CAPTURE CROSS SECTIONS FOR THE ASTROPHYSICAL P PROCESS
By
Stephen J. Quinn

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2015

ABSTRACT
CAPTURE CROSS SECTIONS FOR THE ASTROPHYSICAL P PROCESS
By
Stephen J. Quinn
This dissertation includes the design and development of the Summing NaI (SuN) 4π
γ-ray detector at the National Superconducting Cyclotron Laboratory to measure proton
and α radiative capture reactions relevant in the astrophysical p process. Discussions of
p-process nucleosynthesis, the relevant nuclear reaction theory, experimental details, and
analysis procedures are included. All reaction measurements were performed at the Nuclear
Science Laboratory of the University of Notre Dame. The commissioning experiments in
both regular and inverse kinematics were done using known resonances in the 27 Al(p,γ)28 Si
and 58 Ni(p,γ)59 Cu reactions, and the results agree well with previous literature values. The
success of these proof-of-principle measurements marks the first time that the γ-summing
technique has been implemented in inverse kinematics. Furthermore, in an effort to investigate the synthesis of the light p-process nuclei, the 74 Ge(p,γ)75 As, 74 Ge(α,γ)78 Se, and
90,92Zr(α,γ)94,96Mo

reactions were measured and compared to theoretical calculations using

the nuclear statistical model. It was found that the new 74 Ge(p,γ)75 As measurements cause
an enhancement in the overproduction of 74 Se in p-process models, and that the updated
90 Zr(α,γ)94 Mo

reaction rate seems to confirms the p-process branching point at 94 Mo. Fi-

nally, the 58 Ni(α,γ)62 Zn reaction was measured for its role in nucleosynthesis in type Ia
supernovae. The measurements here lower the reaction rate used in astrophysical models,
which leads to a 5% reduction in the calculated abundances of several isotopes. All of the
measurements in this dissertation greatly reduce the uncertainty in the reaction cross section.

When the Yankees beat the Red Sox all is right in the world.

iii

ACKNOWLEDGMENTS

This is my favorite section of all because it gives me an opportunity to thank everyone
who contributed to this thesis. First of all, special thanks must go to my advisor, Artemis
Spyrou, for her guidance, kindness, and support during my time in graduate school. I
absolutely could not have worked with a better professor, a more brilliant scientist, or on a
more interesting project, and I am extremely grateful for the many opportunities that she
has given me that are far too numerous to list here. Thank you, Artemis! It has been a
pleasure working alongside all of the SuN group members, including Jorge, Farheen, Alex,
Debra, Alicia, and the many brilliant undergraduate students. In particular, I would like
to single out Anna Simon for her countless contributions to this thesis. She has been an
incredible source of information when I had little experience, as well as serving as a wonderful
mentor and friend. Overall, it has been a pleasure to take part in the creation of the SuN
group and it is wonderful to see the SuN detector shine as brightly as it is today.
The experiments comprising this thesis would not have been possible without the hard
work of many scientists and they all deserve recognition. It is a large fear of mine that I will
forget someone! At the NSCL, thanks to John Yurkon for his vast knowledge and patience
in teaching me the art of target making, and thanks to Renan Fontus for designing the beam
pipe for the SuN detector. I would also like to thank Sean Liddick, Jeromy, Chris, Nicki, and
Scott for their time and assistance with DDAS; they are true superstars. At Notre Dame,
thanks to the many graduate students who helped operate the accelerator, the technicians
who assisted in the setup, and everyone who makes Notre Dame such a welcoming place for
external experimenters. A special thanks is in order for Man¨oel Couder, Dan Robertson, Ed
Stech, Wanpeng Tan, and Antonios Kontos for going above and beyond the call of duty, and
iv

whose help was critical for a successful experimental campaign. At Hope College, I express
my gratitude to Paul DeYoung, Graham Peaslee, and Dave Daugherty for their time and
expertise with target thickness measurements. Additionally, I would like to thank Thomas
Rauscher and Eduardo Bravo for their collaboration and contribution to this work.
I greatly appreciate the assistance of Professors Ed Brown, Jim Linnemann, Filomena
Nunes, and Michael Thoennessen through their knowledge, questions, and, most of all, generosity in agreeing to give up their valuable time to serve on my guidance committee. I also
would like to acknowledge the National Science Foundation, the Joint Institute for Nuclear
Astrophysics, and the Michigan State University Graduate School for financial support.
Truthfully, my time in graduate school has been wonderful and I am very fortunate to
have many friends that have made it so enjoyable. I will miss my officemates, roommates,
teammates, and everyone from the daily lunches, Happy Hours, tailgates, and parties that
have kept me laughing since I arrived at MSU. A “merci beaucoup” to Yari and Bazzy for
their support and motivation during the more difficult first few years of trying to balance
classes, exams, teaching, and research. Finally, and most importantly, I would like to thank
my family. I have been incredibly blessed with unquestionably the greatest family of all
time, and I would like to thank them for putting up with me. Thanks Mom and Dad for
being such awesome role models, and thanks Lassu and Mollister for being my two favorite
people in the world. As Yogi Berra would say, “I just want to thank everyone for making
this day necessary.”

v

TABLE OF CONTENTS

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

ix

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

xi

Chapter 1 Introduction . . . . . . . . . . . . . . .
1.1 Elements . . . . . . . . . . . . . . . . . . . . .
1.2 Abundances . . . . . . . . . . . . . . . . . . .
1.3 Nucleosynthesis . . . . . . . . . . . . . . . . .
1.3.1 Big Bang Nucleosynthesis . . . . . . .
1.3.2 Quiescent Stellar Burning . . . . . . .
1.3.3 Supernovae . . . . . . . . . . . . . . .
1.3.4 Nuclear Statistical Equilibrium . . . .
1.3.5 Nucleosynthesis of the Heavy Elements
1.4 Production of the p Nuclei . . . . . . . . . . .
1.4.1 νp process . . . . . . . . . . . . . . . .
1.4.2 rp process . . . . . . . . . . . . . . . .
1.4.3 p process . . . . . . . . . . . . . . . .
1.4.4 Uncertainties . . . . . . . . . . . . . .
1.5 Looking Ahead . . . . . . . . . . . . . . . . .

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1
2
4
6
7
7
8
10
11
14
14
15
15
21
25

Chapter 2 Nuclear and Astrophysical Quantities
2.1 Cross Section . . . . . . . . . . . . . . . . . . .
2.2 Stellar Reaction Rate . . . . . . . . . . . . . . .
2.3 Astrophysical S factor . . . . . . . . . . . . . .
2.4 Gamow Window . . . . . . . . . . . . . . . . .
2.5 Q value . . . . . . . . . . . . . . . . . . . . . .

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27
27
29
31
33
35

Chapter 3 Theoretical Considerations
3.1 Resonant Reactions . . . . . . . . .
3.1.1 In Stars . . . . . . . . . . .
3.1.2 In the Laboratory . . . . . .
3.2 Nuclear Statistical Model . . . . . .
3.3 Reciprocity Theorem . . . . . . . .

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37
37
38
39
44
49

Chapter 4 Experimental Techniques
4.1 Measuring the Cross Section . . .
4.1.1 Activation . . . . . . . . .
4.1.2 γ-Induced Reactions . . .
4.1.3 In-Beam Methods . . . . .

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53
53
54
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55

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vi

4.2

4.3

4.1.3.1 Angular Distributions Method .
4.1.3.2 γ-Summing Technique . . . . .
4.1.4 Techniques in Inverse Kinematics . . . .
Target Production and Characterization . . . .
4.2.1 Evaporation . . . . . . . . . . . . . . . .
4.2.2 Thickness Measurements . . . . . . . . .
4.2.2.1 Experimental Details . . . . . .
4.2.2.2 Calibrations . . . . . . . . . . .
4.2.2.3 RBS Analysis . . . . . . . . . .
4.2.2.4 ERD Analysis . . . . . . . . . .
Experimental Setup . . . . . . . . . . . . . . . .
4.3.1 Beam Production . . . . . . . . . . . . .
4.3.2 Acceleration . . . . . . . . . . . . . . . .
4.3.3 Ion Selection . . . . . . . . . . . . . . .
4.3.4 Experimental Endstation . . . . . . . . .

Chapter 5 The SuN Detector
5.1 Design . . . . . . . . . . .
5.2 Detection . . . . . . . . .
5.3 Data Acquisition . . . . .
5.3.1 PMT signals . . . .
5.3.2 NSCL DDAS . . .
5.3.3 External Triggering
5.4 Radiation Source Testing .
5.5 GEANT4 Simulation . . .

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57
58
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82

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. 84
. 84
. 86
. 89
. 90
. 91
. 96
. 98
. 101

Chapter 6 Analysis . . . . . . . . . .
6.1 Gain Matching and Calibration
6.2 Thresholds . . . . . . . . . . . .
6.3 Sum Peak Analysis . . . . . . .
6.3.1 Isomeric states . . . . .
6.3.2 Doppler reconstruction .
6.4 γ-Summing Efficiency . . . . . .

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107
107
114
117
119
121
123

Chapter 7 27 Al(p,γ)28 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Chapter 8 74 Ge(p,γ)75 As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Chapter 9 58 Ni(α,γ)62Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 10 Additional (α,γ) Measurements
10.1 90 Zr(α,γ)94Mo Results . . . . . . . . . .
10.2 92 Zr(α,γ)96Mo Results . . . . . . . . . .
10.3 74 Ge(α,γ)78 Se Results . . . . . . . . . .
10.4 Discussion . . . . . . . . . . . . . . . . .
vii

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158
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165
166

10.4.1 90 Zr(α,γ)94Mo
10.4.2 92 Zr(α,γ)96Mo
10.4.3 74 Ge(α,γ)78 Se .
10.5 Conclusions . . . . . .

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168
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173

Chapter 11 Measurements in Inverse Kinematics . . . . . . . . . . . . . . . . 175
Chapter 12 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 182
APPENDICES
. . . . . . . . . . . . . . . . . . . .
Appendix A Energy Filter Algorithm . . . .
Appendix B GEANT4 Detector Construction
Appendix C Creating SuN’s ROOT files . . .
REFERENCES

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187
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

viii

LIST OF TABLES

Table 4.1:

4 He

beam energies and silicon surface barrier detector energy calibrations used at HIBAL over the course of five different experiments. .

69

Table 4.2:

Target thickness values. . . . . . . . . . . . . . . . . . . . . . . . . .

73

Table 5.1:

Standard deviation (σ) of a Gaussian function fit to various experimental γ-ray peaks, along with the corresponding energy resolution.

103

Table 6.1:

Energy and source of γ-ray transitions used to calibrate SuN’s segments.113

Table 6.2:

Gain matching multiplication factors for each PMT in the three different experiments at the University of Notre Dame. . . . . . . . . . 114

Table 6.3:

Energy calibrations of the form E = Ax2 + Bx + C for each segment
in the three different experiments at the University of Notre Dame. . 115

Table 6.4:

Average emission angle in radians for the different segments of the
SuN detector calculated from the geometrical center of the NaI crystals, by weighting the angles by the path-length through the crystals,
and from SuN’s GEANT4 simulation. . . . . . . . . . . . . . . . . . 122

Table 6.5:

Doppler correction factors used for each segment in this thesis. . . . 122

Table 7.1:

Measured resonance strengths for the 27 Al(p,γ)28 Si reaction for the
SuN detector and previous results. . . . . . . . . . . . . . . . . . . . 133

Table 8.1:

Cross sections and astrophysical S-factors for the 74 Ge(p,γ)75 As reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Table 8.2:

Reaction rates for the 74 Ge(p,γ)75 As reaction from TALYS 1.4 calculations with the JLM optical model potential and BSFG nuclear level
density which provide the most accurate reproduction of experimental
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Table 9.1:

Cross sections for the 58 Ni(α,γ)62 Zn reaction. . . . . . . . . . . . . . 150

ix

Table 9.2:

Stellar reactivities for the 58 Ni(α,γ)62 Zn reaction. . . . . . . . . . . 155

Table 9.3:

REACLIB parameters for the 58 Ni(α,γ)62 Zn reaction. . . . . . . . . 155

Table 9.4:

Changes to the nucleosynthesis in SNIa models. (1) Chandrasekharmass delayed detonation model with ρDDT = 3.9 × 107 g/cm3 . (2)
Explosion of a sub-Chandrasekhar white dwarf of 1.025 M⊙ C-O core
surrounded by a 0.055 M⊙ He envelope. . . . . . . . . . . . . . . . . 156

Table 10.1:

Cross sections and S-factors for the 90 Zr(α,γ)94Mo reaction. . . . . . 162

Table 10.2:

Cross sections and S-factors for the 92 Zr(α,γ)96Mo reaction. . . . . . 163

Table 10.3:

Cross sections for the 74 Ge(α,γ)78Se reaction.

Table 10.4:

Reaction rates for the 90 Zr(α,γ)94Mo reaction. . . . . . . . . . . . . 170

Table 11.1:

Resonance strength measurements . . . . . . . . . . . . . . . . . . . 179

x

. . . . . . . . . . . . 166

LIST OF FIGURES

Figure 1.1:

Figure 1.2:

Figure 1.3:

Figure 1.4:

Figure 1.5:

Figure 1.6:

Figure 1.7:

Figure 1.8:

Figure 1.9:

Nuclear chart with proton number along the y-axis and neutron number along the x-axis. Stable nuclei are indicated with black boxes and
the observed radioactive isotopes are indicated with shaded boxes.
There are also thousands of nuclei beyond what is plotted here that
are expected to exist and are yet to be discovered. The lines on the
chart indicate nuclear magic numbers. . . . . . . . . . . . . . . . . .

3

Abundances of the nuclides in the solar system based on the data
from Lodders [1]. The abundances are normalized to 106 silicon atoms.

6

Simplified picture of a massive star that has undergone all of the
burning stages in the core. The burning layers are not drawn to scale.

9

Nuclear chart showing the pathways for the s and r neutron capture
processes, with a zoomed in view on the bottom. The s and r processes produce the majority of the stable isotopes heavier than iron,
except for the 35 p nuclei which are shielded from β − decay by the
valley of stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure from Ref. [2] showing the temperature and density of the O/Ne
layer as the shockwave passes through. The curves represent different
depths of the O/Ne layer. . . . . . . . . . . . . . . . . . . . . . . . .

16

Section of the nuclear chart showing an example reaction flow for the
p process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Normalized overproduction factor for the calculated abundance of
each p nucleus plotted against mass number. Plot is taken from
Ref. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figures from Ref. [3] showing the abundances of the s nuclei at the end
of the helium burning stage and how they depend on (a) the intial
metallicity of the star (b) the reaction rates used in the s-process
calculations. The intial metallicities used were solar and 10% solar,
and the different rates used were from Ref. [4] and Ref. [5]. . . . . .

22

Figure from Ref. [2] showing the overproduction factor of selected
p nuclei as a function of the peak temperature of the stellar layer
reached during the supernova explosion. . . . . . . . . . . . . . . .

23

xi

Figure 1.10: Figure from Ref. [6] showing the normalized overproduction factor of
the p nuclei for supernova explosions of stars with mass in the range
between 13-25 times the mass of the Sun. The vertical bar for each
nucleus indicates the spread in p-nuclei abundances as a result of the
various stellar masses. . . . . . . . . . . . . . . . . . . . . . . . . .

23

Figure 1.11: Figure from Ref. [2] showing the ratio of p-nuclei abundances calculated with rates modified by a factor of three up and down compared
to the standard rate. (a) Modified proton-induced reactions and their
inverse reactions. (b) Modified α-induced reactions and their inverse
reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Figure 2.1:

Figure 2.2:

Figure 2.3:

Figure 2.4:

Figure 3.1:

Figure 3.2:

Figure 4.1:

Front view of uniformly placed circular targets. The large grey circle
represents the area in which projectiles strike the targets. . . . . . .

27

A number of particles N1 and velocity v impinging on a number of
particles N2 within a volume that has area A and depth x. . . . . .

30

The tail of the Maxwell-Boltzmann distribution of particle energies
exp(−E/kT ) combined with the probability of charged particles to
penetrate through the Coulomb barrier exp(−2πη) gives rise to the
Gamow peak (not drawn to scale). The Gamow peak defines the
energy window over which the majority of nuclear reactions take place
inside of stellar environments. . . . . . . . . . . . . . . . . . . . . .

34

Binding energies per nucleon based on the data from [7]. The maximum occurs around the iron region. . . . . . . . . . . . . . . . . . .

35

A Breit-Wigner lineshape with resonance energy Er and width Γ
(top panel). Expected excitation functions for the measurement of
the resonance with different target thicknesses (bottom panel). . . .

42

General level scheme for a forward and reverse reaction with a certain
reaction Q value. The nuclei have discrete states at lower excitation
energy and a continuum of states and higher excitation energy. . . .

51

In a capture reaction, the projectile and target form an excited state
in the produced nucleus equal to the Q value of the reaction plus the
center of mass energy of the projectile and target system. The populated state is often at high excitation energies where many resonances
overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

xii

Figure 4.2:

Difference in γ-ray spectra between a (a) small-sized detector and (b)
large 4π detector for a simplified level scheme. The result of using
a large 4π detector is the summation of all γ rays into a single sum
peak whose intensity is directly related to the number of reactions
that occurred. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 4.3:

Schematic drawing of the RBS technique with (a) an ion beam backscattering off of target nuclei and (b) the energy spectrum of backscattered particles detected by the detector. . . . . . . . . . . . . . . . . 66

Figure 4.4:

General layout of the HIBAL scattering chamber. (a) The RBS detector is located at θ = 168.20◦ , a distance 4.0” from the target, with
a 0.2” diameter collimator in front of it. (b) The rotating detector
was placed at a scattering angle of θ = 30.0◦ , a distance 2.75” from
the target, with a 0.2” diameter collimator in front of it. The target
was then rotated 75.0◦ and a 15.0 µ aluminum foil placed in front of
the detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Comparison of the SIMNRA fit to an experimental gold-on-glass measurement. The precise beam energy of Eα = 3804 keV was determined
by fitting the front edge of the gold peak in the spectrum. . . . . . .

70

SIMNRA fits to RBS spectra of nine of the targets used in this dissertation. The resulting thicknesses are given in Table 4.2. . . . . . .

71

SIMNRA fit to the RBS spectrum of enriched 74 Ge on a tantalum
backing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

SIMNRA calculations compared to experimental data for (a) RBS
and (b) ERD spectra of the titanium hydride foil. . . . . . . . . . .

74

Diagram dipicting how wrinkles in the titanium hydride foil may lead
to a seemingly larger thickness in the ERD spectra than in the RBS
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 4.10: Same as Fig. 4.8 with the minimum additional titanium hydride material needed to match the ERD spectrum. . . . . . . . . . . . . . .

76

Figure 4.11: Layout of the Nuclear Science Laboratory at the University of Notre
Dame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure 4.12: Experimental seup for the cross section measurements performed in
this dissertation. The dimensions are not to scale. . . . . . . . . . .

82

Figure 4.5:

Figure 4.6:

Figure 4.7:

Figure 4.8:

Figure 4.9:

xiii

Figure 5.1:

The Summing NaI (SuN) detector. . . . . . . . . . . . . . . . . . . .

85

Figure 5.2:

Typical signal from one of SuN’s PMTs. . . . . . . . . . . . . . . . .

89

Figure 5.3:

Diagram of SuN’s data acquistion system. . . . . . . . . . . . . . . .

90

Figure 5.4:

Illustrating the use of a digital trapezoidal filter using a signal step
function. For every digitized point, the average value of points in the
“−” tpeak region get subtracted from the average value of points in
the “+” tpeak region. The two tpeak regions are separated by a time
tgap . For a step function, the response is a trapezoidal shape and the
height of the trapezoidal filter contains the energy information of the
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

(a) Digitized representation of a signal from SuN’s PMT after passing
through 100 MSPS ADC. (b) Response of SuN’s trigger filter. The
arrival time of the pulse is obtained when the trigger filter passes
above a user-defined threshold. (c) Response of SuN’s energy filter.
The energy extraction is performed at a time tpeak + tgap = 800ns
after the arrival of the pulse, with a correction to the decay constant
applied as in Ref. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Simplified decay scheme showing the dominant decay radiation for
the radioactive sources 137 Cs (left) and 60 Co (right). . . . . . . . . .

98

Experimental spectrum for a segment of SuN taken with a 137Cs
source with room background subtracted. Also shown is the result of
a Gaussian fit to the 661.7 keV γ-ray line. . . . . . . . . . . . . . . .

99

Figure 5.5:

Figure 5.6:

Figure 5.7:

Figure 5.8:

Experimental spectrum of SuN taken with a 60 Co source with room
background subtracted. The result for (a) a segment of SuN and (b)
all eight segments are in excellent agreement with GEANT4 simulations. When including the entire volume of SuN, the two sequential γ
rays get summed together and the sum peak dominates the spectrum. 100

Figure 5.9:

Figure from Ref. [9] showing the effect of the location of a 60 Co source
inside of SuN on the γ-summing efficiency. . . . . . . . . . . . . . . 101

Figure 5.10: Visualization of the GEANT4 SuN detector simulation. The lines
inside of SuN are several γ-ray tracks. The one track that scatters
outside of SuN does not deposit all of its energy inside of the detector.102

xiv

Figure 5.11: Standard deviation in the detected energy of the SuN detector as a
function of the γ-ray energy. The points correspond to experimental
data and the curve is the best fit function that was implemented in
the GEANT4 simulation. . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 5.12: γ-summing efficiencies for various configurations of the CAESAR detector (standard configuration inset), a solid cylindrical detector, and
the SuN detector for the resonance at a proton energy of Ep =
1118keV in the 27 Al(p,γ)28 Si reaction. This resonance deexcites
through the emission of 3.2 γ rays on average. . . . . . . . . . . . . 106
Figure 5.13: SuN’s efficiency for the detection of a single γ ray as a function of
energy both with and without the beam pipe and target holder used
in experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 6.1:

(a) Room background and (b) 137 Cs source spectra from the three
PMT’s in a single segment of SuN after gain matching the voltages
applied to each PMT. Since the 137Cs source emits γ rays from the
center of SuN, the outer two PMTs show a double peak in their
spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 6.2:

Two-dimensional plot showing the response of the left PMT against
the response of the right PMT for a 60 Co source. The two bands
in the spectrum around channel 1000 in each PMT correspond to
the two γ-ray transitions in the decay of 60 Co. The slope of these
bands indicates that the response of the outer two PMTs is affected
by where the γ ray deposits energy in the crystal. . . . . . . . . . . 109

Figure 6.3:

Result of applying gain matching multiplication factors in software
to align the central PMT in a segment to the left (first row), center
(second row), and right (third row) of the double peaks in spectra of
the outer two PMTs. This was done for a 137 Cs source (left column),
a 60 Co source (central column), and the 40 K background line (right
column). After summing the three PMTs in a segment together, the
end result is that there is no noticeable difference in the method of
gain matching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 6.4:

The result of gain matching SuN’s segments using a 60 Co source. The
label T is for the top of the detector and B is for the bottom of the
detector. Based on the spectra it can be deduced that the source was
place off-center under the third segment of SuN and closer to the top
than the bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xv

Figure 6.5:

Experimental data and GEANT4 simulation for one of SuN’s central
segments for a 60 Co source. In the analysis, a hard cut at 160 keV
was applied for each segment in order to use the same threshold in
both experiment and simulation. . . . . . . . . . . . . . . . . . . . . 116

Figure 6.6:

Zoomed in view of a typical sum peak from the γ-summing technique
with the SuN detector, in this case for the 60 Ni(α,γ)64 Zn reaction.
A Gaussian fitting function is used to define a sum-peak region of
(EΣ -3σ,EΣ +3σ) and then a linear background is determined from an
average of the region’s boundary values. The sum peak integral is
taken to be the number of counts above the linear background. . . 118

Figure 6.7:

Cross-sectional view of the SuN detector showing the different angles
used in the Doppler reconstruction. The different angles are to the
geometrical center of the NaI crystals, to the weighted center based
on the path-length through the crystals, and from SuN’s GEANT4
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Figure 6.8:

Histogram of the number of SuN’s segments that detect energy for
events in the sum peak (the “hit pattern”). The spectra are from
simulations of the deexcitation of a 10 MeV state with γ-ray multiplicities of <M>= 2 and <M>= 3 with the emitted γ rays having
equal energy. The spectra are fit with a Gaussian function to determine the hit pattern centroid, and it is shown that the higher the
γ-ray multiplicity the higher the average number of SuN’s segments
that detect energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 6.9:

Method of determining the experimental γ-summing efficiency for the
sum peak shown in Fig. 6.6. Panel (a) shows the hit
pattern spectrum for events in the sum peak region and the corresponding Gaussian fit to determine the hit pattern centroid. Panel
(b) shows the efficiency and hit pattern centroids from GEANT4 simulations with various γ-ray multiplicities and transitions. The best
fit line is also shown, which is used to determine the experimental
efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
60 Ni(α,γ)64 Zn

Figure 6.10: Values of (a) hit pattern centroid and (b) γ-summing efficiency plotted against sum-peak energy for the 60 Ni(α,γ)64 Zn reaction. For
the reactions measured in this thesis, the hit pattern centroid and
therefore the average γ-ray multiplicity increase with energy, and the
γ-summing efficiency decreases with energy. . . . . . . . . . . . . . . 127

xvi

Figure 7.1:

Total γ-summed spectrum for the Ep = 2517.7 keV resonance in the
reaction. Plotted are the experimental spectrum taken
with the SuN detector and the result of GEANT4 simulations. . . . 129
27 Al(p,γ)28 Si

Figure 7.2:

Level scheme of 28 Si for the Ep = 2517.7 keV resonance in the
27 Al(p,γ)28 Si reaction. For simplicity, only the energy levels and
intensities for the primary γ-ray transitions are labeled. The unlabeled levels also participate in the cascades based on their transition
probabilities. The most dominant cascade is highlighted and consists
of the emission of γ rays with energy 9394, 2839, and 1779 keV. . . 130

Figure 7.3:

Spectrum from one segment of the SuN detector for events in the sumpeak region of the Ep = 2517.7 keV resonance in the 27 Al(p,γ)28 Si
reaction. Plotted are the experimental spectrum taken with the SuN
detector and the result of GEANT4 simulations. . . . . . . . . . . . 131

Figure 7.4:

Measured excitation function in the region Ep = 2 − 4 MeV of the
27 Al(p,γ)28 Si reaction. The inset shows a zoomed in view around the
resonance at Ep = 3674.9 keV and a comparison to the theoretical
Breit-Wigner yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 7.5:

Resonance strengths of the 27 Al(p,γ)28 Si reaction measured with the
SuN detector compared to previous results. Overall good agreement
is achieved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 8.1:

A zoomed in section of the nuclear chart showing the dominant reaction flow producing 74 Se in the p process. . . . . . . . . . . . . . . . 136

Figure 8.2:

Sum peak spectra for three different proton energies Ep normalized
to the same number of incoming protons. The location of the sum
peak and the metastable peak are indicated for Ep = 3400 keV. . . 137

Figure 8.3:

Total γ-summed spectrum for the 74 Ge(p,γ)75 As measurement at
Ep = 3400 keV. Also plotted is the linear background with and without including the 304 keV isomer. . . . . . . . . . . . . . . . . . . . 138

Figure 8.4:

Cross section vs. center of mass energy plot for the 74 Ge(p,γ)75 As
reaction. The measurements of this dissertation (solid circles) and
previous data of Ref. [10] (open triangles) are compared to theoretical
calculations using NON-SMOKER and TALYS 1.4 nuclear reaction
codes. The most accurate reproduction of the data is the TALYS
back-shifted Fermi gas model (BSFG). . . . . . . . . . . . . . . . . 140

xvii

Figure 8.5:

Ratio of reaction rates in Table 8.2 to those presented in Ref. [10]
and to the standard REACLIB [11] rates. The shaded area indicates
the p-process relevant temperatures of 1.8 - 3.3 GK. . . . . . . . . . 143

Figure 8.6:

Cumulative mass fraction of 74 Se from a model of the p process in a
type II supernova, plotted as a function of the maximum temperature
of the mass layer included. See text for details. . . . . . . . . . . . 145

Figure 9.1:

A zoomed in section of the nuclear chart showing the 58 Ni(α,γ)62 Zn
reaction and the isotopes most influenced by its reaction rate in SNIa
(62 Ni, 63 Cu, 64 Zn). . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 9.2:

Experimental spectra from the SuN detector for measurements at
Eα = 7.7 MeV. The spectra correspond to 58 Ni (solid black), thick
tantalum backing (dotted blue), and normalized room background
(dot-dashed red). The inset shows a zoom around the sum-peak region of the 58 Ni(α,γ)62 Zn reaction. . . . . . . . . . . . . . . . . . . 149

Figure 9.3:

Cross section of the 58 Ni(α,γ)62 Zn reaction for the present work
(black circles), previous data of Ref. [12] (red triangles), and theoretical calculations from the SMARAGD code [13]. A good description of
the data was obtained by modifying the α width and the γ-to-proton
width ratio (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . 152

Figure 9.4:

Absolute values of the sensitivity of the 58 Ni(α,γ)62 Zn cross section
as function of energy, when separately varying γ, neutron, proton,
and α widths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 10.1: REACLIB reaction rates [11] as a function of temperature for the
photodissociation of 94 Mo through the (γ,n), (γ,p), and (γ,α) channels. The width of the curves corresponds to a factor of 10 uncertainty in the (γ,α) reaction rate and a factor of 5 uncertainty in
the (γ,n) and (γ,p) rates. Within the p-process window of 1.8 − 3.3
GK, the 94 Mo(γ,α)90 Zr reaction may have a higher rate than the
94 Mo(γ,n)93 Mo reaction and could therefore be a branching point in
the p process [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Figure 10.2: Total γ-summed spectrum for an α beam impinging onto the enriched
90 Zr target at E
90
94
c.m. = 9.94 MeV, with the Zr(α,γ) Mo sum peak
at 12 MeV. The peak around 9 MeV in the spectrum comes from
neutrons released in the 90 Zr(α,n)93 Mo reaction. Also plotted is the
normalized room background. . . . . . . . . . . . . . . . . . . . . . 162

xviii

Figure 10.3: Total γ-summed spectrum for an α beam impinging onto the enriched
92 Zr target at E
92
96
c.m. = 11.4 MeV, with the Zr(α,γ) Mo sum peak
at 14.2 MeV. The peak around 12 MeV originates from neutrons emitted in the 92 Zr(α,n)95 Mo reaction. Also plotted are the normalized
room background and the beam-induced background from the tantalum backing. The inset shows a zoomed in view of the sum peak and
the linear background used when integrating the sum peak. . . . . . 164
Figure 10.4: Total γ-summed spectrum for an α beam impinging onto the enriched
74 Ge target at E
74
78
c.m. = 8.5 MeV, with the Ge(α,γ) Se sum peak at
14.5 MeV. The neutron-induced peak around 10.5 MeV comes from
neutrons emitted in the 74 Ge(α,n)77 Se reaction. Also plotted is the
normalized room background spectrum. . . . . . . . . . . . . . . . . 165
Figure 10.5: Experimental cross sections for the 90 Zr(α,γ)94Mo reaction compared
to TALYS 1.6 and NON-SMOKER calculations. The three TALYS
curves correspond to the upper limit, lower limit, and best-fit calculations (see text for details). . . . . . . . . . . . . . . . . . . . . . . 169
Figure 10.6: Experimental cross sections for the 92 Zr(α,γ)96Mo reaction compared
to TALYS 1.6 and NON-SMOKER calculations. The three TALYS
curves correspond to the upper limit, lower limit, and best-fit calculations (see text for details). . . . . . . . . . . . . . . . . . . . . . . 171
Figure 10.7: Experimental cross sections for the 74 Ge(α,γ)78Se reaction compared
to TALYS 1.6 and NON-SMOKER calculations. The three TALYS
curves correspond to the upper limit, lower limit, and best-fit calculations (see text for details). . . . . . . . . . . . . . . . . . . . . . . 173
Figure 11.1: Doppler corrected γ-summed spectra for the p(27 Al,γ)28 Si reaction
plotted with the normalized room background. The inset shows a
zoomed in view of the sum peak at EΣ = 12541 keV. Additionally, the
improvement due to applying the Doppler correction is demonstrated
and the excellent agreement with GEANT4 simulations is shown. . 177
Figure 11.2: Yield curve for the p(27 Al,γ)28 Si reaction with the experimental data
shown in solid circles. The spectrum is dominated by the Ec.m. = 956
keV resonance, with other large contributions from the Ec.m. = 988
keV and Ec.m. = 1078 keV resonances. In this energy region there
are additional contributions from the Ec.m. = 856, 890, 904, 966, 1051,
and 1058 keV resonances [15]. The solid line shows the total yield
from all resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xix

Figure 11.3: Doppler corrected γ-summed spectra for the p(58 Ni,γ)59 Cu reaction
with a comparison to the normalized room background. The inset
shows a zoomed in view of the sum peak at EΣ = 4819 keV as well
as the summed spectrum without Doppler corrections. An excellent
agreement with GEANT4 simulation is also observed. . . . . . . . . 180
Figure 11.4: Yield curve for the p(58 Ni,γ)59 Cu reaction. The spectrum is dominated by the Ec.m. = 1400 keV resonance, with an additional large
contribution from the Ec.m. = 1352 keV resonance. In this energy
region there are additional contributions from the Ec.m. = 1496 and
1514 keV resonances [16]. The solid line shows the total yield from
all resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Figure A.1:

A digitized signal from the SuN detector plotted along with its baseline. The lengths of the peaking time (L0 and L1 ) and gap time (LG )
are also drawn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Figure A.2:

The response of SuN’s tau-corrected energy filter compared to the
uncorrected energy filter. By correcting for the exponential decay
of the signal, the tau-corrected energy filter does not take on any
negative values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xx

Chapter 1
Introduction
It is fascinating to think that the chemical elements that make up our bodies and the world
around us have extraterrestrial origins. The lightest elements, hydrogen and helium (along
with trace amounts of lithium), can be traced as far back as a few seconds to a few minutes
after the Big Bang approximately 13.8 billion years ago, when the universe first cooled. If it
were not for the creation of stars around 100 million years after the Big Bang, there would
be no heavier elements. Instead, stars act as enormous nuclear reactors that combine the
hydrogen and helium building blocks into all of the elements that exist today. Not only do
stars generate the elements, but they may also spew them out into the neighboring universe
during their lifetimes. The ejected material can then be used in the formation of the next
generation of stars. It is through this cycle of stellar birth and death that the elements of
the universe are continually produced and mixed together and, ultimately, have contributed
to the existence of life as we know it today.
The motivation of this thesis work is one of origin, specifically improving the understanding of the origin of the elements. This is a fundamental question, but one that is complex
in nature and involves the interplay of several diverse research fields. The present dissertation represents a small contribution to the ongoing effort at a complete description of the
synthesis of the elements.

1

1.1

Elements

In total, there are less than 100 naturally occurring chemical elements. Each element is
comprised of small units known as atoms, and atoms are in turn comprised of a dense
nuclear core of positive protons and chargeless neutrons surrounded by a negative cloud
of electrons. An element is defined by the number of protons in its nucleus, and there are
different versions, called isotopes, that differ in the number of neutrons. Approximately 3000
isotopes have been discovered and it is thought that there may be over 7000 total possible
combinations of protons and neutrons that could make up a nucleus [17]. Remarkably, out
of these 7000 possible nuclei, less than 300 are known to be stable. The other nuclei, if given
enough time, eventually decay into a stable nucleus. Thus, the complex world that we live
in is made up of less that 300 nuclear building blocks along with their electrons. Nuclei
can be conveniently displayed in a chart as in Fig. 1.1, with each nucleus plotted according
to its unique number of protons and neutrons. The stable nuclei are commonly referred to
as the valley of stability. The notation for labeling each isotope is A
Z XN , where X is the
chemical symbol of the element, Z is the number of protons, N is the number of neutrons,
and A=Z+N is the atomic mass number. Each isotope can be uniquely identified in the
form A X, for example the carbon isotope with six protons and six neutrons is labeled 12 C.
The nuclei of 1 H and 4 He often receive special nicknames with 1 H being labeled p because it
consists of only one proton, and 4 He being labeled an α-particle for historical classification
reasons.
Experimental observations indicate that isotopes with certain “magic numbers” of protons and neutrons exist in an especially stable configuration compared to their neighboring
isotopes [18]. For example, magic nuclei are more tightly bound and therefore require larger

2

Proton Number (Z)

stable nuclei
observed nuclei

Neutron Number (N)
Figure 1.1: Nuclear chart with proton number along the y-axis and neutron number along
the x-axis. Stable nuclei are indicated with black boxes and the observed radioactive isotopes
are indicated with shaded boxes. There are also thousands of nuclei beyond what is plotted
here that are expected to exist and are yet to be discovered. The lines on the chart indicate
nuclear magic numbers.
amounts of energy to excite or completely remove a proton or neutron from the nucleus.
The experimentally observed magic numbers at 8, 20, 28, 50, 82, and 126 were successfully
predicted by the nuclear shell model in 1949 [19, 20]. In the same way that electrons fill
atomic shells with closed shells correspond to the noble gases, in the nuclear shell model
protons and neutrons fill nuclear shells with magic numbers appearing at the closed shells.
The location of the magic numbers on the nuclear chart is indicated in Fig. 1.1.

