20MG β DECAY AND THE 15O(α, γ)19Ne(p, γ)20Na REACTION SEQUENCE IN TYPE

I X-RAY BURSTS

By

Brent Glassman

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Physics - Doctor of Philosophy

2020

ABSTRACT

20MG β DECAY AND THE 15O(α, γ)19Ne(p, γ)20Na REACTION

SEQUENCE IN TYPE I X-RAY BURSTS

By

Brent Glassman

Constraining the 15O(α, γ)19Ne(p, γ)20Na reaction rate is critically important for accu-

rately simulating x-ray burst light curves, our only x-ray burst observable from Earth. Both

the 15O(α, γ)19Ne and 19Ne(p, γ)20Na reaction rates have been studied using a multitude

of experimental techniques, yet only upper limits had been determined at the outset of the

present work. The 15O(α, γ)19Ne reaction in particular has been singled out as the most

important reaction rate of all currently unknown rates to measure.

Utilizing the β decay of 20Mg to populate 20Na excited states, we further constrain the

19Ne(p, γ)20Na reaction rate by searching for γ decays from the most important resonance

at Ex=2647 keV. Additionally, by populating 20Na excited states high above the proton

threshold, we populate the most important 19Ne resonance for measuring the 15O(α, γ)19Ne

reaction rate at Ex=4.03 MeV. Herein, the results of this study are reported including:

The full 20Mg β-delayed γ ray spectrum and decay scheme; the upper limit on the β decay

feeding of the 20Na 2647 keV state; the Doppler broadening analysis of nine γ ray peaks

from six excited states in 19Ne; the intensity of 20Mg(βpγ) to the 4.03 MeV state in 19Ne;

the measurement of the 20Na excitation energy at 7.44+0.25−0.22 MeV feeding the 4.03 MeV state
in 19Ne; and the measurement of the center of mass proton energy for the feeding of the

important 4.03 MeV state in 19Ne at 1.23+0.25−0.22 MeV.

Copyright by
BRENT GLASSMAN
2020

This dissertation is dedicated to my mother Mary Kay and my father Howard for instilling
in me my love for math and problem solving, and to Glenn LaFerriere for always providing

me challenging problems to struggle over.

iv

ACKNOWLEDGMENTS

This work would not be possible without the help and guidance of my advisor Chris

Wrede. Chris’s mentorship has been invaluable in all aspects of the scientiﬁc process even

going beyond by assisting me through my biggest challenge in writing scientiﬁc papers and

of course my dissertation. It has been a pleasure to work alongside Chris during the many

large projects undertaken by his research group and the skills I am able to take from this

experience are incalculable. Thank you for all of the work that you’ve put in and for all

of the opportunities I’ve been aﬀorded just by working alongside such a talented research

group.

It takes a great deal of work to run a successful experiment with only the minimum

amount of heart-attacks and all the credit has to go to the members of my research group.

In particular, David P´erez-Loureiro provided a frankly oﬀensive amount of help during the

setup of the experiment and coding expertise for the data analysis, which I began to suspect

amounted to billable hours rivaling a sizeable law-ﬁrm. Mike Bennett, Cathleen Fry, and

Sarah Schwartz also contributed a tremendous eﬀort during and after the experiment, and

I’m thankful for their support throughout my graduate tenure. As postdocs joining the

research group for brief periods of time, Moshe Friedman and Lijie Sun have been incredibly

helpful with the most challenging aspects of my data analysis. Finally I’d like to thank all

of the members of the collaboration who came to assist during the experiment and for their

continued support during the writing process.

In addition to my advisor Chris, I also have to thank the members of my PhD committee;

Heiko Hergert, Hironori Iwasaki, Jim Linnemann, and Artemis Spyrou. Their insights have

provided me with useful avenues of analysis which had previously gone unconsidered and

v

expertise which was incredibly helpful during my analysis. A special thanks has to go to

Hironori Iwasaki for ﬁlling a role on my committee when another member was unavailable.

It would be impossible to make it through graduate school without all of my friends to

help me keep my sanity. A shout out to the ﬁrst-year all-nighter crew who, after working

through a weeks worth of problem sets in a night, deﬁnitely promised to never ever do that

again. Twas always great to see your bright and shining faces that very next week. What

else can be said about The Killers other than we deﬁnitely won that one softball game, and

I can’t really recall how the other ones went but probably really good. To my workout crew,

we got absolutely diced. Congrats on the sick new bods! Thank you to everyone who made

the tough times go by quicker, I could not have accomplished this without you. Note: In

the event of future arbitration, let this document hereby certify that I Brent Glassman

Frank Reynolds do not

have Donkey Brains.

My sister Alyssa and mother Mary Kay can not be thanked enough for their support

over the past six years. The many late night calls providing a calm voice to counter all

of my doubts of success and their patience throughout graduate school has been incredibly

meaningful to me.

vi

TABLE OF CONTENTS

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

ix

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

x

KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Introduction to Nuclei

Chapter 1 Overview of Pertinent Nuclear Physics

. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2 Nuclear Shell Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Nuclear Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
β-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to Isospin . . . . . . . . . . . . . . . . . . . . .
1.3.1.1
1.3.1.2
β decay selection rules . . . . . . . . . . . . . . . . . . . . .
γ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2
1.3.3 Nucleon Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.1

2.1.2.1 Hot-CNO cycles and Breakout

Chapter 2 Motivations for studying the 15O(α, γ)19Ne(p, γ)20Na Reaction
Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Nuclear Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Nuclear Reactions in stars . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 X-ray Bursts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
2.1.3 Mathematical Formalism . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
2.2 The 15O(α, γ)19Ne Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Previous studies of 15O(α, γ)19Ne . . . . . . . . . . . . . . . . . . . .
2.2.2 Purpose of This Work . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Previous studies of 19Ne(p, γ)20Na
. . . . . . . . . . . . . . . . . . .
2.3.2 Purpose of This Work . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.3.1 Direct and Resonant Capture Reactions

2.3 The 19Ne(p, γ)20Na Reaction Rate

Chapter 3 Experiment E14066 at the National Superconducting Cyclotron
Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Data Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1
1
3
7
9
10
11
16
18

20
20
21
25
27
28
34
38
38
39
40
40
42

43
43
44
46
54
58
58

Chapter 4

4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indirect study of the 19Ne(p, γ)20Na reaction using 20Mg(βγ)20Na 63
63
63
68
69
75
77

Chapter 5 Toward the 15O(α, γ)19Ne Reaction Rate using Doppler Broad-
ening in 20Mg(βpγ)19Ne . . . . . . . . . . . . . . . . . . . . . . . .
78
5.1 Doppler Broadening Line Shape Analysis Technique . . . . . . . . . . . . . .
78
5.1.1 Monte Carlo Simulation of broadened 19Ne γ lines . . . . . . . . . . .
79
5.1.1.1 Treatment of Center of Mass Energy . . . . . . . . . . . . .
80
5.1.1.2 Treatment of Excited State Lifetime (τ ) and Stopping Power
80
5.1.1.3
. . . . . . . . . . . . . . . . . . .
83
5.1.1.4 Angular Correlations . . . . . . . . . . . . . . . . . . . . . .
87
5.1.1.5
Summary of Monte Carlo Procedure . . . . . . . . . . . . .
88
5.1.1.6 Treatment of Background . . . . . . . . . . . . . . . . . . .
90
. . . . . . . . . . . . .
91
5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
19Ne 1507 keV 5/2− state . . . . . . . . . . . . . . . . . . . . . . . .
95
19Ne 1536 keV state
. . . . . . . . . . . . . . . . . . . . . . . . . . .
99
19Ne 1615 keV state
. . . . . . . . . . . . . . . . . . . . . . . . . . . 102
19Ne 4.03 MeV state . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
19Ne 238 and 275 keV states . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1.2 Doppler Broadening Systematic Uncertainties

SeGA Detector Response

5.2.1
5.2.2
5.2.3
5.2.4
5.2.5

Chapter 6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1 The 15O(α, γ)19Ne Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Doppler Broadening in β-delayed particle emission . . . . . . . . . . . . . . . 111

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

viii

LIST OF TABLES

Table 1.1:

Selection rules governing β-decay. Columns 1-5 refer to the type of
transition, orbital angular momentum carried oﬀ by β-decay particles,
change in isospin between initial and ﬁnal states of the nucleus, the
change in total angular momentum between initial and ﬁnal state of
. . . . . . . . . .
the nucleus, and the change in parity respectively.

Table 4.1:

Upper limits on the intensities of 20Mg β-delayed γ-ray transitions
through the 2647-keV 20Na state with 90% conﬁdence. Final energy
level values were adopted from Seweryniak et al. [52]. The search for
each γ-ray encompassed a range of ± 6 keV.
. . . . . . . . . . . . .

14

73

Table 4.2:

Gamow-Teller strengths B(GT) and associated log ft values calcu-
lated for 20Mg β-decay to 1st and 2nd excited 1+ states in 20Na
using the sd shell model with various interactions described in the text. 75

Table 5.1:

Column one reports the 19Ne excited-state energies populated by
20Mg(βp), and were determined by applying recoil corrections to the
measured γ ray energies in the lab-frame (column-four). Column two
reports the measured lifetimes of 19Ne excited states. Column three
reports the intensity of 20Mg(βp) feedings to each excited state, where
each feeding is determined by adding all γ ray decays originating from
each state and subtracting feeding from higher lying states. Column
four reports the measured lab frame energies of each γ ray branch.
Column ﬁve reports the total intensity of each γ-ray transition per
20Mg decay. Column six reports the γ ray branching ratios for each
19Ne excited-state. Column seven reports the measured CoM proton
energies feeding 19Ne excited states.
. . . . . . . . . . . . . . . . .

Table 5.2:

Piechaczek [42] and Lund [44] absolute % proton feeding intensities
to 1536 keV state per 20Mg β-decay. The quoted relative uncertainty
for all intensities measured by Piechaczek is 12%. . . . . . . . . . . .

94

99

ix

LIST OF FIGURES

Figure 1.1: Chart of nuclides. The y-axis represents the number of protons Z
in the nucleus and the x-axis represents the number of neutrons N .
Boxes in black denote stable nuclei whereas colored boxes denote
unstable nuclei. [4]
. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.2: A Woods Saxon Potential, V (r) =

, is often used as in-

−Vo

1+exp( r−R
a )

put into the nuclear wave function and describes the forces on each
nucleon, where Vo ≈ 50 MeV describes the depth of the potential,
a ≈ 0.5 fm describes the so-called surface thickness of the nucleus,
and R ≈ 1.25A1/3 fm approximates the radius of the nucleus.
. . .

Figure 1.3: The energy levels on the left deﬁne shells via the principal quantum
number N = n + 1, given by a nuclear wave function derived from a
harmonic oscillator potential. The addition of a spin-orbit term to the
nuclear potential yields the splitting of these levels and approximates
shell closures which give rise to the magic numbers. Figure Credit:
Bakken (GPL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.4: An approximate level scheme for the diﬀerent Isospin projections in
the A = 20, T = 2 multiplet, where neutrons and protons are degen-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
erate.

Figure 1.5: An approximate level scheme for the diﬀerent Isospin projections in
the A = 20, T = 2 multiplet where the degeneracy of the proton and
neutron is broken by the electromagnetic force. . . . . . . . . . . . .

Figure 2.1: The Herzprung-Russel Diagram relates the luminosity and temper-
ature of a star and classiﬁes the star based on its position on the
diagram. The largest portion of stars lays within the main sequence
category as well as a few within various giant subgroups and white
dwarfs which are remnants of main sequence stars. Only stars which
have an observable luminosity are shown on the diagram and there-
fore stars like neutron stars do not appear in this ﬁgure. [ﬁgure credit
- https://commons.wikimedia.org/wiki/File:HRDiagram.png]
.

x

2

5

6

12

13

22

Figure 2.2: The graph displays the binding energy per nucleon for various mass
number A nuclei. The product of nuclear fusions up to iron releases
energy, and the ﬁssion of heavy elements whose fragments are more
tightly bound are also exothermic. These processes give oﬀ energy by
converting mass into energy since nuclei with higher binding energy
have less mass per nucleon. . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3:

Simpliﬁed diagram of a compact star (black dot) with a hydrogen rich
binary companion (red circle). Each star is surrounded by an imag-
inary surface called the Roche lobe, which indicates the point where
each star’s gravity dominates. The point where the Roche lobe of
each star meets is called the inner Lagrangian point and at this point
the eﬀects of gravity and rotations cancel Credit: [Cornell, http://
hosting.astro.cornell.edu/academics/courses/astro201/roche_
lobe.htm]

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

Figure 2.4: The three Hot CNO cycles shown here occur during the second phase
of thermonuclear runaway in Type I x-ray bursts. The gray boxes
represent the stable isotopes and the arrows show the reactions and
β decays that proceed within the diﬀerent cycles.
[Figure adapted
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
from [11]]

Figure 2.5: Example Gamow window (red) which is created from the convolution
of the Maxwell-Boltzman distribution (black) with the penetration
factor Pl(E) (blue). The Gamow peak is enhanced by a factor of 100
. . . . . . . . . . . . . . . . . . . .
to make the peak more visible.

Figure 2.6: This simple ﬁgure illustrates a direct capture reaction of a nucleus
X and a proton. The resulting nucleus Y emits a γ-ray and settles
into a bound state. The dotted line denotes the separation energy
of a proton Sp, or the energy required to remove a proton from the
nucleus.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.7: This simple ﬁgure illustrates the famous triple α resonant reaction.
In this case the α particle is captured onto an excited state in 12C
where the nucleus can either decay via γ-rays to a bound state or
the α particle could be re-emitted. The resonance energy Er is the
diﬀerence between the excitation energy in 12C and the α particle
separation energy Sα . . . . . . . . . . . . . . . . . . . . . . . . . .

23

26

29

33

34

36

xi

Figure 3.1: A schematic of the K500 and K1200 coupled cyclotron feeding into
the A1900 magnetic separator. The 24Mg stable primary beam is
accelerated through the couple cyclotrons to ∼ 1/2 the speed of light
and impinged onto the 9Be target. The fragments of the secondary
beam are separated using 4 dipole magnets (green) and an aluminum
degrading wedge (yellow). The secondary beam is then delivered to
our experimental setup in the S2 vault.
. . . . . . . . . . . . . . . .

Figure 3.2: A birds-eye view of the experimental setup drawn using computer-
aided design (CAD) software. . . . . . . . . . . . . . . . . . . . . . .

Figure 3.3: TOP: An example particle ID spectrum taken in between full beam
runs. The energy lost to the Si PIN detector is plotted against the
time of ﬂight (TOF) between the XFP scintillator and the Si PIN
detector. The large blob in the top right is the 20Mg and represents
∼ 40% of the secondary beam implanted in the plastic scintillator.
Bottom: The total beam rate was increased in later runs which sac-
riﬁced 20Mg purity (∼ 34%) for an increase in 20Mg pps implanted
. . . . . . . . . . . . . . . . . . . . . . .
into the plastic scintillator.

Figure 3.4: The pneumatic drive utilizes a gas cylinder to retract the PIN detector
from the beam-line when full beam is on and releases the PIN detector
. . . . . . . . . . . . . .
into the beam-line during particle ID runs.

Figure 3.5: The secondary beam exits through a thin kapton window after which
it travels 20 cm in air before implanting into the EJ200 plastic scin-
tillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.6: To hold the plastic scintillator in front of the thin kapton window
an aluminum mount was used which connected to the top of the
SeGA support structure and clamped onto the end of the scintillator’s
photomultiplier tube.
. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.7: A total spectrum of EJ200 scintillator energies detected during E14066
plotted vs γ-ray energies detected in SeGA. The region below 10000
channels is comprised of β-decay events and the γ-ray lines above
20000 channels are related to implant events which we want to re-
move by placing a gate (only looking at speciﬁc events) on the β-decay
energy region.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.8: An example trace from the silicon PIN detector. The decay time of
signals from the Si PIN detector is similar to the SeGA detectors
(∼45 µs) and EJ200 plastic scintillator traces decay much faster (∼1
µs). The program used to view the trace as well as test diﬀerent
signal parameters is called NScope.
. . . . . . . . . . . . . . . . . .

45

46

48

49

50

51

52

55

xii

Figure 3.9:

Simpliﬁed schematic of E14066 electronics. The blue boxes denote the
raw input signals taken from the experimental setup as well as the
RF signal and XFP scintillator signal taken from the beam operators.
The white boxes denote the NIM modules which shape the signals
and produce timing signals which are input into DDAS. The orange
denotes the output of data to either the NSCL DAQ computer or the
patch panel which ports signals to the NSCL Data-U.
. . . . . . . .

Figure 3.10: The blue line shows a total γ-ray spectrum without gating on the
low energy portion of the scintillator. By gating on the β decay
energies we reduce the noise in our γ spectrum (green) by a factor of
2 and remove many of the background γ-ray peaks caused by nuclear
reactions as well as natural sources of radiation.
. . . . . . . . . . .

Figure 3.11: SeGA γ-ray energies vs time produced by subtracting the timestamp
of a germanium detector signal from the timestamp of the scintillator
signal. The X axis values are oﬀset by adding 10000. Each clock tic
on the DDAS crate represents 10 ns and is the smallest increment of
time which can be used. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.12: A paper holder was made with a small pocket for the 125Sb calibra-
tion source and the holder ﬁt over the face of the plastic scintillator
to mimic as closely as possible the eﬃciency from beam particles
implanted into the plastic scintillator. During calibration runs the
holder was placed on the face of the scintillator.
. . . . . . . . . . .

Figure 4.1: Relevant low-lying states in 19Ne and 20Na labeled with J π and en-
ergy in keV. Energies below the proton threshold in 20Na as well as
J π are adopted from Seweryniak et al. [52] and energies above the
. . . . . . . . . . . .
proton threshold are from Wallace et al. [53].

Figure 4.2: Cumulative spectrum of 20Mg β-delayed γ-rays acquired by SeGA,
. . . .

in coincidence with β-decay events in the plastic scintillator.

Figure 4.3: Measured 90% C.L. upper limits on the intensity of 20Mg β delayed
γ-rays as a function of energy in the 2647 keV search region. The
. . . . . . . . . . . . . . . .
most conservative limit is at 2645 keV.

56

59

60

61

64

65

66

xiii

Figure 4.4: Upper panel: Photopeak eﬃciency of SeGA as a function of en-
ergy. Circles correspond to eﬃciencies from the Eu calibration source.
Squares correspond to eﬃciencies from a Geant4 simulation of the
experimental setup. Fit function (4.1) was used to model the Geant4
simulation eﬃciency. The ﬁt was then scaled by a constant factor to
line up with the source calibration eﬃciency (red line). Lower panel:
The residual eﬃciency is the relative diﬀerence between the calibra-
tion source data and the scaled ﬁt of Geant4 simulation eﬃciencies.

Figure 4.5: Each panel shows a magniﬁed region of the spectrum of 20Mg β-
delayed γ-rays acquired by SeGA, in coincidence with β-decay events
in the plastic scintillator (Fig. 4.2). The regions of interest are deter-
mined by the possible transition energies from the 2647-keV state in
20Na (Table 4.1). The blue points represent the data with statistical
error bars and the black lines a linear ﬁt of the background. The
red dotted lines represent the background ﬁt plus the 90% conﬁdence
upper limits for each possible transition energy. For each transition
the ﬁt shown corresponds to the energy within the ±6 keV search
. . . . . . . . . . . .
range that yielded the maximum upper limit.

