ASTROPHYSICAL APPLICATIONS OF GAMOW-TELLER

STRENGTHS IN THE A = 78 − 100 REGION

By

Rachel Charlotte Taverner Titus

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics - Doctor of Philosophy

2019

ABSTRACT

ASTROPHYSICAL APPLICATIONS OF GAMOW-TELLER STRENGTHS IN THE

A = 78 − 100 REGION

By

Rachel Charlotte Taverner Titus

Electron-captures on medium-heavy nuclei play an important role in the late evolution

of core-collapse supernovae. Sensitivity studies of simulations of these supernovae indicate

a region of nuclei near the N = 50 shell closure, just above doubly magic 78Ni, that have a

significant effect on key characteristics of the evolution, such as the lepton fraction, electron

fraction, entropy, stellar density and in-fall velocity. However, the vast majority of the

nuclei in this region do not have dedicated weak-rate calculations; instead their rates are

approximated via a method known for its overestimation of rates in heavy, neutron-rich

nuclei. In order to guide the development of microphysical theoretical models, which yield

Gamow-Teller strength distributions, from which more accurate electron-capture rates can

be deduced, experimental data is needed to verify the validity of current models and to guide

the development of improved models. Charge-exchange reactions at intermediate energies

are an excellent tool with which to extract Gamow-Teller strength distributions model-

independently, because of the well-established proportionality between the reaction cross

section and the Gamow-Teller strength.

In an attempt to obtain more accurate nuclear

physics inputs for astrophysical simulations, an experimental campaign was performed at

the National Superconducting Cyclotron Laboratory to study nuclei near the N = 50 shell

closure. 86Kr lies immediately adjacent to the region of interest for core-collapse supernovae;

it was probed using the (t,3He+γ) with the S800 magnetic spectrometer and the γ-detection

array, GRETINA. The Gamow-Teller strength distribution, and the associated electron-

capture rates, were extracted from the experimental data, to compare with theoretical models

and to determine the validity of the current nuclear physics inputs for core-collapse supernova

simulations.

This thesis is dedicated to Guinevere and Daiba-Washi: you light up my life.

iv

ACKNOWLEDGMENTS

The work presented in this thesis was made possible by many people, too many to properly

name in this limited space. However, there are several people, without whom, I would have

crashed and burned long ago.

First and foremost, I would like to thank Remco for all of the guidance and support for

the past five years. Your endless patience and humor in the face of my constant questions

and background subtraction agonies has been greatly appreciated.

To my committee members: Alex Brown, Sean Couch, Morten Hjorth-Jensen and Artemis

Spyrou. Thank you for your helpful discussions and suggestions during our yearly meetings.

To all of the operators and staff of the NSCL. Thank you for all of the hard work making

e16006 possible. Special thanks to Jorge Periera, for the assistance in setting up our krypton

target system; Daniel Bazin, for being the S800 expert; Elaine Kwan, for all of the support

during the beam time; and Peter Bender and Dirk Weisshaar, for explaining all things

GRETINA-related.

To the postdocs of the Charge-Exchange Club: Shumpei, Ken, Juan and Bingshui. Thank

you for showing me the ropes and allowing me to learn by following your example.

To my groupmates: Sam, Chris and Jaclyn. I could not have asked for better colleagues

and I genuinely enjoyed learning alongside you. Thank you for all of the laughs.

To my mom and dad: Thank you for all of your support over the years. You always

pushed me to follow my dreams, and gave me every opportunity for success.

And finally, to Ben: You have been with me every step of the way and I never would have

made it this far without you. Thank you for your constant support and encouragement.

v

TABLE OF CONTENTS

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

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

ix

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Electron-Capture Rates
2.2 Weak-Rate Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Comparison to Theoretical Calculations
. . . . . . . . . . . . . . . .
2.2.3 Contributions to Electron-Capture Rates . . . . . . . . . . . . . . . .
2.3 Study of the High-Sensitivity Region . . . . . . . . . . . . . . . . . . . . . .
2.4 Sensitivity Study Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

pf/sdg-shell Interactions

Introduction to Charge-Exchange Reactions
3.1.1

Chapter 3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Reaction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 DWBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Calculation inputs
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Gamow-Teller Strength Distribution . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Krypton Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3
S800 Magnetic Spectrometer . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 GRETINA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 PID Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Mask Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Drift Velocity Correction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4
Scattering Angle Correction . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Acceptance Correction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Missing Mass Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Target Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

1
1
4

5
5
5
8
12
12
13
17
22
27

29
29
32
37
37
41
45

48
48
48
51
53
55
59
59
61
62
63
66
69
72

4.3.2 Cross Section Calculation . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Carbon Angular Distribution . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Nitrogen Angular Distribution . . . . . . . . . . . . . . . . . . . . . .
4.4 Target Window Event Subtraction . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Empty Cell Subtraction . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Krypton Cell Subtraction . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Krypton Angular Distributions
. . . . . . . . . . . . . . . . . . . . .
4.5 DWBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Multipole Decomposition Analysis . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Gamma-Ray Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74
76
78
80
82
85
88
88
92
95

Chapter 5 Weak-Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1 B(GT) Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Electron-Capture Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 High-Sensitivity Region Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Chapter 6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . 125

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

vii

LIST OF TABLES

Table 3.1: Optical potential parameters used in the DWBA calculations for nuclei in
the pf /sdg-shells. These values were determined from 3He elastic scattering
on 90Zr at 443 MeV [113]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Table 4.1: Ground state Q-value of (t,3He) reactions on krypton and background nuclei. 71

Table 4.2: Location of states in 12B [133], converted to 86Br excitation energy. . . . .

Table 4.3: Location of states in 14C [134], converted to 86Br excitation energy. . . . .

81

81

Table 4.4: Location of states in 16N [135], converted to 86Br excitation energy.

. . .

82

Table 4.5: Energy loss and angular smearing values for the empty cell and krypton cell
subtraction analysis. ∆E represents the energy loss correction between the
two target foils. σE represents the energy loss correction due to straggling
within the target foils. σΘ is the scattering angle smearing due to the target
foils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Table 4.6: Selection rules for monopole, dipole and quadrupole transitions. . . . . . .

90

Table 4.7: γ-rays from de-excitations of 86Br. These lines appear in the γ spectrum
from the krypton data set, and are not associated with de-excitations of
the residual nuclei from reactions on the target windows, for example 12B,
14C or 16N. Also listed are the J π assignments for each state, when known
(tentative assignments are given in parentheses), the excitation energy of
events associated with each γ-ray, as determined from the (t,3He) data,
and the type of transition, determined from examining the shape of the
calculated angular distribution. . . . . . . . . . . . . . . . . . . . . . . . .

98

Table 4.8: Gamow-Teller strength associated with observed γ-rays from GRETINA.
Due to the uncertainties associated with the measurement, these extracted
values provide only an upper limit for the strength from 86Kr. . . . . . . . 113

viii

LIST OF FIGURES

Figure 2.1: Nuclear species currently included in the weak-rate library. The original
tables included in the library by Sullivan et al. were FFN [45], ODA [55],
LMP [69, 56] and LMSH [26, 54]. The Pruet and Fuller tables [70], the
QRPA diamond rate table, and the Suzuki table [33], as an update to
the Oda rates, were added to the library most recently. This figure was
adapted from [10].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: Contributions to the change in electron fraction (Ye), as a function of
neutron and proton number, in late stages of stellar core-collapse, based
on the work by Sullivan et al. Nuclei in this high-sensitivity region around
N = 50 (diamond outlined region), at and above 78Ni, and N = 82 region,
are particularly important contributors to the change in Ye and strongly
affect the dynamical evolution of the supernova. This figure was taken
from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3: A comparison of Gamow-Teller strengths calculated following equation
(2.3) and (2.4) [77] for nuclei on the N = 50 line (blue triangles), for
nuclei on the Z = 38 line (red squares) and for the approximate method,
which uses a fixed value of B(GT ) = 4.6 (black line). This figure was
taken from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.4: Gamow-Teller strength distribution for 86Kr, calculated using the jj44pna
interaction [78] in the SNE model space [76, 75]. The cumulative Gamow-
Teller strength calculated from the shell-model, shown in the bottom
panel, is comparable to the value obtained from occupancy approxima-
tion of the Gamow-Teller strength, shown in figure 2.3. Therefore, this
simple model is reasonable estimate of the total Gamow-Teller strength
around the N = 50 shell closure.
. . . . . . . . . . . . . . . . . . . . . .

Figure 2.5:

(top) Phase-space factors as a function of electron-capture Q-value at stel-
lar electron densities. (middle) Plot of a sample Gamow-Teller strength
distribution for 86Kr. (bottom) The resulting electron-capture rates ob-
tained by multiplying the top and middle curves. At low densities, the
electron-capture rate calculation is extremely sensitive to the low-lying
Gamow-Teller states (black line), but as the density increases, the total
strength becomes more important to the calculation, as opposed to the
details of the strength distribution (green line).
. . . . . . . . . . . . . .

11

12

16

17

20

ix

Figure 2.6: Comparison between electron-capture rates estimated in a hybrid shell-
model RPA calculation (LMSH) [26, 54] and based on the approximation
of equation (2.2) for nuclei in the high-sensitivity region near N = 50. A
density of 1010 g/cm3 and a temperature of 1 MeV was used in the rate
. . . . . . . . . . . . . . .
calculations. This figure was taken from [10].

Figure 2.7: Diagrams of lepton fraction versus central density during the core-collapse
process for different scalings of electron-capture rates for all nuclei (in red)
and for nuclei in the high-sensitivity region (in blue). Scaling factors of
10, 0.1 and 0.01 are applied. The base simulation in which no rates are
scaled is represented by the black line. This figure was taken from [10].
.

Figure 2.8: Electron fraction, entropy, density and matter velocity at t = tbounce
(defined when the entropy reaches 3.0) for two sets of simulations. The
bands indicate the range of values obtained when the electron-capture
rates are scaled in the high-sensitivity region (red) or across the entire
chart of nuclides (black). This figure was taken from [10]. . . . . . . . . .

Figure 3.1: Diagram of charge exchange reactions involving 86Kr. (n,p)- or (t,3He)-
type reactions involve an isospin change of ∆Tz = +1 and yield a daugh-
ter nucleus that is more neutron-rich than the parent nucleus. (p,n)- or
(3He,t)-type reactions involve an isospin change of ∆Tz = −1 and yield a
daughter nucleus that is less neutron-rich than the parent nucleus. . . . .

Figure 3.2:

Isospin in charge-exchange reactions. The isospin of the ground state of
the parent nucleus is equal to (N − Z)/2.
In the (n,p) direction, only
transitions from T0 to T0 + 1 states are available. Also shown in the figure
are the Clebsch-Gordan coefficients corresponding to each transition; in
the (p,n) direction, transitions to states with increasing isospin are sup-
pressed compared to transitions to states with the same isospin. Figure
adapted from [98, 99].
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.3: A representation of Pauli-blocking for heavy nuclei. Transitions with low
excitation energy are blocked because the necessary orbitals are filled;
therefore, the available transitions are required to have higher excitation
energy. Because transitions with higher excitation energy are energetically
unfavorable and, therefore, less likely to occur, the Gamow-Teller strength
is subsequently diminished.
. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.4: Configuration of orbitals in the nuclear shell-model up to the sdg shell.
Figure adapted from [100]. . . . . . . . . . . . . . . . . . . . . . . . . . .

22

24

26

32

33

34

35

x

Figure 3.5: Simple explanation of the model space truncations used in OXBASH cal-
culations. The panel on the left shows all possible Gamow-Teller tran-
sitions, assuming nucleons are free to move to any orbitals. The middle
panel shows the application of a 78Ni core, filling all of the neutron or-
bitals up to the g9/2 orbital, and then eliminating those particles from
the set that are available to participate in transitions. The panel on the
right forces the proton orbitals above the g9/2 to remain empty, because
it is unlikely for protons to exist there. Both of these restrictions lead to
. . . . . . . . . . . . . .
a single remaining transition: π0g9/2 to ν0g7/2.

Figure 3.6: Definition of the spatial coordinate system used in the DWBA formalism.
This figure was adapted from [105]. . . . . . . . . . . . . . . . . . . . . .

Figure 4.1: Schematic of the Coupled Cyclotron Facility, the A1900 fragment separa-
tor and the beamline to the S3 vault and the S800 magnetic spectrometer.
Image credit to the NSCL. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.2: A visual explanation of dispersion matching within a high-resolution spec-
trometer. a) shows the focus mode, in which the beam is focused on the
target, leading to a large dispersion of ejectiles in the focal plane. b) and
c) show dispersion matching with only lateral, and both lateral and an-
gular dispersion-matching, respectively. By tuning the beam to have a
dispersion that is the same as that of the spectrometer, the energy res-
olution in the focal plane of the spectrometer is improved greatly. This
figure was taken from reference [119]. . . . . . . . . . . . . . . . . . . . .

Figure 4.3: An image of the beam spot on the target viewer during dispersion-matched
tuning. The dispersion of the tall beam matches that of the spectrometer,
such that this beam will become focused to a single point in the focal
. . . . . .
plane of the S800. Photo courtesy of the NSCL A1900 group.

Figure 4.4: Left: The kapton foils curing on a vacuum ring to ensure the seal is strong.
Right: The freshly-glued and assembled krypton gas cell attached to the
gas handling and temperature regulation system.
. . . . . . . . . . . . .

Figure 4.5: A diagram of the event tracking performed by the CRDCs in the focal
plane of the S800. The horizontal position of the hit is determined by
the pattern of pads that fire during the event; the vertical position is
determined by the drift time of the electrons in the chamber. This figure
. . . . . . . . . . . . . . . . . . . . . . . . . . .
was obtained from [123].

36

40

49

50

51

53

55

xi

Figure 4.6: Diagram of the pedestal data for the first CRDC; these data were taken
with no beam in the spectrometer. The peaks are characterized and a
pedestal is extracted for each pad, in order to subtract the pad noise from
the experimental data, yielding the most accurate energy measurement
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
for each event.

Figure 4.7: A photo of GRETINA set up with the S800 spectrometer. Photo credit
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

to S. Noji from FRIB.

Figure 4.8: The efficiency curve of the GRETINA detector array. The 152Eu data
points, measured prior to the experiment, are fit with a curve so that the
efficiency of the detector at all γ energies can be determined. . . . . . . .

Figure 4.9: Left: Photo of the single-hemisphere GRETINA configuration with the
custom beam pipe and krypton gas target system installed. Right: Photo
of the krypton cell inside the beam pipe; the 30 degree angle of the target
system, necessary in order to accommodate the germanium detectors, is
evident.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.10: The Particle Identification plot shows the energy deposited in the plastic
scintillator on the y-axis versus the time of flight of the particle on the x-
axis. The large, bright spot consists of 3He ejectiles from charge-exchange
reactions. After applying corrections to the focal plane position, scattering
angle and energy deposition, the 3He spot in the PID plot is much sharper
and easier to separate from the various background events. . . . . . . . .

Figure 4.11: The image of the mask in the first CRDC. The positions of the holes and
slits are calibrated in order to determine the exact position of the CRDC
in space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.12: This plot shows the x- versus y-position of the ejectiles in the focal plane of
the spectrometer. Because of differences in scattering energy, the ejectiles
are separated by the spectrometer into a number of bands. The curved
line on the left represents reactions on hydrogen nuclei; the straight lines
in the middle are from reactions on carbon, oxygen and nitrogen nuclei.
Any ejectiles from reactions on krypton are lost in the background at this
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
point.

Figure 4.13: Non-dispersive scattering angle versus excitation energy. Before any scat-
tering angle corrections are applied, as shown on the left, the hydrogen
kinematic line is notably curved and offset from zero. After the angular
offset was applied, along with a scaling factor, the hydrogen line is now
vertical and centered on zero on the y-axis. . . . . . . . . . . . . . . . . .

56

57

58

59

60

63

64

65

xii

Figure 4.14: Acceptance cut in YtaBta. Non-dispersive scattering angles greater than
0.08 radians and less than -0.08 radians are excluded; beyond these angles
the acceptance of the events falls off rapidly. Additionally, based on the
size of the target cell, the non-dispersive target position can be restricted
to a range of -10.0 mm to 15.0 mm. . . . . . . . . . . . . . . . . . . . . .

Figure 4.15: Acceptance cut in DtaAta. Dispersive scattering angles greater than 0.05
radians and less than -0.05 radians are excluded; beyond these angles
the acceptance of the events falls off rapidly. The acceptance cuts are
more restrictive than the shape yielded by the spectrometer because the
fringe events typically require large acceptance corrections and introduce
. . . . . . . .
unnecessary uncertainties into the calculated cross section.

Figure 4.16: Plot of the scattering angle versus scattered energy at the target position
for the upstream target foil. For the upstream foil, on left, the hydrogen
kinematic line appears complete for both positive and negative scattering
angles. For the downstream foil, the hydrogen kinematic line shows the
loss of events for positive scattering angles, indicating a difference in the
S800 acceptance when compared to the upstream foil. . . . . . . . . . . .

Figure 4.17: A diagram of the possible interaction with the empty target cell (left) and
krypton target cell (right). With the center of the cell at vacuum, incoming
tritons could interact with either the upstream (A) or downstream (B)
kapton target windows. However, when the target is filled with 86Kr gas.
interactions may also occur in the center of the target. The thicknesses of
. . . . . . . . . . . . . . . . . . . . . . . . .
the targets are not to scale.

Figure 4.18: Excitation energy spectrum for the downstream target foil. This spec-
trum uses carbon kinematics, meaning that the peak at 0 MeV represents
reactions on 12C that yield 12B in its ground state. The energies at which
reactions on various target window nuclei begin to appear are shown. If
86Kr were present, those reactions would appear at approximately -2.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MeV in this plot.

Figure 4.19: Cross section for reactions on the empty target cell for the 0 to 1 degree
angular bin, corrected for the acceptance of the spectrometer. Reactions
on the carbon ground state sit at 0 MeV, due to the use of carbon kine-
matics in this calculation. The contribution from hydrogen reactions are
at negative excitation energies. These contributions need to be subtracted
from the krypton data in order to see the underlying reactions of interest.

66

67

68

70

71

75

xiii

Figure 4.20: A comparison the angular distribution of the carbon ground state with
past experimental data. The agreement between the blue points (past
experiment [116, 131]) and the orange points (current data) indicates that
the calibrations and corrections applied to the data have been successful
in replicating this known angular distribution. . . . . . . . . . . . . . . .

Figure 4.21: A comparison the angular distribution of the carbon ground state, derived
from the upstream (orange points) and downstream (green points) target
foils. After correcting for the difference in the acceptance for each target
window, both contributions match the previously-extracted 12C angular
distribution [131, 116].
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.22: Excitation energy spectrum from the empty cell data, zoomed in on the
region containing 14N events. The 14N events of interest are in the peaks
at 7 MeV and 8.5 MeV. The third peak on the right is composed of 16O
events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.23: 14N Gamow-Teller strength distribution, compared to experimental (d,2He)
. . . . .

data [132]. No error bars were presented with the (d,2He) data.

Figure 4.24: Empty cell subtraction for the 1-2 degree bin. The contributions from the
upstream and down stream target windows are shown in red and blue,
respectively, in the top plot. By subtracting the model spectrum from
the empty cell data (black line), the bottom spectrum is obtained. The
spectrum is consistent with zero (blue horizontal line) indicating that the
target window events were subtracted successfully. . . . . . . . . . . . . .

Figure 4.25: Differential cross section before (top) and after (bottom) the target win-
dow events were subtracted for the 1 to 2 degree angular bin. The bottom
. . . . . . . . . . . . . .
plot shows the extracted krypton cross section.

Figure 4.26: Krypton cross section with target window background events subtracted
following the method detailed in section 4.4. The absolute normalization
factor was obtained by comparing the carbon cross section measured in
this experiment with the 12C cross section obtained in previous work [131].
Scattering angles from 0 to 4 degrees in the center of mass frame are shown,
as the background subtraction began to break down at larger angles. The
error bars on the data are a combination of statistical and systematic
uncertainties.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.27: Krypton angular distributions extracted for each energy bin (0.5 MeV)
up to 5 MeV. Forward peaking distributions indicated a possible Gamow-
Teller component in the cross section. . . . . . . . . . . . . . . . . . . . .

77

78

79

80

84

86

87

89

xiv

Figure 4.28: DWBA calculation results for 86Kr(t,3He) at 140 MeV/u. The calculation
was performed using the FOLD software package [102], and shows the three
most likely transitions to appear in the experimental data. The curves
shown here are normalized so that their peaks have the same magnitude.
The shapes of the ∆L = 0 (blue), ∆L = 1 (yellow) and ∆L = 2 (orange)
components are the important result for the analysis.
. . . . . . . . . . .

Figure 4.29: The MDA results, here, show the experimental data (black points) and the
linear combination of multipoles that best fit the data in each energy bin.
The total combination (red curve) is the sum of each of the components:
∆L = 0 (blue), ∆L = 1 (yellow) and ∆L = 2 (green). . . . . . . . . . . .

Figure 4.30: The Gamow-Teller strength distribution for 86Kr extracted from the re-
sults of the MDA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.31: The γ decay spectra for each excitation energy bin. The γ-lines associated
with marked at the top of each plot: 86Br in red, 14C in blue, 16N in green,
and lines associated with no known nucleus in magenta. The peak at 511
keV in the top three panels originates from electron-positron annihilation.

91

93

94

96

Figure 4.32: Eγ = 77 keV. A gate on Eγ was made from 74 keV to 80 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak
at Ex = 0 MeV, a gate was made from -0.5 MeV to 1 MeV and the angular
distribution found in the bottom right of the figure was extracted. Because
the angular distribution does not appear to peak at forward angles, a
further MDA was not performed on this angular distribution.

. . . . . . 102

Figure 4.33: Eγ = 207 keV. A gate on Eγ was made from 205 keV to 209 keV. Two
peaks appeared in the resulting gated excitation energy spectrum. For
the first peak, at Ex = 3 MeV, a gate was made from 2.5 MeV to 3.5
MeV and the angular distribution found in the bottom right of the figure
was extracted. Because the angular distribution does not appear to peak
at forward angles, a further MDA was not performed on this angular
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.34: Eγ = 207 keV. A gate on Eγ was made from 205 keV to 209 keV. Two
peaks appeared in the resulting gated excitation energy spectrum. For the
first peak, at Ex = 4 MeV, a gate was made from 3.5 MeV to 4.5 MeV
and the angular distribution found in the bottom right of the figure was
extracted. Because the angular distribution appears to peak at forward
angles, a further MDA was performed on this angular distribution.

. . . 104

xv

Figure 4.35: Eγ = 382 keV. A gate on Eγ was made from 380 keV to 384 keV. Two
peaks appeared in the resulting gated excitation energy spectrum. For the
peak at Ex = 2.5 MeV, a gate was made from 2 MeV to 3 MeV and the
angular distribution found in the bottom right of the figure was extracted.
Because the angular distribution does not appear to peak at forward an-
gles, a further MDA was not performed on this angular distribution.

