NUCLEAR DATA TO QUANTIFY URCA COOLING IN ACCRETING NEUTRON STARS

By

Rahul Jain

A DISSERTATION

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

Physics—Doctor of Philosophy
Computational Mathematics, Science, and Engineering—Dual Major

2024

ABSTRACT

Neutron stars in Low Mass X-ray Binaries (LMXBs) can accrete matter onto their surface from

the companion star. Transiently accreting neutron stars go through alternating phases of active

accretion outbursts and quiescence. X-ray observations during the quiescence phase show a drop

in X-ray luminosity with the time in quiescence. This is also inferred as the drop in surface

temperature or the cooling of accreting neutron stars in quiescence. Analyzing these cooling

curves reveals a great deal of information about the structure and composition of neutron stars.

However, model-observation comparisons of such cooling curves are challenging - partly due to

observational uncertainties, and partly due to incomplete knowledge of heating mechanisms during

accretion outbursts. This situation is further exacerbated by the recent discovery of Urca cooling

in the neutron star crust. These are cycles that alternate between electron-capture and 𝛽-decay to

produce a large flux of neutrinos and anti-neutrinos. These freely stream out of the star and carry

energy with them, essentially cooling the neutron star crust without changing the composition. As

a result, it is necessary to accurately quantify the strength of Urca cooling to constrain the heat

sources in neutron star crusts and facilitate better model-observation comparisons of the cooling

curves.

Urca cooling is effective only for a certain subset of nuclei with specific properties. One of

the required conditions is a strong ground-state to ground-state 𝛽-decay transition strength for a

nucleus. Previous studies have shown 33Mg to be a strong Urca cooling agent for neutron star

crusts composed of X-ray burst ashes. This is attributed to a 37% strong ground-state branch in

the 𝛽-decay of 33Mg inferred from high-resolution 𝛽-delayed 𝛾 spectroscopy. This is, however,

a first-forbidden transition and the strong ground-state branch seems anomalously high compared

to theoretical calculations. A goal of this dissertation is to remeasure this transition strength

using Total Absorption Spectroscopy with the SuN detector. 𝛽-delayed neutron branching ratio

is also measured with the NERO detector. A combination of SuN and NERO helps mitigate

the Pandemonium effect, which is shown to systematically overestimate low-energy branchings in

high-resolution 𝛾 spectroscopy.

The ground-state branch for the 𝛽-decay of 33Mg → 33Al was measured to be 0.7(24)%

corresponding to a log-ft value of 7.0+∞

−0.7. This is significantly lower than the previous measurement

and is consistent with the first-forbidden nature of the transition arising from the recently confirmed

negative parity ground state of 33Mg. It further translates into a substantially reduced intrinsic Urca

cooling luminosity of L34 = 60.0. This highlights the importance of Total Absorption Spectroscopy

and motivates future experiments with this technique to refine calculations of Urca cooling.

The rate of Urca cooling (L34) is extremely sensitive to electron-capture thresholds (QEC) and

is proportional to |(QEC)|5. These electron-capture thresholds depend on nuclear masses. Several

of the potential Urca cooling candidates are neutron-rich exotic nuclei whose masses have not been

measured experimentally and Urca cooling calculations have to rely on theoretical mass predictions.

However, theoretical model predictions diverge as they move away from the stable nuclei and do

not have uncertainties. A global nuclear mass model with quantified uncertainties is also developed

as a part of this dissertation using Bayesian Gaussian Process Regression and Bayesian Model

Averaging (BMA). Updated neutron star crust calculations with the BMA mass model change not

only the magnitude of Urca cooling but also the depth at which it happens. This has important

implications for the overall thermal profile of the accreting neutron star crust.

Copyright by
RAHUL JAIN
2024

To Maa –
Who sacrificed her dreams,
so that she could fuel mine.

v

ACKNOWLEDGEMENTS

The challenging endeavor of getting a Ph.D. would certainly not have been possible without the

help and support of many that I received along the way. I am extremely grateful to everyone who

helped me in whatever way possible, no matter how small. While I try my best to recognize as

many people as I can, I apologize in advance if I inadvertently miss someone. Please know that I

sincerely appreciate your time and help.

First and foremost, I would like to thank my advisor, Prof. Hendrik Schatz, for his excellent

mentorship and guidance throughout these years. I could not have asked for a better graduate school

experience than under his advising. A true expert in the field of Nuclear Astrophysics, every single

interaction with you is filled with tremendous learning. I just wish I ‘walked’ into your office more

often. Thank you for always being open to and appreciative of my ideas and providing resources to

try them out. Along with the academic opportunities, the extra-curricular opportunities I received

as a graduate student helped me gain a big-picture perspective of the field and ultimately shaped

me into a better scientist.

I also want to thank Prof. Witek Nazarewicz, who gracefully agreed to advise me on the CMSE

part of my dissertation and welcomed me into the ‘Nuclear Theory’ group at FRIB. I am fascinated

by your passion for nuclear physics and hope that it rubbed off on me too. I would also like to thank

other members of my committee - Prof. Artemis Spyrou for collaborating with us on my thesis

experiment along with her group, supporting me in her role as the Associate Director for Education

at NSCL, and eventually writing recommendation letters during job searching process; Prof. Ed

Brown for all his help with astrophysics and introducing me to his seemingly simple yet powerful

back-of-the-envelope calculations for neutron stars; and Prof. Michael Murillo for introducing me

to Gaussian Process and Machine Learning. I am also grateful to all my committee members for

taking time out of their busy schedules for annual meetings and ensuring I was on track and making

progress. Additionally, I would also like to thank Prof. Sean Liddick for teaching me the basics of

detector setup and data acquisition systems in his BCS lab.

A big thanks to Wei-Jia Ong and Kirby Hermansen for helping me through all the stages of my

vi

thesis experiment. I would never have been able to do this successfully without you both. Your

help on multiple fronts prevented me from reinventing the wheel on so many occasions. Thanks

also to Andrea Richard, Mallory Smith, and Caley Harris for all your help in the S2 vault with

SuN during the experiment in the middle of the COVID-19 pandemic, and Nabin Rijal for helping

with the vacuum systems. I am also thankful to the entire NSCL operations group for successfully

delivering the desired rare isotope beams for the experiment. In addition, I would like to thank

Hannah Berg and Adriana Sweet for helping with GEANT4 simulations, Artemis Tsantiri and

Eleanor Ronning for helping with RAINIER simulations, and Sudhanva Lalit for helping with crust

model calculations. I would also like to thank Leo Neufcourt for helping me with the formalism of

Bayesian Machine Learning and letting me use his code for the nuclear masses project and Samuel

Giuliani for helpful discussions.

I am extremely grateful for the additional mentoring I received as a graduate student. I want

to thank Fernando Montes for his valuable guidance on the professional development front and for

always being open to talk about anything. I also want to thank Ana Becerril-Reyes for supporting

me in all my student leadership roles within JINA and IReNA. Special thanks to Jaspreet Singh

Randhawa who has been an incredible mentor and is always looking out for me. Thanks also to

Steven Thomas and the entire AGEP community for providing a valuable space to rewind and relax

from the hustle of graduate school. Finally, I would also like to thank all members of the Nuclear

Astrophysics Group as well as the Nuclear Theory Group at FRIB who made it fun to work at the

laboratory.

I have also benefited from the support of the wonderful community in East Lansing. Thank

you Larry Lee for being a great mentor and helping me navigate the American way of life. Thank

you also for many of my ‘first-ever’ traditions and for giving me some truly unimaginable life

experiences. I am also thankful to Shay Manawar and Anum Mughal for being dear friends and

helping me in times of crisis. Finally, thanks to all members of the squash community at the

Michigan Athletic Club for helping me continue playing squash and keep improving my game.

This has been a huge factor in preserving my sanity through all these years.

vii

I am exceptionally lucky to have the kind of love and support I get from my family. Mom and

Dad, you are amazing role models and with you by my side, I can confidently chase any goal I

set for myself. Thank you Chimu for your tough love and humbling me whenever needed. Finally

and most importantly, thank you Karishma for always being there for me. You inspire me to keep

pushing my limits and be the better version of myself, every single day. I hope I make you proud.

viii

TABLE OF CONTENTS

CHAPTER 1

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

.

1

CHAPTER 2

EXPERIMENTAL MEASUREMENT . . . . . . . . . . . . . . . . . . . 22

CHAPTER 3

DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

CHAPTER 4

EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . 72

CHAPTER 5

QUANTIFIED NUCLEAR MASS MODEL . . . . . . . . . . . . . . . . 91

CHAPTER 6

IMPACT ON ASTROPHYSICAL MODELS . . . . . . . . . . . . . . . 109

CHAPTER 7

CONCLUSIONS AND FUTURE DIRECTIONS . . . . . . . . . . . . . 114

BIBLIOGRAPHY . .

. .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

APPENDIX A

LIST OF MY CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 134

ix

CHAPTER 1

INTRODUCTION

Neutron stars are the densest stellar objects in the universe. They were discovered in 1967 as

pulsars with radio telescopes [1]. They weigh as much as about 1 – 2 times the mass of our sun

(M⊙) but measure only a few kilometers (10 – 12 km) in their radii [2]. As a result, they have

densities of more than 1017 kg/m3. When massive stars (M ≥ 8 – 10 M⊙) are nearing the end of

their evolution, they form an inert iron core at their center that cannot undergo nuclear fusion any

further to form heavier elements. This is because iron is the most stable element with the highest

binding energy per nucleon. After a certain point, the iron core becomes so massive that is no

longer able to support itself against its gravitational pull, resulting in a core collapse. The core is

compressed to nuclear saturation densities until it begins to feel the repulsive nature of the strong

nuclear force. This creates a shockwave that propagates outward and the infalling matter bounces

off this rigid core, resulting in a core-collapse supernova [3]. The remnant of these explosions is

a proto-neutron star. At this point, if the neutron degeneracy pressure is enough to support the

compact object remnant, it becomes a neutron star. If not, the proto-neutron star collapses into a

black hole.

Neutron stars are interesting sites to study the properties of matter under extreme conditions.

Unlike a black hole where gravity turns everything into a singularity, all four fundamental forces

— the strong force, the weak force, the electromagnetic force, and the gravitational force — play a

role in the interaction of matter in neutron stars [4–6]. They are also prime candidates for multi-

messenger astronomy [7]. Astrophysical sites habitating a neutron star are sources for gravitational

waves [8], neutrino signals [9], X-ray observations, radio wave signals, etc. Recently, neutron star

mergers have also been identified as prominent sites for the production of heavy elements in the

universe via the rapid neutron capture process, or the r-process [10]. UV, visible, and infrared

spectroscopy of such sites also help determine the abundance of each heavy element formed in the

r-process [11]. All these characteristics make neutron stars a prime object of interest to a wide

range of physicists and astronomers [12].

1

Figure 1.1 Accretion onto a neutron star via the mechanism of Roche lobe overflow. Matter is
funneled into an accretion disk around the neutron star through the L1 Lagrange point.Figure
modified from Ed Brown (private communication).

1.1 Accreting Neutron Stars

Up to 85% of the stars in our universe are born in binary systems, with some in triple or even

higher multiple systems [13]. For binary systems with stars having unequal masses, each star

evolves on a different timescale. Massive stars evolve much faster than lighter stars. If the lighter

companion star can withstand the supernova explosion of the heavier star that is nearing the end

of its evolution and if the binary system remains bound, a compact object binary is formed. It is a

binary system with a regular low mass star and a compact object. If the compact object is a neutron

star, it is called a neutron star binary.

Each star in a compact object binary has an imaginary envelope, called the Roche lobe, where

2

matter present within the envelope will be gravitationally bound to that star. The Roche lobes of

two stars in a binary system meet at a point called the L1 Lagrange point (Refer to Figure 1.1). Due

to the internal evolutionary changes in the companion star or changes in it’s orbit, it can expand

and fill up its Roche lobe. At this point, matter is funneled through the L1 Lagrange point [14] into

an accretion disk around the compact star. Figure 1.1 explains the process of accretion through

the mechanism of Roche lobe overflow. An accretion disk forms around the neutron star because

angular momentum still has to be conserved. The angular momentum, however, is eventually lost

due to friction in the disk, and the matter gets deposited onto the surface of the neutron star.

The accreted matter releases about 200 MeV/u of gravitational potential energy while falling

onto the surface of a neutron star. All this energy is radiated away in the form of X-rays. As a result,

an accreting neutron star binary that is accreting matter from a sun-like companion star is also

called a Low Mass X-ray Binary (LMXB). Such systems have been observed and monitored with

modern space-based X-ray telescopes for a few decades now [15–18]. LMXBs are significantly

important to understand the properties of the crust of neutron stars as we will see in the following

sections.

1.1.1 Quasipersistent X-ray Transients

Accretion onto a neutron star can be unstable and it can pause after several months or years.

Such transient systems undergo periodic episodes of accretion, i.e., they alternate between periods

of active accretion and quiescence. Accretion outburst periods are characterized by persistently

high X-ray luminosity (1035 − 1037 erg/s), whereas the quiescence phase is characterized by a 2 –

3 orders of magnitude drop in persistent X-ray flux. While it is agreed that this drop in luminosity

is due to the halting of accretion, the exact mechanism for pausing and resuming accretion on

timescales of just a few years is unknown. Modern sensitive X-ray telescopes allow tracking of

such quasipersistent transients even in their quiescence phase. So far, about a dozen such systems

have been discovered and monitored [19]. Some of them have also been monitored for multiple

outbursts and quiescent periods [20, 21].

Fitting the spectrum of X-ray photon energies to a black body spectrum, we can deduce the

3

Figure 1.2 X-ray observations of the surface temperature of the neutron star in system MXB
1659–29 as a function of the number of days spent in quiescence. Different lines correspond to
different models of the cooling curves. Solid arrows show examples of insights drawn from
modeling the cooling curves. Figure modified from Ed Brown (private communication).

surface temperatures of accreting neutron stars. Tracking transient X-ray binaries in quiescence

reveals a drop in surface temperature with the number of days spent in quiescence. This cooling

of the crust over timescales of a few years is very different from the cooling of the entire neutron

star that happens over timescales of thousands of years. The observed cooling curves, as they are

called, for transiently accreting neutron stars provide insights into the structure and composition

of the crusts of neutron stars [22–25]. As the neutron star spends more time in quiescence, the

surface temperature probes successively deeper layers of the crust. Figure 1.2 shows the observed

cooling curve for the system MXB 1659–29 after 2.5 years of active accretion and possible insights

drawn about the properties of its crust from cooling curve models [25]. The relatively fast cooling at

earlier times reveals a well-ordered lattice structure of the outer crust with high thermal conductivity

[23–25]. The bend in the cooling curve after a few hundred days in quiescence is interpreted as

a phase transition marked by the onset of neutron superfluidity in the inner crust [24, 25]. The

cooling curves may also carry evidence for the existence of nuclear pasta at the crust-core transition

[26, 27]. Molecular dynamics simulations of nuclear pasta predict high impurity scattering which

4

impacts the thermal conductivity [26].

During accretion outbursts, hydrogen and helium-rich matter from the outer layers of the

companion star is deposited on the surface of neutron stars. This matter undergoes either stable

or explosive nuclear burning in the envelope, depending on the composition of accreted matter,

the rate of accretion, and the properties of the neutron star. [28]. The ashes of this burning are

submerged into the crust as the infalling matter from continued mass accretion pushes them deeper.

For the typical accretion rates in Low Mass X-ray Binaries (LMXBs), it is estimated that the entire

crust of the neutron star will be replaced by accreted matter in about ∼10,000 years. Since most

LMXBs are expected to be older than 1 – 10 Myr [29], their crusts are completely replaced by

accreted matter.

As the ashes are pushed deeper into the crust, the rising pressure and density induce nuclear

reactions (Section 1.1.2) [30–32]. The energy generated by nuclear reactions heats the crust and

brings it out of thermal equilibrium with the core. Once accretion stops and the system enters the

quiescent phase, the crust relaxes until it reaches thermal equilibrium with the core again. This is

the origin of observed decrease in surface temperature during the quiescence phase for accreting

neutron stars. The fraction of heat that flows to the surface vs.

the fraction that flows to the

core depends on the distribution of heat sources within the crust, the composition of the crust,

and the thermal conductivity of the different layers of the crust. The composition of the crust,

in turn, depends on the nuclear reactions that process the accreted matter. As a result, a detailed

understanding of all the different nuclear reactions that can occur throughout the neutron star is

important in the modeling of the cooling curves of accreting neutron stars.

1.1.2 Nuclear Reactions during Accretion

A neutron star has a structure, composed of different layers, analogous to that of the Earth

(Refer to Figure 1.3). It has a thin atmosphere of gaseous atomic plasma, followed by an ocean

consisting of liquid plasma with free electrons and ionized nuclei. Beneath this is a solid crust,

composed of a crystallized lattice of nuclei and degenerate electrons. The inner crust also consists

of free and superfluid neutrons. The mantle is composed of nuclear pasta which are structures of

5

Figure 1.3 Nuclear reactions that take place in the different layers of an accreting neutron star.
H/He-rich accreted matter from a companion star is processed by these reactions as it is buried in
the star. Note that the thicknesses of different layers are not drawn to scale.

pure nucleonic matter. Finally, we have the core which is thought to be composed of exotic nuclear

matter. All these layers cover a very wide range in temperature, pressure, and density conditions.

For example, density runs from around 10−4 g/cm3 in the atmosphere to 1015 g/cm3 and beyond in

the core. As a result, many different types of nuclear reactions are enabled in different layers of

the neutron star that change the composition of accreted matter as it is buried in the neutron star.

Figure 1.3 shows the different layers and possible nuclear reactions at those depths.

Nuclear reactions begin altering the composition of accreted material even before it is deposited

on the surface of the neutron star. As the accreted material slows down in the atmosphere, it is

exposed to a flux of high energy protons, resulting in spallation reactions. These reactions reduce the

CNO abundance, and result in hydrogen being present at ignition. The spallation of heavy elements

6

also helps replenish some of the CNO elements [33]. This alters the rate of hydrogen burning in

the atmosphere which primarily proceeds through the CNO cycle, thus requiring heavy elements

as catalysts. Stable hydrogen burning produces helium, whereas stable helium burning through the

3-𝛼 reaction produces carbon. Once the temperature rises beyond ∼ 108 K, the accelerated rate

of nuclear reactions results in a thermonuclear runaway, triggering a type-I X-ray burst powered

by the rapid proton capture or the rp-process [34]. The thermonuclear runaway causes a breakout

from the CNO cycle and allows for the formation of heavy elements through proton captures and

𝛽+-decay. (p, 𝛼) reactions create cycles in the rp-process. The ashes of nuclear burning in the

atmosphere are deposited in the ocean, and may contain significant amounts of carbon. If certain

conditions are met, the carbon layer can be ignited and undergo explosive burning triggered by

carbon fusion, resulting in a superburst [35]. These are similar to the type-I X-ray bursts but are a

thousand times more energetic and last much longer. Recently, a hyperburst powered by explosive

oxygen and neon fusion in the ocean was also proposed to explain the anomalously hot crust in

MAXI J0556-332 [36].

In the crust, the densities are greater than 106−8 g/cm3. At these densities, the Fermi energies

of electrons are sufficiently high to overcome the electron-capture threshold. As a result, a series

of electron-captures occur along isobaric chains. Electron-captures often proceed in pairs of two

because of the odd-even staggering of nuclear masses, and hence, electron capture thresholds. The

reverse 𝛽-decay is usually blocked but can happen under certain conditions as described in Section

1.2 [37]. When nuclei hit the neutron dripline, they emit neutrons that can be captured by light

elements. Recently, another channel for exchanging neutrons, i.e.

the n-transfer reactions was

proposed [38, 39]. At the high densities in the neutron star crust, the wave functions of neighboring

nuclei overlap. As a result, a weakly bound neutron can quantum mechanically tunnel from one

nucleus to another. This changes the abundances in different isobaric mass chains. Even deeper in

the crust, the zero-point energy vibrations overlap and result in density-induced fusion, also called

pycnonuclear fusion [40–42].

All these reactions deposit heat at different locations in the crust during active accretion. xnet

7

is a nuclear reaction network that accounts for all possible reactions in the crust and calculates the

composition and nuclear energy deposition profile in the crust. It accepts an initial composition as

input which depends on nuclear burning stages outside the crust. It then calculates the steady-state

composition of the crust in a planar geometry, where the crust is approximated to be very thin which

leads to a constant surface gravity across the crust. It tracks the changes in the composition of an

accreted fluid element due to increasing pressure, 𝑃 = (cid:164)𝑚𝑔𝑡, where (cid:164)𝑚 is the local mass accretion

rate, and 𝑔 is the constant local surface gravity. The density, 𝜌, is calculated using an equation of

state as prescribed in [43]. Increasing time is a proxy for increasing pressure, density, and depth in

the neutron star crust. Reference [32] lists more details about the nuclear physics properties used

in xnet.

Recently, an alternative approach has been suggested in the literature [44–46] to calculate the

heat deposited during accretion in the neutron star crust. Availability of free neutrons in the inner

crust leads to the diffusion of neutrons, violating the condition of hydrostatic equilibrium and the

assumption of a constant red-shifted neutron chemical potential (𝜇∞

𝑁 ). This leads to a very different

Equation of State (EOS) for the accreted crust that is close to catalyzed EOS in the inner crust

[44]. There are no pycnonuclear fusion reactions in the crust as discussed above. As a result,

the total heat deposited in the crust is only 10–30% of previous calculations with pycnonuclear

fusion reactions [46], although the complete reaction pathways are yet to be identified. The heat

deposition only depends on the pressure 𝑃𝑜𝑖 at the outer-inner crust interface that is different from

the traditional neutron-drip pressure [45].

1.1.3 Shallow Heat Source

The nuclear energy output from the above network calculations is not sufficient to explain the

quiescent X-ray observations. Almost all models of accreting neutron star systems need to account

for an additional artificial heat source at relatively shallow depths to match the observed cooling

curves at early times [25, 47–50]. The physical origins of this mysterious heat source is still

unknown and is one of the most challenging open questions in neutron star crust physics. While

most systems require a few MeV/u of additional shallow heating to match the observations, some

8

Figure 1.4 Modeled cooling curves in quiescence for the quasipersistent transient KS 1731–260
with and without an artificial shallow heat source. Shallow heating of 1.23 MeV/u is required by
the models to match quiescent X-ray observations.

require no shallow heating [51], whereas some require ∼ 10 MeV/u [52]. Even the same systems

over different cycles of outburst and quiescence can require either the same [20] or different [21]

levels of shallow heating. Further, the inferred properties of the shallow heat source also depend

on the composition of the envelope and crust of the neutron star. Figure 1.4 shows how accounting

for additional shallow heating of 1.23 MeV/u in KS 1731-260 helps match the cooling curves

models with observations. Several mechanisms ranging from nuclear reactions to astrophysical

processes for the origin of shallow heating have been proposed. They include residual accretion

during quiescence, viscous heating caused by accretion-induced shear [53], convective mixing,

nuclear fusion of light neutron-rich elements [54], electron captures in the crust [43, 55], pion-

induced heating [56], uncertainties in pycnonuclear fusion [42], gravity-wave transport of angular

momentum [57]. However, none of these mechanisms can reproduce observations and the origin

of shallow heating remains unknown.

