31
   CL β-DELAYED PROTON DECAY AND CLASSICAL NOVA NUCLEOSYNTHESIS
                                        By
                              Tamas Aleksei Budner
                               A DISSERTATION
                                    Submitted to
                            Michigan State University
                    in partial fulfillment of the requirements
                                 for the degree of
                        Physics – Doctor of Philosophy
                                       2022


                                               ABSTRACT
Classical novae occur in binary star systems involving a compact white dwarf and a low-mass stellar
companion. In these events, material siphoned from the donor star forms an accretion disk around
the white dwarf. This hydrogen-rich fuel is compressed, heated, and mixed with the outer layers
of the underlying white dwarf until it eventually ignites in a thermonuclear runaway. These violent
explosions eject freshly synthesized nuclear material into the interstellar medium, contributing to
the chemical evolution of the galaxy.
     Nova sensitivity studies involving the most massive oxygen-neon (ONe) white dwarfs have
identified the thermonuclear rate of the     30 P(𝑝, 𝛾)31 S reaction to be the largest remaining source
of nuclear physics uncertainty associated with modeling nucleosynthesis for intermediate-mass
elements in these highly energetic events. Over the past two decades, considerable experimental
effort has been devoted to determining the rate of this proton-capture reaction, but until now, it has
remained essentially unconstrained. A recent 31 Cl 𝛽-delayed 𝛾 experiment revealed the existence
of a crucial low-energy, ℓ = 0 resonance that could potentially dominate the total 30 P(𝑝, 𝛾)31 S rate.
At the National Superconducting Cyclotron Laboratory (NSCL), on the campus of Michigan State
University, we developed the Gaseous Detector with Germanium Tagging (GADGET) system in
order to measure the weak, low-energy, 𝛽-delayed proton decay of 31 Cl through this astrophysically
important resonance. In GADGET’s first dedicated science experiment, we measured the weakest
𝛽-delayed charged-particle decay ever reported for resonances below 400 keV. Combining our
experimentally determined proton branching ratio with shell-model calculations of the state’s
lifetime and with past work on other resonances, we computed the total thermonuclear rate for
the  30 P(𝑝, 𝛾)31 S reaction across peak classical nova temperatures. Our new, recommended rate
was used in fully hydrodynamic nova model simulations to predict the elemental and isotopic
abundances in ONe nova ejecta.
     In this dissertation, we will discuss the experimental methods and analysis employed to achieve
our scientific results and investigate their astrophysical impact by comparing to observations from
astronomy. Furthermore, we present the first-ever detailed look at the 31 Cl(𝛽𝑝𝛾)30 P decay scheme,


reporting preliminary energies and intensities for previously unobserved 𝛽-delayed proton transi-
tions to 30 P excited states.


                                    ACKNOWLEDGEMENTS
I would like to acknowledge everyone in my research group at Michigan State University with whom
I have had the pleasure of collaborating over the past six years. Specifically, I would like to thank
my Ph.D. advisor, Prof. Christopher Wrede, for his patient guidance and support throughout my
graduate career. He has been working on the astrophysical problem motivating this research for the
better part of two decades and was the principal investigator of the Department of Energy (DOE)
grant proposal that funded this thesis project. I would also like to thank our former postdoctoral
research associate, Prof. Moshe Friedman, who was essential in the technical development of the
GADGET system. Furthermore, I want to acknowledge fellow graduate students Jason Surbrook
and Tyler Wheeler, as well as postdoctoral researcher Lijie Sun, for their specific contributions to
necessary simulations and data analysis for this project.
     In addition, I would like to credit other members of our collaboration, including Prof. Alex
Brown and Prof. Jordi José for their theoretical calculations, David Pérez-Loureiro and Emmanuel
Pollacco for their contributions to detector development, as well as everyone who participated in
NSCL experiment 17024 to ensure that it was a success. I want to also extend this appreciation to
all the NSCL staff for their administrative and technical support; none of the experimental research
at the lab would be possible without their diligent efforts. Lastly, I want to give thanks to all the
friends I have made during my time in Michigan, whose love and support have kept me somewhat
sane throughout graduate school.
                                                  iii


                                  TABLE OF CONTENTS
CHAPTER 1 INTRODUCTION AND BACKGROUND . . . . . . . . . . . . . . . . . .                         1
   1.1 A Brief Overview of Nuclear Structure and Radiation . . . . . . . . . . . . . . . .        1
   1.2 On the Origin of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      9
CHAPTER 2 THE 30 P(𝑝, 𝛾) 31 S REACTION . .          . . . . . . . . . . . . . . . . . . . . . .  20
   2.1 Effects on Astronomical Observables . . .    . . . . . . . . . . . . . . . . . . . . . .  20
   2.2 Thermonuclear Reaction Rate Formalism        . . . . . . . . . . . . . . . . . . . . . .  28
   2.3 Indirect Methods for Reaction Studies . .    . . . . . . . . . . . . . . . . . . . . . .  38
CHAPTER 3 EXPERIMENTAL INVESTIGATION . . . . . . . . . . . . . . . . . . . . 43
   3.1 GADGET: Gaseous Detector with Germanium Tagging . . . . . . . . . . . . . . . 43
   3.2 NSCL Experiment 17024: 𝛽-Delayed Proton Decay of 31 Cl . . . . . . . . . . . . . 55
CHAPTER 4 DATA ANALYSIS AND SCIENTIFIC RESULTS                      . . . . . . . . . . . . . .  71
   4.1 The Proton Branching Ratio . . . . . . . . . . . . . . .     . . . . . . . . . . . . . .  71
   4.2 Calculating the Total Thermonuclear Rate . . . . . . . .     . . . . . . . . . . . . . .  91
   4.3 Astrophysical Impact . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . 110
CHAPTER 5 THE 31 CL(𝛽𝑝𝛾)30 P DECAY SCHEME               . . . . . . . . . . . . . . . . . . . . 121
   5.1 Populating 30 P Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
   5.2 𝛾-Ray Detection Efficiency with SeGA . . . .     . . . . . . . . . . . . . . . . . . . . 128
   5.3 Analyzing the Cumulative Proton Spectrum .       . . . . . . . . . . . . . . . . . . . . 135
   5.4 Preliminary Results . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . 149
CHAPTER 6     CONCLUSIONS AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . 153
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
                                               iv


                                             CHAPTER 1
                             INTRODUCTION AND BACKGROUND
The central component of the doctoral thesis project discussed in this dissertation revolves around
the quantification of one property, of a single quantum state, within a particular atomic nucleus,
which is relevant to a specific astrophysical environment. More specifically, the purpose of this
research is to determine the probability that an excited and unstable sulfur (S) nucleus will emit a
proton, in order to calculate the rate at which radioactive phosphorus (P) nuclei will capture protons
in the thermonuclear explosions that occur on the surface of certain dead stars. In this first chapter,
we will provide a short primer on some basic nuclear physics and define terminology that will be
used throughout. In addition, we wish to contextualize the scientific motivation for this research
within the broader field of modern nuclear astrophysics.
1.1   A Brief Overview of Nuclear Structure and Radiation
The Atom and Nuclear Notation
Almost all matter with which we interact on a daily basis is comprised of atoms. Every atom
contains a nucleus and at least one electron (𝑒 − ), which are held together by the electrostatic
attraction between the negatively charged electrons and the positively charged nucleus. Figure
1.1, although not drawn to scale, offers a comparison between the diameters of the atom and its
nucleus. While the region in which electrons orbit around the nucleus makes up most of an atom’s
volume, over 99.9% percent of the atomic mass is contained within its nucleus. The nucleus itself
is composed of positively-charged protons (𝑝) and electrically neutral neutrons (𝑛). The number of
protons in a nucleus is given by the atomic number Z and determines the chemical properties of that
particular element. Atomic nuclei of the same chemical element with different neutron numbers N
are the various isotopes of that element. The total number of nucleons in a given nucleus is its mass
number 𝐴 = 𝑍 + 𝑁. Standard nuclear notation will sometimes refer to a specific nucleus using the
                                                    1


Figure 1.1: Cartoon diagram of a helium (He) atom. Nucleons are bound systems of three
fundamental particles called quarks; protons have two “up” quarks and one “down” quark, while
neutrons have one up and two down quarks. Figure credit: Contemporary Physics Education
Project, Lawrence Berkeley National Laboratory.
shorthand 𝑍𝐴 𝑋, where 𝑋 is the chemical symbol for a given element, but usually the redundant 𝑍
subscript is omitted.
                                                2


Stable and Unstable Nuclei
Nucleons are tightly bound by the residual effects of the strong interaction, which describes the
behavior of the elementary particles within each nucleon over very small distances and is the
strongest of the four fundamental forces in nature. Stable nuclei are less massive than the sum of
their individual nucleons alone, and their nuclear binding energy is defined as the energy required
to break up a given nucleus into its constituent protons and neutrons. Using perhaps the most
famous equation in all of physics, 𝐸 = 𝑚𝑐2 , which relates the energy of an object to its mass at
rest, we can express the binding energy of a particular nucleus with mass 𝑚(𝑁, 𝑍) as
                            BE(𝑁, 𝑍) = 𝑍𝑚 𝑝 𝑐2 + 𝑁𝑚 𝑛 𝑐2 − 𝑚(𝑁, 𝑍)𝑐2 ,                            (1.1)
where 𝑐2 = 8.98755179 × 1016 m2 /s2 is the speed of light squared, 𝑚 𝑝 = 938.27203(8) MeV/𝑐2
is the mass of the proton, and 𝑚 𝑛 = 939.56536(8) MeV/𝑐2 is the mass of the neutron [1]. In the
energy regime relevant for nuclear physics, it is often convenient to express energies in units of
mega- (MeV) or kiloelectron-volts (keV), where 1 MeV = 103 keV = 106 eV.
    Similar to how the periodic table organizes chemical elements by increasing order in 𝑍, the
chart of nuclides shown in Figure 1.2 plots 𝑍 versus 𝑁 for every known nucleus; rows and columns
are outlined at the so-called magic numbers, which correspond to the “closure” of nuclear shells.
Similar to, although distinctly different from, how electrons fill atomic orbitals, individual protons
and neutrons will occupy the lowest-energy configuration available, following the Pauli exclusion
principle. Despite the fact that vast majority of atoms on Earth are stable, Figure 1.2 clearly shows
that most known nuclei are not, and the stability of an atomic nucleus is determined by its relative
number of protons and neutrons. Only isotopes with the largest binding energy per nucleon (BE/𝐴)
are to be found in the so-called valley of stability.
    Unstable nuclei are said to be radioactive and undergo nuclear decay by emitting radiation, of
which there are a variety of types. Nuclear decay is a fundamentally random process, but it follows
a simple, exponential trend. The number of radioactive nuclei 𝑁 at time 𝑡 is given by
                                                    3


Figure 1.2: The chart of nuclides. Stable nuclei are represented by black squares, while
radioactive nuclei are color-coded by their primary decay modes: 𝛽+ or electron capture (blue), 𝛽−
(pink), 𝛼 (yellow), fission (green), and spontaneous proton (orange) or neutron (purple) emission.
Figure credit: National Nuclear Data Center (NNDC), Brookhaven National Laboratory.
                                           𝑁 (𝑡) = 𝑁0 𝑒 −𝑡/𝜏 ,                                    (1.2)
where 𝑁0 = 𝑁 (𝑡 = 0) is the number of radioactive nuclei present when they are first counted, and
𝜏 is the lifetime of the radioactive nucleus. The lifetime is simply the average time that a nucleus
takes to decay. Typically, when discussing how radioactive a nucleus is, we quote its half-life
𝑡1/2 = ln 2𝜏, which is the time required for half of the radioactive nuclei within an isotopically pure
sample decay [𝑁 (𝑡 1/2 ) = 𝑁0 /2].
                                                   4


Weak Decays
Radioactive nuclei located below this valley in Figure 1.2 are neutron-rich and typically undergo
𝛽− decay, whereby a neutron turns into a proton, emitting an electron and antineutrino (𝜈) in the
process:
                                         𝐴
                                         𝑍𝑋
                                               𝐴
                                             →𝑍+1   𝑋 ′ + 𝑒− + 𝜈                                  (1.3)
Electrons and neutrinos (𝜈) are both fundamental particles called leptons and have a lepton number
of 𝐿 = 1, while their antiparticles have 𝐿 = −1. Thus, emission of the antineutrino during 𝛽− decay
conserves this quantity (Δ𝐿 = 0). Similarly, on the proton-rich, or neutron-deficient, side of the
chart, nuclei typically undergo 𝛽+ decay, in which a proton turns into a neutron:
                                         𝐴
                                         𝑍𝑋
                                               𝐴
                                             →𝑍−1   𝑋 ′ + 𝑒+ + 𝜈                                  (1.4)
In this case, the electron’s antiparticle the positron (𝑒 + ) is emitted instead, along with a neutrino
this time, again conserving lepton number. A related decay process is electron capture, in which a
proton-rich nucleus captures one of its innermost electrons and converts a proton into a neutron:
                                         𝐴
                                         𝑍𝑋 + 𝑒 − →𝑍−1
                                                     𝐴
                                                          𝑋′ + 𝜈                                  (1.5)
These three decay processes are all caused by the other short-range nuclear force, appropriately
named the weak interaction, due to its relatively weak strength compared to the to the strong nuclear
and electromagnetic forces.
Alpha Decay and Nuclear Fission
Not all decay modes are mediated by the weak interaction. For example, 𝛼 decay typically occurs
in heavy nuclei and is governed by competition between the two stronger forces. Even though the
attractive nuclear force is generally much more intense over short ranges than the electric repulsion
between the protons, the strength of the former drops quickly over the distance of ≈ 10−15 m, while
                                                   5


the latter increases in proportion to 𝑍 2 and has an unlimited range. Thus, 𝐴 > 210 nuclei have such
large radii that the strong nuclear force can barely counterbalance the proton-proton repulsion [2].
Qualitatively, the emission of a tightly bound 4 He nucleus, or 𝛼 particle, can be thought to occur
as a means of increasing the stability and reducing the size of the nucleus:
                                             𝐴      𝐴−4 ′
                                             𝑍𝑋  →𝑍−2   𝑋 +𝛼                                       (1.6)
    While it is energetically possible for a heavy nucleus to reduce the amount of energy stored in
its mass via 𝛼 decay, the kinetic energy dissipated by this disintegration is quite small compared
to the large potential energy barrier it must overcome in order to escape the nucleus. This is only
possible due to a phenomenon called quantum tunneling, which we will discuss further in Chapter
2. A similar process is responsible for the effect of nuclear fission, in which a very heavy nucleus
splits into two smaller, more tightly bound nuclei.
Nuclear Reactions and Nucleon Emission
As seen in Figure 1.3, the binding energy per nucleon begins to decline with increasing mass
number beyond iron (Fe) and nickel (Ni), which are the most tightly bound nuclei. This means
a nuclear reaction that fuses two light nuclei with a combined mass of 𝐴 < 58 can result in the
release of energy. In fact, it is this nuclear fusion process that drives the energy production of stars
like our Sun. Nuclear processes, such as fusion reactions and radioactive decay, cause a change in
the amount of energy that stored in the nuclear masses. The difference between the final binding
energies and the initial binding energies of the nuclei involved in these processes is called the
Q-value:
                ∑︁                    ∑︁                ∑︁                   ∑︁
            𝑄=      BE(𝑁 𝑓 , 𝑍 𝑓 ) −     BE(𝑁𝑖 , 𝑍𝑖 ) =    𝑚𝑖 (𝑁𝑖 , 𝑍𝑖 )𝑐2 −    𝑚 𝑓 (𝑁 𝑓 , 𝑍 𝑓 )𝑐2 (1.7)
                  𝑓                    𝑖                 𝑖                    𝑓
    Spontaneous decays always have a positive 𝑄-value, otherwise the nucleus would be stable and
remain unchanged. The 𝑄-value of a nuclear reaction can be positive or negative. Reactions for
                                                     6


Figure 1.3: Binding energy per nucleon plotted over mass number. Nuclear processes that result
in nuclei becoming more stable tend to release energy. Figure credit: HyperPhysics, Carl Rod
Nave.
which 𝑄 > 0 release energy and are said to be exothermic, while endothermic reactions (𝑄 < 0)
require energy from the environment to be added into the nuclear system in order to proceed. The
energy required to remove a single nucleon from the nucleus is called the one-nucleon separation
energy. The proton separation energy is simply defined as
                             𝑆 𝑝 = −𝑄 𝑝 = BE(𝑁, 𝑍) − BE(𝑁, 𝑍 − 1),                          (1.8)
and the neutron separation energy is
                             𝑆𝑛 = −𝑄 𝑛 = BE(𝑁, 𝑍) − BE(𝑁 − 1, 𝑍).                           (1.9)
Extremely proton-rich nuclei for which 𝑆 𝑝 < 0 are said to be proton unbound; extremely neutron-
rich nuclei for which 𝑆𝑛 < 0 are said to be neutron unbound. Exotic nuclei of this kind are only
                                                7


found far away from stability at the nuclear drip lines, beyond which the addition of an extra neutron
or proton would simply “drip” away, not forming a bound nucleus.
Nuclear Levels and Gamma Decay
This brings us to arguably the last piece of introductory knowledge necessary for understanding a
majority of the nuclear physics discussed throughout this dissertation: the concept of excited states.
So far, we have implicitly assumed that the nucleus in question is occupying its ground state, the
lowest possible energy configuration of its nucleons. However, just as the electrons in an atom can
be excited to higher atomic energy levels, so too can the nucleus be excited. Moreover, driving a
current through a chemically pure gas at low pressure, like in a neon (Ne) light fixture, results in
electronic excitations, which upon their decay, result in the emission of photons of visible light,
whose characteristic color is indicative of the energy they radiate away as the atom descends the
discrete levels. Usually, these atomic states decay within 10 ns to 10 ms (10−9 − 10−4 s) [3].
     Analogously, a nucleus that is the product of a reaction, the daughter of radioactive decay, or
simply participates in a sufficiently energetic scattering interaction, will initially populate a nuclear
configuration with a discrete energy level well above its ground state. However, the excitation
energies required to populated these various nuclear levels are on the order of 105 − 106 times
higher than that of atomic energy levels. Thus, the photons they emit, known as 𝛾 rays, are well
outside the visible spectrum and are usually given in keV or MeV units. Furthermore, their lifetimes
are frequently ≲ 10−12 s. By convention, nuclear levels are usually referenced by their lifetime
compared to the half-lives quoted for charged-particle decays.
     The lifetime of a nuclear level will often depend on its total angular momentum 𝐽 = 𝐿 + 𝑆, where
𝐿, or ℓ, is the orbital angular momentum and 𝑆 is the spin. These are conserved, discrete quantities
for which the analog in classical physics is not particularly useful in understanding their quantum
interpretation. Nuclear levels also have either even or odd parity (𝜋 = ±1). If the wave function
corresponding to a particular nucleon configuration is identical under coordinate transformation,
it is said to have even parity (𝜋 = +), but if it is a mirror image under reflection then it has odd
                                                    8


parity (𝜋 = −). The total parity of the quantum state of a nucleus is the product of over all nucleons
𝜋 = Π𝑖 (−1) ℓ𝑖 .
    Lighter nuclei have fewer possible nuclear configurations and thus have sparse and relatively
simple level structures. However, massive nuclei with many nucleons often have a high density of
nuclear levels and very complicated decay schemes, where highly excited states will emit multiple
𝛾 rays in a cascade of transitions between levels of decreasing energy until they reach the ground
state. If an nuclear level is populated such that its excitation energy is greater than the nucleon
separation energy for that nucleus (𝐸 𝑥 > 𝑆 𝑝,𝑛 ), there is some probability that this level will decay by
spontaneous nucleon emission instead. This is particularly relevant for understanding the research
topic of this dissertation specifically, as well as nuclear reactions in stars more broadly.
1.2   On the Origin of Elements
Nuclear astrophysics is an interdisciplinary field which includes areas of study in computational
astrophysics, observational astronomy, as well as both experimental and theoretical nuclear physics.
Each of these different academic disciplines provide unique insights that further our understanding
of the universe on the largest and smallest of scales. Because of the diversity of research interests
between these fields, collaboration and open exchange is extremely important in order to determine
what information is needed from its individual contributors to best serve the scientific interests of
the community as a whole.
    Ultimately, the goal of the nuclear astrophysics community is to address fundamental questions
about the universe which include, but are not limited to the following: Where did the chemical
elements that constitute the natural world originate? What are the individual contributions of
various astrophysical environments? How do the properties of nuclei affect the life cycles of stars?
How might we utilize radioactive nuclei in the cosmos to better understand the evolution of our
galaxy? What is the underlying structure of neutron stars, and what is the nature of dense nuclear
matter? The astrophysical motivation for the original research contained in this thesis dissertation
pertains to the chemical and isotopic abundances produced in certain stellar explosions. Thus, in
this section, we will focus primarily on the first two questions, in order to contextualize our current
                                                     9


understanding of the universe’s chemical evolution.
In the Beginning
The process of nucleosynthesis refers to the nuclear mechanisms by which the chemical elements
and their various isotopes were forged. Nearly all of the complex chemistry that permeates our
nearly 14-billion-year-old universe was made possible due to the nucleosynthesis occurring in
stars. Immediately following the Big Bang, the universe was too hot for any nuclei to form and was
instead likely filled with a hot, dense mixture of electrons, quark-gluon plasma, and potentially other
elementary particles. As the universe rapidly expanded and cooled, the first protons, or hydrogen
(H) nuclei, were able to condense, colliding with the high-electrons to form neutrons once the
universe reached a balmy ∼10 billion degrees (K) [4]. By the time it cooled to a temperature of
∼1 billion K, the first nucleosynthesis occurred between 1 H and free neutrons to form deuterium
nuclei (2 H), also known as deuterons (𝑑). These quickly fused to form the extremely stable 4 He.
Some other reactions participating in early Big Bang nucleosynthesis involved 𝑝, 𝑛, 3 He, and 4 He
produced trace amounts of the first lithium (Li) nuclei. However, given that the half-life of the
neutron is only about 15 minutes [5; 6], all the unreacted neutrons quickly decayed away, leaving
only the remaining H and He nuclei to form the first stars. As shown in Figure 1.4, basically all the
H and He we observe in our Solar System today were produced shortly after the Big Bang. Since
then, about 2% of those primordial nuclei have been processed into the all the heavier chemical
elements we observe today [7; 8].
                                                    10


11
     Figure 1.4: The periodic table. Chemical elements are color-coded to indicate the relative contribution from each nucleosynthesis site.
     Figure credit: Ref. [9].


The Life of Stars
The first stars did not for some ∼100 million years after the inception of the universe. The average
temperature of the gas was too high for gravity, the weakest of the four fundamental forces of nature,
to overcome the thermal pressure of the gas. At first, this prevented gas clouds from clumping
together and collapsing to form stars, but eventually, once the universe reached ∼100 K, the first
generations of stars were able to form. This is was the only time period in our universe’s history
in which star formation favored stellar masses greater than that of our Sun. This is because the
diversity of heavier chemical elements in our modern universe, with their higher degrees of freedom
in atomic and molecular transitions, allow kinetic energy that would otherwise be conserved in the
random motion of simpler atoms to be radiated away, cooling down the gas clouds more rapidly.
This meant that in the early universe, more material was needed to clump together in order for the
gravitational force to be strong enough to collapse a gas cloud into a star.
    Eventually, when the first star did form, only H and He were available for nucleosynthesis,
which proceeded quite slowly. Essentially the only reaction available was 𝑝 + 𝑝 → 𝑑 + 𝑒 + + 𝜈𝑒 ,
which requires the weak interaction to turn a proton into a neutron and is extremely unlikely to
occur. Since this 𝑝-𝑝 fusion does not produce enough energy to power the star, the star begins
to collapse in on itself [10]. However, in the process of stellar contraction, the nuclear fuel heats
up, causing any He produced via H fusion process to be converted relatively quickly to carbon (C)
through the triple-𝛼 reaction [11]. Once C was present, this kicked off the carbon-nitrogen-oxygen
(CNO) cycle, acting as a catalyst for the processing of 1 H into 4 He, as shown in Figure 1.5. The
CNO cycle prevents stellar collapse and begins producing more N and O as well [12].
    The core of a star will continue to fuse H into He until all of the H fuel has been exhausted. This
happens much more quickly in high-mass stars because, despite having more H nuclei to fuse than
low-mass stars, their stronger gravity produces higher temperatures in their interior, accelerating
the rate at which these thermonuclear reactions occur. This makes high-mass stars more luminous
and shorter-lived than low-mass stars. H-burning in a star with the mass of our Sun (𝑀⊙ ) lasts for
10 billion years, while a star with ten times that of a solar mass will exhaust its H reserves in 25
                                                   12


                          Figure 1.5: The CNO cycle. Figure credit: Borb.
million years; a 30-𝑀⊙ star will complete the H-burning stage in only 6 million years [13]. Once
H burning in the core ceases, the star can not longer support its own weight and begins to contract,
causing the core to heat up and increasing the probability for He fusion reactions to take place.
At ∼10 K, He burning ignites to synthesize C and O; the star stops contracting once the outward
thermal radiation pressure and the inward force of gravity reach equilibrium [9].
The Death of Stars: With a Bang
Provided that the star is massive enough (≳8 𝑀⊙ ), this cycle of nuclear burning in the core until fuel
is exhausted, followed by stellar contraction, and the ignition of a new source of nucleosynthesis, can
                                                   13


repeat for progressively heavier elements. All the while, lighter elements in the shells surrounding
the core can continue to burn as well. Once 12 C nuclei start fusing together, the nuclear reactions
become more varied, forming not only 24 Mg, but also 23 Na+𝑝 and 20 Ne+𝛼, among others [14]. At
the next stage, highly energetic photons can begin to break apart Ne nuclei via photodisintegration,
which facilitates the 𝛼-capture reaction 20 Ne(𝛼, 𝛾)24 Mg. During the phase of C and Ne burning,
leftover O from the He-burning stage remains inert, but once the stellar interior reaches temperatures
≳1 billion K, it too ignites [10; 14].
    Once the last of the O fuel has been exhausted, what is left in the core is mainly silicon (Si).
The star contracts once again, bringing the core to a temperature of 3 billion K, but even at these
extreme temperatures, the electrostatic repulsion from the 𝑍 = 14 nuclei prevents the Si nuclei from
fusing together. The photodisintegration reactions (𝛾, 𝑝) and (𝛾, 𝛼) involving intermediate-mass
nuclei within the stellar interior produce 𝑝 and 𝛼 particles, which are then able to overcome the
Coulomb barrier of Si and other intermediate-mass elements, like sulfur (S). These (𝑝, 𝛾) and
(𝛼, 𝛾) reactions drive the Si-burning stage towards the iron peak elements, including Fe and Ni
[15]. In order of increasing binding energy per nucleon, 56 Fe, 58 Fe, and 62 Ni are the mostly tightly
bound nuclei in nature [16]; beyond this mass region, nucleosynthesis is endothermic.
    Each successive burning stage takes less time to complete than the previous. This is because
nuclear reactions become less exothermic, and more of the energy that is released will be carried
away from the core by neutrinos. The thermonuclear reactions that produce the thermal radiation
pressure necessary to the support the star under the crushing weight of its own gravity, increasingly
consume more nuclear fuel. In the case of a 15-𝑀⊙ star, core H burning lasts on the order of
millions of years, while C burning lasts only a few thousand years. By the time the star reaches the
stage of O burning, it ends within a few weeks, and the Si burning that follows only lasts several
days [17]. Near the end of its life, the massive star has accumulated a 1.4-𝑀⊙ Si-burning core,
surrounded by a series intermittent burning shells: O, Ne, C, He, and H, listed in order of increasing
radial distance from the core. Most of these shells contain at least half a Sun’s worth (≳0.5 𝑀⊙ ) of
processed, nucleosynthetic material ready to be injected into the cosmos [9].
                                                  14


     As the Si-burning stage terminates near the end of a massive star’s life cycle, the stellar core
can no longer produce enough energy from nuclear reactions to oppose its collective gravity; the
collapse is inevitable. The core contracts slowly at first, but as the iron-peak nuclei photodisintegrate
and the temperature reaches that of the early universe, protons and high-energy electrons can
start synthesizing neutrons again. These processes dissipate energy from the core, accelerating
its collapse. The details surrounding the physical processes governing these extremely massive,
energetic, and complicated events called core-collapse supernovae (CCSNe) are an ongoing subject
of research in computational astrophysics [18; 19; 20]. Nevertheless, it is believed the rapid change
in density causes a shock wave to propagate out through the outer shells, driving one last burst of
nucleosynthetic activity in the final moments of the massive star’s violent death [21] and ejecting
all that enriched nuclear material processed over a stellar lifetime into space.
     What remains of the core depends on its mass at the time of the explosion. Most stars with
initial masses >30 𝑀⊙ should lead to black holes [22], localized regions of spacetime so dense
that their immense gravity prevents any matter or light from escaping [23]. Black holes do not
directly contribute to the chemical evolution of galaxy but actively delete nuclear matter from
existence. However, only about ∼10% of supernovae come from stars with initial masses >20 𝑀⊙
[24]. The consensus is that the majority of stars in the mass range 8𝑀⊙ < 𝑀 < 20𝑀⊙ will result
in a supernova that leaves behind an ultradense remnant of the collapsed core called a neutron star
[22], which likely play a prominent role in the nucleosynthesis of the heaviest elements found in
nature called the rapid neutron-capture process, or r-process.
The Death of Stars: With a Whimper
In stars with masses ≲8 𝑀⊙ , the nucleosynthesis reactions powering their evolution typically cease
after the production of C and O, or perhaps Mg and Ne, in slightly more massive stars; these
elements are tightly bound gravitationally to the stellar core [9]. Since energy is no longer being
generated in the core to oppose the inward pull of gravity, the core contracts, but the temperature
never gets high enough to ignite the next stage of nuclear burning. Eventually, this contraction
                                                  15


is halted by the degeneracy pressure. The stellar core becomes so compressed that the fermions
cannot be packed any more tightly without violating the Pauli exclusion principle. Having shrunk
to a fraction of its original size, its gravitational hold on the outer layers of the star gradually
weakens. As this happens, the stellar envelope is slowly ejected over the course of >100 thousand
years, along with the He, C, and N synthesized in its shell burnings [9], leaving behind a hot, dense
remnant of the stellar core; these objects are called white dwarfs.
     This is the ultimate fate of stars like our Sun. Without nuclear reactions to power their evolution
or energetic particles to dissipate their energy, white dwarfs remain dormant indefinitely. Although
not active, they are still very hot and will cool extremely slowly, emitting their thermal energy
via emission of black-body radiation. This process will take longer than the current age of the
universe, but eventually, once a white dwarf radiates away the last of its energy, it will reach thermal
equilibrium with whatever still remains of the interstellar medium and become invisible within the
void of space: a black dwarf.
     While low-mass stars do not undergo such dramatic deaths as their more massive counterparts,
they still play an important role in processing nuclear material. For one, they live much longer
than massive stars do, and they are much more common in our modern universe [25]. In fact, they
are responsible for synthesizing a substantial fraction of the heavier elements in the universe. This
is possible because temperatures only need to be high enough for nuclear reactions that release
neutrons, not for nuclei with large numbers of charged particles to merge [9]. Neutrons in low-
mass stars are not produced by high-energy collisions between protons and electrons, but dying
low-mass stars can produce free neutrons in their He-burning shells under certain circumstances
[25]. Through the process of convection, fresh material from the unreacted, H-rich envelope can
be mixed into the He-burning shell, and the combination of 4 He, 1 H, and 12 C enables a series of
nuclear reactions that produce 16 O, releasing a neutron in the process.
     These free neutrons readily fuse with Fe and other seed nuclei forged by the generation of stars
that preceded them. A seed nucleus typically captures a neutron every few weeks to months [26].
If a neutron-capture results in a product nucleus that is radioactive, it will almost certainly undergo
                                                     16