3

1.2

Abundances

In order to understand the synthesis of the elements, it is necessary to know what elements
exist in the universe and their relative quantity compared to each other. Any potential
description of the production of the elements must match the observed abundance pattern,
and any feature present in the abundance pattern is a clue to the stellar processes that are
taking place.
The most complete and accurate elemental abundances are from our solar system and
are known as the solar abundances. The solar abundances are similar to other sites in
the universe, but many differences exist due to different evolutionary histories. The solar
abundances serve as the standard to which all abundance patterns are compared. Since
the solar system formed from the collapse of a gaseous nebula of approximately uniform
composition, all objects are expected to have approximately the same elemental abundance
pattern. However, planets like the Earth have undergone many chemical separation processes
since their formation [21] and solar abundances are instead derived from two main sources;
observations of the Sun by astronomers and analysis of meteorites by cosmochemists.
Astronomers deduce abundances from stars by identifying and analyzing the electromagnetic waves emitted at unique frequencies by the elements near the stellar surface. From
these spectral lines, models of the stellar atmosphere and interior are used to calculate
abundances [22]. For the Sun these observations are primarily taken from the photosphere
because this is the deepest layer of the Sun that can be observed directly, but additional
information on a few elements can be deduced from events like solar winds, solar flares, solar
energetic particles, and through helioseismology (see Ref. [1] and references therein). On
the other hand, cosmochemists measure elemental abundances directly by chemical analysis

4

of meteorites. Many of the meteorites that have been recovered on the Earth have undergone significant changes since their formation 4.5 billion years ago, but there have been 5
meteorites discovered, known as CI carbonaceous chondrites, which are expected to have
maintained the majority of their initial chemical composition [23].
Having two separate and complementary sources of solar abundances is important because
more information is available than just using one source alone. For example, meteorites are
expected to be lacking some of their initial composition of elements like hydrogen, oxygen,
and the noble gases as they form gases which escape from the meteorite, and the Sun
is expected to be lacking in elements like lithium which get processed by the Sun. For
the elements which can be compared in both meteorites and the Sun, their abundances
typically agree remarkably well to within 10%. The comparison is done by normalizing
the abundances from both meteorites and the Sun to 106 silicon atoms. Additionally, solar
isotopic abundances are usually derived from the elemental abundances by using the isotopic
ratios found on the Earth, for example using data from Ref. [24]. However, there have been
small variations in isotopic composition observed between the Earth and other parts of the
solar system due to mass fractionation, and care must be taken when assigning solar isotopic
abundances [25].
There are several compilations of solar abundances from the first detailed work [26] to
more recent works [25, 1] (along with subsequent updates), and a plot based on the data
from Lodders [1] is displayed in Fig. 1.2. The lightest elements, hydrogen and helium, are
by far the most abundant elements in the solar system, while the next lightest elements,
lithium, beryllium, and boron, have relatively low abundances. The elements from carbon
to scandium have a general trend of decreasing abundance with increasing mass, and then
there is a spike in the abundances around iron which is known as the iron peak. The elements
5

Figure 1.2: Abundances of the nuclides in the solar system based on the data from Lodders [1]. The abundances are normalized to 106 silicon atoms.
beyond the iron peak have very low abundances and show maxima in the ranges of mass
110-150 and 180-210.

1.3

Nucleosynthesis

The creation of the chemical elements in the universe is called nucleosynthesis. The modern
view of nucleosynthesis began with two seminal papers by Burbidge, Burbidge, Fowler, and
Hoyle (B2FH) [27] and Cameron [28] in 1957, which were based on existing ideas and the
available abundance patterns of the time. In the nearly 60 years since, the interplay of the
diverse research fields of astronomy, cosmochemistry, theoretical astrophysics, and nuclear
physics have advanced our understanding of the origin of the elements. There are many
astrophysical processes, thousands of nuclear reactions, and countless details that will be

6

glossed over in this section in favor of generality. In the following Section 1.4, more details
will be given on the specific nucleosynthesis process that is most directly related to this
dissertation.

1.3.1

Big Bang Nucleosynthesis

The first second of the universe was a period of remarkable expansion and change with
extreme energies causing particles to pop into and out of existence. After a few seconds,
the universe had cooled enough for quarks to form free protons and neutrons, and a few
minutes later the universe had cooled enough to allow protons and neutrons to combine
into heavier nuclei. The free neutrons were quickly scooped up into nuclei (mostly helium)
creating abundances of roughly 75% hydrogen and 25% helium, which are approximately
what is observed in the universe today. Trace amounts of lithium and beryllium were also
produced, but no heavy elements could be built up at this point because of the lack of stable
isotopes with mass numbers 5 and 8 and the insufficient densities preventing much nuclear
fusion from taking place [29]. This is contrary to an earlier proposal that all the elements
were produced as a result of the Big Bang [30].

1.3.2

Quiescent Stellar Burning

Over the course of 100 million years after the Big Bang, gas clouds in the universe began to
collapse to form the first stars, a process that still takes place today. As material collects, the
gravitational pressure leads to increased temperature and density at the core and eventually
nuclear fusion begins. The energy released as a result of the nuclear fusion opposes the
gravitational pressure of the star in hydrostatic equilibrium. This is known as quiescent

7

stellar burning. At first, hydrogen is converted into helium in the core of the star via the
pp chain or the CNO cycle. Stars spend the majority of their lifetime in this main sequence
stage, but eventually the hydrogen fuel runs low and the gravitational pressure causes the
star to collapse and the core to increase in temperature. The mass of the star determines its
evolution and if the star is massive enough, high enough temperatures will be reached that
the next stage of stellar burning in the core occurs, which converts helium into carbon via
the triple-α reaction in the red giant stage. Depending on the mass of the star, the sequence
of burning stages in the core at increasingly higher temperatures and shorter timescales
continues with carbon burning, neon burning, oxygen burning, and silicon burning. The
previous burning stages do not stop but continue in outer shells of the star in an onion-like
structure as shown in Fig. 1.3. The quiescent burning stages of stars can produce elements
up to those around iron where the sequence stops because the nuclei in the iron region are
energetically the most tightly bound (see Sec. 2.5). Additional fusion reactions beyond the
iron region are mostly endothermic and do not release energy to withstand the gravitational
pressure of the star [31].

1.3.3

Supernovae

A supernova is the explosion of a star, which causes the release of huge amounts of energy
and particles into the universe. Supernovae are classified into two groups based on the
presence of hydrogen absorption lines in their spectra, with type I having no hydrogen
lines and type II having hydrogen lines. There are several subcategories of these two main
classification groups, and this thesis focuses on nucleosynthesis in supernovae type Ia (SNIa)
and supernovae type II (SNII).
SNII occur at the end of quiescent stellar burning of a massive star. At first the inert
8

H
He
C
Ne
O
Si
Fe

Figure 1.3: Simplified picture of a massive star that has undergone all of the burning stages
in the core. The burning layers are not drawn to scale.
iron core is supported by the pressure of the degenerate electrons. However, once the core
reaches approximately 1.4 times the mass of the Sun, electron degeneracy pressure can no
longer support the star and the gravitational pressure causes the core to collapse under its
own weight (the Chandrasekhar limit). For this reason SNII are also referred to as core
collapse supernovae. The collapsing core causes the outer layers of the star to fall inwards.
Once the infalling material strikes the collapsed core it rebounds outward in a shock wave.
The outward propagating shock wave disrupts the star into a supernova and causes explosive
nucleosynthesis to occur. Most of the outer layers get blown off in the explosion, leaving
behind a neutron star or a black hole [32].
Whereas SNII involve only a single star, SNIa are thought to occur in a binary star
system, where at least one of the stars is a white dwarf made out of carbon and oxygen.
Carbon-oxygen white dwarves are formed from stars which are not massive enough to ignite
carbon burning in their cores, and therefore end their quiescent stellar burning lifetimes
after helium burning. Because there is limited energy released from nuclear reactions to
9

counteract the gravitational pressure, white dwarves are incredibly dense objects supported
by electron degeneracy pressure. White dwarves by themselves do not become supernovae,
but if they have a nearby star as a companion they may accrete mass from the neighboring
star until they reach the Chandrasekhar limit of approximately 1.4 times the mass of the
Sun. At this point, the carbon ignites and the burning quickly spreads throughout the star
causing it to explode as a supernova and leave no remnant behind. It is not yet known
whether SNIa consist of a white dwarf and a different type of star (single degenerate), two
white dwarfs that collide (double degenerate), or possibly some other scenario. As with all
types of supernovae, SNIa involve several explosive nucleosynthesis processes [33].

1.3.4

Nuclear Statistical Equilibrium

At high enough temperatures, the material inside a star can quickly reach a point where all
forward and reverse reactions are in equilibrium, except for the reactions involving neutrinos,
which are chargeless subatomic particles that typically escape from the star. For example,
in equilibrium the rate that species A captures a proton to form species B is exactly equal
to the rate that species B is photodissociated into species A plus a free proton. These type
of reactions, which will be used throughout the remainder of this thesis, can be denoted as
follows:
A + p −→ B + γ

A(p, γ)B

(1.1)

B + γ −→ A + p

B(γ, p)A

(1.2)

When all species are in equilibrium with each other, this is known as nuclear statistical
equilibrium (NSE). Typical temperatures to achieve NSE in stellar environments are larger
than 5 GK and such extreme temperatures are reached in the innermost layers of stars when
10

they explode as supernovae. While in NSE the abundance of each isotope can be determined
from a relatively small number of parameters, including nuclear properties like mass and
spin, and stellar properties like temperature, density, and neutron-to-proton ratio of the
material [34]. In NSE, the highest temperatures favor free nucleons and lower temperatures
favor isotopes that are energetically the most bound (have the highest binding energy).
Therefore, in a supernovae the material that reaches NSE first breaks down into free protons,
neutrons, and α particles due to the initial high temperatures. As the material expands and
cools, the free nucleons combine to form mostly iron-peak nuclei due to their high binding
energy. The neutron-to-proton ratio of the stellar material determines which isotopes in the
iron region are the most abundantly produced [31]. Because supernovae spew the majority
of their mass into the universe, the iron-peak nuclei existing in our solar system are nearly
all a result of NSE.
If the expansion of the material undergoing NSE in the supernova occurs very rapidly,
then the triple-α reaction can drop out of equilibrium before all the free α particles can
combine into heavier nuclei. Such a scenario is known as α-rich freeze-out of NSE, and it
leaves a quasi-equilibrium group of heavier nuclei [35]. The final composition of the material
is iron-peak nuclei and free α particles. Further discussion of α-rich freeze-out of NSE is
contained in Chapter 9.

1.3.5

Nucleosynthesis of the Heavy Elements

The elements beyond the iron peak cannot all be produced through nuclear fusion reactions
and must be produced in some other fashion. Since neutrons are chargeless and not repelled
by the positive nucleus, the majority of these heavy nuclei are produced by neutron captures
on existing nuclei inside of stellar environments. There are two main neutron capture pro11

cesses that contribute: the slow (s) neutron capture process [36] and the rapid (r) neutron
capture process [37]. As their names indicate, the two processes are distinguished by the
speed of neutron capture. For a lower abundance of neutrons, the capture reactions occur
slowly enough that any unstable nucleus that is produced has time to decay back into a
stable nucleus before the next neutron capture. This is known as the s process and is expected to occur in the helium-shell burning, red giant phase of a star’s life. For a large flux
of neutrons, as is expected in a supernova explosion or the collision of two neutron stars, the
neutron capture reactions occur very rapidly and produce very neutron-rich radioactive nuclei before decaying back into stable nuclei. This is known as the r process. Fig. 1.4 includes
the nucleosynthesis pathway of the s and r processes. The spikes in the abundance pattern
in Fig. 1.2 around mass 110 − 150 and 180 − 210 are signatures of the s and r processes.
While the s and r processes synthesize the majority of the heavy isotopes, there exists a
group of stable isotopes that cannot be produced by either neutron capture process. These
“p nuclei” lie on the neutron-deficient side of the valley of stability ranging from 74 Se up to
196 Hg

(see Fig. 1.4). There are 35 p nuclei in total, but several of the isotopes are expected

to have contributions from the s and r processes and are not always classified as p nuclei.
With the exception of 92,94Mo and 96,98Ru, the isotopic abundances of the p nuclei are very
low compared to their more neutron-rich neighbors, typically contributing less than 1% to
the total elemental abundance. Several potential astrophysical sites have been proposed for
the creation of the p nuclei and a discussion of these scenarios follows in Sec. 1.4.

12

(a)
196Hg

74Se

s process
r process
p nuclei

(b)

s process
r process decays
p nuclei

Figure 1.4: Nuclear chart showing the pathways for the s and r neutron capture processes,
with a zoomed in view on the bottom. The s and r processes produce the majority of the
stable isotopes heavier than iron, except for the 35 p nuclei which are shielded from β −
decay by the valley of stability.

13

1.4

Production of the p Nuclei

In the influential B2FH paper [27], the authors first noticed the need for an alternative
astrophysical process to explain the production of the p nuclei. They termed it the “p
process” and proposed the scenario of proton capture reactions on preexisting s- and rprocess nuclei at high temperatures inside the hydrogen shell of a massive star when it
explodes as a type II supernova. However, it was later shown that the high temperatures
and densities required to produce the p nuclei in this fashion are not reached in the hydrogen
shell [38]. More recently, two alternative explosive processes involving proton captures were
identified as likely candidates for synthesizing some of the less massive p nuclei; the νp
process [39, 40, 41] and the rp process [42].

1.4.1

νp process

The νp process is expected to occur in a type II supernova, when a massive star explodes
ejecting its outer layers and leaving behind a neutron star [39]. The innermost layers of
the ejecta are proton rich and undergo a series of proton captures as the material expands.
The material expands very rapidly, however, and this provides only a short timescale for
reactions to occur. Thus, the creation of elements with heavier mass gets inhibited by
waiting point nuclei, for which proton capture is inhibited and the reaction flow has to wait
for the relatively long decay time before continuing. However, in the process of forming the
neutron star, neutrinos are radiated outward and can be captured by the large number of
free protons to create neutrons. It is the capture of these newly produced neutrons on the
waiting point nuclei which allows the flow of nucleosynthesis to bypass the long decays and
produce elements with heavier mass. It has been shown that the νp process may produce

14

nuclei up to mass 152 under certain supernova conditions [43]

1.4.2

rp process

The rp process requires a hydrogen-rich environment at high temperatures to lead to rapid
proton captures that synthesize higher mass nuclei. Such conditions can be reached in a
binary star system with a donor star accreting hydrogen and helium from its atmosphere
onto its neutron star partner. As the material builds up on the neutron star’s surface,
instabilities can cause an X-ray burst, which leads to high enough temperatures for the rp
process to occur [44]. The rp process can reach up to mass 110 at the endpoint in the protonrich isotopes of tellurium [45], but it is not clear whether the material in the X-ray burst
escapes the gravitational field of the neutron star to contribute to the changing chemical
history of the universe [46].

1.4.3

p process

While the νp process and rp process can synthesize some of the less massive p nuclei, there is
an astrophysical scenario that can produce the p nuclei across the entire mass range. Instead
of proton captures creating more massive nuclei, the dominant reactions are photodisintegrations that knock out nucleons from existing nuclei and produce the p nuclei as a result.
The preexisting nuclei in the star, referred to as seed nuclei, can originate from the s- and rprocess material expelled from other stars or can be freshly produced through the s process
in the star of interest. All objects emit radiation depending on their temperature through
blackbody radiation, and for temperatures higher than about 1.8 GK the blackbody radiation from the star is at high enough energies to knock out protons, neutrons, and α-particles

15

Figure 1.5: Figure from Ref. [2] showing the temperature and density of the O/Ne layer as
the shockwave passes through. The curves represent different depths of the O/Ne layer.
in (γ,p), (γ,n), and (γ,α) reactions, respectively. It is important that the temperatures do
not reach higher than about 3.3 GK, however, so that the heavier nuclei are not completely
photodissociated into iron peak nuclei.
The conditions of having existing seed nuclei and temperatures between 1.8 and 3.3
GK are reached naturally in the oxygen and neon (O/Ne) layers of a massive star (see
Fig. 1.3). For the more massive stars the temperatures may be already high enough for
photodissociations to occur in the presupernova phase [47, 48], but the largest effect is
expected to occur in the type II supernova when the shockwave passes through the O/Ne
layers [49]. The term “p process” from the original B2FH publication has stuck, although
(perhaps more appropriately) it is also referred to as the γ process in literature because of
the prevalence of photodisintegration reactions. Nice reviews of the p process can be found
in Ref. [3] and Ref. [50].
Fig. 1.5 shows the temperature and density of the O/Ne layer as the shock wave propagates through. There is a very rapid rise followed by an exponential decrease in temperature
and density. A maximum temperature between 1.8 and 3.3 GK is reached for various depths

16

(Ȗ,n)
+
(Ȗ,p) ȕ

Z
(Ȗ,Į)

N
Figure 1.6: Section of the nuclear chart showing an example reaction flow for the p process.
in the O/Ne layer. Fig. 1.6 shows a simplified reaction flow of the p process for only one seed
nucleus. In reality the flow proceeds through a complex network of reactions that compete
at different temperatures. As mentioned, the seed nuclei have already been produced by
a previous s- or r- process event. At first, the most probable reactions are (γ,n) reactions
which cause the flow of mass from the stable seed nuclei into unstable nuclei. For each (γ,n)
reaction that occurs, the nuclei become increasingly neutron deficient and eventually the
(γ,p) or (γ,α) reactions will occur at a higher rate. (γ,α) reactions are typically more significant for the heavier elements while (γ,p) reactions are key contributors to the formation
of the lighter p nuclei [3]. There are also proton capture reactions that play a role for the
lighter masses. The isotope in each elemental chain where the (γ,p) or (γ,α) reaction rate
is comparable to the (γ,n) rate is called the branching point, and its location is critical for
understanding the flow of mass in the p process [14]. The branching point shifts the reaction
flow to a new elemental chain where the photodisintegration reactions continue. Once the
temperature cools and the photodisintegration reactions slow, the competition of the β + decays of unstable nuclei also become important. Eventually all unstable nuclei will β + decay
back to stability.
17

It should be emphasized that Fig. 1.6 represents the initial reaction flow from only one
seed nucleus. Inside of the supernova, all preexisting s and r nuclei are taking part in the
photodisintegrations. In total, the p process includes over 20,000 reactions on approximately
2000 nuclei. These include (γ,p), (γ,n), and (γ,α) reactions, their inverse capture reactions,
and β + decays. It is also important to note that the majority of the reactions involve
radioactive isotopes.
Reaction network calculations of the p process have been performed by various authors
for a range of stellar masses and compositions [51, 6, 3, 2, 52]. The stellar parameters used in
these reaction networks are determined from hydrodynamical simulations of the environment
of massive stars (see Ref. [6] and references therein). In these p-process models, the massive
star is divided into mass layers and reaction network calculations are performed in each
layer. For example, Ref. [2] divided the O/Ne of the star into 14 layers. It is also necessary
to use accurate initial seed abundances, stellar parameters such as temperature and density,
and nuclear properties such as reaction rates, masses, and decay rates. Any uncertainty in
the input physics contributes to the uncertainty in the production of the p nuclei in the
models. A discussion of these uncertainties is presented in Sec. 1.4.4. After performing the
calculation, it is desirable to compare the abundances of the p nuclei produced in the model
to the solar abundances. However, this comparison cannot be done directly because the
solar abundances contain a mixture of many nucleosynthesis processes and the models are
focused only on the p process. Instead, what is typically done is to check whether or not the
models produce the p nuclei in the correct ratios [6]. First, the total mass of each p nucleus
produced in the calculation is compared to what its total mass would be if it was present in

18

the correct solar abundance. This ratio is known as the overproduction factor

Fi =

mi
Mmodel Xi⊙

(1.3)

where mi is the produced mass of a p nucleus over all the mass layers the model, Mmodel
is the total mass of all the mass layers in the model, and Xi⊙ is the solar abundance mass
fraction of the p nucleus. Then, the average overproduction factor for the 35 p nuclei is
calculated by
1
F0 =
35

35

< Fi > .

(1.4)

i=1

Comparing the overproduction factor of each p nucleus to the average overproduction factor is
a convenient way to check if the model reproduces the p nuclei in the correct solar abundance
ratios. This quantity < Fi > /F0 is called the normalized overproduction factor, and if all
the p nuclei were produced in the correct ratio, the normalized overproduction factor for
each p nucleus would be equal to 1. The p nuclei with values above 1 are overproduced
by the calculation compared to the other p nuclei, and p nuclei with values below 1 are
underproduced.
Fig. 1.7 shows a typical result of the normalized overproduction factor for each p nucleus
from a calculation by Ref. [2], and all other models show a similar pattern. Overall, the
majority of the calculated p-nuclei abundances agree to within a factor of 2-3 of the observations. The most notable discrepancy is the underproduction of 92,94 Mo and 96,98Ru, which
is a problem because they are the most abundant of the p nuclei and make up to 14% of the
total elemental abundance. If the p process scenario is correct it should be able to predict
the correct abundances for the most abundant isotopes. However, as previously mentioned
the νp process (Sec. 1.4.1) and rp process (Sec. 1.4.2) may be able to contribute to the pro19

Figure 1.7: Normalized overproduction factor for the calculated abundance of each p nucleus
plotted against mass number. Plot is taken from Ref. [2]
duction of the lighter p nuclei and may help explain this discrepancy. Also underproduced
by the p process calculation are 113 In, 115 Sn, and 138 La. Possible explanations are that
113 In

and 115 Sn may have contributions from the s and r processes [53, 54], and 138La can

be produced through neutrino reactions [55, 56]. While the majority of the light p nuclei are
underproduced, 74 Se is surprisingly overproduced. Thus far the overproduction of 74 Se has
remained a mystery and a nuclear physics measurement related to the 74 Se abundance can
be found in Chapter 8.
In addition to type II supernovae, the p process is expected to occur in type Ia supernovae [57]. In this scenario, material enriched in s nuclei is accreted from a companion star
onto a white dwarf star. Once the white dwarf reaches its maximum mass, known as the
Chandrasekhar mass, it explodes as a type Ia supernova and some of the material reaches
the correct temperatures for the p process to occur. Earlier models of this scenario saw
the same underproduction in light p-process nuclei as in the O/Ne layer of type II super-

20

novae [58, 59]. However, a more recent attempt using a two-dimensional model matches the
solar abundances of the light p-process nuclei by using large enhancements of the s nuclei
prior to the explosion [60].

1.4.4

Uncertainties

There are several sources of uncertainty in the production of the p nuclei via the p process,
including uncertainties in the stellar environment and nuclear physics data both before and
during the p process. Even before the supernova occurs, it is necessary to understand the
development of the pre-supernova star, especially the abundances of the seed nuclei. These
abundances depend on the initial composition of the star, the nuclear reactions contributing
to the s process, and how the seed nuclei get mixed and transformed prior to the explosion. Fig. 1.8 shows the spread in abundances of the s nuclei as a result of some of these
uncertainties. Both varying the initial composition of the star (left panel) and the reaction
rates (right panel) in a s-process simulation lead to different distributions of the seed nuclei.
Additionally, it has been shown that the uncertainties in the 22 Ne(α,n)25 Mg [61, 62] and
12 C(α,γ)16 O

[6] reactions have a significant impact on the result of p-process calculations

due to their effect on the initial composition of the p-process layers of massive stars. The
important contributions of both reactions take place during the helium burning stage. The
22 Ne(α,n)25 Mg

reaction is the main source of neutrons for the s process during helium burn-

ing, while the 12 C(α,γ)16 O reaction determines the composition of the carbon-oxygen core
and its later evolution. Reducing the uncertainty on both of these reactions will lead to more
accurate p-process calculations.
In addition to the uncertainty in the abundances of seed nuclei, the stellar environment during the supernova also contributes uncertainties to the p process. For example,
21

(a)

(b)

Z = ZO
Z = 0.1ZO

Caughlan et. al.
NACRE

Figure 1.8: Figures from Ref. [3] showing the abundances of the s nuclei at the end of the
helium burning stage and how they depend on (a) the intial metallicity of the star (b) the
reaction rates used in the s-process calculations. The intial metallicities used were solar and
10% solar, and the different rates used were from Ref. [4] and Ref. [5].
the temperature and density profile of the stellar layers during the explosion are critical
for determining which p nuclei are produced. This can be seen in Fig. 1.9 from Ref. [2],
which shows how sensitive the p process is to the maximum temperature that a stellar layer
reaches during the explosion. Shifting the maximum temperature of a layer by a tenth of a
GK can have a huge effect on which p nuclei get produced due to the change in rate of photodisintegration reactions. Because the mass of the star and its composition are key factors
in determining the properties of the supernova explosion, the uncertainty in initial stellar
masses and metallicities are also important considerations. Fig 1.10 from Ref. [6] shows how
stellar masses ranging between 13-25 times the mass of the Sun lead to very different results
for the p process. In the universe, there is a distribution of massive stars with different
masses and metallicities which undergo the p process and mix different quantities and ratios
of the p nuclei into the interstellar medium. Because the solar abundances are a mixture of
all these different sources of p nuclei, another uncertainty in the production of the p nuclei
is the distribution of the various massive stars [6].

22

Figure 1.9: Figure from Ref. [2] showing the overproduction factor of selected p nuclei as a
function of the peak temperature of the stellar layer reached during the supernova explosion.

Figure 1.10: Figure from Ref. [6] showing the normalized overproduction factor of the p
nuclei for supernova explosions of stars with mass in the range between 13-25 times the mass
of the Sun. The vertical bar for each nucleus indicates the spread in p-nuclei abundances as
a result of the various stellar masses.

23

(a)

(b)

Figure 1.11: Figure from Ref. [2] showing the ratio of p-nuclei abundances calculated with
rates modified by a factor of three up and down compared to the standard rate. (a) Modified
proton-induced reactions and their inverse reactions. (b) Modified α-induced reactions and
their inverse reactions.
A final crucial uncertainty in the p proccess is the nuclear physics reaction rates used in
the p process models. As previously mentioned, the p process involves over 20,000 reactions
on 2000 nuclei with the majority of the reactions involving radioactive isotopes. Thus far,
very little experimental data exists for reactions relevant to the p process [50] and instead
all astrophysical models rely heavily on theoretical calculations of reaction rates. These
theoretical rates can have uncertainties of up to a factor of three. Fig. 1.11 shows the effect
of varying all proton-induced (left panel) and α-induced (right panel) reaction rates and
their inverse rates up and down by a factor of three. The uncertainties in reaction rates
for reactions involving protons have the largest impact on the production of the lighter p
nuclei, and the uncertainties in reactions involving α particles have the largest impact on
the production of the heavier p nuclei.
Reducing the uncertainty in the nuclear physics reaction rates to improve p-process calculations is the main focus of this dissertation. Since it is not realistic to measure all 20,000
reactions, an interested experimentalist has two goals. The first goal is to measure as many

24

p process reactions as possible at the relevant astrophysical energies to put constraints on
the theoretical calculations. Ideally the constraints would allow the theoretical calculations
to predict the other unknown reactions more accurately. The second goal is to identify
which reactions are the most critical for the production of the p nuclei. While there are
many reactions occurring, only a few have a large impact on the final result. There have
been two publications which have identified some of the most important reactions in the p
process. Rapp et al. [2] performed a sensitivity study which varied reactions up and down by
a factor of 3 (approximately equal to the theoretical uncertainty) and investigated which reactions led to the largest change in the abundances of the p nuclei. The other publication by
Rauscher [14] contains a model independent study identifying the location of the p-process
branching points which are critical to know for accurately understanding the flow of the p
process to lighter mass. This dissertation contains the development of a new detector and
expansion of an existing experimental technique in an effort to measure some of these critical
reactions and improve the accuracy of p-process calculations.

1.5

Looking Ahead

The remainder of this dissertation contains the story of the development of a new detector to
measure reactions relevant to the production of the p nuclei as well as results from the first
experiments with the detector. In Chapter 2 important quantities in nuclear physics and
astrophysics will be introduced. In Chapter 3 the most relevant nuclear reaction theories for
this work will be presented. In Chapter 4 an overview of available experimental techniques
and the methods used for this project will be provided. In Chapter 5 the design and testing
of the primary detector will be discussed. In Chapter 6 the steps which were taken in the

25

analysis of the data will be demonstrated. In Chapters 7, 8, 9, 10, and 11 the results for the
measurements of individual nuclear reactions will be presented, including the extension of
the experimental technique to measurements in inverse kinematics. Finally, in Chapter 12 a
summary and outlook for this dissertation will be provided.

26

Chapter 2
Nuclear and Astrophysical Quantities

2.1

Cross Section

This dissertation contains the measurement of the reaction cross section for several different
nuclear reactions. To introduce this important nuclear physics quantity let’s start with the
classical treatment using the carnival game of throwing a dart at a wall of balloons as our
example. If you hit a balloon, it will hopefully pop and you will win the prize of your
choosing. Let’s assume that all balloons are spheres of the same size and that you have
decent aim so that your throws are always within the wall of balloons. We will say that your
throws hit the wall within a circle of area A which is less than the total area of the wall (see
Fig. 2.1). The probability that a balloon is hit is given by the ratio of the area covered by

Figure 2.1: Front view of uniformly placed circular targets. The large grey circle represents
the area in which projectiles strike the targets.

27

the balloons to the area A:

Psuccess =

area covered by balloons
N πr 2
N
= b
= b σ
area covered by darts
A
A

(2.1)

where Nb is the number of balloons within your throwing accuracy, Nb /A is the areal density,
and σ = πr 2 is the cross-sectional area covered by one balloon. If you throw a large number
of darts (Nd ) then the number of prizes you can expect to win is

Nprizes = Nd Psuccess = Nd

Nb
σ .
A

(2.2)

Thus, the cross section of the balloons can be calculated by

σ =

Nprizes
Nreactions
=
Nd Nb /A
Nprojectiles Ntargets /A

(2.3)

where we made the change in variables from darts to projectiles, balloons to targets, and
prizes to reactions. This same equation will be used throughout the dissertation to calculate
the nuclear cross sections. Before moving on there are two important things to notice from
Eq. 2.1; the cross section is directly related to the probability of an interaction occurring,
and the cross section quantity has units of area.
In this example, the cross section is simply equal to the physical area that a balloon
takes up σ = πr 2 . Thus in the classical mechanics understanding, the cross section of a
nuclear reaction would be given by the cross-sectional area of the nucleus, which for uranium
is approximately equal to 10−24 cm2 . Scientists have designated this value of 10−24 cm2 as
1 barn (b). However, at the small nuclear scale there are quantum mechanical effects and

28

instead of the simple value of πr 2 the cross section has an energy dependence

σ ∝ π¯
λ2 = π

h
¯2
2µE

(2.4)

where λ
¯ is the reduced de Broglie wavelength, h
¯ is the reduced Planck constant, µ is the
reduced mass equal to m1 m2 /(m1 + m2 ), and E is the center of mass energy of the projectile
and target system. In addition to the expected 1/E energy dependence of the cross section,
there are several other factors that may inhibit the nuclear reaction from taking place. For
example, the Coulomb barrier related to the nuclear charge and the centrifugal barrier related
to the nuclear angular momentum are important factors in determining the cross section. In
conclusion, the reaction cross section may be thought of as the effective area of the nucleus
in a reaction. Typical cross sections measured in this thesis range from 10−6 b (1 µb) to
10−3 b (1 mb).

2.2

Stellar Reaction Rate

In order to understand the energy generation and synthesis of the elements inside of stellar
environments, it is necessary to know the rate at which nuclear reactions are taking place.
The stellar reaction rate depends on the abundance of each type of nuclei inside of the star,
the reaction cross section between them, and their relative energy. If we consider a stellar
gas of particles in a volume (say it has area A and depth x), along with a flux of particles
with relative speed v (see Fig. 2.2) then we can derive an expression for the reaction rate
per unit volume per time starting from Eq. 2.2:

NR = N1

N2
σ
A

−→
29

NR
N N
x
= 1 2 σ
V t
V Ax t

(2.5)

Figure 2.2: A number of particles N1 and velocity v impinging on a number of particles N2
within a volume that has area A and depth x.
where we first divided by the volume V and the time t and then multiplied by a convenient
factor of x/x = 1. The quantity Nx /V is the number density of the interacting particles
with units of number of particles per unit volume, and can be written simply as nx . Since
the cross section depends on energy and therefore on velocity, it immediately follows that
the reaction rate is equal to
r12 = n1 n2 σ(v) v .

(2.6)

Inside of stellar environments the nuclei do not all have the same relative velocities but
instead have a distribution of velocities, described by the probability function φ(v). Folding
in this distribution as well as taking into account reactions involving identical particles, the
reaction rate can be written

r12 =

n1 n2
1 + δ12

∞

φ(v) σ(v) v dv

(2.7)

0

where the Kronecker symbol δ12 is 0 for reactions between different nuclei and 1 for reactions
between identical nuclei. The number density of reaction pairs is given by n1 n2 /(1 + δ12 )

30

and the reaction rate per particle pair is given by
∞

σv =

φ(v) σ(v) v dv .

(2.8)

0

In a stellar plasma, nuclei are nondegenerate and nonrelativistic particles and thus move
with velocities given by the Maxwell-Boltzmann distribution

φ(v) = 4πv 2

3/2
µ
exp
2πkT

−µv 2
2kT

(2.9)

where k is the Boltzmann constant and T is the temperature of the stellar plasma. Combining
the Maxwell-Boltzmann distribution with Eq. 2.8 and making the change of variables from
relative velocity to center of mass energy E = µv 2 /2, the reaction rate per particle pair can
be written
σv =

2.3

8
πµ(kT )3

∞

σ(E) E exp

0

−E
kT

dE .

(2.10)

Astrophysical S factor

The nuclear reactions in stellar environments take place at energies which are typically too
low for charged particles to get close enough for the attractive nuclear strong force to fuse the
particles together. Instead the electric repulsion of the two positive nuclei create a Coulomb
barrier
VC =

Z1 Z2 e2
r

(2.11)

with Zx e equal to the charge of the nuclei. For relative energies below the Coulomb barrier,
charged particle reactions would not occur except for the quantum mechanical tunneling

31

effect. At energies well below the Coulomb barrier the probability of tunneling through the
barrier is given by

Ptunnel = exp(−2πη)

with

η =

Z1 Z2 e2
h
¯v

(2.12)

where η is called the Sommerfeld parameter.
The effect of the Coulomb barrier and the expected 1/E dependence of the cross section
as introduced in Eq. 2.4 motivate the writing of the nuclear reaction cross section as

σ(E) =

1
exp(−2πη) S(E)
E

(2.13)

where S(E) is the astrophysical S factor. Because much of the energy dependence of the
cross section has been factored out, S(E) represents all of the nuclear effects in the cross
section and is a smoothly varying function of energy. Due to the reduced energy dependence
of S(E) compared to the cross section it is much more useful to use S(E) to extrapolate
to lower energies. Nuclear reactions taking place in stellar environments typically occur at
low energies, where the cross section for charged particle capture is very low due to the low
probability of tunneling through the Coulomb barrier. Experimental data is often obtained
at higher energies where the higher cross section makes the measurement more feasible, and
then the data is extrapolated down to the lower astrophysical energies. It is more useful to
extrapolate using S(E) than σ(E).

32

2.4

Gamow Window

Plugging the S factor (Eq. 2.13) into the reaction rate (Eq. 2.10) gives

< σv > =

∞

8
πµ(kT )3

exp(−2πη) S(E) exp

0

−E
kT

dE .

(2.14)

Changing the Sommerfeld parameter to a function of energy via E = µv 2 /2 and introducing
the constant term EG = 2µ(πZ1 Z2 e2 /¯h)2 the integral becomes

< σv > =

8
πµ(kT )3

∞
0

exp −

EG
E

S(E) exp

−E
kT

dE .

(2.15)

Since S(E) is a smoothly varying function of energy, the energy dependence of the integrand
is dominated by the two exponential terms. The term arising from the probability to penetrate through the Coulomb barrier approaches zero as the energy approaches zero, and the
term arising from the tail of the Maxwell-Boltzmann distribution approaches zero for large
energies. Thus, when multiplying the two terms together there is a sweet spot in the middle
which creates a peak in the integrand. This is known as the Gamow peak and it gives rise
to the range of energies, known as the Gamow window, over which the majority of nuclear
reactions take place inside of stellar environments. The Gamow peak can be seen in Fig. 2.3.
The energy at the maximum of the Gamow peak can be found by solving for the point at
which the derivative of the integrand of Eq. 2.15 equals zero. Before taking the derivative,
S(E) is taken to be constant and factored outside of the integral. This leads to a value of

Emax =

EG (kT )2
4

33

1/3

.

(2.16)

Maxwell-Boltzmann
Coulomb Barrier

Probability

Gamow Peak

Energy

Figure 2.3: The tail of the Maxwell-Boltzmann distribution of particle energies exp(−E/kT )
combined with the probability of charged particles to penetrate through the Coulomb barrier
exp(−2πη) gives rise to the Gamow peak (not drawn to scale). The Gamow peak defines
the energy window over which the majority of nuclear reactions take place inside of stellar
environments.
The width of the Gamow peak can be estimated by approximating the shape as a Gaussian
function. The 1/e width of the Gaussian is found to be [31]:

∆ = 4

Emax kT
.
3

(2.17)

It has been pointed out by Ref. [63], that the energy and width of the Gamow window in
the two previous equations rely on the cross section being dominated by the probability to
penetrate through the Coulomb barrier. However, this is not always the case and a more
accurate Gamow window can be determined by evaluating the integrand of Eq. 2.10.

34

Figure 2.4: Binding energies per nucleon based on the data from [7]. The maximum occurs
around the iron region.

2.5

Q value

The mass of a nucleus is one of its most fundamental properties. Since the 1920’s [64, 65]
it has been known that the measured mass of an atomic nucleus is less than the sum of
its building blocks. From Einstein’s mass-energy equivalence [66], the missing mass of a
nucleus is equivalent to the energy released when building up the nucleus from free nucleons.
Alternatively the missing mass is equal to the energy required to break a nucleus into free
nucleons, thus it is referred to as the binding energy. The binding energy of a nucleus is
given by
B.E./c2 = Zmp + Nmn − mx

(2.18)

where mp , mn , and mx are the masses of the protons, neutrons, and nucleus, respectively.
The plot of binding energy per nucleon in Fig. 2.4 shows that the maximum binding occurs
around the iron region. Energy can be released by combining nuclei lighter than the iron
35

region in fusion or breaking up nuclei heavier than the iron region in fission.
For a nuclear reaction 1 + 2 → 3 + 4, the energetics can be quantified by the reaction Q
value
Q = (minitial − mf inal )c2 = (m1 + m2 )c2 − (m3 + m4 )c2 .

(2.19)

Reactions with Q > 0 are exothermic and release energy, while reactions with Q < 0 are
endothermic and require an input of energy to occur.