9

1.2 Urca cooling

Nuclear reactions not only heat the crust but can also cool the crust. This is possible through

a process known as Urca cooling, where successive electron captures and beta-decays alternating

between the same nuclei emit a large flux of neutrinos and anti-neutrinos, respectively, that carry

energy away with them. An Urca type process was first theorized by George Gamow and Mário

Schenberg while visiting Cassino da Urca in Urca, Rio de Janeiro. Even though they named

the process Urca cooling to commemorate their visit, they later backronymed it to UnRecordable

Cooling Agent. Urca cooling is the dominant mechanism for the cooling of very hot, newly born

neutron stars [58]. This happens in the neutron star core via the direct-Urca process where free

protons and neutrons cycle as:

𝑝 + 𝑒− → 𝑛 + 𝜈𝑒

𝑛 → 𝑝 + 𝑒− + ¯𝜈𝑒

Depending on the proton and neutron densities in the core that set their Fermi momenta, bystander

nucleons may also participate in the process to enable momentum conservation. This leads to a

modified Urca process which is less effective. Urca cooling in the crust happens via a new type of

crust Urca process [37] where neutron-rich radioactive nuclei undergo this cycle as:

𝑍 𝑋 + 𝑒− → 𝐴
𝐴

𝑍−1

𝑌 + 𝜈𝑒

𝐴
𝑍−1

𝑌 → 𝐴

𝑍 𝑋 + 𝑒− + ¯𝜈𝑒

Urca cooling was initially expected to not play a role in the cooling of accreting neutron star crusts

since the properties of the crust were calculated using zero-temperature limit approximation. This

is valid since the crust has high density and the Boltzmann temperature is negligible compared to

the electron Fermi energy. However, it was later shown that heat deposited by nuclear reactions

during accretion is enough to raise the temperature to 𝑇 ∼ 109 K where Urca cooling can affect

X-ray observations [59]. Hence, it is important in the modeling of quiescent cooling curves.

Electrons in the crust form a degenerate electron gas and the electron population follows Fermi-

Dirac statistics. The probability distribution for a given energy state (E) to be filled is given

10

Figure 1.5 Formation of an Urca cooling layer in the crust of an accreting neutron star at a depth
where the electron Fermi energy is equivalent to the electron-capture threshold for a neutron-rich
isotope. e−-capture/𝛽−-decay cycles generate a flux of neutrinos and anti-neutrinos that stream
freely out of the star since the crust is not dense enough to trap the neutrinos. This effectively
cools the star, which is also known as Urca cooling.

by:

𝑓 (𝐸) =

1
𝑒(𝐸−𝐸𝐹 )/𝑘 𝐵𝑇 + 1

(1.1)

where 𝐸𝐹 is the Fermi energy of the electron gas, 𝑘 𝐵 is the Boltzmann constant, and T is the

temperature. At zero temperatures, the probability distribution is a step-function, where all the

energy states below the Fermi energy are filled and none of the energy states above the Fermi

energy are occupied. This zero-temperature limit approximation was used in earlier reaction

network calculations that find sharp composition changes when Fermi energy reaches the electron

capture threshold. The reverse 𝛽−-decay is blocked since there is no phase space available for the

emitted electron as all the states below Fermi energy are filled.

However, at finite temperatures, there is a diffusion of occupied states at the Fermi energy. A

few electrons jump from states just below to Fermi level to states just above the Fermi level. The

higher the temperature, the higher the diffusion. Now the electrons from the states above Fermi

11

energy can be captured before the Fermi energy reaches the electron capture threshold. There are

also unoccupied states below Fermi energy for the electron from subsequent 𝛽−-decay. Essentially,

this is opening up the phase space for reverse 𝛽−-decays in a very thin shell at the boundary

between two composition layers. This Urca shell contains both the nuclides that rapidly undergo

e−-capture/𝛽−-decay cycles and enable Urca cooling.

Urca cooling pairs are mostly formed by odd-mass nuclei. Even mass nuclei have either both

even or odd numbers of neutrons and protons. The odd-even staggering in the nuclear binding

energy leads to a lower electron capture threshold on odd-odd nuclei than on even-even nuclei. As

a result, the electron captures mostly proceed in steps of two in even mass nuclei, and Urca cooling

is inhibited. If a ground state capture is forbidden and the capture occurs into excited states, the

captures could proceed in single steps, but when the electron Fermi energy exceeds the ground state

threshold, beta decay is also blocked. Odd mass nuclei, on the other hand, can either be odd-even

or even-odd nuclei and allow for e−-capture/𝛽−-decay cycles at each step.

The rate of Urca cooling is extremely sensitive to temperature. The higher temperature opens a

wider phase space for all participating particles which leads to a ∼ 𝑇 5 dependence on temperature

[60]. Additionally, higher temperatures allow for a wider diffusion of the electron occupancy

probability function given by Fermi-Dirac statistics (Eqn. 1.1) leading to a thicker Urca shell.

Several odd mass open-shell nuclei are also deformed which leads to many low-lying excitations.

Any excited state within ∼ 𝑘 𝐵𝑇 can also contribute to Urca cooling. This leads to a thermal

decoupling of the layers below and above the Urca shell. If a heat source is present just below the

Urca shell, most of the energy it deposits will just be carried away by neutrinos from enhanced

Urca cooling. This puts a very strong constraint on the location of the shallow heat source in the

crust. By extension, it is very important to account for accurate Urca cooling while modeling the

cooling curves of accreting neutron stars in quiescence.

1.3 Nuclear Data for Modeling

The total neutrino luminosity from Urca cooling for a given Urca pair can be approximated

analytically as:

12

𝐿𝜈 ≈ 𝐿34 × 1034 ergs/s × 𝑋 ( 𝐴)𝑇 5
9

(cid:16) 𝑔14
2

(cid:17) −1

𝑅2
10

(1.2)

where 𝑇9 is the temperature in GK, 𝑔14 is the surface gravity in 1014 g/cm2, 𝑅10 is the neutron

star radius in 10 km, X(A) is the mass fraction of the nuclear isotope of mass A undergoing Urca

process, and 𝐿34 is a constant depending on the nuclear properties of that isotope as:

𝐿34 = 0.87

(cid:18) 106𝑠
𝑓 𝑡

(cid:19) (cid:18) 56
𝐴

(cid:19) (cid:18) |𝑄𝐸𝐶 |
4MeV

(cid:19) 5 (cid:18) ⟨𝐹⟩∗
0.5

(cid:19)

(1.3)

where 𝑓 𝑡 is the nuclear 𝑓 𝑡 value representing the comparative half-life of the weak transition (refer

Section 1.3.1), A is the nuclear mass number, 𝑄𝐸𝐶 is the electron-capture Q-value, and ⟨𝐹⟩∗ is the

Coulomb factor given by

with

⟨𝐹⟩∗ =

𝐹+𝐹−
𝐹+ + 𝐹−

𝐹± =

2𝜋𝛼𝑍
1 − 𝑒∓2𝜋𝛼𝑍

(1.4)

(1.5)

where 𝛼 = 1

137 is the electrodynamic fine structure constant, and Z is the atomic charge number.

The rate of Urca cooling depends on the equation of state of dense matter that sets the mass and

radius of the neutron star, which also sets the surface gravity. In addition, it is also very sensitive to

the temperature as discussed in the previous section. The nuclear physics parameters of importance

are the mass fraction, comparative half-life, and the absolute Q-value of the weak transition. Mass

fraction mainly depends on the dynamics of nuclear burning of accreted matter outside the crust and

the composition of ashes formed that get buried in the crust. As a result, the comparative half-life

that depends on the weak transition strength, and the Q-value that depends on nuclear masses are

the key pieces of nuclear data that are required to calculate the rate of Urca cooling in accreting

neutron star crusts.

1.3.1 Weak Interactions

The transition strengths between nuclear levels in the parent and the daughter nuclei are required

to calculate the strength of Urca cooling. These are extracted from the laboratory 𝛽-decay process.

13

This section explains how the laboratory 𝛽-decay half-life and the feeding intensities relate to the

transition strengths.

𝛽-decays are mediated by the weak nuclear force. The transition rate, in general, treating the

decay-causing interaction as a weak perturbation is given by Fermi’s Golden Rule [61] as:

𝜆 =

2𝜋
ℏ

|𝑉 𝑓 𝑖 |2𝜌(𝐸 𝑓 )

(1.6)

where the matrix element 𝑉 𝑓 𝑖 is the integral of interaction V between the initial and final states

given by

∫

𝑉 𝑓 𝑖 =

𝜓∗

𝑓 𝑉 𝜓𝑖𝑑𝑣

(1.7)

and 𝜌(𝐸 𝑓 ) is the density of states with final energy 𝐸 𝑓 . Since 𝛽-decay is a three-body process with

the nucleus, electron, and antineutrino in the final state, the matrix element can be written as:

∫

𝑉 𝑓 𝑖 = 𝑔

[𝜓∗

𝑓 𝜙∗

𝑒𝜙∗

𝜈] O𝜒 𝜓𝑖𝑑𝑣

(1.8)

where 𝜓∗

𝑓 , 𝜙∗

𝑒, 𝜙∗

𝜈 are the final states of the nucleus, electron, and the antineutrino, respectively, O𝜒

is the mathematical operator for the interaction, and 𝑔 is the 𝛽-decay strength constant with

𝑔 = 0.88 × 10−4𝑀𝑒𝑉 . 𝑓 𝑚3

(1.9)

The kinetic energy released in the decay will be shared by the recoiling nucleus, the electron,

and the antineutrino. Let 𝑝 be the final momentum of the electron and 𝑞 be the final momentum of

the antineutrino. 𝑝 and 𝑞 will be related as:

𝑞 =

𝑄 − 𝑇𝑒
𝑐

=

𝑄 − √︁𝑝2𝑐2 + 𝑚2

𝑒𝑐4 + 𝑚𝑒𝑐2

𝑐

(1.10)

where 𝑄 is the Q-value of the decay and 𝑇𝑒 is the kinetic energy of the electron which depends

on 𝑝. The number of final states with the momentum of electron between 𝑝 and 𝑝 + 𝑑𝑝 and the

momentum of antineutrino between 𝑞 and 𝑞 + 𝑑𝑞 is given by:

𝑑2𝑛 = 𝑑𝑛𝑒𝑑𝑛𝜈 =

(4𝜋)2𝑉 2 𝑝2𝑑𝑝𝑞2𝑑𝑞
ℎ6

(1.11)

14

Substituting Eqn. 1.11 and Eqn. 1.8 in Eqn. 1.6, we can write the partial transition rate as:

where

𝑑𝜆 =

2𝜋
ℏ

𝑔2|𝑀 𝑓 𝑖 |2(4𝜋)2

𝑝2𝑑𝑝 𝑞2
ℎ6

𝑑𝑞
𝑑𝐸 𝑓

∫

𝑀 𝑓 𝑖 =

𝜓∗

𝑓 O 𝜓𝑖 𝑑𝑣

(1.12)

(1.13)

is the nuclear matrix element.

Integrating Eqn. 1.12 over all the possible values of electron

momenta, and substituting for 𝑞 from Eqn. 1.10, we can write the total transition rate as:

𝜆 =

𝑔2|𝑀 𝑓 𝑖 |2
2𝜋3ℏ7𝑐3

∫ 𝑝𝑚𝑎𝑥

0

𝐹 (𝑍′, 𝑝) 𝑝2 (𝑄 − 𝑇𝑒)2 𝑑𝑝

(1.14)

where 𝐹 (𝑍′, 𝑝) is the Fermi function which accounts for the influence of the nuclear Coulomb

field. Eqn. 1.14 can be rewritten as:

𝜆 =

𝑓 (𝑍′, 𝐸0)𝑔2𝑚5
2𝜋3ℏ7

𝑒𝑐4|𝑀 𝑓 𝑖 |2

where

1
(𝑚𝑒𝑐)3(𝑚𝑒𝑐2)2
is the Fermi Integral. The half-life of 𝛽-decay is related to the transition rate as:

𝐹 (𝑍′, 𝑝) 𝑝2 (𝐸0 − 𝐸𝑒)2 𝑑𝑝

𝑓 (𝑍′, 𝐸0) =

0

∫ 𝑝𝑚𝑎𝑥

𝑡1/2 =

𝑙𝑛(2)
𝜆

Substituting for 𝜆 and rearranging the Fermi Integral, we get

𝑓 𝑡1/2 = 𝑙𝑛(2)

2𝜋3ℏ7
𝑒𝑐4|𝑀 𝑓 𝑖 |2

𝑔2𝑚5

(1.15)

(1.16)

(1.17)

(1.18)

This is also known as the comparative half-life for the weak transition. This 𝑓 𝑡 value depends only

on the nuclear matrix element, whereas all other effects are embedded in the Fermi Integral. This

helps in comparing the overlap between different sets of initial and final nuclear wave functions.

Since the 𝑓 𝑡 values range in many orders of magnitude, they are often quoted as log-ft values for a

transition. The square of the nuclear matrix element |𝑀 𝑓 𝑖 |2 is referred to as the transition strength.

In the 𝛽-decay process, along with the energy and momentum, angular momentum should also

be conserved. Since quantum mechanical particles participate in the transition, the total angular

15

Transition

Fermi (“Superallowed”)
Gamow-Teller (“Allowed”)
First-forbidden
Second-forbidden
Third-forbidden
...

L

0
0
1
2
3
...

Δ𝐽

Δ𝜋

0
0,1
0,1,2
1,2,3
2,3,4
...

0
0
1
0
1
...

Table 1.1 Table listing the quantum mechanical selection rules for different degrees of
allowed/forbidden weak transitions. L denotes the angular momentum carried by the decay
particle (in units of ℏ), whereas (Δ𝐽, Δ𝜋) denotes the change in spin-parity of the initial and final
nuclear states. Note that Δ𝐽 is dependent on L.

momentum is quantized and is given by the vector sum of orbital angular momentum and spin

angular momentum. Naturally, there occur some selection rules between the spin-parity of initial

and final nuclear states (Δ𝐽, Δ𝜋), that must be followed as listed in Table 1.1. The difference in

angular momentum between nuclear states is carried by the decay particles. This difference, along

with the change in parity characterizes the ‘allowedness’ of the weak transition. Allowed decays are

where the decay particles carry no orbital angular momentum (𝐿 = 0) whereas forbidden decays

are where the decay particles carry the difference in angular momentum (𝐿 > 0).

It also sets

the degree of ‘forbiddenness’ of a transition. Forbidden transitions do occur but each degree of

forbiddenness slows down the rate of transition by a few orders of magnitude. For the electron

to carry away the difference in angular momentum, it should be emitted off-center of the nucleus

where the wavefunction drops off exponentially.

1.3.2 Nuclear Masses

The amount of energy released in a nuclear reaction or decay (also called as Q-value) is related

to the total mass loss by Einstein’s mass-energy equivalence principle,

𝐸 = 𝑚𝑐2.

(1.19)

As a result, the mass of a nucleus is an important quantity in any calculation involving nuclear

reactions. The Q-value is given by the difference in the mass of reactants and the mass of products.

For nuclear decays mediated by the weak force, there are three possible types of 𝛽-decays in the

16

Figure 1.6 The nuclear masses landscape on the chart of nuclides. The colored region is for nuclei
whose masses have been experimentally measured as of AME2020 database [62]. The grey
region is the extent of nuclear existence, i.e. the region between the proton and the neutron
driplines. Points in the grey area that aren’t colored are nuclei whose masses remain to be
measured experimentally.

laboratory:

𝛽− : 𝐴

𝑍 𝑋 →𝐴

𝑍+1

𝑌 + 𝑒− + ¯𝜈𝑒

𝛽+ : 𝐴

𝐸𝐶 : 𝐴

𝑍 𝑋 →𝐴

𝑍−1
𝑍 𝑋 + 𝑒− →𝐴

𝑌 + 𝑒+ + 𝜈𝑒

𝑌 + 𝜈𝑒.

𝑍−1

The Q-values for each of these reactions are given by:

𝑄(𝛽−) : [𝑚( 𝐴

𝑍 𝑋) − 𝑚( 𝐴

𝑌 )]𝑐2

𝑍+1

𝑄(𝛽+) : [𝑚( 𝐴

𝑍 𝑋) − 𝑚( 𝐴

𝑍−1

𝑌 ) − 2𝑚𝑒]𝑐2

𝑄(𝐸𝐶) : [𝑚( 𝐴

𝑍 𝑋) − 𝑚( 𝐴

𝑍−1

𝑌 )]𝑐2 − 𝐵𝑛.

Note that these are atomic masses (that in turn depend on the nuclear mass), so the electron mass

cancels for 𝛽−-decay but adds up for 𝛽+-decay. The neutrino mass is considered to be negligible

17

here. 𝐵𝑛 is the binding energy of the captured n-shell electron. To calculate Urca cooling strength

as per Equation (1.3), the |𝑄𝐸𝐶 | term is the Q-value for 𝑒−-capture. However, since the electron is

captured from the degenerate electron gas, 𝐵𝑛 can be neglected. Thus 𝑄′(𝐸𝐶) = 𝑄(𝛽−) is used to

estimate the Urca cooling luminosity. Consequently, it depends on the nuclear masses.

In crusts of accreting neutron stars, electron captures can happen all the way up to the neutron

dripline, where nuclei in their ground state are no longer neutron bound. As a result, we need to know

accurate nuclear masses for all the neutron-rich nuclei to calculate the total Urca cooling luminosity

throughout the crust. Figure 1.6 shows the status of current experimental mass measurements and

theoretical approximates of the extent of driplines on the nuclear chart. It is immediately clear that

many of the nuclear masses for neutron-rich nuclei still need to be measured. Even after considerable

experimental efforts, many of these nuclear mass measurements remain out of our reach for the

foreseeable future and we need to rely on theoretical mass model predictions. There is, however, a

variety of mass models to choose from [63, 64]. Each theoretical model has different assumptions

and approximations and is calibrated on different subsets of experimental data. Moreover, most of

the predictions lack uncertainties, and extrapolating nuclear masses in the very neutron-rich region

yields completely different trends. This becomes a problem for crust cooling calculations since we

do not know apriori which mass model is correct and should be relied on. As a result, there is

a dire need for a quantified global nuclear mass model that can accurately predict nuclear masses

throughout the nuclear chart with quantified uncertainties. Chapter 5 outlines the procedure to

create such a mass model for use in astrophysical models by averaging the prediction of multiple

theoretical mass models using Bayesian Model Averaging (BMA).

1.4 33Mg - 33Al Transition for Urca Cooling

Effective Urca cooling requires a large abundance of nuclei in the Urca pair and a strong ground-

state to ground-state 𝛽-decay/𝑒−-capture transition strength. Xnet calculations starting with ashes of

typical Type-I X-ray bursts powered by rp-process show significant Urca cooling. Figure 1.7 shows

neutrino cooling luminosity in such calculations as a function of the mass number. Transitions in

the isobaric mass chain with A = 33 provide the strongest Urca cooling. The dominance of mass A =

18

Figure 1.7 Urca cooling luminosity in a particular isobaric mass chain for a neutron star crust
composed of typical Type-I X-ray burst ashes powered by rp-process as a function of mass
number. These calculations are performed with the Xnet nuclear reaction network for a crust with
a constant temperature of 0.5 GK. Mass A = 33 chain, circled in red, has the highest Urca cooling
luminosity.

33 cooling stems primarily from two Urca pairs with strong ground-state to ground-state transitions:

33Si ↔ 33Al [65], 33Al ↔ 33Mg [66]. However, there are questions in the literature about the strong

ground-state to ground-state transition strength of 33Mg ↔ 33Al [67]. The measured nuclear log-ft

value of 5.2 [66] is anomalously low for the first-forbidden nature of this decay transition. As a

result, addressing the nuclear physics uncertainties in this transition is critical for a quantitative

prediction of Urca cooling strength.

1.4.1 Anomaly in Nuclear Data

The strong ground-state to ground-state transition strength in 33Al ↔ 33Mg has been inferred

in the 𝛽-decay direction with high-resolution 𝛽-delayed 𝛾-ray spectroscopy. A strong ground-state

branch of 37(8)% was determined by Tripathi et al. [66]. This translates into a comparative half-life

of log-ft = 5.2. They did this by subtracting the feeding to all excited states inferred from 𝛾-ray

intensities from the various de-excitation cascades. Since the spin-parity of the ground state of 33Al

is 5/2+, they conclude that the strong ground-state branch is only consistent with a positive parity

of the ground state in 33Mg. They assigned a ground-state spin-parity of 3/2+ to 33Mg, so that the

33Mg ↔ 33Al ground-state to ground-state transition is an allowed transition as per selection rules

19

outlined in Table 1.1. However, Yordanov et al. measured a negative magnetic moment for the

ground-state of 33Mg [68] and assigned a 3/2− ground-state spin-parity for 33Mg to be consistent

with their observations. This is further supported by Coulomb dissociation measurements [69] and

spectroscopy of 33Mg with knockout reactions [70] and is typically adopted in the literature [71].

This makes the 33Mg ↔ 33Al ground-state to ground-state transition a first-forbidden transition.

Even the Evaluated Nuclear Structure Data Files (ENSDF) states that log ft = 5.2 is small for

first-forbidden decay. Yordanov et al., in their comment on Tripathi et al. [67] provide a possible

explanation stating “the large Q-value of 13.4 MeV opens a wide window for 𝛽-decay up to the

5.5 MeV neutron-separation energy. With the highest reported level at 4.73 MeV, a gap of nearly

1 MeV is left for non-observed states. In such cases, much of the 𝛽-intensity remains undetected,

a problem known as Pandemonium.” Furthermore, there are also questions about the 𝛽-delayed

neutron branching ratio, or the (𝑃𝑛) value of 33𝑀𝑔 decay. The eXperimental Unevaluated Nuclear

Data List (XUNDL), unlike ENSDF presents a reanalysis of the 𝛽-delayed 𝛾-rays from the neutron

emission daughter 32Al measured by Tripathi et al. Along with the likely incomplete neutron

spectra from an experiment at the GANIL facility in France published only in a short conference

proceeding [72], they conclude that P𝑛 = 14% as reported by Langevin et al. [73] seems too low by

almost a factor of 2. This leads to the following possibilities or a combination of these possibilities

as an explanation for the anomalously high ground-state to ground-state transition strength in the

decay of 33Mg:

1. The high transition strength with a low log ft value measured by Tripathi et al. in the 𝛽-decay

of 33Mg is due to nuclear structure effects from the deformed 33Mg nucleus being in the

island of inversion near the N = 20 closed neutron shell.

2. The level scheme of 33Mg is likely incomplete and there are unobserved weak 𝛾-transitions

from higher energy levels that are currently unaccounted for.

3. The P𝑛 value is higher than currently reported, which will reduce the inferred 𝛽-decay

ground-state branching.

20

1.5 Dissertation Goals and Outline

This dissertation aims to improve the quality of nuclear data to help accurately quantify the

strength of Urca cooling in accreting neutron star crusts. It tackles this problem on two fronts:

experimental and computational. For my dissertation, I have performed nuclear physics experiments

in the laboratory and also used state-of-the-art statistical analysis methods in nuclear theory to refine

the quality of nuclear data that goes into nuclear reaction network calculations of accreting neutron

stars. More specifically, the goals of this dissertation are two-fold.

1. To infer the correct ground-state to ground-state 𝛽-decay transition strength in the decay of

33Mg by remeasuring the complete 𝛽-delayed 𝛾-intensity distribution using Total Absorption

Spectroscopy (TAS) that is free of the Pandemonium effect [74] (Refer Section 2.1.3). In

addition to 𝛽-feeding intensities in 33Al, this will also require the measurement of 𝛽-delayed

neutron-branching ratio (P𝑛 value).

2. To create a global nuclear mass model with quantified uncertainties that agree with current

experimental mass measurements and reliably extrapolates nuclear masses to the neutron-rich

regions on the nuclear chart.

Chapter 2 presents the experimental technique and the methods used for the 𝛽-decay measure-

ment of 33Mg. The experimental data analysis procedure is outlined in Chapter 3 and the results

from this experiment are presented in Chapter 4. Chapter 5 then switches gears and presents

the statistical framework for building a quantified nuclear mass model using Bayesian Machine

Learning techniques. The astrophysical impact of experimental measurement as well as the refined

nuclear mass model is discussed in Chapter 6. Finally, the dissertation ends with conclusions and

future outlook in Chapter 7.

