COULOMB EXCITATION OF NEUTRON-RICH SULFUR ISOTOPES

By

Brenden Robert Longfellow

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics — Doctor of Philosophy

2020

ABSTRACT

COULOMB EXCITATION OF NEUTRON-RICH SULFUR ISOTOPES

By

Brenden Robert Longfellow

Understanding how the structure of nuclei is modified far from stability has become one of

the major goals in nuclear science. While the nuclear shell model was successful in explaining

the magic numbers observed for the stable and near-stable nuclei available for study at

the time, subsequent research on nuclei with extreme proton-to-neutron ratios has revealed

surprising changes in nuclear structure. For example, the conventional magic number of

neutrons N = 28 has been shown to break down in the region of neutron-rich nuclei centered

around 42Si and 44S known as the N = 28 island of inversion. In this work, predictions made

by the shell-model effective interaction SDPF-MU, which has been successful in describing

the evolution of collectivity for nuclei in this region, were put to the test using the selectivity

of intermediate-energy Coulomb excitation in an experiment utilizing the scintillator array

CAESAR and the S800 magnetic spectrograph at the National Superconducting Cyclotron

Laboratory at Michigan State University.

In the even-even neutron-rich sulfur isotopes

38,40,42,44S, B(E2) strengths from the ground states to multiple 2+ states were measured

allowing a detailed comparison to theoretical predictions by the SDPF-MU Hamiltonian.

For 43S, excited states built on top of both the ground state and the isomeric state at

320 keV were excited, allowing the collective nature of these shape-coexisting structures to

be characterized.

ACKNOWLEDGMENTS

I am deeply grateful for the support I have received over the course of my graduate career

from friends, colleagues, and mentors. The following list of acknowledgments is certainly

not exhaustive. First and foremost, I would like to thank my advisor, Alexandra Gade, for

her tireless guidance and enthusiasm. Thank you for everything you have taught me about

physics and for encouraging me to pursue opportunities ranging from travel to collaborate on

measurements in Japan to writing proposals for experiments. Your mentorship and advice

have helped me grow tremendously as a scientist. I would also like to thank Dirk Weisshaar

for sharing his knowledge of γ-ray detectors, electronics, and the liquid nitrogen fill system,

answering questions about data analysis, and helping me develop as an experimentalist.

Next, I would like to thank the members of my guidance committee for their support and

encouragement. Thank you to Daniel Bazin for tuning the S800 spectrograph during the

sulfur Coulomb excitation experiment that this dissertation results from and for providing

expertise during gamma group meetings. Thank you to Alex Brown, who I was lucky to be

able to work with on a project exploring Thomas-Ehrman shifts in sd-shell nuclei. Thank you

to Wade Fisher for providing the outside perspective of a high-energy experimentalist to my

committee. Finally, I thank Sean Liddick for his work with the Nuclear Science and Security

Consortium (NSSC), which I am grateful to for funding, and for giving me the opportunity

to expand my knowledge by collaborating on a few β-decay experiments. I would also like

to thank the NSSC giving me the chance to work with Aaron Gallant and Nick Scielzo at

Lawrence Livermore National Laboratory on a summer project.

During my time as a graduate student, I was lucky to have the chance to learn from

postdoctoral researchers including Peter Bender, Joe Belarge, Mark Spieker, Jing Li, and

iii

Sayani Biswas. You have had an indelible impact on my scientific career.

I would also

like to thank the graduate students in the gamma group who I had the privilege to work

alongside: Eric Lunderberg, (upstairs) Brandon Elman, Daniel Rhodes, Matt Hill, and Peter

Farris. These graduate students and the graduate students in the lifetime group have my

appreciation for sharing the burden of the filling system pager.

I would also like to thank the staff at the National Superconducting Cyclotron Labo-

ratory (NSCL). Without them it would not have been possible to run the sulfur Coulomb

excitation experiment. I have enjoyed my time at the NSCL and am looking forward to the

science results that will come from the Facility for Rare Isotope Beams. I am thankful for

the friendships I have made at Michigan State University and for the friendship of Jason

Surbrook, who followed me to the NSCL from the University of North Carolina at Chapel

Hill.

I was glad to be able to play soccer with the foot-pounds, go to trivia nights, and

attend MSU sporting events. Finally, I would like to thank my parents, Russell and Cathie,

and my brothers, Jerrod, Darec, and Rion, for their unwavering support.

iv

TABLE OF CONTENTS

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

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Atomic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Nuclear Shell Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Collective Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Configuration Interaction Method . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Breakdown of the N=28 Magic Number . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
1.7 Motivation for Studying the Neutron-Rich Sulfur Isotopes

Chapter 2

Intermediate-Energy Coulomb Excitation . . . . . . . . . . . . .
2.1 Alder-Winther Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Reduced Transition Probabilities and Lifetimes
. . . . . . . . . . . . . . . .

Chapter 3 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Beam Production and Purification . . . . . . . . . . . . . . . . . . . . . . . .
3.2 S800 Magnetic Spectrograph . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Particle Identification and Trajectory Reconstruction . . . . . . . . .
3.2.2
Isomer Tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In-Beam γ-Ray Spectroscopy with CAESAR . . . . . . . . . . . . . . . . . .
3.3.1
Interaction of γ Rays with Matter . . . . . . . . . . . . . . . . . . . .
3.3.2 Doppler Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Nearest-Neighbor Addback . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 CAESAR Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Chapter 4 Data Analysis and Results

. . . . . . . . . . . . . . . . . . . . . .
4.1 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Results for Even-Even Neutron-Rich Sulfur Isotopes . . . . . . . . . . . . . .
Intermediate-Energy Coulomb Excitation of 38S . . . . . . . . . . . .
Intermediate-Energy Coulomb Excitation of 40S . . . . . . . . . . . .
Intermediate-Energy Coulomb Excitation of 42S . . . . . . . . . . . .
Intermediate-Energy Coulomb Excitation of 44S . . . . . . . . . . . .
4.3 Results for 43S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43S Incoming Beam Isomeric Content . . . . . . . . . . . . . . . . . .
Intermediate-Energy Coulomb Excitation of 43S . . . . . . . . . . . .

4.2.1
4.2.2
4.2.3
4.2.4

4.3.1
4.3.2

1
1
6
10
12
16
17
22

28
30
36
38

39
39
43
45
49
51
51
56
57
58

64
64
69
69
77
82
89
94
94
96

Chapter 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

v

Chapter 6

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 113

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vi

LIST OF TABLES

Table 3.1: Production target, momentum acceptance dp/p, and energy as inferred
from the magnetic rigidity setting of the segment of the analysis line just
before the 492 mg/cm2 209Bi reaction target in front of the S800 spectro-
graph for each secondary beam setting.
. . . . . . . . . . . . . . . . . . .

