𝑓 𝑡)𝑖 𝑗 =

(1.4)

coupling constant. 𝐾 has the value of 6144 ± 2 s and 𝑔𝐴 has the value of 1.2694 ± 0.0028 [54].

Additionally, ground-state ( 𝑓 𝑡)𝑖 𝑗 in the EC direction can be calculated from the ground-state ( 𝑓 𝑡)𝑖 𝑗

in the 𝛽− direction using the following relationship:

( 𝑓 𝑡)EC =

( 𝑓 𝑡)𝛽
2𝐽𝛽 + 1

· (2𝐽EC + 1)

(1.5)

where 𝐽𝐸𝐶 and 𝐽𝛽 are the spins for parent ground states.

For electrons, the phase space integral takes the following form [45]:

12

∑︁

𝑖 𝑗

𝑓𝑖 𝑗 (𝑇, 𝜌, 𝑈𝐹) =

∫ ∞

𝑤𝑙

𝑤2(𝑞𝑅 + 𝑤)2𝑆−(𝑇, 𝑈𝐹 (𝜌))𝑑𝑤,

(1.6)

where 𝑤 = 𝜖𝑒/𝑚𝑒𝑐2 and 𝜖𝑒 is total energy of the electron. The EC threshold energy is 𝑤𝑙 = |𝑞𝑔.𝑠 |

with 𝑞𝑔.𝑠. = 𝑄𝑔.𝑠./𝑚𝑒𝑐2, where 𝑄𝑔.𝑠. = 𝑚( 𝐴, 𝑍, 𝑋) − 𝑚( 𝐴, 𝑍 − 1, 𝑌 ) − 𝑚𝑒𝑐2 and 𝑚𝑁 refers to the

nuclear mass excess. 𝑞𝑅 is obtained in the units of 𝑚𝑒 from Ref. [45]:

𝑞𝑅 = 𝑄 𝑅/𝑚𝑒𝑐2 = [𝐸𝑥 𝑗 − 𝐸𝑥𝑖 − 𝑄𝑔.𝑠.]/𝑚𝑒𝑐2,

𝑆− is the Fermi-Dirac distribution function and can be given as the following:

𝑆− =

(cid:18)

exp

(cid:19)

(cid:18)𝑈 − 𝑈𝐹
𝑘𝑇

(cid:19) −1

+ 1

(1.7)

(1.8)

where 𝑈 is the kinetic energy and 𝑘 is the Boltzmann constant. Since direct study of neutron-

rich nuclei through EC is not feasible, an indirect method must be employed to investigate the GT

transitions of excited states.

1.4 Charge-Exchange Reactions

Charge-exchange (CE) reactions provide an indirect method for extracting B(GT)s. Charge

is transferred between the projectile and target, i.e. a neutron can be converted into a proton in

the daughter nucleus, or vice versa. At intermediate beam energies (𝐸𝑏𝑒𝑎𝑚 ≳ 100 MeV/A), CE

reactions proceed through a single-step, direct reaction mediated by the strong nuclear force. As

discussed below, by using CE reactions, GT transition strengths can be extracted, from which the

EC rates are calculated (see Eq. 1.3).

CE reactions do not have a 𝑄-value limitation, unlike EC or 𝛽− decay, and they connect the

same initial and final states as these processes. CE reactions in the (𝑛, 𝑝) direction will involve the

same parent and daughter nuclei as EC, as shown in Fig. 1.9. These reactions also share similar spin

and isospin transfer operators [55]. In combination with a well established proportionality between

B(GT) and CE cross sections [56] (see Sec. 1.4.1), CE reactions provide a valuable alternative to

EC and 𝛽− decay studies for the purpose of extracting B(GT). They reveal important information

13

on transitions to states at excitation energies inaccessible through EC/𝛽− decay studies, which is

important for astrophysical applications.

Figure 1.9 Electron-capture and charge-exchange reactions are mediated by the weak force and
(a) In the charge-
strong force, respectively, but they connect the same initial and final states.
exchange reaction shown using a (𝑑,2He) probe, a deuteron beam is used on a target resulting in
recoiling nucleus with one less proton but same mass number.
(b) An electron is captured by
a nucleus and a proton is converted to a neutron releasing a neutrino and the mass number is
conserved again.

CE reactions are characterized by an isospin transfer and are isovector transitions (Δ𝑇 = 1).

Fermi (Δ𝑆 = 0) and GT (Δ𝑆 =1) transitions are differentiated by spin transfer and both can be

studied with CE reactions. Both transitions see no change in angular momentum (Δ𝐿 = 0). Our

interest lies in calculating EC rates that are dominated by GT transitions, characterized by the

B(GT). Therefore, this work will focus only on GT transitions and the associated strength, B(GT).

1.4.1 Gamow-Teller Strength

The B(GT) can be experimentally extracted for both the (𝑛, 𝑝) and ( 𝑝, 𝑛) directions. GT

transitions are described by the following operator:

𝑂 (𝐺𝑇) = 𝑔𝐴

∑︁

𝑗

(cid:174)𝜎𝑗 𝜏± 𝑗 ,

(1.9)

where 𝜎 represents the Pauli-spin matrix, and 𝜏 is the isospin transition matrix [55]. The following

equation shows the relationship between the operator and the B(GT):

𝐵(𝐺𝑇)± =

1
2𝐽 + 1

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

∑︁

𝑗

⟨𝜓𝐹 ||𝜎𝑗 𝜏±

𝑗 ||𝜓𝐼⟩

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

(1.10)

14

(a) Charge-Exchange Reaction(b) Electron Capturewhere 𝜓𝐼 and 𝜓𝐹 are the wave functions of the initial and final nuclear states, respectively, and 𝐽 is

the total angular momentum of initial state.

As mentioned previously, the atomic number changes in charge-exchange reactions, whereas

the mass number is conserved, which translates to a change in isospin projection, Δ𝑇𝑧, of 1 or -1.

The isospin projection of the target nucleus can be calculated by the following equation:

𝑇𝑧 =

𝑁 − 𝑍
2

(1.11)

For example, in the case of 32Mg with 𝑁 = 20 and 𝑍 = 12, the isospin projection is 𝑇𝑧 = 4. In a

(𝑝, 𝑛)-type reaction (Δ𝑇𝑧 = −1), the final nucleus becomes 32Al with 𝑁 = 19 and 𝑍 = 13, resulting

in 𝑇𝑧 = 3. Conversely, in an (𝑛, 𝑝)-type reaction (Δ𝑇𝑧 = +1), the final nucleus becomes 32Na with

N = 21 and Z = 11, leading to 𝑇𝑧 = 5. For a parent nucleus in the ground state with isospin 𝑇0,

reactions in the (𝑛, 𝑝) direction can populate states in the daughter nucleus with 𝑇 = 𝑇0 + 1, as this

is the minimum allowed isospin for the residual nucleus at the new 𝑇𝑧. However, for reactions in

the (𝑝, 𝑛) direction, states with 𝑇 = 𝑇0 − 1, 𝑇 = 𝑇0, or 𝑇 = 𝑇0 + 1 can be populated.

Through a detailed investigation, Tadeucci et al. [56] established a proportional relationship

between B(GT) and ( 𝑝, 𝑛) cross-section at zero momentum transfer (𝑞 = 0):

𝑑𝜎
𝑑Ω

(𝑞 = 0) = ˆ𝜎𝐺𝑇 𝐵(𝐺𝑇),

(1.12)

where 𝑑𝜎

𝑑Ω (𝑞 = 0) is the differential cross-section at zero momentum transfer (𝑞 = 0) and ˆ𝜎𝐺𝑇
is the unit cross-section. The ˆ𝜎𝐺𝑇 can be calibrated from transitions with known B(GT). In the

Eikonal approximation, the ˆ𝜎𝐺𝑇 is a combination of three independent terms:

ˆ𝜎𝐺𝑇 = 𝐾 𝑁 𝐷 |𝐽𝜎𝜏 |2

(1.13)

𝐾 =

𝐸𝑖 𝐸 𝑓 𝑘 𝑓
(ℏ2𝑐2𝜋)𝑘𝑖

is a kinematical factor, where 𝐸𝑖 and 𝐸 𝑓 are the reduced energies of incoming and

outgoing channels, and 𝑘𝑖 and 𝑘 𝑓 are the linear momenta of the projectile and ejectile, respectively.

𝑁 𝐷 is the distortion faction and |𝐽𝜎𝜏 |2 is the volume integral of the central 𝜎𝜏 interaction.

15

The proportionality relationship between the CE differential cross-section and the B(GT) holds

true under specific conditions: (i) the differential cross-section has to be extracted at 𝑞 = 0 and

(ii) intermediate beam energies (𝐸𝑏𝑒𝑎𝑚 ≳100 MeV/u) are required to ensure a single step reaction

mechanism [57, 58]. The proportionality has shown to worsen for weaker transition strengths,

caused by the interference between the transition amplitudes mediated by the 𝜎𝜏 and 𝑇 𝜏 (tensor-𝜏)

components of the NN interaction [56, 59, 54].

The model-independent Ikeda sum-rule provides a relationship between the sum of B(GT) in

either direction as follows [60]:

𝑆(𝛽−) − 𝑆(𝛽+) = 3(𝑁 − 𝑍),

(1.14)

where S(𝛽−) and S(𝛽+) are the summed B(GT)s in 𝛽− and 𝛽+ directions, respectively. Experimen-

tally, only about 60% of the GT sum-rule strength is observed due to a phenomenon known as

quenching [61, 62]. Several studies have established quenching for 𝑝, 𝑠𝑑, and 𝑝 𝑓 shells that are

summarized in table 1.1:

Table 1.1 The quenching factors for 𝑝, 𝑠𝑑, and 𝑝 𝑓 shells. When applying the quenching factor to
extract B(GT), the values from the table need to be squared. For example, for the 𝑝 𝑓 shell, the
applied quenching factor should be 0.7442 = 0.555.

Region

𝑝-shell
𝑠𝑑-shell
𝑝 𝑓 -shell

Quenching Factor
1 − 0.19 (cid:16) 𝐴

(cid:17) 0.35

[63]

16
0.76 [64]
0.744 [65]

Several charge-exchange probes have been developed to extract B(GT) in the 𝛽+ and 𝛽− di-

rections. In the 𝛽+ direction, (𝑛, 𝑝), (𝑑,2He), and (𝑡,3He) reactions have been primarily used to

constrain EC rates. Experiments have primarily been performed in forward kinematics in which the

target is the probed nucleus and the light particle serves as the beam, as illustrated in Fig. 1.10(a).

The nuclei relevant for astrophysical studies are often unstable and short-lived that cannot be used

as targets. As a result, the experiment has to be performed in inverse kinematics where the light

particle is used as the target and the probed rare isotope is used as the beam as shown in Fig. 1.10(b).

16

It is not feasible to make a sufficiently dense neutron target, so pursuing the (𝑛, 𝑝) reaction in inverse

kinematics is not realistic. Additionally, inverse reactions with light charged particles at 𝑞 ∼ 0

results in an ejectile that is emitted at very low kinetic energy (∼ 1 MeV) and easily stops in solid

target. The (7Li,7Be) reaction has been studied in inverse kinematics without the detection of 7Be

ejectile directly [66]. However, as the 7Be ejectile is not detected, the measurement is limited

to only relatively light nuclei (A<35) and excitation energies up to particle separation energies.

The (𝑑,2He) reaction in inverse kinematics does not have the aforementioned limitations and is in

principle a suitable tool to study rare isotopes across the nuclear chart. It was successfully used to

study 14O and 13N to extract B(GT) in a pilot experiment at the National Superconducting Cyclotron

Laboratory (NSCL) in 2020 [67, 68].

Figure 1.10 (a) In forward kinematics, the heavy isotope is used as the target and the light isotope
which in this case is the deuteron is used as the beam. (b) In inverse kinematics, the unstable heavy
isotope is used as the beam and impinged on the lighter isotope used as a target.

1.4.2 Application of (𝑑,2He) Probe in Inverse Kinematics

The (𝑑,2He) probe was first used at RIKEN for studies of light nuclei [69]. Using a deuteron

beam energy of 𝐸𝑑 = 260 MeV, the data demonstrated the one-step nature of the reaction and

established a proportionality between 𝐵(GT) values and (𝑑,2He) cross-sections at 0◦. A resolution

of approximately 1 MeV was achieved in this study, primarily limited by the target thickness. The

reaction probe was later successfully utilized at the Texas A&M Cyclotron Institute [70] and the

Kernfysisch Versneller Instituut (KVI) [71]. At Texas A&M, three 𝑠𝑑-shell nuclei and one 𝑝 𝑓 -shell

nucleus were investigated using a deuteron beam energy of 𝐸𝑑 = 125.2 MeV, achieving a resolution

17

(a) Forward Kinematics(b) Inverse Kinematicsof as low as 600 keV. At KVI, experiments conducted with a beam energy of 𝐸𝑑 = 170 MeV

achieved an excitation energy resolution of as low as 150 keV. The (𝑑,2He) reaction probe has been

well established in forward kinematics and is currently regarded as one of the most effective tools

for studying B(GT)s in the 𝛽+ direction.

The 2He in the (𝑑,2He) reaction probe refers to a pair of unbound protons. The use of (𝑑,2He)

probe requires the measurement of the two protons in order to reconstruct the 2He particle.

If

the relative energy between the two protons (𝜖 𝑝 𝑝) is small, it primarily has a 1𝑆0 configuration

and contributions from higher partial waves are negligibly small [72]. The incident deuteron is

predominantly in the 3𝑆1 state, so a pair of protons coupled to the 1𝑆0 state will ensure the reaction

proceeds almost exclusively by spin transfer (Δ𝑆 = 1). Thus, by constraining 𝜖 𝑝 𝑝 to small values,

it is possible to maximize the spin-transfer nature of the (𝑑,2He) reaction probe.

Figure 1.11 Differential cross-sections for the (a) 14O(𝑑,2He) and (b) 13N(𝑑,2He) reactions are
depicted for 𝜃c.m. ≤ 8◦. The dashed lines in the figure represent separation energy for different
decay channels, while the different colors indicate which residual particle was detected in the S800,
as labeled. Figure has been taken from Ref. [68].

The (𝑑,2He) probe was used in inverse kinematics in a pilot experiment in 2020 at NSCL to

study rare isotopes [67, 68]. This experiment utilized the Active-Target Time Projection Chamber

18

SnSαSdθcm< 8°θcm< 8°014N013C013N012C010BAllAll013C012C(AT-TPC) positioned at the target location of the S800 magnetic spectrometer. Figs. 1.11(a)

and 1.11(b) show the differential cross-sections for for 14O(𝑑,2He) and 13N(𝑑,2He), respectively,

plotted as a function of their excitation energies obtained from the analysis of the two primary

channels. Their decay products detected in the focal plane of the S800 were also analyzed to obtain

the cross-sections up to high excitation energy.

For the 14O channel, B(GT) values were determined up to an excitation energy of 22.5 MeV. The

dominant peak at 3.95 MeV, shown in Fig. 1.11(a), has a known B(GT) of 2.73 from 𝛽-decay [73] and

was used to determine the ˆ𝜎GT. The B(GT) for the ground state in 14N, resulting from 14O 𝛽-decay,

is known to be 2 × 10−4 [73, 74], which falls below the threshold for the proportionality between

B(GT) and charge-exchange cross-sections [56, 75]. B(GT)s were obtained for the excitation energy

range of 10–22.5 MeV, with a detailed discussion provided in Ref. [67].

For the 13N channel, ˆ𝜎GT was determined using the peak at 3.68 MeV, shown in Fig. 1.11(b).

The B(GT) for this state was extracted as 1.37 from the 13C(3He,𝑡) experiment [76, 77]. The

B(GT) for the ground state in 13C was calculated to be 0.27 ± 0.07. Additionally, B(GT) values

for the excitation energy regions 7-12 MeV and 13-17 MeV were determined to be 0.93 ± 0.05 and

0.19 ± 0.15, respectively. Further details of these results can be found in Ref. [68].

The results from both channels showed good agreement with previous experimental data and

shell-model calculations. Following the successful implementation of the first (𝑑,2He) experiment

in inverse kinematics, a second experiment using the same setup was conducted to investigate 33Al

and 32Mg isotopes. The experimental and analytical methods for the second (𝑑,2He) experiment

are similar to those of the first experiment and are detailed in Secs. 3 and 4.

19

CHAPTER 2

THEORETICAL CALCULATION

This chapter focuses on the theoretical calculations used in this thesis serving two main purposes:

(1) to compare with experimental results in order to constrain and benchmark the theoretical

models, and (2) to use these models to extract angular momentum components of interest from the

experimental data- specifically, using theoretical cross-sections for transitions with different orbital

angular momentum transfers to isolate the monopole (Δ𝐿 = 0) contribution from the experimental

differential cross-section through a multipole decomposition analysis (MDA) using the following

equation [78, 79]:

𝑑𝜎
𝑑Ω

= 𝑎0

𝑑𝜎
𝑑Ω

(cid:12)
(cid:12)
(cid:12)
(cid:12)Δ𝐿=0

+ 𝑎1

𝑑𝜎
𝑑Ω

(cid:12)
(cid:12)
(cid:12)
(cid:12)Δ𝐿=1

+ 𝑎2

𝑑𝜎
𝑑Ω

(cid:12)
(cid:12)
(cid:12)
(cid:12)Δ𝐿=2

+ ...

(2.1)

where 𝑎0, 𝑎1, and 𝑎2 are fitting parameters. The left-hand side of this equation is the differential

cross-section from the experimental data and on the right-hand side, the differential cross-sections

are obtained from simulation for various Δ𝐿 contributions. By fitting data with simulation, we

extract the Δ𝐿 = 0 component of differential cross-section from experimental data.