36

Chapter 3
Theoretical Considerations

3.1

Resonant Reactions

Nuclei live in the quantum world and therefore have discrete physical states at discrete
energies with a lowest energy ground state and higher lying excited states. The excited states
typically live for less than 10−12 seconds before releasing their energy through the emission
a γ ray or nucleons from the nucleus. Excited states that live longer before deexciting are
called metastable or isomeric states. Due to the Heisenberg uncertainty principle in quantum
mechanics, the excited states have energy fluctuations about their mean energy with a certain
width (Γ), and this width is related to the lifetime (τ ) of the state by Γ = h
¯ /τ
In a nuclear reaction, two nuclei collide to form a new nucleus often in an excited state.
The state is populated through the entrance channel and deexcites through the exit channel.
The notation that will be used in the remainder of this chapter is

A + a −→ B + b

(3.1)

where A + a is the entrance channel and B + b is the exit channel. Like the state itself, the
entrance and exit channel of a reaction have widths, which are related to the energy range
over which the reaction can occur. These widths are called partial widths Γi and they are
related to the total width by Γ = Γ1 + Γ2 + Γ3 + .... The probability that the state decays
37

through a particular reaction channel is given by the branching ratio Γi /Γ.
If the energetics of a nuclear collision match the energy of an excited state of the new
nucleus, then the reaction probability is enhanced in a resonance. The presence of a resonance
in the reaction is also influenced by the angular momentum of the excited state. The shape
of the resonance in the reaction cross section is given by the Breit-Wigner lineshape (see the
top panel of Fig. 3.1):
σ(E) =

Γ1 Γ2
λ2
ω
.
4π (E − Er )2 + (Γ/2)2

(3.2)

The variable λ is the de Broglie wavelength, Er and Γ are the energy and width of the
resonance, Γ1 and Γ2 are the partial widths of the entrance and exit channel, and ω is the
statistical factor of the entrance channel for total nuclear angular momentum (J) given by

ω =

3.1.1

2Jr + 1
.
(2JA + 1)(2Ja + 1)

(3.3)

In Stars

We can see how a resonance influences the stellar reaction rate by plugging Eq. 3.2 into
Eq. 2.10 to obtain

< σv > =

8
πµ(kT )3

∞
0

Γ1 Γ2
λ2
ω
E exp
4π (E − Er )2 + (Γ/2)2

−E
kT

dE .

(3.4)

Noting that λ2 /(4π) = π¯h2 /(2µE), ignoring any energy dependence of Γi , and noticing that
the narrow Breit-Wigner resonance will be like a delta function so that we can replace the

38

energy term in the exponential with the resonance energy Er , the reaction rate becomes

< σv > =

−Er
kT

2π
h
¯ 2 ω Γ1 Γ2 exp
(µkT )3

∞
0

1
(E − Er

)2

+ (Γ/2)2

dE .

(3.5)

The integral is equal to
∞
0

1
(E − Er

)2

+ (Γ/2)2

dE =

2π
Γ

(3.6)

−Er
kT

(3.7)

so that the final reaction rate is given by

< σv > =

2π 3/2 2
h
¯ ωγ exp
µkT

with the resonance strength ωγ defined as

ωγ =

Γ1 Γ2
2Jr + 1
.
(2J1 + 1)(2J2 + 1) Γ

(3.8)

Thus the reaction rate due to a resonance in a stellar environment depends on the resonance strength, the resonance energy, and the temperature of the stellar material. This
dissertation contains the experimental measurement of several resonance strengths. The
method used to obtain this quantity will be introduced in the following section.

3.1.2

In the Laboratory

In the laboratory, we cannot reproduce the conditions inside of a stellar environment and
instead experiments are done by impinging a nuclear beam of particles onto target nuclei.

39

The yield of the reaction is determined by

Y =

number of reactions
.
number of beam particles

(3.9)

For a thin target with thickness dx, the yield can be calculated as a simple extension of
Eq. 2.3:
Y = σ

Nt
= σ nt dx
A

(3.10)

where Nt /A is the areal target density and nt is the volume target density in nuclei per unit
volume. As the nuclear beam travels through the target, the charged projectiles interact
with the target nuclei and electrons and slow down due to a loss in energy. The amount of
energy loss depends on the initial energy of the beam as well as the abundance and Z of the
target nuclei. Thus, a convenient way to quantify the energy loss is through the definition
of the stopping power
ζ(E) =

1 dE
.
nt dx

(3.11)

For a thick target, the yield can be found by integrating the expression for a thin target
of Eq. 3.10 over each slice dx. If the initial energy of the beam is E0 and the energy loss
through the target is ∆E, then the yield of a thick target is given by

Y =

σ nt dx =

E0
E0 −∆E

σ(E)

1
dE .
ζ(E)

(3.12)

For a Breit-Wigner resonance (Eq. 3.2) the expression for the yield is

Y =

E0

1
Γ1 Γ2
λ2
ω
dE .
2
2
(E − Er ) + (Γ/2) ζ(E)
E0 −∆E 4π

40

(3.13)

In our consideration of a narrow resonance, the de Broglie wavelength, partial widths, and
stopping power can be taken as their resonance values and factored out of the integral. Thus,
the integral has an analytic solution

Y =

λ2r ωγ
arctan
2π ζr

E0 − Er
Γ/2

− arctan

E0 − Er − ∆E
Γ/2

.

(3.14)

In the case of an infinitely thick target (∆E → ∞), the maximum yield for the resonance is
reached
Ymax =

λ2r ωγ
.
2 ζr

(3.15)

The bottom panel of Fig. 3.1 shows the resulting excitation function for different target
thicknesses using Eq. 3.14. In an experiment, the excitation function is created by changing
the beam energy between different experimental runs and plotting yield against energy. This
dissertation contains experiments measuring resonances with both thin targets (Chap. 7) and
thick targets (Chap. 11). Typically it is easiest to extract resonance strengths using a target
whose energy loss is much larger than the width of the resonance so that the thickness
appears infinite. In this way, the resonance strength can be calculated by determining the
maximum yield of the excitation function and using Eq. 3.15.
There is an additional way to extract the resonance strength using the area under the
excitation function. Here I will follow the derivation of Iliadis [31]. First we need to introduce
two normalized functions to take into account some experimental factors. The first function
g(E0 −Ei ) is to account for the beam energy resolution and it is the probability that a particle
has energy Ei when the mean beam energy is E0 . The second function f (Ei − E, E ′ ) is to
account for beam energy straggling inside of the target and it is the probability that when a
particle hits the target with energy Ei it will have an energy E at a depth E ′ . Folding these
41

Cross Section

Γ

Thinner

Yield

Thicker
Infinite

Er

Energy
Figure 3.1: A Breit-Wigner lineshape with resonance energy Er and width Γ (top panel).
Expected excitation functions for the measurement of the resonance with different target
thicknesses (bottom panel).

42

functions into Eq. 3.12, we get the general yield equation

Y (E0 ) =

E0
E0 −∆E

dE ′

∞
0

dEi

Ei

dE σ(E)

0

1
g(E0 − Ei )f (Ei − E, E ′ )
ζ(E)

. (3.16)

To calculate the area underneath the excitation curve we put this entire expression inside of
another integral:
AY =

∞
0

Y (E0 ) dE0 .

(3.17)

First, we factor out the stopping power assuming it is constant over the resonance. Second,
the integral over E0 is unity because from g(E0 − Ei ), the probability of finding a beam
particle with energy between 0 and ∞ is 1. Third, the integral over Ei is unity because
from f (Ei − E, E ′ ), the probability that a beam particle with initial energy Ei has energy
between 0 and ∞ in the target is 1. This leaves
AY =

1
ζr

E0
E0 −∆E

dE ′

∞

dE [σ(E)] =

0

λ2
1
∆E r ωγ .
ζr
2

(3.18)

Thus, the area underneath the excitation function due to a resonance is independent of
beam energy resolution and straggling. In fact, by substituting that the areal target density
is equal to nt = ∆E/ζ, it is not necessary to know the stopping power directly. The final
result is
λ2
AY = nt r ωγ .
2

43

(3.19)

3.2

Nuclear Statistical Model

At low excitation energies the levels of a nucleus are well spaced and their properties may
be experimentally known. However, for capture reactions the states that are populated are
typically at high excitation energies where there are many levels that overlap. Thus, instead
of treating each level separately it becomes advantageous to take a statistical average over
the resonances. Averaging over an energy region ∆E that contains many Breit-Wigner
resonances, an energy averaged cross section can be calculated by

σ12 (E) =

1
∆E

E
E−∆E

i

λ2
Γ1 Γ2
dE ′
ω
′
4π (E − Er )2 + (Γ/2)2

(3.20)

where the sum is over the number of Breit-Wigner resonances (nres ) of a specific total angular
momentum J and parity Π. The sum over all J and Π will be put in at the end for the full
solution. Manipulating this equation a little gives

σ12 (E) =

λ2 nres
ω
4π ∆E

E
E−∆E

(E ′

Γ1 Γ2
dE ′ .
− Er )2 + (Γ/2)2

(3.21)

First we can make note that the average level spacing is equal to the energy region divided
by the number of resonances (D = ∆E/nres ). Also, because the width of the resonances is
much narrower than the energy region we are safe to take the limits of integration from 0 to
∞ and use the result of Eq. 3.6 to obtain
σ12 (E) =

λ2 2π
ω
4π D

44

Γ1 Γ2
Γ

.

(3.22)

It is important that the energy averaged cross section does not depend on the average
of the individual widths Γi but depends on the average quantity Γ1 Γ2 /Γ . Because of
correlation between the widths of the entrance and exit channel it is necessary to introduce
a width fluctuation correction defined by

Γ1 Γ2
Γ

= W12

Γ1 Γ2
Γ

(3.23)

with Γ = Σi Γi . This leaves an energy averaged cross section
λ2 2π
Γ Γ2
ω
W12 1
.
4π D
Γ

σ12 (E) =

(3.24)

We are headed towards making a connection between the total reaction cross section
obtained from averaging over many Breit-Wigner resonances to the cross section obtained
from solving the nuclear scattering problem. To do this we need to first sum Eq. 3.2 over all
exit channels Σi Γi = Γ and take the energy average. The result is the total reaction cross
section
σ1tot (E)

λ2 2π Γ1
ω
.
=
4π
D

(3.25)

Next, for the sake of comparison we take the projectile and target to have J = 0 so that the
statistical factor is simply related to the orbital angular momentum of the resulting state
ω = 2ℓ + 1. This gives
σ1tot (E)

=

λ2
2π Γ1
(2ℓ + 1)
.
4π
D

(3.26)

As mentioned, the total reaction cross section can also be obtained by solving the Schr¨odinger
equation describing the interaction of the projectile nucleus and target. Much like light can be

45

refracted or absorbed upon a striking a medium, the projectile can be scattered or absorbed
by the target. This motivates the writing of the nuclear potential in the form of an optical
model with a Coulomb term, a real term for elastic scattering, and an imaginary term for all
other reaction channels. Solving the Schr¨odinger equation with this nuclear optical model
potential gives the total reaction cross section as

σ1tot (E)

=

λ2
(2ℓ + 1) Tℓ
4π

(3.27)

where T is the transmission coefficient specifying the probability that the projectile will
penetrate the potential barrier. Comparing the two equations we see that the relationship
between the transmission coefficient and average partial width is given by

Ti =

2π Γi
.
D

(3.28)

Thus we can rewrite the average cross section of Eq. 3.24 to obtain the Hauser-Feshbach [67]
formula with width fluctuation corrections:

σ12 (E) =

T T
λ2
ω W12 1 2 .
4π
Σi Ti

(3.29)

The Hauser-Feshbach reaction model is based on the independence hypothesis of Bohr [68],
in which he proposed that the entrance channel and the exit channel are completely separate
from each other. The idea is that the projectile and the target form a compound nucleus
in an excited state and the nucleons of the compound nucleus share the energy. Due to
the sharing of energy amongst the nucleons, compound nucleus reactions take place over a
longer timescale than direct reactions and the nucleus “forgets” how it was formed. The
46

deexcitation of the nuclear state is therefore independent of the entrance channel and only
depends on the exit channel branching ratios. However, this independence hypothesis is only
true when the width fluctuation factor is equal to 1. There are several different models that
estimate values of W12 and usually W12 differs from 1 only at low energies where only a few
reaction channels are open [69]. At high excitation energies where many reaction channels
are open, the cross section of Eq. 3.29 is safely factored into a production cross section and
a decay probability given by the Hauser-Feshbach branching ratio T2 /Σi Ti .
As mentioned, nuclei have discrete experimentally known states at lower excitation energies and a continuum of states at higher energy. For a reaction channel to a discrete level
the transmission coefficient simply equals T , but for a reaction channel in the continuum
there is an effective transmission coefficient about a region ∆E given by
E+∆E/2

T

=
E−∆E/2

T (E ′ ) ρ(E ′ ) dE ′

(3.30)

where ρ is the density of levels within the energy region with a specific J and Π. Thus to
account for all energetically available states, the total transmission coefficient is given by
E max

ν

T =

i

Ti +

j

Eν

Tj (E ′ ) ρ(E ′ ) dE ′

(3.31)

where ν is the level up to which the experimental knowledge is complete and the integral
goes up to the maximum state allowed. The reaction channels include the emission of
particles and γ rays and their transmission coefficients are found using different methods.
The transmission coefficients for particles are calculated by solving the Schr¨odinger equation
with the proper nuclear optical model potential, and the transmission coefficients for emission

47

of γ rays are calculated using a radiative strength function fXλ (Eγ ):

TXλ (Eγ ) = 2π fXλ (Eγ ) Eγ2λ+1 .

(3.32)

The variable X describes the type of transition and is labelled E for an electric transition
and M for a magnetic transition, and λ is the transition multipolarity equal to the amount of
angular momentum that the γ ray carries away. There are various models for the radiative
strength function and the models relevant to this thesis are described in Chapter 10.
So far we have limited ourselves to resonances with only a specific total angular momentum and parity. The result of Eq. 3.29 should include a sum over all possible resonances in
the reaction to get the full cross section in the nuclear statistical model. Also, we will now
introduce the explicit sums over angular momentum ℓ and spin s states available using the
selection rules of angular momentum coupling. This gives the final Hauser-Feshbach formula
with width fluctuation corrections:

σ12 (E) =

λ2
4π

ω W12
Jπ

[Σℓs T1 ] Σℓ′ s′ T2
.
Σi Σℓ′′ s′′ Ti

(3.33)

There are many inputs that go into the Hauser-Feshbach formulism, including nuclear
properties such as masses, radii, and level densities, as well as the additional inputs of the
optical model parameters and the radiative strength functions. Three of the most critical
inputs for the production of the p nuclei in this thesis are the optical model potentials,
nuclear level densities, and radiative strength functions, which all have an influence on the
calculation of transmission coefficients. Further discussion of these three key quantities can
be found in the chapters presenting the results of individual nuclear reactions.

48

3.3

Reciprocity Theorem

As mentioned in Sec. 1.4, the majority of reactions relevant for the production of the p
nuclei are (γ,n), (γ,p), and (γ,α) reactions. While there are scientists who measure the
photodisintegration reactions directly using photon beams [70], it is often simpler and more
advantageous to experimentally measure the inverse capture reactions (n,γ), (p,γ),and (α,γ).
The cross sections of the forward and reverse reactions are related through the energy and
number of states available in each channel. If we consider an arbitrary nuclear reaction
A + a → B + b, the relationship is given by the reciprocity theorem [71] also known as
detailed balance:

(2JB + 1) (2Jb + 1) 2
(2JA + 1) (2Ja + 1) 2
kAa σAa→Bb =
kBb σBb→Aa
1 + δAa
1 + δBb

(3.34)

where k is the wave number related to the de Broglie wavelength by k = 2π/λ. For a γ ray,
the factor (2Jγ + 1) is equal to 2 because there are two polarization states for photons.
In a typical experiment the target and projectile are both in their ground states, meaning
that only the ground state cross section and the ground state reaction rate (Eq. 2.10) are
measured. However, in the stellar environment particles may also be found in any of their
low-lying excited states. The probability of finding a nucleus in an excited state i is given
by the Boltzmann distribution

Pi =

(2Ji + 1) e−Ei /kT
−Ej /kT

j 2Jj + 1 e

=

(2Ji + 1) e−Ei /kT
G

(3.35)

where G is the partition function. Using this Boltzmann distribution, the stellar reaction
rate σv ∗ can be calculated from the experimentally measured ground state reaction rate
49

σv g.s. using theoretical models such as the nuclear statistical model (Sec. 3.2) [72]. The
ratio is given by the stellar enhancement factor

f∗ =

σv ∗
.
σv g.s.

(3.36)

After using the stellar enhancement factor to turn the experimentally measured ground
state reaction rate into the stellar reaction rate, detailed balance can be used to calculate
the desired inverse reaction rate. The result is [31, 72]:
σv
σv

∗
Bb→Aa
∗
Aa→Bb

(2JA + 1) (2Ja + 1) (1 + δBb ) GnA Gna
=
(2JB + 1) (2Jb + 1) (1 + δAa ) GnB Gnb

MA Ma 3/2 −Q/kT
e
MB Mb

(3.37)

where Mi is the mass, Q is the reaction Q-value, and Gni is the partition function normalized
g.s.

to the ground state spin Gni = Gi /(2Ji

+ 1). For reactions where one of the particles is a

photon, the decay constant λγ for the photodisintegration reaction is related to the inverse
capture reaction by
λ∗Bγ→Aa

σv ∗Aa→Bγ

=

(2JA + 1) (2Ja + 1) GnA Gna
(2JB + 1) (1 + δAa ) GnB

MA Ma 3/2
MB

µAa kT 3/2 −Q/kT
e
2π¯h2

(3.38)

where µAa is the reduced mass of channel A + a.
A general level scheme of a capture reaction is shown in Fig. 3.2. The nuclei consist of
discrete energy levels at low excitation energy and many levels in the continuum at high
excitation energy. The ground state of the nuclei in the entrance and exit channel are offset
by the Q value of the reaction. As previously mentioned, in the stellar environment nuclei
can be present in any of their excited states. However due to the Q value of the reaction,
more excited states are energetically accessible to the nuclei in the B + b channel than the
50

A+a

Q

B+b

Figure 3.2: General level scheme for a forward and reverse reaction with a certain reaction Q
value. The nuclei have discrete states at lower excitation energy and a continuum of states
and higher excitation energy.
nuclei in the A + a channel. This means that the stellar enhancement factor (Eqn. 3.36) will
be larger for reactions in the direction of negative Q value. Since experimental measurements
are done on nuclei on the ground state, it is more advantageous to measure reactions in the
direction of postive Q value where the ground state contribution to the stellar reaction rate
is largest. For reactions in the p process, the capture reactions typically have positive Q
values and the photodisintegration reactions have negative Q values. Therefore, measuring
the p process capture reactions provide a closer representation of the stellar rate, and then
the inverse photodisintegration rate can be determined by Eqn. 3.38.
Another factor contributing to the stellar enhancement factor being closer to 1 in the
direction of postive Q value is the relative energy of the particles after photodisintegration.
The lower the relative energy of the particles, the higher the excitation energy of the level in
the daughter nucleus that is populated. However for (γ,p) and (γ,α) reactions, there must
be a high enough relative energy to penetrate through the Coulomb barrier. This means that

51

the proton or α particle carries away most of the energy, and the ground state of nucleus
A is populated more often than any of the excited states. By measuring (p,γ) and (α,γ)
reactions from this ground state of nucleus A, an accurate representation of the stellar rate
can be obtained in laboratory.

52

Chapter 4
Experimental Techniques
Although the production of the p nuclei has remained a major open question in the field
of nuclear astrophysics for nearly 60 years, very little experimental data exists for reactions
relevant to the p process. Less than 40 (p,γ) and 20 (α,γ) reaction cross sections have been
measured to date, and all of these measurements have been performed on stable nuclei [50].
For an improved understanding of the p process, additional data on (p,γ) and (α,γ) reactions
and their inverse photodisintegration reactions is required, especially for reactions involving
radioactive nuclei. This chapter contains a discussion of existing methods for measuring
capture cross sections followed by the experimental techniques and procedures used in this
thesis work.

4.1

Measuring the Cross Section

As discussed in Sec. 2.1, the cross section of a reaction is calculated by

σ =

Nreactions
.
Nprojectiles ntargets

(4.1)

Methods of determining both the number of projectiles impinging on the target and the
areal target density will be discussed later in this chapter, and for now we will focus on the
various techniques for measuring the number of reactions that occurred.

53

4.1.1

Activation

The vast majority of the measured (p,γ) and (α,γ) reaction cross sections relevant for the
p process have been performed using the activation technique (for example Ref. [73]). In
activation, determining the number of reactions that occurred is done through the offline
detection of the γ rays or X rays [74] emitted when the reaction products decay. This
means that the target material is first irradiated with a beam of protons or α particles to
create radioactive reaction products and then moved to a separate setup to detect the decay
radiation. As evident by its popular use, there are many benefits to using the activation
technique to measure a capture cross section. First, because the detection is done outside
of the irradiation there is no beam induced background or complicated deexcitation level
schemes to consider. Also, the detectors can be placed in a closely packed geometry around
the target material to maximize detection efficiency and shielded by lead walls to reduce
the background radiation. Finally, each reaction product decays with radiation at unique
energies, so it is possible to do several cross section measurements in one irradiation [75, 76].
Although the activation technique has been very successful, it is limited in its application
to reactions that meet a certain set of requirements. First of all, it is necessary to use target
nuclei that are stable or long-lived so that they can be properly irradiated. This constraint
alone is enough to rule out measuring many of the important reactions relevant for the p
process. It is also required that the reaction products are radioactive and that their decay
radiation is well-known with intense enough γ rays or X-rays to allow for detection. In addition, the half-lives of the reaction products must be long enough to allow for the transition
from irradiation to detection but short enough that decays occur within the timescale of
the experiment. If the half-life of the reaction products is too long for traditional detec-

54

tion methods, the number of reaction products can still be determined by combining the
activation technique with accelerator mass spectrometry (AMS), for example Ref. [77].

4.1.2

γ-Induced Reactions

Experimental measurements of reaction cross sections relevant to the p process can also
be performed using γ-ray beams to measure the photodisintegrations directly. The γ rays
may be produced by Compton scattering laser light off of an electron beam to create an
approximately monoenergetic beam of γ rays [78] or by decelerating an electron beam in
a target to create Bremsstrahlung radiation. [79]. Due to the difficulty of measuring the
protons or α particles freed by photodissociation, the number of reactions that occurred is
typically determined by offline counting as is done in the activation technique [80].
The photodissociations that take place in the laboratory occur on nuclei in their ground
state, which may be only a small contribution to the stellar environment where nuclei may
photodissociate from any of their low lying excited states. As mentioned in Sec. 3.3, the
ground state contribution to the stellar rate is much higher in the direction of positive Q
value and therefore it is oftentimes more advantageous to measure the capture reactions
instead of the photodisintegration reaction in laboratory. However, measuring reactions
with γ beams still provides constraints for theoretical models.

4.1.3

In-Beam Methods

The number of capture reactions that occurred can also be measured by surrounding the
target with γ-ray detectors to detect the prompt γ rays emitted in the reaction. Hence
the term in-beam methods. When the capture occurs, the projectile and target form a new

55

Ec.m.

A+a

Q

B
Figure 4.1: In a capture reaction, the projectile and target form an excited state in the
produced nucleus equal to the Q value of the reaction plus the center of mass energy of the
projectile and target system. The populated state is often at high excitation energies where
many resonances overlap.
nucleus in an excited state with energy Ex = Ec.m. + Q, where Ec.m. is the center of mass of
the projectile and target system and Q is the reaction Q value (Eqn. 2.19). As discussed in
Chap. 3, the excited state will deexcite in less than 10−12 seconds through the emission of
γ rays and possibly particles if it is at an energy above the particle emission threshold. The
deexcitation will continue until the final nucleus is in its ground state or long-lived isomeric
state. By surrounding the target position with γ-ray detectors, it is possible to deduce how
many times the reaction product of interest was produced.
For (p,γ) and (α,γ) reactions relevant to the p process, the populated excited state is
typically at high energies where many resonances overlap (see Fig. 4.1). The deexcitation
of the populated excited state occurs through any of the available γ-ray cascades, and the
energy and intensity of all the γ rays are typically not known beforehand. If there is one

56

particular transition that all the γ-ray cascades pass through or if the percentage of cascades
that pass through a particular transition is known, then simply detecting and analyzing
this dominant transition will provide how many times the reaction product of interest was
produced [81]. However, this is most often not the case and instead it is necessary to detect
all of the transitions to deduce the number of reactions.

4.1.3.1

Angular Distributions Method

One in-beam method of calculating the (p,γ) and (α,γ) reaction cross section is by determining the number of transitions originating from the entry state, or alternatively determining
the number of transitions populating the ground state. Because of incomplete knowledge
of the γ-ray cascades before performing the experiment, it may be necessary to analyze all
of the γ-ray transitions to build up a level scheme of nuclear exctied states. This analysis
can be quite complex as a typical spectrum often has a few hundred transitions to consider,
including room background and beam-induced γ rays [10]. Once all ground-state transitions are identified, the angular distribution of each transition is fit with a sum of Legendre
polynomials given by
cl Pl (cosθ)

A0

(4.2)

l

where Pl are the Legendre polynomials and cl are their coefficients. The A0 term is unique
for each transition to the ground state and is used to scale the fitting function to match the
data. Once all scaling factors are determined, the reaction yield is calculated by

Nreactions
=
Y =
Nprojectiles

57

N

Ai0
i=1

(4.3)

where the sum i corresponds to the transition from excited state i to the ground state. The
angular distribution method has been successfully used in several measurements [81, 10, 82]
but it relies on the detection and correct assignment of each γ-ray transition. Any transition
that is not detected or misidentified may lead to an incorrect value of the cross section. This
may be especially problematic for high energy γ rays emitted from the entry state which
have low efficiency for being detected.

4.1.3.2

γ-Summing Technique

In this dissertation, the reaction yield was experimentally determined by an in-beam method
known as the γ-summing technique [83, 9]. As opposed to using several smaller detectors
that can distinguish individual γ-ray transitions, the γ-summing technique implements a
large volume γ-ray detector that covers as much of the 4π solid angle around the target as
possible. The large size and angular coverage of the detector provide a high efficiency for
detecting γ rays so that when the populated state deexcites (see Fig. 4.1), any γ ray that
is emitted has a high probability of being detected. The goal of the γ-summing technique
is to detect each γ ray regardless of the deexcitation cascade and angular distribution, and
sum them up to the full energy of the populated state. This creates a “sum peak” at an
energy EΣ = Ec.m. + Q. Fig. 4.2 shows the difference between using a small-sized detector
(top panel) and a large 4π detector (bottom panel) for a simplified level scheme. By using
a large volume detector, the individual transitions are summed into a single sum peak that
can be analyzed. The number of counts in the sum peak is directly related to the number
of reactions that occurred through the detector’s γ-summing efficiency. A discussion of the
γ-summing efficiency can be found in Chap. 6.

58

Number of Counts

(a)

(b)

Energy

EΣ

Figure 4.2: Difference in γ-ray spectra between a (a) small-sized detector and (b) large
4π detector for a simplified level scheme. The result of using a large 4π detector is the
summation of all γ rays into a single sum peak whose intensity is directly related to the
number of reactions that occurred.

59

4.1.4

Techniques in Inverse Kinematics

Techniques using proton, α, and γ beams are limited to reactions involving long-lived isotopes
that have chemical properties favorable for creating a target. To overcome this limitation,
reactions can be performed in inverse kinematics with heavy isotope beams impinging onto
targets containing protons, α particles, or virtual γ rays in the field of a heavy target nucleus (Coulomb dissociation [84]). With the advancement of accelerator technologies and
the development of radioactive isotope beams, measurements in inverse kinematics are now
feasible and crucial for extending the experimental scope of capture reaction measurements.
Whereas in regular kinematics the low energy protons or α particles lead to a negligible
recoil velocity of the produced nucleus after capture, in inverse kinematics the momentum
of the heavy projectile leads to a significant recoil velocity.
Thus far, the majority of proton and α capture reactions measured in inverse kinematics
have been measured using recoil mass separators [85, 86, 87, 88, 89]. In this technique,
the heavy isotope beam passes through a target, typically a hydrogen or helium gas, and a
recoil separator comprised of a series of magnets and electric fields is used to separate the
reaction products from the incoming beam. Successful (p,γ) and (α,γ) measurements using
recoil separators have been performed for lighter masses where mass separation is easier to
achieve. Only recently have efforts been made to expand the capability to measurements on
heavier nuclei [90].
Additionally, storage rings have been used to measure (p,γ) and (α,γ) relevant to the p
process in inverse kinematics [91]. This technique involves injecting the heavy isotope beam
into a ring that maintains the nuclei in a large orbit. Somewhere along the storage ring there
is a proton or α target where the capture reaction may occur. Since the reaction products

60

have a different charge and mass than the incoming beam they are not suited for orbiting
the ring and can be detected with the proper placement of a detector.
The in-beam γ-ray detection methods discussed in Sec. 4.1.3 can also be used to measure
(p,γ) and (α,γ) reactions in inverse kinematics. As in regular kinematics, the reaction
product will be formed at the position of the target and the deexcitation γ rays promptly
emitted. However, due to the recoil velocity of the produced nucleus in inverse kinematics,
the γ rays are emitted from a moving source. As with any waves emitted from a moving
source, the detected γ ray frequencies, and hence their energies, are subject to the Doppler
effect. A γ ray detected in the forward direction has a higher energy than if emitted at rest,
and a γ ray detected in the backward direction has a lower energy than if emitted at rest.
The difference in energies is given by the Doppler shift

E0 =

1 − β cos θ
1 − β2

E

(4.4)

where E0 is the energy of the emitted γ ray, E is the detected energy in the laboratory,
β = v/c is the velocity of the recoil compared to the speed of light, and θ is the angle
between the direction of the recoil velocity and the detector. Taking the velocity of the
recoil and the position of the detector into account, it is possible to convert the detected
energy into the actual emitted γ-ray energy.

61

4.2
4.2.1

Target Production and Characterization
Evaporation

This dissertation contains measurements of (p,γ) and (α,γ) reactions on 11 different targets.
Most of the targets used were borrowed from previous nuclear science experiments and did
not have to be fabricated. However, the 27 Al targets and isotopically enriched 74 Ge target
did not exist beforehand and were produced through the process of evaporation. The general
procedure of evaporation involves the heating of material until it evaporates and travels in
gaseous form to a surface where it cools and condenses into a thin film. Before performing
the procedure it is necessary to put the evaporation chamber under vacuum by pumping out
the gas molecules to obtain a low pressure. This allows the evaporated material to travel
without interaction to the deposition surface. Typical pressures used in this dissertation
were 10−6 Torr.
When performing the evaporation, it is advantageous to track the amount of material
that is being deposited to control the thickness of the target being created. To do this,
evaporators are typically equipped with thickness monitors that deduce the thickness of
deposited material based on the oscillation frequency of a quartz crystal. The general idea
is that as evaporated material builds up on the quartz crystal, the oscillation frequency of
the crystal decreases. The exact mathematical relationship between the change in frequency
and change in mass depends on the properties of quartz and the area of the crystal, as well
as the density and acoustic impedance of the evaporated material [92, 93]. The acoustic
impedance is a property of the material itself and quantifies the opposition of the material
to wave propagation. If the acoustic impedance of the material is different from the quartz
crystal, there will be reflection of the wave which will impact the oscillation frequency. Thus,
62

for accurate target thickness measurements the user should input the density of the material
and the ratio of the acoustic impedances between the material and quartz to the thickness
monitor. There are useful tables of density and acoustic impedance ratios (Z-ratios), for
example in Ref. [94].
The oscillation frequency of the quartz crystal provides a measure of the amount of material deposited on the crystal, but this may be different than the thickness of the evaporated
target. To obtain the correct target thickness, the quartz crystal may be placed the same
distance from the evaporation source as the target, or a calibration “tooling” factor can be
used to make the correction. The tooling factor can be calculated by assuming the evaporation is occurring at a point source and estimating the ratio in solid angle covered by the
quartz crystal and the target.
The evaporation of the 27 Al targets was performed at the NSCL’s detector lab. The
evaporation was done onto 3” by 2” glass microscope slides coated with a mixture of 50%
water and 50% Teepol. Aluminum pellets were placed in the evaporator crucible and heated
with an electron gun until they melted and evaporated. Passing electric current through a
tungsten filament causes the filament to heat and emit electrons. The electrons are then
accelerated and directed in an arc onto the aluminum material through the use of electric
and magnetic fields. Water cooling was used outside of the crucible to carry away excess
heat. As mentioned in the preceding paragraphs, the target thickness was monitored with
an oscillating quartz crystal taking into account the properties for aluminum with a density
of 2.7 g/cm3 and Z-ratio of 1.08. The evaporation was done at an approximately steady rate
of 8 ˚
A/s until the desired thickness of 3 k˚
A was reached, at which point the current to the
filament was turned off.
After venting the evaporation chamber with nitrogen air and removing the glass slides,
63

it was necessary to transfer the thin layer of aluminum from the glass slides to the target
frames. The target frames used in this thesis were aluminum rectangles, approximately
20mm by 17mm in size with a 12mm diameter hole in the center. Transferring the thin
evaporated aluminum layer onto the target frames was done by floating the aluminum layer
onto the surface of distilled water and then “fishing” them out with the target frames. First,
a sharp blade is used to divide the evaporated aluminum into the desired size rectangles
while still on the glass microscope slides. Then, slowly inserting the glass slide at an angle
into a container of distilled water, the aluminum gradually peels back from the glass slide
and floats on the water’s surface. The reason the aluminum does not stick to the glass slide
is because of the thin layer of Teepol underneath coating the slide. While performing the
floatation, it is helpful to be patient and use a stand to hold the glass slide in place and a
lab jack to slowly raise the water level. Once the aluminum square breaks free along the cut
lines and is floating on the surface of the water, it can be guided to a target frame, mounted,
and removed from the water. In order to avoid ripping the aluminum, it is helpful to remove
it at an approximately 90 degree angle from the water’s surface. It is also important to try
to avoid wrinkling the aluminum as it is mounted on the target frame to maintain a layer of
uniform thickness. Upon drying, the aluminum will cling to the target frame and the result
is a self-supporting aluminum target for use in experiments. Ref. [95] contains more details
and pictures of the use of an electron gun evaporator and the floatation of targets.
On the other hand, the 74 Ge target was produced from 97.55% enriched germanium
powder using the University of Notre Dame’s evaporator. Due to the limited supply of
enriched powder, the procedure was first performed multiple times with natural germanium
powder to test the setup. Approximately 100 mg of the germanium powder was placed in
a tungsten boat and heated by passing electric current through the tungsten material. The
64

germanium was simultaneously evaporated onto aluminum, carbon, and tantalum backings.
The evaporation was carried out until all of the germanium powder was evaporated and based
on the change in oscillation frequency of the quartz crystal, the thickness was estimated to be
approximately 470 µg/cm2 . A more accurate and detailed analysis of the target thicknesses
is discussed in the following section.

4.2.2

Thickness Measurements

It is necessary to know the number of target nuclei per unit area in order to determine a
reaction cross section. This quantity can be determined by impinging an ion beam onto the
target. As ions travel through the target material, they will lose energy through interactions
with electrons in the target as well as by scattering off of target nuclei. By measuring the
outgoing energy of the incident ions or scattered particles at a well-defined scattering angle,
it is possible to deduce what elements are present in the target, how many atoms of each
element there is, and how the elemental abundances change with depth.
The energy loss due to electrons depends on the initial ion energy as well as the number
of electrons in the target. Since the number of electrons depends on a combination of the
Z and abundance of nuclei in the target, a measurement of the energy loss of the incident
ion or scattered particle provides information on how many target nuclei are present. On
the other hand when scattering off of a target nucleus, the incident ion will impart some of
its momentum to the nucleus, and both nuclei will recoil at an angle. After the collision,
the energy of both particles is related to the two masses of the colliding nuclei and the
scattering angle. As expected, a head-on collision will result in a larger transfer of momentum
than a more glancing collision. Thus, the energy of each particle after a collision provides
information on the mass of the target nucleus. Finally, the scattering cross section is different
65

(a)

(b)

ion beam
front
back

detector
back

front

Figure 4.3: Schematic drawing of the RBS technique with (a) an ion beam backscattering
off of target nuclei and (b) the energy spectrum of backscattered particles detected by the
detector.
for each element, so the number of scatters off of each element provides information on the
relative abundance of various components in the target.
The most widely used technique of ion beam analysis is Rutherford Backscattering Spectrometry (RBS) [96], named for the famous experiments by Ernest Rutherford and collaborators in which they deduced the presence of the positively charged nucleus by measuring
the backscattering of α particles off of metal foils [97]. In the same way, RBS consists of
bombarding a target with an ion beam with MeV energy and measuring the energy of the
backscattered ions. As mentioned, the backscattered ions have an energy related to the
energy immediately before the collision multiplied by a kinematical factor determined by
the masses of the two nuclei participating in the collision and the scattering angle. Ions
that backscatter off of the front surface of the target will be detected at the maximum
backscattered energy, but ions that backscatter off of nuclei in the interior of the target will
be detected at lower energy because the ions lose energy both on the way in and out of the
target, mostly through interactions with target electrons. Therefore, there will be a spread
in detected energies from the maximum corresponding to backscattering off of nuclei at the
66

front of the target to a minimum corresponding to backscattering off of nuclei at the rear of
the target. The width of the energy spread is the energy loss of the ion through the target
and provides the thickness of the target. See Fig. 4.3 for a schematic drawing of the RBS
technique.
One drawback of the RBS technique is that it is not very sensitive to light elements in
the target. This is due to the kinematics of the collision, in which heavier incident ions do
not backscatter off of light nuclei. Instead of using backwards angles, light target nuclei can
be characterized by measuring the incident ions and scattered particles at forward scattering
angles in the elastic recoil detection (ERD) technique [98]. By rotating the target so that the
ion beam has a small incident angle to the surface of the target (see Fig. 4.4), the greatest
depth sensitivity can be achieved with scatterings at the front of the target having maximum
detected energy and scatterings at the rear of the target having minimum detected energy.