𝛽− decay before it can capture another nucleon, thus slowly synthesizing heavier nuclei close to
the valley of stability. Due to the long timescales over which this process occurs, it is called the
slow neutron-capture process, or s-process. However, this tedious mechanism is not the only way
that low-mass stars can contribute to the chemical enrichment of the galaxy. In fact, even their
white-dwarf corpses can play an important role in explosive nucleosynthesis, provided that they
are found in a binary star.
Classical Novae
Depicted in Figure 1.6, a classical nova is a luminous eruption that occurs in a binary system
that includes a white dwarf and a nondegenerate stellar companion, which orbit around a common
center of mass. As the active companion star evolves, its radius swells, and as it begins to overflow
its Roche lobe, the white dwarf can begin to siphon material from the H-rich donor to form an
accretion disk around its surface [27]. As this accreted layer builds up on the surface of the compact
white dwarf, the density and temperature of the compressed material rise, increasing the rate of
nuclear burning. The stability of this nuclear burning process is very sensitive to the mass of the
underlying white dwarf as well as the rate of accretion. Once a critical mass is reached, the layer
becomes unstable, triggering a thermonuclear runaway event [28; 29; 30]. The energy released
causes the H-rich envelope to expand tremendously, leading to its ejection and often dredging up
dense nuclear material from deeper layers of white dwarf in the process [31].
    The standard paradigm of classical nova models is one driven exclusively by thermal emission
from the hot white dwarf. After the thermonuclear runaway, the ejected shell of gas expands
into the surrounding environment at ∼102 − 103 km/s [31]. They are most identifiable as optical
transients, where the intensity of their visible light is observed to rise rapidly to a maximum and
then decay over the timescale of days or months [32]. However, as their ejecta expand and dilute,
they become increasingly transparent to radiation of shorter wavelengths, and their spectral energy
distribution shifts to the predominantly ultraviolet (UV) part of the spectrum [33]. Eventually the
photosphere recedes sufficiently inward that the white dwarf is visible in the low-energy part of
                                                  17


  Figure 1.6: Artist’s depiction of a classical nova. Figure credit: astroart.org, David A. Hardy.
the X-ray spectrum, sustained by residual nuclear burning for weeks or potentially even years [34].
Many novae also form dust grains in their ejecta, which are revealed by sudden increases in their
infrared (IR) emission [35]. This dust formation occurs rapidly and nova grains grow to large sizes
(∼1 𝜇s) compared to dust from the intersteller medium [36; 37].
    Derived from the Latin stella nova, meaning “new star,” the misnomer originates from Danish
astronomer Tycho Brahe’s 1573 book entitled De nova stella, in which he reports the observation
of what appeared to be a new star in the night sky; this stellar event has since been identified as a
supernova, named B Cassiopeiae (SN 1572) [38]. Unlike supernovae, which typically result in the
destruction of their stellar host, nova events will often recur on the timescale of years or decades.
Although estimates of the galactic nova rate have varied widely over the years [39; 40], by all
accounts, these events are common. Their frequency is ≈ 20 − 70 eruptions per year in the Milky
                                                  18


Way, with about a dozen observed annually [41]. This detection rate has increased in recent years
and is likely to continue to do so with more time dedicated to finding them using state-of-the-art
telescopes [42]. Still, many of these observations are reported by amateur astronomers as classical
novae are among the brightest transients in the night sky.
    Despite their storied role in the history of astronomy, the abundance of observational nova
data, and significant progress in computational ability, many open questions surrounding classical
novae remain. We cannot begin to address them all here, but in this dissertation, we will provide
meaningful constraints on nuclear uncertainties associated with modeling explosive nucleosynthesis
in the thermonuclear runaway of classical novae.
                                                 19


                                              CHAPTER 2
                                     THE 30 P(𝑝, 𝛾) 31 S REACTION
2.1    Effects on Astronomical Observables
Nova Nucleosynthesis
Classical novae are the second-most common type of thermonuclear eruptions in the galaxy,
following Type I X-ray bursts which occur in stellar binaries involving a neutron star instead of
a white dwarf [43]. In spite of this, they only process a small fraction of the interstellar matter
throughout the Milky Way, contributing very little in the way of galactic dust condensed from
stellar outflows [44]. Nevertheless, simulations and observational evidence both suggest that novae
are responsible for the overproduction of certain nuclei such as 7 Li [45; 46],     13 C, 15 N, and 17 O
[29; 47; 48; 49; 50; 51], perhaps accounting for a significant fraction of their galactic content [52].
Radioactive 𝛾 emitters are also produced in novae [53; 54; 55; 49; 56; 51]. For example, most novae
are thought to produce the long-lived radionuclide 26 Al and could be responsible for up to 30% of
its presence in the Milky Way [57]. Some novae are also expected to produce 22 Na, whose shorter
half-life might enable the detection of nova sites via this radionuclide’s unique 𝛾-decay signature,
depending on how much of it survives explosive nucleosynthesis [58]. Even heavier species like
31 P, 32 S, 33 S, and 35 Cl have been reported to originate in novae as well [49; 51], but the accuracy
and precision of modeling the chemical composition of their ejecta is constrained by uncertainties
in the thermonuclear rates of the nuclear reactions that participate in nova nucleosynthesis.
     Hydrodynamic simulations are used to model explosive nucleosynthesis in classical novae.
During the thermonuclear runaway, nuclei participate in a complex reaction network of competing
weak interactions, such as 𝛽+ decay and electron capture, versus proton- and 𝛼-capture reactions,
as shown in Figure 2.1. The rates at which these nuclear processes occur affect the path of
nucleosynthesis in classical novae, which is close enough to stability such that many of these decay
                                                    20


Figure 2.1: Path of nucleosynthesis in an ONe nova involving a 1.35-𝑀⊙ white dwarf, assuming
a 50% mixing ratio between the H-rich envelope and the underlying white dwarf material.
Nuclides shown in dark blue are stable, while light blue squares correspond to radioactive nuclei.
Figure credit: Ref. [59].
and reaction rates can be determined experimentally. For this reason, many of the relevant nuclear
reactions in novae have been sufficiently constrained for modeling the chemical composition of
their ejecta. However, sensitivity studies of ONe novae involving the most massive white dwarfs
indicate that 30 P(𝑝, 𝛾)31 S is the largest remaining source of nuclear uncertainty for the synthesis of
intermediate-mass elements [60; 61].
    The 2.5-minute half-life of 30 P is on the order of the thermonuclear runaway event, making it
a waiting-point nucleus. Thus, the      30 P(𝑝, 𝛾)31 S reaction serves as a bottleneck to the production
of 𝐴 > 30 in ONe novae. Furthermore, its thermonuclear rate affects the chemical and isotopic
abundances of the processed nuclear material that ONe novae contribute to the galaxy. These
                                                      21


observables can also inform our understanding of individual novae and their properties.
Presolar Grains
Perhaps the most interesting effect of the 30 P(𝑝, 𝛾) 31 S reaction on the isotopic ratios of nova ejecta
is related to the classification of a handful of unidentified presolar grains recovered from certain
meteorites; an example is depicted in Figure 2.2. These are tiny bits of stardust, often only a few
microns in length, that were ejected as hot plasma from their original astrophysical source into
interstellar space before cooling and condensing into crystalline granules. These grains were then
incorporated into the molecular cloud and mixed with the surrounding material, which eventually
condensed to form the Solar System [62]. Thus, they are said to be “presolar” because they were
forged before the formation of the Solar System and can be identified by isotopic ratios that are well
outside the expected range for terrestrial rocks and meteoroids formed in the early Solar System.
The discovery and detailed analysis of these grains has opened up a new field within astronomy,
providing unique insights into the formation of the Solar System as well as the chemical evolution
of the Milky Way [63; 64; 65]. Several different types of presolar grains have been identified,
including silicon carbide (SiC), graphite (C), diamond (C), silicon nitride (Si3 N4 ), and various
silicate and oxide compounds [66], but to date, none have been confirmed originating in classical
novae.
     SiC grains are among the most well-studied presolar grains and can be classified into distinct
populations according to their stellar birthplace [67]. Almost all of these grains come from
the outflows of asymptotic giant branch (AGB) stars, with the 93% “mainstream” population
originating in low-mass stars of this variety [68; 69; 70]. Another 4 − 5% are classified as AB
grains, characterized by low    12 C:13 C ratios, and while several types of stars have been proposed
as progenitors of AB grains, the majority likely originated in C-rich, so-called J-type stars [71].
Y and Z grains make up about 1% each, whose origins are linked to low-metallicity AGB stars
[72; 73; 74], while X grains account for the remaining 1% and are distinguished by their large
excesses in 44 Ca and 28 Si, evidence they formed in the ejecta of supernovae [75; 76]. However, ion
                                                    22


Figure 2.2: Electron microscope image of a SiC presolar. Photo credit: J. Huth, A. Besmehn, and
J. Kodolányi, Max Planck Institute for Chemistry.
microprobe isotopic analysis of several rare SiC grains, as well as a few anamolous graphite grains,
has revealed isotopic signatures indicative of potential nova origins [77; 78].
    Classical novae have previously been implicated in some of the isotopic anomalies found in
meteorites, despite the fact that they are not major contributors to the chemical evolution of the
galaxy. Although about 30 − 40% of novae are known to produce dust some 20 − 100 days after
their initial outburst [79], the argument for the existence of nova grains has relied primarily on low
20 Ne:22 Ne  ratios in certain graphite grains, which could be attributed to the decay 22 Na, potentially
indicative of nova nucleosynthesis [80; 81; 82]. In contrast, these unusual grains have low 12 C:13 C
ratios, extremely low      14 N:15 N ratios, and high  26 Al:27 Al ratios, all qualitatively consistent with
theoretical predictions for both CO and ONe nova models [77]. The C and N ratios are plotted in
Figure 4.20 for grain data taken from the Presolar Database at Washington Unviersity in St. Louis
[83; 84].
    These candidate nova grains also exhibit large excesses in 30 Si. The Si isotopic ratios of various
presolar grains are shown in Figure 2.4. Generally speaking, 29 Si:28 Si and 30 Si:28 Si ratios of nova
ejecta increase with white dwarf mass. Thus, all CO nova models predict close-to or lower-than
solar  29 Si:28 Si ratios as well as close-to solar  30 Si:28 Si ratios. While nucleosynthesis is mostly
                                                      23


Figure 2.3: Isotopic ratios for plotted for C and N. Candidate nova grains are circled in magenta.
Figure credit: Ref. [85].
constrained to the CNO mass region for CO novae, the higher peak temperatures in ONe novae
and the more abundant “seed” nuclei from beyond the CNO mass region lead to the synthesis of
intermediate-mass elements. So while all ONe nova models predict close-to or lower-than solar
29 Si:28 Si ratios and models for white dwarf masses ≤1.15 predict close-to or lower-than solar
30 Si:28 Si ratios, ONe nova models for white dwarf masses ≥1.25 𝑀⊙ predict large excesses in 30 Si
[60].
    The conclusion that these unidentified grains are from ONe novae ejecta, however, is controver-
sial for two main reasons. First, not only are ONe novae less common than CO novae, they are also
less prolific dust producers. Second, the isotopic abundance of  30 Si in ONe nova ejecta is vastly
                                                 24


Figure 2.4: Isotopic ratios for plotted for Si. Candidate nova grains are circled in magenta.
Figure credit: Ref. [85].
over-predicted by simulations compared to grain data. To quantitatively reproduce the observed
30 Si:28 Si ratios, one must assume some unknown mixing process between the freshly synthesized,
30 Si-enriched   material and over 10 times as much unprocessed, isotopically close-to-solar material
before the grains formed [60]. Because of these complications, in recent years, there have been
efforts to explain SiC grain formation in the ejecta of CO novae without invoking this dilution
process [79; 86]. While one-dimensional (1D), hydrodynamic simulations represent the current
state-of-the-art capabilities of nova modeling and generally agree with astronomical observations
of elemental abundances, large systematic uncertainties remain [60]. For one, the parameter space
of all relevant input variables is quite large and includes factors such as the composition of the white
dwarf, the peak temperature and density, the explosive timescales, any possible dilution of ejecta
after the outburst, as well as potentially the biggest unknown, how the accreted material from the
donor star mixes with the outer layers of the WD [79]. So far, multidimensional investigations into
this mixing problem have not been successful in reproducing the gross properties of nova outbursts
                                                     25


[77], and to make matters worse, the lack of convective mixing in these parameterized calculations
leads to overestimating the influence of nuclear uncertainties [60]. These theoretical challenges
cannot be directly eliminated by nuclear physics experiments, but these measurements can be
employed to reduce uncertainties associated with reaction rates that confound model predictions.
    Currently, most theoretical models rely on the Hauser-Feshbach statistical method for evaluating
the 30 P(𝑝, 𝛾)31 S rate. When estimating the effect of nuclear uncertainties on their predictions,
modelers will often vary this rate by arbitrary factors of 10. Tightening the error bars on this rate
could substantially constrain the range of predicted     30 Si:28 Si ratios for candidate nova grains. If
proton-capture on 30 P is particularly rapid in ONe novae, this will enable increased nucleosynthesis
of 𝐴 > 30 nuclides. However, if this reaction proceeds relatively slowly on the timescales of
the thermonuclear runaway, the radioactive      30 P will primarily 𝛽+ decay to stable    30 Si, leading to
excesses in ONe nova ejecta.
Nuclear Thermometers and Mixing Meters
Observational studies using IR, UV, and optical spectroscopy are able to identify elemental abun-
dances within the ejecta shells surrounding nova sites [44; 87; 88]. The ejected gas and dust from
nova outbursts consist of white dwarf matter and the accreted material from the companion star,
which have been processed by explosive H-burning. The chemical makeup of this ejecta provides
information about the composition of the underlying white dwarf as well as the thermonuclear
runaway event, including the peak temperature achieved and the expansion rate of the accreted
envelope. Thus, chemical abundances can be used to constrain models of stellar explosions [89].
Hydrodynamic simulations using the 1D code shiva were performed for a range of white dwarf
masses suggest that several abundance ratios are quite sensitive to temperature and may be useful
nuclear thermometers for constraining the peak temperatures achieved in novae [52]. The hydro-
dynamic models were used to calculate temperature-density profiles across all mass zones of the
envelope. This profile served as input for Monte Carlo post-processing nuclear reaction network
calculations to assess the impact of thermonuclear rate uncertainties.
                                                    26


Figure 2.5: Ratios of elemental abundances, given in terms of their mass fractions, plotted as
function of peak temperature achieved in hydrodynamic nova model simulations. Figure credit:
Ref. [90].
    The best candidate thermometers exhibit steep, monotonic dependence on peak temperature
and include N:O, N:Al, S:Al, O:Na, Na:Al, O:P, and P:Al. These are plotted in Figure 2.5. The
sensitivity of these thermometers to variation in thermonuclear rates has been investigated using
post-processing nucleosynthesis calculations, and while N:O, N:Al, O:Na, and Na:Al are robust to
these uncertainties, O:S, S:Al, O:P, and P:Al strongly influenced by the poorly constrained rate of
the 30 P(𝑝, 𝛾)31 S reaction. Thus, determining this thermonuclear rate would reduce uncertainties
associated with these nuclear thermometers and allow a more precise, accurate determination of
peak nova temperatures.
    The peak temperature achieved in a nova strongly depends on the mass of the underlying white
dwarf and its initial luminosity. It also is influenced by the mass, metallicity, and accretion rate
of the disk [90]. The fact that both the processed, H-rich fuel and material from the outer layers
                                                  27


of the white dwarf can be observed in nova ejecta suggests that there must be mixing between the
two stellar sources before the thermonuclear runaway, but the mechanism by which this occurs is
not well-understood [91; 92; 93]. However, the extent to which the envelope mixes with the outer
layers of the dense white dwarf affects the chemical abundances of the observed nova ejecta. Thus,
like thermometers, certain elemental abundances can be used as nuclear mixing meters if there
is a steep, monotonic relationship between the abundance ratio and the mixing fraction, which is
defined as mass of the outer white dwarf matter that has been mixed as a fraction of the envelope
[94].
    Nova simulations have been performed for three different mixing fractions 25 − 75% at four
different white dwarf masses (1.15 − 1.35 𝑀⊙ ) with a fixed initial luminosity and accretion rate.
Just as in the studies investigating nuclear thermometers, hydrodynamic simulations were used to
evaluate the temperature-density across the active burning regions in the nova. Nuclear reaction
network calculations were used to quantify the affect of reaction rate uncertainties on the final
ejecta abundances. The results of these calculations, as shown in Figure 2.6, identified several
useful mixing meters for ONe novae, including ΣCNO:H, Ne:H, Mg:H, Al:H, and Si:H. While
most of these ratios were found to be robust with respect to thermonuclear rate uncertainties, again
the 30 P(𝑝, 𝛾)31 S reaction was implicated in the uncertainty of Si:H ratios. The error bars on each
elemental abundance are largely unchanged between the top and bottom panels in Figure 2.6, with
the exception of Si, since it is highly sensitive to the rate of proton capture on 30 P. This further
motivates the experimental determination of this critical thermonuclear rate for constraining ONe
nova models.
2.2    Thermonuclear Reaction Rate Formalism
Here, we follow a derivation provided in Christian Iliadis’s Nuclear Physics of Stars [15] to arrive
at a general formula for the total thermonuclear rate of a non-specific nuclear reaction. A more
detailed description of reaction theory and its applications to nucleosynthesis can be found there,
as well as in Nuclear Reactions for Astrophysics by Thompson and Nunes [95].
    In nuclear physics, often the most fundamental quantity used to describe the probability that
                                                  28


Figure 2.6: Elemental abundances, given in terms of their mass fraction relative to H, predicted
in hydrodynamic nova model simulations. The three different marker shapes/colors represent the
mixing ratios between the underlying white dwarf material and the H-rich envelope. Top panel:
Error bars only include variation due to changes in peak temperature. Bottom panel: Error bars
include variation due to changes in peak temperature as well as uncertainties in thermonuclear
reaction rates. Figure credit: [94].
                                                29


two nuclei will participate in a reaction is the cross section 𝜎. Consider the simple case of beam
particles impinging on a target area 𝐴 at a constant rate 𝑁 𝑏 /𝑡. The cross section is defined as
the ratio between the number of reactions per unit time 𝑁𝑟 /𝑡 and the product of the number of
nonoverlapping target nuclei 𝑁𝑡 times the beam flux:
                                                       𝑁𝑟 /𝑡
                                           𝜎≡                     .                              (2.1)
                                                  𝑁𝑡 · 𝑁 𝑏 /(𝑡 𝐴)
Thus, we can write the rate at which nuclear reactions occur within some volume 𝑉 in terms of the
cross section and the relative velocity between the beam particle and the target nucleus:
                                      𝑁𝑟           𝑁𝑡 𝑁 𝑏         𝑁𝑡 𝑁 𝑏
                                           =𝜎·             = 𝜎𝑣          .                       (2.2)
                                     𝑡 ·𝑉         𝑡𝐴 · 𝑉          𝑉 𝑉
In a stellar environment, neither the target nor the projectile particles are ever truly at rest, and
the nuclear reaction cross section depends on the relative velocity between the two nuclei. For a
reaction involving four species 1 + 2 → 3 + 4, we can define our reaction rate 𝑟 12 ≡ 𝑁𝑟 /(𝑉𝑡) and
write it in terms of the number densities for the target nucleus (𝑁1 ≡ 𝑁𝑡 /𝑉) and projectile nucleus
(𝑁2 ≡ 𝑁𝑡 /𝑉) as
                                           𝑟 12 = 𝑁1 𝑁2 𝑣𝜎(𝑣).                                   (2.3)
    In principle, the relative velocity between projectile and target nuclei can be any positive real
                                                                                    ∫∞
number, which we can represent as an arbitrary probability distribution 𝑓 (𝑣), where 0 𝑓 (𝑣)𝑑𝑣 = 1.
Generalizing Equation (2.3) over this velocity distribution, we can define the reaction rate per
particle pair as
                                         ∫  ∞
                            𝑟 12 = 𝑁1 𝑁2       𝑣 𝑓 (𝑣)𝜎(𝑣)𝑑𝑣 ≡ 𝑁1 𝑁2 ⟨𝜎𝑣⟩12 .                    (2.4)
                                          0
However, in practice, the most useful quantity for describing the frequency of nuclear reactions in
stellar environments is the thermonuclear reaction rate 𝑁 𝐴 ⟨𝜎𝑣⟩12 , whose units are cm3 /mol/s and
where 𝑁 𝐴 is Avogadro’s number.
                                                     30


     In most astrophysical cases, particles within a stellar plasma are not degenerate and move at
nonrelativistic speeds [96]. Thus, their motion in thermal equilibrium can be described using the
Maxwell-Boltzmann velocity distribution
                                             𝜇  3/2
                                                          𝑒 −𝜇𝑣 /(2𝑘𝑇) 4𝜋𝑣 2 𝑑𝑣,
                                                                2
                                𝑓 (𝑣)𝑑𝑣 =                                                          (2.5)
                                             2𝜋𝑘𝑇
where 𝜇 = 𝑚 1 𝑚 2 /(𝑚 1 + 𝑚 2 ) is the reduced mass of the two-body system, 𝑘 = 8.6173 × 10−5 eV/K
is the Boltzmann constant, and 𝑇 is the temperature of the stellar plasma. It can be shown that if
the velocity distributions of both species of interacting particles are Maxwellian, then the relative
velocities between the two species must also follow this distribution [14]. In the nonrelativistic
limit, 𝐸 = 𝜇2 /2, and thus 𝑑𝐸/𝑑𝑣 = 𝜇𝑣, allowing us to write the energy distribution as1
                                           𝜇  3/2                         √︂
                                                          −𝐸/𝑘𝑇      2𝐸 𝑑𝐸      𝜇
                             𝑓 (𝐸)𝑑𝐸 =                  𝑒         4𝜋
                                           2𝜋𝑘𝑇                       𝜇 𝜇 2𝐸
                                                                                                   (2.6)
                                           2      1 √ −𝐸/𝑘𝑇
                                       =√                  𝐸𝑒         𝑑𝐸,
                                             𝜋 (𝑘𝑇) 3/2
which we can then plug back into Equation 2.4. After multiplying by our normalization constant
𝑁 𝐴 , we arrive at a generalized thermonuclear rate for particle-induced reactions:
                                         √︄                ∫   ∞
                                             8 𝑁𝐴
                          𝑁 𝐴 ⟨𝜎𝑣⟩12 =                            𝐸𝜎(𝐸)𝑒 −𝐸/𝑘𝑇 𝑑𝐸.                 (2.7)
                                            𝜋𝜇 (𝑘𝑇) 3/2      0
The Gamow Window
Consider a beam of protons impinging on a target of 30 P. According to Equation 2.1, if the beam rate,
target area, and total number of target nuclei are known, an experiment measuring the number of
induced 30 P(𝑝, 𝛾)31 S reactions over a fixed period of time could be performed to determine the total
cross section, and the total thermonuclear rate could be calculated using Equation 2.7. However,
the 2.5-min half-life of unstable 30 P means using it as a target in a stable beam experiment is nearly
impossible. Inverse kinematics experiments involving a radioactive beam of           30 P incident on a
hydrogen target are also problematic because, due to their refractory chemical nature,       30 P beams
                                                     31


are difficult to produce with sufficient intensity at the relevant energies using either fragmentation
or isotope separation on-line (ISOL) techniques. Due to the energy dependence of the nuclear cross
section and the statistical velocity distribution of the stellar plasma, there is a relatively small range
of energies that are relevant for determining a thermonuclear rate; this is known as the Gamow
window.
    The transmission coefficient 𝑇ˆ describes the probability of a quantum particle tunneling through
a potential barrier and can be determined by solving the Schrödinger equation. For the simple case
of an s-wave (ℓ = 0) scattering on a time-independent potential, the radial equation becomes
                                     𝑑 2 𝑢(𝑟) 2𝑚
                                               + 2 [𝐸 − 𝑉 (𝑟)]𝑢(𝑟) = 0.                               (2.8)
                                       𝑑𝑟 2      ℏ
In the case of charged-particle reactions, over short distances, the attractive nuclear force between
the target and projectile can be described as a square-well with some width 𝑅0 and depth 𝑉0 , but
for 𝑟 > 𝑅0 , there exists a repulsive Coulomb potential as shown in Figure 2.7. Using this potential
to solve Equation 2.8 in the low-energy limit, the leading order term for the s-wave transmission
coefficient is known as the Gamow factor
                                                   √︂              
                                                2𝜋      𝜇
                                  𝑇ˆ ≈ exp −               𝑍0 𝑍1 𝑒 ≡ 𝑒 −2𝜋𝜂 ,
                                                                  2
                                                                                                      (2.9)
                                                 ℏ     2𝐸
where we define the Sommerfeld parameter 𝜂. The Gamow window describes the overlap between
this probability density function and the Maxwell-Boltzmann distribution as shown in Figure 2.8.
The Gamow peak is the energy at which the thermonuclear reactions are most likely to occur and
can be determined by taking the derivative of the product of these two functions:
                                  √︂                      
                    𝑑            2𝜋      𝜇         2     𝐸
                         exp −              𝑍0 𝑍1 𝑒 −                 =0
                   𝑑𝐸             ℏ 2𝐸                  𝑘𝑇 𝐸=𝐸0
                                  √︂                        √︂                                  (2.10)
                               2𝜋       𝜇         2    𝐸0 𝜋 𝜇                2 −3/2  1
                     =exp −                𝑍0 𝑍1 𝑒 −                  𝑍0 𝑍1 𝑒 𝐸 0 −      .
                                ℏ 2𝐸 0                 𝑘𝑇 ℏ 2                       𝑘𝑇
Thus, the peak of the Gamow window occurs at
                                                       32


Figure 2.7: Diagram of the potential energy diagram for a charged particle with energy 𝐸
approaching a nucleus as a function of radial distance 𝑉 (𝑟). At long ranges (𝑟 > 𝑅0 ), the
projectile experiences repulsive Coulomb force, while over short ranges (𝑟 < 𝑅0 ), the residual
nuclear force dominates and results in an attractive potential. A projectile need not have energy
𝐸 > 𝑉𝐶 to overcome the Coulomb barrier given the possibility of quantum tunneling. Figure
credit: Ref. [15]
                                  h 𝜋                 𝜇         i 1/3
                           𝐸0 =          (𝑍0 𝑍1 𝑒 2 ) 2     (𝑘𝑇) 2
                                     ℏ                   2
                                                                 1/3                        (2.11)
                                              2 2 𝑚0𝑚1         2
                               = 0.1220 𝑍0 𝑍1                𝑇            [MeV].
                                                    𝑚0 + 𝑚1 9
Given a temperature 𝑇9 in units of gigakelvin (GK), this numerical expression returns the resonance
energy in units of MeV at which the Gamow distribution is at a maximum for a given thermonuclear
reaction.
    While this distribution is asymmetric, it can be approximated with a Gaussian function of the
same maximum amplitude. Rewriting the Gamow distribution in terms of 𝐸 0 ,
                                                       33


Figure 2.8: Arbitrary Gamow window. Dashed curves correspond to the Maxwell-Boltzmann
distribution and the probability density for penetrating the Coulomb barrier. The Gamow window,
plotted as a solid black curve, represents a product of these two probability distributions. Figure
credit: Ref. [15].
                          √︂                                    3/2
                                                                           !
                        2𝜋     𝜇               𝐸              2𝐸         𝐸
                 exp −            𝑍0 𝑍1 𝑒 2 −       = exp − √ 0 −
                         ℏ    2𝐸               𝑘𝑡               𝐸 𝑘𝑇 𝑘𝑇
                                                                       "         2#        (2.12)
                                                             3𝐸 0            𝐸 − 𝐸0
                                                    ≈ exp −          exp −              ,
                                                               𝑘𝑡             Δ/2
and enforcing the condition that the curvatures must agree at 𝐸 = 𝐸 0 ,
                                             3/2
                                                        !
                                  𝑑 2 2𝐸 0           𝐸                 3
                                     2
                                         √        +            =
                                 𝑑𝐸         𝐸 𝑘𝑇 𝑘𝑇 𝐸=𝐸           2𝐸 0 𝑘𝑇
                                                       2
                                                             0                                 (2.13)
                                         𝑑 2 𝐸 − 𝐸0                8
                                     =                         =     .
                                        𝑑𝐸 2      Δ/2 𝐸=𝐸0 Δ
Solving for the width of the Gamow window peak, we find
                                                     34


                                     √︂
                                        𝐸 0 𝑘𝑇
                              Δ=4
                                           3                                                        (2.14)
                                                   𝑚 0 𝑚 1 5 1/6
                              Δ=   0.2368(𝑍02 𝑍12          𝑇 )      [MeV].
                                                  𝑚0 + 𝑚1 9
    The peak temperature range for ONe classical novae is often reported to be 𝑇peak = 0.1 − 0.4
GK [97], so even across a wide range of temperatures, the Gamow peak for 30 P(𝑝, 𝛾)31 S lies within
a relatively small range at low resonance energies: 159 − 399 keV. Still, even for 𝑇 = 0.4 GK,
this resonance energy is much higher than 𝑘𝑇 = 34 keV, which suggests most of the particles
participating in this reaction have relative velocities sampled from the high-energy tail of Maxwell-
Boltzmann distribution.
    However, the Gamow window is only a rough guide for identifying the energy regime most
likely to be important for constraining a nuclear reaction rate. This is because nuclear reactions
can proceed in various ways, and the existence of a resonance state may greatly enhance the total
thermonuclear rate, even if the resonance energy lies outside the Gamow window.
Radiative Proton Capture
The process of radiative proton capture involves some target nucleus 𝐴 𝑍 capturing a proton to form
a 𝐴+1 (𝑍 + 1) ∗ product nucleus in an excited state. That excitation energy promptly radiates away
via a cascade of one or more 𝛾 ray emissions, leaving behind the       𝐴+1 (𝑍 + 1) nucleus in its ground
state configuration. Radiative proton capture reactions occur when the projectile and target nuclei
get close enough such that they can interact via the attractive nuclear potential shown in Figure
2.7. In a stellar plasma, this typically happens when the center-of-mass energy of the two bodies is
within the Gamow window. A nonresonant, direct capture reaction occurs when the target nucleus
captures a proton into a bound state of the product nucleus. If a nonresonant reaction does occur,
a photon is simultaneously emitted, with a 𝛾 energy equal to the difference between the excitation
energy of the newly-formed      𝐴+1 (𝑍  + 1) ∗ bound state and the summed kinetic energy of both the
projectile and target nucleus prior to the reaction. In general, the cross section for this direct capture
                                                    35