This chapter will introduce the important computational tools used in the analysis: the shell-

model code, NuShellX; the normal-mode code normod; and the Coupled-Channels Born Ap-

proximation code, accba. Rather than going into the detailed theoretical background of these

codes, the focus will be on how they were applied in this study. For more detailed explanations of

these tools, readers can refer to Ref. [80] for NuShellX, Ref. [81] for normod, and Ref. [82] for

accba.

The theoretical differential cross-sections utilized in this analysis are obtained from accba.

One-body transition densities (OBTDs) serve as inputs in the accba code and are obtained from

NuShellX for Δ𝐿 = 0 and Δ𝐿 = 2 transitions and normod for Δ𝐿 = 1 transitions. The OBTD

connects the initial and final nuclear states through a one-body operator [83]. From the OBTDs,

NuShellX also provides the shell-model predictions for the B(GT) distributions, which we compare

our data with.

20

2.1 Shell-Model

The code NuShellX has been designed for performing shell-model calculations and is capable

of handling Hamiltonian matrix calculations with very large basis dimensions. The formalism be-

hind the code can be found in Ref. [80]. We used the shell-model code NuShellX to perform OBTD

calculations for Δ𝐿 = 0 and Δ𝐿 = 2 transitions in the 33Al(𝑑,2He)33Mg and 32Mg(𝑑,2He)32Na re-

actions.

Conventional shell-model calculations often truncate configurations within the 𝑝 𝑓 shell [84, 35,

85, 86], and therefore, these calculations cannot not include mixing between normal and intruder

states [35, 86]. However, because the nuclei we are studying belong to the 𝑁 = 20 island of

inversion (see Sec. 1.2), Y. Utsuno used the 𝑠𝑑𝑝 𝑓 − 𝑚 interaction within the 𝑠𝑑𝑓 7𝑝3 model space,

which allows for mixing between these states. More information about the calculation of this

interaction can be found in Ref. [87].

Fig. 2.1(a) and Fig. 2.1(b) show the shell-model predictions for the B(GT) transition from 33Al

to 33Mg and 32Mg to 32Na, respectively. The shell-model calculations are performed by Y. Utsuno

using the 𝑠𝑑𝑝 𝑓 − 𝑚 interaction [88]. The model was truncated to include up to 6p-6h for 33Mg

and 6p-5h for 32Na. The calculations were performed using the Lanczos strength function method.

The shell-model calculations predict a strong GT state for both 33Mg and 32Na below 2 MeV.

21

Figure 2.1 Shell-model calculation of the B(GT) for (a)33Mg and (b)32Na performed by Y. Utsuno
using 𝑠𝑑𝑝 𝑓 − 𝑚 interaction. The phenomenological quenching factor (see Section 1.4.1), which
would reduce the strength to 55% of what is shown, has not been applied.

2.2 Normal-Mode calculation

For the dipole transitions involving a change in parity, the OBTD calculations were performed

within the normal-mode formalism, as described in Ref. [81]. The normod code [89] was utilized

for these calculations. In the normal-mode formalism, OBTDs of 1p-1h transitions are constructed

that are connected by a particular transition operator such that all these transitions interfere con-

22

(a)(b)33Al33Mg32Mg32Nastructively and the transition strength is maximized. Therefore, the transition strength is associated

with one excitation only, which serves as input for the ACCBA calculation. As we are only inter-

ested in the angular distribution of the dipole transition, which does not depend on the details of

the wave functions, this relatively simple approach is sufficient.

For open-shell nuclei, occupation numbers must be properly considered. The states below the

last closed shell are fully occupied, and the sum of the occupation numbers for the states above the

last closed shell must match the number of available particles [90].

2.3 CCBA calculations

For calculating the differential cross sections, the Coupled-Channels Born Approximation was

implemented using the code accba. accba has been written specifically for (𝑑,2He) reactions and

has been well tested in forward-kinematics experiments. As noted previously, the 2He system in

these reactions is, in reality, a pair of protons, and the resulting three-body final state presents

significant theoretical challenges.

In Ref. [91], Okamura et al. employed a one-step distorted-wave Born approximation (DWBA)

to reproduce the low-lying discrete levels of the residual nucleus in (𝑑,2He) reactions. While the

results were reasonably successful, some ambiguity remained due to the arbitrariness of the final

scattering wave function, primarily caused by uncertainties in using a 2He optical potential. To

address this, Okamura et al. introduced a parameter-free method in Ref. [82], which employed an

adiabatic approximation for treating the three-body dynamics. This approach simplifies the three-

body dynamics by replacing the Hamiltonian of the two-proton system by the relative energy of the

two protons. The coupling of 1S and 1D waves in the proton-proton (𝑝 − 𝑝) system is included in

the calculation, along with contributions from the deuteron D-state. This coupling of partial waves

is the basis of the "coupled channel" approach used in this analysis. This formulation resolves some

of the ambiguities in previous treatments by providing a more robust framework for describing the

final state interactions. This is especially important in (𝑑,2He) reactions where precise modeling

of the final state significantly impacts the interpretation of experimental data. The accba code

calculates the differential cross-section as a function of the relative energies of the protons and

23

the center-of-mass scattering angle. This is crucial for performing acceptance corrections in the

current work, as the cross-section depends sensitively on these variables.

2.3.1 Components of the ACCBA Code

This section describes the various inputs of the accba code that need to be provided. The

primary inputs of this code are comparable to that of other DWBA codes used for CE reactions at

intermediate energies.

Fig. 2.2 shows a version of the code that was run for the 32Mg(𝑑,2He) reaction. Lines 1-8 in Fig.

2.2 set the number of bins and integration limits for 𝜖 𝑝 𝑝. Line 10 is for the directory of the output

file. Lines 13-46 are the skeleton of this code and need to be modified for each transition. All the

numeric values in these lines are in free format and "\" can be used to write any comment. If any

parameters are given after using a "\", it will be set to zero. Inputs for lines 13-46 are described

below:

• Line 13: This line is for the title and is limited to 80 characters.

• Line 14: This line has three inputs: DWBA, point- projection, and triple-differential cross-

section. For each of these parameters, there are only two options: T or F.

– DWBA- Using "T" will treat the 2He as a single particle and it is assumed that the

optical potential acts on the center-of-mass of the 𝑝 − 𝑝 system. Using "F" will generate

the final scattering wave from proton optical potential by solving adiabatic coupled-

channels equation.

– Point-projection- "T" will treat the 2He particle as a point particle and perform two-

body calculation. "F" will treat the 2He particle as unbound pair of protons and perform

a three-body calculation.

– Triple differential cross-section- "T" will provide a triple differential cross-section

and "F" will generate double differential cross-section only.

24

• Line 15: This line sets the parameters for the angular distributions and also requires three

inputs. The first and second input are the minimum and maximum angles, respectively, in

degrees. The third input is the angular step size, also in degrees.

• Line 16: This line requires seven inputs detailing the reaction. The first three inputs are

the mass number, atomic number, and spin of the target. As previously mentioned, accba

was written assuming forward kinematics, so the target is 32Mg, as if it were in forward

kinematics, even though for the experiment discussed here, 32Mg was the beam. All cross-

sections are calculated in the center-of-mass system making the distinction between target

and beam irrelevant. The last four inputs in this line are of the residual nucleus (32Na) spin,

beam energy, 𝑄 value of the reaction, and the excitation energy of the final state. All the

energy inputs in this line have to be in the units of MeV.

• Line 17: This line requires four inputs. The first parameter is the parity transfer. It can

be either +1 or -1. The second parameter indicates the angular momentum transfer (Δ𝐿).

The third and fourth parameters are for the number of cases (n) with different total spin

transfer (Δ𝐽) a calculation must be performed. For example, we can obtain results for Δ𝐿 = 0

transition with the four parameters: +1 1 1 1 (Δ𝜋 Δ𝐿 𝑛 Δ𝐽).

• Line 18: This line consists of three parameters, which control details of the calculations. The

first two parameters pertain to the integration range and the step size of the calculation over

the distance between the target and the center-of-mass of 𝑝 − 𝑝 system. The first parameter is

the number of division for the range and must be even, and the second parameter is the step-

size in units of fm. The third parameter on this line refers to the maximum number of partial

waves in the DWBA calculation. This line can usually be left unchanged for 𝐸𝑏𝑒𝑎𝑚 ∼ 100

MeV/𝜇.

• Line 19: This line consists of five parameters. The first three parameters pertain to the

integration range over the relative coordinate in the 𝑝 − 𝑝 system. The first two parameters,

similar to those in line 18, are for the number of divisions and the step size. The number of

25

divisions is used for integrating the reaction 𝑇 matrix, and the product of these two parameters

determines the maximum radius for integration. The third parameter specifies the number of

divisions for interpolation within the integration range. The product of this parameter with

the first one is used in solving the 𝑝 − 𝑝 scattering wave function. The last two parameters

control whether the D-wave contribution should be included for the incident deuteron (which

is the target in this experiment) and the outgoing 𝑝 − 𝑝 system. Both parameters can take

values of either 0 or 2. For the fourth and fifth parameters, 0 excludes the D-wave contribution

from the deuteron and 𝑝 − 𝑝 system, respectively, while 2 includes the D-wave contribution

for both. This line can be left unchanged unless the inclusion of the D-wave contributions is

desired.

• Line 20: This line provides three parameters related to 𝜖 𝑝 𝑝. The first parameter is for the

step-size in 𝜖 𝑝 𝑝 and the second and third parameters are for the integration minimum and

maximum for 𝜖 𝑝 𝑝. The values for all these parameters are provided in MeV. The range of 𝜖 𝑝 𝑝

was selected based on the detector system’s acceptance for this experiment.

• Lines 21-23: The first line is for the optical potential for the deuteron-target and residual-

proton systems taken from the Koning-Delaroche phenomenological potential [92], where

the extended parametetrizations in the code talys were used [93]. Since in the adiabatic

approximation, it is not possible to include the spin-orbital potential, the conventional optical

potential was used to generate the incident-channel wave function [82].

The eight parameters in line 21 are the potential depth (MeV), reduced radius (fm), and the

diffuseness of the real potential (fm), depth of the volume (MeV), depth of the surface (MeV),

the reduced radius (fm), and the diffuseness of the imaginary potential (fm), and the reduced

radius of charge distribution (fm), respectively. Line 22 has three parameters:

the depth

(MeV), the reduced radius (fm), and the diffuseness (fm) of the spin-potential orbital. If the

first parameter in line 14 is chosen as F, the optical potential must not include the spin-orbit

part of the potential. The eight parameters in line 23 are the same ones as line 21, but for the

26

proton optical potential.

• Lines 24-30: These are the parameters of the Love-Franey effective nucleon-nucleon inter-

action based on nucleon-nucleon scattering observables. [58, 94].

• Line 31: This line details the parameters used for the potentials, which are used to generate

the single-particle wave functions. The four parameters are the reduced radius (fm) and

diffuseness (fm) of the single-particle potential, depth of the spin-orbit part of the potential

(MeV), and the reduced radius of the charge distribution (fm). This line can be left unchanged.

• Lines 32-38: These lines provide the single-particle wave functions for the reaction. Each

line has four parameters. The first parameter specifies the orbit and consists of four values.

The first value denotes whether it is a neutron or proton (‘n’ for neutron and ‘p’ for proton), the

second value provides the number of nodes of the wave function (starts at 0), the third value

is for the orbital angular momentum, and the last value is for the total angular momentum

expressed in fraction. On line 32, ‘p0d5/2’ corresponds to n=0, l=2, and j=5/2 orbit for the

proton. The second parameters in these lines can either be ‘V’ or ‘E’. ‘V’ implies the function

is generated for fixed potential depth, 𝑉0, and ‘E’ implies the wave function is generated by

fixing the binding energy (BE) and varying the depth. We selected a fixed potential depth

of 50 MeV for the system although using the "E" option did not alter the results. Since the

potential depth is fixed, the fourth parameter becomes irrelevant in this case.

• Lines 39-46: These lines provide the one-body transition densities, which contain essential

information that connect the initial and final states by the action of one-body operators. Of the

three parameters, the first two are the orbits of the hole and the particle, respectively, and are

written in the same manner as mentioned before. The last parameter is the OBTD. The value

for these parameters are also obtained from the shell-model calculation using NuShellX.

27

Figure 2.2 Input file for ACCBA.

28

2.3.2 Application of the ACCBA code

For the MDA, we used angular distributions for transitions with angular momentum transfer

Δ𝐿 = 0, Δ𝐿 = 1, and Δ𝐿 = 2. The input shown in Fig. 2.2 was used to obtain the theoretical

cross section for the Δ𝐿 = 2 transition in the 32Mg(𝑑,2He) reaction. We ran the accba code for the

Δ𝐿 = 0 and Δ𝐿 = 1 transitions for 32Mg(𝑑,2He) reaction after making the necessary modifications

in the other inputs. Fig. 2.3 shows the differential cross-section calculated in the accba code for the

Δ𝐿 = 0, 1, and 2 transitions of the 32Mg(𝑑,2He)32Na reaction at one particular excitation energy.

From the accba calculations, we observed a forward-peaking behavior for the Δ𝐿 = 0 transition.

The Δ𝐿 = 1 transition showed a peak between 4◦ and 5◦, while the angular distribution for the

Δ𝐿 = 2 transition appeared largely flat over the angular range covered. We repeated the same steps

for the Δ𝐿 = 0, 1, and 2 transitions for the 33Al(𝑑,2He)33Mg reaction.

Figure 2.3 The differential cross-section obtained from ACCBA code for Δ𝐿 = 0, Δ𝐿 =1, and Δ𝐿 =
2 components for the 32Mg(𝑑,2He)32Na reaction.

29

∆L = 0∆L = 1∆L = 2(arb. units)CHAPTER 3

EXPERIMENTAL SETUP AND PROPERTIES

The 33Al, 32Mg(𝑑,2He) experiment in inverse kinematics was performed at the Facility of Rare

Isotope Beam (FRIB) at Michigan State University. This experiment was performed after the

successful execution of the 14O, 13N(𝑑,2He) experiment [67, 68] at the NSCL at Michigan State

University to study B(GT) distribution in the 𝛽+ direction. The experimental setups for both the

(𝑑,2He) experiments were identical.

3.1 Beam Production

A cocktail beam composed of primarily 33Al (71%) and 32Mg (22%), with minor quantities of

other contaminants, was produced from a 150 MeV/u 48Ca primary beam at FRIB impinged on a

12C production target with a thickness of 5.0 mm. Two aluminum wedges with thicknesses of 1.5

and 2.8 mm were inserted into the Advanced Rare Isotope Separator (ARIS) fragment separator

[95] to purify the components of the beam [96]. The beams were sent to the S800 mangnetic

spectrometer [97], located in the S3 vault at FRIB, with a rigidity of 3.68 Tm. The 33Al and 32Mg

beams had energies of 96.5 MeV/u and 88 MeV/u, respectively. The beam intensities used for this

experiment were between 100 kpps and 140 kpps. During the experiment, the beam intensity was

monitored in a scintillator placed at the object which is at the entrance of the beamline towards

the S800. However, the transmission from the object scintillator to the AT-TPC was not 100%.

The transmission of the beam to the AT-TPC was measured by comparing the event rate in the

object scintillator to the event rate in the S800 focal plane scintillator when delivering the beam

all the way to the S800 focal plane (unreacted beam setting). Since the acceptance of the S800 is

large compared to that of the beamline from the object scintillator to the AT-TPC, this provides an

accurate estimate of the transmission, which was determined to be 31.2%.

3.1.1 ARIS

The A1900 fragment separator [96] was used for the first (𝑑,2He) experiment, but ARIS was

used for the second (𝑑,2He) experiment described in this work. ARIS has been designed to have a

30

large acceptance and transmission along with high suppression for undesired contaminants. This

allows ARIS to provide high-intensity rare-isotope beams with desired purities. This fragment

separator works in three stages (see Fig. 3.1). The first stage of the ARIS fragment separator,

known as the preseparator, receives the beam from the production target and performs an initial

separation of desired fragments from contaminants, delivering the rare-isotope beam to the second

stage for further purification. This stage vertically bends the beam from the underground primary

beamline to the above-ground levels of stages 2 and 3, using superferric magnetic dipoles for

bending and magnetic quadrupoles for focusing, with higher-order magnetic elements correcting

image aberrations. In stage 2, the beam undergoes horizontal separation and purification, where

magnetic fields and a momentum-loss achromatic technique further separate fragments by mass-

to-charge ratio, aided by a wedge degrader that compresses the momentum spread. Finally, in

stage 3, a slit system in the focal plane reduces unwanted beam components, ensuring that only the

rare-isotope beams of interest reach the experimental area with high purity.

Figure 3.1 Schematic layout of ARIS consisting of three stages. The preseparator stage takes place
underground whereas the second and third stage take place at ground. Figure has been taken from
Ref. [95].

31

3.2 S800 Magnetic Spectrometer

The beam is transported to the S3 experimental vault through the transfer line after selection of

the cocktail beam in the fragment separator. The S800 spectrometer and analysis line are located

in the S3 vault (see Fig. 3.2). The S800 has two dipole and two quadrupole magnets that direct the

reaction products towards the focal plane of the S800.

Figure 3.2 Schematic layout of the S800 analysis line and spectrometer in the S3 vault at FRIB.
Figure adapted from Ref. [97].

The focal plane of the spectrograph has a number of detectors, shown in Fig 3.3, that are used for

determining hit position, angle, energy loss, and time-of-flight (𝑇 𝑜𝐹). The momentum, scattering

angle, and particle identity are determined from these parameters.

32

Figure 3.3 Schematic of the S800 analysis line and spectrometer in the S3 vault at FRIB. Figure
has been taken from Ref. [98].

