CONSTRAINING NUCLEAR WEAK INTERACTIONS IN
ASTROPHYSICS AND NEW MANY-CORE ALGORITHMS FOR
NEUROEVOLUTION
By
Christopher James Sullivan

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics — Doctor of Philosophy
Computational Mathematics, Science and Engineering — Dual Major
2018

ABSTRACT
CONSTRAINING NUCLEAR WEAK INTERACTIONS IN ASTROPHYSICS AND
NEW MANY-CORE ALGORITHMS FOR NEUROEVOLUTION
By
Christopher James Sullivan
Weak interactions involving atomic nuclei are critical components in a broad range of astrophysical phenomenon. As allowed Gamow-Teller transitions are the primary path through
which weak interactions in nuclei operate in astrophysical contexts, the constraint of these
nuclear transitions is an important goal of nuclear astrophysics.
In this work, the charged current nuclear weak interaction known as electron capture is
studied in the context of stellar core-collapse supernovae (CCSNe). Specifically, the sensitivity of the core-collapse and early post-bounce phases of CCSNe to nuclear electron capture
rates are examined. Electron capture rates are adjusted by factors consistent with uncertainties indicated by comparing theoretical rates to those deduced from charge-exchange and
-decay measurements. With the aide of such sensitivity studies, the diverse role of electron
capture on thousands of nuclear species is constrained to a few tens of nuclei near N ⇠ 50
and A ⇠ 80 which dictate the primary response of CCSNe to nuclear electron capture. As
electron capture is shown to be a leading order uncertainty during the core-collapse phase
of CCSNe, future experimental and theoretical e↵orts should seek to constrain the rates of
nuclei in this region.
Furthermore, neutral current neutrino-nuclear interactions in the tens-of-MeV energy
range are important in a variety of astrophysical environments including core-collapse supernovae as well as in the synthesis of some of the solar systems rarest elements. Estimates for
inelastic neutrino scattering on nuclei are also important for neutrino detector construction

aimed at the detection of astrophysical neutrinos. Due to the small cross sections involved,
direct measurements are rare and have only been performed on a few nuclei. For this reason, indirect measurements provide a unique opportunity to constrain the nuclear transition
strength needed to infer inelastic neutrino-nucleus cross sections. Herein the (6 Li, 6 Li0 ) inelastic scattering reaction at 100 MeV/u is shown to indirectly select the relevant transitions for
inelastic neutrino-nucleus scattering. Specifically, the probes unique selectivity of isovectorspin transfer excitations ( S = 1,

T = 1,

T z = 0) is demonstrated, thereby allowing the

extraction of Gamow-Teller transition strength in the inelastic channel.
Finally, the development and performance of a newly established technique for the subfield of artificial intelligence known as neuroevolution is described. While separate from the
physics that is discussed, these algorithmic advancements seek to improve the adoption of
machine learning in the scientific domain by enabling neuroevolution to take advantage of
modern heterogeneous compute architectures. Because the evolution of neural network populations o✏oads the choice of specific details about the neural networks to an evolutionary
search algorithm, neuroevolution can increase the accessibility of machine learning. However,
the evolution of neural networks through parameter and structural space presents a novel divergence problem when mapping the evaluation of these networks to many-core architectures.
The principal focus of the algorithm optimizations described herein are on improving the
feed-forward evaluation time when tens-to-hundreds of thousands of heterogeneous neural
networks are evaluated concurrently.

Copyright by
CHRISTOPHER JAMES SULLIVAN
2018

ACKNOWLEDGMENTS

There are many students, sta↵, faculty, friends, family, and others whom I regret having not
listed in these acknowledgments by name. To each and everyone of you, thank you for the
role you have played in this season of my life.
I must begin by thanking my advisor, Remco Zegers. Remco’s leadership has been
unparalleled, knowing not only when to direct and when to withdraw, but also recognizing
potential and encouraging growth in ways that were genuinely for my benefit alone. His
scientific expertise and vision for the field of nuclear astrophysics is as accessible as it is
energizing. Under his guidance I have been able to cultivate my identity as a physicist and
as a computational scientist and have been supported uniformly along both paths. Remco,
your passionate commitment to science, service, and stewardship is remarkable and I am
truly honored to have been exposed to your level of professional investment on a daily basis.
I also owe the other members of the charge-exchange group a debt of gratitude. Every
member of that group has aided in the e↵orts described in this work. Thank you all for your
insight over the years, your willingness to travel with me to Japan for the 6 Li measurement,
and for the intentionality each of you brought to making the group a friendly and welcoming
place to grow.
I must also thank each of the members of my thesis committee, Drs. Dirk Colbry, Sean
Couch, Morten Hjorth-Jensen, Sean Liddick, Hendrik Schatz, (and Ed Brown in my first few
years). They, as much as I, have seen my interests grow and evolve over the past several
years. Thank you for your continued guidance and for providing me with the space to mature
in those areas which I am most energized. Thank you Morten for recognizing my passion
for computational science and suggesting I pursue it as part of my degree. Your relentless
v

optimism and encouragement is contagious.
Many thanks are also in order for Evan O’Connor, whose willingness to partner with an
experimental group to achieve the sensitivity study outlined in this work was exceptional.
His mentorship during our time at CITA and thereafter were some of my first experiences
in collaborative scientific software development. Thank you for your helpful explanations of
the physics and numerics and for being an excellent collaborator.
I am also particularly thankful for the sense of community among the MSU and NSCL
graduate students, and for a number of close relationships I’ve developed over the years. I
had the great fortune of joining MSU in 2012 with Charles Loelius, Eric Lunderberg, and
Sam Lipschutz, who have been constant companions throughout the past six years. It has
been a pleasure to learn, laugh, and complain with each of you. To my officemates, it
was an honor to see you grow, to work beside you, and to be consistently inspired by your
stewardship to our communities. In response to the thesis acknowledgments of Chris Morse:
in fact, suggesting that you pursue a Physics degree at Westmont was likely one of the most
lucrative actions of my professional career. You have been an always willing and capable
source of advice and mentorship over the past decade, and I am thankful for your close
friendship.
The unwavering love and support of my mother, Theresa Sullivan, and my grandparents,
Ronald and Martha Sullivan, cannot be overstated. You have invested in me and my education in ways that only very few people experience. While your involvement in my Ph.D.
has looked di↵erent than when I was young, it was equally foundational. I also must thank
my mother and father in-law, Chris and Jim Hansen, who have been extraordinary in their
selfless support and encouragement. To both the Hansens and the Sullivans, your radical
generosity and love have brought genuine vitality to Rachel and I during many long months
vi

of my degree, and for that I am eternally grateful.
Finally, to Rachel Sullivan, my spouse and closest friend, I owe nearly all of my accomplishments over these past years in Michigan to you. In a period of life that is statistically
plagued with doubt, anxiety, and depression, you have filled my life with continuous joy and
purpose. I am beyond privileged to have had your unceasing love and support at every stage
of my academic career, from secondary school until now. Thank you for your love, your
willingness to uproot with me, and for your encouragement to reach past what was easy and
well beyond what I would have ever been capable of otherwise.

vii

PREFACE
The fence we walked between the years
Did balance us serene
It was a place half in the sky where
In the green of leaf and promising of peach
We’d reach our hands to touch and almost touch the sky
If we could reach and touch, we said,
’Twould teach us, not to, never to, be dead
We ached and almost touched that stu↵;
Our reach was never quite enough.
If only we had taller been
And touched God’s cu↵, His hem,
We would not have to go with them
Who’ve gone before,
Who, short as us, stood as they could stand
And hoped by stretching tall that they might keep their land
Their home, their hearth, their flesh and soul.
But they, like us, were standing in a hole
O, Thomas, will a Race one day stand really tall
Across the Void, across the Universe and all?
And, measured out with rocket fire,
At last put Adam’s finger forth
As on the Sistine Ceiling,
And God’s hand come down the other way
To measure man and find him Good
And Gift him with Forever’s Day?
I work for that
Short man, Large dream
I send my rockets forth between my ears
Hoping an inch of Good is worth a pound of years
Aching to hear a voice cry back along the universal mall:
We’ve reached Alpha Centauri!
We’re tall, O God, we’re tall!
Ray Bradbury
NASA Mariner 9 Symposium
Nov. 12, 1971, Caltech

viii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Nuclear weak interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Accessible machine learning via neuroevolution . . . . . . . . . . . . . . . . .

1
3
7

Chapter 2
2.1
2.2
2.3

2.4

2.5

2.6

The sensitivity of core-collapse supernovae to nuclear electron
capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Astrophysical weak interaction rates . . . . . . . . . . . . . . . . . . . . . . .
Core collapse and the role of electron capture . . . . . . . . . . . . . . . . .
Codes & Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 NuLib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 GR1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Neutrino emission via electron capture . . . . . . . . . . . . . . . . .
Sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Reference simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Species dependent sensitivity . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Systematic variations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3.1 Progenitor model sensitivity . . . . . . . . . . . . . . . . . .
2.4.4 Monte-Carlo variations . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Most important nuclei . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Impact of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Goals for future studies . . . . . . . . . . . . . . . . . . . . . . . . . .
Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Implementation procedure for future updates . . . . . . . . . . . . . .

Chapter 3 The (6 Li, 6 Li0 + ) reaction . . . . . . . . . . . .
3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The CAGRA+Grand Raiden Coincidence Setup .
3.1.1.1 Grand Raiden Spectrometer . . . . . . .
3.1.1.2 CAGRA . . . . . . . . . . . . . . . . . .
3.2 Methods & Analysis . . . . . . . . . . . . . . . . . . . .
3.2.1 Data acquisition and analysis framework . . . . .
3.2.2 Grand Raiden calibrations . . . . . . . . . . . . .
3.2.2.1 Sieve-slit scattering angle reconstruction
3.2.2.2 Energy calibration . . . . . . . . . . . .
3.2.3 CAGRA calibrations . . . . . . . . . . . . . . . .
3.2.3.1 Energy calibration . . . . . . . . . . . .
ix

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

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

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

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

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

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

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

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

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

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

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

12
16
23
25
25
26
27
30
30
33
38
45
48
51
52
53
54
54
57
59
63
65
65
66
70
70
73
73
75
77
77

3.2.4
3.2.5
3.2.6
3.2.7

3.3

3.4

Invariant missing mass reconstruction . . . . . . . . . . . .
Doppler reconstruction . . . . . . . . . . . . . . . . . . . .
Excitation energy resolution . . . . . . . . . . . . . . . . .
Cross-section calculations . . . . . . . . . . . . . . . . . .
3.2.7.1 Normalization of the number of incident particles
3.2.7.2 Multipole Decomposition Analysis . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The 12 C(6 Li, 6 Li0 + ) measurement . . . . . . . . . . . . .
3.3.2 The (6 Li,6 Li’) unit cross-section . . . . . . . . . . . . . . .
3.3.3 Competition with isoscalar resonances . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4
4.1
4.2
4.3

4.4
4.5
4.6

Many-core algorithms for topologically divergent
works . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Applications in physics . . . . . . . . . . . . . . . . . . . . . .
Network Structures & Evaluation . . . . . . . . . . . . . . . .
Directed acyclic graph concurrency . . . . . . . . . . . . . . .
4.3.1 Topological connection sort . . . . . . . . . . . . . . .
4.3.2 Neural network operations . . . . . . . . . . . . . . . .
4.3.2.1 Neural network node lifecycle . . . . . . . . .
4.3.3 Generalized concurrent neural network evaluation . . .
Composite population evaluation . . . . . . . . . . . . . . . .
CUDA implementation . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Performance results . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

. 79
. 82
. 83
. 86
. 88
. 89
. 95
. 95
. 103
. 108
. 112

neural net. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .

114
118
121
124
125
130
131
132
135
138
140
144

Summary & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 146

BIBLIOGRAPHY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

x

LIST OF TABLES

Table 2.1: Density, temperature and mass ranges for the compiled weak rate set . . .

18

Table 4.1: Neural network connection dependency comparator . . . . . . . . . . . . . 129

xi

LIST OF FIGURES

Figure 1.1: Diagrams of the weak interactions studied in this work, and the corresponding strong-force reactions that were utilized to indirectly extract the
relevant nuclear transitions. As can be seen in the Charged-Current (CC)
column of the diagram, even though nuclear charge-exchange is mediated
by a ⇡ + meson instead of the weak W + boson for electron capture, the operator responsible for the nuclear transition, Ô(GT+ ), is the same in both
reactions. This implies that charge-exchange can be used as a surrogate or
indirect measurement of the nuclear matrix element relevant for electron
capture. Similarly, in the Neutral-Current (NC) channel, nuclear probes
can be used to estimate the response of nuclei to inelastic excitation via
neutrinos, if the GT0 , component of the M1 operator can be extracted.
This is the case for certain proton inelastic scattering measurements, and
in chapter 3, 6 Li inelastic scattering is shown to be directly sensitive to
these weak NC transitions. . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 2.1: Chart of the nuclear species depicting the nuclei included in each rate
tabulation as well as the full reach of the weak rate library presented in
this work. The table to which a species belongs is given by the color and
legend in the figure. The Oda set contains rates for lower mass sd-shell
nuclei (light blue), the LMP set contains rates for the intermediate mass
pf -shell nuclei (green), and the LMSH set contains rates for the heavier
mass pf g/sdg-shell nuclei near stability (red). The FFN tabulation provides rates across the sd and pf -shells (dark blue). Squares individually
bordered in black are stable nuclei. The tables are mutually exclusive
except for FFN which spans many nuclear shells. To distinguish between
nuclei with rates from FFN and another table, the border of the FFN set
has been outlined with a black and white line. . . . . . . . . . . . . . .

19

Figure 2.2: Panel (a): Q-value dependence of electron-capture rates at two points
along a core-collapse trajectory. The scattered points are tabulated (shellmodel and SMMC) rates for each electron-capture reaction, while the
black points are the approximate rates given by Eq. 2.1. Panel (b): The
residual di↵erences between log10 of the shell-model rates and the approximate rates for each nucleus in the weak-rate library. An example residual
is indicated on panel (a). When the density and temperature of a simulation evolve outside the range of the rate tables (see Table 2.1), rates are
calculated via the approximate routines in order to avoid an artificial cut
o↵ imposed by the table boundaries. Rates are estimated between density
and temperature grid points via monotonic cubic-spline interpolation as
described by Ste↵en [63]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

xii

Figure 2.3: The average nuclear mass (divided by two), charge, and electron-capture
rate versus central density for the three EOS utilized in this study. The
colors indicate di↵erent EOS, while the line style indicate which quantity
is plotted. All three EOS have nearly identical abundance distributions
up to densities of 2·1012 g cm 3 . Beyond this point the TMA EOS has a
heavier and slightly more neutron rich mass distribution compared to both
SFHo and DD2, but maintains a comparable average electron capture rate
overall. These simulations each utilize the s15WW95 progenitor. . . . . .
Figure 2.4: The contribution of nuclear electron capture to the change of the matter
electron-fraction with time. The contours are the binned sums of |Ẏe | for
each species in several mass regions. For reference, the central density at
t tb = -20, -10, -5, -2, and -1 ms is 1.41 · 1011 , 4.06 · 1011 , 1.42 · 1012 ,
8.49 · 1012 , 3.30 · 1013 g cm 3 respectively. . . . . . . . . . . . . . . . .

32

35

Figure 2.5: Top 500 electron capturing nuclei with the largest absolute change to the
electron fraction up to neutrino trapping. The color scale indicates |Ẏe |
integrated up to the trapping time, occurring when ⇢c ⇠ 2 · 1012 g cm 3 ,
such that the total electron-fraction at this point is equal to its initial
value less the sum of Ye , the plotted quantity, over all nuclides. Calculations are based on the s15WW95+SFHo reference simulation. The black
contour that runs parallel to the valley of stability on the neutron-rich side
is the boundary between measured [77] and theoretical masses used in the
approximate rate estimates of Eqs. 2.1 and 2.2. The rectangular outline
indicates the size of the sampling region used in the statistical resampling
study, and also the set of nuclei which exhibited the largest changes to
the simulations when excluded from the electron-capture calculations. . .

36

Figure 2.6: Projection of the (trapped) lepton fraction at ⇢c = 4 · 1012 g cm 3 as
a function of the electron-capture rate scaling factor for progenitor+EOS
reference simulation. In all the cases the lepton fraction begins to increase
if the capture rate becomes too high because of a dramatic increase in
the electron neutrino absorption cross section. The asymmetry seen here
indicates that those quantities which depend on Yl/e are likely to be more
sensitive to a reduction of the electron-capture rates due to a systematic
overestimate in the base rates, than they are to an increase due to an
underestimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Figure 2.7: Comparison of the central electron, electron-neutrino (left) and lepton
(right) fractions in which the nuclear electron-capture rate for every species
has been scaled by factors shown in the legend. Warmer colors indicate a
higher overall electron capture rate, and cooler colors indicate a lower rate.
The dashed black line indicates the reference s15WW95+SFHo simulation. 41

xiii

Figure 2.8: The electron fraction, entropy, density, and velocity as a function of enclosed mass at four times during a core-collapse simulation, spanning 6
ms around bounce, including the collapse phase just after the onset of
neutrino trapping (a), core bounce (b), and 1 ms (c) and 5 ms (d) after
bounce during which the shock has begun its outward trajectory. The
reference simulation (s15WW95+SFHo) is shown in dashed black. Simulations shown in color have electron-capture rates scaled systematically
for all species by factors of 10, 4, 2, 0.5, 0.25, and 0.1. Core bounce
(t tb = 0) shown in panel (b) is defined as when the entropy at the
shock front exceeds 3.0 kb /baryon. . . . . . . . . . . . . . . . . . . . . .

42

Figure 2.9: The neutrino luminosity as measured at a radius of 500 km as a function
of time after bounce in the s15WW95+SFHo simulation set. Electroncapture rate scaling factors are shown in the legend, where contours with
warmer colors have higher rates, and cooler colors have lower rates. While
the peak electron-neutrino luminosity is considered particularly stable
across core-collapse simulations, it varies significantly with variations of
the electron-capture rates on medium-heavy nuclei. When the rates are
at their lowest (⇥0.1 case), the shock reaches the neutrinosphere more
quickly than in the other simulations. This results in a larger luminosity
in the peak electron-neutrino burst because more ⌫e s are able stream out
of the core at early times. The opposite is true when the rates are higher,
the neutrinosphere and shock converge much more slowly, and so the neutrinos spend more time di↵using out of the inner core, reducing the peak
luminosity but distributing it out to later times. . . . . . . . . . . . . . .

44

Figure 2.10: The full range of sensitivity of the PNS inner-core mass, central entropy,
and central temperature at bounce as well as the peak ⌫e -luminosity, the
peak average ⌫e energy, and the average ⌫e energy prior to neutrino trapping, owing to variations of the progenitor model and electron-capture
rates. Thirty two progenitors were utilized from the WH07 model set of
Woosley and Heger [79] for producing the progenitor bars (red) in the
figure. Each bar of the electron-capture rate variations derives from simulations where the rates have been systematically scaled by factors of 10,
4, 2, 0.5, 0.25, and 0.1. The horizontal tick represents the value of the
reference simulation for the tested Progenitor + EOS combination. The
window ranges are chosen so that the progenitor sensitivity bars are of
equal size across each of the plotted parameters. . . . . . . . . . . . . .

46

xiv

Figure 2.11: Lepton fraction band as a function of central density for the Monte Carlo
(MC) described in the text. The band represents data points from seven
MC simulations based on the s15WW95+SFHo reference, where the band
width represents the min to max range of Yl . The rates are varied by
drawing from the Q-value dependent residual function shown in Figure 2.2
after binning it into 2.5 MeV energy bins. This binning was chosen as it
best tracks the Q-value dependence of the residual distribution. . . . . .

50

Figure 2.12: Updated version of Figure 2.1 showing the addition of the recently added
rate tabulations since the libraries first implementation in [2]. . . . . . .

55

Figure 3.1: A simplified level diagram of 6 Li based on Ref. [103]. The ↵ decay of the
J ⇡ = 0+ , T = 1 state at Ex = 3.56 MeV is parity-forbidden, and thus this
state mainly decays to the ground state via emission. . . . . . . . . .

63

Figure 3.2: The primary facility beam line layout for the Research Center for Nuclear
Physics (RCNP) located at Osaka University, Japan. In this experiment,
the 6 Li beam was transported from the ring cyclotron (shown in green)
into the WS hall. It was then transported achromatically to the Grand
Raiden target position. See text for details. . . . . . . . . . . . . . . . .

64

Figure 3.3: Top panel: Layout of the Grand Raiden Spectrometer which was used to
momentum analyze the inelastically scattered 6 Li nuclei. The spectrometer was configured with seven magnets in the QSQDMDD configuration
(where Q, D, S, and M are Quadrupole, Dipole, Sextupole, and Multipole
magnets, respectively). Bottom panel: configuration of the spectrometer
end station and focal-plane detectors. Figure adapted from [106, 95]. See
text for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 3.4: The CAGRA array of HPGe clover detectors. CAGRA consisted of two
rings of clover detectors with 8 detector slots at 90 degrees and 4 at 135
degrees. There were also 4 forward slots for LaBr3 detectors at 45 degrees,
but they were not used in this work due to their large angular coverage
which made Doppler reconstruction challenging. . . . . . . . . . . . . .

68

Figure 3.5: The range of energies the 3.56 MeV -ray from 6 Li in the laboratory frame.
The -ray emitted from the de-excitation of the 6 Li[0+; T = 1; 3.56M eV ]
state is 3.56 MeV in the rest frame of 6 Li. In the laboratory frame this
gamma is Doppler boosted due to the momentum of the 6 Li ( ⇠0.43),
according to its emission angle in the lab-frame. Shown in red are the
clover crystal positions in the CAGRA array. Each clover has crystals at
two distinct azimuthal angles. Thus, the two rings of clover detectors span
for separate angular ranges. For the purpose of Doppler reconstruction,
the centroid of the clover crystal is used as the gamma interaction point.

69

xv

Figure 3.6: The analysis software stack that was used in the RCNP CAGRA+Grand
Raiden campaign. Dashed blue lines show application and library boundaries. Solid arrows that cross these boundaries represent communication
by local memory, shared memory or disk I/O. See text for details. . . .

71

Figure 3.7: The image of the sieve slit in the focal plane for 3 separate magnetic
rigidity settings (left) and the reconstructed scattering angles at the target
using a least squares fit to Eqs. 3.6 & 3.7 (right). . . . . . . . . . . . . .

74

Figure 3.8: The time-dependence of the Grand Raiden energy calibration for the centroid of the 12 C[2-;T=1;19.4 MeV] state in the 12 C(6 Li,6 Li’) excitation
energy spectrum. Due to the coincident excitation of the 6 Li ejectile to
its 3.56 MeV state, this isovector state in 12 C appears at ⇠23 MeV (=19.4
MeV + 3.56 MeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Figure 3.9: Rate dependence of the CAGRA clover crystals. Panel (a) shows an example count rate spectrum as a function of time for the CAGRA array.
Panel (b) details the rate dependence of the energy calibration o↵set for
a specific HPGe crystal in the array. This trend was established for each
crystal and then used to correct the energy calibration on a second-bysecond basis. Panels (c) and (d) show the rate dependent gamma-ray
energy and (corrected) rate independent gamma-ray energy for the 511
keV line, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Figure 3.10: The kinematics of the 6 Li reaction.

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

80

Figure 3.11: (a) A comparison of the contribution of the intrinsic energy resolution,
opening angle uncertainty, and velocity spread of the ejectile, to the overall uncertainty in the Doppler reconstructed gamma-ray energy for the
detection of the 3.56 MeV de-excitation gamma from 6 Li in CAGRA. The
uncertainty in the emission angle of the -ray is the dominant component
of the overall resolution. (b) The resulting estimated resolution of the
reconstructed invariant 6 Li ejectile given the resolution of the measured
de-excitation gamma. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

xvi

Figure 3.12: Comparison of the simulated reconstruction of the 12 C[1+;T=1;15.1 MeV]
excited state with the measured result. In the left panel, the purple line
represents the (simulated) best-case scenario for the missing mass reconstruction of the 15.1 MeV peak, if the invariant 6 Li[0+;T=1;3.56 MeV] is
ignored. By comparing this with the data in the right panel, it is apparent
that the data follows a much broader distribution. In order to reproduce
the data, the simulated missing mass reconstruction of this state in 12 C
(purple) had to be smeared with a gaussian of a variety of widths. The
shape of the smeared state was interpolated and fit to the data using a
normalization factor. The best fit was found with a gaussian smearing
of FWHM=1.75 MeV (shown by the black scatter crosses and red line in
the left and right panels, respectively). The smearing that was required
suggests that the intrinsic beam energy resolution was 1.75 MeV. . . . .

85

Figure 3.13: Calculated angular distributions for isovector spin-transfer excitations of
di↵erent orbital angular momentum transfer for the 12 C(6 Li,6 Li0 ) inelastic scattering measurement. Each panel is the angular distribution for a
particular angular momentum multipole, and the di↵erent color lines are
the excitation energy (Q-value) dependence of the angular distribution.

93

Figure 3.14: The gamma-ray energy and target excitation energy matrix for the 12 C(6 Li, 6 Li0 +
) reaction. In the top panel, the gamma energies shown are those measured in the detectors, without any transformation applied and in the
bottom panel, the gamma energies have been Doppler reconstructed according to the clover crystal angular positions. Both panels are drawn as
a function of the excitation energy in 12 C which is calculated via the missing mass formalism. The red arrow in the top panel indicates the width
of the 12 C(15.1 MeV) state owing to the momentum kick that is imparted
to the 6 Li and which is not accounted for in the missing-mass–only reconstruction. This width is analogous to the FWHM of the excitation
energy spectrum shown in Figure 3.12. By comparing the two panels, the
e↵ect of the doppler reconstruction can be seen, where the photopeak and
Compton edge owing to the 3.56 MeV gamma from 6 Li is more prominent in the bottom panel after reconstruction. In addition, gamma decay
from the strong isoscalar resonance (occurring above 20 MeV in excitation
energy) is clearly evident. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Figure 3.15: The Doppler-corrected -ray energy gated on the 12 C[15.1 MeV; T=1]
state for emphasis. The 3.56 MeV -line from the 6 Li[0+; T=1] excited
state is clearly evident. The resolvability of this -ray is crucial for tagging
isovector spin-transfer reactions in the target nucleus. The inset plot
compares the signal to noise achieved with the clover detectors to what is
expected were a gamma-ray tracking detector was utilized. See text for
details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

97

Figure 3.16: The reconstructed excitation energy spectrum of 12 C, for di↵erent angular
bins, when applying the two gates shown in the Doppler-reconstructed
gamma energy spectrum (Fig. 3.15). The red spectrum corresponds to
the 3.56 MeV photopeak gate, and the blue spectrum corresponds to the
sideband, representative of the gamma background in the photopeak gate.
The 12 C[1+;T=1;15.1 MeV] state, which is populated via an isovector
spin-transfer reaction, is clearly seen in the photopeak gated spectra, but
not in the sideband data. . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 3.17: Comparison of the 12 C inelastic scattering singles and coincidence excitation energy dependent double di↵erential cross section for the 12 C(6 Li, 6 Li0 )
and the 12 C(6 Li, 6 Li0 + ) reactions, respectively. The coincidence spectrum shown is the result of the subtraction of the di↵erential cross section
resulting from a gate on the photopeak of the 3.56 MeV -ray and a sideband in the region directly above this gamma (shown as the red and blue
cross-hatched regions in Fig 3.15, respectively). The coincidence spectrum
corresponds to the isovector spin-transfer excitations in 12 C, whereas the
singles spectrum contains all types of transitions but is primarily dominated by isoscalar excitations. See text for details. . . . . . . . . . . . . 100
Figure 3.18: Double di↵erential cross section for the 12 C(6 Li, 6 Li0 + ) reaction as a
function of excitation energy and center of mass scattering angle, in which
the 3.56 MeV state was measured in coincidence. Due to conservation of
spin and isospin, this coincidence measurement extracts the isovector spintransfer response of 12 C. In addition, a multipole decomposition analysis
(MDA) has been performed utilizing DWBA angular distributions to extract the components of orbital angular momentum transfer. Shown in
the right panel are the angular distributions for states in 12 C dominated
by Gamow-Teller transitions (top panel) and spin-dipole transitions (bottom panel). The absolute scale has a systematic uncertainty of ±20% due
to uncertainty in the beam normalization which is not depicted in the
statistical error bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xviii

Figure 3.19: The mass dependence of the Gamow-Teller unit cross-section for 6 Li calculated from theoretical estimates of the inelastic GT strength and DWBA
estimates for the cross section of several nuclei (black). Filled circles indicate that the underlying GT strengths were estimated via shell-model
or normal-modes calculations. A postulated unit cross section massdependence for heavy-ion reactions (Eq. 3.36) is fit to these theoretical unit
cross-sections and is shown in the plot (dashed black line). In addition,
the experimentally derived Gamow-Teller UCS for the inelastic (6 Li,6 Li’)
reaction established in this work is shown alongside an estimated unit
cross section from data available for the (6 Li,6 He) charge-exchange reaction [135]. The error bar shown for the inelastic UCS includes systematic
and statistical uncertainties, while no error on the charge-exchange data
was available. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 3.20: Comparison of the achievable signal-to-noise for the 3.56 MeV -ray emitted from inelastic scattering of 6 Li on 12 C (top panel), 24 Mg (middle
panel), and 93 Nb (bottom panel). The Doppler reconstructed gamma is
resolvable in the 12 C measurement but not in the 24 Mg other measurements. The purple and red line-shapes are the simulated response of the
clover and tracking detectors, respectively. In the 12 Ccase, the purple line
shape is fit to the data, and used to estimate the number of counts in the
3.56 MeV peak. The excitation energy was gated on the 15.1 MeV state
in 12 C which has a B(GT)⇠1. Assuming a comparable B(GT) in 24 Mg
and 93 Nb, over an equal-width range in excitation energy, and accounting
for the change in the unit cross-section with mass, the purple and red lineshapes have been scaled to reproduce the expected number of isovector
counts in the other two measurements. This clearly illustrates that resolving the 6 Li 3.56 MeV gamma becomes increasingly difficult in heavier
systems where the isoscalar giant resonances remain strong. Decay from
these resonances contribute a significant background in the region of the
3.56 MeV -ray from 6 Li, and this background increases in magnitude
with increasing nuclear mass. . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 3.21: The di↵erential cross sections for the 24 Mg(6 Li,6 Li0 ) and the 93 Nb(6 Li,6 Li0 )
singles measurements. The cross sections in both cases are dominated by
isoscalar giant resonances which become larger with heavier mass systems. 110
Figure 3.22: Mass dependence of the calculated zero-degree di↵erential cross section for
the isoscalar giant monopole and dipole resonances (IVGMR and IVGDR,
respectively) corresponding to 100% exhaustion of the energy weighted
sum-rule. These calculations illustrate the increase in the isoscalar crosssection as a function of nuclear mass number and were provided at the
courtesy of Umesh Garg [139]. . . . . . . . . . . . . . . . . . . . . . . . 111

xix

Figure 4.1: The basic components an artificial neural network. Each node (shown as
the circle in the figure) accumulates inputs modulated by the weights of
its inbound connections. After all of the inbound connections have been
accumulated into the node, a non-linear activation function, such as a
logistic sigmoid, transforms the nodes value. The transformed node value
is then used as an output along outbound connections to other nodes in
the network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 4.2: (a) A common feedforward neural-network structure with a single hidden
layer. x0i , x1i , and x2i correspond to the input, hidden and output layers
0 , 1 refer to the connection weight matrices between
respectively, while wij
two layers. (b) An unstructured neural network in which the connections
are not constrained to specific layers. Such network structure is common
in neuroevolution populations. Each connection in (b) is labeled with
an integer indicating the lock-free set in which it can be evaluated. See
section 4.3.1 for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Figure 4.3: An illustration of the structural divergence than can exist in populations
of neural networks evolved via neuroevolution. In this case, each network
has a fixed number of input and output nodes (3 and 1, respectively). . 123
Figure 4.4: An example neural network and two directed acyclic dependency graphs
for the evaluation of the neural network’s connections. The left dependency graph includes only logical dependencies whereas the graph on the
right includes both logical & memory dependencies. Each circle in the
dependency graphs represent connections in the top neural network figure. The arrows in the two dependency graphs indicate the dependency
relationship between the connections . For example, the arrow between
A1 and B1 indicates that connection A1 must be evaluated prior to B1.
By inspecting the neural network, this relationship is evident as connection B1 relies on node 1, which in turn relies on connection A1 (as well as
other connections). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 4.5: Prototype of sorting connections into concurrent lock-free sets. For n
connections, scales as O(n2 ). . . . . . . . . . . . . . . . . . . . . . . . . 128

xx

Figure 4.6: The full directed acyclic dependency graphs for two structurally di↵erent
neural networks. The di↵erent symbols in the graph, squares, circles,
and triangles, represent the three neural network operations: zero-node,
apply-connection, and activate-node, respectively. Even though the two
networks have very di↵erent structure, the algorithm proposed in this
work organizes the network evaluation graphs into evaluations steps that
are component-wise parallel. On the right of the figure, each evaluation
step is shown with the number of SIMD operations that can be performed
simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Figure 4.7: Top panel: A representative example population of neural networks consisting of three members. In classic neuroevolution simulations, each of
these neural networks are separately evaluated, and the correctness of
their outputs is used to judge their fitness. Bottom panel: The same population as in the top panel, but constructed as nodes and connections of
a single composite neural network. On the left, the resulting composite
network in which each of the constituting sub-networks requires heterogeneous inputs for evaluation. On the right, a composite network in which
each of the sub-networks will process the same inputs. . . . . . . . . . . 136
Figure 4.8: NVIDIA CUDA kernels for the proposed concurrent neural network evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 4.9: GPU speed up factor when evaluating a population of N evolved neural
networks concurrently. The ratio of time taken for the sequential evaluation of each network on a single CPU thread to the time for transferring
inputs to the device, evaluation on the GPU, and transferring the outputs
back to the host. With a population of approximately 65k networks, a
performance boost of 50x relative to the serial implementation is observed. 140

xxi

Chapter 1
Introduction
Advances in computational capabilities, astrophysical observations, nuclear theory and experimental methods over the past two decades have enabled a new era of multidisciplinary
research in the fields of nuclear science and computational astrophysics. For example, we
now have the capability of probing the impact of specific nuclear processes to large-scale astrophysical events via sensitivity studies, which can direct new experimental and theoretical
e↵orts [1, 2, 3].
Ultimately, the interdisciplinary nature of these e↵orts is fundamentally self-sustaining.
Experimental measurements for all nuclei relevant to a specific astrophysical context are
often unfeasible due to the large number of nuclei that are involved. Therefore, nuclear
theory must be relied upon for the majority of the relevant nuclear information, where key
measurements can help to constrain theoretical calculations. Then, astrophysical sensitivity
studies–which probe the impact of specific nuclear reactions to astrophysical simulations–can
reveal the key nuclei that should be studied experimentally and theoretically. These new
measurements and calculations then feed back into the astrophysical simulations, constraining the uncertainty of the nuclear physics inputs to the astrophysical phenomenon. Finally,
with high fidelity astrophysical simulations, predictions about multi-messenger astronomical observations from such events can be made and verified at earth based electromagnetic,
neutrino, and gravitational wave observatories.