       Figure 2.9: Cartoon diagram of resonant proton capture for the 30 P(𝑝, 𝛾)31 S reaction.
process varies smoothly as a function of energy across the Gamow window, but cross sections may
sharply increase at particular energies due to resonant capture.
     In this context, resonances correspond to nuclear levels with excitation energies above the
proton separation energy (𝐸 𝑥 > 𝑆 𝑝 ), such that they are proton unbound. The presence of these
resonances states can enhance the synthesis or facilitate the destruction of certain nuclei more
efficiently, depending our their properties and the stellar environment. Each resonance state has a
characteristic mean energy and width. The resonance energy is defined by the difference between
the excitation energy of the proton-unbound state and the proton separation energy (𝐸𝑟 = 𝐸 𝑥 − 𝑆 𝑝 ).
Due to the Coulomb barrier, as resonance energy increases, so does the proton decay partial width
Γ𝑝 relative to the partial 𝛾 decay width Γ𝛾 . These quantities are proportional to the probability of a
proton-unbound state decaying via proton emission or 𝛾 emission, respectively. The sum over all
                                                            Í
decay channels is the total decay width of the state Γ = 𝑖 Γ𝑖 and is inversely proportional to the
lifetime Γ = ℏ/𝜏.
     When the combined center-of-mass energy of a free proton and nearby 𝐴 𝑍 nucleus are close
                                                  36


to the resonance energy of a     𝐴+1 (𝑍 + 1) ∗ state, the nuclear reaction cross section is significantly
enhanced. In this case, as shown in Figure 2.9, the proton and target nucleus form a compound
nucleus which can either re-emit the proton or 𝛾 decay to lower energy levels of the product nucleus,
having undergone radiative capture. If there are other decay modes available, it is also possible for
the nucleus to proceed through those channels instead. Using the one-level, Breit-Wigner formalism
to describe narrow, isolated resonances, we can express the resonant reaction cross section as
                                      𝜆2       (2𝐽𝑟 + 1)            Γ𝑝 Γ𝛾
                         𝜎BW (𝐸) =                                               ,                 (2.15)
                                      4𝜋 (2𝐽 𝑝 + 1)(2𝐽𝑡 + 1) (𝐸𝑟 − 𝐸) 2 + Γ2 /4
                  √︁
where 𝜆 = 2𝜋ℏ/ 2𝜇𝐸 is the de Broglie wavelength, 𝐽𝑟 is the spin of the resonance, 𝐽 𝑝 is the spin
of the projectile, and 𝐽𝑡 is the spin of the target nucleus. For the sake of brevity, we can define
𝜔 = (2𝐽𝑟 + 1)/[(2𝐽 𝑝 + 1)(2𝐽𝑡 + 1)]. This Breit-Wigner model is only valid for narrow, isolated
resonances. Resonances are said to be “narrow” if their partial widths are approximately constant
across the total width of the resonance, and they must be “isolated” in the sense that their amplitudes
cannot overlap significantly with any nearby resonances. Since both the nuclear level density and
average resonance width tend to increase as a function of excitation energy, this model is most
useful for relatively low-energy resonances just above the proton separation energy.
    Plugging the Breit-Wigner cross section from Equation 2.15 into Equation 2.7 for the total
thermonuclear reaction rate and simplifying,
                                      √          ∫ ∞
                                        2𝜋ℏ2                  Γ𝑝 Γ𝛾
                     𝑁 𝐴 ⟨𝜎𝑣⟩12  =             𝜔                          𝑒 −𝐸/𝑘𝑇 𝑑𝐸.              (2.16)
                                    (𝜇𝑘𝑇)  3/2
                                                   0    (𝐸𝑟 − 𝐸) + Γ /4
                                                                 2   2
Assuming the resonance is sufficiently narrow, the Maxwell-Boltzmann factor should be constant
over the total width of the state. Thus, it can be pulled out of the integrand with the approximation
𝐸 = 𝐸𝑟 , allowing us to evaluate the remaining integral analytically:
                                                     37


                                    √                              ∫ ∞
                                      2𝜋ℏ2 −𝐸𝑟 /𝑘𝑇              2                Γ/2
                    𝑁 𝐴 ⟨𝜎𝑣⟩12  =            𝑒         𝜔Γ𝑝 Γ𝛾
                                  (𝜇𝑘𝑇)  3/2                   Γ 0 (𝐸𝑟 − 𝐸) 2 + Γ2 /4
                                    √
                                      2𝜋ℏ2       −𝐸𝑟 /𝑘𝑇 Γ 𝑝 Γ𝛾
                                =            𝜔𝑒          𝜔        2𝜋                              (2.17)
                                  (𝜇𝑘𝑇) 3/2                  Γ
                                        3/2
                                     2𝜋
                                =             ℏ2 𝑒 −𝐸𝑟 /𝑘𝑇 𝜔𝛾.
                                    𝜇𝑘𝑇
Defining the quantity 𝛾 ≡ Γ𝑝 Γ𝛾 /Γ, we can now express the thermonuclear reaction rate in terms of
the resonance strength 𝜔𝛾 for a single level. The contributions of multiple narrow resonances can
be summed incoherently to yield the total thermonuclear rate:
                                                    3/2
                                                2𝜋           ∑︁
                              𝑁 𝐴 ⟨𝜎𝑣⟩12  =               ℏ2      (𝜔𝛾)𝑟 𝑒 −𝐸𝑟 /𝑘𝑇 .               (2.18)
                                               𝜇𝑘𝑇            𝑟
    Due to relatively low level density of     31 S and limited peak temperatures achieved in classical
novae, the thermonuclear rate of the 30 P(𝑝, 𝛾)31 S reaction is expected to be dominated by radiative
proton captures into a small number of low-energy resonances. For this reason, over the past two
decades, significant experimental and theoretical effort has been dedicated to identifying these
resonances and constraining their properties.
2.3    Indirect Methods for Reaction Studies
Because it is not possible to measure 30 P(𝑝, 𝛾)31 S directly at present, many experimental attempts
to determine the thermonuclear rate have employed clever, indirect techniques for constraining
the properties of   31 S levels within the relevant energy regime near the Gamow window. In the
years since this reaction was first identified as being particularly important for modeling nova
nucleosynthesis, numerous experimental investigations have been performed, and it is likely that
all of the most important resonances have been observed. Some have been sufficiently constrained
for utilization in a resonant reaction rate calculation. The current status of all relevant 31 S excited
states is discussed in Chapter 4. Here we will introduce some of the indirect methods employed in
to study the 30 P(𝑝, 𝛾)31 S reaction.
                                                       38


Single-Nucleon Transfer Reactions
As the name suggests, this type of reaction involves the transfer of a single nucleon between the light,
projectile nucleus and the heavy, target nucleus such that the ejectile emitted from the compound
nucleus differs in mass number from the incoming projectile by Δ𝐴 = ±1. For example, in the case
of 31 S, excited states have been populated using the forward reactions 32 S(𝑝, 𝑑)31 S [98; 99; 100],
32 S(3 He,𝛼)31 S [101], and 32 S(𝑑, 𝑡)31 S [102; 103].
     The resulting spectra from the charged ejectiles correspond to the energy transferred between
the reactants in the formation of the compound nucleus, allowing the excitation energies of the
levels populated in the 31 S* to be deduced. The energies of these ejectiles can be measured by a
Si detector array, like in the cases of Refs. [98; 100], or with the use of magnetic spectrographs,
which measure the kinetic energy of the charged particles by the magnitude of their deflection
through a strong magnetic field, as employed in Refs. [101; 102; 103; 99; 100]. By performing
these reaction measurements at many different angles, the relative intensity of the observed peaks
reveal an angular distribution. To investigate the spin and parity assignments of the final state in
the recoil nucleus, Distorted-wave Born approximation (DWBA) calculations are often utilized to
fit the data for multiple values of orbital angular momentum ℓ, constraining possible values of 𝐽 𝜋
for a given level.
     Single-nucleon transfer reaction measurements can also be employed using inverse kinematics,
by impinging a heavy beam on a light target. The reaction            30 P(𝑑, 𝑛)31 S was used to measure
the angle-integrated cross sections for levels above the proton emission threshold and deduce
spectroscopic factors of these states [104].
Charge-Exchange Reactions
Charge-exchange reactions are similar to single-nucleon transfer reactions in that both the incoming
projectile and outgoing ejectile are slightly different light nuclei. However, instead of the total mass
number of the reacting nuclei changing, protons and neutrons are effectively swapped in the process.
This occurs in the reaction    31 P(3 He,𝑡)31 S, which has been measured to probe some of the same
                                                    39


resonances of astrophysical interest [105; 102; 106; 107]. The angular distribution of the ejected
triton, 𝑡 in this case, as well as angular momentum selection rules can be used to constrain the spin
and parity of resonance states populated through charge exchange.
In-Beam Gamma-Ray Spectroscopy
Experiments utilizing 𝛾-ray spectroscopy involve populating highly excited states in the nucleus of
interest and observing as many 𝛾 rays as possible that are emitted during the cascade of 𝛾 decay
transitions between different energy levels before reaching the ground state. This requires a high
detection efficiency in order to collect as many statistics as possible, as well as a precise detector
resolution in both energy and time. This is important for the purposes of differentiating 𝛾 rays of
similar energies as well as accurate identification of coincidences between 𝛾 events.
     Experiments performed at the same accelerator facility using the same Gammasphere array
employed two different reactions,      12 C(20 Ne, 𝑛)31 S [108; 109] and  28 Si(𝛼, 𝑛)31 S [110; 111], to
populate high-spin states in 31 S. Both double- (𝛾-𝛾) and triple-coincidences (𝛾-𝛾-𝛾) were used to
construct detailed decay schemes, which were then compared to the level structure of the mirror
nucleus 31 P for the purpose of constraining spin and parity assignments.
Beta Decay Spectroscopy
As discussed in Chapter 1, 𝛽 decay is a form of radioactivity mitigated by the weak interaction
that results in the emission of either an 𝜈 𝑒 -𝑒 − or 𝜈𝑒 -𝑒 + pair. It also follows the selection rules
for Fermi and Gamow-Teller transitions [14; 112; 113]. The emission of two spin-1/2 particles
may cause a change in angular momentum of Δ𝐽 = 0, ±1 between the final and initial state; larger
changes in angular momentum are said to be “forbidden” and, although technically not impossible,
are highly suppressed. In the case of       31 Cl, which has a 𝐽 𝜋 = 3/2+ ground state, its 𝛽+ decay
almost exclusively populates 𝐽 𝜋 = 1/2+ , 3/2+ , 5/2+ levels in 31 S. This is useful for placing strong
spin-parity constraints on newly observed states and is particularly relevant for the astrophysically
                                                    40


important 30 P(𝑝, 𝛾)31 S reaction, since these form the ℓ = 0, 1 resonances expected to dominate the
total thermonuclear rate in ONe classical novae.
    The process of 𝛽-delayed particle emission occurs when a radioactive nucleus undergoes 𝛽
decay and populates an excited state in the daughter nucleus with an excitation energy above the
particle-emission threshold, which can then spontaneously decay via particle emission. The first
observation of 31 Cl 𝛽-delayed proton decay was reported over 40 years ago [114], the implications of
which were presented in further detail by Refs. [115; 116]. However, it was not until 2006 that 31 Cl
𝛽+ decay was investigated on the basis of astrophysical motivations [117]. Using three double-sided
Si strip detectors (DSSDs) to measure proton energies in the laboratory frame, Kankainen et al.
observed all of the same 𝛽-delayed proton decays reported by Ref. [116], including several more.
In addition, inclusion of a high-purity Ge detector (HPGe) allowed for the conclusive identification
of the  31 Cl ground state’s isobaric analog in   31 S through measurement of its 𝛾 decay. Another
experiment, also using solid-state Si detectors, measured the intensities and lab-frame energies of
31 Cl 𝛽-delayed proton decays; these results were reported in a 2011 doctoral thesis dissertation but
were never published in a peer-reviewed journal [118]. However, this measurement has provided the
best-resolution, highest-statistics spectrum of the 31 Cl(𝛽𝑝)30 P decay sequence yet, and its reported
intensities and energies (converted to the center-of-mass frame) were essentially adopted unaltered
in the latest evaluation of recommended literature values for 𝛽-delayed charged-particle decays
[119].
    Prior to the original research contained within this dissertation, the most recent investigation
into the 30 P(𝑝, 𝛾)31 S reaction using 𝛽+ decay spectroscopy involved a 𝛽-delayed 𝛾 ray measurement
performed at NSCL. A radioactive beam of 31 Cl was implanted into a plastic scintillator, surrounded
by the Clovershare Array, which consisted of nine HPGe detectors. Subsequent analysis of the
data collected during this experiment resulted in the most comprehensive study of the 31 Cl(𝛽𝛾)31 S
decay scheme to date [120]. In addition to the discovery of many new        31 S levels and previously
unobserved 𝛾 transitions, Bennett et al. crucially identified the existence of a proton-unbound state
with spin-parity 𝐽 𝜋 = 3/2+ and an excitation energy of 𝐸 𝑥 = 6390.2(7) keV [121].
                                                   41


     The most recent atomic mass evaluation reports that the proton emission threshold for         31 S
to be 𝑆 𝑝 = 6130.65(24) keV [16], which means that the resonance energy is 𝐸𝑟 = 6390.2(7) −
6130.65(24) keV = 259.6(7) keV, placing this resonance in close proximity to the Gamow peak
for ONe nova temperatures during thermonuclear runaway. In addition, since the ground state of
30 P is 𝐽 𝜋 = 1+ [122], proton capture to a 𝐽 𝜋 = 3/2+ level forms an ℓ = 0 resonance, which means
30 P(𝑝, 𝛾)31 S is be unimpeded by a centrifugal barrier through this channel. For these reasons,
we hypothesize that the ℓ = 0, 260-keV resonance will dominate the thermonuclear rate for this
reaction of interest in ONe novae. However, in order to calculate the strength of this resonance
and, by extension, its contribution to the total rate, we need to determine its proton branching ratio
Γ𝑝 /Γ.
     The reason why 31 Cl 𝛽-delayed proton decays through this resonance have not been observed in
the previous measurements is likely due to their use of off-the-shelf Si diodes. Solid-state detectors
made of semiconducting material are plagued by the large 𝛽+ backgrounds that are especially
prominent at low energies. This has motivated the development of a new detector for measuring
weak, low-energy, 𝛽-delayed protons decays. This experimental system and its application to the
scientific problem at hand is the topic of the next chapter.
                                                   42


                                           CHAPTER 3
                              EXPERIMENTAL INVESTIGATION
3.1   GADGET: Gaseous Detector with Germanium Tagging
The GADGET system was specifically developed to measure weak, low-energy, 𝛽-delayed, charged-
particle decays for the purpose of constraining thermonuclear reaction rates relevant to the study of
explosive nucleosynthesis. It was designed for conducting radioactive beam experiments using the
exotic ion beams provided by the Coupled Cyclotron Facility at NSCL. GADGET’s namesake refers
to the coupling of the existing Segmented Germanium Array (SeGA) at NSCL to the new Proton
Detector, a gaseous, proportional counter. The former consists of an array of high-resolution,
high-efficiency 𝛾 ray detectors. The latter was designed, built, tested, and commissioned at NSCL
as part of Phase I of the GADGET system.
Micro-Pattern Gaseous Detectors
Since the invention of the Geiger-Müller counter in 1928, for almost a century, gas-filled radiation
detectors have been employed in a wide variety of applications. The MICRO-MEsh GAs Structure
(MICROMEGAS) was developed for high-rate experiments and proved to be a cost-effective
solution to the spatial-resolution problem encountered by the multi-wire proportional chambers
(MWPC) that preceded them. MICROMEGAS detectors were also able to provide higher signal
amplification than the micro-strip gas chambers (MSGC) built to solve the same problem [123].
The interactions of charged particles with matter are quite complicated, but the semi-classical
Bethe-Bloch expression is often used to approximate the stopping power of charged particles in a
gas. Qualitatively, we can express the differential energy loss per unit distance as
                                             Δ𝐸       𝑍
                                                 ∝ − 2,                                         (3.1)
                                             Δ𝑥       𝛽
                                                 43


where 𝑍 is the charge of the particle, and 𝛽 is its velocity as a fraction of the speed of light [124].
This can be calculated using the relation
                                                √︄
                                                        1
                                           𝛽=      1−     ,                                       (3.2)
                                                       𝛾2
where the relativistic 𝛾 can be written in terms of a particle’s kinetic energy 𝐸 𝑘 and its rest mass
𝑚0:
                                                  𝐸𝑘
                                           𝛾=          + 1.                                       (3.3)
                                                𝑚 0 𝑐2
    For minimum ionizing radiation such as muons or 𝛽 particles, a kinetic energy of 1 MeV
implies a speed of 94% the speed of light, while for the much heavier proton of the same energy
𝛽 ≈ 0.046. This implies that faster, lighter 𝛽 particles hardly deposit any energy in the gas at all
over short distances and have much longer tracks, compared to the slower, heavier protons and
𝛼 particles. Because fast-moving 𝛽 particles do not deposit their full energy in the volume of a
gas-filled proportional counter, their contribution to the background is suppressed in 𝛽-delayed
proton decay experiments that utilize such devices. Furthermore, a high-gain, gaseous amplifier
based on the MICROMEGAS design results in high detection efficiency for low-energy protons,
while maintaining good resolution. This was demonstrated in the measurement of 23 Al 𝛽-delayed
proton decays at ≈ 200 keV with a full width at half maximum (FWHM) resolution of 7% using the
AstroBox instrument at Texas A&M University’s Cyclotron Insitute [125]. The Proton Detector
at NSCL was designed in collaboration with the developers of AstroBox in order to replicate its
technical performance as well as to adapt the gaseous detection system such that it fits snugly
inside the existing SeGA structure. The ability of GADGET to acquire high statistics particle-𝛾
coincidences is useful for determining which states in 30 P are populated by 31 Cl 𝛽-delayed proton
decay, which is important for understanding nova nucleosynthesis.
                                                  44


Principle of Operation
The schematic drawing in Figure 3.1 depicts the basic operating principles of the Proton Detector.
Once the radioactive beam particles have been produced from the accelerator facility and delivered
to the experimental setup, they exit the beam line and implant inside the Proton Detector. After
stopping in the fill gas, they quickly thermalize, diffusing via Brownian motion until they decay
from essentially at rest. Usually, 𝛽 decay is promptly followed by one or more 𝛾 decays, which can
be detected by SeGA, as the daughter nucleus deexcites to its ground state. However, if the 𝛽 decay
populates a proton-unbound state in the daughter nucleus, a proton may be emitted. The decay
proton and the recoil nucleus deposit almost all their kinetic energy into the fill gas by colliding
with neutral atoms to create electron-ion pairs. Under the influence of a uniform electric field, these
primary ionization electrons drift at a constant velocity towards the amplification region, where
they encounter an electric field on the order of 100 times stronger than the field in the drift region.
This causes a Townsend avalanche of secondary electrons which induce a detectable signal on the
detector pads [126; 127]. Thus, the size of the readout voltage is proportional to the center-of-mass
energy of the proton decay. The fast-moving 𝛽 particles have long tracks and deposit only some of
their energy within the detector’s active volume, suppressing their contribution to background in
the final 𝛽-delayed proton spectrum.
Proton Detector Design
Figure 3.2 contains a partially labeled diagram of the Proton Detector. Most of the detector volume
is contained within a cylindrical tube made of stainless steel. The chamber is ≈ 49 mm long and
16.5 cm in diameter. On the upstream end of the detector, relative to the direction of the incoming
beam, a 1.5-𝜇m-thick Kapton® window with a 50.8-mm diameter allows the beam particles to enter
the drift tube. Also attached to that end cap are four outlet lines, which allow gas to flow out of the
detector, as well as a high-voltage electrical feedthrough, which is used to apply a negative bias to
the cathode. The Proton Detector’s cathode provides the large voltage needed to create an electric
field along the length of the drift tube. The uniformity of the electric field is preserved by a field
                                                  45


                Figure 3.1: Diagram of the Proton Detector’s principle of operation.
cage.
     The field cage consists of a thin, flexible polyimide film wrapped into a cylinder, onto which
101 equipotential copper rings have been printed along the inside. The first, most-upstream ring
is in electrical contact with the cathode, and the electric potential is stepped down over the length
of the cage through a series of resistors. The field cage is surrounded by an insulating polyether
ether ketone (PEEK) material, with cutouts allowing space for the resistors. At the downstream
end of the field cage, there is an electronic gating grid that protects sensitive electrical components
from the large charges produced during beam implantation. Between the gating grid and the
MICROMEGAS detector pad plane, there is a pair of copper rings separated by insulating PEEK
spacers. The ring closer to the gating grid is in electrical contact with the last ring of the field
cage on the low-voltage end, while the downstream ring is grounded. This preserves the electric
                                                   46


                  Figure 3.2: Mechanical design drawing of the Proton Detector.
drift field between the gating grid and the MICROMEGAS, which is responsible for amplifying the
detector signal.
    On the downstream end of the detector, there is a flange with a 22.9-cm diameter to which the
other end cap connects. Similar to the upstream end, this flange has four, thin tubes connected to
it which allow for filling the detector with gas. In addition to the gas inlets, a single, thick tube is
connected to the downstream flange for more efficient vacuum pumping as well as for diagnostic
and safety purposes. An additional flange is attached to this end which contains the electrical
feedthroughs necessary for biasing the positive anode and the gating grid. The end cap that seals
the downstream end of the detector is the printed circuit board for routing electronic signals and
the MICROMEGAS.
    Designed and manufactured at CERN, the MICROMEGAS consists of a stainless steel, mi-
                                                  47


Figure 3.3: Diagram of the detector pad plane. The MICROMEGAS is segmented into 13
charge-sensitive pads labeled A-M. The dashed blue circle represents the projection of the beam
entrance window onto the pad plane, and the red arrows correspond to proton tracks. The size of
the arrows, labeled by their energies in units of keV, are scaled in proportion to their calculated
ranges in P10 gas at a pressure of 800 Torr.
croscopic mesh structure, which is electrically grounded, and is supported by insulating pillars of
PEEK material that separate it from the positively biased anode plane. This 128-𝜇m gap is the
amplification region that provides the detector with adjustable gain. A schematic drawing of the pad
plane is shown in Figure 3.3. The anode itself is segmented into 13 gold-plated, copper electrodes.
The five inner pads measure the energy of charged-particle decays, while the eight outer pads are
used to veto events whose full energy is not measured by the active, inner pads.
    The gating grid was installed in the Proton Detector between the field cage and the MI-
CROMEGAS structure, covering the entire area of the charge-sensitive pad plane. The grid is
made of an insulating ring, on which 60 gold-plated, copper wires have been soldered, each 20 𝜇m
in diameter and spaced in 2-mm increments. When the grid is operated in the transparent mode,
the wires are held at the same potential (-225 V) as the final copper ring in the field cage before
the MICROMEGAS, allowing ionization electrons to enter the amplification region. In the opaque
                                                  48


Figure 3.4: The MICROMEGAS being operated in “pancake” mode. By shortening the active
volume of the gas, we were able to use an 55 Fe X-ray source to characterize the performance of
individual detector pads.
mode, the wires are held at a positive potential (+150 V) relative to the micro-mesh, which reverses
the direction of the drift field between the gating grid and the MICROMEGAS. This prevents the
large ionization currents created during beam implantation from damaging the channels in the
preamplifier, which is necessary for signal processing. When running experiments, the gating grid
is operated in cycles synchronized with beam delivery, alternating between opaque mode during
beam implantation, and transparent mode, when beam is not being implanted.
Gas Handling System
A used gas handling system, provided by the NSCL detector lab, was initially utilized during the
testing stages of the Proton Detector development. This was necessary in order to characterize
the individual MICROMEGAS pads using a “pancake” version of the detector, depicted in Figure
3.4. We also used this gas handling system during the the GADGET commissioning experiment.
However, gas purity affects the quality of the detector response and, by extension, the resolution
of the final proton spectrum. Before the first dedicated science experiment, a new, customized gas
                                                  49


Figure 3.5: Gas handling systems used to operating the Proton Detector. Left: Older version of
the gas handling system borrowed from the NSCL detector lab. Right: Custom-built gas handling
system for GADGET.
handling system was constructed to be used in the    31 Cl 𝛽-delayed proton decay measurement as
well as all subsequent GADGET experiments. It was designed to fit on the same portable rack used
to house the electronics modules and remote controls system for GADGET. A comparison between
the old and new gas handling systems is shown in Figure 3.5.
     In collaboration with the mechanical engineering department at NSCL, this new gas handling
system was developed for functionality during radioactive beam experiments as well as during
offline systematic tests. Figure 3.6 provides a detailed schematic drawing of the gas handling
system. While GADGET does not operate by mixing gases in situ, two gas sources can be
connected to the gas handling system simultaneously. Typically, one gas line is connected to a
bottle containing P10, a mixture of 90% Ar and 10% CH4 , which is commonly used to fill gaseous
amplification detectors. Another gas line containing N2 is usually connected to the detector; this
feature was included in order to flush the detector and gas handling system with an inexpensive,
inert gas. This was both for safety purposes when operating the systems with marginally flammable
gases as well as for keeping the detector clean and pressurizing the system when it is not in use.
                                                 50


51
     Figure 3.6: Schematic drawing of the custom-built gas handling system for GADGET.


    The gas pressure inside the detector chamber is regulated by a 𝜋PC PC99 integrated downstream
pressure controller with a mass flow meter from MKS Instruments. The current operating procedure
for controlling this device involves communicating with the mass flow controller over Ethernet
connection through an Internet Explorer session on a Windows laptop. The MKS Instruments
software interface was used to set and record the gas pressure as well as for monitoring the rate of
gas flowing through the meter. The flow rate can only be changed by manually adjusting a metering
value, which is physically attached to the gas handling system. At the end opposite to the gas
sources, an electric vacuum pump causes gas to flow through through the system and disposes of
the spent gas into the experimental vault’s exhaust line, which is then vented to an outdoor area.
    A rough pumping line is connected to the downstream flange of the Proton Detector. This allows
for more efficient pumping of the detector when placing the system under vacuum. Connected to
this rough pumping line is a pressure gauge which gives an independent reading of the pressure
closer to the detector chamber. There is also a safety relief value that will vent over-pressured gas
into the experimental vault. The valve is set manually at a nominal pressure above the operating
pressure of the Proton Detector. For the commissioning and first dedicated science experiment
with GADGET, we chose to operate the system above atmospheric pressure at 800 Torr. Thus, any
small leaks in the experimental setup would cause gas to flow out of the system and prevent air in
the atmosphere from contaminating the vessel.
    Additionally, a radioactive source-holder is connected to an isolated loop in the plumbing of
the gas handling system. By inserting a radioactive sample, such as 232 U or 228 Th, which contain
220 Rn gas in their decay chains, the 𝛼-emitting isotope can diffuse out of the source holder and
into the Proton Detector. Detecting these charged-particle decays has been useful in testing the
experimental setup without the use of a radioactive beam. A more detailed discussion of the design,
testing, and commissioning of the Proton Detector and the GADGET system can be found in Ref.
[128].
                                                 52


SeGA: Segmented Germanium Array
SeGA is a high-resolution, 𝛾-ray detection system with good geometric efficiency. The array
consists of 18 segmented, coaxial high-purity germanium (HPGe) crystals. Ge is a semiconductor,
meaning its atomic structure causes the bulk material to have an electrical conductivity somewhere
between that of a true, metallic conductor and a non-metallic insulator. It has a small band gap,
meaning less energy is required to excite valence electrons to the conduction band, and a large
atomic number, which increases the probability of a radiation interacting with the crystal. An
energetic photon like a 𝛾 ray can ionize atoms in the crystal lattice, creating an electron-hole pair.
In the presence of an external electric field, this causes a pulse of current to flow through the Ge.
    The SeGA detectors are coaxial in the sense that the semiconducting Ge material is cylindrical
in shape with a smaller cylindrical cavity inside along the same axis. The central hole bores
through almost the entire length of the crystal, making these close-ended coaxial HPGe detectors.
An electrical contact in the central hole, as well as all along the outer surface of the of the
HPGe cylinder, provides the requisite bias voltage to allow charge carriers to flow through the
semiconducting material. The end of the central contact is also rounded, or bulletized, to eliminate
low-field regions in the crystal [126]. The amount of current measured in the central contact is
proportional to the amount of energy deposited in the detector by the 𝛾 ray. Unlike conductors,
whose electrical conductivity increases as the metal is cooled, conductivity in a semiconductor
increases with warmer temperatures. Thermal excitations can cause electrons to cross the band gap
and produce leakage currents. This results in noisy electronic signals and degrades the resolving
power of the detector. For this reason, the HPGe crystals must be kept cool at all times using liquid
N2 , which has a temperature ≤ 77 K at atmospheric pressure.
    The HPGe crystals are segmented into eight discs, which are each further subdivided into four,
planar quadrants [129]. This segmentation allows for determining the point of initial interaction
between a 𝛾 ray and the detector as well as increased angular resolution. For GADGET experiments,
we do not need to not collect data from the individual segments of the HPGe crystals, and instead
only measure the current on the central contacts. We have implemented 16 of the SeGA detectors
                                                   53


Figure 3.7: The betaSeGA, or “barrel,” configuration of the Segmented Germanium Array set up
in the experimental vault. This employs 16 HPGe detectors in two rings around the beam line with
a diameter of 17.3 cm. The large, blue-green containers are cryogenic storage dewars that hold the
liquid N2 for cooling the HPGe crystals located in the center of the array. Photo credit: National
Superconducting Cyclotron Facility / Facility for Rare Isotope Beams, Dirk Weisshaar, Alexandra
Gade.
                                                54


                 Figure 3.8: The fully assembled GADGET system in the S2 vault.
in their “barrel” configuration, as shown in Figure 3.7, inside which the Proton Detector was
designed to fit. The angular distribution of 𝛾 emission is irrelevant for the purposes of our scientific
measurement, and we prefer to maximize geometrical detection efficiency for this measurement.
3.2    NSCL Experiment 17024: 𝛽-Delayed Proton Decay of 31 Cl
A radioactive beam experiment (e17024) at NSCL was performed in November 2018 to measure
the 𝛽-delayed proton decay of 31 Cl. The primary goal of this measurement was to determine the
proton branching ratio of the 𝐽 𝜋 = 3/2+ , 6390-keV excited state in 31 S. This is the same resonance
state that had been identified in a previous 31 Cl 𝛽-delayed 𝛾-ray experiment at NSCL. In preparation
for e17024, the GADGET system was assembled in the S2 experimental vault, as pictured in Figure
3.8. Meanwhile, data collection and monitoring of the system was performed remotely in the
NSCL Data-U. Here, we discuss the supporting infrastructure and operating procedure during the
first dedicated scientific experiment using GADGET.
                                                    55