21

CHAPTER 2

EXPERIMENTAL MEASUREMENT

This chapter focuses on the experimental measurement technique for the ground-state to ground-

state 𝛽-decay transition strength from 33Mg – 33Al. This quantity is being remeasured to resolve

the conflict in the literature arising from an anomalously high ground-state branch [67, 75]. This

measurement is performed using Total Absorption Spectroscopy (TAS) to avoid any systematic

errors, especially those arising from the Pandemonium effect [74]. Subsequent sections discuss the

experimental methodology and 𝛽-delayed spectroscopy, production and delivery of 33Mg beam,

details about the BCS, NERO, and SuN detectors used during the experiment, and aspects of the

data acquisition system to record and store the data from the experiment.

2.1 Experimental Technique

𝛽-decay is a three body process involving the nucleus undergoing 𝛽-decay, the emitted electron,

and the emitted neutrino. Energy from the Q-value of the decay is predominantly shared by the

electron and the neutrino. A very small fraction also goes to the recoiling nucleus to conserve

momentum. Due to the three-body nature of 𝛽-decay, the energy of the electron from 𝛽-decay forms

a continuous spectrum. Transitions to each excited state in the daughter nucleus would contribute to

the electron energy spectrum and deconvoluting it to get the contribution from individual transitions

would be very difficult. It is, therefore, not very useful to measure the electron energy from the

𝛽-decay. Instead, the contribution to excited states is estimated by detecting the 𝛽-delayed 𝛾 rays,

as the 𝛾 rays have precise energies depending on the level scheme of the daughter nucleus. 𝛽-decay

transitions to the ground state in the daughter nucleus will not have any 𝛽-delayed 𝛾 radiation. The

ground-state branch can then be inferred by subtracting the contributions of all the excited states

from the total transition strength. In addition, since we are dealing with very neutron-rich nuclei,

the neutron separation energies are lower than the 𝛽-decay Q-value.

If the 𝛽-decay transition

happens to an excited level above the neutron separation energy in the daughter nucleus, it can

deexcite either via 𝛾-decay or neutron emission.

If there is a 𝛽-delayed neutron emission, the

𝛽-delayed neutron branch also needs to be measured and subtracted. In summary, the experimental

22

Figure 2.1 Schematic of the experimental technique to measure the ground-state to ground-state
𝛽-decay transition strength of 33Mg. Feeding intensities to excited states in 33Al are measured by
detecting 𝛽-delayed 𝛾 rays with SuN detector whereas 𝛽-delayed neutron branching ratio to 32Al
is measured by counting neutrons with NERO detector. Subtracting both from the total transition
strength yields the ground-state branch.

technique to measure the ground-state to ground-state 𝛽-decay transition strength is to subtract the

contributions from excited states by counting 𝛽-delayed 𝛾 rays and 𝛽-delayed neutrons. Figure 2.1

shows the schematic for this experimental technique.

2.1.1

𝛽-decay Kinetics

Nuclear decay is kinetically a first-order transition, where the number of decays at time 𝑡 is

directly proportional to the total number of nuclei in a sample at time 𝑡. This can be represented as

𝑑𝑁 (𝑡)
𝑑𝑡

= −𝜆𝑁 (𝑡)

(2.1)

The proportionality constant, 𝜆 is the rate of 𝛽-decay. Solving the differential Equation 2.1 yields

the solution for the number of parent nuclei in the sample at time 𝑡 as

𝑁 (𝑡) = 𝑁0𝑒−𝜆𝑡

(2.2)

23

where 𝑁0 is the number of parent nuclei in the sample at time 𝑡 = 0. The rate 𝜆 is related to the

half-life 𝑇1/2 as

The number of daughter nuclei in a sample at time 𝑡 is equal to the number of decays of parent

𝜆 =

𝑙𝑛(2)
𝑇1/2

(2.3)

nuclei and is given by

𝑁′(𝑡) = 𝑁0(1 − 𝑒−𝜆𝑡)

(2.4)

Equation 2.4 is only valid when the daughter nucleus is stable against further nuclear decay.

However, for experimental measurements on exotic nuclei in the laboratory, the daughter nuclei are

often unstable against 𝛽-decay, and Equation 2.4 has to be appropriately modified. For example,

in the 𝛽-decay of 33Mg, the daughter 33Al has an even shorter half-life than the parent and starts

undergoing 𝛽-decay as soon as it is produced. Considering that the daughter nucleus decays with

rate 𝜆′, the differential equation can be rewritten as

𝑑𝑁′(𝑡)
𝑑𝑡

= 𝜆𝑁 − 𝜆′𝑁′

and the solution for the number of daughter nuclei in a sample is given by

𝑁′(𝑡) = 𝑁′
0

𝑒−𝜆′𝑡 − 𝑁0

𝜆
𝜆′ − 𝜆

(cid:2)𝑒−𝜆′𝑡 − 𝑒−𝜆𝑡 (cid:3)

(2.5)

(2.6)

Note that 𝑁′ depends on both 𝜆 and 𝜆′. For a nuclear decay chain involving multiple transitions,

the amount of any 𝑘 𝑡ℎ nucleus in the decay chain at time 𝑡 is described by Bateman equations [76],

and the analytical solutions [77] are given by

𝑁𝑘 (𝑡) =

𝑘
∑︁

𝑗=1

𝐴 𝑗 𝑒−𝜆 𝑗 𝑡

𝑙=𝑘−2
(cid:214)

𝑙= 𝑗

−𝜆𝑙+1
𝜆𝑙+1 − 𝜆 𝑗

(2.7)

where 𝐴 𝑗 are constants relating to the initial quantity of the 𝑗𝑡ℎ species, and 𝜆 𝑗 are the corresponding

decay constants.

For an experimental decay curve (histogram of decay times for a nucleus) based on the time

of electron detection that does not have 100% efficiency, even the first registered electron could

have been produced from the 𝛽-decay of any nucleus in the decay chain. As a result, to extract

24

the half-life of the parent nucleus from the experimental decay curve, it has to be fit to the sum of

contributions from all decaying species in the decay chain. The half-lives of all daughter nuclei

have to be either known or estimated from a multi-parameter fit to the decay curve.

2.1.2

𝛽-delayed Spectroscopy

2.1.2.1

𝛽-delayed 𝛾 decay

An excited nucleus predominantly deexcites by emitting 𝛾 rays, unless other competing channels

are present (for example, light particle emission). The lifetimes of these excited states are usually

so short that the decay can be considered almost instantaneous for all practical purposes (except

when an isomer is present). The energy of the emitted 𝛾-ray photon is equal to the difference

between the energies of the initial and the final level in a nucleus. Since the photon also carries

some angular momentum depending on its multipolarity (L), the total angular momentum should

also be conserved. Let (𝐼𝑖, 𝜋𝑖) be the total angular momentum and parity of the initial excited state

and (𝐼 𝑓 , 𝜋 𝑓 ) be the total angular momentum and parity of the final deexcited state. For a multipole

radiation carrying angular momentum L,

−→
𝐼𝑖 =

−→
𝐿 +

−→
𝐼 𝑓

(2.8)

L = 0 is not allowed since there is no monopole radiation. L = 1 corresponds to dipole radiation,

L = 2 is the quadrupole radiation, and so on. Whether the emitted radiation is of the electric or

magnetic type is determined by the change in parity (Δ𝜋) between the initial and final levels. (Δ𝜋

= no) corresponds to even parity for the radiation field. This is valid for electric transition where

L = even and magnetic transition where L = odd. On the other hand, (Δ𝜋 = yes) corresponds to

odd parity for the radiation field which is given by electric transition where L = odd and magnetic

transitions where L = even. These selection rules can be summarized as follows:

|𝐼𝑖 − 𝐼 𝑓 | ≤ 𝐿 ≤ |𝐼𝑖 + 𝐼 𝑓 |

(𝐿 ≠ 0)

Δ𝜋 = no:

even electric, odd magnetic

Δ𝜋 = yes:

odd electric, even magnetic

25

Consider, for example, a nucleus deexciting from a 5/2+ state to a 3/2− state. The possible values of

L would range from |5/2 - 3/2| = 1 to |5/2 + 3/2| = 4. And since the parity is changing, there will be

electric transitions with odd L and magnetic transitions with even L. As a result, a 𝛾-deexcitation

from 5/2+ → 3/2− will happen via E1, M2, E3, and M4 radiation. According to Weisskopf estimates

for the transition rates for each electromagnetic radiation type, the lower order multipolarities are

preferred and the transition rate drops down by about five orders of magnitude for one unit increase

in polarity. For the same order of multipolarity, electric radiation is two orders of magnitude more

likely than magnetic radiation. An excited state often has multiple choices for deexcitation, and the

fastest transitions are always preferred in nature. As a result, most practical purposes consider only

E1, M1, and E2 transitions.

2.1.2.2

𝛽-delayed neutron emission

For exotic neutron-rich nuclei, the neutron separation energies can be lower than the 𝛽-decay

Q-value. In that case, 𝛽-decay can populate excited levels in the daughter nucleus above the neutron

separation threshold. This opens an additional channel for the excited daughter nucleus to deexcite

via 𝛽-delayed neutron emission. This neutron emission can happen either into the ground state or

to excited states in 𝛽-delayed neutron daughter, if energetically possible. The energy of the emitted

neutron, 𝐸𝑛, is given by:

𝐸𝑛 = 𝐸𝑥,𝑖 − 𝐸𝑥, 𝑓 − 𝑆𝑛

(2.9)

where 𝐸𝑥,𝑖 is the excitation energy in the 𝛽-decay daughter nucleus, 𝐸𝑥, 𝑓 is the excitation energy

of the 𝛽-delayed neutron daughter (𝐸𝑥, 𝑓 = 0 for neutron emission into the ground state), and S𝑛 is

the neutron separation energy of the 𝛽-decay daughter. The sum of neutron emission probabilities

from all excited states is referred to as the total 𝛽-delayed neutron emission branching ratio, or the

P𝑛 value.

2.1.3 Pandemonium Effect and Total Absorption Spectroscopy

The 𝛽 feeding intensity to an excited state in the daughter nucleus is measured by detecting the

𝛾 ray intensity from that level. However, deexcitation of higher energy levels can also occur via

that intermediate state. The total 𝛾 intensity (𝐼𝑇 ) from that level will then be the sum of 𝛽 feeding

26

Figure 2.2 Schematic explanation of the Pandemonium effect [74]. Weak 𝛽-delayed 𝛾-transitions
from higher excitation energies are hard to detect with high-resolution 𝛾-ray detectors. The
inferred 𝛽-feeding intensities are then wrongly attributed to lower excitation energies. This is a
known source of systematic error in data analysis. Figure adapted from [78].

intensity (𝐼𝛽)and the feeding intensities from higher energy levels (Σ𝐼𝑖).

𝐼𝑇 = 𝐼𝛽 + Σ𝐼𝑖 ⇒ 𝐼𝛽 = 𝐼𝑇 − Σ𝐼𝑖

(2.10)

To apply this formalism in practice, a complete level scheme of the daughter nucleus should be

known to accurately assign observed 𝛾 rays to corresponding transitions between excited states. To

this end, a high-resolution germanium detector is used to isolate individual 𝛾-rays. However, this

high resolution comes at the cost of low efficiency where weaker transitions go undetected. This

is true, especially for 𝛽-decays with high Q-value and daughter nuclei having high level densities.

Each additional 𝛾 ray in a high multiplicity cascade diminishes the likelihood of detecting all the

transitions. Due to this, Σ𝐼𝑖 is underestimated and 𝐼𝛽 is overestimated. This causes systematic

errors in the measurement of 𝛽 feeding intensities, a problem known as the Pandemonium effect.

27

Figure 2.3 Comparison of 𝛾-ray spectra with high-resolution Ge detectors and high-efficiency
TAS detectors for a hypothetical decay to two feeding states in daughter nucleus. While Ge
detectors show peaks at individual transitions, TAS detectors sum the energies in each event, such
that the peaks correspond to excitation energies. The high efficiency of TAS detectors then helps
eliminate the Pandemonium effect shown in Figure 2.2. Figure adapted from [78].

Figure 2.2 gives a schematic explanation of this problem. This was first observed in the Monte

Carlo simulation of 𝛽-decay in a fictional nucleus called ‘Pandemonium’ [74].

This problem is addressed in this work by a method called Total Absorption Spectroscopy

(TAS). In the trade-off between resolution and efficiency of 𝛾-ray detection, TAS employs very

high efficiency (∼100%) scintillator detectors with close to 4𝜋 solid angle coverage that are on the

opposite end of the trade-off spectrum from Ge detectors. High efficiency allows for the detection

of even the weakest branch and the energies of all transitions in a single cascade are summed on an

event-by-event basis. Unless there is an isomer present in the daughter nucleus, a peak in the TAS

spectrum will directly correspond to an excited level in the daughter nucleus. Figure 2.3 shows

the difference between a 𝛾 spectrum from Ge detector v/s TAS spectrum for a hypothetical decay

scheme. In the TAS method, instead of worrying about systematic errors in Σ𝐼𝑖, the 𝛽 feeding

intensity is simply 𝐼𝛽 = 𝐼𝑇 . This is why TAS measurements are free of the Pandemonium effect.

28

Figure 2.4 Beamline at National Superconducting Cyclotron Laboratory (NSCL) for producing
and delivering 33Mg to the experimental end-station in S2 vault. Shown here are the coupled
cyclotrons - K500 and K1200, 9Be fragmentation target, A1900 fragment separator, and detectors
for particle identification. Figure adapted from the NSCL website.

2.2 Beam Delivery

A beam of 33Mg fragments was delivered by the National Superconducting Cyclotron Labo-

ratory (NSCL) facility. NSCL (now decommissioned) used the in-flight fragmentation method to

produce exotic beams of radioactive nuclei. In this method, an accelerated heavy-ion primary beam

undergoes fragmentation in its flight path and produces a cocktail beam with many different parti-

cles. The fragment of interest is separated using a fragment separator that is typically comprised of

electromagnetic optical elements. By carefully tuning the separator and making appropriate cuts,

the beam rate and purity of the fragments delivered to user experiments can be optimized. Figure

2.4 shows the different components of the NSCL beamlines.

For this experiment, a completely stripped primary beam of 48Ca20+ was accelerated to 140

MeV/u with the help of the K500 and K1200 coupled cyclotrons. This beam was then impinged on

a 9Be fragmentation target with a thickness of 846 mg/cm2. Fragmentation of 48Ca produced 33Mg

along with many other isotopes. The 33Mg was separated using the A1900 fragment separator

[80, 81]. The A1900 fragment separator consists of four magnetic dipoles that deflect different

29

Figure 2.5 LISE++ [79] simulations for the optimized range distribution of 33Mg12+ in the
implantation silicon detectors for both parts of the experiment. All the fragments of interest are
completely deposited in the center with minimal contaminants. Silicon detectors also have enough
thickness to detect the 𝛽-decay electron.

30

beam components based on their momentum-to-charge ratio. It also has 24 magnetic quadrupoles

for focusing the beam throughout its flight path. Additionally, an aluminum wedge-shaped degrader

with a 0◦ thickness of 450 mg/cm2 was placed at I2 image position to further separate the beam

components by energy loss and subsequent energy dispersion. An eXtended Focal Plane (XFP)

scintillator was used for the time of flight measurement for particle identification as discussed in

Section 2.2.1. Finally, another aluminum degrader with variable thickness (based on the angle)

was placed just before the experimental end-station to ensure that the fragments stopped in the

implantation detectors. All the parameters for beam delivery were optimized for the rate and purity

of 33Mg fragment based on LISE++ [79] simulations. Figure 2.5 shows the range distribution for

all fragments implanted in the silicon implantation detectors for both parts of the experiment (refer

to Section 2.3 for an overview). 33Mg12+ is deposited at the center of the silicon implantation

detectors with minimal contaminants.

2.2.1 Particle Identification

Even though the contaminants in the beam delivered to the experimental end station are minimal,

they can still contribute to the background and make the data analysis difficult. Therefore, particle

identification (PID) is done on an event-by-event basis, and only the events that pass the PID cuts

are considered for data analysis. There are a few ways to perform particle identification. In this

experiment, the 𝛿E-TOF method was used. For a particle with atomic number 𝑍, atomic mass

number 𝐴, and charge 𝑄, the energy loss in a thin detector 𝛿𝐸 is proportional to 𝑍 2, and the time

of flight TOF between any two points which depends on the velocity of the particle is proportional

to 𝐴/𝑄. Plotting both quantities on a 2-dimensional plot leads to a PID plot with each blob

corresponding to a particular nuclide as seen in Figure 3.5. The energy loss was measured in a thin

silicon PIN detector just upstream of the SuN detector. The time of flight was measured between

the XFP scintillator and the silicon PIN detector. Once the 33Mg blob on the PID plot is identified,

only events that pass the cuts for that blob are used for analysis.

31

Figure 2.6 Picture of the (a) NERO and (b) SuN detector set-up at the experimental decay station
in S2 vault.

2.3 Detector Setup

The experiment utilized an ensemble of detectors to measure the charged particles (33Mg

implants, electrons), neutrons, and 𝛾 rays. Additional detectors were also used for particle identi-

fication, to veto light particle background, to veto punchthrough events, etc. The experiment was

performed in two parts:

1. BCS-NERO part focused on measuring the 𝛽-delayed neutron branching ratio, i.e.

the P𝑛

value. It utilized the Beta Counting System (BCS) for detecting the 33Mg implant and decay

electron and the NERO detector for 𝛽-delayed neutrons.

2. SuN-miniDSSD part focused on measuring the 𝛽-feeding intensities to excited states in 33Al.

It utilized the miniDSSD for 33Mg implant and decay electron and the SuN detector for

𝛽-delayed 𝛾 rays.

2.3.1 BCS-NERO

BCS or the Beta Counting System is a stack of silicon detectors to detect charged particles. The

main detector is a Double-sided Stripped Silicon Detector (DSSD). It is a 979 𝜇m thick Micron

Semiconductor BB1 detector with an active area of 39.90 × 39.90 mm2. The front side of the

DSSD is segmented into 40 vertical strips and the back side is segmented into 40 horizontal strips

32

Figure 2.7 Side view and front view of the NERO detector with BCS chamber inside. 33Mg
fragments are implanted onto the DSSD and 𝛽-delayed neutrons are detected by concentric 3He
and BF3 rings embedded in a high-density polyethylene (HDPE) matrix. This design is optimized
to have a constant neutron detection efficiency over a wide range of neutron energies. Figure
reproduced from [82]. Refer to Figure 2.6(a) for the actual set-up.

Detector

Thickness (𝜇m) Bias Voltage (V)

Gain

PIN1
PIN2
PIN3
DSSD
SSSD
Plastic Scintillator

525
1041
488
979
982
∼25400

40
75
50
280
-210
-850

Low
Low
Low
Low and High
High
High

Table 2.1 Properties of detectors for particle identification and detecting implantation and decay
events in the BCS-NERO part of the experiment.

which together create 1600 pixels of 1 mm2 to localize the interaction vertex of charged particles

detected. Each channel is read out by a Multi-Channel Systems CPA-16 dual-gain preamplifier for

a total of 160 channel readouts. The 80 low-gain channels are used to detect the heavy ion implants,

whereas the 80 high-gain channels are used to detect 𝛽-decay electrons. A coincidence between a

signal from a front strip and a back strip of the DSSD was required for a valid signal to be read out.

This was achieved using DDAS breakout modules. A combination of OR signals from the 40 front

strips AND a combination of OR signals from the 40 back strips was used as an external trigger for

the DSSD.

In addition to the DSSD, a Single Sided Stripped Silicon Detector (SSSD) was placed just

behind the DSSD to act as a veto for punchthrough events. In such events, the implant punches

33

Detector

Type Bias Voltage (V)
3He
Quad A 1–4
Quad A 5–15 BF3
3He
Quad B 1–4
Quad B 5–15 BF3
3He
Quad C 1–4
Quad C 5–15 BF3
3He
Quad D 1–4
Quad D 5–15 BF3

1130
2620
1130
2620
1080
2620
1350
2620

Table 2.2 Properties and types of neutron long counter detectors embedded in the NERO matrix
to detect 𝛽-delayed neutrons and estimate the neutron branching ratio.

through without stopping in the DSSD. These have to be discarded as the decay electron will not

be registered in the DSSD and successful implant-decay correlation cannot be done. Silicon PIN

detectors are placed upstream of the DSSD for particle identification as discussed in Section 2.2.1.

PIN1 and PIN2 were placed high upstream in the experimental setup. A third detector PIN3 was

also placed in the BCS stack to check the transmission from the PINs to the BCS stack. Additionally,

the beam may contain light particles that, when passing through the DSSD, will register a signal

similar to the decay electrons in the high-gain channels. To detect and discard such events, a

plastic scintillator is placed at the end of the stack that is used to distinguish the light particles and

electrons. The BCS stack is embedded in the NERO matrix as shown in Figure 2.7. Table 2.1 lists

the properties of detectors in the BCS stack along with their bias voltages.

The Neutron Emission Ratio Observer (NERO) is a neutron long counter system to detect the

𝛽-delayed neutrons [82]. It consists of 44 BF3-filled tubes and 16 3He-filled tubes inserted into a

High-Density Polyethylene (HDPE) matrix. The total dimensions of the matrix are 60 × 60 × 80

cm3 and the density of the matrix is 𝜌 = 0.93(1) g/cm3. It is divided into 4 quadrants, each housing

11 BF3 tubes and 4 3He tubes as listed in Table 2.2. It also lists the operating bias voltages for these

tubes. The matrix has a cylindrical bore of radius 11.4 cm to accommodate the BCS chamber. All

the tubes are arranged in three concentric circles with 3He tubes occupying the innermost circle as

shown in Figure 2.7. The choice of this design is to optimize the efficiency of neutron detection

and to have a constant efficiency over a wide range of neutron energies. 3He tubes have a relatively

34

Figure 2.8 Schematic of the SuN detector (left) along with its cross-sectional view (right). SuN
comprises of NaI(Tl) scintillator crystals and has a total of 8 segments (T1-T4 on the top, B1-B4
on the bottom) with three photomultiplier tubes (PMTs) for each segment. Figure adapted from
[83]. Refer to Figure 2.6(b) for the actual set-up.

higher neutron detection efficiency and the fastest neutrons will interact first with the tubes in the

innermost rings. The neutrons are moderated and slowed down while interacting with the tubes in

the outer rings.

𝛽-delayed neutrons are moderated to thermal energies in the HPDE matrix with moderation

times of ∼ 200𝜇s. HDPE is an ideal choice due to the high density of protons since protons require

the least amount of scattering events to thermalize the neutrons. The neutrons induce 3He(n,p)3H

reaction in the 3He tubes and 10B(n,𝛼)7Li reaction in the BF3 tubes. The charged particles from

these reactions ionize the gases in the tubes and the electron cascades are detected due to the

applied voltages across the tubes, indicating a neutron detection. The energy information of the

neutron is completely lost in this method of detection. The detector has been designed for constant

efficiencies up to ∼ 2 MeV, so as long as neutron energies are below that, the neutron energy need

not be known. NERO achieves a neutron detection efficiency of ∼ 35%.

35

Detector

Thickness (𝜇m) Bias Voltage (V)

Gain

PIN1
PIN2
miniDSSD
Si Surface barrier

525
1041
1030
500

40
75
110
60

Low
Low
Low and High
High

Table 2.3 Properties of detectors for particle identification and detecting implantation and decay
events in the SuN-miniDSSD part of the experiment.