4.2.2.1

Experimental Details

The target thickness measurements were carried out at the Hope College Ion Beam Analysis
Laboratory (HIBAL) on five separate days over the course of two years. Both RBS and ERD
methods were used, and the general layout of the two methods can be seen in Fig. 4.4. A
4 He

ion beam with current between 1.5 and 20 enA was accelerated to energies between 2.9

and 5.4 MeV using the Pelletron accelerator and impinged onto the surface of the target at
the center of the HIBAL scattering chamber. The scattered particles were detected by two
50 mm2 silicon surface barrier detectors, one fixed at a backwards angle of θ = 168.20◦ a
distance of 4” from the target, and one that was rotated to a forward angle of θ = 30.0◦ a
distance of 2.75” from the target. Collimators 0.2” in diameter were placed in front of both
detectors to reduce the solid angle coverage as well as to shield the edge of the silicon detectors
67

(a) RBS
ɽ

ion beam

(b) Forward Scattering
ɽ

ion beam

Figure 4.4: General layout of the HIBAL scattering chamber. (a) The RBS detector is
located at θ = 168.20◦, a distance 4.0” from the target, with a 0.2” diameter collimator in
front of it. (b) The rotating detector was placed at a scattering angle of θ = 30.0◦ , a distance
2.75” from the target, with a 0.2” diameter collimator in front of it. The target was then
rotated 75.0◦ and a 15.0 µ aluminum foil placed in front of the detector.
where the response may be irregular. The RBS detector had a resolution of approximately
30 keV and the ERD detector had a resolution of approximately 40 keV in the energy region
of interest.
The RBS technique was used to characterize all of the targets, whereas the ERD technique
was only used for the titanium hydride foil. For the RBS measurements only scattered helium
nuclei are detected as all other scatterings are kinematically prohibited. However, for the
ERD measurements of the titanium hydride foil, forward scattering of helium, titanium, and
hydrogen into the detector is possible. Therefore, a 15 µm thick aluminum foil was placed in
front of the silicon detector to completely stop the helium and titanium nuclei, only allowing
hydrogen nuclei knocked out of the titanium hydride foil to be detected. In order to provide
a profile of how the H concentration in the foil was changing with depth, the foil was rotated
75◦ so that the beam was incident at 15◦ to the surface.
The analysis of the data was performed using SIMNRA software [99]. SIMNRA was

68

Table 4.1: 4 He beam energies and silicon surface barrier detector energy calibrations used
at HIBAL over the course of five different experiments.
Experiment
I
II
III
IV
V
V

Beam Energies (keV)
2946
2977, 3484, 5195
2969
3486, 5282
3465, 4059, 3804
3465, 4059, 3804

Detector
RBS
RBS
RBS
RBS
RBS
ERD

Slope (keV/ch)
5.889
5.930
5.928
5.930
5.892
6.612

Offset (keV)
6.854
8.881
-0.301
3.837
-0.576
0.245

specifically developed for ion beam analysis of thin foils and contains the necessary kinematics, energy loss, energy straggling, and scattering cross sections to fit the data. The
user simply needs to input the experimental information such as the scattering geometry,
beam properties, detector response, and a best guess of the composition and thickness of the
target. Then SIMNRA can be used to fit the simulated composition and thickness to the
experimental data using a chi-square minimization routine.

4.2.2.2

Calibrations

The energy calibration of the silicon detectors was done with a quadruple α source consisting
of α particles emitted from the decay of 148 Gd, 239 Pu, 241 Am, and 244 Cm, with energies of
3.183, 5.157, 5.486, 5.805 MeV, respectively. The α calibration measurements were typically
performed the morning before and evening after the RBS and ERD measurements. The
slope and offset of the calibrations used in the analysis can be found in Table 4.1.
After performing the detector energy calibration, the 4 He beam energy was precisely
determined by measuring the scattering off of a heavy target with known composition. The
majority of the time the beam energy determination was performed with a layer of gold on
glass, but for convenience it was once performed with a thick piece of tantalum. By using a

69

Counts

103
experiment
simulation
Au
Si
O

102

10

100

200

300

400

500

600

Channel
Figure 4.5: Comparison of the SIMNRA fit to an experimental gold-on-glass measurement.
The precise beam energy of Eα = 3804 keV was determined by fitting the front edge of the
gold peak in the spectrum.
known heavy nucleus, the maximum energy of the scattered ions off of the front surface is
well known and can be used to deduce the beam energy. This method is more precise than
reading off the terminal voltage of the electrostatic accelerator. An example SIMNRA fit for
a gold-on-glass run is shown in Fig. 4.5, where the beam energy was determined by fitting
the front edge of the gold peak in the spectrum. The beam energies used in the various
experiments are contained in Table 4.1.

4.2.2.3

RBS Analysis

After obtaining the beam energy, the RBS spectra can be fit to obtain the target thickness
and composition. Spectra for nine of the targets can be seen in Fig. 4.6. The spectra
are dominated by a large plateau corresponding to the target layer of interest, and the
width of the plateau determines the thickness of the target. Some of the spectra also have
contributions from other layers or impurities in the target. It was determined that many of
70

Counts

181Ta

16O

27Al

181Ta

181Ta

Channel
Figure 4.6: SIMNRA fits to RBS spectra of nine of the targets used in this dissertation. The
resulting thicknesses are given in Table 4.2.
the targets have a very thin layer of oxygen and carbon on their surface, for example the
spike at lower energies in the 62 Ni spectrum corresponds to 4 He scattering off of 16 O on the
front surface of the target. Other contaminants seen were 27 Al and 181 Ta, for example in
the 58 Ni, 92 Zr, and 93 Nb spectra.
One of the more difficult spectra to fit was the RBS measurement taken with the enriched
74 Ge layer

evaporated onto a tantalum backing. The thickness of the germanium layer caused

the germanium plateau to overlap with the contribution from tantalum in the spectrum. Also
there was an unexpected plateau corresponding to tungsten, which was attributed to the
accidental evaporation of some of the tungsten boat during the evaporation of the enriched
74 Ge

powder. The best fit to the spectrum can be seen in Fig. 4.7 and was obtained with
71

experiment
total
Ge
Ta
W
O

Counts

10000

5000

0

100

200

300

400

500

Channel
Figure 4.7: SIMNRA fit to the RBS spectrum of enriched 74 Ge on a tantalum backing.
a tantalum backing, a layer of germanium, and thin layer of germanium and tungsten on
the surface. This indicates that the evaporation of the tungsten boat occurred towards the
end of the evaporation procedure. The final values for all the target thicknesses used in this
thesis can be found in Table 4.2.

4.2.2.4

ERD Analysis

The composition and thickness of the titanium hydride foil was determined by using both
RBS and ERD techniques. The RBS measurements are more sensitive to the number of
titanium nuclei in the target and the ERD measurements are more sensitive to the number of
hydrogen nuclei in the target, so combining the two techniques provides the full composition
of the target. The measurements were taken at three different locations on the foil and
then added together for maximum statistics. By taking into account the difference in solid
angle of the two detectors, the same target thickness and composition should be able to
describe both the RBS and ERD spectra. The solid angle coverage of one of the detectors is
approximately Ω ≈ A/d2 , where A is the active area of the detector and d is the distance of
72

Table 4.2: Target thickness values.
Isotope
27 Al
58 Ni
60 Ni
61 Ni
62 Ni
64 Ni
74 Ge
90 Zr
92 Zr
93 Nb
1H

Number of Nuclei in Target Total Target Thickness
(×1018 nuclei/cm2 )
(µg/cm2 )
1.574 ± 0.079
70.5 ± 3.5
9.665 ± 0.483
930 ± 47
6.693 ± 0.335
666 ± 33
5.060 ± 0.253
512 ± 26
15.748 ± 0.787
1619 ± 81
5.260 ± 0.263
558 ± 28
2.703 ± 0.135
340 ± 17
6.472 ± 0.324
966 ± 48
6.199 ± 0.310
946 ± 47
21.485 ± 1.074
3315 ± 166
3.581 ± 0.358
232 ± 23

the detector from the target. Since the detectors have identically sized collimators in front of
them, they have identical active areas and the difference in solid angles of the two detectors
is approximately
ΩERD ≈

d2RBS
d2ERD

ΩRBS

(4.5)

where the experimental distances of dRBS = 4” and dERD = 2.75” provide a scaling factor
of 2.1157.
Fig. 4.8 shows the resulting fit for both the RBS and ERD spectra. To achieve this fit,
the target was broken down into 4 layers with different ratios of titanium and hydrogen. The
outside layers have a hydrogen concentration down to 49.5% and the inner layers have a hydrogen concentration up to 64%, indicating that there is more H in the interior of the target.
The total hydrogen thickness from the fit is 3.581×1018 nuclei/cm2 . To determine an uncertainty in the number density of hydrogen nuclei, the standard uncertainty from SIMNRA
of 5% was combined with the uncertainty from the experimental beam current between the

73

Counts

400

300

(a) 400

5000

300

4000

200
3000

100
200

0

400

450

2000

experiment
SIMNRA

100

0

(b)

100

200

300

1000

400

0

500

Channel

50

100

150

200

250

300

Channel

Figure 4.8: SIMNRA calculations compared to experimental data for (a) RBS and (b) ERD
spectra of the titanium hydride foil.
RBS and ERD measurements. This extra uncertainty arises because the experimental setup
at Hope College at the time of the measurements could only record one detector at a time,
so the RBS and ERD measurements had to be taken at separate times. This in itself is not a
problem, but there was no direct measure of the beam current so comparison of the RBS and
ERD measurements could have been difficult if there were large fluctuations in the beam.
However, there was multiple signs of beam current stability during the measurements. One
indication of beam stability was that two gold-on-glass runs at Eα = 3804 keV taken approximately 40 minutes apart show only a 2.5% difference in the beam current. Also, there was
charge collection and integration performed at the back of the scattering chamber which had
a maximum difference of 3.0% for the RBS measurements. Therefore, even though the first
RBS and last ERD measurements were taken approximately 40 minutes apart, we do not
expect the beam current to change by more than 5%. Fitting the spectra for the expected
beam current ± 5% led to a hydrogen thickness of 3.580 ± 0.360 ×1018 nuclei/cm2 .
As seen in Fig. 4.8, the fit of the ERD spectrum matches the height and shape of the
experimental data well, but does not match the data at lower energies at all. As previously
74

(a) RBS

(b) Forward Scattering

Figure 4.9: Diagram dipicting how wrinkles in the titanium hydride foil may lead to a
seemingly larger thickness in the ERD spectra than in the RBS spectra.
mentioned, the width of the spectrum is determined by the energy loss through the target,
which in this case is dominated by the energy loss due to the titanium. Due to the fit of
the RBS spectrum, the amount of titanium in the target is well known and thus the width
of the spectrum in the ERD spectrum is well constrained. Therefore, the difference between
simulation and experimental data for the ERD spectrum is expected to be due to another
hydrogen source.
A possible explanation for the extra hydrogen contribution in the ERD spectrum is that
the many wrinkles in the foil, which are clearly visible to the naked eye, and the small incident
angle of the beam combine to make an effectively thicker target. Instead of scattering off
of a smooth target and producing the expected result, the beam scatters off a target with
many creases that has a larger effective thickness. Figure 4.9 shows how the wrinkles and
small scattering angle of ERD measurements may add extra contributions to the spectrum.
Because the RBS data is taken with the beam at 90◦ to the target, the wrinkles in the
target do not change the RBS spectrum. Simply by adding a titanium hydride layer that
is less than the deduced thickness of the foil, the result in Figure 4.10 is obtained. Adding

75

Counts

400
300

5000

(a)
experiment
SIMNRA

(b)

4000
3000

200

2000

100
0

1000
100

200

300

400

0

500

50

Channel

100

150

200

250

300

Channel

Figure 4.10: Same as Fig. 4.8 with the minimum additional titanium hydride material needed
to match the ERD spectrum.
additional titanium hydride does not change the result.

4.3

Experimental Setup

Since the goal of this dissertation was the measurement of (p,γ) and (α,γ) reactions at
astrophysical energies, it was necessary to use a facility that provides ion beams at astrophysical energies. Although Michigan State University has recently developed the capability
to provide lower energy beams, at the time of these measurements the only beams available
traveled at approximately 40% the speed of light and would be far too energetic for capture
reactions in astrophysics. The idea of slowing these fast beams down with a series of degraders was considered, but the spread in both beam position and energy as a result of using
the degraders was deemed too large for a measurement. Therefore, the experiments were
performed at the Nuclear Science Laboratory of the University of Notre Dame instead. A
layout of the laboratory at Notre Dame can be seen in Fig. 4.11 with the beams originating
from the ion sources (labels 1 and 2), accelerated by the Tandem Van de Graaff accelerator

76

Figure 4.11: Layout of the Nuclear Science Laboratory at the University of Notre Dame.
(label 3), and sent into the experimental area where the capture reactions were measured
(label 6). The important operating principles of these main experimental components will
be discussed in the rest of this chapter.

4.3.1

Beam Production

Two different ion sources were used at the University of Notre Dame. One is exclusively
used to produce beams of helium nuclei, while the second ion source is used for all other
available beams. For injection into the FN Tandem accelerator, the nuclei must initially be
negatively charged in order to be accelerated towards the positive terminal. The two sources
achieve these negative ion beams through different methods.
The Helium Ion Source (HIS) consists of a cavity known as the duoplasmatron and a
charge exchange canal filled with lithium. First, neutral helium atoms are fed into the
77

duoplasmatron from a gas cylinder. Also inside of the duoplasmatron is a tungsten filament
that emits electrons when heated as a result of passing electric current through it. The
electrons emitted from the filament ionize the helium into positive ions, 3 He+ and 4 He+ ,
which are extracted and sent into the lithium charge exchange region. The lithium exchange
region is filled with a lithium reservoir which is heated to produce lithium vapor. Since
lithium has a low ionization energy and gives up its valence electron easily, some of the
3 He+

and 4 He+ ions that pass through the lithium vapor pick up two electrons to become

negative ions 3 He− and 4 He− . These negative ions can then be injected and accelerated by
the FN Tandem accelerator.
On the other hand, the Multi-Cathode Source of Negative Ions by Cesium Sputtering
(MC-SNICS) [100] is used for all other nuclear beams. The desired material is first packed
into a small disk cavity known as the cathode. In principle, any material that can be packed
into a cathode can be accelerated and hence the wide variety of beams that originate from
this source. As many as 40 cathodes can be loaded at one time and rotated into place when
needed. As the source’s name indicates, the negative ions are produced by cesium sputtering.
This is accomplished by heating a reservoir of cesium to produce cesium vapor in the same
region as the cathode. Like lithium, cesium has a low ionization energy and loses its valence
electron easily to form a positive ion. These positive cesium ions are accelerated towards
the cathode and cause some of the cathode material to sputter upon striking it. A thin
layer of cesium condenses and builds up on the cathode so that when the cathode material is
sputtered it passes through the cesium layer and can pick up an electron to become negatively
charged. The negatively charged ions can then be injected into the FN Tandem accelerator.
The beams produced with this ion source in this dissertation were 1 H, 27 Al, and 58 Ni.

78

4.3.2

Acceleration

Once the ions of interest leave the source, they are accelerated to the desired energies using
the FN Tandem accelerator. The accelerator consists of a metal electrode called the terminal
at its center which can be charged up to a maximum of +10 MV, surrounded by a large
tank of gas which helps to insulate the terminal to prevent electrical sparking. The initial
acceleration is from the negatively charged ions being attracted to the large positive terminal
voltage and accelerating towards it. Once inside of the terminal, the ions pass through a
thin carbon foil which strips some of the electrons and leaves a positively charged ion beam.
The positively charged ion beam is repelled by the large positive voltage of the terminal and
this creates the secondary acceleration. The dual acceleration towards and away from the
terminal is why it is referred to as a “tandem” accelerator. Initially the ion has a charge of
−1 and it leaves the accelerator in some charge +q so that the total energy after the FN
Tandem accelerator is
E = Esource + 1V + qV

(4.6)

where Esource is the small injection energy from the ion source and V is the terminal voltage.
The charge +q can take on a range of values from a charge state distribution as the ion beam
passes through the carbon stripper foil.
The charging of the terminal is done with Pelletron chains that move electrically isolated
metal pellets in a loop from a terminal pulley to a grounded drive pulley located near the
outer part of the accelerator. By charging the metal pellets by induction both at the terminal
pulley and at the drive pulley, it is possible to create a system where the metal pellets are
positively charged as they enter the terminal and negatively charged when they leave the
terminal. This allows for more efficient charging of the terminal. The FN Tandem accelerator

79

uses two Pelletron chains on either side of the terminal to create a large positive voltage. A
more detailed description of the Pelletron charging system is found in Ref. [101].
The large voltage drop from the positively charged terminal to the grounded outer tank
is broken down into steps through the use of resistors with large resistances of 300 MΩ that
are evenly spaced. Because they are evenly spaced, there is a uniform voltage drop and the
ions experience a uniform acceleration as they move along the acceleration tubes. During
the experiment, it is necessary to maintain a constant terminal voltage so that there are no
fluctuations in beam energy. This is done with the use of metal needles called corona points
that can be brought closer or farther from the terminal to control the amount of charge
on the terminal. The distance of the corona points to the terminal can be automatically
adjusted to maintain constant terminal voltage. More details can be found in Ref. [102].

4.3.3

Ion Selection

After leaving the FN Tandem accelerator, the beam consists of a mixture of ions due to
the various materials originating from the ion source, as well as different charge states due
to the effect of passing through the carbon stripper foil. The selection of the ion beam of
interest is done through the use of an analyzing magnet. For any charged particle moving
perpendicular to a uniform magnetic field, the Lorentz force deflects the particle into a
circular arc. Combining the Lorentz force (F = qvB) and centripetal force (F = pv/r), the
radius of the circular arc is given by
r=

p
qB

(4.7)

where r is the radius, p is the momentum of the ion, q is the charge of the ion, and B is
the strength of the magnetic field. To make it to the experimental endstation, the ion must

80

travel the radius defined by the geometry of the beam pipe passing through the analyzing
magnet. Since the momentum of the ion is given by the desired beam energy, and the charge
state is typically selected to maximize the number of ions, the strength of the magnetic field
must be adjusted so that the ion can make it to the experimental endstation. Once the
field of the analyzing magnet is set, ions with the correct momentum to charge ratio make
it through, while ions with too low or too high of a ratio get bent in a circular arc that is
too small or too large, respectively. Slits at the end of the analyzing magnet can be open
and closed to help select the ion of interest as well as to control the total number of beam
particles making it to the experimental endstation.
At the University of Notre Dame, the magnetic field strength of the analyzing magnet is
measured with Nuclear Magnetic Resonance (NMR) [103]. NMR works on the principle that
when particles with magnetic moments and angular momentum are placed in a magnetic
field, their magnetic moment precesses around the applied magnetic field with a frequency
called the Larmor frequency given by
f = γB

(4.8)

where f is the precession frequency, γ is the gyromagnetic ratio, and B is the magnetic field.
With a gyromagnetic ratio of 42.576 MHz/T, protons are ideal for measuring magnetic field
strengths up to 2.0 T (for example Ref. [104]), and that is what was used for this dissertation.
Measuring the proton NMR frequency in the field of the analyzing magnet provides the very
precise beam energy equal to (assuming c = 1 units)

KE =

p2 + m2 − m

and

p=

f rq
γ

(4.9)

where Eqn. 4.7 and Eqn. 4.8 were combined to determine the momentum p, and m is the
81

Faraday cup 1

Faraday cup 2

4ʌ detector
ion beam

collimator 2
collimator 1

isolating
flange

target

lead wall

Figure 4.12: Experimental seup for the cross section measurements performed in this dissertation. The dimensions are not to scale.
rest mass of the ion beam. Typical uncertainties in this beam energy with the FN Tandem
accelerator and analyzing magnet are around 2 keV.

4.3.4

Experimental Endstation

Upon leaving the analyzing magnet, the ion beam is directed into the experimental hall
along any one of the experimental beam lines. For all of the measurements contained in this
thesis, the ion beam was sent to the beam line where the “R2-D2” scattering chamber is
housed (label 6 of Fig. 4.11 ). The experimental setup can be seen in Fig. 4.12 with the ion
beam moving through the various components from left to right. For beam tuning purposes,
two circular collimators were used. The collimator at the center of the R2-D2 scattering
chamber was on an adjustable drive with options for a collimation to a beam diameter of 3,
6, or 10 mm. Further downstream was the second collimator with options for diameters of
3 or 10 mm. By reading the beam current deposited on the collimators it was possible to
steer the beam until the minimum amount of current was deposited on both collimators. In
82

this scenario, the beam would be passing through the center of both collimators and thus
would also hit the center of the target.
Besides measuring the beam current on the collimators, the beam current was also measured with two Faraday cups. The first Faraday cup was located immediately after the
second collimator and could be inserted and removed from the beam line to both stop the
ion beam if necessary as well as measure the current. The second Faraday cup was the entire
last 1.2 m of the beam line, and therefore was electrically isolated with an insulating flange
from the rest of the beam line. When the ion beam impinged on the target, electrons were
emitted from the target position so that any charge collected only from the target frame
was not equal to the charge carried in the beam. However, by using a long Faraday cup,
all of the charge including the secondary electrons was collected to ensure an accurate beam
current reading. The collected charge from the second Faraday cup was sent to the input of
a digital current integrator that emitted an output pulse every time that 10−10 C in charge
was deposited by the beam. Using the output of the digital current integrator and knowing
the charge state of the ion beam, it is possible to calculate the total number of beam nuclei
by
Nprojectiles = Npulses

10−10 C 1 projectile
1 pulse
q∗e

(4.10)

where q is the charge state of the beam and e is the elementary charge.
As previously mentioned, the reaction yield was determined by surrounding the target
with a large 4π detector to implement the γ-summing technique. More details on this
detector will be discussed in Chap. 5. To shield the detector from beam induced γ rays
emitted from the beam striking an upstream collimator or the downstream beam stop, two
lead walls were constructed on either side of the detector.

83

Chapter 5
The SuN Detector

5.1

Design

The major goal of this dissertation was the development of a new experimental program
to measure (p,γ) and (α,γ) reactions using the γ-summing technique (Sec. 4.1.3.2) at the
National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. Although there were already several existing γ-ray detectors at the NSCL, through GEANT4
simulations (see Sec. 5.5) it was determined that no configuration of the existing detectors
was ideal for implementing the γ-summing technique. Thus, it was decided to design a new
detector to meet the requirements.
When designing the detector, several important factors needed to be considered. First,
the γ-summing technique relies on having a high γ-ray detection efficiency. This means
having a large volume of material that can be used to detect γ rays while covering as much
of the 4π solid angle as possible. Also, because many of the reactions of interest involve
radioactive nuclei, it was necessary to have the capability of performing the measurements
in inverse kinematics where the γ rays are emitted in-flight due to the velocity of the produced
nucleus. As mentioned in Sec. 4.1.4, this creates a Doppler shift in the γ-ray energy with the
energy of the γ rays emitted in the forward direction getting enhanced by the Doppler factor
and the energy of the γ rays emitted in the backward direction getting reduced. Therefore,
the second constraint on the detector was that it was necessary to introduce segmentation
84

Figure 5.1: The Summing NaI (SuN) detector.
in the design to identify which direction a γ ray was emitted to perform a successful Doppler
reconstruction of the energy. The third constraint was simply to stay within the project’s
budget.
Although other materials have better energy resolution, higher intrinsic efficiency, and
better timing properties, sodium iodide with trace amounts of thallium is a highly-efficiency,
cost-effective material for the detection of γ-rays and it can be built into unique shapes
and sizes [105]. Cesium iodide has similar efficiency at a similar cost, however there is not
much experience in making large volume crystals of CsI. Therefore, the final design was
built with NaI(Tl), which is similar to other 4π detectors that have been used in nuclear
physics experiments before (for example Ref. [106]). Balancing segmentation with detection
efficiency, it was decided to have a total of eight segments with four on the top and four
on the bottom. The detector was named the Summing N aI (SuN ) detector, constructed by
SCIONIX, and arrived at the NSCL in February 2011.

85

As seen in Fig. 5.1, SuN is a cylindrical detector with 16 inches of NaI(Tl) in length
and 16 inches in diameter. There is a 4.5cm borehole along the axis so that the beam
can enter and impinge on a target mounted at the center of the detector. It is comprised
of eight large semicylindrical crystals that are each completely surrounded by a reflective
layer of polytetrafluoroethylene 0.25 mm in thickness. Each crystal is also encased in an
aluminum frame with the verticle plates having a thickness of 0.5 mm and the horizontal
plates having a thickness of 0.75 mm. These dimensions mean that the four crystals in each
half of the detector are separated by a total distance of 1 mm, and the flat surfaces of the
crystals in the top and bottom halves of the detector are separated by a total distance of 2
mm. Therefore, each of the eight segments is optically isolated from its neighbor and can
operate as an individual detector. The crystals are read out by three photomultiplier tubes
to maximize light collection, which creates a total of 24 photomultiplier tubes in the design.
As mentioned, the detector consists of a top half and a bottom half, which allows for easier
mounting onto the beam line for experiments.

5.2

Detection

The nearly 4π angular coverage of SuN allows the vast majority of γ rays emitted from the
target to interact with one or more of the eight crystals. Upon interacting with a crystal,
the γ-ray energy is turned into electron energy in the NaI(Tl) in one of three ways: the
photoelectric effect, pair production, or Compton scattering. At low energies, it is most
probable that photoelectric absorption will occur, where the γ-ray energy is completely used
to free a bound electron from an atom and give the electron some kinetic energy. The
empty orbital left by the free electron is quickly filled, and an X-ray or Auger electron is

86

emitted. At energies larger than twice the rest mass of an electron, it is most probable that
pair production will occur, where all of the γ-ray energy is used to create an electron and
positron pair plus their kinetic energy in the electric field of the nucleus. The positron will
quickly annihilate with an electron and emit two 511 keV γ rays which may or may not also
interact with the crystals. Lastly, at intermediate energies, it is most probable that Compton
scattering will occur, where the γ ray scatters off an electron at a random angle and imparts
some of its energy to the electron. Both the electron and scattered γ ray can continue to
interact with the crystal. In any case, the end result of all three important interactions is
the transfer of γ-ray energy to the kinetic energy of free electrons. Ideally, all of the γ-ray
energy will be deposited in SuN’s crystals but sometimes energy is lost, for example if an
annihilation or Compton scattered γ ray leaves SuN undetected, or if a γ ray interacts with
the inactive reflector and aluminum layers.
As the free electrons move around the crystal, their negative charge causes them to
interact with the atoms in the crystal and lose kinetic energy. Additional electrons are
excited from their bound sites in the lattice (valence band) to an energy high enough to
freely move about the crystal (conduction band). This process is known as exciting electrons
across the band gap and it leaves a positively charged hole in the lattice. By doping the
sodium iodide crystal with thallium, impurities are introduced to the crystal that have
lower ionization energies than the other sites in the lattice. Because these impurities lose
their valence electrons more easily, any hole in the lattice drifts to the site of the impurity.
When an electron in the conduction band recombines with the hole in the valence band in
a neutral configuration, visible light is emitted. In NaI(Tl) the visible light is emitted with
a distribution in wavelengths centered at 415nm. Because the states in the impurity are at
a lower energy than the other sites in the lattice, the light is at a wavelength that passes
87

through the crystal mostly without interaction.
Because NaI(Tl) turns radiation into visible light, it is classified as a scintillator. This
important property was discovered in 1948 [107] and since that time it has remained an
important material in the detection of radiation. As with many other scintillators, there is an
interesting feature of NaI(Tl) detectors known as phosphorescence. This feature arises when
instead of immediately emitting light upon recombination, the electron and hole recombine
into an excited electronic state which cannot immediately transition to the ground state.
Instead, these states rely on thermal excitations to move the electron up to a higher state
that can transition to the ground state. Thus, phosphorescence increases the total detection
time and it can be a problem in high count rate experiments.
By surrounding the NaI(Tl) crystals with a reflective layer, the light travels about the
crystal until it impinges upon one of the three photomultiplier tubes (PMTs). The PMTs
are capped with a thin material called the photocathode that emits electrons when hit by the
light via the photoelectric effect. By applying a voltage to create an electric field inside of
the PMT, the newly freed electrons can be accelerated to hit a dynode which emits several
more secondary electrons. Using a series of dynodes allows a few electrons produced in
the photocathode to be multiplied to a large number of electrons. The SuN detector uses
Photonis’ XP6342 PMTs which have 10 stages of electron multiplication. For the average
voltage applied to the XP6342 PMT a gain of 2.4×105 is achieved. Thus, by collecting the
electrons after the last multiplication stage, there is a significant electrical current that can
be processed by the acquisition system.

88

Signal Voltage (mV)

0

-100

-200
1000

2000

3000

4000

5000

Time (ns)
Figure 5.2: Typical signal from one of SuN’s PMTs.

5.3

Data Acquisition

The main goal of any data acquisition system is to extract and record the important information carried in each detector signal, most notably the time and deposited energy of
an event in the detector. A data acquisition system should also provide the capability to
group information that corresponds to the same physical event together, and may include
the possibility to reject information that does not meet a user-defined requirement. As with
all detectors, the electric signal from each of SuN’s PMTs is a continuous distribution of
voltage as a function of time. When there is no energy deposited in SuN the signal is at
its baseline value. However, there are always some fluctuations from the baseline value,
referred to as noise. Noise can have a number of sources, for example from random thermal
emission of electrons inside of the PMT. When some form of radiation deposits energy in
SuN, scintillation light will be produced that is collected by the PMT and converted into a
large number of electrons. The current of electrons causes the voltage of the signal to rise

89

Feedthrough
Panel
Amplifier

Splitter
Summing
Module

High
Voltage

DDAS

CFD

Computer

Figure 5.3: Diagram of SuN’s data acquistion system.
in a “pulse” (see Fig. 5.2). The signal pulse has a characteristic rise time followed by an
exponential decay back to the baseline value. In SuN the pulse rise time is approximately
80ns and the decay constant of the exponential decay is approximately 230ns. Both the
height and area underneath the pulse depend on the amount of energy deposited in SuN,
and the front edge of the pulse contains important timing information. A general schematic
for SuN’s data acquisition system is shown in Fig. 5.3, and the processing of SuN’s signals
with this data acquistion system will now be discussed in more detail.

5.3.1

PMT signals

First of all, the voltage to each of the 24 photomultiplier tubes is applied through a WIENER
MPOD crate with two ISEG high voltage modules. The crate is connected to an ethernet
cable with a unique IP address so that it can be controlled through a webpage using SNMP
commands. Communicating to the crate through a webpage is user friendly and provides
90

detailed information on each channel. The use of the webpage is also advantageous because
it provides a convenient way for the user to power the crate on and off, change the voltage
of individual PMTs, and control the voltage ramping speed. The use of SNMP commands
allows for the use of automatic procedures to control the voltages applied to the PMTs, for
example an automatic gain matching procedure. Although SuN’s PMTs can withstand a
maximum voltage of +1200V, typical experimental values for this thesis were approximately
+800V.
The signals from each PMT get sent to an amplifier built by Pico Systems. However,
before reaching the amplifier the 24 individual cables pass through a feedthrough panel with
the purpose of sending the signals into two ribbon cables. It is most convenient to arrange
the setup so that signals from the top half of SuN pass through one ribbon cable and signals
from the bottom half of SuN pass through the other. These ribbon cables are plugged into
the input of the amplifier and the signals get split into three different outputs. One of the
outputs provides prompt unattenuated signals, while the remaining two outputs provide
signals that are delayed and attenuated. In this thesis, one of the outputs was fed directly
into the NSCL Digital Data Acquisition System (DDAS), while another output was used to
create an external trigger signal.

5.3.2

NSCL DDAS

As mentioned, signals from the amplifier get fed directly into the NSCL Digital Data Acquisition System (DDAS). In this thesis, both the attenuated and unattenuated outputs were
used at different times for this purpose. A nice description of the NSCL DDAS can be found
in Ref. [108] and the most relevant information will be repeated here. The DDAS setup for
SuN consists of three PIXIE-16 modules that are housed in an XIA crate. Each module
91

has 16 channels that take the continuous signal coming from a PMT and converts it into its
digital representation through the use of an analog-to-digital convertor (ADC). The ADCs
in SuN’s setup are Analog Devices AD9432 12-bit 100 MSPS ADCs. The acronym MSPS
stands for mega-samples per second and it means that the voltage of the signal gets converted
to a digital number 100 million times every second, or equivalently that the conversion takes
place every 10×10−9 seconds (10ns). The ADCs have two gain settings that allow for a
maximum voltage height of 0.50V and 2.22V [109]. Since the ADC has 12-bits, the digital
voltage value can take on any one of 212 = 4096 different possible values between 0V and
0.50V or 2.22V depending on the gain setting. For maximum resolution, it is best to have
the signal pulses take up as much of the ADC range as possible, which is why SuN uses the
0.50V full-scale gain setting. NSCL DDAS requires positive signals for processing, so the
original negative polarity signal from SuN first gets inverted.
Signal triggering and energy extraction in NSCL DDAS is done using digital trapezoidal
filters. The general idea of extracting the energy carried in a signal pulse using a trapezoidal
filter is shown in Fig. 5.4 using a signal step function. Because the height of the voltage
step is directly related to the energy deposited in the detector, the goal is simply to find
the height of the pulse. However, due to random fluctuations in the signal, better energy
resolution is achieved by averaging over many digitized points on the top of the voltage step
and subtracting an average over many digitized points at the bottom of the voltage step.
Thus, there are three time windows to consider; the time range to average over on the top
of the voltage step, the time range to average over at the bottom of the voltage step, and
the time gap between the two regions. In NSCL DDAS, the user can define two timing
parameters tpeak and tgap , so that the averaging over the top and bottom of the voltage step
take place in a time window tpeak and they are separated by the time tgap . For any digitized
92

300

Amplitude (Arb. Units)

Detector Signal
Trapezoidal Filter
200

100

0
- tpeak
-100

0

1000

tgap

+ +
tpeak

2000

3000

Time (ns)

Figure 5.4: Illustrating the use of a digital trapezoidal filter using a signal step function. For
every digitized point, the average value of points in the “−” tpeak region get subtracted from
the average value of points in the “+” tpeak region. The two tpeak regions are separated by
a time tgap . For a step function, the response is a trapezoidal shape and the height of the
trapezoidal filter contains the energy information of the signal.
point i, the difference in average voltage in the two tpeak regions is calculated as


∆Vi = 

i

j=i−P

Vj −

i−P −G
j=i−2P −G



Vj  /P

(5.1)

where P is the number of digitized points in time window tpeak and G is the number of
digitized points in time window tgap . This calculation gets implemented for every point in
the detector signal and for a step function the result is the trapezoidal shape in Fig. 5.4,
hence the name trapezoidal filter. The energy extraction is done by recording the value of
the trapezoidal filter at some point on the flat top.
The signal pulses from SuN are not a step function; instead they exponentially decay
with time constant τ = 230ns. Thus, even though the digital filters implement the same
algorithm as Eq. 5.1, they are not trapezoidal in shape. Fig. 5.5 shows the response of SuN’s
trapezoidal filters to a typical pulse. The filter parameters used for the experiments in this
93

thesis were tpeak = 100ns and tgap = 30ns for the trigger filter, and tpeak = 600ns and
tgap = 200ns for the energy filter.
The arrival time of each pulse is determined by the trigger filter. When the value of the
trigger filter is above a user-defined threshold the time is recorded. The energy extraction
from the energy filter occurs at a time tpeak + tgap = 800ns later based on the value of the
energy filter at that time. By setting the user-defined τ parameter for the decay constant
of the pulse, NSCL DDAS corrects for the exponential decay to obtain a corrected value of
the energy using the algorithm of Ref. [8] (see Appendix A). Additionally, if another pulse
satisfies the trigger filter threshold within tpeak + tgap = 800ns of the first pulse, both are
rejected as pile-up. NSCL DDAS monitors the number of pulses accepted and number of
pulses rejected due to pile-up to return a value for the data acquisition dead-time. In this
thesis, typical values of the dead-time were less than 2% of the total acquisition time, but
this small factor was taken into consideration in the analysis nonetheless.
Obtaining the time when a certain threshold level is reached is known as leading-edge
triggering and it leads to time variation that depends on the height of the pulse. Larger pulses
reach the threshold level sooner than smaller pulses, even though they occur at the same
time. Thus, NSCL DDAS also implements more accurate timing using a constant fraction
discrimination (CFD) algorithm. The general idea is that even though pulses have various
heights, all pulses reach a certain fraction of their maximum height at the same time. Thus,
by recording the time at which a pulse reaches a set fraction of its maximum height, more
accurate timing information can be obtained. Since timing resolution was not critical to the
experiments in this thesis, no further discussion will be included here. More information on
the CFD algorithm implemented in NSCL DDAS can be found in Ref. [109, 110].
The end result of using NSCL DDAS is a raw experimental file which contains a time94

(a)

(b)

threshold

(c)
Energy Extraction

tpeak + tgap

Figure 5.5: (a) Digitized representation of a signal from SuN’s PMT after passing through
100 MSPS ADC. (b) Response of SuN’s trigger filter. The arrival time of the pulse is obtained
when the trigger filter passes above a user-defined threshold. (c) Response of SuN’s energy
filter. The energy extraction is performed at a time tpeak + tgap = 800ns after the arrival of
the pulse, with a correction to the decay constant applied as in Ref. [8].

95

ordered list of events with the energy, time, PIXIE-16 module number, and channel number
that was triggered. Through the use of C++ computer coding, the events are extracted from
this raw experimental file one-by-one and grouped according to their time. In this thesis a
300ns time window was used to put events into a group. This means that the first event opens
a 300ns time window and every other detector that fired in that time window corresponds to
the same physical event. In the case of measuring a reaction using the γ-summing technique,
the physical event was a γ-ray cascade. The first detector that fired outside of the 300ns
time window starts a new 300ns grouping, and so on and so forth. When studying the arrival
time of pulses from SuN’s PMTs, pulses originating from the same γ-ray cascade typically
arrive within 10ns of each other, so the 300ns is a very conservative number for grouping the
PMTs together. However if a γ-ray cascade populates a isomeric state that is longer lived
than 300ns, its deexcitation will occur outside of the 300ns time window and it will not be
grouped with the other γ rays of the cascade. More details on the impact of isomeric states
for the γ-summing technique will be discussed in Chap. 6.