Figure 4.6: Each panel shows a magniﬁed region of the spectrum of 20Mg β-
delayed γ-rays acquired by SeGA, in coincidence with β-decay events
in the plastic scintillator (Fig. 4.2). The regions of interest are deter-
mined by the possible transition energies from the 2647-keV state in
20Na (Table 4.1). The blue points represent the data with statistical
error bars and the black lines a linear plus Gaussian ﬁt for the back-
ground. The red dotted lines represent the background ﬁt plus the
90% conﬁdence upper limits for each possible transition energy. For
each transition the ﬁt shown corresponds to the energy within the ±6
. . . . . .
keV search range that yielded the maximum upper limit.

Figure 4.7: A magniﬁed region of the spectrum of 20Mg β-delayed γ-rays ac-
quired by SeGA, in coincidence with β-decay events in the plastic
scintillator (Fig. 4.2). The blue points represent the data with sta-
tistical error bars. The black line represents the background Monte
Carlo plus continuum background model described in the text. The
red dotted line represents the background Monte Carlo ﬁt plus the
90% conﬁdence upper limits for the 1298 keV transition energy. The
ﬁt shown corresponds to the energy within the ±6 keV search range
. . . . . . . . . . . . . . . .
that yielded the maximum upper limit.

68

70

71

72

xiv

Figure 4.8: Upper limits on the intensity of the 20Mg β-decay transition to the
2647 keV state of 20Na compiled from diﬀerent sources [41, 42, 35].
All of the upper limits from previous work (dashed lines) are incom-
plete. The upper limits in grey include only the proton branch to
the ground state of 19Ne. The upper limit by Piechaczek includes
the proton branch to the ground state and selected γ-ray branches.
The present work includes proton branches to the g.s. and 1st excited
state in 19Ne [35, 40] as well as all energetically possible γ branches
in 20Na, making it the only complete limit.
. . . . . . . . . . . . .

Figure 5.1: The stopping power is treated iteratively over many time steps which
are determined by the lifetime of the 19Ne excited state. The stars
represent the sum of the electric and nuclear stopping powers of the
scintillator at diﬀerent recoil energies and the line represents each
time-step traversed by the algorithm, updating the recoil energy and
. . . . . . . . . . . . . . . . . . . . . .
stopping power continuously.

Figure 5.2: The σ parameter energy dependence for input to the exponentially-
modiﬁed-gaussian response function for each SeGA detector. Each
data point corresponds to the value of the σ parameter for a par-
ticular calibration peak. The σ parameter is ﬁt using a line and
the conﬁdence band (red online) shows 1 standard deviation uncer-
tainty:(a) Detector 1; (b) Detector 2; (c) Detector 3; (d) Detector 4;
(c) Detector 5; and (d) Detector 6; . . . . . . . . . . . . . . . . . . .

Figure 5.3: The σ parameter energy dependence for input to the exponentially-
modiﬁed-Gaussian response function for each SeGA detector. Each
data point corresponds to the value of the σ parameter for a par-
ticular calibration peak. The σ parameter is ﬁt using a line and
the conﬁdence band (red online) shows 1 standard deviation uncer-
tainty:(a) Detector 7; (b) Detector 8; (c) Detector 9; (d) Detector 10;
(c) Detector 11; and (d) Detector 12;
. . . . . . . . . . . . . . . . .

Figure 5.4: The σ parameter energy dependence for input to the exponentially-
modiﬁed-Gaussian response function for each SeGA detector. Each
data point corresponds to the value of the σ parameter for a particular
calibration peak. The σ parameter is ﬁt using a line and the conﬁ-
dence band (red online) shows 1 standard deviation uncertainty:(a)
Detector 13; (b) Detector 14; (c) Detector 15; and (d) Detector 16; .

74

82

84

85

86

Figure 5.5: Example Cumulative Distribution Function for a 1232 keV γ. . . . .

87

xv

Figure 5.6:

Figure 5.7:

19Ne level scheme from 20Mg(βpγ)19Ne decay deduced from the present
work. The γ ray transition intensities are denoted by the thick-
nesses of the arrows, which are proportional to their intensities. The
20Mg(βp) feeding intensities are denoted by the arrows on the right.

93

(color online) Upper panel: The ﬁt of the 1232 keV γ ray peak is
produced by using 4.3 ps lifetime as well as proton feeding intensities
from Piechaczek et al. [42]. The solid gray line represents the data,
the dot-dashed green line denotes the background, the dotted lines
denote the diﬀerent contributions of each proton feeding, and the
dashed red line denotes the total ﬁt. The ﬁt has a χ2
ν=1.07. Lower
panel: The Residual plot shows the data subtracted from the ﬁt
function.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.8:

χ2 values determined by simulating the lifetime of the 1507 keV 19Ne
state for many values and comparing the simulation to the data over
447 degrees of freedom. The minimum is found at 4.3 ps.
. . . . . .

96

97

98

Figure 5.9:

Figure 5.10:

(color online) Upper panel: The γ ray spectrum above contains two
20Mg(βpγ) peaks from diﬀerent excited states in 19Ne. The data
are represented by the solid black line and, the dot-dashed green line
denotes the background, the dotted pink lines denote the diﬀerent
contributions of proton feedings to the 1507 keV state, the dotted
black lines denote the diﬀerent contributions of proton feedings to
the 1536 keV state, and the dashed red line denotes the total ﬁt
which has a χ2
ν = 1.11. Lower panel: The Residual plot shows the
data subtracted from the ﬁt function.
. . . . . . . . . . . . . . . . .

(color online) Fits of the 1298 keV γ ray peak above are produced
using a 16 fs lifetime.
(a) The ﬁt is produced using the relative
proton feeding intensities, measured by Lund [44], from Table 5.2.
The data are represented by the solid gray line, the dot-dashed green
line denotes the background, the dotted lines denote the diﬀerent
contributions of each proton feeding, and the dashed red line denotes
the best total ﬁt. (b) The Residual plot shows the data subtracted
from the ﬁt function in (a). (c) The ﬁt is produced using the relative
proton feeding intensities, measured by Piechaczek [42], from Table
5.2. Similarly to panel (a) the data are represented by the solid line,
the dot-dashed line denotes the background, the dotted lines denote
the diﬀerent contributions of each proton feeding, and the dashed
line denotes the best total ﬁt. (d) The Residual plot shows the data
subtracted from the ﬁt function in (c). . . . . . . . . . . . . . . . . . 101

xvi

Figure 5.11:

(color online) Upper panel: The ﬁt of the 1340 keV γ ray peak is
produced by using a 93 fs lifetime and a CoM energy of 2.7 MeV
between the proton and recoiling 19Ne. The solid gray line represents
the data, the dot-dashed green line denotes the background and the
dashed red line denotes the background+simulated peak. The ﬁt
has a χ2
ν=1.00. Lower panel: The Residual plot shows the data
subtracted from the ﬁt function.

. . . . . . . . . . . . . . . . . . . . 104

Figure 5.12: The ﬁt of the 1377 keV γ ray peak is produced by using a 93 fs lifetime
and a CoM energy of 2.7 MeV between the proton and recoiling 19Ne.
The black line represents the data, the green line denotes the linear
background and the red line denotes the background+simulated peak.
Lower panel: The Residual plot shows the data subtracted from the
background.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 5.13: The ﬁt of the 1616 keV γ ray peak is produced by using a 93 fs lifetime
and a CoM energy of 2.7 MeV between the proton and recoiling 19Ne.
The black line represents the data, the green line denotes the exponen-
tial background and the red line denotes the background+simulated
peak. Lower panel: The Residual plot shows the data subtracted
from the background.

. . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 5.14:

(color online) Upper panel: The ﬁt of the 4.03 MeV peak is produced
by simulating the broadened peak with an 7 fs lifetime and CoM
energy of 1.21 MeV and has a χ2
ν = 0.94. All 16 SeGA detectors
are used to produce this spectrum. The solid gray line represents the
data, the dot-dashed green line denotes a ﬁt of the background and
the dashed red line denotes the total ﬁt using the optimal 1.21 MeV
CoM energy. A simpliﬁed linear background model was applied for
this relatively low statistics case. Lower panel: The Residual plot
shows the data subtracted from the ﬁt function.

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

Figure 5.15: Each χ2 value is determined by simulating a diﬀerent CoM proton
energy feeding the 4.03 MeV excited state in 19Ne and comparing
each simulation to the peak at 4034 keV. χ2 values are determined
from ﬁts with 157 degrees of freedom. The minimum determines the
most likely CoM energy. . . . . . . . . . . . . . . . . . . . . . . . . . 109

xvii

KEY TO ABBREVIATIONS

• NSCL - National Superconducting Cyclotron Laboratory (East Lansing MI, USA)
• CoM - Center of Mass
• DDAS - Digital Data Acquisition System
• SeGA - Segmented Germanium Array
• PDF - Probability Density Function
• CDF - Cumulative Distribution Function

xviii

Chapter 1

Overview of Pertinent Nuclear

Physics

1.1

Introduction to Nuclei

An important discovery about the makeup of the atom was made in 1911. The prevail-

ing model of the atom at the time was that it consisted of positive and negative charge

particles clumped together like a plum pudding [1]. Experiments to test this hypothesis

were conducted by Ernest Rutherford and involved bombarding a thin gold foil with helium

atoms (α-particles). If the pudding model were true, they expected all of the α particles to

simply pass through the foil. However, over many trials the experimenters were surprised

to ﬁnd that some of these α particles were being deﬂected by the foil at angles as though

they were balls bouncing oﬀ a wall [2]. This discovery meant that all of the positive charge

and most of the mass must be contained at the center of the atom. The discovery of the

neutron in 1932 [3] built a more complete picture of the atom where protons and neutrons

were clumped together in the nucleus, bound by the strong force, and were surrounded by

orbiting electrons.

A nucleus is conventionally identiﬁed by its element name and total number of protons

and neutrons (A) in the nucleus, referred to as the mass number. The names of elements

are determined by the number of protons (Z) in the nucleus, each of which is displayed on

1

Figure 1.1: Chart of nuclides. The y-axis represents the number of protons Z in the nucleus
and the x-axis represents the number of neutrons N . Boxes in black denote stable nuclei
whereas colored boxes denote unstable nuclei. [4]

the periodic table of elements. For example, a nucleus of oxygen having 8 neutrons and 8

protons is denoted by 16O.

Nuclei with the same number of protons and diﬀerent numbers of neutrons (N ) are

referred to as isotopes. Similarly, nuclei with the same number of neutrons and diﬀerent

numbers of protons are referred to as isotones. All of the assorted nuclei that have been

studied and many that have yet to be studied are displayed on the chart of nuclides in

Figure 1.1.

The discovery of the nucleus led to many questions about the origin of the elements

and produced the ﬁeld of nuclear astrophysics in the 1950s through the work of Burbidge,

Burbidge, Fowler, and Hoyle [5] which sought to tackle this question. As the ﬁeld of nuclear

2

astrophysics has developed many questions have arisen such as: how do stars generate their

energy, what is the life cycle of a star, where are the elements created in nucleosynthesis,

and what are the corresponding stellar events that occur to synthesize these isotopes?

In order to tackle these questions it is important to understand the properties of both sta-

ble and unstable nuclei (nuclear structure) and how they interact with other nuclei (nuclear

reactions).

1.2 Nuclear Shell Model

Like atoms whose electrons orbit the nucleus in a number of shells, similar to the planets

orbiting the sun, nuclei can exist in a number of quantized energy states. Protons and

neutrons separately ﬁll the lowest energy shells to produce a nucleus in a conﬁguration

called the ground state. If proton(s) or neutron(s) occupy orbitals of higher energy than

other available orbitals, these are referred to as nuclear excited states. The mass Mexc of a

nucleus can be found by summing the mass of the nucleus in its ground state, Mg.s. with

the excitation energy Eexc (Eq. 1.1), where BE is the binding energy of a nucleus with N

neutrons and Z protons and c represents the speed of light.

Mexc = Mg.s. + Eexc/c2

Mg.s. = Zmp + N mn − BE(Z, N )/c2

(1.1)

(1.2)

In the atomic shell model each shell is ﬁlled with a speciﬁc number of electrons, in ac-

cordance with the Pauli Exclusion principle. The ﬁlled shells are considered to be inert and

3

the valence electrons determine the atomic properties in nuclei [6]. Both protons and neu-

trons independently ﬁll orbitals with so-called magic numbers that correspond to quantum

properties such as total angular momentum j, orbital angular momentum l, and number

of nodes n in the wave function (Fig. 1.3), which can be well-described using a harmonic

oscillator potential or Woods-Saxon potential (Fig. 1.2). Because protons and neutrons are

fermions (spin 1/2 particles) these particles must abide by the Pauli Exclusion Principle,

meaning that no 2 protons or neutrons in a nucleus can have the same quantum numbers.

This property eﬀectively restricts the size of each shell since each nucleon requires a diﬀerent

set quantum numbers.

The ﬁrst shell, called the 1s-orbital is deﬁned to have l = 0 angular momentum and

protons are spin s = 1/2 particles which have total angular momentum j = 1/2. In general,
there are 2j+1 projections of the total angular momentum or jz = ±1/2 for the 1s-orbital

and this deﬁnes the number of protons or neutrons allowed in the shell. The 1s-orbital can

contain 2 such nucleons and this deﬁnes the ﬁrst so-called magic number. Magic numbers

denote the amount of nucleons to completely ﬁll a nuclear shell. The p-shell is deﬁned by

l = 1 orbital angular momentum and can contain up to 6 protons or neutrons.

The magic numbers, derived from these quantum numbers, deﬁne shells which can hold

2, 8, 20, 28, 50, 82, 126 nucleons (Fig. 1.3). Since the the nuclear shells are determined

from quantum numbers nlj, the shell model is often used to determine the spin-parity (J π)

of a nucleus in its ground state, where J describes the total coupled angular momentum
and π describes the parity deﬁned to be (−1)l. For example, 20Mg has 12 protons and 8

neutrons. The ﬁrst 8 protons and neutrons ﬁll up the p-shell and can be thought of as an

inert 16O core with j = 0 angular momentum and the remaining 4 protons form two spin-up

and spin-down pairs which sit in the ﬁrst d5/2-shell and each of these pairs couple together

4

Figure 1.2: A Woods Saxon Potential, V (r) =
, is often used as input into
the nuclear wave function and describes the forces on each nucleon, where Vo ≈ 50 MeV
describes the depth of the potential, a ≈ 0.5 fm describes the so-called surface thickness of
the nucleus, and R ≈ 1.25A1/3 fm approximates the radius of the nucleus.

1+exp( r−R
a )

−Vo

5

Figure 1.3: The energy levels on the left deﬁne shells via the principal quantum number
N = n + 1, given by a nuclear wave function derived from a harmonic oscillator potential.
The addition of a spin-orbit term to the nuclear potential yields the splitting of these levels
and approximates shell closures which give rise to the magic numbers. Figure Credit: Bakken
(GPL)

6

to give a total of J = 0 angular momentum. Since the protons in the unﬁlled shell sit in the

sd-shell, the parity will be positive giving the ground state of 20Mg a J π = 0+.

While this model does not provide accurate predictions in every case, it does have rela-

tively good predictive power. Evidence for nuclear shells can be seen in nuclei with magic

numbers of N or Z or both tend to be more stable than similar other isotopes or isotones.

In principle, the shell model can be used to predict many important nuclear properties,

such as excited state energies, spin-parity assignments and, probabilities of transitioning

to lower energy states. However, unlike the atomic shell model, the nuclear shell model’s

states are not ordered simply by the quantum number n since spin-orbit interactions and

orbital angular momentum cause the splitting of states. This makes calculating the diﬀerent

many-body as opposed to single-particle excited states diﬃcult since taking into account

all possible interactions between nucleons scales factorially with the number of nucleons!

Diﬀerent eﬀective interactions such as USD, USDA, and USDB [7] can be used in shell

models to simplify the problem by reducing the interactions between nucleons to just the

valence nucleons in the sd-shell and an inert core. More about these models will be discussed

in subsequent chapters where the results from these models are compared to experimentally

obtained values.

1.3 Nuclear Decay

It can be seen from the chart of nuclides (Fig. 1.1) that most nuclei are not stable. These

unstable nuclei will undergo some form of decay until the nucleus becomes stable. In general,

the further a nucleus is from the stable nuclei on the chart of nuclides, the more unstable it

becomes, but what does it mean to be more unstable?

7

It is impossible to predict when an unstable nucleus will decay, however, a large sample

of the same nucleus will follow the exponential law of radioactive decay (Eq. 1.3):

N (t) = N0e−λt

(1.3)

where N is the number of nuclei in the sample left after time t and τ = 1/λ represents

the statistical lifetime (the average time for a nucleus to decay) of the nucleus [6]. The

decay-constant (λ) for very unstable or rare nuclei will be large compared with nuclei that

are close to stability.

There are a few diﬀerent ways a nucleus can decay in order to become more stable. Light

nuclei decay primarily via β-decay (pronounced beta-decay), γ-decay (pronounced gamma-

decay), and particle emission (p, n, α). Heavier nuclei also decay by these same modes, but

additionally have the possibility to spontaneously ﬁssion, where the nucleus breaks apart

into two roughly similar sized lighter nuclei (Fig. 2.2). This mainly happens in much heavier

nuclei with mass A > 232 because the binding energy per nucleon of heavy nuclei tends to be

smaller than their lighter products, and this process can release energy whereas this process

would require an energy input for light nuclei.

In principle a nucleus can decay to more than one diﬀerent state (each transition is

referred to as a decay branch) and the total lifetime of the parent nucleus will be:

1(cid:80)

i λi

τtotal =

(1.4)

However, it is often more important to know the individual λi as these can be used to

represent the branching ratios.

8

1.3.1 β-Decay

β-decay occurs via the weak interaction and is responsible for converting protons to neutrons

or neutrons to protons inside the nucleus, moving the nucleus closer to the valley of stability

[8]. This process conserves mass number A, as well as total charge, and lepton number.

One type of β-decay involves nuclei below the stable nuclei on the chart of nuclides which
are considered neutron-rich. This form of decay is called β−-decay (pronounced beta-minus)
and involves converting a neutron to a proton and emitting an electron e− and electron

anti-neutrino from the nucleus (Eq. 1.5), where AX is referred to as the parent nucleus and

AY is referred to as the daughter nucleus. The state of the ﬁnal nucleus in β-decay will be

more tightly bound than the initial state or in other words have slightly less overall mass.

This small diﬀerence in mass is converted into kinetic energy which is carried oﬀ mainly by

the β particle and neutrino and is referred to as the Q-value.

A

Z X −→A
Qβ− = [m(A

Z+1 Y + e− + νe
Z X) − m(A

Z+1Y )]c2

(1.5)

Another form of decay, β+-decay (pronounced beta-plus or positron), occurs for proton-rich

nuclei above the stable nuclei on the chart of nuclides, and involves the conversion of a proton

in the nucleus to a neutron and the emission of a positron e+ and an electron neutrino (Eq.
1.6). The atom will emit an electron e− in addition to the positron to conserve charge, and

therefore β+ decay can only proceed if the Q-value is greater than 1.022 MeV, the mass of

two electrons.