At the focal plane, there are two Cathode Readout Drift Chambers (CRDC) that are used for

measuring the vertical and horizontal positions and angles of the fragments at the focal plane. The

CRDCs consist of an active area of 60 cm in the dispersive direction by 30 cm in the non-dispersive

direction [97]. The first CRDC is placed at the nominal optical focal plane and the second CRDC

is located downstream separated from the first one by 1073 mm [99]. They are filled with 20%

isobutane and 80% carbon tetrafluoride. Particles passing through the CRDCs ionize the gas

mixture and the free electrons drift towards the cathode pads along the dispersive (x) axis. The

image charge distribution on the pads allows determination of the dispersive position, as shown

on Fig. 3.4. The drift time of the electrons, obtained from the difference between the S800 DAQ

trigger time and the CRDC anode signal, provide information regarding the non-dispersive (y)

position. The dispersive and non-dispersive angle measurements are obtained by combining the

position measurements in the two CRDCs.

33

PlasticScintillatorCRDCsHodoscopeIonizationChamberBeamFigure 3.4 Schematic of the two CRDCs with the inset showing integrated image charge detected
by the pads. An example event trajectory through the detectors is also shown. Figure was taken
from Ref. [100].

.

A multi-segmented ionization chamber, filled with 90% argon and 10% methane, is placed after

the CRDCs. The energy loss of the particles passing through the chamber allows for the indirect

measurement of atomic number. The energy loss is proportional to the square of the charge of the

particle according to the Bethe-Bloch formula [101]:

𝑑𝐸
𝑑𝑥

=

4𝜋𝑒4𝑍 2
𝑚0𝑣2

𝑛𝑎𝑏𝑠𝑧𝑎𝑏𝑠

(cid:18)

ln

2𝑚0𝑣2
𝐼

(cid:18)

1 −

− ln

(cid:19)

𝑣2
𝑐2

−

(cid:19)

,

𝑣2
𝑐2

(3.1)

where 𝑍 is the charge of the incident particle. 𝑛𝑎𝑏𝑠, 𝑧𝑎𝑏𝑠, and 𝐼 are the number density, atomic

number, and average ionization potential of the absorbing material, respectively, 𝑚0 is the electron’s

rest mass, and 𝑣 is the velocity of the incident particle. Followed by the ionization chamber is a

3-mm thick plastic scintillator that provides the trigger for the S800 data acquisition system and is

also used for particle identification by 𝑇 𝑜𝐹 measurement [97]. There is also a hodoscope at the

focal plane of the S800 that measures the total kinetic energy of the ejectiles to identify different

34

Pad NumberCentral Ray010002000300040005000Integrated Image ChargeEvent Trajectory(X2, Y2)(X1, Y1)Figure4.4:LayoutofCRDCdetectorsandcartoonofcathodepositiondetermination.Figurecharge states, but it was not used for this experiment.

3.3 Active-Target Time Projection Chamber

The active-target mode in Time Projection Chambers (TPC) are ideal for experiments with

rare-isotope beams in inverse kinematics as the gas in the TPC serves as the tracking medium and

the target with large solid-angle coverage and low-energy detection thresholds. The AT-TPC at

FRIB has a cylindrical geometry with a length of 100 cm and a diameter of 50 cm [102]. The gas

volume is enclosed by a field cage. A 3 cm diameter hole is made in the pad plane to allow the

beam to enter. The active region of the AT-TPC is separated from the vacuum of the beam line

and spectrometer by 12 um thick window. The AT-TPC has a symmetrical geometry and the beam

comes in perpendicular to the pad plane on the side.

Figure 3.5 Mechanical drawing of the Active-Target Time Projection Chamber (AT-TPC) showing
all the major components. Figure is taken from Ref. [102].

.

The charged particles from the beam pass through the chamber, stripping electrons from the gas

and ionizing it. The electrons are then driven towards the anode end of the detector by a uniform

35

electric field applied along the beam axis. There is a sensory plane at the anode end consisting of

10240 triangular conductive pads. The plane is arranged in a hexagonal inner region of small pads

with 0.5 cm triangular height surrounded by outer region of large pads with 1 cm triangular height.

A Micromegas detector is installed on the pad plane that consists of a very fine conductive mesh.

The mesh is biased with respect to the electrodes to create a relatively large electric field between

the mesh and the pads. This divides the detector into a large drift region above the mesh and a

small multiplication region below the mesh. When the electrons from the ionized gas pass the

mesh and enter the multiplication region, the much larger electric field there cause the formation

of an electron avalanche, amplifying the signal.

The AT-TPC uses advanced digital electronics developed by the Generic Electronics for Time

Projection Chambers (GET) collaboration [103], providing a fully digital data acquisition (DAQ)

system capable of digitizing and recording traces for all 10,240 channels in the detector. The system

is organized into a hierarchy of modules, with the base level consisting of custom application-

specific integrated circuits (ASICs) known as AGETs (ASIC for GET). An ASIC (Application-

Specific Integrated Circuit) is a custom-designed chip tailored for specific applications. The AGET

amplifies incoming signals using a variable-gain charge-sensitive preamplifier and performs pole-

zero correction. Each AGET reads out 64 physics channels from the detector.

Each AsAd (ASIC Support and Analog-to-Digital Conversion) board houses four AGETs and

includes a four-channel, 12-bit analog-to-digital converter (ADC) that digitizes sample outputs

from the AGET chips and transmits the data through a serial link [104] when a trigger is issued.

These AsAd boards are connected directly to the sensor plane to minimize the distance that analog

signals must travel before being digitized, reducing capacitance and potential noise in the data. Ten

Concentration Boards (CoBos) in the AT-TPC are each connected to four AsAd boards, collecting

data, applying time stamps, and sending events over a 10 Gb/s fiber-optic link to a storage cluster

[105]. The MuTAnT (Multiplicity, Trigger, And Time) board synchronizes CoBos and manages

clock synchronization and global triggers. Fig. 3.6 represents a schematic of the GET electronics

system.

36

Figure 3.6 Schematic of GET electronics system. Only one out of 40 AsAd boards is shown here
for clarity. Figure has been taken from Ref. [105].
.

At higher beam intensities, space-charge effects can be observed inside the AT-TPC [106]. The

positive ions inside the AT-TPC volume move at a much slower pace compared to the electrons

after the ionization has taken place. As a result, positive charge is accumulated in the beam region

over time. The magnitude of this space charge is directly proportional the beam rate. For a constant

beam rate, 𝑅, electron-ion pairs per unit volume have a production rate of [107]:

(cid:164)𝑁 =

(cid:18) 𝑑𝐸
𝑑𝑥

(cid:19) 𝑅
𝑤 𝐴

,

(3.2)

where

(cid:19)

(cid:18)

𝑑𝐸
𝑑𝑥

is the beam stopping power in the gas, 𝐴 is the beam spot area and 𝑤 is the mean

ionization energy of the gas. For 𝐷2 gas, 𝑤 = 37 eV. The space charge produces an electric field

and affect the electron drift. Many electrons interact with nearby ions and recombine causing a loss

in collected charge.

In this experiment, the AT-TPC was placed at the target location of the S800 (see Fig. 3.7) to

allow measurements of heavy residues. The volume of the AT-TPC was filled with pure D2 gas at a

pressure of 530 Torr, corresponding to a target thickness of 11 mg/cm2 and shielded from beamline

37

CoBosCoBosMuTAnTµTCA BackplaneGigabit Ethernet SwitchControl &10 Gb/s Fiber-optic linkAsAdAGETAGETAGETAGETADCIsolation circuitDAQ ClusterLong-term storageSensor planeand S800 vacuum with PPTA windows. The deuterium gas served as the target and tracked the

two protons from the unbound 2He. A uniform electric field of 500 V/cm was applied to guide the

electrons towards the Micromegas detector.

Figure 3.7 (a) Schematic layout of the AT-TPC at the target location of the S800 used for the
experiment. Figure has been taken from Ref. [102]. (b) Picture of the AT-TPC at the target location
of the S800 after installation of the cabling system prior to the start of the experiment.

.

The beam-like particles were transmitted through the AT-TPC to the S800 focal plane by 3 cm

38

(a)AT-TPC(b)exit hole in the cathode plane. Therefore, the particles, fast fragments, and their decay particles

were not detected in the AT-TPC. Particle identification capabilities of the S800 allows the isolation

of reactions with specific exit channels. The S800 has a clock that provides a timestamp, which

is injected into one of the CoBos capable of accommodating an external clock. Meanwhile, the

other clocks in the system use a common internal clock from the MuTAnt board, which records

the timestamps of events in the AT-TPC. These timestamps from both systems are then matched

to correlate the events recorded by the S800 and the AT-TPC. The trigger from the S800 is

used to register events for the entire system. So, in principle, the number of events recorded by

both DAQs should be identical. However, due to occasional event loss in either DAQ, precise

timestamp matching becomes crucial for ensuring accurate data synchronization. Fig. 3.8 shows

the comparison of AT-TPC and S800 timestamps for a sample of events from one run. The data

points forming a line with a slope of 1 confirm that the timestamps are synchronized. The AT-TPC

provided information for the two proton tracks correlated with the (𝑑,2He) reaction and decay

products in the focal plane of the S800 from which the momentum of the 2He particle can be

reconstructed. The reconstruction process is discussed in detail in Sec. 4.7.

Figure 3.8 Correlation between the AT-TPC and S800 timestamps for a subset of events from a
specific run. The data points align along a smooth line with a slope of 1, indicating a consistent
match between the two sets of timestamps.

.

39

3.4 Magnetic Rigidity Setting

The S800 was tuned to an optimal magnetic rigidity (B𝜌) setting for the recoil and decay

products of the two major components of the beam, 33Al and 32Mg. The recoil products of 33Al and

32Mg are 33Mg and 32Na, respectively, which decay to 32Mg and 31Na through neutron emission.

By gating on the one-neutron decay product within the chosen magnetic rigidity setting, we were

able to reconstruct the excitation energy spectra of 33Mg and 32Na up to their two-neutron separation

energies without losing any information. Table 3.1 provides the one- and two-neutron separation

energies, as well as the separation energies for one proton, two protons, alpha, and deuteron for

both nuclei.

Table 3.1 Separation energies for one neutron (𝑆𝑛), two neutrons (𝑆2𝑛), one proton (𝑆 𝑝), one
deuteron (𝑆𝑑), two protons (𝑆2𝑝), and one alpha particle (𝑆𝛼) for 32Na and 33Mg.

Reaction
32Mg(𝑑,2 He)32Na*
33Al(𝑑,2 He)33Mg*

𝑺𝒏 (MeV) 𝑺2𝒏 (MeV) 𝑺 𝒑 (MeV) 𝑺 𝒅 (MeV) 𝑺2 𝒑 (MeV) 𝑺𝜶 (MeV)

1.68
2.28

5.98
8.06

19.80
20.97

19.98
22.64

44.90
40.80

45.49
26.93

As the excitation-energy spectra are obtained by measuring the recoil products and decay

products up the two neutron separation energy, it is possible to study the main features of the low-

lying B(GT)s in the distribution. Fig. 3.9(a) illustrates the B𝜌 settings optimized for the reaction

products and their decay products from the 33Al(𝑑,2He) and 32Mg(𝑑,2He) reactions. The total

rigidity spread of the 4 ejectiles is 5.2% and the rigidity acceptance of the S800 is 5.8%, so it was

possible to study all the ejectiles of interest at the B𝜌 setting of 3.904 Tm. However, a fraction of

the reaction product 32Na was outside of the B𝜌 acceptance, which was taken into account during

the analysis to obtain the differential cross-section. This occurred because the yield of reaction

products was low and the cut-off in the 32Na distribution was not identified during the experiment.

40

Figure 3.9 The 𝐵𝜌 ranges for CE reaction products and decay products from the (𝑑,2He) reaction
on 33Al and 32Mg isotopes are shown. Horizontal bars represent the full width of 𝐵𝜌 distributions.
The blue band shows the 𝐵𝜌 acceptance (±2.9%) centered at 𝐵𝜌 = 3.904 Tm.

41

CHAPTER 4

DATA ANALYSIS

4.1 S800 CRDC Calibration

The CRDC detectors at the focal plane of the S800 had to be calibrated in order to accurately

determine the particle’s position and angle. This task was accomplished by remotely inserting a

"mask", which is a metal plate, in front of each CRDC. The mask is made of tungsten and has holes

and slits at known locations as shown in Fig. 4.1(a). Once the mask was placed, only the particles

making it through the holes are detected in the CRDCs and are used in the calibration process.

The mask calibration is done by relating channel numbers to actual physical distances (in mm)

for the CRDCs. Fig. 4.1(a) represents the mask with holes and slits used for calibration and

Fig. 4.1(b) shows the particles detected at the CRDCs making it through the hole. The 𝑥-position is

determined from the charge induced on the pads and the slope is fixed to 2.54 mm/pad, according

to the width of the pads. The 𝑦-position depends on the drift velocity of the particles, which varies

with gas composition, pressure, and time. The slope in the y-direction was determined by fitting a

Gaussian to the y-axis projection of the particles that passed through the mask shown in Fig. 4.1(b).

The following equations were used to obtain the offsets in x-direction and y-direction, respectively:

𝑥1,2(𝑚𝑚) = 𝑚1,2(𝑚𝑚/𝑝𝑎𝑑) × 𝑥1,2( 𝑝𝑎𝑑) + 𝑏1,2(𝑚𝑚)

𝑦1,2(𝑚𝑚) = 𝑛1,2(𝑚𝑚/𝑛𝑠) × 𝑦1,2(𝑛𝑠) + 𝑐1,2(𝑚𝑚)

(4.1)

(4.2)

Table 4.1 Parameter values obtained from the masked calibration of two CRDCs.

𝒎1
2.54

𝒃1
-278.90

𝒏1
-0.12048

𝒄1
140.24068

𝒎2
2.54

𝒃2
-278.60

𝒏2
0.10638

𝒄2
-133.93590

42

Figure 4.1 (a) Mask used for calibration of the CRDCs. (b) Position measurements for CRDC1
mask calibration. The measurements associated with the known hole positions in the mask are
used to measure the offset and slope of the y-position as explained in text.

The values obtained from the calibration of both the CRDCs are listed in Table 4.1. The angles

and positions in the focal plane can be calculated from these calibrated values using the following

equations:

𝑎 𝑓 𝑝 = 𝑎𝑡𝑎𝑛

(cid:19)

(cid:18) 𝑥2 − 𝑥1
1073

43

(4.3)

[mm][mm]𝑏 𝑓 𝑝 = 𝑎𝑡𝑎𝑛

(cid:19)

(cid:18) 𝑦2 − 𝑦1
1073

𝑥 𝑓 𝑝 = 𝑥1

𝑦 𝑓 𝑝 = 𝑦1

(4.4)

(4.5)

(4.6)

where 𝑎 𝑓 𝑝 and 𝑏 𝑓 𝑝 are the dispersive and non-dispersive angles, respectively, and 𝑥 𝑓 𝑝 and

𝑦 𝑓 𝑝 are the x and y positions, respectively. The distance between the two CRDCs is 1073 mm.

These parameters are used to reconstruct the trajectory and properties of particles at the reaction

vertex by applying an inverse transfer map matrix generated through a COSY INFINITY [108]

calculation. A ray-tracing calculation is employed to determine four parameters at the location of

the scattering event: position in the non-dispersive direction (yta), scattering angle in the dispersive

plane (ata), scattering angle in the non-dispersive plane (bta), and measure of the energy of the

scattered particle (dta), where 𝐸 = 𝐸0(1 + 𝑑𝑡𝑎) with 𝐸0 being the central energy of the focal plane.

The ata and bta distributions were obtained from the unreacted beam setting. As shown in

Fig. 4.2, these distributions are used in the simulation (see Sec. 4.6) to randomly draw the beam

angles in the x and y directions. Since the ray-tracing depends on the distance along the z-position

of the beam and the reaction can take place anywhere in the AT-TPC, we performed the ray-tracing

using a set of ray-trace maps, which are interpolated to cover all positions along the length of the

AT-TPC.

44

Figure 4.2 (a) Dispersive scattering angle (ata) and (b) non-dispersive scattering angle (bta) distri-
butions obtained from ray-tracing calculation from beam runs sent directly to the S800 focal plane
without any reaction with the target.

The magnetic rigidity spread for the reaction products 33Mg and 32Na were obtained from the

dta distribution and are illustrated in Fig. 4.3.

Figure 4.3 Measured 𝐵𝜌 distributions for 33Mg and 32Mg from the 33Al(𝑑,2He) and for 32Na and
31Na from the 32Mg(𝑑,2He) reaction at the central 𝐵𝜌 setting of 3.904 Tm.

45

0.04-0.02-00.020.04 ata (mrad)010002000300040005000 Counts0.04-0.02-00.020.04 bta (mrad)0100020003000400050006000 Counts4.2 Beam Gate

The cocktail beam mainly consisted of 33Al and 32Mg with small quantities of contaminants.

A gate was used in the analysis to isolate events associated with each incoming beam. ToF

measurements between a plastic scintillator positioned at the exit of ARIS and the object scintillator

were used to accurately gate the desired beam within the cocktail beam. The absolute ToF scale for

the S800 was calibrated using the beam velocities and the measured distance between two plastic

scintillators. The beam velocities, calculated using LISE++, were 12.7 cm/ns for the 33Al beam and

12.2 cm/ns for the 32Mg beam [109]. The distance between the two scintillators is 34.28 m. Fig. 4.4

illustrates the separation of the beam components, which was achieved using Time-of-Flight (ToF)

measurement between the two scintillators. We applied an 1D gate on the ToF measurement for

the beam of interest.

Figure 4.4 ToF (in ns), shown on logarithmic scale, between the two plastic scintillators placed at
the exit of ARIS fragment separator and the beam entrance line of the S800 showing event-by-event
incoming beam particles identification. The red and green dashed lines represent the boundaries
of the gates drawn for 33Al and 32Mg beams, respectively.

4.3 Particle Identification

The particles of interest in this experiment were the recoil product 33Mg and its decay product

32Mg from the 33Al beam, as well as 32Na and its decay product 31Na from the 32Mg beam. Fig. 4.5

46

390395400405410415420S800 ToF (ns)010000200003000040000CountsAl33Mg31Al34Mg32Na30illustrates the reaction processes and decay products of the various beam components, recorded at

the focal plane of the S800. The particle identification (PID) was performed using the Δ𝐸-ToF

method.