1

In addition, the dramatic increase in computational capabilities and the increasingly
large amounts of data that have enabled the above cycle are also fueling growth in the
fields of artificial intelligence and machine learning. Specifically, the development of novel
algorithmic techniques in machine learning, which take advantage of these computational
advancements, are beginning to provide new and less expensive solutions to challenging
scientific problems [4, 5].
The research described in this work embodies these principle advancements. In the
following sections, the motivation of the projects that compose this body of work are outlined.
Each subsequent chapter will focus on the specific implementation of what is presented here,
as well as the the scientific results and conclusions that can be drawn. The three research
directions that will be discussed are,
Chapter 2. An investigation into the sensitivity of core-collapse supernovae (CCSNe) to
the nuclear reaction known as electron capture,
Chapter 3. The establishment of a new experimental method, the (6 Li,6 Li’) reaction,
which can provide access to astrophysically relevant neutrino-nucleus reactions,
Chapter 4. The development of many-core algorithms for topologically divergent neural
networks which better enable neuroevolution to compete in the automated machinelearning ecosystem.
While these are quantitatively distinct focuses, each of these projects principally explore
new ways to accelerate the process of scientific discovery: sensitivity analyses can direct
experimental and theoretical e↵orts; new experimental techniques can replace or supplement
previously difficult-to-study avenues of research; and novel high-performance automated

2

machine-learning algorithms can reduce the implementation barrier of artificial intelligence
in the scientific domain.

1.1

Nuclear weak interactions

The physics motivation of the research presented in this work is largely rooted in the topic
of weak interactions with atomic nuclei. The weak interaction is one of the four fundamental
physical forces in nature. Weak interactions that involve nuclei include ± -decay (positron/electron emission), forward and inverse neutrino interactions (nuclear capture of neutrinos
and heavy leptons, respectively), and neutrino-nucleus scattering.
In a terrestrial context, weak-processes play an important role in modern technology.
For example, beta emitting isotopes are commonly used in medical diagnostics, nuclear
medicine, imaging (positron emission tomography; PET-scans), portable energy generation,
natural gas production, and radioactive dating [6].
In astrophysics, however, the influence of the weak interaction is pervasive. This is historically evident in that nearly every new insight that has arisen about the weak interaction
over the course of the past century has been coupled with significant changes in our understanding of astrophysical processes [7]. For example, after the existence of the neutrino was
proposed by Pauli and the first weak-interaction theory was developed by Fermi [8], it was
not long before it was suggested and established that the production of weakly-interacting
neutrinos is the primary source of energy loss in stars [9]. Following the theory of neutralcurrent neutrino interactions with nucleons [10], it was recognized that neutrino scattering in
core-collapse supernovae would create the phenomenon known as neutrino-trapping, which
prevents neutrinos from escaping the collapsing core of a massive star [11, 12] and has a

3

dramatic impact on the evolution of the supernovae.
The class of weak interactions focused on in this work (those involving interactions
with atomic nuclei) are known as semi-leptonic weak interactions, and they fall into two
categories: charged- and neutral-current interactions. Subatomic particles interacting via
charged-current (CC) and neutral-current (NC) weak interactions are mediated by the W ±
and Z 0 bosons of the standard model, respectively. Both types of interactions are studied
in this work, where electron-capture (a CC reaction) is discussed in chapter 2, and inelastic
neutrino-nucleus scattering (a NC reaction) is discussed in chapter 3.
Whereas the unified model for the electro-weak theory can accurately predict weakinteraction cross-sections involving elementary particles, for semi-leptonic weak interactions
involving nuclei, the problem is more complex. By treating the weak-interaction with nuclei
perturbatively, the calculation of weak-interaction cross-sections can be reduced to the manybody nuclear-structure problem well within the domain of nuclear physics. Unfortunately, the
difficulties that arise when solving the many-body nuclear-structure problem often introduce
large uncertainties into the weak reaction rates that are important for astrophysics [7, 13].
This suggests that experimental input is necessary. While measurements cannot be made
for all of the nuclear weak interactions that are important, they can provide invaluable
constraints on theoretical models. This will be discussed in detail in chapter 2.
The complication in the nuclear weak-interaction (the semi-leptonic case) arises principally from the need for detailed information about the nuclear transitions and states that
are involved. However, this relationship also implies that if one is able to attain information about the nuclear many-body configurations and transitions involved in the weak
reaction, the weak-interaction cross-section and reaction rate can be indirectly constrained.
Experimentally, this means that if the direct weak-interaction measurement is difficult or
4

impossible, as is the case for many astrophysically relevant weak reactions, an indirect measurement which probes the same nuclear transitions can be utilized. The ability to indirectly
extract the necessary information to constrain weak reaction rates and cross-sections form
the basis for the experimental programs discussed in the following two chapters.
Nuclear charge-exchange and inelastic scattering, as shown in Figure 1.1, are two independent accelerator-based experimental techniques which are capable of inducing the same
nuclear transitions as CC and NC weak interactions, but are induced via reactions mediated
by the strong force. Also shown in the figure are the two weak reactions studied in this work,
electron capture and inelastic neutrino-nucleus scattering.
The similarity between reactions mediated by the weak and the strong force is evident
when considering the interaction diagrams, also shown in Figure 1.1. In the case of the
electron-capture process, the mediating particle is the W+ boson, and in charge-exchange
reactions the mediating particles are mesons (such as the ⇡ + meson, and others). In both
cases, the interacting nucleus undergoes the same transition between initial and final states,
and so both processes probe the same nuclear matrix elements, albeit with separate couplings
(weak vs strong). An analogous argument can be made for the inelastic neutrino scattering
and the hadronic inelastic scattering shown in the figure; that is, instead of a mediating
Z0 boson, a ⇡ 0 meson (for example) is exchanged and the same nuclear transition occurs.
However, a few subtleties arise in the NC channel that are reserved for discussion in chapter 3.
These indirect techniques are of immense value primarily because the direct measurements are often extremely challenging. For example, very few neutrino-inelastic scattering
measurements on nuclei have been performed to date because of the very small cross-sections
involved (⇠ 10 42 mb/sr). This specific example is the primary motivation for the development of the (6 Li,6 Li0 ) reaction probe, and is discussed at length in chapter 3. Even so,
5

Charged-Current (CC)
Weak interaction

(A,Z-1)*

Neutral-Current (NC)
(A,Z)*

νe
W

O(GT+)

Z0

+

O(GT0)

e-

(A,Z)

(A,Z)

Electron capture

Neutrino inelastic scattering

A *
ZX

A
ZX

A *
ZX

A
ZX

e-

Strong interaction

(A,Z-1)*

(A,Z)*

p

p

+

0

O(M1)

O(GT+)
n

(A,Z)

Proton inelastic scattering

Charge-exchange
A
ZX

n

p

(A,Z)

A *
ZX

A
ZX

p

p

A *
ZX

p

Figure 1.1: Diagrams of the weak interactions studied in this work, and the corresponding
strong-force reactions that were utilized to indirectly extract the relevant nuclear transitions. As can be seen in the Charged-Current (CC) column of the diagram, even though
nuclear charge-exchange is mediated by a ⇡ + meson instead of the weak W + boson for
electron capture, the operator responsible for the nuclear transition, Ô(GT+ ), is the same
in both reactions. This implies that charge-exchange can be used as a surrogate or indirect measurement of the nuclear matrix element relevant for electron capture. Similarly,
in the Neutral-Current (NC) channel, nuclear probes can be used to estimate the response
of nuclei to inelastic excitation via neutrinos, if the GT0 , component of the M1 operator
can be extracted. This is the case for certain proton inelastic scattering measurements, and
in chapter 3, 6 Li inelastic scattering is shown to be directly sensitive to these weak NC
transitions.

6

measurements of all the astrophysically relevant nuclei for which weak reactions are important is an impossible task. Thus, those nuclei which are the most important in astrophysical
simulations should be measured, providing the best possible constraints for the theoretical
methods which are needed to estimate the majority of the weak reactions. It is for this reason
that in chapter 2, a sensitivity analysis of core-collapse supernovae to nuclear electron-capture
rates is described, and the important weakly interacting nuclei are identified.
While chapters 2 & 3 have fundamentally di↵erent focuses, together they represent a
coordinated e↵ort to constrain the nuclear weak-response of nuclei through both the charged
and neutral current channels. As weak interactions with nuclei are critical components of
many astrophysical processes, these constraints strongly impact the field of nuclear astrophysics.

The final chapter of this work departs from discussion of weak interactions in nuclear
astrophysics, and instead focuses on algorithmic advancements in artificial intelligence research. This work aims to lower the accessibility barrier of machine learning, and in chapter 4 evidence will be provided that suggest the application of these methods will improve
the applicability of modern neural-network classification techniques in experimental nuclear
astrophysics.

1.2

Accessible machine learning via neuroevolution

Many e↵orts in recent years have been directed toward increasing the accessibility of machine
learning, through new university programs, internet-based educational opportunities, as well

7

as the development of high-level programming interfaces that abstract away the mathematical details of building machine-learning models [14, 15, 16, 17]. These e↵orts have been
driven by an increased demand for machine-learning tools and models that can be applied
by users with limited machine-learning knowledge, as a result of its numerous successes in a
variety of industrial, scientific, and engineering domains.
Unfortunately, while machine learning has the capability to provide novel insights for
an extremely diverse set of problem domains, most applications require in-depth knowledge
of machine learning techniques in order to arrive at suitable solutions. The reason for this
is that the application of machine learning to any problem domain requires at least three
principle choices:
1. The choice of the optimal machine learning model for the application;
2. How to pre-process the data for consumption by the chosen model;
3. How the model training parameters (known as the hyperparameters) should be set for
a particular dataset.
For the general domain scientist and engineer that wishes to focus on their field of expertise, (1-3) can present a significant obstacle [18]. Even if the scope of (1) is limited specifically
to artificial neural networks (ANNs; hereafter simply referred to as neural networks), picking
the correct network hyperparameters such as the neural-network topology, learning rate, and
others, is a task for which there is no scientifically rigorous method [19]. Instead, hyperparameter searches are often performed manually via rules-of-thumb and by testing large
sets of models with a predefined grid of parameter values [20]. While grid-based searches
are e↵ective, they are very computationally expensive, and manual searches by hand often
require considerable expertise and can lead to poor reproducibility [21]. As methodological
8

reproducibility is a core tenant of the scientific method, such implementations of machine
learning are not viable in scientific applications.
The essence of the hyperparameter problem can be captured in a rudimentary description
of supervised learning with neural networks. Supervised learning is a class of machinelearning algorithms which make use of a set of training data (x, y), where the vector of
inputs x are transformed into output estimates ŷ that approximate the true outputs y. The
process is supervised by the provision of data where the expected output response is known,
and thus the model itself can be adjusted so that the output response is closely representative
of the true desired output response. Mathematically, the problem can simply stated as,

F(x, w; C) = ŷ,

(1.1)

where F is the neural network (or other machine learning model), w are learned parameters
of the neural network which are adjusted so that ŷ approaches the true outputs y, and C
are the model hyperparameters (e.g. the fixed structure of the network, the learning rate,
et cetera).
Typically, supervised learning with neural networks follows three steps, a learning phase
with training data, a validation phase with testing data, and a prediction phase. In the
learning phase, a set of training data is used to solve an optimization problem, where the
set of learned parameters are adjusted so that a loss or error function–which represents the
di↵erence between the true output response (y) and the inferred output response (ŷ)–is
minimized. After this process is complete, two conclusions and consequent actions can be
reached: (1) if the neural network performs well on the training data (the loss function is
sufficiently minimized), it is then validated with the testing data to see if the model has been

9

over fit; or (2) the neural-network was unable to sufficiently reproduce the desired output
response of the training data (could not minimize the loss function sufficiently). Ascertaining
the reason for training failure, as in (2), is a particularly difficult problem to asses, as it may
have occurred because of bias in the training data, or the machine-learning model of choice
may not be well suited to the application, or the model hyperparameters were ill-chosen,
such as the choice of a non-optimal neural-network structure.
In this context, the role of a data scientist [22] is clear: identifying the machine-learning
models which will generalize well to a given dataset, and knowing how to increase the model
complexity in cases of training failure. However, for the domain scientist who wishes to
employ machine-learning as a tool, these choices are not immediately obvious. Furthermore,
the process of trial and error is arduous because of the long feedback cycles: in most applications, the training phase takes a non-trivial amount of time. The potentially long delay in
feedback regarding the model adjustments makes manually searching the model complexity
space excessively burdensome for new users.
The concept of automated machine-learning seeks to remove these complexities, by automating the process of model choice, data processing, and hyperparameter tuning [23]. By
doing so, automated methods seek to lower the machine-learning implementation barrier for
novice users.
One method that is beginning to see more use in the automated machine-learning space
is Neuroevolution [24, 25]. Neuroevolution is a form of artificial intelligence that utilizes evolutionary genetic algorithms to evolve neural networks for specific applications [26]. Instead
of requiring human input to refine a single neural-network model, neuroevolution automatically performs the search process by allowing a population of many neural networks to
evolve, where only those networks which improve in performance survive. In this way, man10

ually tuned hyperparameters of a neural network are transformed into learned parameters.
The application of neuroevolution as a method for automated machine learning has a few
important consequences. Because the neural network model complexity search is done across
a population of many networks, the speed to convergence for an optimal solution scales with
the number of neural networks in the population. The more networks that are employed,
the faster the space can be searched. This thereby increases the computational complexity
significantly. If population sizes are in the tens to hundreds of thousands, the evaluation complexity is large enough to motivate the transition to modern multi- and many-core compute
architectures. As will be described in chapter 4, one of the hallmarks of neuroevolution–its
ability to evolve neural networks of diverse structure–has so far prevented the development
of generalized many-core algorithms which scale well with the network population size. This
is primarily because the diversity in the evolved neural network structure leads directly to
divergent evaluation graphs for these networks. On the one hand, many-core architectures–
which employ single-instruction-multiple-data (SIMD) processors–perform optimally when
execution branching is minimal. On the other hand, evolving neural networks with di↵ering
topologies has the opposite e↵ect, execution branching is enhanced.
Presented in chapter 4 is a novel many-core algorithm for the concurrent evaluation of
entire populations of topologically-divergent neural networks. This work represents the first
general-purpose mapping of large numbers of heterogeneous neural networks to many-core
architectures. The achieved evaluation speed up is a step toward enabling neuroevolution to
play a more competitive role in the automated machine-learning space. Given the abundant
computational resources available in most scientific laboratories, this work has the potential
to lower the barrier of entry for domain scientists interested in achieving meaningful results
from machine learning with neural networks.
11

Chapter 2
The sensitivity of core-collapse
supernovae to nuclear electron
capture
The study of interactions mediated via the weak nuclear force is of importance to a large
number of fields in physics. However, it is of particular importance to the field of astrophysics
because of the longer timescale on which weak interactions operate as compared to the
strong and electromagnetic interactions. This is evidenced by the impact that new insights
into weak reaction physics have on astrophysical models [7]. Specifically, electron-capture
reactions play a prominent role in high-density environments such as those found in the late
stages of massive star evolution [27, 28], thermonuclear [29, 30] and core-collapse supernovae
(CCSNe) [31, 32], neutron stars [33, 34], and compact object merger events [35]. Realistic
simulations of these environments rely on accurate nuclear physics inputs including electroncapture rates.
Electron-capture rates depend sensitively on allowed Gamow-Teller (GT) transitionstrength distributions in the + direction. These transition strengths characterize nuclear
excitations in which a single unit of spin and isospin are transferred ( S =

12

T = 1), with

no transfer of orbital angular momentum ( L = 0).1 While the main component of electron
capture occurs on the ground state configuration of a nucleus, in high temperature stellar
environments electron captures on thermally-populated excited states of the parent nucleus
can also contribute significantly to the overall rate [36]. Unfortunately, it is difficult to obtain information about transitions from excited states in the laboratory. Compounding the
problem is the fact that in order to accurately include electron capture in simulations, one
must include electron captures on a wide range of nuclei. Hence, in general one must rely on
theoretical models for a complete description of stellar electron-capture rates. On the other
hand, measurements of Gamow-Teller strength distributions in a representative set of nuclei
are important for the development and benchmarking of robust theories. At the same time,
it is critical that theoretical and computational e↵orts provide guidance to experimenters on
which measurements to perform.
Presently, configuration-interaction (shell-model) calculations are the primary method
for producing reliable GT strength distributions near stability in the sd- and pf - shells
(8 < [N, Z] < 20 and 20 < [N, Z] < 40, respectively) for electron capture on both ground
and excited states [37, 38]. Quasi-particle random-phase approximation (QRPA) calculations have also been utilized to estimate GT strengths for large sets of nuclei, but only
where transitions from the ground state are considered [39, 40, 41, 42, 43]. Furthermore,
comprehensive sets of electron-capture rates (as a function of density and temperature) for
a large number of nuclei based on QRPA calculations have not been published.
Direct and indirect experiments, such as

-decay and charge-exchange (CE) measure-

ments respectively, provide robust benchmarks for theoretical GT strengths and therefore
1 Transitions which follow these selection rules are referred to as “allowed” transitions, and those which
do not–for example, transitions with L > 0–are known as “forbidden” transitions which typically proceed
at a much reduced rate as compared to allowed transitions.

13

are crucial to understanding astrophysical electron-capture rates. Unfortunately, electroncapture and -decay experiments can only access states in a limited Q-value window. Furthermore, for neutron-rich nuclei

-decay only provides information in the

direction,

which is of limited use for electron-capture studies. Intermediate energy (& 100 MeV/u) CE
reactions in the + direction, however, connect the same initial and final states as electron
capture, providing information about transitions up to high excitation energies, and are thus
well suited to study the full Gamow-Teller strength distribution of interest. At these energies, CE measurements have been emperically established to be accurate at the ⇠10% level
and are therefore able to provide rigorous tests of theoretical Gamow-Teller strengths and
derived electron-capture rates [13].
Recently, the results from (n,p), (d,2 He), and (t,3 He) CE reactions on nuclei in the
pf -shell were systematically compared [13, 44, 45] with shell-model calculations using the
KB3G [46] and GXPF1a [47] e↵ective interactions in the pf -model space, and with calculations based on the QRPA formalism of Möller and Randrup [40]. The authors compared shell-model and QRPA derived electron-capture rates against those derived from CE
measurements. It was found that the QRPA calculations systematically overestimate the
electron-capture rates (⇠100-3000%, depending on density and temperature), whereas the
shell-model estimates produce rates similar to those measured experimentally (⇠1-50%) [13].
Unfortunately, shell-model calculations are computationally challenging for nuclei beyond the
pf -shell, and therefore weak rates used in high-density astrophysical calculations most commonly rely on less accurate methods. In each of these cases, systematic and random error
exist, and it is therefore important to understand the sensitivity of astrophysical simulations
to uncertainties in these rates.
Sensitivity studies are useful tools for guiding theoretical and experimental e↵orts because
14

they highlight nuclei that should be given particular focus, and they indicate the accuracy
with which the parameters of interest need to be known. They also illustrate how strongly
the current parameter uncertainties a↵ect the outcome of the astrophysical simulations. In
this chapter, results from a recent sensitivity study are examined in which ⇠ 150 collapse
simulations were performed with systematic and statistical variations of the electron-capture
rates [2]. The impact these rate variations have on the collapse, bounce and pre-explosion
phases of core-collapse supernovae simulations for a range of presupernova progenitors and
equations of state (EOS) will be described.
As part of this work, a modular and open-source weak reaction rate library1 was developed for use in astrophysical simulations. For the simulations described in this work it was
implemented in the stellar core-collapse code GR1D [48], but has also since been used in the
multi-dimensional CCSNe codes FLASH [49, 50], and COCONUT [51, 52]. 2
In the following sections, the inner core of the protoneutron star (PNS) and the observable
peak neutrino-luminosity from core bounce are shown to depend sensitively on the electroncapture rates of neutron-rich nuclei (+16/-4 % and ±20%, respectively). As variations on
this level are not easily reproduced from uncertainties in other inputs to the simulations,
they motivate the development of new theoretical models for electron-capture rates as well
as relevant measurements, which together will constrain these and other key parameters
discussed in this work.

2 http://www.jinaweb.org/weakrates

15

2.1

Astrophysical weak interaction rates

The weak nuclear force is important for a number of processes that influence the evolution
of massive stars [7]. For example, interactions mediated by the weak force are important
ingredients for nucleosynthesis and also for the internal structure of evolving stars, as they
sensitively determine the electron-to-baryon ratio Ye and the iron-core mass just prior to corecollapse [27]. Unlike the conditions present during quasi-static stellar evolution, however,
in the core of a collapsing star the density and temperature are high enough that nuclear
and electromagnetic reactions equilibrate [53]. This is not the case for weak reactions which
operate much more slowly and thus continue to a↵ect the nuclear composition, the neutrino
emission, and ultimately the dynamics of the entire event.
As compared to other semi-leptonic weak interactions (weak interactions with nuclei),
electron capture has a particularly remarkable impact on the core-collapse environment [54].
In the final stages of a star’s life, the nuclear-energy generation rate of the core that normally
sustains a star against gravitational collapse is absent because the core is composed of highly
stable iron-group nuclei. Instead, at these late times the electron-degeneracy pressure provides the primary stability against collapse. It is therefore apparent that electron captures
that remove electrons from the system will have dramatic consequences for this environment.
Furthermore, the electron chemical potential µe is sufficiently large to overcome Q-value3
restrictions, and so the electron-capture rates are significant.
Just prior to and during the early moments of collapse, other weak interactions can also
play a role. Martı́nez-Pinedo et al. [55] have shown that

decay can temporarily compete

3 The Q-value refers to the rest mass energy di↵erence between the final and initial nucleus, Q = M

f Mi ,
and describes either the energy required for the reaction to occur (Q < 0), or the energy deposited by the
reaction (Q > 0). For stellar electron-capture on neutron rich nuclei, Q < 0.

16

with electron capture when Ye =0.42-0.46, which can occur during Si shell burning and the
early stages of collapse. However, as collapse ensues, µe quickly becomes large enough that
-decay electrons are energetically blocked due to degeneracy. Similarly,

+

decay can

also compete with electron capture for nuclei with Qe+ > 2me c2 , but in the core-collapse
environment neutron-rich conditions are favored, and Qe+ is below this threshold.
The importance of reactions mediated by the weak nuclear force, and specifically electron
capture, as it pertains to core-collapse was first demonstrated by Bethe et al. [56]. Not long
after, the theory of stellar electron capture was formalized by Fuller, Fowler, and Newman
(FFN) [57]. In their pioneering work they published the first tabulation of weak interaction
rates ( ± -decay and e± -capture) considering presupernova conditions where allowed Fermi
and Gamow-Teller (GT) transitions dominate. Since then, advancements in computational
resources have allowed for detailed nuclear shell-model calculations that have increased the
accuracy of the weak-interaction theory first outlined by FFN. Major weak-interaction rate
tabulations that derive from a combination of experimental data and shell-model e↵ective
interactions are the Oda et al. [37] and Langanke and Martı́nez-Pinedo [36] tabulations for sd(A=17-39) and pf -shell (A=45-65) nuclei respectively. For heavier nuclei, where full shellmodel calculations are computationally unfeasible, the Shell Model Monte Carlo (SMMC)
approach has been employed which preserves nuclear properties in very large model spaces.
Langanke et al. [38] have combined this method with an RPA technique to estimate electroncapture rates at densities and temperatures relevant during core-collapse for nuclei in the
pf g/sdg-shell (A=65-112), which have come to be known as the LMSH rates. Juodagalvis
et al. [58] have also produced a set of more than 2200 additional rates based on the same RPA
technique but utilizing a Fermi–Dirac parameterization instead of the more computationally
expensive SMMC calculations. The individual rates were not released, but instead these
17

rates were averaged over NSE abundances and reported along a characteristic core-collapse
(⇢, T, Ye ) trajectory. However, such a prescription is not suitable for sensitivity studies in
which the detailed response of simulations to individual nuclei is desired.
The rate tabulations listed in Table 2.1 have been implemented into the weak rate library
used in this work. Together these tabulations contain 445 rates for 304 unique nuclei over a
large density and temperature grid. This library has been built as a standalone module and
has also been implemented into the neutrino-interaction library NuLib [59] for use in neutrinotransport routines employed by the spherically-symmetric, general-relativistic stellar collapse
code GR1D [59]–see Section 2.3 for more information. Details on the density and temperature
range for each of the included rate tabulations are shown in Table 2.1. The mass coverage
of each rate table is shown in Figure 2.1.
Table 2.1: Density, temperature and mass ranges for the compiled weak rate set
Model space
Table

s

FFN
ODA
LMP
LMSH
Approx.

x
x
x
x

p

sd

pf

x
x

x

pf g/sdg

T (GK)

Log10 (⇢Ye g cm 3 )

Ref.

x
x

0.01 - 100
0.01 - 30
0.01 - 100
8.12 - 39.1
-

1.0 - 11
1.0 - 11
1.0 - 11
9.22 - 12.4
-

Fuller et al. [57]
Oda et al. [37]
Langanke et al. [38]
Hix et al. [31], Langanke et al. [60]
Langanke et al. [38]

x
x

x

x

The LMP+LMSH rates were first implemented into a spherically-symmetric core-collapse
simulation by Hix et al. [31]. They compared simulations with this set of shell-model based
electron-capture rates against simulations that utilized the Bruenn [61] prescription for electron capture. The evolution of the core-collapse phase and the structural di↵erences in the
core at bounce seen in that work were significant. In light of the di↵erences that exist
between theoretical estimates for electron-capture rates and those inferred from CE experiments, these results motivate the need for a detailed sensitivity study.

18

70

60

50

50

=

Z

28

40

N

Proton Number

50

20

30

28

82

8

20

Stable
FFN
ODA
LMP
LMSH
Approx.
No rate

20

2

10

8
2

0
0

20

40

60
Neutron Number

80

100

120

Figure 2.1: Chart of the nuclear species depicting the nuclei included in each rate tabulation
as well as the full reach of the weak rate library presented in this work. The table to which a
species belongs is given by the color and legend in the figure. The Oda set contains rates for
lower mass sd-shell nuclei (light blue), the LMP set contains rates for the intermediate mass
pf -shell nuclei (green), and the LMSH set contains rates for the heavier mass pf g/sdg-shell
nuclei near stability (red). The FFN tabulation provides rates across the sd and pf -shells
(dark blue). Squares individually bordered in black are stable nuclei. The tables are mutually
exclusive except for FFN which spans many nuclear shells. To distinguish between nuclei
with rates from FFN and another table, the border of the FFN set has been outlined with
a black and white line.

19

To handle the large number of nuclei not included in the tables, Hix et al. [31] utilized an
average electron-capture neutrino emissivity for all nuclei which lacked a shell-model based
rate. In this work, instead of performing averaging, the approximate routine of Langanke
et al. [38] is employed, which is based on the parameterization of the electron-capture rate
as a function of the ground state to ground state Q-value. This approximation was first
described by Fuller et al. [62] and was later parameterized and fit to shell-model calculations
in the pf -shell by Langanke et al. [38]. In this approximation, the electron-capture rate is,
ln2 · B
EC =
K

✓

◆5
T
[F4 (⌘)
m e c2

2 F3 (⌘) + 2 F2 (⌘)]

(2.1)

2 F4 (⌘) + 2 F3 (⌘)],

(2.2)

and the neutrino-energy loss rate is,
ln2 · B
⌫e =
K

✓

◆6
T
[F5 (⌘)
me c2

where me is the electron mass, K = 6146 s, Fk are Fermi integrals of rank k and degeneracy
⌘,

= (Q

E)/T , ⌘ =

+ µe /T , and T and µe are the temperature and electron chemical

potential. B (= 4.6) and

E (= 2.5 MeV) are fit parameters taken from Langanke et al. [38]

and respectively represent e↵ective values for the transition strength and energy di↵erence
between final and initial excited states.
Figure 2.2 compares the rate estimates from this approximation and from the shell-model
rate tabulations in Table 2.1. As is easily seen from the figure, the variance of the shell-model
rates depends sensitively on the density of the environment. At lower densities, where the
electron chemical-potential and electron capture Q-value are comparable (µe ⇡ QEC ), the
location of excited states in the daughter nucleus, and the associated Gamow-Teller transition

20

10 6

(a)

10 4

EC

(s-1)

10 2

10 0

2

10

4

Log Residuals

10

4
2
0

Residual
log10
e) = 11.0, T9 = 11.8 GK
log10
e) = 9.61, T9 = 8.19 GK

(b)

-2
-4
-20 -15 -10

-5

0

5

10

15

20

Qg.s. (MeV)

Figure 2.2: Panel (a): Q-value dependence of electron-capture rates at two points along a
core-collapse trajectory. The scattered points are tabulated (shell-model and SMMC) rates
for each electron-capture reaction, while the black points are the approximate rates given by
Eq. 2.1. Panel (b): The residual di↵erences between log10 of the shell-model rates and the
approximate rates for each nucleus in the weak-rate library. An example residual is indicated
on panel (a). When the density and temperature of a simulation evolve outside the range
of the rate tables (see Table 2.1), rates are calculated via the approximate routines in order
to avoid an artificial cut o↵ imposed by the table boundaries. Rates are estimated between
density and temperature grid points via monotonic cubic-spline interpolation as described
by Ste↵en [63].
21

strength, sensitively determine the total electron-capture rate for a nucleus. New final states
in the daughter nucleus become accessible as the electron Fermi-energy, which scales with
µe , increases beyond the energy required to populate them via allowed electron capture. The
large scatter of the electron-capture rates at lower densities (Fig. 2.2) is because the Fermi
energy is comparable to the excited state energies of the daughter nuclei and the internal
structure of each nucleus varies significantly. During collapse, as the density increases and the
material becomes more neutron rich due to successive electron captures, both the magnitude
of the average electron capture Q-value and µe increase. However, µe increases more quickly
with density than the reaction Q-values do, and eventually µe

QEC implying that the

majority of the electron-capture channels are open. In this regime the rate is less sensitive to
the excitation energy spectrum of the daughter nucleus, and instead depends more strongly
on the total GT strength across all possible final states. The decrease in the variance of the
shell-model electron-capture rates in the higher density case of Figure 2.2 is a result of this.
While the parameters of Eqs. 2.1 and 2.2 were originally fit from the LMP nuclei, there is
reasonable agreement of the approximation with the other tabulated rates. Outside of these
tables, significant deviations from the estimates of this approximation may exist, specifically for heavier neutron rich nuclei [64]. But for the purpose of a sensitivity study, this
approximation—from which the majority of the rates are calculated—is used as a base estimate o↵ which the electron-capture rates may be varied. Given this set of rates, it is shown
in section 2.4.2 that the simulations are most sensitive to intermediate mass neutron-rich
nuclei. However, electron-capture rates developed from sophisticated theoretical models do
not exist for individual nuclei in this region, and thus cannot be benchmarked against experimental measurements. Therefore, the estimates provided by the approximation of Eqs. 2.1
and 2.2 may be systematically o↵ by a considerable amount. As will be shown, changes in the
22

predicted rates for these nuclei have significant consequences for the simulations, motivating
the need for experimental and theoretical e↵orts to constrain the rates of these species.

2.2

Core collapse and the role of electron capture

Just prior to collapse, the temperature of the stellar core becomes high enough (T & 0.5 MeV)
that the photon gas has sufficient energy to photodissociate nuclei into alpha particles and
free nucleons. However, the density is also high (⇢ & 109 g cm 3 ) resulting in large nuclear
reaction rates that rapidly form nuclei from these light particles. The balance reached
between these competing processes is known as Nuclear Statistical Equilibrium (NSE). If
the entropy is sufficiently low and the mass fraction of free nucleons is small compared to
that of nuclei, the most abundant nucleus in NSE is the species with the highest binding
energy for a given electron-fraction, Ye (= Z/A) [55]. A broad distribution of abundant nuclei
forms due to finite temperatures which distribute the abundances around these peak nuclei.
As collapse ensues and the central density increases through the first few decades, electron
captures are the primary engine of deleptonization. Electrons are removed from the system
and the produced electron-neutrinos (⌫e ) are able to freely stream out of the core, decreasing
both Ye and total lepton fraction Yl . As Ye decreases, peak abundances move toward neutronrich nuclei, and the core begins to cool as ⌫e ’s carry away energy and entropy. Electron
captures continue to dominate the neutrino transport during collapse until the last few
milliseconds before core bounce. In these final moments, the central density reaches a few
times 1012 g cm 3 , which is large enough that the neutrino mean free path begins to shorten
due to coherent scattering on nucleons and heavy nuclei. This increase in the ⌫e -scattering
cross section results in a neutrino di↵usion time that exceeds the collapse time, thereby

23

trapping the electron neutrinos in the inward flow of matter. After this occurs, the conversion
of electrons into electron-neutrinos via electron captures no longer removes leptons from the
core. Instead, further electron captures increase the electron-neutrino fraction Y⌫e in order
to conserve the now constant lepton fraction and bring the system of electrons and electron
neutrinos into equilibrium.
Prior to the work of Langanke et al. [60] it was believed that electron captures on free
protons were of greater importance than captures on nuclei during collapse. The main
considerations involved were that electron capture on free protons has a higher rate owing
to a smaller in magnitude Q-value. Additionally, nuclei with neutron number N

40 have

full pf -shell single particle states and so the addition of another neutron via an allowed
electron-capture transition would be Fermi-blocked. Langanke et al. [60] recognized that the
many-body nuclear states have mixed configurations and do not follow a simple Hartree-Fock
filling of single particle orbitals. They also suggested that thermal excitation of nucleons to
the g9/2 orbital creates vacancies in the pf -shell, and together with configuration mixing,
electron capture on bound protons is unblocked. Furthermore, because of the low entropy
in the core, and the neutron-rich conditions, the abundance of heavy nuclei is several orders
of magnitude higher than that of free protons, resulting in a higher overall electron-capture
rate. Thus, because electron captures on nuclei dominate, it is of great value to future
experimental and theoretical e↵orts to investigate the contribution each species has, and to
identify which nuclei are most important, as these e↵orts would provide direct constraints
to CCSNe simulations.

24

2.3
2.3.1

Codes & Methods
NuLib

In addition to electron-capture rates, other rates are needed to perform core-collapse simulations. The collection of rates used is contained in NuLib [59], an open-source, neutrinointeraction library.1 NuLib contains routines for calculating electron-type neutrino/antineutrino charged-current absorption opacities on nucleons with corrections for weak magnetism
and nucleon recoil based on the formalism of Burrows et al. [65] and Horowitz [66]. Neutrino emissivities for these processes are determined via Kirchho↵’s law which equates the
absorption rate of a equilibrium neutrino distribution to the emission rate of the underlying
matter. Elastic scattering of neutrinos on nucleons, and coherent scattering of neutrons on
alpha particles and heavy nuclei is also included in NuLib. For the former, corrections for
weak magnetism and nucleon recoil are included, and for the latter, corrections from ion-ion
correlations [67], electron polarization, and the nuclear form factor are employed. Inelastic
scattering of neutrinos on electrons is included based on the expressions of Bruenn [61].
Emissivities of heavy-lepton neutrino/antineutrino pairs via electron-positron annihilation
and nucleon-nucleon Bremsstrahlung are computed ignoring final state neutrino blocking.
For neutrino-antineutrino annihilation, instead of computing the non-linear absorption opacity during the simulation, an e↵ective absorption opacity is used which has been shown to
be an excellent approximation for core-collapse supernovae [59]. 4

4 http://www.NuLib.org

25

2.3.2

GR1D

The electron-capture rate implementation described in the previous sections is used to study
the sensitivities of the core-collapse phase to these rates using the code GR1D [48, 59]. GR1D
is an open-source spherically-symmetric general-relativistic neutrino-transport and hydrodynamics code used for studying stellar collapse and the early stages of a core-collapse supernova. For details of the hydrodynamics module of GR1D see reference [48]. The neutrino
transport is handled though a general-relativistic, energy-dependent two-moment formalism
for which extensive details can be found in O’Connor [59]. The employed scheme numerically
solves for the time evolution of the first two moments of the neutrino distribution function:
the neutrino energy density and the neutrino momentum density. The simulations utilize 18
energy groups logarithmically spaced between 0 and 250 MeV. Only electron type neutrinos
are evolved until the central density reaches 1012 g cm 3 , after which electron anti-neutrinos
and a characteristic heavy lepton neutrino are included. However, these latter two neutrinos
do not become important until core bounce has occurred. Spatial fluxes of the neutrino
moments are treated explicitly. Inelastic neutrino-electron scattering is handled explicitly
until the central density reaches 1012 g cm 3 at which point an implicit treatment is used.
Simulations in spherical symmetry (one dimension) such as those facilitated by GR1D,
a↵ord the computational resources needed for the inclusion of a highly detailed and complete set of microphysics. The same is not true for multidimensional simulations, as the
computational requirements are more extensive, forcing the inclusion of only a small subset
of the physical interactions. Fortunately, the core-collapse and early post-bounce phases of
core-collapse supernovae are well represented in one dimension due to the lack of turbulent
convection in the core. Thus, together NuLib and GR1D are able to provide a robust and

26

extendable code base for CCSNe simulations with state of the art microphysics.