2.3.2 SuN-miniDSSD

The second part of the experiment was to detect the 𝛽-feeding intensities to excited states in

33Al by detecting the 𝛽-delayed 𝛾 rays in the SuN detector. A miniDSSD was used to detect the

implants and decay electrons instead of the BCS. This was because the cylindrical bore in SuN

to allow the beam to pass through was much smaller than in NERO. The miniDSSD is a Micron

Semiconductor BB8 detector that is 1030 𝜇m thick and has an active area of 20.0 × 20.0 mm2. Both

sides of the miniDSSD are orthogonally segmented into 16 strips of 1.25 mm pitch, forming a total

of 256 pixels. It is also read out by a Multi-Channel Systems CPA-16 dual-gain preamplifier. The

low-gain channels detect 33Mg implants and the high-gain channels detect electrons. Coincidence

between a signal from a front strip and a back strip was also required to read out a valid signal

from miniDSSD. This was again achieved with an external trigger using DDAS breakout modules,

similar to the BCS-DSSD. Punchthrough events to be vetoed were identified with a Silicon surface

barrier detector placed downstream of the miniDSSD. It was also used to veto events from light

particles in the beam that mimic decay events in the miniDSSD. Particle identification detectors

are the same as used in the BCS-NERO part of the experiment, except no PIN3 is required. Table

2.3 lists the properties of different detectors along with their bias voltages for this part of the

experiment.

Summing NaI or the SuN detector is a Total Absorption Spectrometer composed of NaI(Tl)

scintillator crystals. These have high intrinsic efficiency in detecting 𝛾 rays. Moreover, owing to

an almost 4𝜋 solid angle coverage of SuN, the geometric efficiency is also high. Combined, this

leads to a very high 𝛾 detection efficiency which is a primary requirement for Total Absorption

36

Spectroscopy. NaI is a commonly used scintillator material due to the ease of producing large

monocrystalline structures, increasing the interaction volume and hence, the efficiency. It is doped

with thallium (Tl) to introduce states between the conduction and valence bands in pure NaI.

These intermediate states produce photons whose wavelengths overlap well with the response of

photomultiplier tubes.

SuN consists of a total of 8 half-annular segments, 4 at the top (T1-T4) and 4 at the bottom

(B1-B4) as shown in Figure 2.8. It forms a cylindrical barrel 16 inch in length and diameter. Lead

bricks are placed on the upstream side to provide shielding from beam particles hitting multiple

components in the beamline. Each segment is read out by three photomultiplier tubes (PMTs) to

make sure all the photons produced in the segment are collected. All three PMTs must be fired

within a coincidence window of 2000 ns for a signal in the SuN segment to be recorded. Multiple

SuN segments can also be fired in a single physics event either due to scattering of the 𝛾 ray into

another segment or because of different 𝛾 rays in a cascade. In such events, the energies from

all the segments are summed and a single count corresponding to the sum is added to the total

absorption (TAS) spectrum. Additionally, multiple counts corresponding to energies deposited in

a single segment are added to the singles spectrum (also known as the ‘sum of segments’ or the SS

spectra). The multiplicity of an event registers the number of SuN segments that fired in an event

and ranges from 1 – 8. Together, all these spectra provide strict constraints for the 𝛾-ray cascade

under study.

2.4 Data Acquisition

Raw data from the detectors in this experiment are time series data of voltage pulses that encode

information about the energy deposited. For digital data acquisition systems (DDAS), the pulse is

discretized. This pulse is processed with some sort of a digital filter that extracts energy information

before a hit in a particular channel is recorded (along with a timestamp and energy deposited). A

combination of hits from different channels based on pre-specified constraints is packaged into an

event for further analysis.

37

Figure 2.9 Detector signal processing in Pixie-16 modules with the Digital Data Acquisition
System (DDAS). (a) shows the raw preamplified signal from the detector, (b) shows the digital
Constant Fraction Discriminator (CFD) and trigger-filter response, and (c) shows the energy-filter
response. Figure reproduced from [84].

38

2.4.1 Digital Pulse Processing

Signals from all the detectors were processed using the NSCLDAQ digital data acquisition

system [84]. It is a suite of software pipelines for data processing and online analysis. It utilizes

XIA digital gamma finder class of electronics with high frequency (100 – 250 MHz) Pixie-16

digitizer modules. Each channel is FPGA programmable and can independently record pulse-

arrival time and pulse-processed energy. A total of three crates were used to house all the Pixie

modules and their clocks were synchronized. (Differences in processing times were determined

using a split pulser signal sent to all three crates and were accounted for in the analysis).

Each input trace is continuously analyzed using a short filter, also known as a trigger filter or

fast filter. It is a trapezoidal filter whose purpose is to register the arrival of an incoming pulse. A

trapezoidal filter is characterized by two windows of width L (𝑇𝑃𝑒𝑎𝑘 ) and G (𝑇𝐺𝑎 𝑝). The height H

of the pulse at point 𝑘 is calculated as

𝐿𝐻𝑘 =

𝑘
∑︁

𝑘−𝐿−𝐺
∑︁

𝐻𝑖

𝐻𝑖 −

𝑘−𝐿+1

2𝐿−𝐺+1

(2.11)

where 𝑘 is a signal point and 𝐻𝑖 is the value of trace at point 𝑖. If the height of a pulse passes a preset

trigger filter threshold, it is timestamped and the trace then passes through a long filter also known

as the slow filter or the energy filter. This is also a trapezoidal filter with longer L and G such that

H represents the energy deposition. L, G, and the trigger filter threshold can be set independently

for each channel, both for the trigger filter as well as the energy filter. These parameters affect the

energy as well as timing resolution and are set as per the following guidelines:

1. 𝑇𝐺𝑎 𝑝 should be equal to or larger than the rise time of the signal.

2. 𝑇𝑃𝑒𝑎𝑘 for the trigger filter depends on the signal-to-noise ratio. A longer 𝑇𝑃𝑒𝑎𝑘 averages

the signal over a longer time which reduces the likelihood of triggering on noise, but also

increases the ‘dead time’ of the system.

3. 𝑇𝑃𝑒𝑎𝑘 for the energy filter can be larger than 𝑇𝑃𝑒𝑎𝑘 for the trigger filter. The energy filter is

applied only when the trigger threshold is crossed, so the dead time is less of an issue.

39

Figure 2.10 Example of an event recorded with the NSCLDAQ/FRIBDAQ. The upper panel
shows the event as it is recorded in the binary .evt file. The lower panel shows the same event
commented with what each word represents. Figure adapted from [85].

Figure 2.9 shows the raw detector signal, and the outputs of the trigger and the energy filter.

Additionally, the decay time of the signal 𝜏 is also an important parameter to be set. For very high

trigger rates in an experiment, there could be a pileup of pulses. In that case, the decay time gives

an idea of how much the previous pulse has decayed so that an accurate baseline subtraction can

be done. The decay time also leads to a slope in the plateau which can be corrected to get more

energy accuracy.

2.4.2 Event Building

NSCLDAQ is highly suitable for processing and merging data from multiple sources. Each

of the three Pixie crates had its individual computer (spdaq) device onboard. Data from each

spdaq is fed into a ring buffer called ‘raw ring’. A data packet in the raw ring contains the crate

number, module number, channel number, timestamp, and the energy recorded. The NSCLDAQ

online event builder accesses data from the raw rings and creates a ‘built ring’ of merged data.

The merging of raw data into events is done in a time-ordered fashion within a user-defined time

window. All the detector hits occurring within the event building time window are bundled together

40

into a single event. An event contains a group of triggers that are supposed to arise from the same

physics event. For example, an implantation event of 33Mg will register a hit in the XFP scintillator,

PIN detectors, and the low-gain channel of the DSSD (or miniDSSD) whereas a 𝛽-decay event will

only register a hit in the high gain channel of the DSSD (or miniDSSD). Depending on the event

correlation window, the 𝛽-delayed neutrons and 𝛾s from the NERO and SuN detector, respectively,

can also be included in the 𝛽-decay event. An example built event is shown in Figure 2.10 in the

way it is stored in a binary ‘.evt’ file. It contains information on the length of the event which

designates the number of detector channels fired in that event, followed by the contents of raw data

for each hit.

The event building window is selected as per the requirements of the particular experiment.

For the BCS-NERO part of the experiment, the event building window was chosen to be 200 𝜇s

to allow for the inclusion of 𝛽-delayed neutrons in the decay event following the moderation of

neutrons in the NERO matrix before detection. On the other hand, the SuN-miniDSSD part of the

experiment had the event building window set to just 2000 ns to avoid the high rate of background

𝛾s to be included in the built events. The event building time window can also influence the rate

of beam particles that can be accepted by the experimental set-up. A primary consideration is that

successive physics events should be separated by at least the event building time window apart from

other instrumentation constraints specific to an experiment.

41

CHAPTER 3

DATA ANALYSIS

This chapter focuses on the experimental data analysis techniques. It covers how the data recorded

from various detectors is mapped onto the nuclear quantities measured.

It includes setting up

the data acquisition and signal processing parameters, detector thresholds, efficiency and energy

calibrations, correlating implant events with decay events, and simulations of detectors. The

experiment was performed in two parts - BCS-NERO part and the SuN-miniDSSD part. Analysis

of the datasets from each part is presented in separate sections.

3.1 BCS - NERO Dataset

This part of the experiment aimed to measure the 𝛽-delayed neutron branching ratio or the P𝑛

value. The set-up included XFP scintillator and the PIN detectors, BCS stack with DSSD, SSSD,

and plastic scintillator, and the NERO matrix with 3He and BF3 long counters as discussed in

Section 2.3.1.

Detector

Channel

Trigger Filter

Energy Filter

𝑇𝑃𝑒𝑎𝑘 (𝜇s) 𝑇𝐺𝑎 𝑝 (𝜇s) Threshold 𝑇𝑃𝑒𝑎𝑘 (𝜇s) 𝑇𝑃𝑒𝑎𝑘 (𝜇s)

PIN1
PIN2
PIN1-XFP Time of flight
PIN3
DSSD (low gain)
DSSD (high gain)
SSSD
Plastic Scintillator
NERO 3He tubes
NERO BF3 tubes

5.12
5.12
0.48
5.12
1.34
6.16
3.04
0.8

2.0
2.0

3.2
3.2
0.08
3.2
1.2
4.0
2.4
0.24

0.8
0.8

∼6400
∼6400
∼600
∼6400
∼2000
∼1400
∼4000
∼1000

∼1000
∼800

9.52
9.52
1.28
9.52
2.3
9.2
7.76
0.8

4.0
4.0

0.64
0.64
1.28
0.64
0.24
0.64
0.48
0

2.0
1.04

Table 3.1 Digital Signal Processing Parameters optimized for each detector channel for the
BCS-NERO part of the experiment. For detectors with multiple channels, each channel was
optimized separately and only the average value is shown here.

42

3.1.1 Digital Signal Processing

The digital signal processing parameters were set for each detector channel individually, by

analyzing the signal trace with an oscilloscope. The time windows, as described in Section 2.4.1,

were chosen to maximize the energy resolutions. Trigger thresholds were chosen to go as low in

energy as possible without hitting the low-energy noise peak. Table 3.1 lists the parameters for all

the detectors in this setup. Parameters for PIN detectors, high gain channels of the DSSD, and the

SSSD were set before the experiment using a 241Am source that emits 𝛼 particles with an energy

of 5.48 MeV. The parameters for the low gain channel of the DSSD and the plastic scintillator were

set online with the beam during the experiment.

3.1.2 NERO Thresholds And Calibrations

3.1.2.1 NERO Thresholds

Figures 3.1(a) and 3.1(b) show the neutron response for a sample 3He tube and a BF3 tube,

respectively from a 252Cf source. The lower and upper thresholds designated by the red lines

define the region of true neutron events. These thresholds are implemented in the software and are

different from the trigger thresholds listed in Table 3.1. These are also determined individually for

all 60 tubes in the NERO matrix. Integrating the region between these limits gives the total number

of neutrons detected in a single counter. Summing the contributions from all counters gives the

total number of neutrons detected by NERO.

3.1.2.2 NERO Efficiency Calibrations

Determining the NERO efficiency is crucial for estimating the total number of neutrons emitted

during the experiment. It is benchmarked using a 252Cf source with known activity. 252Cf has a

half-life of 2.645 years and produces neutrons via spontaneous fission. It has a fission branching

ratio of 3.102% and produces 0.116 neutrons per decay. The remaining 𝛼 decays to 248Cm. The

composition of the Z7153 source as a mixture of Cf isotopes on December 1, 1989 is listed in Table

3.2 as supplied by the manufacturer. The amounts of each isotope along with their 𝛼 daughters on

September 4, 2020 i.e. the day of the calibration are also listed in another column. This translated

43

Figure 3.1 Sample neutron response spectra for (a)3He and (b)BF3 long counters in the NERO
matrix. The low energy noise peaks are visible in both spectra. Red lines designate the upper and
lower energy thresholds. Only the events in between are considered valid neutron events.

44

Isotope Amount (atom %) Amount (atom %) Neutron Activity (s−1)

December 1, 1989 September 4, 2020

249Cf
250Cf
251Cf
252Cf
245Cm
246Cm
247Cm
248Cm

Total

7.32
13.11
4.55
75.02
0
0
0
0

6.89
2.57
4.44
0.02
0.43
10.52
0.11
75.02

0
36.42
0
70.85
0
0.12
0
3.85

111.24

Table 3.2 Isotopic composition and the corresponding neutron activity of a 252Cf source (Z7153)
used for NERO efficiency calibrations. The amounts on December 1, 1989 are supplied by the
manufacturer of the source and the amounts on September 4, 2020 (the day of efficiency
calibrations) is calculated.

to a neutron activity of 111.24 neutrons per second for the 252Cf source on the day of efficiency

calibration.

The rate of neutron background in the room is measured to be 2.0914 neutrons per second.

This is subtracted from the total rate observed during the calibration run to get the rate of detected

neutron from the 252Cf source. Dividing this by the neutron activity of the source gives the NERO

efficiency. Table 3.3 lists the contributions from each quadrant separately. The obtained value for

the efficiency of 31.3(4)% is in agreement with the previously benchmarked value of 31.7(2)% [82]

for the 252Cf source. Subsequently, the same scaling factor as in [82] was used to get the efficiency

for low-energy neutrons which is relevant for estimating the P𝑛 value. The scaled NERO efficiency

used for the P𝑛 value calculations is 36.5(50)%.

3.1.3 BCS Thresholds And Calibrations

3.1.3.1 DSSD Thresholds

The electrons from 𝛽-decay form a continuous energy spectrum from zero energy up to the Q-

value. To identify as many decay events as possible, it is important to successfully detect electrons

with low energies. However, the electronic noise takes over below a certain limit and it becomes

impossible to distinguish the low-energy electrons from the noise. The low energy thresholds are

45

NERO
Quadrant

A
B
C
D

Total

Scaled

Room Background 3He tubes (1–4) BF3 tubes (5–15)
efficiency (%)

efficiency (%)

(neutrons/s)

0.6832
0.5619
0.4526
0.3937

2.0914

2.63
3.04
3.14
2.96

4.73
5.09
4.81
4.89

Total
efficiency (%)

7.36
8.13
7.95
7.85

31.3(4)

36.5(50)

Table 3.3 Neutron detection efficiencies for each quadrant and the total NERO efficiency. The
estimated room background has to be subtracted from the total neutron counts.

Figure 3.2 Energy spectrum for a sample BCS DSSD strip from a 90Sr calibration run. This is to
set the low energy thresholds (red line) for the high gain channel of the DSSD. The threshold is set
individually for each strip at the dip between the noise peak and the low energy electrons peak.
This allows for the highest signal-to-noise ratio.

46

Figure 3.3 Energy spectrum for a sample BCS DSSD strip from a 228Th energy calibration run.
Six characteristic 𝛼-peaks from the decay chain of 228Th are shown in the figure. The centroids
for Gaussian fits to the peaks were used for a linear energy calibration.

determined using the 𝛽 electrons spectrum from a 90Sr source. Thresholds are set at the dip between

the noise peak and the electrons peak as shown by the red line in Figure 3.2. These are determined

for each detector channel and implemented in the analysis software.

3.1.3.2 DSSD Energy Calibrations

Each strip of the DSSD in the high gain channel is calibrated using a 228Th source. The

complete decay chain of 228Th leads to 𝛼 particle emission with six distinct energies of 5.42 MeV,

5.69 MeV, 6.05 MeV, 6.29 MeV, 6.78 MeV, and 8.78 MeV. Each peak can be identified as shown

in Figure 3.3. All the peaks are fit to a Gaussian and the resulting centroids are linearly mapped to

the corresponding alpha energies. Residuals for each strip from the linear fit were below 20 keV.

This calibration for the high-gain channels can be roughly translated to the low-gain channels by

adjusting for the appropriate gain factor. It is useful to check the energy of the incoming beam

which can be confirmed with the LISE++ simulations as discussed in Section 2.2.

47

Figure 3.4 Plastic scintillator spectrum for a sample run with 33Mg beam. It is placed
downstream of the BCS stack to veto light particle events that can mimic decay events in the
DSSD. The red line shows the discriminator threshold between electrons and light particles. The
33Mg beam doesn’t contain any light particles.

3.1.3.3 Plastic Scintillator Threshold

A plastic scintillator was placed downstream of the BCS stack to distinguish between light

particles in the beam that mimic decay events in the high-gain channel of the DSSD. However,

these light particles as a part of the beam have large forward momentum and deposit relatively

higher energy in the plastic scintillator. These high-energy events in the scintillator can then be

vetoed. Figure 3.4 shows the energy spectrum for the plastic scintillator for a single run with 33Mg

beam. The red line shows the threshold beyond which all the events are vetoed. This threshold is

also implemented in the analysis software. It can be seen that the 33Mg beam had very high purity

and negligible light particle contaminants.

48

Figure 3.5 Particle Identification (PID) plot for the 𝛿E-TOF method as described in Section 2.2.1.
Energy loss is from the PIN1 detector and the time of flight is between the XFP scintillator and the
PIN1 detector. The A1900 fragment separator settings were optimized for the 33Mg fragment to
be delivered to the experimental end-station with the least contaminants. Only the events in the
33Mg blob were used for further analysis.

3.1.4 Particle Identification

Figure 3.5 shows the Particle Identification (PID) plot for the 33Mg beam using the 𝛿E-TOF

method as discussed in Section 2.2.1. The energy loss 𝛿E is measured in the PIN1 detector whereas

the time of flight TOF is measured using a Time-to-Amplitude Converter (TAC) between the XFP

scintillator at I2 position in the A1900 fragment separator and the PIN1 detector. The central

blob corresponds to the 33Mg fragments. A tight square gate is implemented around this blob and

only the events that fall within that square gate are considered as valid 33Mg implants. No further

corrections to the PID were required since the relatively tight gate had enough statistics for the

analysis.

Any 𝛽-delayed spectroscopy measurements of 33Mg will always have background contributions

49

Figure 3.6 PID plots for the 33,32Al beams delivered to the experimental end-station to estimate
the background from 𝛽-daughter and 𝛽-delayed neutron daughter, respectively. Refer to the
caption of Figure 3.5 for description.

50

from the further decay of the 33Al daughter and 32Al 𝛽-delayed neutron daughter. As a result, the

complete experiment was also performed using 33,32Al beams. This allowed background estimation

on the same experimental setup to cancel out systematic errors, if any. Figures 3.6(a) and 3.6(b)

show the PID plots for 33Al and 32Al beams, respectively.

3.1.5

Implant and Decay Correlations

Measuring the 𝛽-decay half-life of an isotope requires measuring the time difference between

the implantation of a particle of an isotope and the detection of the electron from its 𝛽-decay. This is

achieved by classifying the physics events recorded by the data acquisition system into implantation

events and decay events and subsequently correlating them in the analysis software. A physics

event can be classified into an implantation event upon satisfying the following conditions:

1. Energies in PIN1, PIN2, and PIN3 are greater than zero. This ensures the event occurred due

to a beam particle.

2. At least one of each of the front and back strips in the low gain channel of the DSSD registers

an energy greater than zero. This allows for spatial localization of the implant on the DSSD.

When multiple strips fire, the strip with the highest energy deposition (usually the one in the

middle of the group of strips fired) is chosen as the position of the implant.

3. No energy deposition in the SSSD downstream of the DSSD. This is to ensure that the

incoming beam particle was implanted in the DSSD and did not just deposit some energy

while punching through.

4. No energy deposition in the plastic scintillator.

5. Additional conditions can be placed on the PID to select implantations of a particular isotope.

To be classified as a decay event, a physics event needs to satisfy the following conditions:

1. No energy deposition on either of the PIN detectors. This ensures that no beam particle was

involved in the event.

51

2. No signal in the low gain channels of the DSSD. The decay electrons can have a maximum

energy deposition of only a few MeV, which is below the thresholds of the low gain channels.

3. Energies in at least one of each of the front and back strips high-gain channel of the DSSD

to be greater than the low energy thresholds applied in the software. This allows for spatial

localization of the decay event and spatial correlations with the implant event during analysis.

4. Energy deposition of either zero or less than the light-ion threshold in the plastic scintillator.

This ensures that the physics event involves an electron.

Once a list of implantation and decay events is obtained, they are spatially and temporally

correlated. Spatial correlation requires that the implantation and decay events occur within a pre-

defined correlation field, i.e. the distance between the pixels for these events on the DSSD should

be less than the correlation window. A correlation window of 3 × 3 was found to be optimal for

the BCS DSSD. Temporal correlation requires that the time between the implantation and decay

events is less than a pre-defined correlation time. The correlation time window should at least be

a few times the half-life of the implanted particle. It was chosen to be 1000 ms for all isotopes

in this experiment. All pairs of implants and decay events that satisfy both temporal and spatial

correlations are considered for further analysis. It should be noted that a single decay event can

be correlated to multiple implantation events and vice versa. These multiple correlations include

true correlations such as daughter or granddaughter decays as well as random correlations from the

background. The rate of random correlations can be estimated with the reverse time correlator as

described in Section 3.1.5.1.

3.1.5.1 Reverse Time Correlations

The correlation logic described in Section 3.1.5 is run forward in time, i.e. the decay events

happen after implantation events. As discussed earlier, this includes true as well as random

correlations. A similar correlation logic can also be run backward in time where the decay events

occur before implantation events. Since this is physically not possible, the reverse time correlator

52

Figure 3.7 The complete decay chain of 33Mg with contributions from the decay of daughter
(33Al), granddaughter (33Si), and 𝛽-delayed neutron daughter (32Al). All other isotopes (33P, 32Si,
31Si) are considered ‘pseudo-stable’ for the correlation time window of 1000 ms.

will include only the random correlations. This allows for the estimation of the rate of random

correlations which constitute the background.

3.1.5.2 Half-life Measurement

Once all the implant-decay correlation pairs are identified, a histogram of the time difference

between the implantation and decay events is generated, also known as the decay curve. This curve

resembles the probability distribution of the time an implanted isotope survives before undergoing

𝛽-decay. Due to nuclear decay being a first-order kinetic transition, the decay curve for a single

nuclide parent decay is exponentially decreasing. This decay curve can be fit to Equation 2.2, to

extract the decay constant, which can be converted into half-life using Equation 2.3. In practice,

however, the decay curve has multiple components from the decay of daughter, granddaughter,

𝛽-delayed neutron daughter, etc. when working with exotic nuclei far from stable isotopes. As a

result, the decay curve is fit with a modified version of Equation 2.7. The modifications include

accounting for 𝛽-delayed neutron branchings and background from random correlations. The half-

lives of all other isotopes in the decay chain should either be known, or a multi-parameter fit can

53

be performed.

Figure 3.7 shows the decay chain for 33Mg along with 𝛽-delayed neutron branchings. The

decay curve for 33Mg will include contributions from the decays of 33Al, 32Al, and 33Si. All other

isotopes are considered ’pseudo stable’ for the 1000 ms correlation time window considered here.

The half-lives of 33Al and 32Al were measured using the same experimental setup and technique

with beams of 33Al and 32Al, respectively. The half-life for 33Si was acquired from the Nudat

database by the National Nuclear Data Center (NNDC) [86]. The P𝑛 values for 33Mg, 33Al, and

32Al were estimated from the NERO data as described in Section 3.1.6. With these inputs, the

decay curve for 33Mg implants was fit to Equation 2.7, and the half-life was extracted.