The energy loss, Δ𝐸, was measured by the ionization chamber at the focal plane, and its

dependence on the particle’s charge 𝑍 2 is described by the Bethe-Bloch formula (see Eq. 3.1). The

measurement of Δ𝐸 allowed identification of the charge of the particle, thereby determining the

element. However, this method alone could not distinguish isotopes of the same element. The time-

of-flight (ToF) was determined by measuring the time difference between the object scintillator and

the focal plane scintillator. The ToF measurement provided crucial information about the particle’s

velocity, 𝑣, for the known flight path, 𝐿. For a constant magnetic rigidity, 𝐵𝜌, and known flight

path 𝐿, the ToF measurement relates the particle’s mass-to-charge ratio, 𝑚

𝑞 , to its velocity as shown

below:

𝐵𝜌 = 𝛾

𝑚𝑣
𝑞

−→

1
𝑣

= 𝑐𝑜𝑛𝑠𝑡. ×

𝑚
𝑞

𝑇 𝑜𝐹
𝐿

=

(4.7)

Figure 4.5 Particle identification at the focal plane of the S800. 33Mg and 32Na are the reaction
products from the 33Al(𝑑,2He) and 32Mg(𝑑,2He) reactions, respectively. Following neutron decay,
33Mg and 32Na produce 32Mg and 31Na, respectively, as their decay products. The 33Al and 32Mg
charge states represent the beam particles that picked up an electron from the AT-TPC gas or
windows, but did not undergo a nuclear reaction.

47

𝟑𝟐𝐌𝐠 Charge State𝟑𝟑𝐀𝐥 Charge State𝟑𝟐𝐌𝐠 𝟑𝟑𝐌𝐠 𝟑𝟏𝐍𝐚 𝟑𝟏𝐍𝐚 The gates around the blobs in the PID plot were defined using 3𝜎 cuts from Gaussian fits to

the projections on both the x and y axes. As an example, Fig. 4.6 shows the projection of the 32Mg

fragment, identified in Fig. 4.5, along the x-axis (S800 time of flight).

Figure 4.6 Projection of the 32Mg fragment on the S800 time of flight (ToF) axis. The red curve
represents a Gaussian fit used to define the gate for fragment identification, with the gate set around
3𝜎 from the fit to ensure accurate selection of events.

A similar approach was used for the y-axis projection to ensure accurate fragment identification,

as shown in Fig. 4.7, also for the case of 32Mg.

48

245-240-235-230-225-220-215-S800 ToF (arb. units)020406080CountsFigure 4.7 Projection of the 32Mg fragment on the Δ𝐸 axis. The red curve represents a Gaussian
fit used to define the gate for fragment identification, with the gate set around 3𝜎 from the fit to
ensure accurate selection of events.

In cases where a 3𝜎 gate would overlap with another fragment’s gate, we reduced the gate

width and adjusted the corresponding cross-sections accordingly. It is noted that there is some

background on the PID plot of Fig. 4.5. However, the requirement of a (𝑑,2He) event, associated

with 2 proton tracks from a single vertex in the AT-TPC (see Sec. 4.4), removes the background not

associated with CE events.

4.4 Event Reconstruction in the AT-TPC

A point cloud, consisting of a set of hit points defined by distinct spatial coordinates, was

generated for each event occurring within the AT-TPC gas volume (see Fig. 4.8). The x and y

coordinates of these points were obtained from the hit pattern on the pad plane of the AT-TPC

and the z coordinate was determined from the drift length of the electrons. The drift length was

calculated from the drift time and drift velocity. The drift velocity depends on certain factors and

requires correction, detailed in Sec. 4.5. After obtaining accurate coordinates for the hit points,

the RANSAC (Random SAmple Consensus) algorithm was utilized to analyze the point clouds for

all events. The routine identifies the hits that belong to a track and reject the noise points. Then

49

120130140150160170180190E (arb. units)D0100200300Countsthe tracks are fitted by a 3D line using 𝜒2 minimization. Ref. [110] describes the details of the

algorithm as applied to the analysis of AT-TPC data.

Figure 4.8 Visualization of the beam entering the AT-TPC through the entrance hole, with the
residual and decay products exiting through the opposite side after interacting with the deuterium
gas. The visualization also includes a representation of fitting the particle tracks onto a point cloud,
based on data from a real event in the 33Al(𝑑,2He) reaction.

.

Inside the AT-TPC, the 𝛿 electrons are the primary source of background. However, the signals

from 𝛿 electrons can be significantly reduced by applying a threshold, as the charge induced on the

pad plane by the interaction of 𝛿 electrons is much lower compared to that of protons or other ion

tracks. The threshold value is optimized empirically by testing a range of values and selecting the

ones that effectively suppress 𝛿 electron signals without affecting the detection of proton signals.

Fig. 4.9 illustrates a real (𝑑,2He) event with fitted tracks using the RANSAC algorithm. There are

some signals in the plot not attached to any particular track which are most likely the result of

electronic noise or remaining signal from 𝛿 electrons produced in the AT-TPC.

50

033Al,032Mg033Mg,032Naor decay products Figure 4.9 Real event from a (𝑑,2He) reaction visualized inside the active volume of the AT-TPC.
(a) The two tracks in 3D are formed from the hits using RANSAC algorithm. (b) Closest distance
between the two tracks, vertex reconstruction, projection of the last points on the vertex, and angles
of the tracks are shown for the same run and event as (a). (c) The same event as (a) and (b) where
the two tracks are seen to be intersecting at the vertex from a different perspective to show the
vertex is reconstructed at the closest distance in 3D.

.

For events where at least two tracks are identified, a vertex is determined by the center of the

closest distance between the two fitted tracks. Due to the insensitivity of the central region of the

AT-TPC, the vertex location was determined based on extrapolated tracks from the fitting process.

The vertex position was weighted by the number of hits on each track for tracks with less than

50 hit points. Without applying this weighting, the distribution of the closest distance between

the two tracks broadened because shorter tracks will have a larger uncertainty in the reconstructed

angle. This distance followed a roughly half-normal distribution, as shown by the fit of the distance

51

(a)(b)(c)between the two reconstructed tracks at the vertex in Fig. 4.10. The scale parameter 𝜎 of the

half-normal distribution ranged from 7 to 7.5 mm during the experiment. Events where the closest

distance exceeded 3𝜎 were excluded from the analysis.

Figure 4.10 Distribution of the closest distance between two reconstructed tracks at the vertex, fitted
with a half-normal Gaussian (red curve) for 33Al beam. The events on the right of dotted line were
excluded from the analysis.

.

In Fig. 4.9(b), the two straight lines in red were the two fitted tracks and the square in pink

was the reconstructed vertex for this event. The length of the tracks were defined as the distance

between the vertex and the last point on the fitted tracks, as labeled in Fig. 4.9(b). The last point is

the hit with the largest radius from the vertex. The angle of the tracks, labeled as 𝜃1 and 𝜃2, were

determined between the beam axis and the direction of the tracks. Events in which either or both

tracks escaped the active volume of the AT-TPC were excluded from the analysis since accurate

reconstruction of these events is not feasible.

As discussed in Sec. 3.3, space-charge effects can distort the tracks close to the beam line in

central region of the AT-TPC (see Fig. 4.11). This can impact the quality of the reconstructed

tracks. To address this issue, a correction factor was incorporated into the analysis code. Here, we

describe the process for correcting the hit points.

52

051015202530Closest Distance (mm)020406080CountsFigure 4.11 The two reconstructed tracks based on hit points. The 3D hit cloud illustrates the
deviation from a straight line near the beam region, where hits are drawn inward due to space
charge effects. Figure has been taken from Ref. [111].

.

If we treat the space charge as a long thin wire centered at (0,0), the electric field can be given

by:

(cid:174)𝐸 =

𝜆𝑧
2𝜋𝜖0𝜌𝐿

ˆ𝜌

(4.8)

where 𝜆 is the magnitude of the line charge at the window, 𝑧 is the distance from the pad plane, and

𝐿 is the length of the drift volume. There is a closed form solution to the Langavin equation in the

z direction for this model given by [111]:

𝑓 − 𝜌2
𝜌2

𝑖 =

𝜆𝑧2
2𝜋𝜖0𝐸𝑧 𝐿

(4.9)

where 𝜌2

𝑓 is the distance from space charge line after correction and 𝜌2
𝑖

is the distance from the

space charge line before correction. We can rewrite Eq. 4.9 as:

(𝜌 𝑓 + 𝜌𝑖) (𝜌 𝑓 − 𝜌𝑖) =

𝜆𝑧2
2𝜋𝜖0𝐸𝑧 𝐿

(4.10)

Here, we can define (𝜌 𝑓 − 𝜌𝑖) = Δ𝜌 and (𝜌 𝑓 + 𝜌𝑖) = 𝑟. If we assume 𝜌 𝑓 is really small, then 𝑟 ≈ 𝜌𝑖.

Therefore, the correction factor we need to apply to our hit points simply becomes:

Δ𝜌 =

𝑐𝑧2
𝜌𝑖

53

(4.11)

where 𝑐 = 𝜆/(2𝜋𝜖0𝐸𝑧 𝐿). In cylindrical coordinates, the hit points in our point cloud will have the

coordinates:

𝜌 =

√︁𝑥2 + 𝑦2

𝜃 = arctan (cid:16) 𝑦
𝑥

(cid:17)

𝑧 = 𝑧

The cylindrical coordinates of the hit points after correction becomes:

𝜌′ = 𝜌 + Δ𝜌 =

√︁𝑥2 + 𝑦2 + Δ𝜌 =

𝑐𝑧2
𝜌𝑖

𝜃′ = 𝑎𝑟𝑐𝑡𝑎𝑛

𝑦
𝑥

Now transforming the cylindrical units back into cartesian ones we get:

𝑥′ = 𝜌′𝑐𝑜𝑠𝜃 = (√︁𝑥2 + 𝑦2 +

𝑦′ = 𝜌′𝑐𝑜𝑠𝜃 = (√︁𝑥2 + 𝑦2 +

𝑐𝑧2
√︁𝑥2 + 𝑦2

𝑐𝑧2
√︁𝑥2 + 𝑦2

)𝑐𝑜𝑠(𝑎𝑟𝑐𝑡𝑎𝑛

)𝑠𝑖𝑛(𝑎𝑟𝑐𝑡𝑎𝑛

𝑦
𝑥

)

𝑦
𝑥

)

𝑧′ = 𝑧

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

In this analysis, we treat 𝑐 as a correction factor, with the primary variable being 𝜆, while all

other parameters remain constant. Thus, varying 𝑐 effectively corresponds to varying 𝜆. In our

code, we run the analysis for several values of 𝑐 to determine the optimal correction for the best

track fit quality. To identify the best fit, we compute the average 𝜒2 of the fits for two tracks and find

which value of 𝑐 yields the lowest 𝜒2. Once the optimal 𝑐 is identified, the subsequent calculations

54

proceed using that value. This method enables us to correct for track curvature on an event-by-event

basis.

In this (𝑑,2 He) experiment, the track curvature was minimal. Fig. 4.12 shows the count

distribution for the optimal 𝑐 values used in the analysis (with the y-axis on a logarithmic scale). It

is evident that for most events, the lowest 𝜒2 was achieved when 𝑐 = 0. We tested 𝑐 values from

0 to 1 × 10−4 in steps of 1 × 10−5, which causes the highest value of 𝑐 to be biased. If we had

extended the range, the final bin would likely have a similar count to the preceding bins. The events

with non-zero 𝑐 values likely arise from random coincidences. Since space-charge corrections did

not significantly improve the fits, they were not applied in the analysis.

Figure 4.12 Distribution of the optimal curvature parameter 𝑐 used in the track fitting analysis
for the 33Al(𝑑,2 He)33Mg reaction, plotted on a logarithmic scale. The majority of events exhibit
minimal track curvature, as indicated by the peak at 𝑐 = 0. The corresponding low 𝜒2 values show
that space-charge corrections did not significantly improve the fits, leading to the decision not to
apply them in the final analysis.

.

Another approach to improve the track fitting was to eliminate sections of the tracks close to

the beam axis, where most of the bending occurs due to space charge accumulation. We ran the

analysis multiple times, cutting off different initial track lengths. Fig. 4.13 presents the average 𝜒2

55

00.020.040.060.080.13-10·Optimal c110210310Countsof the fits, scaled by the total charge, for two tracks in three different cases: the black line represents

the average 𝜒2 with no portion of the tracks removed, the green line represents the average 𝜒2

after removing the first 10 mm to eliminate bending, and the red line shows the average 𝜒2 after

removing the first 40 mm.

While cutting off the first 40 mm resulted in fewer statistics due to excluding short tracks or

having too few remaining hits to reconstruct both tracks, no significant difference in statistics was

observed between the cases with no cut and with the 10 mm cut. In all trials, the charge-scaled

𝜒2 distributions were comparable, as observed in Fig. 4.13, indicating that removing portions of

the tracks did not impact the fit quality. Therefore, we decided to continue the analysis without

eliminating any parts of the tracks.

Figure 4.13 Distribution of the average 𝜒2 values scaled by total charge for two tracks under three
different conditions: no track section removed (black line), first 10 mm of the tracks removed
(green line), and first 40 mm of the tracks removed (red line). While removing the first 40 mm
reduced the overall statistics, the 𝜒2 distributions remain comparable across all cases, indicating
that eliminating portions of the tracks near the beamline does not significantly impact the quality
of the fits.

.

In the experiment, the alignment of the AT-TPC’s axis with the beam axis was not perfect.

56

00.010.020.030.040.050.060.072CAverage 050100150200Counts0 mm10 mm40 mmSince this impacts the kinematic reconstruction, this angle was determined and used in the further

analysis. The angles were calculated based the reconstructed vertices from 2 track events, including

ones in which one or both tracks escaped the chamber.

Fig. 4.14(a) illustrates a scatter plot of the vertex positions in the x-axis (vertexX) versus the

longitudinal axis (vertexZ) inside the AT-TPC. The trendline fitted to these points gives the beam’s

angle relative to the x-axis, along with the corresponding offset at the entrance window. Similarly,

Fig. 4.14(b) shows a plot of the vertex positions in the y-axis (vertexY) against vertexZ, with the

fitted line providing the beam’s angle and offset relative to the y-axis.

Figure 4.14 Reconstructed vertex locations from two-track events in the (a) xz-plane and (b) yz-
plane. The red lines are the fits and indicate the beam axis.

57

These angles and offsets are essential for aligning the experimental setup with the beam axis

and are summarized in Table 4.2, which reports the exact values for the beam’s angular deviation

and the corresponding offsets in both the x and y directions. This correction ensures the precision

of the particle tracking data and helps improve the overall accuracy of the analysis.

Table 4.2 Angles and offsets of the beam in 𝑥 and 𝑦 directions at the entrance window of AT-TPC.
The offset Δ is the offset at 𝑧 = 0.

𝜽beam,𝒙 (deg) 𝚫beam,𝒙 (mm)

𝜽beam,𝒚 (deg) 𝚫beam,𝒚 (mm)

-0.393

-0.89

-0.026

-1.807

4.5 Drift Velocity Correction

The drift velocity of particles inside the AT-TPC gradually decreased over time, possibly due

to the accumulation of contaminants, necessitating a correction. Since the drift length of particles

within the AT-TPC volume is 1000 mm, we expect the reconstructed vertex Z positions to be

constrained within the physical limits of the AT-TPC. Specifically, the vertex distribution along the

Z-axis should span from 0 to 1000 mm. If the vertex distribution stretches beyond this range or ends

before reaching 1000 mm, it indicates that a drift velocity correction is required. To address this,

we apply a straightforward correction procedure on run-by-run basis. The drift velocity correction

was performed with one-track events to improve the accuracy of the correction.

The drift velocity corrections applied during the first and second (𝑑,2 He) experiments in

inverse kinematics were not identical. In the first experiment, for the analysis of the 14O(𝑑,2 He)

and 13N(𝑑,2 He) reactions, the one-track events were analyzed under the assumption of a fixed drift

velocity, which was calculated using the computational program Magboltz [112]. The vertex Z

distribution for each run was then fit with a Fermi function to determine the edge of the distribution,

which provided an approximation of the drift length based on the assumed fixed drift velocity. The

updated drift velocity was subsequently calculated using the following relationship:

New Drift Velocity =

Fixed Drift Velocity × Drift Length in the AT-TPC
Edge of vertex Z distribution from fit

58

The corrected drift velocity was then applied to more accurately reconstruct the vertex Z

positions. After applying this correction, the edge of the vertex Z distribution should align closely

with 1000 mm as shown in Fig. 4.15, which corresponds to the maximum drift length inside the

AT-TPC volume.

Figure 4.15 Maximum drift length as a function of the run number for the 14O,13N(𝑑,2He) experi-
ment, before (red) and after (blue) the drift velocity correction. The uncertainties shown represent
the uncertainties in fitting the maximum drift distance.

For the second (𝑑,2 He) experiment using 33Al and 32Mg beams, the limited statistics made

drift velocity correction more challenging. In several runs, the data were insufficient to accurately

determine the edge of the vertex Z distribution, making it difficult to apply run-by-run corrections.

As in the first (𝑑,2 He) experiment, one-track events were processed using a pre-determined fixed

drift velocity. However, with this fixed velocity, the vertex Z distribution extended beyond the

physical boundaries of the AT-TPC, and the distribution’s edge showed a linear decline with

increasing run numbers, indicating the need for a correction (see Fig. 4.16(a)). Due to the lack of

sufficient statistics, run-by-run corrections could not be performed.