2.3.3

Neutrino emission via electron capture

Electron capture is associated with the emission of electron neutrinos and so the electroncapture rate is proportional to the integrated spectrum of ⌫e emitted per second. The rate
for a particular nuclide, as tabulated in the implemented rate tables, is defined as the sum
of the rates for each of the individual nuclear transitions

=

X

ij ,

(2.3)

ij

where indices i and j correspond to levels in the parent and daughter nucleus respectively.
The spectra of emitted neutrinos from the electrons capturing on nuclei, described by the
matter temperature T and electron chemical potential µe , will vary based on the initial and
final states involved owing to a di↵erent reaction Q-value,

QEC
ij = Qg.s. + Ei

Ej

(2.4)

where Qg.s. is the atomic mass di↵erence of the initial and final nuclei, and Ei and Ej are the
excitation energies of the populated states in the parent and daughter nucleus respectively.
The most comprehensive solution to constructing neutrino spectra would be to coherently
sum the spectra of neutrinos emitted from each nuclear transition. However this would rely
upon rate tabulations for individual transitions which are not presently available. Thus, an
e↵ective neutrino spectra is implemented in terms of a single reaction Q-value, q, that is
chosen to constrain the average energy of the spectrum to match that from the tabulated

27

rates [68],

n(E⌫ , q) = E⌫2 (E⌫

hE⌫ i =

R1
0

q)2

N
1 + exp{(E⌫

E⌫ n(E⌫ , q)dE⌫

R1

=
n(E⌫ , q)dE⌫

q

µe )/kT }

⌫
EC +

+

,

(2.5)

(2.6)

0

where n(E⌫ , q) is the neutrino distribution function and is normalized to the total electroncapture rate for a particular nuclear species.

⌫,

+,

and EC are the neutrino energy

loss, positron emission, and electron-capture rates respectively. Eq. 2.6 is solved numerically
for the e↵ective Q-value, q = qe↵ , which then defines the e↵ective neutrino spectrum for
the electron-capture reaction of interest at a given ⇢, T , and Ye . The approximate neutrino
spectra generated in this way are unable to reproduce complex structure such as double
peaking in the true neutrino distribution, which may occur when there is a resonant allowed
transition (QEC
ij ⇠ 0) between an excited parent state and the daughter-nucleus ground
state. However, it approximates singly-peaked neutrino distributions quite well [68]. The
spectrum is normalized to the total electron-capture rate via Gaussian-Legendre quadrature
with an adaptive algorithm, developed in this work, that adjusts the range of integration to
the full width of the spectrum.
Utilizing these spectra, the electron-capture neutrino emissivity for a given nuclear species
is calculated as

⌘i (Ek ) =

1
E n n(Ek , qe↵ ),
4⇡ k i

28

(2.7)

where the ⌫e ’s are assumed to be emitted isotropically, ni is the number density for the
i-th nucleus, n(Ek , qe↵ ) is the neutrino spectra evaluated at the e↵ective Q-value that solves
Eq. 2.6, and Ek indicates the energy of energy group k. Evaluation of the emissivity is done
point wise at the centroid of each energy bin, and has units erg/(cm3 ·sr·s·MeV).
For estimates of the NSE number densities used above, several EOS from Hempel and
Scha↵ner-Bielich [69] were used. In particular, the SFHo EOS and internally consistent NSE
distribution developed by Steiner et al. [70] was the primary EOS employed. Results were also
compared against the DD2 [71] and TMA [72] EOS, each with self-consistent, but di↵erent,
NSEs. The SFHo and DD2 EOS were chosen because they currently best satisfy both
nuclear and astrophysical constraints [73]. Instead of meson self-interactions, the DD2 EOS
implements density-dependent meson-nucleon couplings which have been used successfully
to describe nuclear structure in a wide region of the nuclear chart and have also been tested
in heavy-ion collisions [71]. For nuclear masses, the SFHo and DD2 NSE distributions rely
on the Finite Range Drop Model (FRDM) from [74] and [75], whereas the TMA EOS utilizes
a mass table calculated by Geng et al. [76]. Both mass tables incorporate experimentally
determined masses from Audi et al. [77] and only resort to theoretical estimates where no
experimental measurements are available. For consistency, in addition to NSE abundances,
these mass distributions are also utilized in the calculation of reaction Q-values for use in
Eq. 2.1.

29

2.4
2.4.1

Sensitivity study
Reference simulations

In order to establish reference simulations o↵ which variations are performed, the widely
studied 15 solar mass, solar metallicity progenitor star s15WW95 [78] was used, as well
as s12, s20, and s40 from WH07 [79], which span the range of stellar compactness ⇠2.5 [80]
in this model set. The core compactness, ⇠M , is the ratio of the mass enclosed M to its
enclosing radius, and for this model set, s12 (s40) has the least (most) compact core. This
variety in compactness is an important factor for assessing the global sensitivity of corecollapse supernovae to detailed microphysics, as progenitors with similar compactness will
likely have similar sensitivities to microphysical variations. More details on the progenitor
model set utilized in this work can be found in the progenitor sensitivity subsection 2.4.3.1.
For each simulation the SFHo EOS is utilized, and in addition, for simulations with the
s15WW95 progenitor star the DD2 and TMA EOS, and NSE distributions, are employed.
A full complement of neutrino-interaction microphysics is incorporated via NuLib in each
reference simulation, which includes the newly implemented weak rates library described
here. The weak-rate tables were included using the following priority hierarchy: LM P >
LM SH > Oda > Approx., ensuring that rates from sources with higher priority are utilized
where rate estimates from multiple sources exist. Approx. indicates the parameterized rate
approximation of Eqs. 2.1 and 2.2, which is used for nuclei not included in the tables and
for regions of density and temperature which are beyond the limits found in Table 2.1. For
consistency, only tables that derive from shell-model calculations are utilized.
For each progenitor and EOS collapse simulations are performed in GR1D and follow the
evolution until at least ⇠ 100 ms after bounce. The collapse proceeds as described in Section
30

2.2. Di↵erences in the collapse evolution for di↵erent progenitors stem from the hydrostatic
conditions in the cores of these massive stars at the onset of collapse. For stars with large ⇠2.5 ,
larger central5 temperatures are needed to balance gravity. This gives lower central densities,
and therefore less electron capture during the final stages of stellar evolution. The range
of initial central Ye goes from ⇠0.422 for the s15WW95 model to ⇠0.447 for the s40WH07
model, or a range of ⇠6%. After neutrino trapping sets in, a range of trapped leptonfraction of ⇠0.288 – 0.297 is observed, where s40WH07 and s12WH07 have the minimum
and maximum trapped Yl , respectively. The overall higher deleptonization rate for the more
compact progenitors is due to both longer collapse times and larger matter temperatures,
which enhance the electron-capture rates.
Simulations utilizing di↵erent EOS, while holding all else constant, demonstrate only
small variations in the density, temperature, and Ye central-zone trajectories up to bounce.
Figure 2.3 details the abundance distributions for each EOS, as well as the resulting average
electron capture rate along a collapse trajectory. The NSE distributions of all three EOS
are largely similar early on, but di↵erences in the mass table of the TMA EOS cause it to
diverge from the others starting around 1

2 ⇥ 1012 g cm 3 . However, di↵erences are seen in

the electron-capture rate only after central densities of 2 ⇥ 1012 g cm 3 , where any e↵ect on
the evolution is suppressed because of neutrino trapping. Near nuclear saturation density,
however, the di↵erences in EOS begin to play a more important role. The density-dependent
couplings of the DD2 EOS, for instance, result in higher central temperatures at bounce.
However, since the average rate in simulations utilizing each of the EOS are nearly identical,
they result in a di↵erence of trapped lepton-fraction of only a fraction of a percent. For
more information on the sensitivity of the electron and lepton fractions to the EOS during
5 Central used in this context refers to the inner most zone of the core-collapse simulation.

31

160

107

SFHo
DD2
TMA

106

140
105

EC

120
104
103

EC

60

102

40

101

20
0
1010

> (s-1)

80

EC
EC

,

100

100
1011

1012
c

1013

1014

(g cm-3)

Figure 2.3: The average nuclear mass (divided by two), charge, and electron-capture rate
versus central density for the three EOS utilized in this study. The colors indicate di↵erent
EOS, while the line style indicate which quantity is plotted. All three EOS have nearly
identical abundance distributions up to densities of 2·1012 g cm 3 . Beyond this point the
TMA EOS has a heavier and slightly more neutron rich mass distribution compared to both
SFHo and DD2, but maintains a comparable average electron capture rate overall. These
simulations each utilize the s15WW95 progenitor.

32

collapse see Fischer et al. [73].
Together, these reference calculations span a wide range of progenitor and EOS dependences that ensure a configuration-independent assessment of the core-collapse sensitivity to
electron capture on nuclei, and furthermore demonstrate the universality of collapse. In what
follows, results are discussed for variations on the s15WW95+SFHo reference simulation but
any significant di↵erences in relation to variations on the other progenitor+EOS reference
simulations are pointed out.

2.4.2

Species dependent sensitivity

To understand the sensitivity of core-collapse to di↵erent regions of electron capturing nuclei,
the central zone collapse profile from the reference simulation is used to decompose the
change of the electron fraction with time, Ẏe , into the electron captures of each nuclear
species. While using only the central zone is an approximation, it is justified by noting the
observation by Liebendorfer [81] that the electron fraction profiles typically correlate quite
well with density during the collapse phase. Therefore, matter will generally have the same
electron capture history. The rate of change of the electron fraction with time, Ẏe , that is
estimated accounts for ⌫e re-absorption in an energy-dependent way,

Ẏei

4⇡↵ X
=
⇢Na
k

✓
✏k · ⌘i (✏k )
· 1
✏k

Ek
Bk

◆

(2.8)

where Ẏei is the time derivative of the electron fraction due to electron captures on the
ith nuclear species, ↵ accounts for the general relativistic time dilation, Na is Avogadro’s
constant, ⇢ is the density, ✏k is the energy of the kth energy bin,
bin width, ⌘i is the emissivity of species i and 1
33

Ek
Bk

✏k is the kth energy

is the neutrino blocking factor that

accounts for re-absorption as collapse approaches weak equilibrium. Along with a hydrodynamical correction due to advection of electrons into the central zone, the time integral
of Eq. 2.8 added for all nuclei reproduces the full time dependent Ye profile of the central
zone during collapse, indicating that electron captures on heavy nuclei singularly drive the
deleptonization of the central zone.
With this method, the deleptonization history due to each nucleus can be individually
investigated. At these densities and temperatures, NSE diversifies the abundant nuclei,
ensuring that no single nucleus dominates the deleptonization. There are, however, subsets
of nuclei that contribute more than others to the reduction of Ye . A nuclear-mass dependence
can be studied by binning the the contribution to |Ẏe | from each nuclide into nuclear mass
bins and tracking the evolution of each region up to neutrino trapping. Figure 2.4 plots the
deleptonization rate in the core for di↵erent nuclear mass bins, as the central Ye progresses
from its progenitor value to its value when weak equilibrium is achieved, just prior to bounce.
Early on, before the collapse becomes strongly dynamical, nuclei in both the mass range
25<A<65 (sd+pf -shell) and those in the 65<A<105 (pf g/sdg-shell) comprise the main
component of the deleptonization. However, during the strongest push toward neutron-rich
conditions, where Ye rapidly changes from ⇠0.41 to ⇠0.28, nuclei with mass A >65 dominate
the evolution as seen by the red and light blue curves in Figure 2.4. Unfortunately, the most
precise electron-capture rate estimates fall below this region and instead, the rates are set
primarily by the approximation of Eq. 2.1.
It is also useful to understand the specific nuclei that have the largest integrated contribution to core deleptonization up to neutrino trapping. Shown in Figure 2.5 are the 500 nuclei
with the largest integrated |Ẏe | from t = 0 to the trapping time—when densities are in excess
of 2 · 1012 g cm 3 . This reveals the channel through which the bulk of electron captures
34

<-100 -50 -40 -30

10 1

-20

t-tb (ms)

-10

(a)

Ye (sec-1)

10 0
10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

0.42

Total
5 < A 25
25< A 65
65< A 105
185< A 225
105< A 145
145< A 185

0.40

0.38

0.36

Ye

0.34

0.32

0.30

0.28

Figure 2.4: The contribution of nuclear electron capture to the change of the matter electronfraction with time. The contours are the binned sums of |Ẏe | for each species in several mass
regions. For reference, the central density at t tb = -20, -10, -5, -2, and -1 ms is 1.41 · 1011 ,
4.06 · 1011 , 1.42 · 1012 , 8.49 · 1012 , 3.30 · 1013 g cm 3 respectively.

35

(b)

50

28

=

Z

40

N

Proton Number

50

30

log10

20

20

20

82

28

e

|

50

30

40

50

60

70

80

90

Neutron Number
Figure 2.5: Top 500 electron capturing nuclei with the largest absolute change to the electron
fraction up to neutrino trapping. The color scale indicates |Ẏe | integrated up to the trapping
time, occurring when ⇢c ⇠ 2 · 1012 g cm 3 , such that the total electron-fraction at this
point is equal to its initial value less the sum of Ye , the plotted quantity, over all nuclides.
Calculations are based on the s15WW95+SFHo reference simulation. The black contour
that runs parallel to the valley of stability on the neutron-rich side is the boundary between
measured [77] and theoretical masses used in the approximate rate estimates of Eqs. 2.1
and 2.2. The rectangular outline indicates the size of the sampling region used in the
statistical resampling study, and also the set of nuclei which exhibited the largest changes
to the simulations when excluded from the electron-capture calculations.

36

operate. The central electron-fraction at the trapping density is reproduced by subtracting
the sum of this quantity over all nuclear species from the initial electron-fraction,

Ye (t = ttrapping ) ' Ye (t = 0)
where

X

Yei ,

(2.9)

i

Yei is shown in Figure 2.5, and the component of Ye due to advection of electrons

into the central zone (otherwise making this relation exact) is left out for simplicity. Within
the pf g and sdg-shells the primary contributors to the deleptonization phase of collapse are
neutron rich nuclei near the N=50 and N=82 closed neutron shells (see the dashed vertical
lines in the figure).
To confirm these results, they are gauged against the sensitivity of the collapse phase to
localized groups of nuclei by employing a statistical resampling technique where sets of nuclei
are removed from the simulation. This method is based on well known statistical resampling
methods such as bootstrap and jackknife resampling [82]. Specifically, a rectangular region
centered on a nucleus and spanning all nuclei within ±3 isobars and ±5 isobaric chains is
removed from the calculation of the electron-capture neutrino emissivity. An example of
such a removed region is drawn on Figure 2.5. This technique is employed in 48 simulations
with resampling performed uniformly across the nuclear chart.6 Using this technique the
simulations are found to be most sensitive to nuclei in the mass range 74-84 with Z/A (= Ye )
between 0.36-0.44, corresponding to nuclei near 78 Ni, 79 Cu, and 79 Zn. These results agree
with the Ẏe calculations performed above, and indicate that species near the N=50 magic
number have the largest contribution in magnitude to the change in the electron fraction
overall. The impact of removing these species from the simulation corresponded to a change
6 The resampling regions were allowed to overlap, but were chosen such that uniform coverage accross
the nuclear chart was achieved in the resampling study.

37

of inner-core mass at bounce of & 10%, whereas resampling in other regions resulted in
variations of only a few percent.
The electron-capture rates for these nuclides rely entirely on the approximation of Eqs. 2.1
and 2.2, which were fit originally to rates of lower-mass mid-shell nuclei near stability. Therefore, in the region indicated by the above two studies, the approximation is largely uncertain
and may be systematically o↵ by a significant amount. For instance, these estimates do not
account for nuclear structure e↵ects that may occur near the N=50 closed neutron shell.
Depending on the nuclear configurations, thermal excitations, and increasing dependence on
forbidden transitions, Pauli blocking7 may considerably reduce the electron-capture rates in
this area. Given that the change of inner-core mass at bounce was largest when the rates
of these nuclides were decreased to zero as compared to any other set, and that without
any evaluative measurements the uncertainties in these rates remain large, experimental and
theoretical work should focus here. Any substantial changes to the electron-capture rate estimates for these nuclei will likely have a relatively large impact on simulation predictions for
the PNS formation, and will therefore help to constrain important collapse and pre-explosion
phase quantities.

2.4.3

Systematic variations

To study the strongest impact of variations in the electron-capture rates, simulations in
which the rate for each A>4 nuclide is systematically scaled by factors of 10, 4, 2, 0.5,
0.25, and 0.1. In this way, the structure of the rates as seen in Figure 2.2 is preserved (the

7 Pauli blocking arises as a result of the Pauli exclusion principle in which identical fermions are unable
to occupy the same quantum mechanical state. In the case of neutron-rich nuclei, N=50 represents a g9/2
orbital fully occupied by neutrons. Unless the electron capturing nuclei are highly excited such that there
are holes in this or other lower energy orbitals, allowed Gamow-Teller transitions cannot occur.

38

s15WW95+SFHo
s15WW95+DD2
s15WW95+TMA
s12WH07+SFHo
s20WH07+SFHo
s40WH07+SFHo
Reference sim.

Yltrapped (ρc = 4 .1012 g cm-3)

0.32

0.31

0.30

0.29

0.28
0.1

0.25

0.5
1.0
2.0
systematic scaling factor

4.0

10.

Figure 2.6: Projection of the (trapped) lepton fraction at ⇢c = 4 · 1012 g cm 3 as a function
of the electron-capture rate scaling factor for progenitor+EOS reference simulation. In all
the cases the lepton fraction begins to increase if the capture rate becomes too high because
of a dramatic increase in the electron neutrino absorption cross section. The asymmetry seen
here indicates that those quantities which depend on Yl/e are likely to be more sensitive to
a reduction of the electron-capture rates due to a systematic overestimate in the base rates,
than they are to an increase due to an underestimate.

39

lower panel of residuals is una↵ected), but the distribution of rates is shifted to larger or
smaller values depending on the scaling factor. Systematic shifts of the rates emphasize
the role of electron capture as a regulator for entropy and temperature in the simulations.
By increasing the rates, more neutrinos are emitted and escape during the initial stages
of collapse, thereby increasing the evaporative neutrino-cooling. Furthermore, because the
dominant source of matter pressure is electron degeneracy, increased electron-capture rates
accelerate the collapse. This impacts the matter profiles outside the shock in the early postbounce phase. Decreasing the rates has the opposite e↵ect, the entropy, temperature, and
electron fraction of the core are significantly higher because less cooling takes place.
The evolution prior to and right at ⇢c = 2 · 1012 g cm 3 (which is the density that defines
neutrino trapping) is what sets the final value of the trapped lepton and electron fractions,
which are important due to their direct impact on the formation of the PNS. For all the
reference simulations, a minimum in the trapped lepton fraction was found to occur with a
systematic scaling factor of approximately four. The minimum that forms can be seen in
Figure 2.6. Scaling by ten slightly reverses the downward trend, and increases the trapped
lepton-fraction from its minimum value. This behavior is the result of electron-neutrino
capture on heavy nuclei becoming the primary source of opacity, exceeding what is typical
as a result of coherent ⌫e -scattering. When the rates have been enhanced by a factor of
ten, the ratio of the absorption and scattering opacities, a /s , surpasses unity already by
central densities of 3 · 1011 g cm 3 and a ⇠ 4s by the time ⇢c = 1012 g cm 3 . Absorption
cross sections are then large enough to trigger an early onset of neutrino trapping at densities
lower than what is found for the reference rates. The consequence is that electron capture
has a smaller window of deleptonization, leading ultimately to a higher Yl overall.
The range of electron fractions near core bounce is commensurate with the range of
40

0.40

x2.0
x4.0
x10.

Base
x0.50
x0.25
x0.10

1012

1013

Ye

0.35

Yl

0.30

0.25

Yνe

0.06
0.04
0.02
0.00
1010

1011

ρc (g/cm3)

1014

0.42
0.41
0.40
0.39
0.38
0.37
0.36
0.35
0.34
0.33
0.32
0.31
0.30
0.29
1010

x2.0
x4.0
x10.

1011

Base
x0.50
x0.25
x0.10

1012

1013

ρc (g/cm3)

1014

Figure 2.7: Comparison of the central electron, electron-neutrino (left) and lepton (right)
fractions in which the nuclear electron-capture rate for every species has been scaled by
factors shown in the legend. Warmer colors indicate a higher overall electron capture
rate, and cooler colors indicate a lower rate. The dashed black line indicates the reference
s15WW95+SFHo simulation.
trapped lepton-fractions so far described, see Figure 2.7. As mentioned above, variations of
Ye (and Yl ) on this level are of importance due to its direct impact on the formation of the
PNS and the supernova shock. Electron fraction, entropy, density and velocity profiles are
shown in Figure 2.8 for s15WW95+SFHo at -1, 0, 1, and 5 milliseconds relative to bounce.
Of particular interest, the mass of the forming PNS inner-core at bounce, seen as the mass
behind the steep velocity gradient in panel (b), was found to vary on the order of ⇠0.1 M ,
and up to ⇠0.2 M five milliseconds after bounce. The asymmetry observed in the trappedlepton fraction, where scaling the rates by 0.1 had a more dramatic e↵ect than scaling by
10, translates directly to the variation of the inner-core mass at bounce (+16/-4 % from the
reference). The result is that the forming PNS has a lower bound on the inner-core mass at
bounce over the range of electron-capture rates explored. Because the rates are already high,
and therefore the absorption opacity is already almost comparable to the scattering opacity,
41

(b)

(c)

(d)

Electron Fraction

0.50 (a)
0.45
0.40
0.35
0.30
0.25
0.20
0.15

Entropy (kb / baryon)

6.0
5.0
4.0
3.0
2.0
1.0
0.0
14

Density (g / cm3)

10

13

10

12

10

11

10

10

10

10

Velocity (104 km/s)

10

9
8

1
0
1
2
3
4
5
6
Core bounce
7
t-tb = -1.0 ms
t-tb = 0.0 ms
t-tb = +1.0 ms
8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0

x10.
x4.0
x2.0
x1.0
x0.50
x0.25
x0.10

t-tb = +5.0 ms

0.2 0.4 0.6 0.8 1.0 1.2

Enclosed Mass (M )

Figure 2.8: The electron fraction, entropy, density, and velocity as a function of enclosed
mass at four times during a core-collapse simulation, spanning 6 ms around bounce, including
the collapse phase just after the onset of neutrino trapping (a), core bounce (b), and 1 ms
(c) and 5 ms (d) after bounce during which the shock has begun its outward trajectory.
The reference simulation (s15WW95+SFHo) is shown in dashed black. Simulations shown
in color have electron-capture rates scaled systematically for all species by factors of 10, 4,
2, 0.5, 0.25, and 0.1. Core bounce (t tb = 0) shown in panel (b) is defined as when the
entropy at the shock front exceeds 3.0 kb /baryon.

42

the range of inner-core mass at bounce comes mainly from simulations with decreased rates
relative to the base simulation.
Such variations in the homologous inner-core mass will translate directly to the kinetic
energy of the emergent shock that eventually detonates the star. Furthermore, the one to
two orders of magnitude range of density outside the shock (see Fig. 2.8) will likely play
an important role during the shock propagation and explosion phases of multidimensional
CCSNe simulations. Therefore, further investigations into the impact of these variations in
the late stages of two and three dimensional CCSNe simulations are warranted.
In addition to the direct impact on core dynamics and structure, the neutrino emission
at bounce is found to be very sensitive to these variations. Figure 2.9 shows the neutrino
luminosity 500 km from the center for the di↵erent neutrino species as a function of time.
Prior to bounce the ⌫e -luminosity begins to rise from electron captures on bound protons in
nuclei, but is quickly regulated by neutrino trapping, causing a down turn in the luminosity.
During this time the core is very sensitive to the nuclear electron-capture rates as the entropy
is low enough that heavy nuclei dominate the available mass. Scaling the rates for each
nucleus using the same systematic factors results in a 40% variation of the ⌫e -luminosity
before bounce. During bounce, the electron-neutrino burst—seen as the peak luminosity
in the left panel of Figure 2.9—is powered primarily by electron capture on free protons.
The core-bounce and shock liberates nucleons from their bound states and the entropy rises
causing a significant increase in the nucleon and light particle abundances. That said, while
the electron-capture rate on free protons, EC
p , is not adjusted in these simulations, a range
of ±20% relative to the reference peak ⌫e -luminosity is observed.
These dramatic variations of the peak electron neutrino luminosity are a result of alterations to the neutrinosphere and shock convergence-timescale. Specifically, when electron
43

600

Base
x2.0
x0.50
x4.0
x0.25
x0.10
x10.

550
500

100

80

Luminosity (erg s-1)

450

70

400

60

νe

350

50

300
250

40

νe

200
150

30

νx

νe

20

100
50
0

90

νx

10

νe

10 5 0 5 10 15 20 25 30 40

t-tb (ms)

60

80 100 120

0

Figure 2.9: The neutrino luminosity as measured at a radius of 500 km as a function of
time after bounce in the s15WW95+SFHo simulation set. Electron-capture rate scaling
factors are shown in the legend, where contours with warmer colors have higher rates, and
cooler colors have lower rates. While the peak electron-neutrino luminosity is considered
particularly stable across core-collapse simulations, it varies significantly with variations of
the electron-capture rates on medium-heavy nuclei. When the rates are at their lowest (⇥0.1
case), the shock reaches the neutrinosphere more quickly than in the other simulations. This
results in a larger luminosity in the peak electron-neutrino burst because more ⌫e s are able
stream out of the core at early times. The opposite is true when the rates are higher, the
neutrinosphere and shock converge much more slowly, and so the neutrinos spend more time
di↵using out of the inner core, reducing the peak luminosity but distributing it out to later
times.

44

captures on nuclei are weaker (scaling by 0.1), the inner-core mass that forms at bounce is
significantly larger. This results in more kinetic energy transferred to the shock, allowing
it to sweep up mass more quickly. In Figure 2.8 this can be seen by the broadening of the
distribution of shock locations in mass between the di↵erent simulations in the velocity plot
5 ms after bounce (bottom-right) as compared to t

tb = 0. Also, with a weaker overall rate

the opacity will be lower, allowing the neutrinosphere to move in to lower radii more quickly.
The combination of these e↵ects result in the shock and neutrinosphere radii converging
earlier for the simulations with lower electron-capture rates, and later for simulations with
higher rates, up to a di↵erence on the order of 3.5 ms. Thus, electron capture on protons
liberated by the shock produce neutrinos that are able to reach the neutrinosphere earlier
and freely stream away, contributing to a larger ⌫e peak luminosity when the nuclear electron
capture rate is systematically lower. On the other hand, when the nuclear electron capture
rate is high, the emitted neutrinos di↵use more slowly through the core, and reach the neutrinosphere at later times, thus strongly quenching the peak luminosity but spreading out
the emission to later times. Due to the high luminosity of the electron-neutrino burst near
the time of bounce, it is a candidate for detection from a galactic core-collapse supernovae
in Earth-based detectors sensitive to electron neutrinos, e.g. those with a detector volume
composed of liquid Argon. And while such measurements are not presently of high enough
precision to resolve each variation seen here, they may indicate the total amount of electron
capture occurring at core bounce.

2.4.3.1

Progenitor model sensitivity

In order to evaluate the significance of the electron-capture systematic sensitivity studies,
they were tested against a study of the progenitor dependence of the core-collapse phase.
45

Sensitivity to
Inner-core
Mass (M )

2.0
1.5
1.0
0.5
0.0
30

Tcc (MeV)

0.55

sc (kb / baryon)

Progenitor variations

Electron capture rate variations

0.50
0.45

20
10
0
700
600
500
400
16
15
14
13
10
8
6
A
M
+T
95
W
2
D
5W
+D
s1
95
o
W
FH
5W
+S
s1
95
o
W
FH
5W
s1
+S
07
o
H
FH
0W
s4
+S
07
o
H
FH
0W
s2
+S
07
H
2W
s1
2)
(3 7
rs 0
ito WH
en 20
og s1
Pr 12s

Figure 2.10: The full range of sensitivity of the PNS inner-core mass, central entropy, and
central temperature at bounce as well as the peak ⌫e -luminosity, the peak average ⌫e energy,
and the average ⌫e energy prior to neutrino trapping, owing to variations of the progenitor
model and electron-capture rates. Thirty two progenitors were utilized from the WH07 model
set of Woosley and Heger [79] for producing the progenitor bars (red) in the figure. Each bar
of the electron-capture rate variations derives from simulations where the rates have been
systematically scaled by factors of 10, 4, 2, 0.5, 0.25, and 0.1. The horizontal tick represents
the value of the reference simulation for the tested Progenitor + EOS combination. The
window ranges are chosen so that the progenitor sensitivity bars are of equal size across each
of the plotted parameters.

46

Drawing from the larger set of progenitors from which the reference progenitors of the
electron-capture study belong, the 2007 non-rotating solar-metallicity single-star model set
from the stellar evolution code KEPLER [79] was utilized. This model set contains the presupernova configuration of 32 stars ranging in zero-age-main-sequence (ZAMS) mass from
12 M

to 120 M – s12WH07 and s120WH07 respectively. Simulations of these progenitors

exhibit a ⇠3.5% range of trapped lepton-fraction (0.288 - 0.298), a ⇠4% range of inner-core
mass at bounce (0.473 – 0.491M ), and a ⇠9% range of electron-neutrino peak luminosity
(5.19

5.65 · 1053 erg s 1 ) during the neutrino flash occurring just after core bounce.

Figure 2.10 compares the stellar progenitor model and electron-capture rate dependence
of several structural and neutrino quantities during collapse. The range of inner-core mass
and peak ⌫e luminosity seen from employing the WH07 progenitor model set are each approximately a factor of 5 smaller than the ranges seen from varying the electron-capture rates
across all progenitor+EOS references. On the other hand, the range of central entropies and
temperatures at bounce are comparable between the two sensitivity studies. The ⌫e average energies just prior to neutrino trapping and during the deleptonization burst are also
compared in Fig. 2.10. The neutrinos emitted during the luminous burst just following core
bounce are of higher energy than those emitted earlier because they arise primarily from
electron capture on free protons. They also decouple from the core at a much hotter and
denser neutrinosphere than prior to bounce, yielding higher energy neutrinos. In both of the
sensitivity studies, variations of the electron-capture rates and of the initial stellar models,
the range of average neutrino energy during peak emission is comparable (⇡ ±0.5 MeV).
While captures on free protons contribute only marginally to deleptonization in the central zone, further out in the iron core, where the densities are lower (and Ye ’s are higher),
electron captures on protons contribute to the deleptonization, especially in cases where
47

electron captures on nuclei are suppressed. The capture of electrons on these free protons
produces neutrinos of a higher average energy, commensurate with the large spread seen in
the bottom panel of Fig. 2.10 (which is taken when the central density is 3 ⇥ 1011 g cm 3 ,
but present from the onset of collapse). Another contribution to the energy spread is the
systematic shift of electron captures to more neutron rich nuclei as the electron-capture rates
are increased and the matter becomes more neutron rich. These neutron-rich nuclei have
more negative Q-values, yielding lower energy neutrino emission. Both of these e↵ects result
in a dispersion of average neutrino energies early on that is several factors larger than what
is seen in the progenitor simulations.
Finally, note that while the peak luminosity is only weakly dependent on the progenitor model, the post-bounce pre-explosion luminosity of all six neutrino species have strong
progenitor dependences [83]. On the other hand, the pre-explosion luminosities investigated
here are much less sensitive to the nuclear electron-capture rates comparatively–see panel
(b) of Figure 2.9. The diverging of the luminosities seen at t

tb = 120 ms is due only to

the di↵erence in collapse times between the simulations which carries over to the evolution
of the mass accretion rate after bounce.

2.4.4

Monte-Carlo variations

In addition to the possibility of systematic errors in the electron-capture rates, the e↵ect
of statistically distributed variations are also explored. Such an investigation is of great
importance if the e↵ect of an approximation such as Eq. 2.1 is to be understood. The main
flaw in a continuous function for rate estimation across many nuclear species is the loss
of structure, which would otherwise serve to statistically distribute the rates on a reaction
by reaction basis (see Fig 2.2). To study this e↵ect, Monte-Carlo (MC) variations of the
48

electron-capture rates were performed. Using an analytic description of the electron-capture
rate distributions, such as a Gaussian or Poisson distribution, is likely to be inaccurate.
Instead, the approximate rate is adjusted via MC for each species by adding to its log10 ( EC )
a value randomly chosen from a distribution created from the residuals of the tabulated rates
and the approximate rates, i.e.

i,table

log10 ( EC

i,Eq.1

)

log10 ( EC

)

(2.10)

where i is an index running over all the tabulated reactions.
In constructing this distribution, it is important also to preserve the Q-value dependence
of the residuals that can be seen in Figure 2.2b. This is done by separating the residual
distribution into subsets so that the reaction-rate residuals in each subset have similar Qvalues. To do so a Q-value binning of 2.5 MeV was chosen, but this method was also tested
with binnings of 5.0 MeV and 10.0 MeV which resolve the Q-dependence less, but have more
counts per bin from which to sample. From these samples, pseudo electron-capture rates
were MC generated such that they retain the Q-value dependence of Eq. 2.1, but statistically
distribute the approximate rate according to the variance of the rates calculated in the shellmodel. Seven simulations for each binning were performed.
As mentioned before, at low densities the electron-capture rate depends strongly on
the energy levels of the initial and final nuclides because the electron chemical potential
is comparable to the excitation energies of the allowed Gamow-Teller transitions. As the
electron chemical potential increases, it encompasses a larger range of excitation energies
which results in the electron-capture rate becoming sensitive primarily to the total strength.
In the low-density case of Figure 2.2a the approximation of Eq. 2.1 while appearing to

49

0.42

Monte-carlo range
Reference sim.

0.41
0.40
0.39
0.38
0.37

Yl

0.36
0.35
0.34
0.33
0.32
0.31
0.30
0.29
10

10

11

10

12

10

13

10

14

10

ρc (g cm-3)

Figure 2.11: Lepton fraction band as a function of central density for the Monte Carlo (MC)
described in the text. The band represents data points from seven MC simulations based
on the s15WW95+SFHo reference, where the band width represents the min to max range
of Yl . The rates are varied by drawing from the Q-value dependent residual function shown
in Figure 2.2 after binning it into 2.5 MeV energy bins. This binning was chosen as it best
tracks the Q-value dependence of the residual distribution.

50

decently reproduce the mean of the shell-model rates, actually has a mean approximately a
factor of two lower than the tabulated rates. As the density increases, this di↵erence between
the mean electron-capture rate estimated by the approximation and the shell-model rates
decrease. Thus, the approximation better reproduces the mean rate in the high density
case of Figure 2.2a. Because the MC simulations are based on residual distributions of the
tabulated rates and the approximation, the average rate produced in each MC trial also has
this bias.
Plotted in Figure 2.11 is the min-to-max band representing the range of lepton fraction
observed from all of the MC simulations. The band drawn corresponds to a 2.5 MeV binning
of the residual distributions from which the MC sampling was performed. For the reasons
just described, the band has lower electron and lepton fractions than the reference at low
densities, but becomes more statistically distributed around the reference at higher densities,
near 5·1011 g cm 3 . The lepton fraction band width varies from about a half percent initially,
to its largest value of ⇠2.5% just before neutrino trapping, and then decreases back to ⇠1.5%
before bounce. Altogether, no significant impact on the core dynamics or the neutrino
transport were observed and therefore it can be conclude that any statistically distributed
scatter in the estimations of the electron-capture rates, such as those seen in Figure 2.2, will
likely not impact the models.