3.1.6 Calculating P𝑛 Value

The event-building time window for defining a physics event during the NERO-BCS part of the

experiment is set to 200 𝜇s. This allows for the 𝛽-delayed neutrons detected in NERO to be a part

of the decay event. The total number of neutrons in the correlated decay events can then be used to

calculate the P𝑛 value. The P𝑛 value is calculated as:

𝑃𝑛 =

𝑁𝑛 − 𝑁𝑑
𝜖𝑛𝑁𝛽

(3.1)

where N𝑛 is the number of neutrons in decay events from forward time correlated decay curve

minus the number of neutrons in decay events from reverse time correlated decay curve, N𝑑 is the

number of 𝛽-delayed neutrons expected from the decay of all other isotopes in the decay chain

except the parent isotope, N𝛽 is the number of forward time correlated decay events minus the

number of reverse time correlated decay events, and 𝜖𝑛 is the NERO neutron detection efficiency.

In terms of 𝛽 and neutron correlations, we need true 𝛽 particles to be correlated with true

neutrons. True here means particles from a valid 𝛽-delayed neutron decay. However, there are

three major sources of background for 𝛽-n correlations. These are accounted for in the P𝑛 value

calculations as follows:

1. Random 𝛽 correlating with a true neutron. This is accounted for when subtracting the reverse

time correlator components in N𝛽.

54

2. True 𝛽 correlating with a random neutron. The rate of random neutrons is about 2.1 neutrons

per second. So the probability of a random neutron correlating with a true 𝛽 within a 200 𝜇s

event building time window is tiny and negligible.

3. Random 𝛽 correlating with a random neutron. This factor is also tiny but accounted for via

the reverse time correlator components subtraction.

The P𝑛 values for 33Al and 32Al were calculated with the respective particle beams implanted

in the BCS-DSSD. The N𝑑 for these isotopes is 0 since none of their daughters undergo 𝛽-delayed

neutron decay. Using these experimental measurements, the P𝑛 value for 33Mg was determined

with an iterative procedure. The number of neutrons from the daughters’ decay N𝑑 depends not

only on the daughters’ P𝑛 value but also on the parent P𝑛 value that is being determined. N𝑑 is

assumed to be 0 at the beginning and then solved recursively until it converges.

3.2 SuN - miniDSSD Dataset

This part of the experiment aimed to measure the 𝛽 feeding intensities into excited states in

33Al by detecting the 𝛽-delayed 𝛾 rays. The set-up included XFP scintillator and the PIN detectors

for particle identification, miniDSSD and silicon veto detector for the implants and electrons, and

the SuN detector for the 𝛾 rays as discussed in Section 2.3.2.

3.2.1 Digital Signal Processing

Digital signal processing parameters were set for each channel individually in a similar manner

as done for the BCS-NERO part. The signal trace was analyzed on an oscilloscope and the timing

and threshold parameters were chosen to reduce noise and maximize energy resolution. Table 3.4

lists the parameters for all detectors in this setup.

3.2.2 miniDSSD Thresholds And Calibrations

The detector thresholds and energy calibrations for each strip of the miniDSSD were determined

similarly to the BCS DSSD using a 90Sr source and a 228Th source, respectively. Figure 3.8 shows

the energy spectrum from a sample miniDSSD strip for a 90Sr source. The low energy noise peak

is barely visible and the thresholds designated by the red line were set such that the entire electron

55

Detector

Channel

Trigger Filter

Energy Filter

𝑇𝑃𝑒𝑎𝑘 (𝜇s) 𝑇𝐺𝑎 𝑝 (𝜇s) Threshold 𝑇𝑃𝑒𝑎𝑘 (𝜇s) 𝑇𝐺𝑎 𝑝 (𝜇s)

PIN1
PIN2
PIN1-XFP Time of flight
miniDSSD (low gain)
miniDSSD (high gain)
Si Surface Barrier Veto

SuN PMTs

5.12
5.12
0.48
0.96
0.96
1.664

0.16

3.2
3.2
0.08
0.512
0.512
0.64

0.048

∼11200
∼6400
∼600
∼2000
∼1500
∼3000

∼20

9.52
9.52
1.28
4.928
4.928
4.096

0.48

0.64
0.64
1.28
3.2
3.2
2.56

0.16

Table 3.4 Digital Signal Processing Parameters optimized for each detector channel for the
SuN-miniDSSD part of the experiment. For detectors with multiple channels, each channel was
optimized separately and only the average value is shown here.

Figure 3.8 Example energy spectrum for a sample miniDSSD strip from a 90Sr calibration run.
The red line designates low energy thresholds for the high gain channel of the miniDSSD. Refer to
the caption of Figure 3.2 for more details.

56

Figure 3.9 Example energy spectrum for a sample miniDSSD strip from a 228Th energy
calibration run. Each peak corresponds to an 𝛼-energy from the 228Th decay chain as shown in
the figure. The high gain channels for each strip were linearly calibrated with these peaks.

peak is included in the analysis. Figure 3.9 also shows the energy spectrum of a 228Th source for

one of the miniDSSD strips. All six 𝛼 peaks can be seen and centroids from their Gaussian fits were

used for a linear energy calibration. Thresholds and calibrations were determined individually for

all 32 strips (16 on the front and 16 on the back) of the miniDSSD.

3.2.3 SuN Gain Matching and Energy Calibrations

The sun detector consists of 8 NaI crystals, or segments (4 at the top and 4 at the bottom).

Light generated in each segment due to interaction with the incoming 𝛾 ray is collected by three

photomultiplier tubes (PMTs) as discussed in Section 2.3.2. These PMTs have to be gain-matched

before adding their energies to get the total energy deposited in each SuN segment. This is achieved

with a 1460 keV 𝛾 ray that exists as a room background from the decay of 40K. Radioactive 40K is

found in trace amounts in the concrete present in the walls of the experimental room. Figure 3.10

shows the 1460 keV line in one of the SuN PMT. A Gaussian peak is fit onto a decaying background.

57

Figure 3.10 Room background spectrum for a sample SuN PMT for gain matching. The PMTs
have to be gain-matched since the energies of 3 PMTs are summed to get the energy deposition in
a SuN segment. This is done using the 1460 keV 𝛾 ray from the room background that originates
from the decay of 40K found in trace amounts in the concrete present in the walls of the room.

Gain-matching factors for each PMT are chosen such that the centroids from the Gaussian fits of

all PMTs match up. Once the PMTs are gain-matched, the energy deposition in each SuN segment

can be determined.

All of the 8 SuN segments still have to be calibrated to establish a relation between channel

numbers and 𝛾 energy. Four 𝛾 lines - 59.5 keV from 241Am source, 662 keV from 137Cs source,

and 1173 keV and 1332 keV from 60Co were used for SuN energy calibrations. Figure 3.11 shows

three of the four 𝛾 lines from 137Cs and 60Co in one of the SuN segments. In addition, a known

4730 keV 𝛾 line from the 𝛽-decay for 33Mg was used as a reference in the higher energy region. A

quadratic calibration was performed with a very small quadratic factor of the order of 10−7. All the

residuals were found to be less than 10 keV.

58

Figure 3.11 Example energy spectrum of a sample SuN Segment (with gain-matched PMTs) from
137Cs and 60Co calibration runs. The photopeaks are labeled in the figure corresponding to the 𝛾
energy. 60Co calibration run also shows the sum peak.

59

3.2.4 Particle Identification

Particle identification for this part of the experiment was similar to the NERO-BCS part as the

beam delivery was the same and only the downstream experimental setup was changed. Beams of

33Mg for the experiment and 33Al and 32Al for the background estimation were delivered to the

SuN-miniDSSD experimental setup. Refer to Section 2.2.1 and Section 3.1.4 for more details on

particle identification.

3.2.5

Implant and Decay Correlations

Similar to the BCS DSSD, implantation and decay events have to be correlated in the miniDSSD

too. The 𝛽-delayed 𝛾 rays detected in SuN are expected to be a part of the decay event. Since the

SuN PMTs have a very fast timing response, the physics event-building time window in the data

acquisition system can be set to 2000 ns (compared to 200 𝜇s for the NERO-BCS part required

for neutron moderation). The shorter window also allows for background suppression as the SuN

PMTs have relatively high count rates from cosmic ray events.

Since the experimental setup is different, the conditions for a physics event to be classified as an

implantation event and a decay event are also different. To be classified as an implantation event, a

physics event must satisfy the following conditions:

1. Energies in PIN1 and PIN2 are greater than zero. This ensures implantation from a beam

particle.

2. At least one of each of the front and back strips in the low gain channel of the miniDSSD

registers an energy greater than zero. If multiple strips fire, the strip with the highest energy

deposition is chosen as the position of the implant.

3. No energy deposition in the Si veto detector downstream of the miniDSSD. This condition

helps discard punchthrough events as they shouldn’t be correlated with decay events.

To be classified as a decay event, the following conditions must be satisfied:

1. No energy deposition on either PIN1 or PIN2.

60

2. No signal in the low gain channels of the DSSD.

3. Energies in at least one of each of the front and back strips high-gain channel of the DSSD

to be greater than the low energy thresholds applied in the software.

4. No energy deposition in the Si veto detector.

Implantation and decay events are temporally and spatially correlated using the same logic as

that for the BCS DSSD. However, since the miniDSSD has smaller dimensions and only 16 strips

on each side, a spatial correlation window of 1 × 1 was optimal, i.e. the implant and decay had to

be in the same pixel to be successfully correlated. Along with a forward time run, the correlator

was run in reverse time to get an estimation of the background from random correlations.

3.2.6

𝛽-delayed 𝛾 Spectra

The SuN detector as a whole has a very high efficiency for detecting 𝛾 rays. However, a 𝛾

ray can deposit partial energies in multiple segments due to scattering across segments. Multiple

𝛾 rays in a cascade from the same physics event can also be detected in different segments of

SuN. As a result, two types of spectra are created with the data from SuN. A total absorption

spectrum (TAS) represents the sum of total energy deposited in all segments in a single physics

event. A singles spectrum, also known as Sum of Segments (SS), represents the energy deposited

in a single segment. Spectra from all eight segments are added together to get the SS. Note that

this summation is not on an event-by-event basis like the TAS but it is the summation of complete

spectra. Additionally, the multiplicity spectrum (Mul) represents the number of segments fired in

a single physics event. Together, these spectra constrain the complete 𝛾 ray cascade from a physics

event.

𝛾 rays from only the decay events that are successfully correlated with implants form the

SuN 𝛾 spectra (TAS, SS, and Mul). However, these include background from random implant-

decay correlations. To account for this, similar 𝛾 spectra from correlated decay events with the

reverse time correlator are subtracted from the forward correlated 𝛾 spectra. This also removes the

contribution from random 𝛾 rays correlating with the decay events. Reverse correlator subtracted

61

Figure 3.12 GEANT4 visualization of SuN detector in the left panel and the miniDSSD
implantation detector in the middle and the right panel. Green lines are for visualizing the 𝛾 ray
tracks whereas the red lines correspond to electron tracks. The green track outside the SuN
detector volume represents partial energy deposition in the detector. Figure adapted from [87].

spectra still contain contributions from the decay of daughter (33Al) and 𝛽-delayed neutron daughter

(32Al) nuclei. As a result, similar spectra are created with the 33,32Al beam implantations and are

proportionately subtracted, depending on the number of their decays within the correlation time

window as estimated with the Bateman Equation 2.7(modified to account for neutron branchings).

This step also accounts for any granddaughter, neutron granddaughter, etc. decay as they will also

be present in the daughter and neutron daughter decay, and be subtracted off. The resulting spectra

are the experimental 𝛽-delayed 𝛾 spectra for just the parent decay (33Mg). The 𝛾 rays in this spectra

will correspond to excited states in 33Al and 32Al as shown in Figure 2.1.

3.3 GEANT4 Simulations of SuN

𝛾 rays interact with the detector material in three principal ways:

1. Photoelectric absorption where the complete 𝛾 energy is deposited in the detector material.

2. Compton Scattering where the 𝛾 ray scatters off an electron and deposits partial energy in

the detector material.

3. Pair production where the incoming 𝛾 ray produces an electron-positron pair. The positron

can further annihilate with an electron in the detector material and produce two 511 keV 𝛾

rays.

62

Figure 3.13 SuN photopeak energy resolution in keV calculated as a function of the incident 𝛾 ray
energy. The resolution function used is from [87]. Also shown here is the pesudolevels spacing
used for RAINIER and GEANT4 simulations. In these simulations, each pseudolevel represents the
contribution from all the energy levels within that energy bin. As long as the spacing is smaller
than SuN resolution, the resulting simulated spectra from any level and its corresponding
pesudolevel should be nearly identical.

The probability of each type of interaction depends on the incoming 𝛾 ray energy. As a result,

the SuN spectra corresponding to multiple 𝛾 cascades from excited levels in the daughter nucleus

(33Al) cannot be read simply as counting the number of 𝛾 rays with a particular energy. Instead,

the detector response is simulated for each possible cascade, and a linear combination of such

simulated spectra reproduces the experimental spectra. The weight of each simulated cascade at a

particular excitation energy represents the 𝛽 feeding intensity at that excitation energy.

The SuN detector response is simulated with the GEANT4 package [88]. It is a versatile tool to

model the interaction of any type of radiation with matter. It provides great flexibility for choosing

detector materials and geometries, characterizing and tracking emitted radiation, recording total

energy deposited in pre-defined volumes, etc. Previous works [83, 87, 89] have explored the process

of simulating SuN detector response in GEANT4 in great detail and have successfully validated the

SuN detector efficiency and resolution.

63

A collection of simulated spectra (TAS, SS, and Mul) for the feeding of the daughter nucleus

at a single excitation energy is referred to as a template. All the excited levels in the daughter

nucleus will have their own templates. At low energies, templates are created for discrete energy

levels depending on the 𝛾 branching cascades from that level. These can either be known from

previous measurements or determined from the experimental dataset using a combination of TAS

and SS as discussed in Section 4.3. At higher energies, especially above the neutron separation

energy, the level density becomes high and it is impractical to create a template for all levels. In

that region, referred to as the quasi-continuum, a statistical approach is used. Pseudolevels are

placed at finite intervals that represent the contribution from all levels in that interval. The spacing

between pesudolevels should be less than the SuN energy resolution so that templates from an

energy level and its corresponding pesudolevel are nearly identical. Figure 3.13 shows the SuN

energy resolution function and the pseudolevel spacings used in this work. The spacings used are

50 keV up to 2 MeV, 100 keV up to 3.9 MeV, 150 keV up to 5.4 MeV, 200 keV up to 8 MeV, 250 keV

up to 10 MeV, and 300 keV up to 14.5 MeV. The 𝛾 branchings from pseudolevels are calculated

using RAINIER as described in Section 3.4.

3.3.1 Particle simulations in SuN

While the SuN detector is primarily designed to detect 𝛾 rays, other particles produced in a

decay event such as an electron, a neutron, or a neutrino can interact with the SuN detector and

change the observed spectra. As a result, these interactions have to be accounted for in the GEANT4

simulated spectra. The neutrinos have a negligible chance of any interactions and can be ignored.

The electrons and neutrons interactions, however, deposit additional energy in SuN and shift the

observed spectra. The low-energy electrons up to ∼400 keV are stopped in the miniDSSD and

electrons up to energies of ∼1 MeV are stopped in the beam pipe. However, the Q-value of the

33Mg 𝛽-decay is 13.477 MeV, and electron energies can go as high as that. Moreover, higher-energy

electrons will lose less energy in the miniDSSD and the beam pipe and deposit an even higher

fraction of their total energy in SuN, shifting the resulting spectra considerably. This is accounted

for in GEANT4 simulations of SuN.

64

Figure 3.14 GEANT4 simulated SuN Total Absorption Spectrum for monoenergetic 1 MeV
neutrons emitted isotropically from the center of the detector. The inelastic scattering peaks on
127I and 23Na and the 7826 keV neutron capture peak on 127I (Q-value = 6826 keV) can be
identified.

The neutron interaction with SuN is a bit more complex. In addition to depositing their kinetic

energy in the detector through elastic scattering, neutrons can also undergo inelastic scattering via

23Na(n,n’𝛾)23Na or 127I(n,n’𝛾)127I as well as neutron capture on 127I(n,𝛾)128I. Figure 3.14 shows

the simulated total absorption spectrum for monoenergetic 1 MeV neutrons. The 58 keV peak

corresponds to 𝛾 rays from inelastic scattering on 127I and the 7.8 MeV peak is due to 𝛾 rays from

neutron capture on 127I. 7.8 MeV corresponds to the Q-value of 127I(n,𝛾)128I (6.826 MeV) plus the

1 MeV neutron energy. Although SuN detector does detect neutrons, the efficiency is not high and

the neutron capture peak is submerged in the 𝛾 rays from the quasi-continuum. Overall, the neutron

response has to be accounted for in the analysis but it is not well-benchmarked and hence, NERO

data is used to constrain the 𝛽-delayed neutron branching.

65

BSFG

CTM

𝑎 (MeV−1) 𝐸1 (MeV) 𝑇 (MeV) 𝐸0 (MeV)

33Al
32Al

4.96
4.59

-0.51
-2.05

1.44
1.52

-1.13
-2.67

Table 3.5 Level Density parameters for the Back Shifted Fermi Gas (BSFG) model and the
Constant Temperature Model (CTM) calculated as described in Section 3.4.1 for both 33Al and
32Al.

3.4 RAINIER Simulations

RAINIER [90] is a software tool used for a probabilistic determination of the 𝛾-ray branchings

from a pseudolevel in the quasi-continuum while accounting for level, width, and cascade fluc-

tuations. It takes user-defined energy, spin, and parity of the psesudolevel as inputs and creates

𝛾 branching cascades. To do this, it requires a prescription for level density (LD) and 𝛾 strength

function (GSF). These are discussed in the following sections.

3.4.1 Level Density Parameters

No experimental data exists for the level densities in 33,32Al. As a result, an empirical prescrip-

tion from [91, 92] was used to determine the level density prescriptions. It is based on extrapolations

from nuclei where experimental data exists. Two models were used for the description of level

density - the Back-Shifted Fermi Gas model (BSFG) and the Constant Temperature Model (CTM).

The level density as a function of excitation energy for a nucleus with 𝑍 protons and 𝑁 neutrons

for BSFG is given by

𝜌(𝐸) =

√

12

√

𝑒2

𝑎(𝐸−𝐸1)

2𝜎𝑎1/4(𝐸 − 𝐸1)5/4

(3.2)

where 𝑎 is the level density parameter, 𝐸1 is the energy backshift, and 𝜎 is the spin cut-off parameter.

The level density for CTM is given by

𝜌(𝐸) =

1
𝑇

𝑒(𝐸−𝐸0)/𝑇

(3.3)

66

Figure 3.15 Comparison of level densities per MeV calculated using the Back Shifted Fermi Gas
(BSFG) model and the Constant Temperature Model (CTM) as described in Section 3.4.1 for both
33Al and 32Al.

67

where 𝑇 is the temperature, and 𝐸0 is the energy backshift. All the parameters in both models are

calculated as follows.

𝑎 = (0.199 + 0.0096𝑆′) 𝐴0.869

𝐸1 = −0.381 + 0.5𝑃′
𝑎

𝜎2 = 0.391𝐴0.675(𝐸 − 0.5𝑃′

𝑎)0.312

𝑇 = 𝐴−2/3/(0.0597 + 0.00198𝑆′)

𝐸0 = −1.004 + 0.5𝑃′
𝑎

(3.4a)

(3.4b)

(3.4c)

(3.4d)

(3.4e)

where 𝐴 is the atomic mass number, 𝑆′ is the modified shell correction factor and 𝑃′

𝑎 is the deutron

pairing energy. 𝑆′ is related to 𝑃′

𝑎 as

𝑆′ = 𝑆 + 0.5𝑃′
𝑎

(3.5)

where S is the shell correction factor. It is the difference between the experimental value for the

mass excess and the theoretical prediction by the liquid drop model that doesn’t account for shell

effects.

𝑆 = 𝑀𝑒𝑥 𝑝 − 𝑀𝐿𝐷

The mass excess for the liquid drop model is related to the liquid drop binding energy as

𝑀𝐿𝐷 = 𝑁 ∗ 𝑀𝑛 + 𝑍 ∗ 𝑀𝑝 − 𝐵𝐸𝐿𝐷

which is given by

𝐵𝐸𝐿𝐷
𝐴

−

= 𝑎𝑣𝑜𝑙 + 𝑎𝑠 𝑓 𝐴−1/3 +

3𝑒2
5𝑟0

𝑍 2 𝐴−4/3 + (𝑎𝑠𝑦𝑚 + 𝑎𝑠𝑠 𝐴−1/3)

(cid:19) 2

(cid:18) 𝑁 − 𝑍
𝐴

(3.6)

(3.7)

(3.8)

The constants for the liquid drop model are - 𝑎𝑣𝑜𝑙 = -15.65 MeV, 𝑎𝑠 𝑓 = 17.63 MeV, 𝑎𝑠𝑦𝑚 = 27.72

MeV, 𝑎𝑠𝑠 = -25.60 MeV, and 𝑟0 = 1.233 fm. Finally, the deutron pairing energy can be written in

terms of mass excess as

𝑃′

𝑎 ( 𝐴, 𝑍) =

1
2

(cid:2)𝑀 ( 𝐴 + 2, 𝑍 + 1) − 2𝑀 ( 𝐴, 𝑍) + 𝑀 ( 𝐴 − 2, 𝑍 − 1)(cid:3)

(3.9)

Table 3.5 lists the calculated level density parameters that are provided in the RAINIER input file

and Figure 3.15 shows the calculated level densities for BSFG and CTM for both 33Al and 32Al.

68

Figure 3.16 𝛾-ray strength function in mb for the E1, M1, and E2 transitions calculated as
described in Section 3.4.2 for both 33Al and 32Al. E1 is calculated using Generalized Lorentzian
functional form whereas E2 and M1 are calculated using Standard Lorentzian functional form.