Table 3.2: Energy resolution parameters for each of the 10 rings of CAESAR for en-
ergies in keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.1: Corrected number of sulfur isotopes Nbeam and average livetimes for par-
ticle triggers in the S800 LTS800 and S800-CAESAR coincidences LTcoinc
for each of the settings.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

63

67

Table 5.1: B(E2) strengths for the neutron-rich sulfur isotopes studied in this work
compared to the results of previous intermediate-energy Coulomb excita-
tion experiments and theoretical calculations using the SDPF-MU Hamilto-
nian with effective proton and neutron charges of 1.35 and 0.35, respectively.108

vii

LIST OF FIGURES

Figure 1.1: Central projection of the nucleon-nucleon interaction. Outside the repul-
sive core, the potential is attractive with rapidly diminishing strength.
Over longer ranges, the central nuclear force can be modeled through
exchange of massive particles called mesons. Figure modified from [1]. . .

Figure 1.2: The chart of the nuclides with neutron number on the x axis and pro-
ton number on the y axis. Terra Incognita refers to the region of the
chart with experimentally unobserved nuclei that are predicted to exist.
The dotted lines indicate the traditional magic numbers of neutrons and
protons discussed in the text. Figure adapted from [2].
. . . . . . . . . .

Figure 1.3: Differences between experimental binding energies and binding energies
calculated using the liquid drop model for nuclei with Z ≥ 8 as a function
of neutron number. The peaks at N = 28, 50, 82, and 126 indicate that
nuclei have internal structure. Figure taken from [3] with data from [4]. .

Figure 1.4: Energies of neutron single-particle orbitals calculated for 208Pb using
different mean field potentials:
three-dimensional harmonic oscillator,
Woods-Saxon, and Woods-Saxon plus spin-orbit. The number in brackets
is the maximum number of neutrons allowed in the orbital by the Pauli
principle and the number outside the brackets is the sum of total allowed
nucleons through this orbital. Orbitals are labeled by their quantum num-
bers. Figure is adapted from [5].
. . . . . . . . . . . . . . . . . . . . . .

Figure 1.5: Energies of the first 2+ excited state and ratios R4/2 of energies for the
first 2+ and 4+ states in even-even nuclei across the chart of nuclides. The
neutron and proton magic numbers are denoted by dashed lines. Figure
taken from [6].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.6: Experimental energies of the first 2+ states [7] (top) and B(E2; 0+

1 → 2+
1 )
strengths [8] (bottom) for Si, S, Ar, and Ca isotopes showing the evolution
of collectivity around the conventional neutron magic numbers N = 20
and N = 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.7: Schematic illustration of the tensor force. The repulsive interaction be-
tween the neutron 0f7/2(j>) orbital and proton 0d5/2(j>) and the at-
tractive interaction between the neutron 0f7/2(j>) orbital and proton
0d3/2(j<) orbital reduce the Z = 14 sub-shell gap. Figure modified from
[9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3

5

9

12

19

21

viii

Figure 1.8: SDPF-MU neutron 1p3/2 occupancies for 0+ (red) and 2+ states (blue)
below 4.5 MeV in 38,40,42,44S. The increase in occupancy for the second
2+ state in 42S relative to 38S explains the observed change in γ-decay
pattern. Figure modified from [10].
. . . . . . . . . . . . . . . . . . . . .

Figure 1.9: SDPF-MU predictions for B(E2; 0+

1 → 2+) strengths in e2fm4 for levels
in 38,40,42,44S. Arrow widths are proportional to transition strength. 2+
states above 4 MeV with B(E2) values less than 10 e2fm4 are omitted
in this figure. Effective proton and neutron charges of 1.35 and 0.35,
respectively, were used in the SDPF-MU calculations. . . . . . . . . . . .

Figure 2.1: Schematic of intermediate-energy Coulomb excitation. The projectile
(target) nucleus is inelastically excited in the Coulomb field of the target
(projectile) nucleus.
In analysis, a maximum center-of-mass scattering
angle θ (minimum impact parameter b) is considered to ensure no nuclear
contribution to the cross section. Figure adapted from [11]. . . . . . . . .

Figure 3.1: Layout of the Coupled Cyclotron Facility at the National Superconduct-
ing Cyclotron Laboratory showing the ion sources, K500 and K1200 cy-
clotrons, and A1900 fragment separator [12]. Figure adapted from [13].
.

Figure 3.2:

Illustration of the S800 magnetic spectrograph including the analysis line.
The target position was surrounded by the CsI(Na) array CAESAR. Fig-
ure taken from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.3: S800 focal plane diagram showing the cathode readout drift chambers
(CRDCs), ionization chamber, and E1 plastic scintillator. Figure modified
from [13].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.4: Typical calibrated mask run for CRDC1 (upstream) taken from the 38S
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

setting.

Figure 3.5: Particle identification plot utilizing energy loss and time of flight for the
44S setting. The energy loss is measured in the S800 ionization chamber
and is proportional to Z2 and the time of flight is measured between the
. . . . . . .
OBJ and E1 plastic scintillators and is proportional to A/Z.

Figure 3.6:

Illustration of the arrangement of CAESAR into 10 rings of detectors.
Left: cross-sectional view of rings F and J perpendicular to the beam axis.
Right: the 3 × 3 × 3 inch detectors occupy the two most upstream (A,B)
and two most downstream (I,J) rings while the 2 × 2 × 4 inch detectors
occupy the six middle rings (C-H). The target position is shown in red.
Figure taken from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

27

29

40

44

45

48

50

52

ix

Figure 3.7: Photograph of the 192 CsI(Na) detectors composing CAESAR in front of
the S800 spectrograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Figure 3.8:

Illustration of the response of a γ-ray detector to many monoenergetic γ
rays showing characteristic features including the full-energy peak, Comp-
ton edge, Compton continuum, and single and double-escape peaks. Fig-
ure taken from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.9: Energies after calibration for the 192 CAESAR detectors for the 88Y
source. The 898-keV and 1836-keV transitions are aligned for all detectors.
The 2614-keV transition from room background is also visible. . . . . . .