A

Z X −→A

Z−1 Y + e+ + νe

Qβ+ = [m(A

Z X) − m(A

Z−1Y ) − 2me]c2

9

(1.6)

A third type of decay, called electron capture, also occurs for the proton-rich nuclei. Instead

of creating an electron or positron like in the ﬁrst two processes, electron capture absorbs

an electron from one of the atom’s inner electronic shells in order to convert a proton in the

nucleus into a neutron while conserving charge (Eq. 1.7).

A

Z X + e− −→A
Z X) − m(A

Z−1 Y + νe
Z−1Y )]c2 − BE(e−)

QEC = [m(A

(1.7)

Because β-decay proceeds via the weak interaction, there are rules that dictate the transitions

which are allowed to occur to states in the daughter nucleus. These rules are discussed in

more detail in section 1.3.1.2.

1.3.1.1 Introduction to Isospin

The theory of isospin was developed following the discovery of the neutron [9]. Isospin was

able to describe the approximate symmetry between proton and neutron, in that they felt

approximately the same force with regard to the strong interaction and they had very nearly

the same mass. By this symmetry the proton could be thought of as the same particle as a

neutron but in a diﬀerent state (isospin-up and isospin-down).

Nucleons are fermions which have intrinsic spin 1

2 and isospin t = 1

2, a vector that obeys

the same rules as angular momentum. Therefore, the number of isospin projections for a
given isospin is deﬁned by −T < Tz < T , yielding 2T + 1 projections. By convention the
isospin projection of a proton is Tz = − 1

2 and the neutron has an isospin projection of

10

Tz = + 1

2, and the total isospin projection of a ground state nucleus is:

Tz =

(N − Z)

1
2

(1.8)

A 20Mg nucleus with 12 protons and 8 neutrons will have an isospin projection for the ground
state of Tz = −2. If the degeneracy of the proton and neutron was not broken by the elec-

tromagnetic force, we would expect the ground states and excited states to approximate the

level scheme in Figure 1.4. However because the degeneracy is broken by the electromagnetic

force, a more realistic picture of the level scheme will cause a splitting of these states (Fig.

1.5).

This has consequences for β-decay since the vector (V ) and axial vector (A) components

of the weak interaction operate on the isospin projection by raising or lowering it. Certain

selection rules will therefore govern which states in the daughter nucleus are allowed to be

populated in β-decay.

1.3.1.2

β decay selection rules

Both the Vector and Axial Vector components of the weak interaction contribute to the

β-decay rate R given by Equation 1.9 where Ko is a constant, f depends on the β-decay

Q-value, gV and gA are the vector and axial vector coupling constants, and B(F ) and B(GT )

are the Fermi and Gamow-Teller transition strengths.

Ri,f =

f
Ko

V Bi,f (F±) + g2
[g2

ABi,f (GT±)]

(1.9)

The Fermi and Gamow-Teller transitions describe diﬀerent ways in which β-decay can

proceed. For so-called allowed transitions the emitted β-decay particles carry L = 0 angu-

11

Figure 1.4: An approximate level scheme for the diﬀerent Isospin projections in the A = 20,
T = 2 multiplet, where neutrons and protons are degenerate.

12

Figure 1.5: An approximate level scheme for the diﬀerent Isospin projections in the A =
20, T = 2 multiplet where the degeneracy of the proton and neutron is broken by the
electromagnetic force.

13

lar momentum. β-decays where L > 0 is carried by the emitted particles are considered

forbidden and are not truly impossible but proceed at a much slower rate than allowed

decays.

The operator for a Fermi transition is given by OF =(cid:80)

a ˆτa± and this operates over all
nucleons (a) in the nucleus and ˆτa± serves as the raising/lowering operator for the isospin

projection Tz. For allowed Fermi transitions the spins of the emitted leptons couple together

for a total spin of S = 0, and it follows that the initial and ﬁnal angular momentum of the

nucleus will not change since no angular momentum is carried oﬀ, ∆J = 0 and also isospin

∆T = 0.

The operator for a Gamow-Teller transition is given by OGT = (cid:80)

a (cid:126)σaˆτa± and also

operates over all nucleons (a), where (cid:126)σa is the vector the Pauli-spin matrix which operates

on angular momentum J such that ∆J = 0, 1. Allowed Gamow-Teller transitions occur for

β-decays where the spin of the emitted particles couple together such that S = 1. In this
form of decay the isospin projection is raised/lowered by 1, ∆Tz = ±1, like in Fermi decay
but also has the properties (cid:126)Tf = (cid:126)Ti + (cid:126)1 and (cid:126)Jf = (cid:126)Ji + (cid:126)1, since is a vector operator. From

the addition of angular momentum this produces the rules for allowed Gamow-Teller decay,

∆J = 0, 1 and ∆T = 0, 1 (Table 1.1).

Table 1.1: Selection rules governing β-decay. Columns 1-5 refer to the type of transition,
orbital angular momentum carried oﬀ by β-decay particles, change in isospin between initial
and ﬁnal states of the nucleus, the change in total angular momentum between initial and
ﬁnal state of the nucleus, and the change in parity respectively.

Fermi

Transition Type L ∆T
0
0,1
0,1,2
2,3
3,4

0
0
Gamow-Teller
1
First-forbidden
Second-forbidden 2
3
Third-forbidden

∆J ∆π
0
no
no
0,1
yes
0,1,2
1,2,3
no
yes
2,3,4

14

The consequences of the selection rules become clear for most β-decay processes. The

constraints on the Fermi transition only allow for decay to a single isobaric analog state, and

the decay rate will tend to be much higher for this speciﬁc transition. This type of transition

is also referred to as super-allowed because of its high transition rate. The constraints on

Gamow-Teller decay are not as stringent as for Fermi decay and Gamow-Teller decays will

be split between many allowed transitions.

First forbidden transitions are not considered impossible, but rather far less likely than

the allowed type of decays previously discussed. This is because it is very unlikely for the

β particle and neutrino to be emitted in states with l > 0. Typically, ﬁrst-forbidden decays
are four orders of magnitude (10−4) less likely than allowed β-decay and each higher order

of forbiddenness is suppressed by another four orders of magnitude due to conservation of

angular momentum.

Because the rate of decay occurs at vastly diﬀerent rates for allowed and forbidden tran-

sitions, the selection rules can be applied to determine the J π of states experimentally by

measuring the partial decay rate R. Reorganizing equation 1.9 and taking the decay rate
R ∼ 1
t1/2

, where t1/2 is the partial half-life, we get:

f t1/2 =

Ko
V Bi,f (F±) + g2
[g2

ABi,f (GT±)]

(1.10)

The value log10f t1/2 is used to compare transitions of diﬀerent energy to each other.

Super-allowed fermi transitions tend to have log10f t1/2 values between 3-4, allowed transi-

tions 4-6, ﬁrst-forbidden transitions 6-10, and second forbidden transitions 10-13.

15

1.3.2 γ decay

Following β-decay or particle emission, such as particle emission of protons, neutrons, or α

particles, the resulting nucleus is often left in a nuclear excited state and can de-excite to

other excited states or the ground state by emission of γ-rays. Gamma-rays are photons

emitted by the nucleus at energies typically 100 keV to 10 MeV.

During the γ decay process, the energy ∆E is released in the form of a γ-ray carrying

energy Eγ, as well as a small amount of recoil energy KErec imparted on the nucleus of mass

Mf . This is because photons carry momentum pγ = Eγ/c and the total momentum of the

system must be conserved (Eq. 1.11):

(cid:126)pγ = −(cid:126)prec

∆E = Ei − Ef = Eγ + KErec

∆E = Eγ +

E2
γ

2Mf c2

(1.11)

Measuring γ-ray energies in the lab is an extremely useful tool for scientists to build

a picture of the excited state energies of a nucleus because of the ability to measure their

energies with very high precision.

Like β-decay, γ decay also has its own set of selection rules for angular momentum and

parity. γ-ray transitions emit via electric or magnetic type radiation, and this is determined

by the multipole order given by the quantum number l of the photon. The type of transition

is electric when the change in parity between initial and ﬁnal nuclear states follows πf =
πi(−1)l and magnetic when πf = πi(−1)l+1. The angular momentum of the initial and
ﬁnal states is given by (cid:126)Jf = (cid:126)Ji + (cid:126)l and since the intrinsic spin of a photon is 1, there are
no transitions of type 0 → 0. From the parity selection rules above and the conservation of

16

angular momentum the parity selection rules can be written as 5.7, where ∆π is the change

in parity between initial and ﬁnal state:

|Ji − Jf| ≤ l ≤ Ji + Jf

∆π = no: even electric, odd magnetic

(1.12)

∆π = yes: odd electric, even magnetic

Approximations for the electric multipole transition probabilities can be calculated from

equation 1.13 and magnetic multipole transition probabilities can be calculated from equation

1.14.

λ(EL) ≈ 8π(L + 1)
L[(2L + 1)!!]2

(cid:16) E

(cid:17)2L+1(cid:16) 3

(cid:17)2

e2

4πo¯hc

¯hc

L + 3

cR2L

(1.13)

λ(M L) ≈ 8π(L + 1)
L[(2L + 1)!!]2

(cid:16)

µp − 1

L + 1

(cid:17)2(cid:16) ¯h

(cid:17)2

e2

(cid:16) E

(cid:17)2L+1(cid:16) 3

(cid:17)2

mpc

4πo¯hc

¯hc

L + 2

cR2L−2 (1.14)

These equations are known as the Weisskopf estimates [6] and help to determine which type

of radiation should be more likely in a transition. From an examination of these estimates,

the γ-ray transition rates are dominated by the lower order multipole transitions E1 (electric

dipole), M1 (magnetic dipole), and E2 (electric quadrupole).

17

1.3.3 Nucleon Emission

Similar to β-decay, the Q-value for nucleon emission must be positive for this type of decay

to occur 1.15.

Sn = −Qn = BE(N, Z) − BE(N − 1, Z)
Sp = −Qp = BE(N, Z) − BE(N, Z − 1)

(1.15)

These are referred to as the neutron separation energy Sn and proton separation energy

Sp. Moving away from the valley of stable nuclei on the chart of nuclides we end up at what

is called the drip-line, where nuclei are very proton-rich or neutron-rich, and the boundaries

of Sp and Sn go from positive to negative values. Any nucleon we try to add is not bound

to the nucleus and simply drips oﬀ.

It is possible for nucleon emission to occur where the values of Sp and Sn are positive.

If the nucleus is in an excited state where Eexc. > Sn or Eexc. > Sp, then this opens up

the possibility for nucleon emission which can occur following β-decay where the resulting

nucleus is left in an excited state. This process is referred to as β-delayed nucleon emission.

It is also possible for an α particle or other particles such as a deuteron (2H) to be emitted

if the particle separation energies are low enough.

The nucleus 20Mg can β decay to excited states of 20Na above the proton separation

energy. In this work, we observed the process of 20Mg β-delayed proton emission (denoted

as 20Mg(βp)19Ne) to a number of excited states in 19Ne via the γ-ray emission. Some of

the 19Ne excited states populated by β-delayed proton decay are expected to be above the

α-particle separation energy and can emit an α particle following this decay chain. This

decay sequence is of great interest to many nuclear astrophysicists studying Type I x-ray

18

bursts, which will be discussed in more detail in section 2.1.2.

We have seen that nuclei transition in many ways to become more stable and will con-

tinue to do so if unimpeded (Fig. 1.1). However, the situation becomes more complex in

stars where nuclei regularly interact with one another in very hot environments. For highly

energetic stellar events, the possibility of particle capture to produce more massive nuclei

and the excitation of nuclei via nuclear reactions also occurs and this will be addressed in

the next chapter.

19

Chapter 2

Motivations for studying the

15O(α, γ)19Ne(p, γ)20Na Reaction

Sequence

2.1 Nuclear Astrophysics

A host of observations and experiments conducted in the 1950s helped to deﬁnitively prove

that nucleosynthesis occurs in stars. In 1952 an important discovery of technetium spectral

lines from red giant stars helped solidify this notion [10]. Since there are no stable isotopes

of technetium and the longest lived isotope is much shorter than the life of the universe, the

element must have been created in the star itself!

These observations helped to solidify one of the fundamental principles of nuclear astro-

physics that the Big Bang created the elements Hydrogen, Helium, and Lithium, and the

rest of the elements are created in stars as well as a small amount via cosmic ray spallation

[11].

Observations of the luminosity, or the amount of energy radiated over time, from thou-

sands of stars has helped us to categorize the many diﬀerent types of stars that exist and

build a model of the life cycles that stars follow from birth to death. The luminosity can vary

20

widely from star to star. When the color (which is determined by the surface temperature)

of a star is plotted against the luminosity a clear grouping emerges of the diﬀerent categories

of stars that exist (Fig. 2.1).

The question of where elements are created and how they are distributed throughout

the universe is still open. An initial assumption was that our own solar system isotopic

abundance would be similar to abundances in other stars in the universe. However, within

our own solar system we see elemental abundances that diﬀer quite substantially from other

places in the universe. Our understanding of elemental abundance must therefore proceed

by modeling stellar evolution and the extreme stellar events which synthesize and expel

these elements. Additionally, modeling these extreme stellar events can also complement

observational information about the properties of certain stars such as mass, radius, and

elemental composition.

2.1.1 Nuclear Reactions in stars

To understand nuclear reactions, it is ﬁrst important to understand what is meant by the

term. In order for a nuclear reaction to occur, one or more of the nuclei involved in the initial

collision must be converted into a diﬀerent nucleus. A typical nuclear reaction with target

nucleus X, projectile nucleus a, and reaction products b and Y is written like Equation 2.1,

or X(a, b)Y for shorthand. In the shorthand equation everything to the left of the comma

deﬁnes the reactants and everything to the right are reaction products.

a + X −→ Y + b

(2.1)

21

Figure 2.1: The Herzprung-Russel Diagram relates the luminosity and temperature of a star
and classiﬁes the star based on its position on the diagram. The largest portion of stars lays
within the main sequence category as well as a few within various giant subgroups and white
dwarfs which are remnants of main sequence stars. Only stars which have an observable
luminosity are shown on the diagram and therefore stars like neutron stars do not appear in
this ﬁgure. [ﬁgure credit - https://commons.wikimedia.org/wiki/File:HRDiagram.png]

22

Figure 2.2: The graph displays the binding energy per nucleon for various mass number A
nuclei. The product of nuclear fusions up to iron releases energy, and the ﬁssion of heavy
elements whose fragments are more tightly bound are also exothermic. These processes give
oﬀ energy by converting mass into energy since nuclei with higher binding energy have less
mass per nucleon.

Some examples of nuclear reactions include fusion X(a, γ)Y , breakup X(a, bc)Y , and transfer

reactions X(a, b)Y . If none of the nuclei are transformed in the collision, this is called nuclear

scattering.

During the stellar burning phase of stars, fusion reactions fuel the output of energy by

converting mass into thermal energy. Because the binding energy per nucleon for nuclei

increases with mass all the way up to the iron mass region, nuclei can fuse and convert a

small amount of their mass into kinetic energy in these nuclear reactions (Figure 2.2).

Stellar masses are described in solar mass terms relative to the sun, where the sun is
described as having 1M(cid:12). The mass of a star strongly inﬂuences its lifespan and the evolution

23

of its composition. Stars tend to be grouped into four categories and include red dwarfs
0.08M(cid:12) ≤ M ≤ 0.4M(cid:12), low mass 0.4M(cid:12) ≤ M ≤ 2M(cid:12), intermediate mass 2M(cid:12) ≤ M ≤
11M(cid:12), and massive 11M(cid:12) ≤ M [11] (Fig. 2.1). Stars with higher masses burn their nuclear

fuel more quickly due to the intense gravitational pressure causing high temperatures which

facilitate faster reaction rates.

The end stages of stars also depend strongly on their initial mass. Red dwarf to interme-

diate mass stars will collapse into white dwarfs with their composition depending on their

initial mass (Helium, Carbon/Oxygen, Oxygen/Neon). However, cores of most massive stars

will collapse into either neutron stars or black holes.

Following the collapse of a massive star, the resulting neutron stars which form have a
mass of ≈ 1.5M(cid:12), a radius of 10-15 km, and have a density of ≈ 1014 g/cm3. The star is

composed almost entirely of neutrons and is kept from collapsing by its neutron degeneracy

pressure as well as the repulsive strong nuclear force. Once they are formed they no longer

generate their own heat through fusion. However, they play an important role in binary

systems which can generate outbursts called Type-I x-ray bursts under the right conditions.

Stellar events which synthesize elements in thermonuclear runaway reactions include clas-
sical novae and x-ray bursts which can synthesize isotopes up to mass A ≈ 40 [12] and
A ≈ 100 [13] respectively as well as Type Ia SNe supernovae and neutron star mergers which

are likely responsible for the creation of heavy elements beyond A = 100. These events
are characterized by the rapid capture of neutrons followed by β−-decays and require an

understanding of a host of diﬀerent reaction rates and decay rates which contribute to the

outbursts in order to accurately model these events.

The 15O(α, γ)19Ne(p, γ)20Na reaction sequence, studied in this work, is important for

the understanding of Type-I X-ray burst energy generation, and the understanding of this

24

reaction rate has a large impact on the accuracy of x-ray burst models.

2.1.2 X-ray Bursts

Many stars are not like our own star in that ≈half of all stars tend to have a companion star

and reside in a gravitationally bound binary system. A Type I x-ray burst can occur when
a binary star system, consisting of a neutron star having mass ≈ 1.4M(cid:12) and hydrogen-rich
star with mass ≤ 1.5M(cid:12), orbit close enough that matter from the hydrogen-rich star is

accreted onto the surface of the neutron star [14].

The two stars in the binary system are surrounded by an imaginary surface called the

Roche lobe which marks the point where each star’s gravity is dominant. The point where

the Roche lobe of each star meets is called the inner Lagrangian point and at this point the

eﬀects of gravity and rotations cancel. A star which evolves into a red giant star may expand

beyond its Roche lobe allowing material to freely ﬂow to its companion.

The surface of the neutron star is hot enough to fuse the hydrogen-rich fuel that falls

onto its surface, donated by its companion star, into helium. The surface eventually becomes

hot enough to fuse 3 helium atoms by resonant capture (see Fig. 2.7) into 12C, known as

the triple -α process (discussed in section 2.1.3.1). This ﬁrst phase of Type-I x-ray bursts

is know as the helium ﬂash and ignites the surface of the neutron star in thermonuclear

runaway. The ﬁnal phase of this outburst is a likely site of the rapid proton capture process

[15].

Following the triple-α reaction, the burning phase enters the hot CNO cycles or HCNO,
followed by the α − p process and ﬁnally the rapid proton capture or rp-process where

temperatures in this explosive environment can rise above 109 kelvin.