To address this issue, we first determined the vertex Z distribution edge for the runs where

sufficient statistics were available. These data points were then used to fit a linear relationship

between the edge of the distribution and the run number. This linear fit was applied to estimate

the drift velocity correction across all runs, allowing for a more consistent correction despite the

59

Range [in mm]Run No.lower statistics in some runs. After applying the corrections, the edge of the vertex Z distribution

aligns at 1000 mm. Fig. 4.16 shows the vertex Z distribution (a) before, and (b) after correcting

for drift velocity across all runs in the experiment. A 2% change in the drift velocity shifts

the excitation energy spectrum by approximately 100 keV. For runs 80–112, we applied a lower

charge threshold to prevent missing good events, resulting in a higher number of recorded events.

However, these additional events were found not to contain additional (𝑑,2He) events and were

primarily background from 𝛿 electrons. As a result, we reverted to our initial charge threshold

setting from run 113.

60

Figure 4.16 The vertex Z distribution across all run numbers: (a) The distribution before any
corrections are applied. (b) The distribution after correcting for drift velocity. The dashed black
line represents where the edge of the vertex Z distribution was determined to be.

4.6 Simulation

Simulations play a crucial role in understanding the detector response and correcting for the

limited acceptance.

It also enables benchmarking and a deeper interpretation of experimental

data. The attpcroot package [113, 114], written in C++ programming language and developed

within the fairroot framework [115], was employed. This package provides comprehensive

61

Run NumberRange (mm)tools for both simulation and data analysis. The simulation process is divided into three main

stages: event generation, digitization, and reconstruction. Notably, the same analysis code used for

reconstructing simulated data is applied to experimental data. This reduces systematic uncertainties

with application of acceptance corrections determined in the simulation to the analysis of the data.

A brief description of each simulation stage is provided below, with more detailed information

available in Ref. [107].

4.6.1 Event Generator

The first stage of the simulation focuses on generating realistic (𝑑,2He) events, comparable to

the process of obtaining raw data during the actual experiment. This simulation stage is executed

using the Virtual Monte-Carlo (VMC) package [5], which serves as an interface to the geant4

toolkit [6]. Within this simulation framework, the geometry of the detection system, beam optics,

and all other essential components are thoroughly defined. The event generator class produces

the particles, which are then transported through the system by geant4, and the corresponding hit

information is stored in a class that handles simulated data points.

In the simulation, we define the AT-TPC as having a cylindrical geometry filled with D2 gas,

consistent with the actual experiment setup. The beam parameters-such as mass, charge, and

energy-are specified, alongside a detailed implementation of beam optics. This includes the beam

angular spread (see Sec. 4.1), the beam momentum spread, and the beam size. These aspects of the

beam optics are important for accurately simulating the transport of the beam and its interaction

with the target gas inside the AT-TPC.

Additionally, the simulation incorporates the in-flight decay of two protons from 2He. The

consistency of the (𝑑,2He) reaction mechanism is maintained by integrating relativistic three-body

kinematics for generating particles in the final state. The acceptance of 𝜖 𝑝 𝑝, the relative energy

between the two protons, depends on the scattering angle of 2He. The simulation uses theoretical

differential cross-sections from accba as inputs (see Sec. 2.3). These cross-sections correspond

to different angular momentum components (Δ𝐿 = 0, 1, 2). To analyze the data, the simulation

is run separately for each Δ𝐿 value. This step is important for isolating the Δ𝐿 = 0 component

62

from the experimental cross-section using the multipole decomposition analysis. The (𝑑,2He) event

generator creates events based on the reaction cross-section, ensuring a realistic representation of

the process. The number of counts in each bin of 𝜃𝑐𝑚 and 𝜖 𝑝 𝑝 depends on the cross-section for that

bin and the total number of events generated in the simulation.

4.6.2 Event Digitizer

This stage follows the event generator and is comparable to the unpacking of raw data during

the experimental phase. Its primary objective is to simulate the detector’s response to the tracks

generated in the previous stage. The stage consists of two key subroutines.

The first subroutine simulates electron diffusion, where the number of ionization electrons

produced for each hit and the drift time of the electrons are calculated. Electron diffusion inside the

AT-TPC volume introduces uncertainty in the reconstructed excitation energy. A larger diffusion

parameter causes broader proton tracks, which subsequently degrade the precision of the track

fitting. For the digitization stage of simulated events in our analysis, we utilized parameters that

reflect the realistic conditions achieved with the AT-TPC filled with deuterium gas (D2). Several

experimental parameters such as the electric field, gas pressure, the average gain of the micromegas

detector, the gain of the GET electronics, and the electron drift velocity are directly obtained from

the experimental setup to ensure consistency. In addition to these, we employed realistic values for

key gas properties such as the ionization energy of D2, gas density, and the Fano factor of the gas,

which is crucial for modeling ionization fluctuations. These values are crucial for simulating the

electron transport properties and signal formation within the AT-TPC. Table 4.3 shows the values

used for these parameters in the digitization process. Fig. 4.17 shows a projection of a simulated

event with two tracks on the pad plane, providing a visual representation of how the diffusion and

digitization stages influence the track shapes.

63

Table 4.3 Parameters and their corresponding values used in the digitization of events in the AT-
TPC.

Parameter
Electric Field
Gas Pressure
Average Gain of Micromegas Detector
Gain of GET Electronics
Electron Drift Velocity
Ionization Energy of D2
Gas Density
Fano Factor of D2
Longitudinal Coefficient of Electron Diffusion
Transverse Coefficient of Electron Diffusion

Value
5000 V/m
530 Torr
1000
1000
0.89 cm/𝜇s
13.6 eV
0.1143 kg/m3
0.24
0.0038 1/m
0.0014 1/m

The second subroutine models the electronic response. The high gain required in the electronics

for detecting proton tracks in the AT-TPC significantly increases the likelihood of detecting 𝛿-rays,

as previously discussed. To mitigate this, the output of this subroutine is coupled to an electronics

response module, which simulates the signal for each pad. This is essential for realistic signal

processing in the analysis pipeline.

64

Figure 4.17 Projection of simulated (𝑑,2He) event on the Micromegas plane. Beam-like particles
pass through the 3 cm aperture in the central region. Figure obtained from Ref. [107].

4.6.3 Reconstruction

Once the events are generated and the detector response is accounted for, the reconstruction

stage of the simulation mirrors the process used for experimental data. Similar to experimental

data, the RANSAC algorithm is employed to fit the particle tracks for all events, and to identify

the (𝑑,2 He) reactions. The momenta of the two recoiling protons are extracted, allowing for

the reconstruction of 𝜖 𝑝 𝑝 through an invariant-mass calculation. A missing-mass calculation is

performed to reconstruct the excitation energy spectrum of the ejectile. Mathematical details of

the invariant-mass and missing-mass calculations are provided in Sec. 4.7. Fig. 4.18 illustrates the

reconstructed excitation energy for the ground state in 33Mg. The simulated ground state has an

energy resolution of approximately 0.5 MeV.

65

Figure 4.18 Simulated reconstructed excitation energy of 33Mg ground state with a full width at
half maximum of about 0.5 MeV.

4.6.4 Acceptance Corrections

The simulation is used for the acceptance corrections for the detection of two protons from

the 2He particle and the residual nucleus in the focal plane of the S800 are crucial for extracting

accurate differential cross-sections.

It is insightful to look at the correlation between 𝜖 𝑝 𝑝 and 𝜃𝑐.𝑚. in the first (𝑑,2 He) experiment

for the 14O beam channel to understand how the acceptance varies with the reaction kinematics.

Fig. 4.19 illustrates the correlation between 𝜖 𝑝 𝑝 and 𝜃𝑐.𝑚. for the transition from 14O(0+; ground

state) to 14N(1+; 3.95 MeV), which was the strongest transition observed in the first (𝑑,2He)

experiment. The ground state 𝑄 value of this reaction is 3.702 MeV, which is relatively small. As

a result, the momentum transfer during the reaction remains small at low center-of-mass scattering

angles (𝜃𝑐.𝑚. < 2◦). At low values of 𝜖 𝑝 𝑝(< 1 MeV), the protons produced in the reaction do not

have sufficient energy to escape the central, insensitive region of the AT-TPC. Therefore, higher

𝜖 𝑝 𝑝 values are required for both protons to be detected. On the other hand, at higher 𝜃𝑐.𝑚., the

momentum transfer increases, leading to higher-energy protons. When this occurs, one or both

of the protons may escape the active volume of the AT-TPC, making their reconstruction difficult

or impossible. Consequently, for larger angles (𝜃𝑐.𝑚. > 4◦), the analysis is restricted to lower 𝜖 𝑝 𝑝

values (< 1.5 MeV), as can be observed in Fig. 4.19.

66

Figure 4.19 Correlation between the relative proton energy 𝜖 𝑝 𝑝 and the center-of-mass scattering
angle 𝜃c.m. for the experimental data from the 14O(𝑑,2He)14N reaction inside the sensitive region
of AT-TPC. The figure has been taken from Ref. [68].

The 𝑄-value for the 33Al(𝑑,2He) reaction is -14.902 MeV for the ground state resulting in a

higher momentum transfer for the reconstructed 2He in the 33Al reaction. As a result, for 𝜃𝑐.𝑚. ≈ 0◦,

protons from the 2He decay with 𝜖 𝑝 𝑝 > 0.5 MeV are able to escape the central, insensitive region

of the AT-TPC, unlike in the 14O(𝑑,2He) reaction where higher 𝜖 𝑝 𝑝 is required for this to occur.

However, at moderately high values of 𝜖 𝑝 𝑝 > 2 MeV and 𝜃𝑐.𝑚. > 6◦, the protons are more likely to

escape the active volume of the AT-TPC due to the larger momentum transfer, as seen in Fig. 4.20.

This explains why for the 14O(𝑑,2He) reaction, events can be detected up to 𝜖 𝑝 𝑝 ≈ 3.5 MeV, whereas

in the case of the 33Al(𝑑,2He) reaction, detection is constrained to 𝜖 𝑝 𝑝 ≈ 2 MeV. Similarly, the

center-of-mass angle 𝜃𝑐.𝑚. can reach up to 7◦ for the 14O(𝑑,2He) reaction, but is mostly limited to

6◦ for the 33Al(𝑑,2He) reaction.

67

εpp(MeV)Θc.m. (deg)Figure 4.20 Correlation between the relative proton energy 𝜖 𝑝 𝑝 and the center-of-mass scattering
angle 𝜃c.m. for the experimental data from the 33Al(𝑑,2He)33Mg reaction inside the sensitive region
of AT-TPC.

Simulations are crucial for accounting the differences in detector acceptance arising from the

specific kinematics of each reaction. Fig. 4.21(a) and 4.21(b) show the correlation between 𝜖 𝑝 𝑝 and

𝜃𝑐.𝑚. from simulations run with an input energy of 8 MeV for the monopole (Δ𝐿 = 0) and dipole

(Δ𝐿 = 1) cross-sections from accba, respectively.

68

0123456789Figure 4.21 Simulated correlation between 𝜖 𝑝 𝑝 and 𝜃𝑐.𝑚. for 8 MeV input energy. (a) Shows the
correlation for monopole transitions (Δ𝐿 = 0) and (b) for dipole transitions (Δ𝐿 = 1), based on
cross-sections from accba.

Additionally, the correlation observed in Fig. 4.19 and 4.20 is impacted by the excitation energy

of the residual nucleus. As the excitation energy increases, it results in greater momentum transfer

to the 2He system and an increase in the energy of the two protons. These effects are included in

the simulation.

When the excitation energy surpasses the threshold for particle emission, the momentum-kick

from the emitted particles affect the residual particle’s momentum, causing a broadening in both

momentum and angular distributions. This broadening can lead to a loss of events due to the limited

69

(a)(b)momentum and angular acceptances of the S800. The AT-TPC exit hole has a diameter of 4 cm,

which also contributes to event loss due to the expanded angular distribution. When determining

the cross section, the S800’s acceptance must be taken into account. Two key factors contribute to

losses: (1) a portion of the momentum distribution lies outside the S800’s momentum acceptance,

and (2) some beam residues are lost due to the limited size of the AT-TPC exit hole. This loss is

more significant for lighter residues because they receive a larger momentum kick from neutron or

proton decay. In the first (𝑑,2He) experiment conducted in inverse kinematics with 14O and 13N

beams, more events were lost at higher excitation energies, as many did not reach the S800 focal

plane. Specifically, 31% of 32Na fragments were lost due to the S800 acceptance limits, while all

other fragments were fully transmitted to the focal plane.

4.7 Kinematic Event Reconstruction

The energies of the protons in the (𝑑,2He) events were determined from their path lengths using

an energy-loss look-up table calculated using srim[116], as illustrated in Fig. 4.22.

Figure 4.22 Energy loss as a function of proton path lengths calculated using srim.

The path lengths and track angles were used to reconstruct the momentum vectors of the two

protons, (cid:174)𝑃𝑝1 and (cid:174)𝑃𝑝2. The momentum of the 2He were then calculated from the proton tracks using

the momentum conservation:

70

05001000150020002500Range (mm)012345E (MeV)D(cid:174)𝑃2𝐻𝑒 = (cid:174)𝑃𝑝1 + (cid:174)𝑃𝑝2,

(4.20)

The total mass of the 2He system, 𝑀2𝐻𝑒, was obtained using an invariant-mass calculation:

√︃

𝑀2𝐻𝑒 =

(𝐸 𝑝1 + 𝐸 𝑝2)2 − 𝑃2

2He

,

(4.21)

where 𝐸 𝑝1 and 𝐸 𝑝2 are the energies of the two protons. The relative energies between the two

protons, 𝜖 𝑝 𝑝, can then be calculated using the following equation:

𝜖 𝑝 𝑝 = 𝑀2𝐻𝑒 − 2𝑚 𝑝,

(4.22)

where 𝑚 𝑝 is the mass of a proton. The 𝜖 𝑝 𝑝 is used in the missing-mass calculation, which is then

used for the excitation-energy reconstruction. In order to find the missing mass, it is important to

first find the missing energy and the missing momentum. The missing energy, M.E., is calculated

by:

𝑀.𝐸 . = 𝐸beam + 𝑚beam + 𝑚𝑑 − (𝐸2He + 𝜖 𝑝 𝑝 + 𝑚2He + 0.511),

(4.23)

where 𝐸beam and 𝑚beam are the kinetic energy and mass of the projectile beam, 𝑚𝑑 is the mass of

target deuteron, 𝐸2He is the kinetic energy of the reconstructed 2He, 𝑚2He is the mass of 2He, and

0.511 is the rest energy of an electron in MeV. The kinetic energy of the 2He particle is calculated

using:

√︃

𝐸2He =

(cid:174)𝑃2
2He + 𝑀 2

2𝐻𝑒 − 𝑀2𝐻𝑒

The missing momentum, 𝑀.𝑃., can be simply calculated using:

√︃

𝑀.𝑃. =

beam − (cid:174)𝑃2
(cid:174)𝑃2
2He

(4.24)

(4.25)

where (cid:174)𝑃beam is the momentum of the projectile beam. We calculate the missing mass, 𝑀.𝑀.:

71

𝑀.𝑀. =

√︁𝑀.𝐸.2 − 𝑀.𝑃.2

Finally, we can calculate the reconstructed excitation energy:

𝐸𝑥 = 𝑀.𝑀. − 𝑚prod

(4.26)

(4.27)

where 𝑚prod is the rest mass of the reaction product.

The theoretical and experimental kinematics of the 33Al(𝑑,2He) reaction are shown in Fig. 4.23.

The theoretical calculations (represented by circles) are calculated with 𝜖 𝑝 𝑝 = 0 MeV, while the

experimental data (represented by diamonds) span a range of 𝜖 𝑝 𝑝 values. This variation in 𝜖 𝑝 𝑝 leads

to an upward shift in the experimental data compared to the theoretical predictions.

Figure 4.23 Theoretical (circles) and experimental (diamonds) kinematics of the 33Al(𝑑,2He) re-
action. The theoretical kinematics are calculated for 𝜖 𝑝 𝑝 = 0 MeV, while the experimental data
correspond to a range of 𝜖 𝑝 𝑝 values, leading to the observed upward shift relative to theory.

4.8 Event Selection

Fig. 4.24 illustrates the excitation energy spectrum derived from the calculation after applying

a gate on the 33Al beam. In this plot, noticeable background contributions appear at both negative-

72

01234560102030405060708090Elab (2He) [MeV]ϴlab (2He) [deg]Data Ex=0-5 MeVData Ex=10-15 MeVData Ex=5-10 MeVand high-energy regions. These background contributions are attributed to various sources. It is

important to apply additional gates to filter out these background events.

Figure 4.24 Excitation energy spectrum of the 33Al(𝑑,2He)33Mg reaction. The background contri-
butions arise from various sources, with elastic scattering events being the predominant factor.

Fig. 4.25 shows a plot of the scattering angles of two proton tracks (𝜃1 and 𝜃2) relative to the

beam axis. The red squares represent events with excitation energies above 0 MeV in Fig. 4.24,

while the blue squares represent those below 0 MeV. A band near 1.57 radians appears in both 𝜃1

and 𝜃2. This pattern is consistent with elastic scattering of deuterium off the beam, which peaks

at 0◦ in the c.m.

frame and near 90◦ in the laboratory frame. As the cross section for elastic

scattering is very high, there is a chance that two of such events happen within a short time and

sufficiently close in distance to create a random 2-track coincidence event creating background for

the (𝑑,2He) channel. A second cluster of events is visible near 0 radians for 𝜃2. These events are

due to a high event rate in a few pads of the pad plane, associated with the beam hitting these

pads, or noisy/sparking pads, introducing background. However, by setting a gate on the angular

correlation spectrum of Fig. 4.25, this background can be removed.

73

80-60-40-20-02040Mg) [MeV]33 (xE020406080100CountsFigure 4.25 Correlation between the proton track angles, 𝜃1 and 𝜃2. The red triangles represent
events with excitation energies greater than 0 MeV, while the blue squares correspond to events with
excitation energies below 0 MeV. A distinct pattern is evident, particularly with bands at 𝜃1 ≈ 1.57
rad and 𝜃2 ≈ 1.57 rad indicating the presence of elastic scattering events.