Figure 3.10: Alignment of the corrected CAESAR timings for all 192 detectors after
calibration gated on γ-ray energies above 500 keV in the laboratory frame
for one of the 38S runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.11: Full-energy peak efficiency of CAESAR as a function of energy as deter-
mined from direct measurement of a low-activity 207Bi source and from
coincidence measurements with a LaBr3(Ce) detector for 60Co, 88Y, and
207Bi sources compared to predictions from GEANT4 simulations. The ef-
ficiency curves here were produced treating the 192 detectors of CAESAR
independently (no addback applied).
. . . . . . . . . . . . . . . . . . . .

Figure 4.1: Doppler-corrected energy versus corrected CAESAR timing gated on 40S.
As seen from the 904-keV peak, the prompt time-energy cut (red) removes
random background counts without affecting the prompt γ decays that
occur after Coulomb excitation.
. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.2: Top: Doppler-corrected energy spectrum for 38S gated on scattering an-
gles smaller than 40 mrad in the laboratory frame. The blue curves are
the individual components of the fit function derived from GEANT4 simu-
lations along with a double exponential background. The red curve is the
total fit function. Bottom: Low-energy portion of the Doppler-corrected
energy spectrum for 38S. The large background at low energies is due to
bremsstrahlung and other beam-correlated background. . . . . . . . . . .

Figure 4.3: Doppler-corrected coincidence matrix for 38S events with exactly two γ
rays detected in CAESAR. Both the x and y axes have 16-keV wide bins.
The matrix is filled symmetrically. The 1292-keV and 1513-keV transi-
tions are in coincidence and the 1292-keV and 2508-keV transitions are in
coincidence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: Background-subtracted, Doppler-corrected γ-ray spectrum for 38S counts
in coincidence with the 1292-keV transition. The 1513-keV and 2508-keV
peaks are visible.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

59

60

62

68

71

72

73

x

Figure 4.5: Counts in the 1292-keV peak as a function of scattering angle in the
laboratory frame. Using the nominal safe scattering angle of 55 mrad
would lead to an underestimation of the angle-integrated cross section. A
maximum safe scattering angle of 40 mrad was considered in the analysis.

Figure 4.6: Top:

feeding-subtracted angle-integrated cross section as a function of
choice of scattering angle cut derived from the experimental data for the
2+
1 state in 38S (black). The red curve is the expected cross section assum-
1 → 2+
ing a B(E2; 0+
1 )
strength calculated using Alder-Winther theory for different choices of
maximum scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 ) strength of 229 e2fm4. Bottom: B(E2; 0+

1 → 2+

Figure 4.7: Doppler-corrected energy spectrum for 40S gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the
individual components of the fit function derived from GEANT4 simula-
tions along with a double exponential background. The red curve is the
total fit function.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: Doppler-corrected coincidence matrix for 40S events with exactly two γ
rays detected in CAESAR. Both the x and y axes have 16-keV wide bins.
The matrix is filled symmetrically. The 904-keV and 2452-keV transitions
are in coincidence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.9: Background-subtracted, Doppler-corrected γ-ray spectrum for 40S counts
in coincidence with the 904-keV transition. A peak at 2452 keV is visible.

Figure 4.10: Top:

feeding-subtracted angle-integrated cross section as a function of
choice of scattering angle cut derived from the experimental data for the
2+
1 state in 40S (black). The red curve is the expected cross section assum-
1 → 2+
ing a B(E2; 0+
1 )
strength calculated using Alder-Winther theory for different choices of
maximum scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 ) strength of 284 e2fm4. Bottom: B(E2; 0+

1 → 2+

Figure 4.11: Doppler-corrected energy spectrum for 42S gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the
individual components of the fit function derived from GEANT4 simula-
tions along with a double exponential background. The red curve is the
total fit function.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.12: Doppler-corrected energy spectrum for 42S for events where only one γ ray
was detected by CAESAR. Nearest-neighbor addback was utilized. The
3000-keV transition from the 2+
2 state to the ground state is visible. . . .

xi

74

76

78

79

80

81

82

83

Figure 4.13: Doppler-corrected coincidence matrix for 42S events with exactly two γ
rays detected in CAESAR. Both the x and y axes have 16-keV wide bins.
The matrix is filled symmetrically. The 903-keV transition is in coinci-
dence with peaks at 2100 keV (more clearly observed in the projection of
. . . . . . . . . . . . . . . . . . . . . . . . .
Figure 4.14) and 4150 keV.

Figure 4.14: Background-subtracted, Doppler-corrected γ-ray spectrum for 42S counts
in coincidence with the 903-keV transition. . . . . . . . . . . . . . . . . .

Figure 4.15: Top:

feeding-subtracted angle-integrated cross section as a function of
choice of scattering angle cut derived from the experimental data for the
2+
1 state in 42S (black). The red curve is the expected cross section assum-
ing a B(E2; 0+
1 ) strength of 326 e2fm4. The blue data points are the
experimental data with the correction for scattering angle explained in the
text. Bottom: B(E2; 0+
1 ) strength calculated using Alder-Winther
theory for different choices of maximum scattering angle. . . . . . . . . .

1 → 2+

1 → 2+

Figure 4.16: Doppler-corrected energy spectrum for 44S gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the
individual components of the fit function derived from GEANT4 simula-
tions along with a double exponential background. The red curve is the
total fit function.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.17: Doppler-corrected coincidence matrix for 44S events with exactly two γ
rays detected in CAESAR. Both the x and y axes have 16-keV wide bins.
The matrix is filled symmetrically. The 1329-keV transition is in coinci-
dence with the 949-keV transition.
. . . . . . . . . . . . . . . . . . . . .

Figure 4.18: Background-subtracted, Doppler-corrected γ-ray spectrum for 44S counts
. . . . . . . . . . . . . . . .

in coincidence with the 1329-keV transition.

Figure 4.19: Top:

feeding-subtracted angle-integrated cross section as a function of
choice of scattering angle cut derived from the experimental data for the
2+
1 state in 44S (black). The red curve is the expected cross section assum-
1 → 2+
ing a B(E2; 0+
1 )
strength calculated using Alder-Winther theory for different choices of
maximum scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 ) strength of 221 e2fm4. Bottom: B(E2; 0+

1 → 2+

xii

84

85

87

90

91

92

93

Figure 4.20: Top: Doppler-corrected energy spectrum for 43S gated on scattering an-
gles smaller than 40 mrad in the laboratory frame. The blue curves are
the individual components of the fit function derived from GEANT4 simu-
lations along with a double exponential background. The red curve is the
total fit function. Bottom: Low-energy portion of the Doppler-corrected
energy spectrum for 43S. The wide feature present in the 43S data that
is not present for 38S is due to the continuous decay of the 320-keV iso-
mer at different points along the beam line. The large background at low
energies is due to bremsstrahlung and other beam-correlated background.