. . 105

Figure 4.36: Eγ = 932 keV. A gate on Eγ was made from 928 keV to 933 keV. Two
peaks appeared in the resulting gated excitation energy spectrum. For
the first peak, at Ex = 1 MeV, a gate was made from 0.5 MeV to 2
MeV and the angular distribution found in the bottom right of the figure
was extracted. Because the angular distribution does not appear to peak
at forward angles, a further MDA was not performed on this angular
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.37: Eγ = 932 keV. A gate on Eγ was made from 928 keV to 933 keV. Two
peaks appeared in the resulting gated excitation energy spectrum. For
the second peak, at Ex = 2.5 MeV, a gate was made from 2 MeV to 3
MeV and the angular distribution found in the bottom right of the figure
was extracted. Because the angular distribution does not appear to peak
at forward angles, a further MDA was not performed on this angular
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 4.38: Eγ = 942 keV. A gate on Eγ was made from 941 keV to 945 keV. One
peak appeared in the resulting gated excitation energy spectrum. For
the peak at approximately Ex = 1.5 MeV, a gate was made from 1 MeV
to 2 MeV and the angular distribution found in the bottom right of the
figure was extracted. Because the angular distribution does not appear to
peak at forward angles, a further MDA was not performed on this angular
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 4.39: Eγ = 1427 keV. A gate on Eγ was made from 1420 keV to 1430 keV. One
peak appeared in the resulting gated excitation energy spectrum. For the
peak at approximately Ex = 2.3 MeV, a gate was made from 1.5 MeV
to 3 MeV and the angular distribution found in the bottom right of the
figure was extracted. Because the angular distribution does not appear to
peak at forward angles, a further MDA was not performed on this angular
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 4.40: Eγ = 1753 keV. A gate on Eγ was made from 1750 keV to 1755 keV. One
peak appeared in the resulting gated excitation energy spectrum. For the
peak at approximately Ex = 1.7 MeV, a gate was made from 1.25 MeV to
2.25 MeV and the angular distribution found in the bottom right of the
figure was extracted. Because the angular distribution appears to peak at
forward angles, a further MDA was performed on this angular distribution.110

xvi

Figure 4.41: Eγ = 2361 keV. A gate on Eγ was made from 2359 keV to 2363 keV. One
peak appeared in the resulting gated excitation energy spectrum. For
the peak at approximately Ex = 2.3 MeV, a gate was made from 2 MeV
to 3 MeV and the angular distribution found in the bottom right of the
figure was extracted. Because the angular distribution appears to peak at
forward angles, a further MDA was performed on this angular distribution.111

Figure 4.42: Multipole decomposition analysis results for the γ-gated angular distribu-
tions. Because of the reduced number of experimental data points, only
the ∆L = 0 (yellow) and ∆L = 2 (green) were used in the fitting pro-
cedure. The small contributions of the monopole in these fits provide a
stricter upper limit on the 86Kr strength compared to the result presented
in figure 4.30.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 5.1: Gamow-Teller strength distribution obtained from theoretical calculations
and experimental results. The black points indicate the upper limit ob-
tained from analysis of the singles data; the green points are the upper
limit from the γ-ray analysis. Also shown: a shell-model calculation (red)
using the jj44pna interaction [78] and a QRPA calculation (blue) [84].

. . 115

Figure 5.2: Gamow-Teller strength distribution obtained from theoretical calculations
and experimental results. To better compare the theoretical models and
the experimental upper limit for the strength, the result of the singles
analysis is omitted here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 5.3: A comparison of the experimentally-determined electron-capture rates, at
10 GK over the range of stellar densities relevant for the late stages of
core collapse, for 86Kr and several theoretical models. . . . . . . . . . . . 118

Figure 5.4: Plot of the lepton fraction versus core density during deleptonization and
neutrino trapping. The results from the original weak-rate library (black
dashed line), the weak-rate library with an updated approximate method
(green) and the weak-rate library with the new high-sensitivity region
rates (blue) are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xvii

Chapter 1

Introduction

1.1 Motivation

Although they are separated by many orders of magnitude, astrophysics and nuclear physics

are inextricably linked. Astrophysics is the study of phenomena on a cosmic scale, including

the life cycle of stars and their catastrophic deaths. Nuclear physics, conversely, is concerned

with nuclei on a microscopic scale; however, the interactions between these nuclei determine

the observed characteristics of supernovae and other astronomical phenomena.

Nuclear physics combines theoretical and experimental work to draw conclusions about

the structure of nuclei and their interactions with each other. Theoretical models are devel-

oped in conjunction with experimental data, and make predictions that are further tested

by an array of experiments. Those theories that are supported by nuclear physics data are

used to provide numerical inputs, such as reaction rates (the rate at which a reaction occurs,

commonly dependent on the density and temperature of the stellar environment) [1, 2] or

spectroscopic factors (which provide information about the occupancy of relevant orbitals,

and its effect on the reaction rate of interest) [3], to astrophysical models, yielding predictions

about the behavior of the universe on a much larger scale.

There are many types of reactions in the nuclear physics repertoire, all of which have their

uses, depending on the topic and conditions of interest. To examine a reaction, a beam of

1

nuclei (projectile) is impinged on a target and the reaction products (ejectiles and residuals)

are measured in detectors in order to evaluate the energy and trajectory of the particles.

Selecting a reaction to study, in a particular energy regime, can yield information about

the mass, shape, size or structure of a nucleus; such results increase the body of nuclear

physics knowledge, but can also be applied to other, related fields. For example, transfer

reactions transfer nucleons between the target and projectile, and are often used to examine

the outer shell structure of nuclei, especially when performed at moderate energies [4]. In

addition, these reactions are a popular tool for nuclear spectroscopy, the results of which have

direct applications to astrophysics [5]. Another type of reaction, which provides both nuclear

physics information and an astrophysical connection, is the charge-exchange reaction. Such

reactions, performed at intermediate energies, are the main focus of this work.

Charge-exchange reactions are characterized by the exchange of a proton and a neutron

between the target nucleus and the projectile nucleus, yielding a change in isospin (∆T =

1) [6]. Such reactions are able to excite a number of transitions, including Gamow-Teller

and Fermi transitions (angular momentum transfer (∆L)= 0), dipole transitions (∆L = 1)

and quadrupole transitions (∆L = 2), and have been used as successful probes of nuclear

structure for almost 40 years [7, 8]. Of particular interest to nuclear astrophysics are Gamow-

Teller transitions, which are characterized by a spin transfer, an isospin transfer and zero

angular momentum transfer. Such transitions are closely related to β-decay and electron-

capture reactions, which are essential inputs to astrophysical simulations, because the initial

and final nuclear states of these reactions are the same. Therefore, obtaining the strengths

of these important astrophysical reactions is possible when the Gamow-Teller strength is

known via charge-exchange reaction measurements.

Core-collapse supernova simulations make use of weak rates, including electron-capture

2

and β-decay rates, in order to determine the process by which a massive star collapses and
explodes, ending its life [9]. β−-decay reactions occur when a nucleus emits an electron and

an anti-neutrino, and converting the parent nucleus into a nucleus with the atomic number

increased by 1. Electron-capture (or β+-decay) reactions occur when the parent nucleus

absorbs an electron (or emits a positron), and converting the parent nucleus into a nucleus

with the atomic number reduced by 1. Many of the nuclei that participate in these processes

have not been studied experimentally. Therefore, it falls to theoretical models to provide

an alternative source for nuclear physics inputs. Because good experimental data for the

heavy mass region of the chart of nuclides is sparse, theoretical models often extrapolate

predictions from more well-studied regions, and lack the benchmarking necessary to ensure

that the results provided are accurate for more exotic nuclei. Therefore, in an effort to study

heavy, neutron-rich nuclei and their associated electron-capture rates in more depth, the

nucleus 86Kr (Z = 36, N = 50) was chosen for experimental study. This nucleus lies on the

edge of a region of nuclei found to be important to the behavior of core-collapse supernovae

(see chapter 2 for details) [9, 10, 11], and also lies on the N = 50 shell closure, which provides

insight into the effect of orbital filling on electron-capture rates.

Experimentally, there are a number of charge-exchange probes that can be used, depend-

ing on the goal of the experiment. (p,n) and (n,p) reactions are indirect probes of β-decay

and electron-capture reactions. There are also composite probes available, including (d,2He),

(3He,t), (t,3He) and heavy ion charge-exchange reactions, depending on the energy resolu-

tion needs of the experiment [12, 13]. The A1900 Fragment Separator [14], in conjunction

with the Coupled Cyclotron Facility, at the National Superconducting Cyclotron Laboratory

(NSCL) produces a secondary triton beam at intermediate energies, which is ideally suited to

performing (t,3He) experiments on a wide range of targets [15]. Utilizing both the capabilities

3

of the cyclotron facility and the S800 magnetic spectrometer [16], the conditions are excellent

for measuring charge-exchange reactions on 86Kr, and using those data to extract the desired

Gamow-Teller strength. The gains from this experiment are two-fold: yielding information

on the structure of the nucleus and providing the opportunity to calculate the associated

electron-capture rates, which are of interest to the astrophysics community[17, 18, 19].

The goal of this and future experimental work will be to validate and benchmark current

theoretical models that are used in astrophysical simulations. Determining which nuclear

physics inputs are accurate, and where improvements can be made, will lead to the develop-

ment of more robust models, facilitating a better understanding of galaxies, stellar evolution

and the energetic phenomena that occur throughout the cosmos.

1.2 Organization

This work is divided into 5 chapters. Chapter 2 presents the astrophysical motivation for the

86Kr charge-exchange experiment. Chapter 3 discusses charge-exchange reaction theory and

calculation framework. Chapter 4 describes the experimental set-up and analysis process

of the 86Kr(t,3He+γ) experiment. Chapter 5 re-visits the astrophysical motivation and

examines the experimental results in the context of weak reaction rates. Finally, Chapter 6

discusses the significance of both the experimental and astrophysical work, and the outlook

for future nuclear physics and astrophysics projects.

4

Chapter 2

Astrophysics

2.1 Core-Collapse Supernovae

2.1.1 Evolution

Core-collapse supernovae occur at the end of the life cycle of massive stars, typically in those

between 8 and 40 solar masses. For the vast majority of a star’s life, it is in the hydrogen

burning stage, in which hydrogen nuclei in the core are fused, predominantly via the CNO

cycle for massive stars, into helium [20]. After the hydrogen fuel is spent, helium burning

begins, converting helium nuclei into carbon nuclei. For the most massive stars, further

burning stages can be ignited, beginning with carbon burning, and then continuing through

neon, oxygen and silicon burning phases. The internal structure of the star in the end is

concentric layers of ashes from each of the burning stages surrounding an iron core [21].

Throughout these main burning stages, the gravitational pull of the star’s mass, attempting

to collapse it into a smaller volume, is balanced by the thermal pressure produced by the

fusion of nuclei during the burning stages. Once the iron core of the star becomes degenerate,

electron degeneracy pressure also contributes to the outward pressure, supporting the outer

layers of the star.

The balance between the gravitational force pulling inward and radiation pressure pushing

outward is maintained for the entirety of the star’s lifetime. Although some stars experience

5

thermal pulsing [22], the general equilibrium is preserved such that the star neither collapses

to a point, nor flies apart; the mass limit for this state is the Chandrasekhar mass [23], and

is equal to approximately 1.4 solar masses. For massive stars, the iron core gains mass as the

silicon burning phase progresses. As the mass of the core increases, the free electrons within

the star exert additional outward pressure because they cannot be compressed further, as

they are fermions and subject to the Pauli exclusion principle [24]. In order to remain non-

degenerate, the electrons in the core become more and more energetic, eventually reaching

relativistic velocities, allowing the progressively more massive iron core to be supported

against collapse. However, once the mass of the core surpasses the Chandrasekhar mass, the

fusion energy combined with the degeneracy pressure is no longer sufficient to balance the

gravitational force inward [20, 24].

Matter from the outer layers of the star and the core press inward, causing the core of

the star to begin to collapse [21]. The core of the star becomes extremely dense as the

collapse continues, reaching the point at which further compression is not possible in its

current composition, as this would cause electrons to occupy the same energy states, and

would violate the Pauli exclusion principle. Instead the free electrons fuse with protons or

capture onto heavy nuclei, reducing the lepton fraction in the core. Because free electrons are

being removed from the nuclear matter, this causes a decrease in the degeneracy pressure,

further exacerbating the imbalance with the inward gravitational force. Additionally, when

an electron captures onto a nucleus, a neutrino is released; these neutrinos are able to escape

the star easily, and carry away energy that was otherwise working against the gravitational

pressure inward [21]. Hence, as the energy streams away with the neutrinos, the outward

pressure is further decreased, leading to less resistance to gravitation collapse [20]. Then, as

the rate of collapse increases, the rate at which electrons are captured increases, releasing

6

more neutrinos and causing more energy loss in the core of the star.

The cycle of electron capture and neutrino energy loss, along with the energy loss asso-

ciated with the photodisintegration of iron nuclei, leads to a run-away process, causing the

collapse of the star to happen faster and faster until the density of the matter reaches 1012

g/cm3. At that point, the opacity of the matter increases to such a point that neutrinos

can no longer escape from the center of the star and are trapped [21, 25]. After neutrino

trapping, the star continues to collapse until the density reaches approximately 1014 g/cm3,

at which point the nuclear matter of the inner core, which is now classified as a proto-neutron

star, is saturated and cannot be compressed further. The outer layers of the core decelerate

and bounce off of the inner core, sending a shock wave moving outward through the layers

of the star.

The general mechanics of the core-collapse are known [1, 24, 26], however, understanding

the explosion mechanism, which ultimately destroys the star, is a work in progress. There

are a number of theories available [21, 27, 28], but the exact process and details are still

being determined through simulations and observations. For this work, the main period

of interest is during the late stages of core-collapse, from the beginning of the period of

deleptonization until core bounce. It is then that electron-capture rates play a significant

role in the hydrodynamic evolution of the supernova, a facet about which nuclear experiments

can provide more information in order to improve the understanding of the mechanisms at

work.

Core-collapse supernovae, in addition to releasing energy in the form of visible light, also

produce large quantities of neutrinos, as mentioned previously. Additionally, these events are

predicted to be the origin of gravitational waves that can be measured [29, 30]. Through all of

these signals, core-collapse supernovae appear to be excellent candidates for multi-messenger

7

studies, coordinating the observation and interpretation of various messenger signals from

a single astrophysical event [31]. By combining the information obtained from a variety

of sources, it is possible to gain a better understanding of these, and other, astrophysical

processes. However, accurate and detailed knowledge of nuclear physics processes is needed

in order to interpret the astrophysical observations and to draw robust conclusions that can

be applied to our understanding of the universe as a whole.

2.1.2 Electron-Capture Rates

Electron-captures on nuclei play an important role in astrophysical phenomena, such as

during carbon burning and later stages in massive stars [32, 33], especially the growth of

the iron core [34, 35] and the subsequent collapse of the iron core leading to a core-collapse

supernova [1, 9, 21, 26, 30, 36], thermonuclear supernovae [37, 38, 39, 40], neutron stars and

their crusts, [41, 42] and neutron-star mergers [43]. Because of their diverse applications,

it is important to obtain accurate nuclear-physics inputs that can be used in astrophysical

simulations, including sets of electron-capture and β-decay rates across a range of stellar

temperatures and densities for a large number of nuclei.

Electron-capture rates depend sensitively on the Gamow-Teller transition (∆L = 0, ∆S =

1, ∆J = 1) strength distribution in the β+ direction [17, 44, 45, 46, 47]:

(cid:88)

λEC = ln(2)

fij(T, ρ, UF )B(GT )ij.

(2.1)

ij

The rate is obtained by summing over the Gamow-Teller strength distribution, B(GT ), and

is weighted by the phase-space factor, f (T, ρ, UF ), which depends on the stellar density, ρ,

temperature, T , and chemical potential, UF . β+ and electron-capture decay experiments

8

yield the Gamow-Teller strength distributions necessary to calculate electron-capture rates

[17, 48, 49, 50]. However, these experiments are energetically limited to states within a

specific Q-value window, if energetically possible at all. On the neutron-rich side of the valley
of stability, decay proceeds in the β− direction and the Gamow-Teller transitions required

to calculate electron-capture rates cannot be measured at all, unless they are ground-state

to ground-state transitions.

Charge-exchange experiments, which provide an indirect method for the extraction of

Gamow-Teller strengths, are not limited by this Q-value window, thus yielding informa-

tion about transitions to higher excitation energies. Because of the known proportional-

ity between Gamow-Teller strength and charge-exchange reaction cross section [12, 51, 52],

Gamow-Teller strengths can be extracted model-independently. Therefore, charge-exchange

experiments in the β+ direction have become an important tool for extracting Gamow-Teller

strengths of relevance for the determination of astrophysical electron-capture rates.

Unfortunately, it is not possible to measure Gamow-Teller strength distributions on all

nuclei that play a role in the aforementioned astrophysical phenomena.

In addition, in

high-temperature stellar environments, low-lying states in parent nuclei can be thermally

populated, and electron-captures can take place on these excited states [53]. Moreover, at

high temperatures, shell orbits that would be Pauli-blocked at T = 0, can become par-

tially unblocked [54], which changes the Gamow-Teller strength distribution. Because these

effects are difficult to study in the laboratory, electron-capture rate calculations and astro-

physical simulations can use rates determined on the basis of experiment, where available

and appropriate, but in general must rely on theoretical estimates. An important role of

the experiments is then to guide and benchmark the theoretical calculations and to provide

measures for uncertainties in the theories.

9

The earliest set of theoretical weak-rates, based on a single-particle estimate, were pro-

vided by Fuller, Fowler and Newman (FFN) [44, 45, 46, 47]. With the advent of increased

computational power, electron-capture rate sets based on shell-model calculations became
feasible, starting with nuclei in the sd-shell (N, Z = 8−20; see section 3.1.1 for an explanation

of the shell structure) [55]; these rate calculations were recently updated [33]. Weak-rate sets
for nuclei in the pf -shell (N, Z = 20 − 40) were generated [39, 53, 56] based on the KB3G

interaction [57] and, recently, the GXPF1 family of interactions [58, 59, 60]. For heavier

nuclei, shell-model calculations are computationally challenging, although calculations with

severely constrained model spaces, such as the 78Ni or 88Sr core, used in reference [61], have

been used to estimate weak rates for astrophysical purposes. Consequently, other theoretical

techniques, such as QRPA [62, 63, 64, 65, 66] or shell-model Monte-Carlo [67, 68], are used.

Such calculations cannot reproduce the details of the Gamow-Teller strength distributions

as well as shell-model calculations can; the relationship between stellar density and the im-

portance of the details of the Gamow-Teller strength distribution will be discussed further

in section 2.2.3.

In order to consolidate all available tables of weak rates for use in astrophysical simula-

tions, a weak-rate library was developed by Sullivan et al [9]. As shown in figure 2.1, weak

rates based on microscopic calculations of varying origin cover low- and medium-mass nuclei

primarily around the valley of stability. For the vast majority of nuclei, especially those

far from stability, an approximate method is used to generate these electron-capture rates

for use in simulations [1, 2]. The weak rate library and this approximate method will be

discussed in section 2.2.1.

Sensitivity studies, in which electron-capture rates are varied within uncertainties, are

helpful for gaining a better understanding of the impact of these rates on the evolution of the

10

Figure 2.1: Nuclear species currently included in the weak-rate library. The original tables
included in the library by Sullivan et al. were FFN [45], ODA [55], LMP [69, 56] and LMSH
[26, 54]. The Pruet and Fuller tables [70], the QRPA diamond rate table, and the Suzuki
table [33], as an update to the Oda rates, were added to the library most recently. This
figure was adapted from [10].

late stages of stellar core-collapse [9, 11, 33]. Additionally, these studies assist in identifying

the rates that have a significant impact on the astrophysical model. In the work by Sullivan

et al., it was shown that nuclei in the upper pf - and pf g/sdg-shells, particularly in the

region around N = 50 at and above 78Ni (see figure 2.2), have the largest impact on the

change in electron density (Ye) and, thus, on the dynamical evolution of the collapse. This

region, marked in dark green in figure 2.1, was denoted the high-sensitivity region or the

high-sensitivity diamond, and it is upon this group of nuclei that effort is focused in order

to demonstrate the importance of constraining the electron-capture rates in this region.

11

Figure 2.2: Contributions to the change in electron fraction (Ye), as a function of neutron
and proton number, in late stages of stellar core-collapse, based on the work by Sullivan et
al. Nuclei in this high-sensitivity region around N = 50 (diamond outlined region), at and
above 78Ni, and N = 82 region, are particularly important contributors to the change in Ye
and strongly affect the dynamical evolution of the supernova. This figure was taken from
[9].

2.2 Weak-Rate Library

2.2.1 Approximate Method

The weak-rate library [9, 71], mentioned previously,

includes several tables of electron-

capture and β-decay rates (see figure 2.1) over a large density and temperature grid. It can

be used independently for calculations or used in conjunction with the neutrino-interaction

library, NuLib [72]. NuLib uses the electron-capture rates in the library to calculate neu-

trino/antineutrino charged-current absorption opacities for use in astrophysical simulations.

The simulations of core-collapse supernovae described here are performed using the general

12

relativistic, spherically-symmetric stellar collapse code, GR1D [72, 73], but the library has

also been used in two-dimensional simulations of core-collapse supernovae [30].

Many of the neutron-rich nuclei in the high-sensitivity region do not yet have an electron-

capture rate based on a microscopic model calculation included in the weak-rate library and,

consequently, an approximate method is used to calculate the electron-capture rate at the

appropriate density and temperature. The approximate electron-capture rate is given by

[47, 69]:

λEC =

ln(2) · B

K

(cid:32)

T

mec2

(cid:33)5(cid:2)F4(η) − 2χF3(η) + χ2F2(η)(cid:3),

(2.2)

where me is the electron mass, K = 6146 s, Fk are Fermi integrals of rank k and degeneracy
η, χ = (Q− ∆E)/T , η = χ + µe/T , and T and µe are the temperature and electron chemical

potential, respectively. B, the effective Gamow-Teller transition strength, is fixed for all

isotopes to 4.6. ∆E, the effective excitation energy, is fixed for all isotopes to 2.5 MeV.

These values were determined in a fit to shell-model calculations for nuclei in the pf -shell

[1] relatively close to the valley of stability. There is also a new version of this approximate

method implemented in the library, which improves the accuracy of the rates by adjusting

the effective excitation energy based on neutron and proton numbers [2] (while leaving the

effective transition strength fixed).

2.2.2 Comparison to Theoretical Calculations

For neutron-rich nuclei, including those in the high-sensitivity region, the use of the approxi-

mation with parameters fit to microscopic calculations for nuclei close to stability constitutes

an extrapolation to a region of the chart of nuclei in which different shells play a role. Hence,

the uncertainties when applying the approximation can be large, as first detailed in refer-

13

ence [68], in which occupation numbers were estimated based on predictions in shell-model

Monte-Carlo calculations. At and above the N = 50 shell closure, Gamow-Teller transitions

are associated with transitions in which the added neutron occupies an orbital above νg9/2
[74]. Therefore, the strongest transition is πg9/2 → νg7/2 [75, 76]; this conclusion will be

discussed in greater detail in section 3.1.1. If protons are present in the πg9/2 orbital, the

Gamow-Teller transition strength is significant; however, with increasing neutron number

beyond N = 50, the νg7/2 orbital becomes Pauli-blocked and the transition strength de-

creases. Similarly, as the nucleus becomes less proton-rich and the occupancy of πg9/2 is
lowered, the transition strength will decrease. For N < 50, πg9/2 → νg9/2 transitions also

contribute to the Gamow-Teller transition strength.

In order to examine the effect of filling these orbitals on the Gamow-Teller strength,

it is beneficial to employ a simple model, using the occupancies of the πg9/2, νg7/2 and

νg9/2 orbitals. Following the work of Macfarlane [77], the Gamow-Teller strength in the β+

direction can be estimated using:

S+ = 3

(cid:0)Cjj(cid:48)

nl

(cid:1)2(cid:0)1 − nN

nlj

(cid:1)nP

nlj(cid:48),

(cid:88)

nljj(cid:48)

where nN

nlj is the average occupancy of the initial neutron orbital, nP

nlj(cid:48)

(2.3)

is the average

occupancy of the final proton orbital, and

Cjj(cid:48)
nl = [2(2j + 1)(2j(cid:48) + 1)]1/2W (l 1

2 j1; j(cid:48) 1
2),

(2.4)

in which j is the total orbital angular momentum quantum number for the initial state, j(cid:48)

is the total orbital angular momentum for the final state and W is the Racah W-coefficient.