Type I X-ray bursts can be observed in the galaxy using space-based telescopes. The

25

Figure 2.3: Simpliﬁed diagram of a compact star (black dot) with a hydrogen rich binary
companion (red circle). Each star is surrounded by an imaginary surface called the Roche
lobe, which indicates the point where each star’s gravity dominates. The point where the
Roche lobe of each star meets is called the inner Lagrangian point and at this point the
eﬀects of gravity and rotations cancel Credit: [Cornell, http://hosting.astro.cornell.
edu/academics/courses/astro201/roche_lobe.htm]

26

burst of X-rays typically lasts for less than one minute and can recur at irregular time

intervals of days to months depending on the masses of the stars, distance between the stars,

accretion rate, and chemical composition of accreted matter. A typical X-ray burst will show

rapid increase in x-rays over the ﬁrst 10 seconds followed by exponential decrease over the

next 100s of seconds. The so-called x-ray burst light-curve is the only observable quantity

which can be detected from earth, and attempts to reproduce this light-curve accurately in

computational models provide information about the neutron star and the explosion.

The deep gravitational potential well of the neutron star is great enough that the elements

synthesized in the burst will fall back onto the neutron star, aﬀecting the evolution of its

crust. Therefore, it is unlikely that x-ray bursts contribute much to the chemical evolution

of our galaxy. The bursts can, however, provide information about the properties of the

neutron star, such as its mass, size, and elemental abundance.

2.1.2.1 Hot-CNO cycles and Breakout

During stellar burning in stars the CNO cycle, or carbon-nitrogen-oxygen cycle, is a catalytic

cycle which eﬀectively fuses 4 hydrogen atoms into a helium atom through a multi-step
process of proton captures and β-decays on CNO isotopes. For stars with mass M > 1.3M(cid:12),

this cycle dominates the production of energy. At these temperatures the proton capture

reactions proceed at a much slower rate than β-decay lifetimes and the cycle is said to be

reaction limited.

During a Type-I X-ray burst at thermonuclear temperatures the reactions begin to pro-

ceed at a much faster pace than the β-decays, which are independent of temperature, and

the CNO process becomes β limited and this is referred to as the Hot CNO cycle.

There are multiple Hot-CNO cycles which can occur following the helium ﬂash stage.

27

Figure 2.4 shows the diﬀerent cycles which occur that eﬀectively fuse four protons into a

helium atom. During this cycle proton captures and (p, α) reactions compete with β-decays.

If we consider the HCNO1 cycle, for instance, the β-decay half-life of 14O is 102 seconds

and 15O has a half-life of 176s whereas the reactions proceed almost instantaneously with

respect to the β decays.

The bottleneck for proceeding to higher masses is caused by the long β-decay lifetimes

of 14O and 15O which are considerably longer than a typical Type-I x-ray burst. However,
if an x-ray burst achieves temperatures T ≥ 0.5 GK, reaction sequences which break out

of HCNO cycles into the A > 20 become more likely. The proposed breakout sequences

15O(α, γ)19Ne, 14O(α, p)17F(p, γ)18Ne(α, p)21Na, and 14O(α, p)17F(γ, p)16O(α, γ)20Ne all

involve capture of an α particle on these bottleneck oxygen isotopes. The ﬁrst reaction

of the ﬁrst sequence 15O(α, γ)19Ne is expected to be the most important reaction to study,

kicking in to allow HCNO breakout ﬁrst as the temperature rises, in order to more accurately

model the x-ray burst light curve [16].

2.1.3 Mathematical Formalism

Nuclear reactions in relatively low temperature stellar environments, such as our sun, tend to

occur between nuclei in their ground states. These reactions can be computed by using three

variables. The ﬁrst is the cross-section, σ which characterizes the probability for two nuclei

to undergo a reaction. The second is the abundance of the reacting nuclei, and the third

is the relative speed between the interacting particles. The cross section can be determined

experimentally from the measured reaction rate R,

R = σIaN

28

(2.2)

Figure 2.4: The three Hot CNO cycles shown here occur during the second phase of ther-
monuclear runaway in Type I x-ray bursts. The gray boxes represent the stable isotopes and
the arrows show the reactions and β decays that proceed within the diﬀerent cycles. [Figure
adapted from [11]]

29

where Ia represents the ﬂux of incoming particles and N the number of target nuclei per

area.

In a thermonuclear environment where explosive burning occurs, nuclei can be excited
from their ground state to excited states by absorbing γ-rays X + γ −→ X∗ or through
inelastic scattering, i.e. X(a, a(cid:48))X∗ where X∗ denotes a nucleus in an excited state. This

opens additional channels for nuclear reactions to occur and must be taken into account to

model the astrophysical process accurately.

Below we will provide a short derivation of the thermonuclear reaction rate formalism

used for Type-I x-ray burst environments and discuss both direct and resonant capture.

Following that we will brieﬂy discuss calculating the reaction rate by observing the reverse

reaction.

From the description of the cross section in equation 2.2, we can obtain (Eq. 2.3,

R12 = N1N2 < vσ >

(2.3)

where N1 and N2 are the interacting particles, v is the relative velocity between N1 and N2,

and σ(v) is the reaction cross section.

The bracket notation < vσ > in equation 2.3 notes that this quantity is averaged since

the relative velocity v is described by a continuous distribution in stellar environments.

In principle the cross section σ(v) can be described by a geometric factor, the interaction

matrix element, and a penetrability factor. Classically this geometric factor is related to

the area of the projectile, but quantum mechanically the geometric factor is related to
the deBroglie (λp = h√
p. The
interaction matrix element is given by M =< f|H|i > where i describes the wave functions

) wavelength of the projectile instead, and given by πλ2

2mE

30

of the initial projectiles N1 and N2 and f describes the wave functions of the reaction

products. Finally, the penetrability Pl(E) describes the likelihood for the projectile to reach

the target nucleus. Two eﬀects which strongly reduce the penetrability are the Coulomb

barrier and the angular momentum barrier. For charged particles the nuclei must overcome

a repulsive Coulomb force which peaks at the nuclear radius. Since angular momentum

values must be discrete in quantum mechanics, L = (cid:112)l(l + 1)¯h, any incoming projectile

with nonzero impact parameter will have less radial kinetic energy available to overcome the

central potential and this manifests itself as the angular momentum barrier. Using these

three factors the cross section can be written as

σ ∝ πλ2

p · | < f|H|i > |2 · Pl(E)

(2.4)

where E is the energy of the incoming projectile. At astrophysical temperatures, the pene-

trability factor Pl(E) is dominated by the l = 0 term and can be expanded to be dependent

only on the Sommerfeld parameter η:

Pl(E) ∝ e−2πη, η =

(cid:114) µ

Z1Z2e2

¯h

2E

(2.5)

Z1 and Z2 describe the charge of the target and projectile nuclei, e is the elementary charge

carried by a single proton, and µ is the reduced mass of the two nuclei.

Because the velocities are non-relativistic and particles are in thermal equilibrium, the

Maxwell-Boltzmann distribution can be used to represent the statistical velocities:

(cid:16) µ

(cid:17)3/2

2πkT

−µv2
2kT

v2e

f (v) = 4π

31

(2.6)

The variables represented in equation 2.6 are µ =

m1m2
m1+m2

, denoting the reduced mass, k

the Boltzmann constant, T the temperature of the explosive environment, and v the relative

velocity between the reaction nuclei. By substituting in the Maxwell-Boltzmann distribution

for the relative velocities of the particles in equation 2.3 we obtain the reaction rate:

R12 = 4πN1N2(

µ

2πkT

)3/2

−µv2
2kT dv

v3σ(v)e

(2.7)

(cid:90) ∞

0

The center of mass energy E, rather than their relative velocities, is often used to char-

acterize the kinetic energy between two particles and it is helpful to transform equation 2.7

into the center-of-mass frame and is often written as the average rate per particle pair:

(cid:114) 8

πµ

R12
N1N2

R12 =

< σv >12=

(cid:90) ∞

N1N2
(kT )3/2

(cid:114) 8

0
1

πµ

(kT )3/2

0

Eσ(E)e−E/kT dE
(cid:90) ∞

Eσ(E)e−E/kT dE

(2.8)

From the derivation of the reaction rate in equation 2.8, we can see that the cross section

and the energy of the particles both contribute in diﬀerent ways to the overall reaction rate.

The cross-section of rapid proton capture in thermonuclear environments is proportional

to σ(E) ∝ 1/e−√

E. Graphing both the cross-section and Maxwell-Boltzman distributions

as a function of energy shows that at very low energies, the particles cannot overcome the

penetrability factor determined by the cross-section, but there is also a low probability for

nuclei to have energies, given by the Maxwell-Boltzman distribution, which can overcome

the penetrability. Figure 2.5 also shows the convolution of these two distributions which

determines the probability as a function of energy for charged particle reactions to occur.

The spike in probability is referred to as the Gamow-window and represents the energy at

which most charged-particle reactions take place in thermonuclear burning.

32

Figure 2.5: Example Gamow window (red) which is created from the convolution of the
Maxwell-Boltzman distribution (black) with the penetration factor Pl(E) (blue). The
Gamow peak is enhanced by a factor of 100 to make the peak more visible.

33

Energy (arb. units)cross section (arb. units)Figure 2.6: This simple ﬁgure illustrates a direct capture reaction of a nucleus X and a
proton. The resulting nucleus Y emits a γ-ray and settles into a bound state. The dotted
line denotes the separation energy of a proton Sp, or the energy required to remove a proton
from the nucleus.

While this is one consideration experimentalists take into account when performing re-

action experiments, another important consideration involves resonance reactions, which

like the Gamow window can also dramatically increase the rate at which reactions proceed

depending on the kinetic energies of the nuclei.

2.1.3.1 Direct and Resonant Capture Reactions

From equation 2.8 we have deﬁned what is known as a direct thermonuclear reaction rate

for charged particles. To understand what this means it is helpful to reference a diagram of

the particle capture with respect to the excited states in the resulting nucleus.

34

In Figure 2.6 the total mass of the reactants will be slightly larger than the ground state

mass of the ﬁnal nucleus and this is how energy is generated in stars. The particle, in this

case a proton, is captured to what is called a bound state which is deﬁned to be below the

separation energy Sp, or the energy it takes to remove that particle from the ﬁnal nucleus.

The excess energy is released via γ radiation.

In principle a nucleus has a multitude of

separation energies for diﬀerent particles such as neutrons, protons, and α particles which

are all relevant to nucleosynthesis.

In the case where the product nucleus has an excited state which is close to the center

of mass energy of the reaction particles, the two particles can proceed via resonant capture.

Perhaps one of the most famous resonant capture reactions is the triple-α process [5]. There

are no stable nuclei with 5 or 8 nucleons, and this process is physically necessary in order to

bypass this gap in stability in order for carbon to be produced in stars.

In Figure 2.7 we see that while 8Be is unstable it can exist long enough to fuse with an α

particle to create 12C. Rather than capture a particle to a stable state of the product nucleus,

resonant capture to the Hoyle state, an excited state in 12C predicted by Hoyle at 7.7 MeV,

will result in the capture of a particle to an unbound state. It is possible for this excited

state to re-emit the particle or it can de-excite to a bound state of 12C via γ-emission.

The probabilities that the α particle will be re-emitted by 12C or de-excite via γ decay

are related to the partial widths Γγ and Γα of the excited state, where the total width

Γ = Γγ + Γα. The width of a resonance is expressed in units of energy and the probabilities

of each decay are determined by the branching ratios Γγ/Γ and Γα/Γ.

For resonant capture, the cross section σ(E) in equation 2.8 is given by the Breit-Wigner

35

Figure 2.7: This simple ﬁgure illustrates the famous triple α resonant reaction. In this case
the α particle is captured onto an excited state in 12C where the nucleus can either decay
via γ-rays to a bound state or the α particle could be re-emitted. The resonance energy Er
is the diﬀerence between the excitation energy in 12C and the α particle separation energy
Sα

36

formula:

σ(E) =

λ2
4π

ω

Γ1Γ2

(E − Er)2 + (Γ/2)2 , ω =

2Jr + 1

(2Ja + 1)(2Jb + 1)

(2.9)

where λ lambda is the DeBroglie wavelength of the incoming projectile, Γ1 and Γ2 are

the partial widths related to the two modes of decay, E and Er are the center of mass energy

and the resonance energy respectively, and Ja, Jb, and Jr are the spins of the reaction and

product nuclei.

By making the assumption that the width of the resonance is narrow and isolated (Γ (cid:28)

Er), which is the case for both the 15O(α, γ)19Ne and 19Ne(p, γ)20Na reactions considered

in the present work, the formula for the resonance reaction rate associated with a single

resonance can be reduced to:

(cid:16) 2π

(cid:17)3/2

kT µ

< σv >=

¯h2e−Er/kT ωγ,

γ =

Γ1Γ2

Γ

(2.10)

Assuming Er is known, all factors in equation 2.10 are reduced to a constant except

ωγ (not to be confused with gamma-ray, γ) and this is called the resonance strength. This

means we can experimentally measure the branching ratios of speciﬁc decay channels (from

an astrophysical resonance), in addition to the spins and lifetimes of the relevant states,

my measuring the reaction rate, rather than trying to recreate the direct reaction! This is

extremely important for experimentalists because it is sometimes very diﬃcult or impossible

to measure the cross section of a resonant reaction directly, especially when one of the

reactants is unstable.

37

2.2 The 15O(α, γ)19Ne Reaction Rate

In the study of Type I x-ray bursts there are a few reaction bottlenecks whose unknown or

highly uncertain rates can have large eﬀects on simulated burst proﬁles. The most important

reaction rate to determine is the 15O(α, γ)19Ne Hot CNO cycle breakout reaction which

heavily aﬀects the onset of the burst [16]. A single resonance is expected to dominate the

reaction rate and corresponds to an excitation energy of Ex(19Ne) = 4.03 MeV.

It is not possible with current facilities to directly measure the 15O(α, γ)19Ne reaction

rate because an 15O rare isotope beam of suﬃcient intensity is not available. However, the

resonance strength can be indirectly determined from measurements of the spin, lifetime,

and branching ratio Γα/Γ of the 4.03 MeV state. Currently, the spin is known to be 3/2+

[17] and the lifetime has been measured [18, 19, 20] to suﬃcient precision leaving the α-decay

branch the only quantity left to be measured.

2.2.1 Previous studies of 15O(α, γ)19Ne

Many diﬀerent transfer reaction methods have been utilized in order to populate the 4.03

MeV state in 19Ne and subsequently search for the α branching ratio. The ﬁrst transfer
reaction p(21Ne,3H)19Ne∗ utilized inverse kinematics (heavy beam and light target) to pro-
duce the 19Ne∗ and tritons (3H) [21]. The combination of tritons and low energy α particles

provides a unique energy signature when searching for the α branch. The background in

this experiment was small and well measured, but they were not sensitive to the α branch

from the 4.03 MeV state due to insuﬃcient statistics and only determined an upper limit of
Γα/Γ < 4.3 × 10−4. A second transfer reaction by Rehm et al. 3He(20Ne, α)19Ne∗ was used

and produced no background in the search region for the α branch, but were also not sensi-

38

tive to the low energy α particles and also produced an upper limit of Γα/Γ < 6× 10−4 [22].
Finally, the 19F(3He, t)19Ne∗ transfer reaction was used by Tan et al. [23] which produced a

much higher background than the other two experiments but with increased statistics. From
this experiment an α branching ratio of Γα/Γ = 2.9 ± 2.1 × 10−4 was extracted. However,

this value is controversial because it is nearly consistent with 0 and the modeling of the

background is not well determined at the lowest energy α branches [24].

Other experiments also employed transfer reactions to populate 19Ne∗ states above the

α separation energy, Sα [25, 26, 27] but were not sensitive the α branch from the 4.03 MeV

state.

Using transfer reactions to populate the 4.03 MeV state in 19Ne and subsequently measure

Γα/Γ has thus far proved elusive [28, 29]. Transfer reactions produce a lot of diﬀerent by-

products and can create a signiﬁcant amount of statistical noise, suppressing the ability to

detect α particles above background. Additionally, α particles are emitted at very low energy
from the 4.03 MeV state because the resonance ≈500 keV is very close to the α separation
energy and the α particle will also lose ≈ 20% of its kinetic energy via the recoiling 15O due

to conservation of momentum.

2.2.2 Purpose of This Work

Rather than relying on transfer reactions to populate the 4.03 MeV state in 19Ne, a recently

proposed technique for measuring Γα/Γ employs the decay sequence 20Mg(βpα) for which

the last step is α-particle emission to 15O, the inverse of α capture [30]. An important
component to identifying p − α coincidence events of interest is the unknown energy of the

proton(s) emitted from the excited states in 20Na which feed the 4.03 MeV state in 19Ne.

In the present work, we employ a γ-ray spectroscopy technique to measure the proton

39

energy. When a nucleon is emitted from a nucleus, following β decay, the momentum of

the system must be conserved so the daughter nucleus will recoil with equal and opposite

momentum as the ejected nucleon. If a γ ray is emitted before an excited daughter nucleus

has time to stop the resulting γ ray will be Doppler shifted. The resulting γ ray line shape

will be broadened. The broadened feature preserves information about the energies of the

emitted nucleons, which is modeled using a computer simulation.

2.3 The 19Ne(p, γ)20Na Reaction Rate

The 19Ne(p,γ)20Na reaction (Q=2190 keV) [31, 32] is expected to be dominated by the only

resonance in the Gamow window at 457 keV (Ex=2647 keV) above the proton threshold

(Fig. 4.1) [33]. The matter of its spin and parity has been debated for nearly 30 years

because these quantities are crucial to determining the resonance strength. The most likely

values for a spin and parity assignment have been suggested to be 1+ or 3+ [34]. A J π of 1+

would result in a signiﬁcantly slower reaction rate than a 3+ assignment [33, 35] and would

be more likely to cause the 19Ne (J π = 1/2+) to return to the hot-CNO cycles rather than

breaking out to higher mass nuclei.

2.3.1 Previous studies of 19Ne(p, γ)20Na

Many diﬀerent types of experiments have been carried out in the past to extract information

about the 2647 keV state including stable beam reaction experiments, rare isotope beam

experiments and β-decay experiments.

Early experiments focused mainly on stable beam transfer reactions in order to populate
the state. Kubono et al. discovered the ≈2647 keV 20Na state via the 20Ne(3He,t)20Na [36]

40

reaction and subsequently Lamm et al. remeasured the same reaction and argued for a 1+

assigment based on a DWBA analysis [37]. Smith et al. also measured the 20Ne(3He,t)20Na

reaction and improved the precision of the 19Ne(p,γ)20Na resonance energies [38]. A theo-

retical study of the mirror energy levels in 20F and 20Na by Brown et al. determined from

the available data that a 3+ assignment of the 2647 keV state was more likely by pairing

it with the 2966 keV mirror state in 20F [33]. The state has also been observed via the

20Ne(p,n)20Na reaction through which a 3+ assignment was made based on the angular

distribution of the neutrons [39]. Concurrent with the present work, Belarge et al. used a

rare-isotope beam of 19Ne to measure the 19Ne(d,n)20Na proton transfer reaction and were

able to populate the three lowest energy resonances and observe their proton emissions to

both the ground state and excited states [40], arguing for a J π of 3+ for the 2647 keV state

based on the analysis of the reconstructed neutron angular distribution.