Figure 4.21: Doppler-corrected energy spectrum for 43S produced from one-neutron
removal for addback events with exactly one or two detected γ rays. The
labeled peaks were also observed in the high-resolution 9Be(44S,43S)X
data presented in [15].
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

99

Figure 4.22: Background-subtracted, Doppler-corrected energy spectrum for 43S mea-
sured with CAESAR in coincidence with 320-keV γ rays detected on the
hodoscope at the S800 focal plane from the decay of the isomer. The
CAESAR spectrum is for addback multiplicity one.

. . . . . . . . . . . . 100

Figure 4.23: Top:

feeding-subtracted angle-integrated cross section as a function of
choice of scattering angle cut derived from the experimental data for the
977-keV state in 43S (black). The red curve is the expected cross section
assuming a B(E2) strength of 91 e2fm4 from the ground state to the
977-keV level. Bottom: B(E2) strength calculated using Alder-Winther
theory for different choices of maximum scattering angle. . . . . . . . . . 101

Figure 4.24: SDPF-MU predictions for B(E2) strengths in e2fm4 for low-lying levels in
43S compared with experimental results. Arrow widths are proportional
to transition strength.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 4.25: Doppler-corrected energy spectrum for 45Cl gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the
individual components of the fit function derived from GEANT4 simula-
tions along with a double exponential background. The red curve is the
total fit function.

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

Figure 5.1: B(E2; 0+

1 → 2+) strengths in the even-even sulfur isotopes for the 2+

states measured in this work compared to predictions from the SDPF-MU
Hamiltonian for the first five 2+ states. . . . . . . . . . . . . . . . . . . . 107

xiii

Chapter 1

Introduction

1.1 Atomic Nuclei

The atomic nucleus is a quantum many-body system made up of neutrons and protons.

Collectively, the electrically-neutral neutrons and positively-charged protons are known as

nucleons. The nucleus consisting of N neutrons and Z protons, with a total of A = N + Z

nucleons, is commonly referred to using the notation AX(Z) where X(Z) is the chemical

symbol for the element with Z protons. Since protons carry a positive charge, they repel

each other via the Coulomb interaction, which is proportional to 1/r2 where r is radial

separation. Despite this repulsion, the nucleons in a nucleus are bound together due to the

strong nuclear force.

Nucleons are composed of three valence quarks: two down quarks and one up quark (neu-

trons) or two up quarks and one down quark (protons). The strong interaction described by

quantum chromodynamics confines the quarks that constitute the nucleons and is mediated
by the exchange of gluons. At a distance of 1 fm = 10−15 m, which is on the order of the

nucleon size, the strong force is roughly 100 times stronger than the Coulomb repulsion.

While quarks and gluons carry color charge, nucleons are color neutral. The strong nuclear

force that binds nucleons together within the nucleus is a residual of the fundamental strong

interaction, analogous to the van der Waals force in chemistry between neutral atoms or

1

molecules which is a residual of the fundamental Coulomb interaction [16].

Figure 1.1 shows the central part of the nucleon-nucleon interaction. The strong nuclear

force has a repulsive core at short distances but outside of the core is strongly attractive

over a short range. At longer distances, the nucleon-nucleon potential can be modeled as

the exchange of light mesons (systems composed of one quark and one antiquark) [17]. In

addition to the scalar central part, the nucleon-nucleon interaction has a complex dependence

on spin that can be decomposed into vector (spin-orbit) and tensor components. Nucleon-

nucleon scattering experiments indicate that the strong nuclear force is approximately charge

independent [18].

Figure 1.1: Central projection of the nucleon-nucleon interaction. Outside the repulsive core,
the potential is attractive with rapidly diminishing strength. Over longer ranges, the central
nuclear force can be modeled through exchange of massive particles called mesons. Figure
modified from [1].

The interplay between the strong nuclear force and the Coulomb repulsion is apparent

in the chart of the nuclides shown in Figure 1.2. The chart of the nuclides displays neutron

2

2001000-100r (fm)V (MeV)0.00.51.01.52.0Repulsivecore2π, ρ, ω, σexchangeπexchange2.5number on the x axis and proton number on the y axis with each square representing a

different nucleus. Nuclei with the same Z, N , and A are referred to as isotopes, isotones,

and isobars, respectively. Of the approximately 7000 nuclei predicted to exist [19], around

3000 have been observed experimentally. The stable nuclei (black squares) are situated in

the region of the chart known as the valley of stability. As the proton number increases, the

valley of stability bends away from the N = Z line, where neutron-proton pairing dominates,

toward the neutron-rich side as a result of the repulsive Coulomb force between protons.

Figure 1.2: The chart of the nuclides with neutron number on the x axis and proton number
on the y axis. Terra Incognita refers to the region of the chart with experimentally unobserved
nuclei that are predicted to exist. The dotted lines indicate the traditional magic numbers
of neutrons and protons discussed in the text. Figure adapted from [2].

One early framework for describing the nucleus is the liquid drop model. By approxi-

3

Neutron number020406080100120140160180120100806040200Proton numberTerra IncognitaObserved nucleiStable nucleiN=Zmating the nucleus as an incompressible fluid of nucleons, the nuclear binding energy can be

calculated [20]. Due to the attractive and short-range nature of the nucleon-nucleon interac-

tion, the interior matter density for nuclei is approximately constant (saturation). Therefore,

the droplet energy, which increases linearly with volume, is proportional to the number of

nucleons A. The nucleons at the surface, however, have fewer neighbors to interact with,

so a correction term proportional to the surface area, which goes as A2/3, must be added.

The Coulomb energy is approximately proportional to Z2/A1/3. Since both neutrons and

protons are fermions with intrinsic spin s = 1/2, they each obey the Pauli exclusion prin-

ciple. Consequently, there is a term that penalizes neutron-proton asymmetry. Finally, the

observed systematic odd-even staggering in binding energy is accounted for using a pairing

term. Altogether, the semi-empirical mass formula for binding energy BE is:

BE = aV A − aSA2/3 − aC

Z2
A1/3

− aA

(N − Z)2

A

+ δ(N, Z),

(1.1)

where



aP A−1/2

N and Z even

0

A odd

−aP A−1/2 N and Z odd.

δ(N, Z) =

(1.2)

The values of the coefficients for the volume term aV , the surface term aS, the Coulomb

term aC , the asymmetry term aA, and the pairing term aP are determined from fits to

experimental mass data and the signs are chosen here to yield positive coefficients. The

liquid drop model successfully reproduces the observed bend in the valley of stability shown

in Figure 1.2 and is accurate to within 10 MeV for most nuclei while the average binding

energy is about 8 MeV per nucleon.