Figure 3.9: Diagram of the Coupled Cyclotron Facility and A1900 fragment separator at NSCL.
Figure credit: Ref. [130].
Beam Delivery
A radioactive ion beam of 31 Cl was produced via fragmentation at the Coupled Cylcotron Facility
by accelerating a stable beam of   36 Ar up to 150 MeV/u and impinging it on a 1645-mg/cm2 Be
production target. The A1900 fragment separator, depicted in Figure 3.9, removed most of the
undesirable nuclear species from the cocktail beam using magnetic rigidity and a 150-mg/cm2 Al
wedge. Magnetic rigidity causes the higher-energy beam particles to pass through the thicker end
of the wedge, slowing them down closer to the speed of the lower-energy beam particles which
interact with less Al material on the thinner end.
     The secondary beam was further purified using the Radio Frequency Fragment Separator
(RFFS), which applied an oscillating electric field to deflect the 31 Cl ions through a set of collimating
slits, meanwhile deflecting all other nuclear fragments such that they do not pass through the
collimator. Ultimately, we were delivered a 65% pure beam of           31 Cl at 6375 pps. In decreasing
order of intensity, the main beam contaminants were       28 Si, 30 S, and  29 P, but fortunately, none of
these nuclide are 𝛽-delayed particle emitters.
                                                  56


The Diagnostics Cross
In preparation for radioactive beam experiments, a six-way, ConFlat® cross, supported by an Al
stand, is installed at the end of the beam line in the S2 vault. The cross structure is used for housing
Si detectors as well as a rotatable Al degrader foil and is physically connected to the beam line
using an ultrahigh vacuum seal and a series of hand-tightened clamps. From the reference point
of the beam, the flange on the left side of the cross is connected to an electric turbo pump, which
is backed by an oil roughing pump. Prior to operating diagnostic tools in the cross, the oil pump
removes most of the air in the cross, and only after the pressure in the cavity reaches ≲ 1 Torr, the
turbo pump is switched on to reach the highest vacuum possible. Attached to the flange on one of
the faces parallel to the beam axis is a manual valve for venting the cross when bringing the cavity
back up to atmospheric pressure as well as a pressure sensor. Connected to the downstream flange
of the cross is a short beam line extension with a thin beam exit window. This limits the distance
the beam particles travel through the air to ≈ 15 cm after leaving the beam line but before entering
the Proton Detector.
PIN Detector
The top flange of the cross is connected to a pneumatic drive which, when connected to a source
of pressurized air in the vault, can control the position of a 300-𝜇m-thick PIN detector, shown
in Figure 3.10. A p-i-n (PIN) detector is a commonly used semiconductor diode, named for its
three distinct doping regions. The i-type region in the middle of the detector refers to a chemically
pure, undoped intrinsic semiconductor material, which in this case is Si. This intrinsic region is
sandwiched between p-type doping and n-type doping regions. The p-type region is doped with
elemental impurities that have a higher electron affinity (e.g. P, As, and Sb), and are thus more
likely to accept electrons. Conversely, dopants with a lower electron affinity (e.g. B, Al, Ga) are
used in the n-type region, which serve as electron donors.
    At the beginning of and periodically throughout the experiment, diagnostic runs were taken for
the purposes of particle identification. During these diagnostic measurements, the beam current
                                                    57


Figure 3.10: Si PIN detector mounted on PEEK spacers before being inserted into the six-way
cross chamber.
from the accelerator was attenuated and the PIN detector was inserted into the path of the ion beam
in order to measure its energy loss Δ𝐸 in the detector. To protect the Proton Detector electronics
during PID runs, the bias voltage on the MICROMEGAS anode was turned off, such that beam need
not be cycled on and off, allowing a constant beam intensity be impinged on the PIN detector for
several minutes. As beam particles pass through the PIN detector, their Δ𝐸 is measured, as shown
in Figure 3.11. This was compared to the expected energy loss of beam particles passing through
300 𝜇m of Si using version 1.2 of the atima program in lise++. This allowed us to verify that 31 Cl
and 28 Si were the two primary nuclear fragments being delivered to GADGET. After a diagnostic
run was completed, the beam is stopped; the pneumatic drive retracts the PIN detector from the
path of the ion beam; beam pulsing resumes, and only then can the MICROMEGAS voltages be
reapplied before beginning a new measurement run.
Beam Energy Degrader
Attached to the bottom flange of the cross is a second pneumatic drive as well as a stepper motor
for controlling the position and angular orientation of the beam energy degrader, respectively.
During a data production run, when beam is being implanted in the gaseous detector chamber and
                                                58


Figure 3.11: Energy loss Δ𝐸 spectrum measured in the PIN detector during particle
identification runs. The ratio between the peak positions is consistent with the ratio between
calculated energy losses in the equivalent thickness of Si.
𝛽-delayed proton decays are being measured, a thin, highly pure Al plate is inserted into the beam
path to degrade the energy of incoming ions before they reach the Proton Detector. Detailed beam
calculations were performed with lise++ to determine 0.75 mm as the ideal degrader thickness for
slowing the ≈ 50-MeV/u 31 Cl ions leaving the RFFS such that they stop in the middle of the Proton
Detector’s drift region.
    By rotating the degrader foil from between 0◦ , when the beam is perpendicular to the broad
side of the Al plate, and 90◦ , when the beam is parallel with the plane of the thin foil, the effective
thickness of the degrader can be adjusted. At 0◦ , beam particles pass through the minimum amount
of Al possible without removing the degrader entirely, but with increasing degrader angle, ions
must traverse more material in the degrader plate, further slowing down the beam particles and
shortening their implantation range in the Proton Detector.
    In principle, the stepper motor allows for remote control of the degrader plate’s angular ori-
entation inside the cross. However, in preparation for the    31 Cl 𝛽+ decay measurement, utilizing
the stepper motor induced substantial electronic noise in the Proton Detector. While this technical
challenge was later resolved, for the duration of e17024, the Al degrader plate was removed from
                                                  59


Figure 3.12: Overhead view of the Al degrader foil mounted on a circular protractor between the
entrance window to the Proton Detector (on the left side of the image) and the exit window from
the beam line (on the right side of the image). The degrader angle was set manually and could be
determined to within 1◦ precision.
the vacuum cross and positioned in air, directly in front of the Proton Detector’s beam entrance
window. Thus, in between beam implantation runs, the angle of the degrader plate was adjusted
manually in the experimental vault until an operating angle was determined. Figure 3.12 depicts
how the degrader angle was measured over the course of the experiment.
    Online analysis of scalars from the upstream and downstream SeGA rings allowed us to estimate
the relative, longitudinal position of the origin for 𝛽-delayed 𝛾 decays occurring in the Proton
Detector. These scalars are proportional to the the number of 𝛾 ray counts detected in either the
eight upstream or the eight downstream SeGA detectors; we assumed that when the scalar readings
from both SeGA rings were approximately equal, the      31 Cl beam distribution should be roughly
centered within GADGET longitudinally. Offline analysis using 𝛽-𝛾 coincidences allowed a more
precise determination of the longitudinal beam distribution.
    The constant drift velocity for electrons within the field cage implies the total drift time in
the Proton Detector is proportional to the distance between the MICROMEGAS pad plane and
                                                 60


Figure 3.13: Drift times recorded for Proton Detector events observed in coincidence with 𝛾 ray
events detected by SeGA. This plot allows us to visualize the longitudinal beam distribution
within the drift chamber.
the position where 𝛽+ decay occurred along the beam axis. The maximum possible drift time
for primary ionization electrons in a 40-cm drift region is about ≈ 8 𝜇s, assuming an 800-Torr
P10 fill gas and a 150-V/cm drift field. This defines the length of our time window for proton-
𝛾 coincidences, since 𝛽-delayed 𝛾 events will be detected in SeGA almost immediately, while
charged-particle events resulting from the same decay will take several microseconds to be detected
by the MICROMEGAS. The measured difference in time between when the 𝛾 and charged-particle
events are detected yields drift times of the ioniziation electrons, plotted in Figure 3.13.
Electronics
A detailed electronics diagram for e17024 is presented in Figure 3.14. In summary, two modules
were used to provide all biases for the Proton Detector system as well as the PIN detector. Setting
voltages can be done remotely over the lab network via a Linux terminal, and the currents in all
preamplifier channels can be monitored through the same terminal. The bias voltages applied to
the HPGe crystals are set and controlled by designated SeGA electronics, which are maintained by
                                                  61


the Gamma Group at NSCL.
                         62


                                                                                                                                              Synchronizer
                                                                                                                                   LeCroy 222        LeCroy 222
                                               Proton Detector
                                                                                                                                      Start             Start
                                          HV      detector HV                                          to DAQ                         NIM
                                                                                                    to gating grid                    TTL
     CAEN HV = -6000 V
                                                                                                                                     delay              delay
                                      gnd       detector ground
                                                                                                                                      Start             Start
                                                                                                                                      NIM               NIM
                                                                                                                                                                   to cyclotron
                                                  GG positive
                                                                                  Mesytec power
                                                                                     supply                                          delay
                                                  GG negative
      Mesytec HV                                                                                                       DDAS
                                                                      Data x13
                                                                                                         x13
                                                  Micromegas                     MPR 16L (preamp)                    DDAS #0-12
       HV0 = +380 V
                                                                       gnd
63                                                                                     bias
                                               pos 1                                                                 DDAS #13-14
                                                                                                         x2
       HV1 = +40 V                             neg 1 (jumper
                                               TTL 1 horizontal)                                                                                             computer
                                                                                                    x1
                                               com 1                                                                  DDAS #15
       HV2 = -225 V                               switch
                                               com 2                                                                 DDAS #16-30
                                               pos 2                                                x15
       HV3 = +150 V                            neg 2 (jumper
                                               TTL 2 perpendicular)                                                   DDAS #31
                         Synchronizer
                         (see to right)
                                               to cyclotron
                                                     SeGA
                                                     PIN
             Figure 3.14: Diagram of GADGET electronics configuration for NSCL experiment 17024.


Drift Field
An N1470B programmable high voltage power supply from CAEN is used to bias the Proton
Detector cathode through a safe high voltage (SHV) cable connection. In order to generate the
150-V/cm electric field in the drift region, a large, negative voltage (-6 kV) is required. Once the
Proton Detector is filled to its operating pressure with P10 gas, the bias is slowly ramped to the
set-point voltage, where it remains for the duration of the measurement. It is important to avoid
applying high voltages to electrical components when the surrounding gas has not yet reached
operating pressure because this can induce sparking in the detector. The relationship between the
breakdown voltage and gas pressure is described by Paschen’s law [131]. Qualitatively, it implies
that the voltage required to cause electric discharge in near-vacuum or near-atmospheric pressures
is very high, but around pressures of 1 − 10 Torr, the breakdown voltage is only on the order of 100
V, depending on the chemistry of the gas.
Detector Gain
The other high voltage power supply is the MHV-4 module from Mesytec. The first of its four
channels supplies the positive voltage necessary for inducing a strong electric field in the ampli-
fication region. This 380-V bias is applied through the preamplifier, which is directly connected
to the 13 charge-sensitive anode pad plane. The MPR-16-L is a 16-channel, charge-integrating
preamplifier, which acts as a charge-to-voltage converter before the electronic detector signals can
be digitized and saved to memory on a lab network computer. The second channel in the MHV-4
unit applies a 40-V bias to the PIN detector via another another preamplifier borrowed from the
NSCL electronics pool. Both preamps are powered by an MNV-4. This multi-channel NIM power
distribution and control module, as well as the two preamps, were all manufactured by Mesytec.
The two remaining channels in the MHV-4 module are used to bias the gating grid.
                                                  64


Gating Grid
The electronic gating grid was designed to protect the preamplifier during beam implantation.
When beam is implanted in the gaseous Proton Detector, large quantities of ionization electrons
are liberated, which can induce strong currents on the MICROMEGAS anode plane. This rapid
charge deposition can saturate the preamp and damage individual channels, preventing signals
induced on the corresponding detector pad from being recorded in data. To avoid charge saturation,
beam experiments using GADGET run in two-step cycles of beam implantation followed by decay
measurement, with which the gating grid is synchronized. When beam is implanting in the Proton
Detector, a positive voltage is applied to the gating grid, causing it to appear “opaque” to free
electrons. Systematic tests of the gating grid were performed before e17024, and a 150-V bias was
found to be most effective.
    After beam implantation stops, the voltage on the gating grid switches to a negative bias at
equipotential with the last copper ring in the field cage, which was calculated to be -225 V. This
causes the gating grid to appear “transparent” to primary electrons, allowing ionization created
from 𝛽-delayed, charged-particle decays to enter the amplification region and produce detectable
signals. In order to utilize beam time as efficiently as possible, the length of these implant-decay
cycles is chosen with respect to the half-life of the nuclear species being studied. Considering the
190-ms half-life of 31 Cl, the length of the implantation time per cycle was set to be ≈300 ms, while
the optimal measurement time per cycle was about 200 ms.
    It is important that the electronics connected to the MICROMEGAS are protected by the gating
grid over the entire beam implantation period. For this reason, the gating grid is switched to opaque
mode slightly before the beam restarts implantation, and it only switches back to transparent mode
once the beam implantation period has ended, as depicted in Figure 3.15. These two timing signals
need to be sent to the beam operators and the gating grid to ensure synchronization. The logic
signals used to communicate this information were constructed using the two LeCroy Model 222
Dual Gate and Delay Generators in Figure 3.16.
    The gating grid logic signal must be sent to a CGC Instruments NIM-AMX500-3 switch, shown
                                                   65


               8
                      GG TTL
               7
               6      beam NIM
               5
                            GG opaque
voltage (V)
               4
               3             300 ms                200 ms
               2
               1
                                               GG transparent
               0
                            beam on
              −1
                                                 beam off
              −2
              −0.5   −0.4     −0.3      −0.2   −0.1         0    0.1   0.2      0.3     0.4      0.5
                                                      time (s)
Figure 3.15: Logic signals represented as voltage over time for the gating grid (blue) and the ion
beam (red). The amplitude of the voltages are related to differences in logic signal families. The
dual gate/delay generator sends a transistor-transistor logic (TTL) signal to the switch,
communicating which bias voltage should be applied to the gating grid. When a TTL signal has a
voltage amplitude of 1.5 − 5 V, this corresponds to a digital signal of “1,” while a signal voltage of
0 − 0.7 V logically implies “0.” Similar timing information is communicated via a Nuclear
Instrumentation Module (NIM) signal, which is sent to the beam operators and the digital data
acquisition system. When a NIM signal voltage is exactly 0 V, this implies a digital “0,” but a
NIM signal voltage between -0.8 and -1 V corresponds to a binary value of “1.”
in Figure 3.17, which is responsible for alternating the source bias between the negative and positive
voltages. SHV cables connect the third and fourth channels in the MHV-4 module to the switch’s
“Neg1” and “Pos1” feedthroughs, respectively. A T-shaped, SHV connector is attached to the
switch’s “Com1” feedthrough, to which two SHV cables directly connect the switch and the gating
grid via the electrical feedthroughs in one of the Proton Detector’s downstream flanges. The same
timing signal is also sent directly to the digital data acquisition system to record the gating grid
status over the course of the entire experiment.
                                                      66


Figure 3.16: Dual gate/delay generators in the Data-U electronics rack. These modules produced
the TTL and NIM timing signals sent to the gating grid and beam operators, respectively.
Figure 3.17: All electronics modules utilized on the rack in the S2 vault during NSCL
experiment e17024.
                                               67


Digital Data Acquisition System
All experimental data measured with GADGET was obtained using NSCL’s digital data acquisition
system (DDAS). Traditionally, data acquisition systems for experimental nuclear physics involved
passing detector signals through a series of analog NIM modules, which are designed to shape
electronic pulses, as well as correct for any timing discrepancies between signals that should arrive
in coincidence but are misaligned in time due to small differences in delay between channels. Once
signals are well-shaped and synchronized, they can then be digitized and read into a computer or
written directly to magnetic tape. In contrast, DDAS eliminates the need for many of the analog
signal-processing modules in its pulse-shape analysis of digitized signals.
    Primarily, DDAS was developed to work in conjunction with SeGA for the purposes of in-beam
studies that combine 𝛾-ray detection with identification of reaction residues [132]. This streamlined,
digitized system was implemented in SeGA for determining the first interaction position of a 𝛾 ray in
one of the segmented crystals and subsequent 𝛾-ray tracking in the detector. This has significantly
improved the ability to measure the angular distribution of 𝛾 emission and has allowed for more
precise Doppler corrections in fast beam experiments. For stopped-beam measurements involving
beam implantation followed by charged-particle decay, like NSCL experiment 17024, DDAS allows
for zero dead time measurements of rare decays [133].
    For NSCL experiment 17024, we utilized two 250-MHz, 16-channel PXI Digital Pulse Pro-
cessors in a PXIE-CRATE-P16X14 from XIA, which was connected to one of the lab network’s
data acquisition computers. DDAS employs a pair of trapezoidal filters for leading-edge triggering
and pulse-amplitude measurements as well as a constant-fraction discriminator (CFD) algorithm to
determine the timestamp and energy of detector signals [134]. As depicted in Figure 3.14, the first
13 DDAS channels were connected to all the MICROMEGAS detector pads through the MPR-16-L
preamp. Another channel was dedicated to measuring the NIM signal from the dual gate/delay
generator modules for recording the status of the gating grid mode throughout the experiment. Each
of the 16 individual SeGA detectors were connected to their own DDAS channel in the XIA crate,
and a single channel was used to record events from the PIN detector in the cross. A more complete
                                                  68


 Figure 3.18: Example of the visualization windows used during online analysis using SpecTcl.
description of hardware implementation for DDAS in experimental applications at NSCL is given
by Ref. [134].
Operating Procedure and Online Analysis
Detailed documentation on the architecture of the NSCL data acquisition software is provided in
the user guide [135], but in short, each detector signal triggered in DDAS is assigned a timestamp
and written to the ring buffer by the Readout program. The ring buffer is divided into events with a
characteristic time window, which we set to 4 𝜇s for this experiment. Data in the ring buffer can be
saved by writing to a binary event file (“.evt”), accessed for online analysis, or both simultaneously.
The ReadoutShell program was used for collecting and saving GADGET events. Experimental data
were generally procured in hour-long measurement runs. The purpose of relatively short runs was
to reduce the probability of the Readout program from crashing in the middle of a measurement,
which would result in the loss of all data from a single run.
    Before beginning of a standard measurement run, the gas flow and pressure inside the Proton
Detector was ensured to be stable. We verified that the beam was off, that the timing between the
gating grid and the beam cycling was appropriately synchronized, and that the PIN detector was in
the retracted position. Then, detector biases were slowly ramped until they reached the set-point
voltage. Once beam was allowed to start entering the Proton Detector again, data collection could
                                                   69


resume, and all the diagnostic information about the present status of the experimental setup would
be recorded. At the end of the hour, the ReadoutShell program was halted to prevent further data
collection. During a diagnostic run for particle identification, which were performed aperiodically
multiple times per day, we would typically attenuate the beam, remove the amplification bias on the
anode, and turn off the beam pulse cycling. The Si detector would then be inserted into the path of
the beam via the pneumatic drive, which is controlled remotely, and a new Readout session would
be used for the PIN detector. All GADGET data was visualized during the experiment using the
SpecTcl program, as depicted in Figure 3.18.
    After a measurement run was terminated and all events were written to file, the “ddasdumper”
program was used to convert “.evt” files to “.root” files. A sorting code was used to extract the
relevant information from each DDAS channel and fill histograms corresponding to each detector
channel in the GADGET system. A second sorting code was used to apply the veto conditions
on the individual Proton Detector pads and build all the histograms storing particle-𝛾 coincidence
information. The time window for building coincidence histograms was extended to 8 𝜇s because
this is the maximum possible drift for ioniziation electrons created by decays near the cathode.
These processed “.root” files could then be analyzed using the ROOT analysis framework to provide
a more detailed understanding of dataset during the experiment.
                                                 70


                                             CHAPTER 4
                         DATA ANALYSIS AND SCIENTIFIC RESULTS
4.1    The Proton Branching Ratio
The primary scientific goal for the analysis of the 31 Cl 𝛽-delayed proton decay data obtained during
NSCL experiment 17024 was to determine the proton branching ratio Γ𝑝 /Γ of the 6390-keV level
in 31 S. This information is needed to evaluate the equation for the resonance strength:
                                               (2𝐽𝑟 + 1)       Γ𝑝 Γ𝛾
                                    𝜔𝛾 =                             .                          (4.1)
                                          (2𝐽 𝑝 + 1)(2𝐽𝑡 + 1) Γ
From a previous    31 Cl 𝛽+ decay measurement, Bennett et al. determined the 𝛽-delayed 𝛾 decay
intensity through this state to be 𝐼𝛾 (abs.) = 3.38(16)% [121]. In the case of this resonance state,
𝛾 and proton emission are the only two open decay channels. Therefore, the sum of the branching
ratios is simply (Γ𝛾 + Γ𝑝 )/Γ = 1. This allows us to express the proton branching ratio in terms of
the 𝛽-delayed radiation intensities through the 6390-keV level as
                                             Γ𝑝        𝐼𝑝
                                                 =           .                                  (4.2)
                                             Γ      𝐼𝛾 + 𝐼 𝑝
Gain Matching and Sorting
The spectra referenced in this dissertation contain data from over 86 hours of on-target beam time.
Each hour-long run was automatically divided into a number of sub-runs, determined by how much
data was acquired in a particular acquisition session. After completing a measurement run, the
raw event files of the recently acquired data were converted to a file format that allowed for more
detailed offline analysis using the root framework [136]. An automated analysis script was used to
provide a preliminary gain-matching of the five inner MICROMEGAS pads. This involved fitting
the largest proton peak in the spectrum to extract the mean analog-to-digital converter (ADC)
channel of the intense 1-MeV 𝛽-delayed proton transition, modeling the proton distribution as a
                                                   71


Gaussian function. Because the statistics for any one particular acquisition run were quite low,
especially at the beginning of the experiment, only the 1-MeV proton peak could be consistently fit
automatically. For the sake of expediting our analysis while still collecting data, our zeroth-order
calibration procedure assumed a linear mapping between ADC channel and center-of-mass decay
energy with no offset, estimating the gain of a single detector pad as the ratio between the 1-MeV
decay energy and the mean ADC channel. After applying this rough gain-matching procedure for
each of the active detector pads, we were able to combine statistics from all inner pads to observe
the presence of the weak, low-energy 𝛽-delayed proton decay near 260 keV before the end of NSCL
experiment 17024.
    The sorting routine used to accomplish this task applied veto conditions to remove Proton
Detector events with long tracks which deposited energy across multiple detector pads. By applying
these anti-coincidence cuts, histograms were generated for each of the five, inner detector pads
labeled A-E in Figure 3.3. The procedure is as follows: If all ionization electrons generated from
a single proton event are detected in only one MICROMEGAS channel, the energy of this event
is added to the one-dimensional histogram spectrum for that detector pad. The 𝛽-delayed proton
spectra for all individual pads A-E are plotted in Figure 4.1. However, if a proton decay deposits
all of its energy within the active region of the detector, but this charge is shared among multiple
inner pads, this event will not be recorded in any single detector channel’s histogram. Instead, this
decay energy is added to the event-level combined spectrum as shown in Figure 4.2. Events that
deposit all their energy in only one of the five inner pads are also included in this cumulative proton
spectrum as well. A more detailed gain-matching procedure was performed for each individual pad
on a run-by-run basis after the experiment during offline analysis. This utilized the two strongest
proton peaks in the spectrum to determine a linear mapping, instead of just using the single 1-MeV
peak and assuming zero offset.
                                                   72


                                                                                                    Pad A
                  5
                10
                                                                                                    Pad B
                104                                                                                 Pad C
 Counts / keV
                                                                                                    Pad D
                103
                                                                                                    Pad E
                  2
                10
                10
                  1
                      0                 500            1000             1500             2000          2500
                                                            Proton energy [keV]
Figure 4.1:               31 Cl   𝛽-delayed proton spectra as measured by each of the five inner pads individually.
                106
                105
 Counts / keV
                104
                103
                102
                10
                  1
                   0                   500           1000            1500         2000          2500          3000
                                                            Proton energy [keV]
Figure 4.2: Combined energy spectrum for 31 Cl 𝛽-delayed proton decays with event-level
summing over all pads.
                                                                73


Preliminary Energy Calibration with Central Pad
All previous reports of 31 Cl 𝛽-delayed proton decays assumed that particles emitted from proton-
unbound states in 31 S populated the ground state of 30 P. If this is also the case for the only visible
proton peak below 300 keV in Figs. 4.1 and 4.2, it should be obvious that these protons were
emitted from the astrophysically important 6390-keV level; no other states within 100 keV have
been observed to be populated by     31 Cl 𝛽+ decay. However, in order to say for sure that we have
observed proton decays from the 260-keV resonance of interest, we must demonstrate that decay
energy we measure is consistent with Ref. [121].
    It did not take long in the process of analyzing the 31 Cl 𝛽-delayed proton data to realize that the
spectrum above 700 keV is much more complicated than had previously been assumed. For this
reason, we did not feel comfortable using all high-energy proton peaks and their reported literature
energies in our initial energy calibration procedure. Instead, we only used the decay energies
of the three strongest 𝛽-delayed proton decays as our calibration standards. We confirmed that
these are ground-state transitions since they are not observed in coincidence with any 30 P 𝛾 rays.
Their decay energies as measured in previous      31 Cl 𝛽-delayed proton experiments [117; 118] are
in agreement with the well-known 31 S excitation energies measured in many reaction experiments
[137; 138; 101; 105; 102; 103; 99]. They also have the added benefit of being the decay energies
closest to our 260-keV resonance of interest. For our first attempt at an energy calibration, we
adopted the evaluated center-of-mass energies 𝐸 𝑝 (c.m.) = 806, 906, and 1026 keV, each with an
associated 2-keV uncertainty [119].
    We chose to use the pad A proton spectrum, comprised of events only measured by the central
MICROMEGAS pad, for our first calibration attempt when reporting the energy of the low-energy
peak. Its location in the detector means that pad A has relatively high statistics compared to the
other four inner pads, but it is also has a smaller area than all other active pads. This limits
the amount of energy 𝛽+ particles can deposit in this pad before being vetoed, pushing the 𝛽+
background to lower energies and reducing the effect of summing with energy from proton decays.
Thus, the resulting proton peaks are the sharpest in the central pad spectrum and allow for a more
                                                   74


Figure 4.3: A 𝜒2 -minimized fit of the calibration peaks as measured by the central pad of the
Proton Detector. The three proton peak distributions were modeled as Gaussian and the
background as linear.
precise energy calibration. The calibration peaks and the low-energy resonance peak of interest
were all fit with Gaussian distributions and a first-order polynomial function to describe the mostly
flat background in the energy range 300 - 700 keV, as shown in Figs. 4.3 and 4.4, respectively.
     The fraction of the proton’s kinetic energy that goes into ionizing the fill gas varies as a function
of decay energy. The discrepancy between the measured amount of ionization in the detector and
the total energy of the resonance center-of-mass decay energy is known as the pulse height defect.
This was calculated for the protons and recoils associated with each of the calibration peaks as well
as for resonances up to 2 MeV using trim [139]. Since the 806- and 1026-keV proton peaks were
used to gain match all proton spectra, calibrating relative to the three peaks in this energy region
should already account for the effect of any small pulse height defect; the difference between their
center-of-mass decay energies and the amount of ionization they created in an 800-Torr P10 gas
is 6-8 keV. However, extrapolating this defect down to lower energies may cause the low-energy
proton peak to appear shifted in the spectrum.
     To account for this, we plotted the means of three calibration peak’s fitted Gaussian distributions
                                                   75


Figure 4.4: Fit of the weak, low-energy proton peak and background region in the pad A
spectrum.
as the independent variable and the calculated total ionization in the gas as the dependent variable.
Figure 4.5 shows this data fit with a line and the associated uncertainty band. We then used this
calibration function to evaluate the amount of ionization the low-energy proton decay should have
produced [268.2(8.5) keV] and then calculated its resonance energy to be 𝐸 𝑝 (c.m.) = 273(10)
keV based on what the decay’s pulse height defect should be in order to produce the observed
amount of ionization in the detector. The total uncertainty in the resonance energy comes from
summing in quadrature the 8.5-keV uncertainty associated with extrapolating the energy calibration
down to low energies, the uncertainty associated with the 5-keV pulse height defect, and the 1-keV
statistical error from fitting the small peak. While not expected to be as accurate or precise as other
methods for determining the resonance energy, we are consistent with Ref. [121] to within 1.4
standard deviations. For this reason, we refer to this resonance as the 260-keV decay throughout this
document. A more robust energy calibration of the proton spectrum over all energies is provided
in Chapter 5 of this dissertation.
                                                   76


Figure 4.5: Central detector pad energy calibration. Top panel: Linear fit mapping positions of
three calibration peaks in Pad A spectrum to known decay energies. Bottom panel: Plotting
residuals between data and linear fit model.
Proton-𝛾 Coincidences
In addition to the  31 Cl 𝛽-delayed proton decay data, we simultaneously collected substantial 𝛽-
delayed 𝛾 ray statistics with SeGA over the course of e17024. This information was extremely
useful for diagnostic purposes and ultimately for constructing the most detailed 31 Cl(𝛽𝑝𝛾)30 P decay
scheme to date, as will be discussed in Chapter 5. However, the primary astrophysical importance
of the 𝛾 rays from this experiment was the use of coincidence analysis to conclusively determine
the initial and final states of the 260-keV proton transition. In order to visualize this 𝛾 data, we
produced a variety of different histograms.
    Ungated 𝛾 ray events were detected by all 16 of SeGA’s HPGe crystals. We produced a gain-
matched energy spectrum histogram for each detector channel using the energies of well-known
sources of background radiation as calibration points. This included 𝛾 rays emitted in the decays
of naturally occuring 214 Bi,  228 Ac, 40 K, and 208 Tl in the environment [140]. We also utilized the
31 Cl 𝛽-delayed 𝛾 at 6279 keV in order to extrapolate our linear gain-matching function to higher
                                                    77


                106
                105
                104
 Counts / keV
                103
                102
                10
                  1
                      0   1000      2000      3000          4000      5000    6000      7000      8000
                                                     γ energy [keV]
Figure 4.6: 31 Cl 𝛽-delayed 𝛾 ray singles spectrum including all events from 15 of the 16 SeGA
detectors observed to be in coincidence with non-vetoed, charged-particle radiation in the Proton
Detector.
energies. Not unlike the proton spectra, each SeGA detector’s energy spectrum was gain-matched
on a run-by-run basis by fitting these calibration peaks with Gaussian functions, recording the mean
ADC channel of the 𝜒2 -minimized distributions, plotting known the 𝛾 ray energies as a function
of these means, and fitting the data with a linear calibration function. Combining histograms from
15 of the 16 HPGe detectors produced a “singles” spectrum with maximum 𝛾 ray statistics; one of
the SeGA detectors exhibited poor energy resolution and was excluded from subsequent analysis.
           However, without enforcing any gating conditions, we cannot be sure that the decays measured
by SeGA occurred within the active region of the detector. This makes it challenging to compare
the intensities of 𝛽-delayed proton and 𝛾 emissions. For this reason, we set an 8-𝜇s time window to
record proton-𝛾 coincidences since this is the maximum possible drift time for primary ionization
electrons originating from decays near the cathode. Thus, every 𝛾 ray detected in SeGA that occurs
within the 8 𝜇s prior to a recorded Proton Detector event is added to the one-dimensional proton-
gated 𝛾 ray spectrum shown in Figure 4.6. In addition to this proton-gated 𝛾 singles spectrum, we
                                                       78