2.5

Conclusion

Nuclear electron capture has long been understood to play an important role in the dynamics
of core-collapse supernovae and large e↵orts have been undertaken to produce reliable estimates of electron-capture rates for astrophysical contexts. Although significant progress has

51

been made in benchmarking theoretical electron-capture rates by comparison with chargeexchange experiments (especially using shell-model calculations) [13], large uncertainties
remain for neutron-rich nuclei and nuclei beyond A=65. Furthermore, sophisticated shellmodel estimates for electron-capture rates exist only for a small subset of the large number
of nuclei that contribute strongly. The implications of uncertainties in the electron-capture
rate estimates for the core-collapse and early post-bounce phases of fully self consistent, general relativistic, core-collapse supernova simulations with comprehensive neutrino transport
are explored in this work.

2.5.1

Most important nuclei

For the reference simulation, the contribution of each nucleus to core deleptonization is
calculated, and a statistical resampling study is also performed. Both of these studies identify
the nuclear species whose rate should be known most precisely due to their significance in
the simulations. With the given set of electron capture rates—from shell-model estimates
to the approximate estimates of Eqs. 2.1 and 2.2—the simulations are found to be most
sensitive to neutron rich nuclei in the upper pf and pf g/sdg-shells.
Specifically, in these simulations nuclei near the A⇠80, N⇠50 closed neutron shell contribute the bulk of core deleptonization, and when removed from the simulations result in
noticeable changes to the protoneutron star formation, with a significantly larger impact
than when any other group of nuclei are removed. However, because sophisticated estimates
from nuclear theory are not available for individual nuclei in this region, the electron-capture
rates for these species have been accounted for in the past via simple averaging techniques
and in this work via an approximation that has been fit to the LMP rate set. While this
approximation reasonably reproduces the average electron capture rate for sd and pf shell
52

nuclei near stability, rates for heavier neutron-rich nuclides will likely diverge from what is
predicted by this parameterization.

2.5.2

Impact of uncertainties

The impact such uncertainties may have are evaluated by varying the electron-capture rates
for more than 6000 nuclei statistically, about the approximate prediction, and also systematically. On one hand, it is found that statistical variations of electron-capture rates e↵ect
the overall dynamics and neutrino emission only weakly, producing marginal changes to the
simulations. These findings indicate that the lack of structural variation that distributes the
rate estimates from one species to the next is not crucial to the simulations.
On the other hand, the average electron capture rate across a region of nuclei strongly
determines the overall impact of those constituent nuclei. By systematically varying the
electron-capture rates by factors between 10 and 0.1, dramatic variations in the inner-core
mass (+16/-4 %) and the electron-neutrino luminosity (± 20%) at and near bounce, respectively, are observed. Comparing with 32 simulations utilizing di↵erent progenitor models,
this range of inner-core mass and peak neutrino-luminosity is found to be 5 times as large
as that seen when varying the progenitor models.
In addition, the nuclear electron-capture rates are found to be already large enough in
the reference simulations that increasing them beyond their base values has a considerably
smaller e↵ect than decreasing them. This has compelling implications. Rates for A⇠80
nuclei near the N=50 shell gap, which have been shown to be the primary contributors
to the overall impact of electron captures during core collapse, may be overestimated by
Eq. 2.1 due to Pauli blocking at the closed neutron shell. Combined with a greater overall
sensitivity to the systematic decrease in electron-capture rates, changes to the collapse and
53

early post-bounce phases of the simulations may be as significant as those seen in this study
if the current rates of these nuclei are found to be overestimated.

2.5.3

Goals for future studies

For these reasons, it is important that experimental and theoretical e↵orts be aimed at
nuclei which span the region on the chart of isotopes between stability and the neutron
drip line in both the pf and pf g/sdg model spaces, and further expand on the work that
has been carried out for (near-)stable nuclei in the pf -shell. Since data from (n,p)-type
charge-exchange experiments for nuclei in the pf g/sdg-shell and for neutron-rich nuclei in
the pf and pf g/sdg-shell are scarce, new experiments are required to obtain a sufficient
set of data to benchmark current and future theoretical estimates. To this end, presently
feasible experiments on neutron-rich nuclei at and near stability with 60<A<120 should add
to the few cases that have been measured in this region. With the higher beam intensities
that will be available at next generation rare isotope facilities, future experimental programs
should focus on the neutron-rich component of the primary electron-capture channel shown
in Figure 2.5. In particular, investigation of nuclei in the A⇠80 and N⇠50 region should
take precedence, as changes to their electron-capture rates will significantly constrain the
core-collapse dependence on nuclear electron-capture.

2.6

Updates

The weak-interaction rate library which was developed in this work in order to characterize
the sensitivity of CCSNe to nuclear electron capture, has seen a number of updates and
continues to be an active research component of the Joint Institute for Nuclear Astrophysics

54

70

60

50

28

40

=Z

50

N

Proton Number

50

20

30

28

82

8

20

20

Stable
FFN
ODA
LMP
LMSH
Raduta et al.
No rate

Pruet & Fuller
Suzuki-Honma (gxpf1j)

2

10

8
2

0
0

20

40

60
Neutron Number

80

100

120

Figure 2.12: Updated version of Figure 2.1 showing the addition of the recently added rate
tabulations since the libraries first implementation in [2].
– Center for the Evolution of the Elements (JINA-CEE). Recently released rate tabulations
for nuclei in the sd- and pf -shells have been added to the weak rate library. The sd-shell
tabulation includes rates calculated on nuclei with A=17-28 using the USDB configurationinteraction including coloumb e↵ects [84]. The pf -shell tabulation was developed using the
GXPF1J configuration-interaction by Honma et al. [85] for nuclei with masses between A=4265. In addition, rates for a large number of proton-rich nuclei from the older empirically
derived single-particle model calculations of Pruet and Fuller [86] have also been added to the
weak rate library. The addition of these new tables can be seen in the updated library mass
coverage of Figure 2.12. In addition to these rate tables, the approximate electron-capture
rate parameterization described by Eq. 2.1–which is used for the majority of the electroncapture rate estimates in the sensitivity study–has been updated. Raduta et al. [87] extended
this simple analytic parameterization to allow for temperature, electron density, isospin, and

55

nuclear odd-even dependencies. They found that the extra degrees of freedom introduced
by the inclusion of isospin and odd-even e↵ects considerably improved the agreement with
electron-capture rate calculations from large-scale microscopic rate calculations [87]. While
these advancements may improve the estimates of electron capture rates near the pf -shell,
shell e↵ects such as Pauli blocking–which as mentioned previously, can significantly suppress
the rate–are not accounted for in the updated model. These methods have been incorporated
and are currently maintained in the official version of the weak rate library.
In addition to the above theoretical updates that have been incorporated, a few subsequent computational and experimental investigations involving electron capture in corecollapse supernovae have been completed.
A follow up sensitivity study investigating the importance of the N=50 nuclei, described
in section 2.4.2, was recently undertaken by Titus et al. [88]. In that work it was demonstrated that even though thousands of electron-capturing nuclei are included, variations in
key characteristics of the evolution, such as the lepton fraction, electron fraction, entropy,
stellar density, and in-fall velocity are about 50% due to uncertainties in the electron-capture
rates on nuclei in the region above 78 Ni near N=50. Because of the significance of early results
of this study, two of the first charge-exchange measurements in this region were performed:
the 86 Kr(t, 3 He) and 88 Sr(t, 3 He) measurements. As 86 Kr and 88 Sr are the two most proton
deficient, stable, N=50 nuclei, the results of these measurements will give a good indication
of the allowed Gamow-Teller strength in nuclei with a full g9/2 neutron shell and will help
to constrain the electron capture rates in this region.
Concurrent with the above e↵orts, Richers et al. [3] investigated the impact various
equations of state have on gravitational waves generated from axisymmetric rotating corecollapse simulations. As part of that work, various systematic variations of the electron
56

capture rates (comparable to what was performed in this work) were also utilized. It was
found that the gravitational waves from post-bounce core oscillation had near equivalent
dependence on the nuclear equations of state and the electron capture rates on heavy nuclei.
Moreover, both influenced the expected gravitational wave signals that can be detected
in current-generation gravitational wave detectors. If the electron capture rates were to be
better constrained, detection of such gravitational waves may be able to untangle information
about the nature of the nuclear matter equation of state.
As both the early post-bounce neutrino emission and gravitational-wave multi-messenger
signals will be significantly constrained should improved experimental and theoretical estimates for the relevant electron capture rates become available, continued e↵ort along this
axis should be a top priority in the field of nuclear astrophysics.

2.6.1

Implementation procedure for future updates

The weak-rate library codebase is maintained on the web based Git version control repository
hosting service GitHub. It is currently implemented as a Git submodule in the neutrinointeraction library NuLib. This organization allows for updates to the weak rate library to
be easily propagated to NuLib without source code duplication. When future updates to the
weak rate library are desired, the following steps should be followed,
1. The most up-to-date GitHub repository for the weak rate library (currently
https://github.com/csullivan/weakrates) should be forked to a di↵erent user account.
2. Development and updates can be made using Git on this new GitHub fork of the weak
rate library.
3. To propagate these changes to NuLib, a fork should be made of the most up-to-date
57

repository of NuLib (currently https://github.com/evanoconnor/NuLib).
4. This new GitHub fork of NuLib should then be updated so that its weak rate library
submodule is pointing to the remote commit of the new weak rate library fork created
above in step 2.
5. Finally, a GitHub pull request should be made to bring this change, of the weak rate
library submodule link, into the main NuLib repository. This way NuLib users can
simply update the NuLib submodules and receive the new weak rate library code.
After following these steps, the officially hosted version of the weak rate library will become
the newly created fork, and NuLib will depend on it, rather than the csullivan/weakrates
fork.

58

Chapter 3
The (6Li, 6Li0 + ) reaction
Nuclear excitations by neutral-current neutrino scattering play an important role in neutrinonuclear astrophysics. The accurate knowledge of neutral-current neutrino-nucleus cross sections is also important for the observation of astrophysical neutrinos in earth-based neutrino
detectors [89].
In a recent review of weak interactions in core-collapse supernovae (CCSNe) and nucleosynthesis, Langanke and Martnez-Pinedo [90] discuss the importance of the comparison of
shell-model and RPA calculations with inelastic neutrino-nucleus scattering (INNS) cross
section estimates from experimental data. Because astrophysical models rely on theoretical calculations of these kind, and because the available data for comparison is limited at
present, the establishment of a new indirect technique to constrain these important theoretical estimates is of high value.
INNS has been argued to be a dissipative mechanism by which neutrinos deposit their energy in nuclear matter during CCSNe [91]. Furthermore, successful CCSNe produce a strong
neutrino signal in the tens-of-MeV range which can be detected via the products of chargedcurrent (CC) and neutral-current (NC) weak interactions with nuclei in various detector
media. However, measurements relevant for supernovae neutrino detection do not exist for
most nuclei, and are highly uncertain where available due to their small cross sections [89].
One method for studying neutrino-nucleus reactions is via direct measurements through neu-

59

trino spallation at reactor [92] and synchrotron [93] facilities. Only a few measurements have
been performed so far, such as the neutrino irradiation of 12 C [94, 93].
An alternative approach is via indirect measurements that involve inelastic scattering of
other probes, such as (p, p0 ) [95, 96, 97] and (e, e0 ) [98, 99]. Such measurements are much
more feasible, and have been used to infer neutral-current neutrino inelastic scattering cross
sections in the past [100].
This inference is possible principally because inelastic scattering cross-sections are sensitive to the probe-independent nuclear transition strength. For example, the dominant
component of the INNS cross section at astrophysical energies is directly related to the
Gamow-Teller (GT) transition strength. The cross section for a transition from an initial (i)
and final (f ) state is given by [101]

i,f (E⌫ )

=

GF 2
(E⌫
⇡

Ef i )2 B(GT0 )f i ,

where GF is the Fermi constant, and E⌫ and

(3.1)

Ef i are the energy of the scattered neutrino

and the di↵erence between final and initial nuclear energies, respectively. B(GT0 )f i is the
reduced GT strength in the inelastic channel ( S = 1,

B(GT0 )f i =

T = 1, and

2
gA
|hf kÔ(GT0 )kii|2 ,
2Ji + 1

Tz = 0),

(3.2)

where gA is the axial vector coupling constant, Ji the spin of the initial nucleus. Ô(GT0 ) is
the corresponding GT operator,

Ô(GT0 ) =

1X
ˆ (k)ˆ
⌧0 (k),
2
k

60

(3.3)

where ˆ = 2ŝ and ⌧ˆ = 2t̂ are the spin and isospin operators, respectively, and the sum
runs over all nucleons in the target. Thus, the allowed component of neutrino-induced
nuclear excitations are isovector spin-transfer excitations, with no change in orbital angular
momentum.
On the other hand, reactions mediated by hadronic inelastic scattering induce M1 transitions whose operator is given by

Ô(M1) =

r

3 X
ˆ + 1 gs (k)ˆ (k)]µN ,
[g` (k)`(k)
4⇡
2

(3.4)

k

where `ˆ is the orbital angular momentum operator, and g` (gs ) is the orbital (spin) gyromagnetic factor. Thus, both isovector and isoscalar transitions contribute, as well as non–spintransfer transitions (transitions that involve only change in orbital angular momentum).
The isovector (IV) component of the M1 operator can be rewritten as

Ô(M1)IV =

X
k

r

3
4⇡

✓

ˆ ⌧0 (k) +
g`IV `(k)ˆ

with the IV gyromagnetic factors g↵IV = (g↵n

◆
1 IV
g ˆ (k)ˆ
⌧0 (k) µN ,
2 s

(3.5)

p

g↵ )/2 (↵ = ` or s). The spin part of the

above isovector M1 operator (Eq. (3.5)) is the same as that of the GT0 operator (Eq. (3.3))
except for a constant factor. This indicates that the GT0 strength, which is needed to
infer the inelastic neutrino-nucleus cross-sections, can be extracted from hadronic probes
such as (p, p0 ) under certain circumstances. Specifically, because the (p, p0 ) probe constitutes
a transition of Ji⇡ = 1/2+ ! Jf⇡ = 1/2+ , and Ti = 1/2 ! Tf = 1/2, it can induce
isovector transitions ( T = 1) as well as isoscalar transitions ( T = 0), both with various
contributions of total spin transfer. Therefore, (p, p0 ) is capable of extracting GT0 strength

61

where the orbital and isoscalar contributions are negligible [100]. These assumptions are
valid primarily for spherically symmetric nuclei with weak or separable isoscalar responses.
In general, it would be ideal to have a probe which is capable of extracting the GT0 strength
from inelastic excitations, without consideration of these properties. In this work, a di↵erent
reaction probe is tested in which isovector spin-transfer excitations can be directly isolated.
The (6 Li, 6 Li0 [T = 1, Tz = 0, J ⇡ = 0+ , 3.56 MeV]) reaction, first suggested by Austin
et al. [102], provides direct access to the GT0 response of nuclei in an unambiguous manner,
as the quantum numbers of the initial and final states guarantee the induced transition of
S = 1,

T = 1, and

Tz = 0. This selectivity is a distinct feature of the (6 Li, 6 Li0 )

probe. A simplified level diagram of 6 Li is shown in Fig. 3.1, which is drawn based on
Ref. [103]. This reaction channel can be identified by tagging the de-excitation
E = 3.56 MeV. Although the ↵ threshold is located below this state (Q↵ =

ray with
1.47 MeV),

the ↵ decay from the 3.56 MeV state is forbidden, unlike for other excited states, as it
violates parity invariance [103]. Instead, this state decays directly to the ground state via
emission. Since it has J ⇡ = 0+ , the branching ratio to the 3+ state at 2.19 MeV, as well
as feeding from other higher excited states is negligible [103]. Therefore, the coincidence
measurement with 3.56-MeV

rays provides clean identification of the reaction.

In order to establish (6 Li, 6 Li0 ) as a viable indirect probe of INNS cross sections, direct
measurements are a necessary benchmark. For this purpose, the 12 C(6 Li, 6 Li0 ) measurement
was performed as long-baseline direct neutrino measurement data exists [93]. Furthermore,
measurements on 24 Mg and 93 Nb were also performed. For these latter measurements the
3.56 MeV de-excitation -rays were not resolvable. As the tagging of this gamma is the basis
for the proposed technique, the gamma background due to the decay of the isoscalar giant
resonances must not dominate. In this work it is demonstrated that for heavier nuclei, this
62

1

(keV)
3562.9

M

J ⇡ ;T
0+ ;1

parity
-forbidden
3+ ;0

2186
1474.3
4
He + d
0+ ;0 1+ ;0

1+ ;0

0.0
6

Li

Figure 3.1: A simplified level diagram of 6 Li based on Ref. [103]. The ↵ decay of the J ⇡ = 0+ ,
T = 1 state at Ex = 3.56 MeV is parity-forbidden, and thus this state mainly decays to the
ground state via emission.
background is much stronger than the isovector signal. Thus, the proposed method is likely
only applicable for light nuclei due to the proportionally significant contribution of gamma
emission from the isoscalar giant resonances in heavier systems.

3.1

Experiment

The 12 C(6 Li, 6 Li0 ) measurement was carried out at the Research Center For Nuclear Physics,
Osaka University, Japan. A layout of the RCNP facility is shown in Fig. 3.2. A 100-MeV/u
6 Li

beam was accelerated via the coupled operation of the azimuthally varying field (AVF)

and ring cyclotrons with a measured resolution of ⇠1.75 MeV in FWHM. The beam energy
of 100-MeV/u is well suited for the present study of the spin-isospin response because at
this energy the reaction mechanism is simple with small momentum transfer at 0 scattering
angle [104].
The 6 Li beam was transported achromatically to the WS experimental hall (see Fig. 3.2)

63

LAS

Grand Raiden

BLP 2

ws
BLP 1

RING Cyclotron

SOL2
SOL1

AVF Cyclotron

Figure 3.2: The primary facility beam line layout for the Research Center for Nuclear Physics
(RCNP) located at Osaka University, Japan. In this experiment, the 6 Li beam was transported from the ring cyclotron (shown in green) into the WS hall. It was then transported
achromatically to the Grand Raiden target position. See text for details.

64

where it was impinged upon a 15.21 mg/cm2 nat C target oriented at 22.5 relative to the
horizontal plane (yielding an e↵ective thickness of 16.46 mg/cm2 ). The beam intensity was
measured to be ⇠1 pnA.

3.1.1

The CAGRA+Grand Raiden Coincidence Setup

3.1.1.1

Grand Raiden Spectrometer

The 6 Li ejectiles were momentum-analyzed and detected at the focal plane of the Grand
Raiden spectrometer [105] in the 0 setting. Figure 3.3 illustrates the layout of the Grand
Raiden spectrometer and the end station focal plane detectors. The Grand Raiden focal
plane consisted of two Multi-Wire Drift Chambers (MWDCs) and three plastic scintillators
of thickness 3 mm, 10 mm, and 10 mm, respectively. Positioned between the two 10-mm
plastic scintillators was a 12-mm aluminum plate (shown in red in the bottom panel of
Fig. 3.3), which was used to stop the residual 6 Li nuclei so that the final scintillator could
serve as a veto on the d and 4 He products from 6 Li breakup. Each MWDC featured two
anode wire planes so that positions in both the horizontal and vertical directions could be
determined, while the horizontal and vertical angles were inferred using the positions from
both MWDCs together. The entire set of focal plane detectors were oriented at a 45 angle
relative to the beam axis.
The unreacted beam was dumped through the holes of the MWDCs into the 0-degree
Faraday cup located at ⇠12 m downstream of the focal plane and was shielded to reduce
background for the -ray measurement at the target position. The distance between the
center of the hole and the edge of the active area of the MWDC was ⇠20 cm. This distance
corresponds to a momentum range which could not be measured with respect to the unre-

65

acted beam at 0 degree and was ⇠1%, corresponding to ⇠10 MeV for 100 MeV/u 6 Li. The
resulting excitation-energy spectrum was studied up to 40 MeV.
The plastic scintillators were used for energy-loss measurements and served as a time-offlight (TOF) start signal, with the cyclotron RF as the stop. Together, the energy loss in
the scintillators and the TOF provided the particle identification with which the d, 4 He, and
6 Li

could be separated.
As the horizontal scattering angle at the target is determined primarily from the hori-

zontal incident angle at the focal plane, a slight under-focus mode was employed, in which
the ion-optical focus of Grand Raiden was placed upstream of the focal-plane MWDCs. In
focused mode, the small vertical angular magnification in Grand Raiden results in poor resolution of the reconstructed vertical scattering angle. However, by slightly under-focusing
the ion optics the vertical scattering angle at the target becomes strongly dependent on the
vertical position in the MWDCs. As the under-focusing is increased, the vertical position
range in the focal-plane detectors will span a smaller range of scattering angles at the target and thereby improve the angular resolution [95]. Further details regarding the nominal
acceptance, magnification, and resolution of the Grand Raiden spectrometer can be found
in reference [106].

3.1.1.2

CAGRA

The target was placed in a scattering chamber which was surrounded by 11 high-purity Germanium (HPGe) clover detectors in the “Clover Array Gamma-ray spectrometer at RCNP
for Advanced research”, CAGRA. The CAGRA array of HPGe detectors were of critical
importance for the success of the measurement, as they detected the 3.56 MeV -ray from
6 Li

which is emitted when an inelastically induced GT0 transitions occur in the target. The
66

MX

D2
D1

Dump-Q

Q2
SX
Q1

100 MeV/u
6
Li beam

Grand Raiden (GR)
Spectrometer

GR End Station

Focal plane
detectors

2

1

C

C

D

D

W

W

M

M

P
P S
PS S2 1
3

Al

Shield block
um

in

um

0 deg. Faraday cup

pl
e

at

Q-magnets

Primary
beam
12 m

GR End Station
Figure 3.3: Top panel: Layout of the Grand Raiden Spectrometer which was used to momentum analyze the inelastically scattered 6 Li nuclei. The spectrometer was configured with
seven magnets in the QSQDMDD configuration (where Q, D, S, and M are Quadrupole,
Dipole, Sextupole, and Multipole magnets, respectively). Bottom panel: configuration of
the spectrometer end station and focal-plane detectors. Figure adapted from [106, 95]. See
text for details.
67

Figure 3.4: The CAGRA array of HPGe clover detectors. CAGRA consisted of two rings of
clover detectors with 8 detector slots at 90 degrees and 4 at 135 degrees. There were also 4
forward slots for LaBr3 detectors at 45 degrees, but they were not used in this work due to
their large angular coverage which made Doppler reconstruction challenging.

68

5.5

Elab (MeV)

5.0
4.5
4.0
3.5

84.3o

3.36 MeV

95.8o

3.08 MeV

3.0

129.0o o
2.53 MeV
140.5
2.42 MeV

2.5
0

50

100

150

lab

Figure 3.5: The range of energies the 3.56 MeV -ray from 6 Li in the laboratory frame. The
-ray emitted from the de-excitation of the 6 Li[0+; T = 1; 3.56M eV ] state is 3.56 MeV in
the rest frame of 6 Li. In the laboratory frame this gamma is Doppler boosted due to the
momentum of the 6 Li ( ⇠0.43), according to its emission angle in the lab-frame. Shown
in red are the clover crystal positions in the CAGRA array. Each clover has crystals at two
distinct azimuthal angles. Thus, the two rings of clover detectors span for separate angular
ranges. For the purpose of Doppler reconstruction, the centroid of the clover crystal is used
as the gamma interaction point.
HPGe clover detectors were configured in the CAGRA array with a ring of 8 detector slots
at 90 and 4 at 135 , as shown in Figure 3.4 . Only seven of the eight 90 slots were utilized
in this measurement.
Because the 6 Li ejectile was moving at a velocity of

⇠ 0.43, the energy of the -ray,

which is 3.56 MeV in the rest frame, varied from 2.25 MeV to 5.63 MeV in the laboratory
frame as shown in Fig. 3.5. Each ring of clovers had two forward crystals and two backward
crystals. Thus, the entire array spanned four scattering-angle ranges, the centroids of each
(coinciding with the HPGe crystal center) were used as the gamma-ray interaction point for
the purpose of Doppler-reconstruction, and are shown as red points in Fig. 3.5. The angular
coverage of each clover crystal in the CAGRA array was ⇠ 12 , and the distance between
69

the target and a germanium crystal was ⇠20.8 cm.

3.2
3.2.1

Methods & Analysis
Data acquisition and analysis framework

In this measurement, Grand Raiden and CAGRA were instrumented with separate data
acquisition (DAQ) systems. The Grand Raiden DAQ was run as the master trigger, and
thus all singles data were saved, while the CAGRA DAQ was run as a slave, only validating
when a trigger in both CAGRA and Grand Raiden were present.
The Grand Raiden DAQ is an analog system that was developed and maintained by Tamii
et al. [107]. Signals from the left and right photomultiplier tubes (PMTs), attached to either
side of the focal plane plastic scintillators, were divided into two signals in order determine
the pulse height (for energy loss information), as well as time discrimination via a CFD
(constant fraction discriminator). The output of the CFD was then split and used for timeof-flight information, as well as for a coincidence trigger with the other PMT associated
with the same plastic scintillator. The measured live-time ratio for the Grand Raiden DAQ
during the 12 C(6 Li,6 Li0 ) measurement was ⇠0.82.
The CAGRA DAQ was newly formed for the campaign of measurements, in which the
6 Li

experiment was a part, the details of which will be described in a forthcoming pub-

lication. The CAGRA DAQ was instrumented with the same 14bit, 100MHz flash ADCs
(analog-to-digital) utilized by the GRETINA tracking-detectors [108] developed at Lawrence
Berkeley National Laboratory (LBNL). However, in this campaign, these digitizers were instrumented with customized firmware by Argonne National Laboratory’s (ANL) Gammasphere team [109]. This firmware was designed to be highly customizable in order to interface
70

RCNP detector systems and HPC center
Grand Raiden

HPC cluster

CAGRA

...

...

UD

Map

MPIEventLoop

Reduce

Analysis framework software stack

1

2

Timestamp

Timestamp

Binary
Payload

Binary
Payload

Disk

Disk

Tamii-Analyzer

Data thread

Build thread

Unpack thread

Write
thread

Analysis
thread

User
Defined
Read data
from source

Time correlate
raw events

raw event unpacked event
(binary) (c++ object)

GUI

GRUTinizer

Figure 3.6: The analysis software stack that was used in the RCNP CAGRA+Grand Raiden
campaign. Dashed blue lines show application and library boundaries. Solid arrows that
cross these boundaries represent communication by local memory, shared memory or disk
I/O. See text for details.
with a variety of di↵erent detector systems, and to be able to handle very high count rates.
The total deadtime of the CAGRA DAQ was measured to be ⇠4.5µs.
The separate DAQs were synchronized using a MyRIAD module developed at ANL,
which was inserted into the Grand Raiden DAQ. The MyRIAD (Multipurpose -ray Interface
to Auxiliary Detectors) module provided a general purpose interface and clock that could
be used in concert with the digital DAQ system utilized by CAGRA. After synchronizing
the MyRIAD clock in the Grand Raiden DAQ with the clock in the CAGRA digitizers,
timestamps from the separate data streams could be correlated with a sampling resolution
of ten nanoseconds.

71

Since the data streams from each system were written to disk separately, a modular data
analysis framework that could accommodate raw data events from a variety of sources was
developed. The block-diagram software stack used in this work is shown in Fig. 3.6. The
core components were,
· Tamii-analyzer: A Grand-Raiden raw event processing and analysis framework [110]
· GRUTinizer: Online and o✏ine framework for time-sorting, unpacking, and analyzing
events from generic sources [111, 112]
· MPIEventLoop: An MPI mapreduce implementation for distributed data processing.
The GRUTinizer analysis framework, which was originally developed for online and o✏ine
analysis at the National Superconducting Cyclotron Laboratory (NSCL), was used for the
primary event correlation and unpacking of data from both CAGRA and Grand Raiden.
First, raw events were extracted directly from CAGRA and indirectly from Grand Raiden
(via the Tamii-analyzer) and were time ordered by the data thread of GRUTinizer. These
heterogeneous event types were then stored in a thread-safe queue from which the next
thread, the building loop, could access. The building thread then time-correlated events
according to a user-defined time window. Then, these time-correlated events were passed
to the unpacking thread (via a threadsafe queue) which was responsible for transforming
the binary time-correlated data into user-defined C++ objects. These unpacked events were
then written to disk via an I/O thread, and then processed by an analysis thread. The
results of which could be displayed via a graphical user interface in real time.
Alternatively, the I/O thread could be used to distribute the processing of the unpacked
events to the high performance computing (HPC) center at RCNP. In this mode of operation,
the events were mapped to an allocation of CPU nodes which each maintained its own
72

analysis thread that reduced the events according to a user-defined analysis library. This
map-reduce functionality was achieved using the Message Passing Interface (MPI) and was
built into a generic library for distributed event processing, known as MPIEventLoop. In
addition to this analysis framework, MPIEventLoop has also been implemented into the
ATTPCROOTv2 analysis framework for distributed processing of events from the active target
time projections chamber (ATTPC) at NSCL.
Utilizing the above described DAQ and analysis framework, the following sections describe the analysis techniques and transformations used to extract physical quantities from
raw experimental observables.

3.2.2

Grand Raiden calibrations

3.2.2.1

Sieve-slit scattering angle reconstruction

In order to infer trajectories of the 12 C recoil and 6 Li ejectile at the target position, the
scattering angles at the interaction point must be reconstructed from the image in the Grand
Raiden focal-plane detectors. Because no calculable ion-optical inverse map from the position
and angles in the focal plane exists for Grand Raiden, a calibration measurement with a sieve
slit was performed. A perforated block (known as a sieve slit) with regularly placed holes was
inserted 60 cm downstream from the target position. This ensured that scattered 6 Li nuclei
of only very specific scattering angles would be transported to the focal plane detectors.
Because the holes in the sieve slit are known, the scattering angles at the target can then be
reconstructed from their image in the focal plane.
The horizontal (dispersive) and vertical (non-dispersive) scattering angles at the target,
✓x and ✓y , were determined via a multivariate fit that depended on the horizontal and vertical

73

Figure 3.7: The image of the sieve slit in the focal plane for 3 separate magnetic rigidity
settings (left) and the reconstructed scattering angles at the target using a least squares fit
to Eqs. 3.6 & 3.7 (right).
positions (xfp , yfp ) and angles (✓fp , fp ) in the focal plane as,

✓x = c0 + c1 xfp + c2 ✓fp

✓y =

3 X
1 X
1 X
1
X

j k l
bijkl xifp yfp ✓fp
fp

(3.6)

(3.7)

i=0 j=0 k=0 l=0

where coefficients ci and bijkl are the adjusted fit parameters.
Measurements of the sieve-slit image in the focal plane of Grand Raiden were done at
several magnetic rigidities, so that the central ray of the elastically scattered 6 Li would be
incident on di↵erent horizontal positions (xfp ) in the focal plane. The left panel of Figure 3.7
is an illustration of the sieve-slit image at various horizontal positions in the focal plane.
Using the hole centroids of the horizontal and vertical positions and scattering angles in the
focal plane, a global least-squares fit was performed using Eqs. 3.6 & 3.7. The coefficients
of the best fit were then used in these equations and the result was applied on an event-by-

74

event basis to the primary measurement data in order to deduce the scattering angles at the
target. An image of the reconstructed horizontal and vertical angles at the target for the
sieve-slit measurements is shown in the right panel of Figure 3.7. From these, the scattering
angles ✓ and

at the target can be deduced from,

✓ = tan

1

✓q
◆ q
tan(✓x )2 + tan(✓y )2 ⇡ ✓x2 + ✓y2
= tan

1

✓

tan(✓y )
tan(✓x )

◆

⇡

✓y
✓x

(3.8)

(3.9)

where the small-angle approximation is used for the RHS of Eqs. 3.8 & 3.9.
With the scattering angles calibrated, the angular resolutions for the horizontal and
vertical scattering angles at the target were found to be 2.8 mrad and 10.3 mrad, respectively.
This yielded a resolution of ⇠10.6 mrad in the scattering angle ✓, suggesting that half-degree
bins are a reasonable choice for the angular distributions discussed in later sections.

3.2.2.2

Energy calibration

The slope of the momentum calibration for the dispersive (horizontal) position in the focal
plane was performed by measuring the elastic scattering peak from the 93 Nb(6 Li,6 Li) reaction
at several magnetic rigidities. The o↵set was then calibrated using known states in 12 C.
In addition, it was found that the energy calibration of Grand Raiden varied as a function
of time as can be seen in Figure 3.8. The centroid of the isovector 12 C[2-;T=1;19.4 MeV]
state was analyzed in 15 minute intervals and it was found that it varied by 500 keV (on
average) over the course of the measurement, as shown in the top panel of Figure 3.8. This
time-varying o↵set in the energy calibration was corrected as shown in the bottom panel.

75

Run divisions
uncorrected

corrected

Sub-run interval (~900 sec.)
Figure 3.8: The time-dependence of the Grand Raiden energy calibration for the centroid
of the 12 C[2-;T=1;19.4 MeV] state in the 12 C(6 Li,6 Li’) excitation energy spectrum. Due to
the coincident excitation of the 6 Li ejectile to its 3.56 MeV state, this isovector state in 12 C
appears at ⇠23 MeV (=19.4 MeV + 3.56 MeV).

76

3.2.3

CAGRA calibrations

3.2.3.1

Energy calibration

When instrumented with the ANL firmware, the CAGRA digitizers perform minimal analysis, instead choosing to o✏oad processing such as pole-zero and asymptotic baseline corrections to software. The primary energy information saved for each digitizer firing are the
pre and post-rise integration windows. In the case of high rates, the gamma-ray energy is
calculated as,
E = slope ⇤

⌃post

⌃pre · pz
+ o↵set(rate),
M

(3.10)

where slope and o↵set are the energy calibration fit parameters, the ⌃’s are the pre and
post-rise ADC integration sums, pz is the constant pole-zero correction to the pre-rise energy
integration, and M is the shaping time or the width of the integration regions (3.5µs in this
measurement). The energy calibration slope, o↵set, and pole-zero corrections were initially
determined with standard calibration sources such as (60 Co, 152 Eu and 56 Co).
However, it was found that during production runs with beam, the o↵set in the energy
calibration of Eq. 3.10 varied linearly with the count rate in each crystal. This rate dependence of a specific HPGe crystal signal is demonstrated in Figure 3.9. Panel (a) shows an
example count rate as a function of time for the CAGRA array. The centroid of the 511 keV
-ray was tracked for various count rates and plotted as a function rate in panel (b). Panels
(c) and (d) show the response of a specific crystal in the array to variations in the rate. A
least squares fit of a linear trend for the gamma energy calibration o↵set as a function of rate
was performed for each HPGe crystal in the array, and then applied on a second-by-second
basis to determine the correct gamma-ray energy calibration o↵set during the production
runs. With this method, the rate dependence was corrected as shown in panel (d), and the
77

x104

90

Calibration offset (keV)

counts / 4 sec.

30

20

10

500

1000 1500

2000 2500

Gamma Energy (keV)

60
50
40
30
20

(b)

CAGRA
crystal #0

0

3000

0 100 200 300 400 500 600 700

Time (sec.)
x10

70

10

(a)
0
0

80

y = 0.0805x + 17.175
R = 0.9763

Crystal count rate (Hz)

single crystal (uncorrected)

single crystal (corrected)

5.3

5.2
511 keV

5.1

5.0
0

(c)

(d)
500 1000 1500 2000 2500 3000

Time (sec.)