4

Figure 1.3: Differences between experimental binding energies and binding energies calcu-
lated using the liquid drop model for nuclei with Z ≥ 8 as a function of neutron number.
The peaks at N = 28, 50, 82, and 126 indicate that nuclei have internal structure. Figure
taken from [3] with data from [4].

However, systematic deviations between the experimental and liquid-drop binding ener-

gies as a function of neutron number can be seen in Figure 1.3, suggesting that nuclei have

internal structure that is not accounted for in this simple picture. The experimental nuclear

binding energy is larger than predicted by this model for nuclei with neutron numbers 28,

50, 82, and 126. These neutron numbers, along with 2, 8, and 20, which are less clearly

visible in Figure 1.3, are known as magic numbers. The same phenomenon can be observed

for nuclei with magic numbers of protons. Nuclei with either a magic number of neutrons or

5

0Exp. Binding Energy - Liquid Drop Energy (MeV)Neutron Number-10-505101520406080100120140protons are referred to as singly-magic while nuclei with a magic number of both neutrons

and protons are called doubly-magic.

1.2 Nuclear Shell Model

Efforts to explain the existence of the neutron and proton magic numbers lead to the de-

velopment of the nuclear shell model [21, 22]. The shell model considers the motion of a

single nucleon in a mean field generated by all the other nucleons in the nucleus. By solving

the Schr¨odinger equation using this mean potential, the available single-particle orbitals and

their corresponding quantized energy levels are obtained. These orbitals are filled according

to the Pauli exclusion principle and the magic numbers correspond to large energy gaps

between successive groups of close-lying orbitals called shells.

Neglecting the Coulomb interaction, the Hamiltonian for a nucleus with A nucleons can

be written as:

A(cid:88)

i=1

A(cid:88)

i>k=1

p2
i
2mi

+

H =

Vik(ri − rk),

(1.3)

assuming only two-body interactions between nucleons. Here pi and mi are the momentum

and mass of each nucleon and Vik is the two-body interaction between two nucleons.

In

the mean-field approximation of the shell model, the A nucleons do not interact with each

other strongly and the system can be treated as A non-interacting nucleons in an external

single-particle potential:

(cid:32)

 A(cid:88)

i=1

H =

(cid:33) +

 A(cid:88)

i>k=1

Vik(ri − rk) − A(cid:88)

i=1

 = H0 + Vres.

Ui(r)

(1.4)

p2
i
2mi

+ Ui(r)

The first term H0 describes the motion in the mean field while the second term Vres is known

6

as the residual interaction.

A convenient starting guess for the mean-field potential, the form of which is not known

exactly, is the three-dimensional harmonic oscillator:

U (r) =

µω2r2,

1
2

(1.5)

where µ is the reduced mass of the system and ω is the oscillator frequency. The energy

levels for H0 in this case depend only on the radial quantum number n, the orbital angular

momentum quantum number l, and the choice of ω:

(cid:18)

Enl =

2n + l +

ω =

(cid:19)

3
2

(cid:18)

N +

(cid:19)

3
2

ω.

(1.6)

Here,  is the reduced Planck constant. The three-dimensional harmonic oscillator success-

fully reproduces the first few magic numbers (2, 8, and 20) but fails to reproduce the other

magic numbers, as shown Figure 1.4 for the neutron single-particle energy levels of 208Pb.

Unlike the short-range nucleon-nucleon interaction, the harmonic oscillator potential goes

to infinity as the radial distance increases. A more realistic form for the mean-field generated

by all the nucleons is the Woods-Saxon potential:

U (r) =

−U0

1 + e(r−R)/a

,

(1.7)

where U0 is the potential depth, the nuclear radius is R = r0A1/3, and a is the diffuseness.

Although the Woods-Saxon potential breaks the degeneracy of orbitals with the same prin-

ciple harmonic oscillator quantum number N = 2n + l, Figure 1.4 shows that this potential

also fails to reproduce the experimentally-observed magic numbers larger than 20.

7

Adding a strongly attractive spin-orbit coupling term [21, 22],

Uso(r,(cid:126)l, (cid:126)s) = −Uso(r)((cid:126)l · (cid:126)s),

(1.8)

to the Woods-Saxon potential lifts the degeneracy for states with the same orbital angular

momentum quantum number l and the gaps in energy between shells formed by the close-

lying orbitals shown in Figure 1.4 successfully explain the experimentally-observed magic

numbers. Similar results are obtained for the proton single-particle orbitals.

The single-particle orbitals are labeled by their quantum numbers n, l, and j using the

notation nlj. The radial quantum number n = 0, 1, 2, . . . counts the number of radial

nodes in the wavefunction. The orbital angular momentum quantum number l is often

referred to using spectroscopic notation where the labels l = s, p, d, f, g, h, . . . represent

l = 0, 1, 2, 3, 4, 5, . . . , respectively. The maximum value of l is the principle quantum number

N . The nucleon spin (cid:126)s couples with the orbital angular momentum (cid:126)l to the total angular
momentum (cid:126)j = (cid:126)l + (cid:126)s, giving the total angular momentum quantum number j = l ± 1/2.
Each orbital has 2j +1 magnetic substates, running over mj = −j, . . . , j in integer steps, and

accordingly can hold a maximum of 2j + 1 neutrons or protons due to the Pauli exclusion

principle. Note that neutrons and protons are distinguishable and can share the same set of

quantum numbers.

The states of the nucleus itself are characterized by their spin-parities J π. The parity π

of a single-particle state indicates whether the wavefunction changes sign under inversion of
all the spatial coordinates and is given by (−1)l. For the nucleus itself, the total parity is

the product of the parities of the occupied single-particle levels. Similarly, the total angular

momentum J of the nucleus is found by coupling the spins (cid:126)j of all the occupied single-

8

Figure 1.4: Energies of neutron single-particle orbitals calculated for 208Pb using different
mean field potentials: three-dimensional harmonic oscillator, Woods-Saxon, and Woods-
Saxon plus spin-orbit. The number in brackets is the maximum number of neutrons allowed
in the orbital by the Pauli principle and the number outside the brackets is the sum of
total allowed nucleons through this orbital. Orbitals are labeled by their quantum numbers.
Figure is adapted from [5].