To determine the appropriate size of the gate in the correlation between 𝜃1 and 𝜃2, we used the

simulation framework. Fig. 4.26 illustrates the simulated proton angles, along with the gate drawn

around the valid (𝑑,2He) events. This gate was be applied to the experimental data to largely filter

out the background events.

74

Figure 4.26 (a) Correlation between proton track angles, 𝜃1 and 𝜃2, from simulated events across
different excitation energy ranges. Red corresponds to the 0-4 MeV range, blue to the 4-8 MeV
range, and green to the 8-12 MeV range. The black gate highlights the region used in the analysis
to select only good (𝑑,2 He) events. (b) The gate obtained from simulated data has been applied to
the data.

4.9 Excitation Energy Spectra and Differential Cross-Sections

Fig. 4.27 shows the excitation energy spectra for: (a) 33Mg, obtained by selecting events with

both 33Mg recoil and its decay product, 32Mg; and (b) 32Na, created by selecting events with 32Na

recoil and its decay product, 31Na. The excitation energy was calculated using Eq. 4.27.

75

(a)(b)0-4 MeV4-8 MeV8-12 MeVFigure 4.27 Excitation energy spectra of (a) 33Al(𝑑,2 )He) and (b) 32Mg(𝑑,2 )He) reactions.

The differential cross-sections are calculated for energy bins using the following formula:

𝑑𝜎
𝑑Ω

=

𝑁𝑏𝑖𝑛
𝑁𝑏𝑒𝑎𝑚 · ΔΩ · 𝑁𝑡𝑎𝑟𝑔𝑒𝑡 · 𝜀

(4.28)

where 𝑁𝑏𝑖𝑛 is the number of counts in each energy bin, 𝑁𝑏𝑒𝑎𝑚 is the number of beam particles,

ΔΩ is the solid angle, 𝑁𝑡𝑎𝑟𝑔𝑒𝑡 is the number of target particles. 𝜀 accounts for the S800 acceptance

of residuals and the acceptance of the two protons inside the AT-TPC, combined with corrections

applied due to all the gates used in the analysis. The beam particle count, 𝑁𝑏𝑒𝑎𝑚, is weighted by the

transmission efficiency of 31.2% from the object scintillator to the focal plane. 𝑁𝑡𝑎𝑟𝑔𝑒𝑡 is calculated

as:

𝑁𝑡𝑎𝑟𝑔𝑒𝑡 =

𝜌 𝐴 · 𝑁 𝐴
𝑀𝐷2

(4.29)

where 𝜌 𝐴 = 11.69 mg/cm2 is the surface density, 𝑁 𝐴 = 6.02 × 1023 is the Avogadro’s number, and

𝑀𝐷2 = 4028 mg/mol is the molar mass for 𝐷2 gas.

76

(a)(b)Fig. 4.28 shows the integrated cross-sections as a function of 𝐸𝑥 for 𝜃𝑐.𝑚. less than 6◦: (a) for

the 33Al(𝑑,2He)33Mg reaction and (b) for the 32Mg(𝑑,2He)32Na reaction.

Figure 4.28 Differential cross-sections for: (a) 33Al(𝑑,2He)33Mg and (b) 32Mg(𝑑,2He)32Na reac-
tions. The dashed lines represent the separation energies. (a) and (b) show differential cross-section
up to 12 MeV and 8 MeV, respectively, but may have missing cross-section above 𝑆2𝑛 as 31Mg and
30Na fragments were not detected in the S800 focal plane for the chosen 𝐵𝜌 setting.

Differential cross-sections were also calculated as a function of 𝜃𝑐.𝑚.. Fig. 4.29 shows angular

distributions for the 33Al(𝑑,2He)33Mg reaction. Subfigures correspond to the following energy

ranges: (a) -1 to 1 MeV, (b) 1-4 MeV, (c) 4-6 MeV, (d) 6-8 MeV, (e) 8-10 MeV, and (f) 10-12 MeV.

77

𝑆𝑛𝑆𝑛𝑆2𝑛𝑆2𝑛(a)(b)Figure 4.29 Angular distributions of differential cross-sections for the 33Al(𝑑,2He)33Mg reaction
for different energy ranges.

Similarly, Fig. 4.30 shows angular distributions for the 32Mg(𝑑,2He)32Na reaction. Subfigures

correspond to the following energy ranges: (a) -1 to 1 MeV and (b)-(f) 1-MeV-wide intervals up to

78

-1 < Ex [MeV] 1 (a)(b)(c)(d)(e)(f)1 < Ex [MeV] 4 4 < Ex [MeV] 6 6 < Ex [MeV] 8 8 < Ex [MeV] 10 10 < Ex [MeV] 12 6 MeV.

Figure 4.30 Angular distributions of differential cross-sections for the 32Mg(𝑑,2He)32Na reaction
for different energy ranges.

The vertical error bars in all differential cross-section plots represent statistical uncertainties.

The horizontal bars in the angular distribution plots indicate the bin widths of the angular intervals.

The systematic uncertainty from all gating conditions was negligible compared to the statistical

uncertainty and was therefore excluded from the analysis.

79

-1 < Ex [MeV] 1 (a)(b)(c)(d)(e)1 < Ex [MeV] 2 3 < Ex [MeV] 4 4 < Ex [MeV] 5 5 < Ex [MeV] 6 64.10 Multipole Decomposition Analysis

As described in Chap. 2, the differential cross-section from the experimental data can have

contributions from various Δ𝐿 components. We use the MDA to extract GT component (Δ𝐿 = 0)

of the differential cross-section using Eq. 2.1.

4.10.1 33Al(𝑑,2He)33Mg Reaction

A state is observed at 𝐸𝑥 (33Mg) ∼ 0 MeV (see Fig. 4.29). The angular distribution of this state

peaks at 0◦, as illustrated by Fig. 4.29(a). This is indicative of a Δ𝐿 = 0 transition, and thus a GT

transition. As discussed in Sec. 1.2, a GT transition from the (5/2)+ ground state to the ground

state in 33Mg is highly unlikely, as current evidence points to the ground state of 33Mg having a

spin-parity of 3/2− [39, 40, 41]. States at 484 keV and 546 keV are tentatively identified as having

spin-parities of (5/2−) and (3/2−) [117], respectively, although Ref. [41] suggested a positive parity

for the 484 keV state. A state at 705 keV has tentatively been assigned a spin-parity of (3/2, 5/2+).

Since the uncertainty in the reconstructed excitation energy due to uncertainty in drift velocity is

∼100 keV (see Sec 4.5), it is unlikely that the observed state is associated with any of the excited

states at 484 keV or above. We concluded that the observed state is likely a low-lying excited state

previously unobserved. However, since the statistical uncertainties are large, it was also investigated

whether a misidentification is possible, and if it is a dipole transition to the ground state.

Due to the properties of the nucleon-nucleon interaction, transitions to final states that can be

associated with Δ𝐿 = 0, also have Δ𝐿 = 2 contributions. Fig. 4.31(a) shows a fit of the angular

distribution with Δ𝐿 = 0 and Δ𝐿 = 2 contributions, resulting in reduced 𝜒2 of 0.29. Fig. 4.31(b)

shows a fit with a Δ𝐿 = 1 angular distribution, resulting in a reduced 𝜒2 of 0.95. While we cannot

rule out that the transition proceeds with Δ𝐿 = 1, it is more likely that there is a strong Δ𝐿 = 0

component.

80

Figure 4.31 MDA fit of the experimental angular distribution for the 33Mg between -1 and 1 MeV
in the 33Al(𝑑,2He) reaction and the simulated angular distribution at 0 MeV.

The experimental angular distributions for other energy ranges in Fig 4.29(b)-(f) were also fit

with simulated angular distributions. Ref. [41] proposed spin-parities of 1/2− and 3/2− for the

states at 1.24 MeV and 1.85 MeV. In our analysis, the angular distribution in the 1 to 4 MeV range

is dominated by dipole (Δ𝐿 = 1) contributions, as illustrated in Fig. 4.32, although minor other

contributions are not excluded.

81

(a)(b)-1 < Ex [MeV] 1 ∆𝐿 = 0∆𝐿 = 1∆𝐿 = 2FitDataFigure 4.32 Fit of the experimental angular distribution for the 33Al(𝑑,2He) reaction between 1 and
4 MeV and the simulated Δ𝐿 = 1 angular distribution at 2.5 MeV.

For all angular distributions with 𝐸𝑥 > 4 MeV, the angular distributions can be reproduced

by assuming dominant quadrupole transitions (Δ𝐿 = 2). Fig. 4.33 shows the fit for the following

energy ranges: (a) 4-6 MeV, (b) 6-8 MeV, (c) 8-10 MeV, and (d) 10-12 MeV. There is no prior

experimental information available for states at these energies, and based on the fit results, it is not

possible to assign any B(GT) to states at these excitation energies.

82

1 < Ex [MeV] 4 ∆𝐿=1DataFitFigure 4.33 MDA fit of the experimental angular distribution for the 33Al(𝑑,2He) reaction for the
energy ranges: (a) 4-6 MeV, (b) 6-8 MeV, (c) 8-10 MeV, and (d) 10-12 MeV with the simulated
angular distribution for: (a) 5 MeV, (b) 7 MeV, (c) 9 MeV, and (d) 11 MeV.

4.10.2 32Mg(𝑑,2He)32Na Reaction

The strongest peak in the 32Mg(𝑑,2He)32Na reaction differential cross-section spectrum occurs

between 3 and 4 MeV, with the peak occurring between 3 and 3.5 MeV (see Fig. 4.27). As the

width of this peak is close to the experimental resolution of 0.5 MeV, it is likely that this peak

belongs to a single excited state. The state peaks at forward angles indicating a Δ𝐿 = 0 transition.

We fit the experimental differential cross section in this energy range with simulated differential

cross-sections for Δ𝐿 = 0 and Δ𝐿 = 2 components at 𝐸𝑥 = 3.5 MeV, shown in Fig. 4.34.

83

(a)(b)(c)∆𝐿=2Data(c)(d)4 < Ex [MeV] 6 6 < Ex [MeV] 8 8 < Ex [MeV] 10 10 < Ex [MeV] 12 Figure 4.34 MDA performed for the 32Mg(𝑑,2He) reaction in the 3 to 4 MeV excitation energy
range, with simulated angular distributions for Δ𝐿 = 0 and Δ𝐿 = 2 at 3.5 MeV.

The forward peaking behavior in the angular distribution was also observed between 1 and 2

MeV in Fig. 4.30. To extract the GT transition in this region, we fit the experimental differential

cross section in this energy range with simulated differential cross-sections for Δ𝐿 = 0 and Δ𝐿 = 2

components, shown in Figs. 4.35(a) and (b), respectively, at 𝐸𝑥 = 1.5 MeV.

Figure 4.35 MDA performed for the 32Mg(𝑑,2He) reaction in the 1 to 2 MeV excitation energy
range, with simulated angular distributions for Δ𝐿 = 0 and Δ𝐿 = 2 at 1.5 MeV.

For the energy ranges -1 to 1 MeV and 4 to 5 MeV, the angular distributions are likely dominated

84

∆𝐿=2DataFit∆𝐿=03 < Ex [MeV] 4 ∆𝐿=2DataFit∆𝐿=01 < Ex [MeV] 2 by the quadrupole (Δ𝐿 = 2) contribution, as shown in Figs. 4.36(a) and (b). However, the statistical

uncertainties are so large it is difficult to be sure about the nature of transitions in these excitation

energy ranges.

Figure 4.36 Fit of the experimental angular distribution for the 32Mg(𝑑,2He) reaction with the
Δ𝐿 = 2 for the energy ranges:
(a) -1 to 1 MeV and (b) 4 to 5 MeV with simulated angular
distribution for: (a) 0 MeV and (b) 4.5 MeV.

Finally, the angular distribution for the 5-6 MeV range appears to be dominated by dipole

(Δ𝐿 = 1) transition, as shown by Fig. 4.37. The statistical uncertainties are too large to draw a

definitive conclusion.

85

-1 < Ex [MeV] 1 4 < Ex [MeV] 5 ∆𝐿=1Data(a)(b)Figure 4.37 MDA fit of the experimental angular distribution for the 32Mg(𝑑,2He) reaction with
simulated angular distributions for: (a) -1 to 1 MeV and (b) 5 to 6 MeV.

4.11 B(GT) Extraction

The differential cross-sections were extrapolated to values at zero momentum transfer (𝑞 = 0)

and 𝜖 𝑝 𝑝 < 1 MeV. This extrapolation was carried out using DWBA calculations with the accba

code and the simulation. The relationship used in the extrapolation is given by:

𝑑𝜎
𝑑Ω

(cid:0)𝜃 ≈ 0◦, 𝑞 = 0, 𝜖 𝑝 𝑝 < 1 MeV(cid:1) =

(cid:32) 𝑑𝜎
𝑑Ω

(cid:0)𝜃 ≈ 0◦, 𝑞 = 0, 𝜖 𝑝 𝑝 < 1 MeV(cid:1)
(cid:0)𝜃 ≈ 0◦, 𝑞, 𝜖 𝑝 𝑝 < 1 MeV(cid:1)
𝑑𝜎
𝑑Ω
(cid:32) 𝑑𝜎
𝑑Ω

(cid:0)𝜃 ≈ 0◦, 𝑞, 𝜖 𝑝 𝑝 < 1 MeV(cid:1)
(cid:0)𝜃, 𝑞, 𝜖 𝑝 𝑝(cid:1)

(cid:33)

𝑑𝜎
𝑑Ω

(cid:33)

accba

(cid:18) 𝑑𝜎
𝑑Ω

(cid:0)𝜃 ≈ 0◦, 𝑞, 𝜖 𝑝 𝑝(cid:1)

(cid:19)

.

exp

×

×

Simulation

(4.30)

In this equation, the third term of the product is the extracted experimental differential cross section

at the lowest 𝜃 bin. The second term extrapolates the experimental differential cross section to

𝜖 𝑝 𝑝 < 1 MeV based on the simulated acceptance. And the first term extrapolated the differential

cross section to 𝑞 = 0.

86

5 < Ex [MeV] 6 ∆𝐿=1DataFitThe B(GT) was calculated from the extrapolated differential cross section at low scattering

angles and 𝜖 𝑝 𝑝 < 1 MeV using the proportionality relationship in Eq. 1.12. The value of ˆ𝜎 used for

calculating B(GT) from the extrapolated cross-section was obtained from the 32S(𝑑,2He) reaction

study for 𝜖 𝑝 𝑝 < 1 MeV [118]. For different CE reactions, it has been shown that the unit cross

section varies smoothly with mass number 𝐴 [119], therefore, using a known unit cross section

for 𝐴 = 32 is reasonable for the systems studied here. Even though there is limited experience

with nuclei far away from the valley of stability, which could introduce systematic uncertainties,

these uncertainties are significantly smaller than the statistical uncertainties reported here. For the

32S(𝑑,2He) reaction, the unit cross section could be determined from 4 transitions, at 𝐸𝑥 = 1.15

MeV, 4.06 MeV, 5.41 MeV, and 6.65 MeV, with B(GT) = 0.344±0.004, 1.005±0.006, 0.087±0.002,

and 0.068 ± 0.023, respectively. To determine the unit cross section for the reactions studied here,

we only included the transition with the high B(GT)s, since for states with low B(GT), a larger

systematic uncertainty is expected [75]. The extracted unit cross section was 1.868 ± 0.100.

For the 33Al(𝑑,2He)33Mg reaction, the B(GT) could only be extracted for the transition to the

lowest lying state. The extracted B(GT) was = 0.062 ± 0.041. Fig. 4.38 compares the extracted

B(GT) value with shell-model predictions (see Sec. 2.2). The vertical error bar indicates the

uncertainty in B(GT). The horizontal error bar reflects the uncertainty in excitation energy, arising

from the peak resolution of 200 keV and 100 keV uncertainty due to the drift velocity correction

(see Sec. 4.5). In other energy regions, the fit results showed that the angular distributions were

dominated by dipole or quadrupole contributions and given the statistical uncertainties, no B(GT)

could be extracted.

87

Figure 4.38 Comparison of extracted B(GT) values with shell-model predictions for the
33Al(𝑑,2He)33Mg reaction. A phenomenological quenching factor of 0.55 has been applied to
the shell-model calculations.

For the 32Mg(𝑑,2He)32Na reaction, the B(GT) was extracted in the 1-2 MeV and 3-4 MeV

excitation energy ranges. We obtained B(GT) = 0.06 ± 0.05 for the 1–2 MeV range and B(GT)

= 0.07 ± 0.06 for the 3–4 MeV range. The B(GT) was not extracted for other excitation energies

because dipole and quadrupole contributions dominated. Fig. 4.39(a) compares the extracted B(GT)

values with shell-model predictions for this reaction. The vertical error bars in Fig. 4.39(a) represent

the uncertainty from the angular distributions fits, consistent with the analysis of 33Al(𝑑,2He)33Mg.

The horizontal error bars indicate the excitation energy ranges analyzed for the GT transition.

Fig. 4.39(b) shows the summed B(GT) up to 6 MeV from this work and the shell-model calculations.

88

Shell ModelThis workFigure 4.39 (a) Extracted B(GT) values from this work compared with shell-model predictions for
the 32Mg(𝑑,2He)32Na reaction. (b) Summed B(GT) from this work compared with shell-model
predictions. A phenomenological quenching factor of 0.55 has been applied to the shell-model
calculations.

4.12 B(GT) Comparison with Theory

For the GT strength from 33Al, the shell-model calculations shown in Fig. 4.38 do not predict

a low-lying state near 0 MeV. However, the predicted B(GT) for a state at roughly 1.3 MeV is

comparable to the experimental B(GT) extracted for state near 0 MeV. Furthermore, the shell model

predicts additional states above 1.5 MeV with non-zero B(GT) values. These states could not be

identified in the data, but given the small predicted strengths and the significant contribution from

transitions with Δ𝐿 > 0 in the experimental data, the presence of some GT strength at higher

excitation energy cannot be ruled out.