0

500 1000 1500 2000 2500 3000

Time (sec.)

Figure 3.9: Rate dependence of the CAGRA clover crystals. Panel (a) shows an example
count rate spectrum as a function of time for the CAGRA array. Panel (b) details the rate
dependence of the energy calibration o↵set for a specific HPGe crystal in the array. This
trend was established for each crystal and then used to correct the energy calibration on a
second-by-second basis. Panels (c) and (d) show the rate dependent gamma-ray energy and
(corrected) rate independent gamma-ray energy for the 511 keV line, respectively.

78

energy resolution was greatly improved. After having applied this technique, sta↵ at ANL
suggested an alternative corrective method that applie the rate dependence to the post- and
pre-rise sums individually [113]. This method was applied and was found to provide results
comparable to the method described in this work.

3.2.4

Invariant missing mass reconstruction

The reconstruction of the 12 C excitation energies for isovector requires two steps: an invariantmass reconstruction, and a missing-mass reconstruction. Figure 3.10 illustrates the full kinematics for the system, as well as the two intermediate steps. Physically, this occurs because
after inducing an isovector transition in the 12 C target nucleus, the 6 Li will itself be excited into the 0+; T=1 excited state at 3.56 MeV. This intermediate particle, known as
the invariant will decay via gamma emission into the 6 Li ground state, which will receive a
momentum-kick from the de-excitation -ray. Thus, the first step in the reconstruction is to
combine the measurements of 6 Li ejectile and its de-excitation -ray to infer the momentum
four-vector of the 6 Li0 invariant as,
6 Li0 (inv.)

Pµ

6

= Pµ Li + Pµ ,

(3.11)

where each of the four-vectors are in the laboratory reference frame. By inferring the 6 Li0
invariant in this way, it can then be combined with knowledge of the 6 Li incident beam to
perform a missing-mass reconstruction of the excited 12 C recoil. The missing mass of the
12 C0

is found from the missing energy and missing momentum as,
q
2
mmissing = Emissing
79

p2missing .

(3.12)

6

Li reaction kinematics
12C*

6Li,

12C

600MeV

6Li
6Li*

[Ex = 3.56 MeV]

measured
unmeasured (reconstructed)

[2.2 - 5.5 MeV]

Invariant-mass
reconstruction

Missing-mass
reconstruction

6Li

12C*

6Li*
12C
6Li
6Li*

Figure 3.10: The kinematics of the 6 Li reaction.
The excitation energy of the recoil is then derived as,

Ex (12 C) = mmissing

m(12 C).

(3.13)

The missing momentum and missing energy are calculated from the 6 Li invariant of Eq. 3.11
and the incident beam four-vector via standard conservation laws,

6 Li(beam)

Pµ

12 C(target)

+ Pµ

12 C0 (missing)

= Pµ

80

6 Li0 (inv.)

+ Pµ

.

(3.14)

However, as the incident beam is not monitored on an event by event basis, uncertainty in
the energy and the incident angles of the 6 Li primary beam contribute to the resolution of
the reconstructed missing-mass excitation energy and scattering angle. From Eq. 3.14 the
missing energy and momentum are apparent,

Emissing = Ebeam + mtarget

Einv.

(3.15)

and,
pmissing = pprojectile

pinv. .

(3.16)

As the invariant 6 Li0 is inferred via the measurement of the 6 Li ejectile and the de-excitation
-ray, the missing excitation energy of the 12 C is fully determined.
It should be noted that in the 12 C(6 Li,6 Li0 )12 C0 measurement, only transition in 12 C
which are isovector in nature will cause 6 Li to emit the characteristic 3.56 MeV

-ray.

Therefore, the above mentioned invariant-mass + missing-mass spectroscopy only applies to
isovector transitions. For the case of isoscalar transitions, the excitation energy in 12 C can
be deduced via a standard missing-mass technique without consideration of any intermediate
invariant. (Because isovector transitions in the target necessitate that the 3.56MeV state in
6 Li

was also populated, energies of the isovector states will be shifted in energy relative to

the isoscalar transitions by 3.56 MeV.)

81

Figure 3.11: (a) A comparison of the contribution of the intrinsic energy resolution, opening
angle uncertainty, and velocity spread of the ejectile, to the overall uncertainty in the Doppler
reconstructed gamma-ray energy for the detection of the 3.56 MeV de-excitation gamma from
6 Li in CAGRA. The uncertainty in the emission angle of the -ray is the dominant component
of the overall resolution. (b) The resulting estimated resolution of the reconstructed invariant
6 Li ejectile given the resolution of the measured de-excitation gamma.

3.2.5

Doppler reconstruction

The Doppler-reconstructed -ray energy in the rest frame (cm) of the incident particle, E cm ,
was obtained from that in the laboratory frame (lab), E lab , as

E cm = (1

where

cos ✓lab )E lab ,

(3.17)

is the velocity of the projectile, and ✓lab is the -ray emission angle in the laboratory

frame. This reconstructed -ray energy peak is broadened compared with the intrinsic energy

82

resolution of the detector as,
E cm
E cm

!2

=

E lab
E lab

!2

+

sin ✓lab
1

cos ✓lab

!2

(

✓lab )2 +

2(

cos ✓lab )
cos ✓lab

1

!2 ✓

◆2

.

(3.18)

The first term in Eq. 3.18 corresponds to the intrinsic energy resolution of the HPGe clover
detectors and the third term corresponds to the velocity spread of the ejectile, which arises
from the unknown reaction point in the target thickness, and is

/ ⇡ 0.03%. Both of these

contributions to the uncertainty are marginal compared to the uncertainty of the detection
angle due to the finite size (corresponding to an angular coverage of

✓lab ) of a single crystal

in the CAGRA clover detectors. Figure 3.11a compares the estimated contributions of each
component of Eq. 3.18 to the total resolution. Panel (b) of the same figure shows the net
e↵ect on the energy resolution of the reconstructed 6 Li invariant.
The e↵ect this resolution has on the signal-to-noise of the Doppler-reconstructed deexcitation gamma is significant. As will be described in later sections, the resolvability of
this transition above a significant high-energy background due to statistical -ray emission
from the isoscalar excitations (in particular, the isoscalar giant monopole/dipole resonances
(ISGMR/ISGDR)) necessitates high precision angular resolution for the gamma detectors.
The large angular coverage of each clover crystal in the CAGRA array (⇠ 12 ), significantly
limited the achievable signal-to-noise in this measurement.

3.2.6

Excitation energy resolution

As the energy of this de-excitation gamma is a component in the reconstruction of the
invariant (as described by Eq. 3.11), it thereby also contributes to the resolution of the
reconstructed excitation energy in 12 C. If one ignores the momentum-kick imparted to the
83

6 Li0

ejectile during its decay, the best resolution that can be achieved is analogous to the

full width of the Doppler reconstructed gamma energy,

E=5.5-2.2=3.3 MeV.1 This was

confirmed via a monte-carlo Geant4 simulation, which included,
• Realistic beam profile and inelastic scattering reaction mechanism
• Nuclear inelastic excitation of the target/recoil 12 C and projectile/ejectile 6 Li systems
• The decay of the 6 Li0 [3.56 MeV] invariant inflight to the 6 Li[0.0 MeV] ground state
and the emission of the [3.56M eV ].
By analyzing the simulated reaction data and ignoring the decay of the invariant (the
momentum-kick of the de-excitation gamma, as mentioned above) the purple line in the left
panel of Figure 3.12 represents the simulated best-case reconstruction of the 12 C[15.1 MeV]
excited state. The square shape is a direct result of uniform emission of the de-excitation
⇣
⌘
-ray’s according to cos ✓lab .
However, when compared with the shape of the missing-mass reconstructed excitation

energy in the 12 C(6 Li,6 Li’) measurement (right panel of Fig. 3.12), the data was found to
have a much broader shape. To reproduce this shape, a gaussian beam energy resolution of
1.75 MeV in FWHM was required to smear out the simulated shape (shown as the black
crosses in the left panel of Fig. 3.12). This width was determined by fitting simulations
with di↵erent beam energy resolutions to the data. The red line in the left and right panels
of the same figure produced the best fit to the data and corresponds to an intrinsic beam
energy resolution of 1.75 MeV. Prior to the measurement, the beam energy resolution was
expected to be on the order of a few hundred keV, but due to problems with the cyclotrons
and difficulty tuning the 6 Li beam, the resolution was much worse.
1 The square purple line shape in Fig. 3.12 has exactly this width.

84

Simulation vs Smeared Simulation

Smeared Simulation Fit to Data

x103

x102

counts / 100 keV

12
C[15.1 MeV]
+6Li[3.56 MeV]
Q=18.66 MeV

Q (MeV)

Q (MeV)

[1]: Simulated 12C Missing Mass Ex
[2]: FWHM=1.5 MeV Gaussian Smearing of [1]
[3]: Interpolation of [2]

Exp. (missing mass Ex)
Fit of const.*[3] to exp. data

Figure 3.12: Comparison of the simulated reconstruction of the 12 C[1+;T=1;15.1 MeV]
excited state with the measured result. In the left panel, the purple line represents the
(simulated) best-case scenario for the missing mass reconstruction of the 15.1 MeV peak,
if the invariant 6 Li[0+;T=1;3.56 MeV] is ignored. By comparing this with the data in the
right panel, it is apparent that the data follows a much broader distribution. In order to
reproduce the data, the simulated missing mass reconstruction of this state in 12 C (purple)
had to be smeared with a gaussian of a variety of widths. The shape of the smeared state
was interpolated and fit to the data using a normalization factor. The best fit was found
with a gaussian smearing of FWHM=1.75 MeV (shown by the black scatter crosses and red
line in the left and right panels, respectively). The smearing that was required suggests that
the intrinsic beam energy resolution was 1.75 MeV.

85

3.2.7

Cross-section calculations

The final step in the data analysis procedure was to evaluate the excitation-energy and
center-of-mass scattering-angle dependent di↵erential cross sections for states populated in
12 C.

The di↵erential cross section for the 12 C(6 Li,6 Li’) singles and coincidence data was

calculated as,
d
Y
=
,
d⌦
Nb Nt L⌘d⌦✏

(3.19)

where Y is the yield (integrated number of counts over a given energy and angular range),
Nb,t are the number of incident 6 Li beam particles and 12 C nuclei in the target, L is the
average live-time of the DAQ (82%), ⌘ is the efficiency of the MWDCs, d⌦ is the solidangle corrected for the angular acceptance of Grand Raiden, and ✏ is the efficiency of the
CAGRA array at detecting the E = 3.56 MeV -ray in the laboratory frame (note that the
CAGRA efficiency was only used in the the calculation of the coincident 12 C(6 Li,6 Li’+ )
cross-section). The evaluation of each component in Eq. 3.19 is detailed in the following list,
• Y : The yield was measured by integrating the number of counts in each energy and
angular bin after applying the 6 Li gamma-coincidence of the prompt events (promptrandom ratio of 3.3±0.3) and background subtraction. The background subtraction is
described in section 3.3.1.
• Nt : The number of 12 C nuclei in the 15.21 mg/cm2 nat C target, positioned at an angle
of 22.5 degrees, was calculated as,

Nt =

15.21mg/cm2 98.9%(12 C/nat C)
= 8.17 · 1020 cm 2
cos(22.5 ) 1.9926 · 10 20 mg/12 C

(3.20)

• Nb : The normalization of the number of incident 6 Li beam particles was the leading
86

order uncertainty in this measurement and is described in section 3.2.7.1.
• d⌦: The acceptance corrected opening angle for each angular bin was calculated as,
d⌦ = 2⇡ C ·

Z ✓
f
✓i

sin✓ · d✓

(3.21)

where ✓i,f are the boundaries of the center-of-mass angular bin. C is the angle dependent acceptance correction factor for the Grand Raiden spectrometer. The nominal
acceptance of Grand Raiden in the horizontal and vertical angles is ±20 mrad and
±70 mrad, respectively. Therefore, in order to extract angular distributions beyond
20 mrad in laboratory scattering angle, an acceptance correction to d⌦ is required.
Because the angular distributions are azimuthally invariant (constant with respect to
), and because the acceptance in the vertical (non-dispersive) angle was much greater
than the horizontal angle, the full acceptance angular distribution along ✓ = ✓y (where
✓x ⇡ 0) was used to determine the shape of the angular distribution. From this shape,
the required acceptance correction could be determined for horizontal and vertical
scattering angles where the acceptance was cut.
• L: The average live-time ratio of the Grand Raiden DAQ determined from the readout;
the live-time ratio was 82% for the 12 C measurement.
• ⌘: The voltage of the four anode wire planes in the MWDCs was optimized for peak
efficiency using a pilot 6 Li beam at the beginning of the experiment. The total efficiency
⌘ was then estimated as ⌘ = ⌘1 ⌘2 ⌘3 ⌘4 = 0.74 where ⌘i was between 91-93% for the
approximate efficiency of each anode plane.
• ✏ : The efficiency of the CAGRA HPGe clover array to the 6 Li 3.56 MeV de-excitation
87

-ray, which is Doppler boosted to the energies shown in Fig. 3.5 for the four distinct
crystal positions in the CAGRA array. The energy dependency of the efficiencies for
each clover were determined via a Geant4 simulation and were then fit to absolute
efficiency measurements made with the CAGRA array using calibrated sources. The
measured intensity was corrected for the deadtime of the clover crystal readout (4.5µs),
which was only a ⇠1.5% e↵ect given a rate of 3.4 kHz. The non-uniform angular
distribution of the gamma-emission from the moving frame was also accounted for on
a per crystal basis using Geant4. Altogether, the uncertainty in the estimated gamma
efficiency calibrations were much smaller than that of the incident beam normalization.
The aggregated efficiency of the CAGRA array to the 3.56 MeV gamma was found to
be ⇠(0.44±0.03)%.
3.2.7.1

Normalization of the number of incident particles

Following the primary measurement, absolute normalization measurements were made in
which the incident beam intensity was measured with a Faraday cup at the entrance of the
WS experimental hall. The di↵erential cross section for the singles data in this normalization
run was then used to normalized the summed spectra of all the 12 C(6 Li,6 Li’) data. This
method was chosen because it was believed that the transmission to the faraday cup at 0
was not constant over the course of measurement, nor during the normalization runs, and
thus a run-by-run, or instantaneous normalization using the scalar data would not have been
reliable.
The uncertainty in the normalization produced in this way was dominated by the uncertainty in the 6 Li beam intensity during the normalization runs. The primary component of
this uncertainty came from the relatively low rate which was near the tolerance of the beam
88

monitor, and which appeared to fluctuate on the ⇠20% level between normalization runs.
The uncertainty in the incident particle normalization, as well as the time-dependent
energy variation discussed in the previous section, together lend evidence that variations in
the 6 Li beam integrity may have been present throughout the experiment.

3.2.7.2

Multipole Decomposition Analysis

Various multipole contributions to the cross sections were extracted via a multipole decomposition analysis (MDA) [114]. In the MDA, a linear combination of theoretical angular
distributions, each with di↵erent units of angular momentum transfer, were fit to the measured di↵erential cross section,



d
d
= c1 ·
d⌦ Total
d⌦



d
+ c2 ·
d⌦
L=0

L=1

+ c3 ·



d
d⌦

.

(3.22)

L=2

The theoretical di↵erential cross sections for the (6 Li, 6 Li0 ) reactions that are utilized in
the MDA have been estimated by means of Distorted Wave Born Approximation (DWBA)
calculations. The theoretical angular distributions chosen for the MDA were the combination
of

L = 0, 1 and 2 components that provided the best overall fit. The microscopic DWBA

code fold/dwhi [115] was used to estimate di↵erential cross-sections by double-folding a
nucleon-nucleon (N N ) interaction, in the form of a t-matrix, over transition densities of
the projectile-ejectile and target-residual systems. This calculation requires five principle
components,
1. An e↵ective nucleon-nucleon (N N ) interaction which describes the coupling of the
projectile (6 Li) and target (12 C) nuclei.
2. One-body transition densities (OBTDs), which describe the overlap of the initial and
89

final nuclear many-body states in the projectile/ejectile (6 Li-6 Li) and target/recoil
(12 C-12 C) systems.2
3. Single particle radial wave functions of the participant nucleons in the presence of a
many-body core.
4. Optical model potentials which describe the distortion of the incoming and outgoing
nuclear wave functions by the target/recoil system.
5. Kinematics and spin variables describing the incident projectile energy, the reaction
Q-value, as well as the spin, isospin, and parity quantum numbers of the projectile and
target systems.
By the virtue of performing the inelastic scattering measurement at 100 MeV/u, many of
the complicated reaction mechanisms known to obscure the nucleon-nucleus interaction are
circumvented via the impulse approximation [104]. The requirement of calculation inputs
1. – 5. above can been seen in a straightforward derivation of the N N t-matrix.
For elastic scattering, the exact t-matrix can be written as,

T =

where

⌦ +
V

↵

(3.23)

is the full incoming wave function, + is an outgoing distorted plane-wave, and V

is the interaction potential (+ and

indicate outgoing and incoming, respectively). In first

order DWBA, the full incoming wave function is approximated by an incoming plane-wave

2 One-body transition densities weight the contribution of one-particle one-hole (1p-1h) transitions between single-particle orbits, and thus provide a way to connect the single-particle wave functions (and nucleon
operators) to the many-body system.

90

that is distorted by an optical model potential (4. from above),

T =

⌦ +
V

↵

.

(3.24)

However, for inelastic scattering and charge-exchange reactions in which isospin transfer
occurs, the elastic scattering potential V must be replaced by a form factor that contains
information about the structure of the participant nuclei (OBTDs, 2. from above) and the
nature of the N N -interaction (1. from above). The t-matrix is then,

T =

⌦ +
F [R]

F [R] = hB, b| V |A, ai

↵

(3.25)
(3.26)

where F [R] is the form-factor, and V is a many-body potential acting on the many-body
states of the projectile (a), target (A), ejectile (b) and recoil (B). In general, many-body
operators, such as V, can be expanded in second-quantized form to make its action on each
nucleon in the composite many-body state explicit,

V=

X⌦

p

Ve↵

pq

q

↵ †
âp âq .

(3.27)

Here, Ve↵ is the e↵ective N N interaction, the ’s are the single particle wave functions of the
†

nucleons in the reaction system (3. from above), âq /âp are 1p-1h creation and annihilation
operators, and the sum runs over all 1p-1h excitations.
By applying second quantization as in Eq. 3.27 for the two-body operator which acts on
the target/recoil and projectile/ejectile systems, the form-factor in Eq. 3.26 can be written

91

as,
F [R] = hB, b|

XD

B b
p q

Ve↵

A a
r s

pqrs

E

† †

âp âq âr âs |A, ai ,

(3.28)

where the sum over indices p, q, r, and s refers to 1p-1h excitations in the recoil, ejectile, target
and ejectile systems, respectively This second quantized expression for the form-factor can
then be factored into densities, which represent the overlap of the initial and final many-body
states (the one-body transition densities, OBTDs),

F [R] =

X

†

pqrs

=

X

†

hB| [âp âr ] |Ai hb| [âq âs ] |ai

D

OBTD(Aq Bp )OBTD(as br )

pqrs

B b
p q

D

Ve↵

B b
p q

Ve↵

A a
r s

E

A a
r s

(3.29)
E

,

(3.30)

where the OBTD(Aq Bp ) and OBTD(as br ) are the one-body transition densities for the
target/recoil and projectile/ejectile systems, respectively. Letting ⇢pq = OBTD(p, q) p q
it’s then readily apparent that this form-factor is simply double folding the target-recoil and
projectile-ejectile densities over the nucleon-nucleon e↵ective interaction,

F [R] =

X

pqrs

=

h⇢AB | Ve↵ |⇢ab i

XZ Z

pqrs

d⇠AB · d⇠ab · ⇢AB (⇠AB )Ve↵ ⇢ab (⇠ab ),

(3.31)
(3.32)

where ⇠ab (⇠AB ) are the relative spatial coordinates between the projectile and the ejectile
(target and recoil).

92

ΔL = 0

ΔL = 1

ΔL = 2

Q (MeV)

40

0

Figure 3.13: Calculated angular distributions for isovector spin-transfer excitations of different orbital angular momentum transfer for the 12 C(6 Li,6 Li0 ) inelastic scattering measurement. Each panel is the angular distribution for a particular angular momentum multipole,
and the di↵erent color lines are the excitation energy (Q-value) dependence of the angular
distribution.
Finally, the optical-model potentials are used to calculate the incoming and outgoing
distorted waves

, and are then used with the double-folded e↵ective N N t-matrix as,

Tf i =

D

+ (~
~
f kf , R)

F [R]

E
~
~
i (ki , R) ,

(3.33)

where ki,f is the initial and final momentum vector for the distorted wave (5. from above).
The angular distributions are calculated via the standard t-matrix formalism as,
d
µ 2 kf
=(
)
|Tf i |2 ,
2
d⌦
ki
2⇡~
where µ is the reduced mass of the outgoing particle.
Thus, items 1. – 4. in the above list are necessary and sufficient for DWBA estimates
that can be used to extract the multipole components of the 12 C(6 Li,6 Li0 ) di↵erential cross
section. For this particular analysis, these items were chosen as follows,
• NN-interaction: The e↵ective interaction used in the t-matrix parameterization was
93

the e↵ective NN interaction by Franey and Love at 140 MeV [104], which has been
very successfully applied to heavy-ion induced charge-exchange and inelastic scattering
reactions in the past [116, 117, 118].
• OBTDs: The one-body transition densities for the 6 Li and 12 C systems were calculated with the configuration-interaction shell-model code oxbash [119]. The CohenKurath (6-16)CKI interaction [120] was used in the p-shell model space to calculate the
transition densities for 6 Li, and the WBP interaction [121] was used in the spsdpf -shell
model space to calculate the transition densities for the 12 C system.
• Single-particle wave functions: The single-particle radial wave functions were calculated with the program WSAW [115] by matching the binding energy of a single-particle
wave-function outside a closed-core, to the single-particle binding energies calculated
in oxbash with the subprogram dens. These calculations employed the SkX skyrme
interaction [122].
• Optical Model Potential: The optical model potential (OMP) used to compute the
6 Li 12 C

entrance and exit channels were fit from 12 C(6 Li,6 Li) elastic scattering data

taken at 600 MeV [123]. The OMP parameters were fit utilizing the ecis [124] code
and the resulting best fit parameters were -60.94 MeV, 1.3725 fm, and 0.9142 fm for
the depth (V ), radius (rv ), and di↵useness (ar ) of the real Wood-Saxon potential and
-22.529 MeV, 1.610 fm, and 0.693 fm for the depth (W ), radius (rw ), and di↵useness
(aw ) of the imaginary Wood-Saxon potential.
• Kinematics & quantum numbers The reaction calculations were performed using
an incident total kinetic energy of 600 MeV/u for the 6 Li projectile. Furthermore, the
projectile-ejectile system underwent the transition from J ⇡ = 1+ (T = 0) ! 0+ (T =
94

1), while the target-recoil system underwent the J ⇡ = 0+ (T = 0) ! 1+ (T = 1)
transition. Since parity (⇡) is unchanging in these transitions, the change in total
angular momentum J is due purely to spin-transfer reactions

S = 1. The quantum

numbers for J ⇡ and T for each system were the final component necessary for the
reaction calculations described in this section.
Angular distributions for isovector spin-transfer excitations of di↵erent orbital angular
momentum transfer for the 12 C(6 Li,6 Li0 ) inelastic scattering reaction are shown in Figure 3.13. Angular distributions for separate angular momentum multipoles are shown, and
the calculations are performed over a range of excitation energies, corresponding to the
excitation energy range measured in the experiment. The multipole decomposition of the
measured cross-section with these angular distributions is discussed in the following section.

3.3
3.3.1

Results
The

12

C(6 Li, 6 Li0 + ) measurement

The coincident particle–gamma-ray spectroscopy measurement achieved via the use of the
Grand Raiden spectrometer and the CAGRA clover array is illustrated in Figure 3.14 in
which the coincident gamma-rays are shown as a function of excitation energy in 12 C. Both
the laboratory and center-of-mass frame gamma-ray energies are shown (top and bottom
panels, respectively), from which a comparison of the 15.1 MeV state in 12 C reveals the
Compton edge and photopeak of the 3.56 MeV gamma-ray emitted in-flight from 6 Li. Without performing the doppler reconstruction, gammas from this state are spread out in the lab
frame as can be seen in the top panel of Fig. 3.14.

95

lab frame

14

Elab (keV)

12
10
8
6
4
2
5

cm frame
Edoppler (keV)

4

3

2

1

10

15
12

20

25

30

35

40

45

C Excitation Energy (MeV)

Figure 3.14: The gamma-ray energy and target excitation energy matrix for the
12 C(6 Li, 6 Li0 + ) reaction. In the top panel, the gamma energies shown are those measured in the detectors, without any transformation applied and in the bottom panel, the
gamma energies have been Doppler reconstructed according to the clover crystal angular
positions. Both panels are drawn as a function of the excitation energy in 12 C which is calculated via the missing mass formalism. The red arrow in the top panel indicates the width
of the 12 C(15.1 MeV) state owing to the momentum kick that is imparted to the 6 Li and
which is not accounted for in the missing-mass–only reconstruction. This width is analogous
to the FWHM of the excitation energy spectrum shown in Figure 3.12. By comparing the
two panels, the e↵ect of the doppler reconstruction can be seen, where the photopeak and
Compton edge owing to the 3.56 MeV gamma from 6 Li is more prominent in the bottom
panel after reconstruction. In addition, gamma decay from the strong isoscalar resonance
(occurring above 20 MeV in excitation energy) is clearly evident.

96

Li [3.56 MeV; T=1]
sideband
sim. line-shape
background

clover-crystal
tracking-detector

counts / 4 keV

counts / 4 keV

6

2500

3500
k

k
Figure 3.15: The Doppler-corrected -ray energy gated on the 12 C[15.1 MeV; T=1] state for
emphasis. The 3.56 MeV -line from the 6 Li[0+; T=1] excited state is clearly evident. The
resolvability of this -ray is crucial for tagging isovector spin-transfer reactions in the target
nucleus. The inset plot compares the signal to noise achieved with the clover detectors to
what is expected were a gamma-ray tracking detector was utilized. See text for details.
The Doppler reconstructed -ray energy spectra in coincidence with the population of
the 12 C[15.1 MeV; T=1] state is shown in Figure 3.15. The yellow line shape shown in
the spectrum is the simulated response of the CAGRA array to the 3.56 MeV Doppler
reconstructed gamma, fit to the measured data. The blue line is the double exponential
background component of the fit. Shown in the red cross-hatched region is the 3.56 MeV
photopeak. Just below this region in energy, a convolution of the Compton distribution and
escape peaks for the 3.56 MeV gamma can be clearly seen. The four distinct peaks in the
region below the photopeak are from a single gamma transition at rest, in the target, that
has been distributed in energy according to the four angular ranges covered by the CAGRA
array.

97

counts / 500 keV

x102
10
8
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0

6

Li [3.56 MeV; T=1]
sideband

15

20

25

30

35

Ex (12C) (MeV)
Figure 3.16: The reconstructed excitation energy spectrum of 12 C, for di↵erent angular
bins, when applying the two gates shown in the Doppler-reconstructed gamma energy spectrum (Fig. 3.15). The red spectrum corresponds to the 3.56 MeV photopeak gate, and the
blue spectrum corresponds to the sideband, representative of the gamma background in the
photopeak gate. The 12 C[1+;T=1;15.1 MeV] state, which is populated via an isovector spintransfer reaction, is clearly seen in the photopeak gated spectra, but not in the sideband
data.

98

By gating on the region shown in red (Ecm = 3.4 MeV - 3.8 MeV) and subtracting a
scaled sideband representative of the background contribution (shown in blue; Ecm = 3.9
MeV - 4.3 MeV) 3 , the coincident invariant missing-mass excitation energy for pure isovector
( T = 1) spin-transfer ( S = 1) transitions in 12 C can be extracted. The excitation energy
spectrum for each of these gamma gates is shown in Fig. 3.16. Even without performing
the sideband subtraction, the Gamow-Teller transition to the (1+;T=1;15.1 MeV) state and
the isovector spin-dipole transition to the (2-; T=1; 19.4 MeV) can be easily seen as excess
counts in the 6 Li[0+;T=1;3.56 MeV] gated spectra (red) over the sideband (blue).
Figure 3.17 compares inelastic scattering for the 12 C(6 Li, 6 Li0 ) singles cross section to
the 12 C(6 Li, 6 Li0 + ) coincident cross section (the subtraction of the red and blue spectra in
Fig. 3.16). The singles data is dominated by the isoscalar resonances in 12 C, shown in grey.
Making the coincidence measurement with the

S =

T = 1 transition in 6 Li by gating

on the corresponding de-excitation gamma, the isovector spin-transfer excitations in 12 C are
identified (black).4
The novelty of the 6 Li probe is its ability to separate the proportionally weak isovector
spin-transfer excitations from the strong isoscalar excitations. Figure 3.17 displays this
selectivity. Whereas the 12 C0 [1+; T = 1; 15.1 MeV] is located in the shoulder of the isoscalar
resonances in the singles data, this state is easily resolved using the gamma-coincidence tag
3 A wide sideband of ⇠ 400 keV, as shown in Fig. 3.15, was often not practical due to contamination
from other gamma lines in this region. In practice, a narrower sideband from 3.8 - 3.9 MeV was used.
Furthermore, the counts from the sideband must be scaled to match the expected background in the region
of the 3.56 MeV photopeak. This was done by using the ratio of the integral of the double exponential
background fit, an example of which can be seen as the blue dashed line in Fig. 3.15. As this ratio changes
with the amount of background gamma contamination (which is a function of excitation energy of 12 C), an
excitation energy dependent sideband scaling factor was utilized.
4 The di↵erence in energy scales shown in Fig. 3.17 is an o↵set of 3.56 MeV, where the reaction Q-value
for isovector spin-flip transitions includes the the excitation energy of the target nucleus and the commensurate excitation of the 6 Li nucleus. Therefore, in order to recover the excitation energy spectrum for the
12 C(6 Li, 6 Li0 + ) coincident measurement, the 6 Li0 [0+; T = 1; 3.56 MeV] invariant must be reconstructed
prior to the missing mass reconstruction (see section 3.2.4).

99

Ex [12C(6Li,6Li’)] (MeV)
12

C(6Li,6Li’)
12
C(6Li,6Li*[3.56 MeV]) x 2

x2

Ex [12C(6Li,6Li’[3.56 MeV])] (MeV)
Figure 3.17: Comparison of the 12 C inelastic scattering singles and coincidence excitation energy dependent double di↵erential cross section for the 12 C(6 Li, 6 Li0 ) and the
12 C(6 Li, 6 Li0 + ) reactions, respectively. The coincidence spectrum shown is the result
of the subtraction of the di↵erential cross section resulting from a gate on the photopeak
of the 3.56 MeV -ray and a sideband in the region directly above this gamma (shown as
the red and blue cross-hatched regions in Fig 3.15, respectively). The coincidence spectrum
corresponds to the isovector spin-transfer excitations in 12 C, whereas the singles spectrum
contains all types of transitions but is primarily dominated by isoscalar excitations. See text
for details.

100

proposed in this work.
With the methods described in section 3.2.7.2, theoretical cross sections for the 6 Li-12 C
system were calculated for excitations of isovector, spin-transfer reactions ( T = 1, S = 1)
with multipoles of

L =0,1, and 2. The calculated angular distributions with distinct orbital

angular momentum transfer are shown in Fig. 3.13. To decompose the entire excitation
energy spectrum of 12 C, angular distributions for each of these multipole components were
calculated in a range from 0-40 MeV (indicated by the colored lines in Fig. 3.13). These
energy dependent theoretical cross-sections were then fit to the angular distributions for each
excitation energy bin of the data, according to equation 3.22.
The multipole-decomposed double di↵erential cross section for the 12 C(6 Li, 6 Li0 + )
reaction as a function of excitation energy in 12 C and center-of-mass scattering angle, is
shown in Fig. 3.18. The di↵erential cross sections were corrected for the acceptance of
Grand Raiden, the detector live-time ratios, as well as the efficiency of the CAGRA array
for the 3.56 MeV

detection (see section 3.2.7). The angular distributions for the 15.1 MeV

Gamow-Teller transition and the 19.4 MeV spin-dipole transition are shown in the right
panel of Fig. 3.18. With the aide of the multipole decomposition analysis described above,
the di↵erent multipole components of the 12 C excitation energy are extracted.
Shown in red in left panel of Fig. 3.18 are the ( T = 1, S = 1, L = 0) components
of the 12 C spectra at di↵erent scattering angles. As expected, the strong transition to the
12 C[15.1

MeV; T=1] state can be clearly identified and is dominated by

L = 0. This

is also clearly seen in the top right panel of Fig. 3.18 which illustrates that the angular
distribution for this state is primarily

L = 0. Thus, the 6 Li probe’s unique selectivity

together with a multipole decomposition can uniquely identify states populated via GT0
transitions ( T = 1, S = 1, L = 0). The extracted angular distribution for this state
101

12

C(6Li,6Li’[3.56 MeV])
E = 100 MeV/u

(d)

[13.1 MeV < Ex < 17.1 MeV]

10

(a)

/

.

12

8
6

Ex = 15.1 MeV

/

4
2
0

(b)

7.5

(e)

[17.1 MeV < Ex < 22.1 MeV]

Ex = 19.6 MeV

5.0

2.5

(c)

IV

x

12

C

Figure 3.18: Double di↵erential cross section for the 12 C(6 Li, 6 Li0 + ) reaction as a function of excitation energy and center of mass scattering angle, in which the 3.56 MeV state
was measured in coincidence. Due to conservation of spin and isospin, this coincidence
measurement extracts the isovector spin-transfer response of 12 C. In addition, a multipole
decomposition analysis (MDA) has been performed utilizing DWBA angular distributions
to extract the components of orbital angular momentum transfer. Shown in the right panel
are the angular distributions for states in 12 C dominated by Gamow-Teller transitions (top
panel) and spin-dipole transitions (bottom panel). The absolute scale has a systematic uncertainty of ±20% due to uncertainty in the beam normalization which is not depicted in
the statistical error bars.

102

is shown in the upper right panel of Fig. 3.18 in which the three components of the MDA
fit, and their sum, are shown. This state is almost purely

L = 0, with a small

L=2

component at larger scattering angles.
This result compares well to the analogous angular distribution excited through chargeexchange reactions on 12 C into 12 B populated via the 12 C(t,3 He) reaction [125]. For the
case of N=Z nuclei such as 12 C, charge-exchange provides equivalent selectivity. In general,
however, it is difficult to select the analogous inelastic isovector spin-transfer excitations
via charge-exchange for N6=Z nuclei. This is because the analog states in the chargedexchange channel ( Tz = ±1) are suppressed due to the large Clebsch-Gordan coefficients
that occur when the ground state isospin (which scales roughly with the isospin projection
T ⇡ Tz = 12 (N

Z)) is non-zero.

From the results for 12 C shown in Fig.

3.18, is it clear the (6 Li,6 Li⇤ [3.56MeV]) reac-

tion is suitable for isolating the isovector-spin excitation energy spectrum in the inelastic
channel which establishes this probe as the inelastic analog to spin-transfer charge-exchange
reactions. Furthermore, with comparison to the direct 12 C(⌫, ⌫ 0 ) neutrino measurement of
Ref. [126], we see that the (6 Li,6 Li⇤ [3.56MeV]) reaction populates the same states thereby
confirming this probes utility as an indirect technique for constraining INNS cross sections.