Figure 4.7: Two-dimensional coincidence matrix between 𝛾 rays detected in SeGA and
charged-particle events in the Proton Detector.
also generated two-dimensional histograms that plot 𝛾 energy versus proton energy for all proton-𝛾
coincidences, as depicted in Figure 4.7. Similarly for multiple 𝛾 rays detected in coincidence with
each other, assuming their detection in SeGA precedes a non-vetoed Proton Detector event, these
events are added to another coincidence matrix, shown in Figure 4.8, where both axes correspond
to 𝛾 energy.
    By applying a coincidence cut on 260-keV proton events and plotting all the detected 𝛾 rays in
that are observed within that same 8-𝜇s time window, we can check to see if this low-energy, 𝛽-
delayed proton decay is truly populating the ground state of 30 P. This is relevant because only proton
captures on the ground state of 30 P are relevant for 30 P(𝑝, 𝛾)31 S in ONe nova nucleosynthesis. For
comparison, we can place another coincidence gate on a featureless background region of the Proton
Detector data over the energy range 350 − 700 keV. This roughly linear background contribution
decreases gradually across the entire spectrum and likely originates from higher-energy proton
events that do not deposit their full energy in the active region. This continuum should be a
reasonable representation of the background under the 260-keV peak.
    To ensure a proportional comparison between the two regions, the background-gated 𝛾 ray
                                                  79


Figure 4.8: Two-dimensional 𝛾-𝛾 coincidence matrix for events occurring within the same 8-𝜇s
window preceding charged-particle radiation observed in the Proton Detector.
spectrum should be scaled down by the ratio of counts in each region of the proton singles
spectrum. We can see in Figure 4.9 that there is no evidence of 𝛾 emissions in coincidence with the
260-keV protons above background level. The fact that there are slight excesses across most bins
in the 260-keV-proton-gated spectrum likely reflects the fact that some of the background under the
peak is the result of 𝛽+ particles, which we know are likely to populate 𝛾-emitting states in       31 S.
Meanwhile, sampling from the background region 350−700 keV almost exclusively includes proton
events, so any coincidence 𝛾 rays must be either accidental or true coincidences from higher-energy
protons populating   30 P excited states. Still, there is no statistically significant difference in the
number of the photopeak between the two particle-gated histograms, suggesting that we are, in fact,
observing 𝛽-delayed proton decay through the 260-keV resonance of interest.
Combined Pads Detector Response
The random, statistical variation associated with measuring some quantity with a particular central
value can usually be described by a Gaussian distribution, assuming the number of observed
events is large. Since the Proton Detector and SeGA are designed to measure the charge or
                                                   80


Figure 4.9: Comparison between 260-keV proton-gated 𝛾 rays and background-gated 𝛾 ray
spectrum. The 𝛾 rays in coincidence with background are sampled from the energy range
350 − 700 keV in the Proton Detector spectrum and have been scaled down by the ratio of counts
in the two gating regions in the proton singles spectrum. The arrows point to the energies where 𝛾
events would be observed if the 260-keV 𝛽-delayed proton decays populate excited states in 30 P;
their colors correspond to the energy levels from which the 𝛾 rays are emitted.
current induced by stochastic processes, which are proportional to the energies of discrete nuclear
transitions, the corresponding spectral peaks exhibit shapes that can be represented by Gaussian
functions. However, the complicated physical interactions between radiation and the detection
medium degrade the symmetry of a normal statistical distribution.
    In the case of the Proton Detector, 𝛽-delayed proton decays result in the emission of three
ionizing, charged particles: the 𝛽+ particle, the proton, and the recoil nucleus. Because the proton
and recoil nucleus decay back-to-back over a relatively short range, moving in opposite directions
due to conservation of linear momentum, depositing comparable fractions of their energy into
ionizing the detector’s fill gas, the peak shape associated with the center-of-mass energy of the
proton decay alone is basically symmetric. However, the 𝛽+ particles inevitably deposit some of
their energy in the gas as well, resulting in an observed decay energy which is slightly higher than
the proton-emitting resonance energy. This 𝛽+ summing effect causes the 𝛽-delayed proton decay
                                                  81


peaks to be slightly skewed to the right. The shape of the large 𝛽+ background that exists at low
energies (<200 keV) is well fit with a decaying exponential function, and peaks corresponding to
𝛽-delayed proton decays can be understood as a sum of these energies. For this reason, we model
the detector response of each Proton Detector pad as an exponentially modified Gaussian (EMG)
function with a high-energy tail. In terms of the independent variable 𝑥, which is proportional to
energy in our spectra, the shape can be expressed using the equation
                                                                                
                                           𝜆 𝜆 (2𝜇+𝜆𝜎2 −2𝑥)         𝜇 + 𝜆𝜎 2 − 𝑥
                        𝑓 (𝑥; 𝜇, 𝜎, 𝜆) = 𝐴 𝑒 2              erfc       √           ,              (4.3)
                                           2                             2𝜎
where 𝜇 represents the centroid of the function’s symmetric Gaussian component, 𝜎 is proportional
to the width of the peak, 𝜆 determines how asymmetrically skewed the distribution is, and the
normalization constant 𝐴 represents the area under the curve [141]. The complementary error
function (erfc) is defined in terms of the error function (erf) as
                                       erfc(𝑥) = 1 − erf(𝑥)
                                                      ∫ ∞                                         (4.4)
                                                  2
                                                            𝑒 −𝑡 𝑑𝑡.
                                                                2
                                               =√
                                                   𝜋 𝑥
    The proton peaks in the individual pad’s histograms have a unique shape and resolution. The
extent to which proton peaks are skewed in each pad is primarily due to the relationship between
detector pad surface area and the transverse distribution of the radioactive beam. Detector pads with
a larger surface area can collect more charge from 𝛽+ particles, resulting in a more right-skewed
distribution function. Similarly, the beam distribution in the detector affects how likely the 𝛽+
particles are to deposit their maximal energy into ionizing the fill gas. Furthermore, event-level
summing over all five inner pads also results in a unique detector response, which although having
worse energy resolution, has the benefit of much higher statistics and a better, more precise detection
efficiency.
    Many of the peaks in the 31 Cl 𝛽-delayed proton spectra are quite close together at higher proton
energies since the level density of proton-emitting states increases with excitation energy. To
accurately quantify the number of counts in any single peak using an analytical fit function, it is
                                                   82


often necessary to fit multiple peaks in the spectrum over a range of several hundred keV using
a sum of several EMG distributions. We refer to the ratio between a spectral peak’s full width at
                            √
half maximum (FWHM = 2 2 ln 2𝜎 ≈ 2.355𝜎) and its corresponding decay energy as the relative
resolution, which should, in principle, decrease as a function of energy. However, the absolute
value of the FWHM is likely to increase both due to detector physics and because the total decay
width of quantum states tends to increase as a function of energy, and some of these levels are likely
to be intrinsically broad. The observed peak widths over a relatively small energy range (≈ 300
keV) are basically constant, but they tend to increase slightly from 700 keV up to 2.5 MeV. For this
reason, when fitting multiple proton peaks, we assume the width varies smoothly as a function of
energy and thus parameterize the shape of our EMG distributions by the relation 𝜎(𝜇) = 𝑏 + 𝑚𝜇.
This constrains the peak width to be a linear function of decay energy, the slope and intercept of
which we determine via 𝜒2 -minimization. Similarly, we assume the skew parameter 𝜆 to vary
smoothly as a function of energy and let this parameter be constant for each EMG distribution over
a given fit range.
    The mostly featureless region over ≈ 300 − 700 keV in the event-level combined spectrum is
extremely flat. We suspect this background results from higher-energy decays near the upstream
end of the detector chamber, whose decay protons are emitted in the direction of the cathode. This
“wall effect” likely causes protons to deposit some fraction of their energy in the active detection
region before being absorbed by the cathode and are thus not rejected by the veto pads. This
hypothesis agrees with the observation that the individual pad spectra have slightly steeper slopes
in this background region since increasing proton energy heightens the probability of triggering
the veto condition more sensitively when the effective pad detection area is smaller. While much
of the spectra are populated with many features at decay energies above 700 keV, in the regions
between obvious peaks, the background level clearly decreases with energy, which also agrees
with the presented explanation for this effect. Thus, we believe it is reasonable to assume a linear
model to describe our background, while fitting the 𝛽-delayed proton peaks with EMG functions.
We estimated the systematic uncertainty in our detector response model by fixing and varying all
                                                  83


Figure 4.10: Fit of the 260-keV resonance peak summing proton events over all active detector
pads.
parameters given in Equation 4.3 and applying these fits over various energy ranges across the
entire proton spectrum.
Normalization Procedure
Having parameterized our detector response function at higher proton energies background, we
determined the observed number of protons emitted from the 6390-keV level in       31 S to be 𝑁 𝛽𝑝 =
2731(203). As shown in Figure 4.10, this was accomplished by fitting the small, low-energy
protons in the event-level combined spectrum with the sum of an EMG distribution for the peak;
a linear function was used to describe the flat background region above the resonance energy and
an exponential function for the 𝛽+ background below the resonance energy. The total uncertainty
in the number of counts was determined by adding the statistical error of the fit Δ𝑁stat. = ±200
in quadrature with the systematic uncertainty Δ𝑁sys. = ±36 associated with varying the fit range
and constraints on the response function parameters. However, in order to determine the proton
branching ratio of interest, we must normalize this 𝛽-delayed proton decay intensity to some known
                                                 84


Figure 4.11: Fit of the three calibration peaks in the all-pads-combined spectrum. We do not
extend the fit above 1.1 MeV in this case because we determined that there are previously
unobserved 𝛽-delayed proton decays in this region. These transitions had not been considered
when evaluating the intensity of the 1-MeV proton peak, and thus for the purposes of normalizing
the literature values, we exclude them as well.
value.
    We considered a variety of different normalization strategies, including normalizing to known
𝛽-delayed 𝛾 decay intensities, counting individual 31 Cl ions by measuring their energy deposition
in the detector without the amplification field, as well as using proton-𝛾 coincidences that indicate
population of 30 P excited states. However, each of these methods had their own unique challenges
that prevented us from achieving an unambiguous normalization standard. Ultimately, we chose to
adopt the evaluated literature intensity [𝐼 𝑝 (abs) = 1.31(2)%] of a strong, 𝛽-delayed proton emission
with the center-of-mass decay energy 𝐸 𝑝 (c.m.) = 1026 keV [119]. We determined the number of
1-MeV proton counts to be 𝑁 𝑝1026 = 3.16(2) × 106 by fitting the spectrum from 350 to 1100 keV
with three EMG distributions to model proton peaks at 806, 906, and 1026 keV on top of a linear
background as shown in Figure 4.11.
    By taking the ratio between the number of counts in the small, 260-keV resonance peak of
                                                    85


interest and the total counts in the large, 1-MeV normalization peak, we can determine the 𝛽-
delayed proton intensity needed to evaluate Equation 4.23. However, we must also take into
account the effect of proton detection efficiency 𝜖 𝑝 at the relevant energies. This relationship can
be expressed as
                                                𝑁 𝑝260 /𝜖 𝑝260
                                      𝐼 𝑝260 =                  𝐼 𝑝1026 .                        (4.5)
                                               𝑁 𝑝 /𝜖 𝑝
                                                 1026      1026
Proton Detector Efficiency Simulations
In the process of designing the Proton Detector, Monte Carlo simulations were used to model
detector response and evaluate the detection efficiency as a function of proton energy. These initial
simulations were performed using the GEometry ANd Tracking (geant4) framework, which is a
detailed, object-oriented platform designed at CERN to simulate the passage of particles through
matter and their interactions with variety of detection media [142; 143; 144]. The exact geometry
of the Proton Detector was specified in code as well as the chemistry of the gas and the specific
energies of known proton decays. The only experimental input to the geant4 simulations was the
three-dimensional distribution of the 31 Cl gas inside the detector chamber. Assuming the beam spot
on the detector pad plane is well-described by a two-dimensional Gaussian function, the transverse
distribution of the radioactive source was deduced from the observed hit pattern over the course of
the measurement.
    Using the multiplicity of proton events above 700 keV detected by each of the five inner pads, a
𝜒2 -minimization procedure was implemented to approximate the x- and y-coordinates of the beam
spot’s origin as well as the radial width for the transverse component of the beam distribution. In
the initial geant4 calculations, we also enforced the condition that the beam spot be truncated at a
radial distance of 25.4 cm from the center of the pad plane, since this corresponds to the size of the
window through which the beam enters the Proton Detector; Figure 4.12 shows the transverse beam
distribution after allowing for one half-life of radial beam diffusion. The longitudinal distribution
                                                    86


Figure 4.12: Beam spot used to represent the transverse distribution of 31 Cl 𝛽-delayed proton
decays in the Monte Carlo efficiency simulations.
in the efficiency simulation was specified in terms of the observed drifted times, plotted in Figure
3.13, for Proton Detector events observed in the 8-𝜇s window following the detection of a 𝛾 ray in
SeGA. However, this model did not take into account the effect of electron diffusion in the P10 fill
gas.
    The GADGET system has recently been upgraded to operate as a time projection chamber
(TPC), resulting in a need for improved spacial granularity. The number of detection channels
has been increased from 13 to 1024, contained within basically the same pad plane area. This
means that each detector pad is now much smaller, and the effect of ionization electrons spreading
out radially in time as they drift toward the anode is readily apparent. Upon further investigation,
this effect was actually found to be much larger than originally anticipated, which motivated us
to incorporate this electron diffusion into our efficiency calculations. In the interest of time, we
developed a simple, geometrical model of the efficiency first to estimate the size of this effect.
When neglecting electron diffusion, this Monte Carlo method with simplified physics was able
to reproduce the proton detection efficiency from the geant4’s much more detailed simulation
of physical interactions to within 3% for proton energies up to 1.4 MeV, as shown in Figure ??.
                                                 87


Ultimately, we decided to use this simplified geometric model to calculate our final proton detection
efficiencies after adding in the diffusion of ionization electrons.
     The simplified, geometric Monte Carlo model assumes the same transverse and longitudinal
beam distributions as described for the geant4 simulations. The x- and y-position coordinates at
which a 31 Cl beam particle undergoes 𝛽-delayed proton decay are randomly sampled from the beam
spot histogram. The z-coordinate of the decay is generated by converting a randomly selected drift
time to its analogous spatial location, since the average z-component of the ionization electrons’
drift velocity is constant under a uniform electric field. A total of 104 proton decay events were
simulated for every proton energy corresponding to a decay peak in our spectrum, resulting in a
1% statistical uncertainty. The proton stopping power 𝑑𝐸/𝑑𝑟 was calculated for a range of energies
using srim for a 808-Torr gaseous medium of 90% Ar and 10% CH4 .
     In the simulation, components of the emitted proton’s direction are generated isotropically in
all three spatial dimensions. The deposition of proton energy in the fill gas is iteratively calculated
as
                                                     𝑑𝐸
                                              Δ𝐸 =       Δ𝑟                                       (4.6)
                                                     𝑑𝑟
for step sizes of Δ𝑟 = 0.01 mm over the entire length of the track. Because the recoiling 30 P nuclei
have such short ranges in the gas, we only consider the energy carried away by the proton in the
laboratory frame, just like in the geant4 simulations. At each 0.01-mm step along the proton track,
we generate a cloud of ionization by randomly sampling the x- and y-positions of electrons from a
two-dimensional Gaussian distribution, whose width is proportional to the drift time of the decay
event
                                                    √︁
                                           𝜎(𝑡) =      2𝑛𝐷 𝑒 𝑡,                                   (4.7)
where 𝑛 = 2 dimensions, since only diffusion in the transverse plane affects detection efficiency
[127]. The electron diffusion coefficient 𝐷 𝑒 = 9116(273) cm2 /s was calculated using magboltz
[145], and the primary component of the uncertainty in 𝐷 𝑒 comes from the gas pressure, while the
                                                   88


uncertainty from the drift field is negligible. The number of electrons simulated on each iterative
step is proportional to the energy loss of the proton in the gaseous volume
                                              𝑁 𝑒 ≈ Δ𝐸/𝑊,                                          (4.8)
where 𝑊 = 26 eV/ion pair is the energy required to form a single electron-ion pair in P10 gas
[126; 146]. If the number of electrons sampled from outside the active region of the detector, that
               √︁
is for which 𝑥 2 + 𝑦 2 > 4 cm, exceeds some trigger threshold, the corresponding proton decay
event is vetoed. The veto threshold is set in software to be the same for all pads in units of ADC
channels. However, the five active pads all have slightly different gains, corresponding to an energy
threshold range of 5 − 20 keV. The number of electrons was calculated using Equation 4.8 for 5-,
10-, and 20-keV thresholds, and simulations were performed at each trigger sensitivity level in
order to quantify the systematic uncertainty associated with the veto condition. The electron veto
trigger threshold accounts for a 3 − 4% uncertainty on both the upper and lower limits of the relative
detection efficiency 𝜖1026 /𝜖260 .
    The other sources of systematic uncertainty associated with modeling the ratio of detection
efficiencies are related to the initial spatial distribution of the beam particles when they decay as
well as the ranges of the protons in the gas. The latter is a small effect, for which srim documentation
quotes a 4% uncertainty in the stopping powers for protons in gas [139]; this translates to a 1%
uncertainty in the lower limit of 𝜖1026 /𝜖 260 and a 2% effect on the upper limit. The former
uncertainty related to the distribution of    31 Cl in the gaseous volume is a bit more complicated.
Using the 𝜒2 -minimization procedure to determine the x- and y-components of the beam spot’s
centroid as well as the width of the distribution, we estimate that these parameters can be realized to
within ±6 mm. Varying each of these parameters by 6 mm in simulation produces a 2% uncertainty
on the lower limit and a 5% uncertainty on the upper limit of the relative detection efficiency.
    Because the beam enters the Proton Detector through a thin window with a 25.4-mm radius,
the initial efficiency model assumed a hard cutoff at this boundary, meaning no 31 Cl decays were
simulated further than 25.4 mm away from the origin of the pad plane. However, it is possible for
                                                     89


beam particles to diffuse under Brownian motion before undergoing 𝛽-delayed proton decay. The
amount of outward radial diffusion from the entrance aperture was calculated using Equation 4.7
for zero, one, and two 31 Cl half-lives to evaluate the upper limit, central value, and lower limit of
𝜖1026 /𝜖260 , respectively. The diffusion constant for 31 Cl in Ar gas was calculated using the equation
                                                     √︂
                                              𝐴𝑇 3/2      1      1
                                        𝐷𝑎 =                  +     ,                              (4.9)
                                              𝑝𝜎12 Ω     𝑀1 𝑀2
where 𝑀1 and 𝑀2 are the molar masses (g/mol) of the 31 Cl and Ar, 𝑝 is the pressure (atm) inside
the chamber, 𝑇 is room temperature (K), and 𝐴 is just an empirical coefficient [147]. Values
for the dimensionless, temperature-dependent collision integral Ω are tabulated in Ref. [148] but
are usually on the order of unity, which we adopt. The average collision diameter (Å) is simply
𝜎12 = (𝜎1 + 𝜎2 )/2, where 𝜎1 and 𝜎2 are twice the Van der Waals atomic radii for Cl and Ar,
adopted from Ref. [149]. The calculated value of 𝐷 𝑎 for           31 Cl in Ar gas agrees well with the
empirical self-diffusion coefficient for Ar as reported in Ref. [148]. Ultimately, like the uncertainty
associated with parameterizing the transverse beam distribution, the systematic error from beam
particle diffusion from the aperture is 2% for the lower limit of 𝜖1026 /𝜖260 and 5% for the upper
limit. After performing numerous Monte Carlo simulations over a variety of input parameters and
having tested all the systematic effects in our efficiency model, we determined that the ratio of
proton detection in efficiencies in our experiment was 𝜖1026 /𝜖260 = 0.73+0.09
                                                                             −0.05
                                                                                   .
    Since originally reporting this number in literature [150], we have updated the efficiency
simulation to include the wall effect and improved the sampling procedure. The latest version
yields consistent, albeit slightly different numerical results. The preliminary numbers reported in
Chapter 5 for all 31 Cl 𝛽-delayed proton intensities reflect these changes. The Monte Carlo results
of the simplified, geometric model and the geant4 simulations are compared in Figure 4.13.
Final Result
Substituting the results from relative detection efficiency simulations into Equation 4.5 along with
the adopted literature intensity of the 1-MeV normalization peak and the observed counts in both
                                                    90


Figure 4.13: Efficiency to detect protons with GADGET as calculated by geant4 simulations
assuming no electron diffusion (blue), the simplified, geometric model without electron diffusion
(green), and the simplified, geometric model including the electron diffusion effect (red).
spectral peaks, we determined that the 𝛽-delayed proton decay intensity through the low-energy
resonance of interest was 𝐼 𝑝260 = 8.3+1.2     −6
                                      −0.9 × 10 . After perusing the most recent literature evaluation
of 𝛽-delayed charged-particle decays [119], we determined that our result represents the weakest
𝛽-delayed proton decay ever measured for resonances below 400 keV. Using Equation 4.23, we
have calculated the desired proton branching ratio to be Γ𝑝 /Γ = 2.5+0.4        −4
                                                                      −0.3 × 10 .
4.2   Calculating the Total Thermonuclear Rate
Having determined the crucial proton branching ratio, the only remaining piece of information
needed to evaluate Equation 4.21 for the 260-keV, 𝐽 𝜋 = 3/2+ resonance is the 𝛾 partial width Γ𝛾 .
Since 𝐼 𝑝 ≪ 𝐼𝛾 for the 31 S excited state at 6390 keV, we can make the approximation Γ𝛾 /Γ ≈ 1 =⇒
Γ𝛾 ≈ Γ. Because the total width of the nuclear level is inversely proportional to its lifetime Γ = ℏ/𝜏,
there have been several attempts to measure the lifetime 𝜏 of this short-lived state, but so far, a finite
value has eluded experimental determination. Thus, we must rely on theory to calculate this value
in collaboration with Prof. Alex Brown at Michigan State University. Discussion of the nuclear
                                                   91


shell model in this section also borrows heavily from his lecture notes [1].
    In addition, the 6390-keV level discussed extensively in this dissertation is not the only resonance
state that contributes to the   30 P(𝑝, 𝛾)31 S reaction, although it is expected to dominate the total
thermonuclear rate. Here, we will also evaluate past work on other resonances in literature,
calculate a new recommended rate, and compare the resonant reaction formalism to Hauser-
Feshbach statstical model calculations.
The Nuclear Shell Model
The nuclear shell model was first developed by Eugene Paul Wigner, Maria Goeppert Mayer,
and Johannes Hans Daniel Jensen, for which they earned the 1963 Nobel Prize. As fermions,
protons and neutrons follow the Pauli principle and fill their respective orbitals independently. The
shell model assumes that a given nucleon moves in an effective attractive potential created by
the presence of all other nucleons in the nucleus. Although realistically, due to the short-range
nature of this interaction, the potential can be approximated by the mean field produced by only the
nearest neighboring nucleons. Protons and neutrons are almost identical in term of their nuclear
interactions, but the net positive charge of the proton means introduces an electrostatic interaction
as well, as illustrated in Figure 4.14.
Woods-Saxon Potential
While the negatively-charged electrons in an atom are bound to the positively-charged nucleus
exclusively by an attractive, long-range Coulomb potential 𝑉𝑐 (𝑟) , the Woods-Saxon potential
describes the interaction for a single nucleon with the mean field produced by the nucleus and can
be written as
                                  𝑉 (𝑟) = 𝑉𝑤𝑠 (𝑟) + 𝑉𝑠𝑜 (𝑟) ℓ® · 𝑠® + 𝑉𝑐 (𝑟),                      (4.10)
where 𝑉𝑤𝑠 (𝑟) = 𝑉𝑤𝑠 𝑓𝑤𝑠 (𝑟) is the spin-independent, central Woods-Saxon potential. The average
proton-neutron potential is stronger than the average proton-proton or neutron-neutron potential.
                                                     92


Figure 4.14: Effective interaction potentials seen by the proton and neutron as a function of the
distance between the nucleon and the nucleus. The effect of alignment between spin and orbital
angular momentum on the potential is also shown. Figure credit: HyperPhysics, Georgia State
University, Carl Rod Nave.
Thus, for protons
                                                  (𝑁 − 𝑍)
                                     𝑉𝑝 = 𝑉𝑤𝑠 +            𝑉1 ,                              (4.11)
                                                     𝐴
and for neutrons
                                                  (𝑁 − 𝑍)
                                     𝑉𝑛 = 𝑉𝑤𝑠 −            𝑉1 .                              (4.12)
                                                     𝐴
The spin-orbit component of the potential 𝑉𝑠𝑜 describes the coupling between the spin of the
nucleon and its orbital angular momentum:
                                                   1 𝑑𝑓𝑠𝑜 (𝑟)
                                     𝑉𝑠𝑜 (𝑟) = 𝑉𝑠𝑜            .                              (4.13)
                                                   𝑟 𝑑𝑟
In both cases, the Fermi component of these functions have the form
                                                93


                                                             1
                                          𝑓𝑖 (𝑟) =                      ,                       (4.14)
                                                      1 + 𝑒 (𝑟−𝑅𝑖 /𝑎𝑖 )
while the Coulomb term in the Woods-Saxon potential describes the electrostatic interaction over
two different ranges with the boundary conditions
                                             
                                                 2
                                              𝑍𝑒𝑟 ,                  𝑟 ≥ 𝑅𝑐
                                             
                                             
                                             
                                  𝑉𝑐 (𝑟) =                                 .
                                              𝑍𝑒 2        𝑟 2
                                              2𝑅𝑐 3 − 𝑅𝑐2 , 𝑟 ≤ 𝑅𝑐
                                             
                                             
                                             
     The radii 𝑅𝑤𝑠 , 𝑅𝑠𝑜 , and 𝑅𝑐 are often expressed in terms of the mass number 𝑅𝑖 = 𝑟𝑖 𝐴1/3 .
One can usually assume 𝑟 𝑤𝑠 = 𝑟 𝑠𝑜 = 1.25 fm and similarly for the “surface thickness” of the
nucleus 𝑎 𝑤𝑠 = 𝑎 𝑠𝑜 = 0.65 fm. However, the Coulomb radius 𝑟 𝑐 = 1.20 fm is a bit smaller. The
interaction strengths 𝑉0 , 𝑉1 , and 𝑉𝑠𝑜 determine the depth of the potential well, and as with all the
other numerical constants, they can vary depending on the region of the nuclear chart.
Nuclear Orbitals and Shell Structure
Solving the Schrödinger equation for the Woods-Saxon potential with the addition of spin-orbit
coupling results in a set of quantized energy levels as depicted in Figure 4.15, which compares this
more realistic model to the relatively simple solution for the isotropic harmonic oscillator potential
𝑉 (𝑟) = 𝜇𝜔2𝑟 2 /2. These nuclear orbitals are denoted by the combination of quantum numbers 𝑛ℓ 𝑗
that define a state. The principle quantum number 𝑛 specifies the number of nodes in the wave
function, ℓ is the orbital angular momentum, and 𝑗 is the total angular momentum. Thus, the 1𝑠
orbital, for example, has ℓ = 0 and can only contain two nucleons of the same type. Since the total
angular momentum is 𝑗 = 1/2, within a pair of identical fermions, if one is spin-up ( 𝑗 𝑧 = +1/2),
the other must be spin-down ( 𝑗 𝑧 = −1/2). Furthermore, the 1𝑝 orbital, with its ℓ = 1, can hold up
to six identical particles: two in the 1𝑝 1/2 state ( 𝑗 = 1/2; 𝑗 𝑧 = ±1/2) and four in the 1𝑝 3/2 state
( 𝑗 = 3/2; 𝑗 𝑧 = ±3/2, ±1/2).
     Unlike in atomic theory, nuclear shells do not strictly correspond to the specific quantum
numbers of their orbitals. Instead, shells represent clusters of nuclear orbitals which are separated
                                                       94


Figure 4.15: Neutron single-particles states in 208 Pb using three different potentials. The addition
of each term further breaks the level degeneracy. Bracketed numbers indicate the number of
neutrons contained in each orbital, while the following number is the running sum total. Figure
credit: Ref. [1].
                                                 95


Figure 4.16: Deviation in neutron separation energy plotted as a function of neutron number,
where the line at zero represents expected binding energy according to the semi-empirical mass
formula. Figure credit: HyperPhysics, Georgia State University, Carl Rod Nave.
by large gaps in energy as shown in Figure 4.15. While the 𝑠-shell is just the 1𝑠1/2 orbital and the
𝑝-shell only contains the 1𝑝 3/2 and 1𝑝 1/2 orbitals, the 𝑠𝑑-shell is comprised of the 1𝑑5/2 , 2𝑠1/2 ,
and 1𝑑3/2 orbitals; the clustering of orbitals with different quantum numbers only becomes more
complicated as the number of nucleons increases. The magic numbers correspond to the total
number of nucleons needed to fill the nuclear shells. As depicted in Figure 4.16, the fact that
the energy required to remove a neutron from the nucleus peaks at shell closures then drops off
dramatically with the addition of extra neutron, provides strong experimental evidence for shell
model theory.
     In addition to this, the shell model can predict the spin and parity of the ground state for a given
nucleus. For example, in the case of 30 P, both its 15 protons and its 15 neutrons fill their respective
1𝑠, 1𝑝, and 1𝑑 orbitals. This leaves the remaining proton and neutron in their 2𝑠 orbitals. Thus,
we can think of the 30 P nucleus as an inert core consisting of 14 protons and 14 neutrons with two
extra valence nucleons, each with spin 𝑗 = 1/2. The inert core has a total angular momentum of
𝑗 = 0 since it fills the 1𝑑5/2 orbital, and the sum total angular momentum of the valence nucleons
is 𝑗 = 1. Because the 2𝑠 orbital is an ℓ = 0 level and since Π = (−1) ℓ , this means the parity of the
30 P ground state is positive, which agrees with the observation that 𝐽 𝜋 = 1+ .
     While the shell model allows us to infer ground state spin-parity from the properties of individual
                                                   96


valence nucleons, this is not trivial for excited states. However, the existence of mirror nuclei allows
us to deduce information about the level structure of a nucleus with mass number 𝐴, proton number
𝑍1 , and neutron number 𝑁1 from a different nuclide with mass number 𝐴 but for which the proton
number is 𝑍2 = 𝑁1 and the neutron number is 𝑁2 = 𝑍1 . Due to their symmetry, the excited states
in one mirror nucleus can often be matched to an analogous state in the other mirror nucleus, and
assuming we know the properties of one mirror state, we can assign spin and parity to the other
mirror state. This reflects a useful property of the nucleus called isospin.
Isospin Mixing
The isospin model proposes that because the proton and neutron are nearly identical, instead of
treating them as distinct nucleons, they can be understood as projections in isospin space of a single
nucleonic particle. This isospin doublet still has spin 𝑗 = 1/2, but in addition, it also possesses
isospin 𝑇 = 1/2, the projections of which correspond to 𝑇𝑧 = +1/2 for the neutron and 𝑇𝑧 = −1/2 for
the proton. The total isospin projection of a particular nuclide can be expressed as 𝑇𝑧 = (𝑁 − 𝑍)/2,
which is the same across all excited states of that nucleus. However, each nuclear level will not
have the same total isospin 𝑇. The isospin model is formulated similarly to angular momentum in
the sense that for a given total isospin 𝑇 there will be 2𝑇 + 1 projections spanning −𝑇 < 𝑇𝑧 < 𝑇.
Thus, the analogous levels in mirror nuclei can be thought of as nearly identical states with the
same spin-parity and quantum number 𝑇 but with different projections in isospin space; these are
called isobaric analog states (IAS).
     Of course, protons and neutrons are not identical particles, and the difference in the strength
of the Coulomb interaction between mirror nuclei perturbs the energy level spacing between the
nuclides. In addition to this, Coulombic perturbations can result in isospin mixing between nearby
states of the same spin and parity. In the simple case of two-level mixing, the quantum states |𝜓1 ⟩
and |𝜓2 ⟩ can be written as a linear combination of isospin basis states:
                                       |𝜓1 ⟩ = cos 𝜃 |𝑇1 ⟩ + sin 𝜃 |𝑇2 ⟩                          (4.15)
                                                     97