3.4.2 𝛾-ray Strength Function Parameters

The 𝛾 strength functions (GSF) for different transitions were calculated similarly to the pre-

scription used in TALYS [93]. GSF for type 𝑋 (E or M) and multipole (ℓ) is modeled as a Standard

Lorentzian given by

where

𝑓𝑋ℓ (𝐸𝛾) = 𝐾𝑋ℓ

𝜎𝑋ℓ𝐸𝛾Γ2
𝑋ℓ

(𝐸 2

𝛾 − 𝐸 2

𝑋ℓ)2 + 𝐸 2

𝛾Γ2
𝑋ℓ

𝐾𝑋ℓ =

1
(2ℓ + 1)𝜋2ℏ2𝑐2

69

(3.10)

(3.11)

33Al

32Al

Energy E Width Γ Strength 𝜎 Energy E Width Γ Strength 𝜎
(MeV)

(MeV)

(MeV)

(MeV)

(mb)

(mb)

E1
M1
E2

21.23
12.78
19.64

8.90
4.00
5.71

40.57
6.59
0.53

21.39
12.91
19.84

9.03
4.00
5.73

39.19
6.81
0.55

Table 3.6 Energy, width, and strength 𝛾-ray strength function parameters for the E1, M1, and E2
transitions calculated as described in Section 3.4.2 for both 33Al and 32Al.

and 𝐸 𝑋ℓ is the energy of the resonance, Γ𝑋ℓ is the width of the resonance, and 𝜎𝑋ℓ is the strength

of the resonance. For a nucleus with mass number A and atomic number Z, the parameters for E1

transition are given by

𝐸𝐸1 = 31.2𝐴−1/3 + 20.6𝐴−1/6 MeV

Γ𝐸1 = 0.026𝐸 1.91

𝐸1 MeV

𝜎𝐸1 = 1.2 × 120𝑁 𝑍/( 𝐴𝜋Γ𝐸1) mb

The parameters for E2 transition are given by

𝐸𝐸2 = 63𝐴−1/3 MeV

Γ𝐸2 = 6.11 − 0.012𝐴 MeV

𝜎𝐸2 = 0.00014𝑍 2𝐸𝐸2/( 𝐴−1/3Γ𝐸2) mb

For the M1 transition, the energy and width are given as

𝐸𝑀1 = 41𝐴−1/3 𝑀𝑒𝑉

Γ𝑀1 = 4.0 𝑀𝑒𝑉

(3.12a)

(3.12b)

(3.12c)

(3.13a)

(3.13b)

(3.13c)

(3.14a)

(3.14b)

The strength for M1 is back-calculated using Equation 3.10 from the value of GSF at 7 MeV given

by

𝑓𝑀1(7 MeV) = 1.58 × 10−9 𝐴0.47

(3.14c)

70

These three are the only types of electromagnetic transitions considered for the GSF. E2 and M1

are calculated using the Standard Lorentzian functional form described in Equation 3.10 whereas

E1 uses the Generalized Lorentzian functional form [94] given by

𝑓𝐸1(𝐸𝛾, 𝑇) = 𝐾𝐸1

(cid:20)

𝐸𝛾 ˜Γ𝐸1(𝐸𝛾)
)2 + 𝐸 2

𝛾 ˜Γ𝐸1(𝐸𝛾)2

(𝐸 2

𝛾 + 𝐸 2
𝐸1

+

(cid:21)

0.7Γ𝐸14𝜋2𝑇 2
𝐸 5
𝐸1

𝜎𝐸1Γ𝐸1

(3.15)

where the energy-dependent damping width ˜Γ𝐸1(𝐸𝛾) is given by

˜Γ𝐸1(𝐸𝛾) = Γ𝐸1

𝐸 2

𝛾 + 4𝜋2𝑇 2
𝐸 2
𝐸1

and T is the nuclear temperature given by

𝑇 =

√︂ 𝐸 − Δ − 𝐸𝛾
𝑎

(3.16)

(3.17)

where 𝐸 is total excitation energy, Δ is the pairing correction, and 𝑎 is the level density parameter.

Table 3.6 lists the calculated GSF parameters for E1, M1, and E2 transitions for both 33,32Al. Figure

3.16 shows the actual GSF as a function of 𝛾 energy. It can be confirmed that E1 is the strongest

transition followed by M1, followed by E2.

71

CHAPTER 4

EXPERIMENTAL RESULTS

This chapter lists the results from the experimental measurement of the ground state to ground

state 𝛽-decay transition strength of 33Mg for both, the BCS-NERO and the SuN-miniDSSD parts

of the experiment.

It includes the half-life measurements, 𝛽-delayed neutron branching ratio

measurement, and the 𝛽 feeding intensities into the excited levels of 33Al. The corresponding

nuclear log-ft values and Gamow-Teller transition strengths are also listed.

It also includes a

section on new levels proposed in 33Al based on the experimental total absorption spectrum (TAS).

Finally, the results from implementing two different nuclear level density models for treating the

quasi-continuum in 33,32Al are discussed.

4.1 Half-life Measurements

Following the procedure outlined in Section 3.1.5, the decay curve for 33Mg implants is gen-

erated for a correlation time window of 1000 ms. A spatial correlation window of 3 × 3 is used

for the BCS DSSD and 1 × 1 for the miniDSSD. Figure 4.3 shows the decay curves for both.

The characteristic shape of the decay curves with a hump at early times is a feature of Bateman

Equation 2.7 and can be attributed to the fact that the daughter 33Al has a shorter half-life than the

parent 33Mg. The SuN-miniDSSD dataset has a high rate of background since the miniDSSD is

small so the incoming beam is focused on a smaller region, which increases the chances of random

coincidences. The decay curves for 33Mg contain decays from the following components:

1. Decay of parent 33Mg.

NNDC Reported BCS-NERO Dataset SuN-miniDSSD Dataset

Isotope
33Mg
33Al
32Al

T1/2 (ms)

89.4(10)
41.7(2)
31.9(8)

T1/2 (ms)

92.0(4)
42.0(2)
31.7(1)

T1/2 (ms)

93.2(6)
42.6(3)
31.5(1)

Table 4.1 Experimentally measured half-lives for 33Mg, 33,32Al in this work compared to the
half-lives reported by the National Nuclear Data Center (NNDC) based on previous measurements.

72

Figure 4.1 Corner plot for posterior distributions from the 33Mg decay curve fit for the
BCS-NERO dataset. Since the posteriors are nearly Gaussian, the median is reported to be the
parameter fit whereas the 16th and the 84th percentile constitute the 1-𝜎 uncertainty.

73

Figure 4.2 Corner plot for posterior distributions from the 33Mg decay curve fit for the
SuN-miniDSSD dataset. Since the posteriors are nearly Gaussian, the median is reported to be the
parameter fit whereas the 16th and the 84th percentile constitute the 1-𝜎 uncertainty.

74

Figure 4.3 33Mg decay curve fits for the (a)NERO-BCS and the (b)SuN-miniDSSD datasets.
Half-lives are reported in Table 4.1.

75

2. Decay of daughter 33Al.

3. Decay of neutron daughter 32Al.

4. Decay of granddaughter 33Si.

5. Flat background from random implant-decay correlations.

The half-lives of 33,32Al were experimentally determined using the same setup and technique (See

Figures 4.4 and 4.5). The P𝑛 value for 33Mg was determined from the NERO dataset as reported

in Section 4.2. The rate of random background coincidences was estimated from the time reverse

correlated decay curve over the same correlation time window. With all these components included

in the Bateman Equation 2.7 model (modified to allow for the neutron branchings), a multi-

parameter Bayesian fit was performed to fit the model to the experimental decay curve. Posterior

distributions of the fit parameters for the BCS-NERO dataset and the SuN-miniDSSD dataset can

be seen in the corner plots in Figures 4.1 and 4.2, respectively. Gaussian priors were chosen for the

P𝑛 value and the half-lives of 33,32Al based on experimental measurements in this work. Uniform

non-informative priors were chosen for the half-life of 33Mg and the initial activity (normalization

factor). Half-life measurements for 33Mg from the fits from both the datasets are reported in Table

4.1. These are in agreement with each other but slightly larger than the 33Mg half-life reported on

NNDC based on previous measurements [66].

A similar procedure was also performed for 33,32Al beam implantations. The decay curves and

the fits for these are shown in Figure 4.4 for 33Al and Figure 4.5 for 32Al. The decay curve for

33Al had contributions from the parent (33Al), daughter (33Si), and neutron daughter (32Si) decay,

whereas the decay curve for 32Al had contributions from just the parent (32Al) and the daughter

(32Si) decay. Half-life measurements for both these isotopes for both datasets are also reported in

Table 4.1 along with those reported by the NNDC based on previous measurements [65, 66, 95].

The half-life measurements for 33Al and 32Al are consistent with previous measurements but the

measured half-lives for 33Mg from both the datasets are higher than the previous measurement [66]

by more than 2 standard deviations.

76

Figure 4.4 33Al decay curve fits for the (a)NERO-BCS and the (b)SuN-miniDSSD datasets.
Half-lives are reported in Table 4.1.

77

Figure 4.5 32Al decay curve fits for the (a)NERO-BCS and the (b)SuN-miniDSSD datasets.
Half-lives are reported in Table 4.1.

78

NNDC Reported NERO Dataset

Isotope
33Mg
33Al
32Al

P𝑛 (%)

14(2)
8.5(7)
0.7(5)

P𝑛 (%)

7.9(13)%
9.0(13)%
0.42(9)%

Table 4.2 Experimentally measured P𝑛 values for 33Mg, 33,32Al with the BCS-NERO dataset.
Measurements for 33,32Al listed here were also used in the determination of the P𝑛 value for 33Mg.

4.2 P𝑛 Values

The experimental 𝛽-delayed neutron branching ratio (P𝑛 value) is determined by counting

the number of neutrons in the decay events correlated with implants. Section 3.1.6 outlines the

procedure to determine the P𝑛 value while accounting for various sources of background. Table

4.2 lists the experimentally determined P𝑛 values for 33Mg, and 33,32Al along with those reported

by NNDC. The P𝑛 value measurements for 33,32Al are consistent with the literature values reported

by the National Nuclear Data Center (NNDC). The P𝑛 value measurement of 7.9(13)% for 33Mg is

significantly lower than the literature value of 14(2) % reported by NNDC [72].

The major contribution to the reported experimental uncertainty comes from the NERO effi-

ciency uncertainty. The NERO efficiency is fairly constant up to neutron energies of ∼ 2 MeV. The

uncertainty on NERO efficiency, (with the assumption a flat efficiency) increases if higher energy

neutrons are expected [82]. Although the energies of the 𝛽-delayed neutrons can range anywhere

from 0 to Q𝛽 - S𝑛 ≈ 8 MeV, the expected neutron energies as per the neutron branchings reported

in [72] are less than 2.5 MeV, so a flat efficiency can be used. Nonetheless, the NERO efficiency

uncertainty still contributes the most to the uncertainty on the P𝑛 value measurement.

4.3 Nuclear Structure of 33Al

The nuclear structure of 33Al as reported by the NNDC is based on two experimental results

[66, 72] from the 𝛽-decay of 33Mg with high-resolution 𝛾-spectroscopy. These works also report

the 33Mg 𝛽 feeding intensities into excited states in 33Al. Based on this information, a 𝛾 template

consisting of total absorption spectrum (TAS), singles spectrum (SS), and multiplicity (Mul) is

generated with GEANT4 simulations of the SuN detector and compared with the experimental

79

Figure 4.6 Nuclear scheme for 33Al from the 𝛽-decay of 33Mg constructed in this work. This is
compared to the current level scheme reported by NNDC based on two experimental results by
Angélique et al. and Tripathi et al. [66, 72]. Red levels and 𝛾-rays are deleted from the current
level scheme and blue levels and 𝛾-rays are new additions to the current level scheme based on
this work. The black levels and 𝛾-rays from the current level scheme were left unchanged.

template obtained with the SuN-miniDSSD dataset. Figure 4.7 shows the comparison. The

experimental TAS has a few peaks missing and also has two new peaks previously unreported. SS

matches well at low energies of up to ∼ 2.5 MeV but not quite at higher energies. The literature

template has way more events with a single multiplicity than seen in the experimental spectrum.

This is the impact of the Pandemonium effect. Based on this information, a few changes were

made to the level scheme in 33Al before extracting the experimental 𝛽-feeding intensities from the

SuN-miniDSSD dataset. Figure 4.6 show a summary of these changes.

80

Figure 4.7 Comparison of the 𝛽-delayed 𝛾 experimental template from the 𝛽-decay of 33Mg
consisting of the Total Absorption Spectrum (TAS), Singles Spectrum (SS), and the multiplicity
(Mul) from the SuN-miniDSSD dataset with the literature template obtained from GEANT4
simulations of the SuN detector based on 𝛽-feeding intensities reported by the National Nuclear
Data Center (NNDC). Two new levels identified in 33Al are highlighted in the TAS.

81

Figure 4.8 Singles Spectrum (SS) gated on the 3983 keV peak in the Total Absorption Spectrum
and multiplicity (Mul) 2 events for the 33Mg 𝛽-delayed 𝛾 rays from the SuN-miniDSSD dataset.
Decay of the new proposed 3983 keV level via the 1618 keV level in 33Al can be confirmed.

4.3.1

2365 and 3983 keV states

NNDC reports a 7% 𝛽-feeding intensity into the 2365 keV level in 33Al. However, the 2365

keV peak was found missing in the experimental TAS. Instead, a new peak at 3983 keV was found

in the experimental TAS. Plotting the SS gated on the new 3983 keV peak reveals a 2365 keV 𝛾

from this level. Further restricting it to multiplicity 2 events confirms a 𝛾 cascade consisting of

2365 keV and 1618 keV from the 3983 keV level. Figure 4.8 shows the SS gated on 3983 keV

TAS peak and Mul 2. The proposed cascade can be easily identified. In conclusion, all the feeding

previously assigned to the 2365 keV level should actually be to the 3983 keV level.

4.3.2

2765 keV state

Angélique et al. [72] found a 2761 keV 𝛾 ray from the 𝛽-decay of 33Mg which they attributed

to an excited level in 33Al. Tripathi et al. [66] also report a 2765 keV 𝛾 ray which they attribute

completely to an excited state in 32Al. As a result, the decay scheme from NNDC doesn’t include

a 2765 keV excited state for 33Al. A 2765 keV 𝛾 peak was also found in the experimental TAS.

However, the 2765 keV was also found in the SS gated on the 4730 keV TAS peak which is a known

excited state in 33Al. Figure 4.9 shows the SS gated on the 4730 keV TAS peak and the 2765 keV

82

Figure 4.9 Singles Spectrum (SS) gated on the 4730 keV peak in the Total Absorption Spectrum
for the 33Mg 𝛽-delayed 𝛾 rays from the SuN-miniDSSD dataset. Decay via the 2765 keV level in
33Al can be confirmed.

peak can be clearly identified. Additionally, assigning 100% of the 2765 keV peak intensity to an

excited level in 32Al would conflict with the significantly lower P𝑛 value measurement from the

BCS-NERO dataset. As a result, the 2765 keV excited level was included in both 33Al and 32Al for

the analysis. This is also reflected in the relatively larger 𝛽-feeding uncertainty to this level in 33Al.

4.3.3

5361 keV state

A new peak at 5361 keV was found in the experimental TAS beyond the last reported excited

state of 4730 keV in 33Al. It was also found in the SS confirming a direct decay to the ground state.

In addition, a 3 𝛾 cascade decay was also found via the 3714 kev level and the 1618 keV level.

Figure 4.10 shows the experimental SS gated on the 5361 keV TAS peak and Mul 3. A 1647 keV

𝛾 ray to the 3714 keV level, followed by a 2096 keV 𝛾 ray to the 1618 keV level, and a 1618 keV 𝛾

ray to the ground state can be identified.

4.4

𝛽 Feeding Intensities

The experimental 𝛽-delayed 𝛾 template (TAS, SS, and Mul) for the 𝛽-decay of 33Mg from

the SuN-miniDSSD dataset was fit as a linear combination of simulated templates generated

from GEANT4 simulations of SuN. Simulated templates were generated for feeding into all the

83

Figure 4.10 Singles Spectrum (SS) gated on the 5361 keV peak in the Total Absorption Spectrum
and multiplicity (Mul) 3 events for the 33Mg 𝛽-delayed 𝛾 rays from the SuN-miniDSSD dataset.
Decay of the new proposed 5361 keV level via the 3714 keV level and the 1618 keV level in 33Al
can be confirmed.

State (keV) in 33Al Angélique et al. [72] Tripathi et al. [66] This work

1618
1838
2096
2365
2765
3264
3714
3983
4310
4730
5361
Quasi-continuum
P𝑛

Ground state

21.3(2)
4.8(2)
8.6(7)
-
8.4(8)
-
-
-
-
18(2)
-
-
14(2)

-

-5(3)
6(2)
-
7(2)
-
2(2)
12(3)
-
13(2)
15(5)
-
-
-

37(8)

-
3.7(4)
-
-
7.1(11)
0.7(1)
1.3(3)
6.9(1)
2.5(2)
28.9(4)
10.7(2)
29.6(15)
7.9(13)

0.7(24)

Table 4.3 Experimental measurements of the 𝛽-feeding intensities (%) in this work for all excited
states in 33Al from the 𝛽-decay of 33Mg compared to two previous measurements [66, 72].
Quasi-continuum refers to the contribution for all levels above the neutron separation energy. P𝑛
refers to the 𝛽-delayed neutron branch. Subtracting everything from unity yields the ground-state
branch.

84

Figure 4.11 Experimental 𝛽-delayed 𝛾 template (TAS, SS, and Mul) for the 𝛽-decay of 33Mg
from the SuN-miniDSSD dataset (black) fit with templates from the GEANT4 simulations of SuN
(blue) using Bayesian Ridge Regression as described in Section 4.4. The light blue band around
the fit shows the 1-𝜎 uncertainty on the posterior distribution of the fit parameters propagated to
the simulated template.

85

discrete energy levels in 33Al and all pseudolevels in the quasi-continuum ranging from the neutron

separation energy up to the 𝛽-decay Q-value. The spacing between pesudolevels was determined

from the SuN energy resolution functions as discussed in Section 3.3. The 𝛾 decay scheme for

each pseudolevel was determined using RAINIER simulations for two nuclear level density models

as described in Section 3.4. Additional templates were generated for 𝛽-delayed neutron decay from

each pesudolevel in the quasi-continuum of 33Al to discrete levels in 32Al. The neutron energy 𝐸𝑛

for all these templates was determined as

𝐸𝑛 = 𝐸𝑙 − 𝑆𝑛 − 𝐸𝑥

(4.1)

where 𝐸𝑙 is the energy of the pesudolevel in 33Al, 𝑆𝑛 is the neutron separation energy of 33Al,

and 𝐸𝑥 is the excitation energy of discrete level in 32Al. All the simulated templates were fit to

the experimental template using Bayesian Ridge Regression [96] which is a probabilistic model of

linear regression. Due to its Bayesian nature, it is found to be more robust to ill-posed problems.

The main advantage of this algorithm is that the function predictions are treated as a random variable

by adding a Gaussian error term with mean 0 and variance 𝜆2 to the deterministic predictions. 𝜆 is

also treated as a free parameter and inferred from the data. The parameters of the fit are normalized

to sum up to 100 which then constitute the experimental 𝛽-feeding intensities. Since the posterior

distributions are nearly Gaussian, 1-𝜎 uncertainties on the posteriors are scaled with the same

normalization factor to report the experimental uncertainties. Figure 4.11 shows the fit obtained

for simultaneously fitting the TAS, SS, and Mul and Table 4.3 lists the corresponding 𝛽-feeding

intensities. A few levels from previous works [66, 72] are not considered here since there wasn’t a

TAS peak in the experimental template and the corresponding 𝛾 rays were identified as a part of

the higher multiplicity cascade. A significant fraction of the 𝛽-feeding intensity is found to be in

the 3 – 5 MeV excitation energy region in 33Al which is consistent with previous measurements.

However, the actual intensities are very different. Table 4.3 compares the 𝛽-feeding intensities from

this work with previous measurements [66, 72].

86

State (keV) in 33Al Feeding Intensity (%)

Ground state
1838
2765
3264
3714
3983
4310
4730
5361

0.7(24)
3.7(4)
7.1(11)
0.7(1)
1.3(3)
6.9(1)
2.5(2)
28.9(4)
10.7(2)

log-ft
7.0+∞
−0.7
6.24(5)
5.48(7)
6.38(7)
6.02(11)
5.24(2)
5.61(4)
4.45(2)
4.73(2)

Table 4.4 Nuclear log-ft values corresponding to the 𝛽-feeding intensities measured in this work
for all excited states in 33Al from the 𝛽-decay of 33Mg.

4.4.1 Nuclear log-ft Values

The nuclear log-ft values from the 𝛽-decay of 33Mg were calculated for all the discrete energy

levels including the ground state in 33Al using the LogFT web tool provided by the National Nuclear

Data Center (NNDC). These are listed in Table 4.4 alongside the 𝛽-feeding intensities for each level.

Since the 𝛽-feeding intensity for the ground state is very small, only a 1 − 𝜎 lower limit is provided

for the log-ft value for the ground state, the upper limit for the log-ft value being zero feeding

intensity and infinite log-ft value.

4.4.2 Gamow-Teller Transition Strengths

The Gamow-Teller transition strength B(GT) is defined as the square of the nuclear matrix

element |𝑀 2

𝑓 𝑖 | connecting the initial and the final states in the parent and daughter nuclei. It is

related to the nuclear log-ft value according to Equation 1.18. B(GT) was calculated for all states

in 33Al from the experimentally determined log-ft values. These are plotted in Figure 4.12 as a

function of the excitation energy in 33Al up to the neutron separation energy.

The experimental B(GT) values are also compared to the B(GT) values calculated with the

Quasi-Random Phase Approximation (QRPA) [97] using a folded-Yukawa potential, a Lipkin-

Nogami pairing model, and including a residual Gamow-Teller interaction. A spheroidal defor-

mation of 𝜖2 = 0.110, 𝜖4 = -0.040, and 𝜖6 = -0.040 determined by the minimum of the nuclear

potential energy surface defined by FRLDM macroscopic-microscopic model [98] was used in this

87

Figure 4.12 Comparison of the experimentally determined Gamow-Teller Transition strengths
B(GT) for the 𝛽-decay of 33Mg with Quasi-Random Phase Approximation (QRPA) calculations.
Experimental B(GT) values can only be reliably determined up to the neutron separation energy
S𝑛 = 5469 keV.

calculation. Figure 4.12 also shows the theoretically calculated B(GT) values using QRPA.

4.5 Level Density Model Comparison

RAINIER and GEANT4 simulations of the pesudolevels templates in the quasi-continuum in

33Al require a prescription for the nuclear level density as a function of the excitation energy.

Two models of the nuclear level densities were considered in this work - the Back-Shifted Fermi

Gas model (BSFG) and the Constant Temperature Model (CTM) as described in Section 3.4.1.

The experimental results for the 𝛽-feeding intensities were insensitive to the nuclear level density

model chosen. This is evident from Figure 4.13 that compares the experimental results from using

different level density models. A correlation coefficient of 0.98 is found between the weights of fit

templates using BSFG and CTM models. Figure 4.14 shows the cumulative 𝛽-feeding intensities in

the quasi-continuum for both the models and Figure 4.15 shows the fit to the experimental TAS at

higher energies (on a log scale) for both the models. Both the figures confirm the non-dependence

of the experimental results on the choice of nuclear level density model. Finally, based on the

marginally better fit to the data, experimental results obtained using BSFG model for the nuclear

88

Figure 4.13 Scatter plot for comparison of the experimentally determined 𝛽-feeding intensities
into excited levels (and pseudolevels in quasi-continuum) in 33Al using different level density
models to treat the quasi-continuum in 33Al. Both BSFG and CTM models lead to very similar
experimental results and a correlation coefficient of 0.98 is found between the 𝛽-feeding
intensities obtained by both the models.

Figure 4.14 Experimental results for cumulative 𝛽-feeding intensities in the quasi-continuum for
both the level density models. Quasi-continuum ranges from the neutron separation energy (5.469
MeV) to the Q-value (13.477 MeV). Both models lead to similar feeding in the quasi-continuum.

89

Figure 4.15 Comparison of simulated TAS using different level density models with the
experimental TAS on the log scale. Colored bands represent 1-𝜎 uncertainties on the posteriors of
fit parameters propagated to the simulated spectra. Both the fits are nearly identical.

level density are reported.

90

CHAPTER 5

QUANTIFIED NUCLEAR MASS MODEL

Nuclear masses are an important piece of data to accurately quantify the strength of Urca cooling in

accreting neutron star crusts. As stated in Section 1.3, Urca cooling luminosity sensitively depends

on the electron-capture threshold, or the Q-value, to the fifth power, which in turn depends on

the nuclear mass. Of the 7000 nuclei predicted to exist in nature [99, 100], less than 3000 have

their masses experimentally measured to date [62]. Even after considerable experimental efforts

ongoing worldwide, many of the exotic nuclei would still be out of reach in the foreseeable future.

Moreover, since the nuclei undergoing Urca cooling are typically neutron-rich, away from the valley

of stability on the nuclear chart, an even smaller fraction of them will have existing experimental

mass measurements. As a result, we have to rely on theoretical predictions which have their own

unique set of challenges. A goal of this dissertation is to build a global nuclear mass model with

quantified uncertainties and covariances and that will be the focus of this chapter.

