PROTON-CAPTURE CROSS-SECTION MEASUREMENTS FOR THE ASTROPHYSICAL
GAMMA PROCESS: FROM STABLE TO RADIOACTIVE ION BEAMS

By

Artemis Tsantiri

A DISSERTATION

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

Physics—Doctor of Philosophy

2025

ABSTRACT

One of the key questions in nuclear astrophysics is understanding how elements heavier than iron

are forged in the stars. Heavy element nucleosynthesis is primarily governed by the slow and

rapid neutron capture processes. However, a relatively small group of naturally occurring, neutron-

deficient isotopes, known as p nuclei, cannot be formed by either of those processes. These ∼ 30

nuclei are believed to be synthesized in the 𝛾 process, where preexisting r- and s-process seeds

are “burned" through a sequence of photodisintegration reactions. The astrophysical sites where

such conditions occur have been a subject of controversy for more than 60 years, and is currently

believed that the 𝛾 process can take place in the O/Ne layers of core collapse supernovae, and in

thermonuclear supernovae.

Reproducing solar p-nuclei abundances through nuclear reaction networks requires input on

a large number of mostly radioactive isotopes. However, as experimental cross sections of 𝛾-

process reactions are very limited, and almost entirely unknown for radioactive nuclei, the related

reaction rates are based on Hauser-Feshbach (HF) theoretical calculations and therefore carry large

uncertainties. Therefore, it is crucial to develop techniques to accurately measure these reactions

within the astrophysically relevant Gamow window with radioactive beams. The SuN group at the

Facility for Rare Isotope Beams (FRIB) has been developing such a program for the past decade.

This thesis focuses on implementing a technique to measure reaction cross sections in inverse

kinematics with a radioactive beam. Specifically, this work presents data analysis from the proof-

of-principle stable beam experiment for the 82Kr(p,𝛾)83Rb reaction, along with the measurement

of the 73As(p,𝛾)74Se reaction in our first radioactive beam experiment. The latter reaction is

particularly significant for the final abundance of the lightest p nucleus, 74Se, since the inverse

reaction 74Se(𝛾,p)73As is one of the primary destruction mechanisms of 74Se.

The experiments were conducted at FRIB at Michigan State University using the ReA facility.

The 82Kr and 73As beams were directed onto a hydrogen gas cell located in the center of the Summing

NaI(Tl) (SuN) detector and the obtained spectra were analyzed using the 𝛾-summing technique. In

addition to the total cross section measurements, this thesis also presents the development of an

analysis technique to extract statistical properties of the compound nucleus (nuclear level density

and 𝛾-ray strength function) through a series of simulations. This approach enables the extraction

of an experimentally constrained cross section across the entire Gamow window of the 𝛾 process.

Finally, the experimentally constrained reaction rate for the 73As(𝑝, 𝛾)74Se reaction is used in Monte

Carlo one-zone network simulations of the 𝛾 process to explore its impact on the production of the

74Se.

Copyright by
ARTEMIS TSANTIRI
2025

Στη μάνα μου, που μου έδωσε φτερά να πετάξω.

v

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest gratitude to my thesis supervisor, Artemis

Spyrou, for accepting me as a graduate student and for being an incredible mentor over the past

five years. Artemis generously provided the means for an excellent education with great emphasis

on professional development, encouraging me to attend multiple summer schools, conferences and

workshops. Through her guidance, I learned how to work within a research group and a larger

collaboration, and I am especially grateful for her support in helping me build my own professional

network by introducing me to her collaborators and providing opportunities to establish connections

that will be invaluable throughout my career. Most importantly, I deeply appreciate the trust she

placed in me by allowing me to chose my thesis topic, and her hands-off approach to supervision

that gave me the freedom to explore beyond my thesis, develop my own ideas, and grow both

scientifically and personally. Looking back, I am certain that my decision to move halfway across

the world to work with her has been the best academic decision I have ever made.

I would like to thank Hendrik Schatz and Sean Liddick, for all their help and guidance during

my graduate studies and my thesis project. I also want to express my graditude to the rest of my

supervisory committee, Scott Bogner, Kendall Mahn and Brian O’Shea for their advice, questions,

and offering their valuable time to serve on my guidance committee.

I would like to thank all past and present members of the SuN group for their support and

camaraderie. I am especially grateful to the former and current postdocs — Andrea, Erin, Mallory,

and Sivi — for their guidance and friendship, with special thanks to Erin for her support and help

during the execution of my experiment, and for teaching me to keep moving forward with a smile,

calmness, and lightheartedness when things don’t go as planned. I also want to acknowledge former

graduate students Alicia, Steven, and Alex, whose work laid the foundation for this research. To

my fellow current graduate students — Hannah, Jordan, Caley, Kostas, Amal, and Sydney —thank

you for making this journey both productive and enjoyable, especially to Hannah who has been

my academic sister, and best office and travel mate. I wish all of them the best in their next stages

of their career. I also extend my gratitude to the members of the 𝛽 group and everyone I have

vi

interacted with through the SuN detector. This group has truly felt like a family, and I thank them

for all the great memories and adventures.

I would also like to thank all the staff physicists and faculty at FRIB who contributed to the

execution of my thesis experiment and this project. I am especially grateful to Jorge Pereira and

Remco Zegers for their support with the hydrogen gas handling system, Aaron Chester for teaching

me the ins and outs of the FRIB DAQ and helping me set up the experiment’s software, and Ana

Henriques, Sam Nash, and Chandana Sumithrarachchi for assisting with the setup in ReA3 and

providing such a beautiful—yet tricky—arsenic beam. I am also very grateful to Pablo Giuliani for

his generous help on the mathematical description of the scores for the astrophysical calculations

and for allowing me to bother him so often.

Additionally, I want to thank the members of the NuGrid Collaboration for welcoming me into

their community and enabling the astrophysical calculation portion of my dissertation. In particular,

I am grateful to Pavel Denissenkov and Falk Herwig for hosting me during my internship at the

University of Victoria, introducing me to the NuGrid codes, and engaging in many interesting

discussions during my visit. I also appreciate the support of Thanassis Psaltis and Lorenzo Roberti

for their help in setting up this impact study, sharing their codes, and contributing valuable ideas

and discussions. Finally I am thankful for Umberto Battino and Claudia Travaglio for providing

the trajectories for the Type Ia impact study.

I am also grateful for the support of IReNA throughout my graduate studies, including the

generous funding that allowed me to attend numerous conferences and summer schools, as well as

the Visiting Fellowship that made my research stay at UVic possible.

I am deeply grateful for all my close friends, both near and far. To Pely, Nikos, Yannis, and

Emily — thank you for the memories, adventures, and for making Lansing feel like home. To

my friends from afar—Christos, Christina, Anna, Orestis, and Eva—your unwavering support has

meant the world to me, no matter the distance. You have always been with me wherever I go. A

very special thank you to my partner, best friend, roommate, and colleague, Jordan. Meeting you

has been the best part of this journey, and I can’t wait to see where our next adventures take us.

vii

Finally, I would like to express my deepest gratitude to my family—my parents, Despoina and

Nikos, and my sister, Christina. I cannot overstate my appreciation for their love, support, and

the countless sacrifices they have made. They have always encouraged me to pursue my goals, no

matter how far they took me. Especially to my mother, to whom I dedicate this thesis—I doubt

many people can say their mother followed them to a scientific conference because Austria was the

closest they had been to home in a while.

And, of course, to my cat, who has been a constant source of comfort, chaos, and companionship

through it all.

viii

CHAPTER 1

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

.

1

TABLE OF CONTENTS

CHAPTER 2

NUCLEAR PHYSICS FOR ASTROPHYSICS . . . . . . . . . . . . . .
2.1 Nuclear Masses and Binding Energies . . . . . . . . . . . . . . . . . . . . . .
2.2 Energetics of Nuclear Reactions
. . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Reaction Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Nuclear Reactions .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Nuclear Reactions in Stars

2
3
5
6
7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

.

CHAPTER 3

.
.

ASTROPHYSICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1 Abundances . .
. .
3.2 Stellar Evolution .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Stellar Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Production of the p Nuclei
3.5 Nuclear Networks and Uncertainties . . . . . . . . . . . . . . . . . . . . . .
. 45
3.6 The Lightest p Nucleus, 74Se . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

CHAPTER 4

EXPERIMENTAL SETUP &TECHNIQUES . . . . . . . . . . . . . . . 51
4.1 Beam Delivery in ReA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Hydrogen Gas Target
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
.
4.3 The SuN Detector .
4.4 The SuNSCREEN Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

CHAPTER 5

ANALYSIS .
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
.
5.1 Effective Energy .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Beam Particle Number
5.3 Experimental Yield .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Target Particle Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Detection Efficiency .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Theoretical Investigation with Rainier and Talys . . . . . . . . . . . . . . . . 79
5.7 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CHAPTER 6

RESULTS &DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1 The 82Kr(𝑝, 𝛾)83Rb cross section . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 The 73As(𝑝, 𝛾)74Se cross section . . . . . . . . . . . . . . . . . . . . . . . . . 93

CHAPTER 7

ASTROPHYSICAL IMPACT . . . . . . . . . . . . . . . . . . . . . . . 98
. . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1 Core-Collapse Supernova - SNII
7.2 Type Ia Supernova - SNIa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

CHAPTER 8

SUMMARY & CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . 106

BIBLIOGRAPHY .

. .

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

ix

APPENDIX

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

x

CHAPTER 1

INTRODUCTION

Carl Sagan said, “we are made of star stuff". But how is this “star stuff" made? The carbon in our

cells, the calcium in our bones, and the oxygen in our blood are all forged during the life of a star.

But the silver and gold in our jewelry, the platinum in our cars, and the tungsten in our LED lights

were all made during a star’s death.

The origins of nuclear astrophysics trace back over a century to Eddington’s 1920 manuscript,

which, based on Aston’s demonstration that the mass of helium is less than four times that of the

proton [1], proposed that an unknown process in the Sun’s core converts hydrogen into helium,

releasing energy [2]. Nearly 40 years later, the groundbreaking review by Burbidge, Burbidge,

Fowler, and Hoyle (B2FH) presented a detailed framework of stellar nucleosynthesis, describing

how the elements are synthesized in stars [3]. These discoveries highlighted the need for a field that

combines astronomy, astrophysics, and nuclear physics to explore how the elements in the universe

came to be, the interdisciplinary field of nuclear astrophysics.

The following thesis aims to offer but a small contribution to this ongoing effort, by focusing

on the nucleosynthesis of a particular isotope, namely 74Se. Given the interdisciplinary nature

of nuclear astrophysics, the necessary theoretical foundations are introduced in separate chapters

covering nuclear physics and astrophysics. Then, the experimental study of the destruction mecha-

nism of 74Se is presented, followed by analysis of the experimental data. Finally, the impact of the

measurement is investigated through astrophysical network calculations, aiming to constrain the

final production of 74Se in stars.

1

CHAPTER 2

NUCLEAR PHYSICS FOR ASTROPHYSICS

As per Rutherford’s description, the majority of the atom’s volume is comprised of empty space and

electrons that surround a tiny, dense, positively charged central core called the nucleus. Composed

of protons and neutrons, the nucleus is characterized by the mass number, 𝐴, which represents

their total number. The number of protons is called the atomic number 𝑍, and the neutron number

𝑁 is defined as 𝑁 = 𝐴 − 𝑍. Different elements are distinguished by the different atomic number

𝑍, whereas nuclei of the same element with different neutron number 𝑁 are called isotopes. The
𝑍 𝑋𝑁 , or simply 𝐴 𝑋, where 𝑋 represents the chemical symbol of

notation for different isotopes is 𝐴

the element.

Similarly to chemicals elements in the periodic table, isotopes are represented in the chart of the

nuclides. This two-dimensional chart, shown in Fig. 2.1 displays the number of protons, 𝑍, on the

Figure 2.1 The chart of nuclides. The black squares correspond to stable nuclei, while the gray to
radioactive. The white lines indicate proton and neutron magic numbers. Data from IAEA [4].

2

y-axis, and the number of neutrons 𝑁 on the x-axis. Each square represents a different isotope with

different properties. The black squares correspond to stable isotopes that can be found in nature,

while the gray are radioactive, meaning that after some time they will lose energy by emitting

radiation. The white lines correspond to proton and neutron magic numbers, which according

to the nuclear shell model developed Mayer in 1948 [5], refer to specific numbers of protons or

neutrons within a nucleus that result in significantly increased stability. The magic numbers for

nuclei are 2, 8, 20, 28, 50, 82, and 126.

2.1 Nuclear Masses and Binding Energies

One might initially assume that the nuclear mass, 𝑚nuc, equals 𝑍 𝑚 𝑝 + 𝑁 𝑚𝑛, where 𝑚 𝑝 and 𝑚𝑛

are the masses of protons and neutrons that comprise the nucleus, respectively. However, Aston’s

experiments in the 1910s revealed that the mass of helium (4He) is less than 4 times the mass of

hydrogen (1H) [1]. This mass difference is called the mass excess or mass defect, Δ𝑚, and is a

direct result of the binding energy that holds the nucleus together. The binding energy, 𝐵(𝑍, 𝑁),

represents the energy required to break a nucleus into its constituent 𝑍 protons and 𝑁 neutrons, and

can be expressed as

𝐵(𝑍, 𝑁) = Δ𝑚 · 𝑐2 = (𝑍 𝑚 𝑝 + 𝑁 𝑚𝑛 − 𝑚nuc) 𝑐2

(2.1)

[6], where 𝑐2 often includes a unit conversion factor so that 𝑐2 = 931.50 MeV/u, and thus the

binding energy is expressed in atomic mass units.

A useful nuclear property derived from the binding energy is the particle separation energy,

which corresponds to the energy required to remove a proton or a neutron from a nucleus. Hence

the proton separation energy is equal to the difference between the binding energy of 𝐴

𝑍 𝑋𝑁 and

𝐴−1
𝑍−1 𝑋𝑁 :

𝑆 𝑝 = 𝐵(𝑍, 𝑁) − 𝐵(𝑍 − 1, 𝑁)

= [𝑚( 𝐴

𝑍 𝑋𝑁 ) − 𝑚( 𝐴−1

𝑍−1 𝑋𝑁 ) + 𝑚(1H)] 𝑐2

and similarly the neutron separation energy is:

𝑆𝑛 = 𝐵(𝑍, 𝑁) − 𝐵(𝑍, 𝑁 − 1)

= [𝑚( 𝐴

𝑍 𝑋𝑁 ) − 𝑚( 𝐴−1

𝑍 𝑋𝑁−1) + 𝑚𝑛] 𝑐2

3

(2.2)

(2.3)

The separation and binding energies carry important information about the stability and structure

of the nuclides.

To allow for a systematic study of the nuclear binding energy, it is common to display the

average binding energy per nucleon, 𝐵/𝐴. Figure 2.2 shows 𝐵/𝐴 as a function of mass number 𝐴

[7]. A few notable features shown in Fig. 2.2 include that, aside from the light nuclei, the average

Figure 2.2 The binding energy per nucleon, 𝐵(𝑍, 𝑁)/𝐴, as a function of the mass number, 𝐴.
Figure from [8].

binding energy is around 8 MeV/u. The most bound nuclei, those with the maximum 𝐵/𝐴, are

found in the mass range of 𝐴 = 50 − 65. This is the so-called iron peak, with the most tightly bound

nuclides being 62Ni, 58Fe, and 56Fe [9]. It follows that there are two ways to release energy through

nuclear processes: for nuclei lighter than the iron peak energy is released by fusion, the assembly

of light nuclei into heavier species, while for nuclei heavier than iron, energy is released by fission,

the breaking of heavy nuclei into lighter ones [7]. As will be described later in Ch. 3, these are the

two main mechanisms for energy generation in a stellar environment.

4

2.2 Energetics of Nuclear Reactions

A binary nuclear interaction is written as

𝑎 + 𝑋 → 𝑌 + 𝑏

or

𝑋 (𝑎, 𝑏)𝑌

(2.4)

where 𝑎 and 𝑋 are the two colliding nuclei (entrance channel), and 𝑏 and 𝑌 are the reaction products

(exit channel). Typically, 𝑎 is an accelerated projectile of lighter mass and 𝑋 is a stationary heavy

target, while 𝑏 is the light ejectile that can be directly measured and 𝑌 is the heavy recoil nucleus

that stays in the target and is not observed. The various classifications of nuclear reactions are

discussed in Sec. 2.4.

As with every other interaction in nature, nuclear reactions are governed by fundamental

conservation laws, which provide a basis for deriving various characteristic quantities to describe

the system. For example the conservation of total energy and linear momentum allows us to

deduce the energy of the undetected recoil nucleus from the known energies of the reactants and

the measured energy of the ejectile. Other conserved quantities include the angular momentum,

proton and neutron number (or baryon number), and parity [7].

The conservation of total relativistic energy for a reaction of the form shown in Eq. 2.4 yields

𝑚 𝑋 𝑐2 + 𝐸 𝑋 + 𝑚𝑎 𝑐2 + 𝐸𝑎 = 𝑚𝑌 𝑐2 + 𝐸𝑌 + 𝑚𝑏 𝑐2 + 𝐸𝑏

(2.5)

where 𝐸𝑖 are the kinetic energies in the center-of-mass system and 𝑚𝑖 the rest masses. The energy

available in the system for this reaction is defined as the Q value, and represents the difference in

mass energy of the system before and after the reaction.

𝑄 = (𝑚initial − 𝑚final) 𝑐2

= (𝑚 𝑋 + 𝑚𝑎 − 𝑚𝑌 − 𝑚𝑏) 𝑐2

or in terms of the excess of kinetic energy

𝑄 = 𝐸final − 𝐸initial

= 𝐸𝑌 + 𝐸𝑏 − 𝐸 𝑋 − 𝐸𝑎

5

(2.6)

(2.7)

If the 𝑄 is positive, then the reaction releases energy, and is called exoergic or exothermic.

If

𝑄 is negative, then energy is consumed for the reaction to occur, and it is called endoergic or

endothermic [7].

The particle separation energy introduced in Sec. 2.1 corresponds to the 𝑄 value for particle

emission. The proton separation energy, 𝑆 𝑝, corresponds to the 𝑄 value for proton emission, while

the neutron separation energy, 𝑆𝑛, corresponds to that for neutron emission. This will become

important later in Ch. 3, as these quantities define the limits of the nuclear landscape.

2.3 Reaction Cross Section

One of the most important quantities characterizing a nuclear reaction is the cross section,

𝜎, which can be broadly understood as the probability for an interaction to occur. Consider the

geometry illustrated in Fig. 2.3. An incident beam of 𝐼𝑎 particles per unit time impinges on a

Figure 2.3 Illustration of a typical nuclear physics cross section measurement, showing an incident
beam, target and detector. Figure recreated based on [6, 7].

target of 𝑁𝑡 particles per unit area. A detector is positioned at an angle (𝜃, 𝜙) with respect to the

beam axis, and its surface covers a solid angle 𝑑Ω. If the rate of outgoing ejectile particles is 𝑅𝑏,

6

then the reaction cross section is defined by

𝜎 =

=

(interactions per unit time)
(incident particles per unit time) (target nuclei per unit area)
𝑅𝑏
𝐼𝑎 𝑁𝑡

(2.8)

By this definition, the cross section has dimensions of area, but is proportional to the reaction

probability and is typically measured in barns, where 1 b = 10−24 cm2.

The detector of Fig. 2.3 covers but a small solid angle 𝑑Ω, and therefore could not have possibly

detected all outgoing particles. The ejectiles are emitted in a non uniform manner, and if we assume

an angular distribution 𝑟 (𝜃, 𝜙) for the emitted ejectiles, then the fraction of ejectiles detected would

be 𝑑𝑅𝑏 = 𝑟 (𝜃, 𝜙) 𝑑Ω/4𝜋. The illustrated geometry would allow for the measurement of the

differential cross section, 𝑑𝜎/𝑑Ω = 𝑟 (𝜃, 𝜙)/(4𝜋 𝐼𝑎 𝑁𝑡). The total reaction cross section can be

calculated by integrating 𝑑𝜎/𝑑Ω over all angles, where 𝑑Ω = sin 𝜃 𝑑𝜃 𝑑𝜙.

𝜎 =

∫ 𝑑𝜎
𝑑Ω

𝑑Ω =

∫ 𝜋

0

sin 𝜃 𝑑𝜃

∫ 2𝜋

0

𝑑𝜙

𝑑𝜎
𝑑Ω

(2.9)

In this work, as described later on in Ch. 4, the detector geometry has a solid angle of 4𝜋.

Therefore this work regards to a total cross section measurement, and the analysis follows Eq. 2.8,

with the addition of a detection efficiency term, 𝜖. The efficiency accounts for the fraction of

the outgoing ejectiles that enter the active volume of the detector, but are not recorded, so that

𝑅𝑏 = 𝑌 /𝜖, where 𝑌 is the experimental yield, meaning the total number of particles detected. The

analysis presented in Ch. 5 follows:

𝜎 =

𝑌
𝐼𝑎 𝑁𝑡 𝜖

(2.10)

2.4 Nuclear Reactions

Nuclear interactions of the form 𝑋 (𝑎, 𝑏)𝑌 are categorized based on the nature of the species

involved, as well as the mechanisms governing the process. If the reactants 𝑎 and 𝑋 are identical to

the reaction products, the interaction is referred to as scattering. Scattering is classified as elastic if

the products remain in their ground state and as inelastic if they are in an excited state. Otherwise,

the reactants and products are distinct species and a nuclear reaction occurs. If particle 𝑏 is a 𝛾 ray,

7

the reaction is called radiative capture, whereas if particle 𝑎 is a 𝛾 ray, it is a photodisintegration.

In cases where particles 𝑎 and 𝑏 are identical but an additional ejectile is present (resulting in

three final products), the reaction is referred to as a knockout reaction.

If one or two nucleons

are exchanged between the projectile and target, this is classified as a transfer reaction. Transfer

reactions can be further categorized as pick-up reactions, where the projectile acquires nucleons

from the target, or stripping reactions, where the target removes nucleons from the projectile [7, 10].

Lastly, if the projectile exchanges a proton for a neutron or vice-versa, the process is known as a

𝑐ℎ𝑎𝑟𝑔𝑒 − 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒 reaction.

Another important way to classify nuclear reactions is based on the governing mechanism,

which determines the timescale of the interaction and the extent to which the target nucleus is

affected. Imagine you’re running through a forest. If you run slowly, you have the time to observe

each tree, interact with them, and maybe even touch their leaves. The entire forest feels your

presence as you pass through it. But if you’re running really fast, you barely notice the trees. All

the forest notices is a blur, like a bullet, and you might only interact with a single tree if you hit it

directly. Most of the forest remains unaffected by your passage. Nuclear reactions work in a similar

way.1 At low energies, the incoming particle has a large de Broglie wavelength, comparable to the

size of the whole nucleus. This allows it to interact with the entire nucleus, forming a compound

nucleus. For example, a 1 MeV proton has a de Broglie wavelength of about 4 fm, which is equal

to the average radius of the Fe nucleus. At higher energies, the particle’s de Broglie wavelength

becomes much smaller, and it’s more likely to interact with individual nucleons. A 50 MeV proton,

for instance, with a de Broglie wavelength of about 0.6 fm, is more likely to perform a direct

reaction. In between these two mechanisms are the pre-equilibrium reactions, in which the system

of the reactants breaks up before it reaches statistical equilibrium.

2.4.1 Direct Reactions

Direct reactions, also called peripheral, involve the interaction of one or very few particles from

the target with the projectile. These reactions occur on a timescale of approximately 10−22 seconds

1This metaphor was first introduced to me by my undergraduate supervisor, Mike Kokkoris.

8

and primarily affect the surface of the target, leaving the remaining nucleons largely unaffected

as spectators. An example of such reactions are transfer reactions, which are commonly used to

study the structure of nuclei. They may insert or remove a particle from a specific state within the

nucleus, leaving the rest of the system unperturbed. Such experiments allow to probe particle states

of specific angular momentum, spin and parity, by detecting ejectile particles at different angles,

and most often regard to the measurement of differential cross sections [7, 11].

2.4.2 Compound Nucleus Reactions

The bound nuclear states studied in direct reactions are stable against particle emission. There-

fore, their lifetimes, 𝜏, are very long, and have a narrow width, Γ, corresponding to a small

uncertainty in their energy, based on Heisenberg’s uncertainty principle Γ = Δ𝐸 = ℏ/𝜏 [12]. On

the opposite side is the compound nucleus mechanism, in which the incoming particle 𝑎 and target 𝑋

merge, populating an excited state of a compound nucleus 𝐶∗. The captured particle remains in the

compound system for an extended period, typically on the order of 10−16 to 10−18 seconds. Unlike

direct reactions, this timescale allows the incoming particle to interact randomly with a very large

number of nucleons, sharing its energy across the entire system. The resulting compound nucleus

has undergone so many small interactions that loses any memory of its formation mechanism. As a

result, the entrance and exit channels of the system can be treated independently, an idea described

by Bohr’s independence hypothesis [13, 14].

Compound reactions can populate either the resonance region or the statistical region (also

known as continuum), depending on the excitation energy of the compound nucleus and the number

of available states. The resonance region corresponds to discrete nuclear states, while the statistical

region consists of numerous, closely spaced states that overlap.

2.4.2.1 Resonance Reactions

In resonance reactions, the incoming particle becomes “quasibound" to a nuclear state with

a very high probability of formation, resulting in a very large cross section. These states of the

compound nucleus often have small widths and low excitation energies, and will decay either by

emitting 𝛾 rays or by re-emitting the incident particle, as in scattering.

9

The cross section 𝜎(𝐸) for a resonance with energy 𝐸𝑅 and width Γ is described through the

Breit-Wigner formula as

𝜎(𝐸) = 𝜎0

Γ2/4
(𝐸 − 𝐸𝑅)2 + 1

4 Γ2

(2.11)

where 𝜎0 is the cross section value at the maximum of the resonance peak [10]. An illustration of

a resonance reaction is shown in Fig. 2.4, where the incident particle 𝑎 is captured by the target

𝑋 and populating the 𝐸𝑅 state of the 𝑌 compound nucleus. The right-hand-side of the illustration

shows the cross section for this capture. The resonant state then decays by either emitting 𝛾 rays,

or re-emitting particle 𝑎.

Figure 2.4 Illustration of a resonance reaction. Figure recreated based on [6, 15].

Resonant reactions are particularly important in nuclear astrophysics, as the existence of such

resonances enables reactions that would otherwise hinder nucleosynthesis, due to very low cross

sections, as described in Sec. 2.5.

2.4.2.2 Statistical Model of Compound Nucleus Reactions

As the excitation energy, 𝐸 𝑋, of the compound nucleus populated by the reaction increases,

the number of available nuclear states grows almost exponentially. At higher energies, the states

become so numerous that their spacing is much smaller than their width, leading to significant

overlap and resembling a structureless continuum. Under these conditions, the resonance reaction

mechanism becomes inadequate, and the reaction is instead described by the statistical model of

10

compound reactions, initially developed by N. Bohr in 1936 [13].

The most widely used implementation of Bohr’s independence hypothesis in the statistical

model is the Hauser-Feshbach (HF) theory [16], that results from averaging over a large number of

Breit–Wigner resonances. The central quantities in the HF formalism are the averaged transmission

coefficients 𝑇, that reflect the probability for a particle’s wavefunction to cross an obstacle. In this

context, the transmission coefficients, instead of a resonance behavior, they describe the formation

of the system as an absorption of the incident particle’s wavefunction in the nuclear potential.

The cross section 𝜎 of the reaction 𝑎 + 𝑋 → 𝐶∗ → 𝑌 + 𝑏 (proceeding via compound nucleus

𝐶), is expressed as a summation over all possible spin and parity states of the compound system as

𝜎𝑎𝑋→𝑌 𝑏 (𝐸𝑎𝑋) =

𝜋ℏ2/(2𝜇𝑎𝑋 𝐸𝑎𝑋)
(2𝐽𝑋 + 1)(2𝐽𝑎 + 1)

∑︁

𝐽,𝜋

(2𝐽 + 1)

𝑎 (𝐸, 𝐽, 𝜋)𝑇𝑌
𝑇 𝑋

𝑏 (𝐸, 𝐽, 𝜋)

𝑇tot(𝐸, 𝐽, 𝜋)

𝑊 𝑎𝑋→𝑌 𝑏

(2.12)

where 𝐸𝑎𝑋 the center mass energy, 𝜇𝑎𝑋 the reduced mass, 𝐽 and 𝜋 the spin and parity, and 𝑇 𝑋

𝑎 and

𝑇𝑌
𝑏 the transmission coefficients for the entrance and exit channels respectively. The summation
includes all individual transitions to all 𝐽𝜋 states accessible by particle or photon emission from

the same compound nucleus C, accounting for the quantum mechanical spin/parity selection rules.

The final term 𝑊 𝑎𝑋→𝑌 𝑏 is the width fluctuation correction (WFC) and describes non-statistical

correlations between the widths of the entrance and exit channels and is close to unity [14, 17].

The transmission coefficient for the entrance channel 𝑇 𝑋

𝑎 is typically calculated numerically by

solving the Schrödinger equation with an optical nucleon-nucleus potential, which represents the

average nuclear potential. The development of optical model potentials that accurately describe

the complexity of the potential caused by the strong nuclear force has been an active field of study

for decades. As a detailed discussion of these potentials exceeds the scope of this thesis, further

information can be found in Refs. [18, 19, 20].

The transmission coefficient for the exit channel can be described by assuming all possible

bound and unbound states 𝜈 in all energetically accessible exit channels.