The β-decay of 20Mg (J π=0+) to 20Na has been utilized because it constrains the J π of

the 2647 keV state experimentally. If the J π of the state were 1+ then its β-decay feeding

by 20Mg would be allowed by β-decay selection rules. However, if the state were 3+ then the

β transition would be second-forbidden and very weak. An early study of 20Mg β-delayed

proton decay by G¨orres et al. determined an upper limit on the feeding of the 2647 keV state
to be 2 × 10−3 under the assumption that proton emission dominates over γ-ray emission

[41]. Piechaczek et al. also conducted a 20Mg β-decay study which yielded an upper limit
on the β-decay feeding of 1 × 10−3 to the 2647 keV state in 20Na [42]. The experiment

by Piechaczek et al. searched for protons and also for selected γ-rays but did not ﬁnd any

evidence for the 2647 keV state being populated at their level of sensitivity. A recent study

of 20Mg β-delayed proton emission by Wallace et al. yielded the most stringent upper limit
on the partial β-decay branch to the 2647 keV state of 2× 10−4 leading to a partial log f t of

41

>6.9 [35], making a 1+ assignment unlikely under the implicit assumption that the proton

branching ratio is large. Recent experiments by Sun et al. [43] and Lund et al. [44] searched

for both protons and γ-rays but did not have enough statistics to improve upon the previous

constraints on J π.

Rare isotope beams of 19Ne have been used to search for the important resonance directly

at astrophysical energies. Two experiments yielded upper limits on the resonance strength

of 21 meV [45] and 15 meV [46] with 90% conﬁdence. A recent experiment concurrent with

the present work conducted at TRIUMF utilized a high intensity 19Ne beam and was able

to measure the resonance strength directly for the ﬁrst time [47] to be 17+7−5 meV.

2.3.2 Purpose of This Work

Previous analysis of 20Mg β-decay to the 2647 keV state has relied heavily on the protons

emitted from this state and has rarely addressed the γ-ray branch.

In fact, a complete

experimental upper limit on the β-decay feeding of this state has never been set. Either 1+

or 3+ assignments should yield a γ-decay branching ratio Γγ/Γ on the order of 10% [45].

This value is based on theoretical estimates and could be larger. Therefore, it is important

to measure the β-delayed γ-decay branch or limit it experimentally. In the present work, we

describe a search for these 20Mg β delayed γ-rays to complement the previously measured

upper limit [35] on the proton branch. This work was done concurrently with the direct

reaction rate experiment at TRIUMF [47].

42

Chapter 3

Experiment E14066 at the National

Superconducting Cyclotron

Laboratory

Experiment E14066 was carried out at the National Superconducting Cyclotron Laboratory

at Michigan State University in May 2015. The experiment lasted for ﬁve days beginning on

May 8 and ending on May 13. Hour long data runs were taken during the experiment to en-

sure consistent measurements from our assortment of detectors. Additional runs were taken

before and after the experiment for calibration purposes and to characterize the background

(see section 5.1.1.6 for treatment γ-ray of background).

3.1 Purpose

We utilize 20Mg β-decay in order to populate excited states in 20Na and 19Ne. As stated in

the previous chapter, the purpose for this work is two fold.

First we search for γ-rays emitted from the excited state resonance in 20Na at 2647 keV

to determine Iβγ, the intensity of 20Mg β decays which feed the 2647 keV state and de-

excites followed by γ-ray de-excitation. This measurement can be used in conjunction with

previous searches for the protons emitted from this state, which yielded an upper limit on Iβp

43

measured by Wallace et al. [35]. This combined upper limit can be used to help determine

the J π of this state and constrain the astrophysical reaction rate important in the breakout

of the HCNO cycle in Type I x-ray bursts.

Secondly, we search for and characterize γ-rays emitted from the 4.03 MeV excited state

in 19Ne. By utilizing 20Mg(βp), we can determine if this reaction sequence can populate the

4.03 MeV excited state suﬃciently to search for the α-particles in a future experiment.

3.1.1 Beam Production

Stable Primary beam particles are stripped of their electrons and injected into the center of

the ﬁrst cyclotron where a transverse magnetic ﬁeld causes these charged particles to move

in a circular motion. A rapidly changing electric ﬁeld is applied in the direction of the ions,

causing them to accelerate. The radius of the charged particles in the cyclotron will continue

to increase along with their speed until they reach the desired electrostatic channel used to

extract ions, where they exit along the beam line before heading to the second cyclotron to

pick up more energy.

The primary beam for this experiment was 24Mg and was accelerated through the K500
and K1200 CCF at the NSCL to an energy of ∼170 MeV/u. The intensity of the primary

beam was 60 pnA (particle nanoamperes). The 24Mg beam was then impinged on a 9Be

target with a thickness of 987 mg/cm2. This fragmentation procedure knocks out nucleons

from the nuclei of some ions in the primary beam that hit the target and produces a cocktail

beam of both stable and radioactive isotopes including the desired 20Mg.

The fragmentation of the primary beam produces a lot of contaminants in the secondary

beam, which we do not want to deliver to the target. The A1900 magnetic separator is used

to remove the bulk of these unwanted particles. The A1900 includes 4 dipole magnets and a

44

Figure 3.1: A schematic of the K500 and K1200 coupled cyclotron feeding into the A1900
magnetic separator. The 24Mg stable primary beam is accelerated through the couple cy-
clotrons to ∼ 1/2 the speed of light and impinged onto the 9Be target. The fragments of the
secondary beam are separated using 4 dipole magnets (green) and an aluminum degrading
wedge (yellow). The secondary beam is then delivered to our experimental setup in the S2
vault.

wedge degrader. The dipole magnets separate out fragments according to the Lorentz force

mv/q = Bρ where m is the mass of the ion, v the velocity, q the charge of the nucleus,

and Bρ represents the magnetic rigidity. The magnets in the A1900 are tuned in order to

keep the 20Mg at the center of the beam-line and ions with diﬀering magnetic rigidity will

travel at diﬀerent radii and become separated from the secondary beam. Ions with diﬀerent

masses may still share the same rigidity as the desired 20Mg, so a wedge is placed after the

ﬁrst two dipole magnets which slows down ions proportionally with the square of their mass

A2 in order to change the unwanted ions’ magnetic rigidity. The wedge degrader is made

of aluminum and has an eﬀective thickness of 594 mg/cm2 for the 20Mg path of travel. A

diagram of the beam production can be seen in ﬁgure 3.1.

The secondary beam then exits the A1900 magnetic separator through the focal plane

where collimating slits are used to block excess unwanted ions. A 1 mm thick scintillator is

45

Figure 3.2: A birds-eye view of the experimental setup drawn using computer-aided design
(CAD) software.

also placed at the focal plane of the A1900 which is used to identify particles and will be

discussed in the next section. After passing this A1900 X-focal plane (XFP) scintillator the

secondary beam is delivered to our experimental setup in the S2 vault of the NSCL.

3.1.2 Experimental Setup

There are three types of detectors included in our experimental setup. These detectors

consist of a silicon PIN detector (Fig. 3.4), 16 high-purity germanium detectors (SeGA),

and a plastic scintillator (Fig. 3.5).

The silicon PIN (positive-intrinsic-negative) detector was located ∼1 m upstream of the

46

Experiment 14066-Wrede    Existing Beamline w/ Thin Kapton Window XYZ SeGA Stand Base Beta SeGA w/ 2” ext. bars on each side to make room for GeDSSD plastic scintillator and germanium detectors. The PIN detector consists of a p-type semi-

conductor (positive charge from electron hole), intrinsic-type semiconductor (neutral), and

n-type semiconductor (negative charge) sandwiched together. A reverse bias (voltage) ap-

plied to the PIN detector will produce an electric ﬁeld which outputs a signal when charged

beam particles pass through and create charge carriers, where the signal strength is propor-

tional to the square of the charge Z2 of the ion. In our case a voltage of 22.0 V was applied to

the 300-µm-thick Si PIN detector while under vacuum. This detector is used in conjunction

with the A1900 XFP scintillator to produce a timing signal proportional to the time of ﬂight

between the scintillator and the PIN detector. The time-of-ﬂight was measured over a path

of 25 m using the Si detector and a scintillator at the focal plane of the A1900. Together,

the signal from the PIN detector can be used with the timing signal to produce a particle

identiﬁcation plot (Fig. 3.3) using the ∆E-TOF method. The ﬁnal beam consisted of 34%

20Mg (QEC =10.7 MeV), 24% 18Ne (QEC =4.4 MeV), 12% 17F (QEC =2.8 MeV), 22% 16O

(stable) and 8% 15N (stable).

While the PIN detector is important for general particle identiﬁcation, the detector can-

not be used when the full beam is being delivered to the setup since the high beam current

can radiation damage the detector. Therefore, particle ID runs were done intermittently be-

tween full beam runs with 30x attenuated beam, and used to ensure the beam composition

delivered to the experimental setup was consistent throughout the experiment. A pneumatic

drive was used to lower the PIN detector into the beam-line during particle ID runs and

removed during production runs. A computer aided design drawing of this drive system is

shown in ﬁgure 3.4

The beam then travels through a thin kapton window (Fig. 3.5), after which it continues

through 20 cm in air before implanting into the EJ200 plastic scintillator. The plastic

47

Figure 3.3: TOP: An example particle ID spectrum taken in between full beam runs. The
energy lost to the Si PIN detector is plotted against the time of ﬂight (TOF) between the
XFP scintillator and the Si PIN detector. The large blob in the top right is the 20Mg and
represents ∼ 40% of the secondary beam implanted in the plastic scintillator.
Bottom: The total beam rate was increased in later runs which sacriﬁced 20Mg purity
(∼ 34%) for an increase in 20Mg pps implanted into the plastic scintillator.

48

Figure 3.4: The pneumatic drive utilizes a gas cylinder to retract the PIN detector from
the beam-line when full beam is on and releases the PIN detector into the beam-line during
particle ID runs.

scintillator is a 5.0 x 5.0 by 2.5 cm block and has a photomultiplier tube mounted onto the

back. The scintillator was supported mechanically by clamping the phototube onto a long

piece of aluminum mounted onto the top of the SeGA frame (Fig. 3.6) so that the face of

the scintillator is situated 20 cm front of the kapton window, centered on the beam axis.

The photomultiplier tube mounted on the plastic scintillator was biased to 1000 V during

the experiment. A scintillator will produce light when the radiation interacts with it. The

photomultiplier tube then ampliﬁes the light signal which is sent to the data acquisition

system. The energy resolution of the scintillator is not useful except to distinguish between

an implant and a decay since the energies of these events are vastly diﬀerent. A spectrum of

the scintillator energy is shown in ﬁgure 3.7 where the energies below 10,000 ADC channels

are related to β decays and above 20,000 ADC channels are due to implantation events.

The plastic scintillator is surrounded by 16 high purity segmented germanium detectors

49

Figure 3.5: The secondary beam exits through a thin kapton window after which it travels
20 cm in air before implanting into the EJ200 plastic scintillator.

50

Figure 3.6: To hold the plastic scintillator in front of the thin kapton window an aluminum
mount was used which connected to the top of the SeGA support structure and clamped
onto the end of the scintillator’s photomultiplier tube.

51

Figure 3.7: A total spectrum of EJ200 scintillator energies detected during E14066 plotted
vs γ-ray energies detected in SeGA. The region below 10000 channels is comprised of β-decay
events and the γ-ray lines above 20000 channels are related to implant events which we want
to remove by placing a gate (only looking at speciﬁc events) on the β-decay energy region.

(SeGA) in two rings of 8 detectors, one ring downstream and one upstream. Germanium

detectors are also positive-intrinsic-negative type detectors and can detect ionizing radiation

such as γ-rays between 100 keV and 10 MeV with high resolution. The germanium detectors

must be kept cool, due to a lower band gap than in Si, to obtain a consistent energy resolution
and avoid damage to the detectors so liquid nitrogen (77◦ K) is pumped into dewars attached

to each segmented germanium detector. These detectors were used to search for γ-ray

transitions in 20Na and 19Ne.

Three distinct processes can occur when γ-rays interact with matter. The ﬁrst is the

photoelectric eﬀect where an incident γ-ray is fully absorbed and transfers all of its energy

to electrons (called photoelectrons). The photoelectrons are collected at the edge of the

semiconductor and produce a signal proportional to the γ-ray energy. The second process

is called Compton scattering where the incident γ-ray imparts some of its energy onto an

52

electron and scatters away. In principle a scattered γ-ray can also produce photoelectrons

or it can escape the detector volume without being detected. This interaction causes a

characteristic Compton background for a single γ-ray energy. The third type of interaction

is called pair production. If a γ-ray has energy greater than 1.022 MeV (the mass energy of

two electrons) in the presence of nuclei, then the photon’s energy can be converted to particle
mass by producing an electron positron pair (e+− e−) [48]. The e+ will then annihilate with
another e− to produce to 511 keV gamma rays. The probability of pair production occurring

increases as a function of the incident γ-ray energy. This process produces a characteristic

escape and double escape peak where the energy detected is 511 keV and 1022 keV less than

the full photopeak energy.

During the experiment we noticed a much higher rate of γ-ray background at high energies

than was expected and it was proposed that some β particles from the β-decays in the

scintillator were entering the germanium detectors which can also produce ionizing radiation.

To test this hypothesis the plastic scintillator was surrounded with a thick piece of steel

to shield the germanium detectors from any escaping βs. There was no reduction in noise,

however this helped conﬁrm another process which produced γ-rays in the scintillator. When

electrons or positrons are deﬂected by a charged particle, they will lose energy which is

converted into a photon. This process is known as bremsstrahlung, or braking radiation,

and produces a continuous background in the γ-ray spectrum up to higher energies. The

background contribution due to bremsstrahlung was borne out by MCNP simulations using

a set of β endpoint energies produced by the β-decays in the scintillator [49, 50]. The total

γ-ray background at high energies does not aﬀect our ability to search for γ-ray peaks of

interest.

53

3.1.3 Data Acquisition

The digital data acquisition system (DDAS) is supported by the NSCL and is built around

the Pixie-16 digitizer which has the ability to quickly process signals from a variety of

diﬀerent detectors . Each module has 16 channels and is placed in a computing crate where

multiple modules can be connected. Prior to running the experiment diﬀerent parameters

for processing signals can be input into the software such as a threshold trigger to ﬁlter out

low energies as well as a rise time and decay time of signal. Pulse processing is then applied

to a trace (Fig. 3.8) and outputs the area of the pulse as a binary number which is stored

along with a time-stamp for the signal in an event ﬁle. A more detailed application of the

NSCL digital data acquisition system (DDAS) can be found in reference [51].

During the experiment all signals detected within a certain time frame are stored in

a bucket called an event. The event window deﬁnes the length of time where signals are

recorded and placed in a single event. During the experiment an event window of 1 µs
was used for events with full beam with ∼ 12000 particles per second implanted into the

scintillator comprised of particles proportionate with the PID runs in ﬁgure 3.3. This event

length was chosen to avoid pile-up events where multiple signals are detected in the same

detector. Particle ID runs required an extended event window of 10 µs.

Events processed by the NSCL digital data acquisition are managed by the Readout

program. Each signal processed by DDAS is placed into a buﬀer where Readout buckets

these signals into events deﬁned by the event window. Each of these events are stored in

event (.evt) ﬁles managed by Readout which is also responsible for stopping and starting

runs and producing scalers related to the event frequency in each detector for each event ﬁle.

The energy signals from the germanium detectors as well as the scintillator and PIN

54

Figure 3.8: An example trace from the silicon PIN detector. The decay time of signals from
the Si PIN detector is similar to the SeGA detectors (∼45 µs) and EJ200 plastic scintillator
traces decay much faster (∼1 µs). The program used to view the trace as well as test diﬀerent
signal parameters is called NScope.

55

Figure 3.9: Simpliﬁed schematic of E14066 electronics. The blue boxes denote the raw input
signals taken from the experimental setup as well as the RF signal and XFP scintillator signal
taken from the beam operators. The white boxes denote the NIM modules which shape the
signals and produce timing signals which are input into DDAS. The orange denotes the
output of data to either the NSCL DAQ computer or the patch panel which ports signals to
the NSCL Data-U.

detector can be directly sent to the DDAS module for processing, however, the timing signals

need to be ﬁltered before being processed by DDAS which is optimized to process preampliﬁer

pulses. The electronics used in this experiment were relatively simple and are displayed in

the electronics diagram (Fig. 3.9).

A description of the modules used in E14066 are as follows:

1. Ortec 142 Preampliﬁer - Ampliﬁes singal from Si PIN detector.

2. TC 241s Ampliﬁer - The ampliﬁer takes the preampliﬁed signal from the Si PIN de-

tector and shapes it into a unipolar gaussian-like waveform.

56

3. Canberra QCFD - The Quad Constant Fraction Discriminator (QCFD) is used to

extract time information from a detector. In the case of E14066 the input signal comes

from the TC 241s preamp with input from the Si PIN detector. The signal is split into

two signals where one is then inverted, attenuated, and delayed. The two signals are

then recombined to create a bipolar wave and the point where the signals sum to zero

is taken as the pulse time.

4. Time Amplitude Converter (TAC) - The TAC takes two input signals, which are con-

sidered the start and stop timing signal. The diﬀerence in the start and stop times

of the signals are converted into a voltage pulse with amplitude proportional to the

start-stop time.

5. Delay Box - The delay box is a box with a very long length of wire which serves to

delay the time for a signal to reach its destination.

6. 2mm To LEMO Converter - The standard 2mm cables used for the NSCL DDAS are

not compatible with the output to the patch panel and require a converter to LEMO

cables.

7. Fast Trigger - A module which produces a logic signal when a threshold is met by the

input signal.

8. Fan In Fan Out FIFO - This module is capable of taking in many signal inputs and

outputting to many sources.

9. TTL to NIM - The module translates the TTL logic signal (positive square wave) from

the fast triggers and SeGA OR into NIM signals (negative fast pulse).

57

3.1.4 Data Procedure

In principle each detector channel in the SeGA array will output the same value for a given

energy input. However, over time the output values may drift slightly and this is known as

gain drift. In order to protect against gain drifts, a new run is started every hour so that

each run can account for these drifts over time in the energy calibration procedure. It is

also practical to start new data runs regularly, minimizing the amount of data lost due to

crashes of Readout, which stores DDAS signals in a buﬀer.

A majority of the analysis was done oﬄine by converting the evt ﬁles output by Readout

into ROOT ﬁles which store the time and area of each signal within an event in a tree

structure. These trees can be eﬃciently looped over to sort the data into diﬀerent spectra

that are used for analysis.

3.1.4.1 Data Reduction

For experiment E14066 we have two methods to reduce the amount of background in our

data. Each signal stored in an event might be detected in coincidence with the other signals.

Since we are looking to detect γ-rays following β-decays we may only want to look at γ-rays

which correlate with β-decay energies in the scintillator. By gating on events with these low

β-decay energies in the scintillator we can produce a β-delayed γ-ray spectrum. From ﬁgure

3.7 a reasonable gate can be placed below 10000 ADC channels to encompass a majority of

low-energy scintillator events. By doing this we reduce γ-ray signals unrelated to β-decay

events by a factor of 2 in the γ-ray spectrum and eliminate peaks which are produced by

natural sources of γ radiation (Fig. 3.10).