For the 32Mg(𝑑,2He)32Na reaction, the experimentally extracted B(GT) between 1 MeV and

89

(a)(b)This workShell ModelThis workShell Model2 MeV agrees well with the shell-model prediction for a state near 1.6 MeV, despite the large

uncertainties arising from low statistics (see Fig. 4.39(a)). The shell-model calculations have

some strength at higher excitation energy, but it is rather fragmented. We observe a transition to

a state at ∼3.5 MeV that is likely associated with a GT transition (see Fig. 4.28(b)). As shown

in Fig. 4.39(b), the summed B(GT) measured in this work is in reasonable agreement with the

shell-model predictions, although the error bars are large.

The observation of a GT transition from 33Al to 33Mg highlights the importance of including

𝑝 𝑓 -shell configurations in describing nuclei in this region, and confirms that 33Al ground state has

a significant 𝑝 𝑓 −shell component in its wavefunction. Meanwhile, 32Mg is already established

to be part of the island of inversion, and the extracted B(GT)s for the 32Mg(𝑑,2He)32Na reaction

provides further evidence for this.

In addition, further refinements to the shell model, along

with experiments offering improved statistics, would help confirm these findings and reduce the

associated uncertainties.

90

CHAPTER 5

NEUTRON STAR CRUST SIMULATION

We ran simulations for neutron star crust using xnet, a nuclear reaction network that accounts for all

possible reactions in the crust and has been used to simulate the heating and cooling in neutron star

crusts in Ref. [11]. xnet calculates the EC and the 𝛽− decay rates from a set of input files provided

by the user. First, we ran xnet with inputs calculated with QRPA calculations from Ref. [14]. We

then repeated the simulation by replacing these inputs with shell-model calculations for 𝑠𝑑- and

𝑝 𝑓 -shell nuclei, using USDB [120] and GXPF1 [121] interactions, respectively. In addition, we

ran another version of the simulation where we incorporated 𝛽− decay experimental data for the

𝑠𝑑- and 𝑝 𝑓 - shell nuclei where available from the Nudat database by National Nuclear Data Center

(NNDC). For the calculation of EC rates, experimental data from charge-exchange reactions were

also used in the input files if available.

In cases where no experimental data were available to

calculate EC rates, we used Eq. 1.5 to determine the ground-state EC rates. To ensure consistency,

everything except the inputs for calculating EC and 𝛽− decay rates remained unchanged across all

three versions used in the simulation.

The neutron star simulation uses the dynamic model described in Ref. [7]. The neutron star

structure is adopted from Ref. [122] and the thermal structure is based on the calculations from

Ref. [123]. The model incorporates EC rates, neutron-capture rates, and pycnonuclear-fusion rates,

which are coupled to a thermal model of the neutron star crust. It also includes important processes

such as neutrino cooling and heat transport. In addition to the energy lost via neutrinos emitted

during the Urca process, neutrino pair bremsstrahlung is generated when electrons scatter off the

Coulomb field of nuclei in the crust, providing an additional cooling mechanism [124]. The reaction

network used in the model includes approximately over 3000 isotopes and spans from the valley of

stability to regions beyond the neutron-drip line.

In the model, the crust is a plane-parallel slab in a local Newtonian frame, where Newtonian

mechanics are a valid approximation, with constant gravity 𝑔, set to the value of 1.85 × 1014

cm/s2. The composition changes of an accreted fluid element is induced by the increasing pressure

91

𝑃 = (cid:164)𝑚𝑔𝑡, where (cid:164)𝑚 is the local accretion rate and 𝑡 is the time. It determines the steady-state of the

composition of the crust. The mass density is calculated using an equation of state 𝑃 = 𝑃(𝑇, 𝜌, 𝑌𝑖)

where 𝑇 is the temperature, 𝜌 is density, and 𝑌𝑖 is the nuclear abundance [125]. 𝑌𝑖 includes the

neutron abundance as well.

The astrophysical parameters used in the simulation are realistic for a neutron star and the same

as those used in Ref. [11]. The mass and radius of the neutron star are fixed at 1.4 solar mass and

10 km, respectively. The mass accretion rate, (cid:164)𝑀, is fixed at 0.3 (cid:164)𝑚Edd, where (cid:164)𝑚Edd is the Eddington

mass accretion rate- defined as the rate at which the gravitational energy released by the in-falling

matter produces the maximum luminosity for a star in hydrostatic equilibrium. The local accretion

rate per unit area, 𝐴 (cid:164)𝑀, is set to 2.637 × 104 g s−1 cm−2. We started all of our calculations at a

density of 𝜌 = 1.4 × 109 g cm−3.

Nuclear masses are a critical input for the reaction network, influencing the calculation of

phase space for EC and 𝛽− decay rates, as well as photodisintegration and neutron-capture rates.

Neutron-capture rates are computed assuming a Maxwell-Boltzmann distribution [7] and the pho-

todisintegration rates are derived using a detailed balance [126]. Additionally, nuclear masses

determine the 𝑄 value of reaction processes, significantly impacting the phase space of the elec-

trons.

In the neutron star crust simulation, theoretical masses are taken from the Finite Range

Droplet Mass Model (FRDM) [?], which is a macroscopic model augmented by microscopic cor-

rections for shell and pairing effects. To maintain consistency, the FRDM is used for calculating EC

and 𝛽− decay strength functions for nuclei with 𝑍 > 7. For nuclei with 𝑍 ≤ 7 and very neutron-rich

isotopes, the analytical Hilf mass formula [127], a semi-empirical atomic mass model, is applied.

Where available, experimental nuclear masses were also incorporated.

Building on this framework, we focus on heating and cooling in the neutron star crust. As

discussed in Sec. 1.1, two-step ECs are among the strongest heating sources, while the Urca

process is the main cooling mechanism. The strength of the transitions in these reactions is given

by the 𝑓 𝑡 value thant can be calculated from B(GT) using Eq. 1.4. The strength of an Urca pair

can be estimated from its 𝛽− decay flow, which measures how quickly 𝛽− decay transforms one

92

nucleus into another during nucleosynthesis. In this work, we compare heating generated from ECs

and different Urca cycles’ contributions to cooling. The Urca cooling strength is calculated using

time-integrated reaction flow, measured relative to the strongest Urca pair.

We ran three versions of the simulation, each using different input files for EC and 𝛽− decay and

will be referred to as Model 1, 2, and 3 from this point onward. The inputs used in these models

are as follows:

• Model 1: All inputs from QRPA.

• Model 2: Shell-model inputs for 𝑠𝑑 and 𝑝 𝑓 shells, with all other inputs from QRPA.

• Model 3: Data incorporated into shell-model inputs in Model 2 where available, otherwise

kept unchanged from Model 2.

Fig. 5.1 shows the nuclear chart with the inputs used for each isotope in the reaction network.

To ensure consistency, we used the same input model for both EC and 𝛽− decay rates within each

Urca pair. However, different inputs were allowed for the same isotope for EC and 𝛽− decay.

For example, 18O and 18F form a potential Urca pair where 18O undergoes 𝛽− decay to 18F, and

18F undergoes EC to regenerate 18O. For this pair, only shell-model inputs were used to ensure

consistency. On the other hand, 18F and 18Ne form another potential Urca pair, but shell-model

inputs were unavailable for 18Ne EC rates. In this case, we used QRPA inputs for EC and 𝛽− decay

rates of 18Ne and 18F, respectively, maintaining input consistency within the pair even though

different models were allowed for EC and 𝛽− decay rates of the same isotope.

93

Figure 5.1 Nuclear chart illustrating the input models used for each isotope in the simulation.
Green squares represent isotopes where only QRPA calculations were used for Models 1, 2, and
3; red squares indicate the use of shell-model inputs for Models 2 and 3; and yellow squares
highlight isotopes where experimental data were incorporated in Model 3. Squares with mixed
colors represent isotopes where different models were used for EC and 𝛽− decay rates for the same
isotope for Models 2 and 3.

5.1 Using the Neutron Star Crust Simulation Code

The EC and 𝛽− decay rates were calculated in the simulation dynamically using input files

that we provide to the simulation framework. For EC rates, the calculation relies on a table of

B(GT)s as a function of excitation energy, with the assumption that the parent nucleus is always

in its ground state. For relatively low temperatures (𝑇 <1 GK) in the neutron star crust, it is a

reasonable assumption for neutron stars [128]. Four specific input files are required to compute the

EC rates within the framework. The naming and formatting conventions for these files are detailed

in Ref. [129]. The four input files primarily contain the following information:

1. Parent nucleus ground state spin

2. List of daughter states

94

Proton Number (Z)Neutron Number (N)Quasiparticle Random Phase ApproximationShell ModelShell Model and Data3. Log( 𝑓 𝑡) values for transitions from the parent ground state to the daughter states

4. Nuclear properties of both the parent and daughter nuclei

In contrast, only a single input file is needed to calculate 𝛽− decay rates. This file contains

information about the atomic and mass numbers, the excitation energy of the parent, the excitation

energy of the daughter nucleus, and the log( 𝑓 𝑡) value of the transition.

The initial composition for crust model depends on the system and depends on the composition

of the accreted matter and the characteristics of surface burning processes such as X-ray bursts

and superbursts. We can fix the abundance, 𝑌 , for each initial isotope, 𝑖, and the abundances are

automatically normalized to:

Table 5.1 lists the three key output files generated from the simulation along with their functions.

𝐴𝑖𝑌𝑖 = 1

∑︁

𝑖

(5.1)

Table 5.1 Description of key input files used in the simulation.

Function
Records the abundances of isotopes over time.

Input File
abuntime
totalenergy Provides the energy at each timestep.
flux_movie Details the reaction flows at each timestep.

5.2 Calculation at 𝑇 = 0.5 GK

In the simulation, we started the evolution at time 𝑡 = 1.4×108 s, with a temperature 𝑇 = 0.5 GK,

a mass density 𝜌 = 1.45 × 109 g/cm3, and electron Fermi energy 𝐸𝐹 = 4.0 MeV. Fig. 5.2 illustrates

the initial isotopic abundances for superburst ashes as reported in Ref. [130]. This figure was

generated from the " 𝑓 𝑙𝑢𝑥_𝑚𝑜𝑣𝑖𝑒" output file produced by the simulation, and the initial composition

was identical across all three models.

95

Figure 5.2 Initial composition at the start time of 𝑡 = 1.4 × 108 s for all three versions of data. The
isotopic abundances are depicted on a logarithmic color scale ranging from blue (lowest) to green,
yellow, orange, and red (highest). The lower cutoff is at 1.0 ×10−7. Thick black squares indicate
stable nuclei, while thin black squares represent unstable but particle-bound nuclei. Squares without
outlines correspond to particle-unbound nuclei, and gray-shaded squares denote nuclei not included
in the reaction network. The red diagonal lines represent EC reactions, while the blue diagonal
lines indicate 𝛽− decay reactions. Nuclei pairs connected by both red and blue lines simultaneously
are identified as Urca pairs.

The deposition of integrated heat energy as a function of time for the different input sets are

shown in Fig. 5.3, generated from the "totalenergy" file. The deposited heat energy per nucleon is

calculated using the following equation:

𝑑𝑄 = 𝑑𝐸𝑛𝑢𝑐 + 𝑑𝐸𝑒 − 𝑑𝐸𝜈

(5.2)

where 𝑑𝐸𝑛𝑢𝑐 = (cid:205)𝑖 𝑑𝑌𝑖 𝐵𝐸𝑖 is the nuclear energy generation calculated from the abundance change

𝑑𝑌𝑖 and binding energy 𝐵𝐸𝑖 of each isotope 𝑖. 𝑑𝐸𝑒 = 𝐸𝐹 𝑑𝑌𝑒 and 𝑑𝐸𝜈 is the energy released as

neutrinos from EC. In Fig. 5.3, the red line represents energy changes using Model 1, the green line

corresponds to Model 2, and the blue line represents inputs for Model 3. Time = 0 s marks the start

of accretion, but in Fig. 5.3 the time axis begins at 1.4E08 s, which corresponds to the occurrence

of the superburst. In this figure, increases in energy are interpreted as heating and decreases as

96

2820882028Neutron Number (N)Proton Number (Z)cooling. Because the code calculates composition and energy generation as functions of depth

under a steady state approximation, increasing time represents progression into deeper layers of

the neutron star crust, where the pressure is higher. The upward trend observed in all three input

models reflects increasing heating and 𝐸𝐹 as the material moves into these deeper layers.

Figure 5.3 Energy evolution in the neutron star crust as a function of time for three different input
sets. Time on x-axis is shown on a logarithmic scale. The red line corresponds to energy changes
using Model 1 inputs, the green line represents inputs from Model 2, and the blue line illustrates the
case for Model 3 inputs. Distinct regions of interest in the energy curve are labeled and highlighted.

Fig. 5.3 is divided into six regions where significant heating, cooling, or differences between the

models are observed. To analyze these regions, the nuclei responsible for heating and cooling were

identified by examining their abundances and reaction flows at specific timesteps. The input files

for these isotopes were then reviewed to determine which parameters may be driving the observed

variations in heating and cooling. Discussion on different regions identified in Fig. 5.3 is as follows.

97

00.50.10.150.21×1091×1010Region 1Region 2Region 3Region 4Region 5Region 6Model 1Model 2Model 3Heat Energy (MeV)Time (s)6.7×1094.1×1010Density (g/cm3)5.2.1 Region 1

Region 1 spans the time interval from 𝑡 = 6.2 × 108 s to 𝑡 = 7.5 × 108.

In this region, no

difference in deposited heat energy is observed between Models 2 and 3. However, significant

heating is observed for Model 1. Fig. 5.4 illustrates the abundances of nuclei and the reactions

occurring in Region 1 for Model 1.

Figure 5.4 Isotopic abundances and reaction flows at 𝑡 = 7.2 × 108 s in Region 1, calculated using
inputs from QRPA.

In Model 1, the primary contributor to heating is a two-step EC process: 56Fe → 56Mn → 56Cr.

Significant heat is deposited in the crust as both EC transitions occur for relatively highly excited

states, leading to substantial energy release during the de-excitation of 56Mn and 56Cr, as illustrated

in Fig. 5.5(a).

In contrast, Models 2 and 3 involve only a single EC transition from 56Fe to 56Mn. Moreover,

in these models, the EC from 56Fe proceeds to a low-lying state in 56Mn, which deposits minimal

energy into the crust upon de-excitation to the ground state. Notably, Model 3 incorporates

experimental data for the 56Fe → 56Mn transition from Ref. [131], resulting in inputs that closely

align with those of Model 2. It is concluded that the heating in Model 1 is artificial.

98

2820882028Neutron Number (N)Proton Number (Z)Figure 5.5 Energy level schemes in Region 1: (a) A two-step EC reaction from 56Fe to 56Cr in
Model 1. (b) A single-step EC reaction from 56Fe to 56Mn in Model 2.

5.2.2 Region 2

Region 2 spans the time interval from 𝑡 = 1.2 × 109 s to 𝑡 = 2.5 × 109 s. Heating is observed

across all three models. Model 1 exhibits lower heating that starts earlier in comparison to Models

2 and 3. Significant cooling is observed for Model 1 whereas the cooling in Models 2 and 3 are

minimal. Fig. 5.6 shows the isotopic abundance and reaction flow for Model 2 at 𝑡 = 1.4 × 109 s.

99

Ex=2.52 MeV (log𝑓𝑡=3.32) 𝛾Model 156Fe56MnEC𝛾56CrECEx=3.18 MeV (logft  = 4.29) (a)|QEC|=3.7MeV |QEC|=1.63 MeV EF=6.3 MeV EF=6.3 MeV Ex=0.063 (logft  = 4.16)Model 2𝛾56Fe56MnEC(b)|QEC|=3.7MeV EF=6.3 MeV Figure 5.6 Isotopic abundances and reaction flows at 𝑡 = 1.4 × 109 s, calculated for Model 2 in
Region 2.

In Fig. 5.6 at 𝑡 = 1.4 × 109 s, the significant source of heating is a two-step EC from 54Cr to

54Ti across all three models. Fig. 5.8 illustrates the energy level scheme for the 54Cr→54V→54Ti

transition for Models 1 and 2. No data was available for these nuclei, therefore, these inputs were

kept unchanged from Model 2 for Model 3. In Model 1, EC from 54Cr happens for a 0.07 MeV

state in 54Mn, and then in the next step it occurs in the ground state of 54Ti. In Model 2, EC from

54Cr takes place at a 0.309 MeV state in 54Mn followed by a 2.46 MeV state in 54Ti. As a result,

Model 2 adds more heat to the crust.

100

2820882028Neutron Number (N)Proton Number (Z)Figure 5.7 Energy level schemes shown for (a) Model 1 and (b) Model 2 showing the difference
between heat deposited in the crust in Region 2.

In Region 2, we see cooling for Model 1 inputs indicating formation of Urca pairs. Fig. 5.6

shows the isotopic abundance and reaction flow for Model 2 at 𝑡 = 2 × 109 s.

101

Model 1ECEC(a)54Cr54V54TiEx=0.07 MeV(logft = 5.68) |QEC|=7.04 MeV |QEC|=4.27 MeV EF=7.5 MeV EF=7.5 MeV 𝛾Model 2ECEC(b)54Cr54V54Ti|QEC|=7.04 MeV Ex=0.309 MeV (log𝑓𝑡 = 6.17)𝛾Ex=2.456 MeV (logft  = 6.33) 𝛾EF=7.5 MeV EF=7.5 MeV |QEC|=4.27 MeV Figure 5.8 Isotopic abundances and reaction flows at 𝑡 = 2 × 109 s in Region 2, calculated using
inputs from QRPA.