𝑇𝑌
𝑏 (𝐸, 𝐽, 𝜋) =

𝜈𝑚∑︁

𝜈=0

𝑇 𝜈
𝑏 (𝐸, 𝐽, 𝜋) +

∫ 𝐸𝑖

∑︁

𝐸 𝜈𝑚

𝐽,𝜋

𝑇𝑏 (𝐸, 𝐽, 𝜋, 𝐸𝑖, 𝐽𝑖, 𝜋𝑖) × 𝜌(𝐸𝑖, 𝐽𝑖, 𝜋𝑖) 𝑑𝐸𝑖

(2.13)

11

The first term on the right-hand side represents a summation over all experimentally known discreet

states, 𝜈𝑚. The second term is an integration that accounts for the transmission coefficient of all

excited states above the highest experimentally known state, 𝜈𝑚, weighted by the nuclear level

density 𝜌(𝐸𝑖, 𝐽𝑖, 𝜋𝑖). This density corresponds to the number of available spin-parity states within

an energy region 𝑑𝐸𝑖. If the ejectile 𝑏 is a particle, 𝑇𝑏 is calculated in a similar manner as 𝑇 𝑋
𝑎 ,

which requires knowledge of the optical potential that the ejectile must overcome in order to escape

the compound system. In the case of radiative capture, however, particle 𝑏 is a 𝛾 ray, and the exit

channel will be described by the 𝛾-ray transmission coefficient 𝑇𝛾, which is directly proportional to

the 𝛾-ray strength function, and represents the escape probability of a 𝛾 ray that is stuck inside the

volume of a nucleus. The nuclear level density and 𝛾-ray strength function will be further discussed

in the following paragraphs.

2.4.2.3 Nuclear Level Density

The nuclear level density (NLD) corresponds to the available quantum levels Δ𝑁 at a specific

excitation energy 𝐸𝑥, spin 𝐽, and parity 𝜋, and is defined as:

𝜌(𝐸𝑥, 𝐽, 𝜋) =

Δ𝑁 (𝐸𝑥, 𝐽, 𝜋)
Δ𝐸𝑥

(2.14)

where Δ𝐸𝑥 is the energy interval considered, typically 1 MeV. Summing over all possible spin and

parity values gives the total level density 𝜌(𝐸𝑥) [21].

For the statistical-model formalism to apply, the total level density needs to be sufficiently

high. While this criterion is somewhat relative, an accuracy of 20% in the description of the level

densities with numerical calculations is achievable when 𝜌 ≳ 10 MeV−1 (non-overlapping, narrow

resonances) [22]. For nuclei with mass 𝐴 > 60, excitation energies above approximately 4 MeV

have sufficiently high level density for the statistical model to be applicable.

The first theoretical description of level density was proposed by Bethe in 1936 [23], treating

the nucleus as a gas of non-interacting fermions (protons and neutrons). While simplistic, this

approach captured all the essential information, apart from the influence of the pairing between

the nucleons, that was realized and described almost twenty years later by Bardeen, Cooper and

Schrieffer [24]. This pairing was then introduced in the description of the level density as a simple

12

constant energy shift, that was later on found to be too large of a correction. This lead to the

back-shifted Fermi Gas formula (BSFG) proposed by Gilbert and Cameron in 1965 [25]:

𝜌(𝑈) =

√

√

𝜋
12

1

2𝜋𝜎2

√

exp(2

𝑎𝑈)

𝑎1/4𝑈5/4

(2.15)

where 𝑈 = 𝐸𝑥 − Δ is the shifted excitation energy. The energy shift Δ is an empirical parameter

closely associated with the pairing energy, accounting for odd-even effects in nuclei. The concept

behind Δ is that nucleon pairs must first be separated before their individual components can be

excited. In practice, Δ serves as an adjustable parameter to reproduce observables.

The 𝑎 term in Eq. 2.15, referred to as the level density parameter, is, in its most simplistic form,

given by 𝑎 = 𝜋

6 (𝑔𝑝 + 𝑔𝑛), where 𝑔𝑝 and 𝑔𝑛 is the spacing of the proton or neutron single-particle
states near the Fermi energy. Recognizing that 𝑎 should include energy-dependent shell effects,

more sophisticated expressions for 𝑎 have been developped [26, 27, 28].

The spin cut-off parameter, 𝜎2, of Eq. 2.15 represents the width of the angular momentum

distribution of the level density. The description of 𝜎2 is based on the observation that the nucleus

possesses collective rotational energy, and the spin cut-off parameter is related to the moment

of inertia of the undeformed nucleus 𝐼0, and the thermodynamic temperature 𝑡 = √︁𝑈/𝑎, so that

𝜎2 = 𝐼0 𝑡. Similarly to the parameter 𝑎, energy-dependent shell effects are often included in more

advanced models for 𝜎2 [22, 29, 30].

An alternative analytical description of the level density is the Constant Temperature (CT)

model, introduced by Ericson in 1959, who described it as incorporating “a temperature 𝜏 which

is somewhat different from the ordinary nuclear temperature 𝑇, defined by the level density", [31].

This temperature 𝜏 is related to the nuclear temperature 𝑇 by:

𝑑
𝑑𝐸

log 𝜌(𝐸) =

1
𝑇

1
𝜏

=

(cid:18)

1 −

(cid:19)

𝑑𝜏
𝑑𝐸

and the level density is then described as

𝜌(𝐸 𝑋) =

exp[(𝐸 𝑋 − 𝐸0)]/𝜏
𝜏

(2.16)

(2.17)

where in practice, 𝐸0 and 𝜏 are parameters used to adjust the formula to experimental discreet

levels. Since the BSFG model diverges as 𝑈 → 0, a common practice is to use the CT model at low

13

energies and the BSFG model at higher excitation energies, with parameters to ensure a smooth

transition between the two models.

Additional approaches include the phenomenological Generalized Superfluid Model (GSM) [32,

33], which incorporates nucleon pairing correlations according to the Bardeen-Cooper-Schrieffer

theory [24], along with various microscopic models grounded in first principles and fundamental

interactions. These microscopic descriptions of the NLD can capture intricate details of nuclear

structure that are beyond the capabilities of analytical expressions. One microscopic approach is

the shell model Monte-Carlo by Alhassid [34], as well as the approach based on mean-field theory

by Demetriou and Goriely [35]. Additional examples of microscopic models that will be included

in the analysis in the next chapters include the calculated NLD by Goriely from Hartree-Fock

calculations [36], parity-dependent NLD based on the microscopic combinatorial model by Hilaire

[37], as well as temperature-dependent Hartree-Fock-Bogoliubov calculations using the Gogny

force [38].

2.4.2.4 Radiative Decay, Transmission Coefficients and 𝛾 Strength Function

Gamma rays emitted from an excited nucleus must follow selection rules to conserve the angular

momentum and parity. They are classified with an electric (𝐸) or magnetic (𝑀) character, along

with a multipolarity, based on the angular momentum 𝐿 they carry, and the parity change Δ𝜋

between the initial 𝑖 and final state 𝑓 :

|𝐼𝑖 − 𝐼 𝑓 | ≤ 𝐿 ≤ 𝐼𝑖 + 𝐼 𝑓

(𝐿 ≠ 0)

Δ𝜋 = no :

even electric, odd magnetic

(2.18)

Δ𝜋 = yes :

odd electric, even magnetic

For instance, a transition from an initial state of 𝐽 𝜋

𝑓 = 0+ involves
angular momentum 𝐿 = 2 without a change of parity, making it an 𝐸2 transition. When many

𝑖 = 2+ to a final state of 𝐽 𝜋

multipolarities are possible, the lower multipoles are significantly more likely to occur. For example

the transition from 𝐽 𝜋

𝑓 = 5/2+ permits 𝑀1, 𝐸2, 𝑀3 and 𝐸4 transitions. Among these,
𝑀1 transition is typically a thousand times more probable than 𝐸2, 𝐸2 a thousand times more

𝑖 = 3/2+ to 𝐽 𝜋

likely than 𝑀3, and so forth [7].

14

However, even if a photon can be emitted according to the selection rules, its probability of

escaping the volume of the nucleus is much smaller than the probability of being reflected back. This

escape probability is described by the 𝛾-ray transmission coefficient 𝑇𝛾, a quantity that characterizes

the average electromagnetic properties of excited states, and can be described through the 𝛾-ray

strength function 𝑓 (𝐸𝛾) (also called photon strength function or radiative strength function) as:

𝑇𝑋 𝐿 (𝐸𝛾) = 2𝜋 𝐸 (2𝐿+1)

𝛾

𝑓𝑋 𝐿 (𝐸𝛾)

(2.19)

where 𝑋 denotes the character (𝐸 or 𝑀), 𝐿 the multipolarity, and 𝐸𝛾 the energy of the 𝛾 ray.

Photon strength functions are important for the description of all transitions involving 𝛾 rays, but

their significance is even more apparent in (𝑛, 𝛾) and (𝛾, 𝑛) reactions, as neutrons are not affected

by the Coulomb force of the nucleus, and photon strength functions directly govern the reaction

cross section. They are distinguished by the upward 𝛾-strength function

−−→
𝑓XL, associated with the

average photo-absorption, and the downward strength function

←−−
𝑓XL, related to the 𝛾 decay. The

treatment of photon strength functions involves two key assumptions. First, the strength function

is assumed to be independent of 𝐽 and 𝜋 [39], an approximation valid when the initial and final

state have high excitation energies, and therefore overlap with many states of the same energy and

different 𝐽𝜋 values. Second, the upward and downward strength functions are assumed to be equal,

implying that the photo-absorption cross section on an excited state will have the same shape as the

photo-absorption on the ground state. This assumption is known as the Brink hypothesis [40].

In calculations of the 𝛾-ray transmission coefficient for astrophysics, at least the most dominant

𝐸1 and 𝑀1 transitions have to be considered. Similar to the level density, there is a plethora of

models, both analytical and microscopic, to describe the dipole (𝐸1 and 𝑀1) strength functions.

The 𝐸1 transitions are calculated on the basis of the Lorentzian representation of the giant dipole

resonance (GDR), that has been observed throughout the periodic table to strongly influence the

strength function. Macroscopically, this strong resonance is described as a vibration of the charged

(proton) matter in the nucleus against the neutral matter (neutrons).

The magnetic dipole (𝑀1) strength function is also commonly described by Lorentzian reso-

nance-like structures that are much smaller in magnitude compared to the GDR. Depending how

15

deformed the many-body system is, collective excitations can appear as enhancements in the 𝑀1

strength function, such as the scissors mode around 3 MeV, or the spin-flip strength around 5-9

MeV [41, 42].

Examples of phenomenological models to describe the photon strength function that are widely

used in astrophysics are the Standard Lorentzian function by Brink [40] and Axel [43], and the

Generalized Lorentzian model of Kopecky and Chrien [44] and Kopecky and Uhl [45]. Mi-

croscopic models to describe 𝐸1 and 𝑀1 radiation include, but are not limited to, large-scale

calculations based on the quasi-particle random-phase approximation (QRPA) model combined

with the Hartree–Fock–Bogoliubov (HFB) method [46, 47, 48, 49], and the relativistic mean-field

approach (RMF) [50, 51, 52, 53].

Additionally to the Lorentzian resonance-like structures that comprise the form of the strength

function, an enhancement at low transition energies and excitation energies in the statistical region

has been experimentally observed [54, 55]. This feature is called the low energy enhancement or

upbend, and even though it is not clear whether it correspond to the electric or magnetic radiation

[56, 57], it is believed to be of dipole character [58]. The upbend is parameterized in the form of

an exponential tail as

𝑓upbend(𝐸𝛾) = 𝐶 exp(−𝜂 𝐸𝛾)

(2.20)

where 𝐶 and 𝜂 are adjustable parameters [55]. The existence of the upbend has shown to have

significant impact on capture reaction cross sections [59, 60], and its intensity appears to be

dependent on the nuclear structure [61].

2.4.2.5 Statistical Model Calculations with Talys

A software package for simulations and predictions of nuclear reactions that will be extensively

used in the analysis of the following chapters is Talys [62]. A variety of nuclear reactions can

be simulated using Talys including direct reactions, compound nucleus model, pre-equilibrium

reactions and fission. In the context of this thesis, Talys will be used for calculating (𝑝, 𝛾) reaction

cross sections based on the statistical model for compound nucleus reactions. As discussed in the

previous sections, main ingredients of the HF formalism include the optical model potential (OMP),

16

the nuclear level density (NLD) and 𝛾-ray strength function (𝛾SF). In this paragraph the models

used for the description of these quantities will be listed along with their respective references.

Regarding the proton-OMP, the default option used in Talys is the phenomenological param-

eterisation of Koning and Delaroche [63]. In addition to the default p-OMP option, a so-called

“jlm-type" potential (based on the work of Jeukenne, Lejeunne, and Mahaux [64, 65, 66, 67] with

later modifications by Bauge et al. [68, 69]) is utilized.

The various models of the NLD anf 𝛾 SF used in Talys were discussed in Sec. 2.4.2.3 and

2.4.2.4 and are listed in Tables 2.1 and 2.2.

Table 2.1 The available models for the nuclear level density in Talys [62].

Talys Keyword Model

ldmodel 1
ldmodel 2
ldmodel 3
ldmodel 4

ldmodel 5

ldmodel 6

Constant Temperature & Fermi Gas Model
Back-shifted Fermi Gas Model
Generalized Superfluid Model
Skyrme-Hartree-Fock-Bogolyubov level densities from nu-
merical tables
Skyrme-Hartree-Fock-Bogolyubov
densities from numerical tables
Temperature-dependent Gogny-Hartree-Fock-Bogolyubov
combinatorial level densities from numerical tables

combinatorial

level

Ref.

[31]
[25]
[32, 33]
[36]

[37]

[38]

Finally, the width fluctuation correction (WFC) from Eq. 2.12 takes into account that there

are correlations between the incident and outgoing wave functions. By default, Talys applies a

WFC using the formalism of Moldauer (so-called “widthmode 1") [70, 71]. A much stronger

WFC is obtained for the approach of Hofmann, Richert, Tepel, and Weidenmüller (HRTW ap-

proach, “widthmode 2") [72, 73, 74], leading to significantly lower calculated (𝑝, 𝛾) cross sections,

especially at low energies.

2.5 Nuclear Reactions in Stars

In the previous sections, we introduced the concept of nuclei and nuclear reactions, describing

the probabilities and mechanisms through which these reactions occur. As nuclear reactions can

transform nuclei while releasing energy, they play a crucial role in understanding both the production

of energy and the nucleosynthesis of elements in stars. While the various stellar environments and

17

Table 2.2 The available models for the 𝛾-ray strength function in Talys [62].

Talys Keyword Transition Model

strength 1
strength 2
strength 3
strength 4

strength 5

strength 6

strength 7

strength 8

strength 9
strengthM1 1

strengthM1 2
strengthM1 3
upbend y/n

𝐸1
𝐸1
𝐸1
𝐸1

𝐸1

𝐸1

𝐸1

𝐸1

𝐸1
𝑀1

𝑀1
𝑀1
𝑀1

Skyrme-Hartree-

Kopecky-Uhl Generalized Lorentzian
Brink-Axel Standard Lorentzian
Skyrme-Hartree-Fock BCS model with QRPA
Skyrme-Hartree-Fock-Bogoliubov model with
QRPA
Hybrid model (Lorentzian model with energy
and temperature dependent width)
Temperature-dependent
Fock-Bogoliubov model with QRPA
Temperature-dependent Relativistic Mean
Field Model
Gogny-Hartree-Fock-Bogoliubov model with
QRPA by based on the D1M version of the
Gogny force
Simplified Modified Lorentzian Model
Standard Lorentzian Model as parameterized
in RIPL3 Library
𝑀1 normalized on 𝐸1 as 𝑓𝐸1/(0.0588𝐴0.878)
Addition of spin-flip and scissors mode
Flag to include upbend or not

Ref.

[45]
[75, 76]
[46]
[48]

[77]

[48]

[53]

[49]

[78]
[79]

[41, 80, 42]
[55, 54, 59, 61]

astrophysical processes will be explored in detail in Ch. 3, it is helpful to introduce the key concepts

needed to bridge the theory of nuclear physics discussed earlier with stellar nucleosynthesis.

The energy dependence of the cross section 𝜎(𝐸), can be interpreted as velocity dependence

𝜎(𝑣), where 𝑣 represents the relative velocity between the projectile and target nucleus. Instead

of projectile beam and stationary target we can consider nuclear species being part of a stellar gas,

where the kinetic energy available for the reaction comes from thermal movement. The reactions

initiated by such motion are called thermonuclear reactions [6].

In a stellar gas that consists of 𝑁𝑎 nuclei per cubic centimeter of species 𝑎, and 𝑁𝑋 nuclei per

cubic centimeter of species 𝑋 the reaction rate 𝑟 between species 𝑎 and 𝑋 is given by:

𝑟 = 𝑁𝑋 𝑁𝑎 𝑣 𝜎(𝑣)

(2.21)

where 𝑟 is in reactions per cubic centimeter per second. The velocities of gas particles vary over

a wide range of values, described by a probability distribution 𝜙(𝑣) that is normalized to unity,

18

𝜙(𝑣)𝑑𝑣 = 1. Averaging the product 𝑣𝜎(𝑣) over this distribution gives the reaction rate per

∫ ∞
0
particle pair:

⟨𝜎𝑣⟩ =

∫ ∞

0

𝜙(𝑣)𝑣𝜎(𝑣)𝑑𝑢

The total reaction rate 𝑟 then becomes:

𝑟 = 𝑁𝑋 𝑁𝑎 ⟨𝜎𝑣⟩

(2.22)

(2.23)

Stellar matter is normally non-degenerate, and nuclei move non-relativistically. Therefore, in

most cases, the velocities of nuclei can be described by the Maxwell-Boltzmann velocity distribu-

tion:

or in terms of energy

𝜙(𝑣) = 4𝜋𝑣2 (cid:16) 𝑚
2𝜋𝑘𝑇

(cid:17) 3/2

exp

(cid:19)

(cid:18)

−

𝑚𝑣2
2𝑘𝑇

𝜙(𝐸) ∝ 𝐸 exp (−𝐸/𝑘𝑇)

(2.24)

(2.25)

where 𝑇 refers to the temperature of the gas, 𝑚 the mass of the nucleus of interest, and 𝑘 the

Boltzmann constant. As shown in Fig. 2.5 at low energies the function increases almost linearly

Figure 2.5 The Maxwell-Boltzmann energy distribution of a gas at temperature 𝑇.

with 𝑇 until it reaches its maximum value at 𝐸 = 𝑘𝑇. At higher energies, the function decreases

exponentially [15].

19

In a stellar gas, the velocities of both species 𝑎 and 𝑋 follow the Maxwell-Boltzmann distribution.

By combining Eq. 2.22 and 2.24 we obtain:

⟨𝜎𝑣⟩ =

(cid:19) 1/2

(cid:18) 8
𝜋𝜇

1
(𝑘𝑇)3/2

∫ ∞

0

𝜎(𝐸)𝐸 exp

(cid:19)

(cid:18)

−

𝐸
𝑘𝑇

𝑑𝐸

(2.26)

where 𝐸 is the center-of-mass energy and 𝜇 the reduced mass [15].

2.5.1 Reactions at elevated temperatures

In stellar plasma at elevated temperatures, nuclei are thermally excited, so a significant fraction

of reacting nuclei will not be in their ground state. The fraction of nuclei in an excited state 𝜇 is

given by the Boltzmann distribution:

𝑁𝜇
𝑁

=

(2𝐽𝜇 + 1)𝑒−𝐸 𝜇/𝑘𝑇
(cid:205)𝜇 (2𝐽𝜇 + 1)𝑒−𝐸 𝜇/𝑘𝑇 =

(2𝐽𝜇 + 1)𝑒−𝐸 𝜇/𝑘𝑇
𝐺

(2.27)

where 𝐺 is the partition function that reflects the Boltzmann factor, and 𝐽𝜇 and 𝐸𝜇 the spin and

excitation energy of the 𝜇 state, respectively.

The ratio of the reaction rate involving thermally excited nuclei, ⟨𝜎𝑣⟩∗, to the reaction rate

involving only the ground state ⟨𝜎𝑣⟩, is known as the stellar enhancement factor (SEF):

SEF ≡

⟨𝜎𝑣⟩∗
⟨𝜎𝑣⟩

=

(cid:205)𝜇 (2𝐽𝜇 + 1)𝑒−𝐸 𝜇/𝑘𝑇 (cid:205)𝜈 ⟨𝜎𝑣⟩ 𝜇→𝜈
𝐺 (cid:205)𝜈 ⟨𝜎𝑣⟩g.s.→𝜈

(2.28)

Here, the summations over 𝜇 and 𝜈 include all the possible excited states of the target nucleus and

all available states of the final nucleus, respectively [15].

2.5.2

Inverse Reactions

At low stellar temperatures, a nuclear reaction requires a positive 𝑄 value to proceed. However,

as the temperature increases, the number of particles with energy exceeding the 𝑄 value also

increases, allowing the inverse process to become energetically possible. Thus, when calculating

the total reaction rate 𝑟, contributions from both reactions 𝑎 + 𝑋 ↔ 𝑏 + 𝑌 should be considered:

𝑟 = 𝑟𝑎𝑋 − 𝑟𝑏𝑌 =

𝑁𝑋 𝑁𝑎
1 + 𝛿𝑎𝑋

⟨𝜎𝑣⟩𝑎𝑋 −

𝑁𝑌 𝑁𝑎𝑏
1 + 𝛿𝑏𝑌

⟨𝜎𝑣⟩𝑏𝑌

(2.29)

where 𝛿𝑖 𝑗 is the Kronecker delta, and the term (1 + 𝛿𝑖 𝑗 ) accounts for identical particles.

20

If the reaction rate ⟨𝜎𝑣⟩𝑎𝑋 is known, the reaction rate for the inverse reaction ⟨𝜎𝑣⟩𝑏𝑌 can

be calculated using the reciprocity theorem, which relies on the invariance of the strong and

electromagnetic interactions under time-reversal symmetry, meaning they are independent of the

direction of time. As long as the cross sections depends on these two interactions, the ratio of the

two cross sections can be written as:

𝜎𝑎𝑋
𝜎𝑏𝑌

=

𝑚𝑏𝑚𝑌 𝐸𝑏𝑌 (2𝐽𝑏 + 1) (2𝐽𝑌 + 1) (1 + 𝛿𝑎𝑋)
𝑚𝑎𝑚 𝑋 𝐸𝑎𝑋 (2𝐽𝑎 + 1) (2𝐽𝑋 + 1) (1 + 𝛿𝑏𝑌 )

Then, by using Eq. 2.26 and the relation 𝐸𝑏𝑌 = 𝐸𝑎𝑋 + 𝑄 (𝑄 > 0), we obtain:

⟨𝜎𝑏𝑌 ⟩
⟨𝜎𝑎𝑋⟩

=

(2𝐽𝑎 + 1)(2𝐽𝑋 + 1) (1 + 𝛿𝑏𝑌 )
(2𝐽𝑏 + 1)(2𝐽𝑌 + 1) (1 + 𝛿𝑎𝑋)

(cid:18) 𝜇𝑎𝑋
𝜇𝑏𝑌

(cid:19) 3/2

exp

(cid:19)

(cid:18)

−

𝑄
𝑘𝑇

which, replacing back in Eq. 2.29 leads to:

(cid:34)

𝑟 =

⟨𝜎𝑎𝑋⟩
1 + 𝛿𝑎𝑋

𝑁𝑎 𝑁𝑋 − 𝑁𝑏 𝑁𝑌

(2𝐽𝑎 + 1) (2𝐽𝑋 + 1)
(2𝐽𝑏 + 1) (2𝐽𝑌 + 1)

(cid:18) 𝜇𝑎𝑋
𝜇𝑏𝑌

(cid:19) 3/2

exp

(cid:19)(cid:35)

(cid:18)

−

𝑄
𝑘𝑇

(2.30)

(2.31)

(2.32)

It is important to note that Eq. 2.32 refers to ground state contributions. For an accurate astrophysical

calculation the stellar enhancement factor described in Sec. 2.5.1 needs to be accounted for [15].

2.5.3 Neutron-Induced Reactions

Neutrons play an important role in stellar nucleosynthesis, however due to their short lifetime

of about 10 minutes, they only exist in stellar environments in which they can be produced.

Some important neutron producing reactions in stars are the 13C(𝛼,n)16O, 18O(𝛼,n)21Ne, and

22Ne(𝛼,n)25Mg. Neutrons produced in stars are very quickly thermalized due to elastic scattering,

so their velocities are described by the Maxwell-Boltzmann distribution. Being electrically neutral,

neutrons do not experience the Coulomb barrier of nuclei. Additionally, for angular momentum

𝑙 = 0 (s-wave), they do not encounter a centrifugal barrier, meaning their penetrability depends

solely on their velocity. As a result, neutron capture is mainly proceeding with s-wave neutrons,
and the cross section is approximated by the 1/𝑣 law:

𝜎𝑛 (𝐸𝑛) ∝

1
𝑣𝑛

(2.33)

It follows that, in the absence of resonances, the reaction rate per particle pair ⟨𝜎𝑣⟩ is approximately

constant [15].

21

Figure 2.6 Cross section for thermal s-wave neutrons follows the 1/𝑣 law.

2.5.4 Charged-Particle-Induced Reactions

As will be discussed in more detail in Chapter 3, stars consist mainly of hydrogen and helium.

Therefore proton and 𝛼 (4𝐻𝑒) capture reactions are some of the most common and important

reactions that can happen in a star. Unlike the case of neutrons, charged particles need to overcome

the repulsive Coulomb barrier of nuclei to be captured. The Coulomb barrier is of the form:

𝑉𝐶 =

𝑍1𝑍2𝑒2
𝑟

(2.34)

where 𝑍𝑖 the atomic number, 𝑒 the electron charge and 𝑟 the distance between nuclei. It is apparent

that the closer to the nucleus, the larger the Coulomb barrier encountered. Combined with the

attractive potential caused by the strong nuclear force leads to the effective potential shown in

Fig. 2.7.

Classically, a charged particle reaction can occur only when the projectile energy is sufficient to

overcome the Coulomb barrier. However, as will be discussed in Ch. 3, the temperatures reached

in stellar environments during the majority of a star’s lifetime are not sufficient to thermally excite

nuclei to such energies. Fortunately, quantum mechanics provides a solution: charged particles

can penetrate the Coulomb barrier through quantum tunneling, allowing stellar nucleosynthesis to

begin at lower temperatures, an idea first proposed in 1929 [81].

22

Figure 2.7 Schematic of the combined Coulomb and nuclear potential. The incident projectile
needs to penetrate the Coulomb barrier to be captured in the nucleus. Classically, the nearest
allowed distance it would reach is the turning point. Figure recreated based on [15].

The probability for tunneling through the Coulomb potential can be found by solving the

Schrödinger equation as

𝑃 = 𝑒−2𝜋𝜂

(2.35)

where 𝜂 = 𝑍1𝑍2𝑒2

ℏ𝑣

2𝜋𝜂 = 31.29𝑍1𝑍2

is the Sommerfeld parameter. The exponent can be numerically calculated as
√︁𝜇/𝐸. The cross section, being proportional to exp(−2𝜋𝜂), decreases sharply

below the Coulomb barrier. To make experiments feasible, cross section measurements are typically

performed at higher energies and then extrapolated to the lower energies relevant to astrophysical

temperatures. However, the rapid decline of the cross section below the Coulomb barrier signifi-

cantly limits the accuracy of this extrapolation. To address this issue, it is more useful to express

the cross section in terms of the astrophysical S-factor, 𝑆(𝐸), defined as:

𝜎(𝐸) =

1
𝐸

exp(−2𝜋𝜂)𝑆(𝐸)

(2.36)

23

The S-factor has much smoother variations in the non-resonant region, and contains only the nuclear

information, eliminating the rapidly decreasing energy-dependent factors.

Calculating the charged-particle reaction rate involves combining the probability of a particle

being in a specific energy, meaning the Maxwell-Boltzmann rate from Eq. 2.26, with the probability

of tunneling through the Coulomb barrier. These probabilities overlap within a narrow energy range,

𝐸0 ± Δ𝐸0/2, where the S-factor, 𝑆(𝐸0), can be considered approximately constant. Therefore,

substituting Eq. 2.36 into Eq. 2.26 yields:

⟨𝜎𝑣⟩ =

(cid:19) 1/2

(cid:18) 8
𝜋𝜇

1
(𝑘𝑇)3/2

𝑆(𝐸0)

∫ ∞

0

exp

(cid:18)

−

𝐸
𝑘𝑇

−

(cid:19)

𝑏
√

𝐸

𝑑𝐸

(2.37)

where 𝑏 = (2𝜇)1/2𝜋𝑒2𝑍1𝑍2/ℏ = 0.989𝑍1𝑍2𝜇1/2 MeV1/2. The quantity 𝑏2 is called the Gamow

energy. The peak shown in Fig. 2.8 is formed by the overlap of the two probability distributions

Figure 2.8 The convolution of the Maxwell-Boltzmann energy distribution and the quantum me-
chanical tunneling function through the Coulomb barrier produce the Gamow peak at energy
𝐸0 ± Δ𝐸0/2. Figure from [82].

at energy 𝐸0, and is called the Gamow peak. It represents the energy range in which the reaction

can happen in a star [15]. The width of the Gamow peak can be approximated by

Δ =

4
√

3

√︁𝐸0𝑘𝑇 = 0.2368 (cid:16)

0 𝑍 2
𝑍 2

1 𝜇𝑇 5
9

(cid:17) 1/6

(MeV)

(2.38)

24

where 𝜇 is the reduced mass, 𝑍𝑖 the atomic number and 𝑇9 the stellar temperature in GK [6].