5.1 Current status

Nuclear reaction network calculations generally rely on the choice of a particular theoretical

mass model. For example, Xnet (refer Section 1.1.2) uses nuclear mass predictions from the Finite

Range Droplet Model (FRDM2012) [63]. A common metric used for making this choice is the root-

mean-squared standard deviation from the experimental dataset. However, theoretical mass models

with many parameters are prone to overfitting and may not generalize well beyond the experimental

dataset. They also lack quantified uncertainties for their predictions. Figure 5.1 shows the binding

energy predictions for the calcium isotopic chain for several theoretical mass models (normalized

to a random model) along with experimental measurements. It can be seen that the spread in their

predictions is least where experimental masses are measured and quickly diverges beyond that, both

in the neutron-rich and neutron-deficient regions. This is why generalizable predictions with honest

uncertainty quantification are very important. With the rise of Machine Learning (ML) in recent

years, especially physics-informed ML, several attempts have been made to tackle this problem.

They try to either optimize the parameters of theoretical models [101–103], learn mass residuals

91

Figure 5.1 Binding Energy predictions for the Calcium isotopic chain for several theoretical mass
models considered here (See Section 5.2.2). The binding energies are normalized to UNEDF2 for
plotting. Note the increased divergence as one moves away from the experimentally measured
region.

[104–115], or even directly predict the masses [116, 117]. More phenomenological approaches

that directly predict masses from neighboring measurements are also possible [118]. (See recent

reviews [119], [120] for more information). All of these highlight the importance of incorporating

domain knowledge in this exercise which can be done in very different ways. Each approach,

however, has a trade-off between complexity, extrapolation accuracy, and computational costs. The

model presented in the following sections uses Gaussian Process Regression for reduced complexity

and improves extrapolation accuracy using Bayesian Model Averaging over the predictions of many

theoretical approaches to utilize their collective wisdom.

5.1.1 Residuals Modeling

Any nuclear mass model that has to be used in nuclear reaction network calculations or astro-

physical simulations should have the following characteristics.

1. It should be based on controlled theoretical formalism such that it uses quantified nuclear

92

Figure 5.2 Nuclear Binding Energy Residuals, i.e. the difference between experimental
measurements and theoretical predictions for all theoretical mass models considered here (See
Section 5.2.2). The residuals have a characteristic structure that provides clues for potential
shortcomings of the underlying theoretical model.

93

properties as inputs.

2. It should generalize beyond the region of known nuclear masses and should cover the entire

nuclear region.

3. Along with nuclear masses, it should also be consistent with other nuclear observables, for

example, separation energies, Q-values, shell gaps, etc.

4. It should have a measure of confidence in its predictions so that proper uncertainty quantifi-

cation can be done.

Keeping this in mind, data-driven Machine Learning (ML) methods, with their ability to capture

the topology of nuclear mass surfaces, are a great tool of choice. However, the nuclear force is quite

complex, and its exact parameterization is unknown. And training a complex ML model requires

a lot of data. Since we only have experimental mass measurements for less than 3000 nuclei, it

puts a restriction on the complexity of ML model and makes it difficult to directly predict accurate

nuclear masses. This is where physics-informed ML models can help solve this issue.

A popular way to incorporate known physics in the modeling process is to model the residuals,

i.e.

the difference between experimental mass measurements and theoretical mass predictions.

Figure 5.2 plots the residuals for a suite of physics-based mass models. The residuals are non-zero

since each model is based on a different set of assumptions and approximations, and has a different

parameterization. The models are also calibrated on different subsets of experimental data. Some

models perform better on the neutron-rich side whereas others perform better on the proton-rich

side. There is, however, a characteristic structure of these residuals that shows where the physics-

based models lack. Modeling and extrapolating just the residuals and adding them back to base

theoretical model predictions gives us a way to build physics-informed data-driven mass models.

These have the desired characteristics and outperform currently available mass models. A set of

such models can be further combined to form a single mass model that combines the predictive

power of each underlying model. This is done via Bayesian Model Averaging (BMA).

94

Figure 5.3 Nuclear chart showing the various nuclei in training set (AME2003), validation set
(AME2016-AME2003) and evidence set for BMA (89Ru, 91Se, 97Kr, 160Nd, 161Pm, 166Gd, 208Hg,
234Ra, 252Md). The choice of these 9 evidence nuclei is intended to cover all mass regions as well
as both the neutron-rich and the proton-rich side of the training set.

5.2 Binding Energy Datasets

The nuclear binding energy is directly related to nuclear masses as

−BE (Z, N) = [𝑍𝑚 𝑝 + 𝑁𝑚𝑛 − M (Z, N)]𝑐2,

(5.1)

where BE (Z,N) is the binding energy of a nucleus with Z protons and N neutrons, 𝑚 𝑝 is the mass

of the proton, 𝑚𝑛 is the mass of the neutron, and M (Z,N) is the mass of the nucleus. It is the energy

associated with the strength of the binding of the nucleons. The higher the binding, the lower is the

mass, and vice versa. Due to this one-on-one correspondence, binding energy is used as a proxy for

nuclear masses. Both experimental data and theoretical predictions are required for the modeling

of residuals.

5.2.1 Experimental Data

The experimental values are the total binding energies taken from the Atomic Mass Evaluation

datasets, AME2003 [121] and AME2016 [122]. Only the experimentally measured values are

95

considered and the empirical predictions referred to as trends of mass surfaces (TMS) are ignored.

Additionally, a new dataset called NEW2018 was also created by augmenting the AME2016 dataset.

This included some newer mass measurements [123–129] after the compilation of AME2016. A

few masses published in KROLL2020 [130] beyond the NEW2018 dataset were also used as

evidence for BMA. Whenever there is a conflict of mass measurement for a particular isotope, the

most recent value is used.

AME2003 was used for initial training followed by AME2016-AME2003 (isotopes with their

experimental mass measurements compiled between 2003 and 2016) for validation. Rather than

selecting a random subset of training data for validation, this method of choosing the validation

set seeks to gauge the extrapolating power of these models and reduce overfitting with the limited

amount of data in possession. After fixing the hyperparameters, the predictions with the model

trained on AME2003 were also used to calculate the BMA weights, although the evidence nuclei

were selected from NEW2018 and KROLL2020. This was to ensure that a comprehensive set of

evidence nuclei is available that are not included in the training. Figure 5.3 shows the nuclear chart

with nuclei contained in training, validation, and evidence sets. The final predictions are made

using the NEW2018 dataset and the BMA weights from the previous step.

AME2020 [62] has also been recently published. However, it would have little impact on the

final global mass model at the moment. Future AME publications should still be used to update

this model periodically.

5.2.2 Theoretical Mass Predictions

A total of 11 theoretical mass models were considered for modeling residuals, and to subse-

quently perform BMA. Seven of these models were based on Skyrme Energy Density Functionals

(EDFs) for the nuclear Density Functional Theory (DFT): SkM* [131], SkP [132], SLy4 [133],

SV-min [134], UNEDF0 [135], UNEDF1 [136], UNEDF2 [137]. Two models were based on

Gogny EDFs: BCPM [138] and D1M [139]. Two additional models were also considered that are

generally used currently for astrophysics simulations: the microscopic-macroscopic FRDM2012

model [63] and the Skyrme-HFB model HFB-24 [64]. The deviations for these models from exper-

96

imental values, measured as the root mean square over all the isotopes range from a few MeV for the

DFT-based models to around 600 keV for the FRDM2012 model. Following statistical treatment

as described in Section 5.3.1, and adding back the residuals, this RMS deviation is reduced to less

than 200 keV for all mass models.

5.3 Statistical Methods

5.3.1 Gaussian Process Regression

The residuals between experimental and theoretical binding energies are modeled as a Gaussian

Process following [109]. Let 𝑥𝑖 be the feature vector describing the nucleus and 𝑦𝑖 be the corre-
sponding residual to be modeled. The residual 𝑦𝑖 = 𝑦𝑒𝑥 𝑝
𝑖 − 𝑦𝑡ℎ (𝑥𝑖) dataset is constructed for each

nuclear mass model. The mapping from 𝑥𝑖 to 𝑦𝑖 can be represented as:

𝑦𝑖 = 𝑓 (𝑥𝑖, 𝜃) + 𝜎𝜖𝑖 ,

(5.2)

where 𝜖𝑖 is a standard Gaussian variable and 𝜎 is the noise scale factor. If 𝑓 is deterministic, this

term accounts for the stochastic nature of the experimental data. Here, the function 𝑓 (𝑥𝑖, 𝜃) itself

is stochastic and is modeled as a Gaussian Process:

𝑓 (𝑥𝑖, 𝜃) ∼ GP (𝜇, 𝑘𝜂,𝜌 (𝑥, 𝑥′)) ,

(5.3)

with mean 𝜇 and covariance kernel 𝑘. The covariance kernel is chosen to be the squared exponential

kernel such that

𝑘𝜂,𝜌 (𝑥, 𝑥′) := 𝜂2𝑒

(cid:205)𝑖 −

𝑖 )2

( 𝑥𝑖 −𝑥′
𝜌𝑥𝑖

,

(5.4)

where 𝑖 goes over the features. 𝜂 is the variance and 𝜌𝑥𝑖 corresponds to the lengthscale for each

feature. Thus, the parameter vector 𝜃 comprises of {𝜇, 𝜂, 𝜌𝑥𝑖 }. Additionally, a white noise kernel

with a noise level of 0.0235 MeV was added to the squared exponential kernel to account for the

experimental errors. This noise level was chosen based on the average experimental uncertainty and

was held constant considering its small value. Effectively, it allows the GP a leeway to pass within

0.0235 MeV of the training points to get the best fit. The feature vector is comprised of {𝑍, 𝑁 } i.e.

the proton number and the neutron number, respectively. Two additional features, promiscuity and

97

pairing factor were also considered to aid the learning process as described in Section 5.4.2. As a

Bayesian model, we draw samples for the parameters 𝜃 from their posterior distribution given by

the Bayes equation

𝑝(𝑦|𝜃) :=

𝑝(𝜃|𝑦)𝜋(𝜃)
∫ 𝑝(𝜃|𝑦)𝜋(𝜃)𝑑𝜃

,

(5.5)

where 𝑝(𝜃|𝑦) is the statistical model’s likelihood and 𝜋(𝜃) the prior on its parameters. Gaussian

likelihood with an inverse-gamma function for priors was used for modeling. Samples were drawn

from the posterior using Markov Chain Monte Carlo (MCMC) after the stationary state was reached.

Finally, the GP predictions are added back to the theoretical masses to get the statistically corrected

mass models and the GP uncertainties represent the uncertainties for the statistically corrected

models. These models can now be averaged using BMA as discussed in the next section.

5.3.2 Bayesian Model Averaging

The predictions of statistically corrected mass models are combined using Bayesian Model

Averaging (BMA). The BMA predictions are an average of the predictions of the individual model,

weighted by the posterior probability for that model to be the hypothetical ’true’ model given priors

and data. These can be written as:

𝑤 𝑘 = 𝑝(M𝑘 |𝑦) =

𝑝(𝑦|M𝑘 )𝜋(M𝑘 )

(cid:205)𝑝

ℓ=1

𝑝(𝑦|Mℓ)𝜋(Mℓ)

,

(5.6)

where 𝜋(M𝑘 ) is the prior weight of model M𝑘 , and the evidence 𝑝(𝑦|M𝑘 ) is obtained by integrating

the likelihood equation over the parameter space. For the GP emulators, this gives

∫

𝑝(𝑦|M𝑘 ) =

𝑝(𝑦|𝜃 𝑘 , M𝑘 )𝜋(𝜃 𝑘 , M𝑘 )𝑑𝜃 𝑘 .

(5.7)

A uniform, non-informative prior distribution is used for the weights. The data 𝑦 are the

experimental mass measurements of a subset of “evidence” nuclei. As a result, our BMA weights

are also sensitive to the choice of the evidence nuclei. Their choice in our study and the resulting

weights are discussed in Section 5.4.4.

Once the weights are calculated, the final BMA predictions and uncertainties can be obtained

98

Figure 5.4 The absolute change in Binding Energy residuals (Δ𝐵𝐸) predicted by the GP emulator
with 10 additional samples drawn as a function of the number of samples. This is a proxy for
MCMC convergence as (Δ𝐵𝐸 → 0) means additional samples do not significantly change the
predictions.

as:

𝑦(𝑥) =

∑︁

𝑘

𝑤 𝑘 𝑦 (𝑘) (𝑥) ;

𝜎2

𝑦 (𝑥) =

∑︁

𝑘

𝑤 𝑘 (𝑦 (𝑘) (𝑥) − 𝑦(𝑥))2 +

∑︁

𝑘

𝑤 𝑘 𝜎2

𝑦𝑘 (𝑥) ,

(5.8a)

(5.8b)

where 𝑦𝑘 are the individual model predictions, 𝜎(𝑦𝑘 ) are the model uncertainties and 𝑤 𝑘 are the

model weights [140]. The two terms in the equation for uncertainties correspond to the uncertainty

on model choice and the uncertainty on individual model parameters, respectively. The first term

therefore represents a part of the uncertainty that is not accounted for when using predictions

derived from a single model. This highlights the role of BMA in proper uncertainty quantification.

5.4 Model Aspects

5.4.1 Sample Convergence

For most practical applications of Bayesian analysis, the exact integrals of the posterior dis-

tributions for making final predictions are often intractable and we have to resort to sampling

methods. As a result, Markov Chain Monte Carlo MCMC sampling methods were used to sample

99

Model

Raw GP_Zero GP_Con GP_Dis GP++_Zero GP++_Con GP++_Dis

SkM*
SkP
SLy4
SV-min
UNEDF0
UNEDF1
UNEDF2
BCPM
D1M
FRDM2012
HFB-24

Average

7.74
3.84
6.01
3.37
1.73
2.58
2.51
2.13
5.40
0.79
0.70

3.35

3.59
2.11
3.47
1.88
1.09
1.51
1.51
1.27
2.74
0.62
0.66

1.86

3.81
2.04
2.76
1.49
1.09
1.46
1.43
1.25
1.01
0.63
0.67

1.60

3.24
1.33
1.34
0.97
0.89
1.31
1.24
1.07
0.99
0.57
0.72

1.24

1.62
1.06
1.47
1.02
0.70
0.84
0.84
0.90
1.19
0.65
0.67

1.00

1.74
1.12
1.24
0.97
0.73
0.79
1.01
1.05
0.74
0.54
0.67

0.96

1.59
1.00
1.17
1.84
0.82
0.75
1.22
0.90
1.03
0.70
0.69

1.06

Table 5.1 Root-mean-squared (RMS) deviations of binding energies (in MeV) over the
AME2016-AME2003 validation set, for each nuclear mass model, with and without (raw) the GP
corrections for variants GP_Zero, GP_Constant (GP_Con), GP_Discrete (GP_Dis), GP++_Zero,
GP++_Constant (++_Con), GP_++_Discrete (++_Dis) as described in Section 5.4. The average
values in the last row correspond to a simple average over all the nuclear mass models for each
variant. Based on this measure, the GP++_Constant is the best-performing variant overall.

the GP parameters from their posterior distributions. A natural dilemma that arises is the trade-off

between computational costs and the convergence of the final predictions. Figure 5.4 shows Δ𝐵𝐸

i.e. the change in the binding energies when 10 additional samples are drawn. (This is after the

initial 50,000 samples burnt in for the MCMC to stabilize). The quantity plotted is the rms of the

difference over all nuclei. Initially, we see large changes as we draw more samples which nicely

converge as the number of samples becomes large. Occasional jumps in Δ𝐵𝐸 correspond to the

exploration of new parameter space. We decided to stop at 1000 samples since the change in BE at

this stage was about 20 keV for all models which is very small for all practical aspects.

5.4.2 Feature Selection

A unique nucleus can be identified with just the atomic number Z and the number of neutrons

N. However, in the spirit of physics-informed ML, additional aspects of the nuclear force can be

incorporated that can aid in the learning process. Nuclei exhibit shell structure where a certain

number of nucleons corresponding to closed shells are more tightly bound [141, 142]. They also

exhibit pairing effects since nuclei with an even number of nucleons have slightly lower energies

100

than nuclei with an odd number of nucleons. As a result, two additional features were constructed -

the promiscuity factor and the pairing factor. The promiscuity factor 𝑝, accounted for shell effects

and is defined as

𝑝 =

𝜈𝑝𝜈𝑛
𝜈𝑝 + 𝜈𝑛

(5.9)

where 𝜈𝑝 and 𝜈𝑛 are the differences between the actual nucleon numbers 𝑍 and 𝑁 and the nearest

magic number (20, 28, 40, 50, 82 and 126). The pairing factor

𝛿 =

[(−1) 𝑍 + (−1) 𝑁 ]
2

(5.10)

accounts for pairing effects. Models with these two additional features are designated as GP++

in the model validation step (see Table 5.1). The best-performing model was found to be the one

where these two additional features were used in the training.

5.4.3 Choice of GP Mean

There are no hyperparameters to be tuned, given the Bayesian nature of the analysis. The model

predictions essentially integrate over all possible hyperparameters weighted by their posterior

probabilities. However, there is a choice in terms of parameterizing the mean of GP. The mean

of GP is usually set to 0 (GP_Zero), if no significant bias is expected in the training data. This

wasn’t guaranteed for some of the base theoretical models considered since they were calibrated

on different subsets of experimental data. To allow for this, GP_Constant (GP_Con) models had a

constant mean for the GP throughout the nuclear chart. Additionally, the GP_Discrete (GP_Dis)

model had a constant mean value for each subset of nuclei with the same open shell, i.e. all nuclei

between two proton and neutron magic numbers. Combined with the two feature variants, we get

a total of 6 models as shown in Table 5.1.

After training on the AME2003 dataset, all six GP variants were validated against the 281 nuclei

in the AME2016-AME2003 dataset, containing all the nuclei that had their experimental masses

compiled between 2003 and 2016. Table 5.1 shows the root mean squared error over the validation

set for each variant as well as for each theoretical mass model. All the GP variants perform

better than the raw models, with some raw models showing more improvements than others. More

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Corrected Model BMA Weight

SKM*
SkP
SLy4
SV-min
UNEDF0
UNEDF1
UNEDF2
BCPM
D1M
FRDM2012
HFB-24

0.07
0.13
0.08
0.17
0.12
0.11
0.03
0.13
0.10
0.01
0.05

Table 5.2 BMA model weights (rounded to two digits past decimal) for statistically corrected
mass models (see Section 5.3.2 based on evidence nuclei as described in Section 5.4.4. Note that
these weights are for statistically corrected models as per Section 5.3.1 and not the raw theoretical
models.

phenomenological models like HFB24 or FRDM2012 have already captured most of the information

contained in the experimental data in the raw models themselves. The ++ variants outperform their

counterparts, showing definite correlations along 𝑝 and 𝛿, highlighting the importance of shell

effects and pairing effects for predicting the masses of nuclei. The GP_++_Constant performs better

than GP_++_Zero, as already noted before [109, 110, 143]. The most complex GP_++_Discrete

variant, surprisingly doesn’t perform as well as the GP_++_Constant variant. This may be because

of redundancy where the additional features and the discrete mean both try to capture the shell

effects, thereby complicating the training. Based on these observations, the GP_++_Constant was

chosen for all further analysis owing to its accuracy and simplified implementation.

5.4.4 BMA Weights

Calculations of the model weights require a choice of evidence nuclei. In the classical BMA

literature [144], the evidence can be a subset of the training set itself. However, the predictions of

GP models are very close to the training data. As a result, the evidence set is chosen to be comprised

of nuclei outside the training or the validation set. The weights then represent the true extrapolative

power of the models. A total of 9 nuclei chosen in the evidence set are 89Ru, 91Se, 97Kr, 160Nd,

161Pm, 166Gd, 208Hg, 234Ra, 252Md. These are chosen to cover all the regions in the nuclear chart

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(light, medium, and heavy mass regions) on both the neutron and the proton-rich sides. Figure

5.3 shows the intended balanced distribution of evidence nuclei throughout the nuclear chart and

Table 5.2 gives the corresponding weights for each model. Models based on Syrme EDF like

SV-min, SKP, and the BCPM model based on Gogny EDF have large weights as compared to the

phenomenological model like HFB-24. This shows that the statistically corrected DFT models can

better extrapolate nuclear masses than the corrected phenomenological models. Changing the set

of evidence nuclei can change the absolute weights for each model. Our choice here was motivated

by the need for a global mass model. Moreover, it serves as an example for BMA analysis. Future

studies can customize their evidence set depending on the applications one is interested in. For

example, see [110, 145].

The restricted choice of 9 evidence nuclei was limited in part by the availability of additional

data and in part by the computational cost of computing high-dimensional evidence. As more mass

measurements become available in the future, the evidence can be approximated more strongly

for faster computations and increased stability, for instance via the Laplace Approximation [144]

where the posteriors are matched to a Gaussian distribution. For the given evidence set here, the

weights calculated by sampling the posterior were very similar to the weights calculated by Laplace

approximation. Thus, we can include more evidence nuclei in future analysis by using appropriate

approximations.

5.5 Final Mass Model

5.5.1 Binding Energy Predictions

The residuals predicted by the GP++_Constant model are weighted as per Equation 5.8 for

BMA. Figure 5.5 shows the residuals for 6 different isotopic chains. It also shows the need for

interpolations, for example, in Ytterbium and Uranium isotopic chains, along with extrapolations

which is the primary goal of this model. These residuals are then added back to base theoretical

model predictions (also weighted as per BMA) to get the final mass predictions. Figure 5.6 shows

how the final trend in total binding energy per nucleon is extrapolated for the calcium isotopic

chain. The uncertainties are modest near the experimentally accessed region and grow upon

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Figure 5.5 BMA residuals for six isotopic chains. Uncertainties become larger as one moves away
from experimentally accessed regions. Ytterbium and Uranium mass chains also feature the need
for interpolations along with extrapolations.

104

Figure 5.6 The trend in BMA predictions of binding energy per nucleon for the calcium isotopic
chain. 60Ca is the most exotic calcium isotope to be seen in the laboratory so far [146] but its mass
hasn’t been experimentally measured yet. The red star shows BMA prediction along with
uncertainty.

moving further. This is natural since the model is least confident in its predictions in the most

exotic regions. 60Ca is the most exotic isotope made in the laboratory recently [146] whose mass

is yet to be measured. BMA predicts its mass along with uncertainty as shown in Figure 5.6.

5.5.2 Correlations

Neighboring nuclei in the samples of mass tables from posterior distributions show plenty

of correlations along the proton number as well as the neutron number axes. Figure 5.7 shows

the correlation coefficients with neighboring nuclei for 131Eu on the proton-rich side and 60Ca

on the neutron-rich side of the valley of stability. The mass prediction of any nucleus is perfectly

correlated with itself, which is shown by the black square in the center corresponding to a correlation

coefficient of unity. The correlation is strong on an average of up to 3 units in the proton number

axis and up to 5 units in the neutron number axis.

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Figure 5.7 Heatmap of correlation coefficients of BMA predictions of the mass of a nucleus with
its neighboring nuclei. The black square in the center corresponds to a complete correlation of the
mass of a nuclei with itself. BMA predictions have high positive correlations on both sides of the
valley of stability, and along both the proton number as well as the neutron number axes.