                                    |𝜓2 ⟩ = − sin 𝜃 |𝑇1 ⟩ + cos 𝜃 |𝑇2 ⟩                      (4.16)
     If the energy difference between two levels is large, the amount of mixing between then will be
negligible. The mixing angle 𝜃 is given by the relation
                                                         2𝑉
                                              tan 2𝜃 =      ,                                (4.17)
                                                         𝐷
where 𝑉 is the mixing matrix element and 𝐷 is the unperturbed energy difference between the
original states. The relationship between these two quantities and the observed (perturbed) level
                           √
spacing is given by 𝐸 = 𝐷 2 + 4𝑉 2 .
     Using 𝛽 decay measurements, the extent of this mixing can sometimes be determined from
experimental observables. If the 𝛽 decay proceeds via a Fermi transition, the emitted 𝑒 − -𝜈¯ 𝑒 or
𝑒 + -𝜈𝑒 pair will have antiparallel spins, such that the total angular momentum of the nucleus is
unchanged (Δ 𝑗 = 0), and similarly, parity is also conserved (Δ𝜋 = 0). Typically, Fermi decays
of a radioactive parent nucleus will only populate its IAS in the daughter nucleus, but if isospin
mixing occurs between its IAS and another state, the Fermi transition strength 𝐵(𝐹) to the nearby
level will be nonzero. The transition strengths of two states (𝐵1 and 𝐵2 ) can be deduced from the
𝛽-feeding intensities, and their ratio is related to the mixing angle:
                                                         √︂
                                                            𝐵2
                                           𝑅 = tan 𝜃 =          .                            (4.18)
                                                            𝐵1
This allows us to write the unperturbed energy spacing as
                                                     1 − 𝑅2
                                             𝐷=𝐸              ,                              (4.19)
                                                     1 + 𝑅2
and the unperturbed mixing matrix element becomes
                                                       𝑅
                                             𝑉=𝐸             .                               (4.20)
                                                     1 + 𝑅2
                                                    98


USDC Model Calculations
In Mayer and Jensen’s original formulation, they assumed each nuclear level corresponded to a
single valence nucleon in a single-particle orbit around an inert core at shell closure. We now
know that the nucleus is much more complicated than this and that the collective behavior of
quantum many-body interactions is necessary to accurately described the properties of nuclear
excitations. Modern shell-model calculations must incorporate many multinucleon configurations
as a consequence of the fact that valence nucleons can occupy multiple, partially filled, single-
particle states simultaneously [151]. This requires extensive numerical computation in order
to predict experimental observables, and even still, the valence space must often be truncated
simply due to the sheer number of possible configuration-interactions. Nevertheless, the shell
model remains an extremely useful tool for predicting excitation energies, spin-parity assignment,
multipole moments, and matrix elements between nuclear levels.
    In order to perform these calculations, a Hamiltonian is needed to describe the configuration-
interactions within the valence space. Nearly 40 years ago, the “universal 𝑠𝑑-shell” (USD) Hamil-
tonian was proposed to account for the wide variety of observed phenomena in the level structure
of nuclides with mass numbers 𝐴 = 17 − 39 and has since provided realistic 𝑠𝑑-shell wave func-
tions for this region of the nuclear chart [152]. In 2006, its successors USDA and USDB were
introduced using an updated and expanded set of nuclear levels [153; 154]. Recently, several new
isospin-breaking, USD-type Hamiltonians have been developed, including USDC [155], which we
used to evaluate the 𝛾 decay width Γ𝛾 for the 6390-keV level in 31 S.
    For the USDC Hamiltonian, the strongest isospin mixing with the IAS for the 31 Cl ground state
comes from a 𝑇 = 1/2 level in 31 S, which the shell model predicts to be 300 keV below the 𝑇 = 3/2
IAS. The isospin-mixing matrix element for these levels 𝑉theory = 36 keV is in good agreement
with the experimental value 𝑉exp = 41(1) keV [155]. Theory predicts 𝛾-decay widths of 190 meV
for the 𝑇 = 1/2 state and 920 meV for the 𝑇 = 3/2 state. However, the observed 𝑇 = 1/2 state,
corresponding to the 6390-keV level in 31 S, lies above the 𝑇 = 3/2 IAS at 6279 keV. The mixing of
these two states depends on the energy difference between them, which is determined by the strong
                                                  99


Figure 4.17: A comparison of experimental and theoretical values of the M1 matrix elements for
magnetic dipole transitions. Figure credit: Ref. [154].
interaction. Adding a term in the Hamiltonian, which is proportional to the 𝑇ˆ 2 operator, moves
the 𝑇 = 1/2 state up by 410 keV. After this shift, the new partial widths are Γ𝛾6390 = 490 meV and
Γ𝛾IAS = 430 meV. The sum of these two widths is not exactly the same due to interference from
other 𝑇 = 1/2 states that do not demonstrate strong isospin mixing.
    There are two primary sources of systematic uncertainty associated with calculating Γ𝛾 theo-
retically. The first comes from the choice of the USD Hamiltonian. By comparing results from the
four different Hamiltonians derived in Ref. [155], we assume this uncertainty in Γ𝛾 between the
models is relatively small, at about 50 meV. The second source of error in this calculation arises
                                                 100


from the uncertainty in the matrix element of the magnetic dipole (M1) transition. 𝛾 decay of the
𝑇 = 1/2 state is dominated by a 66% branch to the lowest 𝐽 𝜋 = 5/2+ state with an M1 transition
strength of 𝐵(𝑀1) = 0.48𝜇2𝑁 , where 𝜇 𝑁 = 𝑒ℏ/2𝑚 𝑝 is the nuclear magneton. The M1 decay matrix
                  √︁
element is 𝑀 = (2𝐽𝑖 + 1)𝐵(M1) = 1.38𝜇 𝑁 . Comparing to other experimental values of 𝑀, as
shown in Figure 4.17 taken from Ref. [154], we adopt a systematic uncertainty of ≈ 0.4𝜇 𝑁 . This
ultimately leads to a total uncertainty of 280 meV. Thus, we can combine Prof. Alex Brown’s shell-
model calculations for Γ𝛾 = 490(280) meV and our experimentally determined proton branching
ratio Γ𝑝 /Γ = 2.5+0.4
                  −0.3 × 10
                            −4 with the known angular momentum information 𝐽 = 3/2, 𝐽 = 1/2,
                                                                                     𝑟          𝑝
and 𝐽𝑡 = 1 for the resonance, proton, and target nucleus, respectively. Using our definition of the
resonance strength
                                                (2𝐽𝑟 + 1)     Γ𝑝 Γ𝛾
                                    𝜔𝛾 ≡                            ,                             (4.21)
                                           (2𝐽 𝑝 + 1)(2𝐽𝑡 + 1) Γ
we computed a final value of 𝜔𝛾 = 80(48) 𝜇eV for the crucial 260-keV resonance of interest.
Resonant Contributions
The most important resonances that contribute to the total thermonuclear rate typically fall within
±Δ/2 of the Gamow peak as calculated using Eqs. 2.11 and 2.14. For a peak nova temperature of
𝑇 = 0.31 GK, this corresponds to resonance energies 𝐸 0 ± Δ/2 = 337 ± 110 keV for the 30 P(𝑝, 𝛾)31 S
reaction. There have been many experiments dedicated to the study of proton-unbound states in
31 S over this resonance energy range. We believe most of the key resonances within the Gamow
window have been experimentally observed, but the nuclear dataset remains incomplete for many of
these states. For example, the spin-parity assignments for several high-spin states remain tentative,
and certain measurements of spectroscopic factors provide only upper limits. Here, we review
the resonance strengths reported in literature as summarized nicely by Kankainen et al. [104].
Finally, we are able to calculate a total thermonuclear rate for     30 P(𝑝, 𝛾)31 S using the individual
contributions of nine narrow, isolated resonances based on the available experimental information
complemented by theoretical evaluations.
                                                   101


Calculating Proton Partial Widths
For states with known spin-parity and well-constrained spectroscopic factors, it is possible to
calculate proton partial widths with reasonable accuracy. The analytical approximation can be
expressed as
                                                 ℏ2
                                       Γ𝑝 = 2       2
                                                       𝑃𝑐 (𝐶 2 𝑆)𝜃 sp  2
                                                                         ,                       (4.22)
                                                𝜇𝑎 𝑐
where 𝑃𝑐 is the penetration factor, 𝐶 2 𝑆 is the spectroscopic factor, and 𝜃 sp2 is the dimensionless
single-particle reduced width [15]. The interaction radius is 𝑎 𝑐 = 1.25( 𝐴1/3     1/3
                                                                           𝑝 + 𝐴𝑡 ) fm and depends
on the mass of the proton 𝐴 𝑝 and target nucleus 𝐴𝑡 in atomic mass units. The penetration factor is
the probability that a single proton will penetrate the Coulomb and centrifugal barriers and is given
by
                                                       𝑘 𝑐 𝑎𝑐
                                              𝑃=                 ,                               (4.23)
                                                   𝐹𝑐2   + 𝐺 2𝑐
              √︁
where 𝑘 𝑐 =     2𝜇𝐸𝑟 /ℏ is the wave number, while 𝐹𝑐 and 𝐺 𝑐 represent the regular and irregular
Coulomb wave functions, respectively, evaluated at the interaction boundary [156]. To compute
these wave functions, we use the GNU Scientific Library’s numerical calculation.
    The dimensionless quantity for the single-particle reduced width is related to the probability
that the single nucleon, the proton in our case, will appear at the boundary. We calculate this value
using the summation
                                                 ∑︁2
                                            2
                                          𝜃 sp =        𝑐𝑖, 𝑗 𝐴𝑖 𝐸 𝑗 ,                           (4.24)
                                                  𝑖, 𝑗
where here, 𝑐𝑖, 𝑗 are parameters given by Illiadis in Ref. [156], who arrives at these values by fitting
angular distribution data as a function of bombardment energy for protons scattered on various
target nuclei. More formal values calculated from an R-matrix formula are reported by Ref. [157],
but these results agree with Illiadis’s values, as well as with shell-model calculations, to within
                                                    102


about 10% on average. Thus, we assign a 10% systematic uncertainty to the single-particle reduced
width Γ𝑠𝑝 = Γ𝑝 /𝐶 2 𝑆.
    Lastly, the spectroscopic factor 𝐶 2 𝑆 represents the probability that the nucleons involved in the
reaction will arrange themselves in a configuration that corresponds to the final compound state.
This value can often be measured via single-nucleon transfer reaction experiments, but many times
we must rely on shell-model calculations of 𝐶 2 𝑆, which have their own associated uncertainties.
The 6159-keV Level
Ref. [104] reports a spectroscopic factor 𝐶 2 𝑆 = 0.036(13) for this 𝐽 = 7/2 (−) state. Because this
resonance has such an extremely low resonance energy (𝐸𝑟 = 28 keV) and a relatively high angular
momentum transfer, it does not contribute substantially to the overall rate at peak nova temperatures.
However, for completeness, we calculated this state’s proton partial width to determine its resonance
strength and found that it could be potentially dominant at low temperatures (𝑇 < 0.02 GK).
Any resonances below this excitation energy are excluded from the calculation because their
contributions are assumed to be negligible.
The 6255-keV Level
This  31 S excited is known to be an ℓ = 0 resonance state with spin-parity 𝐽 𝜋 = 1/2+ and a
resonance energy of 124 keV. Ref. [104] reports an upper limit on the spectroscopic factor for this
resonance 𝐶 2 𝑆 ≤ 0.19 and references Brown et al., whose shell-model calculations predict a value
of 1.7 × 10−3 [158]. While this resonance lies just below the Gamow window at 𝑇 = 0.4 GK,
this relatively weak resonance could contribute to the total rate at lower temperatures (𝑇 < 0.15
GK). For this reason, we include it in our total thermonuclear rate calculation, adopting Brown’s
theoretical resonance strength 𝜔𝛾 = 9.5 × 10−12 , with a factor of two uncertainty.
                                                  103


The 6279-keV Level
The nuclear level at 6279 keV in 31 S was first definitively identified as the IAS of the ground state in
31 Cl in Ref. [117]. Thus, it is strongly populated by 31 Cl 𝛽+ decay, but because this state has isospin
𝑇 = 3/2, it should not contribute to the total 30 P(𝑝, 𝛾)31 S reaction rate due to isospin conservation.
However, Ref. [121] provides evidence of isospin mixing between the IAS and the 6390-keV level.
Theory predicts the energies of these states should be higher than observed, but by shifting down the
excitation energies such that the difference between the levels agrees with experiment, we can use
the USDC Hamiltonian to calculate the spectroscopic factor 𝐶 2 𝑆 = 0.0036+0.0036  −0.0018 for the 6279-keV
state. Using Equation 4.22, we can compute the proton partial width Γ𝑝 and thus constrain the
resonance strength.
The 6327-keV Level
In the case of this excited state, calculating the resonance strength is straightforward. The spin-parity
is known to be 𝐽 𝜋 = 3/2− [111]. Ref. [104] reports a spectroscopic factor of 𝐶 2 𝑆 = 0.023(12).
Thus, we can simply calculate this resonance’s contribution to the total thermonuclear rate using
our analytical approximation for the proton decay partial width.
The 6357-keV Level
The 𝛾-ray angular distribution data suggests this state has spin 𝐽 = 5/2 [111]. The fact that this state
is not populated in 𝛽 decay indicates negative parity [121], so we adopt 𝐽 𝜋 = 5/2 (−) . Ref. [104]
only reports an upper limit on the level’s spectroscopic factor and thus on its resonance strength.
For this reason, we exclude it from our total thermonuclear rate calculation.
The 6377-keV Level
Similarly to the 6327-keV level, the spin and parity of resonance are known to be 𝐽 𝜋 = 9/2− [111].
Its spectroscopic factor was measured to be 𝐶 2 𝑆 = 0.051(17) [104]. We use this information to
                                                     104


calculate a resonance strength.
The 6390-keV Level
The importance of this level has been discussed at length previously and is the central topic of this
dissertation. It is noteworthy that Ref. [104] only reports an upper limit on the spectroscopic factor
for this state, demonstrating the unique sensitivity of our measurement. Our newly constrained
resonance strength is ≈ 3 times larger than the previous theoretical evaluation provided in Ref.
[121].
The 6393-keV Level
The 6393-keV level is another relatively simple case because its spectroscopic factor has been
measured 𝐶 2 𝑆 = 0.007(3). Ref. [104] reports this as a 𝐽 𝜋 = 5/2 (+) state, implying it is an ℓ = 2
resonance. We calculate its contribution to the total rate.
The 6394-keV Level
Ref. [104] only reports an upper limit on this state’s spectroscopic factor 𝐶 2 𝑆 ≤ 0.002, which is
already quite small. This high-spin 𝐽 𝜋 = 11/2+ state [111] is unlikely to contribute to the total
thermonuclear rate, so we exclude it from our total rate calculation.
The 6402-keV Level
Evidence of the existence of a 6402-keV excited state from a 32 S(𝑑, 𝑡) 31 S reaction experiment was
reported in Ref. [103], which tentatively assigns this state a spin of 𝐽 = (7/2). Beyond this, there
is little experimental information on this candidate state in the literature, and it may not even exist.
For this reason, we exclude it from total rate calculation.
                                                 105


The 6542-keV Level
The spin-parity assignment of the 6542-keV level is ambiguous, but Kankainen et al. adopts
𝐽 𝜋 = 7/2+ from Ref. [107]. Deuteron angular distribution data is fit well with ℓ = 4, 5, 6 transfers,
and thus, the state is potentially consistent with 𝐽 𝜋 = (7/2+ , 9/2− , 11/2+ ) [99]. Ref. [104] sets an
upper limit on the spectroscopic factor and resonance strength. While this resonance is likely not
important for ONe nova nucleonsynthesis, it could play a role at higher temperatures.
The 6583-keV Level
This resonance has a tentative spin-parity assignment 𝐽 𝜋 = (5/2, 7/2) − [111]. Ref. [104] reports
an upper limit on the spectroscopic factor and assumes ℓ = 3 but does not include an upper limit
on the resonance strength. Because of the large angular momentum transfer and position in the
Gamow window, this resonance is unlikely to contribute substantially to the total thermonuclear
rate. For theses reasons, we ignore proton capture on this level.
The 6636-keV Level
Most literature for this state assigns a spin-parity of 𝐽 𝜋 = 9/2− . This is consistent with deuteron
angular distribution data from a 32 S( 𝑝, 𝑑) 31 S reaction experiment, which are fit reasonably well with
𝐽 𝜋 = 7/2+ , 9/2± , 11/2+ assignments [99]. No spectroscopic information is found in literature other
than the arbitrary 𝐶 2 𝑆 = 0.02 according to Ref. [105]. Because of the large angular momentum
transfer and the unknown spectroscopic factor, we choose to neglect this resonance.
The 6720-keV Level
This level is tentatively a 𝐽 = (5/2) state. Due to a lack of other experimental information on this
level, we refer to Ref. [158] for a theoretical calculation of the resonance strength. Uncertainties
associated with this resonance strength are assumed to be a factor of two. Due to its relatively high
                                                     106


energy, this resonance likely does not contribute substantially to the 30 P( 𝑝, 𝛾) 31 S rate at peak nova
temperatures. However, it could play a role at higher temperatures.
The 6749-keV Level
As with the previous resonance, a spectroscopic factor has not been measured, and its high excitation
energy suggests it does not play an important role in the reaction of interest at lower temperatures.
Still, it is a 𝐽 𝜋 = 3/2+ state, making the ℓ = 0 potentially dominant at higher energies 𝑇 > 0.4 GK.
Again, we refer to Ref. [158] for a theoretical evaluation. All known states above this level fall well
beyond the Gamow window for this reaction at nova temperatures and are assumed to be irrelevant.
Total Rate Summed Over All Resonances
A new thermonuclear rate for       30 P(𝑝, 𝛾)31 S was computed using Equation 2.18. The resonances’
properties, including their adopted strengths for this calculation, are tabulated in 4.1. The total rate
and its contributions from each individual resonance are plotted in Figure 4.18; this excludes the
very weak, low-energy 28-keV resonance whose contribution to the total rate is not visible on the
scale of this plot. Figure 4.18 clearly shows that the 260-keV, 𝐽 𝜋 = 3/2+ resonance at the heart of
this work dominates this reaction rate at temperatures 𝑇 > 0.15 GK.
Hauser-Feshbach Statistical Model
In the case of most of the nuclear reactions which are relevant for understanding stellar nucle-
osynthesis, insufficient experimental information prevents the total thermonuclear rate from being
calculated on the basis of individual resonances alone. For this reason, the Hauser-Feshbach
statistical model is often employed as an approximation method, and to date, most nova model
simulations have relied on it to evaluate the 30 P(𝑝, 𝛾)31 S rate. Originally developed in 1952 for cal-
culating the total and differential cross sections of inelastic neutron scattering, the model assumes
that the compound nucleus in question is sufficiently excited that the nuclear levels can be treated
                                                    107


              𝐸 𝑥 (keV)      𝐸𝑟 (keV)      𝐽𝜋             𝐶2𝑆                   𝜔𝛾 (eV)
              6158.5(5)       27.9(6)    7/2 (−)    0.0036(13) [104]        1.1(4) × 10−33
              6255.3(5)      124.7(6)     1/2+        ≤ 0.19 [104]        9.5 × 10−12 [158]
              6279.0(6)      148.4(6)     3/2+        0.0036+0.0036             +2.1 × 10−9
                                                                            2.1−1.1
                                                             −0.0018
              6327.0(5)      196.4(6)     3/2−       0.023(12) [104]        5.1(27) × 10−7
              6357.3(2)      226.7(3)    5/2 (−)      ≤ 0.011 [104]       ≤ 1.4 × 10−6 [104]
              6376.9(4)      246.3(5)     9/2−       0.051(17) [104]        6.1(21) × 10−8
              6390.2(7)      259.6(7)     3/2+          0.016+16
                                                              −8            8.0(48) × 10−5
              6392.5(2)      261.9(3)    5/2 (+)     0.007(3) [104]         5.8(25) × 10−7
              6394.2(2)      263.6(3)    11/2+        ≤ 0.002 [104]                —
               6402(2)      271.4(20)      —               —                       —
              6541.9(4)      411.3(5)     7/2+    ≤ 5.9 × 10−3 [104]      ≤ 1.7 × 10−4 [104]
             6583.1(20)     452.5(20)    (7/2)        ≤ 0.007 [104]                —
              6636.1(7)      505.5(7)     9/2−             —                       —
                6720           589       (5/2)         0.081 [158]         0.072+0.072
                                                                                 0.036
                                                                                        [158]
                6749           618        3/2+      4.5 × 10−3 [158]          0.2+0.2
                                                                                 −0.1 [158]
Table 4.1: All resonances near the Gamow window for 30 P(𝑝, 𝛾)31 S at nova temperatures. We
adopt all measured values reported in Ref. [104] whenever possible. In the case of insufficient
experimental information, we appeal to theoretical evaluation, excluding only resonances for
which spectroscopic factors are unknown and likely do not contribute substantially to the total rate
due to a large centrifugal barrier.
statistically [159]. Using input parameters such as level density and transmission coefficients, a
cross section can be calculated for many resonances and averaged over the whole energy region.
    While this method is usually reliable within a factor of 2 − 3 for proton capture reactions, this is
only true if the density of states is high enough such that the level structure can be approximated as
a continuum. The largest uncertainties associated with this statistical model arise from the nuclear
level density and transmission coefficients. For many cases, especially near shell closures and the
drip lines, radiative capture into narrow, isolated resonances must be considered individually. The
30 P(𝑝, 𝛾)31 S reaction is expected to be near the edge of applicability for the Hauser-Feshbach model
                                                   108


Figure 4.18: Individual resonances contributions and their sum total thermonuclear rate for the
30 P(𝑝, 𝛾)31 S over peak nova temperatures.
because the level density within the Gamow window at nova temperatures is relatively low.
    Using the non-smoker code [160], Rauscher and Thielemann report their calculations of a
variety of reaction types and their inverse reactions for target nuclei with 10 ≥ 𝑍 ≥ 83 performed
over a grid of 24 temperatures 𝑇 = 0.1 − 10.0 GK [161]. For easy evaluation of any astrophysical
rate within a given temperature range, the results of the formal theoretical calculations were fit with
the highly parameterized numerical formula
          𝑁 𝐴 ⟨𝜎𝑣⟩ = exp(𝑎 0 + 𝑎 1𝑇9−1 + 𝑎 2𝑇9−1/3 + 𝑎 3𝑇91/3 + 𝑎 4𝑇9 + 𝑎 5𝑇95/3 + 𝑎 6 ln 𝑇9 ),    (4.25)
where the free parameters 𝑎 0 − 𝑎 6 are tabulated for all reactions of interest and 𝑇9 is the temperature
in GK units. Ref. [161] also tabulates partition functions for all nuclei provided, in order to
account for the effect of nuclear reactions involving radiative capture to excited states on the total
thermonuclear rate. In the case of 30 P, this effect only becomes relevant at 𝑇 > 1 GK, well above
peak nova temperatures. We used the functional form provided in Equation 4.25 to evaluate the
thermonuclear rate of 30 P(𝑝, 𝛾)31 S across peak nova temperatures and compared our experimentally
                                                   109


Figure 4.19: Ratio of thermonuclear rates for the 30 P(𝑝, 𝛾)31 S reaction comparing the narrow
resonances model to the Hauser-Feshbach statistical method. The solid curve represents the
recommended central value of the narrow resonance model, while its upper and lower limits are
plotted as dashed curves.
constrained resonant reaction calculation to the Hauser-Feshbach statistical method. Figure 4.19
shows the ratio between two thermonuclear rate calculations, where the numerator is calculated
from the resonance strengths of individual 31 S levels, and the denominator is the evaluated using
Equation 4.25. There is relatively good agreement between the two models to within about a
factor of 2 for the case of this radiative proton capture reaction at nova temperatures. However, as
Figure 4.18 shows, the 𝐽 𝜋 = 3/2+ , 260-keV resonance is by far the largest contribution to the total
30 P(𝑝, 𝛾)31 S rate, and the Hauser-Feshbach method should not be considered a generally reliable
approximation for thermonuclear reactions dominated by a single resonance.
4.3   Astrophysical Impact
Hydrodynamic Nova Simulations
In collaboration with Prof. Jordi José from the Departament de Física Universitat Politècnica de
Catalunya and the Institut d’Estudis Espacials de Catalunya, simulations of classical nova explosions
                                                  110


were performed to investigate the effect of the newly constrained 30 P(𝑝, 𝛾)31 S rate on the nuclear
abundances of nova ejecta. The one-dimensional, hydrodynamic code shiva was developed to
simulate nova outbursts, starting from the onset of accretion up to the mass ejection stage [47; 162].
The parallelized computer program program solves a set of differential equations for conservation
of mass, momentum, and energy, both radiative and convective transport equations, as well as the
definition of Lagrangian velocity. The hydrodynamics model has been coupled to a network of 370
nuclear reactions involving 100 nuclear species from 1 H to 40 Ca. The chemical composition of the
envelope and the underlying white dwarf mass are among the most impactful input parameters in
the prediction of nova nucleosynthesis [52].
    Modeling the chemical composition of the envelope is a complicated problem and requires
further investigation. Multi-dimensional simulations have now been employed to study the mixing
between the H-rich envelope with the dense outer layers of the underlying white dwarf [91; 92],
but due to computational limits, the state-of-the-art, fully hydrodynamic simulations of explosive
nucleosynthesis in classical novae are still one-dimensional. A much more detailed discussion of
modern computational hydrodynamics for astrophysical simulations can be found in Ref. [163].
The most recent shiva calculations using our new recommended rate for proton capture on             30 P
assumed an a priori 50-50 mixing between the H-rich envelope and ONe white dwarf material.
This 50% value is based on observations of nova ejecta and is meant to mimic results of the complex
mixing process that should occur near the stellar surface. It reflects the mean composition of the
ejecta in ONe novae, which is characterized by a “metal” content of around 𝑍 = 0.5, implying the
ejected material is composed of half H/He and half heavier elements.
    In addition, because of the relatively high mass number (𝐴 > 19) for the seed nuclei necessary to
activate 30 P(𝑝, 𝛾)31 S, the reaction is only expected to be relevant for the heaviest ONe white dwarfs.
The nuclear abundances in nova ejecta were predicted for a 1.35-𝑀⊙ ONe white dwarf using the
upper and lower limits on the constrained reaction rate in addition to the central, recommended
value for the thermonuclear rate. Partial results from these simulations are shown in Table 4.2. The
chemical elements for which abundances are provided here were chosen because of their relevance
                                                    111


to the aforementioned astrophysical motivation for this research. Namely, this includes identifying
the origins of certain presolar grains based on their isotopic ratios as well as the calibration of
nuclear thermometers for classical novae.
Isotopic Ratios in Candidate Nova Grains
For reference, we consider the isotopic ratios of SiC and graphite (C) grains with an inferred nova
origin tabulated in Ref. [60]. Figure 4.20 plots the ratios 12 C:13 C and 14 N:15 N observed in grain
data as well as the results from our simulations. The latest computational values for C and N
ratios are in agreement with previous 1.35-𝑀⊙ ONe nova models for 50-75% mixing fractions. At
12 C:13 C = 2.19+0.20
                −0.06
                      , our calculations slightly underestimate measurements of the C isotopic ratios
in all candidate nova grains. Again for N, our prediction of 14 N:15 N = 0.547+0.003
                                                                                   −0.029 is in agreement
with previous evaluations of classical nova explosions under the same conditions. However, our
theoretical results underpredict the observed N ratios by at least an order of magnitude in all grains
hypothesized to be of nova origin.
    With respect to Si isotopic ratios, as shown in Figure 4.21, we are overpredicting 29 Si and 30 Si
abundances by at least an order of magnitude for all samples in this unidentified class of presolar
grains. Our results overlap with previous uncertainty bands for both 𝛿(29 Si:28 Si) and 𝛿(30 Si:28 Si)
predictions, which are given in units of permille (‰) deviation from Solar isotopic abundances:
                              29,30   29,30   29,30            
                                    Si           Si          Si
                          𝛿 28         =     28 Si
                                                     ÷ 28          − 1 × 1000.                      (4.26)
                                   Si                       Si ⊙
The size of our error bars due to nuclear uncertainties are significantly reduced in comparison to past
predictions as a result of our experimental constraints on the 30 P(𝑝, 𝛾)31 S rate. Previous simulations
yielded near-Solar isotopic ratios for 29 Si:28 Si, with either excesses or even slight deficits as likely
possibilities. In our ONe nova ejecta, we strictly predict excesses of 𝛿(29 Si:28 Si)= 471(8), which
actually brings us out of agreement with observational data on 29 Si abundances in candidate nova
grains.
                                                    112


Figure 4.20: C and N isotopic ratios, comparing grain data to previous ONe nova models for
different white dwarf (WD) masses as well as the most recent shiva calculations using both the
nominal STARLIB rate for 30 P(𝑝, 𝛾)31 S and our experimentally constrained rate. All simulations
assume an intial mixing fraction of 50% between the H-rich envelope and WD material.
    Similarly for 30 Si, which is among the nuclides most dramatically affected by the rate of proton
capture on     30 P, previous estimates on its prevalence in nova ejecta had massive uncertainties.
Abundances calculated using the lower limit of the 30 P(𝑝, 𝛾)31 S rate were a factor of ≈ 6 higher than
results produced from the nominal rate, and at the theoretical upper limit of this rate, simulations
even predicted slight deficits relative to Solar [60]. At 𝛿(30 Si:28 Si)=1.14+0.93−0.35
                                                                                        × 104 , we can
conclusively say that ONe novae involving the most massive white dwarfs should produce enhanced
30 Si:28 Si ratios in their ejecta when compared to Solar abundances. While this result qualitatively
agrees with all candidate nova grains, whose 30 Si:28 Si ratios exceed that of the Solar System’s, we
overpredict this deviation in each case by at least a factor of 10.
    To reconcile this discrepancy, some have invoked an ad hoc mixing of the processed nucle-
osynthetic material with some unspecified source of near-Solar material which dilutes the 29,30 Si
excesses before the grains are able to condense. It is not clear if this can be explained by the
                                                  113