5.3.3

External Triggering

While one output from the amplifier gets processed by NSCL DDAS, another gets used to
create an external trigger signal (see Fig. 5.3). The goal of an external trigger is to cut down
on some of the data that gets saved to disk by checking to see if it satisfies a user-defined
requirement first. Because SuN uses the γ-summing technique, this requirement is simply
that a sum of the signals is above a threshold, which cuts down on the amount of data at
lower energies that has no impact on the sum peak region. If the sum of the signals is large
enough, then the information of all NSCL DDAS channels that fired in coincidence with the
valid external trigger signal is recorded. The use of the external trigger signal allows the
96

threshold of individual PMTs to be lowered to ensure that all γ rays are recorded.
To implement external triggering into SuN’s acquisition system, multiple electronics modules were needed. First, the ribbon cables from the amplifier pass into a splitter where their
signals are divided into 24 individual channels. Taking only the eight channels which correspond to the eight central PMTs of SuN’s segments, the signals are sent into a summing
module. The reason for choosing the central PMT of each segment is that it typically has
the best light collection of the three PMTs and is therefore representative of the total energy
deposited in a segment. As its name indicates, the summing module takes the eight signals
and sums them together into one output signal which is fed into CAEN V812 CFD module.
From the previous section, CFDs determine the arrival time of a pulse when the height
is a certain fraction of its maximum height. The CAEN V812 module takes this value to be
20% of the pulse height at which time a logic signal is emitted. The advantage of using the
CAEN CFD module is that it can be controlled through a computer graphical user interface
to define a reproducible threshold from -1mV to -255mV in 1mV steps. The pulse height
must be above the user-defined threshold level for a logic signal to be emitted. Therefore,
if the height of the signal from the summing module is larger than the threshold, a valid
external trigger signal is sent into NSCL DDAS and the data is recorded to disk.
Sending the signals through a splitter, a summing module, and a CFD takes a finite time,
which causes a time delay between the arrival of SuN’s signals and the potential arrival of
the external trigger signal (if the CFD threshold is satisfied). This time delay must be taken
into account when setting up a successful external trigger system. Conveniently, NSCL
DDAS includes a “fast-trigger delay” variable which is user-defined so that the arrival of
the detector signals is delayed to overlap with the external trigger signal. For this thesis a
fast-trigger delay of 100ns was used.
97

137Cs

60Co

ȕ-

ȕ-

661.7 keV

2505.7 keV

1332.5 keV

0.0 keV

0.0 keV
60Ni

137Ba

Figure 5.6: Simplified decay scheme showing the dominant decay radiation for the radioactive
sources 137 Cs (left) and 60 Co (right).

5.4

Radiation Source Testing

Once the SuN detector arrived at the NSCL and the data acquisition system was set up,
the initial testing of the detector was done using standard radioactive sources. The majority
of the tests were done with two γ-ray sources; 137Cs which β − decays into 137 Ba with the
emission of a single γ ray at 661.7 keV 94.7% of the time, and 60 Co which β − decays into
60 Ni

and then emits two sequential γ rays at 1173.2 keV and 1332.5 keV over 99% of the

time. The general decay scheme of these two sources can be seen in Fig. 5.6.
The first feature of SuN to test was the energy resolution of the NaI(Tl) crystals. In any
detector, the energy resolution quantifies the spread in detected energies for monoenergetic
radiation. The distribution in detected energies is typically a Gaussian shape of the form
(E − E0 )2
G(E) = A exp −
2σ 2

(5.2)

where A is the height, E0 is the centroid, and σ is the standard deviation related to the
width of the Gaussian. The energy resolution is defined as the full width at half maximum

98

A

137

Cs Source

4000

Counts / keV

Gaussian Fit
3000

FWHM = 2.3548σ

2000

1000

0

200

400

600 E
0

800

Energy (keV)
Figure 5.7: Experimental spectrum for a segment of SuN taken with a 137 Cs source with
room background subtracted. Also shown is the result of a Gaussian fit to the 661.7 keV
γ-ray line.
divided by the centroid energy, which for a Gaussian is
√
2 2ln2 σ
2.3548 σ
FWHM
=
≈
.
Resolution =
E0
E0
E0

(5.3)

Fig. 5.7 shows the energy spectrum and Gaussian fit to the 661.7 keV line of a 137 Cs source
for one of SuN’s segments. The creation of the experimental spectrum will be discussed in
Chap. 6. The standard deviation of the 661.7 keV line was determined to be 17.2 keV, which
gives an energy resolution of 6.1% [9]. This is typical for a NaI(Tl) crystal and was the first
proof that the SuN detector worked and met expectations. The energy resolution of SuN
for the 1173.2 keV and 1332.5 keV 60 Co γ-ray lines was determined in a similar manner
with standard deviations of 33.2 keV and 34.8 keV giving resolutions of 6.7% and 6.1%,
respectively.

99

Counts / 2keV

(a)

experiment
GEANT4

400

200

1500

Counts / 2keV

(b)

1000

500

0

500

1000

1500

2000

2500

3000

Energy (keV)

Figure 5.8: Experimental spectrum of SuN taken with a 60 Co source with room background
subtracted. The result for (a) a segment of SuN and (b) all eight segments are in excellent
agreement with GEANT4 simulations. When including the entire volume of SuN, the two
sequential γ rays get summed together and the sum peak dominates the spectrum.
Since the decay of 60 Co emits two sequential γ-rays, it offers a nice test of the γ-summing
technique. The energy spectrum obtained with a 60 Co placed at the center of SuN is shown in
Fig. 5.8. For an individual segment of SuN, the spectrum is dominated by the two individual
γ-ray lines with a smaller sum peak from when both γ rays simultaneously deposit their
energy in the segment. However, when including the full volume of the detector the sum
peak increases in intensity and dominates the spectrum. There are still two smaller peaks
at 1173.2 keV and 1332.5 keV corresponding to when only one of the sequential γ rays is
detected and the other one does not deposit any energy. By integrating the number of counts
in the sum peak, the γ-summing efficiency for 60 Co was determined to be 62.4(1.2)% with
no beam pipe and 51.5(1.0)% with a 1.5 mm thick stainless steel beam pipe.
The effect of the source position on the γ-summing efficiency was also investigated by
100

Figure 5.9: Figure from Ref. [9] showing the effect of the location of a 60 Co source inside of
SuN on the γ-summing efficiency.
moving a 60 Co source to various locations along SuN’s borehole, and the result is shown in
Fig. 5.9. Although a small dip in the efficiency at the very center of SuN was discovered
due to the thin layer of aluminum and reflector between the central segments, overall the
efficiency was fairly constant at the center of the detector. This was encouraging because it
showed that the position of the target in the center of SuN when performing experiments was
not so crucial. As the source was moved further from the center, the γ-summing efficiency
decreased as expected due to the decreased angular coverage and amount of NaI(Tl) near
the source. See Ref. [9] for more details.

5.5

GEANT4 Simulation

In order to analyze the data in this thesis and make predictions of how the SuN detector will
behave under various experimental conditions, it was necessary to develop a working and
reliable computer simulation of the detector. As with many applications that involve the
101

Figure 5.10: Visualization of the GEANT4 SuN detector simulation. The lines inside of SuN
are several γ-ray tracks. The one track that scatters outside of SuN does not deposit all of
its energy inside of the detector.
detection of radiation, the GEANT4 software package was used which implements Monte
Carlo methods to simulate how particles interact with matter [111]. In the case of SuN, this
means repeatedly sampling how a cascade of γ rays interacts with SuN to obtain the detector
response. GEANT4 is an ideal tool for this purpose because it already has the physics of
how γ rays interact with matter built in to its physics libraries, and the implementation has
been shown to be reliable for many other applications.
The successful adaptation of GEANT4 for use with the SuN detector involved a few
important steps. First, SuN was constructed in the simulation by inputting each component with the correct dimensions, composition, and location within SuN’s geometry. The
construction of the SuN detector can be found in Appendix B. It was also important to
include experimental equipment such as the beam pipe and target holder in the design for
maximum accuracy. Second, the capability to specify the location, energy, and direction
for the emission of γ rays was introduced. In most cases the γ rays are simply emitted in
a random direction from the center of SuN with an input file listing the γ-ray cascades to

102

Table 5.1: Standard deviation (σ) of a Gaussian function fit to various experimental γ-ray
peaks, along with the corresponding energy resolution.
E
(keV)
511
1173
1332
1779
2839

σ
(keV)
24.4
33.2
34.8
42.1
54.8

Res.
(%)
11.2
6.66
6.15
5.57
4.55

Source

60 Co
60 Co
27 Al(p,γ)28 Si

E
σ
(keV) (keV)
6129
92.7
9394
119
10509 126
12420 138

Res.
(%)
3.56
2.98
2.82
2.62

Source
19 F(p,αγ)16 O
27 Al(p,γ)28 Si
27 Al(p,γ)28 Si
27 Al(p,γ)28 Si

27 Al(p,γ)28 Si

simulate. An optional Doppler shift in energy was introduced for γ rays emitted in flight
for comparison to experiments performed in inverse kinematics. Third, the simulation was
set up so that for each simulated γ-ray cascade the energy deposited in each segment, the
total energy deposited in the detector, and the number of segments that detected energy
were extracted and saved. A visualization of the GEANT4 SuN detector simulation can be
seen in Fig. 5.10.
Additionally, the resolution of SuN’s NaI(Tl) crystals had to be input into the simulation.
This resolution was experimentally determined by fitting the peak of several individual γ-ray
lines, including measurements taken with a γ-ray sources as well as γ rays emitted from the
deexcitation of well-known resonances in the 27 Al(p,γ)28 Si reaction. Table 5.1 lists the γ-ray
energies and the standard deviation of a Gaussian fit to the peak in the spectrum, which
was used for the resolution function of SuN. This information is also plotted in Fig. 5.11
along with the best fit function that was implemented in the GEANT4 simulation. The
GEANT4 simulation of SuN agrees excellently with the experimental data contained in this
thesis. Validation of the code can be found in comparison to γ-ray sources (Fig. 5.8 and
5.9) and well-known resonances in the 27 Al(p,γ)28 Si and 58 Ni(p,γ)59 Cu reactions (Chap. 7
and 11). Although the resolution function worked well for this thesis and described the
103

200
-15

σ = 4.57868*10

-7

-10

*E4 - 1.09955*10
2

3

*E

-2

+ 4.09992*10 *E + 1.27752*10 *E + 17.7835

σ (keV)

150

100

50

0

0

5000

10000

15000

Energy (keV)

Figure 5.11: Standard deviation in the detected energy of the SuN detector as a function of
the γ-ray energy. The points correspond to experimental data and the curve is the best fit
function that was implemented in the GEANT4 simulation.
experimental data up to γ-ray energies above 12 MeV, it has been seen in more recent SuN
experiments that the extrapolation of σ to low energies of a few hundred keV does not match
experimental values. Thus, future scientists should take note that the resolution function
of SuN was based on the available experimental data at the time and may be altered and
improved with additional γ-ray lines.
As mentioned in Sec. 5.1, GEANT4 simulations were used to aide in creating the final design of SuN. By simulating well-known γ-ray cascades, the γ-summing efficiency was
determined for various possible geometries by taking the ratio of the number of counts in
the sum peak to the total number of cascades simulated. Fig. 5.12 shows sample efficiency
results for the SuN detector, a solid cylindrical detector of the same dimensions, and the
existing CAESAR detector at the NSCL [112]. CAESAR consists of 192 CsI crystals and the
standard configuration of the crystals is shown in the inset of Fig. 5.12. By arranging the 192
crystals of CAESAR into different configurations it was possible to improve the efficiency,

104

however it was still too low for the ideal implementation of the γ-summing technique. It was
decided that it was best to go with a cylindrical detector and that eight segments was the
best compromise between segmentation and efficiency. The simulated efficiency of SuN for
the detection of a single γ ray up to 15 MeV is shown in Fig. 5.13. The efficiency decreases
from 82% for a 1 MeV γ ray to 45% for a 15 MeV γ ray. When using the stainless steel
beam pipe and aluminum target holder in experiments, the efficiencies drop to 71% for a 1
MeV γ ray to 42% for a 15 MeV γ ray.

105

Figure 5.12: γ-summing efficiencies for various configurations of the CAESAR detector (standard configuration inset), a solid cylindrical detector, and the SuN detector for the resonance
at a proton energy of Ep = 1118keV in the 27 Al(p,γ)28 Si reaction. This resonance deexcites
through the emission of 3.2 γ rays on average.

1

Efficiency

0.8

0.6

0.4

SuN

0.2

SuN with beam pipe and target holder
0

0

5

10

15

Energy (MeV)

Figure 5.13: SuN’s efficiency for the detection of a single γ ray as a function of energy both
with and without the beam pipe and target holder used in experiments.

106

Chapter 6
Analysis
The raw output of the experimental measurements with the SuN detector included in this
thesis were files containing a time-ordered list of entries. Each entry contained the PIXIE-16
module number and channel number that fired, along with the time and energy information
extracted from the signal pulse that caused the entry. As discussed in Sec. 5.3.2, the time
and energy were calculated with digital filter algorithms, and entries occurring within the
same 300ns time window were grouped together as originating from a single γ-ray cascade.
This chapter contains the procedures to analyze the data, the bulk of which was performed
using the C++ programming language and the ROOT software package.

6.1

Gain Matching and Calibration

The first step in the analysis procedure is to make sure that SuN’s PMTs have the same
response when collecting the scintillation light originating from γ-rays of the same energy.
This procedure is called gain matching and it can be done in two separate ways. One
method is to adjust the amount of high voltage applied to a PMT to alter the number of
electrons produced in the different electron multiplication stages. In this way, the PMTs
themselves are adjusted to create similar output signals so that the energy extracted by the
data acquisition system is consistent for every PMT. The 1460.8 keV γ-ray line from the
room background decay of 40 K is well suited for this purpose. Therefore, the voltage on

107

2500

600

(a)

(b)

central PMT
left PMT
right PMT

500

2000

Counts

Counts

400
300

1500

1000

200
500

100
0

600

800

1000

1200

0
200

1400

Channel

400

600

800

Channel

Figure 6.1: (a) Room background and (b) 137 Cs source spectra from the three PMT’s in
a single segment of SuN after gain matching the voltages applied to each PMT. Since the
137 Cs source emits γ rays from the center of SuN, the outer two PMTs show a double peak
in their spectrum.
each PMT was adjusted so that the energy extracted from the 1460.8 keV γ-ray line was
approximately the same for each PMT before performing the measurements of this thesis.
Fig. 6.1 shows the result after altering the PMT voltages in one of SuN’s segments for
both a room background measurement and a measurement taken with a 137 Cs source. In
the room background spectra there is a single peak from the 40 K line for each PMT, and
the peaks are aligned to the same channel number as expected. However, the 137Cs spectra
shows a single peak for the 661.7 keV transition only for the central PMT. Instead of single
peaks as one might expect, the outer two PMTs show a double peak in their energy spectra.
It was concluded that the double peak for γ rays emitted from the center of SuN is due
to the geometry of the segments, where the semicylindrical shape of the crystal and the
placement of the three PMTs causes a position dependence on the light collection of the
outer two PMTs. This position dependence is further illustrated in Fig. 6.2, which contains
a plot of the left PMT against the right PMT for a 60 Co source. The two γ-ray transitions
in the decay of 60 Co show up as the two strong bands around channel 1000 in each PMT.
108

Left PMT Channel

2500
103

2000

1500

102

1000
10
500

0

0

500

1000

1500

2000

2500

1

Right PMT Channel
Figure 6.2: Two-dimensional plot showing the response of the left PMT against the response
of the right PMT for a 60 Co source. The two bands in the spectrum around channel 1000
in each PMT correspond to the two γ-ray transitions in the decay of 60 Co. The slope of
these bands indicates that the response of the outer two PMTs is affected by where the γ
ray deposits energy in the crystal.
Depending on where the γ ray interacts with the crystal, the light collection of the outer
two PMTs is affected. For instance, if the γ ray deposits energy near the left PMT, the left
PMT collects more light than the right PMT, and vice versa. This creates a spread in the
amount of light collected by the outer two PMTs for a single energy, which corresponds to
a poor energy resolution. However when adding the two outer PMTs together, good energy
resolution is achieved. Contrary to the outer PMTs, the location of the central PMT allows
for uniform light collection regardless of where the emitted γ ray interacts with the crystal,
and hence the single peak in the spectrum and why is was used to create the external trigger
signal (see Sec. 5.3.3). Since the 40 K γ-ray line originates outside of the detector and deposits
its energy near the outside of the NaI(Tl) crystals, all PMTs receive uniform light collection
and there is a single peak in all PMTs.

109

As mentioned, there is also an additional method of gain matching SuN’s PMTs. The
second method of gain matching is to apply a correction to the PMT energies in software
after the data is taken. By multiplying the energy extracted from a PMT by a numerical
multiplication factor it is possible to more precisely align the energy spectra for every PMT.
In this thesis, different methods of aligning the energy spectra from the three PMTs in a
segment were investigated to see which method provides the best energy resolution when
summing the PMTs together to create a total energy spectrum for the segment. During a
reaction measurement, the γ rays are emitted from the center of SuN so the energy spectra
of the outer two PMTs have a double peak for each γ ray. Three different gain matchings
were considered; aligning the γ-ray peak to the left of the double peaks, to the right of the
double peaks, and in the center of the double peaks. The results are shown in Fig. 6.3 for
a 137 Cs source, a 60 Co source, and the 40 K background line. In this figure, the top row of
three panels are aligning the spectra to the left, the second row is aligning to the center,
and the third row is aligning to the right. The three PMTs in each alignment are summed
together to create a total energy spectrum for the segment, and the result is then calibrated.
The result of calibrating the total segment spectrum is shown in the bottom row of three
panels in the figure. Surprisingly, there is no noticeable difference in resolution between the
three methods. Therefore, it is often easiest to gain match to the center, so that the single
peak in the energy spectrum from the central PMT is aligned at the center of the double
peaks in the energy spectra of the outer two PMTs. This can be done by fitting the 1460.8
keV γ-ray line from the room background decay of 40 K in each PMT with a Gaussian to
determine the centroid, and then calculating the multiplication factor necessary for precise
alignment.
After gain matching the three PMTs in each segment, their energies can be added for each
110

137

Counts

20000

Cs

10000

4000

2000

300

400

500

600

700

20000

Counts

Co

K

100

600

800

1000

1200

4000

200

2000

100

800

1000

1200

800

1000

1200

800

1000

1200

Center
10000

0

300

400

500

600

700

20000

600

800

1000

1200

4000

200

2000

100

Right
10000

0

300

400

500

600

700

600

Channel

800

1000

1200

Channel

Channel

20000

Counts

40
200

Left

0

Counts

60

200

4000

10000

0

100

2000

600

650

700

Energy (keV)

750

1100

1200

1300

1400

Energy (keV)

1300

1400

1500

1600

Energy (keV)

Figure 6.3: Result of applying gain matching multiplication factors in software to align the
central PMT in a segment to the left (first row), center (second row), and right (third row)
of the double peaks in spectra of the outer two PMTs. This was done for a 137Cs source
(left column), a 60 Co source (central column), and the 40 K background line (right column).
After summing the three PMTs in a segment together, the end result is that there is no
noticeable difference in the method of gain matching.

111

Counts

1500

1000

500

2000

T1
T2
T3
T4
B1
B2
B3
B4

2200

2400

2600

2800

3000

Channel

Figure 6.4: The result of gain matching SuN’s segments using a 60 Co source. The label T
is for the top of the detector and B is for the bottom of the detector. Based on the spectra
it can be deduced that the source was place off-center under the third segment of SuN and
closer to the top than the bottom.
γ-ray cascade to create an energy spectrum for the entire segment. As with the individual
PMTs, it is necessary that SuN’s segments have the same response when detecting γ rays
of the same energy. This can be assured by fitting peaks in the energy spectrum of each
segment with a Gaussian function to determine the centroid and then aligning each centroid.
For creating source spectra like Fig. 5.7 and Fig. 5.8, the alignment of each segment was
done using the γ-ray lines from the sources themselves. For example when creating the 60 Co
spectrum, multiplication factors to align the 1173.2 keV and 1332.5 keV γ-ray lines were
calculated for each segment and the average of the two values was used. A typical result of
gain matching all of SuN’s segments for 60 Co is shown in Fig. 6.4. After gain matching each
segment, the energies can be added together to create the total γ-summed spectrum. This
spectrum can then be calibrated so that each γ-ray line shows up at the correct energy in
the spectrum. For source spectra, a linear calibration of the form E = Ax + B is sufficient,
where A is the slope and B is the intercept of the calibration.
112

Table 6.1: Energy and source of γ-ray transitions used to calibrate SuN’s segments.
E (keV)
511
596
1173
1332
1779
2839
4438

Source

E (keV)
6129
74 Ge(α,α’)74 Ge
6668
60 Co decay
8938
60 Co decay
9394
27 Al(p,γ)28 Si
10509
27 Al(p,γ)28 Si
10762
12 C(α,α’)12 C
12420

Source
19 F(p,αγ)16 O, 16 O(α,α’)16 O
27 Al(p,γ)28 Si
27 Al(p,γ)28 Si
27 Al(p,γ)28 Si
27 Al(p,γ)28 Si
p(27 Al,γ)28 Si
27 Al(p,γ)28 Si

Initially when analyzing the (p,γ) and (α,γ) measurements, it was thought that the same
procedure as used to create the source spectra could be used to create the reaction γ-summed
spectra. However, because the reactions measured in this thesis produced compound nuclei
at high excitation energy, it turns out that this method was not satisfactory. A multiplication
factor that gain matched SuN’s segments using γ rays with energies between 1 and 2 MeV
misaligned γ-ray lines at energies above 5 MeV, and vice versa. For the proper summation of
all eight segments in this thesis, the response of each segment needed to be gain matched up to
γ-ray energies of approximately 15 MeV. Therefore, it was decided to calibrate each segment
individually before summing them all together. The calibration points for each segment came
from single γ-ray transitions from standard radioactive sources, the 27 Al(p,γ)28 Si reaction,
and various beam induced reactions. Due to the large energy range, it was necessary to use a
quadratic function of the form E = Ax2 + Bx + C to achieve the best calibration. Table 6.1
lists the γ-ray energies used to calibrate SuN’s segments in this thesis.
In an experiment, it is possible that the PMT responses may drift over time, which causes
a shift in the energy spectrum from a PMT despite the maintaining of a constant applied
voltage. This would lead to different gain matching and calibration values over the course
of an experiment. The drift may be due to factors such as shifting high voltages, fluctuating
113

Table 6.2: Gain matching multiplication factors for each PMT in the three different experiments at the University of Notre Dame.
PMT
T11
T12
T13
T21
T22
T23
T31
T32
T33
T41
T42
T43

Exp. I
1.00453
1.00000
0.99230
1.00607
1.00000
0.98970
0.99193
1.00000
0.99233
0.98590
1.00000
0.99113

Exp. II Exp. III
0.99467 1.02591
1.00000 1.00000
1.00715 1.05147
1.00764 1.04427
0.96076 1.00000
1.00000 1.03669
0.97883 1.02639
1.00000 1.00000
1.01458 1.05110
0.99266 0.98726
1.00797 1.00000
1.00000 0.99374

PMT
B11
B12
B13
B21
B22
B23
B31
B32
B33
B41
B42
B43

Exp. I
1.00014
1.00000
1.00245
0.98698
1.00000
0.98800
1.01521
1.00000
1.00682
1.00403
1.00000
0.99396

Exp. II Exp. III
1.00689 1.04008
0.97160 1.00000
1.00000 1.04778
1.00078 1.00872
0.99070 1.00000
1.00000 0.97929
1.03585 1.01702
1.00000 1.00000
0.98628 0.97929
0.98217 0.95456
1.01543 1.00000
1.00000 0.98351

temperatures or magnetic fields in the experimental area, or perhaps the ageing of a PMT
with use. During the (p,γ) and (α,γ) measurements at the University of Notre Dame, room
background runs were taken at least a few times a day and oftentimes many more. By monitoring the location of the 1460.8 keV γ-ray line in the room background spectrum for each
PMT, it was possible to determine whether or not the response of the PMTs were changing
with time. Over the course of the three separate, week-long experiments, no significant drift
in the PMTs was discovered. Therefore, it was possible to perform a single gain matching
and calibration procedure for each experiment. The values used for the analysis of the data
in this thesis can be found in Table 6.2 and Table 6.3.

6.2

Thresholds

As will be described later in this chapter, the complete analysis of SuN’s experimental data
requires an accurate simulation of the detector that matches existing experimental data

114

Table 6.3: Energy calibrations of the form E = Ax2 + Bx + C for each segment in the three
different experiments at the University of Notre Dame.
Segment

A

B

C

Exp. I

T1
T2
T3
T4
B1
B2
B3
B4

1.08547×10−06

1.39848×10−06
1.17564×10−06
2.18739×10−06
1.41779×10−06
1.11948×10−06
1.51094×10−06

0.47770
0.48932
0.47533
0.48355
0.46264
0.48360
0.46897
0.46840

-38.8211
-45.3902
-23.1084
-28.8541
-2.88832
-43.1425
-46.4133
-23.2460

Exp. II

T1
T2
T3
T4
B1
B2
B3
B4

5.64057×10−06
7.00597×10−06
6.44751×10−06
7.40063×10−06
7.86493×10−06
7.15952×10−06
7.17831×10−06
8.35966×10−06

1.09818
1.14240
1.12545
1.12614
1.11152
1.13639
1.10718
1.09061

-21.0829
-21.6076
-35.0943
-5.59329
-6.65785
-43.1837
-37.0146
-11.0828

Exp III

T1
T2
T3
T4
B1
B2
B3
B4

3.08654×10−06
1.16270×10−06
5.40427×10−06
5.40296×10−06
3.65558×10−06
3.29566×10−06
4.40908×10−06
7.81034×10−06

1.04165
1.08097
1.04891
1.09220
1.06209
1.12088
1.05076
1.07178

-42.4995
-54.4708
-27.7781
-30.9802
-44.4285
-49.5915
-38.5224
-25.6977

1.06741×10−06

115

cut

400

experiment

Counts

simulation

200

0

500

1000

Energy (keV)
Figure 6.5: Experimental data and GEANT4 simulation for one of SuN’s central segments
for a 60 Co source. In the analysis, a hard cut at 160 keV was applied for each segment in
order to use the same threshold in both experiment and simulation.
and makes accurate predictions. The simulation should match the experimental data for
all energies, including the low energy part of SuN’s spectra which is directly affected by
the experimental threshold. As discussed in Sec. 5.3.2, NSCL DDAS applies a threshold
based on the trigger filter response to SuNs PMT signals. After applying the energy filter to
signals satisfying the threshold requirement, the result is not a sharp cut in the PMT energy
spectrum but instead the low energy part of the spectrum has a positive slope.
The GEANT4 simulation of the SuN detector developed in this thesis does not include
the response of individual PMTs, but instead calculates the response of all eight of SuN’s
segments. Therefore it is necessary to compare an experimental spectrum of a segment to
a simulated spectrum of the segment to view the effect of the threshold. Fig. 6.5 shows
the difference between simulation and experimental data for one of SuN’s central segments
for a 60 Co source. Overall the simulated spectrum matches the experimental data well for

116

the energies shown. However for the energies below a few hundred keV, there are some
discrepancies between the two spectra due to the experimental threshold. Because of the
difficulty in correctly implementing the effect of NSCL DDAS’s threshold in simulation, it
was decided to apply a hard cut in both experimental and simulated spectra. A conservative
value of 160 keV was chosen based on the energy spectra of SuN’s 24 PMTs. If the total
energy deposited in a segment was below the hard cut at 160 keV, it was ignored in the
analysis.

6.3

Sum Peak Analysis

After performing the gain matching, calibration, and threshold steps described in the previous sections, the energy deposited in each of SuN’s eight segments was added together to
create a total γ-summed energy. In this way, the detected energy of all the γ-rays emitted
in deexcitation of the produced nuclei in the (p,γ) and (α,γ) reactions were summed together to create a sum peak at EΣ = Ec.m. + Q. At this point in the analysis, the effects
of detection dead time (less than 2% for all measurements) and room background were also
taken into account. Both the total γ-summed reaction spectrum and total γ-summed roombackground spectrum were scaled by their corresponding live-time ratios. After this scaling,
the room background spectrum was further scaled by normalizing to the same run time as
the reaction spectrum. By subtracting the normalized room background spectrum from the
reaction spectrum, the final result was achieved. Fig. 6.6 shows a typical sum peak after the
dead-time corrections and room-background subtraction, in this case for the 60 Ni(α,γ)64 Zn
reaction.
As discussed in the γ-summing technique in Sec. 4.1.3.2, the cross section of a reaction

117

300

Counts

EΣ-3σ

EΣ+3σ

200

100

9500

10000

10500

11000

11500

Energy (keV)
Figure 6.6: Zoomed in view of a typical sum peak from the γ-summing technique with the
SuN detector, in this case for the 60 Ni(α,γ)64 Zn reaction. A Gaussian fitting function is used
to define a sum-peak region of (EΣ -3σ,EΣ +3σ) and then a linear background is determined
from an average of the region’s boundary values. The sum peak integral is taken to be the
number of counts above the linear background.
is directly related to the number of counts in the sum peak. Because this thesis contains the
first ever implementation of the γ-summing technique with the SuN detector, the method
of integrating the number of counts in the sum peak was not yet determined and several
methods were initially considered. When deciding which method to use, it was desirable
to find a technique that was reproducible and provided a reasonable estimate of the background underneath the sum peak. The sum peaks contained in this thesis were typically
fairly large and it was possible to describe the background with a polynomial function if
necessary, although some of the (α,γ) sum peaks did have low statistics. In addition, future
measurements with the SuN detector are expected to utilize low intensity radioactive beams
and low cross sections with a corresponding low count rate in the sum peak. Therefore, it
was decided that a linear background would be easiest to implement for sum peaks of all
shapes and sizes.
118

The γ-summing technique tends to produce sum peaks that are asymmetric with the
low energy side of the peak extending outwards in a “tail”. This low energy tail is a result
of incomplete summation of the γ-ray energy when not all of the energy is deposited into
the detection crystal. Instead, some of the energy may be deposited in an inactive layer of
the detector or scattered outside of the detector. Therefore, the total γ-summed energy for
these events is less than the energy of the sum peak, creating the asymmetric shape. Because
the size and shape of the low energy tail may be different for each sum peak depending on
the number and energy of the γ rays being summed together, it was necessary to use a
technique independent of the low energy side of the sum peak when determining the bounds
of integration. Therefore, only the high energy side of the sum peak was fit with a Gaussian
fitting function (Eq. 5.2) to determine the centroid (EΣ ) and standard deviation (σ) of the
peak. Using the values of EΣ and σ, the sum-peak region was defined from an energy of EΣ 3σ on the left to an energy of EΣ +3σ on the right. To create a linear background beneath
the sum peak an average of the number of counts on both the left and right boundaries of
the sum-peak region was taken. A line was then drawn between the average values. The
number of counts in the sum peak was calculated as the total number of counts in the sumpeak region subtracted by the number of counts in in the linear background spectrum within
the sum-peak region. Fig. 6.6 shows a typical Gaussian fit as well as the linear background
determination from a typical sum peak in the 60 Ni(α,γ)64 Zn reaction. The vertical lines in
the spectrum denote the location of the sum-peak region.

6.3.1

Isomeric states

An important consideration to take into account when performing the sum-peak analysis
is whether or not there are any isomeric states in the produced nuclei. The presence of
119

an isomeric state that has a lifetime close to or longer than the data acquisition system’s
event window requires additional analysis. In this thesis an event window of 300ns was used.
Therefore, if any excited state produced in the (p,γ) and (α,γ) reactions lived longer than
300ns before deexciting, the γ rays emitted from this longer lived state would be outside
of the event window and not grouped with the other γ rays from the same cascade. When
populating an isomeric state, this separate grouping of the γ rays creates a total γ-summed
energy that is less than the total energy of the reaction. Thus, multiple sum peaks would be
observed in the γ-summed spectrum. One peak would correspond to the full energy of the
reaction when the isomeric state is not populated in a γ-ray cascade. A second peak would
also be present, corresponding to when the isomeric state is populated. This second peak
would be found at an energy equal to the full energy of the reaction minus the energy of the
isomeric state.
Since the total radiative capture cross section depends on the total number of γ-ray
cascades, including ones that populate and ones that do not populate the isomeric state,
it is necessary to integrate both peaks in the spectrum. For example, the 74 Ge(p,γ)75 As
reaction discussed in Chap. 8 includes a 304 keV isomeric state in 75 As with a half-life of
17.62 ms [113], which is much longer than the 300ns event window. The 304 keV energy
is small compared to the full energy of the reaction of approximately 10 MeV studied in
Chap. 8, and the integral of events populating the isomeric state was included in the analysis
by expanding the sum-peak region to (EΣ -3σ-304keV,EΣ +3σ). As mentioned, for isomeric
states at higher energies, it may be necessary to completely analyze separate peaks in the
total γ-summed spectrum.

120

6.3.2

Doppler reconstruction

An additional consideration to take into account is that reactions measured in inverse kinematics have deexcitation γ rays emitted from a moving source, which creates a shift in the
detected energy by the Doppler factor (Eq. 4.4). Therefore, the analysis of measurements in
inverse kinematics require the additional step of Doppler reconstruction of the γ-ray energy.
The Doppler reconstruction is done separately for each of SuN’s segments before summing
the energy of the segments together. Since the shift in γ-ray energy depends on the angle of
emission, it is important to choose the correct angle to use for the Doppler reconstruction
for each segment. For the analysis in this thesis, three different angles were investigated for
each segment. The first angle was simply the angle from the center of the target position to
the geometrical center of the NaI crystal. However, it is possible that this angle is incorrect
because it does not take into account that there are different thicknesses of NaI crystal available at different angles in a segment. The greater the thickness of the NaI crystal, the more
likely it is that γ rays emitted at the corresponding angle will interact and deposit energy.
Therefore, the second angle was calculated by weighing each emission angle by the pathlength through the NaI crystal. Lastly, SuN’s GEANT4 simulation was used to determine
the average emission angle for the γ rays that deposit energy in each crystal.
The calculated angles for each segment using the three different methods are listed in
Table 6.4 and drawn in Fig. 6.7. Overall, the three methods return approximately the same
angle to use in the Doppler reconstruction. It was also found that the GEANT4 angles are
closer to the geometrical center of the crystals for the central two segments, but closer to
the weighted center for the outer two crystals. Because the GEANT4 simulation has been
shown to provide reliable results for the SuN detector, it was decided to use the GEANT4

121

Table 6.4: Average emission angle in radians for the different segments of the SuN detector
calculated from the geometrical center of the NaI crystals, by weighting the angles by the
path-length through the crystals, and from SuN’s GEANT4 simulation.
Segment
1
2
3
4

Crystal Center
2.510
1.999
1.142
0.632

Weighted Center
2.559
2.082
1.059
0.582

GEANT4 Center
2.550
2.024
1.118
0.592

Table 6.5: Doppler correction factors used for each segment in this thesis.
Segment
1
2
3
4

27 Al

(β =0.0459) 58 Ni (β =0.0550)
1.0395
1.0476
1.0212
1.0256
0.9800
0.9762
0.9629
0.9558

angles in the analysis. However, it should be mentioned the angles provided by the three
methods are close enough that they do not demonstrate any significant differences in the
sum-peak resolution.
In this thesis, it was necessary to perform Doppler corrections for the p(27 Al,γ)28 Si and
p(58 Ni,γ)59 Cu reactions measured in inverse kinematics (see Chap. 11). The p(27 Al,γ)28 Si
reaction was measured at a resonance energy of Ec.m. = 956 keV, which corresponds to a
27 Al

projectile velocity of β = 0.0459. On the other hand, the p(58 Ni,γ)59 Cu reaction was

measured at a resonance energy of Ec.m. = 1400 keV, which corresponds to a 58 Ni projectile
velocity of β = 0.0550. Using these values of β and the GEANT4 angles listed in Table 6.4,
the Doppler correction factors were obtained. The final values are listed in Table 6.5.

122

Weighted center
GEANT4 center
Crystal center
1

2

3

4
Crystal center
GEANT4 center
Weighted center

ion beam

Figure 6.7: Cross-sectional view of the SuN detector showing the different angles used in
the Doppler reconstruction. The different angles are to the geometrical center of the NaI
crystals, to the weighted center based on the path-length through the crystals, and from
SuN’s GEANT4 simulation.