5.5.3 Quantified Uncertainties with Covariances

BMA doesn’t just predict quantified uncertainties, but also the full covariance matrix. As seen

in Section 5.5.2, masses of neighboring nuclei are correlated. This face should be exploited when

sampling nuclear masses, for example, for a sensitivity study. Moreover, masses are seldomly used

directly in nuclear astrophysics simulations. They are converted into mass filters like separation

energies, Q-values, etc. before being utilized. It then becomes crucial to use the covariances to

properly propagate uncertainties to mass filters. Let’s say we want to calculate the two-neutron

separation energies (𝑆2𝑛) from BMA masses. It can be calculated as -

𝑆2𝑛 (𝑍, 𝑁) = 𝐵𝐸 (𝑍, 𝑁) − 𝐵𝐸 (𝑍, 𝑁 − 2)

(5.11)

where BE is the binding energy, Z is the proton number and N is the neutron number. The

uncertainties without covariances can be calculated as:

√︃

𝛿(𝑆2𝑛) =

𝜎2
(𝑍,𝑁) + 𝜎2

(𝑍,𝑁−2)

(5.12)

This corresponds to the red curves in Figure 5.8. It shows the empirical coverage probabilities

(ECP), i.e. the fraction of validation data that lies within a given standard deviation from the mean

106

Figure 5.8 Empirical Coverage Probabilities (ECP) for two-neutron separation energies derived
from BMA mass predictions (see Section 5.5.3). The red line doesn’t account for the covariance
term in uncertainty calculation whereas the green line accounts for it and as a result, showcases a
more honest uncertainty quantification (closer to the black line).

107

prediction. For the normal uncertainties reported, approximately 68% of validation data should fall

within 1-𝜎, 95% within 2-𝜎, and so on. A perfectly honest uncertainty quantification is shown by

the black lines that correspond to Gaussian quantiles. Rewriting Equation 5.13 with the covariance

term, we get

√︃

𝛿(𝑆2𝑛) =

(𝑍,𝑁) + 𝜎2
𝜎2

(𝑍,𝑁−2) − 2𝜎(𝑍,𝑁)(𝑍,𝑁−2)

(5.13)

where the last term represents the covariance between nuclei (Z,N) and (Z,N-2). Plotting ECPs

with covariances gives the green curves in Figure 5.8. It can be seen that the green curves are

closer to the black curves than the red curves. While not necessarily dishonest, the red curves are

overly pessimistic in their predictions. This highlights the role of covariances in proper uncertainty

quantification and for calculating mass filters that are used in nuclear astrophysics calculations.

108

CHAPTER 6

IMPACT ON ASTROPHYSICAL MODELS

This chapter discusses the astrophysical impact of updated nuclear data from this work on the

strength of Urca cooling in accreting neutron stars. The Xnet nuclear reaction network used for

such calculations is introduced, followed by the changes in Urca cooling due to the revised ground-

state to ground-state 𝛽-decay transition strength in 33Mg → 33Al. The changes in the profile of

Urca cooling due to the updated BMA mass model (Chapter 5) are also discussed.

6.1 Xnet Nuclear Reaction Network

Xnet is a customized nuclear reaction network for simulating the steady-state composition in

accreting neutron star crusts. It follows the compositional changes in an accreted fluid element as

it is pushed deeper into the neutron star by matter from ongoing accretion. It is a comprehensive

reaction network that allows for all possible nuclear reactions that can take place in the crust

including electron capture, 𝛽 decay, neutron capture/emission, thermonuclear and pycnonuclear

fusion, and neutron transfer reactions as discussed in Section 1.1.2. It also tracks the composition

of the fluid element and the amount of nuclear energy generation as a function of the depth.

This gives the steady-state composition of the crust, and the nuclear heating profile in the crust,

respectively. It is, therefore, a great tool to study the effects of changing nuclear properties on the

composition and heat deposition in the accreting neutron star crust.

The amount of Urca cooling in the outer crust is the most sensitive to weak interaction rates

and nuclear masses. The 𝛽-decay and electron capture rates due to weak interactions depend on

external conditions like temperature, density, and electron Fermi energy. As a result, only the

transition strengths that depend on the initial and final nuclear wavefunctions are stored in the Xnet

database. All the weak reaction rates are calculated on the fly for each timestep using a fast phase

space approximation [43]. Experimental data for the transition strengths is used wherever available

whereas theoretically calculated transition strengths with the Quasi-Random Phase Approximation

(QRPA) [97] are used otherwise. Nuclear masses from the latest Atomic Mass Evaluation (AME)

database [122] are used if available along with the Finite Range Droplet Model (FRDM) mass

109

33Mg𝑔.𝑠. → 33Al𝑔.𝑠.

Tripathi et al. [66] This work

Feeding Intensity (%)
log-ft value
Intrinsic Urca Cooling Luminosity (L34)

37(8)
5.2(1)
3847.3

0.7(24)
7.0+∞
−0.7
61.0

Table 6.1 Comparison of the updated parameters for 33Mg𝑔.𝑠. → 33Al𝑔.𝑠. transition from this
work with previous measurements. A two orders of magnitude reduction in the Intrinsic Urca
Cooling Luminosity is found.

Figure 6.1 Total integrated energy deposited by nuclear reactions as a function of pressure in the
accreting neutron star crust (a proxy for depth) as calculated by Xnet for a pure A = 33 mass
chain. Updated log-ft values for 33Mg → 33Al transition from this work leads to a 6.4 MeV/u
reduction in Urca cooling.

predictions [63] where experimental data is not available.

6.2 33Mg - 33Al Transition

The transition strengths for the 𝛽-decay of 33Mg → 33Al had been previously measured by

Tripathi et al.[66], which are included in the National Nuclear Data Center (NNDC) database.

These experimental values are used in Xnet for the crust cooling calculations by default. However,

the ground state to ground state transition strength is found to be significantly lower in this work.

Table 6.1 compares the feeding intensities, nuclear log-ft values, and the corresponding Urca

110

cooling luminosity for previous and current measurements. The lower feeding intensity to the

ground state, and the corresponding larger log-ft value translates into a two orders of magnitude

reduction for the intrinsic Urca luminosity. The updated log-ft value and the transition strengths

from this work were used in Xnet for refining the crust cooling calculations. Xnet was run for a

pure A = 33 mass chain with and without the updated experimental results from this work. Figure

6.1 shows the integrated nuclear energy deposition in the crust as a function of the pressure, which

is a proxy for depth in the neutron star crust. Decreases in energy depositions signify negative net

energy generation and thus, the presence of Urca cooling. The reduced ground state to ground state

transition strength for the 𝛽-decay of 33Mg → 33Al from this work leads to a 6.4 MeV/u reduction

in Urca cooling as shown in Figure 6.1. It should be noted that these simplistic calculations are for

the maximum Urca cooling possible as the entire crust is composed of just mass A = 33 nuclei. In

realistic crusts, the mass fraction of A = 33 nuclei will be folded into the calculations, which is still

significant for the crust composed of X-ray burst ashes.

6.3 Nuclear Masses

Nuclear masses in Xnet were updated from the default FRDM model to the BMA mass model

developed in this work. Changing nuclear masses affects multiple nuclear reaction rates. This

can change the trajectory of the entire network due to a complex non-linear dependence of the

abundances on the reaction rates. Changes in electron capture and 𝛽-decay Q-value change the

depth and strength of electron capture heating as well as Urca cooling. Changes in neutron

separation energies change the sequence of neutron release and the transition from the outer crust

to the inner crust. They also change the neutron transfer reaction rates. This changes the abundance

distribution per mass chain, and again impacts electron capture heating and Urca cooling. Xnet

calculations were performed for a crust composed of superburst ashes [147] with the default FRDM

mass model as well as the updated BMA mass model. Figure 6.2 shows the comparison of nuclear

energy deposition as a function of the pressure or the depth in the neutron star crust for both cases.

Dips in energy generation correspond to Urca cooling. Updating the nuclear masses from the BMA

mass model changes the relevant Urca cooling pairs. 59Ca – 59Sc is the most important Urca pair

111

Figure 6.2 Total integrated energy deposited by nuclear reactions as a function of pressure in the
accreting neutron star crust (a proxy for depth) as calculated by Xnet for crust composed of
superburst ashes. Updating the nuclear masses in Xnet from the BMA model leads to completely
different Urca cooling pairs in the crust and the overall location of Urca cooling is shifted deeper
in the crust.

for mass A = 59 chain instead of the 59Sc – 59Ti Urca pair with the FRDM mass model. The 57Sc –

57Ti Urca pair in the mass A = 57 chain becomes relevant which was not a significant Urca cooling

chain with the FRDM mass model. Thus, changing nuclear masses leads to a very different Urca

cooling profile.

Although the BMA masses lead to a slightly lower Urca cooling overall, it happens at deeper

locations. The depth of Urca cooling has important implications for the thermal structure of the

neutron star crust. Since Urca cooling is extremely sensitive to temperature (∼ T5) [60], it thermally

decouples the regions above and below an Urca cooling layer. If a particular heat source is below

an Urca cooling layer, it will increase the rate of Urca cooling, and most of the energy deposited

will be radiated away by the neutrinos. This puts a strong constraint on the location of heat sources

in the neutron star crust. A previous study [148] explored the effect of nuclear masses on Urca

cooling where new experimental mass measurements led to a cooler crust. Results in this work

112

also confirm this crust cooling sensitivity to nuclear masses. As a result, it is very important to use

an accurate nuclear mass model for crust cooling calculations.

113

CHAPTER 7

CONCLUSIONS AND FUTURE DIRECTIONS

The experimental results presented in this work confirm the forbidden nature of the 𝛽-decay

transition from the ground state in 33Mg to the ground state in 33Al. This can be concluded from the

small 0.7(24)% branching measured to the ground state corresponding to a nuclear log-ft value of

7.0+∞

−0.7. This result was achieved using a combination of high-efficiency 𝛾 detectors for performing
Total Absorption Spectroscopy and a high-efficiency neutron detector to constrain the 𝛽-delayed

neutron branching ratio. The choice of this experimental setup helped mitigate the systematic

errors arising from the Pandemonium effect that could lead to an overestimation of the low-energy

branchings. Based on the 𝛽-decay selection rules, this result is in agreement with a negative parity

assigned to the ground state in 33Mg [69, 70]. The impact of this measurement on the Urca cooling

in accreting neutron star crusts was also investigated. For a crust composed completely of mass A =

33 mass chain, the new ground state to ground state transition strength of 33Mg ←→ 33Al reduces

the Strength of Urca cooling from 55.8 MeV/u to 49.4 MeV/u by 6.4 MeV/u. Previous results from

the 𝛽-decay of 61V [149] and the ongoing analysis from the 𝛽-decay of 55Ca seem to indicate a

trend that both QRPA and experimentally determined Urca cooling rates seem to be over-estimated.

In addition to the ground state transition, the 𝛽-feeding intensities to all the excited states in

33Al were also measured. The strongest feeding of 28.9(4)% was found to the 4730 keV excited

state in 33Al, corresponding to a nuclear log-ft value of 4.45(2). A total of 29.6% feeding was

found to the quasi-continuum region above the neutron separation energy. The 𝛽-delayed neutron

branching ratio was measured to be 7.9(13)%. Additionally, based on the analysis of 𝛾 ray energy

spectra, two new excited states at 3983 keV and 5361 keV were proposed with 𝛽-feeding intensities

of 6.9(1)% and 10.7(2)% corresponding to log-ft values of 5.24(2) and 4.73(2), respectively. The

3983 keV state is found to decay via the 1618 keV state emitting a 2365 keV 𝛾 ray in the cascade.

This intensity was wrongly attributed to a 2365 keV state in previous measurements and the Total

Absorption Spectroscopy technique helped identify the new level.

A new technique for modeling nuclear mass residuals was developed using Bayesian Gaussian

114

Process Regression leading to statistically corrected theoretical mass models. Further, predictions

from multiple statistically corrected models were combined using Bayesian Model Averaging

(BMA) to yield a global nuclear mass model with quantified uncertainties. The BMA mass model

agrees with all experimental mass measurements while also retaining the power to extrapolate the

nuclear mass surface to more exotic nuclei with quantified uncertainties. Accreting neutron star

crust calculation with the updated BMA mass model for a crust composed of superburst ashes leads

to a slightly reduced Urca cooling but deeper in the crust compared to the default calculations with

the FRDM mass model. This sensitivity highlights the need to use accurate nuclear mass models

in such calculations. Previous works [148] also report the correlations between nuclear masses and

the strength of Urca cooling.

7.1 Future Directions for Nuclear Experiments

The experimental results presented in this work highlight the efficacy of the Total Absorption

Spectroscopy to measure the ground state to ground state 𝛽-decay transition strengths in exotic

neutron-rich nuclei. Even for nuclei with experimentally measured log-ft values, the Pandemonium

effect may have systematically skewed the results and there seems to be a trend of QRPA and previous

experiments to overestimate Urca cooling. As a result, it is important to perform more experimental

measurements for all relevant Urca pairs with the technique used in this work. With the Facility

for Rare Isotope Beams (FRIB) coming online in the last couple of years, more exotic Urca cooling

candidates will be within experimental reach in the near future. Detector systems like the FRIB

Decay Station initiator (FDSi) and the FRIB Decay Station (FDS) in the future with their modular

ability to perform high-resolution 𝛾-ray spectroscopy with DEGA, total absorption spectroscopy

with MTAS, and neutron spectroscopy with VANDLE and NEXT in the same experiment provide

the best platform for such studies. Multiple scientific results from FDSi experiments have already

been published [150–152] and a 𝛽-decay experiment of multiple isotopes in the region around 55Ca

motivated by Urca cooling in superbursting systems is currently being analyzed. In addition to

𝛽-decay measurements, the results presented in this work also highlight the importance of nuclear

mass measurements given the sensitivity of Urca cooling to nuclear masses. All such efforts will

115

help improve our understanding and properly quantifying the role of Urca cooling in accreting

neutron star crusts.

7.2 Future Directions for Modeling Neutron Star Crusts

The immediate future direction is to calculate the impact of the experimental results presented

in this work for more realistic crust compositions, including X-ray burst ashes. Xnet is a tool of

choice for such accreting neutron star crust calculations relying on a lot of nuclear data. One avenue

for improvement is to increase the quality of nuclear data in its database and regularly update it

with new experimental results. Another avenue for improvement would be on the computational

side to speed up the calculations. The availability of a mass model like the BMA with quantified

uncertainties and covariances allows for the full propagation of mass uncertainties on the crust

calculations. However, to do this with an MCMC-type sampling process, the reaction network

would need to run much faster than it currently does.

Recently suggested alternate approach [44–46] with the diffusion of free neutrons in the inner

crust would violate the hydrostatic equilibrium condition and the assumption of a constant red-

shifted neutron chemical potential. This would lead to a very different Equation of State (EOS) that

would transform the inner crust into homogeneous nuclear matter. As a result, no pycnonuclear

fusion reactions can take place and the heat deposited would be much lower than current estimates.

Although the reaction pathways for this process have not yet been identified, Xnet with the

appropriate modifications can be used to identify such reaction pathways and improve the accreting

neutron star crust models.

7.3 Future Directions for Modeling Nuclear Masses

The BMA nuclear mass model developed in this work extrapolates the nuclear mass surface

beyond the experimentally measured region on the nuclear chart. The correlation length for mass

predictions is found to be few nucleons in the neutron number axis as well as the proton number

axis. As a result, with newer experimental nuclear mass measurements, the BMA model needs

to be updated periodically to ensure most of the experimental trend is still captured. More mass

measurement will also lead to a larger evidence set that can be used to refine BMA weights. Another

116

promising avenue for further progress would be toward Local Bayesian Model Mixing (LBMM)

[153] where the weights of participating models can vary over the nuclear chart as opposed to being

constant like in BMA. It is a kind of hierarchical model where the parameters of the functions

for model mixing weights are learned from the data. This allows for the development of niche

theoretical models that can be mixed locally.

Nuclear mass residuals modeled with the Gaussian Process emulator contain information about

the shortcomings of a theoretical model. This information can be used to develop high-fidelity

theoretical models and drive progress in nuclear theory. Several frameworks combining data

science and machine learning with the domain knowledge in nuclear physics have been explored

in the past and are actively being explored for many nuclear observables. The Bayesian Gaussian

Process Regression and Bayesian Model Averaging techniques explored in this work for predicting

nuclear masses are excellent tools that can also be applied to other nuclear quantities.

117

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APPENDIX A

LIST OF MY CONTRIBUTIONS

A.1 Publications

A.1.1 Neutron star crust modeling

1. “Impact of Pycnonuclear Fusion Uncertainties on the Cooling of Accreting Neutron Star

Crusts."

R. Jain et al., The Astrophysics Journal 955, 51 (2023).

• Performed and analyzed the sensitivity studies calculations.

• Wrote the first draft for the paper.

2. “The impact of neutron transfer on heating and cooling of accreted neutron star crusts."

H. Schatz et al., The Astrophysical Journal 925, 205 (2022).

• Performed reaction network calculations with neutron transfer reactions included for

multiple crust compositions.

• Discussed and analyzed the outputs of these calculations.

3. “Properties of the Shallow Heat Source in KS 1731–260."

R. Jain et al., (in prep).

• Performed Bayesian fitting for the properties of the shallow heat source in KS 1731–260

for various crust compositions.

• Wrote the first draft for the paper.

4. “Effects of nuclear pasta impurity on the cooling of KS 1731–260."

R. Jain et al., (in prep).

• Inferred the properties of the shallow heat source in KS 1731–260 with and without

nuclear pasta impurity and analyzed the results.

134

5. “Effect of nuclear masses on Urca Cooling in accreting neutron star crusts."

R. Jain, S. Lalit et al., (in prep).

• Provided nuclear masses for the Bayesian Model Averaging mass model.

• Analyzed the effects of changing mass models on numerous Urca pairs.

• Contributed to the writing of the paper.

A.1.2 Nuclear Physics Experiments

1. “𝛽-decay of 33Mg with Total Absorption Spectroscopy."

R. Jain et al., (in prep).

• Set up and performed the experiment, and analyzed the experimental data as my Ph.D.

thesis experiment.

• Wrote the first draft for the paper.

2. “Proton shell gaps in N=28 nuclei from the first complete spectroscopy study with FRIB

Decay Station Initiator."

I. Cox et al., Physical Review Letters 132, 152503 (2024).

3. “𝛽-decay of neutron-rich 45Cl located at the magic number N = 28."

S. Bhattacharya et al., Physical Review C 108, 024312 (2023).

4. “𝛽-decay of 36Mg and 36Al: Identification of 𝛽-decaying isomer in 36Al."

R. S. Lubna et al., Physical Review C 108, 014329 (2023).

5. “Microsecond Isomer at the N = 20 Island of Shape Inversion Observed at FRIB."

T. J. Gray et al., Physical Review Letters 130, 242501 (2023).

6. “Crossing N = 28 Toward the Neutron Drip Line: First Measurement of Half-Lives at FRIB."

H. L. Crawford et al., Physical Review Letters 129, 212501 (2022).

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7. “Evidence of a Near-Threshold Resonance in 11B Relevant to the 𝛽-Delayed Proton Emission

of 11B𝑒."

Y. Ayyad et al., Physical Review Letters 129, 012501 (2022).

8. “57Zn 𝛽-delayed proton emission establishes the 56Ni rp-process waiting point bypass."

M. Saxena et al., Physics Letters B 829, 137059 (2022).

9. “Online Bayesian optimization for a recoil mass separator."

S. A. Miskovich et al., Physical Review Accelerators and Beams 25, 044601 (2022).

10. “First direct measurement of 59Cu(p,𝛼)56Ni: A step towards constraining the Ni-Cu cycle in

the cosmos."

J. S. Randhawa et al., Physical Review C Letters 104, L042801 (2021).

A.1.3 Nuclear mass models

1. “Quantified nuclear mass model using Bayesian Model Averaging."

R. Jain. L. Neufcourt, S. Giuliani, W. Nazarewicz (in prep).

• Performed the Bayesian analysis of nuclear mass models.

• Analyzed the results and wrote the first draft of the paper.

2. “Precision mass measurement of lightweight self-conjugate nucleus 80Zr."

A. Hamaker, E. Leistenschneider, R. Jain et al., Nature Physics 17, 1408-1412 (2021).

• Performed Bayesian analysis.

• Contributed to the writing of the methods section.

A.1.4 Others

1. “Horizons: nuclear astrophysics in the 2020s and beyond."

H. Schatz et al., Journal of Physics G: Nuclear and Particle Physics 49, 110502 (2022).

• Organizer of the JINA Horizons Meeting Junior Workshop that lead to this white paper.

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• Contributed to the writing of Section 6 - Diversity in nuclear astrophysics and Section

7 - Career development: perspectives of early career researchers.

A.2 Presentations

A.2.1

Invited Talks

1. “Nuclear Physics of Accreting Neutron Stars.”

Postdoctoral Candidate Seminar, Lawrence Livermore National Laboratory.

November 13, 2023, Virtual Seminar.

2. “Heating and Cooling of Accreting Neutron Star Crusts.”

Cyclotron Colloquium, Texas A&M University.

October 12, 2023, College Station, Texas.

3. “Quantified Nuclear Mass Model for r-process simulations using Bayesian Machine Learning

Techniques”

The 1𝑠𝑡 IReNA-Ukakuren Joint Workshop “Advancing Professional Development in Nuclear

Astrophysics and Beyond".

August 28 - September 1, 2023, Mitaka, Tokyo, Japan.

4. “Weak Interactions for Modeling Accreting Neutron Star Crusts”

IReNA Workshop on Weak Interactions in Nuclear Astrophysics.

July 10-12, 2023, East Lansing, Michigan.

5. “Nuclear Experiments for Astrophysics: An Overview”

JINA-CEE Frontiers in Nuclear Astrophysics - Junior Researchers Workshop.

May 23-27, 2022, South Bend, Indiana.

A.2.2 Contributed Talks

1. “The role of A = 33 mass chain in Urca Cooling of Accreting Neutron Star Crusts.”

American Physical Society Division of Nuclear Physics Meeting.

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November 26 - December 1, 2023, Waikolao Village, Hawaii.

2. “Nuclear Physics of Accreting Neutron Stars.”

17th International Symposium on Nuclei in the Cosmos (NIC XVII).

September 17-22, 2023, Daejeon, South Korea.

3. “Quantified Nuclear Mass Model for Nuclear Astrophysics Simulations.”

CeNAM Frontiers in Nuclear Astrophysics.

May 21-26, 2023, East Lansing, Michigan.

4. “Inferring properties of a mysterious shallow heat source in accreting neutron star crusts.”

18𝑡ℎ Russbach School on Nuclear Astrophysics.

March 12-18, 2023, Russbach, Austria.

5. “Can Machines Learn using Gaussian Distributions?”

Physics Graduate Organization (PGO) Seminar, Dept. of Physics & Astronomy, Michigan

State University.

February 6, 2023, East Lansing, Michigan.

6. “𝛽-decay of 33Mg using Total Absorption Spectroscopy.”

American Physical Society Division of Nuclear Physics Meeting.

October 27-30, 2022, New Orleans, Louisiana.

7. “𝛽-decay in Neutron Star Crusts.”

Nuclear Physics in Astrophysics - X Conference.

September 4-9, 2022, CERN, Switzerland.

8. “𝛽-decay Experiments for Urca Cooling in Neutron Star Crusts.”

JINA-INT Workshop on Neutron Star Cooling.

February 14-18, 2022, INT, University of Washington, Seattle.

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9. “X-rays from the Space.”

AGEP Student Success Conference.

November 5-6, 2021, Michigan State University, East Lansing, Michigan.

10. “Quantified Nuclear Mass Model for Nuclear Astrophysics Simulations.”

American Physical Society Division of Nuclear Physics Meeting.

October 10-14, 2021, Virtual Meeting.

11. “Pycnonuclear Fusion and the Shallow Heat Source in Accreting Neutron Star Crusts.”

American Physical Society Division of Nuclear Physics Meeting.

October 29 - November 1, 2020, Virtual Meeting.

12. “Extrapolating Nuclear Masses using Bayesian Gaussian Process Regression.”

A.I. for Nuclear Physics Workshop.

March 4-6, 2020, Jefferson Lab, Newport News, Virginia.

13. “Sensitivity Studies of Fusion Reactions in the Crusts of Accreting Neutron Stars”

The 10th European Summer School on Experimental Nuclear Astrophysics.

June 16-23, 2019, LNS, Catania, Italy.

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