Figure 4.21: Si isotopic abundances in plotted in permille deviations from Solar ratios for
29 Si:28 Si and 30 Si:28 Si.
interaction between the ejected shells and the surrounding accretion [66]. It is also possible that
ignorance of relevant details in modeling explosive nucleosynthesis, such as the mixing of the
accreted material with the outer layers of the white dwarf, prevents an accurate prediction of ejecta
composition. Still, others have suggested that these grains did not originate in classical novae at
all but came from supernovae instead [164]. Based on our findings and the current state of the
literature, we cannot draw any definitive conclusions about the cosmic origins of the presolar grains
in question. Nevertheless, from the perspective of experimental nuclear physics, we have done our
part by substantially constraining the reaction rate uncertainties related to this issue surrounding Si
isotopic ratios.
Elemental Abundances for Nuclear Thermometers
As discussed in Chapter 2, the relative amount of different chemical elements in the ejecta shells
from classical nova explosions can be used to constrain the peak temperatures achieved during the
                                                 114


thermonuclear runaway. In contrast to the isotopic ratios discussed in the previous section, which
compare the relative number of atomic nuclei, the elemental abundance ratios presented in Ref.
[90] are given in terms of their mass fraction relative to H. Thus, all ratios mentioned here should
be understood as comparing their proportional contributions to the total mass of the ejecta material.
    Among the various abundance ratios, N:O, N:Al, O:S, S:Al, O:Na, Na:Al, O:P, and P:Al
have been identified as the most useful thermometers, due to the fact that they all exhibit a steep,
monotonic dependence on peak nova temperature. The mass fraction ratios N:O, N:Al, O:Na, and
Na:Al are robust to nuclear uncertainties in the sense that they are not particularly sensitive to the
effects of unconstrained reaction rates. However, the ratios O:S, S:Al, O:P, and P:Al reveal a strong
dependence on both temperature and the thermonuclear rate of 30 P(𝑝, 𝛾)31 S. The ratio between the
upper and lower limits on an astronomical value is sometimes called the variation. For the N and
Na ratios, the variation of final elemental abundance ratios over the range of nova models explored
in Ref. [90] spans factors of 5.59 − 13.4, but for S and P ratios, these factors in variation are
216 − 541. Furthermore, the effect of varying the 30 P(𝑝, 𝛾)31 S rate between its previous upper and
lower limits resulted in factors of 3.36 – 6.44 in the uncertainty of predicted S and P ratios; our
recently constrained reaction rate reduces these uncertainties.
    The peak temperature achieved during thermonuclear runaway is highly correlated with the
mass of the underlying white dwarf. Classical novae involving a 1.35-𝑀⊙ ONe white dwarf are
expected to reach peak temperatures of 𝑇 = 0.312 GK, according to our model. Thus, the most
recent simulation results can be compared to the predicted mass fraction ratios in Ref. [90] for the
hottest peak temperatures. Performing more nova simulations that adopt the new recommended
rate for 30 P(𝑝, 𝛾)31 S with smaller white dwarf masses may reduce uncertainties in the expected
elemental abundances at lower temperatures. However, any reduction in the variation of predicted
abundance ratios at lower temperatures will likely not be as substantial as this effect on modeling
the hottest novae, since proton capture on 30 P is most important for nucleosynthesis involving very
massive ONe white dwarfs.
    By summing the ejected mass fractions in Table 4.2 for all isotopes of a given element, we arrive
                                                  115


                                                                 +0.9            +0.5                 +0.1
at the elemental abundance ratios O:S = 0.86+0.12
                                              −0.04 , S:Al = 15.7−2.2 , O:P = 4.9−0.2 , and P:Al = 2.7−0.3 .
These findings are in good agreement with past abundance predictions and have smaller error bars.
Overlaying our simulation results on the original plot from Downen et al., we can clearly see in
Figure 4.22 that the uncertainty in the predicted mass fractions has been reduced [90]. For some
candidate thermometers, fully hydrodynamic simulations either over- or underpredict elemental
abundance ratios when compared to post-processing calculations. The latter have the advantage
of being less computationally expensive since they merely adopt the temperature-density profiles
produced from hydrodynamic models and use this as input to execute a nuclear reaction network
code for each burning zone in the nova event, but ultimately, fully hydrodynamic simulations should
provide more realistic results.
     By visual inspection at 𝑇peak = 0.31 GK, we estimate that the variation in the four ratios, as
predicted by the post-processing calculations, were factors of about 2 − 3 [90]. Meanwhile, our
expected variation due to the rate of 30 P(𝑝, 𝛾)31 S is within factors of 1.1 − 1.2 for the same peak
temperature across all nova thermometers. As a consistency check and for the sake of completeness,
we note that the mass fractions O:Na and Na:Al are also in good agreement with the hydrodynamic
simulation results. However, the variations in these abundances due to nuclear uncertainties alone,
as evaluated by the post-processing calculations, were already quite small and are unaffected when
varying the 30 P(𝑝, 𝛾)31 S rate.
     Unfortunately, neither Na nor P have yet been observed in ONe nova ejecta. Thus, the otherwise
prime candidates O:Na, Na:Al, O:P, and P:Al are currently useless as thermometers. However, the
presence of N, O, Al, and S have all been identified in the shells of ONe novae. In fact, the N:O and
O:S abundance ratios observed from the nova event V838 Herculis suggest its peak temperature
range is 𝑇peak = 0.30 − 0.31 GK, which corresponds to a white dwarf mass of 𝑀WD = 1.34 − 1.35
𝑀⊙ [90]. Therefore, we are able to directly compare our predicted abundances to the observed
mass fractions of the processed nuclear material ejected by V838 Herculis. As shown in Figure
4.23, our results are in agreement with post-processing calculations. Previous simulations slightly
overestimate the expected ratios N:Al and N:O for V838 Herculis, and since our constraints on
                                                  116


Figure 4.22: Mass fraction ratios plotted as a function of peak nova temperature for all candidate
thermometers. The solid black and red lines represent predicted abundance ratios from
post-processing calculations and fully hydrodynamic simulations, respectively. The dashed lines
form error bands on these predictions as a result of independently varying the rate of relevant
nuclear reactions within their uncertainties. Green markers correspond to our simulation results,
whose error bars are approximately the size of the data point. Left column: abundance ratios for
elements observed in ONe nova shells. Right column: abundance ratios for elements that have not
yet been observed in the ejecta of ONe novae. Figure credit: Ref. [90].
                                                117


30 P(𝑝, 𝛾)31 S do not affect these abundances, the latest shiva predictions are unable to resolve the
discrepancy. However, we do agree with observations of S:Al and O:S from V838 Herculis to the
level of one standard deviation and with substantially smaller uncertainties.
                                                 118


Figure 4.23: Mass fraction ratios comparing observational nova data to theoretical predictions.
The black lines represent abundances and their uncertainties according to post-processing
calculations, while the magenta data point corresponds to our latest hydrodynamic simulation
results using the constrained 30 P(𝑝, 𝛾)31 S rate. Top panel: Al and O abundances plotted relative to
N, where peak nova temperature increases from left to right. Bottom panel: Al and O abundances
plotted relative to S, where peak nova temperature increases from right to left. Figure credit: Ref.
[90].
                                                   119


                                                  Ejected mass fraction
                 Nuclide    ( 𝐴 𝑋)  Lower limit Recommended rate           Upper limit
                       9C          1.59 × 10−31       1.96 × 10−28        5.49 × 10−32
                      11 C         5.64 × 10−10       6.14 × 10−10        5.84 × 10−10
                      12 C         2.22 × 10−2         2.22 × 10−2         2.20 × 10−2
                      13 C         1.13 × 10−2         1.01 × 10−2         1.09 × 10−2
                      12 N         6.39 × 10−33       1.88 × 10−30       4.27 × 10−33
                      13 N         4.33 × 10−3         4.17 × 10−3         4.66 × 10−3
                      14 N         5.49 × 10−2         5.32 × 10−2         5.46 × 10−2
                      15 N         1.07 × 10−1         1.10 × 10−1         1.07 × 10−1
                      13 O         1.30 × 10−34       1.83 × 10−34        1.30 × 10−34
                      14 O         1.13 × 10−6         3.70 × 10−6         2.12 × 10−6
                      15 O         1.48 × 10−4         2.94 × 10−4         2.12 × 10−4
                      16 O         6.06 × 10−3         5.95 × 10−3         6.00 × 10−3
                      17 O         3.98 × 10−2         4.07 × 10−2         4.09 × 10−2
                      18 O         6.63 × 10−6         6.45 × 10−6         6.67 × 10−6
                      23 Al        2.30 × 10−34       4.90 × 10−30        2.30 × 10−34
                      24 Al        2.40 × 10−34       2.08 × 10−25        2.40 × 10−34
                      25 Al        2.10 × 10−31       5.36 × 10−18        7.20 × 10−32
                      26 Al        4.56 × 10−4         4.57 × 10−4         4.53 × 10−4
                      27 Al        3.01 × 10−3         3.02 × 10−3         3.02 × 10−3
                      25 Si        2.50 × 10−34       3.59 × 10−33        2.50 × 10−34
                      26 Si        2.60 × 10−34       9.44 × 10−25        2.60 × 10−34
                      27 Si        8.22 × 10−28       1.47 × 10−18        1.48 × 10−28
                      28 Si        3.04 × 10−2         3.10 × 10−2         3.11 × 10−2
                      29 Si        2.34 × 10−3         2.40 × 10−3         2.42 × 10−3
                      30 Si        2.37 × 10−2         1.38 × 10−2         9.89 × 10−3
                       29 S        2.90 × 10−34       6.53 × 10−32        2.90 × 10−34
                       30 S        3.00 × 10−34       6.00 × 10−27        3.00 × 10−34
                       31 S        1.96 × 10−33       4.59 × 10−21        1.16 × 10−33
                       32 S        4.57 × 10−2         5.33 × 10−2         5.63 × 10−2
                       33 S        6.67 × 10−4         7.98 × 10−4         8.52 × 10−4
                       34 S        3.00 × 10−4         3.60 × 10−4         3.86 × 10−4
                       35 S        3.50 × 10−34       3.50 × 10−34        3.50 × 10−34
                       36 S        3.60 × 10−34       3.60 × 10−34        3.60 × 10−34
Table 4.2: Predicted mass fractions for C, N, O, Al, Si, and S relative to the total mass ejected
(9.043 × 1027 g) during a classical nova explosion on a 1.35-𝑀⊙ ONe white dwarf. Three
different rates for the 30 P(𝑝, 𝛾)31 S reaction were used to quantify the effect of this nuclear
uncertainty on isotopic and elemental abundances.
                                                   120


                                              CHAPTER 5
                              THE 31 CL(𝛽𝑝𝛾)30 P DECAY SCHEME
Having achieved the primary scientific motivation of this research by measuring the proton branch-
ing ratio of the key 260-keV, 𝐽 𝜋 = 3/2+ resonance and interpreting its astrophysical impact, we may
now turn our attention to the higher-energy protons in our decay spectrum. Essentially all previ-
ous measurements of 31 Cl 𝛽-delayed proton decay assumed that these charged-particle emissions
populate the ground state of    30 P. However, using the 𝛾-tagging capabilities of GADGET on the
high-statistics dataset acquired during NSCL experiment 17024, we demonstrate conclusively that
many of these transitions proceed from proton-unbound states in 31 S to excited states of 30 P. In ad-
dition, we have identified several new 𝛽-delayed proton decay transitions and present a preliminary
31 Cl(𝛽𝑝𝛾)30 P  decay scheme for the first time.
5.1   Populating 30 P Excited States
Evidence of level population above the       30 P ground state was first realized, and is perhaps most
obviously illustrated, upon plotting the 𝛾 ray spectrum observed in coincidence with protons in the
region spanning 1170 − 1280 keV in our energy spectrum. Figure 5.1 clearly shows that emitted
protons are strongly populating all four of the lowest-lying excited states in 30 P. For reference, the
𝛾 decay scheme for these low-lying levels is shown in Figure 5.2. By placing coincidence gates on
these observed 𝛾 ray transitions, we have been able to identify previously unobserved proton peaks
that, due to their weak intensities, are otherwise obscured in the cumulative singles spectrum.
Background-Subtracted Coincidence Spectra
At least one of the 𝛾 ray deexcitations for each of the four lowest excited states in 30 P can be observed
in the proton-gated 𝛾 singles spectrum. By placing a coincidence gate on all 𝛾 events that fall
within the energy region that contains the photopeak of a 30 P transition and plotting the resulting
proton spectrum, we can clearly see features emerge that are distinct from the qualitative structure
                                                    121


Figure 5.1: Proton-gated 𝛾 ray spectra. The legend indicates the energy range over which 𝛾
coincidences are sampled for the 1.2-MeV region (blue) and the background region (red), whose
statistics have been scaled by the ratio of counts in the proton singles spectrum within the two
gating regions. Photopeaks corresponding to 30 P transitions are labeled with their 𝛾 energy in keV
units. Labels of the same color indicate 𝛾 transitions from the same excited state.
of the cumulative proton spectrum. In order to determine the decay energies and intensities of
these new 𝛽-delayed protons, it is necessary to remove the background contribution from strong
𝛽-delayed proton decays which are in random coincidence with 𝛾 rays of all energies.
     The most prominent feature in the    31 Cl 𝛽-delayed proton spectrum is the massive peak at 1
MeV. As discussed in Chapter 4, this strong proton branch has been observed in previous 𝛽+ decay
experiments and was assumed to populate the       30 P ground state. Furthermore, the corresponding
31 S level from which this proton is emitted has been observed in single-nucleon transfer and charge-
exchange reaction experiments as well and has an excitation energy of 7156 keV [101; 105; 102; 99].
By gating on this proton peak in our data and analyzing the coincident 𝛾 spectrum, we were able to
determine that the 1-MeV 𝛽-delayed proton decay is, in fact, a ground-state transition.
     We can represent the contribution of random coincidences to the background in our 𝛾-gated
proton spectra as a scaled-down version of the cumulative 𝛽-delayed proton spectrum in Figure
                                                  122


Figure 5.2: Level scheme for the first four excited states in 30 P. Level energy labels (black) are
given in units of keV, and transition labels (colored) represent relative intensities. All of these
transitions are observed in coincidence with 𝛽-delayed proton decay events with energies between
1.1 and 1.2 MeV.
                                                 123


Figure 5.3: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
677-keV 𝛾 rays emitted from the first excited state. The four peaks were fit with EMG
distributions, and the background is modeled as linear. The feature near 1 MeV shows large
statistical fluctuations in each bin due to the subtraction of the largest proton peak in the spectrum;
the integral over this energy range is consistent with zero.
4.2. The scaling factor is simply the ratio between the number of 1-MeV events in 𝛾-gated proton
spectrum and the number of 1-MeV protons in the total, combined spectrum. Taking the difference
bin-by-bin between the scaled “background” spectrum and the proton spectrum that results from
gating on the 𝛾 ray of interest, we can produce a background-subtracted proton spectrum for each
30 P transition. These spectra are shown in Figures 5.3, 5.4, 5.5, 5.6, 5.7, and 5.8 for        31 Cl 𝛽-
delayed protons detected in coincidence with 677-, 709-, 1454-, 746-, 1265- and 1973-keV, 𝛾 rays,
respectively.
     This analysis procedure provides a very clean background-subtraction method for eliminating
accidental coincidences between protons and random 𝛾 rays. However, there are also a number of
“real” coincidences that can also contribute to the background of our 𝛾-gated proton spectra. In
our 𝛾 ray spectra, we can think of the photopeak as corresponding to 𝛾 events that deposit their full
energy in the detector. However, many more 𝛾 rays will scatter off the HPGe crystal, depositing only
a fraction of their energy in SeGA. This results in a substantial Compton background associated
                                                  124


Figure 5.4: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
709-keV 𝛾 decays from the second 30 P excited state to the ground state. The nine peaks were fit
with EMG distributions, and the background is assumed to be constant. Several background
models were applied to fits of all spectra in order to quantify systematic uncertainty. The proton
peak above 2.4 MeV was fit using different constraints over a smaller energy region because it
appears to be intrinsically broad.
with each 𝛾 ray that spans all energies up to the photopeak. Thus, the contribution from Compton
scattering decreases at higher energies since there are fewer 𝛾 rays whose partial energies are high
enough to populate this region of the spectrum. When placing a coincidence gate on 𝛾 ray events
within the photopeak of a known transition, we necessarily include Compton-scattered events from
higher-energy transitions. Those higher-energy 𝛾 transitions that precede the coincidence 𝛾 ray of
interest in the decay scheme are real coincidences, in the sense that protons could be populating
more highly excited states before 𝛾 decaying to the lower levels on which we are applying the
coincidence gate.
    To account for these real coincidences, we set another coincidence gate on a “featureless” back-
ground region of the spectrum. The energy range from which we are sampling proton coincidences
with background 𝛾 events should be higher in energy than the the photopeak of interest. This
ensures we avoid the Compton tail of the 𝛾 decay on which we are gating. Then, we scale down
                                                 125


Figure 5.5: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
1454-keV 𝛾 decays to the 30 P ground state from the third excited state. The six peaks were fit with
EMG distributions, and the background is modeled as linear.
Figure 5.6: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
746-keV 𝛾 decays from the third excited state in 30 P at 1454 keV to the second excited state at 709
keV. This 𝛾 branch is much weaker than the 1454-keV transition to the ground state from the same
level, and the binning reflects the low statistics. Again, the six peaks were fit with EMG
distributions, and the background is constrained to be constant.
                                                   126


Figure 5.7: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
1265-keV 𝛾 decays from the fourth excited state in 30 P at 1973 keV to the second excited state at
709 keV. This is the stronger of the two 𝛾 branches for this level. The three peaks were fit with
EMG distributions, and the background is constrained to be constant.
this background 𝛾-gated proton spectrum by the ratio between the number of counts under the
photopeak and the total number of background 𝛾 counts over their respective energy ranges. We
used these two background subtraction methods for the accidental and real coincidences to quantify
the systematic uncertainty in the number of 𝛽-delayed proton decays to excited states.
    Some of the stronger 𝛽-delayed proton decays found to be populating excited states of 30 P have
been observed before and are clearly visible in the cumulative proton spectrum. However, most of
the smaller peaks are completely obscured by the much larger peaks that correspond to ground-state
transitions. In order to accurately quantify the energy and intensity of the various 𝛽-delayed proton
decays in the singles spectrum, we need to determine how many of these weaker proton decays we
should expect to see in the entire dataset. We can calculate this from the integrals of the 𝛾-gated,
background-subtracted proton peak fits if we also know the absolute detection efficiency of SeGA
at the appropriate 𝛾 ray energies.
5.2   𝛾-Ray Detection Efficiency with SeGA
                                                  127


Figure 5.8: Background-subtracted energy spectrum for 𝛽-delayed protons in coincidence with
1973-keV 𝛾 decays from the fourth 30 P excited state to the ground state. Again, the three peaks
were fit with EMG distributions, and the background is constrained to be constant.
A detailed decay scheme for 31 Cl(𝛽𝛾)31 S is provided in Ref. [120]. After correcting some minor
arithmetic inconsistencies in the 𝛽-delayed 𝛾 intensities reported by Bennett et al., we essentially
adopted these literature values for our efficiency calibration. Using these numbers and fitting well-
separated, high-statistics photopeaks across most of our SeGA spectrum, we are able to determine
the relative detection probability of 𝛾 rays as a function of energy. In order to extract absolute
efficiencies for arbitrary 𝛾 energies, we only need to evaluate the absolute detection efficiency at
one, or ideally a few, intermediate energies. This allows us to normalize our relative efficiency
curve with respect to these fixed values.
Relative Efficiency Curve
We first identified over a dozen photopeaks corresponding to strong 𝛽-delayed 𝛾 ray emissions in
the proton-gated 𝛾 singles spectrum depicted in Figure 4.6. In order use the 𝛾 detection efficiency
to calculate the expected number of 𝛽-delayed proton decays to 30 P excited states that should appear
in the singles spectrum of Figure 4.2, we must ensure that we only consider 𝛾 events that originate
                                                 128


in the active volume of the Proton Detector when evaluating the efficiency of SeGA. This is because
GADGET can only detect        31 Cl 𝛽-delayed proton events that occur within the Proton Detector’s
active region during the 200-ms time window when the gating grid is in its transparent mode.
SeGA, on the other hand, can detect 𝛾 rays originating anywhere in the experimental vault at any
time, including from the decays of 31 Cl nuclei that do not make it into the detector chamber, 31 Cl
decays within the detector during the 300-ms beam implantation period, as well as 31 Cl(𝛽𝑝𝛾)30 P
decays that result in a proton being vetoed. Thus, for the purpose of determining 𝛽-delayed proton
intensities, the absolute 𝛾 detection efficiency must be evaluated for only the 𝛾 rays that are detected
in coincidence with Proton Detector events that survive the veto condition.
    We used a similar functional form to Equation 4.3 to fit the 𝛾 ray photopeaks in our SeGA
spectrum, but instead of utilizing an EMG distribution with a high-energy tail, as we did for
the proton peaks in order to account for the effect of 𝛽-summing, we used the same asymmetric
Gaussian function but skewed in the opposite direction [165; 166]. This skew to the left is useful
for describing the low-energy tail of the photopeak, which results from imperfect charge collection
in some regions of the detector or secondary electron and bremsstrahlung escape from the active
volume [126]. The magnitude of this skew is parameterized by the quantity 𝜆, which should be
independent of 𝛾 energy. The prominent 1248-keV photopeak in Figure 5.9 was fit using our EMG
response function, where 𝜆 was allowed to be a free parameter. Since this is one of the strongest 𝛾
decays in the spectrum that is not obscured by neighboring peaks, we adopted this value of 𝜆 for all
other peak fits. This yielded good fits as well as a consistent interpretation of the peak width and 𝛾
decay intensity.
    Based on our knowledge of 𝛾 ray spectra acquired using HPGe detectors, we assumed the width
of the photopeaks should vary smoothly with energy, such that
                                                              √
                                       𝜎(𝜇) = 𝜎(𝜇 = 0) + 𝑚 𝜎 𝜇.                                    (5.1)
First, all calibration peaks were fit in the spectrum allowing 𝜎 to be a free parameter. After plotting
all 𝜒2 -minimized values of 𝜎 as a function of 𝜇, we fit the data using equation Equation 5.1 to
                                                    129


Figure 5.9: Top panel: Fit of the second-strongest photopeak in the 31 Cl 𝛽-delayed 𝛾 spectrum at
1248 keV. The background is modeled with a quadratic polynomial over the energy region
1220 − 1280 keV. Bottom panel: Residual plot of the difference between the histogram’s bin
content and the value of the fit as a function of 𝛾 energy.
find 𝜎(𝜇 = 0) and 𝑚 𝜎 , as shown in Figure 5.10. Then, we constrained 𝜎 for each photopeak fit to
conform to this energy dependence. The difference in the peak integrals between the constrained
and unconstrained fits represented the largest source of systematic uncertainty associated with the
total number of observed counts for a given 𝛾 ray transition, ranging from < 1% for high-statistics
peaks to 14% for weaker 𝛾 transitions. The efficiencies for each photopeak were then calculated
relative to the 1248-keV peak using the relationship
                                                      𝐼𝛾 𝑁1248
                                          𝜖 𝛾 (rel) =           ,                                     (5.2)
                                                      𝑁 𝛾 𝐼1248
where 𝐼𝛾 is the adopted 𝛽-delayed 𝛾 intensity for a given        31 S excited state transition, and 𝑁 𝛾 are
the total number of counts in the photopeak; 𝐼1248 and 𝑁1248 correspond to the same values but for
the 1248-keV reference peak. Figure 5.11 plots the resulting efficiency curve where all values are
given in arbitrary units relative to 𝜖1248 (rel) = 1. The functional form of the curve typically used
to fit efficiency data can be expressed as
                                                    130


Figure 5.10: Peak width plotted with respect to peak position in the 𝛾 energy spectrum. The
𝜒2 -minimized values of the free 𝜎 parameter are fit with a function proportional to the square root
of the 𝜇 parameter.
                                                 𝑖=5
                                                 ∑︁
                                          ln 𝜖 =     𝐶𝑖 (ln 𝐸 𝛾 ) 𝑖 .                             (5.3)
                                                 𝑖=0
Absolute Efficiency Calibration Points
In certain cases, it is relatively straightforward to determine the absolute detection efficiency at a
specific 𝛾 energy. For example, in the case of       30 P shown in Figure 5.2, the fourth excited state
decays to the second excited state via the 𝛾1265 transition. The decay of the second excited state
promptly follows, resulting in 𝛾709 to be emitted. In this simple decay scheme, the emission of
𝛾1265 is always followed by the 𝛾709 transition to the ground state. Thus, if we place a coincidence
gate on all 𝛾1265 events, the number of 𝛾709 decays in coincidence must be equal to the total number
of 𝛾1265 transitions, assuming perfect efficiency. If these quantities are not equal, then, the ratio
between observed 𝛾709 − 𝛾1265 coincidences and the total number of measured 𝛾1265 events is simply
                                                   131


Figure 5.11: Relative detection efficiency in SeGA as a function of 𝛾 energy for events in
coincidence with 𝛽-delayed proton decays.
the detection efficiency for 𝛾709 . Expressed generally in terms of initial 𝑖 and final 𝑓 transitions,
the relationship is
                                Number of 𝛾 𝑓 − 𝛾𝑖 coincidences observed
                          𝜖𝑓 =                                              .                    (5.4)
                                    Total number of 𝛾𝑖 events detected
    In order to determine absolute efficiencies in this way for the  31 Cl(𝛽𝛾)31 S decay scheme, we
must identify analogous transition sequences where two 𝛾 rays are always emitted one after the
other. Of course, the first two excited states in any level scheme will be populated by many 𝛾
transitions from higher energy levels. However, not all of these transition sequences are useful
for this method. For example, the 1248-keV 𝛾 ray is emitted from the first excited state in       31 S.
According to Ref. [120], this level is populated directly by the decay of several more highly excited
states with transition energies of ranging from 946 to 5901 keV.
    However, a substantial fraction of the background under lower-energy photopeaks will be
contaminated by higher-energy transitions that also populate the 1248-keV level but whose 𝛾 rays
have undergone Compton scattering, depositing only some of their energy in SeGA. In addition,
                                                 132


Figure 5.12: A two-dimensional histogram plotting 𝛾-𝛾 coincidences between the 1248- and
5030-keV transitions.
many of the higher-energy 𝛽-delayed 𝛾 decays are quite weak and had previously been unobserved.
Some of the corresponding photopeaks are visible in our 𝛾 spectrum, but the probability of observing
both transitions independently is reduced by the product of their detection efficiencies, which are
each on the order of a few percent. This makes weak 𝛾 branches impractical for determining SeGA’s
absolute efficiency at 1248 keV. Furthermore, some photopeaks are obscured by other nearby 𝛾
ray peaks, making the integrals extracted from their fits unreliable. Ultimately, we were able to
identify three usable 𝛾 decays to the 1248-keV level with transition energies 2035, 2959, and 5030
keV; Figure 5.12 shows one example of such 𝛾-𝛾 coincidences.
    Fitting the photopeak shown in Figure 5.13 from the proton-gated 𝛾 spectrum yields the
denominator for Equation 5.4, where in this case 𝛾𝑖 is the 5030-keV transition. To determine the
numerator of this ratio, we need to determine the number of instances in which the 1248- and
5030-keV 𝛾 rays are detected together in coincidence. Just as we demonstrated previously with the
𝛾-gated, background-subtracted proton spectra, we can place a coincidence gate on all 5030-keV 𝛾
ray events over several widths of the photopeak and plot the resulting coincidence 𝛾 rays. To account
                                                 133


Figure 5.13: Fit of the 5030-keV photopeak. The integral of this peak is the total number of
5030-keV 𝛾 decays populating the 1248-keV first excited state of 31 S that are detected in
coincidence with non-vetoed Proton Detector events.
for the background, we generate another 𝛾 spectrum gated on coincident 𝛾 rays with energies above
the 5030-keV photopeak. To estimate the contribution of random, accidental coincidences to the
1248-keV peak, we scale the background-gated 𝛾 spectrum by the ratio between the number of
𝛾 events in the energy range of the photopeak gate and the number of counts in the background
gate’s range. For estimating the contribution of real, true coincidences from the Compton tails
of higher-energy transitions also populating the 1248-keV state, we scale the background-gated
𝛾 spectrum by the ratio between the number of background counts under the photopeak and the
number of 𝛾 events over the energy range of the background gate.
    Taking the difference between the photopeak-gated and the scaled-down, background-gated 𝛾
spectra results in Figure 5.14. Lastly, fitting the 1248-keV peak in this background-subtracted
coincidence spectrum allows us to calculate the absolute detection efficiency of SeGA for this 𝛾
energy. This process was repeated for both accidental and real coincidences for three different 𝛾
transitions populating the 1248-keV state in order to quantify our systematic uncertainty. Combining
this with the statistical uncertainty of our median fit value results we arrive at an efficiency of
                                                 134


Figure 5.14: Fit of the 1248-keV photopeak after background subtraction. The integral of this
peak corresponds to the number of 1248-keV 𝛾 events observed in coincidence with the 5030-keV
transition that populates the first excited state.
𝜖1248 = 0.045(5). Similarly, we applied the same analysis procedure to determine an absolute
detection efficiency at 2234 keV using transitions that feed the second excited state via the emission
of 1050-, 1852-, and 2484-keV 𝛾 rays. We found that the probability to detect a 2234-keV 𝛾 in
coincidence with a Proton Detector event in our experiment was 𝜖2234 = 0.027(6). Fixing the
position of these two data points in place, we normalized our relative efficiency curve from Figure
5.11 such that we are able to interpolate the proton-𝛾 detection efficiency of the GADGET system
spanning the energy range 0.7 − 7 MeV.
5.3   Analyzing the Cumulative Proton Spectrum
Having determined the detection efficiency of SeGA for 𝛾 energies spanning 0.7 − 7 MeV, we
may now revisit the analysis of the cumulative spectrum for the higher-energy, 𝛽-delayed proton
decays. Extracting the centroids from fits of the 𝛾-gated, background-subtracted proton peaks
allows us to determine the positions of these weak proton branches in the cumulative energy
spectrum that combines all non-vetoed Proton Detector events. Dividing the peak areas obtained
                                                   135


from the coincidence spectra fits by SeGA’s detection efficiency evaluated at the relevant 𝛾 energies,
we calculated the expected number of total proton counts in the combined spectrum from these
transitions. Upon fixing these parameters for the distributions of smaller proton peaks, we fit
the larger proton peaks in the singles spectrum, allowing their positions and intensities to be
free parameters, in order to determine their decay energies and strengths under these constraints.
This process was repeated iteratively over several energy regimes, relaxing and tightening various
fit parameters for the purpose of quantifying systematic uncertainties in our model of the data.
However, in order to convert the peak positions within the histogram as determined by our fits into
real decay energies, it was necessary for us to establish a robust energy calibration that is reliable
across the entire Proton Detector spectrum.
Energy Calibration Revisited
As discussed in Chapter 4 for the calibration of the central detector pad, we relied on extrapolating
from three data points within 700 − 1100 keV when reporting our measured energy of the low-
energy, 𝛽-delayed proton decay through the key 𝐽 𝜋 = 3/2+ resonance. We plotted the decay
energies of the calibration peaks as evaluated in literature [119] against the means of the three
Gaussian distributions used to fit the corresponding peaks for the strongest 𝛽-delayed proton decays
observed in our spectrum. Fitting these three points with a first-degree polynomial provided a
reasonable energy calibration function over a relatively small range of the spectrum and yielded
a decay energy that was consistent with the hypothesis that we observed 𝛽-delayed proton decay
through the 260-keV resonance of interest. However, this extrapolation results in large uncertainties
when evaluated at energies well above or below 1 MeV. We initially hesitated to incorporate more
data points from higher energies into this fit because we realized that this region of the spectrum
is much more complicated than had been previously assumed. However, after careful subsequent
analysis of the proton-𝛾 coincidence data for all observable 30 P transitions, we have acquired a much
more detailed understanding of the cumulative spectrum’s underlying structure, which allowed us
to produce a precise energy calibration that spans the entirety of the combined-pads spectrum.
                                                  136