14

Here, the jj44pna interaction [78] is used in the SNE model space in which protons can

populate the 0f5/2, 1p3/2, 1p1/2, and 0g9/2 orbitals, and neutrons can populate the 0g7/2,

1d5/2, 1d3/2, 2s1/2 and 0h11/2 orbitals, to estimate the filling of the neutron and proton

orbitals.

Figure 2.3 shows that the Gamow-Teller strength increases with proton number along
the N = 50 line (blue triangles). For Z ≤ 30 (A ≤ 80), protons do not populate the

πg9/2 orbital and Gamow-Teller transitions are completely blocked. The opposite trend is

seen when increasing the neutron number (shown in red squares in figure 2.3 for the case of
krypton (Z = 36) isotopes). Below N = 50 (A = 86), both πg9/2 → νg7/2 and πg9/2 →
νg9/2 transitions can contribute. Above N = 50 (A = 86), πg9/2 → νg9/2 transitions are
completely blocked, and πg9/2 → νg7/2 transitions increasingly so. Hence, the Gamow-Teller

strength decreases, until it is completely blocked at N = 58 (A = 94) when the neutrons

completely fill the g7/2 orbital.

Figure 2.3 also shows the strength used in the approximate method [1, 2], which is higher

than the estimates based on the orbital fillings in this region of the chart of nuclides by

at least an order of magnitude. It is important to note that at high stellar temperatures,

thermally-driven Pauli-unblocking effects will increase the electron-capture rates beyond the

value calculated at T = 0. For nuclei in which the Pauli-blocking effects are strong, the

unblocking can lead to a significant increase of the electron-capture rates [1, 54, 65]. However,

for nuclei in which the Pauli-blocking is incomplete, the effects of the increased temperatures

and additional unblocking are relatively small [54], which is the likely scenario for nuclei in

the high-sensitivity region.

Ultimately, it is preferable to generate electron-capture rates using microscopic calcula-

tions based on realistic strength distributions that are benchmarked against data. In the

15

Figure 2.3: A comparison of Gamow-Teller strengths calculated following equation (2.3) and
(2.4) [77] for nuclei on the N = 50 line (blue triangles), for nuclei on the Z = 38 line (red
squares) and for the approximate method, which uses a fixed value of B(GT ) = 4.6 (black
line). This figure was taken from [10].

pf -shell, rates calculated by using shell-model results are the most accurate based on a com-

parison with charge-exchange data [17]. Although shell-model calculations near N = 50

are a challenge, initial calculations were performed using NUSHELLX [79] using the jj44pna

interaction [78]; an example of such a calculation, for the case of 86Kr, is shown in figure

2.4. The excitation energy of the first transition in the shell-model calculation was chosen to
match the first known 1+ state in 86Br (at 2.45 MeV), because the energy of the 1− ground

state in 86Br cannot be calculated within the chosen model space.

16

80859095Massnumber10−210−1100101B(GT)B(GT)fromapproximationB(GT)forZ=36B(GT)forN=50Figure 2.4: Gamow-Teller strength distribution for 86Kr, calculated using the jj44pna interac-
tion [78] in the SNE model space [76, 75]. The cumulative Gamow-Teller strength calculated
from the shell-model, shown in the bottom panel, is comparable to the value obtained from
occupancy approximation of the Gamow-Teller strength, shown in figure 2.3. Therefore, this
simple model is reasonable estimate of the total Gamow-Teller strength around the N = 50
shell closure.

2.2.3 Contributions to Electron-Capture Rates

It is found that the shell-occupancy approximation, discussed above, estimates a similar

total strength to the shell-model calculation for 86Kr, indicating that the νg7/2 and πg9/2

orbitals are dominant in the shell-model calculations as well. However, in the shell-model

calculations, about half of the total strength is distributed over many states at higher exci-

tation energies, which results in a reduction of the electron-capture rates compared to the

assumption that all strength is contained in a single state at 2.5 MeV, as in the rate ap-

proximation (equation 2.2). This reduction is in addition to that which is caused by the

overestimated total transition strength, due to the lack of Pauli-blocking included in the ap-

17

02468100.0000.0020.0040.0060.0080.0100.0120.014B(GT)GTstengthfor86Kr0246810Excitationenergy(MeV)0.000.010.020.030.040.050.060.07B(GT)CumulativeB(GT)proximation model. Finally, it is well-known that Gamow-Teller strengths calculated in the

shell-model must be quenched by factors of 0.5-0.6 to account for effects not included in the

calculation [80, 81, 82, 83], which reduce the total strengths and, thus, the electron-capture

rates further. The quenching applied to the strength takes the form of a hindrance factor,

1/h, which accounts for the mixing of two-particle two-hole states with oscillator energies of

2¯hω and greater.

The dependence of the electron-capture rate on stellar density is explained by the phase

space energy equation [17, 44]:

UF = 0.511 ×

(cid:34)(cid:18)
1.018(cid:0)ρ6Ye

(cid:1)2/3 + 1

(cid:35)
(cid:19)1/2 − 1

,

(2.5)

where ρ6 is the stellar density divided by 106 gm/cm3 and Ye is the electron fraction. The

Fermi energy, 
F , is related to the phase space energy by the following expression: 
F =

UF + mec2. For reactions in which the Fermi energy far exceeds the Q-value of the reaction,

electron-capture rates will be relatively constant with respect to changes in stellar density.

However, when the Q-value of the reaction is close in magnitude to the Fermi energy, as is
the case for 86Kr, with Q = −7.607 MeV and UF = 10.6 MeV at ρYe = 106 g/cm3, the

electron-capture rates are sensitive to changes in the stellar density, ρ, and electron fraction,

Ye. This behavior occurs because, as the density increases, more excited states in the nucleus

become available to contribute to the overall electron-capture rate.

By examining the two components of the electron-capture calculation (strength distribu-

tion and phase space factor), it is possible to visualize the relationship between the Gamow-

Teller strength distribution and the electron-capture rates. Shown in the top plot of figure

2.5 is the phase space factor, f , as a function of Q-value and normalized to the phase space

18

factor at Q = 0 MeV, for a variety of stellar densities. This factor, as defined in equa-

tion 2.1, weights each of the Gamow-Teller states in the sum that is used to calculate the

electron-capture rate. For relatively low stellar densities (ρ < 1010 g/cm3), the phase space

factor drops off quickly as a function of Q-value, meaning that only the lowest-lying Gamow-

Teller states will contribute significantly to the electron-capture rate calculations. However,

as the stellar density increases to 1011 g/cm3 (red line) or 1012 g/cm3 (green line), the

phase space curve decreases less sharply, indicating that Gamow-Teller states with higher

excitation energies begin to contribute more strongly to the electron-capture rate.

The center plot of figure 2.5 shows the Gamow-Teller strength distribution for 86Kr as

obtained from the QRPA calculation that will be discussed in section 5.1 [84]. Now plotted

as a function of Q-value, it is clear that the majority of the strength sits at or above a Q-value

of 10 MeV. For the other nuclei in the high-sensitivity region, the Gamow-Teller strength

distributions have a similar form. Finally, the bottom plot in figure 2.5 illustrates the

relationship between the electron-capture rates, the Gamow-Teller strength distribution and

the stellar density. The electron-capture rates were calculated by multiplying the Gamow-

Teller strength by the phase space factor, following equation 2.1. Then, the running sum of

electron-capture rates, normalized to the total rate, as a function of Q-value, was plotted,

illustrating the relative contribution of each Gamow-Teller transition to the total, for each

of the stellar densities.

For lower stellar densities (black and blue curves), the first peak in the strength distribu-

tion contributes significantly to the overall electron-capture rate, while all of the successive

states contribute very little. However, with increasing density, as in the cases of 1011 g/cm3

(red line) and 1012 g/cm3 (green line), the higher-lying states begin to contribute more to the

overall electron-capture rate. Because 86Kr has a large, negative Q-value for electron-capture

19

Figure 2.5: (top) Phase-space factors as a function of electron-capture Q-value at stellar
electron densities. (middle) Plot of a sample Gamow-Teller strength distribution for 86Kr.
(bottom) The resulting electron-capture rates obtained by multiplying the top and middle
curves. At low densities, the electron-capture rate calculation is extremely sensitive to the
low-lying Gamow-Teller states (black line), but as the density increases, the total strength
becomes more important to the calculation, as opposed to the details of the strength distri-
bution (green line).

reactions, the details of the strength distribution play an important role in the calculation of

electron-capture rates at high densities. Such a conclusion differs from previous assumptions

that, at high stellar densities, it is only the cumulative Gamow-Teller strength that affects

20

0.00.20.40.60.81.0f / f(Q=0)0123456B(GT)1e305101520Q-value [MeV]0.00.20.40.60.81.0Fraction of total λe109g/cm31010g/cm31011g/cm31012g/cm3the electron-capture rate [17].

It is clear from this result that the electron-capture rates

for nuclei with large, negative Q-values are predominantly dependent on the details of the

Gamow-Teller strength distribution.

When the densities are in excess of 1011 g/cm3, forbidden transitions begin to contribute

to the rate as well [68, 69, 85], but they are not explicitly included in this work. At densities

of 1012 g/cm3, the electron-capture rate could double due to contributions from forbidden

transitions [68]. Although charge-exchange experiments provide information on the strength

distributions for forbidden transitions, a proportional relationship between forbidden transi-

tion strengths and charge-exchange cross sections has not been established, making it difficult

to extract strengths and to quantify the uncertainties in the associated electron-capture rates.

For some of the nuclei in the high-sensitivity region, electron-capture rates have been

estimated by considering thermally-driven Pauli-unblocking effects in a hybrid shell-model

RPA calculation [26, 54]. These rates are included in the weak-rate library in the form of the

LMSH table in figure 2.1. A comparison between the LMSH rates and the rates based on

approximate method for nuclei in the high-sensitivity region is shown in figure 2.6. Because

these rate calculations do consider the effects of Pauli-blocking and have more accurate

estimates for the Gamow-Teller strengths, the electron-capture rates are generally less than

the estimates based on the approximation, with deviations between the two methods reaching

two orders of magnitude in some cases. Still, this is not uniformly the case, because the

electron-capture rate Q-values used in the two methods can be different and may change the

estimated rates significantly. Clearly, guidance from charge-exchange experiments, similar

to the case for nuclei in the pf -shell, is required to constrain the rates, to estimate the

uncertainties in the theory, and to provide data on the basis of which theoretical estimates

can be improved.

21

Figure 2.6: Comparison between electron-capture rates estimated in a hybrid shell-model
RPA calculation (LMSH) [26, 54] and based on the approximation of equation (2.2) for nuclei
in the high-sensitivity region near N = 50. A density of 1010 g/cm3 and a temperature of 1
MeV was used in the rate calculations. This figure was taken from [10].

2.3 Study of the High-Sensitivity Region

As illustrated in figure 2.2, there exists a cluster of nuclei around the N = 50 shell closure at

and above 78Ni that has a large impact on the change in electron fraction in the core-collapse

supernova simulations. In order to determine the sensitivity of the simulation to this region

in more detail, a set of 74 nuclei was chosen, covering both sides of the N = 50 line from

the valley of stability to the neutron drip line. In this sensitivity study, the electron-capture

rates were systematically scaled in two parts. The first was the case in which the rates for

all nuclei were scaled; the second was the case in which which only the rates for nuclei in the

high-sensitivity region were scaled. The scaling factors (x10, x1, x0.1, x0.01) were chosen

22

0.11101001000λelectron capture(1/s)ρ=1010 g/cm3T=1 MeVLMSHApproximation of equation 1to represent the uncertainty in the electron-capture rates based on the comparison between

different models, as discussed above, and the models and experimental data for nuclei in the

pf -shell.

The simulation framework used for the sensitivity study is similar to the work performed

by Sullivan et al. [9]. The open-source spherically symmetric, general-relativistic neutrino-

transport and hydrodynamics code, GR1D, was used to simulate the late stages of core-

collapse [72, 73]. In these simulations, all of the tables included in the weak-rate library,

except for the Suzuki-Honma and QRPA rate tables, were used, with the hope that a larger

set of nuclei would have microphysically-accurate rates available. The electron-capture rates

from the library were integrated into NuLib [72], the open-source neutrino-interaction library

discussed in section 2.2. A well-known 15 solar-mass, solar-metallicity progenitor (s15WW95)

[86] was used in the simulations with the SFHo equation of state and nuclear statistical

equilibrium distributions from reference [87]. It has been shown previously that variations

in the electron-capture rates play a more important role in the simulation than variations in

the progenitor or equation of state [9] therefore, only the electron-capture rates that were

input to the simulations were changed in the sensitivity studies discussed here.

A comparison of the lepton fraction (Yl) as a function of central density (ρc) for simula-

tions in which the electron-capture rates are scaled by various factors is shown in figure 2.7.

The black solid line represents the base calculation, in which the electron-capture rates have
not been scaled. At low densities (ρc ∼ 1010g/cm3), electron-captures are responsible for the

deleptonization of the matter in the core of the star. The opacity of the matter is low, allow-

ing electron neutrinos to escape the star freely, causing the lepton fraction to decrease. When
the central density reaches ρc ≈ 2 × 1012 g/cm3, the lepton fraction saturates (at a value of

Yl = 0.292 in the base calculation) as neutrino trapping prevents further deleptonization. If

23

Figure 2.7: Diagrams of lepton fraction versus central density during the core-collapse process
for different scalings of electron-capture rates for all nuclei (in red) and for nuclei in the
high-sensitivity region (in blue). Scaling factors of 10, 0.1 and 0.01 are applied. The base
simulation in which no rates are scaled is represented by the black line. This figure was
taken from [10].

the electron-capture rates in the library are increased (dashed-dotted lines in figure 2.7), sim-

ulating the situation in which the electron-capture rates in the library are underestimated, a

similar or slightly higher final lepton fraction is attained. This is because the opacity related

to electron-neutrino captures on heavy nuclei becomes stronger than opacities related to the

electron-neutrino scattering, reducing the window for deleptonization [9].

For reasons discussed in previous sections, the scenario in which the electron-capture

24

Scaling in high-sensitivity regionBase calculationScaling all nucleix10x0.1x0.01x10x0.1x0.01rates in the library are overestimated is more likely. In the simulations in which the electron-

capture rates are reduced (dashed (x0.1) and dotted (x0.01) in figure 2.7) the final lepton

fraction significantly increases compared to the base calculation. The blue curves in figure 2.7

represent scenarios in which the electron-capture rates for all nuclei in the library are scaled,

whereas the red curves denote the results in which only the 74 nuclei in the high-sensitivity

region are scaled. Scaling all electron-capture rates in the simulation by a factor of 0.1 (blue
dashed line) produces a final lepton fraction of ∼0.314, an increase of 7.5% compared to the

base simulation. By scaling the electron-capture rates only in the high-sensitivity region (red
dashed line), the final lepton fraction is ∼0.303, a 3.8% increase from the base simulation.

The relatively strong effect of scaling the electron-capture rates in the high-sensitivity

region is also apparent in the simulation results for the electron fraction, entropy, stellar

density and in-fall velocity at core bounce, as a function of enclosed mass, as shown in figure

2.8. Here, the definition of tbounce by the authors of GR1D, namely that it occurs when the

entropy reaches a value of 3.0 [72], is followed. The variation in simulated characteristics

of these parameters, resulting from applying scaling factors of 10, 1, 0.1, and 0.01 to the

electron-capture rates on the nuclei in the high-sensitivity region only is indicated by the red

bands. The variation by scaling the electron-capture rates on all nuclei is indicated by the

black lines. As in the case of the lepton fraction, the effect of only changing the rates for the

nuclei in the high-sensitivity region accounts for about half of the changes observed when

scaling the electron-capture rates on all nuclei. It is likely that the nuclei in the pf -shell and

nuclei N = 82 region account for much of the additional variation observed for the simulation

in which the rates of all nuclei are scaled. Of the 8140 nuclei included in the core-collapse

supernova simulations, fewer than half play a role [88] and, of these, the 74 nuclei in the

high-sensitivity region are most important during the core-collapse and early post-bounce

25

Figure 2.8: Electron fraction, entropy, density and matter velocity at t = tbounce (defined
when the entropy reaches 3.0) for two sets of simulations. The bands indicate the range
of values obtained when the electron-capture rates are scaled in the high-sensitivity region
(red) or across the entire chart of nuclides (black). This figure was taken from [10].

phases.

It is important to note that figure 2.2 highlights two groups of nuclei that contribute

significantly to the change in electron fraction: those in the regions around N = 50 and N =

82. While this is true, electron-captures on the nuclei near N = 50 are much more important

26

0.200.250.300.350.400.450.50ElectronFraction1.02.03.04.05.06.0Entropy (kb/baryon)1081091010101110121013101415Density (g/cm3)0.00.20.40.60.81.01.2EnclosedMass (Msolar)−8−7−6−5−4−3−2−10Velocity (104 km/s)100.15alldiamondbase (x1)for the evolution of core-collapse supernovae. Simulations indicate that the nuclei at the N =

82 shell closure do not contribute significantly until just before core-bounce when the density

is extremely high and neutrino trapping has already occurred. Therefore, it can be concluded

that experimental and theoretical studies of Gamow-Teller strength distributions in the high-

sensitivity region around N = 50 are the most important objective for constraining the

weak-interaction inputs for simulations and for understanding the evolution of core-collapse

supernovae.

2.4 Sensitivity Study Conclusions

From the simulations of core-collapse supernovae performed here and in previous work, it is

clear that accurate electron-capture rates are important for understanding the dynamical evo-

lution of these cataclysmic events. Electron-captures on the 74 nuclei in the high-sensitivity

region around N = 50 just above doubly-magic 78Ni affect key characteristics, such as the

lepton fraction, electron fraction, entropy, stellar density, and in-fall velocity, by as much as

the other thousands of isotopes included in the simulation combined.

The electron-capture rates on nuclei in the high-sensitivity region that are currently

used in the weak-rate library are likely too high because the approximate method used to

estimate such rates is based on microscopically calculated rates for nuclei close to stability,

where Pauli-blocking effects are much weaker. Although Pauli-unblocking occurs at finite

stellar temperatures due to thermally-driven excitations of nuclei present in the star [65, 69],

the overestimation of the electron-capture rates could amount to an order of magnitude

or more. To reduce such large uncertainties, it is important to develop better estimates

which requires a combination of improved theoretical approaches and high-quality data to

27

benchmark new models as they become available. Charge-exchange experiments are used to

test theoretical predictions, but, as mentioned previously, only provide information about

transitions from the ground state of the parent nucleus. Significant uncertainties related to

the effects of finite temperature corrections on electron-capture rates still exist [65], but this

is not a facet that charge-exchange experiments are able to examine.

Because of the astrophysical motivation presented here, an experimental program fo-

cused on using charge-exchange reactions at intermediate energies to measure Gamow-Teller

strength distributions for the nuclei of primary interest is underway at the National Super-

conducting Cyclotron Laboratory. Experiments have been performed on several nuclei on

and around the N = 50 shell closure, including 96Mo(d,2He) [89], 90Zr(n,p) [90, 91, 92],

100Mo(t,3He) [93], and most recently, 88Sr(t,3He) [94] and 93Nb(t,3He) [95]. This thesis

presents the results of the 86Kr(t,3He) experiment, which rounds out the chosen group of

nuclei for this campaign. The experimental Gamow-Teller strength distributions will be

compared with theoretical calculations, such as shell-model and QRPA calculations, in order

to determine the validity of the current electron-capture rate sets, and to benchmark current

and future theoretical developments.

28

Chapter 3

Theory

3.1

Introduction to Charge-Exchange Reactions

Charge-exchange reactions are used to examine a wide variety of phenomena, encompassing

both nuclear structure and astrophysics. In a hadronic model, a charge-exchange reaction

occurs when a neutron in the target nucleus is exchanged with a proton in the projectile

nucleus, or vice versa. This process is mediated by the exchange of pions, or other mesons,

between the projectile and the target when the reaction occurs at intermediate energies [7].

Experimentally, the projectile may be either a single nucleon, as in the case of an (n,p) or

(p,n) reaction, or a composite probe, as in the case of (t,3He), (3He,t) or (d,2He).

Because they are mediated by pion-exchange, the strong nuclear force governs the me-

chanics of this reaction. However, such a probe also provides information about reactions

mediated by the weak nuclear force, namely β-decay, because both reactions populate the

same initial and final states. In β-decay experiments, states can be measured from 0 MeV

of excitation energy up to the Q-value of the reaction, and states with higher excitation

energy cannot be observed via this reaction. In contrast, charge-exchange reactions are not

limited in their scope by the reaction Q-value and can provide a complete spectrum up to

high excitation energies.

Charge-exchange reactions are characterized by an isospin transfer (∆T = 1), and are

29

divided into two groups, based on their spin transfer. Fermi transitions are identified by no

spin transfer (∆S = 0), while Gamow-Teller transitions are denoted by ∆S = 1. Both of

these types of transitions connect orbitals with the same principal quantum number, meaning

that they produce no change in orbital angular momentum (∆L = 0). Therefore, describing

these transitions in terms of the total angular momentum, J, Fermi transitions are denoted

by ∆J = 0 and Gamow-Teller transitions are denoted by ∆J = 1.

The Fermi transition strength is almost completely exhausted by the excitation of the

isobaric analog state (the state in the daughter nucleus with the same structure and isospin

as the ground state of the parent nucleus, but with a different isospin projection). However,

for neutron-rich systems, there is no isobaric analog state because the initial isospin is greater

than 0. Therefore, Gamow-Teller transitions are of most interest in this work because of the

associated spin transfer, which parallels that of β-decay reactions. The total amount of
Gamow-Teller strength, B(GT ), including both the β+ and β− directions, for a particular

nucleus is determined by the Ikeda sum rule [96]:

S(β−) − S(β+) = 3(N − Z),

(3.1)

where N is the number of neutrons present in the nucleus, Z is the number of protons, S(β−)
is the Gamow-Teller strength in the β− direction and S(β+) is the strength in the β+ direc-

tion. Typically, only about 60% of the Gamow-Teller strength is measured experimentally

because of quenching effects [97].

The operator that describes Gamow-Teller transitions is:

−→σjτ±j,

(3.2)

GA

(cid:88)

j

30

in which GA is the axial-vector coupling constant for the weak interaction, σ are the Pauli

spin matrices, and τ is the isospin transition matrix [8]. The relationship between this

operator and B(GT ) is given by the following equation:

B(GT )± =

1

2J + 1

(cid:12)(cid:12)(cid:12)(cid:88)

j

j (cid:107)ψI(cid:105)(cid:12)(cid:12)(cid:12)2

(cid:104)ψF(cid:107)σjτ±

.

(3.3)

Here, ψI and ψF are the initial and final nuclear states, respectively, and J is the total

angular momentum of the initial state.

As mentioned previously, charge-exchange reactions are characterized by an isospin trans-
fer (∆T = 1), meaning that ∆Tz is +1 or −1. The case in which ∆Tz = −1 corresponds to

a (p,n)-type reaction, causing the residual nucleus to become more proton-rich. Conversely,

the case in which ∆Tz = +1 corresponds to an (n,p)-type reaction, in which the residual

nucleus becomes more neutron-rich. A diagram of the directions of these reactions in the

region around 86Kr is shown in figure 3.1. In terms of astrophysical reactions, (n,p) reactions

are in the same direction as electron-capture reactions (β+-decay), shown by the red arrow
in figure 3.1, and (p,n) reactions are in the same direction as β−-decay reactions on the chart

of nuclides, shown by the green arrow.

The isospin of the target nucleus is given by the expression Tz = (N − Z)/2; this value

determines the states that can be populated in the daughter nucleus, as shown in figure

3.2.