It should be noted that the resonances introduced in Sec. 2.4.2.1 play a crucial role in charged-

particle reactions within stellar environments. When a resonance lies within the Gamow peak, it

can contribute significantly to the total reaction rate. As will be discussed in the next chapter, the

existence of such narrow resonances within the Gamow window enables the nucleosynthesis of

elements vital to life, such as carbon.

25

CHAPTER 3

ASTROPHYSICS

In the previous chapter, the fundamental concepts of nuclei, nuclear reactions, and the mechanisms

by which these reactions occur in stellar environments were introduced. This chapter begins with

a discussion of the observed element abundances, as any description of nucleosynthesis processes

must ultimately account for these observations. The evolution of stars is then discussed to provide

the context for the environments where nucleosynthesis takes place. The various nucleosynthesis

processes are then presented, leading to the main topic of this thesis, the astrophysical 𝛾 process.

3.1 Abundances

Understanding the origin of the elements in the universe involves explaining and reproducing

their abundances, meaning the relative amounts of nuclides. Astronomical observations provide

spectra that identify elements found in the interstellar medium and on the surface of stars. Pre-solar

grains that were ejected in stellar winds, can become embedded in meteorites in our solar system.

Many of those grains are carbonaceous materials such as diamond, silicon carbide (SiC), and

graphite, and can offer isotopic ratios of their original environment. By analyzing the patterns of

the elemental abundances, nuclear astrophysics can infer the mechanisms responsible for producing

each element.

Among the elemental abundances across all stars, the solar abundance pattern is the most

extensively studied. The solar system formed from a uniform gaseous nebula that contained

contributions from many generations of stars and explosive events. Today, 280 naturally occurring

isotopes remain in 83 elements [14]. The solar system’s elemental abundances are shown in

Fig. 3.1 normalized to silicon atoms. The most abundant elements, hydrogen and helium, make

up approximately 98% of the solar mass. About 1.5% of the mass is carbon and oxygen, while

all other elements account for the remaining 0.5%. A significant drop is observed near the very

weakly bound elements lithium, beryllium and boron (𝐴 = 5 − 8), known as the mass gaps, and a

peak forms in the region around iron. As discussed in Sec 2.1, this corresponds to the most tightly

bound nuclides in the iron peak. The zig-zag structure reflects the differences in binding energies

26

Figure 3.1 Solar system abundances based on data from Asplund [83]. The data is normalized to
106 Si atoms.

between nuclei with odd and even number of nucleons, due to pairing effects.

3.2 Stellar Evolution

The life of a star begins as a collapsing molecular cloud primarily composed of hydrogen (H)

and helium (He). As the cloud contracts, its temperature rises due to an increase in gravitational

potential energy. This rise in temperature causes the pressure at the cloud’s center to increase,

which counteracts further contraction. Smaller clouds may remain in this balanced state and never

contract further, but larger ones continue collapsing until thermonuclear reactions ignite in the core,

and hydrostatic equilibrium maintains stability.

The different burning phases will be discussed in Sec. 3.3, but they generally follow a pattern.

A primary nuclear “fuel" undergoes fusion in the core, releasing energy that temporarily halts

gravitational contraction. As the fuel depletes, the burning region shifts outward, and the star

transitions from core burning to shell burning. With insufficient energy production in the core to

counteract gravity, contraction resumes, increasing the temperature until the next fuel ignites. The

cycle repeats as long as fusion is possible [14].

The lifetime of a star varies vastly, from several billion years for smaller stars to just a few

million years or less for the most massive ones. Small stars with masses below 0.08 𝑀⊙ (where

1 𝑀⊙ is the mass of the Sun) cannot ignite hydrogen and remain as brown dwarfs, supported by

molecular gas pressure. Stars between 0.6–2.3 𝑀⊙, like the Sun, spend billions of years burning

27

hydrogen. Once the hydrogen in the core is depleted, these stars expand into red giants, burning

hydrogen in a shell. Helium ignition leads to a core He-flash and He-shell burning during the

asymptotic giant branch (AGB) phase. Unable to ignite carbon, they eventually shed their outer

layers, leaving behind a white dwarf. Stars up to 8 𝑀⊙ evolve further, igniting carbon in their C-O

cores. These stars are very luminous and lose most of their mass during the AGB phase through

stellar winds, ending their lives as white dwarfs. Such AGB stars, as will be discussed in the next

section, are responsible for synthesizing almost half of the heavy elements [6, 14].

The evolution of stars with masses larger than 8 𝑀⊙ is fundamentally different and much more

spectacular than the previous cases. These massive stars have significantly shorter lifetimes, lasting

only a few million years. However, within this relatively brief period, stars more massive than

12𝑀⊙ undergo all burning phases, successively fusing hydrogen, helium, carbon, neon, oxygen,

and finally silicon. Each phase becomes progressively shorter, with silicon burning lasting only

about a day. By the time silicon is exhausted in the core, the star has developed an “onion-like"

structure, with layers of elements separated by thin nuclear burning shells, as illustrated in Fig. 3.2.

Stars more massive than 25 𝑀⊙, or those that rotate rapidly, lose a significant fraction of their mass

through strong stellar winds and eruptions, resulting in the loss of most of their outer envelopes.

At this stage, the core primarily consists of iron-peak nuclei, which, as discussed in the previous

chapter, have the highest binding energy and fusion is no longer energetically favorable. Without a

nuclear source to counteract gravity, the core continues shrinking. Once the core’s mass exceeds

the Chandrasekhar limit of 1.4 𝑀⊙, it collapses [6, 14].

3.2.1 Supernovae

Supernovae are among the brightest, most complex and cataclysmic events in the universe.

They are mainly classified based on the type of light emitted and their detailed mechanisms remain

an active area of research.

The endpoint of the evolution of a red supergiant described above, is a core-collapse supernova

(CCSN) known as Type II (SNII), and is characterized by strong presence of hydrogen in the

emitted light spectra (light curves). As the core reaches 𝑇 > 1010 K, photons break apart 56Fe into

28

Figure 3.2 Schematic drawing of the onion shell structure of massive stars of 12 ≲ 𝑀 ≲ 25𝑀⊙ at
the end of their evolution, with the dominant elements indicated. The outer envelopes of stars of
𝑀 > 25𝑀⊙ are stripped before they explode as supernovae. Note that the figure is not to scale.

𝛼 particles and neutrons, an endothermic reaction that accelerates the collapse. Within a fraction

of a second, a core with size of several thousand kilometers collapses to tens of kilometers radius.

Nuclear force at very small distances becomes repulsive supporting the core, along with rising

density that enables electron capture, forming a degenerate neutron gas that can support very high

pressure. If the collapsing core’s mass is below ∼2 𝑀⊙ (the Oppenheimer-Volkoff limit [84]), the

pressure halts the collapse, forming a neutron star. If not, then it continues until a black hole forms

[85].

During the collapse, several phenomena occur, leading to the ejection of the star’s outer

envelopes into the interstellar medium at extreme velocities. The infalling matter encounters

the very compact proto-neutron star and experiences an intense shock, that pushes matter outwards

as it bounces back. The outward moving shock wave compresses and heats the outer layers for short

29

periods of time, giving grounds for explosive nucleosynthesis. The temperature in the collapsing

core is so high that photons can be captured by electrons, creating neutrinos. The neutrinos either

escape or get trapped due to the high density and are captured by the infalling layers further

increasing the temperature. The strong neutrino and antineutrino fluxes drive a continuous flow of

protons and neutrons, known as neutrino-driven wind, that enables further nucleosynthesis to take

place [6].

Another important category is Type Ia (SNIa), that has no hydrogen in the light spectra, but a

lot of silicon. These are created in binary systems, where a white dwarf is accreting mass from a

companion star. Once the white dwarf’s mass approaches the Chandrasekhar limit, carbon ignites

under degenerate conditions, increasing temperature while supporting very high pressure. Once the

degeneracy is lifted, the energy generation rate is so large that an explosion occurs. In the single-

degenerate (SD) scenario the companion star is a red giant, supplying hydrogen and helium-rich

material onto the primary star, and after the explosion a remnant of the secondary star remains. In

the double-degenerate (DD) scenario, both stars are white dwarfs that eventually merge, and the

collapse leaves no remnant behind [6, 85].

Other types include SN Ib and Ic, that are most likely caused by supermassive stars that have

lost almost all their envelope, called Wolf-Rayet stars, and are characterized by absence of hydrogen

and silicon in their light spectra [85].

3.3 Stellar Nucleosynthesis

The previous section described the evolution of stars starting from clouds of dust.

In the

early stages of the universe, these molecular clouds consisted of light elements formed during the

Big Bang. A few seconds after the Big Bang, the universe had cooled down sufficiently for free

protons and neutrons to form. About 20 minutes later, primordial nucleosynthesis established the

abundances of light elements, consisting of 75% 1H, 25% 4He, and traces of the stable deuterium

(2H), 3He, and 7Li, as well as unstable tritium (3H) and 7Be that decayed shortly after [86].

Any elements heavier than A=7 were not formed until hundreds of millions of years later, when

the first nuclear reactions started taking place in stellar cores. As it was pointed out in the previous

30

section, nuclear reactions allow the star to maintain hydrostatic equilibrium, preventing it from

gravitational collapse. Initially, the fuel consists of nuclei with the smallest nuclear charges, as their

low Coulomb barrier allows fusion to occur at lower temperatures. Once this fuel is consumed, the

core contracts, increasing the temperature and enabling fusion of nuclei with progressively higher

Coulomb barriers.

3.3.1 Hydrostatic Burning

The first nuclear fuel is the lightest and most abundant hydrogen. H-burning occurs through

the net reaction:

4𝑝 → 4𝐻𝑒 + 2𝑒+ + 2𝜈𝑒

(3.1)

with a Q-value of 26.731 MeV. The two positrons are immediately annihilated with free electrons

in the stellar plasma. However, the probability of four protons interacting simultaneously is too

small for this reaction to occur directly. Instead, the net reaction is achieved with sequences of

two-particle interactions. There are two main mechanisms for H-burning, the proton-proton (or

pp) chains and the CNO-cycles. Which mechanism dominates depends on the core temperature

and the availability of CNO-cycle nuclei, which act as catalysts, converting 1H into 4He. The pp

chains are dominant in first-generation stars, which formed from primordial material, or in less

massive stars with lower central core temperatures. In contrast, the CNO cycle becomes significant

in more massive, and later-generation stars enriched with heavier elements synthesized in massive

first-generation stars that have already exploded [6, 14, 15].

After the hydrogen fuel has been consumed, the stellar core consists mainly of 4He, as the

creation of heavier elements is blocked by instabilities at the mass gaps 𝐴 = 5 and 8. However, 12C,

the third most abundant element in the universe, is not a product of primordial nucleosynthesis and

must therefore be synthesized in stars. This problem was solved by Öpik and Salpeter [87, 88, 89]

who proposed that 12C is produced through a two-step process known as the triple-alpha process.

In this process, two alpha particles fuse to form an unstable 8Be nucleus, which can occasionally

capture another alpha particle before decaying, resulting in 12C. As the non-resonant tunneling

probability for this reaction is too low to explain the observed abundance of 12C in the universe,

31

Fred Hoyle [90] proposed that the reaction proceeds via a resonance in 12C just above the 8Be+𝛼

threshold, at 𝐸 𝑋 ≃ 7.68 MeV. This resonance, now known as the Hoyle state, has been subject of

extensive research since its discovery and continues to be studied today [91, 92, 93, 94, 95, 96].

After the formation of 12C, further reactions like 12C(𝛼, 𝛾)16O occur, but the subsequent

16O(𝛼, 𝛾)20Ne proceeds at an extremely low rate, blocking significant nucleosynthesis via He-

burning beyond 16O. As carbon-based life forms that depend on the existence of oxygen in the

atmosphere, we should appreciate that it is only through some fortunate nuclear properties of

carbon and oxygen that they are produced so plentifully and survive the red giant phase of stars.

With the exhaustion of helium in the core, the star transitions to helium-shell burning. As

the density increases, thermal pulses, known as helium-shell flashes, can occur. Similar to the

He-core flash, these thermal pulses cause mixing of the material between the He-burning and

H-burning shells. These thermal pulses combined with a complicated convective mixing process

in the inter-shell regions combines the p-rich material of the H-burning shell with 12C from the

He-burning shell, forming 13C through the 12C(𝑝, 𝛾)13N(𝛽−)13C reaction sequence [97]. The

reaction 13C(𝛼,n)16O is an important neutron source for heavy element nucleosynthesis, as will be

discussed in Sec. 3.3.3. Another significant source of neutrons for heavy element nucleosynthesis

is the 22Ne(𝛼,n)25Mg reaction, making the availability of 22Ne very important during He-burning.

The exhaustion of He in the center of the star, leaves a core rich in C and O contracting under

gravity. As discussed in Sec. 3.2, stars with mass above 8 𝑀⊙ will proceed to more advanced burning

stages, while only stars above 12 𝑀⊙ undergo all burning stages. Carbon burning ignites first in

the CO-rich core, followed by neon burning and then oxygen burning. While carbon and oxygen

burning proceed mostly through fusion, Ne-burning mostly proceeds through photodisintegration

reactions on 20Ne, making this phase particularly brief due to its lower energy output. Similarly,

the final burning phase, Si-burning, is based on photodisintegrations of silicon isotopes rather than

fusion, leading to a complex network of reactions.

In this network, many forward and reverse

reaction rates become comparable to the burning timescales, leading to the formation of certain

“clusters" in the nuclear chart where particle captures and photodisintegrations are in equilibrium,

32

a state know as a quasi-statistical nuclear equilibrium (QSE). The nuclear flow is therefore largely

confined within these clusters. The final abundances produced during Si-burning depend on where

those clusters form, which depends on the available neutrons in the system (also known as neutron

excess 𝜂, neutron to proton ratio, or electron fraction 𝑌𝑒). This dependency on 𝑌𝑒 determines the

composition of the core and sets the stage for the subsequent core-collapse [6, 14, 15].

3.3.2 Nucleosynthesis During Core-Collapse SN

The collapse of a stellar core discussed in Sec. 3.2, enables two main mechanisms for nucle-

osynthesis. The first is driven by the outward-moving shock wave during the explosion, and the

second by the large amount of energy released from the core in the form on neutrinos.

As the shock wave moves outward, it heats and compresses the star’s layers almost instanta-

neously. First, it passes through the silicon-burning shell, followed by the oxygen-rich layer, and

finally the region primarily composed of neon, carbon, and oxygen. Each layer undergoes a specific

explosive burning process at varying peak temperatures, producing a range of nuclei. The resulting

abundances depend strongly on the expansion timescale during cooling and the availability of free

particles (𝛼, 𝑝, 𝑛) [6].

The launch of the shock generates a strong flux of electron neutrinos and antineutrinos, which

drive protons and neutrons from the region near the proto-neutron star in what is known as the

neutrino-driven wind [98]. Neutrinos interact with nuclei from infalling layers, populating excited

nuclear levels that decay via particle emission (𝑝, 𝑛, 𝛼). This neutrino-driven nucleosynthesis,

known as the 𝜈 process, depends on the wind’s properties, such as electron fraction 𝑌𝑒 and entropy

(or photon-to-baryon ratio), determining whether the wind is proton-rich or neutron-rich. Final

abundances also depend on the expansion and cooling timescales, as in hotter environments the

wind will consist mainly of neutrons and protons, but in cooler environments protons and neutrons

will combine to 𝛼 particles.

3.3.3 Nucleosynthesis Beyond Iron

As discussed in previous sections, hydrostatic burning stages synthesize elements through

fusion up to the iron region. At very high temperatures, where charged-particle capture reactions

33

are enabled, nucleosynthesis occurs in clusters where nuclear statistical equilibrium is established,

favoring either iron peak nuclei or lighter elements. However, during hydrostatic burning, neutrons

are produced through the 13C(𝛼, 𝑛)16O and 22Ne(𝛼, 𝑛)25Mg reactions. As neutrons are not affected

by the increasingly large Coulomb barrier of heavy nuclei, neutron capture reactions are able to

synthesize elements heavier than iron.

In their pioneering work in 1957, Burbidge, Burbidge, Fowler, and Hoyle (henceforth B2FH)

described two main neutron-capture processes for heavy-element nucleosynthesis [3]. The dis-

tinction between the two lies on the vastly different timescales on which they operate. These

processes are known as the slow and rapid neutron-capture process, or s- and r-process for short.

These two processes are responsible for synthesizing the majority of heavy elements. However,

it is now known that additional neutron-capture processes such as the intermediate (i-) [99, 100]

and n-process [101], are also required to accurately reproduce the observed abundances of stars.

Additional processes are also required for the production of the proton-rich elements, and those

will be discussed in Sec. 3.4.

One of the most compelling pieces of evidence for the two neutron-capture mechanisms is their

ability to naturally explain the existence of peaks in the solar system abundance pattern of heavy

elements. As shown in Fig. 3.3, double peaks appear in the regions of 𝐴 ≃ 84, 138 and 208. These

patterns arise from the neutron magic numbers 𝑁 = 50, 82 and 126. The sharp peaks correspond

to abundances formed by the s-process, whereas the broader peaks about 10 mass units below

reflect r-process enhanced abundances. The s- and r-processes are discussed in more detail in the

following sections, along with the i- and n-process.

3.3.3.1 The s Process

First evidence of the slow neutron-capture process (s-process) nucleosynthesis was found in

spectra of AGB stars, where radioactive Tc was observed [102]. The s-process involves a series

of neutron-capture reactions followed by 𝛽− decays and is responsible for the production of almost

half of the isotopes of heavy elements. The name “slow" reflects the time intervals between

successive captures that are inversely proportional to the neutron capture reaction rates and the

34

Figure 3.3 Decomposition of solar s– (solid line), r– (black circles) and p–abundances (white
squares) relative to silicon. Figure from M. Arnould et al., Physics Reports 450 (2007), with
permission from Elsevier.

neutron flux. In environments of typical neutron density ≈ 107−11 neutrons/cm3, the rate for n-

captures is comparable to that of the 𝛽− decay, and therefore the synthesis path follows closely the

valley of stability in the nuclear chart. The process terminates in Bi, as any heavier elements are

unstable and decay by 𝛼 emission back to stability.

An example of the s-process path is shown in Fig. 3.4, starting from 77Se. The path can be

calculated by comparing the decay rate 𝜆 = ln 2/𝑡1/2, with the neutron-capture reaction rate. If
the two are comparable, such as in 85Kr, a branching point occurs, where the path can follow both

directions, leading to different abundance patterns. There are about 15-20 significant branching

points along the s-process path. Branching points can provide information on the detailed conditions

of the stellar environment, such as neutron density and temperature. However, achieving this

requires accurate knowledge of neutron-capture reaction cross sections, decay half-lives and any

temperature dependence of the rates [6, 14].

The example path of Fig. 3.4 passes through the neutron magic number 𝑁 = 50. This configu-

ration is energetically more favorable than 𝑁 = 51, and therefore the neutron-capture cross section

35

Figure 3.4 An example of the s-process path in the region around 𝐴 = 85. The halflives are obtained
from [4]. The gray and white squares correspond to stable and radioactive isotopes, respectively,
and the circle indicates a branching point.

on nuclei with 𝑁 = 50 will drop significantly, blocking the s-process path from more n-rich nuclei,

and pushing the reaction flow to higher elemental chains. This results in the first peak in the solar

abundance pattern from Fig. 3.3.

The s process is a secondary process, as it requires the existence of iron-peak nuclei to act

as seeds, unlike the hydrostatic burning phases that are primary processes, and do not depend on

preexisting nuclei. As a secondary process, the produced abundances can vary significantly based

on the stellar conditions. As discussed in Sec. 3.3.1, during He-burning in AGB stars, neutrons

are produced by the reactions 13C(𝛼,n)16O and 22Ne(𝛼,n)25Mg. A significant amount of 13C can

be produced in thermally pulsing, low mass (1.5-3 𝑀⊙) AGB stars. A complex mixing of the

intershell, which is the region between the He- and H-burning shells, mixes protons with material

36

rich in 12C and 4He. This enables the sequence 12C(𝑝, 𝛾)13N(𝛽+𝜈)13C, forming a region known as

the 13C pocket [97]. At temperatures near 𝑇 ≈ 0.09 GK, a neutron density in the order of ≈ 107

n/cm3 produced by the 13C(𝛼,n)16O reaction, provides fuel for s-process nucleosynthesis for a

period of ≈ 20 000 years. This is known as the main s-process component and provides about 95%

of the total neutron exposure, synthesizing elements up to Pb. The remaining neutron exposure is

achieved with the 22Ne(𝛼,n)25Mg source. This so-called weak s-process component, is achieved in

massive stars at the end of the convective He-burning core and in the C-burning shell [103].

3.3.3.2 The r Process

The presence of abundance peaks that can’t be explained by the s process, along with the

existence of long-lived isotopes heavier than Bi, such as 232Th and 238U, highlights the need for

an additional neutron-capture process beyond the s process. The environment for such a process

can be found in extreme stellar environments, where neutron fluxes are so high (≈ 1020−22 n/cm3)

that the 𝛽-decay rate of unstable nuclei is small compared the rate of neutron capture.

In this

case, the nucleosynthesis path may move close to the neutron dripline. Only when the neutron flux

terminates, the neutron-rich nuclei decay back to stability through 𝛽− decays. This nucleosynthesis

mechanism is named the rapid (r-) neutron-capture process and is responsible for the synthesis of

approximately the other half of the isotopes of heavy elements [3].

Similarly to the s process, the magic neutron numbers impact the reaction flow, leading to the

creation of two peaks at mass numbers 𝐴 = 130 and 195, which are about 10 mass units below

the s-process peaks near 𝐴 = 138 and 208, as shown in Fig. 3.3. At neutron densities in the order

of 1020−22 n/cm3 neutron captures can drive the r-process flux close to magic neutron numbers.

However, the neutron separation energy 𝑆𝑛 decreases for more neutron-rich nuclei. Therefore in

each isotopic chain (𝑛, 𝛾) and (𝛾, 𝑛) reactions may eventually reach a quasi-statistical equilibrium

(QSE), similar to the clusters formed during Si-burning. The flow toward higher elemental chains

depends on the 𝛽-decay rates of the nuclei in QSE. As these rates are slow compared to the rates in

equilibrium, waiting points may be established, usually one or two per chain in QSE. These waiting

points can determine the timescale of the reaction network and influence the final abundance, as

37

𝛽− decays will follow the isobaric chain (𝐴 = const). As a consequence the r-process peaks are

located in mass regions below the corresponding s-process peaks.

The r-process network can extend to the neutron dripline, where most nuclear properties remain

experimentally unknown, especially for nuclei far from stability. Extensive efforts are underway

to both theoretically and experimentally determine key nuclear properties such as masses, level

schemes, halflives, 𝛽-decay rates, fission rates, and neutron-capture cross sections [21, 104, 105,

106, 107]. Improving our understanding of these quantities is essential for enhancing the predictive

power of r-process models, which are critical for explaining observed abundances in the Sun and

other stars.

The site of the r process has been one of the most significant open questions in the field

for decades, further complicated by the lack of nuclear data for exotic isotopes. Over the years,

numerous potential sites have been proposed, including neutrino-driven core collapse supernovae

[108, 109], electron-capture supernovae [110], magneto-rotational supernovae leading to magnetars

(i.e. neutron stars with very high magnetic fields) [111, 112, 113], collapsars (massive stars that

collapse into a black hole) that produce powerful relativistic jets [114, 115], as well as black hole

- neutron star mergers [116]. Compact binary mergers (NS-NS mergers) have been suggested

as r-process sites since the 1970s [116] with first nucleosynthesis predictions in 1999 [117]. In

August 2017 LIGO and Virgo detected gravitational waves from the NS-NS merger GW170817

[118], providing the first direct evidence of an r-process event. Since then, research on r-process

nucleosynthesis in NS-NS mergers has been exponentially growing [119, 120, 121].

3.3.3.3 Other Neutron-Capture Processes

The solar abundances shown in Fig. 3.3 can be sufficiently explained by a combination of the

s and r processes, but this is not the case for many other stars. The abundance patterns of a

group of very old, carbon-enriched stars known as carbon-enhanced metal poor (CEMP) stars can

instead be explained by an alternative neutron-capture mechanism, operating at intermediate neutron

densities between those of the s and r process. This intermediate (i-) neutron capture process was

first proposed by Cowan in 1977 [99], and was found to match the observed abundances of CEMP

38

stars in 2016 [100]. Since then, many studies have been dedicated to the i process and its potential

sites, with candidates including rapidly rotating white dwarfs accreting material from a companion

red giant [122], and thermal-pulsing AGB stars [123]. The mechanism is similar to that of the s

and r process, but intermediate neutron fluxes of ≈ 1012−15 n/cm3 drive neutron-capture reactions

a few steps away from stability before 𝛽 decays return the nuclear flow to stable species. This

proximity to stability makes the i process particularly promising for experimental studies, as many

of the relevant neutron-capture reactions are accessible with current facilities [124, 125].

Another neutron-capture process, the n process, has been proposed to occur in the He shell after

its composition is modified by the SN shock passage. A neutron flux of ≈ 1018 n/cm3 or higher can

then be produced by the 22Ne(𝛼, 𝑛)25Mg reaction. The n process is able to reproduce anomalous

Mo isotopic abundances measured in SiC meteorites, motivating further study of this mechanism

[101].

3.4 Production of the p Nuclei

As discussed in the previous section, neutron-capture processes dominate heavy element nu-

cleosynthesis. However, these processes cannot produce all isotopes of heavy elements. In their

pioneering work, B2FH identified 35 proton-rich nuclides that are shielded by the valley of stability

and cannot receive contributions from the s or r process. These isotopes were named p nuclei, and

the mechanism responsible for their synthesis, the p process.

The 35 classical p nuclei as identified by B2FH are listed in Table. 3.1, along with their isotopic

fractions within their respective elements and their solar abundances [126]. Subsequent research has

shown that many of these p isotopes also receive contributions from neutron-capture [127, 129] or

neutrino-driven processes [128, 130], meaning they cannot be strictly classified as p-only isotopes.

Additionally, certain unstable nuclides, such as 92Nb, 97,98Tc and 146Sm, though not part of the

original list of classical p nuclei, play a significant role in studies of the p process. These isotopes,

with half-lives comparable to astronomical timescales, are believed to form in the same events as

stable p nuclei and can be used as cosmochronometers, providing important information on the

composition of the early Solar system [131, 132].

39

Table 3.1 The classical p nuclei, their fraction (in %) of the isotopic composition of the elements
and solar abundances (relative to Si=106). Data from Lodders [126].

Isotope Element (%) Solar Abundance Comment

74Se
78Kr
84Sr
92Mo
94Mo
96Ru
98Ru
102Pd
106Cd
108Cd
113In
112Sn
114Sn
115Sn
120Te
124Xe
126Xe
130Ba
132Ba
136Ce
138Ce
138La
144Sm
152Gd
156Dy
158Dy
162Er
164Er
168Yb
174Hf
180Ta
180W
184Os
190Pt
196Hg

0.889
0.362
0.555
14.8
9.25
5.54
1.87
1.02
1.25
0.89
4.29
0.971
0.659
0.339
0.096
0.129
0.112
0.106
0.101
0.186
0.251
0.0902
3.07
0.203
0.056
0.096
0.137
1.61
0.13
0.162
0.0123
0.12
0.0198
0.0136
0.153

5.80 · 10−1
2.00 · 10−1
1.31 · 10−1
3.86 · 10−1
2.41 · 10−1
1.05 · 10−1
3.55 · 10−2
1.46 · 10−2
1.98 · 10−2
1.41 · 10−2
7.80 · 10−3
3.62 · 10−2
2.46 · 10−2
1.26 · 10−2
4.60 · 10−3
6.94 · 10−3
6.02 · 10−3
4.60 · 10−3
4.40 · 10−3
2.17 · 10−3
2.93 · 10−3
3.97 · 10−4
7.81 · 10−3
6.70 · 10−4
2.16 · 10−4
3.71 · 10−4
3.50 · 10−4
4.11 · 10−3
3.23 · 10−4
2.75 · 10−4
2.58 · 10−6
1.53 · 10−4
1.33 · 10−4
1.85 · 10−4
6.30 · 10−4

r-process contribution [127]

r-process contribution [127]

𝜈-process contribution [128]

s-process contribution [129]

s-process contribution [129]

s-process [129] and 𝜈-process contributions [130]

The p process was initially described in B2FH as occurring in the hydrogen-rich layers of core

collapse supernovae, through a series of (𝑝, 𝛾) and (𝛾, 𝑛) reactions on existing s- and r-process

seeds during the passage of the shock wave [3]. In 1978, Woosley and Howard [133] suggested

40

that the required conditions for the process, including high densities, elevated temperatures, and

extended time scales, are unlikely to exist in the hydrogen-rich regions of most stars. Alternatively,

they proposed an explosive nucleosynthesis mechanism based on a series of photodisintegration

reactions on s- or r-process seed nuclei, which was named the 𝛾 process.

The production of p nuclei remains an active area of research. Various mechanisms have been

proposed in different astrophysical environments, involving both explosive and neutrino-driven

nucleosynthesis. The term p process has been retained within the astrophysics community for

historical reasons and now serves as an “umbrella" term that includes these diverse processes. The

following provides an overview of the primary scenarios currently under investigation.

3.4.1 The 𝛾 Process

The 𝛾 process is widely regarded as the main mechanism for the synthesis of the p nuclei. It

occurs in stellar environments of sufficiently high plasma temperatures through particle emission

from thermally excited nuclei. Rather than the hydrogen-rich layer proposed by B2FH, it is thought

to occur in a zone where hydrogen is exhausted and heavy elements are subjected to a “hot photon

bath" [133]. Under these conditions, the most likely reactions are photodisintegrations, meaning

(𝛾, 𝑛), (𝛾, 𝑝) and (𝛾, 𝛼), as shown schematically in Fig. 3.5.