     To begin, we returned to the three largest peaks in our spectrum peaks as a starting point for
our calibration. For the spectrum including data combined from all active pads, we cannot ignore
the effect of 𝛽-summing, as the peaks are clearly not symmetrical. We used EMG distributions to
fit these calibration peaks, also taking into account their small areal contributions from the much
weaker proton decays that populate excited states. Since the response function is asymmetric, we
chose to calibrate our proton energies with respect to the mode of each distribution: the location
along the x-axis of the histogram at which the peak’s fit function is at its maximum value. We
determined the decay energies of all resonances in this calibration via the relation
                                           𝐸𝑟 = 𝐸 𝑥 ( 31 S) − 𝑆 𝑝 − 𝐸 𝑥 ( 30 P),                                 (5.5)
where 𝐸 𝑥 ( 31 S) is the excitation energy of the proton-emitting level in 31 S, 𝑆 𝑝 is the proton separation
energy of 31 S, and 𝐸 𝑥 ( 30 P) is the excitation energy of the 30 P level populated by 𝛽-delayed proton
decay. 𝐸 𝑥 ( 30 P) = 0 for these first three calibration points, since they correspond to proton decays to
the ground state. Using the evaluated level energies and their assigned uncertainties from Nuclear
Structure and Decay Data (NuDat) [167] as well as 𝑆 𝑝 from the latest mass evaluation [16], we
compute 𝐸𝑟 = 806(2), 906(2), 1025(2) keV.1 Fitting these energies as a linear function of the our
distribution modes provides a well-constrained energy calibration within the region between these
points. However, in order to extend this calibration to higher energies, we would like to identify
protons within the energy range 806 − 1025 keV which decay from highly excited states in 31 S that
also emit protons of several different energies.
     Fortunately, we were able to identify one such proton transition populating the third excited state
of 30 P, whose energy falls within the region over which our calibration is well-constrained; applying
our local energy calibration yields a decay energy of 834 keV. After fitting all proton peaks in our
𝛾-gated coincidence spectra, we estimated the excitation energy of their proton-emitting levels
using Equation 5.5. In the case of the 834-keV transition, 𝐸 𝑥 ( 31 S) = 834(3) + 6130.65(24) +
     1 Excitation energies used in the proton spectrum energy calibration are based on evaluated nuclear level energies
as reported in NuDat [167] prior the most recent update in late June 2022. Any future changes to these preliminary
values will likely reflect this.
                                                           137


1454.23(2) keV = 8419(3) keV. The closest level tabulated in NuDat has an excitation energy and
associated uncertainty of 𝐸 𝑥 = 8424(3) keV. This level has reportedly been populated in several
different experiments, including 31 Cl 𝛽-delayed proton decay measurements [117; 118; 168], with
researchers reporting a range of excitation energies. Saastamoinen et al. attributes the observation
of 2.3-MeV protons, which likely correspond to a visible peak of the same energy in our spectrum,
to the decay of a proton-unbound      31 S level located at 𝐸 𝑥 = 8429(3). However, results from
experiments employing a high-resolution magnetic spectrometer for the reactions          31 P(3 He,𝑡)31 S
and  32 P(𝑝, 𝑑)31 S conclude 𝐸 𝑥 = 8418(5) [102] and 𝐸 𝑥 = 8422(2) [99], respectively. This is in
good agreement with our prediction, and we can consider our result an independent measurement
of this level’s excitation energy. Thus, we adopt the 8419-keV level for the purpose of our energy
calibration.
    Gating on the proton peak near 2.3-MeV, we do not observe any evidence of coincident 𝛾
rays from 30 P excited states, suggesting that these events indeed correspond to ground-state proton
transitions and likely originate from the 8419-keV level. Under this reasonable assumption, the
transition should have decay energy of 𝐸𝑟 = 8419(3) - 6130.65(24) keV = 2288(3) keV. We also
identified two more relatively strong 𝛽-delayed proton transitions that populate the first and second
excited states, calculating the predicted excitation energies of their proton-emitting level in     31 S.
While our preliminary, three-peak energy calibration predicted excitation energies for both proton
decays to be ≈ 10 keV less than 8419 keV, this is the only confirmed state within 20 keV. Furthermore,
just as we saw for the case of the lowest-energy resonance in our spectrum, for which we reported
a decay energy of 𝐸𝑟 = 273(10) keV, when extrapolating the three-peak energy calibration > 500
keV above or below the well-constrained energy region, a 10-keV uncertainty in decay energy is
to be expected. Thus, we believe it is reasonable to conclude that these protons are also emitted
from the 8419-keV level, with decay energies of 𝐸𝑟 = 8419(3) - 6130.65(24) keV - 677.01(3) keV
= 1611(3) keV and 𝐸𝑟 = 8419(3) - 6130.65(24) keV - 708.70(3) keV = 1578(3) keV. Figure 5.15
shows the combined-pads singles spectrum with the full set of proton calibration peaks labeled.
    Having determined the decay energies of seven well-separated peaks across the spectrum, we
                                                   138


Figure 5.15: All proton peaks used in the combined-pads energy calibration are color-coded by
the 30 P levels their transitions populate directly. Visible singles peaks corresponding to ground
state transitions (black) and weaker transitions to excited states (colored) are labeled by their
center-of-mass decay energies in units of keV.
applied a revised fit to the modes of the distributions modeling each of these seven peaks. Figure
5.16 shows that the Proton Detector is a good example of a proportional counter, since a linear
function fits the data extremely well over the energy range 0.8 − 2.3 MeV. We utilized this function
to calibrate all 𝛽-delayed proton decays observed in this experiment, and all future references to
proton energies in this document reflect this calibration. This application to the combined-pads
spectrum in the case of the low-energy proton peak corresponding to the astrophysically important
𝐽 𝜋 = 3/2+ resonance results in a decay energy of 𝐸𝑟 = 258(3) keV, which brings us into near-perfect
agreement with Ref. [121]. Furthermore, using Equation 5.5, we computed excitation energies for
the proton-emitting levels in 31 S that would correspond to our observed 𝛽-delayed proton decays.
In almost all cases, our expected excitation energies are consistent with the evaluated level energies
of previously observed excited states, as tabulated in NuDat to within their uncertainties. In the few
instances for which this is not the case, newly proposed 𝛽-delayed proton decays are very weak,
and their status should be considered tentative.
                                                   139


Figure 5.16: Linear energy calibration function mapping the peak position in the combined-pads
spectrum to proton decay energy. The peak positions are determined by the mode of EMG
distributions used to fit the data. Fixed points correspond to proton decays of 31 S levels with
well-known excitation energies.
Constraining Fits Below 1.2 MeV
Figure 5.17 shows one such multi-peak fit over the energy region 575 − 1325 keV. What looks
like only four distinct peaks, which have all been previously observed with center-of-mass decay
energies 806, 906, 1026, and 1225 keV, actually contains data from over a dozen unique 𝛽-delayed
proton transitions following   31 Cl decay. The 𝛾-gated coincidence spectrum in Figure 5.4 reveals
numerous weak proton branches to 30 P excited states, including decays with energies of 870, 958,
1066, 1138, 1224, and 1318 keV. We determined that of these transitions, the three highest-energy
proton decays actually populated the four excited state of      30 P, since they can are observed in
coincidence with both 1973- and 1265-keV 𝛾 rays, as shown in Figs. 5.8 and 5.7, respectively. The
other three, lower-energy proton peaks, which are buried under the most prominent features in the
singles spectrum, are attributed to 𝛽-delayed proton transitions to the 709-keV level of 30 P.
    The background-subtracted proton spectra in coincidence with 𝛾 ray transitions from the third
excited state of 30 P, as shown in Figures 5.5 and 5.6, reveal three additional 𝛽-delayed proton
                                                  140


Figure 5.17: Fit of the cumulative proton spectrum over 575 − 1325 keV, including 11 EMG
distributions and a linear background model. The positions of proton peaks corresponding to
decay energies of 834, 869, 958, 973, and 1066 keV have all been fixed, along with their
efficiency-corrected fit integrals from the background-subtracted, 𝛾-gated coincidence spectra.
The energies and intensities of the stronger proton branches are allowed to vary as free parameters,
allowing us to quantify the uncertainty associated with fixing the intensities of smaller peaks.
emissions populating the 1454-keV level within the same region of the cumulative proton spectrum,
which correspond to center-of-mass energies of 834, 973, and 1229 keV. Since these decays are
relatively close in energy to the aforementioned weak, 𝛽-delayed protons, we considered the
possibility that these were, in fact, the same transitions, populating a single 30 P state of relatively
high excitation energy, which would result in real coincidences between these protons with 𝛾 rays
of several different energies. However, we could not find any evidence of this aside from the 1138-,
1224-, and 1318-keV decays to the 30 P level at 1973 keV, as discussed previously. This was verified
by ruling out several different potential scenarios.
    First, no 𝛾 rays were observed from transitions of 30 P levels above excitation energies of 1973
keV. This implies that all 𝛽-delayed proton decays in our data must populate either the ground state
or one of the first four excited states in  30 P, whose 𝛾 cascade has already been well-established
                                                   141


[169; 170; 171; 172]. Second, if the 834/870-keV, 958/973-keV, and 1224/1229-keV protons really
correspond to the same transitions, then we should expect the number of counts observed in the
coincidence spectrum to be consistent, after correcting for 𝛾 detection efficiency. The 1454-keV
level in 30 P primarily 𝛾 decays to the ground state, but about 5% of the time, it decays to the
second excited state via emission of a 746-keV 𝛾 ray. Thus, 834/870-keV and 958/973-keV protons
in coincidence with 1454-keV 𝛾 rays should be 19 times more common than the same protons in
coincidence with 709-keV 𝛾 rays, if they represent the same proton decays. However, the ratios
between the efficiency-corrected counts are closer to ≈ 3 and ≈ 7, respectively.
    If these ratios were approximately equivalent, we could not rule out the possibility that this
discrepancy reflects systematic uncertainties in our background subtraction method or 𝛾 detection
efficiency, but the fact that these ratios differ by more than a factor of 2 suggests they should
be understood as unique transitions. Thus, in all of our cumulative spectrum over the energy
range ≈ 500 − 1400 keV, we fix both the positions and efficiency-corrected counts of these weak,
𝛽-delayed proton decays at energies 834, 869, 958, 973, and 1066 keV. Conversely, in all cases,
the peak positions and amplitudes of the three strongest 𝛽-delayed proton decays in the spectrum
are allowed to be free parameters, since the fitter should be able to reasonably determine this
information with our newly applied constraints. As in the case of previous fits, we generally
assumed a linear background model but tested other polynomials and functional forms as well.
Peak widths were constrained to be a linear function of energy across the fit range.
    For protons decays above 1.1 MeV in energy, we tried many fits, constraining and relaxing
the positions and intensities of all peaks in various combinations. The difference between the
expected number of efficiency-corrected counts in the 1138-keV peak, as determined from the
coincidence spectra in Figures 5.7 and 5.8, and the integral of cumulative spectrum’s best fit, for
which the distribution’s area and centroid are constrained as shown Figure 5.17, yields an estimate
of the systematic uncertainty associated with using our 𝛾 efficiency to evaluate the weak, 𝛽-delayed
proton intensities in this region. In this case, the expected number of efficiency-corrected, 1138-keV
protons only account for ≈ 70% of the statistics in the total, combined spectrum, but fixing the
                                                    142


position of this peak in our cumulative spectrum fit improves our agreement to ≈ 80%. The multiple
proton contributions to what visually appears to be a single peak at 1.2 MeV were neglected in
Figure 5.17, and for the purposes of getting a reasonable 𝜒2 value for such a highly parameterized
fit, the position and intensity of this feature, as well as the much smaller 1318-keV peak, were
allowed to vary freely in most instances when fitting across the energy region below 1.4 MeV.
Constraining Fits Between 1.2 and 1.4 MeV
As was the case for distinguishing the 834/870- and 958/973-keV, a similar argument can be made
for disentangling the multiple 𝛽-delayed proton contributions to the substantial peak at 1.2 MeV.
Protons at 1138, 1224, and 1319 keV are clearly in coincidence with the 709-, 1973-, and 1265-keV
𝛾 rays. The simplest explanation for this is that these proton decays populate the 1973-keV state of
30 P, which promptly decays to either the 709-keV level or the ground state. There are no known 𝛾
transitions from either the fourth or third excited states to the first excited state, which means the
1218-keV protons in coincidence with the 677-keV 𝛾 rays in Figure 5.3 must be populating this
30 P  level directly. Furthermore, only a weak, tentative, 519-keV 𝛾 branch between the fourth and
third excited states has been proposed [169], and this cannot explain why there are more 1229-keV
protons in coincidence with 1454-keV 𝛾 rays than with the combined statistics of 1265- and 1973-
keV 𝛾 coincidences. Thus, the most reasonable conclusion is that three distinct 1.2-MeV proton
transitions populate the first, third, and fourth excited states of 30 P.
     Within the energy region that spans 1.2 − 1.4 MeV, shown in Figure 5.18, the 𝛽-delayed proton
spectrum contains two more distinct features. There appears to be some visible structure near
1.3 MeV in the cumulative spectrum, but the presence of a clear proton peak at 1319 keV only
becomes obvious when analyzing the coincidence spectra involving deexcitations from the fourth
excited state of 30 P, as previously mentioned. The much more prominent peak near 1.4 MeV has
been reported by Ref. [118] but was assumed to populate the ground state. However, protons in
coincidence with 1454- and 746-keV 𝛾 rays reveal that at least some of this peak’s amplitude must
originate from 1381-keV decays that populate the third excited state of       30 P. Just as we analyzed
                                                   143


Figure 5.18: Fit of the cumulative proton spectrum over 1175 − 1450 keV, including five EMG
distributions and a linear background model. The positions of proton peaks corresponding to
decay energies of 1217, 1224, and 1229 keV have all been fixed, along with their
efficiency-corrected fit integrals of the background-subtracted, 𝛾-gated coincidence spectra. The
positions and intensities of the 1318- and 1381-keV proton decays are allowed to vary freely and
converge well with observations in the coincidence spectra.
the cumulative proton spectrum for energies below 1.2 MeV, we constrained our model of the data
using the information we extracted from the proton-𝛾 coincidence spectra in order to fit this energy
region between 1.2 and 1.4 MeV.
    It is possible that, in addition to decays to excited states, 𝛽-delayed protons that populate the
ground state of 30 P could be contributing to the large peaks at 1.2 and 1.4 MeV as well. In order
to estimate the possible size of this effect, we first assumed only a three-peak model, fixing only
the position of a single EMG distribution for the 1.3-MeV peak, allowing the fitter to determine
the energies and intensities of much larger features and assuming, just as in the previous fit range,
that all decay events near 1.2 MeV can be considered part of a single proton peak. Again, we
constrained the peak widths to be linear as a function of decay energy, our skew parameter to be
constant, and the background beneath the peaks to be linear as well. The areas of the loosely
                                                  144


constrained three-peak fit could then be compared to the efficiency-corrected fit integrals of the
background-subtracted, 𝛾-gated proton peaks. In this case, the expected counts calculated using
our numbers for the SeGA efficiency actually overpredicted the total number of events by factors of
≈ 1.16 − 1.26 for the 1.2-MeV peak, ≈ 1.66 − 1.76 for the 1.3-MeV peak, and ≈ 1.16 − 1.36 for the
1.4-MeV peak. Fixing the position of the highest-energy peak to a 1381-keV decay energy erased
the discrepancy between the two counting methods to within statistical uncertainty for 1319-keV
proton decays and reduced the overcounting factors for the 1.2- and 1.4-MeV peaks to ≈ 1.09 − 1.18
and ≈ 1.15 − 1.25, respectively. Thus, we can reasonably say that there is no need to invoke
ground-state transitions to explain the structure of this region in the proton spectrum.
    Adjusting the fit range for the three-peak model, while varying which parameters were fixed
or free, enabled us to assess the systematic effect of our background model on the uncertainty in
our peak integrals. Furthermore, we introduced a five-peak fit, as shown in Figure 5.18, where
the positions and intensities of the 1218-, 1224-, and 1229-keV peaks are fixed, to ensure that our
model is able to accurately reproduce the data for the large composite peak at 1.2 MeV. This fit
agrees well with the data over the region where the response function’s parameters are heavily
constrained, and it is able to reproduce the energy to within 1 keV, as well as the peak area to
within 1% of the 𝛾-gated, efficiency-corrected proton counts at 1381-keV. Its assignment of the
peak position and integral to the more ambiguous region between the two larger features still agrees
with our predictions to within 10 keV and 50%, respectively. This gives us good confidence in
our response function as a reasonably accurate model of the data as well as in the quality of our
efficiency calibration and our ability to use it for evaluating the weak, 𝛽-delayed protons intensities.
Constraining Fits Between 1.5 and 2 MeV
Continuing to move across the cumulative decay spectrum, we encounter a region with what appears
to be three, broad proton peaks peaks between 1.5 and 2 MeV, shown in Figure 5.19. Again, upon
inspection of the proton-𝛾 coincidence spectra, we see that there are many underlying peaks that
correspond to proton emissions populating excited states in 30 P, some of which are quite intense,
                                                    145


Figure 5.19: Fit of the cumulative proton spectrum over 1450 − 2050 keV, including 10 EMG
distributions and a linear background model. The positions of proton peaks corresponding to
decay energies of 1577, 1610, 1653, 1717, 1827, 1864, and 1872 keV have all been fixed, along
with their efficiency-corrected fit integrals of the background-subtracted, 𝛾-gated coincidence
spectra. The positions and intensities of the 1574-, 1759-, and 1889-keV proton decays are
allowed to vary freely, and their energies are in good agreement with previous measurements.
while others are very small. By far, the two strongest of these 𝛽-delayed proton decays populate the
first and second excited states with center-of-mass energies of 1610 and 1577 keV, respectively. At
first glance, it might appear that a combination of these two proton decays alone might account for
the singular, massive peak visible in this region of the proton spectrum. However, after dividing the
number of counts in the 𝛾-gated proton spectra by the 677- and 709-keV 𝛾 detection efficiencies,
respectively, fixing their positions in the energy spectrum along with their total peak areas, and
applying the multi-peak fit as seen in Figure 5.19, we find that the combined intensity of both of
these decays is still about a factor of 50 too small to account for the rest of the protons in this peak.
Thus, we must conclude, that there is indeed a ground-state proton transition at 1574 keV, which
agrees with the previously reported decay energy [114; 115; 116; 173; 117; 118].
     In fact, contrary to the previous spectral region we considered, none of the three biggest peaks
                                                   146


between 1.5 and 2 MeV are primarily composed of proton decay events populating excited states.
This makes fitting the spectrum over this energy range relatively straightforward. We can simply fix
the energies and intensities of the weaker, excited-state transitions, add in literature values for the
ground-state proton decays while letting the exact positions and integrals of their peaks remain free
parameters before fitting. As always, we constrain our peak shapes to vary smoothly with energy
and apply a linear model to our background, which is much more well-behaved at higher energies
and, coupled with the lower statistics, makes fitting the data significantly easier. Finally, just as
before, small branches close together in energy, such the 1864- and 1872-keV protons, should not
be considered the same decay, as they populate different excited states with no known 𝛾 transitions
between these 30 P levels.
Constraining Fits Above 2 MeV
With increasing excitation energy, fewer states are intensely populated by 𝛽+ decay, as expected
due to the energy dependence of the phase-space factor. This is reflected generally by decreasing
intensities for 31 Cl 𝛽-delayed proton decays above 2 MeV, as observed in past measurements.
This trend can also be seen in our data and is exacerbated by the strong energy-dependence of
the Proton Detector’s efficiency, as shown in Figure 4.13. However, while the statistics in this
relatively high-energy regime are low, we still are able to observe some structure and evidence
of a previously unobserved proton transitions. In coincidence with the 677-keV 𝛾 rays, we can
clearly identify proton events at 2146 keV. This energy is consistent within the uncertainties of the
evaluated literature energies that assume a ground-state transition. For the sake of simplicity, one
would be inclined to think that the proton peak we observe in the cumulative spectrum is composed
entirely of 𝛽-delayed proton events that populate the first excited state of    30 P. The fit shown in
Figure 5.20 is constrained such that the integral of this peak is fixed to the intensity observed in
the 𝛾-gated coincidence spectrum, scaled by SeGA’s efficiency at 677 keV. However, if we relax
this parameter, the 𝜒2 -minimizing fitter prefers a peak amplitude roughly twice the size. Thus,
we cannot currently rule out the possibility of multiple nearby proton peaks, especially since Ref.
                                                147


Figure 5.20: Fit of the cumulative proton spectrum over 1980 − 2700 keV, including three EMG
distributions and a linear background model. The positions of proton peaks corresponding to
decay energies of 2146 and 2494 keV have all been fixed, along with their efficiency-corrected fit
integrals of the background-subtracted, 𝛾-gated coincidence spectra. The positions and intensities
of the 2494-keV proton decays are allowed to vary freely.
[102] reports the existence of a 8268-keV level, which is close to the excitation energy needed for
a state which proton decays to the ground state of 30 P.
    We identified the strongest 𝛽-delayed proton decay in this region to occur at 2290 keV, which
is close to previously reported decay energies for this ground-state transition. When applying a
variety of different constraints on the smaller peaks nearby, we allowed the peak position and area to
vary as free parameters. However, despite large changes in the relative intensities of other features
within the fit range, the number of counts in the 2290-keV peak varied by less than 4% over all
cases. Because any proton peaks in this region will be quite broad, there is substantial overlap
between nearby distributions, and relatively small changes in the amplitude of a large peak will
have a disproportionately large effect on the relative intensities of smaller peaks in the vicinity.
    For this reason, we are unable to conclusively identify the presence of a 2.4-MeV proton peak,
reportedly observed in both Ref. [117; 118]. Based on its quoted intensity and our simulation
                                                  148


results for the Proton Detector efficiency at this energy, we would expect to find a distribution with
an area ≈ 3.4 times smaller than that of the neighboring 2290-keV peak. When we attempt to
enforce this condition on the fitter, we substantially overpredict the data within the energy range
we expect to find the peak. When we relax the areal parameter of this distribution, the fitter
determines that the ratio of their magnitudes should be at least ≳ 9. The relatively low statistics
of this region of the spectrum and the many degrees of freedom for multi-peak distributions that
are not well-constrained leads to overfitting. Therefore, a variety of models return reasonable 𝜒2
values, and our associated systematic uncertainties associated with with quantifying peaks above 2
MeV are substantially higher than at lower energies.
    The last definitive proton peak we were able to identify in our 𝛽-delayed proton spectrum has
a decay energy of 2494 keV. It was observed in coincidence with the 709-keV 𝛾 rays, implying it
directly populates the second excited state of 30 P. In the proton-𝛾 coincidence spectrum, it appears
to be much broader than almost all other peak in our dataset. This could be a result of poor gain
matching at such high energies or simply a function of degrading detector resolution with energy,
but it is worth noting the relative resolution across most of the energy spectrum is reasonably stable
at 4 − 5%. Another possibility is that the corresponding      31 S level, reported in Ref. [102], could
be intrinsically broad. The peak itself in the cumulative spectrum is barely visible and was not
reported in the results of previous 𝛽-delayed proton measurements.
    At present, we cannot draw any quantitative conclusions about the presence of 𝛽-delayed proton
decays above 2.5 MeV in our spectrum. This regime appears to consist of a continually declining
background with some broad, bumpy regions of excess counts. More careful and detailed analysis is
required to provide robust upper limits on 𝛽-delayed proton decays that do not constitute obviously
identifiable peaks in our energy spectrum.
5.4    Preliminary Results
In the analysis discussed throughout this dissertation, we have demonstrated the capabilities of
the GADGET system to measure weak, low-energy, 𝛽-delayed, charged-particles decays as well
as their 𝛾-ray coincidences with high efficiency and good energy resolution. In this chapter, we
                                                   149


have provided the most detailed description to date of the 31 Cl(𝛽𝑝𝛾)30 P sequence, and a tentative
decay scheme is proposed in Figure 5.21. We report definitive observations of 25 unique                           31 Cl
𝛽-delayed proton decay channels for decay energies below 2.5 MeV, compared to the most recent
literature evaluation that lists only 13 over that same energy range [119]. Furthermore, we claim
that 19 of these 𝛽-delayed proton transitions populate excited states of 30 P, none of which have been
previously reported in literature or appeared in evaluated nuclear data tables. However, it is worth
mentioning the Ref. [118] did tentatively suggest that 1.2-MeV 𝛽-delayed protons could populate
the third excited state of       30 P, and we have confirmed this observation. Preliminary values for all
energies and intensities measured during NSCL experiment 17024 are tabulated in Table 5.1.2
    2 All excitation energies listed in Table 5.1 are computed from calibrated proton decay energies, which are subject
to change. Not all unique level energies listed should be considered distinct and require further evaluation before
publication.
                                                          150


                                                                          β+
Figure 5.21: Preliminary, partial decay scheme for 31 Cl(𝛽𝑝𝛾)30 P, including only proton-unbound
levels in 31 S that can result in a proton decay to a 30 P excited state.
                                                   151


                       𝐸 𝑝 (c.m.) [keV]    𝐼 𝑝 (rel) [%]  𝐸 𝑥 (31 S) [keV] 𝐸 𝑥 (30 P)
                             258(3)          0.069(5)         6389(4)           0
                             806(2)           15.4(8)         6937(2)           0
                             834(3)           0.10(1)         8419(3)       1454
                            870(18)            <0.05         7709(18)        709
                             906(2)            6.6(4)         7037(2)           0
                            958(12)            <0.05         7797(12)        709
                             973(2)           0.33(5)         8558(2)       1454
                            1025(2)           100(8)          7156(2)           0
                            1066(8)           0.20(5)         7905(8)        709
                            1138(4)           0.37(9)         9242(4)       1973
                            1218(2)           0.25(5)         8025(2)        677
                            1224(5)           0.31(6)         9328(5)       1973
                            1229(2)            1.4(3)         8814(2)       1454
                            1319(8)          0.61(29)         9422(8)       1973
                            1381(2)          0.68(18)         8966(2)       1454
                            1574(2)          10.8(20)         7705(2)           0
                            1580(3)          0.98(20)         8419(3)        709
                            1611(3)          2.18(45)         8419(3)        677
                            1653(2)          0.81(18)         9238(3)       1454
                            1717(3)          0.84(18)         8557(3)        709
                            1759(2)          2.88(55)         7889(2)           0
                            1822(6)          0.25(65)         9406(6)       1454
                            1864(8)            <0.15          8672(8)        677
                            1872(6)           0.22(6)         8711(6)        709
                            1889(2)           6.9(12)         8020(2)           0
                            2146(4)            0.7(6)         8954(4)        677
                            2148(3)            <0.62          8954(4)        677
                            2288(3)            4.0(7)         8419(3)           0
                            2494(7)            1.0(3)         9333(7)        709
Table 5.1: Preliminary energies and intensities of all 31 Cl 𝛽-delayed proton transitions, including
the excited state of origin in the 31 S parent and the excited state they populate in the 30 daughter
nucleus.
                                                      152


                                              CHAPTER 6
                                CONCLUSIONS AND OUTLOOK
The stated goals outlined in this dissertation have been achieved over the course of this doctoral
thesis project. Our primary scientific objective was to constrain the thermonuclear rate of the
30 P(𝑝, 𝛾)31 S reaction with the intent of reducing the nuclear physics uncertainties associated with
predicting elemental and isotopic abundance ratios in ONe nova ejecta. To do this, we developed
the new radiation detection system GADGET specifically to measure weak, low-energy,               31 Cl 𝛽-
delayed proton decays through the crucial 𝐽 𝜋 = 3/2+ , 6390-keV excited state of 31 S. We successfully
measured the proton branching ratio Γ𝑝 /Γ of this particular resonance. Combining our experimental
result with shell-model calculations for the 𝛾 decay partial width Γ𝛾 of this level along with past
work on other resonances, we calculated the thermonuclear rate for proton capture on            30 P  over
peak nova temperatures. In agreement with our original hypothesis, we demonstrated that the
contribution from this 260-keV, ℓ = 0 resonance dominates the total rate.
    Our new, recommended rate was utilized in state-of-the-art hydrodynamic simulations of clas-
sical nova explosions for a 1.35-𝑀⊙ ONe white dwarf, and we compared the results of these
calculations to observed elemental abundances in the ejected shells of the V838 Herculis nova.
We found our results to be in agreement with the measured O:S and S:Al ratios, and we have
constrained these nuclear thermometers for ONe novae at a peak temperature of 𝑇 = 0.31 GK. We
also compared the nova model predictions to measured isotopic ratios of presolar grains hypothe-
sized to originate in classical novae. The results of the simulations were not able to reproduce the
observed isotopic ratios, but our constrained rate for the 30 P(𝑝, 𝛾)31 S reaction substantially reduced
uncertainties in the expected 30 Si:28 Si ratio for ONe nova ejecta.
    In addition, we determined that our measurement of 31 Cl 𝛽-delayed proton decay through the
resonance of interest represents the weakest 𝛽-delayed charged-particle decay ever reported for
resonances below 400 keV. Furthermore, we tentatively claim to observe a dozen new               31 Cl  𝛽-
delayed proton decay channels and demonstrate conclusively for the first time that many of these
                                                    153


transitions populate excited states of 30 P. We have also constructed a preliminary decay scheme for
31 Cl(𝛽𝑝𝛾)30 P.
     Our future goals related to this work include finalizing the analysis of the decay 31 Cl(𝛽𝑝𝛾)30 P.
Aside from this, we would also like to constrain the lifetime of the 𝐽 𝜋 = 3/2+ , 6390-keV excited
state of 31 S, since this is the largest remaining source of uncertainty in our resonance strength as
well as the last missing piece of experimental information for this nuclear level. A Doppler shift
lifetime experiment has been planned for this measurement. In addition, the GADGET system as
been upgraded to Phase II and is now fully operational as a time projection chamber. We also
have plans to measure the decay 20 Mg(𝛽𝑝𝛼)15 O in order to constrain the 15 O(𝛼, 𝛾)19 Ne reaction in
explosive He-burning for modeling type I X-ray bursts.
                                                  154


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