If the isospin of the ground state of the target nucleus is T0, then reactions in the

(n,p) direction will populate only states in the residual nucleus with T = T0 + 1; this occurs

because the minimum isospin in the residual nucleus is equal to Tz. In the opposite direction,
(p,n)-type reactions can populate states with T = T0 − 1, T0 or T0 + 1.

31

Figure 3.1: Diagram of charge exchange reactions involving 86Kr. (n,p)- or (t,3He)-type
reactions involve an isospin change of ∆Tz = +1 and yield a daughter nucleus that is more
(p,n)- or (3He,t)-type reactions involve an isospin
neutron-rich than the parent nucleus.
change of ∆Tz = −1 and yield a daughter nucleus that is less neutron-rich than the parent
nucleus.

3.1.1 pf/sdg -shell Interactions

Charge-exchange reactions change a neutron into a proton, or a proton into a neutron, de-

pending on the direction of the reaction. Macroscopically, this is likened to exciting a proton-

hole/neutron-particle, in the case of a (p,n) reaction, or exciting a neutron-hole/proton-

particle, in the case on an (n,p) reaction. For stable, heavy nuclei the proton-neutron

asymmetry is high, typically with a large excess of neutrons. Because of this configura-

tion, excitations in the (n,p) direction within the same oscillator shell are impeded due to

the Pauli-blocking effect; this phenomenon is illustrated in the simplified diagram of a nu-

32

Figure 3.2: Isospin in charge-exchange reactions. The isospin of the ground state of the
parent nucleus is equal to (N − Z)/2. In the (n,p) direction, only transitions from T0 to
T0 + 1 states are available. Also shown in the figure are the Clebsch-Gordan coefficients
corresponding to each transition; in the (p,n) direction, transitions to states with increasing
isospin are suppressed compared to transitions to states with the same isospin. Figure
adapted from [98, 99].

cleus above 78Ni in figure 3.3. Protons are unable to transition to the low-lying neutron

orbitals, and, as such, the only available transitions have a higher excitation energy. Be-

cause the overlap between the initial proton orbital and final, higher-lying neutron orbital is

small, the Gamow-Teller strength in the β+ direction is reduced. For the same nucleus, the
Gamow-Teller strength in the β− direction is unaffected, because excitations of a neutron-

hole/proton-particle are available for all orbitals.

Nuclei in the pf /sdg-shell are an excellent example of the effect of such Pauli-blocking

on the Gamow-Teller strength in the (n,p) direction. 86Kr (Z=36) is a stable isotope, and is

singly-magic with 50 neutrons. For nuclei that lie on the N = 50 shell closure, the neutron

shells are filled up to the 0g9/2 orbital (a diagram of the ordering of the orbitals is shown in

figure 3.4). The protons, depending on the excitation of the nucleus, are distributed among

33

Figure 3.3: A representation of Pauli-blocking for heavy nuclei. Transitions with low ex-
citation energy are blocked because the necessary orbitals are filled; therefore, the avail-
able transitions are required to have higher excitation energy. Because transitions with
higher excitation energy are energetically unfavorable and, therefore, less likely to occur, the
Gamow-Teller strength is subsequently diminished.

the 1p3/2, 0f5/2, 1p1/2, and 0g9/2 orbitals, when a 78Ni core (Z=28) is assumed. Then,

Gamow-Teller transitions cause a spin-transfer of ∆L = 0 and ∆S = 1, moving the valence

protons from their original orbital in the target nucleus to a neutron orbital above 0g9/2 in

the residual nucleus.

In the shell-model, this configuration is achieved by using the SNE model space [75, 76],

which includes the orbitals of both the pf - and sdg-shells. The ordering of the levels is

shown in figure 3.4. For calculations in this region, an inert core of 78Ni (Z = 28, N = 50)

is assumed: all of the protons and neutrons in this core are fixed and cannot participate

in any transitions.

Instead, a mean field for the core is taken into account to define the

potential well of the nucleus. Using 78Ni as an inert core is a reasonable assumption because

the Gamow-Teller strength for the nucleus is extremely small [101]. The jj44pna interaction

[78] is used for these calculations within the shell-model calculation framework, NUSHELLX

34

Figure 3.4: Configuration of orbitals in the nuclear shell-model up to the sdg shell. Figure
adapted from [100].

[79].

Because the nuclei in the pf - and sdg-shells have a large number of valence particles and

possible transitions, it is necessary to make truncations to the model space, otherwise the

calculations become too computationally intensive. These constraints and their effect on

the available Gamow-Teller transitions are shown in figure 3.5. The left-hand panel of the

figure shows all available Gamow-Teller transitions in the pf/sdg-shell. The first model space

simplification is in the form of the core used in the calculation, as discussed previously. In the

case of 86Kr, the assumption of the 78Ni core leaves 8 valence protons that are distributed in

orbitals up to the 0g9/2, and are available to participate in Gamow-Teller transitions. There

are no valence neutrons because both 86Kr and 78Ni are N = 50 nuclei; therefore, all of

35

the neutrons are fixed in place by the core in this example. The center panel of figure 3.5

illustrates the reduction in the number of possible transitions when the core of the nucleus

is fixed.

The other truncation concerns orbitals above the sdg-shell; because transitions to higher-

lying orbitals are extremely unlikely, protons are prohibited from occupying them. For these

calculations, proton orbitals above 0g9/2 and neutron orbitals above 1d3/2 are omitted.

The right-hand panel of figure 3.5 shows this final model space truncation, and the further

reduction in available Gamow-Teller transitions in this model space. After all of these
restrictions, only one Gamow-Teller transition is available in the calculations: π0g9/2 →

ν0g7/2.

Figure 3.5: Simple explanation of the model space truncations used in OXBASH calculations.
The panel on the left shows all possible Gamow-Teller transitions, assuming nucleons are
free to move to any orbitals. The middle panel shows the application of a 78Ni core, filling
all of the neutron orbitals up to the g9/2 orbital, and then eliminating those particles from
the set that are available to participate in transitions. The panel on the right forces the
proton orbitals above the g9/2 to remain empty, because it is unlikely for protons to exist
there. Both of these restrictions lead to a single remaining transition: π0g9/2 to ν0g7/2.

36

In order to determine whether these shell-model restrictions are acceptable, it is neces-

sary to compare the Gamow-Teller transition strength distribution obtained from shell-model

calculations to the strength distribution extracted from a charge-exchange experiment [93].

Then, discrepancies between the experimental and theoretical strength distributions, either

in the details of the distribution or the cumulative strength, indicate ways in which theo-

retical models can be improved. Attributes, such as the deformation of the nucleus or the

accuracy of the single-particle energies, incorporated into the theory yield a more robust

the theoretical framework, leading, in turn, to more accurate experimental Gamow-Teller

strength distributions.

3.2 Reaction Theory

Cross section calculations for (t,3He) experiments are performed with the double-folding

code, FOLD [102, 103]. FOLD uses Distorted Wave Born Approximation (DWBA) methods,

in which both the incoming and outgoing waves are distorted by the mean field of the

target nucleus. Inputs to the calculation include single-particle wavefunctions and one-body

transition densities for the reaction in question, a nucleon-nucleon interaction and an optical

potential. The code uses a three step process to calculate the incoming and outgoing distorted

waves, the transition matrix and the cross section.

3.2.1 DWBA

In the DWBA method, the scattering potential is divided into two parts: one with a known

solution, typically an elastic scattering potential, and one that contains any residual inter-

37

actions. Here, the residual interaction is the charge-exchange interaction:

V = Uelastic + Wcharge-exchange

(3.4)

A T-matrix is used to describe the transition between the initial and final states in the

reaction. When the potential is substituted into the T-matrix it is similarly split into two

parts [104, 105]:

Tf i = (cid:104)φ|T +|φ(cid:105) = (cid:104)φ|Uelastic + Wcharge-exchange|ψ+(cid:105)

Tf i = (cid:104)φ|Uelastic|χ+(cid:105) + (cid:104)χ−|Wcharge-exchange|ψ+(cid:105)

(3.5)

(3.6)

where |χ+(cid:105) and |χ−(cid:105) are the incoming and outgoing distorted waves.

The first term in equation 3.6, containing the elastic scattering potential, is neglected,

because the operator is isoscalar and is unable to connect the initial and final states of such

a reaction. Therefore, only the term containing the charge-exchange interaction remains to

be evaluated. The elastic potential is still used, however, to generate the distorted waves.

Including the distorted waves, the simplified transition matrix is now written as [104, 105]:

Tf i = (cid:104)χ+

f ((cid:126)kf , (cid:126)R(cid:48))|F ( (cid:126)R(cid:48))|χ−

i ((cid:126)ki, (cid:126)R(cid:48))(cid:105).

(3.7)

Here, F ( (cid:126)R(cid:48)) is the form factor, which provides information about the interaction between the

nucleons in the projectile and the nucleons in the target. It is a folded transition density, in

which the overlap between the initial and final states is calculated, while taking into account

the projectile-nucleus interaction [104, 105]. The first step of the FOLD code, called WSAW,

38

calculates the radial part of the single particle wavefunctions, using the nucleus charge, core

mass and binding energies of the protons and neutrons in valence orbitals that participate

in the reaction. Then, the second part of the FOLD code, called FOLD, calculates the form
factor, F ( (cid:126)R(cid:48)) from equation 3.7, using these wavefunctions, along with the energy of the

incoming particle, an effective interaction and one-body transition densities. Double-folding,

or integrating over the nucleons in both the target and projectile, is necessary because the

probe (a triton, in this case) is not a point particle.

The form factor, when written in its entirety, can be reduced to two parts: one that

depends only on the spin and isospin coordinates ((cid:126)σ and (cid:126)t), and another that depends

only on the spatial coordinates ((cid:126)r). The spatial coordinates used here are defined in figure
3.6. The potential contained within the form factor, F ( (cid:126)R(cid:48)), is equal to the charge-exchange
portion, as defined in equation 3.4: Wcharge-exchange = V − Uelastic. As stated previously,

because the interaction of interest is isovector, the elastic term, Uelastic, is neglected, as it is

isoscalar. This leaves only the term that depends on V , the sum of all two-body interactions

between the target and projectile, given by:

(cid:88)

V =

V (|(cid:126)ri

(cid:48) − −→rj|, (cid:126)σi

(cid:48)

(cid:48), (cid:126)σj, (cid:126)ti

, (cid:126)tj),

(3.8)

i,j

where i and j refer to the particles in the target and projectile that are being summed over,

σ is the spin and t is the isospin of the particle in question, and the spatial coordinates are

defined in figure 3.6.

If the projectile is assumed to carry no orbital angular momentum, then the two portions

of the wavefunctions (spatial and spin/isospin) can be completely separated by re-writing the

form factor using an effective potential, Vef f , which characterizes the interaction between a

39

Figure 3.6: Definition of the spatial coordinate system used in the DWBA formalism. This
figure was adapted from [105].

nucleon in the target and a nucleon in the projectile at −→r (cid:48) and −→rj

(cid:48), respectively [104, 105]:

(cid:90)

Vef f ( (cid:126)R(cid:48), (cid:126)r, (cid:126)σ, (cid:126)t, (cid:126)σj

(cid:48), (cid:126)tj

(cid:48)

) =

d(cid:126)r(cid:48)d(cid:126)s(cid:48)ff ((cid:126)r(cid:48), (cid:126)s(cid:48))fi((cid:126)r(cid:48), (cid:126)s(cid:48))V (| (cid:126)rj

(cid:48) − (cid:126)r|, (cid:126)σi

(cid:48), (cid:126)σj, (cid:126)ti

(cid:48)

, (cid:126)tj),

(3.9)

where V is the sum of the two-body interactions, as defined in equation 3.8, and fi and ff are

the components of the internal wave function of the projectile and ejectile, respectively, that

depend only on the spatial coordinates. With the effective interaction, defined in equation

3.9, the form factor can be re-written using the following expression:

F ( (cid:126)R(cid:48)) = (cid:104)αeαr|Vef f|αtαp(cid:105),

(3.10)

where αe,r,t,p are the wavefunctions of the ejectile, residual, target and projectile, respec-

tively, and Vef f is the effective interaction between nucleons in the target and projectile.

Combining all of this information for the target and projectile of interest, the transition

matrix element, Tf i, is generated, and is used to calculate the differential cross section for

40

the reaction:

(cid:16) µ

(cid:17)2 kf

2π¯h2

ki

dσ
dΩ

=

|Tf i|2,

(3.11)

where µ is the reduced mass of the system, and ki and kf are the incoming and outgoing wave

numbers. The final part of the FOLD code, called DWHI, calculates the transition matrix

elements, Tf i from equation 3.7, using the previously-calculated form factor (equation 3.10),

along with a suitable optical potential, which is used to generate the distorted wavefunctions.

Finally, with the matrix elements, DWHI calculates the cross section for the reaction, after

equation 3.11.

3.2.2 Calculation inputs

Several inputs are needed to perform DWBA calculations for the 86Kr(t,3He) reaction. First,

an effective interaction is needed to calculate transition matrix elements; such calculations

are complex and depend on a large number of parameters, including the energy of the pro-

jectile and the structure of the target nucleus. Charge-exchange experiments are performed

at intermediate energies, typically greater than 100 MeV/u, and a free nucleon-nucleon in-

teraction is a reasonable approximation. In this assumption, all of the nucleons in the target

and projectile may be distinguished from one another, and non-locality effects are neglected.

The interaction typically used to calculate (t,3He) reaction cross sections is the Love-

Franey interaction at 140 MeV/u [106, 107, 108]. Consisting of three components, the

potential used in this framework contains central, spin-orbit and tensor terms:

V12 = V c(r12) + V LS(r12)

−→
−→
S + V T (r12)S12.
L˙

(3.12)

41

Each of the components is decomposed in terms of Yukawa potentials, Y (x) = e−x

x , which

are similar to the one-pion exchange potential (OPEP):

V C (r) =

V C
i Y (r/Ri)

i=1

NC(cid:88)
NLS(cid:88)
NT(cid:88)

i=1

V LS(r) =

V LS
i Y (r/Ri)

(3.13)

V T (r) =

V T
i Y (r/Ri)

i=1

where the sums run over the ranges of the π, ρ and 2-π meson exchanges, respectively. The

range of the π meson exchange at 140 MeV/u is known to be 1.4 fm, while the ranges of the

ρ and 2-π meson exchanges are assigned a value based on fits to nucleon-nucleon scattering

data [106].

Each of the components in the NN-interaction are represented by two terms: direct and

exchange. The direct term describes the case in which a neutron is turned into a proton,

or vice versa, via meson exchange. The exchange term describes the case in which the

target neutron (proton) is struck and ejected from the nucleus, and the projectile proton

(neutron) is captured and replaces it [8]. While the direct terms can be calculated without

the need for simplifications or approximations, the exchange terms are less straightforward.

These terms contain non-local effects, which are not calculated exactly in FOLD; instead, a

short-range approximation is used [106]. While this approximation is acceptable for simple

charge-exchange probes, such as (n,p) reactions, for more complex probes, such as (t,3He)

reactions, the exchange term is known to be too weak [109, 110, 111]. Because the direct and

exchange components interfere destructively, this leads to an overestimation in the calculated

cross section, typically on the order of 30% [12, 110]. However, because the experimental

42

analysis, discussed in sections 4.5 and 4.6, only uses the shape of the calculated DWBA cross

section, not its magnitude, the validity of the short-range approximation and the effect on

the magnitude of the cross section are not considered in great detail here.

The next nuclear physics input to the DWBA calculation is the optical potential, which is

used to determine the extent to which the incoming and outgoing waves are distorted by the

mean field of the target nucleus. This mean field takes the form of a Woods-Saxon potential

with both a real and an imaginary component, combined with the Coulomb interaction [112]:

V = VC + U + iW.

(3.14)

The Coulomb term of the potential takes the following forms:

VC (r) = ZpZT e2/r
VC (r) = ZpZT e2(3 − r2/R2

for

r > RC

C )/2RC for

r < RC ,

(3.15)

where RC = rC A1/3, and rC is the reduced Coulomb radius. Each of the Woods-Saxon

components describing the potential well depth takes the following form [105]:

V (r) =

V0

1 + [exp(r − rnA1/3)/an]

(3.16)

where V0 is the depth, rn is the radius and an is the diffuseness of the potential. To obtain

values for these parameters, fits of elastic scattering data on the nucleus of interest are made,

and the constants are extracted, based on the shape of the calculated angular distribution.

Elastic scattering data is unavailable for every nucleus, and there is a dearth of available data

in the middle- to heavy-mass region; therefore, at times, it is necessary to use the optical

43

potential for a nucleus that is merely close in mass to the target nucleus. For nuclei such as

86Kr or 88Sr, the optical potential for 90Zr is used [113].

Elastic 3He scattering data on 90Zr at 443 MeV were used to obtain the optical poten-

tial parameters. The extracted values correspond to the 3He channel, which, for a (t,3He)

reaction, is the outgoing channel. In order to obtain the distorted wave for the incoming

triton, the real and imaginary potential depths from the 3He reaction are scaled to 85% of

the original value [114]; the other parameters remain the same. The optical potential values

for both the incoming triton and outgoing 3He waves are presented in table 3.1.

Table 3.1: Optical potential parameters used in the DWBA calculations for nuclei in the
pf /sdg-shells. These values were determined from 3He elastic scattering on 90Zr at 443 MeV
[113].

VR

rR

aR

WI

rI

aI

Incoming triton -26.5 MeV 1.363 fm 0.818 fm -35.8 MeV 1.04 fm 1.06 fm
Outgoing 3He
-32.1 MeV 1.363 fm 0.818 fm -42.1 MeV 1.04 fm 1.06 fm

The final inputs to the FOLD code are one-body transition densities (OBTDs), typi-

cally from shell-model or other microscopic calculations, which provide structure information

about the target and residual nuclei. Additionally, each 1p-1h amplitude that is included in

the calculation is weighted depending on the OBTD used: smaller transition densities indi-

cate a small overlap between the initial and final states, meaning that such a transition is less

probable. As mentioned in section 3.1.1, for the middle-mass nuclei in the pf /sdg-shell, only

one transition contributes to the Gamow-Teller distribution: π0g9/2 to ν0g7/2. One-body

transitions densities for 86Kr and 88Sr were calculated in OXBASH [115] in the SNE model

space [75, 76] using the SNET interaction [76]. For the experimental analysis that uses the

DWBA result, discussed in further detail in section 4.6, the interaction used to generate the

one-body transition densities is less important. It is the shape of the angular distribution

44

that is critical, and can be obtained using one-body transitions densities from any reliable

shell-model interaction.

OBTDs obtained from the shell-model calculation are converted from their original pn-

formalism value to a format to follow the reduced matrix element convention [115]:

Z = OBTD × CG ·(cid:99)dT
(cid:98)Ji ·(cid:99)Tf

(3.17)

where (cid:98)Ji is equal to

√

2Ji + 1, where Ji is the spin of the initial state. (cid:99)dT and (cid:99)Tf are

similarly defined for the change in isospin and isospin of the final state, respectively. CG

refers to the Clebsch-Gordan coefficient for the transition, and is defined by the following

expression:

CG = (cid:104)Ti∆T ; Tim∆Tm|Ti∆T ; Tf Tf m(cid:105),

(3.18)

where Ti,f are the total isospins of the inital and final states, Tim and Tf m are the isospin

projections of the initial and final states, and ∆T and ∆Tm are the changes in the total

isospin and the isospin projection, respectively, for the reaction.

3.3 Gamow-Teller Strength Distribution

There is a well-known proportionality between the Gamow-Teller transition cross section

and B(GT ) [12, 51, 52], given by the following equation:

=(cid:98)σB(GT ).

(3.19)

(cid:34)

dσ
dΩ

(cid:35)

(q = 0)

GT

45

The left-hand side of the equation is the Gamow-Teller cross section at zero momentum

transfer (q = 0), obtained in theoretical calculations by setting the Q-value of the reaction

to 0 MeV and taking the cross section at zero degrees. Experimentally, the cross section that

is measured is a combination of all possible charge-exchange transitions. In order to extract

the ∆L = 0 component, a multipole decomposition analysis is performed by fitting the

experimental angular distribution with a linear combination of multipole curves, as derived

from a DWBA calculation. The ∆L = 0 component obtained from this fit is used to calculate

the cross section at zero momentum transfer (q = 0), which appears on the left-hand side of

equation 3.19.

The constant of proportionality is called the unit cross section and, for Gamow-Teller

transitions, is approximately constant for all transitions. (cid:98)σ is calculated using the following

expression:

(cid:98)σ = KN D|Jστ|2.

(3.20)

Here, K is a kinematic factor, depending on the energies of the incoming and outgoing chan-

nels, the incoming linear momentum of the projectile, and the outgoing linear momentum of

the ejectile. N D is a distortion factor, representing the effect of the mean field of the target

nucleus on the scattering waves. At zero momentum transfer, the distortion factor is defined

as the ratio of the DWBA cross section to the PWBA (partial-wave born approximation)
cross section. Finally, |J| is the integral of the στ component of the effective interaction

between the projectile and target nucleons. For experimental data, the unit cross section

can be calculated using an expression that is dependent only on the mass number of the

target nucleus: (cid:98)σ = 109/A0.65. This equation provides a reasonable value for (cid:98)σ for nuclei

with A ≥ 12, as shown in previous studies [12, 51, 116, 117, 118].

46

The Gamow-Teller strength distribution is a useful result in its own right, yielding infor-

mation about the structure and configuration of the nucleus. When obtained experimentally,

it can be compared to theoretical models in order to discern which models replicate exper-

imental results well, and where improvements can be made. Additionally, the strength

distribution, obtained from theoretical calculations or experimental data, can be used to

calculate weak-reaction rates, which are of great interest in this work, and are discussed in

greater detail in chapter 5. Following the discussion in chapter 2, electron-capture rates for

a nucleus are known to be sensitive to its Gamow-Teller strength distribution. Because so

many nuclei participate in simulations of astrophysical phenomena, and because it is impos-

sible to measure them all experimentally, being able to obtain Gamow-Teller strengths and

electron-capture rates by other methods, while still retaining accuracy, is an invaluable tool.

47

Chapter 4

Experiment

The 86Kr(t,3He+γ) experiment was performed at the National Superconducting Cyclotron

Laboratory at Michigan State University. Production of a triton beam by the facility has

been used extensively by the charge exchange group for studying the Gamow-Teller strength

distribution for nuclei in the β+ direction [15, 48, 50].

4.1 Experimental Setup

4.1.1 Beam Production

The triton beam at the NSCL was produced by initially creating a primary beam of 16O at

150 MeV/u using the K500 and K1200 coupled cyclotrons, and impinging it on a beryllium

production target with a thickness of 3525 mg/cm2. This method was determined to be op-

timal by Hitt et al. [15] to produce a secondary triton beam with an energy of approximately

115 MeV/u. To remove 6He and 9Li contaminants from the triton beam, an aluminum wedge

with a thickness of 195 mg/cm2 was inserted into the A1900 fragment separator [14]. The
slits in the A1900 were set to obtain a momentum acceptance of ±0.25% (0.5% total). After

the beam was separated with the desired purity (>99%), it was transported to the target

position and the S800 magnetic spectrometer [16], located in the S3 vault at the NSCL. The

Coupled Cyclotron Facility, A1900 fragment separator, and beam line leading to the S800

48

spectrometer are shown in figure 4.1.

Figure 4.1: Schematic of the Coupled Cyclotron Facility, the A1900 fragment separator and
the beamline to the S3 vault and the S800 magnetic spectrometer.
Image credit to the
NSCL.

For this experiment, the analysis beam line was operated in dispersion-matched mode to

obtain the best possible resolution in the focal plane of the spectrometer. During dispersion-

matching tuning, the beam was focused in the non-dispersive direction and momentum-

dispersed along the dispersive axis, creating a tall, thin beam spot. The beam line was tuned

to yield a beam that was momentum-distributed across the target with a dispersion equal to
that of the S800 spectrometer (DS = −9.536 cm/%). Then, when the beam particles passed

through the spectrometer, its intrinsic dispersion compensated for the initial dispersion of

49

the beam, focusing all of the beam particles to a single point in the focal plane, as shown

in figure 4.2b. The process of lateral dispersion-matching optimizes the measured energy

resolution of the ejectile particles in the focal plane, improving the energy resolution that is

calculated for the recoil particle.