6.4

γ-Summing Efficiency

SuN’s γ-summing efficiency depends not only on the total energy of the deexcitation γ
rays, but also on the number of γ rays in the deexcitation cascade. SuN’s efficiency of
detecting a single γ ray from the entry state to the ground state is very different than the
efficiency of detecting multiple γ rays whose total energy is equal to the entry state. In
fact, the efficiency even varies depending on how those multiple γ rays split up the total
energy. During an experiment, the capture reaction and subsequent deexcitation of the
produced nucleus is happening many times through many different possible γ-ray cascades.
Therefore, the average γ-ray multiplicity, or the average number of γ-rays that are emitted
in the deexcitation of the entry state, is considered in the analysis. For the majority of
(p,γ) and (α,γ) reactions measured with the SuN detector, the average γ-ray multiplicity is
not known beforehand and correspondingly, the γ-summing efficiency is not known before
123

performing the experiment. Therefore, it is necessary to use the information obtained during
the measurement of a reaction to determine the average γ-ray multiplicity.
In previous applications of the γ-summing technique, the authors made use of the “in/out
ratio” method to determine the average γ-ray multiplicity. This method relies on comparing
the number of sum-peak counts for the full 4π solid angle coverage of the detector (“in”) to
those with only half of the full solid angle coverage (“out”). For a single γ ray, the ratio of
detected counts for half the solid angle compared to the full solid angle is expected to be
1/2. This means that a sum peak with average multiplicity <M> is expected to have an
in/out ratio of (1/2)<M >. While successful, this method can be difficult in low count rate
experiments because a small number of counts in the sum peak becomes even tinier when
considering only half of the solid angle coverage.
Fortunately, the unique design of the SuN detector provides a new method of determining
the average γ-ray multiplicity during an experiment by making use of SuN’s eight segments.
This new method is based on the expectation that the higher the γ-ray multiplicity of a
cascade, the larger the number of SuN’s segments that will detect γ-ray energy on average.
Fig. 6.8 illustrates this concept using GEANT4 simulations of a 10 MeV sum peak for γ-ray
multiplicities of two and three. For these calculations, the total 10 MeV excitation energy is
divided equally among the emitted γ rays. By selecting the events in the sum-peak region, a
histogram of the number of SuN’s segments that detected energy is drawn. This histogram
is known as the “hit pattern” of the sum peak. By fitting the hit pattern plots in Fig. 6.8
with a Gaussian, it is clear that the GEANT4 simulation with a multiplicity of three has a
higher centroid value than the GEANT4 simulation with a multiplicity two. This means that
the average number of SuN’s segments that detected energy for multiplicity three is larger
than the average number of SuN’s segments that detected energy with multiplicity two, as
124

<M> = 2

Counts

15000

<M> = 3

10000

5000

0

1

2

3

4

5

6

7

8

9

Hit Pattern
Figure 6.8: Histogram of the number of SuN’s segments that detect energy for events in
the sum peak (the “hit pattern”). The spectra are from simulations of the deexcitation of
a 10 MeV state with γ-ray multiplicities of <M>= 2 and <M>= 3 with the emitted γ
rays having equal energy. The spectra are fit with a Gaussian function to determine the
hit pattern centroid, and it is shown that the higher the γ-ray multiplicity the higher the
average number of SuN’s segments that detect energy.
expected. Through GEANT4 simulations of the SuN detector, the relationship between
hit pattern centroids and the average multiplicity is well known. This was verified using
resonances of known strength and average multiplicity (see Chap. 7).
When analyzing a reaction, the total γ-summed spectrum is created and the sum-peak
region is identified as described in the previous section. For each event within the sum-peak
region, the number of segments that detected energy is calculated and used to create the
hit pattern spectrum. In cases when the sum peak is sitting on top of a beam-induced
background, it is necessary to subtract this background out of the hit pattern spectrum.
In certain situations, it may also be advantageous to narrow the bounds of the sum-peak
region when creating the hit pattern. Overall, the main goal is to obtain a hit pattern for
SuN that is representative of the events in the sum peak of the reaction of interest. Once
the desired spectrum is obtained, a Gaussian fitting function is used to determine the hit
125

50
5000

(a)

45

Efficiency (%)

3000
2000

35

<M>=2

30

<M>=3

25

simulation

20

linear fit

10
1

2

3

4

5

6

7

8

9

2.5

Hit Pattern

<M>=4

experiment

15

1000
0

(b)

40

4000

Counts

<M>=1

3

<M>=5
3.5

4

4.5

5

Hit Pattern Centroid

Figure 6.9:

Method of determining the experimental γ-summing efficiency for the
sum peak shown in Fig. 6.6. Panel (a) shows the hit pattern spectrum for
events in the sum peak region and the corresponding Gaussian fit to determine the hit
pattern centroid. Panel (b) shows the efficiency and hit pattern centroids from GEANT4
simulations with various γ-ray multiplicities and transitions. The best fit line is also shown,
which is used to determine the experimental efficiency.
60 Ni(α,γ)64 Zn

pattern centroid. The left panel of Fig. 6.9 shows the hit pattern spectrum and its Gaussian
fit for the 60 Ni(α,γ)64 Zn sum peak in Fig. 6.6.
The next step in determining the γ-summing efficiency is to run a series of GEANT4
simulations with the total deexcitation energy equal to the experimental sum-peak energy.
The simulations should be performed for a variety of γ-ray multiplicities and transitions to
provide a good basis for comparison to the experimental data. The γ-summing efficiency
of each simulation is determined by the ratio of number of counts in the sum peak to total
number of γ-ray cascades simulated. In addition, the hit pattern centroid is determined using
the events in the simulated sum-peak region. After determining the γ-summing efficiency
and hit pattern centroid for each simulation, a plot of the results is made. The right panel
of Fig. 6.9 contains a plot of γ-summing efficiency versus hit pattern centroid based on the
results of 30 simulations with the same total energy as the sum peak of Fig. 6.6. On this plot,
the simulations are grouped in order according to their γ-ray multiplicity, with multiplicity 1
126

50

5

(b)

4.5

40

Efficiency (%)

Hit Pattern Centroid

(a)

4
3.5
3

20

10

2.5
2

30

9

9.5

10

0

10.5

Sum Peak Energy

9

9.5

10

10.5

Sum Peak Energy

Figure 6.10: Values of (a) hit pattern centroid and (b) γ-summing efficiency plotted against
sum-peak energy for the 60 Ni(α,γ)64 Zn reaction. For the reactions measured in this thesis,
the hit pattern centroid and therefore the average γ-ray multiplicity increase with energy,
and the γ-summing efficiency decreases with energy.
having a much higher efficiency than multiplicity 5. The various data points in the multiplicity groupings correspond to different possible γ-ray cascades with the same total energy but
different transitions between excited states. Next, all of the GEANT4 simulations are used
to create a linear fit describing the relationship between hit pattern centroid and γ-summing
efficiency. Using the best-fit line and the experimental value of the hit pattern centroid, the
experimental γ-summing efficiency and its uncertainty are immediately obtained. Typical
uncertainties of the γ-summing efficiency using the hit pattern technique are approximately
10%.
Because the majority of (p,γ) and (α,γ) reactions studied in this thesis produce nuclei
in high-lying excited states where many resonances overlap, it can be interesting to look at
systematic trends in the hit pattern centroid (and therefore average γ-ray multiplicity) as well
as the γ-summing efficiency. Typically as the excitation energy increases in a nucleus, the
average γ-ray multiplicity increases and the corresponding γ-summing efficiency decreases.
Results for the 60 Ni(α,γ)64 Zn reaction are shown in Fig. 6.10.
127

Chapter 7
27Al(p,γ)28Si

The first measurements with the SuN detector at the University of Notre Dame were of the
27 Al(p,γ)28 Si

reaction. This reaction was chosen because it has been measured many times

previously and was therefore a well-understood benchmark for comparison of SuN’s results.
The measurements of the 27 Al(p,γ)28 Si reaction served as an excellent proof-of-principle of
the γ-summing technique with the SuN detector, as well as a check on the analysis procedures
of the data.
As discussed in Sec. 4.2, the aluminum target was produced by evaporation at the NSCL
and its thickness of 74.7(3.7) µg/cm2 was measured with the RBS technique at Hope College.
The target was mounted at the center of the SuN detector, and the energy of the proton
beam was changed in small steps, typically 1-2 keV, to scan well-known resonances within
Ep = 2 − 4 MeV of the 27 Al(p,γ)28 Si reaction. Fig. 7.1 shows a typical γ-summed spectrum,
in this case for the resonance at Ep = 2517.7 keV. The spectrum is dominated by the sum
peak at 14.012 MeV with additional peaks in the spectrum corresponding to incomplete
summation of the γ rays. From previous measurements of the Ep = 2517.7 keV resonance,
the corresponding entry state in 28 Si is known to deexcite to the ground state with the
emission of three γ rays of energy 9394, 2839, and 1779 keV 88.2% of the time [114] (see
Fig. 7.2 for a simplified level scheme). Therefore, the smaller peaks observed in Fig. 7.1 are
found at energies 4.618, 11.173, and 12.233 MeV from when two of the γ rays are detected but
the third γ ray escapes undetected. Additionally, a GEANT4 simulation of the Ep = 2517.7
128

400

experiment
simulation

Counts

300

200

100

0

4000

6000

8000

10000 12000 14000

Energy (keV)
Figure 7.1: Total γ-summed spectrum for the Ep = 2517.7 keV resonance in the 27 Al(p,γ)28 Si
reaction. Plotted are the experimental spectrum taken with the SuN detector and the result
of GEANT4 simulations.
keV resonance was performed using the known deexcitation γ-ray cascades of 28 Si. Fig. 7.1
demonstrates that the GEANT4 simulation accurately matches the experimental data for
the SuN detector all the way up to a total energy of above 14 MeV. Excellent agreement
between GEANT4 simulations and experimental data was obtained for all of the measured
resonances, with subtle differences for some of the higher-energy resonances which could be
attributed to incorrect deexcitatation cascades in the literature.
In addition to creating a total γ-summed spectrum, the SuN detector also provides insight
into the deexcitation cascades through the energy spectra of the eight segments. By drawing
the energy spectrum of individual segments for events that have the full sum-peak energy
in the γ-summed spectrum, the individual γ-ray transitions can be identified. For example,
Fig. 7.3 shows the spectrum of a central segment for the Ep = 2517.7 keV resonance for
events that lie in the sum-peak region. The dominant γ-ray transitions at 9394, 2839, and
1779 keV are clearly visible, along with an intense peak at 511 keV from an electron-positron

129

1.9%

0.4%

0.3%

1.3%

0.1%

0.4%

1.9%

0.8%

1.6%

1.1%

88.2%

2.0%

14012

11779
11265
10668 10944
10209 10418
9417

6879
6276

6888

100%

4618

100%

1779

28

0 keV

Si

Figure 7.2: Level scheme of 28 Si for the Ep = 2517.7 keV resonance in the 27 Al(p,γ)28 Si reaction. For simplicity, only the energy levels and intensities for the primary γ-ray transitions
are labeled. The unlabeled levels also participate in the cascades based on their transition
probabilities. The most dominant cascade is highlighted and consists of the emission of γ
rays with energy 9394, 2839, and 1779 keV.

130

100

experiment
simulation

Counts

80
60
40
20
0

2000 4000 6000 8000 10000 12000 14000

Energy (keV)
Figure 7.3: Spectrum from one segment of the SuN detector for events in the sum-peak
region of the Ep = 2517.7 keV resonance in the 27 Al(p,γ)28 Si reaction. Plotted are the
experimental spectrum taken with the SuN detector and the result of GEANT4 simulations.
annihilation occurring in a different segment. Similarly, sometimes a 511 keV γ ray escapes
from the plotted segment, which creates the peaks visible 511 keV below the energy of the
γ-ray transitions at 8883, 2328, and 1268 keV. The experimental spectrum is matched well
by GEANT4 simulation, indicating that the detector response in simulation is correct and
that the intensities of all the γ-ray transitions in literature are accurate.
For each measurement, the sum peak was integrated and the γ-summing efficiency determined by the techniques discussed in Chap. 6. The yield of each measurement was determined
as the ratio of number of times the 27 Al(p,γ)28 Si reaction occurred to the number of incoming protons. Fig. 7.4 shows the measured excitation function for all of the data included in
the analysis. Additionally, the inset of the figure shows a zoomed-in view of the resonance
at Ep = 3674.9 keV and its comparison to the theoretical yield of a Breit-Wigner resonance
of Eq. 3.14. For each resonance in the excitation function, the area was determined and the
resonance strength calculated using Eq. 3.19. Due to lack of experimental data scanning the

131

Yield (counts/proton)

-9
0.4 ×10

×10
0.4

0.3

experiment
theoretical Breit-Wigner yield

0.2

0.2

0

3.66

3.68

3.7

0.1

0

2

2.5

3

3.5

4

Ep (MeV)
Figure 7.4: Measured excitation function in the region Ep = 2 − 4 MeV of the 27 Al(p,γ)28 Si
reaction. The inset shows a zoomed in view around the resonance at Ep = 3674.9 keV and
a comparison to the theoretical Breit-Wigner yield.
full resonance or the overlap of many resonances, not all of the data could be used to extract
a resonance strength. In total, 11 resonances were completely scanned and their resonance
strengths compared to previous measurements. The literature values for comparison came
from Ref. [15] and Ref. [5, 115], which contain compilations and evaluations of all previous
data sets for the 27 Al(p,γ)28 Si reaction.
The final results for the measurement of the 27 Al(p,γ)28 Si reaction with the SuN detector
are listed in Table 7.1 and plotted in Fig. 7.5. The values of Sp shown are calculated by

Sp = ωγ 2Jp + 1 (2Jt + 1)

(7.1)

where Jp = 1/2 is the spin of the proton and Jp = 5/2 is the ground-state spin of the 27 Al
target nuclei. Overall, there is good agreement between the SuN results and the previous
measurements. In the case of the Ep = 2374 keV resonance, there was a discrepancy between
132

Table 7.1: Measured resonance strengths for the 27 Al(p,γ)28 Si reaction for the SuN detector
and previous results.
Ep (keV)
2303.1
2311.9
2359.9
2373.8
2517.7
2675.5
2711.7
3338.4
3674.9
3791.7
3960.8

SuN
3.3 (0.6)
10.1 (1.8)
2.8 (0.5)
3.9 (0.7)
14.6 (2.1)
4.7 (1.0)
16.1 (2.2)
3.6 (0.7)
32.3 (4.5)
11.9 (1.6)
5.4 (0.8)

40

Endt [15]
1.6 (0.48)
6.7 (1.3)
5.4 (1.62)
26 (7.8)
16 (3)
7.2 (1.4)
14 (3)
4.3 (0.9)
35 (7)
7.3 (1.5)
3.3 (0.7)

NACRE [5, 115]
2.6 (1.3)
6.72 (0.84)
1.32 (0.48)
4.2 (1.2)
17.2 (1.9)
7.32 (0.84)
15.5 (3)
4.2 (8)
33.5 (6.7)
7.1 (1.7)
3.2 (0.6)

SuN
Endt 1998

Sp (eV)

30

NACRE Database

20
10
0
2500

3000

3500

4000

Ep (keV)
Figure 7.5: Resonance strengths of the 27 Al(p,γ)28 Si reaction measured with the SuN detector compared to previous results. Overall good agreement is achieved.

133

the resonance strength values listed in Ref. [15] and Ref. [115] and the SuN results agree nicely
with the value given in Ref. [115]. The agreement of the 27 Al(p,γ)28 Si resonance strengths
between SuN results and previous measurements demonstrated the validity of measurements
with the SuN detector. The same techniques that gave these results are also used in the
more astrophysically interesting measurements contained in the following chapters.

134

Chapter 8
74Ge(p,γ)75As

One of the significant results obtained during the SuN experimental campaign at the University of Notre Dame was the measurement of the 74 Ge(p,γ)75 As reaction and its role in
p-process nucleosynthesis. The motivation for measuring the 74 Ge(p,γ)75 As reaction was
that it was previously identified in a p-process sensitivity study by Rapp et al. [2] as playing an important role in the production of the lightest of the p nuclei, 74 Se. The authors
found that varying the 74 Ge(p,γ)75 As reaction rate within its theoretical uncertainty had
a direct impact on the production of 74 Se and therefore deemed this reaction “particularly
important” to experimentally measure.
74 Se

is produced in the high-temperature layers of type II supernovae (SNII) which are

undergoing p-process nucleosynthesis. The higher temperatures of the stellar environment
allow for charged particle capture reactions on less massive seed nuclei, and (p,γ) and (p,n)
reactions can dominate over the typical photodisintegration scenario. The dominant reaction chain for the production of 74 Se is 74 Ge(p,γ)75 As(p,n)75 Se(γ,n)74 Se, which is shown in
Fig. 8.1. The main destruction reaction is 74 Se(γ,α)70 Ge with smaller contributions from
74 Se(γ,p)73 As.

Prior to the measurement, most of the other reaction rates had been well con-

strained by previous experiments. The 74 Se(n,γ)75 Se reaction rate was well known through
several measurements and in particular by the measurement of Dillmann et al. [116] focusing on p-process relevant energies. Similarly, the cross section of the 75 As(p,n)75 Se reaction
had been measured down to the reaction threshold (Q = -1.646 MeV) [117, 118], and the
135

73

Br

72

Br

Se

73

As

72

Ge

71

Ga

70

71

70

69

74

75

76

Br

Se

74

As

73

Ge

72

Ga

71

Br

Se

75

As

74

Ge

73

Ga

72

77

Br

Se

76

As

75

Ge

74

Ga

73

78

Br

Se

77

As

76

Ge

75

Ga

74

Se

As

Ge

Ga

Figure 8.1: A zoomed in section of the nuclear chart showing the dominant reaction flow
producing 74 Se in the p process.
70 Ge(α,γ)74 Se

reaction had been measured at p-process energies covering the majority of

the Gamow window [119]. Therefore, the 74 Ge(p,γ)75 As reaction was the remaining reaction
to measure in the production chain of 74 Se.
Investigating the production of 74 Se is significant because, unlike the majority of the
light p nuclei, 74 Se is surprisingly overproduced by approximately a factor of 3 (see Fig. 1.7).
A similar, but not as pronounced, behavior is observed for the second lightest p nucleus,
78 Kr.

As mentioned, the lighter p nuclei are the only ones influenced by the (p,γ) and

(p,n) reactions taking place at high temperatures, and it is therefore crucial to constrain
the involved nuclear reaction rates and reduce the relevant nuclear physics uncertainties to
provide deeper insight into this alternative reaction mechanism. Since the 74 Ge(p,γ)75 As
reaction was the last unconstrained reaction in the production chain of 74 Se, it could perhaps
explain the overproduction of 74 Se and be an important step towards understanding the
discrepancy in the overall production of the puzzling light p nuclei.
The measurements were performed with proton energies in the range Ep = 1.6 - 4.2 MeV

136

74

5000

sum peak

Ge(p,γ )75As

metastable
peak

Counts / 10 keV

Q = 6901 keV

4000

Ep = 3400 keV
Ep = 2800 keV

3000

Ep = 2200 keV

F(p,αγ )16 O

19

2000
O(p,αγ )15N

18

1000
0

4000

6000

8000

10000

12000

Energy (keV)
Figure 8.2: Sum peak spectra for three different proton energies Ep normalized to the same
number of incoming protons. The location of the sum peak and the metastable peak are
indicated for Ep = 3400 keV.
and an enriched 74 Ge target. As discussed in Sec. 4.2, the target was made by evaporating
97.55% enriched 74 Ge powder onto a thick tantalum backing, and the thickness of 340(17)
µg/cm2 was determined through RBS analysis. For the measurement of the 74 Ge(p,γ)75 As
reaction, the beam intensity was maintained below 10 enA to ensure minimal dead time in
the SuN detector. The resulting total γ-summed spectra from the SuN detector are shown
in Fig. 8.2 for three different energies. Above 3 MeV the spectra are dominated by the sum
peak of the 74 Ge(p,γ)75 As reaction. The shift in location of the sum peak is due to the
changing center of mass energy, and the different intensities of the sum peaks give indication
that the cross section is increasing with energy. Small signatures of the contaminants in the
target can be observed around 7 MeV coming from the 19 F(p,αγ)16 O reaction, and around
5.3 MeV from the 18 O(p,αγ)15 N reaction caused by a small amount of oxygen present in the
target.
137

5000

Counts / 10 keV

4000

3000

2000

1000

8000

isomer
no isomer

9000

10000

11000

Energy (keV)

Figure 8.3: Total γ-summed spectrum for the 74 Ge(p,γ)75 As measurement at Ep = 3400
keV. Also plotted is the linear background with and without including the 304 keV isomer.
When determining the number of counts in the sum peak, it was necessary to take into
account the 304 keV isomeric state in 75 As with a half-life of 17.62 ms [113]. As this half-life
was much longer than the 300 ns event window used in the analysis, any deexcitation to the
isomeric state does not get summed into the sum peak. Instead, these events are found 304
keV below the sum peak energy (EΣ ). To include the events populating the isomeric state in
the analysis, it was decided to expand the sum-peak region to (EΣ - 3σ - 304 keV , EΣ + 3σ).
Once the sum-peak region was defined, a linear background was subtracted and the total
number of counts above the linear background was determined as discussed in Sec. 6.3.
Fig. 8.3 shows the difference in linear backgrounds with and without including the 304 keV
isomeric state. The γ-summing efficiencies (Sec. 6.4) ranged from 28.1(2.3)% at Ep = 1.6
MeV to 21.8(1.7)% at Ep = 4.2 MeV. Using the number of sum-peak counts and SuN’s
summing efficiency, the total number of reaction products was determined and the total
(p,γ) cross sections and astrophysical S-factors were calculated using Eq. 2.3 and Eq. 2.13.
The results are listed in Table 8.1.
138

Table 8.1: Cross sections and astrophysical S-factors for the 74 Ge(p,γ)75 As reaction.
Ec.m.
(MeV)
1.554
1.763
1.961
2.159
2.357
2.554
2.752
2.950
3.049
3.148

σ
(mb)
0.045 ± 0.005
0.19 ± 0.02
0.51 ± 0.06
1.11 ± 0.12
2.17 ± 0.24
3.78 ± 0.41
6.23 ± 0.68
10.5 ± 1.1
11.6 ± 1.3
13.2 ± 1.4

S-factor
(GeV b)
6251 ± 708
6464 ± 714
5796 ± 637
4910 ± 540
4196 ± 463
3534 ± 387
3062 ± 333
2914 ± 317
2468 ± 268
2177 ± 237

Ec.m.
(MeV)
3.246
3.345
3.444
3.543
3.642
3.741
3.839
3.938
4.037
4.136

σ
S-factor
(mb)
(GeV b)
15.5 ± 1.7 2005 ± 218
17.5 ± 1.9 1799 ± 196
15.5 ± 1.7 1279 ± 139
13.0 ± 1.4
875 ± 95
6.08 ± 0.66 334 ± 36
5.46 ± 0.59 247 ± 27
3.68 ± 0.40 138 ± 15
2.96 ± 0.32
94 ± 10
2.93 ± 0.32
78 ± 9
3.13 ± 0.34
71 ± 8

The uncertainties reported in Table 8.1 include both statistical and systematic errors. The
primary sources of error are the 5% uncertainty in target thickness, 5% uncertainty in beam
charge collection, and the 8% uncertainty in summing efficiency. There is also an uncertainty
in the integral of the sum peak due to a possible contribution from the 73 Ge(p,γ)74 As reaction
with a Q-value of 6852 keV, only 49 keV below the Q-value of the 74 Ge(p,γ)75 As reaction.
However, the cross section of the 73 Ge(p,γ)74 As reaction is predicted to be lower by a factor
of 1.5 to 25 throughout the energy range measured [120], and the isotopic composition of
73 Ge

in the target was only 1.9%. Thus, an upper bound contribution of 1.5% to the sum

peak from the 73 Ge(p,γ)74 As reaction was estimated.
As the analysis of the 74 Ge(p,γ)75 As reaction cross section measurement was nearing
completion, a research group in K¨oln, Germany published their results on the same reaction [10]. A comparison of values in this dissertation to the K¨oln data set is shown in
Fig. 8.4. The two separate measurements agree within the uncertainty, with the K¨oln data
giving systematically slightly lower cross sections, 6% to 21% below the present results. This
systematic effect could be attributed to the two different measurement techniques. The K¨oln
139

Cross Section (b)

10-2

10-3
Gamow Window
present work
Sauerwein et al.
NONSMOKER
TALYS CTFG
TALYS BSFG
TALYS GT

10-4

10-5

1.5

2

2.5

3

3.5

4

4.5

Ec.m. (MeV)
Figure 8.4: Cross section vs. center of mass energy plot for the 74 Ge(p,γ)75 As reaction.
The measurements of this dissertation (solid circles) and previous data of Ref. [10] (open
triangles) are compared to theoretical calculations using NON-SMOKER and TALYS 1.4
nuclear reaction codes. The most accurate reproduction of the data is the TALYS backshifted Fermi gas model (BSFG).

140

group performed the measurement using the in-beam method (Sec. 4.1.3), where four highpurity germanium detectors were used to detect the individual 75 As deexcitation γ-rays and
their angular distribution. This method may underestimate the total cross section if there
are low intensity γ-rays that are below the detectable limits or high energy γ-rays that have a
low efficiency for being detected. In Fig. 8.4 both data sets show an increase in cross section
values until an energy around Ec.m. = 3.3 MeV where the (p,n) reaction channel opens. The
SuN measurements extend to both higher and lower energies than the K¨oln data. At the
lower energies this allows for a better characterization of the energy dependence of the cross
section. At the higher energies, above the energy where the (p,n) channel opens, the larger
energy coverage allows for a more sensitive selection of the appropriate nuclear level density
model.
The theoretical calculations of the K¨oln group. [10] were done using the statistical model
code SMARAGD [121]. The authors conclude that the best description of the data was
one which used the standard microscopic optical model potential of Jeukenne, Lejeune, and
Mahaux (JLM) [122] with the proton width multiplied by two. In Fig. 8.4, the solid black
curve represents standard NON-SMOKER [120] statistical model calculations, which is the
predecessor code to SMARAGD and is therefore representative of the K¨oln calculations. The
NON-SMOKER calculations do not reproduce the energy dependence of the cross section,
as an incorrect slope in the curve leads to an underestimation of the cross section at higher
energies and an overestimation of the cross section at lower energies. Scaling the proton
widths by two, as done in Ref. [10], would better match the data at higher energies, but
then further overestimate the cross section at lower energies. This large difference at lower
energies, up to 70% at 1.55 MeV, is only noticeable with the new data presented here.
Therefore, aiming at an improved description of the data, additional theoretical calcu141

lations were performed using the TALYS 1.4 nuclear reaction code [123]. The remaining
three curves in Fig. 8.4 are TALYS calculations for different nuclear level densities. All were
performed using the semimicroscopic optical model of Bauge, Delaroche, and Girod [124],
which is a reparametrization of the JLM optical model to cover a wider energy range of
scattering data. TALYS calculations were performed using level densities from the constant temperature and Fermi gas model (CTFG) [125, 126], back-shifted Fermi gas model
(BSFG) [127], generalized superfluid model (GSM) [128, 129], and the microscopic level densities from Goriely’s table (GT) [130] and Hilaire’s table (HT) [131]. The calculations using
level densities from GSM and HT are left off of Fig. 8.4, as they produce results similar to
CTFG for the present reaction. Default parameter values were used in TALYS 1.4 except
for options for the JLM potential and different nuclear level densities. As seen in Fig. 8.4
the best description of the 74 Ge(p,γ)75 As reaction is achieved when using the back-shifted
Fermi gas model (BSFG). The BSFG model uses the Fermi gas model down to zero energy
with a shift in energy to account for nucleon pairing. This shift attempts to correct for the
fact that the Fermi gas model is based on non-interacting fermions. In TALYS, the BSFG
model uses the modification of Ref. [132] to fix the divergence of the nuclear level density at
zero energy.
Using the BSFG nuclear level density and JLM optical model potential, new astrophysical
reaction rates were calculated with TALYS 1.4 and their values are shown in Table 8.2.
Furthermore, in Fig. 8.5 the ratio of the new reaction rate calculations compared to the
reaction rates presented in the K¨oln paper [10] and to the standard REACLIB [11] rates
used in astrophysical calculations is shown. Within the relevant temperature window for
the p process, 1.8 - 3.3 GK, the new rates are higher than the standard REACLIB rates
with a maximum increase of 27%, but below the rates proposed by the K¨oln group with a
142

Table 8.2: Reaction rates for the 74 Ge(p,γ)75 As reaction from TALYS 1.4 calculations with
the JLM optical model potential and BSFG nuclear level density which provide the most
accurate reproduction of experimental data.
T
(GK)
0.10
0.15
0.20
0.25
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00

(cm3

Rate
mol−1 s−1 )

T
(GK)

4.629 × 10−31
4.310 × 10−21
4.173 × 10−16
4.429 × 10−13
5.130 × 10−11
2.638 × 10−8
1.556 × 10−6
3.055 × 10−5
3.146 × 10−4
2.130 × 10−3
1.073 × 10−2
4.338 × 10−2

1.50
2.00
2.50
3.00
3.50
4.00
5.00
6.00
7.00
8.00
9.00
10.00

(cm3

Rate
mol−1 s−1 )

6.228
1.369
1.101
4.742
1.345
2.845
6.961
9.787
9.502
7.379
5.182
3.600

×
×
×
×
×
×
×
×
×
×
×
×

100
102
103
103
104
104
104
104
104
104
104
104

Figure 8.5: Ratio of reaction rates in Table 8.2 to those presented in Ref. [10] and to the standard REACLIB [11] rates. The shaded area indicates the p-process relevant temperatures
of 1.8 - 3.3 GK.
143

maximum decrease of 20%. The large change in the new reaction rates at low T compared
to the other two data sets can be attributed to the different description of cross section at
low energies. Because the TALYS model gives a more accurate reproduction of the energy
dependence of the experimental data, the reaction rates in Table 8.2 are new recommended
rates for the 74 Ge(p,γ)75 As reaction.
The impact of the experimental cross sections on the final abundance of the p nuclei
was investigated using the post-processing code NucNet Tools [133] to follow the reaction
network of a 25M⊙ SNII when the shock front passes through the O/Ne layer. In this
model, the initial seed abundances and temperature and density profiles were taken to be
the same as Rapp et al. [2]. The p process was then followed through 11 different mass
layers and the final abundances of the p nuclei were summed for all the included layers.
This reaction network uses reactions that are all based on theoretical cross sections from
REACLIB [11], whose uncertainty arises from uncertainties in the optical model potential
and nuclear level density models used. The theoretical uncertainty of the 74 Ge(p,γ)75 As
reaction cross section was investigated using TALYS 1.4, and the spread in predicted cross
sections can be seen in Fig. 8.4. An uncertainty up to a factor of three was determined
based on the difference between the constant temperature Fermi gas model (CTFG) and
the microscopic level densities taken from Goriely’s Table (GT). In Fig. 8.6, the impact of
the 74 Ge(p,γ)75 As reaction on the final abundance of the lightest p nucleus, 74 Se, is shown
for the 25M⊙ SNII model. The cumulative mass fraction for 74 Se is plotted against the
maximum temperature of the mass layer. The maximum temperature of the mass layer
decreases along the abscissa, corresponding to integrating the mass fraction from the inside
layer to the outside layer of the supernova. The majority of 74 Se is produced in the inner
(hotter) layers, as the mass fraction does not increase when including layers with a maximum
144

Figure 8.6: Cumulative mass fraction of 74 Se from a model of the p process in a type II
supernova, plotted as a function of the maximum temperature of the mass layer included.
See text for details.
temperature below 2.6 GK. The solid black line and shaded grey area represent the results
for 74 Se using the standard REACLIB 74 Ge(p,γ)75 As reaction rate and a factor of three
increase and decrease, respectively. Increasing the reaction rate by 3 leads to a 25% increase
in the production of 74 Se, while decreasing the reaction rate by 3 leads to a 23% decrease
in the production of 74 Se. The overall spread in the final cumulative mass fraction of 74 Se
when relying on the theoretical predictions and uncertainties of the 74 Ge(p,γ)75 As reaction
spans from 4.94×10−7 to 8.03×10−7 , which is a factor of 1.6.
Also in Fig. 8.6, the impact of the new experimental reaction rates on the final mass
fraction of 74 Se is shown. The dashed black line and shaded hatched area show the 74 Se
mass fraction when performing the network calculation with experimental values and un-

145

certainties for the 74 Ge(p,γ)75 As reaction. The uncertainty in the production of 74 Se from
the 74 Ge(p,γ)75 As reaction is reduced to a factor of 1.05 with the reaction rates presented
here, with an overall increase in the mean mass fraction from 6.4×10−7 to 7.5×10−7 . As
mentioned previously, 74 Se is strongly overproduced in p-process models, and this overproduction is further enhanced with the experimental results reported here. Thus, the end
result of the 74 Ge(p,γ)75 As measurement contained in this chapter is that the uncertainty
in the production of 74 Se in the p process is greatly reduced, but the general overproduction
of 74 Se compared to its observed abundance remains.

146

Chapter 9
58Ni(α,γ)62Zn
The story of the 58 Ni(α,γ)62 Zn reaction is unique to this thesis in that the astrophysical
motivation for its measurement is different from the other reactions contained here. Rather
than playing an important role in the p process of type II supernovae, the 58 Ni(α,γ)62 Zn
reaction instead contributes to the nucleosynthesis inside of a type Ia supernovae (SNIa).
Its importance was first discovered by Bravo and Mart´ınez-Pinedo [134] when performing a
sensitivity study quantifying the influence of individual reaction rates on the nucleosynthesis
in SNIa. For the study, the authors used a one-dimensional model of a Chandrasekhar-mass
white dwarf and varied the reaction rates up and down by a factor of 10. Overall, the authors
concluded that nucleosynthesis was relatively insensitive to the change of individual reaction
rates, but many reactions were identified as relevant for having a direct impact on the final
abundance of particular isotopes. One such reaction was the 58 Ni(α,γ)62 Zn reaction, which
was selected for its impact on the production of 62 Ni, 63 Cu, and 64 Zn. The relation of the
58 Ni(α,γ)62 Zn

reaction to these isotopes is shown in the nuclear chart of Fig. 9.1.

The astrophysical scenario in which the 58 Ni(α,γ)62 Zn reaction is expected to play an
important role is during the α-rich freeze-out from nuclear statistical equilibrium (NSE) (see
Sec. 1.3.4). In the innermost layers of SNIa, temperatures and densities are sufficiently high
to reach NSE, and a large portion of the material in NSE is expected to undergo α-rich
freeze-out. For the Chandrasekhar-mass white dwarf studied in Ref. [134] it was noticed
that the inner 0.4M⊙ reached NSE, of which 0.24M⊙ underwent α-rich freeze-out. After
147

59

Ga

58

57

Ga

Zn

59

Cu

58

56

55

60

60

Cu

59

57

Co

56

Fe

Ga

Zn

Ni

54

61

61

Cu

60

58

Co

57

Fe

Ga

Zn

Ni

55

62

62

Cu

61

59

Co

58

Fe

Ga

Zn

Ni

56

63

63

Cu

62

60

Co

59

Fe

Ga

Zn

Ni

57

64

64

Cu

63

61

Co

60

Fe

Ga

Zn

Ni

58

65

65

Cu

64

62

Co

61

Fe

Ga

Zn

Ni

59

66

Zn

Cu

Ni

63

Co

62

60

Fe

Ni

Co

61

Fe

Figure 9.1: A zoomed in section of the nuclear chart showing the 58 Ni(α,γ)62 Zn reaction
and the isotopes most influenced by its reaction rate in SNIa (62 Ni, 63 Cu, 64 Zn).
α-rich freeze-out, the composition of the layer is dominated by isotopes in the iron region
and α-particles that did not reassemble into heavier nuclei. Thus it is expected that αinduced reactions on nuclei in the iron region are important for the final abundance pattern,
including the 58 Ni(α,γ)62 Zn reaction.
The 58 Ni(α,γ)62 Zn reaction has been measured three times previously, all over 50 years
ago. Morinaga [135] and Ball et al. [136] performed cross section measurements using the
activation technique (Sec. 4.1.1) with energies Eα = 10.6 - 31.0 MeV. After irradiation, both
measurements included an additional step of chemically separating zinc from other elements
before counting the decay of 62 Zn with Geiger counters. McGowan and collaborators [12]
extended the measurements to lower energies by using thick-target yields from enriched
58 Ni

targets. The yield was determined every 100 keV within the beam energy range of

Eα = 4.9 - 6.1 MeV and the cross section determined by differentiating the yield curve.
The measurement of the 58 Ni(α,γ)62 Zn reaction cross section was performed using the
SuN detector and the γ-summing technique. The FN Tandem Van de Graaff Accelerator at

148

105
200

Counts / 8 keV

104

100

0

103

102

58

8

9

10

11

Ni + α

181

Ta + α

10

room background
58

Ni(α,p)61Cu

2

4

6

8

10

Energy (MeV)
Figure 9.2: Experimental spectra from the SuN detector for measurements at Eα = 7.7
MeV. The spectra correspond to 58 Ni (solid black), thick tantalum backing (dotted blue),
and normalized room background (dot-dashed red). The inset shows a zoom around the
sum-peak region of the 58 Ni(α,γ)62 Zn reaction.
the University of Notre Dame was used to accelerate 4 He2+ nuclei to energies Eα = 5.5 − 9.5
MeV. The beam current was varied between 4 − 60 enA in order to balance count rate with
minimal detection dead time. For the present work the dead time was kept below 1.2%. For
each data run the total charge collected from the 4 He2+ beam was between 7 − 159 µC.
The 58 Ni target was isotopically enriched to 95(5)% and its thickness of 930 ± 46 µg/cm2
was measured using the RBS technique. Trace amounts of carbon and oxygen on the front
and back surfaces of the target were revealed during the RBS measurements. The target
thickness corresponded to a beam energy loss of 0.42 MeV and 0.30 MeV at Eα = 5.5 MeV
and Eα = 9.5 MeV, respectively [137].
A total γ-summed spectrum from the SuN detector for the 58 Ni(α,γ)62 Zn reaction with
Ec.m. = 7039 keV and Q = 3364.27 keV is shown in Fig. 11.3 (solid black line). At

149

Table 9.1: Cross sections for the 58 Ni(α,γ)62 Zn reaction.
max (MeV)
Ec.m.

min (MeV)
Ec.m.

eff (MeV)
Ec.m.