In addition to gating on β decays in the scintillator, we can also gate on the time diﬀerence

between a signal in the scintillator and a signal in a germanium detector. In a β-delayed

58

Figure 3.10: The blue line shows a total γ-ray spectrum without gating on the low energy
portion of the scintillator. By gating on the β decay energies we reduce the noise in our γ
spectrum (green) by a factor of 2 and remove many of the background γ-ray peaks caused
by nuclear reactions as well as natural sources of radiation.

59

Energy (keV)0100020003000400050006000700080009000Counts/keV210310410510610710810Figure 3.11: SeGA γ-ray energies vs time produced by subtracting the timestamp of a
germanium detector signal from the timestamp of the scintillator signal. The X axis values
are oﬀset by adding 10000. Each clock tic on the DDAS crate represents 10 ns and is the
smallest increment of time which can be used.

γ-ray event the signal from the scintillator will be produced before the signal in a germanium

detector. By only accepting events where these signals are detected by DDAS in the correct

order we can also reduce the random noise produced by an accidental coincidence in the

plastic scintillator concurrent with γ-rays. A timing spectrum produced by subtracting these

two quantities shows a very clear peak on β-delayed γ events (Fig. 3.11). A timing gate is

placed on channels 9940 to 10010. This reduces a small amount of background, however,

most of the background is reduced by gating on the low energy portion of the scintillator.

The peak is asymmetric due to a phenomenon called time-walk which causes the time signals

between the scintillator and SeGA detectors to increase for low energy γ-rays. The pulse

shape from the detection of γ-rays is ﬁltered by the digital CFD in DDAS. The lower energy

γ-ray pulses result in the timing signal, picked oﬀ by the digital ﬁlter, to be detected later

and follows an exponential trend as a function of energy.

60

time (Scintillator-SeGA)[timestamp] (10 ns tics)99209940996099801000010020100401006010080 Energy (keV)g210310110210310410510610710Figure 3.12: A paper holder was made with a small pocket for the 125Sb calibration source
and the holder ﬁt over the face of the plastic scintillator to mimic as closely as possible the
eﬃciency from beam particles implanted into the plastic scintillator. During calibration runs
the holder was placed on the face of the scintillator.

61

Following the experiment, data runs were taken with the beam oﬀ to measure back-

ground from long lived beam contaminants as well as natural background γ-ray sources. An

absolutely calibrated mixed source of 125Sb (T1/2 = 1008(2) days), 154Eu (T1/2 = 3141(12)

days), and 155Eu (T1/2 = 1738(4) days) was used following the background runs to measure

the eﬃciency of SeGA for our experimental setup. The source was created in 1988 and the

shorter half-life 125Sb and 155Eu have nearly all decayed away leaving only 154Eu β-delayed

γ-rays from 100 - 1600 keV. A paper holder was made with a small pocket for the calibra-

tion source which ﬁts over the face of the plastic scintillator to mimic as closely as possible

the eﬃciency from beam particles implanted into the plastic scintillator (Fig. 3.12). The

calibration of γ-ray energies and eﬃciencies are discussed in the next chapter alongside the

analysis.

62

Chapter 4

Indirect study of the 19Ne(p, γ)20Na

reaction using 20Mg(βγ)20Na

4.1

Introduction

As discussed in section 2.3.2, the search for a γ-decay branch from the 2647 keV is required

to complement the upper limit which has been placed on the proton branch [35, 40].

4.2 Analysis and Results

A ﬁrst order γ-ray energy calibration of the SeGA detectors was applied to the pulse area

spectra in each run by performing a gain matching using strong room-background lines from
the β-decays of 40K and 208Tl which produced γ-ray energies of 1460.851±0.006 keV [54]
and 2614.511±0.010 keV [55], respectively. Peaks were ﬁt using an exponentially modiﬁed

gaussian function (Eq. 4.2) added to a local background model (usually a linear function)

to determine the centroids. A cumulative spectrum incorporating all 16 SeGA detectors

was generated by applying a coincidence gate in the timing spectrum between SeGA and

scintillator events in order to select the γ-rays originating from a β-decay event (Fig. 4.2).

By applying this timing gate we were able to compare the ratio of photopeak counts in the

ungated spectrum to the counts in the gated spectrum. This yielded a consistant ungated

63

Figure 4.1: Relevant low-lying states in 19Ne and 20Na labeled with J π and energy in keV.
Energies below the proton threshold in 20Na as well as J π are adopted from Seweryniak et
al. [52] and energies above the proton threshold are from Wallace et al. [53].

64

Figure 4.2: Cumulative spectrum of 20Mg β-delayed γ-rays acquired by SeGA, in coincidence
with β-decay events in the plastic scintillator.

to gated ratio of 0.90(1) for 20Na and 20Ne peaks over a broad energy range representing

the constant eﬃciency of the scintillator to detect β particles from 20Mg β-decay. Further

energy calibration was applied to the cumulative spectrum [56] to account for second order

eﬀects, which reﬁned the calibration by less than 1 keV.

The photopeak γ-ray detection eﬃciency of SeGA, up to 1596 keV, was measured using

an absolutely calibrated 154Eu source placed in the center of the front face of the plastic

scintillator (Fig. 4.4). The total number of 20Mg ions implanted into the scintillator (N20M g)

was calculated indirectly using the number of counts in the 984 keV transition peak, the
known 20Mg β-delayed γ-decay intensity of the 984 keV transition in 20Na of 0.697±0.012

[42] (recently conﬁrmed in [44]) and the eﬃciency at this energy, which was determined to
be E984= 0.0653±0.0025 by interpolating the eﬃciency data from the calibration source.

A Geant4 Monte Carlo simulation was used to model the experimental setup and sim-

65

Figure 4.3: Measured 90% C.L. upper limits on the intensity of 20Mg β delayed γ-rays as
a function of energy in the 2647 keV search region. The most conservative limit is at 2645
keV.

66

ulate an independent eﬃciency curve [57]. In order to match our experimental conditions,

monoenergetic γ-rays were emitted isotropically from the center of the scintillator and inter-

acted with the surrounding SeGA array to produce a gamma ray spectrum. The photopeak

eﬃciency was extracted from the generated spectrum and this procedure was repeated over a

wide range of energies from 0 to 8 MeV for comparison with the experimental data. Compar-

ison of Geant4 simulations of γ-rays emitted from the center of the scintillator versus the

front face of the scintillator showed a negligible diﬀerence in total photopeak eﬃciency due
to the ≈1 cm diﬀerence in calibration source position and online source position, allowing

us to treat our calibration source data without any correction for source position.

The discrete Geant4 eﬃciency curve was interpolated with a continuous function of the

form (Eq. 4.1)

i=5(cid:88)

i=0

ln(E) =

Ci(ln(E))i

(4.1)

and compared to the calibration data (Fig. 4.4). The ratio of the Geant4 simulation eﬃcien-

cies to the eﬃciencies from calibration data show that we can scale the Geant4 simulation

by a constant normalization factor of 0.975 to fall in line with the data. We therefore used the

scaled functional form representing the Geant4 simulation to interpolate and extrapolate

the eﬃciency continuously as a function of energy. The systematic uncertainty associated
with extrapolating was estimated to be ≤5% for γ-ray energies < 2.7 MeV [58]. Statistical

uncertainties associated with the calibration data were obtained from the exponentially mod-

iﬁed Gaussian ﬁts. Systematic uncertainties associated with the source were calculated from
uncertainties in the branching ratios and were determined to be ≤1%. A simulation was run

in Geant4 to account for the loss of photopeak counts due to γ-γ summing eﬀects which

resulted in a 2% systematic uncertainty at all energies. All uncertainties were combined in

67

Figure 4.4: Upper panel: Photopeak eﬃciency of SeGA as a function of energy. Circles
correspond to eﬃciencies from the Eu calibration source. Squares correspond to eﬃciencies
from a Geant4 simulation of the experimental setup. Fit function (4.1) was used to model
the Geant4 simulation eﬃciency. The ﬁt was then scaled by a constant factor to line
up with the source calibration eﬃciency (red line). Lower panel: The residual eﬃciency
is the relative diﬀerence between the calibration source data and the scaled ﬁt of Geant4
simulation eﬃciencies.

quadrature to determine a total eﬃciency uncertainty for each γ-ray line from the source.

4.2.1 Analysis

In order to measure the feeding of the 2647-keV 20Na state, we searched for γ-ray lines in

the SeGA spectra corresponding to all possible primary branches by applying exponentially

modiﬁed Gaussian ﬁts of the form

(cid:114) π

2

N σ
λ

(cid:32)

(cid:16)σ

(cid:17)2

λ

1
2

exp

(cid:33)

(cid:32)

erf c

(cid:16) σ

λ

1√
2

x − µ

λ

+

(cid:17)(cid:33)

x − µ

σ

+

(4.2)

f (x; N, µ, σ, λ) =

68

Energy(keV)Efficiency %51015Eu Calibration Run154Geant4 SimulatedExperimental dataEnergy(keV)100020003000400050006000Efficiency %  Residual10-5-0510to the spectrum and ﬁxing the mean (µ) over a range of energies. Based on well known

unbroadened β-delayed γ lines in the spectrum from 20Na and 20Ne [34] we parameterized

the line shape using the width of the Gaussian (σ) and exponential decay constant (λ) as a

function of energy and interpolated these values to each region of interest. N is deﬁned to

be the area below the curve, B is a linear background, and erfc is the complementary error

function.

We determined the intensity of each 20Na γ-ray branch in 20Mg β-decay, using the number

of counts in the photopeak NE at γ-ray energy E, the photopeak eﬃciency of SeGA at energy
E, EE, and the number of 20Mg ions implanted into the plastic scintillator, N20M g, as input

to the following equation:

IE
βγ =

NE

EEN20M g

(4.3)

4.2.2 Results

No signiﬁcant γ-ray peaks were observed to correspond with the eleven possible transition

energies deexciting the 2647-keV state of 20Na. The upper limit on Iβγ for all but three

branches was determined by a single exponentially modiﬁed Gaussian (Eq. 4.2) with linear

background ﬁt over a range of energies (Fig. 4.5). For two of the other three branches a

Gaussian + linear function was used to represent the background (Fig. 4.6) because of a

narrow background peak near the search region. Searching for the peak at 1298 keV was

complicated by the fact that it lies at nearly the same energy as a Doppler-broadened 19Ne

peak [29].

In this special case a Monte Carlo method was used (see section 5.2.2) to ﬁt

the Doppler-broadened 19Ne peak using known proton intensities and energies [42, 44]. In

69

Figure 4.5: Each panel shows a magniﬁed region of the spectrum of 20Mg β-delayed γ-rays
acquired by SeGA, in coincidence with β-decay events in the plastic scintillator (Fig. 4.2).
The regions of interest are determined by the possible transition energies from the 2647-
keV state in 20Na (Table 4.1). The blue points represent the data with statistical error
bars and the black lines a linear ﬁt of the background. The red dotted lines represent the
background ﬁt plus the 90% conﬁdence upper limits for each possible transition energy. For
each transition the ﬁt shown corresponds to the energy within the ±6 keV search range that
yielded the maximum upper limit.

70

Figure 4.6: Each panel shows a magniﬁed region of the spectrum of 20Mg β-delayed γ-rays
acquired by SeGA, in coincidence with β-decay events in the plastic scintillator (Fig. 4.2).
The regions of interest are determined by the possible transition energies from the 2647-keV
state in 20Na (Table 4.1). The blue points represent the data with statistical error bars and
the black lines a linear plus Gaussian ﬁt for the background. The red dotted lines represent
the background ﬁt plus the 90% conﬁdence upper limits for each possible transition energy.
For each transition the ﬁt shown corresponds to the energy within the ±6 keV search range
that yielded the maximum upper limit.

71

Figure 4.7: A magniﬁed region of the spectrum of 20Mg β-delayed γ-rays acquired by SeGA,
in coincidence with β-decay events in the plastic scintillator (Fig. 4.2). The blue points
represent the data with statistical error bars. The black line represents the background
Monte Carlo plus continuum background model described in the text. The red dotted line
represents the background Monte Carlo ﬁt plus the 90% conﬁdence upper limits for the 1298
keV transition energy. The ﬁt shown corresponds to the energy within the ±6 keV search
range that yielded the maximum upper limit.

72

Table 4.1: Upper limits on the intensities of 20Mg β-delayed γ-ray transitions through the
2647-keV 20Na state with 90% conﬁdence. Final energy level values were adopted from
Seweryniak et al. [52]. The search for each γ-ray encompassed a range of ± 6 keV.

Transistion (keV) Energy (keV)

Transition

Iβγ

20Na γ-ray
2647 → 0
2647 → 600
2647 → 799
2647 → 984
2647 → 1032
2647 → 1349
2647 → 1829
2647 → 1837
2647 → 1848
2647 → 1992
2647 → 2060

2647
2047
1848
1663
1615
1298
818
810
799
655
587

< 2.5×10−5
< 3.7×10−5
< 5.5×10−5
< 3.5×10−5
< 1.0×10−4
< 5.3×10−4
< 5.6×10−5
< 6.7×10−5
< 8.9×10−5
< 2.7×10−5
< 8.2×10−5

the simulation, protons were emitted isotropically from the center of the plastic scintillator

and the stopping power was determined by SRIM [59]. The relative proton intensities and

energies were ﬁxed, which determined the peak shape, and the overall normalization was

left as a free parameter. Due to incomplete charge collection associated with the large peak,

we couldn’t model the continuum with a simple linear function. Instead we connected two

linear functions (to represent the distinct backgrounds on either side of the peak) using a

continuous step underneath the Doppler-broadened peak. The Doppler-broadened peak was

added to this function and used to model the total background. An unbroadened 20Na peak

was searched for on top of the background (Fig. 4.7).

Each individual transition was searched for over a ±6 keV range about its nominal value

to cover any potential inaccuracies in the literature energy of the 2647-keV state. The

ﬁtting procedure output the number of counts in the peak and a corresponding uncertainty

represented together by a Gaussian probability density function (PDF). The 90% C.L. upper

limit on the number of counts was determined by integrating the PDF from zero to 90% of

73

Figure 4.8: Upper limits on the intensity of the 20Mg β-decay transition to the 2647 keV state
of 20Na compiled from diﬀerent sources [41, 42, 35]. All of the upper limits from previous
work (dashed lines) are incomplete. The upper limits in grey include only the proton branch
to the ground state of 19Ne. The upper limit by Piechaczek includes the proton branch to
the ground state and selected γ-ray branches. The present work includes proton branches to
the g.s. and 1st excited state in 19Ne [35, 40] as well as all energetically possible γ branches
in 20Na, making it the only complete limit.

the integral from zero to inﬁnity. By applying Equation (2), the most conservative upper

limit on the intensity in the search range was determined for each transition (Table 4.1).

For example, Fig. 4.3 shows the intensity upper limits in the search range for the 2647-keV

γ-ray.

To ﬁnd the total upper limit on Iβγ, the intensity for each transition was calculated from

the central value of the number of counts using Equation (2). The intensities were summed

and the uncertainties were added in quadrature to produce a Gaussian PDF representing Iβγ.

The 90% C.L. upper limit on Iβγ was determined by integrating the PDF from zero to 90%

of the integral from zero to inﬁnity. In this procedure, a single value of the excitation energy
of the 2647-keV state was used for all transitions. By repeating this procedure over the ±6
keV search range, the maximum value of the upper limit was found to be Iβγ < 5.7 × 10−4.
The 984 keV peak is the only γ-ray transition from 20Na that is present in the cumulative

74

spectrum, and we can search for coincident γ-rays in a γ-gated spectrum for transitions from

the 2647 keV state. By looking at the γ-gated spectrum we ﬁnd that there is no evidence

for transitions from the 2647 keV state.

Additionally gates were placed on γ-ray energies in search for γ-γ coincident peaks form

the 2647 keV state, however no peaks were present in these gated spectra above upper limits

determined from the cumulative spectrum.

4.3 Discussion

Wallace et al. determined the proton intensity from the 2647 keV state to be Iβp < 2× 10−4
[35]. However, this limit did not include the recently detected proton emission to the 1st

excited state of 19Ne at 238 keV, which Belarge et al. measured to have approximately the

same branching ratio as emission to the ground state [40]. The inclusion of this new path

leads to Iβp < 4 × 10−4. The underlying PDF representing Iβp was not documented in

Wallace et al. so we assume that their uncertainties were also dominated by statistics and

that they followed a procedure similar to ours. Adding their limit on Iβp with our limit on
Iβγ in quadrature has yielded the ﬁrst complete experimental limit of Iβ < 6.9 × 10−4 and
log ft > 6.4 for the 20Mg β+ decay transition to the 2647-keV 20Na state.

Table 4.2: Gamow-Teller strengths B(GT) and associated log ft values calculated for 20Mg
β-decay to 1st and 2nd excited 1+ states in 20Na using the sd shell model with various
interactions described in the text.

Interaction B(GT) 1+
1

log ft 1+

1 B(GT) 1+
2

4.03
3.96
3.93
3.94
3.67

75

USD
USDA
USDB

USDB-EDF

IMSRG

0.366
0.431
0.460
0.446
0.828

log ft 1+
2

3.59
3.89
3.80
3.82
4.01

0.994
0.494
0.621
0.587
0.377

For comparison to the experimental limit, we calculated theoretical values of log ft for

the transition to the unbound 2nd 1+ state of interest as well as the bound 984 keV 1+ state

utilizing the sd shell model with harmonic oscillator radial wave functions and a quenching

of 0.6 [60] (Tab. 4.2).

In order to assess the uncertainty in this calculation, a variety of

interactions were used including USD [61], USDA [62], USDB [62], USDB-EDF, and IMSRG

[63, 64, 65]. The USDB-EDF calculation was used to assess the uncertainties associated

with the radial wave functions for the loosely-bound states involved and the EDF part was

obtained with the Skx Skyrme functional [66]. All calculated log ft values for the bound 984
keV state were in the range 3.67 − 4.03. This agrees with the measured values of 3.87 [67]
and 3.83 ± 0.02 [42]. Similarly, the range of values we expect for the transition to the 2nd
excited 1+ state in 20Na is 3.59 − 4.01, much lower than the lower limit placed on the log

ft. Therefore, the 2647-keV state does not correspond to the 1+ state from the shell model.

In fact, the 2647 keV state is likely not any 1+ state because conﬁguration mixing should

result in smaller log f t values than observed even for an intruder 1+ state.

Considering a 3+ assignment, the log ft values of second forbidden ∆J = 3 β-decay

transitions in the mass region 9 < A < 27 are all greater than 14 [68]. Our limit of

log ft > 6.4 for the transition from J π = 0+ 20Mg to the 2647 keV state is, therefore,

consistent with a 3+ assignment; however, this does not exclude other possibilities for the

J π assignment.

The 2987 keV 20Na state is most likely the analog of the 1+ 20F state at 3488 keV [33, 53].

log ft values of 4.08(6) and 4.07(3) were measured in the previous work for the feeding of
this 20Na state [42, 44], which is essentially consistent with the range 3.59 − 4.01 predicted

by our shell model calculations for the 2nd 1+ state.