The first reactions to take place are (𝛾, 𝑛), as these dominate the photodisintegration processes

for most stable nuclei [134]. As the nuclear flow progresses to more neutron-deficient nuclei, the

(𝛾, 𝑛) reaction rate decreases. At the same time, the proton-richer the isotope, the less energy is

needed to remove a proton or an 𝛼 particle. Consequently, (𝛾, 𝑝) and (𝛾, 𝛼) reactions take over,

moving the nuclear flow to lower elemental chains and eventually the p nucleus of interest.

The process is highly sensitive to temperature, as higher temperatures or prolonged exposure

would completely photodissociate the seeds into iron peak nuclei. On the other hand, cooler envi-

ronments would not allow thermally excited nuclei to decay by particle emission. The temperature

range for 𝛾-process nucleosynthesis to occur is between 1.8 and 3.2 GK. As these photodisintegra-

tion reactions are strongly influenced by the particle separation energies of the seed nuclei, lighter

p nuclei require higher plasma temperatures (𝑇 ≈ 3–3.2 GK) because their seeds, being closer to

41

Figure 3.5 An schematic illustration of the 𝛾-process path. The gray and white squares correspond
to stable and radioactive isotopes, respectively, and the blue square indicates the produced p nucleus.

the iron peak, are more tightly bound. The heavier p nuclei are synthesized in lower temperatures

(𝑇 ≈ 1.8 − 2 GK) as their seeds are less bound.

Any site capable of sustaining 𝛾-process nucleosynthesis must maintain these temperatures for a

short period of time, while providing an adequate supply of seed nuclei. The most viable candidates

are stellar explosions that involve a rapid expansion and subsequent cooling of the material. As

a result, the 𝛾-process nucleosynthesis is highly sensitive to factors such as the temperature and

density profile, expansion timescales, the initial abundances of seed nuclei, and the hydrodynamic

properties of the explosion. The two main explosive environments where the 𝛾 process is thought

to occur are the oxygen and neon enriched layers of a core-collapse supernovae (O/Ne SNII)

[133, 135, 136], and thermonuclear Type Ia supernovae [137, 138, 139].

The mechanism of SNII, as discussed in Sec. 3.2, involves the propagation of an outward

moving shock wave. There are two main components of the SNII that contribute to the 𝛾 process:

the explosive component during shock wave propagation [133, 135, 136, 140], and the pre-explosive

component [141, 136, 142, 143]. During the explosion, the shock encounters the O/Ne burning

shell, which has been enriched in s-process material. The inner layers reach higher temperatures,

enabling the synthesis of the lightest p nuclei, while the heavier ones are formed in the cooler outer

42

layers [135, 136, 134].

In the final stages of stellar evolution, just before the explosion, C-rich material may be ingested

in the convective O-burning shell, forming a merged convective zone. This zone, known as the

C-O shell merger [136, 142], provides a sustainable environment for synthesizing p nunclei heavier

than Pd [143]. The produced material is mixed throughout the extended C-O shell and as it is not

fully reprocessed by the shock wave, it maintains its pre-supernova abundances in the ejecta.

It is important to note that the 𝛾-process abundances depend strongly on the s-process seed

distributions. Early studies assumed solar s-process distributions [133, 136], however more recent

studies have shown that these seeds may be enhanced. This can occur either through the presence

of additional 13C from the convective C core, which leads to enhanced s-process seed distributions

[144], or through stellar rotation, which enhances the weak s-process abundances driven by the

22Ne(𝛼,n)25Mg neutron source [145].

The other main environment for 𝛾-process nucleosynthesis is thermonuclear Type Ia supernovae,

which was briefly discussed in Sec. 3.2. In the single-degenerate (SD) scenario, a CO white dwarf

(WD) accretes material from a main-sequence or red giant companion. The WD explodes once its

growing mass approaches the Chandrasekhar limit. During the explosion, a broad range of peak

temperatures are reached across different mass coordinates of the WD, including temperatures

sufficient to sustain 𝛾-process nucleosynthesis in the outer layers [138]. However, it is again

essential to determine the available seed distributions in the exploding WD. These seeds can be

provided by the s process during the AGB or TP-AGB phase [138], by the n process from recurring

H-shell flashes [146], or by the i process from recurring He-shell flashes in the WD [147]. Although

there have been efforts to explore the 𝛾 process in a sub-Chandrasekhar helium detonation model,

there were significant uncertainties in the seed distributions, and sufficient p-nuclei abundances

could only be achieved with highly enhanced seed abundances [148]. Additionally, ongoing

research is exploring the potential role of the double-degenerate scenario in Type Ia SNe.

The 𝛾 process is the most established scenario for the production of the p nuclei, as it has so

far been the most successful at reproducing the majority of the observed solar abundances within

43

a factor of three. However, several discrepancies arise, especially in the region near 𝐴 = 95, as

the 92,94Mo and 96,98Ru are systematically underproduced by one order of magnitude compared to

other 𝛾-process nuclei [131, 134, 149]. An example overproduction factor divided by the average

overproduction factor of all 35 p nuclei is shown in Fig. 3.6 by Roberti et al. [143], using a 20 M⊙

SNII model by Ritter et al. [142].

Figure 3.6 p-nuclide overproduction factors divided by their average, from a 20 M⊙ SNII model
[142]. The different color symbols represent nuclei explicitly produced by the
by Ritter et al.
𝛾 process (blue), and nuclei that may have an additional explosive contribution (orange) or an s, r
process, or neutrino-capture contribution. (Figure from Roberti et al. Astronomy & Astrophysics
677, A22 (2023), under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0))

3.4.2 Other Scenarios for the p Process

Such discrepancies in the produced p-nuclei abundances lead to investigate possible contribu-

tions from multiple processes other than the 𝛾 process. Alternative scenarios include the rp, 𝜈p,

and 𝜈r processes, which will be briefly discussed here.

An alternative mechanism for the production of the p nuclei was proposed to take place on the

surface of a neutron star that accretes H- and He-rich material from a companion star. This accretion

leads to a large gravitational energy release called an X-ray burst. Within this environment, some

of the lighter p nuclides can be synthesized through the rp-process (short for rapid proton capture

44

process) [150]. The rp-process path proceeds through a series of proton captures on CNO nuclei

toward progressively heavier nuclei, until the proton dripline is approached and 𝛽+ decay takes place.

While this process can produce the lightest p nuclei, it remains uncertain whether the synthesized

material can escape the strong gravitational pull of the neutron star [151]. Recent studies suggest

that if the neutron star is accreting in binary common envelopes, where the expanding star envelops

its companion, then the material produced by the rp-process may escape the strong gravitational

field and be ejected into the interstellar medium [152].

An additional mechanism involves the neutrino-driven winds produced in CCSN. For values

of the electron fraction 𝑌𝑒 > 0.5, a proton-rich neutrino-driven wind can be obtained, and the

so-called 𝜈p process can occur [153, 154]. The proton-rich environment is constantly supplied by

a small number of free neutrons created by antineutrino captures on free protons. The resulting

nucleosynthesis flow near the heavy elements is similar to the r-process, as it is characterized by

rapid proton captures in a (𝑝, 𝛾)-(𝛾, 𝑝) equilibrium, with (𝑛, 𝑝) reactions connecting the isotonic

(𝑁 = const) chains. This process has been shown to produce the lightest p nuclei [155], however

such calculations depend on large uncertainties in neutrino interaction cross sections, the average

energies associated with different neutrino flavors, the overall neutrino luminosity, and the specific

details of the stellar evolution and explosion models used in the simulations.

A recent research worth highlighting proposed an new nucleosynthesis mechanism that could

contribute to p-nuclei production, called the 𝜈𝑟 process [156]. This process is suggested to take place

on neutron-rich ejecta, where r process occurs. In this scenario, r-process seeds experience strong

neutrino irradiation, and thus the (𝑛, 𝛾) (𝛾, 𝑛) equilibrium is broken by the neutrino interactions

instead of 𝛽 decays. This pushes the nuclear flow toward and beyond the valley of stability,

producing p nuclei. This process is highly dependent on uncertainties on neutrino interactions, and

the specific astrophysical conditions required for such strong neutrino fluxes are still uncertain.

3.5 Nuclear Networks and Uncertainties

As discussed in the previous sections, stellar nucleosynthesis involves various complex pro-

cesses that occur simultaneously in a stellar environment. Simulating these processes requires

45

stellar evolution codes that integrate nuclear physics with the physical mechanisms governing stars,

such as gas properties, hydrodynamics for hydrostatic equilibrium, and energy transport via radia-

tion or convection. An example of such a code is Modules for Experiments in Stellar Astrophysics

(MESA) [157], a 1D stellar evolution code that employs modern numerical and software techniques

to solve stellar structure and composition equations while incorporating nuclear physics.

However, simulating an entire star is computationally intensive, particularly for environments

such as the p or r process, which involve hundreds of thousands of reactions across thousands of

isotopes. To simplify the problem and reduce computational costs, one approach is post-processing.

Post-processing nuclear network calculations requires a prior stellar evolution simulation, such as

one produced by MESA, using a reduced network of isotopes and reactions. These networks

are limited to reactions critical for energy generation and those that significantly alter the star’s

composition. For example, simulating hydrostatic hydrogen burning only necessitates reactions

from the pp-chains and CNO cycles (see Sec. 3.3.1), as these are key to energy production. Reactions

on other light nuclei, while important for accurately reproducing final abundances, can be omitted

during the evolution phase without affecting the star’s structure, density, or temperature.

Once the stellar evolution simulation is complete, the temperature and density profiles as

functions of time (trajectories) are extracted. Post-processing then uses these trajectories along

with the initial abundances of all isotopes and nuclear physics inputs, such as masses, half-lives,

and reaction rates to calculate nucleosynthesis. At this stage, the problem is reduced to solving a

system of ordinary differential equations that describe the production and destruction of each nuclear

species. This decoupling of nucleosynthesis calculations from stellar structure and evolution allows

the inclusion of extensive networks of isotopes and reactions without making the computation

prohibitively expensive.

An example of such a post-processing framework is provided by the nucleosynthesis grid (Nu-

Grid) collaboration to perform both single-zone (PPN) and multizone parallel (MPPNP) simulations

for given thermodynamic conditions [158, 159]. The difference between a single-zone and a multi-

zone model lies in the mass coordinates that the model can simulate. A single-zone model would

46

follow the evolution of a single mass coordinate of the star for one trajectory, while a multi-zone

model can follow multiple zones with different initial abundances and trajectories.

In such complex calculations, uncertainties in input quantities naturally have a significant

impact on the calculated abundances. For the 𝛾-process network, in addition to the astrophysical

uncertainties described in Sec. 3.4.1 regarding the astrophysical site and the distribution of seed

nuclei, numerous nuclear uncertainties affect the network calculations, as there are nearly 20 000

nuclear reactions on almost 2000 nuclei that must be considered [131]. As the nuclides involved in

the 𝛾 process are predominantly stable or moderately unstable proton-rich nuclei, their masses and

corresponding reaction Q-values are generally well-known. Similarly, half-lives are in principle

known, aside from the dependence of electron captures and 𝛽+-decay rates on ionization and

thermal excitation in the stellar plasma that require theoretical corrections [134].

The main uncertainty on the nuclear physics lies in the photodisintegration reaction rates,

which must be determined with high accuracy for many possible reactions. As experimental data

are scarce, uncertainties in the predicted reaction rates increase substantially for nuclei farther from

stability [160]. Given that the 𝛾-process network involves thousands of possible reactions, it is

crucial to focus on those with the most significant impact to address the problem effectively. To

this end, several sensitivity studies have been conducted over the years [160, 161, 162] to identify

reactions whose uncertainties notably influence the production of specific p nuclei, helping in the

planning of nuclear physics experiments.

During the recent decades, significant experimental efforts have been made to measure cross

sections relevant to the 𝛾 process on stable nuclei [163, 164, 165, 166, 167, 168, 169, 170, 171, 172].

However, only one experiment involving a radioactive beam has been conducted to date [173].

For the thousands of reactions yet to be measured experimentally, reaction rates rely primarily

on Hauser-Feshbach (HF) theoretical calculations. The HF model calculations most commonly

adapted for 𝛾-process networks are obtained from the Non-Smoker code [174].

As discussed in Sec. 2.4.2.2, HF cross-section calculations depend on the nuclear optical model

potential (OMP), nuclear level density (NLD), and 𝛾-ray strength functions (𝛾SF). For heavier p

47

nuclei, where lower temperatures are required and the relevant energies lie near the lower end of the

Gamow window, uncertainties in the OMP dominate. In contrast, for lighter p nuclei, which require

higher temperatures and relevant energies correspond to the higher end of the Gamow window,

uncertainties in the NLD and 𝛾SF have a greater impact on the calculated reaction rates.

Developing experimental techniques to directly measure 𝛾-process reactions involving unsta-

ble isotopes is therefore critically important. This thesis focuses on the application of such an

experimental technique, to study the destruction of the lightest p nucleus, 74Se.

3.6 The Lightest p Nucleus, 74Se

74Se is the lightest of the p nuclei, as its production is shielded from the s-process path and

r-process decay chains as shown in Fig. 3.7.

In sensitivity studies of the 𝛾 process during SNII

Figure 3.7 The lightest p nucleus, 74Se, is shielded by the valley of stability from the s-process path
and the r-process decay chain. The gray and white squares correspond to stable and radioactive
isotopes, respectively, and the blue square indicates the 74Se nucleus of interest. The halflives are
obtained from [4].

scenario the 74Se(𝛾, 𝑝)73As reaction has been identified as key reaction rate to impact the final

abundances of 74Se [160, 161]. While stellar models for SNII [136, 176, 142, 177] show some

variation, 74Se is often found to be overproduced compared to solar abundances, as shown in Fig. 3.6.

48

Figure 3.8 The 𝛾-process fluxes producing and destroying 74Se during a SNII. The sum of all
production and destruction fluxes is normalized 100%. Fluxes smaller than 1% are not shown.
Fluxes obtained from [175], using trajectories from [160].

The possible production and destruction mechanisms of 74Se in a SNII, shown in Fig. 3.8, have been

a topic of experimental studies for many decades. Of those reactions the 74Se(𝑝, 𝛾)75Br [178, 179,

180, 181], 70Ge(𝛼, 𝛾)74Se [182], and 74Se(𝑛, 𝛾)75Se [183] have been measured directly, and the

75Se(𝛾,n)74Se can be inferred from the latter through the reciprocity theorem (see Sec. 2.5.2). The

only reaction channels that immediately affect the final 74Se abundance, for which no experimental

data exist are the 74Se(𝛾, 𝑝)73As and the 74Se(𝛾, 𝑛)73Se. This work focuses on the measurement of

the inverse 73As( 𝑝, 𝛾)74Se reaction, that can be used to calculate the ground-state contribution of

the 74Se(𝛾, 𝑝)73As reaction through the reciprocity theorem.

In simulations of SNII [142], the maximum production of 74Se is found in layers with peak

temperature 𝑇 ≈ 3 GK. For such temperature, the Gamow window for the 73As(𝑝, 𝛾)74Se reaction

is located at center-mass-energy ranging from 1.7 to 3.6 MeV. The predicted cross section of the

73As(𝑝, 𝛾)74Se reaction in this energy range is shown in Fig. 3.9. The solid black line corresponds

to standard statistical model calculations using the Non-Smoker code [174], and the blue band

is calculated through Talys [62], using the various possible models for the OMP, NLD, and 𝛾SF

discussed in Sec. 2.4.2.5. It can be seen that the statistical properties of the 74Se nucleus result

in a cross section uncertainty of a factor of 6 at center-mass-energy near 3 MeV. This thesis aims

49

Figure 3.9 The cross section of the 73As(𝑝, 𝛾)74Se reaction using standard statistical model
calculations from the Non-Smoker code [174] (black line) and Talys [62] (blue band). The
Talys calculations include all possible NLD and 𝛾SF options listed in Tables 2.1 and 2.2, as well as
the default and JLM OMP discussed in Sec. 2.4.2.5. The energy range covers the Gamow window
for the 𝛾 process at 𝑇 = 3 GK.

to investigate whether this uncertainty in the reaction rate is responsible for the overproduction of

74Se in SNII models and whether an experimental measurement can help resolve this discrepancy.

Regarding the SNIa scenario, while sensitivity studies suggest that nuclear uncertainties in 74Se

reactions do not significantly affect 74Se production [162], this work aims to provide a quantitative

assessment of their potential role.

The following chapter discuses the experimental setup for the measurement of the 73As( 𝑝, 𝛾)74Se

cross section using a radioactive 73As beam. Chapter 5 presents the analysis from the proof-of-

principle stable beam experiment on the 82Kr(p, 𝛾)83Rb reaction, the measured 73As( 𝑝, 𝛾)74Se

reaction, as well as the development of an analysis method to constrain the NLD and 𝛾SF used

in the statistical model calculations. Chapter 6 presents the results of the two cross section mea-

surements, and in chapter 7 the impact of the measured 73As( 𝑝, 𝛾)74Se reaction cross section is

investigated in a SNII and SNIa scenario.

50

CHAPTER 4

EXPERIMENTAL SETUP & TECHNIQUES

This work regards to two experiments in inverse kinematics using the same experimental setup. The

first experiment, conducted in 2017, served as the proof-of-principle stable beam experiment for the

measurement of the 82Kr(𝑝, 𝛾)83Rb and 84Kr(𝑝, 𝛾)85Rb reaction cross sections. The 84Kr(𝑝, 𝛾)85Rb

reaction has been analyzed and published by a former student of the group [184], therefore this work

focuses only on the 82Kr(𝑝, 𝛾)83Rb measurement [185]. The second experiment is the radioactive

beam experiment for the measurement 73As(𝑝, 𝛾)74Se reaction cross section that took place in

2023. The experiments took place in the ReA post-accelerator of the Facility for Rare Isotope

Beams (FRIB) (formerly known as National Superconducting Cyclotron Laboratory) at Michigan

State University. More details on the facility and the delivered beams are discussed in Sec. 4.1.

An illustration of the experimental setup is shown in Fig. 4.1. The 82Kr and 73As beams

impinged onto a hydrogen gas-cell target described in Sec. 4.2. The 𝛾 rays produced by the

Figure 4.1 The experimental setup with the SuN and SuNSCREEN detectors in ReA3. The beam
enters from the right, as shown by the white arrow. The inset shows a side-section view of the SuN
detector, indicating the location of the hydrogen gas cell.

51

(𝑝, 𝛾) reaction were detected using the Summing NaI (SuN) detector and analyzed using the 𝛾-

summing technique, as discussed in Sec. 4.3. To minimize cosmic-ray background contributions,

the Scintillating Cosmic Ray Eliminating ENsemble (SuNSCREEN) was used, as explained in

Sec. 4.4.

4.1 Beam Delivery in ReA

The Facility for Rare Isotope Beams (FRIB) is a national scientific user facility that can provide

access to thousands of nuclei far from stability [186]. It succeeded the Coupled Cyclotron Facility

at the National Superconducting Cyclotron Laboratory (NSCL) that pursued a very successful

science program with rare isotopes produced by projectile fragmentation until early 2021 [187].

Along with fast and stopped beams, the facility can provide reaccelerated beams in energies ranging

from 300 keV/nucleon to 6 MeV/nucleon, providing unique opportunities for nuclear astrophysics

experiments.

Ion beams can alternatively be generated in the so-called “offline mode”, using

radioactive or stable source samples [188]. Such was the case for the stable 82Kr, and the radioactive

73As beam.

Firstly the source samples were evaporated in an ion source and directed toward the ReAcceler-

ator charge breeder. For the 73As source, the Batch-Mode Ion Source (BMIS) was used (Fig. 4.2),

which is an oven ion source combination commissioned in 2021 [188]. The 73As sample was

inserted inside a cylindrical tantalum oven, and heated up to about 1000 0C. With the oven-ion

source biased to a few tens of kV, the evaporated material exited through a transfer tube and was

directed to the ReAccelerator charge breeder. Details on the preparation of the radioactive 73As

source sample can be found in Ref. [189].

The ReA charge breeder is the Electron Beam Ion Trap (EBIT) [190]. EBIT is a superconducting

magnet in which the evaporated ions were injected and charge bred to high charge states (73As+23 and

82Kr27+). After the EBIT, the beam was mass selected in a charge-over-mass (𝑄/𝐴) separator and

accelerated through the Radio Frequency Quadrupole (RFQ) [191] and superconducting LINAC

(short for linear accelerator) [192]. The ReA LINAC has three accelerating cryomodules that

include superconducting quarter wave resonators (QWR) and superconducting solenoids (SS) that

52

Figure 4.2 The batch mode ion source: (left) internal components of the oven-ion source and (right)
the fully assembled front end.

accelerated the 82Kr and 73As beams.

The beams were delivered to the experimental end station in the ReA3 area, shown in Fig. 4.1,

with energies 3.1, 3.4 and 3.7 MeV/nucleon for the 82Kr beam, and 3.1 and 3.7 MeV/nucleon for

the 73As beam. The measurement of the beam current for the 82Kr experiment was achieved by

electrically isolating the beam pipe with an isolating flange upstream of SuN and using the entire

beamline as a Faraday cup. This way, any electrons produced by ionization in the cell or beam

pipe were still recorded and the total current did not get affected. Due to issues with grounding the

beamline was not properly isolated during the 73As experiment and the beam current was measured

in regular intervals through a Faraday cup upstream.

More details on the current measurements are provided in Sec. 5.2. However, it is worth noting

that while the proof-of-principle experiment was performed with a stable 82Kr beam of intensity

on the order of 107 particles/sec, the radioactive 73As beam was around 105 particles/sec, almost

two orders of magnitude lower. This highlights the inherent challenges of working with radioactive

beams, particularly with a highly toxic element like arsenic. Nevertheless, as will be shown in the

following chapters, the measurement was successfully achieved.

53

4.2 Hydrogen Gas Target

The beam impinged on a hydrogen gas target positioned at the center of the SuN detector. The

target system involved two main components: the gas cell, which held the hydrogen gas, and the

gas handling system.

The gas cell, designed at Hope College, consisted of two plastic halves. A schematic side section

of the cell is shown in Fig.4.3, with construction images in Fig.4.4. A 2-𝜇m thick molybdenum

entrance and 5-𝜇m thick exit window were glued onto the tantalum rings, as shown in the figures.

Figure 4.3 Side-section illustration of the gas cell showing the tantalum rings supporting the
entrance and exit molybdenum windows, along with openings for the gas supply and grounding
cable.

The choice of molybdenum for the target window was based on several criteria. High atomic

number (𝑍) materials are required to ensure that the Coulomb barrier is high enough, keeping the

threshold for fusion-evaporation reactions well above the maximum beam energy. Additionally, the

target window needs to be as thin as possible to minimize beam straggling and energy loss. At the

same time, the material must have high strain tolerance, allowing the very thin foil to withstand the

atmospheric pressure of the hydrogen gas against the vacuum in the beamline without rupturing.

Lastly, it must be available from the manufacturer as leak-tight, so the hydrogen gas remains secured

in the cell.

Ensuring that the components of the gas-cell interacting with the beam do not create significant

54

Figure 4.4 (left) tantalum rings with molybdenum entrance and exit windows during construction
(right) assembled gas cell

beam-induced background is critical. This background can become significant, especially if the

beam interacts with anything plastic, as synthetic polymers consist mostly of carbon and hydrogen

atoms, which can become target for the (𝑝, 𝛾) reaction of interest, or get scattered, affecting the

quality of the data. For these reasons, the inner volume and front face of the cell were lined

with tantalum foil to shield the plastic components from the beam. The tantalum also allowed

charge collection from the entire inner volume of the cell. The back half of the cell contained

two small openings: one for the grounding cable that was connected to the inner lining, which

prevents charging and discharging inside the cell, and another for attaching the gas supply pipe.

Both openings were sealed with epoxy glue after assembly to ensure leak-tight operation.

The gas handling system ensured safe handling and proper disposal of the flammable hydrogen

gas. The system, shown in Fig. 4.5, consisted of a 10 L hydrogen reservoir, flowmeters, and

regulators to control the slow filling and emptying of the cell, preventing rupture of the thin

window foils. It also included a dry nitrogen supply line to purge the hydrogen before venting,

preventing hazardous mixtures with atmospheric oxygen. A series of valves regulated the flow,

while two manometers monitored the pressure of the reservoir and gas cell. The reservoir remained

overpressurized at ≈ 850-900 Torr, to mitigate the risk of atmospheric oxygen entering the small

volume of the container. The cell was filled with 600 Torr of hydrogen during operation, with its

pressure continuously monitored and recorded throughout the experiment. The system featured

55

detailed procedures for pumping down, filling the reservoir and gas cell with hydrogen, purging

hydrogen, and venting. Multiple interlocks were incorporated to ensure that, in the event of target

failure—such as a rupture of the target window—any potential hazards were effectively mitigated.

Figure 4.5 The setup of the 73As(𝑝, 𝛾)74Se experiment in the ReA3 experimental area. The picture
shows the SuN and SuNSCREEN detectors and the hydrogen gas supply system.

4.3 The SuN Detector

Surrounding the hydrogen gas target was the Summing NaI (SuN) 𝛾-ray detector, shown in

Fig. 4.1 and 4.5. SuN is a calorimeter with the shape of a 16 × 16 inch barrel with a 1.8 inch

diameter borehole along its axis. The barrel is segmented into 8 NaI(Tl) crystals, each connected

to three photomultiplier tubes (PMTs) [193, 194]. The large volume of the detector allows for high

𝛾-ray detection efficiency and nearly 4𝜋 solid angle coverage for a source placed in its center.

Sodium iodide (NaI) crystals doped with traces thallium (Tl) are the most common inorganic

scintillation material, widely used for the detection and spectroscopy of 𝛾-ray radiation. Introduced

in 1948 [195], NaI(Tl) crystals are valued for their availability in large volumes at relatively low

56

costs, which often outweighs the advantages of newer inorganic scintillators offering higher light

output, better energy resolution or faster timing capabilities [196]. However, NaI(Tl) is highly

hygroscopic and deteriorates upon contact with atmospheric water. To prevent this, the top and

bottom SuN crystals are encapsulated in aluminum casing. Additionally, the crystals are surrounded

by a reflective polytetrafluoroethylene layer, therefore they are optically isolated from each other

and can function as individual detectors.

When ionizing radiation enters the active volume of the detector, it excites the crystal atoms,

leading to emission of visible light with wavelength of approximately 415 nm during de-excitation.

This light is collected by the PMTs, where it is converted to photoelectrons. These photoelectrons

are accelerated and multiplied through a series of dynodes, creating an electrical signal whose

magnitude is proportional to the incident radiation energy [196]. The signals from the PMTs are

amplified by a pre-amplifier and processed by XIA Pixie-16 digitizers. In the Pixie-16 modules the

analog electronic signal from the PMTs is converted to a digital representation through an analog to

digital converter (ADC). The digitizers are configured, read out and analyzed using the Digital Data

Acquisition System (DDAS) [197], a lab-supported framework built around the Pixie-16 digitizers.

The FRIBDAQ software suite manages the data flow and sets up the analysis pipeline, transforming

the raw data from its hexadecimal format into physically meaningful parameters, such as energy

and time spectra [194].

4.3.1 Summing Technique

The large angular coverage and high detection efficiency of the detector allows for the application

of the 𝛾 -summing technique [198, 193]. In this technique, the spectra obtained by the individual

crystals (segments) provide sensitivity to the individual 𝛾 -ray transitions, whereas the full energy

deposited in SuN provides sensitivity to the populated excitation energies. More specifically, there

are three main types of spectra used in this analysis: the Sum of Segments (SoS), the Total Absorption

Spectrum (TAS), and multiplicity spectrum. The SoS corresponds to the sum of the energy spectra

recorded by each one of the optically isolated segments. This contains all the individual 𝛾-ray

transitions that occurred within the compound nucleus and got detected by SuN. TAS includes the

57

energy deposited in all segments added up on an event-by-event basis. This represents to the full

energy deposited inside the detector and reflects the populated excitation energy in the compound

nucleus. Finally, the multiplicity spectrum indicates how many segments of SuN recorded energy

in each event, which is indicative of the 𝛾-ray multiplicity in a 𝛾 cascade.

An example of the summing technique using the spectra of a 60Co calibration source is shown

in Fig. 4.6. The 60Co nucleus populates excited states in the 60Ni compound nucleus by 𝛽− decay.

(a)

(c)

(b)

(d)

Figure 4.6 SuN spectra from a 60Co nucleus: (a) decay scheme of 60Co, (b) SoS spectra showing
the two characteristic 𝛾 rays, (c) TAS with a sum peak at 2.5MeV and (d) multiplicity spectrum
showing the majority of cascades to have multiplicity two.

58

In most cases (99.88%), the 𝛽− decay populates a state of 𝐸 𝑋 = 2.505 MeV, that will subsequently

decay by the emission of two 𝛾 rays, of energies 1.173 MeV and 1.332 MeV. A small fraction

(0.12%) of the decays populate the 1.332 MeV state and therefore only one 𝛾 ray is emitted. The

SoS spectra (Fig. 4.6b) contain both the 1.173 MeV and 1.332 MeV, with a slightly higher intensity

on the 1.332 MeV 𝛾 ray, as well as a small peak at 2.505 MeV, in case both 𝛾 rays are recorded by the

same segment. The dominant feature of the TAS spectrum (Fig. 4.6c) is a sum peak at 2.505 MeV

corresponding to the excitation energy of the 60Ni compound nucleus, and small peaks at 1.173

MeV and 1.332 MeV, corresponding to the 0.12% chance of populating the 1.332 MeV, as well

as the few instances of incomplete summation. The multiplicity spectrum (Fig. 4.6d) shows the

majority of events with multiplicity 2, and the average multiplicity is 2.06. Events with multiplicity

higher than 2 correspond to scattered 𝛾 rays deposited their energy to more than one segment.