σ (µb)

5.143
5.330
5.517
5.704
6.078
6.452
6.826
7.201
7.574
7.949
8.415
8.884

4.750
4.946
5.143
5.337
5.723
6.112
6.496
6.883
7.268
7.649
8.129
8.606

4.988
5.171
5.360
5.548
5.922
6.298
6.673
7.051
7.428
7.805
8.277
8.749

3.13 ± 0.44
4.70 ± 0.60
6.69 ± 1.04
9.65 ± 1.34
15.3 ± 2.4
22.2 ± 3.5
34.0 ± 6.1
52.4 ± 7.1
66.9 ± 9.7
92.8 ± 15.1
138.8 ± 22.1
158.9 ± 25.2

higher energies, both room background and beam-induced background contributions to the
spectrum are reduced allowing the sum peak to be clearly visible at 10.4 MeV. The source
of room background in the region of the sum peak comes from cosmic rays. During the
experiment the 58 Ni target was mounted in front of a thick tantalum backing. Thus, the
beam induced background was determined by taking data without the 58 Ni target in place
so that the beam was impinging solely onto the tantalum backing. Additional peaks were
visible in the low energy region of the 58 Ni spectrum that originate from the 58 Ni(α,pγ)61 Cu
reaction which has a higher cross section than the 58 Ni(α,γ)62 Zn reaction at this energy by
approximately two orders of magnitude [12, 138]. The additional nickel isotopes have (α,γ)
Q values larger than the 58 Ni(α,γ)62 Zn reaction, and thus do not contribute in the sum-peak
region. Also, these additional nickel isotopes are present in very low amounts in the target
and there was no indication of their (α,γ) reactions in the summed spectra.
The number of counts in the sum peak and the γ-summing efficiency were determined at
each energy step, and then the reaction cross section was calculated using Eq. 2.3. SuN’s γ150

summing efficiency ranged from 26.7(2.8)% at Ec.m. = 4.943 MeV to 17.4(2.3)% at Ec.m. =
8.742 MeV. The results of the cross section calculations are displayed in Table 9.1. In the
table, the first two columns contain the maximum and minimum energies of the beam due
to the thickness of the target. The third column contains the effective energy for each data
point taking into account the variation of the cross section in the target. The last column lists
the calculated cross sections. Of the uncertainty reported, roughly 3% comes from statistical
uncertainties, 5% from the beam charge collection, 5% from the target thickness, 5% from
the target enrichment, and 10% - 15% from the detection efficiency. The uncertainty in
energy from the accelerator is 4 keV at all energies.
A plot of the 58 Ni(α,γ)62 Zn reaction cross section is shown in Fig. 9.3. The present
work is in agreement with the previous results of Ref. [12] and extends the measurements to
higher energies. The increased energy coverage of the experimental cross section allows for
a more sensitive study of the energy dependence of the cross section and provides a better
test for theoretical models. The theoretical calculations of the 58 Ni(α,γ)62 Zn reaction were
performed using the code SMARAGD [13, 139] which is based on the nuclear statistical
model (see Sec. 3.2). According to Ref. [140], the nuclear statistical model is expected to be
valid down to 0.12 GK for the 58 Ni(α,γ)62 Zn reaction. Since this is well below the relevant
temperature range of 2-5 GK for which the 58 Ni(α,γ)62 Zn reaction is expected to play a role
in the nucleosynthesis of SNIa, the use of the nuclear statistical model is well founded. The
Gamow window (see Sec. 2.4) for the 58 Ni(α,γ)62 Zn reaction at 2 GK is from approximately
3 to 5 MeV with a maximum contribution to the rate at 4 MeV, and at 5 GK the Gamow
window is from 4 to 7 MeV with a maximum contribution at 5.25 MeV [63]. As shown
in Fig. 9.3, the experimental values cover only the upper part of this energy window, and
theoretical predictions are required for the lower energies.
151

Cross Section (b)

10-3

10-4

10-5
present work
F.K. McGowan et al.
-6

10

SMARAGD
SMARAGD (rescaled widths)

10-7 4

5

6

7

8

9

10

Ec.m. (MeV)
Figure 9.3: Cross section of the 58 Ni(α,γ)62 Zn reaction for the present work (black circles),
previous data of Ref. [12] (red triangles), and theoretical calculations from the SMARAGD
code [13]. A good description of the data was obtained by modifying the α width and the
γ-to-proton width ratio (dashed line).
The initial SMARAGD calculation systematically overestimated the cross section values
by approximately a factor of 2, as shown with the solid line of Fig. 9.3. Since this theoretical
value is representative of the standard value that gets used in nucleosynthesis calculations,
the previously accepted reaction rate for the 58 Ni(α,γ)62 Zn reaction was overestimated by
approximately a factor of 2. The investigation of how to modify the theoretical calculation
to best describe the data is summarized in Fig. 9.4, which shows the sensitivity of the
58 Ni(α,γ)62 Zn

cross section to variations in partial widths. For the energy range here, the

emission of γ rays, neutrons, protons, and α particles was considered and therefore their
partial widths were varied. The sensitivity is defined as [141]

υ −1
ΩSq = Ω
υq − 1
152

(9.1)

1

Sensitivity

0.8
0.6
0.4

γ
n
p

0.2

α

0
2

4

6

8

10

12

Energy (MeV)
Figure 9.4: Absolute values of the sensitivity of the 58 Ni(α,γ)62 Zn cross section as function
of energy, when separately varying γ, neutron, proton, and α widths.
where q is the quantity being changed and Ω is the resulting quantity. A change in q is
given by the factor υq = qnew /qold and the subsequent change in Ω is given by the factor
υΩ = Ωnew /Ωold . In the current context, the quantity q is an averaged width used in
the reaction model and the resulting quantity Ω is the reaction cross section. Using these
definitions, the sensitivity ΩSq = 0 when there is no change in the cross section after changing
the partial width and ΩSq = 1 when the cross section changes by the same factor as the
change in the partial width.
Below 4 MeV the 58 Ni(α,γ)62 Zn reaction cross section is exclusively sensitive to the α
width with the sensitivity to the α width persisting throughout the energy region plotted.
This low-energy region is also important for the calculation of the astrophysical reaction rate
and reactivity [140]. Conversely, there is very little sensitivity to the neutron width even
for energies above the neutron emission threshold at Ec.m. = 9.526 MeV. The remaining
two parameters, the proton and γ widths, show an increasing effect on the cross section
153

with increasing energy in the region between 4 and 10 MeV. As mentioned, the default
SMARAGD calculation overestimated the experimental data but accurately reproduced the
energy dependence of the cross section. Because the energy dependence was reproduced
well and the sensitivity to the α width was present for all energies, it was expected that
rescaling the alpha width could provide a good description of the data. It was determined
that the α width obtained with the optical potential of [142] has to be scaled by a factor
0.45 to match the low energy data below 6 MeV. However, at higher energies where the
cross section sensitivity to the α width is reduced, there were still small deviations between
the SMARAGD calculation and the experimental data. By increasing the γ-to-proton width
ratio by 10%, an improved agreement with the data at the upper end of the measured energy
range was achieved. The γ- and proton widths cannot be constrained separately with data
from only this reaction and only the increase in ratio can be determined.
The SMARAGD calculation with the rescaled widths is shown as dashed line in Fig. 9.3,
which matches the experimental data very well. Although the scaling factors for the α- width
and γ-to-proton width ratio provide an excellent description of the 58 Ni(α,γ)62 Zn cross section, calculations using these scaling factors underproduce the 58 Ni(α,p)61 Cu experimental
data [12, 138] by a factor of three. Further theoretical work is required to obtain a full
understanding of α-induced reaction cross sections on 58 Ni, which was beyond the scope of
this thesis.
Since there is no indication from the data that the energy dependence of the cross section changes towards even lower energies, the modified widths were used to calculate new
stellar reactivities, which are shown in Table 9.2. These stellar reactivities are dominated
by the ground-state cross sections with only small influence from thermally excited states
in 58 Ni. The ground-state contributions are 100% at 2 GK and 95% at 5 GK and thus the
154

Table 9.2: Stellar reactivities for the 58 Ni(α,γ)62 Zn reaction.
T
(GK)
0.10
0.15
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.50

Reactivity
(cm3 mol−1 s−1 )
6.623
1.971
2.068
1.171
3.546
2.649
2.293
4.941
4.070
1.666
4.020
2.679

×
×
×
×
×
×
×
×
×
×
×
×

10−62
10−50
10−43
10−34
10−29
10−25
10−22
10−20
10−18
10−16
10−15
10−10

T
(GK)

Reactivity
(cm3 mol−1 s−1 )

2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00

2.268 × 10−7
2.045 × 10−5
5.000 × 10−4
5.407 × 10−3
3.399 × 10−2
1.461 × 10−1
4.755 × 10−1
2.812 × 100
9.844 × 100
2.443 × 101
4.755 × 101
7.688 × 101

reactivities are well constrained by experimental data. Table 9.3 contains a fit to the stellar
reactivities using the standard 7 parameter REACLIB format [72], which is commonly used
in astrophysical calculations.
After calculating new reactivities, their effect on the nucleosynthesis of SNIa was investigated. The reduction in the rate of 58 Ni(α,γ)62 Zn is expected to translate into a decrease in the abundance of 62 Zn and other nuclei linked by subsequent reaction chains,
Table 9.3: REACLIB parameters for the 58 Ni(α,γ)62 Zn reaction.
Parameter
a0
a1
a2
a3
a4
a5
a6

Value
−
−
−
−

5.194217
2.314329
2.528868
5.651307
1.088296
1.763373
3.858753

155

×
×
×
×
×
×
×

101
100
101
101
100
10−1
101

Table 9.4: Changes to the nucleosynthesis in SNIa models. (1) Chandrasekhar-mass delayed
detonation model with ρDDT = 3.9 × 107 g/cm3 . (2) Explosion of a sub-Chandrasekhar
white dwarf of 1.025 M⊙ C-O core surrounded by a 0.055 M⊙ He envelope.
υΩ − 1 Delayed detonation
-0.05
-0.04
-0.02
-0.01

Sub-Chandrasekhar

64 Zn

62 Ni

62 Ni
63 Cu, 66 Zn

63 Cu, 66 Zn, 69 Ga
64 Zn, 65 Zn, 73 Ge

e.g. 62 Zn(α,γ)66Ge, 62 Zn(p,γ)63 Ga(p,γ)64 Ge, and so on. After the decay of the radioactive
isotopes, the result is a decrease of the ejected abundances of 62 Ni, 66 Zn, 63 Cu, 64 Zn, and
others. In the temperature range in which α-rich freeze-out takes place, namely from 2-5 GK,
the new reaction rates given by Table 9.3 are smaller than the standard REACLIB [11] rates
by a factor of 0.45. From Ref. [134], these lower rates correspond to an expected decrease in
the ejected abundance of 62 Ni by approximately 6-7%.
Simulations of SNIa were performed with both the standard REACLIB [11] rates and
the new rates following the same methodology and codes as described in Ref. [134]. Calculations were done for both a delayed detonation of a Chandrasekhar-mass white dwarf
and a thermonuclear explosion of a sub-Chandrasekhar white dwarf. The relative change
in the ejected abundances of the most sensitive species is listed in Table 9.4. The results
agree with the prior estimate and have a maximum sensitivity of approximately 5% for the
abundances of 62 Ni and 64 Zn. Additional calculations were performed with different sets
of deflagration-to-detonation transition densities, ρDDT , and initial metallicities, but their
effect is small and the maximum sensitivities never exceed the values reported in Table 9.4.
It was found that in general, the sensitivities increase with metallicity and with ρDDT .
In summary, the 58 Ni(α,γ)62 Zn reaction cross section results contained in this chapter

156

agree well with previous measurements and expand to energies not covered by prior experiments. The standard theoretical calculation by the SMARAGD code overproduced the
measured cross section, but multiplying the α width by a factor of 0.45 accurately reproduced
the data. New reactivities were reported, and the new reaction rates used in nucleosynthesis
calculations for SNIa. It was determined that the new rates have at most a 5% effect on the
ejected abundances of several isotopes, all in cases where a significant portion of the mass
participates in α-rich freeze-out.

157

Chapter 10
Additional (α,γ) Measurements
This chapter contains the first ever measurements of the 90 Zr(α,γ)94Mo, 92 Zr(α,γ)96Mo, and
74 Ge(α,γ)78 Se

reaction cross sections, which are relevant for the production of the lightest

p nuclei. As discussed in Sec. 1.4, this region is especially interesting to study due to the
large discrepancies between astrophysical calculations and the observed abundances. Models
of the p process predict a notable underproduction of the 92,94 Mo and 96,98Ru isotopes of
greater than a factor of 10, while simultaneously predicting a significant overproduction in
the amount of 74 Se by approximately a factor of 3 [2].
In general, there is limited existing experimental data for (α,γ) reactions in the p process,
with less than 20 of the relevant (α,γ) reactions being measured to date [50]. Due to this
lack of experimental data, reaction networks of the p process rely heavily on theoretical
reaction rates which typically have large uncertainties. The 90 Zr(α,γ)94Mo, 92 Zr(α,γ)96 Mo,
and 74 Ge(α,γ)78 Se measurements in this chapter expand the existing experimental database
for (α,γ) reactions and fit into the larger effort aimed at constraining the reaction theory
relevant to the p process through systematic measurements across a large mass and energy
range.
In addition to the global systematic studies, efforts have been made to identify a smaller
list of reactions which have the largest impact on the final abundances of the p nuclei. Such
sensitivity studies are valuable because they inform experimentalists which reactions are the
most crucial to measure. One such sensitivity study by Rauscher [14] took the approach of
158

identifying the isotope of each element where the (γ,p) or (γ,α) photodisintegration rates
are comparable to the (γ,n) rates. As discussed in Sec. 1.4, near stability the (γ,n) reactions
dominate, but as the isotopes become more neutron-deficient, the (γ,p) or (γ,α) reactions
may proceed with a higher rate. Accurately constraining the reaction rates at these “branching points” is critical for correctly modeling the mass flow of the p process. Because the
locations of the branching points at a given temperature rely solely on the nuclear properties, it is possible to identify potentially critical reactions independently of the astrophysical
model.
In Rauscher’s study [14], the branching point in the molybdenum isotopes was determined
to be at 94 Mo, at which point the (γ,α) reaction is expected to proceed at a higher rate
than the (γ,n) reaction for p-process temperatures. However, the two reaction channels
have reaction rates that are within the theoretical uncertainty of each other, which makes
the identification of 94 Mo as a branching point highly sensitive to the individual reaction
rates. This is illustrated in Fig. 10.1, which shows REACLIB reaction rates [11] for the
photodissociation of 94 Mo through the (γ,n), (γ,p), and (γ,α) channels as a function of
temperature. The uncertainty in the rates was taken to be a factor of 10 for the (γ,α)
reaction and a factor of 5 for the (γ,n) and (γ,p) reactions. Within the p-process window
of 1.8 − 3.3 GK, the 94 Mo(γ,α)90 Zr and 94 Mo(γ,n)93 Mo reactions are within the theoretical
uncertainty of each other. If the actual 94 Mo(γ,α)90 Zr reaction rate is towards the lower
end of its uncertainty band, then the (γ,n) reaction would proceed with a higher rate, and
the branching point in molybdenum would be shifted to more neutron deficient isotopes,
potentially increasing the production of the isotope 92 Mo. Therefore, the 90 Zr(α,γ)94 Mo
was identified as one of the critical reactions to investigate experimentally to improve the
understanding of the p-process mass flow in this mass region [14].
159

1012
p process

3

Rate (cm mol-1 s-1)

7

10

102
10-3
94

Mo(γ ,α)90Zr

94

Mo(γ ,n)93Mo

94

Mo(γ ,p)93Nb

-8

10

10-13
1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

T (GK)
Figure 10.1: REACLIB reaction rates [11] as a function of temperature for the photodissociation of 94 Mo through the (γ,n), (γ,p), and (γ,α) channels. The width of the curves
corresponds to a factor of 10 uncertainty in the (γ,α) reaction rate and a factor of 5 uncertainty in the (γ,n) and (γ,p) rates. Within the p-process window of 1.8 − 3.3 GK, the
94 Mo(γ,α)90 Zr reaction may have a higher rate than the 94 Mo(γ,n)93 Mo reaction and could
therefore be a branching point in the p process [14].
The measurements were performed at the University of Notre Dame and consisted of
impinging α particles onto isotopically-enriched targets. The FN Tandem Van de Graaff
accelerator was used to accelerate the 4 He2+ beam to energies between 9.5 to 12.0 MeV in
0.5 MeV steps. The beam intensity was set between 4.3×109 and 2.5×1010 pps as needed to
maximize the count rate and minimize detection dead time. The dead time was below 1.8%
for all measurements reported here. The 90 Zr and 92 Zr targets were self-supporting foils with
thicknesses of 966(48) and 946(47) µg/cm2 , respectively, and both were isotopically enriched
to 98(1)%. In order to eliminate scattered α particles from hitting the beam pipe after
passing through the Zr foils, a tantalum backing was placed directly behind each foil during
the measurements. In the 74 Ge(α,γ)78 Se case, the 74 Ge was made through the evaporation
of 97.55% enriched 74 Ge powder onto tantalum backing and had a thickness of 340(17)
160

µg/cm2 . Additional details on the fabrication of the 74 Ge target and the RBS thickness
measurements can be found in Sec. 4.2.
The (α,γ) reactions on 90 Zr, 92 Zr, and 74 Ge reported here have Q values of 2064.2, 2758.9,
and 6028.4 keV, respectively. When combined with the beam energy of Eα = 9.5−12.0 MeV,
these Q values produce sum peaks in SuN’s total γ-summed spectrum from approximately
11 to 17 MeV. The sum peaks for these (α,γ) reactions had low intensities, and the statistical
error associated with the sum-peak integral was 6 − 14%. The largest source of uncertainty
in the measurements was the γ-summing efficiency, with uncertainty values of up to ±17%.

10.1

90

Zr(α,γ)94Mo Results

The 90 Zr(α,γ)94Mo reaction was measured in the energy range of Ec.m. = 9.0 − 11.4 MeV.
The total γ-summed spectrum for an energy of Ec.m. = 9.94 MeV is shown in Fig. 10.2
along with the normalized room-background spectrum. The majority of low energy peaks
in the spectrum can be attributed to contaminants in the tantalum backing used during the
measurements. At an energy of approximately 12 MeV, the sum peak of the 90 Zr(α,γ)94 Mo
reaction is indicated. Also indicated is the peak around 9 MeV, which originates from
neutrons released in the 90 Zr(α,n)93Mo reaction.
The cross section values (see Eq. 2.3) and corresponding S-factors (see Eq. 2.13) for the
90 Zr(α,γ)94 Mo

reaction are listed in Table 10.1. The effective energy of each data point

was calculated by taking into account the variation in cross section through the 90 Zr target.
In this calculation, the slope of the cross section was taken from the best-fit theoretical
calculation to the data (see Sec. 10.4). The energy loss through the target was between 0.22
and 0.26 MeV for the measurements here.

161

α+

105

90

Zr

Counts / 15 keV

room background
neutron-induced bkgd

104

sum peak
3

10

102

10
2000

4000

6000

8000 10000 12000 14000 16000

Energy (keV)
Figure 10.2: Total γ-summed spectrum for an α beam impinging onto the enriched 90 Zr
target at Ec.m. = 9.94 MeV, with the 90 Zr(α,γ)94 Mo sum peak at 12 MeV. The peak around
9 MeV in the spectrum comes from neutrons released in the 90 Zr(α,n)93Mo reaction. Also
plotted is the normalized room background.

Table 10.1: Cross sections and S-factors for the 90 Zr(α,γ)94Mo reaction.
Ec.m. (MeV)
8.98 ± 0.03
9.46 ± 0.02
9.94 ± 0.02
10.42 ± 0.02
10.90 ± 0.02
11.39 ± 0.02

σ (mb)
0.060 ± 0.018
0.093 ± 0.024
0.155 ± 0.028
0.171 ± 0.028
0.207 ± 0.034
0.307 ± 0.051

162

S (1015 eV b)
15500 ± 4600
6690 ± 1730
3420 ± 620
1260 ± 210
547 ± 60
306 ± 51

10.2

92

Zr(α,γ)96Mo Results

The 92 Zr(α,γ)96Mo reaction was measured at energies Ec.m. = 10.0 − 11.4 MeV. Fig. 10.3
contains the experimental γ-summed spectrum for the measurement taken at Ec.m. = 11.4
MeV. The sum peak at 14.2 MeV is indicated on the plot, and is at an energy where the
room background and beam-induced background are greatly reduced. In this case, the
beam-induced background was determined by removing the 92 Zr target so that the α beam
impinged soley onto the tantalum backing. The neutron-induced signature around 12 MeV is
also indicated on the plot and comes from neutrons released in the 92 Zr(α,n)95 Mo reaction.
The inset of Fig. 10.3 shows a zoomed-in view of the sum-peak region, along with the linear
background used to calculate the integral of the peak. The sum peaks for the 92 Zr(α,γ)96 Mo
reaction had the lowest intensity of the three measured reactions, and therefore carry the
largest statistical uncertainty of up to 14%.
The energy loss through the 92 Zr target was 0.22 − 0.24 MeV, and the small change in
cross section through the target was taken into account in calculating the final energy value.
To perform this effective energy calculation, the slope of the cross section was taken from
the best-fit theoretical calculation (see Sec. 10.4). The final values for the center of mass
energy, the cross sections, and the astrophysical S-factors are reported in Table 10.2.
Table 10.2: Cross sections and S-factors for the 92 Zr(α,γ)96Mo reaction.
Ec.m. (MeV)
9.96 ± 0.03
10.44 ± 0.03
10.92 ± 0.03
11.40 ± 0.03

σ (mb)
0.034 ± 0.008
0.057 ± 0.012
0.090 ± 0.018
0.116 ± 0.024

163

S (1015 eV b)
732 ± 172
410 ± 86
233 ± 47
116 ± 24

α + 92Zr
Ta backing
room background

5

Counts / 15 keV

10

neutron-induced bkgd

104
300

linear bkgd

103

sum peak

200

102

100
0

10

13000

14000

15000

5000

10000

15000

Energy (keV)
Figure 10.3: Total γ-summed spectrum for an α beam impinging onto the enriched 92 Zr
target at Ec.m. = 11.4 MeV, with the 92 Zr(α,γ)96Mo sum peak at 14.2 MeV. The peak
around 12 MeV originates from neutrons emitted in the 92 Zr(α,n)95Mo reaction. Also plotted
are the normalized room background and the beam-induced background from the tantalum
backing. The inset shows a zoomed in view of the sum peak and the linear background used
when integrating the sum peak.

164

10.3

74

Ge(α,γ)78Se Results

The measurements of the 74 Ge(α,γ)78 Se reaction were carried out for energies Ec.m. =
8.5 − 10.9 MeV. The γ-summed spectrum for the measurement at Ec.m. = 8.5 MeV is shown
in Fig. 10.4. The sum peak at 14.5 MeV is clearly visible above the normalized room background, and the peak around 10.5 MeV comes from the effect of neutrons released in the
74 Ge(α,n)77 Se

reaction. After integrating the sum peak and determining the γ-summing

efficiency, the cross sections and the astrophysical S-factors were calculated. The final values
are reported in Table 10.3 along with the center of mass energy of the measurement. The
energy loss through the target was 0.08 − 0.10 MeV for the measured energies.

106

α+

Counts / 15 keV

105

74Ge

room background
neutron-induced bkgd

104
103

sum peak

102
10
2000

4000

6000

8000 10000 12000 14000 16000

Energy (keV)
Figure 10.4: Total γ-summed spectrum for an α beam impinging onto the enriched 74 Ge
target at Ec.m. = 8.5 MeV, with the 74 Ge(α,γ)78 Se sum peak at 14.5 MeV. The neutroninduced peak around 10.5 MeV comes from neutrons emitted in the 74 Ge(α,n)77 Se reaction.
Also plotted is the normalized room background spectrum.

165

Table 10.3: Cross sections for the 74 Ge(α,γ)78 Se reaction.
Ec.m. (MeV)
8.49 ± 0.01
8.97 ± 0.01
9.44 ± 0.01
9.92 ± 0.01
10.39 ± 0.01
10.87 ± 0.01

10.4

σ (mb)
0.048 ± 0.008
0.075 ± 0.012
0.080 ± 0.014
0.113 ± 0.021
0.124 ± 0.026
0.169 ± 0.039

S (1012 eV b)
1010 ± 170
528 ± 84
210 ± 37
116 ± 22
55 ± 11
33 ± 8

Discussion

The experimental (α,γ) reaction cross sections contained in this chapter were compared
to theoretical calculations using the TALYS 1.6 nuclear reaction software package [123].
Because the states populated in the α capture were at high excitation energy where there
are many levels that overlap, the theoretical cross sections were calculated using the nuclear
statistical model. TALYS 1.6 implements Hauser-Feshbach formalism for this purpose [67],
which relies on the calculation of the transmission coefficients in the entrance channel and
all exit channels. A discussion of the nuclear statistical model is contained in Sec. 3.2.
For the reactions considered here there are three ingredients that dominate the uncertainties in the theoretical cross sections; the nuclear optical model potential (OMP), the
γ-ray strength function (GSF), and the nuclear level density (NLD). TALYS 1.6 has multiple options on its menu to choose from, including two OMPs for protons and neutrons, five
OMPs for α particles, five GSFs for the dominant E1 transitions, and six NLDs. For details
on all the models, the reader is directed to the TALYS 1.6 manual [143]. In the following
paragraphs, only a brief description of the models with references to the original publications
will be provided.
The two proton and neutron optical model potentials in TALYS are the phenomenolog166

ical OMP of Koning and Delaroche [144] constrained by experimental data across a large
energy and mass range (KD OMP), and the semi-microscopic OMP of Jeukenne-LejeuneMahaux [122] reparametrized by Bauge, Delaroche, and Girod [124] (JLM OMP). For the α
optical model potentials, the default option in TALYS is to take the KD OMP for protons
and neutrons and combine them appropriately for the α particle. This concept of applying
OMP for single nucleons to describe interactions of more complex nuclei was first done by
Watanabe for deuterons [145]. TALYS also includes the α potential of McFadden and Satchler [142], as well as the three potentials provided by Demetriou, Grama, and Goriely [146]
which differ in their description of the imaginary part of the OMP. The first α potential
of Demetriou et al. used experimental data to constrain the imaginary part consisting of
only a volume component, the second used a surface and volume component, and the third
potential was determined using the dispersion relation to relate the imaginary part of the
OMP to the real part.
The six nuclear level densities in TALYS are: the constant temperature and Fermi
gas model [125, 126], the back-shifted Fermi gas model [127], the generalized superfluid
model [128, 129], the microscopic level densities from Goriely’s tables calculated with a
Skyrme force [130], the microscopic level densities from Hilaires’s table calculated with a
Skyrme force [131], and the microscopic level density from Hilaire’s table calculated with the
Gogny interaction [147].
In TALYS, the γ-ray strength function for all transition types besides E1 are calculated
with the Brink-Axel Lorentzian [148, 149]. However for the dominant E1 transitions, five different models can be used. These models are: the Kopecky-Uhl generalized Lorentzian [150],
the Brink-Axel Lorentzian [148, 149], the microscopic option calculated from the HartreeFock BCS model [143], the microscopic option calculated from the Hartree-Fock-Bogolyubov
167

model [143], and Goriely’s hybrid model [151, 152].
In total, 300 TALYS calculations were performed for each reaction in an attempt to determine which combination of parameters best describes the data. In addition to the TALYS
calculations, the cross section data was also compared to Hauser-Feshbach calculations from
the NON-SMOKER code obtained through the NON-SMOKER web interface [120, 153].
Since the NON-SMOKER reaction rates are used in the REACLIB database [11], they are
the reaction rates that are often used in astrophysical calculations.

10.4.1

90

Zr(α,γ)94 Mo

A plot of the 90 Zr(α,γ)94Mo reaction cross sections as a function of center of mass energy is
shown in Fig. 10.5. The relevant Gamow window is from 4.2 to 9.6 MeV for p-process temperatures of 1.8 − 3.3 GK [154]. Therefore, the data reaches the higher energies of the astrophysically relevant region and extrapolation is required for the lower energies. In addition to
the data points, the result of four different theoretical calculations are shown. The upper and
lower lines correspond to maximum and minimum of the TALYS calculations in this energy
region, which represent the total theoretical uncertainty of up to a factor of 35. The upper
TALYS limit was calculated using the JLM OMP [122], the first Demetriou α potential [146],
the nuclear level density from Hilaire’s tables calculated with a Skyrme force [131], and the
Brink-Axel Lorentzian γ-ray strength function [148, 149]. On the other hand, the minimum
TALYS limit was calculated using the KD OMP [144], the third Demetriou α potential [146],
the nuclear level density from the constant temperature plus Fermi gas model [125, 126], and
the Kopecky-Uhl generalized Lorentzian γ-ray strength function [150]. The data presented
here significantly reduces the uncertainty in the 90 Zr(α,γ)94Mo reaction cross section.
Also plotted in Fig. 10.5 is the NON-SMOKER theoretical cross section. In the en168

Zr(α,γ )94Mo

Cross Section (mb)

90

1

10-1
SuN
TALYS best
TALYS high
TALYS low
NONSMOKER

10-2

8.5

9

9.5

10

10.5

11

11.5

Ec.m. (MeV)
Figure 10.5: Experimental cross sections for the 90 Zr(α,γ)94Mo reaction compared to TALYS
1.6 and NON-SMOKER calculations. The three TALYS curves correspond to the upper
limit, lower limit, and best-fit calculations (see text for details).
ergy region plotted, the NON-SMOKER calculation overestimates the 90 Zr(α,γ)94 Mo cross
section by a factor of 1.6 − 2.3, with a larger discrepancy at the lower energies. Because
NON-SMOKER calculations are used in the REACLIB database, it can be concluded that
the standard REACLIB reaction rate for the 90 Zr(α,γ)94 Mo reaction is too high as well. In
order to extract a new reaction rate, the TALYS calculation that most accurately described
the data was used. In this case, the best fit was achieved with the JLM OMP [122], the
second Demetriou α potential [146], the microscopic level densities from Hilaire’s tables calculated with the Gogny force [147], and the microscopic γ-ray strength function from the
Hartree-Fock BCS model [143]. The calculated reaction rates are listed in Table 10.4. As
expected, these new rates are lower than the REACLIB rate by approximately a factor of
1.9 − 2.2 for p-process temperatures.
In order to investigate whether the reduction in the 90 Zr(α,γ)94 Mo reaction rate has

169

Table 10.4: Reaction rates for the 90 Zr(α,γ)94Mo reaction.
T
(GK)
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.5
2.0
2.5

Rate
(cm3 mol−1 s−1 )
6.385 × 10−42
2.962 × 10−36
2.955 × 10−32
3.762 × 10−29
1.374 × 10−26
2.189 × 10−24
1.867 × 10−22
8.485 × 10−16
5.908 × 10−12
2.662 × 10−9

T
(GK)
3.0
3.5
4.0
5.0
6.0
7.0
8.0
9.0
10.0

Rate
(cm3 mol−1 s−1 )
2.577 × 10−7
8.368 × 10−6
1.235 × 10−4
5.527 × 10−3
5.880 × 10−2
2.429 × 10−1
6.254 × 10−1
1.359 × 100
2.760 × 100

an impact on the 94 Mo branching point in the p process, it is necessary to compare the
inverse 94 Mo(γ,α)90 Zr reaction rate to the 94 Mo(γ,n)93 Mo rate. The reaction rates for both
reactions were taken from the REACLIB database [11] and the 94 Mo(γ,α)90 Zr rate reduced
by a factor of 2 as indicated by the experimental results here. Even with this reduction,
the rate of photodissociating 94 Mo through the (γ,α) channel is still larger than the (γ,n)
channel for temperatures below 2.5 GK. Therefore, 94 Mo appears to be the branching point
in the molybdenum isotopic chain. However, this could change if a future measurement of the
93 Mo(n,γ)94 Mo

reaction indicates that the rate is much higher than the current theoretical

predictions.

10.4.2

92

Zr(α,γ)96 Mo

The cross sections of the 92 Zr(α,γ)96Mo reaction are plotted in Fig. 10.6. The Gamow
window is from 4.2 to 9.6 MeV for p-process temperatures [154], so the data reaches to
just above the astrophysically relevant region. Also plotted in Fig. 10.6 are the results

170

Zr(α,γ )96Mo

Cross Section (mb)

92

10-1

10-2
SuN
TALYS best
TALYS high
TALYS low
NONSMOKER

10-3
9

9.5

10

10.5

11

11.5

12

Ec.m. (MeV)
Figure 10.6: Experimental cross sections for the 92 Zr(α,γ)96Mo reaction compared to TALYS
1.6 and NON-SMOKER calculations. The three TALYS curves correspond to the upper
limit, lower limit, and best-fit calculations (see text for details).
from theoretical calculations with the TALYS and NON-SMOKER codes. In the energy
region plotted, NON-SMOKER describes the energy dependence of the cross section well,
but overestimates the values by a factor of 1.5 − 1.7.
The three TALYS calculations that are plotted represent the upper limit, lower limit,
and best description of the experimental results. The large spread between the maximum
and minimum TALYS calculations show the total theoretical uncertainty of a factor of 6 at
the highest energies and a factor of 29 at the lowest energies plotted. To achieve the upper
limit, TALYS calculations were performed with the JLM OMP [122], the first Demetriou α
potential [146], the nuclear level density from Hilaire’s tables calculated with a Skyrme
force [131], and the Brink-Axel Lorentzian γ-ray strength function [148, 149]. On the
other hand, the lower TALYS limit was calculated using the KD OMP [144], the second
Demetriou α potential [146], the nuclear level density from the constant temperature plus

171

Fermi gas model [125, 126], and the Kopecky-Uhl generalized Lorentzian γ-ray strength function [150]. The experimental data greatly reduces the uncertainty in the cross section of the
92 Zr(α,γ)96 Mo

reaction to approximately 20%. Lastly, the TALYS calculation which most

accurately describes the data was performed with the JLM OMP [122], the first Demetriou
α potential [146], the generalized superfluid level density model [128, 129], and Goriely’s
hybrid γ-ray strength function [151].

10.4.3

74

Ge(α,γ)78 Se

A plot of the 74 Ge(α,γ)78 Se cross section results compared to the TALYS and NON-SMOKER
theoretical calculations is shown in Fig. 10.7. As in the case of the 90,92Zr(α,γ)94,96Mo reactions, the experimental data is lower than the NON-SMOKER calculations. Here, NONSMOKER is a factor of 1.6 − 2.3 larger. The Gamow window for the 74 Ge(α,γ)78 Se reaction
is 3.5 − 8.4 MeV [154] for p-process temperatures, so the data reaches to just above the
astrophysically relevant region.
The upper TALYS limit corresponds to calculations with the JLM OMP [122], the default
α potential [143], the nuclear level density from Hilaire’s tables calculated with a Skyrme
force [131], and the Brink-Axel Lorentzian γ-ray strength function [148, 149]. The lower
TALYS limit corresponds to calculations with the KD OMP [144], the second Demetriou
α potential [146], the nuclear level density from Hilaire’s tables calculated with the Gogny
force [147], and the Kopecky-Uhl generalized Lorentzian γ-ray strength function [150]. The
total theoretical uncertainty between the upper and lower bounds is a factor of 11 at the
higher energies and a factor of 21 at the lower energies plotted. The experimental uncertainty
of 16 − 23% greatly constrains the cross section values in this energy region.
The experimental data was compared to the 300 TALYS calculations to determine what
172

Ge(α,γ )78Se

74

Cross Section (mb)

1

10-1
SuN
TALYS best
TALYS high
TALYS low
NONSMOKER

10-2
8

8.5

9

9.5

10

10.5

11

Ec.m. (MeV)
Figure 10.7: Experimental cross sections for the 74 Ge(α,γ)78 Se reaction compared to TALYS
1.6 and NON-SMOKER calculations. The three TALYS curves correspond to the upper
limit, lower limit, and best-fit calculations (see text for details).
combination of input parameters provided the best match. The best fit to the data was
achieved with TALYS calculations performed with the KD OMP [144], the default α potential [143], the nuclear level densities from Goriely’s tables [130], and the microscopic γ-ray
strength function calculated from the Hartree-Fock-Bogolyubov model [143].

10.5

Conclusions

In an effort to expand the very limited quantity of existing experimental data for (α,γ) reactions relevant in the astrophysical p process, the first ever measurements of the 90 Zr(α,γ)94 Mo,
92 Zr(α,γ)96 Mo,

and 74 Ge(α,γ)78 Se reaction cross sections were performed by implementing

the γ-summing technique with the SuN detector at the University of Notre Dame. The
measurements greatly reduce the uncertainty of the cross section values between energies of
Eα = 9.5 − 12.0 MeV from a theoretical uncertainty of larger than a factor of 10 to an exper173

imental uncertainty of approximately 20%. The nuclear reaction code TALYS was used to
calculate theoretical cross sections to determine which input parameters provided the most
accurate description of the data. The results of the 90 Zr(α,γ)94 Mo reaction seem to confirm
that 94 Mo is a branching point in the p process.

174

Chapter 11
Measurements in Inverse Kinematics
In this chapter, an alternative method to measure (p,γ) and (α,γ) reactions in inverse kinematics by utilizing the γ-summing technique is presented. The aim of this new experimental
technique is to measure reaction cross sections at or near astrophysically relevant energies
involving nuclei that cannot be studied in regular kinematics. The data obtained with this
method, especially for reactions on radioactive nuclei, can help advance the field of nuclear astrophysics by expanding the nuclear physics input used in astrophysical models, particularly
models of the p process. The feasibility of the γ-summing technique in inverse kinematics
was demonstrated through the measurement of well-known resonances in the p(27 Al,γ)28 Si
and p(58 Ni,γ)59 Cu reactions.
The experiments in inverse kinematics were performed at the University of Notre Dame
using beams of 27 Al5+ and 58 Ni10+ with intensities of 2 − 5 × 1010 pps. The beam energies were changed in small energy steps in order to scan well-known resonances in the
p(27 Al,γ)28 Si and p(58 Ni,γ)59 Cu reactions. The target used during the measurements was
a titanium hydride foil produced at Argonne National Laboratory using the method of
Ref. [155]. The characterization of the target using RBS and ERD methods is described
in detail in Sec. 4.2.2.4. The total measured thickness of the titanium hydride foil was
232(14) µg/cm2 with 3.58(36) × 1018 cm−2 hydrogen nuclei.

175

Using the SuN detector, the yield of each experimental run was determined as

Y =

NΣ
Nb εΣ

(11.1)

where NΣ is the number of counts in the sum peak, Nb is the number of beam particles,
and εΣ is the γ-summing efficiency. Because the measurements were performed in inverse
kinematics, the recoil velocity caused the γ rays to be emitted in-flight and led to an energy
broadening of the sum peak. Doppler corrections were applied using the segmentation of
SuN and Eq. 4.4. After applying the Doppler corrections, the standard analysis procedures
of this thesis were followed. Specifically, a Gaussian fitting function was used to define a
sum-peak region of 3σ below to 3σ above the sum peak centroid, and then NΣ was taken as
the number of counts above a linear background in the sum-peak region and εΣ was taken
based on the hit pattern analysis of the sum peak region.
A resonance yield curve was created for each reaction by changing the beam energies in
small increments and determining the yield at each energy. By determining the integral under
the resonance yield curve, the resonance strength can be determined from (see Sec. 3.1.2)
λ2r
AY = nH ωγ .
2

(11.2)

Here AY is the integral under the resonance yield curve, nH is the number of hydrogen nuclei
in the target per unit area, λr is the center-of-mass de Broglie wavelength of the resonance,
and ωγ is the resonance strength.
The first reaction measured was the p(27 Al,γ)28 Si reaction (Q = 11585 keV) at the
resonance energy of Ec.m. = 956 keV. A typical example of the Doppler corrected γ-summed

176

Counts

104

100

103

50

0

102

before Doppler correction
GEANT4 simulation

8000

10000

12000

14000

10
p(27Al,γ ) Si
28

room background

1
2000

4000

6000

8000 10000 12000 14000

Energy (keV)
Figure 11.1: Doppler corrected γ-summed spectra for the p(27 Al,γ)28 Si reaction plotted with
the normalized room background. The inset shows a zoomed in view of the sum peak at
EΣ = 12541 keV. Additionally, the improvement due to applying the Doppler correction is
demonstrated and the excellent agreement with GEANT4 simulations is shown.
spectrum is shown in Fig. 11.1. The sum peak at EΣ = 12541 keV is clearly visible above the
normalized room background. Additional peaks in the spectrum at 1461 keV and 2614 keV
come from the room background decay of 40 K and the decay series of 232 Th, respectively.
The inset of the figure is a comparison of the sum peak with and without Doppler corrections
with the room background subtracted. By correcting for the velocity of the 28 Si recoil, the
resolution of the sum peak is significantly improved. The inset of the figure also contains the
result of GEANT4 simulations based on the known decay scheme of the resonance. Excellent
agreement between the experimental data and simulation is obtained. For this p(27 Al,γ)28 Si
resonance, the summing efficiency was determined to be approximately 31(2)%.
The resonance yield curve from the p(27 Al,γ)28 Si reaction is displayed in Fig. 11.2. In
this plot, the experimental data is compared to estimated contributions from known reso-

177

0.5

×10-9

Yield

0.4

0.3

experimental data
Total
Ec.m. = 956 keV

0.2

Ec.m. = 988 keV
Ec.m. = 1078 keV
other contributing resonances

0.1

0

940

960

980

1000

1020

1040

1060

1080

Ec.m. (keV)
Figure 11.2: Yield curve for the p(27 Al,γ)28 Si reaction with the experimental data shown in
solid circles. The spectrum is dominated by the Ec.m. = 956 keV resonance, with other large
contributions from the Ec.m. = 988 keV and Ec.m. = 1078 keV resonances. In this energy
region there are additional contributions from the Ec.m. = 856, 890, 904, 966, 1051, and 1058
keV resonances [15]. The solid line shows the total yield from all resonances.
nances [15]. Due to the thickness of the target, many resonances can contribute at a given
beam energy. The yield in this energy region is dominated by the Ec.m. = 956 keV resonance that was completely mapped out in this experiment. The integral underneath this
resonance was determined by fitting the height and width on the resonance yield curve. The
uncertainty in the integral of approximately 5% was calculated from the uncertainty in the
fit. Using Eq. 11.2, the resulting resonance strength was determined to be ωγ = 2.05(23) eV.
A comparison between the present work and values in literature is shown in Table 11.1.
The Ec.m. = 956 keV resonance of the p(27 Al,γ)28 Si reaction has been measured many times
previously. Compilations and evaluations of earlier work by Endt [15] and Angulo et al. [5]
provide adopted values of ωγ = 2.00(17) eV and 1.90(10) eV, respectively. In addition,

178

Table 11.1: Resonance strength measurements
Reaction

Ec.m.
(keV)

Present work
ωγ (eV)

27 Al(p,γ)28 Si

956

2.05 ± 0.23

58 Ni(p,γ)59 Cu

1400

0.58 ± 0.07

a

Other values
ωγ (eV)

Ref.