9

Single-Particle Energy (MeV)-40-30-20-100[2]   2 N=0[6]   8 N=1[12]  20 N=2[20]  40 N=3[30]  70 N=4[42] 112 N=5[56] 168 N=6[ 2]   2 0s[ 6]   8 0p[10]  18 0d[2]  20 1s[14]  34 0f[6]  40 1p[18]  58 0g[10]  68 1d[2]  70 2s[22]  92 0h[14] 106 1f[6] 112 2p[26] 138 0i[18] 156 1g[10] 166 2d[2] 168 3s[2]   2 0s1/2[4]   6 0p3/2[2]   8 0p1/2[6]  14 0d5/2[4]  18 0d3/2[2]  20 1s1/2[8]  28 0f7/2[6]  34 0f5/2[4]  38 1p3/2[2]  40 1p1/2[10]  50 0g9/2[8]  58 0g7/2[6]  64 1d5/2[4]  68 1d3/2[2]  70 2s1/2[12]  82 0h11/2[8]  90 1f7/2[10] 100 0h9/2[14] 114 0i13/2[4] 118 2p3/2[6] 124 1f5/2[2] 126 2p1/2[10] 136 1g9/2[12] 148 0i11/2[6] 154 2d5/2[2] 156 3s1/2[8] 164 1g7/2[16] 180 0j15/2[4] 184 2d3/2                                           HarmonicOscillatorPotentialWoods-SaxonPotentialWoods-SaxonPlus Spin-OrbitPotentialNeutron Single-Particle Energies2820285082126particle states. For a completely filled orbital, the nucleons couple to spin-parity 0+. If all

of the orbitals in a shell are full, then the shell is called closed and the J π of the nucleus

is determined by the remaining valence nucleons above the closed shell. For example, 17O

has one valence neutron in the 0d5/2 orbital with respect to doubly-magic 16O and the spin-

parity of its ground state is 5/2+. Furthermore, energy is typically minimized when nucleons

in the same orbital form pairs that couple to 0+. In fact, the spin-parity of the ground state

for all nuclei with even numbers of both neutrons and protons (even-even nuclei) is 0+

[23]. Typically, within this scheme, only nucleons outside of a so-called core are allowed to

contribute to nuclear excitations. Due to the properties of closed shells described above and

the large energy gaps at magic numbers, it is common to choose a core that corresponds to

a doubly-magic nucleus.

The nuclear shell model describes the properties of nuclei in the single-particle picture

and is therefore most successful near shell closures where the structure is dominated by only

a few nucleons outside a closed core. In this framework, nuclear excited states are formed

by rearranging one or a few nucleons in their orbitals. For example, in this simple picture,

the 1/2+ first excited state of 17O arises from the promotion of the valence neutron in the

0d5/2 orbital to the 1s1/2 orbital. However, away from closed shells nuclei exhibit collective

behavior where many valence nucleons contribute coherently to nuclear excitations.

1.3 Collective Models

The excitation patterns in collective nuclei can be explained using macroscopic geometric

models of the nucleus that describe vibrations and rotations. In the vibrational model, even-

even nuclei have evenly-spaced energy levels associated with oscillations of an incompressible

10

liquid-drop nucleus about an average spherical shape. The first excited state has spin-parity

2+ and the energy of a single phonon, the elementary excitation in the quantum-mechanical

treatment of the surface vibrations on the liquid drop. The second excited state is a triplet

of nearly-degenerate levels with J π = 0+, 2+, 4+ at twice the energy of the first excited state

and comes from the coupling of two phonons. The three-phonon states, at three times the

energy of the first excited state, are a nearly-degenerate quintet with J π = 0+, 2+, 3+, 4+, 6+

[23].

If an even-even nucleus is rigidly deformed, the excitation energies can be calculated as

those of a quantum-mechanical rigid rotor:

Erot(J) =

2
2I

J(J + 1),

(1.9)

where I is the moment of inertia of the nucleus and J is even. Note that these rotations

are about an axis perpendicular to the symmetry axis of the deformed nucleus. As shown

in Figure 1.5, even-even nuclei evolve from displaying signs of magicity near closed shells

(e.g. high-lying first-excited 2+ states) to spherically vibrational just outside of closed shells

to deformed and rotational with low-lying first-excited 2+ states near mid-shell. The ratio

R4/2 of energies for the first 2+ and 4+ states provides evidence for the type of collective

motion: R4/2 = 2 for vibrations while R4/2 = 3.33 for rotations from Equation 1.9 [23]. The

collective models described in this section can be extended to systems with odd numbers of

nucleons by coupling the unpaired nucleons to the vibrator or rotor (particle-rotor models).

11

Figure 1.5: Energies of the first 2+ excited state and ratios R4/2 of energies for the first
2+ and 4+ states in even-even nuclei across the chart of nuclides. The neutron and proton
magic numbers are denoted by dashed lines. Figure taken from [6].

1.4 Electromagnetic Transitions

The properties of low-lying bound excited states in nuclei can be used to probe collectivity,

which is a measure of how many nucleons are involved in the excitation. The majority of

nuclear excited states bound to particle emission will decay to a lower-energy level via the

emission of a photon known as a γ ray. Neglecting the small recoil energy of the nucleus,

the energy of this electromagnetic radiation is equal to the difference in energy of the initial

and final states.

Electromagnetic transitions are characterized by their multipolarity λ and parity. Since

the photon is a boson with intrinsic spin 1, the total angular momentum λ it carries cannot

be 0. The allowed multipolarities for the γ ray are constrained by the spins of the initial

12

state (Ji) and final state (Jf ) due to conservation of angular momentum:

(cid:12)(cid:12)Ji − Jf

(cid:12)(cid:12) ≤ λ ≤ Ji + Jf .

(1.10)

Generally, transitions are referred to as 2λpole: λ = 1 is dipole, λ = 2 is quadrupole, λ = 3

is octupole, and so on. In order to conserve parity, the following relation between the parity

of the initial state (πi), the parity of the final state (πf ), and the parity of the γ ray must

be satisfied:

πiπf = πσ(−1)λ.

(1.11)

For electric transitions (σ = E), πσ = 1 and for magnetic transitions (σ = M ), πσ = −1.

Transitions are often labeled using the scheme σλ. For instance, the allowed transitions
for πiπf = 1 are M 1, E2, M 3, E4, . . . and the allowed transitions for πiπf = −1 are

E1, M 2, E3, M 4, . . . for all λ allowed by Equation 1.10.