76

4.4 Conclusion

An intense source of 20Mg and a γ-ray spectrometer with high resolution and eﬃciency were

used to search for population of the 2647 keV state in 20Na via β-decay. For the ﬁrst time,

all possible γ-ray branches from this state were limited in order to complement previous

searches for the proton branch. An upper limit on the β delayed γ decay intensity was

measured completing the limit on the log f t value. The combined results from the present

20Mg(βγ) experiment and past 20Mg(βp) [35] and 19Ne(d, n)20Na experiments [40] make an

assignment of 1+ highly unlikely, in agreement with recent work. This result is consistent

with the concurrent direct measurement [47].

77

Chapter 5

Toward the 15O(α, γ)19Ne Reaction

Rate using Doppler Broadening in

20Mg(βpγ)19Ne

5.1 Doppler Broadening Line Shape Analysis Technique

The analysis of β-delayed nucleon emission has traditionally relied on the direct detection of

the emitted nucleons to measure important nuclear properties such as excited state lifetimes,

excited state transition energies, feeding intensities, and excited state spin-parity. However,

a technique measuring just the line-shape of γ-rays following nucleon emission can provide

information on all of these quantities.

Doppler shifts occur when a γ-ray is emitted from a recoiling nucleus following particle

emission from a nucleus initially at rest. One such example is β-delayed proton emission to

an excited state daughter (Equation 5.1) where Q, R, and S represent diﬀerent isotopes, A

represents the number of nucleons, and Z the number of protons in a nucleus.

Z → R∗A
QA

Z−1 → S∗A−1
Z−2

(5.1)

Due to conservation of momentum (Eq. 5.2) the daughter nucleus, S, will recoil with

78

equal and opposite momentum from the emitted nucleon. The center of mass energy, de-

termined by the conservation of momentum, is calculated using (Eq. 5.3). This implies the

daughter, S, carries mp/mS of the center of mass energy. The recoiling daughter nucleus

will, therefore, preserve information about the energy of the emitted nucleon which can be
used to determine which excited state the parent nucleus, R∗, was in.

mSvS = −mpvp

ECM =

1
2

mSv2

S +

1
2

mpv2
p

(5.2)

(5.3)

An ensemble of such events will give rise to a broadened peak shape. The Doppler

Broadening technique utilizes the information about the shape of the γ peak to reconstruct

information about the daughter and parent nucleus. The recoil from nucleon emission and

the resulting Doppler-shifted γ-rays will contain information about the energy of the emitted

nucleon. The emitted nucleon can also cause the γ-ray to be emitted in a preferential

direction from the direction of recoil. This can give us information of the excited state spin-

parity of the parent and daughter nucleus. Each of these features cause the shape of the

resulting peak to be modiﬁed from an un-broadened line shape, and a reconstruction of the

peak from simulations can be used to extract these quantities.

5.1.1 Monte Carlo Simulation of broadened 19Ne γ lines

The Monte Carlo simulation of a γ peak should take into account the information from

the parent nuclei in addition to information about the experimental setup. Inputs of the

79

simulation for this particular experiment include the Center of Mass (CoM) energy between
the emitted proton and 19Ne∗ state (denoting the 19Ne in an excited state), the lifetime
and excitation energy of the 19Ne∗ state, the stopping power of the implantation material

(Polyvinyltoluene), and the response function of each SeGA detector.

5.1.1.1 Treatment of Center of Mass Energy

The recoiling 19Ne∗ is given an initial kinetic energy based on the CoM energy of the proton

emission from an excited 20Na state which is, to a good approximation, at rest in the plastic

scintillator following β decay. The Monte Carlo simulation works by ﬁrst assuming a lifetime
for a 19Ne∗ state and randomly sampling the exponential decay curve distribution.

A state can be fed by multiple excited states and, with enough sensitivity, can allow for the

measurement of multiple proton feeding intensities. Alternatively, if there is no information

about excited states in the proton emitter an excited state energy can be reproduced in the

simulation by iterating the proton energy.

5.1.1.2 Treatment of Excited State Lifetime (τ ) and Stopping Power

The excited state lifetime of 19Ne∗ is invariably linked to how much the nucleus slows down

inside the implantation material. The longer the lifetime (τ ), the more time the recoiling

nucleus has to slow down in the medium.

The uncertainties in lifetime and stopping power are treated as a systematic uncertainties

in the simulation for the quantity that is being measured. The lifetime tends to be the less

well known quantity of the two in the literature since simulations of stopping power tend to

be accurate to within a few %, however it would be possible to measure the stopping power

of a material given suﬃciently precise measurements of all other inputs into the simulation.

80

The daughter nucleus, 19Ne, will emit Doppler shifted γ rays at an energy determined

from Equation 5.4, where θ is characterized by the angle between the γ-ray and recoiling

19Ne −→v .

(cid:115)

Eγ = Eγo

1 + β
1 − β

, β =

v · cos(θ)

c

(5.4)

In a vacuum the daughter nucleus emits a Doppler shifted γ ray with energy dependent

on its initial kinetic energy and cos(θ). However, in order to analyze γ rays from rest

in 20Mg(βpγ)19Ne decay with signiﬁcant intensity, the 20Mg beam must be stopped in a

material. The recoiling 19Ne nucleus will slow down in the material due to electronic and

nuclear stopping powers.

The stopping power of the material is determined as a function of the speed of the recoil

nucleus as well as the properties of the implantation material and can be simulated using

SRIM [59] (Fig. 5.1) or LISE++ [69]. We therefore, apply the energy loss recursively over

many small time-steps in order to mimic a continuous energy loss.

= −fe(v) − fn(v)

dE
dx

(5.5)

This is implemented by applying equation 5.5 over many small time-steps, where dE

dx is the

energy lost by the recoiling nucleus in the medium and fe(v) and fn(v) are the electric and

nuclear stopping power terms derived from the material of the medium and the velocity v

recoiling nucleus.

The Doppler broadened line shape is most apparent in cases where the excited state of

the daughter nucleus B is short lived.

81

Figure 5.1: The stopping power is treated iteratively over many time steps which are deter-
mined by the lifetime of the 19Ne excited state. The stars represent the sum of the electric
and nuclear stopping powers of the scintillator at diﬀerent recoil energies and the line rep-
resents each time-step traversed by the algorithm, updating the recoil energy and stopping
power continuously.

82

Recoil Energy (MeV)00.020.040.060.080.10.120.140.16Stopping Power (kev/micron)2802903003103203305.1.1.3 SeGA Detector Response

A γ ray that is detected by a SeGA detector will deposit energy dependent on a response

function which is well characterized by an exponentially modiﬁed Gaussian (EMG) Eq. 4.2.

Each detector can have diﬀerent resolution which has the largest eﬀect on the σ parameter

in Eq. 4.2. Therefore, we need to characterize this quantity as a function of energy for each

SeGA detector in order to properly simulate the ensemble of Doppler-Broadened γ-rays

entering our setup.

In order to do this unbroadened β-delayed γ ray peaks were ﬁt over a range of energies

to parameterize σ as a function of energy for each detector. β-delayed γ ray peaks were ﬁt

using Eq. 4.2 at energies 238 (19Ne), 984 (20Mg), 1634 (20Ne), 2312 (14N), 3332 (20Ne), and

6129 (16O) keV (Eq. 4.2). The exponential parameter (λ) was ﬁxed to 0.7 as this ﬁt the left

side tail of the peak well at all energies in all detectors. All other parameters were left free.

The value of σ was plotted as a function of energy and ﬁt using a linear function (Figs.

5.2,5.3,5.4). Each detector has a diﬀerent contribution to the total number of counts in the

peak depending on eﬃciency and the simulation reﬂects this by normalizing the number of

counts simulated in each detector.

83

(a)

(b)

(c)

(e)

(d)

(f)

Figure 5.2: The σ parameter energy dependence for input to the exponentially-modiﬁed-
gaussian response function for each SeGA detector. Each data point corresponds to the
value of the σ parameter for a particular calibration peak. The σ parameter is ﬁt using a
line and the conﬁdence band [red online] shows 1 standard deviation uncertainty:(a) Detector
1; (b) Detector 2; (c) Detector 3; (d) Detector 4; (c) Detector 5; and (d) Detector 6;

Rather than sampling the γ-ray’s ﬁnal energy from the EMG probability distribution, it

is much faster to sample the energy from the normalized cumulative distribution function

Eq. 5.6. From Fig. 5.5 we can generate a random number between 0 and 1 to sample the

84

Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.3: The σ parameter energy dependence for input to the exponentially-modiﬁed-
Gaussian response function for each SeGA detector. Each data point corresponds to the
value of the σ parameter for a particular calibration peak. The σ parameter is ﬁt using a
line and the conﬁdence band (red online) shows 1 standard deviation uncertainty:(a) Detector
7; (b) Detector 8; (c) Detector 9; (d) Detector 10; (c) Detector 11; and (d) Detector 12;

85

Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5(a)

(b)

(c)

(d)

Figure 5.4: The σ parameter energy dependence for input to the exponentially-modiﬁed-
Gaussian response function for each SeGA detector. Each data point corresponds to the
value of the σ parameter for a particular calibration peak. The σ parameter is ﬁt using a
line and the conﬁdence band [red online] shows 1 standard deviation uncertainty:(a) Detector
13; (b) Detector 14; (c) Detector 15; and (d) Detector 16;

86

Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Energy (keV)0100020003000400050006000 EMG parameters0.511.522.533.5Figure 5.5: Example Cumulative Distribution Function for a 1232 keV γ.

output energy of a SeGA detector.

F (x; N, µ, σdet, λ = 0.7) =

(cid:32)

exp

(cid:32)

(cid:16)σ

(cid:17)2

λ

1
2

N
2

x − µ

λ

+

(cid:33)

(cid:32)

erf c

x − µ

(cid:17)(cid:33)
(cid:17)(cid:33)

+ 1 (5.6)

(cid:16) σ

λ

1√
2
− erf c

σ

+

(cid:16)x − µ√

2σ

5.1.1.4 Angular Correlations

Angular correlations between protons and γ rays can have an eﬀect on the overall line-

shape [70]. Generally speaking, the direction of the proton may produce a γ-ray angular

87

Energy (keV)40154020402540304035404040454050Random Number Energy Sample00.20.40.60.81distribution described by a linear combination of even Legendre polynomials [71] in the

center of mass frame (Eq. 5.7).

W (θcm) =

(cid:88)

2κ

AκPκ(cos(θcm))

(5.7)

The highest order Legendre polynomial for each γ ray transition is determined by the spin

of the proton-emitting 20Na state, multipolarity of the γ ray transition, angular momentum
of emitted proton, and spin of 19Ne∗ [72, 73] such that

2κmax ≤ min[(2j20Na), (2L)max, (2l)max, (2j19Ne* − 1)max]

The spin of 20Na states is constrained to be 0+ and 1+ in allowed 20Mg β decay, restricting

the angular correlation function to the P0(cos(θcm)) and P2(cos(θcm)) terms. A ﬁrst order

assumption is made that the isotropic term (P0) dominates and a P2 term will be added if

a good ﬁt can not be achieved with this assumption.

An angle, dependent on the angular distribution function 5.7, is randomly chosen between

the recoiling 19Ne atom and emitted γ ray, to calculate the Doppler shift at the observation

point. This γ ray enters a random detector and the known response function of that detector

is treated as a probability density function which outputs a ﬁnal observed energy. An

ensemble of such events can be used to construct a simulated peak shape for comparison to

the actual data.

5.1.1.5 Summary of Monte Carlo Procedure

1. The lifetime τ and the proton energies which feed the excited state γ transition are

ﬁxed. In principle these may not be known quantities and multiple simulations are

88

done over a set of these quantities. The γ-ray energy is also ﬁxed to the mean value

of the Doppler Broadened line.

2. The stopping power is determined over a large range of energies so that interpolation

can be done to model a continuous energy loss by the recoiling particle. The stopping

power of a number of materials can be obtained using SRIM or LISE++.

3. The decay time of each simulated nucleus is sampled from a distribution determined

by the ﬁxed statistical lifetime.

4. An initial velocity is prescribed to the recoil nucleus depending on the energy imparted

via the emitted proton.

5. The stopping power is continuously sampled over 50 time-steps where the small distance

dx the nucleus traveled is calculated from the current velocity of the nucleus and

multiplied by the current stopping power ( dE

dx (E)) to get the energy loss for each τ /50

time-step.

6. After the ﬁnal velocity is calculated from the previous step, a random projection of

the angle [cos(θ)] between the emitted photon and direction of the nucleus is chosen to

apply a Doppler shift correction to. Because the recoil nucleus is initially at rest there

is no preferential direction for the initial velocity in our case. In principle the recoil

nucleus can polarize the direction of the emitted γ-ray and this should be taken into

account at this step rather than assuming a random cos(θ).

7. Finally, a treatment of the SeGA response function is applied to the γ-ray energy.

8. Repeat steps 3-7 to compare an ensemble of events to the data and to reduce statistical

noise from the simulation.

89

5.1.1.6 Treatment of Background

In the case of a γ ray peak with low statistics, it is suﬃcient to use a linear function to model

the background.

The treatment of background from high statistics peaks requires a more complex model.

In some cases it is possible to use an error function where the slope on the left side of the

peak is similar to the slope on the right side of the peak, but this is not always the case.

Instead a more general model is used to ﬁt the background where the slope on the left and

right side of the peak do not match.

The background on the left side of the peak is ﬁt with a linear function as well as the

right side, fl(x) = m1x + b1 and fr(x) = m2x + b2. A symmetric range around the centroid
of the peak is chosen such that the ﬁt spans [µ − a , µ + a].

c1(x) = (2a − x)/2a

c2(x) = x/2a

N = c1(x)3 + c2(x)3

BG(x) = (fl(x)c1(x)3 + fr(x)c2(x)3)/N

This creates a smooth tanh-like background and can generalize for cases where fl (cid:54)= fr.

The cubic exponents in the background function can be changed to diﬀerent values to assess

the systematic uncertainties in the centroid and area of the peak. In this work systematic

uncertainties introduced by the treatment of the background were much smaller than other

sources of uncertainty when used to ﬁt high-statistics peaks.

90

5.1.2 Doppler Broadening Systematic Uncertainties

In order to extract accurate information from the Doppler broadening of each peak it is

important to ﬁrst quantify how well we know the inputs and how sensitive the simulation

will be to slight changes in each quantity.

The stopping power, which is determined by SRIM, is expected to be accurate to within

10% [59]. The uncertainty in the stopping power is directly related to the uncertainty in the

lifetime and will have a greater systematic eﬀect when the lifetime of the excited state is not

well known or unknown.

The exponentially modiﬁed gaussian response function is well known for all 19Ne γ ray

energies. The σ parameter in the response function has <0.7% uncertainty for each detector

below 1600 keV, however, this uncertainty is larger in the case of the 4.03 MeV γ ray which

lies far away from many of the β-delayed γ rays used to model σ.

The ﬁnal two inputs of the Doppler broadening simulation, 19Ne excited state lifetimes

and the feeding intensities and energies from 20Na excited states, have large literature uncer-
tainties, and in some cases, are unknown. The absolute 20Mg(βp)19Ne∗ feeding intensities are

obtained from the direct proton measurements of Piechaczek et al. and Lund et al. [42, 44]

and used when available. The uncertainty in the better known quantity between the lifetime

and proton feeding energy is used to determine a systematic uncertainty in measurements

of the lesser known quantity, or will be considered a free parameter for χ2 minimization if

there are no prior measurements. In the subsequent subsections this is delineated for each

case.

The systematic uncertainties determined from these quantities are combined in quadra-

ture with statistical uncertainties.

91

In all ﬁts described below, we are able to achieve a minimum in the χ2

ν distribution close

to 1, using an isotropic distribution of γ-rays with respect to proton distribution indicating

that we are not sensitive to angular correlations.

5.2 Results and Discussion

The decay scheme presented in Figure 5.6 is deduced from the γ ray spectrum obtained

in this experiment. Only the 19Ne levels which are populated by 20Mg(βp) are displayed.

The measured 20Mg(βp) intensities and γ ray energies are reported in Table 5.1. The γ

ray intensities per 20Mg β decay (Iβpγ) are determined from the integral of each ﬁt. These

values are corrected for the SeGA eﬃciency and normalized to the number of 20Mg β decays.

We proceed to discuss the individual 19Ne states.

92

Figure 5.6: 19Ne level scheme from 20Mg(βpγ)19Ne decay deduced from the present work.
The γ ray transition intensities are denoted by the thicknesses of the arrows, which are
proportional to their intensities. The 20Mg(βp) feeding intensities are denoted by the arrows
on the right.

93

Table 5.1: Column one reports the 19Ne excited-state energies populated by 20Mg(βp), and were determined by applying
recoil corrections to the measured γ ray energies in the lab-frame (column-four). Column two reports the measured lifetimes
of 19Ne excited states. Column three reports the intensity of 20Mg(βp) feedings to each excited state, where each feeding is
determined by adding all γ ray decays originating from each state and subtracting feeding from higher lying states. Column
four reports the measured lab frame energies of each γ ray branch. Column ﬁve reports the total intensity of each γ-ray
transition per 20Mg decay. Column six reports the γ ray branching ratios for each 19Ne excited-state. Column seven reports
the measured CoM proton energies feeding 19Ne excited states.

Ex(19Ne) (keV)

τ (ps)

238.04(10)
274.96(10)
1507.52(25)

4.3+1.3−1.1

I20Mg(βp)
0.0221(14)
0.0313(15)
0.00278(7)

1535.95(24)

0.01663(45)

1615.24(30)

0.00212(7)

4034.7(16)

0.000149(35)

Eγ (keV)
238.04(10)
274.96(10)
1232.49(22)
1269.47(24)
1260.87(24)
1297.94(22)
1535.90(24)
1340.27(25)
1377.1(3)a
1615.16(30)b
4034.2(16)

Iβpγ

(3.80 ± 0.07stat ± 0.08sys)×10−2
(3.59 ± 0.06stat ± 0.08sys)×10−2
(2.36 ± 0.04stat ± 0.05sys)×10−3
(4.18 ± 0.12stat ± 0.09sys)×10−4
(6.75 ± 0.15stat ± 0.15sys)×10−4
(1.539 ± 0.027stat ± 0.033sys)×10−2
(5.68 ± 0.44stat ± 0.17sys)×10−4
(1.57 ± 0.03stat ± 0.03sys)×10−3
(1.82 ± 0.41stat ± 0.04sys)×10−4
(3.68 ± 0.18stat ± 0.08sys)×10−4
(1.19 ± 0.12stat ± 0.12sys)×10−4

a Value derived from 238, 275, and 1340 keV γ ray peak energies
b Value derived from addition of 275 and 1340 keV γ ray peak energies
c Value adopted from [17]

Branch (%) ECoM (MeV)

100
100

84.9(4)
15.1(4)
4.05(16)
92.53(35)
3.42(29)
74.0(17)
8.6(18)
17.4(9)
80(15)c

2.70(23)

1.21+0.25−0.22

94

5.2.1 19Ne 1507 keV 5/2− state

There are two γ rays which are emitted from this state at 1232.5 keV and 1269.3 keV and

they are expected to have branching ratios of 88(3)% and 12(3)% respectively [74]. The

1507 keV excited state lifetime has been previously measured to be 1.4+0.5−0.6 ps [75], 1.7(3)
ps [19], and 4.1+3.5−1.4 ps [74]. Since there is signiﬁcant tension between the various lifetime
measurements, the lifetime was treated as a free paramater for χ2 minimization.