Building on the simplistic example of the 60Co decay, Fig. 4.7 shows the application of the

summing technique in a more complex scenario, such as a capture reaction experiment. The sum

Figure 4.7 Illustration of the energetics of a 𝐴 𝑋 ( 𝑝, 𝛾) 𝐴𝑌 reaction with the summing technique. The
compound nucleus 𝐴𝑌 is populated at an excitation energy 𝐸 𝑋 = 𝐸CMS + 𝑄 and a sum peak forms
at the TAS spectrum at energy 𝐸 𝑋.

59

peak forms in the TAS spectrum at energy 𝐸 𝑋 = 𝐸CMS + 𝑄, where 𝐸CMS is the center-of-mass

energy and 𝑄 the reaction 𝑄 value. As will be discussed in Sec. 5.5, the efficiency of the sum peak

depends on the multiplicity of the cascade [193], and therefore it is important to take all three types

of spectra into consideration when applying the summing technique.

4.4 The SuNSCREEN Detector

In the energy region of interest for these measurements, the main source of background comes

from cosmic rays, which pose a significant challenge when trying to measure very small cross

sections on the order of millibarns. To address this issue and increase the sensitivity of the

SuN detector, the Scintillating Cosmic Ray Eliminating ENsemble (SuNSCREEN) [199] was

positioned above SuN, as shown in Fig. 4.1 and 4.5. SuNSCREEN is a plastic scintillator detector

array comprised of nine bars, each with two PMTs, forming a roof-like arrangement above the SuN

detector. To reduce the cosmic-ray induced background, SuNSCREEN was used as a veto detector.

For this reason, all events that recorded signals in both PMTs of a SuNSCREEN bar, and at least

one segment of SuN were rejected from the SuN spectra.

During the SuNSCREEN’s commissioning this method was shown to reduce the cosmic-ray

background contributions by up to a factor of four in the SoS spectra and a factor of two in TAS

[199]. Spectra of the background radiation with and without the SuNSCREEN veto gate applied

are shown in Fig. 4.8, where in the region around 10 MeV, which is the relevant region for the

present work, a background reduction of a factor of three is observed in both SoS spectra and TAS.

The spectra also show characteristic background peaks at 1461 keV and 2614 keV from natural

radiation, as well as a small peak at 6.8 MeV in the TAS from neutron capture on the NaI crystal via

the 127I(𝑛, 𝛾)128I reaction with Q-value of 6.8 MeV. Additional background reduction is achieved

by taking advantage of the time structure of the beam as discussed in Sec. 5.3.4.

60

(a)

(b)

Figure 4.8 Typical room background (a) sum of segments and (b) total absorption spectra shown
in red dashed line. The same spectra are shown with black solid line after SuNSCREEN veto
rejection. The lower panels show the ratio before and after rejection.

61

CHAPTER 5

ANALYSIS

As discussed in Ch. 2, the cross section formula used in this analysis is given by Eq. 2.10

𝜎 =

𝑌
𝐼𝑎 𝑁𝑡 𝜖

(5.1)

where 𝑌 the experimental yield, meaning the number of reactions recorded by the detector, 𝐼𝑎 the

total number of beam particles, 𝑁𝑡 the number of target nuclei and 𝜖 the detection efficiency.

This chapter focuses on calculating the cross sections for the 82Kr(𝑝, 𝛾)83Rb and 73As(𝑝, 𝛾)74Se

reactions. Details on the calculation of each parameter, along with the associated uncertainties are

discussed, and the resulting cross section values are presented in Ch. 6.

5.1 Effective Energy

Before calculating the individual parameters of Eq. 5.1, it is useful to determine the center-

of-mass energy at which the cross section is measured. In thin target experiments, where energy

loss through the target is minimal, it is common to assume that the reaction energy corresponds to

the one in the middle of the target. However, for the 4-cm-long gas cell used in this experiment,

the effective center-of-mass energy 𝐸eff needed to be calculated [15]. The effective energy 𝐸eff,

represents the beam energy within the target at which half of the total yield is produced.

For non-resonant reactions the astrophysical 𝑆 factor (Sec. 2.5.4) can be considered nearly

constant over a relatively small energy interval of the target thickness Δ. This can be utilized to

calculate the 𝐸eff from the integral of the cross section from Eq. 2.36 over the target thickness Δ as

∫ 𝐸0

𝐸0−Δ𝐸

1
𝐸

exp(−2𝜋𝜂)𝑑𝐸 = 2

∫ 𝐸0

𝐸eff

1
𝐸

exp(−2𝜋𝜂)𝑑𝐸

(5.2)

where 𝐸0 is the incident projectile energy and Δ𝐸 is the energy loss within the target. Assuming

the cross section decreases linearly between 𝜎1 at 𝐸0 and 𝜎2 at 𝐸0 − Δ𝐸, the effective energy 𝐸eff

can be calculated from Eq. 5.2 as

𝐸𝑒 𝑓 𝑓 = 𝐸0 − Δ𝐸 + Δ𝐸

−

𝜎2
𝜎1 − 𝜎2

+

(cid:34) 𝜎2
1 + 𝜎2
2
2(𝜎1 − 𝜎2)2





(cid:35) 1/2



62

(5.3)

which is a good approximation for 𝜎1/𝜎2 < 10 [15]. The values of 𝜎1 and 𝜎2 are obtained from

Non-Smoker [174], and their ratios are 1.6, 1.7, and 2.0 for initial 82Kr beam energy of 3.7, 3.4,

and 3.1 MeV/u, and 1.4 and 2.0 for the 73As beam at 3.7 and 3.1 MeV/u.

5.2 Beam Particle Number

The design of the experimental setup includes an isolating flange upstream of SuN, and a plastic

feedthrough in the end of the beamline for the gas supply. This way the beamline is electrically

isolated from the rest of the setup and is used as a Faraday cup. The beam current in the 82Kr

experiment was calculated from the ammeter measurement of the beamline. Unfortunately, during

the 73As experiment the isolation was not successful and as shown in Fig. 5.1 a clear charge and

discharge of the beamline can be seen in the measured current in the order of tens of fA.

Figure 5.1 The 73As beam current measured from the ammeter connected to the improperly isolated
beamline as a function of time. The blue highlighted areas correspond to data acquisition times.
The small frequent drops correspond to upstream Faraday cup measurement where no beam is
present. As the baseline varies by tens of fA, whereas the beam current was only a few fA, this
measurement cannot be used for analysis.

As the deposited beam current was in the order of a couple fA, the measured current from the

dowstream ammeter cannot be used for this analysis. Instead, throughout the experiment, Faraday

cup measurements were taken every 20 minutes that interrupted the beam momentarily, as can be

seen from the frequent dips in Fig. 5.1 that reflect the position of the baseline during that time.

63

The beam current calculations are shown in Fig.5.2. Every 20 minutes a Faraday cup interrupted

the beam upstream of the SuN detector, and the current was measured off of that cup. The top

Figure 5.2 Beam current calculation for the 73As experiment. The top panel displays the upstream
Faraday cup measurements taken every 20 minutes, with the inset zooming in on one minute of
data and identifying the plateau region used for the calculation. The bottom panel highlights the
integrated regions, with shaded areas indicating data acquisition periods.

panel shows the current recorded by that collimator, were each peak corresponds to a current

measurement. The beam intensity was considered at the plateau of the peak, shown in the inset.

The baseline was fitted with a linear fit, reflected by the dashed line. The blue solid line in the

bottom panel corresponds to the subtracted baseline from the peaks. The total deposited beam

64

charge, 𝐶tot, was calculated using square integrals between consecutive peaks, shown by the shaded

region in the bottom panel.

The total number of beam particles, 𝐼𝛼, was calculated as

𝐼𝛼 =

𝐶tot
𝑄beam · 𝑞𝑒

(5.4)

were 𝐶tot is the integral of the beam current, 𝑄beam is the charge state of the beam (23 for 73As and

27 for 82Kr), and 𝑞𝑒 the electron charge.

5.3 Experimental Yield

5.3.1 SuN Gainmatching & Energy Calibration

The energy spectra from the SuN detector, acquired using the 12-bit digitizers discussed in

Sec. 4.3, are expressed in 212 = 4096 ADC channels. Each channel corresponds to a specific

voltage height of a PMT output signal. The first step in analyzing these spectra is to ensure

that all 24 PMTs respond consistently to 𝛾 rays of the same energy. This procedure is known as

gainmatching.

To eliminate position dependence of the PMTs within each crystal, gainmatching is typically

performed with 𝛾 rays from natural background radiation, such as 40K. Approximately 10% of the

𝛽− decays of 40K populate an excited state of 40Ar, resulting in the emission of a characteristic 1461

keV 𝛾 ray.

Gainmatching for the SuN PMTs is conducted in two stages. The first stage, hardware gain-

matching, involves adjusting the voltages applied to each PMT before the experiment so that the

1461 keV 𝛾 ray from 40K appears in approximately the same ADC channel in the recorded spectrum.

The second stage, software gainmatching, fine tunes the PMT gains by normalizing the obtained

spectra so the 40K peak appears in the exact same channel for each PMT. Fig. 5.3 shows the spectra

from all PMTs in the 1461 keV region before software gainmatching during the 73As experimemt.

Gaussian fits on a linear background were applied to locate the peak centroids, and the resulting

gainmatching factors, listed in Table 5.1, were calculated to align the peaks to the same ADC

channel, as shown in Fig. 5.4.

65

Figure 5.3 Background spectra of all 24 SuN PMTs before gainmatching in the region near the 1461
keV 40K peak during the 73As experiment. Each panel represents one detector segment, showing
spectra from its three associated PMTs, with black, red, and blue lines corresponding to PMTs 1,
2, and 3, respectively.

Figure 5.4 Same as Fig. 5.3 after applying the gainmatching factors from Table 5.1.

After gainmatching, all PMTs produce consistent spectra, however, the spectra remain in arbi-

trary ADC units. To find the correlation between ADC units and energy, an energy calibration is

performed using sources that emit characteristic 𝛾 rays of known energy. The 𝛾 rays used were the

59.5 keV 241Am peak, the 661.7 keV 137Cs peak, and the 1173.2 keV and 1332.5 keV peaks from

60Co. Additionally, 𝛾 rays from a 228Th source were used, specifically the 238.6 keV peak from

the 212Pb daughter nucleus and the 583.2 keV and 2614.5 keV peaks from 208Tl. The resulting

66

calibrations are shown in Fig. 5.5 and the fit parameters are listed in Table 5.1.

Table 5.1 The gainmatching and calibration factors for SuN for the 73As experiment.

Gainmatching Factors

PMT Factor
1.0075
B11
0.9997
B12
0.9929
B13
1.0005
B21
1.0000
B22
1.0022
B23
1.0116
B31
0.9903
B32
0.9812
B33
1.0018
B41
0.9892
B42
0.9825
B43

PMT Factor
0.9927
T11
1.0141
T12
1.0135
T13
0.9948
T21
0.9873
T22
1.0141
T23
0.9863
T31
1.0003
T32
0.9962
T33
1.0166
T41
1.0059
T42
1.0136
T43

Calibration Factors

Segment
B1
B2
B3
B4
T1
T2
T3
T4

Scale
0.1815
0.1839
0.1801
0.1817
0.1812
0.1837
0.1801
0.1824

Intercept
-28.0864
-29.7079
-27.8599
-29.5897
-27.6117
-26.6570
-26.8405
-27.9716

Figure 5.5 SuN calibration fits. Each panel corresponds to one segment of SuN.

67

5.3.2 The Sum Peak

As discussed in Sec. 4.3, in the summing technique the experimental yield 𝑌 is calculated from

the integral of the sum peak, that forms in the TAS spectrum at energy 𝐸 𝑋 = 𝐸CMS + 𝑄, where

𝐸CMS is the center-of-mass energy and 𝑄 the reaction 𝑄 value.

The summing technique has been successfully applied to a plethora of (𝑝, 𝛾) and (𝛼, 𝛾) reaction

measurements on stable nuclei [198, 200, 201, 202, 203]. These experiments are typically conducted

in regular kinematics, where a 𝑝 or 𝛼 beam impinges on a heavy, solid, stable target. For radioactive

isotopes, however, it becomes necessary to transition to inverse kinematics, as constructing targets

from exotic isotopes with short half-lives is highly challenging.

Inverse kinematics is a widely used approach at many facilities for 𝛾-process measurements

using mass separators [204] and storage rings [166], but there has so far been only one experiment

with a radioactive beam [173]. The summing technique has been successfully applied in inverse

kinematics using a solid target [205]. However, as the beam intensity in radioactive beams decreases

by orders of magnitude compared to stable beams, the introduction of a gas target was necessary to

increase the purity of the target, and allow for more efficient measurements.

Transitioning to inverse kinematics and the introduction of a gas target increases the complexity

of this method. Firstly, due to the experiment being conducted in inverse kinematics, the recoil

nucleus has significant momentum and continues its path along with the unreacted beam, rather

than remaining stationary in the target. Therefore the 𝛾 rays are emitted from a moving source,

and Doppler corrections need to be applied for the detected 𝛾-ray energy [206]. Additionally,

the beam’s passage through the target entrance window introduces energy straggling, resulting in

a range of incident beam energies. This, in turn, populates the compound nucleus at a range of

excitation energies, causing a significant widening of the resulting sum peak.

5.3.3 Doppler-Shift Corrections

The Doppler effect is the change in frequency of a wave emitted by a moving source relative to

a stationary observer, compared to the frequency that would be measured if the source were at rest.

For a relativistic moving particle, such as light, the observed energy will be shifted according to

68

the velocity of the moving source. Therefore the energy of the 𝛾 rays emitted by the moving recoil

nucleus are shifted by

𝐸0 =

1 − 𝛽 cos 𝜃
√︁1 − 𝛽2

𝐸

(5.5)

where 𝐸0 the 𝛾-ray energy emitted by the source, 𝐸 is the detected 𝛾-ray energy by the stationary

observer, 𝛽 = 𝑣/𝑐 the recoil relative velocity and 𝜃 the relative angle between the recoil and the

detector. Assuming the decay of the moving recoil in the center of the detector, each SuN segment

has a different angle corresponding to the center of the segment, as shown in Table 5.2. Table 5.3

shows the different beam energy for the 73As and 82Kr beam in the laboratory frame, the center-of-

mass energy at the center of the target accounting for the energy loss through the entrance window

and the recoil relative velocity 𝛽.

Table 5.2 SuN segment angles
from Ref. [194].

Segment Angle (deg)

1
2
3
4

2.550
2.024
1.118
0.592

Table 5.3 82Kr and 73As beam energies and relativistic ve-
locities.

Beam

82Kr

73As

Energy Lab CoM energy in middle

(MeV/u)
3.7
3.4
3.1
3.7
3.1

of target (MeV/u)
2.98
2.67
2.37
2.95
2.31

𝛽 = 𝑣/𝑐

0.079
0.075
0.071
0.079
0.070

Once all acquired spectra were corrected on a segment-by-segment basis, they were summed to

form a Doppler-shift corrected sum peak.

5.3.4 Background Subtraction

There were two main types of background contributions that could interfere with the sum peak in

the energy region of interest: cosmic-ray background and beam-induced background. Minimizing

those background contributions in the region where the sum peak was expected was important for

the accurate determination of the experimental yield.

The cosmic-ray background, as discussed in Sec. 4.4, was significantly reduced using the

SuNSCREEN veto detector. Any remainder cosmic-ray background contributions, were removed

by utilizing the pulsed structure of the beam. The beam was delivered in 80 𝜇s pulses every 200

69

ms. While data was recorded continuously, two distinct time gates were applied during processing.

The first gate corresponded to the 80 𝜇s beam-on intervals, triggered by a signal from the EBIT

charge breeder. The second gate, applied 100 ms later, captured 800 𝜇s of background data between

pulses. An illustration of this structure is shown in Fig. 5.6. The background data were scaled

Figure 5.6 Illustration of the pulsed-beam structure. Blue boxes correspond to 80 𝜇s beam pulses
occurring every 200 ms, while grey boxes reflect the 100 ms time-shifted gate used to record 800
𝜇s of background between beam pulses. The time axis is not to scale.

down by a factor of ten to account for the shorter recorded time and subtracted from the beam-on

data to remove room-background contributions.

The last form of background originates from the beam and regards to any beam-induced reactions

other than the reaction of interest. This includes beam scattering on any beamline components, or

any interaction with the gas cell and its entrance and exit windows. As discussed in Sec. 4.2 high 𝑍

materials are used to cover all parts of the cell the beam may interact with. Unfortunately, during the

73As experiment, of the two gas cell targets used, one had significant scattering background caused

by epoxy glue residue that had seeped onto the entrance window. Therefore, the data acquired with

that gas target could not be analyzed, as the sum peak was not visible above the background.

Removing beam-induced background contributions requires isolating them from the data cor-

responding to the reaction of interest. For this purpose data are acquired while the gas cell is full

of hydrogen gas, as well as while the cell is empty. The empty cell data are scaled based on the

70

ratio of beam current during full cell and empty cell runs and subtracted from the full cell data.

Fig.5.7 shows the Doppler corrected TAS spectrum for the 3.7 MeV/u 82Kr background subtracted

sum peak, after subtracting cosmic-ray and room background. The black line corresponds to the

full cell data and the red line is the empty cell data scaled based on the total deposited beam current

of the full and empty cell runs. Subtracting the two gives the sum peak shown in the blue line. The

blue band corresponds to the statistical uncertainty from the background subtraction.

Figure 5.7 Doppler-corrected TAS spectra showing the background subtraction for the sum peak
for the 82Kr(𝑝, 𝛾)83Rb reaction at the initial beam energy 3.7 MeV/nucleon. The black histogram
corresponds to the gas cell filled with hydrogen gas, the red histogram corresponds to the empty
gas cell scaled to the beam current, and the blue histogram is the fully subtracted sum peak that
was used for the remaining analysis. The blue band corresponds to the statistical uncertainty of the
background subtraction. (Figure adapted with permission from Tsantiri et al., Phys. Rev. C 107,
035808 2023. Copyright 2023 by the American Physical Society)

The low energy region of the TAS spectrum shows a number of peaks from scattering of the

beam on the molybdenum window. This serves as a useful tool to identify if the position of the

incoming beam remains the same between full and empty cell runs, as the two spectra should

overlap after scaling on the beam current. Similarly in the energy region higher than the sum peak

the spectra should overlap as well. In the case of the 73As experiment, scaling on the beam current

did not result in overlapping scattering peaks or high energies, indicating that the position of the

71

beam had changed during data acquisition, as shown in Fig. 5.8.

For this reason, instead of

Figure 5.8 Doppler-corrected TAS spectra for the 73As(𝑝, 𝛾)74Se reaction at the initial beam energy
3.7 MeV/nucleon. The black histogram corresponds to the gas cell filled with hydrogen gas, the
magenta histogram corresponds to the empty gas cell, and the blue to the empty gas cell scaled
to the beam current. The disagreement of the full cell and scaled empty cell spectra indicates the
beam position on the cell has shifted.

scaling on the beam current, the empty cell data were scaled based on the high energy region of

the spectrum. Specifically, for energies between 13 and 18 MeV, where no contribution of the sum

peak is expected, more than 1000 integrals over different energy ranges and regions were sampled.

This created a distribution of scaling factors as shown in Fig. 5.9. The background subtraction was

performed using the mean scaling factor, and the deviation of the distribution was introduced as an

additional source of uncertainty for the remainder of this analysis.

Figure 5.10(a) shows the background subtraction for the 3.7 MeV/u 73As beam. The challenge

of running a radioactive beam experiment is apparent from the significantly lower statistics and

large statistical uncertainties compared to the stable beam experiment. For the 3.1 MeV/u sum peak

there was an additional challenge faced. Unfortunately, a few minutes after the beginning of full

cell data acquisition in the 3.1 MeV/u beam energy the 2-𝜇m thin molybdenum entrance window

ruptured, allowing only for 30-minutes-worth of data acquisition. Regardless of this unfortunate

72

(a)

(b)

Figure 5.9 Sampling of various high energy regions for empty cell data scaling. Figure (a) shows the
full cell data spectrum with black, the various empty cell scaled data with green and the resulting
background subtracted sum peaks with blue, where each line corresponds to a different scaling
factor. Figure (b) reflects the distribution of scaling factors. More details in text.

event, the data were deemed worth analyzing, as this reaction has never been measured before, and

any new information that may be provided is important. The significantly low statistics accumulated

result in very large statistical uncertainty, as shown by the blue error band in the resulting sum peak

of Fig. 5.10(b). The uncertainty quantification is discussed in detail in Sec. 5.7.

(a)

(b)

Figure 5.10 Same as Fig. 5.7 for the 73As(𝑝, 𝛾)74Se reaction at the initial beam energy of (a) 3.7
MeV/nucleon and (b) 3.1 MeV/nucleon. The blue bands correspond to statistical uncertainty. The
uncertainty introduced by the empty cell scaling methodology is not included here.

73

5.4 Target Particle Density

The next parameter in Eq. 5.1, the target particle density 𝑁𝑡, is calculated based on the average

target pressure. The pressure was recorded throughout the experiment and it can be seen for the

73As experiment in Fig. 5.11.

Figure 5.11 The gas cell pressure recorded during the 73As experiment. The highlighted regions
correspond to the full cell and empty cell runs of the 3.7 and 3.1 MeV/u beam energies.

From the target density, the effective target thickness can be calculated using LISE++ [207], an

FRIB software used primarily for beam production and transmission calculations. It features many

useful calculators, including a physics calculator used in this analysis that calculates effective target

thickness of gas target based on the target material, pressure and width.

The target particle density 𝑁𝑡, can be calculated through:

where 𝑡 is the target thickness in grams/cm3 from LISE++, 𝑁 𝐴 is the Avogadro’s number, and 𝑚𝐻

𝑁𝑡 =

𝑁 𝐴 · 𝑡
𝑚𝐻

(5.6)

the hydrogen mass.

5.5 Detection Efficiency

The final term to be calculated in Eq.5.1 is the detector efficiency, 𝜖, which represents the

fraction of particles recorded relative to the total number emitted. For the SuN detector, a 4𝜋

74

calorimeter, the geometric efficiency is nearly 100% from its design. As a result, the detection

efficiency is primarily governed by the intrinsic efficiency, defined as the ratio of detected particles

to incident particles. Since uncharged radiations such as 𝛾 rays can travel large distances before

interacting, or may interact with materials other than the scintillator, such as the casing, scintillating

detectors are typically less than 100% efficient [196].

The intrinsic efficiency varies with energy, 𝜖 (𝐸), and for a total absorption spectrometer like

SuN, it also depends on the multiplicity of the cascade [193]. For example, the efficiency of

detecting a single 𝛾 ray of 𝐸𝛾 = 10 MeV is higher than detecting two 𝛾 rays of 5 MeV each, emitted

from a state of 𝐸𝑥 = 10 MeV. Beyond this multiplicity dependence, regular kinematics experiments

are characterized by narrow sum peaks, which allow the efficiency to be approximated for a single

excitation energy. However, in this work, the sum peak is significantly broader, as discussed in

Sec.5.3.2. The Doppler shift corrections were applied assuming the angle at the center of each

segment, which is an approximation given the large angular coverage of each segment. Beyond any

incomplete Doppler broadening corrections, substantial energy straggling also occurs as the beam

passes through the target window foil. This results in sum peak widths around 2 MeV, as shown

in Fig.5.7 and 5.10. Consequently, the efficiency must be calculated as a function of all energies

contributing to the sum peak, while also accounting for the multiplicity of the cascades. For this

reason, the detection efficiency is determined through a series of simulations, as outlined in the

following paragraphs.

5.5.1 Rainier Simulations

The first step in calculating the detection efficiency is to simulate the 𝛾-ray deexcitation of the

compound nucleus for all possible excitation energies populated during the reactions. The 𝑄 values

for the reactions of interest are 8.55 MeV for 73As(𝑝, 𝛾)74Se and 5.77 MeV for 82Kr(𝑝, 𝛾)83Rb.

These values are sufficiently high for the excitation energy 𝐸𝑥 of the compound nucleus to be in the

nuclear continuum region, where statistical model calculations are applicable (see Sec. 2.4.2.2). To

simulate the 𝛾-ray deexcitation of the compound nucleus, the Rainier code [208] was used.

Rainier is a Monte Carlo code that simulates the deexcitation of a compound nucleus using

75

statistical nuclear properties. For these simulations, the nuclear level structure of the compound

nucleus is provided as input. The low-energy level schemes for 74Se and 83Rb were taken from

Ref. [79], up to approximately 3.3 MeV and 1.8 MeV, respectively, where the level schemes are

considered complete. The upper portion of the level scheme can be constructed using the analytical

nuclear level density (NLD) models described in Sec. 2.4.2.3, namely the constant temperature

(CT) model [31] and the back-shifted Fermi gas (BSFG) model [25].

Rainier also requires a description of the 𝐸𝑥 and 𝐽𝜋 of the entry state. The values of 𝐸𝑥 were

considered throughout the range that the experimental sum peak extends. For example in the 3.7

MeV/u beam energy, this corresponds to energies between 9.8 and 11.8 MeV for 74Se, and between

7.0 and 8.8 MeV for 83Rb. For the 𝐽𝜋 of the state, 𝑠-wave proton capture (1/2+) on the ground state

of 82Kr (0+) was considered, while for the 74Se compound nucleus, higher order corrections were

needed. The 𝐽𝜋 population was obtained from Talys (Sec. 2.4.2.5) by enabling the outdecay and

outpopulation options, that output detailed information of the population and statistical decay

of the compound nucleus to all possible states. The 𝐽𝜋 distribution used is shown in Fig. 5.12.

Figure 5.12 The 𝐽𝜋 distribution of populated excitation energies in the 74Se compound nucleus,
calculated using Talys .

Once the level scheme is built and the entry state defined, the deexcitation of the nucleus is

governed by the 𝛾-ray strength function (𝛾SF ), where a generalized Lorentzian of the form of

76

Kopecky and Uhl [45] (Sec. 2.4.2.4) was adopted. As Rainier is a Monte Carlo code, for each

excitation energy component, 10 or 20 realizations of 1000 cascades were calculated for the 74Se

and 83Rb compound nucleus, respectively.

5.5.2 Geant4 Simulations

The 𝛾 rays obtained by the deexcitation of each contributing 𝐸𝑥 of the compound nucleus

through Rainier were then input in Geant4 simulations [209] to account for the detector’s response

function. An example of TAS, SoS and multiplicity spectra obtained from Geant4 for the decay a

83Rb compound nucleus at an 𝐸𝑥 = 8 MeV is shown in Fig. 5.13.

(a)

(b)

Figure 5.13 Simulated (a) TAS, (b) SoS and (c) multiplicity spectra of a 𝐸𝑥 = 8 MeV 83Rb state
decay using Geant4.

(c)

77

5.5.3 Chi-Square Minimization

The last step is to determine the contribution of each possible excitation energy in the exper-

imental spectra. For this purpose, a 𝜒2 minimization algorithm was implemented [210]. The 𝜒2

code uses the simulated TAS, SoS, and multiplicity spectra, along with the experimental spectra

gated on the sum peak, to minimize the following global 𝜒2 value

𝜒2
global =

∑︁

∑︁

𝑖

𝑗

(cid:169)
(cid:173)
(cid:173)
(cid:171)

exp − (cid:205)𝑘 𝑓𝑘𝐶𝑖 𝑗
𝐶𝑖 𝑗
sim
𝐶𝑖 𝑗
exp

√︃

2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(5.7)

where the summations are over 𝑖 types of histograms (TAS, SoS and multiplicity) and 𝑗 number of

bins in the 𝑖-th histogram. 𝐶𝑖 𝑗

exp and 𝐶𝑖 𝑗
sim

are the counts of the 𝑗-th bin in the 𝑖-th experimental or

simulated histogram respectively [210]. The simulated histograms are summed over the 𝑘 different

components contributing in the spectra, with 𝑓𝑘 their respective scaling factor. The code utilizes

the Minuit algorithm [211] from the Root data analysis toolkit [212], to assign values to the 𝑓𝑘

factors until the minimum global 𝜒2 is achieved.

Eq. 5.7 assumes that the error in 𝐶𝑖 𝑗

exp is equal to its square root, which is a valid assumption for

spectra that do not carry uncertainty from subtractions. However, in the case of 73As, the error was

dominated by statistical uncertainty from the background subtractions (Sec. 5.3.4). Therefore, for

the 73As analysis, the denominator in Eq. 5.7 was replaced with the statistical uncertainty shown in

Fig. 5.10 for each bin.

The detection efficiency is inherently accounted for in the Geant4 simulations, as the detector’s

material and geometry are included in the model. Therefore, from the 𝜒2 minimization output,

the efficiency can be extracted in the form of a ratio 𝑌 /𝜖, where 𝑌 is the experimental yield. In

more detail, based on Eq. 5.7, the 𝜒2 code calculates scaling factors to match the integral of the

sum peak with a weighted linear combination of the simulated spectra. For example, consider a

sum peak with 100 counts, where two energies contribute to the peak with weights of 60% and

40%. For 10 000 simulated events at each energy, the scaling factors would be 0.006 and 0.004

respectively, so that 0.006 · 10 000 + 0.004 · 10 000 = 100 counts. However, if the detector had

a 50% intrinsic efficiency, only 5 000 of the 10 000 simulated events would contribute to the sum

78

peak.

In that case, the factors would be 0.012 (= 0.006/0.5) and 0.008 (= 0.004/0.5), so that

(0.012 · 10 000 + 0.008 · 10 000) × 0.5 = 100 counts.