1.93
1.90
1.94
2.00

[15]a
[5]
[156]
[157]

±
±
±
±

0.16
0.10
0.07
0.15

0.61
[16]a
0.62 ± 0.10 [158]a
0.687 ± 0.096 [90]

converted to center of mass frame for comparison

the resonance was measured more recently by Chronidou et al. [156] and Harissopulos et
al. [157] using 4π NaI detectors in regular kinematics finding values of ωγ = 1.94(7) eV and
2.00(15) eV, respectively. The present value of ωγ = 2.05(23) eV agrees well with all values.
The second measurement was of the Ec.m. = 1400 keV resonance in the p(58 Ni,γ)59 Cu
reaction. A Doppler corrected γ-summed spectrum is shown in Fig. 11.3. This reaction
has Q = 3419keV which leads to the sum peak found at EΣ = 4819 keV. A zoomed in
view of the sum peak is shown in the inset of the figure. The inset also demonstrates the
improvement in sum-peak resolution after the Doppler corrections were applied, and shows
the excellent agreement between experimental data and GEANT4 simulations. A summing
efficiency of approximately 44(3)% was calculated for each energy and the yield determined
with Eq. 11.1. The resulting experimental resonance yield curve is shown in Fig. 11.4 along
with estimated contributions from known resonances in this energy region [16]. By fitting
the height and width of the Ec.m. = 1400 keV resonance in the yield plot, the integral
underneath the resonance was determined and the resonance strength was calculated using
Eq. 11.2 to be ωγ = 0.58(7) eV.

179

105

100

before Doppler correction
GEANT4 simulation

104

Counts

50

103
0

4000

5000

6000

2

10

p( Ni,γ ) Cu
room background
58

10

2000

59

4000

6000

8000

Energy (keV)
Figure 11.3: Doppler corrected γ-summed spectra for the p(58 Ni,γ)59 Cu reaction with a
comparison to the normalized room background. The inset shows a zoomed in view of the
sum peak at EΣ = 4819 keV as well as the summed spectrum without Doppler corrections.
An excellent agreement with GEANT4 simulation is also observed.
×10-12
80
70

Yield

60
experimental data

50
40

Total
Ec.m. = 1352 keV
Ec.m. = 1400 keV

30

other contributing resonances

20
1380 1400 1420 1440 1460 1480 1500 1520 1540

Ec.m. (keV)
Figure 11.4: Yield curve for the p(58 Ni,γ)59 Cu reaction. The spectrum is dominated by the
Ec.m. = 1400 keV resonance, with an additional large contribution from the Ec.m. = 1352
keV resonance. In this energy region there are additional contributions from the Ec.m. = 1496
and 1514 keV resonances [16]. The solid line shows the total yield from all resonances.
180

The resonance strength of the Ec.m. = 1400 keV resonance of the p(58 Ni,γ)59 Cu reaction
was published three times previously and the results are shown in Table 11.1. Carver and
Jones [16] used the work of Butler and Gossett [159] to deduce ωγ = 0.62 eV, although no
uncertainty was given. The measurement by Tingwell et al. [158] reported a nearly identical
value of ωγ = 0.63(10) eV. More recently Simon et al. [90] measured the resonance in inverse
kinematics with the DRAGON recoil separator and determined ωγ = 0.687(96) eV. Overall,
there is excellent agreement between the measurements of this thesis and the previous values
since the present results are consistent with all literature values within the experimental
uncertainty.
In summary, the first ever measurements utilizing the γ-summing technique in inverse
kinematics were performed using the SuN detector at the University of Notre Dame. As a
proof of principle, well-known resonances in the p(27 Al,γ)28 Si and p(58 Ni,γ)59 Cu reactions
were measured with 27 Al and 58 Ni beams impinging onto a titanium hydride target. The
deduced resonance strengths are in excellent agreement with previous values, illustrating
the feasibility of the technique. The new method expands the experimental possibilities
for measuring (p,γ) and (α,γ) reactions to additional nuclei, such as radioactive isotopes
and stable isotopes that exist in a liquid or gas state. Future measurements with the γsumming technique in inverse kinematics will contribute to an improved understanding of
stellar nucleosynthesis.

181

Chapter 12
Summary and Outlook
Throughout history, humankind has searched for an explanation of the origin the world and
of ourselves. Today we are closer to a full understanding than ever before. We know that
our complex world is made up of less than 100 elements whose abundances in the universe
are evolving due to various nucleosynthesis processes. Hydrogen and the majority of helium
were produced immediately after the Big Bang, and the heavier elements are continually
synthesized in various stellar environments. The bulk of isotopes beyond iron are synthesized
via two neutron capture processes, although there is a group of 35 stable neutron-deficient
p nuclei which must be produced by an alternative “p-process” nucleosynthesis scenario.
Overall, the synthesis of the p nuclei remains a major open question in the field of nuclear
astrophysics, with current state of the art p-process models failing to reproduce the observed
solar abundance pattern.
The most studied scenario of synthesizing the p nuclei is from existing seed nuclei in the
O/Ne layers of a type II supernovae, but this environment suffers from large uncertainties
in the nuclear physics (in addition to the astrophysical uncertainties). A severe lack of
experimental data means that models of this p-process scenario rely heavily on theoretical
reaction rates and their corresponding large uncertainties. To reduce these nuclear physics
uncertainties it is crucial to expand and develop new experimental techniques to measure the
relevant reactions. It is especially important to develop experimental capabilities in inverse
kinematics which allow for the investigation of reactions involving radioactive nuclei which
182

currently have the largest uncertainties of all.
To this end, this dissertation focused on the development of a new experimental program
at the National Superconducting Cyclotron Laboratory (NSCL) to measure proton and α
radiative capture reactions which are relevant for the astrophysical p process. To detect the
radiation released in these reactions, a new γ-ray detector was designed which combines a
large active volume and a nearly 4π angular coverage for the ideal implementation of the
γ-summing technique. The Summing NaI (SuN) detector also was designed to include eight
segments to allow for the Doppler reconstruction of γ-ray energy when performing measurements in inverse kinematics. After SuN’s arrival to the NSCL, a digital data acquisition
system was installed and testing and optimization was performed using standard radioactive
γ-ray sources.
All (p,γ) and (α,γ) reaction measurements in this dissertation were performed at the
University of Notre Dame using the FN Tandem Van de Graaff accelerator. Two of the
targets had to be produced through evaporation, and all of the targets had their thicknesses
measured with the RBS technique. The commissioning of the SuN detector was performed
by measuring 11 known resonances in the 27 Al(p,γ)28 Si reaction. The measured resonance
strengths were in good agreement with previously published values, demonstrating the validity of the experimental techniques and the analysis procedures. A new method of extracting
the γ-summing efficiency was developed, which makes use of the segmentation of the SuN
detector and requires comparison to a well-tested GEANT4 simulation.
Four reactions were measured to investigate the region of the light p nuclei, namely
the 74 Ge(p,γ)75 As, 90 Zr(α,γ)94 Mo, 92 Zr(α,γ)96Mo, and 74 Ge(α,γ)78 Se reactions. The measured cross sections were compared to theoretical calculations using the TALYS and NONSMOKER nuclear reaction codes, and the TALYS parameters that most accurately described
183

the data were determined. New reaction rates were extracted for the 74 Ge(p,γ)75 As reaction
which were approximately 20% higher than the standard REACLIB rates for p-process temperatures. The increase in reaction rates was found to enhance the production of 74 Se in the
p process by 17% when performing calculations with a reaction network of a type II supernova. Additionally, the nuclear physics uncertainty in the production of 74 Se was reduced
from 63% to 5% with the new measurements here. In the case of the 90 Zr(α,γ)94Mo reaction,
the new reaction rates were lower than the standard REACLIB rates by approximately a
factor of 2 for p-process temperatures. Even with the reduction in reaction rate, the inverse
94 Mo(γ,α)90 Zr

reaction was found to have a higher rate than the 94 Mo(γ,n)93 Mo reaction

for temperatures below 2.5 GK, which tentatively confirms that 94 Mo is the branching point
in the p process for the molybdenum isotopic chain. The results of the 92 Zr(α,γ)96Mo and
74 Ge(α,γ)78 Se

reactions greatly reduces the uncertainty in the reaction cross sections from

larger than a factor of 10 to an uncertainty of less than 20%. These measurements expand
the experimental database for (α,γ) reactions relevant for the p process, and can be used to
place contsraints on the nuclear reaction theory.
The 58 Ni(α,γ)62 Zn reaction was measured for its role in the nucleosynthesis in the innermost layers of type Ia supernovae, in which α-rich freeze-out from nuclear statistical
equilibrium is expected to occur. The measurements were approximately a factor of 2 below the default calculations of the SMARAGD code, but good agreement to the data was
achieved by multiplying the α width by a factor of 0.45. The new reaction rates were found
to have up to a 5% effect on the ejected abundances of several isotopes, including 64 Zn and
62 Ni.

The last measurements of this dissertation, the p(27 Al,γ)28 Si and p(58 Ni,γ)59 Cu reactions, were performed in inverse kinematics with 27 Al and 58 Ni beams impinging onto a
184

titanium hydride target. Well-known resonances in both reactions were measured, and the
deduced resonance strengths were found to be in excellent agreement with previous literature values. The success of these measurements served as a proof of principle of using the
γ-summing technique in inverse kinematics. This opens the door to studying (p,γ) and (α,γ)
reactions on radioactive nuclei and other nuclei which are difficult to make targets out of.
Looking ahead, this dissertation is just the beginning of what should be a bright future
of measuring (p,γ) and (α,γ) reactions with the SuN detector and the γ-summing technique.
In the time since the initial measurements were performed with the SuN detector at the
University of Notre Dame, the reaccelerator facility (ReA3) at the NSCL has been completed
and commissioned. The ReA3 facility is just starting to provide radioactive beams with the
intensities necessary to successfully perform cross section measurements. Combining the
SuN detector with ReA3 beams will allow for the measurement of (p,γ) and (α,γ) reactions
on radioactive nuclei at astrophysical energies which were previously not possible. These
future experiments will be the pioneering experiments involving radioactive nuclei in the p
process, and it will be fascinating to see what the findings will be and the impact that these
findings will have on our understanding of the nucleosynthesis of the p nuclei.
Before performing experiments with the SuN detector and ReA3, however, the current
experimental setup should be updated to increase the measurement sensitivity. The increase
in sensitivity is necessary due to the fact that the best-case radioactive beam intensities
provided by ReA3 facility are expected to be at least 3 orders of magnitude below the
beam intensities utilized at the University of Notre Dame. The lower beam intensities
will contribute to a significant decrease in the experimental count rate. To counteract this
reduction, it is possible to increase the number of target nuclei by moving towards a gas
cell. For example, a hydrogen gas cell with a pressure of 1 atm would have proton thickness
185

larger than 10 times that of the titanium hydride foils used in this dissertation.
In addition, most of the background in the total γ-summed spectrum in the region of the
sum peak comes from cosmic rays depositing their energy in the SuN detector. An increase
in the sum-peak signal can be achieved if this cosmic-ray background is somehow reduced.
One solution is to develop an active volume to surround the SuN detector and detect cosmic
rays so that their contribution is removed from SuN’s spectra. Therefore, SuN’s Scintillating
Cosmic Ray Eliminating Ensemble (SuNSCREEN) was recently developed in collaboration
with Hope College. SuNSCREEN consists of 9 plastic scintillator bars arranged in a roof-like
configuration above SuN. It was shown that a reduction of the cosmic-ray background by up
to a factor of 2 in SuN’s total γ-summed spectrum was achieved when using SuNSCREEN.
An additional way to reduce the cosmic-ray background is to remove events in the data
when the beam is not impinging onto the target. The ReA3 facility has the capability to
provide bunched radioactive beams so that there are regular time intervals of no particles
impinging on the target followed by a short burst of beam. By selecting only the small time
intervals when the (p,γ) and (α,γ) reactions of interest might be occurring, the cosmic-ray
background can be significantly reduced. It is recommended that the initial measurements
with the SuN detector and ReA3 should start with the most sensible candidates to check the
sensitivity of the experimental setup before moving onto even challenging measurements.
The best candidates are reactions where the theoretical models predict a relatively large
reaction cross section, as well as reactions with large Q-values that produce sum peaks at
high energies where the background is reduced.

186

APPENDICES

187

Appendix A
Energy Filter Algorithm
This appendix contains the algorithm that NSCL DDAS implements to extract the energy of
a pulse. This algorithm was introduced in Ref. [8] and it takes into account the exponential
decay of the signal. Its implementation was not part of this thesis, but it may be useful for
future experimenters who want to understand how the energy spectrum of the SuN detector
is obtained. Also, Ref. [8] does not include the value of all the coefficients needed.
First of all, Fig. A.1 shows a typical digitized pulse from the SuN detector. Both the
signal and its baseline are drawn. As discussed in Sec. 5.3.2, NSCL DDAS implements digital
trapezoidal filter algorithms, which make use of a user-defined “peaking” time and “gap”
time. In the figure, SuN’s energy filter peaking time of 600ns is equal to L0 and L1 , and
SuN’s energy filter gap time of 200ns is equal to LG . For each digitized point, the following
sum is first calculated:

Filter Sum = a0 (S0 − B0 ) + aG (SG − BG ) + a1 (S1 − B1 )

(A.1)

where Si is the integral of the signal in region i and Bi is the integral of the baseline in
region i. This sum is straightforward to calculate with the only complication coming from
the value of the coefficients. The coefficients depend on the decay constant τ of the signal

188

Signal Amplitude (ADC units)

900
800

Signal (S)

700

Baseline (B)

600
500
400
300

L0

200
1000

LG

L1

2000

3000

4000

5000

Time (ns)

Figure A.1: A digitized signal from the SuN detector plotted along with its baseline. The
lengths of the peaking time (L0 and L1 ) and gap time (LG ) are also drawn.
and the length Li of each region. Their values are

a0 =

exp (−2L0 /τ )
exp (−2L0 /τ ) − 1

aG = 1

a1 =

1
.
exp (−2L1 /τ ) − 1

(A.2)

It should be mentioned that these coefficients are for an NSCL DDAS energy filter range of
1, and the value of the exponentials change slightly for energy filter ranges larger than 1.
After calculating the sum, the value is multiplied by a factor that depends on the decay
constant τ . This factor is equal to

1
.
1 − exp (−1/τ )

(A.3)

A plot of the tau corrected energy filter using algorithm described here is plotted in Fig. A.2
compared to the uncorrected energy filter. It is shown that by using the coefficients in
Eq. A.2, the energy filter successfully accounts for the exponential decay of the signal and

189

Energy Filter Amplitude

200

100

0

-100

Energy Filter
Tau Corrected Filter

-200
1000

2000

3000

4000

5000

Time (ns)

Figure A.2: The response of SuN’s tau-corrected energy filter compared to the uncorrected
energy filter. By correcting for the exponential decay of the signal, the tau-corrected energy
filter does not take on any negative values.
therefore does not take on negative values. The only additional step that NSCL DDAS takes
is to multiply the result by a normalization factor to ensure that the extracted energy falls
within the desired channel range of 0 to approximately 32000.

190

Appendix B
GEANT4 Detector Construction
Below is the code used to build the SuN detector in GEANT4:
/////////////////////////////////////////////////////////////////////////////////////
// Author: Steve Quinn
//
//
//
// Description: This is the file where you build the detector. For now it consits //
//
of the experimental setup for the SuN detector, but it can be modified by
//
//
following the syntax to build and place new materials. There is also many
//
//
useful examples online.
//
//
//
// Steps: 1. Define the elements that you need
//
//
2. Use these elements to define the materials for the experimental setup //
//
3. Create an experimental room of air, vacuum, etc.
//
//
4. Create your experimental setup and place it in the experimental room //
//
5. Apply the color scheme you want for the optional visualization
//
//
//
// Important: The way the code is currently set up, you are required to fill the
//
//
array called detectorName[i] with the name of the detectors you want to save //
//
in your ROOT file. In this example the names are "T1", "B1", etc.
//
//
(see syntax below).
//
/////////////////////////////////////////////////////////////////////////////////////
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include

"DetectorConstruction.hh"
"G4SDManager.hh"
"G4Element.hh"
"G4Material.hh"
"G4Box.hh"
"G4Tubs.hh"
"G4Cons.hh"
"G4Trd.hh"
"G4LogicalVolume.hh"
"G4ThreeVector.hh"
"G4PVPlacement.hh"
"G4SubtractionSolid.hh"
"G4UnitsTable.hh"
"globals.hh"
"G4VisAttributes.hh"
"G4Colour.hh"
<iostream>
<sstream>
"G4String.hh"
"G4ios.hh"
<stdio.h>

191

DetectorConstruction::DetectorConstruction()
: NaI(0), Al(0), N78O21Ar1(0), Cr20Ni8Fe76(0), C2F4(0), C5O2H8(0)
{}
DetectorConstruction::~DetectorConstruction()
{}
G4VPhysicalVolume* DetectorConstruction::Construct()
{ DefineMaterials();
return ConstructDetector();
}
void DetectorConstruction::DefineMaterials()
{
// define Parameters
G4String name, symbol;
G4double a, z, density;
G4int ncomponents, natoms;
// define Elements
a = 22.99*g/mole;
G4Element* Na = new G4Element(name="Sodium" ,symbol="Na" , z= 11., a);
a = 126.90*g/mole;
G4Element* I = new G4Element(name="Iodine" ,symbol="I" , z= 53., a);
a = 204.38*g/mole;
G4Element* Tl = new G4Element(name="Thalium" ,symbol="Tl" , z= 81., a);
a = 26.982*g/mole;
G4Element* elAl = new G4Element(name="element_Aluminum",symbol="elAl",
z= 13., a);
a = 14.00*g/mole;
G4Element* N = new G4Element(name="Nitrogen",symbol="N" , z= 7., a);
a = 16.00*g/mole;
G4Element* O = new G4Element(name="Oxygen",symbol="O" , z= 8., a);
a = 39.95*g/mole;
G4Element* Ar = new G4Element(name="Argon",symbol="Ar" , z= 18., a);
a = 51.996*g/mole;
G4Element* Cr = new G4Element(name="Chromium" ,symbol="Cr" , z= 24., a);
a = 58.69*g/mole;
G4Element* Ni = new G4Element(name="Nickel" ,symbol="Ni" , z= 28., a);
a = 55.847*g/mole;
G4Element* Fe = new G4Element(name="Iron" ,symbol="Fe" , z= 26., a);
a = 12.011*g/mole;
G4Element* C = new G4Element(name="Carbon" ,symbol="C" , z= 6., a);

192

a = 18.998*g/mole;
G4Element* F = new G4Element(name="Fluorine" ,symbol="F" , z= 9., a);
a = 1.008*g/mole;
G4Element* H = new G4Element(name="Hydrogen" ,symbol="H" , z= 1., a);
// define Materials
//..........Stainless Steel..........
density = 8.0*g/cm3;
Cr20Ni8Fe76 = new G4Material(name="Stainless_Steel", density, ncomponents=3);
Cr20Ni8Fe76->AddElement(Cr, natoms=20);
Cr20Ni8Fe76->AddElement(Fe, natoms=76);
Cr20Ni8Fe76->AddElement(Ni, natoms=8);
//..........Polytetrafluorine (PTFE)............
density = 2.20*g/cm3;
C2F4 = new G4Material(name="PTFE", density, ncomponents=2);
C2F4->AddElement(C, natoms=2);
C2F4->AddElement(F, natoms=4);
//..........NaI......................
density = 3.67*g/cm3;
NaI = new G4Material(name="Sodium Iodine", density, ncomponents=3);
NaI->AddElement(Na, natoms=1000);
NaI->AddElement(I, natoms=1000);
NaI->AddElement(Tl, natoms=1);
//..........Al.......................
density = 2.698*g/cm3;
Al = new G4Material (name="Aluminum", density, ncomponents=1);
Al->AddElement(elAl, natoms=1);
//..........Air.....................
density = 1.2927*mg/cm3;
N78O21Ar1 = new G4Material (name="Air", density, ncomponents=3);
N78O21Ar1->AddElement(N, natoms=78);
N78O21Ar1->AddElement(O, natoms=21);
N78O21Ar1->AddElement(Ar, natoms=1);
//..........Acrylic.....................
density = 1.18*g/cm3;
C5O2H8 = new G4Material (name="Acrylic", density, ncomponents=3);
C5O2H8->AddElement(C, natoms=5);
C5O2H8->AddElement(O, natoms=2);
C5O2H8->AddElement(H, natoms=8);
// Print
G4cout
G4cout
G4cout
G4cout
G4cout

out Elements and Materials
<< "\n\n ####----------------------------------------------------#### \n";
<< "\n\n\n\n\t\t #### List of elements used #### \n";
<< *(G4Element::GetElementTable());
<< "\n\n\n\n\t\t #### List of materials used #### \n";
<< *(G4Material::GetMaterialTable());

}

193

G4VPhysicalVolume* DetectorConstruction::ConstructDetector()
{
// EXPERIMENTAL ROOM
G4Tubs* room_tube = new G4Tubs("room",0.0*cm,100.0*cm,300.0*cm,0.0*deg,360.0*deg);
G4LogicalVolume* room_log = new G4LogicalVolume(room_tube,N78O21Ar1,"room",0,0,0);
G4VPhysicalVolume* room_phys = new G4PVPlacement(0,G4ThreeVector(0.0*cm,0.0*cm,
0.0*cm),"room",room_log,NULL,false,0);
// BEAM PIPE
G4double outerR_beam = 19.0*mm;
G4double innerR_beam = outerR_beam - 1.5*mm;
G4double halflength_beam = 720.0*mm;
G4double startAngle_beam = 0.*deg;
G4double spanAngle_beam = 360.*deg;

//radius of the beam pipe
//thickness of the beam pipe
//length of the beam pipe

G4Tubs* beam_tube = new G4Tubs("beam_tube",innerR_beam,outerR_beam,halflength_beam,
startAngle_beam, spanAngle_beam);
G4LogicalVolume* beam_log = new G4LogicalVolume(beam_tube,Cr20Ni8Fe76,"beam_log",
0,0,0);
G4VPhysicalVolume* beam_phys = new G4PVPlacement(0,
G4ThreeVector(0.0*mm,0.0*mm,100.0*mm),beam_log,"beam_phys",room_log,false,0);
// DIMENSIONS OF SuN
**(changing these will scale the whole simulation)**
G4double innerR_scint = 22.5*mm;
// 45mm borehole
G4double outerR_scint = 203.0*mm;
// total of 406mm in diameter
G4double length_scint = 101.5*mm;
// total of 406mm in length
G4double width_Refl = 0.25*mm;
// width of reflector
G4double width_Al_vert = 0.50*mm;
// width of Al between each segment
G4double width_Al_horiz = 0.75*mm;
// width of Al between top and bot
G4double innerR_Al = 21.5*mm;
// 43mm in center
G4double outerR_Al = 222.5*mm;
// thick outer casing
// NaI SCINTILLATOR
G4double halflength_scint = 0.5*length_scint;
G4double startAngle_scint = 0.0*deg;
G4double spanAngle_scint = 180.0*deg;
G4Tubs* scint_tube = new G4Tubs("scint_tube",innerR_scint,outerR_scint,
halflength_scint,startAngle_scint,spanAngle_scint);
G4LogicalVolume* scint_log = new G4LogicalVolume(scint_tube,NaI,"scint_log",0,0,0);
// REFLECTOR
G4double innerR_Refl = innerR_scint - width_Refl;
G4double outerR_Refl = outerR_scint + 2.0*width_Refl;
G4double length_Refl = length_scint + 2.0*width_Refl;
G4double halflength_Refl = 0.5*length_Refl;
G4Tubs* refl_tube = new G4Tubs("refl_tube",innerR_Refl,outerR_Refl,
halflength_Refl,startAngle_scint,spanAngle_scint);
G4SubtractionSolid* refl_sub = new G4SubtractionSolid("refl_sub",refl_tube,
scint_tube,0,G4ThreeVector(0.0*mm,width_Refl,0.0*mm));
G4LogicalVolume* refl_log = new G4LogicalVolume(refl_sub,C2F4,"refl_log",0,0,0);

194

// ALUMINUM
G4double length_Al = length_Refl + width_Al_vert;
G4double halflength_Al = 0.5*length_Al;
G4Tubs* al_tube = new G4Tubs("al_tube",innerR_Al,outerR_Al,halflength_Al,
startAngle_scint,spanAngle_scint);
G4SubtractionSolid* al_sub = new G4SubtractionSolid("al_sub",al_tube,refl_tube,0,
G4ThreeVector(0.0*mm,width_Al_horiz,0.0*mm));
G4LogicalVolume* al_log = new G4LogicalVolume(al_sub,Al,"al_log",0,0,0);

G4double length_Al_side = 13.0*mm;
G4double halflength_Al_side = 0.5*length_Al_side;
G4Tubs* al_tube_side = new G4Tubs("al_tube_side",innerR_Al,outerR_Al,
halflength_Al_side,0.0*deg,360.0*deg);
G4LogicalVolume* al_log_side = new G4LogicalVolume(al_tube_side,Al,
"al_log_side",0,0,0);
// ====================== Placing the volumes ==============================//
G4RotationMatrix* rot_180 = new G4RotationMatrix();
rot_180->rotateZ(180*deg);
G4double
G4double
G4double
G4double
G4double

Pos_x = 0.0*mm;
Pos_y_Al = 0.0*mm;
Pos_y_Refl = width_Al_horiz;
Pos_y_Scint = Pos_y_Refl + width_Refl;
Pos_z = -1.5*width_Al_vert - 3.0*width_Refl - 3.0*halflength_scint;

G4String topName;
G4String botName;
for (int i=1; i<=4; i++)
{
if(i==1)
{ topName = "T1";
botName = "B1"; }
if(i==2)
{ topName = "T2";
botName = "B2"; }
if(i==3)
{ topName = "T3";
botName = "B3"; }
if(i==4)
{ topName = "T4";
botName = "B4"; }
// TOP OF SUN
Pos_y_Al = 0.0*mm;
Pos_y_Refl = width_Al_horiz;
Pos_y_Scint = Pos_y_Refl + width_Refl;

195

//aluminum
G4VPhysicalVolume* al_top = new G4PVPlacement(0,
G4ThreeVector(Pos_x,Pos_y_Al,Pos_z),al_log,"al_top",room_log,false,0);
//reflector
G4VPhysicalVolume* refl_top = new G4PVPlacement(0,
G4ThreeVector(Pos_x,Pos_y_Refl,Pos_z),refl_log,"refl_top",room_log,false,0);
//scintillator
G4VPhysicalVolume* scint_top = new G4PVPlacement(0,
G4ThreeVector(Pos_x,Pos_y_Scint,Pos_z),scint_log,topName,room_log,false,0);
// BOTTOM OF SUN
Pos_y_Al = 0.0*mm;
Pos_y_Refl = -width_Al_horiz;
Pos_y_Scint = Pos_y_Refl - width_Refl;
//aluminum
G4VPhysicalVolume* al_bottom = new G4PVPlacement(rot_180,
G4ThreeVector(Pos_x,Pos_y_Al,Pos_z),al_log,"al_bottom",room_log,false,0);
//reflector
G4VPhysicalVolume* refl_bottom = new G4PVPlacement(rot_180,
G4ThreeVector(Pos_x,Pos_y_Refl,Pos_z),refl_log,"refl_bottom",room_log,false,0);
//scintillator
G4VPhysicalVolume* scint_bottom = new G4PVPlacement(rot_180,
G4ThreeVector(Pos_x,Pos_y_Scint,Pos_z),scint_log,botName,room_log,false,0);
Pos_z = Pos_z + width_Al_vert + 2.0*width_Refl + length_scint;
}
// SIDES OF SUN
Pos_x = 0.0*mm;
Pos_y_Al = 0.0*mm;
Pos_z = 2.0*length_scint + 2.0*width_Al_vert + 4.0*width_Refl + halflength_Al_side;
G4VPhysicalVolume* al_sideA = new G4PVPlacement(0,
G4ThreeVector(Pos_x,Pos_y_Al,Pos_z),al_log_side,"al_sideA",room_log,false,0);
G4VPhysicalVolume* al_sideB = new G4PVPlacement(0,
G4ThreeVector(Pos_x,Pos_y_Al,-Pos_z),al_log_side,"al_sideB",room_log,false,0);
// **********************************************************************************
// TO PROPERLY SAVE THINGS TO ROOT, YOU NEED TO SPECIFY WHAT YOU NAMED THE DETECTORS
// **********************************************************************************
//
The name of the detectors is in the G4VPhysicalVolume command
detectorName[0] = "T1";
detectorName[1] = "T2";
detectorName[2] = "T3";
detectorName[3] = "T4";
detectorName[4] = "B1";
detectorName[5] = "B2";
detectorName[6] = "B3";
detectorName[7] = "B4";
// **********************************************************************************

196

// VISUALIZATION STUFF
room_log->SetVisAttributes (G4VisAttributes::Invisible);
//visualization for scintillators = GREEN
G4VisAttributes *GreenAttr = new G4VisAttributes(G4Colour(0.,1.,0.));
GreenAttr->SetVisibility(true);
GreenAttr->SetForceSolid(true);
//visualization for reflector = PURPLE
G4VisAttributes *PurpleAttr = new G4VisAttributes(G4Colour(1.,0.,1.));
PurpleAttr->SetVisibility(true);
PurpleAttr->SetForceSolid(true);
//visualization for aluminum = GREY
G4VisAttributes *GreyAttr = new G4VisAttributes(G4Colour(0.5,0.5,0.5));
GreyAttr->SetVisibility(true);
GreyAttr->SetForceSolid(true);
//visualization for BLUE
G4VisAttributes *BlueAttr = new G4VisAttributes(G4Colour(0.,0.,1.));
BlueAttr->SetVisibility(true);
BlueAttr->SetForceSolid(true);
//visualization for RED
G4VisAttributes *RedAttr = new G4VisAttributes(G4Colour(1.,0.,0.));
RedAttr->SetVisibility(true);
RedAttr->SetForceSolid(true);
// applying the color scheme
scint_log->SetVisAttributes(GreenAttr);
refl_log->SetVisAttributes(PurpleAttr);
al_log->SetVisAttributes(GreyAttr);
al_log_side->SetVisAttributes(GreyAttr);
beam_log->SetVisAttributes(BlueAttr);
return room_phys;
}

197

Appendix C
Creating SuN’s ROOT files
In the SuN group, the initial experimental data is saved in the .EVT file format. Converting
the initial data into a format suitable for analysis with the ROOT software package currently
involves two steps. First, the “ddasdumper” program written by members of the current
NSCL data acquisition (DAQ) committee is used to create a raw output .ROOT file from
the .EVT file. This raw .ROOT file consists of a time-ordered list of detectors that triggered
above their threshold. The list contains the module number and channel number of the
detector that fired, along with the corresponding timestamp and extracted energy. In the
second step, the raw .ROOT file is used to create a new .ROOT file which hopefully contains
events grouped together properly with the SuN detector. The code for the second step of
the process is listed below:

/////////////////////////////////////////////////////////////////////////////////////////
// Steve Quinn
October 2013
//
/////////////////////////////////////////////////////////////////////////////////////////
// PURPOSE: Take the output ROOT file from ddasdumper and turn it into another ROOT
//
//
file with a pretty SuN tree.
//
//
//
// OPTIONS: Change the variable "timewindow" to the amount of time that you want to
//
//
consider for coincidences. For example a time window of 30 clock ticks means
//
//
that you group everything up to 300ns after the trigger as a single event.
//
//
(This is assuming you are using the 100MSPS modules)
//
//
//
// HOW TO RUN:
//
//
1. To uncompile <terminal> make clean
//
//
2. To compile
<terminal> make
//
//
3. Before compiling, make sure that the timewindow, input file, and output file //
//
are what you want them to be.
//
/////////////////////////////////////////////////////////////////////////////////////////

198

#include
#include
#include
#include
#include
#include
#include

"ddaschannel.h"
"DDASEvent.h"
<iostream>
<cmath>
<TFile.h>
<TTree.h>
<TBranch.h>

int main()
{
// **********************************************************************************
//
THE COINCIDENCE WINDOW (1 tick = 10ns for 100MSPS module)
// **********************************************************************************
int timewindow = 30; //the coincidence window is set to 300ns
//Filenames
TFile *fIn = new TFile("/pathway/to/the/ddasdumper/files/yourInputFile.root");
TFile *fOut = new TFile("/pathway/to/wherevers/best/yourOutputFile.root","RECREATE");
//Variables for input file
int crate;
//event crate number
int slot;
//event slot number
int chan;
//event channel number
int energy;
//event energy
double time;
//event time
double refT=0.; //placeholder for time of previous trigger
double deltaT;
//difference in time between events
int nEntries;
//number of entries
int nEvents;
//number of events per entry
//Variables for output file
int eSuN[2][5][4]={0};
//energy of SuNs PMTs
int eExt=0;
//energy of raw external trigger signal
int eExtG=0;
//energy of external trigger signal, after applying it
int multi=0;
//multiplicity of SuNs segments
int nPMT=0;
//multiplicity of SuNs PMTs
int numPMT[8]={0};
//keeps track of the number of pmts fired in each segment
double tSuN[2][5][4]={0}; //time of SuNs PMTs
double tExt=0.;
//time of raw external trigger signal
double tExtG=0.;
//time of raw external trigger signal, after applying it
double tTrigger=0.;
//trigger time
int counter=10; //counter for status bar
//Create tree and branches for output file
TTree *tOut = new TTree("t","SuN Tree");
tOut->Branch("energy",&eSuN,"E[2][5][4]/I");
tOut->Branch("time",&tSuN,"t[2][5][4]/D");
tOut->Branch("energy_ext",&eExt,"E_ext/I");
tOut->Branch("time_ext",&tExt,"t_ext/D");
tOut->Branch("energy_ext_G",&eExtG,"E_ext_G/I");
tOut->Branch("time_ext_G",&tExtG,"t_ext_G/D");
tOut->Branch("trigger",&tTrigger,"time/D:pos/I:segm:pmt");
tOut->Branch("multiplicity",&multi,"multiplicity/I");

199

//Get DDAS tree from input file
fIn->cd();
TTree *tIn = (TTree*)fIn->Get("dchan");
DDASEvent *dEvent = new DDASEvent();
tIn->SetBranchAddress("ddasevent",&dEvent);
nEntries = tIn->GetEntries();
//Read in data, entry-by-entry
for(int i=0; i<nEntries; i=i+1)
{
tIn->GetEntry(i);
//status bar
if (i % (int)(0.1*nEntries) == 0)
{std::cerr << counter << " ";
if (counter==0) std::cerr << std::endl;
counter--;
}
//Readout now has correlations, so entries may have multiple events
nEvents = dEvent->GetNEvents();
//Read in data event-by-event
for(int j=0; j<nEvents; j++)
{
//Get all the input variables we need
ddaschannel *dchan = dEvent->GetData()[j];
crate = dchan->GetCrateID();
slot
= dchan->GetSlotID();
chan
= dchan->GetChannelID();
energy = dchan->GetEnergy();
time
= dchan->GetTime();
deltaT = time - refT;
//If the new time is outside of the timewindow, calculate the multiplicities,
// fill the tree, save the new timestamp, and set everything back to zero
if (deltaT > timewindow)
{
for(int a=0;a<8;a++)
{ nPMT=nPMT+numPMT[a];
if (numPMT[a]>0) {multi++;}
}
tOut->Fill();
refT = time;
for (int a=0;a<2;a++){
for (int b=0;b<5;b++){
for (int c=0;c<4;c++){
eSuN[a][b][c]=0;
tSuN[a][b][c]=0.0;}}}
for (int a=0;a<8;a++){
numPMT[a]=0;
}

200

eExt=0;
tExt=0.0;
eExtG=0;
tExtG=0.0;
multi = 0;
nPMT=0;
tTrigger = 0.0;
}
//..................Set SuN tree variables to correct values..................//
//Top of SuN
if (crate==0 && slot==2 && chan<12)
{ int seg = (chan / 3) + 1;
int pmt = (chan % 3) + 1;
eSuN[1][seg][pmt] = eSuN[1][seg][pmt] + energy;
tSuN[1][seg][pmt] = time;
numPMT[seg-1]++;
}
//Bottom of SuN
if (crate==0 && slot==3 && chan<12)
{ int seg = (chan / 3) + 1;
int pmt = (chan % 3) + 1;
eSuN[0][seg][pmt] = eSuN[0][seg][pmt] + energy;
tSuN[0][seg][pmt] = time;
numPMT[seg+3]++;
}
//External trigger stuff
if (crate==0 && slot==4 && chan==0)
{ eExt = eExt + energy;
tExt = time;
}
if (crate==0 && slot==3 && chan==15)
{ eExtG = eExtG + energy;
tExtG = time;
}
//Trigger time
tTrigger = refT;
} //finish loop over j=events
}

//finish loop over i=entries

//Write to output file
fOut->cd();
tOut->Write();
fOut->Close();
return 0;
}

201

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