It is important to note that there is very little broadening in the 1232 keV peak due to

a long lifetime and therefore any 20Na states assumed to feed this 19Ne level yield almost

exactly the same peak shape. Therefore, even though the feedings in Lund and Piechaczek

diﬀer substantially, they will both ﬁt the data equally well. The lack of sensitivity to the

proton branches adopted makes it relatively simple to measure the lifetime of the state. The

χ2 is minimized by taking a value of the lifetime long enough that nearly all the recoiling

19Ne ions in this state are stopped before emitting a gamma ray (Fig. 5.7). By minimizing

the χ2 as a function of the lifetime, a value of 4.3+1.3−1.1 ps is measured for the lifetime of
the 1507 keV state (Fig. 5.8). The uncertainty is determined from the χ2 minimization

as well as a systematic uncertainty associated with the σ parameter and stopping power.

This measurement is in agreement with [74] and more precise, but does not agree within 1

standard deviation with the measurements in [75] or [19].

The peak at 1269.3 keV is ﬁt using the lifetime of 4.3 ps, determined by the 1232.5 keV

peak, since the former peak had much higher statistics. This peak sits next to a Doppler

broadened peak from the 1536 keV state of 19Ne that will be addressed in the next section.

The γ ray intensities per 20Mg β decay of the 1232 keV and 1269 keV γ rays are shown in

Table 5.1. We can use these intensities to determine a γ-decay branching ratio from the 1507

95

Figure 5.7: (color online) Upper panel: The ﬁt of the 1232 keV γ ray peak is produced by
using 4.3 ps lifetime as well as proton feeding intensities from Piechaczek et al. [42]. The
solid gray line represents the data, the dot-dashed green line denotes the background, the
dotted lines denote the diﬀerent contributions of each proton feeding, and the dashed red
line denotes the total ﬁt. The ﬁt has a χ2
ν=1.07. Lower panel: The Residual plot shows the
data subtracted from the ﬁt function.

96

counts/0.1 keV12000130001400015000Energy (keV)1210121512201225123012351240124512501255cts/0.1 keV Residual400-200-0200Figure 5.8: χ2 values determined by simulating the lifetime of the 1507 keV 19Ne state for
many values and comparing the simulation to the data over 447 degrees of freedom. The
minimum is found at 4.3 ps.

97

Lifetime (fs)20003000400050006000700080002c475480485490495500505510515520Figure 5.9: (color online) Upper panel: The γ ray spectrum above contains two 20Mg(βpγ)
peaks from diﬀerent excited states in 19Ne. The data are represented by the solid black
line and, the dot-dashed green line denotes the background, the dotted pink lines denote
the diﬀerent contributions of proton feedings to the 1507 keV state, the dotted black lines
denote the diﬀerent contributions of proton feedings to the 1536 keV state, and the dashed
red line denotes the total ﬁt which has a χ2
ν = 1.11. Lower panel: The Residual plot shows
the data subtracted from the ﬁt function.

98

counts/0.1 keV120001250013000Energy (keV)124512501255126012651270127512801285cts/0.1 keV Residual200-0200400Table 5.2: Piechaczek [42] and Lund [44] absolute % proton feeding intensities to 1536 keV
state per 20Mg β-decay. The quoted relative uncertainty for all intensities measured by
Piechaczek is 12%.

Piechaczek

Ex(20Na) MeV

4.7-5.2

6.266(30)
6.521(30)
≈6.92
≈7.44

Iβp
0.7

0.1
0.51
0.02
0.01

Lund
Ex(20Na) MeV

5.604(5)
6.273(7)
6.496(3)

Iβp

0.03(4)
0.33(9)
0.47(7)

keV state. The uncertainties in eﬃciency cancel out and we are only concerned with the

statistical uncertainty for calculating the branching ratio, which is measured to be 84.9(4)%

decay to the 275 keV state and 15.1(4)% decay to the 238 keV state, in agreement with

previous measurement [74].

The total β-delayed proton feeding of the 1507 keV state Iβp−1507 = 2.78(7) ×10−3
is consistent with the value from Piechaczek et al. of Iβp−1507 = 2.5(3) ×10−3 and more
precise but is a factor of 2.7 lower than the value measured by Lund et al. Iβp−1507 = 7.4(21)
×10−3, which has a large uncertainty.

5.2.2 19Ne 1536 keV state

There are two γ rays which have been measured from this state at 1261 keV and 1298 keV

and are expected to have branching ratios of 5(3)% and 95(3)% respectively [74]. In this

work we measure an additional branch decaying to the ground state at 1536 keV for the

ﬁrst time. The lifetime of the state has a recently measured value of 16(4) fs [19] and is

in agreement with the previous evaluation of 28(11) fs [17] so a value of 16 fs is adopted

for the simulation. Clear broadening is apparent for all three of the γ rays emitted and the

99

diﬀerent proton energies and intensities that feed the 1536 keV state become much more

important. For the simulation of each recoil energy, the relatively precise values of Ex(20Na)

from Lund et al. were adopted. The relative branches from both Piechaczek and Lund were

used to separately ﬁt the data and the total number of counts in the peak was left as a free

parameter.

It is easy to see that the relative branches from Lund do not ﬁt the 1298 keV peak

accurately with a χ2

ν = 30.8 (Fig. 5.10). An additional lower-energy proton feeding is

required to ﬁt the data. The relative branches from Piechaczek ﬁt the data much better
ν = 1.14. From the ﬁt of the 1298 keV peak a value of Iβpγ−1298 = (1.54 ±

and return a χ2
0.03stat ± 0.03sys)×10−2 is measured.

A ﬁt of the 1261 keV peak is shown in Fig. 5.9. The simulation for this peak used the

relative proton feedings from Piechaczek as well as the 16 fs lifetime of the state, which ﬁt

the 1298 keV peak well. The feeding of the 1261 keV peak is measured to be Iβpγ−1261 =
(6.75 ± 0.15stat ± 0.15sys)×10−4.

The 1536 keV state has three γ decay paths to the ground state of 19Ne. The two

cascades that do not directly decay to the ground state will yield a small portion of counts

in the 1536 keV peak due to summing in a single γ ray detector. The number of counts

in the 1536 keV peak due to the summing eﬀect is calculated from the number of counts

in the 1298 keV peak and SeGA eﬃciency for a 238 keV γ ray as well as the number of

counts in the 1261 keV peak and SeGA eﬃciency for a 275 keV γ ray. After subtracting the

summing counts from the 1536 keV peak integral we measure an intensity of Iβpγ−1536 =
(5.68 ± 0.44stat ± 0.17sys)×10−4.

From Iβpγ−1261, Iβpγ−1298, and the newly measured Iβpγ−1536 we measure the γ ray

branching ratio from the 19Ne 1536 keV state to be a 4.05(16)% branch to the 275 keV

100

Figure 5.10: (color online) Fits of the 1298 keV γ ray peak above are produced using a 16
fs lifetime. (a) The ﬁt is produced using the relative proton feeding intensities, measured by
Lund [44], from Table 5.2. The data are represented by the solid gray line, the dot-dashed
green line denotes the background, the dotted lines denote the diﬀerent contributions of each
proton feeding, and the dashed red line denotes the best total ﬁt. (b) The Residual plot
shows the data subtracted from the ﬁt function in (a). (c) The ﬁt is produced using the
relative proton feeding intensities, measured by Piechaczek [42], from Table 5.2. Similarly
to panel (a) the data are represented by the solid line, the dot-dashed line denotes the
background, the dotted lines denote the diﬀerent contributions of each proton feeding, and
the dashed line denotes the best total ﬁt. (d) The Residual plot shows the data subtracted
from the ﬁt function in (c).

101

counts/0.1 keV1500020000(a)Energy (keV)12801290130013101320cts/0.1 keV Residual2000-1000-010002000(b)counts/0.1 keV1500020000(c)Energy (keV)12801290130013101320cts/0.1 keV Residual400-200-0200400600(d)state, a 92.53(35)% branch to the 238 keV state, and a 3.42(29)% branch to the ground

state.

5.2.3 19Ne 1615 keV state

There are three γ rays which are emitted from this state with energies of 1340, 1377, and

1615 keV and they are expected to have branching ratios of 70(4)%, 10(3)%, and 20(3)%

respectively [74]. This state has never been observed in 20Mg β-decay before the present

work, so there is no available proton feeding data. It is possible that multiple 20Na states

contribute to the feeding, however, the simplest procedure is to begin by assuming one proton

energy to ﬁt the peak and this CoM energy will be considered a free parameter. A lifetime of

143(31) fs was determined in a data evaluation [17] by combining measurements from [74, 76],

however a more recent value of 80(15) fs was reported in Ref. [19] so we have re-evaluated

the lifetime to be 93(20) fs by taking a weighted average with inﬂated uncertainty.

Using the adopted lifetime of 93(20) fs and interpolated σ parameter to simulate the

broadening of the 1340 keV peak, a CoM energy of 2.7 MeV minimizes the χ2 (Fig. 5.11).

From the χ2 distribution we get an uncertainty in the CoM energy of 100 keV. An additional

systematic uncertainty in the CoM energy of 200 keV from the uncertainty in the lifetime

as well as an uncertainty of 50 keV for the uncertainty in the σ parameter yields a value of

2.70(23) MeV for the CoM energy. From this we determine an excitation energy Ex(20Na)=

6.51(23) MeV for the proton-emitting state. This is consistent with proton emission from

the 20Na isobaric analog state at 6498.4(5) keV [56].

The 1377 and 1615 keV lines both have low statistics and do not provide signiﬁcant

information about the energies of protons feeding the state. We apply the peak shape

corresponding to the proton energies that best ﬁt the higher statistics 1340 keV peak to

102

these two peaks to determine the total intensity of protons feeding the 1615 keV state.

For a ﬁt of the 1377 keV peak a simple linear background was used for this relatively

low statistics case and a broad peak was ﬁt on top of it. In the case of the 1615 keV peak

a linear plus exponential function was used to model the background since the peak sits on

the tail of a very high statistics 1634 keV peak from 20Na(βγ) decay.

Since the 1615 keV state also has two cascades that do not directly decay to the ground

state, a small portion of counts in the 1615 keV peak are due to summing in a single γ ray

detector and must be subtracted. The number of counts in the 1615 keV peak due to this

eﬀect is calculated from the number of counts in the 1340 keV peak and SeGA eﬃciency for

a 275 keV γ ray as well as the number of counts in the 1377 keV peak and SeGA eﬃciency

for a 238 keV γ ray.

The γ ray intensities per 20Mg β decay of the 1340, 1377, and 1615 keV γ rays are shown

in Table 5.1. A measurement of the branching ratios from the 1615 keV state using the

intensities yields a 74.0(17)% branch to the 275 keV state, a 8.6(18)% branch to the 238

keV state, and a 17.4(9)% branch to the ground state of 19Ne, in agreement with and more

precise than previous measurement [74].

5.2.4 19Ne 4.03 MeV state

There are three γ rays which are emitted from this state at 2497, 3758, and 4034 keV and

they are expected to have branching ratios of 15(5)%, 5(5)%, and 80(15)% respectively [17].

In the present experiment, only the 4.03 MeV γ ray is detected above background. For this

case all 16 detectors are used to determine the feeding of the 4.03 MeV state and the shape

of the Doppler broadened feature in order to reduce the statistical uncertainty.

The lifetime of the 4.03 MeV state has been measured to be 13+16−9 fs [19], 11+4−3 fs [18],

103

Figure 5.11: (color online) Upper panel: The ﬁt of the 1340 keV γ ray peak is produced by
using a 93 fs lifetime and a CoM energy of 2.7 MeV between the proton and recoiling 19Ne.
The solid gray line represents the data, the dot-dashed green line denotes the background
and the dashed red line denotes the background+simulated peak. The ﬁt has a χ2
ν=1.00.
Lower panel: The Residual plot shows the data subtracted from the ﬁt function.

104

counts/0.1 keV13500140001450015000Energy (keV)13201330134013501360cts/0.1 keV Residual400-200-0200Figure 5.12: The ﬁt of the 1377 keV γ ray peak is produced by using a 93 fs lifetime
and a CoM energy of 2.7 MeV between the proton and recoiling 19Ne. The black line
represents the data, the green line denotes the linear background and the red line denotes
the background+simulated peak. Lower panel: The Residual plot shows the data subtracted
from the background.

105

Energy (keV)counts/0.1 keV1500016000Energy (keV)1355136013651370137513801385139013951400cts/0.1 keV Residual400-200-0200400Figure 5.13: The ﬁt of the 1616 keV γ ray peak is produced by using a 93 fs lifetime and a
CoM energy of 2.7 MeV between the proton and recoiling 19Ne. The black line represents
the data, the green line denotes the exponential background and the red line denotes the
background+simulated peak. Lower panel: The Residual plot shows the data subtracted
from the background.

106

Energy (keV)counts/0.1 keV400050006000Energy (keV)15901595160016051610161516201625  cts/0.1 keVResidual[D-B]200-0200Figure 5.14: (color online) Upper panel: The ﬁt of the 4.03 MeV peak is produced by
simulating the broadened peak with an 7 fs lifetime and CoM energy of 1.21 MeV and has
a χ2
ν = 0.94. All 16 SeGA detectors are used to produce this spectrum. The solid gray
line represents the data, the dot-dashed green line denotes a ﬁt of the background and the
dashed red line denotes the total ﬁt using the optimal 1.21 MeV CoM energy. A simpliﬁed
linear background model was applied for this relatively low statistics case. Lower panel: The
Residual plot shows the data subtracted from the ﬁt function.

107

counts/ keV3400360038004000Energy (keV)39603980400040204040406040804100 cts/keVResidual100-0100and 6.9±1.7 fs [20]. The most precise lifetime of 6.9 fs was adopted and the uncertainty

is used to determine a systematic uncertainty in the CoM energy which was left as a free

parameter. In this case, where the statistics are relatively low, a simple linear model was

used for the background. Additionally, an assumption is made that only one 20Na excited

state feeds the 4.03 MeV level (Fig. 5.14).

Minimizing the χ2 as a function of CoM energy (Fig. 5.15) yields a CoM energy of

1.21+0.25−0.22 MeV. An additional 0.025 MeV is incorporated into this uncertainty from the
shift in minimum χ2 introduced by moving the lifetime to the limits of uncertainty. This

corresponds to a feeding from an excited state in 20Na at 7.44 +0.25−0.22 MeV, consistent with
the 7.44(10) MeV state observed to be populated in 20Mg β-decay by its proton emission to

lower lying 19Ne states [42].

From this ﬁt the intensity is measured to be Iβpγ−4034 = (1.19±0.12stat±0.12sys)×10−4.
The γ branch from the 4.03 MeV state is expected to be 80(15)% [17]. Therefore, Iβp−4034
= (1.49 ± 0.15stat ± 0.32sys)×10−4. This value is consistent with the one reported in our

initial publication [30] but slightly diﬀerent because the ﬁtting procedure is diﬀerent and a

diﬀerent literature intensity was adopted for the 984-keV 20Na line for normalization.

5.2.5 19Ne 238 and 275 keV states

Both of these lower lying 19Ne states have long lifetimes, and the corresponding 19Ne atoms

are completely stopped in the scintillator before emitting γ rays. Therefore, we do not gain

any information from Doppler broadening analysis. However, the direct feeding of the 238

and 275 keV states from 20Mg(βp) decay can be determined by measuring the intensity of

the γ decays and subtracting the feeding contribution to each of these states from γ decays

of higher lying states in 19Ne. Both of these states are fed by the 1507, 1536, 1615 and

108

Figure 5.15: Each χ2 value is determined by simulating a diﬀerent CoM proton energy
feeding the 4.03 MeV excited state in 19Ne and comparing each simulation to the peak at
4034 keV. χ2 values are determined from ﬁts with 157 degrees of freedom. The minimum
determines the most likely CoM energy.

4034 keV states and these contributions are subtracted to obtain the intensities reported in

Table 5.1. These values are consistent with the previously measured values of Iβp−238 =
(2.29±0.27)×10−2 and Iβp−275 = (3.12±0.37)×10−2 [42] and Iβp−238 = (2.23±0.34)×10−2
and Iβp−275 = (3.69 ± 0.52)×10−2 [44] and more precise.

5.3 Conclusion

We have measured the 20Mg(βp)19Ne feedings and γ ray branches of 6 excited states in

19Ne. We have developed a Monte Carlo simulation to analyze 9 Doppler broadened 19Ne

peaks. We have measured the energy of the proton transition which feeds the astrophysically

important 4.03 MeV state, facilitating future measurements of the α-branch from this state.

109

Energy (keV)10001100120013001400150016002c148148.5149149.5150Additionally we have measured the energy of the proton transition which feeds the 1615 keV

state as well as the lifetime of the 1507 keV state and found a new γ decay branch from the

1536 keV state.

This is the ﬁrst time Doppler broadening analysis has been applied to such high statistics

β-delayed proton-γ peaks, enabling a substantial improvement in sensitivity over [29]. We

have shown this method can be a useful tool to measure excited state lifetimes, proton

branches, and proton energies and can distinguish between conﬂicting decay schemes. The

method is therefore complementary to direct measurements of β-delayed protons and should

prove to be even more useful when applied to β-delayed neutron emission.

110

Chapter 6

Outlook

6.1 The 15O(α, γ)19Ne Reaction

Experiment E14066 has ultimately provided a road-map for future experiments to utilize

20Mg β-decay in the search for α-emission from the 4.03 MeV state in 19Ne. This alternative

method to transfer reactions, which have thus far only provided upper limits of Γα, has the

potential to provide a ﬁnite value for the α-branch.

Both particles in the p− α emission following 20Mg β-decay have well-measured center of

mass energies. The center of mass energy of the proton, measured in this work, is 1.21+0.25−0.22
MeV and the α particle CoM energy has been measured at 505.8 ± 1.0 keV [17, 24, 19],

providing a unique signal for charged particle detectors to search for. A detector system,

such as GADGET [77], devoted to measuring low energy charged particles, may be an ideal

setup for measuring this unique signal.

6.2 Doppler Broadening in β-delayed particle emission

The Doppler Broadening line shape analysis has been incredibly useful in determining many

unknown quantities in β-delayed pγ emission. With the high statistics in our experiment

we have been able to determine branching ratios, excited state lifetimes, proton energies

and can diﬀerentiate between competing decay schemes. While this method has been useful

111

on the proton-rich side, it may be even more useful to implement in β-delayed neutron

emission. The detection of neutrons is generally much more diﬃcult and relies on a kinematic

interaction to directly detect the particle whereas it is much easier to detect charged particles

directly. Since information about the neutron is preserved in β-delayed particle emission,

where excited state lifetimes are short, Doppler Broadening line shape analysis may be an

alternative to directly measuring these particles.

112

BIBLIOGRAPHY

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