Figure 4.2: A visual explanation of dispersion matching within a high-resolution spectrom-
eter. a) shows the focus mode, in which the beam is focused on the target, leading to a
large dispersion of ejectiles in the focal plane. b) and c) show dispersion matching with only
lateral, and both lateral and angular dispersion-matching, respectively. By tuning the beam
to have a dispersion that is the same as that of the spectrometer, the energy resolution in
the focal plane of the spectrometer is improved greatly. This figure was taken from reference
[119].

With the ±0.25% momentum acceptance, which allowed for complete transmission of the

beam through the beamline, a beam spot that was approximately 5 cm tall in the dispersive

direction and approximately 0.5 cm wide in the non-dispersive direction was produced. In

order to direct 3He ejectiles into the focal plane of the S800, the rigidity of the spectrometer

was set to 2.32 Tm. During beam tuning, a viewer, made of an aluminum plate coated in

red phosphorus, which scintillates when the beam impinges upon it, was placed immediately

in front of the target position to check the position of the beam and the shape of the beam

spot. An image intensifying camera was used to see the shape of the beam spot on the viewer

50

during secondary tuning, as shown in figure 4.3.

Figure 4.3: An image of the beam spot on the target viewer during dispersion-matched
tuning. The dispersion of the tall beam matches that of the spectrometer, such that this
beam will become focused to a single point in the focal plane of the S800. Photo courtesy
of the NSCL A1900 group.

4.1.2 Krypton Target

The krypton gas cell was attached to a simplified version of the Ursinus College Liquid

Hydrogen Target [120]. The cell had a diameter of 7 cm and a depth of 3 cm, and was

fabricated specifically for the experiment to accommodate the dispersion-matched beam spot;

it is shown in figure 4.4. For the krypton production runs, the target cell was filled with 86Kr

gas with a purity of 99.952%. The target windows were made of kapton foil (C22H10N2O5)

with a thickness of 125 µm. In past charge-exchange experiments, a CH2 target was used to

assist in the calibration and normalization of the data [48, 116, 121, 122]. In this experiment,

however, the kapton contained carbon in sufficient quantities that calibrations were able to

be performed using the production run data sets.

51

During the beam time, it was found that, due to the wide spacing of the target windows,

events from each foil produce distinct peaks in the spectra, as opposed to experiments that

have thin target foils, in which all of the events from the target windows fall into a single

peak. Such behavior was observed because of the uncertainty of the event locus. The ray-

tracing routine used in the analysis, discussed further in section 4.2.4, assumed that all events

originated in the center of the target; this assumption is typically valid for experiments with

thin targets because the deviation between the actual event position and the assigned position

are within the experimental position resolution. However, because of the thickness of the

target in this experiment, the deviation between the actual event location and the calculated

event location was greater, leading to the double peak in the spectrum, as events from the

upstream foil were placed behind their actual locus, and events from the downstream foil

were placed in front of their actual locus.

Because such a phenomenon complicates the extraction of the krypton signal (section

4.4), two additional sets of data were taken in order to have a better understanding of the

contributions of the target windows to the total measured cross section. The first set had

only the upstream foil of the gas cell present in the target position, and the second had only

the downstream foil of the gas cell present. Because the total cross section from events on

the target windows is the sum of the contributions of each foil with some additional energy

loss corrections, these two additional data sets were used to develop a model of both the

empty target cell (as proof that the subtraction method was reasonable) and of the krypton

cell. The method of subtracting the target window events from the krypton data is discussed

in detail in section 4.4.

It was originally proposed to fill the cell to a pressure of 760 Torr and then cool the cell to

a temperature of 120 K, to increase the target thickness in order to obtain sufficient statistics.

52

However, while testing with the target system with neon gas, it was determined that the

target windows would not hold against the gas pressure at low temperatures, because of the

large surface area of the cell. Instead, the gas cell was kept at a temperature of 295 K, and

then filled with 86Kr gas to a pressure of approximately 1210 Torr. Although there were

some small pressure fluctuations, it can be assumed that the density of the target remained
effectively constant throughout the experiment with a thickness of 20 ± 0.5 mg/cm3.

Figure 4.4: Left: The kapton foils curing on a vacuum ring to ensure the seal is strong.
Right: The freshly-glued and assembled krypton gas cell attached to the gas handling and
temperature regulation system.

4.1.3 S800 Magnetic Spectrometer

The S800 magnetic spectrometer consists of two dipole magnets and two quadrupole magnets

that are used to direct 3He ejectiles into the focal plane. Particles pass through two Cathode

Readout Drift Chambers (CRDCs) providing information enabling the reconstruction of

position and angle measurements for each event. The CRDCs have an active area of 56 cm

53

in the dispersive direction by 26 cm in the non-dispersive direction. CRDC1 is placed at the

optical focal plane and CRDC2 is located 1 m downstream from the focal plane [123].

Each detector is filled with a gas mixture that is 80% CF4 and 20% C4H10; this mixture

is used because of its high drift velocity and low avalanche electron spread [123]. Particles

entering the detectors ionize the gas and produce ion-electron pairs; the electrons drift toward

the anodes in the CRDC, generating a current in the wires closest to the locus of the event.

This current causes a charge to be induced on two sets of cathode pads, placed before and

after the anode wires. The charge distribution is spread across approximately 8 pads, and

a Gaussian fit to the distribution was used to determine the position of the event in the

dispersive direction; the position in the non-dispersive direction was obtained from the drift

time of the electron to the anode wire.

In order to calculate the angle of each particle

track through the CRDCs, the line connecting the position of the event in each CRDC was

calculated and the angle at which the particle was traveling was extracted, as shown in figure

4.5.

A plastic scintillator was placed behind the CRDCs to measure the energy loss of the

ejectile particle after it traveled through the spectrometer. Additionally, the signal in the

scintillator provided a trigger for the S800 data acquisition system, and was used as the

starting time for the time-of-flight measurement; the cyclotron RF signal yielded the stopping

time. The difference between the start and stop times is a measure of the time-of-flight of the

ejectile from the target position to the focal plane of the spectrometer. Both the energy loss

measured in the plastic scintillator and the time-of-flight were used for particle identification,

which will be discussed in more detail in section 4.2.1.

The CRDCs have inherent noise that must be eliminated so that the event amplitude,

used during event fitting to determine the location of the event, is accurate. For this,

54

Figure 4.5: A diagram of the event tracking performed by the CRDCs in the focal plane of
the S800. The horizontal position of the hit is determined by the pattern of pads that fire
during the event; the vertical position is determined by the drift time of the electrons in the
chamber. This figure was obtained from [123].

data were taken when there was no beam present in the spectrometer, and the spectra for

the CRDCs were analyzed. Any peaks in the spectra (figure 4.6) were attributed to noise

in the cathode pads. The pedestal for the spectrometer is a set of constant values, each

corresponding to a pad in the CRDCs; the value indicates how much noise the pad typically

measures. During event fitting, the pedestal value was subtracted from the recorded pad

charge to provide the true energy measurement for the event.

4.1.4 GRETINA

GRETINA is the first stage of the Gamma-Ray Energy Tracking Array (GRETA), built at

Lawrence Berkeley National Laboratory (LBNL) in 2011 and moved to different institutions

(e.g. LBNL, NSCL and Argonne National Laboratory (ANL)) to complete a range of exper-

55

Figure 4.6: Diagram of the pedestal data for the first CRDC; these data were taken with
no beam in the spectrometer. The peaks are characterized and a pedestal is extracted for
each pad, in order to subtract the pad noise from the experimental data, yielding the most
accurate energy measurement for each event.

imental campaigns. When it is complete, the array will cover nearly 4π solid angle with 20

detector modules.

During the experiment, GRETINA consisted of 8 detector modules, each containing four

high-purity germanium crystals and a cryostat. The germanium crystals are segmented

into 36 electrically-isolated elements, and are used because they provide a precise location

of a photon interaction within the detector [124]. With the presently available modules,

GRETINA covers one quarter of a sphere as a result of its closely-packed geometry. Figure

4.7 shows GRETINA installed in the S3 vault at the NSCL with the S800 spectrometer

56

during a previous experimental campaign.

Figure 4.7: A photo of GRETINA set up with the S800 spectrometer. Photo credit to S.
Noji from FRIB.

GRETINA is capable of detecting γ-rays with high resolution, and because of the seg-

mented nature of the germanium detectors, doppler-reconstruction of events is available for

in-beam spectroscopy experiments [125, 126].

In the 86Kr(t,3He+γ)86Br experiment, no

doppler-reconstruction was needed, because the interest was for γ-rays from stopped 86Br

or its decay products. It was expected that the Gamow-Teller strength associated with 86Kr

would be weak, based on the calculations presented in chapters 2 and 3. The array detects

the γ-rays from the decay of the excited residual nucleus, in this case 86Br, and by using

these γ-rays to gate on the excitation energy spectrum, it became easier to differentiate the

events from 86Kr from the other events present in the spectrum. The efficiency of GRETINA

was measured prior to the experiment using a 152Eu source. Figure 4.8 shows the efficiency

57

curve obtained using these data. It is necessary to incorporate the efficiency of the detector

array into the cross section for each γ-ray examined.

Figure 4.8: The efficiency curve of the GRETINA detector array. The 152Eu data points,
measured prior to the experiment, are fit with a curve so that the efficiency of the detector
at all γ energies can be determined.

Using the gas target system and GRETINA in the same experiment presented a unique

challenge because space around the target position is at a premium. All of the germanium

crystal modules were placed in the northern hemisphere so that the southern side of the

target would be clear for the gas target system, shown in the left-hand side of figure 4.9.

Additionally, the target system was affixed to the beam pipe at an angle of 30 degrees from

the vertical and supported by a chain hoist, shown in the left-hand picture in figure 4.9. A

port was machined into the side of the beam pipe in the horizontal plane at an angle of 45

degrees from the beam axis to allow the image-intensifying camera to have a clear view of

the target position. The camera was used during the secondary tuning process so that the

58

Energy [keV]0200400600800100012001400Absolute singles efficiency0.040.060.080.10.120.140.16Efficiency Fitting Functionlocation and shape of the beam spot could be verified, as discussed in section 4.1.1.

Figure 4.9: Left: Photo of the single-hemisphere GRETINA configuration with the custom
beam pipe and krypton gas target system installed. Right: Photo of the krypton cell inside
the beam pipe; the 30 degree angle of the target system, necessary in order to accommodate
the germanium detectors, is evident.

4.2 Experimental Analysis

4.2.1 PID Calibration

The triton beam delivered to the target position could interact with several isotopes, includ-

ing hydrogen, carbon, nitrogen and oxygen in the kapton target windows, and krypton when

the target cell was filled. Additionally, although the experiment aimed to measure charge-

exchange reactions, any type of reaction could occur between the beam and target particles,

59

producing many different ejectiles that traveled to the focal plane of the spectrometer. In

order to select for the proper type of reaction in the analysis, the particle identification

(PID) spectrum was examined. The PID was generated using the energy loss measured by

the plastic scintillator in the focal plane of the S800, and the time-of-flight of the particles,

calculated from the cyclotron RF and the time of a recorded hit in the spectrometer. The

initial, uncorrected PID plot from this experiment is shown on the left in figure 4.10.

Figure 4.10: The Particle Identification plot shows the energy deposited in the plastic scin-
tillator on the y-axis versus the time of flight of the particle on the x-axis. The large, bright
spot consists of 3He ejectiles from charge-exchange reactions. After applying corrections to
the focal plane position, scattering angle and energy deposition, the 3He spot in the PID
plot is much sharper and easier to separate from the various background events.

To ensure that only the charge-exchange reactions were examined in the analysis, a gate

was made on the 3He ejectiles in the focal plane, shown by the bright spot in the right-

hand spectrum in figure 4.10. Corrections, based on the dispersive position and angle of the

particles in the focal plane, were applied to the energy loss and time-of-flight of the particles.

These corrections sharpened the PID, as observed on the right in figure 4.10, improving the

signal-to-noise ratio within the 3He gate and reducing the number of background events

present in the analysis. The gate only separated the charge-exchange reactions from the

60

Time of Flight [ns]20800209002100021100212002130021400215002160021700Energy Loss [MeV]050100150200250300350400020406080100Particle Identification - uncorrectedTime of Flight [ns]20800209002100021100212002130021400215002160021700Energy Loss [MeV]0501001502002503003504000102030405060Particle Identification - correctedbackground events, including unreacted tritons that scattered off of the S800 dipole magnets,

or ejectiles from other triton reactions, such as (t,d) or (t,p), present in the spectrum [127].

A full background subtraction, to separate the 86Kr charge-exchange events from the charge-

exchange events on the target window nuclei, will be discussed in section 4.4.

For this analysis, the exact size and shape of the 3He gate is not of great importance

because of the way in which the data are normalized. Charge-exchange reactions on 12C are

present across all of the data, due to the kapton target windows. Because the cross section

for 12C(t,3He) reactions is well-known, the current 12C data were normalized to previous

experimental data so that all of the information about the 3He gate and the detector efficiency

was already included in the normalization factor. Then, when the 86Kr events were examined,

the normalization factor was based on the previously-determined 12C normalization and,

therefore, included the efficiency information without requiring further analysis.

4.2.2 Mask Calibration

In order to obtain an acceptable position resolution, the position of the CRDCs in the focal

plane of the S800 needed to be calibrated, because the exact location of the focal plane with

respect to the rest of the experimental setup is not known. For this calibration a tungsten

mask with precisely positioned holes and slits was placed in front of each CRDC in turn,

while the empty target cell and viewer were situated at the target position. The mask was

not thick enough to prevent 3He particles from passing through. However, punching through

the mask caused significant energy loss, compared to the particles that passed through one of

the holes in the mask. Ejectiles continued through the focal plane to the hodoscope, placed

downstream of the plastic scintillator, which measured the total remaining kinetic energy of

the implanted nuclei using 32 sodium-doped cesium iodide crystals [128].

61

A gate was made on the hodoscope energy spectrum corresponding to the high-energy

3He particles, neglecting those particles with reduced energy from passing through the mask.

With this method, only the particles that passed through a hole in the mask were included

in the spectrum, and the mask pattern could be seen clearly, as shown in figure 4.11. The

known locations of the holes and slits in the mask were matched to the position in the CRDC

spectrum, yielding calibration parameters to obtain the precise position of the ejectiles in

the focal plane.

The mask runs, one each for the upstream and downstream CRDCs, were taken at the

beginning of the experiment, and provided an initial calibration of the CRDC x- and y-

position offsets. The non-dispersive slopes, as mentioned in section 4.1.3, were determined

from the drift velocity of the electrons in the CRDCs, and could change over time. A drift

velocity correction was applied on a run-by-run basis to account for these fluctuations, and

will be discussed further in the next section.

4.2.3 Drift Velocity Correction

The drift velocity in the CRDCs is the speed at which free electrons travel from their point

of origin to the anode wire. Because the drift velocity changed over time during the exper-

iment, it was necessary to correct for these changes in each production run. Without these

corrections, the position resolution in the focal plane is decreased and larger uncertainties

would be introduced into subsequent steps of the analysis. The correction took the form of

a change in the drift velocity in the non-dispersive direction; the correct velocity was deter-

mined by finding the value that placed the peak of the hydrogen kinematic curve in each

spectrum at the same y-position as the first production run after the mask calibration. After

these corrections were applied to each run, the peaks of the hydrogen curve are aligned, as

62

Figure 4.11: The image of the mask in the first CRDC. The positions of the holes and slits
are calibrated in order to determine the exact position of the CRDC in space.

shown in figure 4.12.

4.2.4 Scattering Angle Correction

A map of the magnetic field of the S800 spectrometer was used to track particles from their

position in the focal plane back to their initial location at the target position. This inverse

map was generated using the ion-optical code, COSY Infinity [129].

Inputs to this code

include the currents in the quadrupole and dipole magnets, the rigidity of the spectrometer,

and the mass and charge of the ejectile of interest; an inversion matrix up to the fifth order

was produced. Initially, the 3He ejectile had an associated dispersive position (xfp), non-

dispersive position (yfp), dispersive angle (afp) and non-dispersive angle (bfp) in the focal

plane of the S800. A ray-tracing calculation implemented in the analysis code used these

parameters, along with the magnetic field map, to determine the non-dispersive position

63

CRDC1 horizontal position [cm]300-200-100-0100200300CRDC1 vertical position [cm]100-80-60-40-20-02040608010000.050.10.150.20.250.30.350.40.450.5CRDC mask patternFigure 4.12: This plot shows the x- versus y-position of the ejectiles in the focal plane of the
spectrometer. Because of differences in scattering energy, the ejectiles are separated by the
spectrometer into a number of bands. The curved line on the left represents reactions on
hydrogen nuclei; the straight lines in the middle are from reactions on carbon, oxygen and
nitrogen nuclei. Any ejectiles from reactions on krypton are lost in the background at this
point.

(yta), dispersive scattering angle (ata), non-dispersive scattering angle (bta) and momentum

of the scattered particle (dta) at the locus of the scattering event.

Due to the size of the 86Kr cell, with the target windows separated by at least 3 cm,

the exact location of the scattering event, calculated via the inverse map, was impossible

to resolve. The target windows were far apart and events could occur in either of them, so

determining which window from which an event originated was not feasible. Instead, the

inverse map was calculated using the approximate position of the center of the target cell.

Because of imperfections in the ray tracing, a noticeable curve was present in the hydrogen

kinematic line, shown on the left in figure 4.13. Hydrogen kinematics are shown, in this

case, because the reconstruction of the excitation energy depends most sensitively on the

64

CRDC1 horizontal position [cm]200-100-0100200CRDC1 vertical position [cm]100-50-0501000510152025303540Drift chamber positionFigure 4.13: Non-dispersive scattering angle versus excitation energy. Before any scattering
angle corrections are applied, as shown on the left, the hydrogen kinematic line is notably
curved and offset from zero. After the angular offset was applied, along with a scaling factor,
the hydrogen line is now vertical and centered on zero on the y-axis.

scattering angle, and small imperfections in the angular reconstruction strongly distort the

correlation. For heavier target nuclei, such effects are less pronounced.

In these experimental data, a slight rotation was present in the non-dispersive scattering

angle spectrum, which was corrected with an angular offset to the non-dispersive scattering

angle to center the spectrum at 0 degrees. Then, in order to correct the curvature of the

hydrogen line, a scaling factor of 0.9 was applied to the non-dispersive scattering angle. The

corrected scattering angle versus excitation energy spectrum is shown on the right in figure

4.13, illustrating that the rotation in the spectrum has been corrected, and that the bright

hydrogen kinematic line at 0 MeV is now centered around zero for all angles. The angular

resolution of the data, with all corrections implemented, was 12 mrad (FWHM) over the
range 0o ≤ θ ≤ 3o.

65

Excitation Energy [MeV]10-5-051015Non-dispersive scattering angle [rad]0.15-0.1-0.05-00.050.10.1505101520253035Uncorrected target positionExcitation Energy [MeV]10-5-051015Non-dispersive scattering angle [rad]0.15-0.1-0.05-00.050.10.1505101520253035Corrected target positionFigure 4.14: Acceptance cut in YtaBta. Non-dispersive scattering angles greater than 0.08
radians and less than -0.08 radians are excluded; beyond these angles the acceptance of the
events falls off rapidly. Additionally, based on the size of the target cell, the non-dispersive
target position can be restricted to a range of -10.0 mm to 15.0 mm.

4.2.5 Acceptance Correction

The acceptance of the S800 must be accounted for in the data analysis because of charge-

exchange events that had large scattering angles, but that were blocked from entering the

spectrometer. These events could be accounted for by weighting the data with an acceptance

matrix. Figure 4.14 shows the non-dispersive scattering angle versus yta; the trend that

appears between these parameters comes from correlations in the beam, as well as possible

imperfections in the ray-tracing calculation. Figure 4.15 shows the dispersive scattering angle

versus dta; hard cut-offs appear in this spectrum, illustrating the limits of the acceptance of

the spectrometer.

The limits chosen for the acceptance (red boxes) were more restrictive than the actual

cut-off of the spectrometer; this way, it was ensured that only the regions with reasonably

66

Non-dispersive target position [cm]20-15-10-5-05101520Non-dispersive scattering angle [rad]0.15-0.1-0.05-00.050.10.1502468101214Acceptance cut - YtaBtaFigure 4.15: Acceptance cut in DtaAta. Dispersive scattering angles greater than 0.05
radians and less than -0.05 radians are excluded; beyond these angles the acceptance of the
events falls off rapidly. The acceptance cuts are more restrictive than the shape yielded by
the spectrometer because the fringe events typically require large acceptance corrections and
introduce unnecessary uncertainties into the calculated cross section.

high acceptance were used in the calculation of the acceptance matrix, and included in the

final analysis. While it may have been possible to loosen the restrictions on the kinematic

parameters and obtain more statistics, the correction factors in the remote regions of the

spectra would become large and introduce greater uncertainties into the final cross section

calculation. A simple Monte Carlo simulation was used to generate the acceptance matrix,

based on the presented figures. The correction factors in the matrix weight each event, taking

into account the probability of that event being accepted into the spectrometer. With this

method, the usable angular range was extended from 3.5 degrees to approximately 6 degrees

in the laboratory frame.

Upon application of the acceptance correction to the data, it was found that there was

67

Energy deviation0.06-0.04-0.02-00.020.040.06Dispersive scattering angle [rad]0.08-0.06-0.04-0.02-00.020.040.060.080510152025303540Acceptance cut - DtaAtaan asymmetry in the non-dispersive scattering angle when comparing the data from the

upstream and downstream target windows, shown in figure 4.16. Such an effect was due

to the acceptance being slightly different for each end of the target, meaning that events

from the upstream foil had a greater acceptance than the downstream foil. The asymmetry
appeared as a loss of events at large scattering angles (∼4 degrees in the center of mass

frame). It was hypothesized that the asymmetry appeared because the beam was slightly

offset, leading to more events from the downstream target window being cut off. Because

it was not possible to correct such an effect on an event-by-event basis, it was decided that

consideration of the asymmetry would be postponed until later in the analysis, where it

would be negated during the target window background subtraction. As such, this issue will

be discussed further in section 4.3.3.

Figure 4.16: Plot of the scattering angle versus scattered energy at the target position for
the upstream target foil. For the upstream foil, on left, the hydrogen kinematic line appears
complete for both positive and negative scattering angles. For the downstream foil, the
hydrogen kinematic line shows the loss of events for positive scattering angles, indicating a
difference in the S800 acceptance when compared to the upstream foil.

68

Energy deviation0.06-0.05-0.04-0.03-0.02-0.01-00.010.020.030.04Non-dispersive scattering angle [rad]0.1-0.05-00.050.10246810121416182022Upstream target foil - DtaBtaEnergy deviation0.06-0.05-0.04-0.03-0.02-0.01-00.010.020.030.04Non-dispersive scattering angle [rad]0.1-0.05-00.050.1024681012141618202224Downstream target foil - DtaBta4.3 Missing Mass Calculation

The excitation energy spectrum was found via a missing mass calculation, as the residual

nucleus from the charge-exchange reaction was never measured. Conservation of energy

and momentum were applied to the incoming energy and momentum of the projectile and

the outgoing energy and momentum of the ejectile. From this, any leftover energy in the

calculation was used to excite the residual nucleus. Combining this energy and momentum

with the mass of the residual nucleus, mr, the excitation energy spectrum was generated,

following the expression:

Ex =

(cid:113)

missing − P 2
E2

missing − mr

(4.1)

The kinematics of the missing mass calculation took into account the masses of the initial

and final particles in the reaction, leading to slight differences in the position and energy

spectra, depending on which target nucleus was specified. Hydrogen kinematics refers to the

case in which the incoming triton interacts with a hydrogen nucleus in the target, yielding

a 3He ejectile and a neutron residual. Similarly, carbon kinematics refers to an interaction

with a 12C nucleus, yielding a 12B residual, nitrogen kinematics refers to an interaction with

a 14N nucleus, yielding a 14C residual, and krypton kinematics refers to an interaction with

a 86Kr nucleus, yielding a 86Br residual.