In Region 2, the strongest Urca pair is 55V ↔ 55Ti for Model 1 (see Fig. 5.9). In this model,

EC happens for the ground state of 55Ti, and because the Fermi energy is nearly equal to the EC

threshold, 𝛽− decay can occur. In Models 2 and 3, the first allowed EC happens for an excited state

at 1.017 MeV in 55Ti, but the Fermi energy is too low for the transition to occur. Thus, an Urca

pair forms in Model 1, but not in Models 2 and 3.

102

2820882028Neutron Number (N)Proton Number (Z)Figure 5.9 (a)Energy level scheme in Region 2 for Urca pair formed between 55V and 55Ti in Model
1. (b) and (c) No Urca pair is formed between 55V and 55Ti in Models 2 and 3, respectively.

Another important cooling contributor in this region is the Urca pair between 57Cr and 57V (see

Fig. 5.10). In all models, EC happens from ground state to ground state. However, Model 1 has

low log 𝑓 𝑡 values for both EC and 𝛽− decay, meaning its transition is the strongest. Model 2 has the

highest log 𝑓 𝑡 value for EC and the weakest cycle, while Model 3 provides slightly more cooling

than Model 2. It is concluded that Model 1 provides less heating and more cooling than Models 2

and 3.

103

55V55Ti|QEC|=7.48 MeV EF=8 MeV ECβ¯(a)Model 1G.S. logft (EC) = 4.54G.S. logft (β−decay) = 5.74 55V55Ti|QEC|=7.48 MeV EF=8 MeV (b)Model 2Ex=0.67 MeV logft (β−decay) = 5.41Ex=1.017 MeV G.S. logft (EC) = 4.7555V55Ti|QEC|=7.48 MeV EF=8 MeV (c)Model 3Ex=0.67 MeV logft (β−decay) = 5.1Ex=1.017 MeV G.S. logft (EC) = 4.75Figure 5.10 Energy level scheme shown for Urca pair formed between 57Cr and 57V in Region 2 in
(a) Model 1, (b) Model 2, and (c) Model 3.

5.2.3 Region 3

Region 3 spans from 𝑡 = 2.9 × 109 s to 𝑡 = 3.8 × 109 s. No significant cooling occurs in this

period, but heating differs among the models: Model 2 heats the most, Model 3 the least, and

Model 1 is in between.

The main heating source here is the two-step EC reaction: 56Cr →56 V →56 Ti. Fig. 5.11

displays the energy level schemes for these transitions. In Model 2, both steps involve excited states,

with the 56V →56 Ti step occurring at 1.21 MeV. This excited state releases a significant amount of

heat when it de-excites to the ground state. In contrast, for Models 1 and 3, the second step happens

from the ground state, so only the EC from 56Cr to an excited state in 56V contributes to heating. In

Model 1, this transition is at 0.05 MeV, while in Model 3 it is at 0.01 MeV. Consequently, heating

in Model 2 is higher than in Model 3.

104

G.S. logft (EC) = 4.8G.S. logft (EC) = 6.63G.S. logft (EC) = 4.3857Cr57V|QEC|=8.11 MeV EF=8.9 MeV ECβ¯(a)Model 1G.S. logft (β−decay) = 4.7657Cr57V|QEC|=8.11 MeV EF=8.9 MeV β¯(b)Model 2G.S. logft (β−decay) = 4.55 57Cr57VEF=8.9 MeV β¯(c)Model 3G.S. logft (β−decay) = 5.11 ECEC|QEC|=8.11 MeV Figure 5.11 Level schemes shown for (a) Model 1, (b) Model 2, and (c) Model 3 for a two-step EC
reaction from 54Cr to 54Ti in Region 3.

5.2.4 Region 4

Region 4 spans from 𝑡 = 5.3 × 109 s to 𝑡 = 6.6 × 109 s. In this period, all three models show the

same patterns of heating and cooling as the inputs are identical. The heating mainly comes from a

two-step EC reaction: 52Ti → 52Sc → 52Ca. The cooling is provided by the two Urca pairs 59Cr

↔ 59V and 57V↔57Ti. Fig. 5.12 displays the energy level scheme for the two-step EC reaction. In

both steps, the transitions occur to excited states in the daughter nuclei, which then release heat as

they de-excite to the ground state.

105

Ex=0.01 MeV logft = 3.9 Ex=0.05 MeV logft = 6.41 Ex=0.01 MeV (logft = 3.9) 𝛾Model 156Cr56VEC56TiECG.S. (logft = 4.71) (a)|QEC|=9.13 MeV |QEC|=6.83 MeV EF=9.3 MeV EF=9.3 MeV 𝛾Model 256Cr56VEC56TiEC(b)|QEC|=9.13 MeV |QEC|=6.83 MeV Ex=1.21 MeV (logft = 4.73) 𝛾Model 356Cr56VEC56TiEC(c)|QEC|=9.13 MeV |QEC|=6.83 MeV G.S. (logft = 5.12) EF=9.3 MeV EF=9.3 MeV EF=9.3 MeV EF=9.3 MeV Figure 5.12 Energy level schemes shown for Model 1 for a two-step EC reaction from 52Ti to 52Ca
in Region 4. The inputs were unchanged for Models 2 and 3, therefore, the level schemes for those
Models are identical.

The energy level schemes for the two strong Urca pairs contributing to the cooling are shown

in Fig. 5.13.

Figure 5.13 Energy level schemes shown for the Urca pairs in Region 4: (a) 59Cr↔59V and (b)
57V↔57Ti.

106

Ex=3.53 MeV (logft = 3.9) Ex=1.82 MeV (logft = 3.51) 𝛾Model 152Ti52ScEC𝛾52CaEC|QEC|=9.03 MeV |QEC|=6.17 MeV EF=10.9 MeV EF=10.9 MeV 59V59Cr|QEC|=10.25 MeV EF=10.8 MeV ECβ¯(a)Model 1G.S. logft 4.53G.S. logft 4.6557Ti57V|QEC|=10.25 MeV EF=10.8 MeV ECβ¯(b)Model 1G.S. logft 5.29G.S. logft 5.085.2.5 Region 5

Region 5 spans from 𝑡 = 9.34 × 109 s to 𝑡 = 1.17 × 1010 s. Heating in this region is dominated

by the two-step EC reaction: 55Ti → 55Sc → 55Ca, shown by Fig. 5.14. The inputs in Models 2

and 3 were kept unchanged from Model 1; therefore, identical heating is observed across models.

Figure 5.14 Energy level scheme for Model 1 for a two-step EC reaction from 55Ti to 55Ca in Region
5.

Cooling in Models 2 and 3 in this region is mainly driven by the strongest Urca pair: 33Al

↔ 33Si. Fig. 5.15 show the energy level schemes of these transitions to illustrate the difference

between the formation of Urca pairs between the models. In Model 1, the first allowed EC transition

occurs at an excited state of 0.37 MeV, while in Models 2 and 3 it takes place at the ground state.

Because the Fermi energy is not high enough for EC to occur at the excited state in Model 1, no

Urca pair forms. In contrast, in Models 2 and 3, EC is allowed, and the low log 𝑓 𝑡 values of the

transitions lead to a strong Urca pair. It is concluded that this is an Urca pair that should be in the

simulations, using the experimental data available.

107

Ex=0.695 MeV Logft = 4.8𝛾Model 155Ti55ScEC55CaECEx=0.0 MeV Logft = 4.21|QEC|=11.51 MeV |QEC|=11.51 MeV EF=12.4 MeV EF=12.4 MeV Figure 5.15 Energy level schemes for (a) Model 1, (b) Model 2, and (c) Model 3 for the Urca pair
33Si↔33Mg in Region 5.

5.2.6 Region 6

In Region 6, spanning from 𝑡 = 1.2 × 1010 s to 𝑡 = 1.5 × 1010 s, cooling is observed for all three

models. The 29Na ↔ 29Mg pair is identified as the strongest transition for all input versions, though

with different predicted strengths. Model 1 predicts the highest strength, resulting in the largest dip

in Region 6. In contrast, Model 2 predicts approximately half the strength compared to Model 1,

while Model 3 reduces the predicted strength by an order of magnitude. Energy level schemes of

the Urca pair 29Mg↔29Na are shown in Fig. 5.16. It is concluded that the 29Mg↔29Na Urca pair

exists, but is weaker than originally assumed.

108

33Al33Si|QEC|=12.02 MeV EF=12.3 MeV (a)Model 133Al33Si|QEC|=12.02 MeV EF=12.3 MeV ECβ¯(b)Model 2Ex=0.37 MeV logft = 3.8G.S. logft = 10.4064G.S. logft = 3.93G.S. logft = 4.133Al33Si|QEC|=12.02 MeV β¯(c)Model 3ECG.S. logft = 4.12EF=12.3 MeV G.S. logft = 4.3Figure 5.16 Energy level schemes for (a) Model 1, (b) Model 2, and (c) Model 3 for the Urca pair
29Mg↔29Na in Region 6.

5.3 Urca Pairs

The Urca pairs are crucial for cooling in the neutron star crust, and accurate inputs are necessary

in the simulation to identify the strongest pairs responsible for cooling. Table 5.2 lists the strongest

Urca pairs integrated over the entire time range for all three models. The strengths of these Urca

pairs, calculated using time-integrated reaction flow, are scaled by the strength of the strongest

Urca pair across all models, 29Na ↔ 29Mg, from Model 1. Urca pairs with scaled strengths lower

than 10−4 are not included in the table, as they contribute minimally to the overall cooling. By

focusing on the pairs with stronger contributions, we can better understand which reactions are

most important for cooling the neutron star crust.

The 29Na ↔ 29Mg pair is identified as the strongest Urca pair in Models 1 and 2, while 33Al

↔ 33Si is the strongest Urca pair for Model 3 results. The strength for the 29Na ↔ 29Mg pair is

highest for Model 1. In Models 2 and 3, the strength is reduced. The 33Al ↔ 33Si pair is the

109

|QEC|=13.28 MeV β¯(b)Model 229Na29Mg|QEC|=13.28 MeV EF=13.31 MeV ECβ¯(a)Model 1G.S. logft = 4.229Mg29Na|QEC|=13.28 MeV β¯(c)Model 329Mg29NaG.S. logft = 4.34G.S. logft = 5.06G.S. logft = 5.06G.S. logft = 5.06G.S. logft = 4.8EF=13.31 MeV ECEF=13.31 MeV ECsecond strongest for Model. However, Model 1 does not identify this as a strong pair. Both of the

above mentioned pairs involve isotopes near the 𝑁 = 20 "island of inversion", and the experimental

results described in this thesis will be helpful for constraining the weak interaction rates.

In contrast, the 55Ti ↔ 55V pair is identified as one of the strongest by Model 1, but neither

Model 2 or 3 support this finding. For the 57Ti ↔ 57V pair, Model 1 inputs were kept unchanged

across the three versions, so all predict identical strengths. The 59V ↔ 59Cr pair is consistently

identified as the fourth strongest across all versions, with minimal variation in strength.

Table 5.2 Comparison of Urca pair strengths for Models 1, 2, and 3 (scaled by the strength of
29Na ↔ 29Mg pair from Model 1).

Urca Pairs
29Na ↔ 29Mg
33Al ↔ 33Si
55Ti ↔ 55V
57Ti ↔ 57V
59V ↔ 59Cr
61V ↔ 61Cr
55V ↔ 55Cr
57V ↔ 57Cr
57Sc ↔ 57Ti
59Ti ↔ 59V
59Cr ↔ 59Mn
57Cr ↔ 57Mn
59Mn ↔ 59Fe

Model 1 Model 2 Model 3
0.433
0.227
–
0.090
0.045
0.011
–
0.017
0.022
0.013
–
0.001
0.001

0.112
0.145
–
0.090
0.045
0.011
–
0.026
0.022
0.013
–
0.001
0.0003

1.000
–
0.129
0.090
0.044
0.041
0.032
0.092
0.026
0.013
0.013
–
–

For pairs like 61V ↔ 61Cr, 57Sc ↔ 57Ti, and 57Cr ↔ 57Mn, no experimental data was available.

As a result, Model 2 inputs were kept unchanged for Model 3, leading to identical strength

predictions across the versions. Models 2 and 3 do not identify the 55V ↔ 55Cr pair as strong,

while Model 1 inputs fail to recognize 57Cr ↔ 57Mn and 59Mn ↔ 59Fe as strong pairs.

Finally, for the 57V ↔ 57Cr pair, the three input versions predict varying strengths. Overall, the

strengths based on experimental data align more closely with the shell-model than with QRPA pre-

dictions. Referring back to Fig. 5.3, it is observed that when the best estimates available for heating

and cooling are used, the magnitudes of heating and cooling tend to become smaller. Whether this

trend is true in other relevant regions in the chart of nuclei, requires further investigation.

110

CHAPTER 6

CONCLUSION AND OUTLOOK

Weak reactions such as electron capture and 𝛽− decay contribute to heating and cooling in neutron

stars. Accurate weak interaction rates are important for realistic neutron star simulations. However,

the reaction networks used in simulations involve over 3,000 isotopes, making it impractical to

obtain experimental weak interaction rates for all nuclei. As a result, theoretical calculations must

be used, and experimental data are necessary to validate and benchmark these models.

Currently, several studies of neutron stars rely on mean-approximation calculations for weak

reaction rates, which have been shown to be not as reliable for predicting transitions to specific low-

lying states, which are most important for accurate simulations on neutron star crust. In contrast, the

shell model is known to perform better for light and medium mass (A< 65) nuclei. Gamow-Teller

(GT) strengths are crucial for determining electron-capture rates, but direct weak-interaction studies

are, if available, limited to ground state to ground state transitions for neutron-rich nuclei due to

𝑄-value constraints. Charge-exchange reactions, governed by the strong interaction, overcome this

limitation as they connect the same initial and final states as electron capture and 𝛽− decay and

allow for the extraction of GT strengths. These features make them an effective tool for studying

weak-interaction rates indirectly.

Many astrophysically relevant nuclei are unstable, preventing their study in forward kinematics

since a target for experimental studies cannot be created. Instead, the charge-exchange reactions

must be performed in inverse kinematics, using unstable nuclei as beams to ensure they do not decay

in-flight. The (𝑑,2He) reaction in inverse kinematics is particularly well-suited for this purpose

due to its spin-selective nature. This approach was applied in two experiments at the National

Superconducting Cyclotron Laboratory (NSCL) and the Facility for Rare Isotope Beams (FRIB)

using the Active-Target Time Projection Chamber (AT-TPC) and the S800 magnetic spectrometer.

The use of the active target and the ability to cleanly select (𝑑,2He) events because of the vertex

reconstruction following the emission of two protons from 2He are critical advantages of the method.

It is, at present, the only way to obtain GT strength in the EC/𝛽+ direction for neutron-rich unstable

111

nuclei.

In this work, two reaction channels were studied: 33Al(𝑑,2He) and 32Mg(𝑑,2He). 32Mg and

33Al are near the 𝑁 = 20 island of inversion, an area important for understanding neutron star

heating, cooling, and nuclear structure. GT strengths were extracted for a state near 0 MeV in the

33Al→33Mg transition and for states between 1–2 MeV and 3–4 MeV in the 32Mg→32Na transition.

The presence of a GT transition shows that the ground state of 33Al has significant 𝑝 𝑓 shell

contribution in its wavefunction. Although statistical uncertainties are large, we find reasonable

correspondence between shell-model calculations using the 𝑠𝑑𝑝 𝑓 − 𝑚 interaction, which included

contributions from 𝑠𝑑− and 𝑝 𝑓 − shell configuration. For the 33Al(𝑑,2He) reaction, we observed a

GT transition at low excitation energies that the shell model predicts, for a state near 1.3 MeV. For

the 32Mg(𝑑,2He) reaction, the GT strength between 1–2 MeV agrees well with the shell model, and

strength at higher exctiation energy is found, albeit in what appears to be a transition to single final

state, where as in the shell-model calculations, the strength is distributed. Due to low statistics in

the experiment, the uncertainties are high and it was difficult to draw strong conclusion about the

strengths at high-excitation energies.

In addition to the experimental work, weak interaction rates and their role in the neutron star

crusts were studied for nuclei in the 𝑠𝑑 and 𝑝 𝑓 shells, for which accurate shell-model calculations

are already available. Neutron star crust simulations were performed using three sets of weak

interaction rates: (1) exclusively from QRPA calculations, (2) shell-model calculations for 𝑠𝑑 and

𝑝 𝑓 shell nuclei combined with QRPA calculations for all others, and (3) experimental data for 𝑠𝑑 and

𝑝 𝑓 shells when available combined with shell-model calculations, otherwise, QRPA calculations

were used. The evolution of heat energy in the crust was studied for each set and reactions that

contributed to heating and cooling at different depths were identified. The rates based on QRPA

calculations tend to overestimate both heating and cooling compared to the models that use the

shell-model calculations and experimental data. Moreover, the deviations between rates based on

experimental data and the shell-model calculation are small. It is concluded that the approach of

using shell-model calculations in combination with experimental data, where available, is a good

112

approach.

To improve neutron star crust simulation results, it will be important to extend shell-model

calculations to include weak interaction rates for nuclei beyond the 𝑠𝑑 and 𝑝 𝑓 shells, including

those in the 𝑁 = 20 island of inversion. The experimental work described in this thesis provides

valuable benchmarks for these improvements. In addition, further experiments using the (𝑑,2 He)

probe in inverse kinematics should be considered for important transitions such as 29Mg ↔29 Na,

61V ↔61 Cr, 57Sc ↔57 Ti, and 57Cr ↔57 Mn. The further increases in beam intensities at

FRIB will be critical for such measurements. In addition, for the AT-TPC to handle higher beam

intensities, it should be considered to operate in a solenoidal field, which would strongly reduce the

background due to 𝛿 electrons. Additionally, it would improve the angular and energy resolution

of the measured proton tracks and allow for the stopping of higher-energy protons, reducing event

loss from tracks exiting the AT-TPC chamber.

113

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