3.3.2

The (6 Li,6 Li’) unit cross-section

The proportionality between Gamow-Teller transition strength and reaction cross sections
at scattering angle ✓ = 0 , has been well established in the context charge-exchange reactions [127, 125]. In a similar way, the inelastic Gamow-Teller (GT0 ) matrix elements can be
deduced from the measured di↵erential cross sections by applying the empirically established
unit cross-sections from charge-exchange under the assumption of isospin symmetry [128].
103

For charge-exchange, this proportionality relation has been written as,

d
(0 ) = ˆGT F (q, !)B(GT),
d⌦

(3.34)

where GT is the unit cross section, F (q, !) is a kinematical factor correcting for non-zero
momentum and energy transfer, and B(GT) is the reduced GT transition strength. This
relationship was originally established for (p, n)/(n, p) reactions [129] and subsequently applied to many inelastic and charge-exchange reactions at medium energy [130]. Analogously,
the corresponding relation for the present (6 Li, 6 Li0 ) reaction is,
6 6 0
d ( Li, Li )
(6 Li,6 Li0 )
(0 ) = ˆGT
F (q, !)B(GT0 ),
0
d⌦
(6 Li,6 Li0 )

with ˆGT
0
the

(3.35)

the Gamow-Teller unit cross section (UCS) for this reaction and B(GT0 ) is

Tz = 0 (inelastic) Gamow-Teller transition strength.
From the -decay data of 12 B and 12 N, the GT transition strengths for the transitions

from the 12 C ground state to the ground states of 12 B and 12 N, which are both analogs of
the transitions from 12 C ground state to the 15.1 MeV state in 12 C, are determined to be
(6 Li,6 Li0 )

0.99 and 0.88, respectively. For the determination of the unit cross section ˆGT
0

, the

average of these measurements was adopted. The Gamow-Teller strength for the transition
to the 15.1 MeV analog state of the 12 B ground state was also calculated via oxbash [119]
using the Cohen-Kurath (8-16)POT interaction in the p-shell-model space [120], and found
to be 0.921, which agrees well with the average strength of the beta-decay measurements.
Utilizing Eq. 3.35, the 12 C(6 Li, 6 Li⇤ [3.56MeV]) unit cross section was found to be 11.3 ± 2.7
mb/sr. The unit cross section was also determined from the DWBA calculation and found

104

to be in agreement (11.325 mb/sr) with the data. Finally, the unit cross section was also
determined from the analog transition in the 12 C(6 Li,6 He) data [? ] with a value of ⇠ 10
mb/sr. Although it was not possible to determine an error from the data presented in Ref.
[? ], this value is also in good agreement with the present results.
Following the established mass dependence of the unit-cross section for (t,3 He) and
(3 He,t) reactions [125], it is postulated here that the 6 Li Gamow-Teller unit cross section
decreases with target mass as
(6 Li,6 Li0 )
(A)
0

ˆGT

= N exp( xA1/3 ).

(3.36)

where N , and x are fit parameters. The original intention of this work was to use the
24 Mg

and 93 Nb measurements to constrain the above fit parameters with data, thereby

determining the mass dependence of the UCS fully. However, it was found that the signalto-noise (S/N) for the 6 Li 3.56 MeV -ray was low enough to be unresolvable for the 24 Mg
and 93 Nb measurements (see section 3.3.3). Thus, to infer the mass dependences of the
Gamow-Teller UCS, the experimentally determined UCS for 12 C was supplemented with
theoretical DWBA calculations.
Theoretical DWBA estimates for 12 C and 26 Mg, as well as four heavier doubly-closed
nuclei, 48 Ca, 78 Ni, 132 Sn, and 208 Pb were made, as well as calculations for their GamowTeller transition strengths. One-body transition densities (OBTDs) for 12 C were calculated
using the same interaction as described previously and were calculated for 26 Mg using the
USDA interaction in the sd-shell-model space [131]. The OBTDs for the heavier nuclei were
obtained via normal-mode calculations with the code NORMOD [132]. Optical-model potentials
(OMP) for the DWBA calculations were taken from Refs. [133, 134].

105

12

10

C
26

Mg
σ(
A

σGT

48

)=

80

Ca
78

.4
*

ex

p(

132
6

6

12

.8
4

Ni

1
12

-0

(1+; g.s.)

* A 1/
3
)

Sn

C( Li, He) N
12
C(6Li,6Li’[3.56 MeV])12C(1+; 15.1 MeV)

10 20 30 50
A

208

Pb

100 150 200250

Figure 3.19: The mass dependence of the Gamow-Teller unit cross-section for 6 Li calculated from theoretical estimates of the inelastic GT strength and DWBA estimates for
the cross section of several nuclei (black). Filled circles indicate that the underlying GT
strengths were estimated via shell-model or normal-modes calculations. A postulated unit
cross section mass-dependence for heavy-ion reactions (Eq. 3.36) is fit to these theoretical
unit cross-sections and is shown in the plot (dashed black line). In addition, the experimentally derived Gamow-Teller UCS for the inelastic (6 Li,6 Li’) reaction established in this
work is shown alongside an estimated unit cross section from data available for the (6 Li,6 He)
charge-exchange reaction [135]. The error bar shown for the inelastic UCS includes systematic and statistical uncertainties, while no error on the charge-exchange data was available.

106

The theoretical estimates for the 6 Li UCS are shown in Figure 3.19. Performing a least
squares fit of equation 3.36 to these calculations, the fit parameters N and x are found to be
80.4 mb/sr and 0.840, respectively, and the fit result is shown as the dashed blue line in the
figure. The theoretical DWBA estimates follow the postulated mass dependence very well.
Furthermore, for 12 C, the theoretical Gamow-Teller unit cross-section (11.3 mb/sr) agrees
well with the measured unit cross-section for the (6 Li,6 Li0 ) probe as can be seen in Fig. 3.19.
In addition, Laurent et al. [135] investigated the 12 C(6 Li,6 He)12 N charge-exchange reaction where they found a cross section of ⇠7.2 mb/sr for the population of the 12 N ground
state at 100 AMeV and (✓ = 0 ± 1.3) . Under the assumption of isospin symmetry, the
charge-exchange transition to the 12 N ground state is analogous to the inelastic population
of the 15.1 MeV state in 12 C that is discussed above.5 Indeed, comparing the chargeexchange di↵erential cross section of [135] with the (6 Li,6 Li0 + ) inelastic di↵erential cross
section for this state at zero-degrees (8.84 mb/sr), we see that the measurements agree within
the systematic uncertainty of the present study. A rough estimate of the unit cross section
for the (6 Li,6 He) reaction is made assuming pure (L=0) and is shown in Fig. 3.19. Given
this agreement, and the good agreement of both measurements with the calculations, the
inelastic Gamow-Teller unit cross section for the (6 Li,6 Li0 + ) reaction is reasonably well
constrained.
Finally, it should be further noted that the conclusion of equation 3.36 and the above
described measurements and calculations, is that the isovector spin-transfer cross-sections
will decay exponentially with the nuclear mass number. This fact, and the discussion in the
next section, illustrate why the 6 Li de-excitation gamma was ultimately not resolvable in

5 The Clebsch-Gordan coefficients are also the same for the 12 C(6 Li,6 Li0 ) and 12 C(6 Li,6 He) measurements.

107

the heavier systems that were investigated.

3.3.3

Competition with isoscalar resonances

The utility of the 6 Li probe relies on the resolvability of the 3.56 MeV -ray from the deexcitation of its 0+;T=1 excited state. As this -ray is Doppler boosted from the decay in
flight, significant improvement in the signal-to-noise (S/N) could be achieved by using a -ray
tracking detector such as GRETINA where the nominal interaction position in the HPGe
crystals can be deduced to within 2-mm [136, 137, 138]. Assuming equivalent efficiency,
shown in the inset plot of Fig. 3.15, the yellow and red line-shapes compare the simulated
response of the CAGRA clover detectors and a tracking detector with 2 mm resolution,
respectively. As illustrated in Eq. 3.18, the resolution of the reconstructed center of mass
gamma energy is directly proportional to the angular resolution of the detector. For this
reason, the resolution of the tracking-detector is estimated to be an order of magnitude better
than what was achievable using the clover-type HPGe detectors in CAGRA, due primarily
to the large angular coverage of the clover crystals. While the S/N in the 12 C measurement
was sufficient to resolve the 3.56 MeV -ray of interest, for the measurements of 24 Mg and
93 Nb

the signal could not be resolved.

The number of resolved counts in the 3.56 MeV peak for the 12 C measurement (⇠19250)
can be used to estimate the S/N that would be observed in heavier systems. Assuming
a B(GT0 ) ⇠1, as is the case for the transition to the 12 C[1+; T = 1; 15.1 MeV] state,
over a similar excitation energy range (5 MeV), only ⇠ 3700 counts are estimated to be
measured in 24 Mg, and only ⇠ 300 in 93 Nb for comparable run times and target thicknesses.
Figure 3.20 compares the expected signal-to-noise for the 3.56 MeV gamma in the 24 Mg and
93 Nb

measurements, given the measured unit cross-section for 12 C. The purple line in the
108

12

C
background
clover-crystal
tracking-detector

16.5 MeV < Ex (12C) < 22.5 MeV

(a)
24

Mg

13 MeV < Ex (24Mg) < 18 MeV

(b)
93

13.5 MeV < Ex (93Nb ) < 18.5 MeV

Nb

(c)

Figure 3.20: Comparison of the achievable signal-to-noise for the 3.56 MeV -ray emitted
from inelastic scattering of 6 Li on 12 C (top panel), 24 Mg (middle panel), and 93 Nb (bottom
panel). The Doppler reconstructed gamma is resolvable in the 12 C measurement but not in
the 24 Mg other measurements. The purple and red line-shapes are the simulated response
of the clover and tracking detectors, respectively. In the 12 Ccase, the purple line shape is
fit to the data, and used to estimate the number of counts in the 3.56 MeV peak. The
excitation energy was gated on the 15.1 MeV state in 12 C which has a B(GT)⇠1. Assuming
a comparable B(GT) in 24 Mg and 93 Nb, over an equal-width range in excitation energy, and
accounting for the change in the unit cross-section with mass, the purple and red line-shapes
have been scaled to reproduce the expected number of isovector counts in the other two
measurements. This clearly illustrates that resolving the 6 Li 3.56 MeV gamma becomes
increasingly difficult in heavier systems where the isoscalar giant resonances remain strong.
Decay from these resonances contribute a significant background in the region of the 3.56
MeV -ray from 6 Li, and this background increases in magnitude with increasing nuclear
mass.

109

24

93

Mg

Ex (24Mg) (MeV)

Nb

Ex (93Nb) (MeV)

Figure 3.21: The di↵erential cross sections for the 24 Mg(6 Li,6 Li0 ) and the 93 Nb(6 Li,6 Li0 )
singles measurements. The cross sections in both cases are dominated by isoscalar giant
resonances which become larger with heavier mass systems.
middle panel shows the expected signal (of ⇠ 3700 counts) for the 24 Mg measurement using
CAGRA. The isovector signal decreases significantly due to the exponential decrease of the
GT unit cross-section as a function of mass. Unfortunately, at the same time the theoretical
estimates of the 0 cross-section for 100% exhaustion of the isoscalar giant resonance energyweighted sum-rule increases monotonically with nuclear mass as can be seen in Figure 3.22.
These estimates are in line with what was observed in the 24 Mg and 93 Nb measurement.
In Figure 3.21, the singles di↵erential cross section for 24 Mg and 93 Nb are shown to both
be larger than that seen for the 12 C measurement (Fig. 3.17). A larger overall cross-section
due to the increase of the isoscalar strength with nuclear mass implies that a proportionally
larger number of de-excitation gamma-rays from the decay of the isoscalar resonances will
be observed. This is clearly the case for 93 Nb, where the background in the region of

110

d2σ/dΩdE (θ = 0o) [mb/sr/MeV]

ISGMR, 100% EWSR
ISGDR, 100% EWSR
103

102

101

0

50

100
150
Mass Number

200

250

Figure 3.22: Mass dependence of the calculated zero-degree di↵erential cross section for
the isoscalar giant monopole and dipole resonances (IVGMR and IVGDR, respectively)
corresponding to 100% exhaustion of the energy weighted sum-rule. These calculations
illustrate the increase in the isoscalar cross-section as a function of nuclear mass number and
were provided at the courtesy of Umesh Garg [139].
the 3.56 MeV gamma increased considerably relative to the gamma-background in 12 C; see
Figure 3.20, bottom panel. At the same time, the estimated Gamow-Teller unit cross-section
for these nuclei decreases exponentially with mass, implying that the gamma-ray signal from
6 Li

should become proportionally weaker with mass.
Unfortunately, this e↵ect can only be partially mitigated by improved detector resolu-

tion. Using the above estimates for 24 Mg and 93 Nb with the available angular resolution
of advanced HPGe tracking detectors, the red line in the bottom two panels of Figure 3.20
demonstrates that the achievable S/N would be barely sufficient for detection for 24 Mg and

111

impossible for 93 Nb. The background is simply too strong in the latter case.6 Therefore,
unless there is a strong localization of isovector strength–in many discrete states–this large
systematic background in the gamma spectrum makes identifying the increasingly weak
isovector signal from the 6 Li 3.56 MeV -ray very challenging.

3.4

Conclusion

In this work, the (6 Li, 6 Li0 [T = 1, Tz = 0, J ⇡ = 0+ , 3.56 MeV]) reaction probe has been
shown to be capable of directly selecting the isovector spin-transfer transitions in 12 C via
inelastic excitation. In this way the 6 Li reaction probe is the neutral-current analog to spin
transfer charge-exchange reactions. The mass-dependent Gamow-Teller unit cross-section for
6 Li

has been inferred from the data and configuration-interaction shell-model calculations.

Hence, Gamow-Teller transition strength in the inelastic channel can now be extracted with
this probe, and used to indirectly infer inelastic neutrino-nucleus scattering cross-sections.
However, it was found that in addition to providing an isovector spin-transfer tag in the
form of the de-excitation 3.56 MeV -ray, 6 Li strongly excites the isoscalar giant resonances.
Specifically, while the isovector cross-section decreases exponentially with mass number, no
commensurate decrease in the isoscalar cross-section was observed. Thus, the resolvability of
the isovector de-excitation -ray from 6 Li is extremely challenging for all but light nuclei due
to a large high-energy -ray background from the decay of the isoscalar resonances. If the
isoscalar and isovector responses were near-equivalent in cross section as a function of nuclear
6 An additional e↵ect introduced via the use of a tracking detector is that instead of discrete angular
positions (e.g. the four angular positions covered by the clover detectors in this measurement, see Fig. 3.5),
an approximately continuous coverage of emission angles is achieved. This has the result of smoothing out
the background from gamma-rays emitted at rest (from the target). This would make a di↵erence for the
lighter nuclei, such as 24 Mg where the background in the sideband and photopeak regions are not smooth
(see the middle panel of Fig. 3.20).

112

mass, this challenge would be overcome by utilizing modern HPGe tracking detectors which
could provide a significant increase in the resolvability of the Doppler-boosted de-excitation
gamma. As this is not the case for the (6 Li, 6 Li0 [T = 1, Tz = 0, J ⇡ = 0+ , 3.56 MeV]) reaction,
it remains a viable isovector spin-transfer probe only for light nuclei.

113

Chapter 4
Many-core algorithms for
topologically divergent neural
networks
Early research into artificial neural networks was inspired by the kinds of computational
processing found in nature [140]. It was motivated by the computational paradox presented
by the human brain: while comparatively inadequate in speed of serial calculations, the
brain is capable of performing highly complex cognitive tasks with trivial e↵ort compared
to that of a modern computer [141]. It was suggested that the primary di↵erence in the
mode of processing performed by the brain is its distributed nature. Consisting of ⇠ 1011
neurons, with a connectivity across ⇠ 1014 synapses [141], the computational power of the
brain appears to come from its massively parallel organization and execution.
Every neuron can be connected to many other neurons, can undergo activation, and can
induce activations in neurons to which it is connected. These fundamental components comprise the core structure and operations within biological neural systems, as well as artificial
neural networks (ANNs). Shown in Figure 4.1 are the typical components of an artificial
neural network. The basic components of a neural network are inbound and outbound
connections, a node, and its corresponding activation function.
114

Artificial neural-network components
inputs

x1

in
co boun
nn d
ec
tio
n

activation
yi = f(ui )

s

node

xj

wij
weights

ui
accumulation
ui = j wij xj

yi

output
to outbound
connections

xn
Figure 4.1: The basic components an artificial neural network. Each node (shown as the circle
in the figure) accumulates inputs modulated by the weights of its inbound connections. After
all of the inbound connections have been accumulated into the node, a non-linear activation
function, such as a logistic sigmoid, transforms the nodes value. The transformed node value
is then used as an output along outbound connections to other nodes in the network.
Input values (xj ) are fed forward via inbound connections, shown as grey arrows on the
left of Fig. 4.1, which modulate the inbound signal by connection weight factors (wij ). These
connection weights represent the strength of each connection and are the principle degrees
of freedom in a neural network. By adjusting these weights, a neural network can be trained
to yield the desired response given the inputs.
The product of the inputs and the weights are accumulated into a floating point number
known as the node. When all of the inbound connections have been fed into the node, a
non-linear function is applied to the node’s value, and it then can be used as the source for
any outbound connections, see the right side of Fig. 4.1. This non-linear transformation is
a unique characteristic of the ANN neuron and is analogous to the activation potential in
biological systems: the capability of a neuron to fire (or not) given its accumulated input
signal. In neural networks, this function is referred to as the activation function and is often
chosen as a smoothly varying function that can represent both final states of the node (active
or inactive)–for example, a logistic sigmoid [142]. Finally, the activated node value (yi in
115

the figure) is then output via the outbound connections which feed into other nodes.
Together, the above described neural-network components form a structure resembling
a biological neuron. Neural networks are formed by combining many such neurons together
into a larger network. Panels (a) and (b) of Figure 4.2 detail two simple neural networks,
where each circle is a network node, and each arrow between nodes is a network connection.
Nodes with no inbound connections are known as input nodes (left most set of nodes in
Fig. 4.2), and they are loaded with the user input data, and output nodes are nodes with no
outbound connections. The output of a neural network are the values of its output nodes.
The network shown in (a) is representative of the kinds of neural networks found in modern
shallow and deep learning applications, and is known as a hidden-layer neural network or a
multi-layer perceptron.
It has been shown [143] that a neural network with a single hidden layer (similar to
network (a) in Fig. 4.2) can approximate any function given arbitrarily many nodes in the
hidden layer. It is because of this capability that neural networks have been applied so
successfully to diverse applications. Furthermore, the use of additional hidden layers, as in
deep neural networks, increases the non-linearity and can capture higher-level abstractions
of the data more easily [144]. However, while these points have been known for the past
decades, it wasn’t until the advent of modern computational capabilities and enormous
datasets, known as “big data”, that it would become possible to train neural networks with
such large numbers of degrees of freedom. As more hidden nodes and hidden layers are
added, more free parameters in the form of connection weights are introduced.
Neuroevolution, on the other hand, is an approach that has developed in parallel to complex layered neural networks, and has instead drawn on biological inspiration to describe the
complexification process of neural networks [26]. In addition to modeling artificial neural
116

network behavior from the way in which complex systems of neurons operate, neuroevolution
seeks to model how these systems are formed. It does so by applying evolutionary algorithms
(EAs) to large populations of neural networks in order to search for the optimal network
structure and hyperparameters [26]. Similar to biological evolution, these networks compete
for limited resources and in each generation, only the best performing neural networks survive. The concept of limited resources in neuroevolution arises from the limited compute
cycles available and is often represented by penalizing networks which evolve large structures that do not benefit the overall performance of the network. In this way, populations
of neural networks with minimally complex structure, evolve new nodes and connectivity
only if the structural changes represent innovations that improve overall performance. This
concept strictly contrasts that of complex layered neural networks. With hidden-layer neural
networks, few a priori constraints are placed on the network size. Even though the number
of connection weights (degrees of freedom) in the these models are large, enormous datasets
are often available which can be used to train these weights, albeit at large computational
cost.
Unfortunately, in many scientific domains, data sizes are not comparable to what is
collected in industrial big data applications, and so the utility of arbitrary network complexification is limited. The method of evolving neural networks through augmenting topologies [145] on the other hand, seeks to find the best performing neural network by minimizing
the search space and only increasing the network complexity when doing so improves the
neural network. For this reason, neuroevolution is able to generate neural networks with
complexities that scale with the size of the training data, making it generally applicable to
scientific applications, regardless of data size.

117

...

2
2
3
1

1

Figure 4.2: (a) A common feedforward neural-network structure with a single hidden layer.
0 ,1
x0i , x1i , and x2i correspond to the input, hidden and output layers respectively, while wij
refer to the connection weight matrices between two layers. (b) An unstructured neural network in which the connections are not constrained to specific layers. Such network structure
is common in neuroevolution populations. Each connection in (b) is labeled with an integer
indicating the lock-free set in which it can be evaluated. See section 4.3.1 for details.

4.1

Applications in physics

While neural networks have been employed in a variety of physics domains for the past few
decades [146], their utility has been increasingly significant in experimental high-energy [147]
and neutrino [148] physics. In these physics applications, neural networks have almost exclusively been employed as classification tools as a means to separate signal and background
events. The most commonly employed classification techniques in these fields are neural
networks with a single hidden layer (shallow ANNs), boosted decision trees, and as of the
past few years, deep neural networks. Until the work of [147], physicists reluctantly accepted the limitations of shallow neural networks, and guided the networks by hand toward
better event selection by manually constructing nonlinear input feature combinations that
helped the training processes. However, with the availability of large datasets in high energy
physics, Baldi et al. [147] showed that recent advances in deep neural networks lift these

118

limitations by automatically discovering nonlinear feature combinations that provide better
event discrimination than shallow neural networks even with manually constructed input
combinations.
The ability of deep neural networks to circumvent the need for manual data preprocessing comes from the large number of hidden layers, and the correspondingly large available
datasets. Without sufficient data, however, the feasibility of deep learning is limited. On
the other hand, as discussed previously, neuroevolution generates neural networks with complexities that scale with the size of the training data, making it more generally applicable. In
this way, complicated data preprocessing can be evolved according to the constraints of the
data, regardless of data size. Indeed, neuroevolution has been previously used in this way
to provide high precision selection of dilepton events for the measurement of the top-quark
mass at CDF [149].
Recently, researchers involved with the NEXT neutrino physics experiments, which employ high pressure xenon time projection chambers (TPCs), have shown that deep convolutional neural networks, presently the state-of-the-art deep machine learning technique for the
analysis of image data, can be 60% more efficient at the selection of neutrinoless double beta
decay events as compared to traditional techniques [150]. Because TPCs produce volumetric
images of particle tracks within the detector, convolutional neural networks are e↵ective at
event selection based on the di↵erence in tracks for signal and background events. However,
as neural networks become increasingly complex, as is the case for convolutional networks,
they are endowed with additional hyperparameters that increase the burden of tuning that
must occur prior to training [151].
Very recently neuroevolution techniques have been employed to evolve minimally complex convolutional neural networks [24]. Similarly, Fernando et al. [25] have shown that by
119

using neuroevolution to evolve network topologies, the resulting network had two orders of
magnitude fewer free parameters and was able to achieve the same accuracy as a comparable
fully connected convolutional network. With far fewer connections, the evolved convolutional
networks have fewer free parameters to be learned, and are therefore able to be trained with
significantly smaller datasets as compared to their fully connected counterparts. Therefore,
neuroevolution extends the utility of deep-learning methods to application domains in which
the datasets are smaller in magnitude than what is required for traditional deep neural
networks.
For this reason, the experimental nuclear astrophysics program discussed in chapters 2
and 3 may significantly benefit from neuroevolution since the available training datasets are
likely to be of intermediate size. In chapter 5, a new experimental technique in which a TPC
is utilized to image recoil particle tracks after they undergo + charge-exchange reactions
will be discussed. Similar to the NEXT experiment previously described, convolutional
neural networks may be capable of efficient event selection for TPC-based charge-exchange
experiments in inverse kinematics. By utilizing neuroevolution and the methods described in
this work, even though the available training data will be smaller than in experimental high
energy physics applications, the evolved networks will be tuned in structure to the available
data and are therefore likely to have comparable performance.1

1 It should also be noted that in experimental physics, a viable option for the application of deep learning
techniques is to utilize physics simulations to generate the required training data. In this case, the performance of networks trained on simulated data will depend significantly on the accuracy of the simulated
data.

120

4.2

Network Structures & Evaluation

One hallmark of neuroevolution is its ability to develop populations of structurally diverse
neural networks, which do not necessarily have strict layer boundaries. For example Fig. 4.2b
demonstrates a neural network that could evolve, wherein the concept of a hidden-layer no
longer applies due to the connections from the two input nodes directly to the output node.
That said, this functionality can be reproduced in a hidden-layer neural network (such as
panel (a) in the same figure) if the connection weights are tuned correctly. While networks
(a) and (b) in Fig. 4.2 are capable of producing the same network output, given the larger
number of nodes and connections involved, the hidden-layer network (a) would be more
computationally expensive to evaluate. This is easily seen when considering the feedforward
evaluation pattern for hidden-layer neural networks which can be written as [152],

=

⇣

W (0) x(0)

,

(4.1)

x(2) =

⇣

⌘
W (1) x(1) + b(1) .

(4.2)

x(1)

+ b(0)

⌘

Here, evaluation is performed by feeding forward the network inputs, x(0) into consecutive
layers by multiplication of the connection weight matrix, W (0) , to this input vector, the
addition of a bias vector, b(0) , and the activation of the result.

is the activation function

and is often chosen to be a smooth function that is bounded between two values, one of
which represents a neuron firing (activated) and the other which represents quiescence (not
active). The result of Eq. 4.1 are the values of the nodes in the next layer, x(1) , which then
become the input vector for the next feedforward iteration, shown in Eq. 4.2. This process is
repeated until the inputs have been fed through to the final output layer. Thus, feedforward

121

evaluation of layered neural networks is simply the repeated application of matrix-vector
multiplication, and so it is evident that the more nodes and connections a network has, the
more floating point operations (FLOPs) will be required.
While network (a) in Figure 4.2 requires more total FLOPs than network (b), the method
of evaluation described by Eq. 4.1 is highly parallelizable as it is simply repeated matrix
operations [153]. Neural networks which do not conform to the layered description, on
the other hand, cannot be described by matrix multiplication. Principally, this is because
the calculation of the neural-network output is no longer divisible into a series of linear
computations. Instead, as for network (b), the evaluation must follow a graph traversal in
which connections are applied consecutively, and node activations occur for each node once
all inputs to the node have been computed.
As mentioned in the beginning of this section, the novelty of neuroevolution is its capability to produce highly topologically diverse neural network structures, which can reproduce
inherent symmetries in the underlying datasets [154]. However, such diversity in the neural
network population implies that the computational graphs of each neural network can be
extremely divergent. At the same time, because the search capability of the evolutionary algorithm scales with the population size, using as many networks as computationally feasible
is ideal. If population sizes are in the tens to hundreds of thousands, the evaluation complexity warrants the use of modern multi- and many-core compute architectures. However,
the divergence that arises in evaluating neural networks of diverse structure, has so far prevented the development of generalized many-core algorithms for neuroevolution which scale
well with the population size. This is because many-core architectures, such as general purpose graphics processing units (GPGPUs), employ single-instruction-multiple-data (SIMD)
vector processors. These kinds of processors perform optimally when execution branching is
122

Figure 4.3: An illustration of the structural divergence than can exist in populations of
neural networks evolved via neuroevolution. In this case, each network has a fixed number
of input and output nodes (3 and 1, respectively).
minimal. Unfortunately, the parallel evaluation of neural networks with di↵ering structures
has the opposite e↵ect, execution branching is enhanced.
Presented in this work is a novel many-core algorithm for the concurrent evaluation of
entire populations of topologically-divergent neural networks. These algorithms represent the
first general-purpose mapping of large numbers (100k+) of heterogeneous neural networks
to many-core architectures.2 By harnessing modern many-core processors, this work aims
to enable neuroevolution to be a more competitive player in the modern machine learning
ecosystem.

2 An open source implementation of the described methods are freely available online at
http://www.github.com/csullivan/Entendre.

123

4.3

Directed acyclic graph concurrency

An example of the structural diversity within populations of neural networks is shown in
Figure 4.3, where each network has exactly 3 input nodes and 1 output node. The hidden
node structure of the networks is then evolved during many generations of neuroevolution.
While all of the networks in the figure have standard connections which feed forward, neuroevolution can also evolve recurrent connections.3 This is worth noting as the algorithm
proposed in this section performs equally well with recurrent neural networks.
The naive feedforward evaluation of a network population, such as that shown in the
figure, would be to distribute each network to a di↵erent processor or thread. For multi-core
processors this is not a necessarily bad approach, particularly if the networks involved are
large in size and can take advantage of the heterogeneous cache-memory layouts of modern
multi-core processors. However, if the population size is much larger than the number
of available processors, or the networks themselves are relatively small, cache-coherence is
likely unattainable. This is because a population of neural networks is essentially an arrayof-structures (AoS) organization of data, and so consecutive neural networks are loaded from
RAM neglecting any benefits of hierarchical memory.
For many-core SIMD coprocessors, distributing neural networks across threads has far
worse performance. There are three principal considerations to be made in SIMD parallelism
by distributing neural networks across threads:
1. Branch divergence from di↵ering neural network topologies (a result of neuroevolution
as previously discussed).
3 Recurrent connections in a neural network are connections which form a directed cycle in the network.
However, instead of actually producing a cycle, they function by acting as external inputs into the network,
which pass the value of a node from the previous feedforward evaluation into the destination node of the
recurrent connection in the current evaluation. In this way, recurrent connections act as a kind of neural
network memory.

124

2. Lack of memory access coalescing requiring each network to be loaded from device
RAM (DRAM).4
3. The input data and output node values must be communicated from the Host (CPU)
to the Device (GPU) for each neural network separately.
The three items above can be summarized as a lack of optimization in instruction execution,
memory access, and communication, respectively.
Instead of seeking to parallelize over individual neural networks, this work establishes
a sorting algorithm that organizes the evaluation of many neural networks into a problem
of parallelism over the directed acyclic dependency graphs of each neural network. By
reformulating the problem in this way, optimizations for items 1–3 are naturally attained:
instruction branching is removed, memory accesses are coalesced, and the communication
between the host and device is minimized.

4.3.1

Topological connection sort

In order to avoid hierarchical data structures, which typically rely on many heap-allocated
blocks of memory and thereby reduce the utility of memory caching, the network structure
is stored as flat arrays of the connection components5 . Here, each connection consists of the
index of the origin node, the destination node, and the weight of the connection. Using an
array of floating point numbers to store the values of the nodes, or their partially computed
intermediate values, each connection then represents a single uniform action to be applied
4 Memory access coalescing, or the lack thereof, is similar to cache-coherence on a CPU. Whereas during
a memory load for a CPU, a block of memory (ABC) is brought from RAM and stored into the CPU cache,
for a GPU, a memory request from DRAM coalesces a cycle of memory to each thread, so that threads 1,
2, & 3 receive floats A, B & C. A lack of memory coalescing indicates that data utilized by threads in the
GPU are not spatially colocated, and so each thread must make individual memory loads [155].
5 A structure of arrays (SoA) memory layout as opposed to an AoS layout

125

Neural Network
A2

recurrent
A1

1

2

B1

C1

3

B2

Connections Directed Acyclic Graph
A1

A2

A1

B2

B1

B2

or

A2

B1

C1

C1

Logical
Dependencies

Logical & Memory
Dependencies
(Lock-Free)

Figure 4.4: An example neural network and two directed acyclic dependency graphs for the
evaluation of the neural network’s connections. The left dependency graph includes only
logical dependencies whereas the graph on the right includes both logical & memory dependencies. Each circle in the dependency graphs represent connections in the top neural
network figure. The arrows in the two dependency graphs indicate the dependency relationship between the connections . For example, the arrow between A1 and B1 indicates
that connection A1 must be evaluated prior to B1. By inspecting the neural network, this
relationship is evident as connection B1 relies on node 1, which in turn relies on connection
A1 (as well as other connections).

126

on the array of nodes. In this way, the neural network is organized by its connections instead
of via its nodes.
The key algorithmic insight in the proposed method is the sorting of neural network
connections into evaluatable sets based on their dependencies. The directed acyclic graph
(DAG) for an example neural network’s connections is shown in Figure 4.4. Every circle
in the DAG represents a connection in the neural network, and every arrow in the DAG
implies a dependency relationship. For example J!K implies that K depends on J, and
so if these represent connections it implies that connection J must be evaluated first in the
neural network. These relationships arise from the fact that connections in a neural network
have direct and transitive dependencies–connections earlier in the neural network must be
evaluated before those that come later. For example, A1 must be evaluated before B1 in the
example neural network shown on the top of panel Figure 4.4. Dependencies of this type can
be referred to as logical dependencies. Memory dependencies can also exist if the employed
algorithm is to be lock-free.6 These type of dependencies occur when two connections which
are not logically dependent, a↵ect the same destination node.
The algorithm described in this work topologically sorts connections into ordered sets that
respect both of these dependencies. That is, within each set, all constituent connections have
di↵erent destination nodes (lock-free) and are logically independent. If the algorithm need
not be lock-free, the adherence to connection memory dependencies can be relaxed. The two
graphs in Figure 4.4 illustrate the relationships between connections in an example neural
network when only logical dependencies are considered (left) as well as when logical-andmemory dependencies are considered (right). Determining the order of connection evaluation

6 Lock-free in this context implies that no memory synchronization or exclusions are required, such that
no two threads will attempt to write to the same memory space simultaneously.

127

1 for ( i=0; i<num_connections ; i++) {
2
for ( j=i+1; j<num_connections ; j++) {
3
a = connections [ i ] ;
4
b = connections [ j ] ;
5
switch ( compare_connections ( a , b ) ) {
6
case a > b :
7
a . num_dependencies++;
8
break ;
9
10
case a < b :
11
b . num_dependencies++;
12
break ;
13
14
// ( a , b ) not a d j a c e n t
15
case Unknown :
16
break ;
17
}
18
}
19 }
20
21 topological_sort ( connections ) ;

Figure 4.5: Prototype of sorting connections into concurrent lock-free sets. For n connections,
scales as O(n2 ).
of the neural network is then straightforward from these dependency graphs, and concurrency
is immediately evident. Consider the left graph in Fig. 4.4. Any connection (circle in the
DAGs) which has no inbound dependencies (shown as the grey arrows), can be evaluated
immediately. Thus, one can see that connections A1 and A2 can be evaluated in parallel at
the first step. The next step is to remove A1 and A2 from the dependency graph, and to
again search for connections which now have no inbound dependencies. After A1 and A2 are
evaluated, we see from the DAG that connections B1 and B2 have no dependencies, and so
are able to be evaluated next and in parallel. Finally connection C1 can be evaluated. This
sorting technique is known as a topological sort of a directed acyclic graph and is often used
in determining the order of tasks [156].
Algorithmically, the dependencies each connection has is determined using the comparator scheme shown in Table 4.1. With this comparison, if two connections are adjacent and

128

Table 4.1: Neural network connection dependency comparator
Type

Condition

Dependency

Connection pair (a,b) with di↵ering output nodes
a recurrent
b recurrent
a normal
b normal

a.origin = b.destination
b.origin = a.destination
a.destination = b.origin
b.destination = a.origin

a
a
a
a

<
>
<
>

b
b
b
b

Connection pair (a,b) with the same output node
a self-recurrent
b self-recurrent
arbitrary choice
arbitrary choice

a.origin = a.destination
b.origin = b.destination
a.origin < b.origin
b.origin < a.origin

a & b not adjacent

-

a
a
a
a

<
>
<
>

b
b
b
b

unknown

The neural network connection dependency comparator. There are three subsections in
the above, when connections a and b (1) are adjacent and have di↵erent output nodes, (2)
are adjacent and have the same output nodes, and (3) are not adjacent. If the algorithm
need not be lock-free (does not respect memory dependencies) then the second subsection
(connections with the same output node) can be dropped and the rules of the first subsection
are used exclusively for adjacent connections. Each connection has an origin and destination
node, which are what is referred to in the Condition column. The less-than and greater-than
symbols in the Dependency column imply the ordering of the two connections; if b > a, then
the evaluation of connection a must come prior to b. In the case of an unknown relationship
between non-adjacent connections, transitive relationships will enforce and ordering if one
exists. This comparator handles both normal and recurrent connections.