The ways in which incoming tritons could interact with these nuclei are shown in figure

4.17. The left-hand figure illustrates the case of the empty target cell, which consists of

two kapton windows, containing mostly 12C, 14N, 16O and 1H. Because the center of the

target was at vacuum when the cell was empty, reactions could only occur in the upstream

(interaction A) or downstream (interaction B) target windows. After the triton interacted

with one of the windows, a 3He ejectile was produced, which continued into the spectrometer.

69

Figure 4.17: A diagram of the possible interaction with the empty target cell (left) and
krypton target cell (right). With the center of the cell at vacuum, incoming tritons could
interact with either the upstream (A) or downstream (B) kapton target windows. However,
when the target is filled with 86Kr gas.
interactions may also occur in the center of the
target. The thicknesses of the targets are not to scale.

When the target cell was filled with 86Kr gas, as is shown on the right in figure 4.17,

interactions could still occur on the target windows (A and B as described previously), but

the tritons could also interact with the target gas (interaction C).

Interactions described in the case of the empty cell accounted for the peaks found in the

excitation energy spectrum, shown in figure 4.18. Carbon kinematics were used to generate

the spectrum, placing reactions on 12C that yielded 12B in its ground state at 0 MeV.

Contributions from other charge-exchange reactions appear in the spectrum, based on their

reaction Q-value. Table 4.1 shows the Q-values for the most common target nuclei in this

analysis. For the case of carbon kinematics, the difference between the chosen nucleus’ Q-

value and 12C Q-value yields the energy at which the reaction products begin to appear in

the excitation energy spectrum.

Reactions on hydrogen (blue) and nitrogen (green) begin to appear at large negative

excitation energies, because their reaction Q-values are smaller in magnitude than the 12C

70

Figure 4.18: Excitation energy spectrum for the downstream target foil. This spectrum
uses carbon kinematics, meaning that the peak at 0 MeV represents reactions on 12C that
yield 12B in its ground state. The energies at which reactions on various target window
nuclei begin to appear are shown. If 86Kr were present, those reactions would appear at
approximately -2.5 MeV in this plot.

Table 4.1: Ground state Q-value of (t,3He) reactions on krypton and background nuclei.

Q-value (MeV)

14N
1H
86Kr
16O
12C

-0.138
-0.764
-7.609
-10.402
-13.35

reaction Q-value. Alternatively, reactions on carbon (orange) and oxygen (purple) enter the

spectrum closer to 0 MeV, because their Q-values are of similar magnitude. If the complete

cell were being examined and 86Kr were present in the system, those reactions would be

expected to appear at approximately -2.5 MeV in figure 4.18.

71

Each of the kinematics settings were useful for different parts of the analysis. Because

hydrogen nuclei are light, the kinematics were extremely sensitive to small changes in the

scattering angle, energy and position in the focal plane. Therefore, this setting was used

when determining corrections to the drift velocity in the CRDCs (section 4.2.3) and to the

scattering angles (section 4.2.4).

Carbon kinematics were used because 12C has been studied extensively by the charge-

exchange group, and its properties and cross section are well-known. As such, the carbon

cross section was used to ensure that all angular, positional and energy corrections were

applied to the data appropriately. Additionally, because carbon and hydrogen were present

in all data sets, due to the kapton windows of the target cell, these contributions were used

to normalize the data, as will be discussed in section 4.3.2. Finally, the krypton kinematics

setting was used to examine events of a triton interacting with the 86Kr nucleus and yielding

a residual nucleus of 86Br, which was the measurement of interest, and will be discussed in

more detail in sections 4.4.2 and 4.4.3.

4.3.1 Target Energy Loss

Using the fragment separator software, LISE++ [130], estimates for the energy loss experi-

enced by tritons and 3He particles as they passed through the target were obtained. In order

to simplify the calculation, the interaction between the beam and the target were assumed

to occur in the center of the target windows, and gas cell, when it was filled.

For the empty cell data, two scenarios for energy loss were possible. The first was the

case in which the incoming triton interacted with the upstream target foil (interaction A in

figure 4.17). In the first half of the first target foil, the energy loss was calculated based on

a triton passing through the kapton. Then, for the last half of the upstream target window

72

and the entire downstream target window, the energy loss was calculated based on a 3He

particle moving through the target. Summing the energy loss contributions for the incoming

and outgoing particles, the total energy loss for this scenario was equal to approximately

0.72 MeV.

The other scenario was the case in which the beam interaction occured in the downstream

target foil (interaction B in figure 4.17). In the first target window and half of the second

target window, the energy loss was calculated based on a triton moving through the kapton.

After the interaction, for the final half of the downstream target foil, the energy loss was

calculated based on a 3He particle moving through the foil. Because energy loss within the

target was dependent on the mass of the particle moving through it, this scenario produced

less energy loss than the first, because much of the path was traveled by the triton. Again,

summing the energy loss contribution for the incoming and outgoing particles, the total

energy loss for this scenario was equal to approximately 0.39 MeV. Determining the difference

in energy loss between the two scenarios yielded a value of 330 keV for the difference in the

energy loss associated with the empty cell data.

A similar exercise was performed for the krypton target cell, but it was necessary to

include an additional source of energy loss, as the tritons and 3He particles passed through

86Kr gas instead of vacuum, causing them to lose more energy as they traveled through the

target. In order to obtain the energy resolution for the krypton data, the energy losses yielded

from interactions A and B were used. With the addition of the krypton gas, the energy loss

associated with the interaction in the upstream target window and in the downstream target

window were equal to 1.06 MeV and 0.47 MeV, respectively. From these values, the difference

in the energy loss associated with the data obtained from the filled krypton cell was 590 keV.

It is also important to obtain a measure of the energy resolution for the experiment. The

73

12C peak present at 0 MeV in figure 4.18 was used to determine that the energy resolution
for the carbon events was approximately 575 keV (FWHM) in the range 0 MeV ≤ Ex ≤

20 MeV. For the krypton events the calculation of the energy resolution was more difficult,

because there were no obvious 86Kr peaks in the excitation energy spectrum, so that simply

measuring the width of the distribution was not possible.

Instead, the coincidence data

were used; by gating on known 86Be γ-rays an excitation energy spectrum for 86Kr events

was obtained (see section 4.7 for the details of this process). The energy resolution for the
krypton data was approximately 400 keV (FWHM) in the range 0 MeV ≤ Ex ≤ 5 MeV.

4.3.2 Cross Section Calculation

The cross section of a reaction is a measure of the likelihood of the reaction occurring at a

particular excitation energy. In order to convert the cross section from counts measured in the

detector to the proper units of a differential cross section, the acceptance and geometry of the

system were taken into account. The expression for the cross section for the 86Kr(t,3He+γ)

experiment is as follows:

dσ
dΩ

=

Y

NbNtdΩ
1

.

(4.2)

Here, Y is the absolute yield of the reaction of interest, Nb is the number of particles in

the beam, Nt is the number of particles in the target, dΩ is the solid angle and 
1 is the

acceptance of the S800 for a particular target position, scattering energy and scattering

angle bin from the acceptance matrix. The solid angle, which describes the area subtended

by the cone formed by the azimuthal scattering angle, is given by dΩ = sinθdθdφ. For

small scattering angles, there are no limitations on the polar angle, φ. However, as the

scattering angle increases, some events begin to be blocked from entering the spectrometer,

74

leading to restrictions on the polar angle; the loss of these events is taken into account with

the acceptance correction, detailed in section 4.2.5. Angular bins for the cross section were

chosen to be 1 degree in the center of mass frame.

Figure 4.19: Cross section for reactions on the empty target cell for the 0 to 1 degree angular
bin, corrected for the acceptance of the spectrometer. Reactions on the carbon ground state
sit at 0 MeV, due to the use of carbon kinematics in this calculation. The contribution
from hydrogen reactions are at negative excitation energies. These contributions need to be
subtracted from the krypton data in order to see the underlying reactions of interest.

The cross section for the empty target cell in carbon kinematics for an angular range of

0 to 1 degrees is shown in figure 4.19; all contributions to this spectrum come from reactions

on nuclei in the target windows. The large hydrogen peak appears at negative excitation

energies because the Q-value for reactions on hydrogen is much smaller than the Q-value

for reactions on carbon. At higher excitation energies, the peaks correspond to reactions

75

on carbon, oxygen and nitrogen, which were also present in the target windows. Because

the signals from reactions on hydrogen and carbon were strong, it was necessary to subtract

these and other target window reaction contributions from the krypton data set in order to

observe the cross section of reactions on 86Kr.

4.3.3 Carbon Angular Distribution

After the cross section calculation was finalized, it was possible to extract an angular dis-

tribution from the data. Initially, the angular distribution of the carbon ground-state peak

was examined because the carbon data were used to normalize the remainder of the data

and to obtain an estimate of the beam intensity. It was possible to obtain reasonable cross

section measurements up to about 6 degrees in the center of mass frame, at which point the

carbon and hydrogen peaks began to overlap, as was illustrated in figure 4.16.

The angular distribution for the carbon data from the empty cell runs is shown in figure

4.20, along with a comparison to a DWBA calculation and 12C experimental data from

previous experiments [116, 131]. A scaling factor was applied here, which is equivalent to

1

, from equation 4.2. Using this normalization factor, the average beam
the luminosity,
intensity, Nb, for the experiment was extracted and found to be approximately 5 × 106

NbNt

particles per second, which is the typical intensity for a triton beam at the NSCL.

Because, as mentioned previously (figure 4.16), the target cell was thick, it was neces-

sary to examine the angular distributions from both the upstream and downstream target

windows, in order to determine whether the acceptance was treated properly for each of the

target windows. The extracted angular distributions for both foils is shown in figure 4.21.

Prior to the acceptance corrections, with increasing scattering angle, the angular distribution

from the downstream window was less than the angular distribution from the upstream win-

76

Figure 4.20: A comparison the angular distribution of the carbon ground state with past
experimental data. The agreement between the blue points (past experiment [116, 131]) and
the orange points (current data) indicates that the calibrations and corrections applied to
the data have been successful in replicating this known angular distribution.

dow by about 25%. This phenomenon was due to events from the upstream target window

having a larger acceptance than those from the downstream window, as was mentioned in

section 4.2.5.

The downstream target window, which had a smaller acceptance, had fewer events suc-

cessfully entering the spectrometer, causing a smaller yield to be measured. To correct this,

the ratio between the cross section from the downstream and upstream windows was taken

for each angular bin, quantifying the magnitude of the asymmetry; it was found that the

effect became more pronounced with increasing scattering angle. Then, a scaling factor was

applied to each angular bin of the downstream cross section, such that the asymmetry was

corrected, and the ratio between the upstream and downstream angular distributions was

equal to 1. With the asymmetry taken into account in the data processing, the angular dis-

77

Figure 4.21: A comparison the angular distribution of the carbon ground state, derived from
the upstream (orange points) and downstream (green points) target foils. After correcting
for the difference in the acceptance for each target window, both contributions match the
previously-extracted 12C angular distribution [131, 116].

tributions from each target window were consistent with the 12C angular distribution from

previous experiments, indicating that the corrections and calibrations for these experimental

data were reasonable.

4.3.4 Nitrogen Angular Distribution

Also present in the data set were charge-exchange reactions on 14N. As marked in figure 4.19,

these events mainly appeared at small negative excitation energies in carbon kinematics.

Zooming in on the excitation energy spectrum, the 14N events are shown in the top plot in

figure 4.22. The events of interest appear in the peaks at 7 MeV and 8.5 MeV; the third

peak shown in the figure is composed of events on 16O.

Following the same procedure as the 12C analysis, and angular distribution was extracted

78

Figure 4.22: Excitation energy spectrum from the empty cell data, zoomed in on the region
containing 14N events. The 14N events of interest are in the peaks at 7 MeV and 8.5 MeV.
The third peak on the right is composed of 16O events.

for the 14N data, and is shown in the bottom plot of figure 4.22. Comparing the extracted

angular distribution to the DWBA calculation (black line), it is clear that there is good

agreement between the shape of the experimental distribution and the theoretical result.

Using the well-known proportionality between the charge-exchange cross section and the

Gamow-Teller strength, as presented in equation 3.19, the strength distribution for the 14N

events was extracted. The DWBA calculation was used to extrapolate the experimental

angular distribution to Θ = 0, and the unit cross section,(cid:98)σ, was obtained using the empirical

expression dependent on the mass number (equation 4.7).

79

Excitation energy [MeV]3456789101112Counts051015202530354045N14N14O16 [deg]CMQ00.511.522.533.54]-1 [mb srWdsd0246810N DWBA14N data14Figure 4.23: 14N Gamow-Teller strength distribution, compared to experimental (d,2He)
data [132]. No error bars were presented with the (d,2He) data.

Figure 4.23 shows the calculated Gamow-Teller strength distribution (green points). Also

shown in blue is a previous measurement of the 14N Gamow-Teller strength using the (d,2He)

reaction [132].

It is shown that there is good agreement between the experimental data

from this work, and from the (d,2He) experiment. Such a result indicates that the current

calibrations and experimental analysis are reasonable up to this point, as both the 14N

data, here, and the 12C data in section 4.3.3, match previously published results, within

experimental uncertainties.

4.4 Target Window Event Subtraction

The experimental data runs intended to measure events on 86Kr also contained events from

the kapton target windows. Because the krypton signal was weak, however, it was hidden

80

underneath the extraneous events from hydrogen, carbon, nitrogen and oxygen. The Q-

values for all expected reactions are presented in table 4.1. The energies of the lowest-lying

states for 12B, 14C and 16N are shown in tables 4.2, 4.3 and 4.4, respectively. The energies

are also converted to the excitation energy in 86Br, in order to determine where those states

would appear when analyzing the spectrum using krypton kinematics. Because the 86Kr

signal was expected to appear at approximately 2.5 MeV, it is clear that mainly events from

14N and 16O were located at the same energy as the krypton.

Table 4.2: Location of states in 12B [133], converted to 86Br excitation energy.

Excitation energy Excitation energy

in 12B [MeV]

in 86Br [MeV]

0

0.0

5.74

Table 4.3: Location of states in 14C [134], converted to 86Br excitation energy.

Excitation energy Excitation energy

in 14C [MeV]

in 86Br [MeV]

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

0.0
6.09
6.59
6.73
7.01
7.34
8.31
9.75
9.80
10.43
10.45
10.50
10.74
11.31
11.40
11.50
11.67
11.73
11.90
12.58

81

-7.48
-1.38
-0.89
-0.75
-0.47
-0.14
0.84
2.27
2.32
2.95
2.97
3.02
3.26
3.83
3.92
4.02
4.19
4.25
4.42
5.11

Table 4.4: Location of states in 16N [135], converted to 86Br excitation energy.

Excitation energy Excitation energy

in 16N [MeV]

in 86Br [MeV]

0
1
2
3
4

0.0
0.12
0.30
0.40
3.34

2.81
2.93
3.10
3.20
6.14

Although it was unfortunate that the target window events were numerous compared

to the krypton signal, the background spectrum was uniform across all data runs, so it

was a simpler process to normalize and subtract it away. During the beam time, it was

determined that simply taking data using the empty gas cell would be insufficient to model

the background signal, due to the thickness of the target and the uncertainties in the energy

loss in the target windows. Therefore, additional data sets were taken, one set with only the

upstream target window present, and one set with only the downstream target window in

place, for use in developing a model of the target window events.

4.4.1 Empty Cell Subtraction

By combining the single-target-window data sets, it was possible to reconstruct the target

window event spectrum and to subtract it from the krypton data set. This method was first

tested with the empty cell data to verify that it would work as expected. The average energy

loss of a triton or 3He passing through a target foil caused a shift in the location of the cross

section peak. Calculated for both of the target windows, the shift was applied to each of

the foils respectively, in order to align the spectrum with the empty cell data. Additionally,

the scattering angle and excitation energy were smeared for each target window spectrum,

to account for energy straggling and differences in the energy resolution.

82

The equation for the smeared excitation energy is given by the following expression:

Exsmeared = Ex + AσE + ∆E

(4.3)

where σE is the energy straggling due to the target foil, ∆E is the energy loss in the foil,

and A is a random number generated from a gaussian distribution. A similar expression was

used to smear the scattering angle:

Θsmeared = Θ + BσΘ

(4.4)

where σΘ is the angular straggling imparted upon the particle traveling through the target

foil, and B is another random number generated from a gaussian distribution. The energy

loss and energy straggling values necessary for the analysis were determined based on figure

4.17, along with the energy loss discussion presented in section 4.3.1. A compilation of energy

loss and angular smearing values used for both the empty cell and krypton cell subtractions,

generated by the fragment separation software, LISE++ [130], is found in table 4.5.

Table 4.5: Energy loss and angular smearing values for the empty cell and krypton cell
subtraction analysis. ∆E represents the energy loss correction between the two target foils.
σE represents the energy loss correction due to straggling within the target foils. σΘ is the
scattering angle smearing due to the target foils.

Empty cell

Reaction in:
Upstream foil
Downstream foil

Krypton cell Upstream foil

Downstream foil

∆E (MeV) σE (MeV) σΘ (mrad)
0.443
0.111
0.7787
0.1949

0.0598
0.0294
0.0993
0.0496

0.658
0.329
1.480
0.741

After these modifications were applied, a composite cross section was generated by sum-

ming the cross sections for the upstream (red) and downstream (blue) foils. Subtracting this

83

constructed cross section from the empty cell cross section (black line) yielded a spectrum

that was consistent with 0, as shown in the bottom plot of figure 4.24; the 1-2 degree bin is

presented here, but similar results are obtained for all of the angular bins up to 4 degrees in

the center of mass frame. From this comparison, it is clear that the cross section obtained

from the background modeling procedure was equivalent to the empty cell cross section,

within the experimental uncertainties.

Figure 4.24: Empty cell subtraction for the 1-2 degree bin. The contributions from the
upstream and down stream target windows are shown in red and blue, respectively, in the
top plot. By subtracting the model spectrum from the empty cell data (black line), the
bottom spectrum is obtained. The spectrum is consistent with zero (blue horizontal line)
indicating that the target window events were subtracted successfully.

From the results shown in figure 4.24, it was illustrated that it was possible to perform a

successful subtraction by summing the cross sections of the upstream and downstream target

foils, along with a treatment of energy loss and energy straggling in the windows, in order

to recreate the complete empty cell cross section. For reasons discussed in the next section,

84

024681012141618Counts02004006008001000120014001600180020002200Modeled Spectrumo 2£ CMQ £ o1Excitation Energy [MeV]024681012141618Counts40000-20000-0200004000060000Subtracted Spectrumonly the data below 5 MeV were used for the krypton analysis. Referring again to tables

4.2, 4.3 and 4.4, it is apparent that, within the 5 MeV window, it was mainly events on 14N

that sat atop the 86Kr signal. 12C states do not appear until higher excitation energies and

there is a gap in the 16O states where the krypton signal was expected to appear.

4.4.2 Krypton Cell Subtraction

Following the technique of the empty cell subtraction, the target window events were then

subtracted from the krypton data. Different energy loss parameters were used for the krypton

cell subtraction because of the additional energy loss introduced by the presence of the 86Kr

gas. Additionally, the new parameters took into account the outward bulging of the target

windows due to the internal gas pressure of the cell. Because this phenomenon introduced

a position-dependent target thickness, the thickness of the 86Kr portion of the target was

modified until the cross section spectra from the background model matched those of the

experimental data, in order to generate an accurate model of the target window events. The

energy smearing parameters (equation 4.3) used for the krypton subtraction are shown in

table 4.5, and were calculated using an average target thickness of 20 mg/cm2.

The cross section for the 1-2 degree angular bin is shown in figure 4.25, for excitation

energies from 0 to 20 MeV. The top figure shows krypton data (black) and the modeled target

window event spectrum (red); then, the krypton cross section, obtained by subtracting the

two histograms, is shown in the bottom figure.

It is clear that the uncertainties in the

parameters used for the subtraction procedure introduced systematic uncertainties into the

subtracted spectrum. Therefore, the subtracted spectra were generated for a series of choices

of the energy loss parameters, in order to estimate these systematic errors. From this study,

the total systematic uncertainty for the subtraction procedure was estimated to be 25% of

85

Figure 4.25: Differential cross section before (top) and after (bottom) the target window
events were subtracted for the 1 to 2 degree angular bin. The bottom plot shows the
extracted krypton cross section.

the total cross section, and was incorporated into the remainder of the analysis.

It was

concluded that above 5 MeV, the systematic errors in combination with the poor signal-to-

noise ratio, and therefore, high statistical errors, made it difficult to extract reliable results

for 86Kr.

The absolute cross section, presented in figure 4.26 for each of the angular bins, was

obtained by applying a normalization factor after the target window event subtraction.

Before the subtraction, the data were corrected for the acceptance of the spectrometer,

and the modeled spectra were normalized to the krypton data set using the number of

counts within the hydrogen peak. To determine the overall normalization, the well-known

magnitude of the 12C(t,3He) cross section [131] was used, because events on carbon nuclei

were present across the entire data set. The scaling factor needed to match the magnitude of

86

05101520]-1 0.5 MeV-1 [mb srdEWds2d0510152025303540Full Spectrumo 2£ CMQ £ o1Excitation Energy [MeV]05101520]-1 0.5 MeV-1 [mb srdEWds2d2-1-0123Subtracted Spectrumthe present 12C cross section to the known value was extracted. Then, combining this factor

with the target thickness and the molar mass of 86Kr yielded the factor needed to normalize

the entire krypton data set. The subtracted and normalized spectra for each of the angular

bins, up 4 degrees in the center of mass frame, are shown in figure 4.26 with error bars that

include both statistical and systematic uncertainties.

Figure 4.26: Krypton cross section with target window background events subtracted follow-
ing the method detailed in section 4.4. The absolute normalization factor was obtained by
comparing the carbon cross section measured in this experiment with the 12C cross section
obtained in previous work [131]. Scattering angles from 0 to 4 degrees in the center of mass
frame are shown, as the background subtraction began to break down at larger angles. The
error bars on the data are a combination of statistical and systematic uncertainties.

Target window contaminants, including 12C and 16O, began to appear with greater mag-

87

Excitation Energy [MeV]012345]-1 0.5 MeV-1 [mb srdEWds2d0.5-00.511.5o 1£ CMQ £ o0Excitation Energy [MeV]012345]-1 0.5 MeV-1 [mb srdEWds2d00.20.40.60.811.2o 2£ CMQ £ o1Excitation Energy [MeV]012345]-1 0.5 MeV-1 [mb srdEWds2d0.1-00.10.20.30.40.50.60.70.8o 3£ CMQ £ o2Excitation Energy [MeV]012345]-1 0.5 MeV-1 [mb srdEWds2d0.1-00.10.20.30.40.50.60.70.80.9o 4£ CMQ £ o3nitude at 5 MeV of excitation energy; see figure 4.18, and tables 4.2 and 4.4, for the location

of the 12C and 16O events in krypton kinematics. The presence of these additional con-

tributions to the cross section yielded large systematic and statistical uncertainties in the

spectra after the event subtraction was completed. Therefore, as was mentioned in relation

to figure 4.25, it was decided that further experimental analysis would be performed only

on the data up to 5 MeV. Additionally, because the background subtraction began to break

down at larger scattering angles, due to the presence of hydrogen events in the spectrum,

angles above 4 degrees were also omitted from the analysis.