Based on the example, the efficiency-corrected yield 𝑌 /𝜖, can be expressed as the product of

the sum of all scaling factors and the number of simulated events as

𝑌
𝜖

∑︁

=

𝑘

𝑓𝑘 · 𝑁sim

(5.8)

where 𝑓𝑘 represents the scaling factor of the 𝑘-th energy contributing to the peak as calculated

through Eq. 5.7, and 𝑁sim the number of simulated events for each energy. T

The number of 𝐸𝑥 components simulated for each case reflects the resolution of the sum peak.

For 83Rb where the sum peak has significant statistics, ∼20 energy components were simulated,

one every 100 keV. For the case of 74Se, where the statistics are small and large energy binning was

required (Fig. 5.10), only 5 energy components were simulated, one for every 400 keV. Simulating

more than 5 components, accounting for the statistical uncertainty of the spectra, resulted in the

minimizer not properly converging.

The segmentation of SuN allows to extract valuable information on the statistical properties

of the compound nucleus. Specifically, the choice of NLD and 𝛾SF model parameters that are

input in Rainier, significantly affects the 𝛾 rays that can be emitted through the deexcitation of

a nuclear level in the continuum. This dependence is mostly apparent in the shape of the SoS

spectra, and as discussed in detail in the next section, it allowed for the development of an analysis

method to constrain the products of NLD and 𝛾SF used in statistical model calculations, and provide

predictions for the cross section in a much larger energy range than the experimentally measured.

5.6 Theoretical Investigation with Rainier and Talys

The shape of the SoS spectrum depends on the product of the NLD and 𝛾SF . The theoretical

models for the NLD (Sec. 2.4.2.3) and 𝛾SF (Se. 2.4.2.4) include multiple adjustable parameters. The

default parameters for the constant temperature (CT) and back-shifted Fermi gas (BSFG) models

for the NLD, as well as the Generalized Lorentzian for the 𝛾SF found in literature [79, 213, 62],

do not appear to reproduce the shape of the SoS spectra, as shown in Fig.5.14.
In particular, the
simulations overestimate high-energy 𝛾 rays while, in the case of 83Rb, underestimating low-energy

79

(a)

(b)

Figure 5.14 The 𝜒2 minimization SoS fits for the (a) 82Kr(𝑝, 𝛾)83Rb and (b) 73As(𝑝, 𝛾)74Se reactions
at initial beam energy 3.7 MeV/u. The black lines are the experimental spectra and the red and blue
lines correspond to the default initial parameters of the BSFG and CT model NLD from Ref. [213]
and 𝛾 SF from Ref. [45, 62]. (Figure (a) adapted with permission from Tsantiri et al., Phys. Rev.
C 107, 035808 2023. Copyright 2023 by the American Physical Society)

ones. These discrepancies could stem from several factors. One possibility is that, for the same

𝛾SF , a higher NLD is needed to reproduce the spectra. In that case, instead of emitting fewer

high-energy 𝛾 rays, the compound nucleus would emit multiple lower-energy ones, de-exciting in

smaller steps. Alternatively, for the same NLD, the 𝛾SF may assign a lower probability to emitting

low-energy 𝛾 rays. An enhancement in the lower-energy region of the 𝛾SF could then resolve the

discrepancy, especially in the case of 83Rb.

Since this method is sensitive only to the product of the NLD and 𝛾SF , rather than their

individual values, this analysis focuses on identifying suitable combinations of these two quantities

that reproduce the experimental spectra, and not absolute values for the two quantities independently.

5.6.1 Constraining the Statistical Properties of 83Rb

For the 82Kr analysis, five NLD and 𝛾SF combinations were identified as listed in Table 5.4.

The first two rows show the literature values for default parameters shown in Fig. 5.14(a).

In

the first three combinations, the NLD is modified using different parameterizations of the CT and

BSFG models (Eqs. 2.17 and 2.15), while the 𝛾SF follows the Generalized Lorentzian model of

Kopecky and Uhl [45]. Specifically combination number 2 includes CT parameters obtained from

80

Ref. [214]. In the last two combinations the NLD is considered to follow the default BSFG model

parameters, while an upbend in the M1 strength function was implemented, following Eq. 2.20.

In combination number 4 the upbend follows data from Ref. [215]. Simulations using those five

different combinations create the bands shown in Fig. 5.15 for the TAS, SoS, and multiplicity

spectra.

Table 5.4 Parameters for modeling the NLD and 𝛾 SF of the 83Rb nucleus. The default parameters
in the first two rows are shown in Fig. 5.14(a) with a red and blue lines. The rest of the parameters
were chosen for the calculation of the ratio 𝑌 /𝜖, and form the band shown in Fig. 5.15. See text
for details on parameters. (Table with permission from Tsantiri et al., Phys. Rev. C 107, 035808
2023. Copyright 2023 by the American Physical Society)

NLD Model

CT default

BSFG default

1.

2.

3.

4.

5.

CT

CT

BSFG

BSFG

BSFG

NLD Model
Details

Upbend in 𝛾 SF

[213]

[213]

[214]

T = 0.824
𝐸0 = -1.16
𝛼 = 10.17
Δ = - 0.54
T = 0.824
𝐸0 = -2.2
T = 0.861
𝐸0 = -3.34
𝛼 = 10.17
Δ = -1.6
𝛼 = 10.17
Δ = -0.54
𝛼 = 10.17
Δ = -0.54

No

No

No

No

No

[215]

a = 1.5
c = 8.7 × 10−8
a = 1.0
c = 1.0 × 10−7

5.6.2 Constraining the Statistical Properties of 74Se

There are infinite possible combinations of NLD and 𝛾SF that produce the same transmission

coefficient (Eq.2.13). In the previous section, only five suitable combinations were identified. Given

the large number of free parameters in this problem, the 74Se analysis aimed not just to find a set

of possible solutions, but to systematically characterize the different model combinations available

in Talys . As seen in Fig. 3.9, the variations in NLD, 𝛾SF and pOMP result in a cross-section

uncertainty of a factor of six at a center-of-mass energy of approximately 3 MeV, which is the

81

(a)

(b)

(c)

Figure 5.15 The 𝜒2 minimization fits for the SoS (a), TAS (b) and multiplicity (c) for the
82Kr(p,𝛾 )83Rb reaction at an initial beam energy 3.7 MeV/nucleon. The black lines are the
experimental spectra and the light blue bands indicate the simulated spectra for the combinations
of the NLD and 𝛾 SF models listed in Table 5.4. In (a) the red and blue lines are the same as in
Fig. 5.14(a). (Figure adapted with permission from Tsantiri et al., Phys. Rev. C 107, 035808 2023.
Copyright 2023 by the American Physical Society)

energy region relevant for the production of 74Se in the 𝛾 process. Thus, aside from extracting the

detection efficiency, the goal of this analysis was to identify suitable NLD and 𝛾SF combinations

within the available Talys models, in an attempt to the constrain the cross section of the reaction

even further.

As shown in Tables 2.1 and 2.2, there are six models available to describe the NLD, nine for

the 𝐸1 SF, three for the 𝑀1, and an option to include an 𝑀1 upbend or not. This produces 324

82

possible combinations. Additionally, there two main models for the 𝑝OMP (Sec. 2.4.2.5), and

while fitting the experimental spectra is only sensitive to the NLD and 𝛾SF , these will be included

in reproducing the experimental cross section in the next chapter, resulting in 648 total possible

combinations. Talys is able to output the NLD and 𝛾SF used in each calculation, and therefore

those tables were directly input in Rainier.

The ability of each Talys model to reproduce the experimental data can be expressed by adapting

the concept of a likelihood function from Bayesian analysis [216] as

(cid:34)

𝑃(𝑌 |𝑚) ∝ exp

−

𝑁
∑︁

𝑗

(𝑦 𝑗 − 𝑓𝑚 (𝑥 𝑗 ))2
2𝜎2
𝑗

(cid:35)

(5.9)

where 𝑃(𝑌 /𝑚) represents the likelihood of model 𝑚 to reproduce the data 𝑌 , the summation goes

over the 𝑁 data-points 𝑦 𝑗 of the dataset 𝑌 , 𝜎𝑗 is the uncertainty of 𝑦 𝑗 , and 𝑓𝑚 (𝑥 𝑗 ) is the prediction

of model 𝑚 at value 𝑥 𝑗 . This is often expressed in terms of the log-likelihood log 𝑃(𝑌 |𝑚).

The log-likelihood was applied in this analysis as

log 𝑃(𝑌 |𝑚) ∝ −

𝑁 𝑏𝑖𝑛𝑠
∑︁

𝑗

(𝐶 𝑗

exp − (cid:205)𝑘 𝑓𝑘𝐶 𝑗

sim(𝑚))2

2𝜎2
𝑗

= −

𝜒2
2

(5.10)

where for each Talys model 𝑚, the three histograms 𝑌 (TAS, SoS, and multiplicity) were compared

with the simulated ones on a bin-by-bin basis. The values 𝐶 𝑗

exp were the experimental counts

per bin 𝑗 and 𝜎𝑗 was the corresponding statistical uncertainty from the background subtraction.

The simulated histograms, as described in Sec. 5.5.3, were a linear combination of all energy

components weighted by 𝑓𝑘 as calculated in Eq. 5.7. As each of the three types of histograms is

treated individually the first summation of Eq. 5.7 is omitted.

A few things should be considered in this analysis. Firstly, treating the TAS, SoS and multiplicity

spectra independently is an assumption. A more accurate approach would require including

correlations between them, but this is beyond the scope of this thesis. Additionally, the log-

likelihood obtained for each of the three types of spectra was normalized to the number of data

points (bins) in the respective spectrum, 𝑁bins. This normalization ensures that each type of

spectrum contributes equally to the overall likelihood, independently of how many bins it contains

83

[217]. The “score" derived by the modified log-likelihood for each type of spectrum is

Score = exp

(cid:19)

(cid:18) log 𝑃
𝑁bins

= exp

(cid:19)

(cid:18)

−

𝜒2
2𝑁bins

(5.11)

The simulated spectra obtained from the various Talys model combinations for initial beam energy

of 3.7 MeV/u, are shown in Fig. 5.16, color-coded based on their score, exp(log 𝑃/𝑁bins). For the

(a)

(b)

(c)

(d)

Figure 5.16 The 𝜒2 minimization fits for the TAS (a), multiplicity (b), and SoS (c) for the
73As(p,𝛾 )74Se reaction at an initial beam energy 3.7 MeV/nucleon. The black lines represent
the experimental spectra and the various green lines correspond to the simulated spectra for the
combinations of the NLD and 𝛾 SF models from Talys . The varying shades of green in each line
reflect different scores from Eq. 5.11. Darker tones represent higher scores, as shown by the color
bar (d).

3.1 MeV/u datapoint, due to the low statistics, the 𝑌 /𝜖 was extracted using the NLD and 𝛾SF model

84

Figure 5.17 Beam energy distributions calculated using Srim for 73As at an initial beam energy
of 3.7 MeV/u. The blue distribution represents the beam energy after the Mo entrance window,
and the red to energy at the end of the cell. The mean of each distribution was used to calculate
the effective energy, while the maximum and mean-3𝜎 values determined the upper and lower
uncertainty, respectively.

combination that produces the lowest 𝜒2 for the 3.7 MeV/u data. No further investigation was

performed for that energy.

5.7 Uncertainty Quantification

The following describes the uncertainties that contribute to the cross-section calculation for all

the quantities discussed in the previous sections.

The uncertainty in the effective energy 𝐸eff is mainly attributed to energy straggling as the beam

passes through the Mo foil and hydrogen gas. The beam’s energy distribution was calculated using

Srim[218]. Fig 5.17 shows an example distribution for 73As at an initial beam energy of 3.7 MeV/u.

The blue distribution represents the energy straggling after the 2-𝜇m-thick Mo entrance window,

while the red corresponds to the energy at the end of the 4-cm-long gas cell filled with 600 torr of

hydrogen. The mean of each distribution was used as 𝐸0 and 𝐸0 − Δ𝐸 in Eq. 5.3. The upper and

85

lower uncertainties in 𝐸eff were determined using the maximum and mean-3𝜎 of each distribution,

respectively. Due to the asymmetrical energy straggling distribution, which has a long low-energy

tail, the errors in the effective energy are similarly asymmetric. An additional 1% uncertainty was

included to reflect the uncertainty in the delivered beam by the facility.

The uncertainty in the total number of beam particles 𝐼𝛼 (Sec. 5.2) in the 82Kr measurement

was considered to be 5% from the beam-charge accumulation. For the 73As measurement in

addition to the 5% uncertainty for the ammeter measurement, there were two methods used for the

integration of the current shown in Fig. 5.2, introducing an additional 2% uncertainty, as well as

a 6% uncertainty from the baseline fits. Overall the uncertainty in the beam current for the 73As

measurement was 8.3%.

The target particle density 𝑁𝑡 was considered to carry an uncertainty of 5%. This includes

systematic uncertainty from the calibration of the manometer to the atmospheric pressure and the

zero offset, as well as the random error from the instrument resolution.

The uncertainty in the 𝑌 /𝜖 for the 82Kr measurement includes statistical uncertainty varying

between 1% and 4%, along with uncertainty from the various parameters chosen for the NLD and

𝛾SF models varying between 26% and 17%, with the latter value corresponding to the smaller

energy. For the 73As measurement the uncertainty is dominated by the statistical uncertainty from

the background subtraction varying between 18% and 72%, for the larger and smaller beam energies,

respectively. An additional 6% uncertainty was assumed for the 2𝜎 deviation of the scaling factors

distribution for the empty cell measurement (Fig. 5.9(b)). Finally the various Talys combinations

produced a range of 𝑌 /𝜖 values. In Fig. 5.18, for the 3.7 MeV/u measurement, the various 𝑌 /𝜖

values produced by all the different fits are shown compared to their total fitting score, defined

as the product of all three scores from Eq. 5.11. The higher scores that reproduce the best fits,

converge near 𝑌 /𝜖 ≈ 6000. By weighing each 𝑌 /𝜖 with the score of the corresponding model, the

distribution shown in Fig. 5.19 was obtained. An uncertainty of 13.5% was obtained from the 3𝜎

deviation of that distribution. The overall uncertainty for 𝑌 /𝜖 of the 73As measurement was 24%

and 74% for the 3.7 and 3.1 MeV/u beam energies.

86

Figure 5.18 The 𝑌 /𝜖 values produced using the various available Talys combinations of NLD ang
𝛾SF compared to the fitting score, defined as the product of all three exp(log 𝑃/𝑁bins) for the three
types of histograms (TAS, SoS and multiplicity). The fits with the highest score converge around
𝑌 /𝜖 ≈ 6000.

Figure 5.19 The distribution of 𝑌 /𝜖 values shown in Fig. 5.18 weighted by their respective score.
The value of 𝑌 /𝜖 used in the analysis was obtained from the best fit (dashed line), and the 3𝜎
uncertainty of the distribution was assumed as model uncertainty for this measurement (dot-dashed
line).

87

CHAPTER 6

RESULTS & DISCUSSION

6.1 The 82Kr(𝑝, 𝛾)83Rb cross section

The cross section for the 82Kr(𝑝, 𝛾)83Rb reaction as calculated using Eq. 5.1 is presented in

Table 6.1 along with all calculated quantities described in Ch. 5. In Fig. 6.1 the cross section is

shown along with statistical model calculations using the Non-Smoker code [174] and Talys , for

all available NLD and 𝛾SF models.

Figure 6.1 The measured cross section of the 82Kr(𝑝, 𝛾)83Rb reaction (black dots) compared
with standard Non-Smoker theoretical calculations [174] (blue solid line) and default Talys 1.96
calculations [62] (orange band). (Figure with permission from Tsantiri et al, Phys. Rev. C 107,
035808 2023. Copyright 2023 by the American Physical Society)

88

Table 6.1 Measured cross section of the 82Kr(𝑝, 𝛾)83Rb reaction. The first column represents the initial beam energy in the lab system,
and the second column the center-of-mass effective energy of the reaction. The third and fourth columns represent the total number of
incident and target particles, and the fifth the total number of reactions that occurred 𝑌 /𝜖. The last two columns represent the efficiency
in detecting 𝛾 -rays from the de-excitation of the 83Rb compound nucleus, and the measured cross section. (Table with permission from
Tsantiri et al., Phys. Rev. C 107, 035808 2023. Copyright 2023 by the American Physical Society)

Initial Beam
Energy (MeV/u)

3.7

3.4

3.1

Effective Energy
𝐸𝑒 𝑓 𝑓 (MeV)
2.99+0.03
−0.06
2.68+0.03
−0.06
2.38+0.02
−0.09

Total Incident
Particles 𝐼𝛼
(2.15 ± 0.11)×1011
(2.05 ± 0.10)×1011
(2.07 ± 0.10)×1011

Total Target
Particles 𝑁𝑡
(7.44 ± 0.39)×1019
(7.43 ± 0.39)×1019
(7.39 ± 0.39)×1019

Number of
Reactions 𝑌 /𝜖
26079 ± 6707
11011 ± 2460
3578 ± 595

Efficiency
𝜖 (%)
51.6 ± 5.6
51.3 ± 5.7
52.6 ± 6.0

Cross Section
𝜎 (mb)
1.63 ± 0.40
0.72 ± 0.16
0.23 ± 0.04

89

The comparison in Fig. 6.1 shows that standard statistical model calculations with default model

parameters generally overestimate the cross section for the 82Kr(𝑝, 𝛾)83Rb reaction relative to the

experimental measurements. The discrepancies between the experimental data and predictions

from the Non-Smoker code range from 23% to 47%, with the largest deviation observed at the

lowest beam energy. A similar trend has been reported by Lotay et al. [173] for the (𝑝, 𝛾) reaction

on the neighboring 83Rb nucleus, as well as by Gyürky et al.

[219] on various proton-rich Sr

isotopes. In both instances, the experimental cross sections were systematically lower than those

predicted by the Hauser-Feshbach (HF) theory, indicating the existence of a trend in this mass

region.

This substantial overestimation by theoretical models motivated further investigation on the

model parameters used in Talys . As discussed in Sec. 5.6.1, specific parameters were identified

to model the NLD and 𝛾 SF of the 83Rb nucleus (Table 5.4). These parameters were selected to

reproduce the experimental spectra, and therefore, should provide a more accurate representation

of the calculated cross section. Interestingly, the various combinations of NLD and 𝛾SF primarily

impact the upper range of the calculated 82Kr(𝑝, 𝛾)83Rb cross sections, while the lower energy

range near the 3.1 MeV/u measurement appears to not be as sensitive. Consequently, a better

description could be obtained by modifying other input quantities in Talys .

As discussed in Sec. 2.4.2.2, the cross section in the statistical model includes the transmission

𝑎 and 𝑇𝑌

coefficients 𝑇 𝑋

𝑏 between the 𝑋 + 𝑎 entrance and 𝑌 + 𝑏 exit channel, and the width fluctuation
correction 𝑊 𝑎𝑋→𝑌 𝑏 (WFC). At the low energies examined in this experiment, the only available

exit channels are the proton and 𝛾 emission. The neutron channel opens just above 5 MeV, and the

𝛼 channel remains negligible throughout the energy range under study, due to the higher Coulomb

barrier. Therefore in this case, Eq. 2.12 simplifies to

𝜎( 𝑝, 𝛾) ∼

𝑇𝑝𝑇𝛾
𝑇𝑝 + 𝑇𝛾

𝑊 𝑝𝛾

(6.1)

where the 𝑇𝑝 corresponds to the proton capture and 𝑇𝛾 to the 𝛾 decay. It is significant to examine

the competition between the available open channels. Fig. 6.2 shows a decomposition of the total

reaction cross section into the different exit channels, expressed in terms of astrophysical 𝑆 factor

90

(Sec. 2.5), for better readability. The dashed lines in Fig. 6.2 correspond to a Talys calculation

Figure 6.2 Decomposition of the astrophysical 𝑆 factor for proton capture on 82Kr. Dashed lines
correspond to a standard talys calculation, while solid lines use an optimized set of parameters.
See text for more details on the optimized parameters. The contribution of the 82Kr(p,𝛼)79Br
reaction is below the scale of the figure. (Figure adapted with permission from Tsantiri et al., Phys.
Rev. C 107, 035808 2023. Copyright 2023 by the American Physical Society)

using the NLD and 𝛾SF parameter set No.5 from Table 5.4. Since these models are fixed, the cross

section remains dependent only to the 𝑇𝑝, meaning the proton optical model potential (pOMP), as

well as the WFC. As discussed in Sec. 2.4.2.5, another option provided in Talys for the description

of the pOMP is the “jlm-type" potential, based on the work of Jeukenne, Lejeunne, and Mahaux

[64, 65, 66, 67] with later modifications by Bauge et al. [68, 69]). This potential predicts a slightly

lower cross section at lower energies, providing a better agreement with the experimental data.

The WFC accounts for correlations between the incident and outgoing wave functions, which

typically enhance the compound-elastic channel. The WFC becomes most significant when only

a few channels are open, which is the case for lower energies. At higher energies, where many

channels are accessible, the relevance of the WFC becomes negligible. The default WFC used

in Talys is based on Moldauer’s formalism [70, 71] (widthmode 1). An alternative approach is

91

that of Hofmann, Richert, Tepel, and Weidenmüller (HRTW, widthmode 2) [72, 73, 74]. This

approach provides a stronger correction, enhancing the compound-elastic channel and reducing the

calculated (𝑝, 𝛾) cross section.

Combining a HWRT-based WFC with a “jlm-type" potential yields the solid lines in Fig. 6.2,

showing an improved agreement with the experimental data, particularly at lower energies. This

combination along with the different NLD and 𝛾SF models of Table 5.4 produce the teal band in

Fig. 6.3.

Figure 6.3 Same as Fig. 6.1, with the addition of the optimized Talys band, resulting from the NLD
and 𝛾SF combinations of Table 6.1 along with a HWRT-based WFC and a “jlm-type" potential.
(Figure with permission from Tsantiri et al., Phys. Rev. C 107, 035808 2023. Copyright 2023 by
the American Physical Society)

This result demonstrates the successful proof-of-principle implementation of the summing tech-

nique in inverse kinematics using this experimental setup and a gas cell target for the measurement

of (𝑝, 𝛾) reactions. Additionally, this study shows that a consistent description of the cross section,

SoS, TAS, and multiplicity spectra can be achieved through a careful choice of parameters in the

statistical model. The ability to simultaneously describe multiple observables provides stronger

92

constraints on the model parameters than a comparison with cross-section data alone, enabling a

more constrained cross section over a broader energy range than the one directly measured.

6.2 The 73As(𝑝, 𝛾)74Se cross section

Similarly to the previous section, the cross section for the 73As(𝑝, 𝛾)74Se reaction as calculated

using Eq. 5.1 is presented in Table 6.2 along with all calculated quantities. Due to the low statistics

of the low-energy data point, the measurement efficiency was assumed to be the value calculated for

the 3.7 MeV/u data point for both energies. The asymmetry in the errors of 𝑌 /𝜖 results from avoiding

negative counts in specific bins. In Fig. 6.4 the cross section is shown along with statistical model

calculations using the Non-Smoker code [174] and Talys , for all available NLD and 𝛾SF models,

and the two pOMP options (default and JLM).

Figure 6.4 The measured cross section of the 73As(𝑝, 𝛾)74Se reaction (black dots) compared with
standard Non-Smoker theoretical calculations [174] shown in a solid line and default Talys 1.96
calculations [62] creating a band.

93

Table 6.2 Same as Table 6.1 for the 73As(𝑝, 𝛾)74Se reaction measurement.

Initial Beam
Energy (MeV/u)

3.7

3.1

Effective Energy
𝐸𝑒 𝑓 𝑓 (MeV)
2.92+0.04
−0.05
2.32+0.03
−0.05

Total Incident
Particles 𝐼𝛼
(2.51 ± 0.21)×1010
(1.67 ± 0.14)×109

Total Target
Particles 𝑁𝑡
(7.94 ± 0.40)×1019
(7.94 ± 0.40)×1019

Number of
Reactions 𝑌 /𝜖
6193 ± 1480
152+112
−95

Efficiency
𝜖 (%)

48.2 ± 6.5

Cross Section
𝜎 (mb)
3.11 ± 0.80
1.15+0.86
−0.73

94

As discussed in the previous chapters, the challenge of running a radioactive beam experiment is

apparent from the large statistical uncertainties compared to the stable beam experiment, especially

in the case of the low-energy datapoint, as the measurement was limited to just 30 minutes

of data acquisition (see Sec. 5.3.4). Nevertheless, this is the first experimental data provided

on the 73As(𝑝, 𝛾)74Se reaction cross section, and as the measurement is performed within the

Gamow window window for the astrophysical 𝛾 process, it is particularly relevant for calculating

the production of 74Se. The calculated cross section is in agreement with the theoretical prediction

by Non-Smoker within the experimental uncertainty, however the higher energy measurement is

∼18% higher than the Non-Smoker calculation.

As with the 82Kr analysis, the 73As(𝑝, 𝛾)74Se cross section measured can be combined with the

description of the TAS, SoS and multiplicity spectra obtained in Sec. 5.6.2, to provide a constrained

Talys band that can be used in network calculations. For this analysis, only the higher-energy data

point will be utilized, as the uncertainty of the low-energy point exceeds that of Talys .

As shown in Sec. 5.6.2 for each of the various Talys models, the three different types of spectra

(TAS, SoS and multiplicity) have been evaluated by a score described by Eq. 5.11. The calculated

cross section from Fig. 6.4 is used to evaluate one last factor that contributes to the score. However,

the likelihood as defined in Eq. 5.9, includes only uncertainty on the y-axis, whereas the cross

section also includes uncertainty in the effective energy. The uncertainty in the effective energy

reflects the energy straggling of the beam in the gas cell and hydrogen gas, and that distribution of

energies has been calculated using Srim (Fig. 5.17). This distribution can be used to weight the

energies on the x-axis as shown in Fig. 6.5. The error energy range is divided into 20 bins, and the

likelihood is calculated for each one of the bins weighted by its normalized probability distribution

as

(cid:34)

𝑃(𝜎exp|𝑚) = exp

−

20
∑︁

𝑗

𝑤 𝑗

(𝜎exp − 𝑓 (𝐸 𝑗 , 𝑚))2
2 · 𝛿𝜎2
exp

(cid:35)

(6.2)

where the summation goes over the 20 energy bins with weight 𝑤 𝑗 shown in Fig. 6.5, 𝜎exp and

𝛿𝜎exp is the experimental cross section with its uncertainty from Table 6.2, and 𝑓 (𝐸 𝑗 , 𝑚) is the

predicted cross section from Talys model 𝑚 at the central energy 𝐸 𝑗 of each bin. Fig. 6.6(top)

95

Figure 6.5 The effective energy uncertainty of the 73As(𝑝, 𝛾)74Se reaction as a function of the
energy straggling distributions from Fig. 5.17.

shows the calculated cross section with the Talys models shown in Fig 6.4, color-coded based

on the likelihood calculated through Eq. 6.2. Combining this likelihood with the total score for

the TAS, SoS and multiplicity spectra from Sec. 5.6.2, produces the combined score shown in

Fig. 6.6(bottom). One can observe that this combination leads to a significantly narrower favored

cross-section band. A complete table of the scores for all Talys models is shown in the Appendix.

This distribution is used to constrain the production of 74Se in network calculations of the 𝛾 process

in the following chapter.

It can be seen that the favored distribution is located slightly higher

than the Non-Smoker calculation shown in red, which is the reaction adopted in astrophysical

calculations, even though this deviation is within the experimental uncertainty. This is consistent

with the experimental measurement being ∼18% higher than the one calculated by Non-Smoker.

96

Figure 6.6 The measured cross section of the 73As(𝑝, 𝛾)74Se reaction in the 3.7 MeV/u initial
beam energy compared to Talys calculations color-coded based on (top) the likelihood calculated
through Eq. 6.2 and (bottom) the combined score to reproduce all types of data (cross section, TAS,
SoS and multiplicity).

97

CHAPTER 7

ASTROPHYSICAL IMPACT

As discussed in Sec. 3.4.1, the two main explosive environments where the 𝛾 process is thought to

occur are the oxygen and neon enriched layers of a core-collapse supernovae (SNII) [133, 135, 136],

and thermonuclear Type Ia supernovae (SNIa) [137, 138, 139]. The impact of the measured cross

section will be examined in both cases using Monte Carlo simulations of both scenarios.

7.1 Core-Collapse Supernova - SNII

For the SNII environment, NuGrid stellar data set II models [142] were used. The most massive

progenitors with solar metallicity available in these models were stars with masses of 15, 20, and

25 𝑀⊙, calculated using MESA [157]. The initial solar composition was from Grevesse and Noels

[220] with isotopic ratios from Lodders [126], while the nucleosynthesis during stellar evolution

and explosion was computed using the Multi-zone Post-Processing Network – Parallel (MPPNP)

code [159]. The nuclear network beyond iron consisted of 74 313 reactions, for which experimental

rates from the KADoNIS compilation [221] were incorporated wherever available, and any missing

reaction rates were obtained from the JINA REACLIB library [222].

Among the three available mass models, the 25 𝑀⊙ model ended in a failed supernova due to

significant fallback, trapping all the material produced during the explosion in the remnant star.

The 15 𝑀⊙ model experienced a C-O shell merger event, significantly enhancing the pre-supernova
production of p nuclei. Since 74Se is expected to be primarily produced by the explosive component,

even in the presence of a C-O shell merger [143], the 20 𝑀⊙ model was chosen as a representative

case, where the explosive component dominates p-nuclei production, similarly to other SNII models

used in 𝛾-process studies [136, 176, 177]. The impact of the measured cross section was expected

to be similar for any SNII model reaching comparable peak temperatures during the shock wave

propagation.