129

one connection depends on the other, the dependent connection’s number of dependencies is
incremented. This comparator can then be applied to any neural network regardless of its
structure and a topological sort, as described previously, can determine the sets of concurrently evaluatable connections. An example implementation of this is shown in Figure 4.5.

4.3.2

Neural network operations

The ordering of connections into concurrently evaluatable sets, as described in the previous
section, provides the sca↵olding o↵ which the rest of the neural network operations are built.
Fundamentally, three operations are required in the feedforward evaluation of an arbitrary
neural network,
1. Apply connection: The application of a connection A1 from node 1 to node 2:
x2 = x2 + wA1 · x1 , where xi is the floating point value of node i, and wA1 is the weight
of the connection A1.
2. Zero/Reset node: Setting the memory address of a node’s floating point value to
zero, xi = 0.
3. Activate node: Applying the activation function to a node’s value, xi = (xi ).
In the previous section, connections within a neural network were grouped into ordered sets,
such that the application of the connections within each set could be performed simultaneously. In this way, parallelism over the directed acyclic dependency graph of a neural
network boils down to the bulk synchronous evaluation of operations 1–3 above on di↵erent connections and nodes within the network. Using the ordered sets of connections, the
zeroing and activation of nodes can be deduced from the following observation: given the

130

above neural network operations, a strict order-of-operations can be defined such that the
neural network’s full directed acyclic dependency graph can be established. Specifically this
operation order is,
Activate origin nodes < Zero destination nodes < Apply connections.
Therefore, given a set of connections which are to be applied, any origin node that has not
been previously fed forward must be activated, any destination node which has not yet been
fed into must be zeroed/reset, and then all connections in the set can be applied.
By building the dependency graph from the connections DAG previously discussed, and
by employing the above order of operations, this algorithm guarantees that at every step,
the performed operations will be homogeneous. That is, many nodes will be activated
concurrently, then many nodes will be zeroed concurrently, and finally many connections
will be applied concurrently. In this way the evaluation of a neural network is factored into
steps that can be easily mapped to SIMD processors. This will be discussed in detail in
section 4.3.3.

4.3.2.1

Neural network node lifecycle

While the order of operations proposed captures the general method requirements of the feed
forward evaluation of arbitrary neural networks, it misses a few subtleties that are apparent
when considering the lifecycle of a node in a neural network.
Thus, let us consider the lifecycle of a single node in a network. At the start of neural
network evaluation, a node holds its activated value as determined in the previous evaluation
of that network. The first step that must be taken is that all recurrent connections must
be applied which use this node’s value as an origin. This is because a recurrent connection

131

is meant to act as a type of memory, feeding information from previous network evaluation
into the current evaluation.
Next, the node must be zeroed out. Following this, there is a phase when the node is
used as a destination, with its value being the weighted cumulative sum of all incoming
connections that have been applied. Once all connections inbound to this node have been
applied, the activation function transforms the node’s value. Then, the node can be used as
an origin for normal connections, and finally, the output value is left untouched, so that it
can be used for any recurrent connections in the next evaluation.
These steps are fundamentally what impose the ordering conditions described in the
previous section,
• The zero/reset must occur after the last use of the node as an origin of a recurrent
connection, and before the first use of the node as a destination.
• The activation function must be applied after the last use of the node as a destination,
and before the first use of the node as an origin of a normal connection.
Thus, these actions are performed between the evaluation of each set of connections.
Together with the connection ordering described in section 4.3.1, we arrive at a general
distribution of tasks which follow the basic life cycle of a node and which are constrained by
the parallelism of the evaluatable connection sets.

4.3.3

Generalized concurrent neural network evaluation

By applying the topological sort of network connections and enforcing the order of operations
described above, the full neural network evaluation graph can be built. Figure 4.6 shows
the directed acyclic dependency graphs for two neural networks with di↵erent structures.
132

Network #2

Network #1

Directed Acyclic Dependency Graph

Neural Networks

A2
A1

1

B1

A2
A3

A1

3
2

1

2

B1

B2

B2

1

1

2

3

A1

A2

A3

1

A1

1

2

B1

B2

Evaluation SIMD
Step Concurrency
#1

4

#2

5

#3

3

3

#4

2

B1

B2

#5

4

2

#6

2

C1

#7

1

3

#8

1

Symbol

Name
Zero node 1

0

Activate node 1

1

1

Apply connection
A1

3

2

2

1

A2

C1

2

A1

Figure 4.6: The full directed acyclic dependency graphs for two structurally di↵erent neural networks. The di↵erent symbols in the graph, squares, circles, and triangles, represent
the three neural network operations: zero-node, apply-connection, and activate-node, respectively. Even though the two networks have very di↵erent structure, the algorithm proposed in
this work organizes the network evaluation graphs into evaluations steps that are componentwise parallel. On the right of the figure, each evaluation step is shown with the number of
SIMD operations that can be performed simultaneously.

133

The neural network label network 2 in the figure is the same network shown in Fig. 4.4.
However, instead of just showing the DAG for the application of connections, it includes
the other operations as well: squares represent the zero-node operation, circles represent the
apply-connection operation, and triangles represent the activate-node operation; see the key
in Fig. 4.6 for more details.
Factoring the neural network evaluation into these component operations, the individual
structure of di↵erent neural networks is no longer a consideration. Each network has some
number of nodes which must be zeroed and activated, as well as some number of connections
that must be applied. Starting at the top of the DAGs shown in Figure 4.6, the first step in
evaluating both networks 1 and 2 is to zero out four nodes (see the axis to the right of both
networks). Therefore, in this first step, an operation concurrency of four is achieved. In the
second step of evaluation, five connections must be applied across the two separate networks.
This process continues until all evaluation steps in the shown DAGs have been performed,
at which point the inputs have been fully fed through both networks, and evaluation is
complete.
Thus, the algorithms presented here are completely agnostic to the macroscopic structure
of the constituent neural networks. At the same time, these algorithms order the tasks to
be performed into homogeneous sets of operations which map very well to SIMD processors.
This e↵ectively resolves the instruction branching problem described in section 4.3. Furthermore, as can be seen in Fig. 4.6, this algorithm will scale with the neural network population
size. Since instruction branching is removed, the more networks that are utilized, the greater
the concurrency at each evaluation step. This indicates that the performance gain should

134

scale at least linearly with the number of neural networks.7

4.4

Composite population evaluation

While the DAG based algorithms described in the previous section solves the problem of
evaluating structurally diverse neural networks on SIMD-based processors, it doesn’t alleviate the lack of optimization in memory access and Host-Device communication described in
section 4.3. In this section, both of these latter issues are resolved via a simple reordering
of the memory layout for the neural network population.
Instead of storing each network individually, a refactoring of a population of N networks
into a single composite network, consisting internally of all the nodes and connections of the
sub-networks, is employed. In this way, the resulting neural network will have at minimum N
connections per evaluation step that can be processed concurrently using the evaluation algorithm described in section 4.3.3. An example of two such refactorizations are demonstrated
in Figure 4.7.
The construction of the composite net is straightforward. First, all network input nodes
are added, then all output nodes, and finally all hidden nodes. Following this, each connection of each sub-network is reindexed according to the position of its nodes in the larger
set of nodes of all networks in the population. After all connections are added, the connection sort described in section 4.3.1 and detailed in Figure 4.5 is applied to each subset of
connections in the composite neural network. As these populations can grow quite large,
it is computationally most efficient to sort based on these unconnected sub-structures (the
7 The performance scaling with the neural-network population size can be less than linear if the population
contains a subset of outlier networks which have much larger network topologies than the majority of the
population. In this case, the majority of networks will finish their final step of evaluation while the evaluation
of the subset will only be partially completed. Thus, in the remaining steps of evaluation for the outlier
networks, the concurrency will be dramatically reduced.

135

NETWORK 1

NETWORK 2

NETWORK 3

SINGLE COMPOSITE NEURAL NETWORK

POPULATION OF INDIVIDUAL NEURAL NETWORKS

HETEROGENEOUS INPUTS

HOMOGENEOUS INPUTS

Figure 4.7: Top panel: A representative example population of neural networks consisting
of three members. In classic neuroevolution simulations, each of these neural networks are
separately evaluated, and the correctness of their outputs is used to judge their fitness.
Bottom panel: The same population as in the top panel, but constructed as nodes and
connections of a single composite neural network. On the left, the resulting composite
network in which each of the constituting sub-networks requires heterogeneous inputs for
evaluation. On the right, a composite network in which each of the sub-networks will process
the same inputs.

136

individual networks) instead of the composite network, as the sorting algorithm shown above
scales as O(N 2 ) with respect to the number of connections. Finally, the connections of the
composite network are globally sorted according to their evaluation set index. Following this
procedure then guarantees that connections which can be evaluated together are colocated
in contiguous memory space. This then allows the origin and destination node indices and
connection weights to be perfectly coalesced from DRAM to the individual threads. The
resulting population-level composite network is then ready to be evaluated on a GPU or
other SIMD coprocessor.
In constructing a composite network from a population of neural networks, an additional
optimization that can be made is to refactor all of the input nodes into a single set of inputs.
This can occur if each network in the population will be evaluated given homogeneous sets of
inputs. If on the other hand, each network will receive heterogeneous inputs, this refactoring
cannot be performed. In the bottom panel of Figure 4.7 both possible composite networks
are shown, one in which all inputs are explicitly specified (heterogeneous inputs), and one
in which a single set of input nodes are fed to the rest of the sub-networks (homogeneous
inputs).
In both cases, to evaluate the resulting neural network, a single memory transfer of
the input node array (consisting of nodes for all the networks) is copied to the device, the
evaluation proceeds concurrently, and finally a single copy of the outputs (again in contiguous
memory) finishes the calculation. This is possible because, the input and output nodes of
all the networks are positioned in continuous memory space.
Since this network is quite large and has many unconnected regions, the DAG-based
algorithm described previously takes maximal advantage of the available component-wise
concurrency in the evaluation. Furthermore, the connection weights and node indices are
137

memory access coalesced for each connection, and the interaction between the host and
device occurs only twice (on the boundaries of the evaluation), minimizing any possible
communication contention. In summary, by combining the DAG-based concurrent evaluation
described in section 4.3.3 with the composite organization of a population of neural networks
into a single composite network, instruction execution, memory access, and communication,
are fully optimized for execution on many-core processors.

4.5

CUDA implementation

The DAG-based connection sort and construction of the composite neural network described
in the previous sections are performed on the host CPU, after which the network connections
and nodes are copied to the many-core device for evaluation. The network evaluation performance is benchmarked using the compute unified device architecture (CUDA) [157]. The
implemented CUDA kernels can be found in Figure 4.8. There are a total of three kernels
for the three necessary actions to be performed: resetting a node to zero, activating a node,
and applying a network connection.
As a result of the presorting of operations on the host, the CUDA kernels are extremely
simple, bringing the possibility of (thread) branch divergence to an absolute minimum. The
only possible divergence that exists is seen on line 43 of Figure 4.8 in which a self-recurrent
connection8 is handled as a special case. In the case of neural networks with no self-recurrent
connections, this algorithm is GPU thread divergence free.

8 A self-recurrent connection is a connection whose origin node is also its destination node. These connections function as memory in the neural network, returning the output the node from the previous network
evaluation.

138

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
31
32
33
34
35
36
37
38
39
40
41

__global__
void device_clear_nodes ( uint32_t ⇤ list , float ⇤ nodes , uint32_t n )
{
int i = threadIdx . x + blockIdx . x ⇤ blockDim . x ;
if ( i<n ) {
nodes [ list [ i ] ] = 0 ;
}
}
__global__
void device_activate_nodes ( uint32_t ⇤ list , float ⇤ nodes , uint32_t n )
{
int i = threadIdx . x + blockIdx . x ⇤ blockDim . x ;
if ( i<n ) {
nodes [ list [ i ] ] =
device_activate ( nodes [ list [ i ] ] ) ;
}
}
__global__
void device_apply_connections (
float ⇤
node
,
uint32_t ⇤ origin
,
uint32_t ⇤ dest
,
float ⇤
weight
,
uint32_t n
)
{
int i = threadIdx . x + blockIdx . x ⇤ blockDim . x ;
if ( i<n ) {
auto& conn_origin = origin [ i ] ;
auto& conn_dest = dest [ i ] ;
auto& conn_weight = weight [ i ] ;
if ( conn_origin == conn_dest ) { // S p e c i a l c a s e : s e l f
node [ conn_origin ] ⇤= conn_weight ;
} else {
node [ conn_dest ] += conn_weight ⇤ node [ conn_origin ] ;
}
}
}

r e c u r r e n t node

Figure 4.8: NVIDIA CUDA kernels for the proposed concurrent neural network evaluation.

139

GPU speedup factor

50
40
30
20
10
0
2 10 2 11 2 12 2 13 2 14 2 15 2 16 2 17 2 18 2 19 2 20
Number of neural networks evaluated concurrently

Figure 4.9: GPU speed up factor when evaluating a population of N evolved neural networks
concurrently. The ratio of time taken for the sequential evaluation of each network on a single
CPU thread to the time for transferring inputs to the device, evaluation on the GPU, and
transferring the outputs back to the host. With a population of approximately 65k networks,
a performance boost of 50x relative to the serial implementation is observed.

4.5.1

Performance results

In the performance tests that follow, the primary machine on which these algorithms were
tested was equipped with a 3800MHz i7-2600 CPU, along with a NVIDIA GTX 1070 GPU.
In addition, these results were also verified on a system with a 2400MHz E5-2680v4 CPU
and a NVIDIA Tesla k80 GPU. The results reported are those for the former system, as the
results from the later system were found to be comparable.
To illustrate the performance of the methods described in this work, a population of neural networks was evolved according to the Neuroevolution of Augmenting Topologies (NEAT)
technique. NEAT is an evolutionary algorithm (EA) introduced by Stanley and Miikkulainen
[145]. The principle components of NEAT are (1) a method of crossover (mating) to efficiently combine network structures into resulting child networks, (2) the introduction of

140

speciation to protect new topological innovations that arise, and (3) incrementally growing
networks from minimal structure, allowing the networks that form to both optimize and
complexify solutions simultaneously [145].
Using NEAT, a populations of neural networks of arbitrary size were evolved. Each of
these populations was seeded with a topologically minimal network: the seed network had
three input nodes, and one output node. Initially, the three input nodes (two inputs for
the XOR plus a bias/o↵set node) were connected directly to the output node making for a
total of 3 connections. These networks were then evolved to solve the exclusive-or operation
(XOR).
The simplest solution to the XOR operation is a network of only 5 nodes and 7 connections, which is quite quickly evolved by NEAT. However for the purpose of this test, the
networks were allowed to continue evolve, thereby increasing in complexity past the minimal solution, so as to have a larger network structures on average. After 30 generations of
evolution, networks with an average of 11±2 connections and 8±1 nodes were produced. As
many as 1 million (220 ) networks were evolved and evaluated.
Each network was then individually evaluated on the host machine, and the evaluation
time tracked for 100 trials. Additionally this population was built into a homogenous-input
composite neural network and evaluated on the GPU for an equivalent number of trials
using the many-core methods described in this work. In Figure 4.9, the ratio of the time
for the separate evaluation on a single CPU core to the composite evaluation on the GPU
is compared. Beginning with small populations of 1024 networks, a modest speedup factor
of ⇠ 2.5 is observed. As the number of neural networks evaluated is increased, the GPU
utilization increases and strong scaling up to ⇠ 65k networks is observed with a maximal
performance speed up of ⇠50x.
141

In Fig. 4.9, the number of neural networks evaluated concurrently (the horizontal axis)
is essentially a metric for operational intensity. The larger the neural network population
(bigger problem size) the more arithmetic operations are performed per DRAM memory load
(burst). This is principally due to the memory access coalescing of the network connections
that is achieved by organizing the population of networks into a single composite network.
Each memory access of the stored connections brings with it the needed information for all
the neighboring GPU threads, minimizing the number of loads and increasing the amount
of floating point operations per memory access.
The behavior observed in Figure 4.9 is well described by the roofline model [158]. The
roofline model sets an upper bound on the performance of a compute kernel (network population evaluation, in this case) depending on the kernel’s operational intensity. Because the
operational intensity scales with the neural network population size, the performance speedup
shown in Fig. 4.9 traces out the e↵ective performance roofline for this system. There are
three identifiable performance regions that are observed. Specifically, the algorithm is
• communication-bound with population sizes between 210 –212 neural networks,
• memory-bound with population sizes between 212 –216 neural networks, and
• compute-bound with population sizes larger than 216 neural networks.
With few neural networks in the genetic population, the peak performance is dominated by
the time it takes to transfer the input and output data from the host to device and back.
Even though only two memory transfers are required, the time required to evaluate the
population of neural networks is comparable to this communication time and is therefore
communication-bound.

142

However, as the number of networks increase, the peak performance gradually becomes
memory-bound by the architecture. That is, the evaluation time is now much greater than
the communication time, but the performance is dominated by the memory bandwidth. In
this region, the more neural networks that are used, the more connections can be memory
access coalesced, which means the GPU threads spend less time waiting for data to arrive
from DRAM. Thus, in this region the performance increases with the operational intensity,
or the number of neural networks evaluated concurrently.
Beyond 216 neural networks, however, the performance reaches the compute-bound roofline
in which it no longer improves with increasing computational load. This is reflective of the
underlying algorithms utilization of the hardware. In this region, the number of operations
per memory load no longer increases with increasing network population size. In this way,
the time each GPU thread waits for connection data to arrive from DRAM has been minimized as much as achievable by the architecture. Thus, increasing the amount of connections
to evaluate in parallel (increasing the population size), no longer improves the performance.
While the roofline or performance-plateau seen in Figure 4.9 is dictated by the architecture, the roofline could potentially be improved (a higher speedup achieved) if the underlying
algorithm were improved to better utilize features of the architectures. For example, while
the DAG-based algorithm described in this work takes great care to ensure memory access
coalescing for the evaluated neural network connections, the application of each connection
requires the load and store of node memory (the connection origin and destination nodes).
Without optimization, these loads and stores are essentially random access, thus requiring a separate memory operation for each node. With consideration of the temporal node
access–by, for example, determining an efficient ordering of nodes used in each evaluatable
connection set–the compute-bound performance roofline could be improved. Furthermore,
143

temporal consideration for both nodes and connections could be used to take advantage of
shared memory on the GPU to further reduce memory latency.

4.6

Conclusion

In this work, new algorithms designed to optimize the concurrent evaluation of topologically divergent neural networks on many-core architectures are introduced. By pre-sorting
the connections into evaluatable sets, and parallelizing over the networks directed acyclic
dependency graphs, nearly all branch divergence is eliminated from execution on SIMD processors. Furthermore, the factorization of many neural networks into a single composite
network optimizes the memory access for network connections, and minimizes the hostdevice communication. When applying these techniques to populations of neural networks
that were evolved via neuroevolution, the feedforward evaluation of these networks was significantly accelerated, with peak performance reaching 50 times the evaluation speed for each
neural network on a single CPU thread. While this algorithm has been targeted principally
for many-core SIMD architectures, multi-core CPUs would likely also greatly benefit from
these methods. The refactorization described in section 4.4 is essentially a transposition of
the population of neural networks from an array-of-structures into a stucture-of-arrays data
organization, which will take maximum advantage of per-core CPU cache. Moreover, the
DAG-based algorithm described in section 4.3 would be well suited to a CPU thread pool
or other task-based concurrency paradigm.
The novel algorithms presented in this work represent the first general-purpose mapping
of large numbers of topologically heterogeneous neural networks to many-core architectures.
Moreover, the achieved feedforward evaluation speedup is of particular importance to the

144

field of neuroevolution. The methods described here enable the networks evolved via neuroevolution to harness the compute capabilities of modern many-core architectures. Evolutionary algorithms are easily distributable across many compute nodes (for example, placing
a population of neural networks on each node), and the algorithms presented here enable the
evaluation of tens-to-hundreds of thousands of structurally diverse neural networks to now
be o✏oaded to many-core accelerators available on these nodes. As the ability to harness
distributed heterogeneous computing is a primary feature of modern machine learning frameworks, this work represents a step forward in enabling neuroevolution to better compete in
this space.

145

Chapter 5
Summary & Outlook
Over the past two decades, there has been an impressive number of experimental programs
studying Gamow-Teller distributions in pf -shell nuclei. Importantly, it has been found that
the nuclear shell model appears to be able to describe the measurements in this region particularly well [13, 90]. This is a significant experimental, theoretical, and computational
achievement, as it implies that allowed weak reactions for most iron-group nuclei near stability can be reliably estimated. As many astrophysical phenomenon rely on weak rates for
these nuclei, significant constraints on the nuclear physics can now be placed, and other
more significant uncertainties pursued. However, weak interaction rates for nuclei beyond
the pf -shell, such as the neutron rich nuclei that have been shown to dominate the total
electron capture rate in core-collapse supernovae, as well as heavier nuclei that also play an
important role, remain poorly constrained. To date, weak interaction rates for these nuclei
are often estimated by simple single-particle methods, and in many cases, phenomenological
parameterizations which reproduce the data in the pf -shell but likely have orders of magnitude error for heavier and more neutron rich systems [31, 38, 64, 2, 88]. The impact that
such uncertainties have on a variety of astrophysical environments remains an important
open question in nuclear astrophysics.
As a logical continuation of the e↵orts of Cole et al. [13], where the authors systematically compared the inferred electron capture rates from many measurements and theoretical

146

calculations for nuclei in the pf -shell, the sensitivity study e↵orts I led investigated how the
errors in this region (and more significantly, those outside the pf -shell) impact the early
phases of core-collapse supernovae. This e↵ort allowed us to understand on which nuclei
experimental and theoretical e↵orts should focus. Furthermore, it is continuing to lend insight on how significantly these nuclei determine key characteristics of the events, such as
the neutrino emission signal at core bounce, and the gravitational wave signal that would be
emitted from a rotating star undergoing core-collapse [2, 3].
Ultimately, this work motivates a systematic study similar to that performed by Cole
et al. [13] but for neutron-rich nuclei in the sdg-shell. As such an investigation relies on
the availability of measurements and theoretical calculations for these nuclei, future e↵orts
should focus in this region. Moving to heavier neutron rich nuclei, however, provides a unique
experimental challenge, as there currently is no established method for studying + GamowTeller distributions for radioactive nuclei. It is therefore of critical importance to develop
new techniques that take advantage of inverse kinematics, where the investigated radioactive
nuclei are formed as secondary beams at fast-fragmentation facilities. A promising method is
the (d, 2 He) charge-exchange reaction, wherein a rare isotope beam is produced and impinged
on an active target time projection chamber (TPC) filled with deuterium gas. In such a
measurement the time projection chamber would be utilized to track the position and energy
of the recoiling protons (which are emitted from the 2 He system). As the (d, 2 He) reaction will
induce + transitions in the neutron-rich rare isotope beam, these astrophysically important
nuclei would be accessible for the first time. Similarly, neutron rich nuclei in the sdg-shell,
as well as the very neutron rich systems on the border of the pf and sdg-shells, represent a
significant theoretical challenge due to the large many-body configuration space.
Moreover, forbidden transitions (non zero orbital angular momentum transfer) also play
147

an important role in astrophysical environments, specifically in the case of core-collapse
supernovae where the electron chemical potential is large enough that electrons can induce non-negligible rates for these transitions [38]. Unfortunately, configuration-interaction
shell model methods for estimating these rates are not presently available, because the extremely large model spaces. Popular alternatives such as quasi-random phase approximations
(QRPA) are able to estimate rates for forbidden transitions in large sets of nuclei, however
there are also no measurements available from which such estimates can be evaluated. Furthermore, QRPA estimates have been shown to reproduce the data for allowed Gamow-Teller
transitions only poorly [13]. Because of the astrophysical importance of forbidden weak interactions in nuclei, significant experimental and theoretical investments should be made in
this area.
Computationally, the weak interaction rate library that has been produced as part of this
work has received several updates since its implementation described in chapter 2. As it is the
first comprehensive set of weak rates across many di↵erent nuclear species, its incorporation
into the simulation of other astrophysical phenomenon–where weak interactions on nuclei
are suspected to play an important role–presents a unique opportunity to learn more about
the impact of the weak interaction in nuclear astrophysics.
In chapter 3, a new method for extracting Gamow-Teller transition strengths in the
inelastic neutral-current channel was presented. This work runs parallel to the above chargedcurrent weak interaction investigations, as it allows for the indirect constraint of the neutralcurrent weak response of nuclei which is also of importance in astrophysics. The (6 Li,6 Li0 + )
reaction has been established as a probe of inelastic isovector-spin transfer excitations in
nuclei. Furthermore, the Gamow-Teller unit cross section for 6 Li inelastic scattering was
extracted and agrees well with estimates from charge-exchange and theoretical calculations.
148

While the method has limited reach due to being experimentally challenging, it enables
the direct extraction of inelastic Gamow-Teller transition strength for light nuclei. As the
Gamow-Teller transition strength is the primary ingredient to inelastic neutrino-nucleus
scattering cross sections, the (6 Li,6 Li0 + ) reaction provides a novel indirect technique that
can supplement, and in some cases replace, the need for multi-year direct measurements with
neutrino beams [159].
In the final chapter of this work, the development and performance of a newly established
technique in artificial intelligence is described. While wholly separate from the physics previously discussed, these algorithmic advancements seek to enable neuroevolution to take
advantage of modern heterogeneous compute architectures, so that scientists with HPC centers at their disposal may better employ machine learning in their respective domains. The
principal focus of the algorithm optimizations described in chapter 4 were on improving the
feed-forward evaluation time when tens-to-hundreds of thousands of heterogeneous neural
networks are evaluated concurrently. The evolution of these neural networks through hyperparameter and structural space presents a novel divergence problem when mapping the
evaluation of these networks to many-core architectures. Because the evolution of neural
network populations o✏oads the model choice and refinement process to an evolutionary
search algorithm, neuroevolution increases the accessibility of machine learning to novice
users, and thus motivates the need for a solution to the above described divergence problem.
The algorithms described in chapter 4 provide a new evaluation paradigm which enables
the processing of many heterogeneous neural networks concurrently on many-core architectures. By pre-sorting the connections into evaluatable sets, and parallelizing over the
network’s directed acyclic dependency graphs, nearly all branch divergence is eliminated.
Principally, this is because the relationships between operations in network evaluation can
149

be ordered so that heterogeneity in network structure is ignored, and that at each step,
SIMD parallelism is maximized. Thus, these methods represent a general-purpose mapping
of large numbers of structurally diverse neural networks, including recurrent neural networks,
to many-core architectures. As the ability to harness distributed heterogeneous computing is
a primary feature of modern machine-learning frameworks, these algorithms represent a few
of the needed steps toward enabling neuroevolution to lower the machine learning barrier for
use in scientific endeavors. Moving forward, the application of this DAG-based method in
reverse-mode di↵erentiation for neural-network training would be the next logical step. Furthermore, as evolutionary algorithms employed in neuroevolution can be easily distributed
across many CPUs, a distributed implementation of neuroevolution which harnesses the heterogeneous network evaluation described in this work should be developed, as it will greatly
increase the search capability of neural-network evolution.

150

BIBLIOGRAPHY

[1] M. R. Mumpower, R. Surman, D.-L. Fang, M. Beard, P. Möller, T. Kawano, and
A. Aprahamian, Physical Review C 92, 035807 (2015).
[2] C. Sullivan, E. OConnor, R. G. T. Zegers, T. Grubb, and S. M. Austin, The Astrophysical Journal 816, 44 (2016).
[3] S. Richers, C. D. Ott, E. Abdikamalov, E. O’Connor, and C. Sullivan, Physical Review
D 95, 063019 (2017).
[4] G. Carleo and M. Troyer, Science 355, 602 (2017).
[5] J. Carrasquilla and R. G. Melko, Nature Physics 13, 431 (2017).
[6] A. Waltar, in Proceedings of the Americas Nuclear Energy Symposium (2004).
[7] K. Langanke and G. Martı́nez-Pinedo, Reviews of Modern Physics 75, 819 (2003).
[8] E. Fermi, Zeitschrift fr Physik 88, 161 (1934).
[9] B. Pontecorvo, Soviet Journal of Experimental and Theoretical Physics 9, 1148 (1959),
[Zh. Eksp. Teor. Fiz.36,1615(1959)].
[10] D. Z. Freedman, Physical Review D 9, 1389 (1974).
[11] T. J. Mazurek, Astrophysics and Space Science 35, 117 (1975).
[12] K. Sato, Progress of Theoretical Physics 54, 1325 (1975).
[13] A. L. Cole, T. S. Anderson, R. G. T. Zegers, S. M. Austin, B. a. Brown, L. Valdez,
S. Gupta, G. W. Hitt, and O. Fawwaz, Physical Review C 86, 015809 (2012).
[14] M. Georgiopoulos, R. F. DeMara, A. J. Gonzalez, A. S. Wu, M. Mollaghasemi, E. Gelenbe, M. Kysilka, J. Secretan, C. A. Sharma, and A. J. Alnsour, IEEE Transactions
on Education 52, 503 (2009).

152

[15] P. Anderson, J. McGu↵ee, and D. Uminsky, in Proceedings of the 45th ACM Technical
Symposium on Computer Science Education, SIGCSE ’14 (ACM, New York, NY, USA,
2014) pp. 705–706.
[16] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau,
M. Brucher, M. Perrot, and E. Duchesnay, The Journal of Machine Learning Research
12, 2825 (2011).
[17] M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat,
G. Irving, M. Isard, M. Kudlur, J. Levenberg, R. Monga, S. Moore, D. G. Murray,
B. Steiner, P. Tucker, V. Vasudevan, P. Warden, M. Wicke, Y. Yu, and X. Zheng, in
Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation, OSDI’16 (USENIX Association, Berkeley, CA, USA, 2016) pp. 265–283.
[18] C. Thornton, F. Hutter, H. H. Hoos, and K. Leyton-Brown, in Proceedings of the 19th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,
KDD ’13 (ACM, New York, NY, USA, 2013) pp. 847–855.
[19] M. Claesen and B. D. Moor, Computing Research Repository abs/1502.02127 (2015),
arXiv:1502.02127 .
[20] G. E. Hinton, “A Practical Guide to Training Restricted Boltzmann Machines,” in
Neural Networks: Tricks of the Trade: Second Edition, edited by G. Montavon, G. B.
Orr, and K.-R. Müller (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012) pp.
599–619.
[21] M. Claesen, J. Simm, D. Popovic, Y. Moreau, and B. D. Moor, Computing Research
Repository abs/1412.1114 (2014), arXiv:1412.1114 .
[22] S. Viaene, IT Professional 15, 12 (2013).
[23] M. Feurer, A. Klein, K. Eggensperger, J. Springenberg, M. Blum, and F. Hutter, in
Advances in Neural Information Processing Systems 28 , edited by C. Cortes, N. D.
Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett (Curran Associates, Inc., 2015)
pp. 2962–2970.
[24] R. Miikkulainen, J. Z. Liang, E. Meyerson, A. Rawal, D. Fink, O. Francon, B. Raju,
H. Shahrzad, A. Navruzyan, N. Du↵y, and B. Hodjat, Computing Research Repository
abs/1703.00548 (2017), arXiv:1703.00548 .

153

[25] C. Fernando, D. Banarse, M. Reynolds, F. Besse, D. Pfau, M. Jaderberg, M. Lanctot, and D. Wierstra, Computing Research Repository abs/1606.02580 (2016),
arXiv:1606.02580 .
[26] X. Yao, Proceedings of the IEEE 87, 1423 (1999).
[27] A. Heger, K. Langanke, G. Martı́nez-Pinedo, and S. E. Woosley, Physical Review
Letters 86, 1678 (2001).
[28] G. Martı́nez-Pinedo, Y. H. Lam, K. Langanke, R. G. T. Zegers, and C. Sullivan,
Physical Review C 89, 045806 (2014).
[29] K. Iwamoto, F. Brachwitz, K. Nomoto, N. Kishimoto, H. Umeda, W. R. Hix, and
F.-K. Thielemann, The Astrophysical Journal Supplement Series 125, 439 (1999).
[30] F. Brachwitz, D. J. Dean, W. R. Hix, K. Iwamoto, K. Langanke, G. Martnez-Pinedo,
K. Nomoto, M. R. Strayer, F.-K. Thielemann, and H. Umeda, The Astrophysical
Journal 536, 934 (2000).
[31] W. R. Hix, O. E. B. Messer, A. Mezzacappa, M. Liebendoerfer, J. Sampaio, K. Langanke, D. J. Dean, and G. Martı́nez-Pinedo, Physical Review Letters 91, 201102
(2003).
[32] H.-T. Janka, K. Langanke, A. Marek, G. Martnez-Pinedo, and B. Mller, Physics
Reports 442, 38 (2007), the Hans Bethe Centennial Volume 1906-2006.
[33] S. Gupta, E. F. Brown, H. Schatz, P. Möller, and K.-L. Kratz, The Astrophysical
Journal 662, 1188 (2007).
[34] H. Schatz, S. Gupta, P. Möller, M. Beard, E. F. Brown, A. T. Deibel, L. R. Gasques,
W. R. Hix, L. Keek, R. Lau, A. W. Steiner, and M. Wiescher, Nature 505, 62 (2014).
[35] S. Goriely, A. Bauswein, O. Just, E. Pllumbi, and H.-T. Janka, Monthly Notices of
the Royal Astronomical Society 452, 3894 (2015).
[36] K. Langanke and G. Martı́nez-Pinedo, Nuclear Physics A 673, 481 (2000).
[37] T. Oda, M. Hino, K. Muto, M. Takahara, and K. Sato, Atomic Data and Nuclear
Data Tables 56, 231 (1994).