4.4.3 Krypton Angular Distributions

Using the normalized cross sections presented in section 4.4.2, an angular distribution was

extracted for each energy bin; these spectra are shown in figure 4.27 in 0.5 MeV energy bins

up to an excitation energy of 5 MeV. A Gamow-Teller transition yields a forward peaking

angular distribution. Despite the large error bars shown, there are some bins that might

contain a Gamow-Teller transition component, for example, the 2.5 to 3.0 MeV bin. A

further examination of this possibility will be discussed in section 4.6, with the multipole

decomposition analysis.

4.5 DWBA

Distorted-Wave Born Approximation (DWBA) calculations, performed using the FOLD soft-

ware package [102], were initially discussed in a theory context in section 3.2. In terms of the

experimental analysis, DWBA calculations had two purposes. The first was to obtain the

shape of the angular distributions of the three dominant multipoles, which were then used in

88

Figure 4.27: Krypton angular distributions extracted for each energy bin (0.5 MeV) up to
5 MeV. Forward peaking distributions indicated a possible Gamow-Teller component in the
cross section.

the multipole decomposition analysis (section 4.6). The other use for the DWBA calculation

was to extrapolate the extracted ∆L = 0 cross section to zero momentum transfer (which

is equivalent to Q = 0, Θ = 0). With this extrapolated cross section, the proportionality

89

between the cross section and the Gamow-Teller strength distribution could be exploited

(equation 3.19).

As mentioned previously, three multipole components were assumed in the krypton data:

monopole, dipole and quadrupole. Although higher order multipoles could contribute, they

are likely small due to the small momentum transfer involved. Additionally, their angular

distributions are similar to that of the ∆L = 2 component at forward scattering angles and,

hence, are difficult to separate from the quadrupole component. Each of the transitions

has a defining set of selection rules, which are detailed in table 4.6. Because Gamow-Teller

transitions are of interest for this work, the ∆J = 1 component is used for the monopole
component. For the dipole components, the ∆J = 1 component was chosen because 2−
transitions, on average, appear at lower excitation energies than 0− and 1− transitions.
Moreover, because the total dipole strength scales with 2J + 1, the 2− component is the

strongest. Finally, for the quadrupole component, the ∆J = 2 component was used because

it tends to be the strongest of the three possible transitions.

Table 4.6: Selection rules for monopole, dipole and quadrupole transitions.

Monopole
Dipole
Quadrupole

∆L ∆S ∆J
0,1
0
0,1,2
1
2
1,2,3 no transfer

Parity
no transfer
transfer

0,1
0,1
0,1

The inputs to the 86Kr DWBA calculation, such as the nucleon-nucleon interaction and

optical potential, were detailed in section 3.2. To designate the multipole to be calculated

by FOLD, one-body transition densities (OBTDs) were input based on the selection rule

of the transition in question. The OBTDs for the monopole and quadrupole transitions

were generated by OXBASH [115] using the SNET interaction [76]. The dipole transition,

however, required a parity transfer, which could not be calculated using the aforementioned

90

interaction; instead a normal modes calculation, using the code NORMOD [136], was utilized

to obtain the proper transition densities. With this information, the general shape of each

of the multipoles was extracted, and the results are shown in figure 4.28. It is important to

note that, because the shape of the angular distributions is not sensitive to the details of

the microscopic input, the method used to obtain the one-body transitions densities is not

of great importance.

Figure 4.28: DWBA calculation results for 86Kr(t,3He) at 140 MeV/u. The calculation was
performed using the FOLD software package [102], and shows the three most likely transitions
to appear in the experimental data. The curves shown here are normalized so that their peaks
have the same magnitude. The shapes of the ∆L = 0 (blue), ∆L = 1 (yellow) and ∆L = 2
(orange) components are the important result for the analysis.

As expected, the monopole component peaks at forward scattering angles, and each

progressive multipole peaks at larger angles.

It was assumed that the experimental data

contained a mixture of these transitions in varying proportions. Therefore, by fitting the

data with a linear combination of the calculated multipoles, the components contributing to

the total experimental cross section were extracted.

91

4.6 Multipole Decomposition Analysis

The multipole decomposition analysis (MDA) sought to determine which features of the ex-

tracted angular distributions were consistent with monopole, dipole or quadrupole transitions

from the target nucleus [137, 138]. A linear combination of the multipoles was constructed

with the following form:

σTotal = Aσ∆L=0 + Bσ∆L=1 + Cσ∆L=2

(4.5)

where σ∆L=0,1,2 are the cross sections obtained from the DWBA calculations (section 4.5)

for the ∆L = 0, 1 and 2 components, respectively. A, B, and C are the constants optimized

in the MDA fitting routine, and describe the relative contributions of each of the multipoles

to the total cross section. The sum of the cross sections was fit to the angular distributions

from figure 4.27, and the results of the MDA fits, up to 5 MeV, are shown in figure 4.29.

From these results, it was apparent that a combination of these transitions described the

cross section as it was measured. However, because the error bars on the experimental data

points were large, the fitting parameters also carried large uncertainties. Nevertheless, it was

possible to use the extracted fitting parameters for the monopole component to generate a

Gamow-Teller strength distribution, and to place an upper limit on the strength present in

86Kr.

The Gamow-Teller strength distribution was extracted using the well-established propor-

tionality with the cross section at zero momentum transfer, discussed in section 3.3. The

92

Figure 4.29: The MDA results, here, show the experimental data (black points) and the linear
combination of multipoles that best fit the data in each energy bin. The total combination
(red curve) is the sum of each of the components: ∆L = 0 (blue), ∆L = 1 (yellow) and
∆L = 2 (green).

equation describing the proportionality is as follows [51]:

(cid:34)

dσ
dΩ

=(cid:98)σB(GT ).

(4.6)

(cid:35)

(q = 0)

GT

93

The left-hand side of the equation is the ∆L = 0 component of the differential cross section,

as measured in the experiment, extrapolated to zero momentum transfer (q = 0) by using

the value of the DWBA calculation with the Q-value of the reaction set to 0 MeV, at a

scattering angle of 0 degrees. On the right-hand side of the equation, (cid:98)σ is the unit cross

section, and is determined empirically using the following expression [12, 51, 116, 117, 118]:

(cid:98)σ = 109 × A−0.65

(4.7)

where A is the mass number of the nucleus. For the transition from 86Kr to 86Br, the unit

cross section is 6.03, with an uncertainty of approximately 5%. Combining the extrapolated

cross section from the experiment with the unit cross section yielded the Gamow-Teller

strength distribution for the 86Kr(t,3He) reaction, shown in figure 4.30.

Figure 4.30: The Gamow-Teller strength distribution for 86Kr extracted from the results of
the MDA.

94

012345Excitation energy [MeV]0.000.050.100.150.20B(GT)The level scheme for 86Br does not indicate that there are any 1+ states below 2.5 MeV;

therefore, it is unlikely to find true Gamow-Teller strength below that energy. Examining
figure 4.30, it is shown that the limit of the sensitivity of this experiment is B(GT ) ∼ 0.05.

Because the data points below 2 MeV are similar in magnitude to the points above 3.5 MeV,

the strength at the higher excitation energies is also unlikely to be Gamow-Teller.

It is

important to note that the error bars on all of the points are large due to the statistical and

systematic uncertainties present in the experimental data. As such, it is possible only to

set an upper limit on the Gamow-Teller strength. Analysis of the γ-ray data, discussed in

section 4.7, provides stricter limits on the strength that was observed in this experiment.

4.7 Gamma-Ray Analysis

Because of the systematic uncertainties in the experiment, as discussed in the previous

section, it was only possible to obtain an upper limit on the Gamow-Teller strength for

86Kr from the singles data. In an effort to improve this limit, the coincidence data from

GRETINA and the S800 were used, as the γ-detection array was capable of measuring weak

transitions with high precision, without subtracting away the target window events.

The decay spectrum of 86Br, the residual nucleus of the 86Kr(t,3He) reaction, is partially

known, though spin and parity assignments are not available for all of the recorded levels

[139]. It was expected that some Gamow-Teller strength would be observed at 2.45 MeV

and 2.67 MeV, which are the locations of two known 1+ states in 86Br. However, none

of the γ-rays associated with de-excitations from those states was observed. Analyzing the

remainder of the γ-ray spectrum, known lines from the target window reaction residuals,

such as decays of 12B, 14C and 16N, or lines from 85Br, which occur when the excitation

95

Figure 4.31: The γ decay spectra for each excitation energy bin. The γ-lines associated with
marked at the top of each plot: 86Br in red, 14C in blue, 16N in green, and lines associated
with no known nucleus in magenta. The peak at 511 keV in the top three panels originates
from electron-positron annihilation.

energy is greater than the single neutron separation energy at 5.128 MeV, were identified

and excluded from the analysis. The γ-decay spectra for each 1 MeV excitation energy bin

are shown in figure 4.31: γ-lines associated with known lines in 86Br are marked in red,

96

 [keV]gE020040060080010001200140016001800200022002400Counts0246810121416182022 1 MeV£ x E£0 MeV  [keV]gE020040060080010001200140016001800200022002400Counts0246810121416 2 MeV£ x E£1 MeV  [keV]gE020040060080010001200140016001800200022002400Counts0102030405060 3 MeV£ x E£2 MeV  [keV]gE020040060080010001200140016001800200022002400Counts020406080100120 4 MeV£ x E£3 MeV  [keV]gE020040060080010001200140016001800200022002400Counts024681012141618 5 MeV£ x E£4 MeV lines from 14C are marked in blue, lines from 16N are marked in green, and lines that are

not associated with any known nucleus (and, thus, are assumed to originate from 86Br) are

marked in magenta.

Several weak lines from 86Br were observed, as well as γ-rays not measured and recorded

in the Nuclear Data Sheets [139]. The γ-ray spectra obtained from the krypton coincidence

data set were examined in close detail in 1 MeV excitation energy bins. Some lines were only

visible at specific excitation energies, while others appeared throughout the excitation energy

spectrum. Each of the observed peaks was compared to known γ-ray spectra of 12B, 14C and

16N, the residuals from reactions on target window nuclei. Lines in the krypton spectrum

that were known to originate from a target window residual were disregarded as background

events, while any other peaks were noted for further analysis. The final list of γ-rays that

were either identified from the 86Br decay scheme, or those that were not identified in any

other spectrum, are compiled in the first column of table 4.7. The assumption was made

that the γ-lines listed here originated from the de-excitation of 86Br, so that they could be

further studied in the search for weak Gamow-Teller strength.

The general steps of the analysis are listed here; more details concerning each of the

states are presented below. The Eγ spectrum, zoomed in on each γ-line, is shown in the top

plot in figures 4.32 through 4.41. A gate with a width of 5 keV was made around the γ-line,

and the excitation energy spectrum of the particles associated with that γ-ray was extracted.

These excitation energy spectra, deduced from the (t,3He) reaction data, are shown in the

bottom left plots in figures 4.32 through 4.41. Finally, a gate was placed on the excitation

energy, encompassing the largest peak in the spectrum, typically with a width on the order

of 1.5 MeV.

The data were re-processed, including both the cuts on Eγ and the excitation energy,

97

Table 4.7: γ-rays from de-excitations of 86Br. These lines appear in the γ spectrum from
the krypton data set, and are not associated with de-excitations of the residual nuclei from
reactions on the target windows, for example 12B, 14C or 16N. Also listed are the J π as-
signments for each state, when known (tentative assignments are given in parentheses), the
excitation energy of events associated with each γ-ray, as determined from the (t,3He) data,
and the type of transition, determined from examining the shape of the calculated angular
distribution.

382
932

Eγ (keV) J π
Ex (MeV)
Transition
(4−)
∆L ≥ 1
∼0.0
77
(1−,2−)
peak 1: ∼3.0 ∆L ≥ 1
207
peak 2: ∼4.0 ∆L = 0, ∆L = 2
(1−,2)
∼2.5
∆L = 1
unknown peak 1: ∼1.0 ∆L ≥ 2
peak 2: ∼2.5 ∆L = 1
unknown ∼1.75
∆L ≥ 1
unknown ∼2.25
∆L = 1
unknown ∼1.75
∆L = 0, ∆L = 2
unknown ∼2.25
∆L = 0, ∆L = 2

942
1427
1753
2361

and using these gated spectra, the angular distribution for the events associated with each

of the listed γ-lines was extracted (shown in the bottom right plots in figures 4.32 through

4.41). The efficiency of the GRETINA detector array was incorporated into the calculation

of the cross section and angular distribution. The efficiency curve, presented in figure 4.8,

was obtained by fitting 152Eu source data taken prior to the experiment. An assumption

for the type of transition that produced each γ-ray, deduced from the shape of the angular

distribution, is presented for each case, and also listed in table 4.7.

77 keV: Figure 4.32 shows the 77 keV γ-ray. γ energies from 74 keV to 80 keV were

selected, generating a gated excitation energy spectrum. A single peak appeared in this

spectrum, just above 0 MeV. A gate on the excitation energy was made from -0.5 MeV

to 1 MeV. Applying both the Eγ gate and the excitation energy gate to the data allowed

for the calculation of the cross section for the events that yielded a 77 keV γ-ray. The

angular distribution that was extracted from the data peaked at approximately 2.5 degrees

98

in the center of mass frame, indicating that the γ-rays most likely came from a dipole or a

quadrupole transition.

207 keV: Figures 4.33 and 4.34 show the analysis of the 207 keV γ-ray. γ energies from

205 keV to 209 keV were selected, generating the gated excitation energy spectrum. Two

peaks appeared in this spectrum: one at 3 MeV and another at approximately 4 MeV. Both

of these peaks were analyzed further. First, an excitation energy gate from 2.5 MeV to 3.5

MeV was made (figure 4.33), and the angular distribution was extracted. The distribution

peaked between 1.5 and 2.5 degrees in the center of mass frame, indicating that the γ-rays

yielded from events in the first excitation energy peak were likely the product of a dipole or

a quadrupole transition. Then, returning to the original excitation energy spectrum, a new

gate was made from 3.5 MeV to 4.5 MeV (figure 4.34). The extracted angular distribution

could not be associated with a particular ∆L value, indicating that these events were likely

from background under the 207 keV γ-ray peak, rather than from a single state.

382 keV: Figure 4.35 shows the 382 keV γ-ray. γ energies from 380 keV to 384 keV

were selected, generating the gated excitation energy spectrum with a single peak at approx-

imately 2.5 MeV. A gate on the excitation energy from 2 MeV to 3 MeV was applied to the

data. The extracted angular distribution peaked at approximately 1.5 degrees in the center

of mass frame, indicating that the γ-rays yielded from these events were likely the product

of a dipole transition, although a higher value of ∆L cannot be ruled out.

932 keV: Figures 4.36 and 4.37 show the analysis of the 932 keV γ-ray. γ energies from

928 keV to 933 keV were selected, and, as with the case of the 207 keV line, two peaks

appeared in the gated excitation energy spectrum, at 1 MeV and 2.5 MeV, respectively.

First, an excitation energy gate from 0.5 MeV to 2 MeV was made (figure 4.36), and the

angular distribution was extracted. The distribution peaked beyond 1.5 degrees in the center

99

of mass frame, indicating that the γ-rays yielded from events in the first excitation energy

peak were likely the product of a quadrupole transition. Then, returning to the original

excitation energy spectrum, a new gate was made from 2 MeV to 3 MeV (figure 4.37). The

extracted angular distribution peaked at approximately 1.5 degrees, indicating the γ-rays

were likely the product of a dipole transition.

942 keV: Figure 4.38 shows the 942 keV γ-ray. γ energies from 941 keV to 945 keV

were selected, yielding a gated excitation energy spectrum with a single peak at 1.75 MeV.

Applying a gate to the excitation energy from 1 MeV to 2 MeV, the angular distribution

was extracted and shown to peak between 1.5 and 2.5 degrees in the center of mass frame,

indicating that the γ-rays were likely the product of a dipole, quadrupole or higher multipole

transition.

1427 keV: Figure 4.39 shows the 1427 keV γ-ray. γ energies from 1420 keV to 1430 keV

were selected, yielding a gated excitation energy spectrum with a single peak at 2.25 MeV.

Applying a gate to the excitation energy from 1.5 MeV to 3 MeV, the angular distribution

was extracted and shown to peak around 1.5 degrees in the center of mass frame, indicating

that the γ-rays were likely the product of a dipole transition.

1753 keV: Figure 4.40 shows the 1753 keV γ-ray. γ energies from 1750 keV to 1755 keV

were selected, yielding a gated excitation energy spectrum with a single peak at approxi-

mately 1.75 MeV. Applying a gate to the excitation energy from 1.25 MeV to 2.25 MeV,

the angular distribution was extracted and seemed to peak at forward angles, indicating a

potential Gamow-Teller transition. As such, the distribution associated with 1753 keV γ-ray

was noted for further analysis.

2361 keV: Figure 4.41 shows the 2361 keV γ-ray. γ energies from 2359 keV to 2363

keV were selected, yielding a gated excitation energy spectrum with a single peak at ap-

100

proximately 2.25 MeV. Applying a gate to the excitation energy from 2 MeV to 3 MeV,

the angular distribution was extracted and shown to peak at forward angles, indicating a

potential Gamow-Teller transition. Therefore, the distribution associated with 2361 keV

γ-ray was noted for further analysis.

Peaks appear in the excitation energy spectrum at 2.25 MeV for both the 1427 keV and

2361 keV γ-lines, indicating a potential decay from the 2361 keV state via the emission

of 1427 keV and 942 keV γ-rays, within the experimental resolution. However, because

excitation energy peaks corresponding to the 942 keV γ-ray do not appear in the spectra of

the 2361 keV and 1427 keV lines, the possibility of this decay was ruled out.

After gating on Eγ and the excitation energy, the angular distributions were assumed to

contain contributions from a single state, making the analysis cleaner. For the majority of

the lines (207 keV, 382 keV, 932 keV (Ex peak 2), 942 keV, 1427 keV), peaks in the excitation

energy spectrum appeared above the energy of the measured γ-ray, indicating that the γ

resulted from a decay in which the residual nucleus passed through several excited states

before emitting the final photon that was measured. For the cases of the 77 keV, 932 keV

(Ex peak 1), 1753 keV and 2361 keV lines, the excitation energy peaked at approximately

the same energy as the γ-ray, within the experimental energy resolution, indicating that the

state decayed directly to the ground state of 86Br, or to one of its low-lying states.

As mentioned in section 4.5, Gamow-Teller transitions are characterized by a forward-

peaking angular distribution. Therefore, for an MDA fit, the distributions would be fit with

either a ∆L = 0+∆L = 2 linear combination, or with a ∆L = 1 curve, described by equation

4.5. Those distributions that were flat or that peaked at backward angles were assumed to

not be Gamow-Teller transitions and were, therefore, discarded from the remainder of the

analysis. As described previously, of the γ-rays listed in table 4.7, only two of them had

101

Figure 4.32: Eγ = 77 keV. A gate on Eγ was made from 74 keV to 80 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak at Ex = 0 MeV,
a gate was made from -0.5 MeV to 1 MeV and the angular distribution found in the bottom
right of the figure was extracted. Because the angular distribution does not appear to peak
at forward angles, a further MDA was not performed on this angular distribution.

102

Figure 4.33: Eγ = 207 keV. A gate on Eγ was made from 205 keV to 209 keV. Two peaks
appeared in the resulting gated excitation energy spectrum. For the first peak, at Ex = 3
MeV, a gate was made from 2.5 MeV to 3.5 MeV and the angular distribution found in the
bottom right of the figure was extracted. Because the angular distribution does not appear
to peak at forward angles, a further MDA was not performed on this angular distribution.

103

Figure 4.34: Eγ = 207 keV. A gate on Eγ was made from 205 keV to 209 keV. Two peaks
appeared in the resulting gated excitation energy spectrum. For the first peak, at Ex = 4
MeV, a gate was made from 3.5 MeV to 4.5 MeV and the angular distribution found in the
bottom right of the figure was extracted. Because the angular distribution appears to peak
at forward angles, a further MDA was performed on this angular distribution.

104

Figure 4.35: Eγ = 382 keV. A gate on Eγ was made from 380 keV to 384 keV. Two peaks
appeared in the resulting gated excitation energy spectrum. For the peak at Ex = 2.5 MeV,
a gate was made from 2 MeV to 3 MeV and the angular distribution found in the bottom
right of the figure was extracted. Because the angular distribution does not appear to peak
at forward angles, a further MDA was not performed on this angular distribution.

105

Figure 4.36: Eγ = 932 keV. A gate on Eγ was made from 928 keV to 933 keV. Two peaks
appeared in the resulting gated excitation energy spectrum. For the first peak, at Ex = 1
MeV, a gate was made from 0.5 MeV to 2 MeV and the angular distribution found in the
bottom right of the figure was extracted. Because the angular distribution does not appear
to peak at forward angles, a further MDA was not performed on this angular distribution.

106

Figure 4.37: Eγ = 932 keV. A gate on Eγ was made from 928 keV to 933 keV. Two peaks
appeared in the resulting gated excitation energy spectrum. For the second peak, at Ex =
2.5 MeV, a gate was made from 2 MeV to 3 MeV and the angular distribution found in the
bottom right of the figure was extracted. Because the angular distribution does not appear
to peak at forward angles, a further MDA was not performed on this angular distribution.

107

Figure 4.38: Eγ = 942 keV. A gate on Eγ was made from 941 keV to 945 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak at approximately
Ex = 1.5 MeV, a gate was made from 1 MeV to 2 MeV and the angular distribution found
in the bottom right of the figure was extracted. Because the angular distribution does
not appear to peak at forward angles, a further MDA was not performed on this angular
distribution.

108

Figure 4.39: Eγ = 1427 keV. A gate on Eγ was made from 1420 keV to 1430 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak at approximately
Ex = 2.3 MeV, a gate was made from 1.5 MeV to 3 MeV and the angular distribution
found in the bottom right of the figure was extracted. Because the angular distribution does
not appear to peak at forward angles, a further MDA was not performed on this angular
distribution.

109

Figure 4.40: Eγ = 1753 keV. A gate on Eγ was made from 1750 keV to 1755 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak at approximately
Ex = 1.7 MeV, a gate was made from 1.25 MeV to 2.25 MeV and the angular distribution
found in the bottom right of the figure was extracted. Because the angular distribution
appears to peak at forward angles, a further MDA was performed on this angular distribution.

110

Figure 4.41: Eγ = 2361 keV. A gate on Eγ was made from 2359 keV to 2363 keV. One peak
appeared in the resulting gated excitation energy spectrum. For the peak at approximately
Ex = 2.3 MeV, a gate was made from 2 MeV to 3 MeV and the angular distribution found
in the bottom right of the figure was extracted. Because the angular distribution appears to
peak at forward angles, a further MDA was performed on this angular distribution.

111

corresponding angular distributions that could have a ∆L = 0 contribution: 1753 keV and

2361 keV.

Figure 4.42: Multipole decomposition analysis results for the γ-gated angular distributions.
Because of the reduced number of experimental data points, only the ∆L = 0 (yellow) and
∆L = 2 (green) were used in the fitting procedure. The small contributions of the monopole
in these fits provide a stricter upper limit on the 86Kr strength compared to the result
presented in figure 4.30.

An MDA was performed on the angular distributions associated with the 1753 keV and

2361 keV lines, the results of which are shown in figure 4.42. Both cases showed a small

∆L = 0 component, although, as with the case of the singles data, the fitting parameters

carried significant uncertainties, due to the large statistical uncertainties present in the data.

112

0.00.51.01.52.02.53.03.54.001234567891e1E = 1753 keVL=0L=2SumData0.00.51.01.52.02.53.03.54.00123456781e1E = 2361 keV0.00.20.40.60.81.0CM [deg]0.00.20.40.60.81.0dd [mb sr1]Using the ∆L = 0 component, and the method discussed in section 4.6, the Gamow-Teller

strength was extracted for each transition; these values are shown in table 4.8.