The produced mass fraction of 74Se as the shock wave propagates through the ONe layer of the

star is shown in Fig. 7.1. The maximum production of 74Se occurs in the inner ONe layer, at mass

coordinate 𝑀 = 2.93𝑀⊙. At this location, the temperature and density profiles are presented in

98

Figure 7.1 The final mass fractions of the 20 𝑀⊙ mass model by Ritter et al. [142] as a function of
mass coordinate. The 16O, 20Ne, 12C, 4He and 1H lines indicate the different layers of the star. 74Se
is produced in the inner ONe layer, where the higher peak temperatures in the Gamow window for
the 𝛾 process are achieved.

Fig. 7.2, where the temperature trajectory exhibits a plateau at 𝑇 = 3.08 GK.

To investigate the impact of the measured cross section under such conditions, the cross section

predictions of the measured 73As(𝑝, 𝛾)74Se reaction from Fig. 6.6 were converted to reaction rates

In Fig. 7.3 the reaction rate of the 73As(𝑝, 𝛾)74Se reaction is shown as a function of
using Talys .
temperature, with models color-coded by the total score calculated in Sec. 6.2. The solid red line

is the Jina-Reaclib [222] reaction rate, while the dotted line marks the temperature of interest,

𝑇 = 3.08 GK. A projection of the total scores along the 𝑇 = 3.08 GK temperature is shown in

Fig. 7.4, as a function of each model’s total score. The various points represent the different

Talys models, and the blue bars are the normalized distribution of those rates grouped in 50 bins,

between the minimum and maximum Talys reaction rate. For Monte Carlo simulations, 10 000

rates were randomly sampled from this distribution, forming the set of rates shown by the magenta

line.

The Monte Carlo one-zone nucleosynthesis simulations were performed using the PPN code

99

(a)

(b)

Figure 7.2 The temperature and density trajectories at mass coordinate 𝑀 = 2.93𝑀⊙ in the 20 𝑀⊙
mass model by Ritter et al. [142]. The arrows in (b) indicate the direction of time.

Figure 7.3 The reaction rate of the 73As(𝑝, 𝛾)74Se reaction as a function of temperature calculated
using Talys . The Talys models are color coded based on their total score, similarly to Fig. 6.2.
The red line indicates the reaction rate of the Jina-Reaclib [222] library, typically used in network
calculations. The dotted line represents the peak temperature for the SNII model from Fig. 7.2a.

from the NuGrid framework [159]. The temperature and density profiles (Fig. 7.2), as well as

the initial abundances were obtained from the 20 𝑀⊙ mass model shown in Fig. 7.1 for the mass

100

Figure 7.4 A projection of the reaction rates calculated by Talys along the 𝑇 = 3.08 GK temperature
of interest, as a function of each model’s total score. The blue bars correspond to the Talys models
grouped in 50 bins between the minimum and maximum rate calculated by Talys . The magenta
line shows the sampled rates used in Monte Carlo simulations of the SNII scenario.

coordinate 𝑀 = 2.93𝑀⊙. The only variable varied in each of the 10 000 realizations of the code was

the multiplication factor for the 73As(𝑝, 𝛾)74Se reaction rate (and its inverse 74Se(𝛾, 𝑝)73As). This

multiplication factor was calculated as the ratio of the sampled reaction rate over the Jina-Reaclib

rate that is used as a default value.

The produced 74Se mass fraction as a function of time is shown in Fig. 7.5. The various blue

lines correspond to the 10 000 runs with varied 73As(𝑝, 𝛾)74Se reaction rate, with the darker colors

reflecting more lines that overlap in that region. The orange line corresponds to the mass fraction

produced using the Jina-Reaclib rate, and the red dashed line are the mass fractions obtained with

the minimum and maximum rate obtained from Talys . Fig. 7.6 shows final mass fraction of 74Se

relative to the default Jina-Reaclib model.

The experimentally measured 73As(𝑝, 𝛾)74Se cross section from Sec. 6.2 is ∼18% higher

than the Non-Smoker rate adopted in the Jina-Reaclib library but still consistent within the

101

Figure 7.5 74Se mass fraction as a function of time calculated using PPN one-zone simulations.
The various blue lines correspond to different 73As(𝑝, 𝛾)74Se reaction rates sampled from the
distribution of Fig. 7.4, and the darker color correspond to more overlapping trajectories. The
orange line corresponds to simulation using the default Jina-Reaclib reaction rate and the red
dashed lines to the minimum and maximum reaction rates by Talys .

experimental uncertainty (Fig. 6.4). Consequently, the final 74Se abundance in SNII models

remains in good agreement with calculations using the Jina-Reaclib rate, though a slightly lower

mean 74Se production is suggested. However, this deviation is too small to indicate that the observed

overproduction of 74Se is driven by uncertainties in this reaction rate. Comparing the full width at

half maximum of the distribution of Fig. 7.8 to the Talys uncertainty shows that the measurement

significantly reduces theoretical uncertainties in reaction rates by approximately a factor of two,

providing a more constrained and reliable input for future sensitivity studies.

102

Figure 7.6 Comparison of the final 74Se mass fraction distribution obtained in the Monte Carlo
simulations varying the 73As(𝑝, 𝛾)74Se reaction rate relative to the default model using the Jina-
Reaclib reaction rate. The dashed lines correspond to the maximum and minimum reaction rate
from Talys .

7.2 Type Ia Supernova - SNIa

For the Type Ia scenario the models by Travaglio et al. [138] were adopted. In their work, they

explored two-dimensional SNIa models of a white dwarf (WD) accreting mass from a companion

star in the single degenerate (SD) scenario. The s-process seed distribution is assumed to be

produced during the AGB phase leading to the formation of the WD, through thermal pulses

that mix material from the H-rich envelope to the C-rich layers, resulting in a 13C pocket (see

Sec. 3.3.3.1). From the different explosion mechanisms studied by Travaglio et al., the model

adopted in this work is the so-called DTT-a. DTT-a is a delayed detonation model assuming the

deflagration-to-detonation criterion of Kasen et al. [223] and a CO-WD structure by Domínguez

et al. [224] with solar metallicity and a progenitor mass of M = 1.5 𝑀⊙.

The nucleosynthesis in such multidimensional simulations was calculated using the tracer parti-

cle method [225, 226] by placing 51 200 tracer particles, uniformly distributed in mass coordinate.

103

For this work, the trajectories of three tracer particles were provided from the external layers of the

model, where the peak temperatures reached allowed for 𝛾-process nucleosynthesis. The isotopic

abundances of elements heavier than Ge were also provided, while for the lighter elements solar

abundances by Asplund [83] were used. The temperature and density profiles for one of the three

tracers (number 2876) is shown as an example in Fig. 7.7. The other two tracer trajectories had

minimal differences, an thus the tracer shown in Fig. 7.7 was chosen as a representative case.

(a)

(b)

Figure 7.7 The temperature (a) and density (b) profiles of tracer 2876 of the DTT-a SNIa model by
Travaglio et al. [138]. The tracer is located on the external layers of the SNIa model, where the
peak temperatures reached allow for 𝛾-process nucleosynthesis.

The peak temperature obtained is 𝑇 = 3.27 GK. Similar to the SNII scenario one-zone nucle-

osynthesis simulations were performed using the PPN code for the Jina-Reaclib rate, the minimum

and maximum rate predicted by Talys models at the peak temperature of interest, and a sample

of the reaction rate distribution weighted by the total score. The produced 74Se mass fraction

as a function of time, shown in Fig. 7.8, confirms that variations in the 73As(𝑝, 𝛾)74Se reaction

rate within the nuclear uncertainty predicted by Talys do not significantly impact the final 74Se

abundance. This is consistent with Ref. [162], indicating that 74Se production it SNIa is affected

by astrophysical conditions rather than nuclear uncertainties in this reaction. As the 73As(𝑝, 𝛾)74Se

104

Figure 7.8 Same as Fig. 7.5 for the SNIa trajectories shown in Fig. 7.7.

reaction is a destruction mechanism for 74Se, it is possible that the material is not exposed to the

peak temperature for a long enough time, for this destruction mechanism to become relevant. As

seen in Fig. 7.7 compared to Fig. 7.2, the material is exposed to peak temperatures for fractions

of a second, while in the SNII scenario the temperature exhibits a plateau that lasts for multiple

seconds, allowing the 73As(𝑝, 𝛾)74Se reaction to destroy the produced 74Se and impact the final

abundances.

105

CHAPTER 8

SUMMARY & CONCLUSIONS

To summarize, the astrophysical 𝛾 process, the most established scenario for the production of

the p nuclei, involves a vast network of primarily radioactive isotopes. Developing experimental

techniques to directly measure photodisintegration reactions using unstable beams is essential, as

reaction rates in network calculations are largely estimated through statistical model calculations,

that carry significant uncertainties away from stability.

This work focused on implementing such a technique to measure reaction cross sections in

inverse kinematics using a radioactive beam, the summing technique and a hydrogen gas target.

The proof-of-principle application of this method was successfully demonstrated with the first

measurement of the 82Kr(𝑝, 𝛾)83Rb reaction cross section. The experimental results suggest a

smaller cross section than theoretically predicted, but a more consistent description of the statistical

model parameters was obtained through simulations to reproduce the experiment spectra. The

ability to simultaneously describe multiple observables provides stronger constraints on the model

parameters than a comparison with cross-section data alone, enabling a more constrained cross

section over a broader energy range than the one directly measured.

The same methodology was applied using a radioactive 73As beam, leading to the first mea-

surement of the 73As(𝑝, 𝛾)74Se reaction cross section. The measured cross section showed good

agreement with the theoretical prediction from Non-Smoker. The cross-section data, along with

experimental spectra, were used to characterize various nuclear level density and 𝛾-ray strength

function models from Talys , allowing for the extraction of an experimentally constrained cross

section across the entire Gamow window of the 𝛾 process. This characterization enabled the deter-

mination of an experimentally constrained reaction rate for the 73As(𝑝, 𝛾)74Se reaction, which was

then used in Monte Carlo one-zone network simulations of the 𝛾 process.

Although the production of 74Se in the Type Ia supernova scenario showed no sensitivity to the

constrained reaction rate, a significant impact on the final 74Se abundance in Type II supernovae

was observed. The uncertainty in the 74Se production was reduced by a factor of two, providing a

106

more constrained and reliable input for future sensitivity studies.

The successful implementation of this technique opens the possibility of extending similar mea-

surements to reactions involving heavier radioactive beams. Such studies would further constrain

reaction rates relevant for the 𝛾 process, improving the accuracy of nucleosynthesis models and

reducing uncertainties in astrophysical simulations.

107

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124

APPENDIX

TALYS SCORES FOR THE 73AS(𝑝, 𝛾)74SE REACTION

Table .1 The Talys model parameters and scores

Model # NLD

𝛾SF

𝛾SF

𝛾SF

E1

M1

upbend

OMP

TAS Score

SoS Score Mul Score

Cross Section

Score (Eq. 6.2))

Total Score

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

JLM

0.4341

default

0.4341

JLM

0.4341

default

0.4341

JLM

0.3805

default

0.3805

JLM

0.3805

default

0.3805

JLM

0.6034

default

0.6034

JLM

0.5111

default

0.5111

JLM

0.6068

default

0.6068

JLM

0.6068

default

0.6068

JLM

0.5904

default

0.5904

JLM

0.5904

default

0.5904

JLM

0.6513

default

0.6513

JLM

0.6650

default

0.6650

JLM

0.1251

default

0.1251

JLM

0.1251

default

0.1251

JLM

0.1196

default

0.1196

JLM

0.1196

125

0.1215

0.1215

0.1215

0.1215

0.1132

0.1132

0.1132

0.1132

0.2326

0.2326

0.1921

0.1921

0.2244

0.2244

0.2244

0.2244

0.2119

0.2119

0.2119

0.2119

0.2892

0.2892

0.2708

0.2708

0.0336

0.0336

0.0336

0.0336

0.0306

0.0306

0.0306

0.1064

0.1064

0.1064

0.1064

0.0982

0.0982

0.0982

0.0982

0.3122

0.3122

0.1852

0.1852

0.2058

0.2058

0.2058

0.2058

0.1643

0.1643

0.1643

0.1643

0.3077

0.3077

0.2538

0.2538

0.0058

0.0058

0.0058

0.0058

0.0049

0.0049

0.0049

0.6233

0.4702

0.6233

0.4702

0.1828

0.1284

0.1828

0.1284

0.6610

0.4984

0.5090

0.3717

0.1150

0.3221

0.1150

0.3221

0.0739

0.2417

0.0739

0.2417

0.1092

0.3127

0.1344

0.3566

0.9894

0.9743

0.9894

0.9743

0.9991

0.9335

0.9991

3.50e-03

2.64e-03

3.50e-03

2.64e-03

7.73e-04

5.43e-04

7.73e-04

5.43e-04

2.90e-02

2.18e-02

9.26e-03

6.76e-03

3.22e-03

9.03e-03

3.22e-03

9.03e-03

1.52e-03

4.97e-03

1.52e-03

4.97e-03

6.33e-03

1.81e-02

6.14e-03

1.63e-02

2.39e-05

2.36e-05

2.39e-05

2.36e-05

1.79e-05

1.68e-05

1.79e-05

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

3

3

3

3

3

4

4

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

Table .1 (cont’d)

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

default

0.1196

JLM

0.1893

default

0.1893

JLM

0.1570

default

0.1570

JLM

0.2687

default

0.2687

JLM

0.2687

default

0.2687

JLM

0.2277

default

0.2277

JLM

0.2277

default

0.2277

JLM

0.3996

default

0.3996

JLM

0.3407

default

0.3407

JLM

0.5923

default

0.5923

JLM

0.5923

default

0.5923

JLM

0.5877

default

0.5877

JLM

0.5877

default

0.5877

JLM

0.6936

default

0.6936

JLM

0.6530

default

0.6530

JLM

0.2837

default

0.2837

JLM

0.2837

default

0.2837

JLM

0.2792

default

0.2792

JLM

0.2792

default

0.2792

126

0.0306

0.0519

0.0519

0.0410

0.0410

0.0755

0.0755

0.0755

0.0755

0.0663

0.0663

0.0663

0.0663

0.1296

0.1296

0.1024

0.1024

0.1996

0.1996

0.1996

0.1996

0.2218

0.2218

0.2218

0.2218

0.3048

0.3048

0.2735

0.2735

0.0897

0.0897

0.0897

0.0897

0.0873

0.0873

0.0873

0.0873

0.0049

0.0171

0.0171

0.0084

0.0084

0.0215

0.0215

0.0215

0.0215

0.0183

0.0183

0.0183

0.0183

0.0739

0.0739

0.0397

0.0397

0.1673

0.1673

0.1673

0.1673

0.2642

0.2642

0.2642

0.2642

0.4075

0.4075

0.3104

0.3104

0.0315

0.0315

0.0315

0.0315

0.0374

0.0374

0.0374

0.0374

0.9335

0.9818

0.9817

0.9993

0.9380

0.9545

0.8232

0.9545

0.8232

0.7658

0.5993

0.7658

0.5993

0.9685

0.8437

0.8989

0.7431

0.9989

0.9471

0.9989

0.9471

0.9084

0.7550

0.9084

0.7550

0.9966

0.9577

0.9926

0.8992

0.9846

0.8793

0.9846

0.8793

0.8439

0.6795

0.8439

0.6795

1.68e-05

1.65e-04

1.65e-04

5.38e-05

5.05e-05

4.17e-04

3.60e-04

4.17e-04

3.60e-04

2.12e-04

1.66e-04

2.12e-04

1.66e-04

3.71e-03

3.23e-03

1.25e-03

1.03e-03

1.98e-02

1.87e-02

1.98e-02

1.87e-02

3.13e-02

2.60e-02

3.13e-02

2.60e-02

8.58e-02

8.25e-02

5.50e-02

4.98e-02

7.90e-04

7.05e-04

7.90e-04

7.05e-04

7.69e-04

6.19e-04

7.69e-04

6.19e-04

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

6

6

6

6

7

7

7

7

7

7

7

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

Table .1 (cont’d)

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

JLM

0.4293

default

0.4293

JLM

0.3489

default

0.3489

JLM

0.2374

default

0.2374

JLM

0.2374

default

0.2374

JLM

0.1838

default

0.1838

JLM

0.1838

default

0.1838

JLM

0.3343

default

0.3343

JLM

0.2825

default

0.2825

JLM

0.2741

default

0.2741

JLM

0.2377

default

0.2377

JLM

0.2891

default

0.2891

JLM

0.2507

default

0.2507

JLM

0.3842

default

0.3842

JLM

0.3022

default

0.3022

JLM

0.3274

default

0.3274

JLM

0.3274

default

0.3274

JLM

0.3239

default

0.3239

JLM

0.3239

default

0.3239

JLM

0.4561

127

0.1508

0.1508

0.1266

0.1266

0.0712

0.0712

0.0712

0.0712

0.0661

0.0661

0.0661

0.0661

0.1170

0.1170

0.0976

0.0976

0.0703

0.0703

0.0656

0.0656

0.0690

0.0690

0.0662

0.0662

0.1075

0.1075

0.0826

0.0826

0.0837

0.0837

0.0837

0.0837

0.0785

0.0785

0.0785

0.0785

0.1359

0.1118

0.1118

0.0697

0.0697

0.0260

0.0260

0.0260

0.0260

0.0234

0.0234

0.0234

0.0234

0.0782

0.0782

0.0486

0.0486

0.0227

0.0227

0.0190

0.0190

0.0234

0.0234

0.0206

0.0206

0.0689

0.0689

0.0311

0.0311

0.0357

0.0357

0.0357

0.0357

0.0291

0.0291

0.0291

0.0291

0.0898

0.9919

0.8962

0.9483

0.8105

0.9990

0.9310

0.9990

0.9310

0.8592

0.6940

0.8592

0.6940

0.9990

0.9435

0.9848

0.8764

0.8199

0.9766

0.8515

0.9869

0.8010

0.9693

0.8335

0.9812

0.7990

0.9696

0.9057

0.9983

0.9823

0.8748

0.9823

0.8748

0.8855

0.7287

0.8855

0.7287

0.9903

7.18e-03

6.48e-03

2.92e-03

2.50e-03

4.39e-04

4.09e-04

4.39e-04

4.09e-04

2.44e-04

1.97e-04

2.44e-04

1.97e-04

3.06e-03

2.89e-03

1.32e-03

1.17e-03

3.59e-04

4.27e-04

2.52e-04

2.92e-04

3.75e-04

4.53e-04

2.85e-04

3.36e-04

2.27e-03

2.76e-03

7.03e-04

7.74e-04

9.62e-04

8.56e-04

9.62e-04

8.56e-04

6.55e-04

5.39e-04

6.55e-04

5.39e-04

5.51e-03

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

9

9

9

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

Table .1 (cont’d)

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

default

0.4561

JLM

0.3966

default

0.3966

JLM

0.7550

default

0.7550

JLM

0.7550

default

0.7550

JLM

0.7579

default

0.7579

JLM

0.7579

default

0.7579

JLM

0.8004

default

0.8004

JLM

0.8017

default

0.8017

JLM

0.7800

default

0.7800

JLM

0.7800

default

0.7800

JLM

0.7604

default

0.7604

JLM

0.7604

default

0.7604

JLM

0.8162

default

0.8162

JLM

0.8090

default

0.8090

JLM

0.3597

default

0.3597

JLM

0.3597

default

0.3597

JLM

0.3119

default

0.3119

JLM

0.3119

default

0.3119

JLM

0.4162

default

0.4162

128

0.1359

0.1131

0.1131

0.3980

0.3980

0.3980

0.3980

0.4332

0.4332

0.4332

0.4332

0.5198

0.5198

0.4793

0.4793

0.4271

0.4271

0.4271

0.4271

0.4153

0.4153

0.4153

0.4153

0.4802

0.4802

0.4547

0.4547

0.1108

0.1108

0.1108

0.1108

0.1065

0.1065

0.1065

0.1065

0.1681

0.1681

0.0898

0.0513

0.0513

0.4213

0.4213

0.4213

0.4213

0.6112

0.6112

0.6112

0.6112

0.7146

0.7146

0.6168

0.6168

0.3763

0.3763

0.3763

0.3763

0.3203

0.3203

0.3203

0.3203

0.4998

0.4998

0.4203

0.4203

0.0350

0.0350

0.0350

0.0350

0.0336

0.0336

0.0336

0.0336

0.1134

0.1134

0.8920

0.9443

0.8052

0.6055

0.8607

0.6055

0.8607

0.9987

0.9278

0.9987

0.9278

0.6417

0.8870

0.7648

0.9549

0.0002

0.0052

0.0002

0.0052

0.0001

0.0028

0.0001

0.0028

0.0003

0.0055

0.0004

0.0068

0.1684

0.4097

0.1684

0.4097

0.2385

0.5084

0.2385

0.5084

0.1836

0.4344

4.96e-03

2.17e-03

1.85e-03

7.66e-02

1.09e-01

7.66e-02

1.09e-01

2.00e-01

1.86e-01

2.00e-01

1.86e-01

1.91e-01

2.64e-01

1.81e-01

2.26e-01

3.10e-05

6.47e-04

3.10e-05

6.47e-04

1.04e-05

2.85e-04

1.04e-05

2.85e-04

5.33e-05

1.08e-03

5.67e-05

1.05e-03

2.35e-04

5.72e-04

2.35e-04

5.72e-04

2.66e-04

5.67e-04

2.66e-04

5.67e-04

1.46e-03

3.45e-03

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

3

4

4

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

Table .1 (cont’d)

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

JLM

0.3803

default

0.3803

JLM

0.5390

default

0.5390

JLM

0.5390

default

0.5390

JLM

0.4906

default

0.4906

JLM

0.4906

default

0.4906

JLM

0.6107

default

0.6107

JLM

0.5955

default

0.5955

JLM

0.7854

default

0.7854

JLM

0.7854

default

0.7854

JLM

0.7968

default

0.7968

JLM

0.7968

default

0.7968

JLM

0.8128

default

0.8128

JLM

0.8179

default

0.8179

JLM

0.6189

default

0.6189

JLM

0.6189

default

0.6189

JLM

0.5958

default

0.5958

JLM

0.5958

default

0.5958

JLM

0.6786

default

0.6786

JLM

0.6584

129

0.1359

0.1359

0.2134

0.2134

0.2134

0.2134

0.2055

0.2055

0.2055

0.2055

0.3340

0.3340

0.2653

0.2653

0.4463

0.4463

0.4463

0.4463

0.5018

0.5018

0.5018

0.5018

0.5518

0.5518

0.5174

0.5174

0.2926

0.2926

0.2926

0.2926

0.2853

0.2853

0.2853

0.2853

0.3997

0.3997

0.3403

0.0534

0.0534

0.0879

0.0879

0.0879

0.0879

0.0891

0.0891

0.0891

0.0891

0.3055

0.3055

0.1452

0.1452

0.4802

0.4802

0.4802

0.4802

0.6083

0.6083

0.6083

0.6083

0.6866

0.6866

0.6195

0.6195

0.1990

0.1990

0.1990

0.1990

0.2310

0.2310

0.2310

0.2310

0.4442

0.4442

0.2727

0.2441

0.5169

0.3580

0.6466

0.3580

0.6466

0.6688

0.9020

0.6688

0.9020

0.3855

0.6773

0.4893

0.7722

0.0965

0.2880

0.0965

0.2880

0.2555

0.5316

0.2555

0.5316

0.1058

0.3068

0.1423

0.3702

0.2232

0.4881

0.2232

0.4881

0.4601

0.7467

0.4601

0.7467

0.2425

0.5156

0.3172

6.74e-04

1.43e-03

3.62e-03

6.54e-03

3.62e-03

6.54e-03

6.01e-03

8.10e-03

6.01e-03

8.10e-03

2.40e-02

4.22e-02

1.12e-02

1.77e-02

1.62e-02

4.85e-02

1.62e-02

4.85e-02

6.22e-02

1.29e-01

6.22e-02

1.29e-01

3.26e-02

9.45e-02

3.73e-02

9.71e-02

8.04e-03

1.76e-02

8.04e-03

1.76e-02

1.81e-02

2.93e-02

1.81e-02

2.93e-02

2.92e-02

6.21e-02

1.94e-02

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

6

7

7

7

7

7

7

7

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

9

9

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

Table .1 (cont’d)

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

default

0.6584

JLM

0.5634

default

0.5634

JLM

0.5634

default

0.5634

JLM

0.5225

default

0.5225

JLM

0.5225

default

0.5225

JLM

0.6292

default

0.6292

JLM

0.6231

default

0.6231

JLM

0.5668

default

0.5668

JLM

0.5088

default

0.5088

JLM

0.5212

default

0.5212

JLM

0.5401

default

0.5401

JLM

0.5994

default

0.5994

JLM

0.5556

default

0.5556

JLM

0.6245

default

0.6245

JLM

0.6245

default

0.6245

JLM

0.6088

default

0.6088

JLM

0.6088

default

0.6088

JLM

0.6654

default

0.6654

JLM

0.6953

default

0.6953

130

0.3403

0.2482

0.2482

0.2482

0.2482

0.2405

0.2405

0.2405

0.2405

0.3374

0.3374

0.2853

0.2853

0.2143

0.2143

0.1901

0.1901

0.2127

0.2127

0.1860

0.1860

0.2829

0.2829

0.2240

0.2240

0.2380

0.2380

0.2380

0.2380

0.2378

0.2378

0.2378

0.2378

0.3563

0.3563

0.2917

0.2917

0.2727

0.1750

0.1750

0.1750

0.1750

0.1976

0.1976

0.1976

0.1976

0.4192

0.4192

0.2759

0.2759

0.1036

0.1036

0.0675

0.0675

0.1029

0.1029

0.0694

0.0694

0.2509

0.2509

0.1045

0.1045

0.1091

0.1091

0.1091

0.1091

0.1166

0.1166

0.1166

0.1166

0.3242

0.3242

0.1659

0.1659

0.6048

0.2049

0.4641

0.2049

0.4641

0.5237

0.8019

0.5237

0.8019

0.2229

0.4909

0.2929

0.5782

0.0409

0.1641

0.0467

0.1790

0.0368

0.1530

0.0421

0.1670

0.0451

0.1757

0.0703

0.2348

0.2357

0.5039

0.2357

0.5039

0.4106

0.6999

0.4106

0.6999

0.2558

0.5318

0.3336

0.6221

3.69e-02

5.02e-03

1.14e-02

5.02e-03

1.14e-02

1.30e-02

1.99e-02

1.30e-02

1.99e-02

1.98e-02

4.37e-02

1.44e-02

2.84e-02

5.15e-04

2.07e-03

3.05e-04

1.17e-03

4.20e-04

1.75e-03

2.93e-04

1.16e-03

1.92e-03

7.48e-03

9.15e-04

3.05e-03

3.82e-03

8.17e-03

3.82e-03

8.17e-03

6.93e-03

1.18e-02

6.93e-03

1.18e-02

1.97e-02

4.09e-02

1.12e-02

2.09e-02

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

4

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

Table .1 (cont’d)

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

JLM

0.6586

default

0.6586

JLM

0.6586

default

0.6586

JLM

0.6616

default

0.6616

JLM

0.6616

default

0.6616

JLM

0.7736

default

0.7736

JLM

0.7523

default

0.7523

JLM

0.7595

default

0.7595

JLM

0.7595

default

0.7595

JLM

0.7561

default

0.7561

JLM

0.7561

default

0.7561

JLM

0.7646

default

0.7646

JLM

0.7734

default

0.7734

JLM

0.2005

default

0.2005

JLM

0.2005

default

0.2005

JLM

0.1736

default

0.1736

JLM

0.1736

default

0.1736

JLM

0.2664

default

0.2664

JLM

0.2530

default

0.2530

JLM

0.3782

131

0.2592

0.2592

0.2592

0.2592

0.2761

0.2761

0.2761

0.2761

0.4157

0.4157

0.3517

0.3517

0.3203

0.3203

0.3203

0.3203

0.3050

0.3050

0.3050

0.3050

0.3777

0.3777

0.3516

0.3516

0.0523

0.0523

0.0523

0.0523

0.0496

0.0496

0.0496

0.0496

0.0819

0.0819

0.0680

0.0680

0.1190

0.2887

0.2887

0.2887

0.2887

0.3993

0.3993

0.3993

0.3993

0.6112

0.6112

0.4410

0.4410

0.2998

0.2998

0.2998

0.2998

0.2614

0.2614

0.2614

0.2614

0.4196

0.4196

0.3426

0.3426

0.0119

0.0119

0.0119

0.0119

0.0111

0.0111

0.0111

0.0111

0.0372

0.0372

0.0195

0.0195

0.0425

0.9991

0.9337

0.9991

0.9337

0.6630

0.4979

0.6630

0.4979

0.9980

0.9497

0.9817

0.8678

0.0124

0.0747

0.0124

0.0747

0.0066

0.0488

0.0066

0.0488

0.0115

0.0715

0.0152

0.0859

0.7192

0.9326

0.7192

0.9326

0.8170

0.9769

0.8170

0.9769

0.6899

0.9182

0.7980

0.9704

0.9250

4.92e-02

4.60e-02

4.92e-02

4.60e-02

4.84e-02

3.63e-02

4.84e-02

3.63e-02

1.96e-01

1.87e-01

1.15e-01

1.01e-01

9.02e-04

5.45e-03

9.02e-04

5.45e-03

3.98e-04

2.94e-03

3.98e-04

2.94e-03

1.40e-03

8.66e-03

1.41e-03

8.01e-03

9.00e-05

1.17e-04

9.00e-05

1.17e-04

7.82e-05

9.35e-05

7.82e-05

9.35e-05

5.59e-04

7.45e-04

2.68e-04

3.26e-04

1.77e-03

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

6

7

7

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

Table .1 (cont’d)