154

[38] K. Langanke, G. Martı́nez-Pinedo, J. M. Sampaio, D. J. Dean, W. R. Hix, O. E. B.
Messer, A. Mezzacappa, M. Liebendörfer, H.-T. Janka, and M. Rampp, Physical
Review Letters 90, 241102 (2003).
[39] N. Paar, G. Colò, E. Khan, and D. Vretenar, Physical Review C 80, 055801 (2009).
[40] P. Möller and J. Randrup, Nuclear Physics A 514, 1 (1990).
[41] A. A. Dzhioev, A. I. Vdovin, V. Y. Ponomarev, J. Wambach, K. Langanke, and
G. Martı́nez-Pinedo, Physical Review C 81, 015804 (2010).
[42] Y. F. Niu, N. Paar, D. Vretenar, and J. Meng, Physical Review C 83, 045807 (2011).
[43] J.-U. Nabi and H. V. Klapdor-Kleingrothaus, Atomic Data and Nuclear Data Tables
88, 237 (2004).
[44] M. Scott, Y. Shimbara, S. M. Austin, D. Bazin, B. A. Brown, J. M. Deaven, Y. Fujita, C. J. Guess, S. Gupta, G. W. Hitt, D. Koeppe, R. Meharchand, M. Nagashima,
G. Perdikakis, A. Prinke, M. Sasano, C. Sullivan, L. Valdez, and R. G. T. Zegers,
Physical Review C 90, 025801 (2014).
[45] S. Noji, R. Zegers, S. M. Austin, T. Baugher, D. Bazin, B. Brown, C. Campbell,
A. Cole, H. Doster, A. Gade, C. Guess, S. Gupta, G. Hitt, C. Langer, S. Lipschutz,
E. Lunderberg, R. Meharchand, Z. Meisel, G. Perdikakis, J. Pereira, F. Recchia,
H. Schatz, M. Scott, S. Stroberg, C. Sullivan, L. Valdez, C. Walz, D. Weisshaar,
S. Williams, and K. Wimmer, Physical Review Letters 112, 252501 (2014).
[46] A. Poves, J. Snchez-Solano, E. Caurier, and F. Nowacki, Nuclear Physics A 694, 157
(2001).
[47] M. Honma, T. Otsuka, B. Brown, and T. Mizusaki, The European Physical Journal
A - Hadrons and Nuclei 25, 499 (2005).
[48] E. O’Connor and C. D. Ott, Classical and Quantum Gravity 27, 114103 (2010).
[49] A. Dubey, K. Antypas, M. K. Ganapathy, L. B. Reid, K. Riley, D. Sheeler, A. Siegel,
and K. Weide, Parallel Computing 35, 512 (2009).
[50] B. Fryxell, K. Olson, P. Ricker, F. X. Timmes, M. Zingale, D. Q. Lamb, P. MacNeice,
R. Rosner, J. W. Truran, and H. Tufo, The Astrophysical Journal Supplement Series
131, 273 (2000).
155

[51] H. Dimmelmeier, J. Novak, J. A. Font, J. M. Ibáñez, and E. Müller, Physical Review
D 71, 064023 (2005).
[52] Dimmelmeier, H., Font, J. A., and Mller, E., A&A 393, 523 (2002).
[53] C. Iliadis, Nuclear Physics of Stars, by Christian Iliadis. ISBN 978-3-527-406029. Published by Wiley-VCH Verlag, Wenheim, Germany, 2007. (Wiley-VCH Verlag,
2007).
[54] K. Langanke and G. Martnez-Pinedo, Nuclear Physics A 928, 305 (2014), special
Issue Dedicated to the Memory of Gerald E Brown (1926-2013).
[55] G. Martı́nez-Pinedo, K. Langanke, and D. J. Dean, The Astrophysical Journal Supplement Series 126, 493 (2000).
[56] H. Bethe, G. Brown, J. Applegate, and J. Lattimer, Nuclear Physics A 324, 487
(1979).
[57] G. M. Fuller, W. A. Fowler, and M. J. Newman, The Astrophysical Journal 252, 715
(1982).
[58] A. Juodagalvis, K. Langanke, W. Hix, G. Martı́nez-Pinedo, and J. Sampaio, Nuclear
Physics A 848, 454 (2010).
[59] E. O’Connor, The Astrophysical Journal Supplement Series 219, 24 (2015).
[60] K. Langanke, E. Kolbe, and D. J. Dean, Physical Review C 63, 032801 (2001).
[61] S. W. Bruenn, The Astrophysical Journal Supplement Series 58, 771 (1985).
[62] G. M. Fuller, W. A. Fowler, and M. J. Newman, The Astrophysical Journal 293, 1
(1985).
[63] M. Ste↵en, Astronomy & Astrophysics 239, 443 (1990).
[64] A. Juodagalvis, J. M. Sampaio, K. Langanke, and W. R. Hix, Journal of Physics G:
Nuclear and Particle Physics 35, 014031 (2008).
[65] A. Burrows, S. Reddy, and T. a. Thompson, Nuclear Physics A 777, 356 (2006).

156

[66] C. J. Horowitz, Physical Review D 65, 043001 (2002).
[67] C. J. Horowitz, Physical Review D 55, 4577 (1997).
[68] K. Langanke, G. Martı́nez-Pinedo, and J. Sampaio, Physical Review C 64, 055801
(2001).
[69] M. Hempel and J. Scha↵ner-Bielich, Nuclear Physics A 837, 210 (2010).
[70] A. W. Steiner, M. Hempel, and T. Fischer, The Astrophysical Journal 774, 17 (2013).
[71] S. Typel, G. Röpke, T. Klähn, D. Blaschke, and H. H. Wolter, Physical Review C 81,
015803 (2010).
[72] H. Toki, D. Hirata, Y. Sugahara, K. Sumiyoshi, and I. Tanihata, Nuclear Physics A
588, c357 (1995).
[73] T. Fischer, M. Hempel, I. Sagert, Y. Suwa, and J. Scha↵ner-Bielich, The European
Physical Journal A 50, 46 (2014).
[74] P. Möller, J. Nix, W. Myers, and W. Swiatecki, Atomic Data and Nuclear Data Tables
59, 185 (1995).
[75] P. Möller, J. Nix, and K.-L. Kratz, Atomic Data and Nuclear Data Tables 66, 131
(1997).
[76] L. Geng, H. Toki, and J. Meng, Progress of Theoretical Physics 113, 785 (2005).
[77] G. Audi, A. Wapstra, and C. Thibault, Nuclear Physics A 729, 337 (2003).
[78] S. E. Woosley and T. A. Weaver, The Astrophysical Journal Supplement Series 101,
181 (1995).
[79] S. Woosley and A. Heger, Physics Reports 442, 269 (2007).
[80] E. O’Connor and C. D. Ott, The Astrophysical Journal 730, 70 (2011).
[81] M. Liebendorfer, The Astrophysical Journal 633, 1042 (2005).
[82] C. F. J. Wu, The Annals of Statistics 14, 1261 (1986).
157

[83] E. O’Connor and C. D. Ott, The Astrophysical Journal 762, 126 (2013).
[84] T. Suzuki, H. Toki, and K. Nomoto, The Astrophysical Journal 817, 163 (2016).
[85] M. Honma, T. Otsuka, T. Mizusaki, M. Hjorth-Jensen, and B. A. Brown, Journal of
Physics: Conference Series 20, 7 (2005).
[86] J. Pruet and G. M. Fuller, The Astrophysical Journal Supplement Series 149, 189
(2003).
[87] A. R. Raduta, F. Gulminelli, and M. Oertel, Physical Review C 95, 025805 (2017).
[88] R. Titus, C. Sullivan, R. G. T. Zegers, B. A. Brown, and B. Gao, Journal of Physics
G: Nuclear and Particle Physics (2017).
[89] K. Scholberg, Annual Review of Nuclear and Particle Science 62, 81 (2012).
[90] K. Langanke and G. Martnez-Pinedo, Nuclear Physics A 928, 305 (2014), special
Issue Dedicated to the Memory of Gerald E Brown (1926-2013).
[91] W. C. Haxton, Physical Review Letters 60, 1999 (1988).
[92] W. R. Hix, A. Mezzacappa, O. E. B. Messer, and S. W. Bruenn, Journal of Physics
G: Nuclear and Particle Physics 29, 2523 (2003).
[93] B. Bodmann, N. Booth, F. Burtak, A. Dodd, G. Drexlin, V. Eberhard, K. Eitel, J. Edgington, E. Finckh, H. Gemmeke, G. Giorginis, A. Glombik, T. Gorringe,
W. Grandegger, T. Hanika, J. Hö l, M. Kleifges, J. Kleinfeller, W. Kretschmer, A. Malik, R. Maschuw, R. Meyer, P. Plischke, J. Rapp, F. Raupp, F. Schilling, B. Seligmann,
J. Wochele, J. Wolf, S. Wölfle, and B. Zeitnitz, Physics Letters B 280, 198 (1992).
[94] R. Imlay, Nuclear Physics A A629, 531 (1998).
[95] A. Tamii, Y. Fujita, H. Matsubara, T. Adachi, J. Carter, M. Dozono, H. Fujita, K. Fujita, H. Hashimoto, K. Hatanaka, T. Itahashi, M. Itoh, T. Kawabata, K. Nakanishi, S. Ninomiya, A. Perez-Cerdan, L. Popescu, B. Rubio, T. Saito, H. Sakaguchi,
Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. Smit, Y. Tameshige, M. Yosoi,
and J. Zenhiro, Nuclear Instruments and Methods in Physics Research Section A:
Accelerators, Spectrometers, Detectors and Associated Equipment 605, 326 (2009).

158

[96] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertulani, J. Carter, M. Dozono, H. Fujita, K. Fujita, K. Hatanaka, D. Ishikawa, M. Itoh,
T. Kawabata, Y. Kalmykov, A. M. Krumbholz, E. Litvinova, H. Matsubara, K. Nakanishi, R. Neveling, H. Okamura, H. J. Ong, B. Özel-Tashenov, V. Y. Ponomarev,
A. Richter, B. Rubio, H. Sakaguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu,
F. D. Smit, T. Suzuki, Y. Tameshige, J. Wambach, R. Yamada, M. Yosoi, and J. Zenihiro, Physical Review Letters 107, 062502 (2011).
[97] I. Poltoratska, P. von Neumann-Cosel, A. Tamii, T. Adachi, C. A. Bertulani, J. Carter,
M. Dozono, H. Fujita, K. Fujita, Y. Fujita, K. Hatanaka, M. Itoh, T. Kawabata,
Y. Kalmykov, A. M. Krumbholz, E. Litvinova, H. Matsubara, K. Nakanishi, R. Neveling, H. Okamura, H. J. Ong, B. Özel-Tashenov, V. Y. Ponomarev, A. Richter, B. Rubio, H. Sakaguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. D. Smit,
T. Suzuki, Y. Tameshige, J. Wambach, M. Yosoi, and J. Zenihiro, Physical Review C
85, 041304 (2012).
[98] D. I. Sober, B. C. Metsch, W. Knüpfer, G. Eulenberg, G. Küchler, A. Richter,
E. Spamer, and W. Ste↵en, Physical Review C 31, 2054 (1985).
[99] O. Burda, N. Botha, J. Carter, R. W. Fearick, S. V. Förtsch, C. Fransen, H. Fujita,
J. D. Holt, M. Kuhar, A. Lenhardt, P. von Neumann-Cosel, R. Neveling, N. Pietralla,
V. Y. Ponomarev, A. Richter, O. Scholten, E. Sideras-Haddad, F. D. Smit, and
J. Wambach, Physical Review Letters 99, 092503 (2007).
[100] K. Langanke, G. Martı́nez-Pinedo, P. von Neumann-Cosel, and A. Richter, Physical
Review Letters 93, 202501 (2004).
[101] T. W. Donnelly and R. D. Peccei, Physics Reports 50, 1 (1979).
[102] S. M. Austin, N. Anantaraman, and J. S. Winfield, Canadian Journal of Physics 65,
609 (1987).
[103] D. Tilley, C. Cheves, J. Godwin, G. Hale, H. Hofmann, J. Kelley, C. Sheu, and
H. Weller, Nuclear Physics A 708, 3 (2002).
[104] M. A. Franey and W. G. Love, Physical Review C 31, 488 (1985).
[105] M. Fujiwara, H. Akimune, I. Daito, H. Fujimura, Y. Fujita, K. Hatanaka, H. Ikegami,
I. Katayama, K. Nagayama, N. Matsuoka, S. Morinobu, T. Noro, M. Yoshimura,
H. Sakaguchi, Y. Sakemi, A. Tamii, and M. Yosoi, Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated
Equipment 422, 484 (1999).
159

[106] H. Matsubara, Isoscalar and isovector spin-M 1 transitions from the even-even, N = Z
nuclei across the sd-shell region, Ph.D. thesis, Osaka University (2010).
[107] A. Tamii, H. Sakaguchi, H. Takeda, M. Yosoi, H. Akimune, M. Fujiwara, H. Ogata,
M. Tanaka, and H. Togawa, IEEE Transactions on Nuclear Science 43, 2488 (1996).
[108] M. Cromaz, V. Riot, P. Fallon, S. Gros, B. Holmes, I. Lee, A. Macchiavelli, C. Vu,
H. Yaver, and S. Zimmermann, Nuclear Instruments and Methods in Physics Research
Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 597, 233
(2008).
[109] J. T. Anderson, M. Albers, M. Alcorta, C. Campbell, M. P. Carpenter, C. J. Chiara,
M. Cromaz, H. M. David, D. Doering, D. T. Doherty, C. R. Ho↵man, R. V. F. Janssens,
J. Joseph, T. L. Khoo, A. Kreps, T. Lauritsen, I. Y. Lee, C. Lionberger, C. J. Lister,
T. Madden, M. B. Oberling, A. M. Rogers, D. Seweryniak, P. Wilt, S. Zhu, and S. Zimmermann, in 2012 IEEE Nuclear Science Symposium and Medical Imaging Conference
Record (NSS/MIC) (IEEE, 2012) pp. 1536–1540.
[110] A. Tamii, Polarization transfer observables for proton inelastic scattering from 12 C at
zero degrees, Ph.D. thesis, Kyoto University (1999).
[111] P. Bender, E. Lunderberg, C. Sullivan, et al., “CAGRA-GrandRaiden/GRUTinizer:
1st Post RCNP CAGRA+GR Experimental Campaign Release,” (2017).
[112] E. Lunderberg, Subbarrier coulomb excitation at REA3, commissioning of the SeGAJanus experimental setup, Ph.D. thesis, Michigan State University (2017).
[113] M. P. C. S. Zhu, private communication (2017).
[114] M. A. Moinester, Canadian Journal of Physics 65, 660 (1987).
[115] J. Cook and J. Carr, Computer program FOLD. Florida State University (unpublished); based on F. Petrovich and D. Stanley, Nuclear Physics A 275:487, 1977; modified as described in J. Cook et al., Physical Review C 30:1538, 1984; R.G.T. Zegers
(2005), unpublished.
[116] R. G. T. Zegers, T. Adachi, H. Akimune, S. M. Austin, A. M. van den Berg, B. A.
Brown, Y. Fujita, M. Fujiwara, S. Galès, C. J. Guess, M. N. Harakeh, H. Hashimoto,
K. Hatanaka, R. Hayami, G. W. Hitt, M. E. Howard, M. Itoh, T. Kawabata, K. Kawase,
M. Kinoshita, M. Matsubara, K. Nakanishi, S. Nakayama, S. Okumura, T. Ohta,
Y. Sakemi, Y. Shimbara, Y. Shimizu, C. Scholl, C. Simenel, Y. Tameshige, A. Tamii,
M. Uchida, T. Yamagata, and M. Yosoi, Physical Review Letters 99, 202501 (2007).
160

[117] G. Perdikakis, R. G. T. Zegers, S. M. Austin, D. Bazin, C. Caesar, J. M. Deaven,
A. Gade, D. Galaviz, G. F. Grinyer, C. J. Guess, C. Herlitzius, G. W. Hitt, M. E.
Howard, R. Meharchand, S. Noji, H. Sakai, Y. Shimbara, E. E. Smith, and C. Tur,
Physical Review C 83, 054614 (2011).
[118] M. Scott, R. G. T. Zegers, R. Almus, S. M. Austin, D. Bazin, B. A. Brown, C. Campbell, A. Gade, M. Bowry, S. Galès, U. Garg, M. N. Harakeh, E. Kwan, C. Langer,
C. Loelius, S. Lipschutz, E. Litvinova, E. Lunderberg, C. Morse, S. Noji, G. Perdikakis,
T. Redpath, C. Robin, H. Sakai, Y. Sasamoto, M. Sasano, C. Sullivan, J. A. Tostevin,
T. Uesaka, and D. Weisshaar, Physical Review Letters 118, 172501 (2017).
[119] B. A. Brown et al., (2004), NSCL report MSUCL-1289 (unpublished).
[120] S. Cohen and D. Kurath, Nuclear Physics 73, 1 (1965).
[121] E. K. Warburton and B. A. Brown, Physical Review C 46, 923 (1992).
[122] B. Alex Brown, Physical Review C 58, 220 (1998).
[123] K. Schwarz, C. Samanta, M. Fujiwara, H. Rebel, R. De Leo, N. Matsuoka, H. Utsunomiya, H. Akimune, I. Daito, H. Fujimura, F. Ihara, K. Ishibashi, Y. Maeda,
T. Yamanaka, H. Yoshida, A. Okihana, T. Yoshimura, P. van Aarle, W. Uijen, M. Ito,
and Y. Sakuragi, The European Physical Journal A 7, 367 (2000).
[124] J. Raynal, Notes on ECIS97 (unpublished).
[125] G. Perdikakis, R. G. T. Zegers, S. M. Austin, D. Bazin, C. Caesar, J. M. Deaven,
A. Gade, D. Galaviz, G. F. Grinyer, C. J. Guess, C. Herlitzius, G. W. Hitt, M. E.
Howard, R. Meharchand, S. Noji, H. Sakai, Y. Shimbara, E. E. Smith, and C. Tur,
Physical Review C 83, 054614 (2011).
[126] B. Bodmann, N. Booth, F. Burtak, A. Dodd, G. Drexlin, V. Eberhard, J. Edgington, E. Finckh, H. Gemmeke, G. Giorginis, A. Glombik, T. Gorringe, W. Grandegger, T. Hanika, M. Kleifges, J. Kleinfeller, W. Kretschmer, A. Malik, R. Maschuw,
R. Meyer, P. Plischke, F. Raupp, F. Schilling, B. Seligmann, J. Wochele, J. Wolf,
S. Wölfle, and B. Zeitnitz, Physics Letters B 267, 321 (1991).
[127] T. N. Taddeucci, L. Alamos, T. A. Carey, C. Gaarde, J. Larsen, D. Horen, O. R.
National, O. Ridge, J. Rapaport, and E. Sugarbaker, 469, 125 (1987).
[128] H. Matsubara, A. Tamii, H. Nakada, T. Adachi, J. Carter, M. Dozono, H. Fujita, K. Fujita, Y. Fujita, K. Hatanaka, W. Horiuchi, M. Itoh, T. Kawabata,
161

S. Kuroita, Y. Maeda, P. Navrátil, P. von Neumann-Cosel, R. Neveling, H. Okamura, L. Popescu, I. Poltoratska, A. Richter, B. Rubio, H. Sakaguchi, S. Sakaguchi,
Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. D. Smit, K. Suda, Y. Tameshige,
H. Tokieda, Y. Yamada, M. Yosoi, and J. Zenihiro, Physical Review Letters 115,
102501 (2015).
[129] T. N. Taddeucci, C. A. Goulding, and T. A. Carey, Nuclear Physics A469, 125 (1987).
[130] M. N. Harakeh and A. van der Woude, Giant Resonances: Fundamental HighFrequency Modes of Nuclear Excitations (Oxford University Press, New York, 2001).
[131] J. Fridmann et al., Physical Review C 74, 034313 (2006).
[132] S. van der Werf, “Normod,” Unpublished.
[133] N. Anantaraman et al., Physical Review C 44, 398 (1991).
[134] A. Nadasen et al., Physical Review C 39, 536 (1989).
[135] H. Laurent, S. Gals, D. Beaumel, G. Crawley, J. Finck, S. Fortier, J. Maison, C. Massolo, D. Mercer, J. Winfield, and G. Yoo, Nuclear Physics A 569, 297 (1994).
[136] S. Paschalis, I. Lee, A. Macchiavelli, C. Campbell, M. Cromaz, S. Gros, J. Pavan,
J. Qian, R. Clark, H. Crawford, D. Doering, P. Fallon, C. Lionberger, T. Loew,
M. Petri, T. Stezelberger, S. Zimmermann, D. Radford, K. Lagergren, D. Weisshaar,
R. Winkler, T. Glasmacher, J. Anderson, and C. Beausang, Nuclear Instruments and
Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment 709, 44 (2013).
[137] T. Lauritsen, A. Korichi, S. Zhu, A. Wilson, D. Weisshaar, J. Dudouet, A. Ayangeakaa,
M. Carpenter, C. Campbell, E. Clment, H. Crawford, M. Cromaz, P. Fallon, J. Greene,
R. Janssens, T. Khoo, N. Lalovi, I. Lee, A. Macchiavelli, R. Perez-Vidal, S. Pietri,
D. Radford, D. Ralet, L. Riley, D. Seweryniak, and O. Stezowski, Nuclear Instruments
and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors
and Associated Equipment 836, 46 (2016).
[138] D. Weisshaar, D. Bazin, P. Bender, C. Campbell, F. Recchia, V. Bader, T. Baugher,
J. Belarge, M. Carpenter, H. Crawford, M. Cromaz, B. Elman, P. Fallon, A. Forney, A. Gade, J. Harker, N. Kobayashi, C. Langer, T. Lauritsen, I. Lee, A. Lemasson,
B. Longfellow, E. Lunderberg, A. Macchiavelli, K. Miki, S. Momiyama, S. Noji, D. Radford, M. Scott, J. Sethi, S. Stroberg, C. Sullivan, R. Titus, A. Wiens, S. Williams,
162

K. Wimmer, and S. Zhu, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 847, 187
(2017).
[139] U. Garg, private communication (2017).
[140] W. S. McCulloch and W. Pitts, The Bulletin of Mathematical Biophysics 5, 115 (1943).
[141] B. Cheng and D. M. Titterington, Statistical Science 9, 2 (1994).
[142] A. J. Turner and J. F. Miller, Evolutionary Intelligence 7, 135 (2014).
[143] S. M. Carroll and B. W. Dickinson, International 1989 Joint Conference on Neural
Networks , 607 (1989).
[144] J. Hu, J. Zhang, C. Zhang, and J. Wang, Neurocomputing 171, 63 (2016).
[145] K. O. Stanley and R. Miikkulainen, Evolutionary Computation 10, 99 (2002).
[146] C. Peterson, Nuclear Physics News 2, 14 (1992).
[147] P. Baldi, P. Sadowski, and D. Whiteson, Nature Communications 5, 4308 (2014).
[148] A. Aurisano, A. Radovic, D. Rocco, A. Himmel, M. Messier, E. Niner, G. Pawloski,
F. Psihas, A. Sousa, and P. Vahle, Journal of Instrumentation 11, P09001 (2016).
[149] T. Aaltonen, J. Adelman, T. Akimoto, M. G. Albrow, B. Álvarez González, S. Amerio, D. Amidei, A. Anastassov, A. Annovi, J. Antos, G. Apollinari, A. Apresyan,
T. Arisawa, A. Artikov, W. Ashmanskas, A. Attal, A. Aurisano, F. Azfar, P. Azzurri, W. Badgett, A. Barbaro-Galtieri, V. E. Barnes, B. A. Barnett, V. Bartsch,
G. Bauer, P.-H. Beauchemin, F. Bedeschi, P. Bednar, D. Beecher, S. Behari, G. Bellettini, J. Bellinger, D. Benjamin, A. Beretvas, J. Beringer, A. Bhatti, M. Binkley,
D. Bisello, I. Bizjak, R. E. Blair, C. Blocker, B. Blumenfeld, A. Bocci, A. Bodek,
V. Boisvert, G. Bolla, D. Bortoletto, J. Boudreau, A. Boveia, B. Brau, A. Bridgeman, L. Brigliadori, C. Bromberg, E. Brubaker, J. Budagov, H. S. Budd, S. Budd,
K. Burkett, G. Busetto, P. Bussey, A. Buzatu, K. L. Byrum, S. Cabrera, C. Calancha, M. Campanelli, M. Campbell, F. Canelli, A. Canepa, D. Carlsmith, R. Carosi,
S. Carrillo, S. Carron, B. Casal, M. Casarsa, A. Castro, P. Catastini, D. Cauz, V. Cavaliere, M. Cavalli-Sforza, A. Cerri, L. Cerrito, S. H. Chang, Y. C. Chen, M. Chertok, G. Chiarelli, G. Chlachidze, F. Chlebana, K. Cho, D. Chokheli, J. P. Chou,
G. Choudalakis, S. H. Chuang, K. Chung, W. H. Chung, Y. S. Chung, C. I. Ciobanu,
163

M. A. Ciocci, A. Clark, D. Clark, G. Compostella, M. E. Convery, J. Conway, K. Copic,
M. Cordelli, G. Cortiana, D. J. Cox, F. Crescioli, C. Cuenca Almenar, J. Cuevas,
R. Culbertson, J. C. Cully, D. Dagenhart, M. Datta, T. Davies, P. de Barbaro,
S. De Cecco, A. Deisher, G. De Lorenzo, M. Dell’Orso, C. Deluca, L. Demortier,
J. Deng, M. Deninno, P. F. Derwent, G. P. di Giovanni, C. Dionisi, B. Di Ruzza, J. R.
Dittmann, M. D’Onofrio, S. Donati, P. Dong, J. Donini, T. Dorigo, S. Dube, J. Efron,
A. Elagin, R. Erbacher, D. Errede, S. Errede, R. Eusebi, H. C. Fang, S. Farrington,
W. T. Fedorko, R. G. Feild, M. Feindt, J. P. Fernandez, C. Ferrazza, R. Field, G. Flanagan, R. Forrest, M. Franklin, J. C. Freeman, I. Furic, M. Gallinaro, J. Galyardt, F. Garberson, J. E. Garcia, A. F. Garfinkel, K. Genser, H. Gerberich, D. Gerdes, A. Gessler,
S. Giagu, V. Giakoumopoulou, P. Giannetti, K. Gibson, J. L. Gimmell, C. M. Ginsburg,
N. Giokaris, M. Giordani, P. Giromini, M. Giunta, G. Giurgiu, V. Glagolev, D. Glenzinski, M. Gold, N. Goldschmidt, A. Golossanov, G. Gomez, G. Gomez-Ceballos,
M. Goncharov, O. González, I. Gorelov, A. T. Goshaw, K. Goulianos, A. Gresele,
S. Grinstein, C. Grosso-Pilcher, R. C. Group, U. Grundler, J. Guimaraes da Costa,
Z. Gunay-Unalan, C. Haber, K. Hahn, S. R. Hahn, E. Halkiadakis, B.-Y. Han, J. Y.
Han, R. Handler, F. Happacher, K. Hara, D. Hare, M. Hare, S. Harper, R. F. Harr,
R. M. Harris, M. Hartz, K. Hatakeyama, J. Hauser, C. Hays, M. Heck, A. Heijboer,
B. Heinemann, J. Heinrich, C. Henderson, M. Herndon, J. Heuser, S. Hewamanage,
D. Hidas, C. S. Hill, D. Hirschbuehl, A. Hocker, S. Hou, M. Houlden, S.-C. Hsu, B. T.
Hu↵man, R. E. Hughes, U. Husemann, J. Huston, J. Incandela, G. Introzzi, M. Iori,
A. Ivanov, E. James, B. Jayatilaka, E. J. Jeon, M. K. Jha, S. Jindariani, W. Johnson,
M. Jones, K. K. Joo, S. Y. Jun, J. E. Jung, T. R. Junk, T. Kamon, D. Kar, P. E.
Karchin, Y. Kato, R. Kephart, J. Keung, V. Khotilovich, B. Kilminster, D. H. Kim,
H. S. Kim, J. E. Kim, M. J. Kim, S. B. Kim, S. H. Kim, Y. K. Kim, N. Kimura,
L. Kirsch, S. Klimenko, B. Knuteson, B. R. Ko, S. A. Koay, K. Kondo, D. J. Kong,
J. Konigsberg, A. Korytov, A. V. Kotwal, M. Kreps, J. Kroll, D. Krop, N. Krumnack,
M. Kruse, V. Krutelyov, T. Kubo, T. Kuhr, N. P. Kulkarni, M. Kurata, Y. Kusakabe,
S. Kwang, A. T. Laasanen, S. Lami, S. Lammel, M. Lancaster, R. L. Lander, K. Lannon, A. Lath, G. Latino, I. Lazzizzera, T. LeCompte, E. Lee, S. W. Lee, S. Leone,
J. D. Lewis, C. S. Lin, J. Linacre, M. Lindgren, E. Lipeles, A. Lister, D. O. Litvintsev,
C. Liu, T. Liu, N. S. Lockyer, A. Loginov, M. Loreti, L. Lovas, R.-S. Lu, D. Lucchesi, J. Lueck, C. Luci, P. Lujan, P. Lukens, G. Lungu, L. Lyons, J. Lys, R. Lysak,
E. Lytken, P. Mack, D. MacQueen, R. Madrak, K. Maeshima, K. Makhoul, T. Maki,
P. Maksimovic, S. Malde, S. Malik, G. Manca, A. Manousakis-Katsikakis, F. Margaroli,
C. Marino, C. P. Marino, A. Martin, V. Martin, M. Martı́nez, R. Martı́nez-Balları́n,
T. Maruyama, P. Mastrandrea, T. Masubuchi, M. E. Mattson, P. Mazzanti, K. S.
McFarland, P. McIntyre, R. McNulty, A. Mehta, P. Mehtala, A. Menzione, P. Merkel,
C. Mesropian, T. Miao, N. Miladinovic, R. Miller, C. Mills, M. Milnik, A. Mitra,
G. Mitselmakher, H. Miyake, N. Moggi, C. S. Moon, R. Moore, M. J. Morello, J. Morlok, P. Movilla Fernandez, J. Mülmenstädt, A. Mukherjee, T. Muller, R. Mumford,
P. Murat, M. Mussini, J. Nachtman, Y. Nagai, A. Nagano, J. Naganoma, K. Nakamura,
I. Nakano, A. Napier, V. Necula, C. Neu, M. S. Neubauer, J. Nielsen, L. Nodulman,
M. Norman, O. Norniella, E. Nurse, L. Oakes, S. H. Oh, Y. D. Oh, I. Oksuzian, T. Okusawa, R. Orava, K. Osterberg, S. Pagan Griso, C. Pagliarone, E. Palencia, V. Pa164

padimitriou, A. Papaikonomou, A. A. Paramonov, B. Parks, S. Pashapour, J. Patrick,
G. Pauletta, M. Paulini, C. Paus, D. E. Pellett, A. Penzo, T. J. Phillips, G. Piacentino,
E. Pianori, L. Pinera, K. Pitts, C. Plager, L. Pondrom, O. Poukhov, N. Pounder,
F. Prakoshyn, A. Pronko, J. Proudfoot, F. Ptohos, E. Pueschel, G. Punzi, J. Pursley,
J. Rademacker, A. Rahaman, V. Ramakrishnan, N. Ranjan, I. Redondo, B. Reisert,
V. Rekovic, P. Renton, M. Rescigno, S. Richter, F. Rimondi, L. Ristori, A. Robson,
T. Rodrigo, T. Rodriguez, E. Rogers, S. Rolli, R. Roser, M. Rossi, R. Rossin, P. Roy,
A. Ruiz, J. Russ, V. Rusu, H. Saarikko, A. Safonov, W. K. Sakumoto, O. Saltó,
L. Santi, S. Sarkar, L. Sartori, K. Sato, A. Savoy-Navarro, T. Scheidle, P. Schlabach,
A. Schmidt, E. E. Schmidt, M. A. Schmidt, M. P. Schmidt, M. Schmitt, T. Schwarz,
L. Scodellaro, A. L. Scott, A. Scribano, F. Scuri, A. Sedov, S. Seidel, Y. Seiya, A. Semenov, L. Sexton-Kennedy, A. Sfyrla, S. Z. Shalhout, T. Shears, R. Shekhar, P. F.
Shepard, D. Sherman, M. Shimojima, S. Shiraishi, M. Shochet, Y. Shon, I. Shreyber,
A. Sidoti, P. Sinervo, A. Sisakyan, A. J. Slaughter, J. Slaunwhite, K. Sliwa, J. R. Smith,
F. D. Snider, R. Snihur, A. Soha, S. Somalwar, V. Sorin, J. Spalding, T. Spreitzer,
P. Squillacioti, M. Stanitzki, R. St.Denis, B. Stelzer, O. Stelzer-Chilton, D. Stentz,
J. Strologas, D. Stuart, J. S. Suh, A. Sukhanov, I. Suslov, T. Suzuki, A. Ta↵ard,
R. Takashima, Y. Takeuchi, R. Tanaka, M. Tecchio, P. K. Teng, K. Terashi, J. Thom,
A. S. Thompson, G. A. Thompson, E. Thomson, P. Tipton, V. Tiwari, S. Tkaczyk,
D. Toback, S. Tokar, K. Tollefson, T. Tomura, D. Tonelli, S. Torre, D. Torretta, P. Totaro, S. Tourneur, Y. Tu, N. Turini, F. Ukegawa, S. Vallecorsa, N. van Remortel,
A. Varganov, E. Vataga, F. Vázquez, G. Velev, C. Vellidis, V. Veszpremi, M. Vidal,
R. Vidal, I. Vila, R. Vilar, T. Vine, M. Vogel, I. Volobouev, G. Volpi, F. Würthwein,
P. Wagner, R. G. Wagner, R. L. Wagner, J. Wagner-Kuhr, W. Wagner, T. Wakisaka,
R. Wallny, S. M. Wang, A. Warburton, D. Waters, M. Weinberger, W. C. Wester,
B. Whitehouse, D. Whiteson, S. Whiteson, A. B. Wicklund, E. Wicklund, G. Williams,
H. H. Williams, P. Wilson, B. L. Winer, P. Wittich, S. Wolbers, C. Wolfe, T. Wright,
X. Wu, S. M. Wynne, S. Xie, A. Yagil, K. Yamamoto, J. Yamaoka, U. K. Yang, Y. C.
Yang, W. M. Yao, G. P. Yeh, J. Yoh, K. Yorita, T. Yoshida, G. B. Yu, I. Yu, S. S. Yu,
J. C. Yun, L. Zanello, A. Zanetti, I. Zaw, X. Zhang, Y. Zheng, and S. Zucchelli (CDF
Collaboration), Physical Review Letters 102, 152001 (2009).
[150] J. Renner, A. Farbin, J. M. Vidal, J. Benlloch-Rodrı́guez, A. Botas, P. Ferrario,
J. Gómez-Cadenas, V. Álvarez, C. Azevedo, F. Borges, S. Cárcel, J. Carrión, S. Cebrián, A. Cervera, C. Conde, J. Dı́az, M. Diesburg, R. Esteve, L. Fernandes, A. Ferreira, E. Freitas, A. Goldschmidt, D. González-Dı́az, R. Gutiérrez, J. Hauptman,
C. Henriques, J. H. Morata, V. Herrero, B. Jones, L. Labarga, A. Laing, P. Lebrun,
I. Liubarsky, N. López-March, D. Lorca, M. Losada, J. Martı́n-Albo, G. Martı́nezLema, A. Martı́nez, F. Monrabal, C. Monteiro, F. Mora, L. Moutinho, M. NebotGuinot, P. Novella, D. Nygren, B. Palmeiro, A. Para, J. Pérez, M. Querol, L. Ripoll,
J. Rodrı́guez, F. Santos, J. dos Santos, L. Serra, D. Shuman, A. Simón, C. Sofka,
M. Sorel, J. Toledo, J. Torrent, Z. Tsamalaidze, J. Veloso, J. White, R. Webb,
N. Yahlali, and H. Yepes-Ramı́rez, Journal of Instrumentation 12, T01004 (2017).

165

[151] M. Jaderberg, V. Dalibard, S. Osindero, W. M. Czarnecki, J. Donahue, A. Razavi,
O. Vinyals, T. Green, I. Dunning, K. Simonyan, C. Fernando, and K. Kavukcuoglu,
(2017), arXiv:1711.09846 .
[152] B. J. Wytho↵, Chemometrics and Intelligent Laboratory Systems 18, 115 (1993).
[153] H. H. B. Sørensen (Springer, Berlin, Heidelberg, 2012) pp. 377–386.
[154] J. Gauci and K. Stanley, in Proceedings of the 9th Annual Conference on Genetic
and Evolutionary Computation, GECCO ’07 (ACM, New York, NY, USA, 2007) pp.
997–1004.
[155] N. Chatterjee, M. O’Connor, G. H. Loh, N. Jayasena, and R. Balasubramonian, in
Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’14 (IEEE Press, Piscataway, NJ, USA, 2014) pp.
128–139.
[156] G. Wang, S. Chen, X. Zhang, and Z. Zhou, Information Processing Letters 112, 823
(2012).
[157] NVIDIA Corporation, NVIDIA CUDA Compute Unified Device Architecture Programming Guide (NVIDIA Corporation, 2007).
[158] S. Williams, A. Waterman, and D. Patterson, Communications of the ACM 52, 65
(2009).
[159] KARMEN Collaboration, Physics Letters B B267, 321 (1991)

166