Table 4.8: Gamow-Teller strength associated with observed γ-rays from GRETINA. Due to
the uncertainties associated with the measurement, these extracted values provide only an
upper limit for the strength from 86Kr.

B(GT )

Eγ
1753 keV 0.045 ± 0.043
2361 keV 0.063 ± 0.046

As with the previously presented results in this chapter, the extracted Gamow-Teller

strength was weak. This γ-ray analysis presented an opportunity to verify the result from

the initial singles analysis, and to further constrain the upper limit of the 86Kr Gamow-

Teller strength. The extracted strength distribution is an interesting result in and of itself,

providing insight into the structure and behavior of 86Kr, and other nuclei like it. How-

ever, further analysis and interpretation in the context of astrophysical phenomena tie these

experimental results to the larger goal of understanding and improving the nuclear physics

inputs for core-collapse supernova simulations.

113

Chapter 5

Weak-Rates

5.1 B(GT) Comparison

One of the goals of the 86Kr(t,3He+γ) experiment was to obtain data for validating and

benchmarking theoretical models. The experimental Gamow-Teller distributions that were

extracted (sections 4.6 and 4.7) are compiled in figure 5.1, with the result from the singles

MDA shown as black points and the result from the γ-ray analysis shown by the green points.

As noted in chapter 4, both data sets provide only upper limits for the Gamow-Teller strength

of 86Kr, due to significant statistical and systematic uncertainties. Also shown in figure 5.1

are two theoretical calculations to compare with the experimental results. For a better visual

comparison, a plot omitting the upper limit obtained from the singles MDA is shown in figure

5.2.

Comparing the Gamow-Teller strength distribution derived from the singles data to the

known 86Br level scheme, it is possible to obtain draw a conclusion about the sensitivity

of the 86Kr experiment. Because there is no known 1+ state below 2.5 MeV in 86Br, it is

unlikely that there is Gamow-Teller strength below this energy. Because of this, it appears

that a strength of 0.05 is the limit to the experimental sensitivity. Examining the rest of

the singles spectrum in figure 5.1, above approximately 3.5 MeV, the extracted strength is

at the 0.05 level. Again, because of the sensitivity limit, it is difficult to determine whether

114

Figure 5.1: Gamow-Teller strength distribution obtained from theoretical calculations and
experimental results. The black points indicate the upper limit obtained from analysis of
the singles data; the green points are the upper limit from the γ-ray analysis. Also shown: a
shell-model calculation (red) using the jj44pna interaction [78] and a QRPA calculation (blue)
[84].

there is strength at excitation energies above 3.5 MeV.

The first theoretical calculation shown is a shell-model calculation (red) performed in

NUSHELLX [79], within the SNE model space [75, 76] using the jj44pna interaction [78]. A

78Ni core was assumed for the calculation, with a valence space of (0f5/2, 1p3/2, 1p1/2, 0g9/2)

for protons, and (0g7/2, 1d5/2, 1d3/2, 2s1/2, 0h11/2) for neutrons. Because the model space

was truncated, it was necessary to apply a hindrance factor to the result of the calculation.

This hindrance factor, 1

h, is a combination of two factors, hhigh and hc.p., in the following

form:

1
h

=

1

hhigh

× 1
hc.p.

.

(5.1)

hhigh was first mentioned in section 2.2.3 and is associated with the mixing of two-particle

115

012345Excitation energy [MeV]0.000.050.100.150.20B(GT)QRPASingles dataConcidence dataShell-modelFigure 5.2: Gamow-Teller strength distribution obtained from theoretical calculations and
experimental results. To better compare the theoretical models and the experimental upper
limit for the strength, the result of the singles analysis is omitted here.

two-hole states with oscillator energies of 2¯hω and greater. It is the typical scaling factor

applied to calculated Gamow-Teller strength distributions and accounts for the well-known

quenching of the Gamow-Teller transition strength [80, 81]. hc.p. takes into account the core

polarization of the 0g orbital, describing the mixing of the 0g9/2 and 0g7/2 orbitals [140]. It

is necessary to include this additional factor because of the truncation of the model space

used in the shell-model calculation. With these two contributions, the total hindrance factor

applied to the calculated Gamow-Teller strength distribution was 1/9.

The second calculation shown in figure 5.2 is a quasiparticle random-phase approximation

(QRPA) calculation (blue).

It was performed using the axially-deformed Skyrme Finite

Amplitude Method (FAM) [141, 142], with the Skyrme functional and single-particle space

from reference [143]. The width of the states in the QRPA calculation was set to 0.59 MeV

116

012345Excitation energy [MeV]0.000.020.040.060.080.100.120.14B(GT)QRPAConcidence dataShell-model(FWHM). For both of the theoretical models, the first peak in the strength distribution

was placed at the excitation energy of the first 1+ state in 86Br, 2.446 MeV. Therefore, it

follows that both distributions peak at approximately the same excitation energy. Comparing

these theoretical results to the strength extracted from the coincidence data, it is apparent

that they are both within the experimental error bars, and thus, are consistent with the

experimental result.

Examining the summed strength up to 5 MeV, both the shell-model calculation and the

QRPA calculation yielded reasonably close results: 0.035 for the shell-model calculation and

0.024 for the QRPA calculation. The cumulative strength obtained from the coincidence
data was 0.108±0.063. Therefore, while the cumulative Gamow-Teller strength up to 5 MeV

obtained from the theoretical models is less than the magnitude of the strength extracted

from the γ-ray analysis, they still fell within the calculated experimental uncertainties.

5.2 Electron-Capture Rates

Using the Gamow-Teller strengths calculated in sections 4.6 and 4.7, as well as the theoretical

calculations shown in section 5.1, it was possible to calculate electron-capture rates for a

variety of stellar densities and temperatures. These calculations were performed using the

ECRATES code [44, 144, 145, 146], which takes the reaction Q-value, excitation energy and

Gamow-Teller strength distribution as inputs. The calculation yielded electron-capture rates

for the nucleus of interest as a function of density and temperature; the grid used in this

work is the same as that of the LMP table mentioned in section 2.2 [56, 69].

The results of several electron-capture rate calculations, at a temperature of 10 GK and

as a function of stellar density, are shown in figure 5.3. The central temperature of a star

117

Figure 5.3: A comparison of the experimentally-determined electron-capture rates, at 10 GK
over the range of stellar densities relevant for the late stages of core collapse, for 86Kr and
several theoretical models.

during the late stages of collapse just prior to core bounce ranges from 9 GK to 11 GK [9];

at the same time, the central density increases from 109 gm/cm3 to 1012 gm/cm3 during

the period of deleptonization until neutrino trapping. In the figure, the black solid line and

the gray shaded region represent the electron-capture rates that are associated with Gamow-

Teller strength determined from the γ-ray analysis in section 4.7; this is the most stringent

upper limit placed on the strength and, by extension, the electron-capture rates. Also shown

are the rates determined from strength distributions from the jj44pna shell-model calculation

with a 78Ni core (red) [79, 78], QRPA calculation (green) [84] and the approximate method

implemented in NuLib (blue) [1, 2, 47].

Both the shell-model calculation and the QRPA calculation were consistent with the ex-

perimental result, as they fell within the experimental error band for all stellar densities.

118

9.09.510.010.511.011.512.0log(ρYe[g/cm3])10-310-210-1100101102103104105106λEC[s−1]10.0GKUpper limit (γ)jj44pnaQRPAApproximate methodThe rates obtained from the approximate method from section 2.2 were the most inaccu-

rate, overestimating the electron-capture rates by about two orders of magnitude. Results

illustrating the overestimation of electron-capture rates by the approximate method have

also been shown for the case of 88Sr [94]. A 88Sr(t,3He) experiment was performed, as part

of the same experimental campaign as the 86Kr experiment, and a Gamow-Teller strength

distribution was extracted; the electron-capture rates calculated from the distribution were

also smaller than those rates provided by the approximate method by approximately two

orders of magnitude [94].

Because it is known that the approximate method accounts for the electron-capture rates

for a large number of nuclei in astrophysical simulations, see figure 2.1, it is likely that it

also overestimates the rates for those nuclei, particularly the ones encompassed by the high-

sensitivity region. The approximate method does not account for Pauli-blocking effects,

which suppress the Gamow-Teller strength of increasingly neutron-rich nuclei, leading to the

rate overestimation seen in the cases of 88Sr and 86Kr. Conversely, in high-temperature

stellar environments, low-lying states in parent nuclei can be thermally populated, and shell

orbits that were previously Pauli-blocked at T = 0 can become partially unblocked [54],

allowing electron-capture reactions to take place on these excited states [53]. These effects

are also neglected in the approximate method, so that the effect of thermal unblocking and

excited state transitions is uncertain. As noted in figure 2.6, including thermal unblocking

and excited state transitions to the rates, as with the case of the LMSH table, can cause large

variations in the electron-capture rates, as compared to the approximated rates. As such, it

was necessary to perform further theoretical calculations to obtain better electron-capture

rate tables for use in core-collapse supernova simulations.

The slope of the electron-capture rate curves, as a function of stellar density, are governed

119

by equation 2.5. For 86Kr, it is clear that the electron capture rates are sensitive to changes

in the density, because the electron-capture reaction Q-value is close to the Fermi energy.

Such a correlation occurs due to the location of the Fermi surface in relation to states in

the Gamow-Teller strength distribution. As the stellar density increases, the Fermi surface

moves to higher excitation energies, and allows more Gamow-Teller transitions to be included

in the electron-capture rate calculation. Therefore, small changes in the stellar density can

lead to significant increases in the electron-capture rate, as is shown in the case of 86Kr in

figure 5.3.

As was discussed in section 2.2.3, figure 2.5 presents the Gamow-Teller strength distribu-

tion for 86Kr, as obtained from the QRPA calculation discussed in section 5.1. It was shown

that, with increasing stellar density, the derived electron-capture rates remained sensitive to

the details of the Gamow-Teller strength distribution at higher excitation energies. Because

the other nuclei in the high-sensitivity region have large, negative Q-values, similar to 86Kr,

it is likely that their electron-capture rates exhibit the same behavior. This is an important

feature, because the highest stellar densities occur during the late stages of stellar core-

collapse, when nuclei in the high-sensitivity region have the largest effect on the dynamics

of the supernova [9]. Such a conclusion highlights the need for theoretical predictions of the

details of the strength distribution in order to obtain high-quality electron-capture rates, as

opposed to merely replicating the cumulative Gamow-Teller strength of the nucleus.

In summary, at the lower stellar densities, for nuclei in the high-sensitivity region, the

details of the Gamow-Teller strength distribution at low excitation energies have a significant

effect on the calculated electron-capture rates. As the stellar density increases, as shown in

figure 2.5, the details of the strength distribution continue to influence the calculated rates.

Such behavior is different from previous assumptions that, at high stellar densities, the

120

total Gamow-Teller strength becomes more important than the way in which the strength

is distributed. For neutron-rich nuclei with large negative Q-values, which are important

in the study of core-collapse supernovae, it has now been shown that the electron-capture

rates are sensitive to both the details of the Gamow-Teller strength distribution, as well

as the magnitude of the strength. Therefore, theoretical models are needed to accurately

reproduce all aspects of the strength distribution for use in electron-capture rate calculations,

and further studies of the high-sensitivity region.

5.3 High-Sensitivity Region Rates

Because the high-sensitivity region has been determined to have a significant effect on the

behavior of the late stages of stellar core collapse [9, 10, 11], it was chosen as the target

for the development of a new electron-capture rate table. In the original weak-rate library,

the rates for that region were predominately obtained via the approximate method, which

overestimates the electron-capture rates for heavy, neutron-rich nuclei. However, because the

nuclei in this region are influential of the progression of stellar core-collapse, such inaccuracies

in the rates lead to large uncertainties in the simulation. Therefore, the electron-capture

rates for the 74 high-sensitivity region nuclei [10], as well as 86Kr, 88Sr, 90Zr and 93Nb, were

calculated, in an attempt to obtain more accurate inputs for astrophysical simulations. The

chosen nuclei are marked in dark green in figure 2.1.

The Gamow-Teller strength distributions used in the electron-capture rate calculations

were obtained from QRPA calculations, which were performed in collaboration with Evan

Ney from the University of North Carolina at Chapel Hill, using the same parameters as

described in section 5.1 [84]. Whereas previous theoretical calculations determined the ex-

121

citation energy scale of the distribution based on the location of the first Gamow-Teller

transition, typically obtained from experimental data, this was not possible for these calcu-

lations. Many of the nuclei lacked comprehensive information of the level structure, including

excitation energy, spin and parity. Therefore, an estimate of the energy of the first Gamow-

Teller transition, as well as the spin of the ground state of the daughter nucleus and the

Q-value of the reaction, were obtained from a Hartree-Fock-Bogoliubov (HFB) calculation,

which was performed to provide inputs to the final QRPA calculation [84]. In order to ob-

tain the most accurate electron-capture rates, the parent spin and Q-value were set to the

experimentally-determined values where they were available, and the theoretical estimates

were used in the cases where no spin assignments existed.

The electron-capture rate calculations were performed following the same methods de-

scribed in section 5.2, using the ECRATES code [44, 144, 145, 146], and then were compiled

into a single table to be added to the weak-rate library [71]. Finally, to determine the effect

of these rates on the late stages of core-collapse, a new simulation was performed with the ad-

dition of the updated rates. This simulation was run using the NuLib [72] and GR1D codes

[72, 73] with the same parameters described in section 2.3. The resulting lepton fraction

from this simulation is shown in figure 5.4.

The dashed black line shows the status of the weak-rate library prior to any changes in

the approximate method or addition of new rates [10]; the green line shows the weak-rate

library with the addition of the isospin-dependent approximate method [2] and updated pf -

shell rate calculations [33]. Finally, the blue line shows the result of the simulation with

the weak-rate library in its current state, with the added high-sensitivity region rate table.

For comparison, there are two simulations from the original sensitivity study, discussed in

section 2.3, illustrating the cases in which the electron-capture rates of the diamond-region

122

Figure 5.4: Plot of the lepton fraction versus core density during deleptonization and neutrino
trapping. The results from the original weak-rate library (black dashed line), the weak-rate
library with an updated approximate method (green) and the weak-rate library with the
new high-sensitivity region rates (blue) are compared.

nuclei were scaled by a factor of 0.1 (red line) and by a factor of 0.01 (orange line) [10].

Introducing more accurate electron-capture rates to the simulation had a significant effect

on the final lepton fraction after 1012 g/cm3. Comparing the blue and green lines, the weak-

rate library without the updated rates (green line) attained a final lepton fraction of 0.294,

while the weak-rate library including the new rates (blue line) reached a final lepton fraction

of 0.312. This constituted a 15% variation in the lepton fraction with the addition of more

accurate rates for 78 nuclei. It is also possible to compare the results of the newest simulation

123

10101011101210131014ρc(g/cm3)0.290.300.310.320.330.340.350.360.370.380.390.400.410.42YlQRPA diamond ratesOriginal weak-rate libraryUpdated approximate methodDiamond rates x0.1Diamond rates x0.01to the sensitivity study that motivated this work, and it is found that the final lepton fraction

for this work (blue) is comparable to the case in which the rates in the diamond region were

scaled by a factor of 0.01 (orange).

With the addition of the newly-calculated electron-capture rates for the high-sensitivity

region nuclei, the core-collapse supernova simulation now provides a more reliable descrip-

tion of the dynamics of the event. Examinations of the lepton fraction during the late stages

of stellar core-collapse illustrated that the electron-capture rates for these nuclei were over-

estimated by approximately 2 orders of magnitude, on average. By introducing a dedicated

table of rates, rather than relying on the approximate method for this region, the dynamics

of the simulation were changed significantly. Now, with improved nuclear physics inputs, all

astrophysical simulations that rely on NuLib and the weak-rate library will yield more accu-

rate results, leading to a better understanding of core-collapse supernovae, and astrophysical

phenomena, in general.

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Chapter 6

Conclusions and Outlook

The evolution of core-collapse supernovae is highly sensitive to electron-captures on medium-

heavy nuclei. In the previous sensitivity study, as discussed in chapter 2, a region of neutron-

rich nuclei surrounding the N = 50 shell-closure was found to have a significant effect on

the dynamics of the late stages of stellar core-collapse [9, 11]. A set of 74 nuclei, above

78Ni were chosen and their electron-capture rates were scaled in a series of simulations. It

was found that, in changing the rates in the simulation by a factor of 0.1, the rates in

the high-sensitivity region were responsible for half of the total variation in dynamics of the

supernova, compared to the thousands of other nuclei contributing [10]. Because the method

used for estimating electron-capture rates in this high-sensitivity region was thought to yield

overestimated rates, it was clear that a more in-depth study of these nuclei was needed.

While it would be preferable to obtain Gamow-Teller strengths and electron-capture rates

for all of these nuclei using experimental data, such as charge-exchange reaction measure-

ments, this is not feasible because it is not time-efficient to perform high-quality experiments

on a group of more than 70 nuclei. Additionally, experimental data cannot provide all of the

necessary information, such as transitions from excited states [53], or thermal unblocking

effects [54], which become relevant at high stellar temperatures. Instead, it is necessary to

rely on theoretical models to provide accurate Gamow-Teller strength distributions for use

in electron-capture rate calculations.

125

These theoretical calculations must be validated and benchmarked by experimental data

to ensure the models provide accurate rates that can be used in the astrophysical simulations.

Experiments also guide theory development, finding those models in need of improvement.

An experimental campaign was performed on nuclei lying near the high-sensitivity region,

including 86Kr, which was studied with the (t,3He+γ) reaction. This nucleus was chosen

because it is the most neutron-rich stable nucleus on the N = 50 shell closure, and is as close

to the high-sensitivity region for core-collapse supernovae as can be studied with current

experimental facilities. The 86Kr experiment was challenging because of the use of a gas

target, which introduced target window events into all of the data sets. In order extract the

signal of interest, a careful background subtraction was performed, introducing significant

systematic uncertainties into the calculated 86Kr(t,3He) cross section.

Using both the multipole decomposition analysis on the singles data and an analysis

of the measured γ-rays, an upper limit for the Gamow-Teller strength was extracted for

86Kr up to an excitation energy of 5 MeV. In the analysis of the coincidence data, γ-rays

associated with the two known 1+ states in 86Br were not observed, leading to a broader

search for Gamow-Teller transitions. To that end, several new γ-lines were identified in 86Br,

two of which were found to potentially have some Gamow-Teller strength associated with

them. For both the singles and coincidence data, the systematic and statistical uncertainties

present in the extraction of the strength distribution yielded a Gamow-Teller strength that

was consistent with zero.

The experimental data analyzed in this work provided new information about the Gamow-

Teller strength of 86Kr, indicating that it is heavily suppressed due to Pauli-blocking in the

nucleus. Such a conclusion translated to electron-capture rates that were different from

those rates originally used in the core-collapse supernova simulation. The simulations were

126

performed using inputs from the weak-rate library, which includes a number of weak-rate

tables covering the valley of stability and lighter-mass nuclei. For neutron-rich, heavy nuclei,

the weak-rate library uses an approximate method for calculating the weak rates, assuming

a single Gamow-Teller peak instead of a strength distribution. It was shown, in previous

sensitivity studies, that this approximation likely overestimates the electron-capture rates for

neutron-rich, heavy nuclei, and the experimental results from the 86Kr experiment confirmed

an overestimation of the rates by approximately two orders of magnitude.

Because of this result, it was clear that different theoretical models were needed to ob-

tain more accurate electron-capture rates. For this, shell-model and QRPA calculations were

performed, and the subsequent Gamow-Teller strength distributions were used to derive sets

of electron-capture rates. Comparing the 86Kr experimental results to the theoretical cal-

culations indicated that both models were consistent with the electron-capture rates for the

range of densities relevant to the late stages of stellar core-collapse, within the experimental

uncertainties. Validating theoretical models against the experimental data opened the way

to performing calculations for additional nuclei, and yielding more accurate electron-capture

rates for a larger section of the chart of nuclides.

Using the QRPA method, the Gamow-Teller strength distributions and electron-capture

rates were calculated for the 74 nuclei in the high-sensitivity region, as well as for 86Kr,

88Sr, 90Zr and 93Nb [84]. Compiling all of the rates into a single rate table, they were

added to the weak-rate library [71] and NuLib [72], to take the place of the approximated

rates. The intention of this final step was to provide more accurate nuclear physics inputs to

the astrophysical simulation, both to observe the effect on the dynamics of the supernova,

and to determine the extent to which the electron-capture rates in this region were changed

with the new calculations. A new core-collapse supernova simulation was performed using

127

GR1D [72, 73], incorporating the new rate table and other updates to the weak-rate library.

The results of the simulation indicated that the electron-capture rates in the high-sensitivity

region were, on average, overestimated by about two orders of magnitude, which parallels

the results of the 86Kr experiment.

As a result of these findings, it is clear that it is important to continually develop better

theoretical models for the more exotic and heavier mass regions of the chart of nuclides. Then,

electron-capture rates with increased reliability will become available for use in astrophysical

simulations, yielding a more accurate picture of the behavior of core-collapse supernovae, and

other cataclysmic phenomena. In the era of multi-messenger astronomy, measurements of

neutrino signals and gravitational waves are combined in order to obtain a more complete

understanding of the astrophysical event. However, it is known that variations in electron-

capture rates, especially in the high-sensitivity region, can lead to large uncertainties in the

expected neutrino luminosity [9] or gravitational wave frequency [29, 30]. Therefore, it is

imperative that astrophysical simulations be as accurate as possible, in order to guide and

inform observations of real astronomical events.

Charge-exchange experiments at intermediate energies will continue to be influential in

the development process, benchmarking theoretical models and providing insight into regions

of the chart of nuclides that require further study. Experimental campaigns at the Facility

for Rare Isotope Beams (FRIB) will allow charge-exchange experiments to probe heavier and

more exotic nuclei with improved statistics. Performing experiments in inverse kinematics

will also be important to reach more nuclei that are of astrophysical interest. New experi-

mental methods, such as (d,2He) reaction in inverse kinematics, are being developed to use

the Active Target-Time Projection Chamber (AT-TPC) [147] in conjunction with the S800

spectrometer. Although this type of experiment relies on the measurement of two protons,

128

this probe was chosen because alternative probes present greater experimental challenges.

(t,3He) experiments in inverse kinematics would require a hydrogen target with significant

containment measures, which would make it difficult for the 3He particle to reach the detec-

tors. The (7Li,7Be) probe [148] would also not be ideal because it would not populate states

with high excitation energy, making the results less useful for astrophysical purposes.

Improvement to experiments in forward kinematics could also be considered. For ex-

ample, in the case of the 86Kr experiment, it could be possible to re-do the measurement,

but with a frozen target. This would allow the target to be thinner, with the same density,

and also remove the need for a background subtraction. With these steps, the statistics for

the measurement would be increased, and the systematic uncertainties would be reduced, in

theory providing a cleaner measurement of the 86Kr Gamow-Teller strength. Knowing the

difficulties of a gas target measurement for nuclei with weak Gamow-Teller strength, future

charge-exchange experiments can be improved in order to obtain higher-quality data, and

yielding, in turn, more accurate electron-capture rates for use in astrophysical simulations.

The synergy between nuclear physics experiments and astrophysical observations is nec-

essary, as discoveries in one field guide the progress of the other. Sharing improved nuclear

physics inputs to astronomers provides accurate simulations that can be used as a template to

which observational data can be compared. In reverse, new observational data indicates the

ways in which astrophysical simulations and nuclear physics data can be improved, guiding

theoretical development and future experimental studies. By collaborating across disciplines,

information about avenues for further study is continually uncovered, leading to a better,

and more detailed, understanding of the universe.

129

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