The electric transition operator can be written as:

A(cid:88)

O(Eλ) =

rλ
i Y λ

µ (θ, φ)eie,

(1.12)

i

where Y λ

µ (θ, φ) are the spherical harmonics, µ is the z-projection of λ, and eie is the electric

charge of the ith nucleon. For free neutrons, en = 0 and for free protons, ep = 1. Effective

charges for the nucleons can also be used to compensate for model space truncation in shell

model calculations [5]. The magnetic transition operator can be written as:

(cid:21)

(cid:126)∇(cid:104)

(cid:105)

rλY λ

µ (θ, φ)

µN ,

(1.13)

A(cid:88)

(cid:20) 2gli

λ + 1

i

O(M λ) =

(cid:126)li + gsi(cid:126)si

13

where gli and gsi are the orbital and spin g-factors for the ith nucleon and µN = 0.105 efm
is the nuclear magneton. The g-factors for free neutrons are gln = 0 and gsn = −3.826 and

the g-factors for free protons are glp = 1 and gsp = 5.586. Effective values for the g-factors

can also be used to take into account model-space truncation [5].

The electromagnetic transition rate W between an initial state with wavefunction |JiMi(cid:105)
and final state with wavefunction |Jf Mf(cid:105) can be calculated as a sum over the allowed electric

and magnetic transition operators using time-dependent perturbation theory:

(cid:18) 8π(λ + 1)

(cid:88)

λ[(2λ + 1)!!]2

σλ

(cid:19) 1



(cid:18) Eγ

c

(cid:19)2λ+1(cid:12)(cid:12)(cid:104)Jf Mf

(cid:12)(cid:12)O(σλ)µ

(cid:12)(cid:12) JiMi(cid:105)(cid:12)(cid:12)2 .

(1.14)

WMi,Mf ,µ =

Here, c is the speed of light in vacuum. Averaging over the initial magnetic substates Mi

and summing over the projections µ and the final magnetic substates Mf , which are not

typically observed individually in experiments, results in:

1

(cid:88)
(cid:88)
(cid:18) 8π(λ + 1)

Mf ,µ

Mi

2Ji + 1

(cid:88)

λ[(2λ + 1)!!]2

σλ

W =

=

WMi,Mf ,µ

(cid:19) 1



(cid:18) Eγ

c

(cid:19)2λ+1(cid:12)(cid:12)(cid:104)Jf ||O(σλ)|| Ji(cid:105)(cid:12)(cid:12)2

2Ji + 1

(1.15)

,

(1.16)

through use of the Wigner-Eckart theorem and the orthogonality properties of the Clebsch-

Gordan coefficients [5]. The last factor in this expression is defined as the reduced transition

probability:

B(σλ; Ji → Jf ) ≡

(cid:12)(cid:12)(cid:104)Jf ||O(σλ)|| Ji(cid:105)(cid:12)(cid:12)2

2Ji + 1

.

(1.17)

The double-bar notation of the Wigner-Eckart theorem for the reduced matrix element
(cid:104)Jf ||O(σλ)|| Ji(cid:105) signifies that there is no dependence on Mi, µ, or Mf .

14

For γ decay, the initial state is higher in energy than the final state. Conversely, through

Coulomb excitation, a projectile nucleus can be inelastically excited due to electromagnetic

interaction with a target nucleus. Consider two states Ja and Jb. From Equation 1.17, the

relation between the reduced transition probabilities in each direction is:

B(σλ; Jb → Ja) =

2Ja + 1
2Jb + 1

B(σλ; Ja → Jb).

(1.18)

From examination of Equation 1.16, it can be seen that the lowest allowed multipolarities

dominate the transition rate W . For a typical γ-ray energy of 1 MeV, the rate for the next

allowed electric (magnetic) multipole, λ + 2, is about seven orders of magnitude smaller

than the rate for the lowest allowed electric (magnetic) multipole λ [5]. For transitions that

can occur via both low-multipole electric and magnetic transitions, it is useful to define a

multipole mixing ratio δ. For example, for electromagnetic transitions that can occur via

M 1 and E2:

δ2(E2/M 1) =

WE2
WM 1

,

and the branching fractions b are:

b(E2) =

WE2

WM 1 + WE2

=

δ2
1 + δ2 ,

b(M 1) =

WM 1

WM 1 + WE2

=

1

1 + δ2 .

(1.19)

(1.20)

(1.21)

For even-even nuclei, the reduced electric quadrupole strength from the first 2+ state

to the ground state, B(E2; 2+

1 → 0+

1 ), is an indicator of collectivity. This strength can be

expressed in terms of the Weisskopf unit, which is an estimate of the B(E2; 2+

1 → 0+

1 ) for a

15

single particle. In general, the Weisskopf units are given by [5]:

(cid:21)2

(cid:19)(cid:20) 3
(cid:18) 1
(cid:21)2
(cid:19)(cid:20) 3
(cid:18)10

4π

3 + λ

π

3 + λ

BW (Eλ) =

(1.2A1/3)2λ e2fm2λ,

BW (M λ) =

(1.2A1/3)2λ−2 µ2

N fm2λ−2.

(1.22)

(1.23)

B(E2; 2+

1 → 0+

1 ) strengths are largest at mid-shell and smallest near closed shells. Accord-

ingly, there is also a link between the deformation in the rotational model and the reduced

electric quadrupole strength. For an axially-symmetric quadrupole deformed nucleus, the

deformation can be expressed using the Hill-Wheeler parameterization [24]:

(cid:16)

(cid:17)

R(θ) = R0

1 + βY 2

0 (θ, 0)

,

(1.24)

where the radius R0 = r0A1/3, Y 2

0 (θ, 0) is a spherical harmonic, and β is the quadrupole

deformation parameter.

If β is positive, the nucleus has prolate deformation and if β is

negative, the nucleus has oblate deformation. The magnitude of the quadrupole deformation

parameter is related to the B(E2; 0+

1 ) excitation strength:

1 → 2+
(cid:113)

4π

3ZeR2
0

|β| =

B(E2; 0+

1 → 2+
1 ).

(1.25)

1.5 Configuration Interaction Method

Going beyond the independent particle picture by including the residual interactions of

Equation 1.4 in the nuclear shell model is required in order to perform predictive calculations

of nuclear properties. However, solving the Schr¨odinger equation for all the nucleons in a

16

nuclear system as large as 42Si, for example, is not feasible computationally. Instead, an

inert core of Acore nucleons, often taken as a nearby doubly-magic nucleus, is utilized to
simplify the problem. Only the A − Acore valence nucleons outside of the core are allowed

to contribute to the nuclear excitations. Furthermore, a model space limiting the available

single-particle orbitals for the valence nucleons must be chosen.