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

default

0.3782

JLM

0.3782

default

0.3782

JLM

0.3534

default

0.3534

JLM

0.3534

default

0.3534

JLM

0.5235

default

0.5235

JLM

0.4456

default

0.4456

JLM

0.7473

default

0.7473

JLM

0.7473

default

0.7473

JLM

0.7784

default

0.7784

JLM

0.7784

default

0.7784

JLM

0.8001

default

0.8001

JLM

0.8018

default

0.8018

JLM

0.4608

default

0.4608

JLM

0.4608

default

0.4608

JLM

0.4846

default

0.4846

JLM

0.4846

default

0.4846

JLM

0.5915

default

0.5915

JLM

0.5581

default

0.5581

JLM

0.3487

default

0.3487

132

0.1190

0.1190

0.1190

0.1071

0.1071

0.1071

0.1071

0.2008

0.2008

0.1513

0.1513

0.3399

0.3399

0.3399

0.3399

0.3781

0.3781

0.3781

0.3781

0.4469

0.4469

0.4171

0.4171

0.1674

0.1674

0.1674

0.1674

0.1600

0.1600

0.1600

0.1600

0.2519

0.2519

0.2189

0.2189

0.1293

0.1293

0.0425

0.0425

0.0425

0.0387

0.0387

0.0387

0.0387

0.1438

0.1438

0.0669

0.0669

0.3579

0.3579

0.3579

0.3579

0.4683

0.4683

0.4683

0.4683

0.6076

0.6076

0.5065

0.5065

0.1082

0.1082

0.1082

0.1082

0.1011

0.1011

0.1011

0.1011

0.2618

0.2618

0.1760

0.1760

0.0772

0.0772

0.9993

0.9250

0.9993

0.9960

0.9108

0.9960

0.9108

0.9036

0.9984

0.9718

0.9886

0.5619

0.8319

0.5619

0.8319

0.8279

0.9811

0.8279

0.9811

0.5356

0.8138

0.6387

0.8866

0.8020

0.9710

0.8020

0.9710

0.9734

0.9876

0.9734

0.9876

0.7745

0.9608

0.8735

0.9937

0.7386

0.9433

1.91e-03

1.77e-03

1.91e-03

1.46e-03

1.33e-03

1.46e-03

1.33e-03

1.37e-02

1.51e-02

4.39e-03

4.46e-03

5.11e-02

7.56e-02

5.11e-02

7.56e-02

1.14e-01

1.35e-01

1.14e-01

1.35e-01

1.16e-01

1.77e-01

1.08e-01

1.50e-01

6.69e-03

8.10e-03

6.69e-03

8.10e-03

7.63e-03

7.74e-03

7.63e-03

7.74e-03

3.02e-02

3.75e-02

1.88e-02

2.14e-02

2.57e-03

3.28e-03

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

7

7

7

7

7

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

9

9

1

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

Table .1 (cont’d)

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

JLM

0.3487

default

0.3487

JLM

0.3188

default

0.3188

JLM

0.3188

default

0.3188

JLM

0.4854

default

0.4854

JLM

0.4194

default

0.4194

JLM

0.4105

default

0.4105

JLM

0.3996

default

0.3996

JLM

0.4020

default

0.4020

JLM

0.3958

default

0.3958

JLM

0.4946

default

0.4946

JLM

0.4639

default

0.4639

JLM

0.5015

default

0.5015

JLM

0.5015

default

0.5015

JLM

0.4754

default

0.4754

JLM

0.4754

default

0.4754

JLM

0.6402

default

0.6402

JLM

0.5221

default

0.5221

JLM

0.7562

default

0.7562

JLM

0.7562

133

0.1293

0.1293

0.1213

0.1213

0.1213

0.1213

0.1975

0.1975

0.1675

0.1675

0.1086

0.1086

0.1008

0.1008

0.1122

0.1122

0.1015

0.1015

0.1639

0.1639

0.1251

0.1251

0.1410

0.1410

0.1410

0.1410

0.1318

0.1318

0.1318

0.1318

0.2289

0.2289

0.1771

0.1771

0.3451

0.3451

0.3451

0.0772

0.0772

0.0760

0.0760

0.0760

0.0760

0.2313

0.2313

0.1235

0.1235

0.0484

0.0484

0.0375

0.0375

0.0512

0.0512

0.0372

0.0372

0.1214

0.1214

0.0604

0.0604

0.0678

0.0678

0.0678

0.0678

0.0686

0.0686

0.0686

0.0686

0.1993

0.1993

0.1053

0.1053

0.5904

0.5904

0.5904

0.7386

0.9433

0.9764

0.9852

0.9764

0.9852

0.7102

0.9301

0.8156

0.9774

0.3808

0.6734

0.4155

0.7073

0.3606

0.6526

0.3942

0.6865

0.3595

0.6538

0.4830

0.7696

0.8351

0.9826

0.8351

0.9826

0.9639

0.9927

0.9639

0.9927

0.8085

0.9743

0.9022

0.9981

0.8579

0.9892

0.8579

2.57e-03

3.28e-03

2.87e-03

2.89e-03

2.87e-03

2.89e-03

1.57e-02

2.06e-02

7.08e-03

8.48e-03

8.22e-04

1.45e-03

6.27e-04

1.07e-03

8.32e-04

1.51e-03

5.88e-04

1.02e-03

3.54e-03

6.43e-03

1.69e-03

2.70e-03

4.01e-03

4.72e-03

4.01e-03

4.72e-03

4.14e-03

4.27e-03

4.14e-03

4.27e-03

2.36e-02

2.85e-02

8.79e-03

9.72e-03

1.32e-01

1.52e-01

1.32e-01

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

Table .1 (cont’d)

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

default

0.7562

JLM

0.7756

default

0.7756

JLM

0.7756

default

0.7756

JLM

0.8119

default

0.8119

JLM

0.7948

default

0.7948

JLM

0.7943

default

0.7943

JLM

0.7943

default

0.7943

JLM

0.7898

default

0.7898

JLM

0.7898

default

0.7898

JLM

0.8233

default

0.8233

JLM

0.8214

default

0.8214

JLM

0.2891

default

0.2891

JLM

0.2891

default

0.2891

JLM

0.2758

default

0.2758

JLM

0.2758

default

0.2758

JLM

0.5295

default

0.5295

JLM

0.3626

default

0.3626

JLM

0.5632

default

0.5632

JLM

0.5632

default

0.5632

134

0.3451

0.3670

0.3670

0.3670

0.3670

0.5291

0.5291

0.4647

0.4647

0.4419

0.4419

0.4419

0.4419

0.4155

0.4155

0.4155

0.4155

0.5187

0.5187

0.4829

0.4829

0.0773

0.0773

0.0773

0.0773

0.0740

0.0740

0.0740

0.0740

0.1570

0.1570

0.1036

0.1036

0.1828

0.1828

0.1828

0.1828

0.5904

0.6593

0.6593

0.6593

0.6593

0.6888

0.6888

0.7332

0.7332

0.5530

0.5530

0.5530

0.5530

0.5038

0.5038

0.5038

0.5038

0.7032

0.7032

0.6197

0.6197

0.0401

0.0401

0.0401

0.0401

0.0341

0.0341

0.0341

0.0341

0.2276

0.2276

0.0697

0.0697

0.1343

0.1343

0.1343

0.1343

0.9892

0.8012

0.6305

0.8012

0.6305

0.8553

0.9895

0.9579

0.9945

0.0002

0.0040

0.0002

0.0040

0.0001

0.0022

0.0001

0.0022

0.0002

0.0040

0.0002

0.0050

0.1750

0.4218

0.1750

0.4218

0.2478

0.5225

0.2478

0.5225

0.1742

0.4226

0.2402

0.5142

0.3434

0.6335

0.3434

0.6335

1.52e-01

1.50e-01

1.18e-01

1.50e-01

1.18e-01

2.53e-01

2.93e-01

2.59e-01

2.69e-01

3.28e-05

7.79e-04

3.28e-05

7.79e-04

1.17e-05

3.64e-04

1.17e-05

3.64e-04

5.09e-05

1.21e-03

5.79e-05

1.24e-03

1.57e-04

3.78e-04

1.57e-04

3.78e-04

1.73e-04

3.64e-04

1.73e-04

3.64e-04

3.29e-03

7.99e-03

6.29e-04

1.35e-03

4.75e-03

8.76e-03

4.75e-03

8.76e-03

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

6

7

7

7

7

7

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

Table .1 (cont’d)

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

JLM

0.5290

default

0.5290

JLM

0.5290

default

0.5290

JLM

0.7825

default

0.7825

JLM

0.6102

default

0.6102

JLM

0.7941

default

0.7941

JLM

0.7941

default

0.7941

JLM

0.8046

default

0.8046

JLM

0.8046

default

0.8046

JLM

0.8181

default

0.8181

JLM

0.8145

default

0.8145

JLM

0.6735

default

0.6735

JLM

0.6735

default

0.6735

JLM

0.6440

default

0.6440

JLM

0.6440

default

0.6440

JLM

0.8053

default

0.8053

JLM

0.7540

default

0.7540

JLM

0.4978

default

0.4978

JLM

0.4978

default

0.4978

JLM

0.5072

135

0.1685

0.1685

0.1685

0.1685

0.3546

0.3546

0.2481

0.2481

0.4369

0.4369

0.4369

0.4369

0.4883

0.4883

0.4883

0.4883

0.5647

0.5647

0.5357

0.5357

0.2304

0.2304

0.2304

0.2304

0.2495

0.2495

0.2495

0.2495

0.4060

0.4060

0.3112

0.3112

0.2143

0.2143

0.2143

0.2143

0.2039

0.1327

0.1327

0.1327

0.1327

0.5466

0.5466

0.2417

0.2417

0.6438

0.6438

0.6438

0.6438

0.7257

0.7257

0.7257

0.7257

0.7118

0.7118

0.7180

0.7180

0.2754

0.2754

0.2754

0.2754

0.3173

0.3173

0.3173

0.3173

0.6619

0.6619

0.4025

0.4025

0.2585

0.2585

0.2585

0.2585

0.2409

0.6498

0.8918

0.6498

0.8918

0.3418

0.6345

0.4509

0.7410

0.1850

0.4370

0.1850

0.4370

0.4489

0.7395

0.4489

0.7395

0.1845

0.4382

0.2521

0.5299

0.2794

0.5614

0.2794

0.5614

0.5540

0.8257

0.5540

0.8257

0.2783

0.5625

0.3726

0.6664

0.2180

0.4843

0.2180

0.4843

0.5508

7.68e-03

1.05e-02

7.68e-03

1.05e-02

5.19e-02

9.63e-02

1.65e-02

2.71e-02

4.13e-02

9.76e-02

4.13e-02

9.76e-02

1.28e-01

2.11e-01

1.28e-01

2.11e-01

6.07e-02

1.44e-01

7.90e-02

1.66e-01

1.19e-02

2.40e-02

1.19e-02

2.40e-02

2.82e-02

4.21e-02

2.82e-02

4.21e-02

6.02e-02

1.22e-01

3.52e-02

6.29e-02

6.01e-03

1.34e-02

6.01e-03

1.34e-02

1.37e-02

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

7

7

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

9

9

1

1

1

1

1

1

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

Table .1 (cont’d)

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

default

0.5072

JLM

0.5072

default

0.5072

JLM

0.7489

default

0.7489

JLM

0.5514

default

0.5514

JLM

0.5204

default

0.5204

JLM

0.4857

default

0.4857

JLM

0.5472

default

0.5472

JLM

0.4750

default

0.4750

JLM

0.7238

default

0.7238

JLM

0.5199

default

0.5199

JLM

0.5734

default

0.5734

JLM

0.5734

default

0.5734

JLM

0.5770

default

0.5770

JLM

0.5770

default

0.5770

JLM

0.7777

default

0.7777

JLM

0.6611

default

0.6611

JLM

0.3036

default

0.3036

JLM

0.3036

default

0.3036

JLM

0.1869

default

0.1869

136

0.2039

0.2039

0.2039

0.3448

0.3448

0.2550

0.2550

0.1786

0.1786

0.1550

0.1550

0.1742

0.1742

0.1525

0.1525

0.2818

0.2818

0.1905

0.1905

0.2016

0.2016

0.2016

0.2016

0.1886

0.1886

0.1886

0.1886

0.3632

0.3632

0.2617

0.2617

0.0750

0.0750

0.0750

0.0750

0.0707

0.0707

0.2409

0.2409

0.2409

0.5994

0.5994

0.3248

0.3248

0.1618

0.1618

0.1033

0.1033

0.1479

0.1479

0.0996

0.0996

0.4360

0.4360

0.1598

0.1598

0.1693

0.1693

0.1693

0.1693

0.1675

0.1675

0.1675

0.1675

0.5763

0.5763

0.2557

0.2557

0.0472

0.0472

0.0472

0.0472

0.0454

0.0454

0.8250

0.5508

0.8250

0.2172

0.4854

0.2952

0.5832

0.0397

0.1618

0.0460

0.1783

0.0357

0.1508

0.0414

0.1664

0.0395

0.1623

0.0653

0.2251

0.2693

0.5484

0.2693

0.5484

0.4619

0.7492

0.4619

0.7492

0.2681

0.5495

0.3599

0.6524

0.4802

0.3451

0.4802

0.3451

0.1260

0.0874

2.06e-02

1.37e-02

2.06e-02

3.36e-02

7.51e-02

1.35e-02

2.66e-02

5.96e-04

2.43e-03

3.58e-04

1.39e-03

5.03e-04

2.13e-03

2.99e-04

1.20e-03

3.52e-03

1.44e-02

1.03e-03

3.56e-03

5.27e-03

1.07e-02

5.27e-03

1.07e-02

8.42e-03

1.37e-02

8.42e-03

1.37e-02

4.37e-02

8.95e-02

1.59e-02

2.89e-02

5.16e-04

3.71e-04

5.16e-04

3.71e-04

7.56e-05

5.25e-05

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

Table .1 (cont’d)

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

JLM

0.1869

default

0.1869

JLM

0.5168

default

0.5168

JLM

0.4158

default

0.4158

JLM

0.5129

default

0.5129

JLM

0.5129

default

0.5129

JLM

0.5509

default

0.5509

JLM

0.5509

default

0.5509

JLM

0.5929

default

0.5929

JLM

0.5875

default

0.5875

JLM

0.0633

default

0.0633

JLM

0.0633

default

0.0633

JLM

0.0572

default

0.0572

JLM

0.0572

default

0.0572

JLM

0.0789

default

0.0789

JLM

0.0726

default

0.0726

JLM

0.1498

default

0.1498

JLM

0.1498

default

0.1498

JLM

0.1403

default

0.1403

JLM

0.1403

137

0.0707

0.0707

0.1586

0.1586

0.1247

0.1247

0.1558

0.1558

0.1558

0.1558

0.1469

0.1469

0.1469

0.1469

0.2068

0.2068

0.1936

0.1936

0.0205

0.0205

0.0205

0.0205

0.0193

0.0193

0.0193

0.0193

0.0344

0.0344

0.0262

0.0262

0.0464

0.0464

0.0464

0.0464

0.0422

0.0422

0.0422

0.0454

0.0454

0.1913

0.1913

0.1039

0.1039

0.1336

0.1336

0.1336

0.1336

0.1113

0.1113

0.1113

0.1113

0.2309

0.2309

0.1912

0.1912

0.0026

0.0026

0.0026

0.0026

0.0024

0.0024

0.0024

0.0024

0.0086

0.0086

0.0044

0.0044

0.0124

0.0124

0.0124

0.0124

0.0117

0.0117

0.0117

0.1260

0.0874

0.5020

0.3589

0.3803

0.2665

0.2266

0.4995

0.2266

0.4995

0.1591

0.4013

0.1591

0.4013

0.2193

0.4910

0.2553

0.5379

0.9907

0.8891

0.9907

0.8891

0.9603

0.8246

0.9603

0.8246

0.9942

0.8983

0.9651

0.8311

0.8608

0.6922

0.8608

0.6922

0.6378

0.4743

0.6378

7.56e-05

5.25e-05

7.87e-03

5.63e-03

2.05e-03

1.44e-03

2.42e-03

5.33e-03

2.42e-03

5.33e-03

1.43e-03

3.62e-03

1.43e-03

3.62e-03

6.21e-03

1.39e-02

5.55e-03

1.17e-02

3.37e-06

3.03e-06

3.37e-06

3.03e-06

2.52e-06

2.17e-06

2.52e-06

2.17e-06

2.31e-05

2.08e-05

7.98e-06

6.88e-06

7.39e-05

5.95e-05

7.39e-05

5.95e-05

4.41e-05

3.28e-05

4.41e-05

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

6

7

7

7

7

7

7

7

7

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

Table .1 (cont’d)

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

default

0.1403

JLM

0.2640

default

0.2640

JLM

0.2071

default

0.2071

JLM

0.4847

default

0.4847

JLM

0.4847

default

0.4847

JLM

0.5033

default

0.5033

JLM

0.5033

default

0.5033

JLM

0.5957

default

0.5957

JLM

0.6019

default

0.6019

JLM

0.2278

default

0.2278

JLM

0.2278

default

0.2278

JLM

0.1900

default

0.1900

JLM

0.1900

default

0.1900

JLM

0.3249

default

0.3249

JLM

0.2584

default

0.2584

JLM

0.1145

default

0.1145

JLM

0.1145

default

0.1145

JLM

0.1045

default

0.1045

JLM

0.1045

default

0.1045

138

0.0422

0.0871

0.0871

0.0709

0.0709

0.1357

0.1357

0.1357

0.1357

0.1501

0.1501

0.1501

0.1501

0.2289

0.2289

0.1969

0.1969

0.0558

0.0558

0.0558

0.0558

0.0566

0.0566

0.0566

0.0566

0.1036

0.1036

0.0821

0.0821

0.0435

0.0435

0.0435

0.0435

0.0416

0.0416

0.0416

0.0416

0.0117

0.0447

0.0447

0.0261

0.0261

0.1175

0.1175

0.1175

0.1175

0.1587

0.1587

0.1587

0.1587

0.3039

0.3039

0.2111

0.2111

0.0215

0.0215

0.0215

0.0215

0.0219

0.0219

0.0219

0.0219

0.0701

0.0701

0.0363

0.0363

0.0119

0.0119

0.0119

0.0119

0.0117

0.0117

0.0117

0.0117

0.4743

0.8758

0.7064

0.7857

0.6094

0.9629

0.8298

0.9629

0.8298

0.7754

0.6008

0.7754

0.6008

0.9704

0.8411

0.9206

0.7637

0.9097

0.7513

0.9097

0.7513

0.7140

0.5422

0.7140

0.5422

0.9218

0.7644

0.8468

0.6738

0.9590

0.8217

0.9590

0.8217

0.7359

0.5606

0.7359

0.5606

3.28e-05

9.01e-04

7.27e-04

3.01e-04

2.34e-04

7.44e-03

6.41e-03

7.44e-03

6.41e-03

9.30e-03

7.20e-03

9.30e-03

7.20e-03

4.02e-02

3.48e-02

2.30e-02

1.91e-02

2.49e-04

2.06e-04

2.49e-04

2.06e-04

1.68e-04

1.28e-04

1.68e-04

1.28e-04

2.17e-03

1.80e-03

6.52e-04

5.19e-04

5.68e-05

4.87e-05

5.68e-05

4.87e-05

3.74e-05

2.85e-05

3.74e-05

2.85e-05

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

9

9

1

1

1

1

1

1

1

1

1

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

Table .1 (cont’d)

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

JLM

0.1738

default

0.1738

JLM

0.1389

default

0.1389

JLM

0.1443

default

0.1443

JLM

0.1746

default

0.1746

JLM

0.1804

default

0.1804

JLM

0.1628

default

0.1628

JLM

0.2219

default

0.2219

JLM

0.2017

default

0.2017

JLM

0.2159

default

0.2159

JLM

0.2159

default

0.2159

JLM

0.2056

default

0.2056

JLM

0.2056

default

0.2056

JLM

0.3073

default

0.3073

JLM

0.2890

default

0.2890

JLM

0.7673

default

0.7673

JLM

0.7673

default

0.7673

JLM

0.7323

default

0.7323

JLM

0.7323

default

0.7323

JLM

0.8081

139

0.0770

0.0770

0.0593

0.0593

0.0449

0.0449

0.0410

0.0410

0.0441

0.0441

0.0403

0.0403

0.0691

0.0691

0.0517

0.0517

0.0514

0.0514

0.0514

0.0514

0.0489

0.0489

0.0489

0.0489

0.0948

0.0948

0.0722

0.0722

0.3867

0.3867

0.3867

0.3867

0.3952

0.3952

0.3952

0.3952

0.5319

0.0464

0.0464

0.0242

0.0242

0.0145

0.0145

0.0128

0.0128

0.0173

0.0173

0.0138

0.0138

0.0439

0.0439

0.0202

0.0202

0.0208

0.0208

0.0208

0.0208

0.0178

0.0178

0.0178

0.0178

0.0729

0.0729

0.0355

0.0355

0.4275

0.4275

0.4275

0.4275

0.5677

0.5677

0.5677

0.5677

0.7389

0.9670

0.8331

0.9137

0.7526

0.9455

0.9972

0.9616

0.9928

0.9343

0.9987

0.9518

0.9958

0.9368

0.9983

0.9861

0.9753

0.9060

0.7473

0.9060

0.7473

0.7648

0.5915

0.7648

0.5915

0.9184

0.7606

0.8425

0.6698

0.3713

0.6677

0.3713

0.6677

0.9942

0.9607

0.9942

0.9607

0.3685

6.00e-04

5.17e-04

1.82e-04

1.50e-04

8.86e-05

9.34e-05

8.80e-05

9.08e-05

1.29e-04

1.38e-04

8.63e-05

9.03e-05

6.31e-04

6.72e-04

2.07e-04

2.05e-04

2.09e-04

1.72e-04

2.09e-04

1.72e-04

1.37e-04

1.06e-04

1.37e-04

1.06e-04

1.95e-03

1.61e-03

6.25e-04

4.97e-04

4.71e-02

8.47e-02

4.71e-02

8.47e-02

1.63e-01

1.58e-01

1.63e-01

1.58e-01

1.17e-01

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

4

4

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

Table .1 (cont’d)

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

default

0.8081

JLM

0.8008

default

0.8008

JLM

0.8341

default

0.8341

JLM

0.8341

default

0.8341

JLM

0.8320

default

0.8320

JLM

0.8320

default

0.8320

JLM

0.8414

default

0.8414

JLM

0.8407

default

0.8407

JLM

0.4530

default

0.4530

JLM

0.4530

default

0.4530

JLM

0.4214

default

0.4214

JLM

0.4214

default

0.4214

JLM

0.5519

default

0.5519

JLM

0.4760

default

0.4760

JLM

0.6271

default

0.6271

JLM

0.6271

default

0.6271

JLM

0.6350

default

0.6350

JLM

0.6350

default

0.6350

JLM

0.7381

default

0.7381

140

0.5319

0.4783

0.4783

0.4911

0.4911

0.4911

0.4911

0.4784

0.4784

0.4784

0.4784

0.5451

0.5451

0.5200

0.5200

0.1323

0.1323

0.1323

0.1323

0.1312

0.1312

0.1312

0.1312

0.2078

0.2078

0.1645

0.1645

0.2633

0.2633

0.2633

0.2633

0.2416

0.2416

0.2416

0.2416

0.3871

0.3871

0.7389

0.6240

0.6240

0.4870

0.4870

0.4870

0.4870

0.4572

0.4572

0.4572

0.4572

0.6200

0.6200

0.5588

0.5588

0.0508

0.0508

0.0508

0.0508

0.0487

0.0487

0.0487

0.0487

0.1850

0.1850

0.0824

0.0824

0.1343

0.1343

0.1343

0.1343

0.1349

0.1349

0.1349

0.1349

0.4340

0.4340

0.6738

0.5039

0.7975

0.0000

0.0006

0.0000

0.0006

0.0000

0.0003

0.0000

0.0003

0.0000

0.0005

0.0000

0.0007

0.0119

0.0751

0.0119

0.0751

0.0203

0.1078

0.0203

0.1078

0.0117

0.0758

0.0180

0.1010

0.0662

0.2270

0.0662

0.2270

0.2051

0.4668

0.2051

0.4668

0.0646

0.2272

2.14e-01

1.20e-01

1.91e-01

1.98e-06

1.11e-04

1.98e-06

1.11e-04

6.53e-07

5.07e-05

6.53e-07

5.07e-05

2.70e-06

1.56e-04

3.26e-06

1.68e-04

3.62e-05

2.29e-04

3.62e-05

2.29e-04

5.48e-05

2.90e-04

5.48e-05

2.90e-04

2.48e-04

1.61e-03

1.16e-04

6.52e-04

1.47e-03

5.03e-03

1.47e-03

5.03e-03

4.25e-03

9.66e-03

4.25e-03

9.66e-03

8.01e-03

2.82e-02

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

4

5

5

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

6

7

7

7

7

7

7

7

7

7

7

7

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

Table .1 (cont’d)

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

JLM

0.6954

default

0.6954

JLM

0.8231

default

0.8231

JLM

0.8231

default

0.8231

JLM

0.8208

default

0.8208

JLM

0.8208

default

0.8208

JLM

0.8222

default

0.8222

JLM

0.8279

default

0.8279

JLM

0.6819

default

0.6819

JLM

0.6819

default

0.6819

JLM

0.6880

default

0.6880

JLM

0.6880

default

0.6880

JLM

0.7840

default

0.7840

JLM

0.7402

default

0.7402

JLM

0.6568

default

0.6568

JLM

0.6568

default

0.6568

JLM

0.6185

default

0.6185

JLM

0.6185

default

0.6185

JLM

0.7514

default

0.7514

JLM

0.7254

141

0.3223

0.3223

0.4822

0.4822

0.4822

0.4822

0.5158

0.5158

0.5158

0.5158

0.5742

0.5742

0.5517

0.5517

0.3159

0.3159

0.3159

0.3159

0.3103

0.3103

0.3103

0.3103

0.4414

0.4414

0.3730

0.3730

0.2637

0.2637

0.2637

0.2637

0.2533

0.2533

0.2533

0.2533

0.3744

0.3744

0.3146

0.2294

0.2294

0.5633

0.5633

0.5633

0.5633

0.6566

0.6566

0.6566

0.6566

0.7216

0.7216

0.6661

0.6661

0.2500

0.2500

0.2500

0.2500

0.2857

0.2857

0.2857

0.2857

0.5535

0.5535

0.3597

0.3597

0.2284

0.2284

0.2284

0.2284

0.2579

0.2579

0.2579

0.2579

0.5157

0.5157

0.3567

0.0967

0.2945

0.0379

0.1559

0.0379

0.1559

0.1507

0.3818

0.1507

0.3818

0.0366

0.1549

0.0551

0.2022

0.0513

0.1920

0.0513

0.1920

0.1620

0.4028

0.1620

0.4028

0.0499

0.1919

0.0752

0.2500

0.0560

0.2004

0.0560

0.2004

0.2423

0.5099

0.2423

0.5099

0.0541

0.1992

0.0811

4.97e-03

1.51e-02

8.47e-03

3.49e-02

8.47e-03

3.49e-02

4.19e-02

1.06e-01

4.19e-02

1.06e-01

1.25e-02

5.27e-02

1.68e-02

6.15e-02

2.76e-03

1.03e-02

2.76e-03

1.03e-02

9.88e-03

2.46e-02

9.88e-03

2.46e-02

9.57e-03

3.68e-02

7.46e-03

2.48e-02

2.21e-03

7.93e-03

2.21e-03

7.93e-03

9.79e-03

2.06e-02

9.79e-03

2.06e-02

7.85e-03

2.89e-02

6.60e-03

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

7

8

8

8

8

8

8

8

8

8

8

8

8

9

9

9

9

9

9

9

9

9

9

9

9

3

1

1

1

1

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

3

3

3

3

Table .1 (cont’d)

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

y

y

n

n

default

0.7254

JLM

0.6343

default

0.6343

JLM

0.6427

default

0.6427

JLM

0.6543

default

0.6543

JLM

0.6266

default

0.6266

JLM

0.7286

default

0.7286

JLM

0.6855

default

0.6855

JLM

0.6802

default

0.6802

JLM

0.6802

default

0.6802

JLM

0.6559

default

0.6559

JLM

0.6559

default

0.6559

JLM

0.7808

default

0.7808

JLM

0.7282

default

0.7282

0.3146

0.2487

0.2487

0.2310

0.2310

0.2480

0.2480

0.2282

0.2282

0.3404

0.3404

0.2694

0.2694

0.2770

0.2770

0.2770

0.2770

0.2671

0.2671

0.2671

0.2671

0.4074

0.4074

0.3324

0.3324

0.3567

0.1493

0.1493

0.1254

0.1254

0.1563

0.1563

0.1266

0.1266

0.3552

0.3552

0.1744

0.1744

0.1628

0.1628

0.1628

0.1628

0.1618

0.1618

0.1618

0.1618

0.4439

0.4439

0.2572

0.2572

0.2586

0.0019

0.0212

0.0022

0.0237

0.0016

0.0193

0.0019

0.0215

0.0018

0.0211

0.0033

0.0313

0.0381

0.1589

0.0381

0.1589

0.0935

0.2856

0.0935

0.2856

0.0372

0.1590

0.0562

0.2083

2.11e-02

4.46e-05

5.00e-04

4.14e-05

4.41e-04

4.18e-05

4.89e-04

3.50e-05

3.90e-04

1.62e-04

1.86e-03

1.05e-04

1.01e-03

1.17e-03

4.87e-03

1.17e-03

4.87e-03

2.65e-03

8.09e-03

2.65e-03

8.09e-03

5.25e-03

2.25e-02

3.50e-03

1.30e-02

142