The excitation energies for nuclear states are calculated through diagonalization of an

effective interaction expressed in matrix form. Omitting higher-body terms, the Hamiltonian

can be written as a series of zero-, one-, and two-body operators in the form H = H0+H1+H2

where H0 is the binding energy of the core, H1 is a one-body operator with matrix elements

given by the single-particle energies multiplied by the orbital occupancies, and H2 is the

two-body part [5]. The two-body matrix elements can be written in terms of the residual
interaction as (cid:104)ab|Vres| cd(cid:105)JT where the nucleons in orbits a, b and c, d, respectively, are

coupled to total spin J and isospin T . The wavefunctions for the nuclear states found by

diagonalizing the matrix are linear combinations of the different configurations of nucleons

possible in the model space given the number of neutrons and protons in the nucleus, giving

rise to the name “Configuration Interaction” for the method. The average number of nucleons

in an orbital for a given shell-model state from the calculation is referred to as the orbital’s

occupancy. Two examples of Hamiltonians used in the configuration-interaction framework

are SDPF-MU [25] and SDPF-U [26] which are described in detail in the following section.

1.6 Breakdown of the N=28 Magic Number

Studies of nuclei far from the valley of stability have revealed surprising changes in nuclear

structure. For example, in some regions of the chart of the nuclides, the conventional magic

17

numbers disappear and new magic numbers emerge [27].

In particular, the neutron-rich

silicon and sulfur isotopes approaching the conventional neutron magic number N = 28 have

attracted much experimental and theoretical attention that has provided great insight into

the evolution of shell structure in exotic nuclei. Recall that magic nuclei characteristically

have high-lying first 2+ excited states and small B(E2; 0+

1 → 2+

1 ) values. The first evidence

for the breakdown of the N = 28 magic number was provided by the observation of moderate

deformation in 40,42S [28] and low-lying quadrupole collectivity in 44S [29]. Figure 1.6 shows
1 → 2+
1 )

the systematic trends in the energies of first 2+ excited states E(2+

1 ) and B(E2; 0+

strengths across the Ca (Z = 20), Ar (Z = 18), S (Z = 16), and Si (Z = 14) isotopic chains.

The Ca isotopes show the tell-tale signs of magicity at both N = 20 and N = 28 while the

Ar isotopes at N = 20 and N = 28 only show small relative peaks in E(2+

1 ). However, for S

and Si near N = 28, the E(2+

1 ) values are low and the B(E2; 0+

1 → 2+

1 ) strengths are large

relative to their values closer to stability near N = 20. This provides evidence that while

the conventional neutron magic number N = 20 holds for these nuclei, N = 28 is no longer

a magic number at Z = 14 and Z = 16.

Reproducing the evolution of the N = 28 shell gap in the vicinity of 42Si has been

a major endeavor for modern nuclear shell model calculations. The mechanism driving

the breakdown of N = 28 as a magic number in this region has been attributed to the

neutron-proton monopole component of the tensor part of the nucleon-nucleon interaction

[30]. Consider a nucleon in a single-particle orbit with spin ja. The single-particle energy of

this nucleon depends on the mean field generated by all the other nucleons in the nucleus,

which changes when a nucleon is added to the orbit with spin jb. The nucleons in the two

orbitals couple to total spin J. The shift in energy of the single-particle orbital for a proton

18

1 → 2+
Figure 1.6: Experimental energies of the first 2+ states [7] (top) and B(E2; 0+
1 )
strengths [8] (bottom) for Si, S, Ar, and Ca isotopes showing the evolution of collectivity
around the conventional neutron magic numbers N = 20 and N = 28.

in the orbital with spin ja due to neutrons in the orbital with spin jb is [9]:

(cid:104) ¯V T =0

(cid:105)

∆
p(ja) =

1
2

ab + ¯V T =1

ab

nn(jb),

(1.26)

where nn(jb) is the number of neutrons in the orbital with jb and ja (cid:54)= jb. Here, ¯V T

ab is the

19

18202224262830Neutron Number0200400600B(E2; 0+1→2+1) (e2fm4)01000200030004000E(2+1) (keV)Si (Z=14)S (Z=16)Ar (Z=18)Ca (Z=20)angle-averaged (monopole) tensor interaction between the orbitals:

(cid:80)
J (2J + 1)(cid:104)jajb |V | jajb(cid:105)JT

(cid:80)

J (2J + 1)

¯V T
ab =

.

(1.27)

Above, T is the isospin quantum number for the coupled nucleons where the projection

Tz = 1

2(N − Z). T = 0 for the neutron-proton interaction and T = 1 for the neutron-

neutron and proton-proton interactions. The T = 0 component of the tensor interaction is

stronger [9].

For a single nucleon with a given value of orbital angular momentum l, there are two

possible values for the total angular momentum j: j< = l − 1
interaction between orbitals with ja = j< and jb = j(cid:48)
attractive while the tensor interaction between orbitals with ja = j< and jb = j(cid:48)
ja = j> and jb = j(cid:48)

2 and j> = l + 1

> is repulsive. From N = 20 to N = 28, the neutron 0f7/2(j>) orbital

> and with ja = j> and jb = j(cid:48)

< is

2. The tensor

< and with

is filled. The tensor force between the neutron 0f7/2(j>) orbital and the proton 0d5/2(j>)

orbital is repulsive while the tensor force between the neutron 0f7/2(j>) orbital and the

proton 0d3/2(j<) orbital is attractive. Therefore, as shown schematically in Figure 1.7, the

Z = 14 sub-shell gap is reduced as the neutron 0f7/2(j>) orbital is filled.

At the same time, removing protons from the filled 0d3/2(j<) orbital in 48Ca causes the

energy of the neutron 0f7/2(j>) orbital to increase since the attractive interaction is weak-

ened. This energy shift is larger than the energy shift due to the weakening of the attractive

interaction between the proton 0d3/2(j<) orbital and the neutron 1p3/2(j>) orbital, result-

ing in a reduction of the N = 28 shell gap. The simultaneous narrowing of the Z = 14

and N = 28 shell gaps leads to mutually-enhanced deformation in the region surrounding

42Si. The origin of this deformation has been argued in the context of a tensor-force-driven

20

Figure 1.7: Schematic illustration of the tensor force. The repulsive interaction between the
neutron 0f7/2(j>) orbital and proton 0d5/2(j>) and the attractive interaction between the
neutron 0f7/2(j>) orbital and proton 0d3/2(j<) orbital reduce the Z = 14 sub-shell gap.
Figure modified from [9].

Jahn-Teller effect [25] and within the context of SU(3)-like models of quadrupole correlations

[26, 31, 32].

The SDPF-MU effective interaction was constructed to describe the shell and shape evolu-

tion in the S and Si isotopic chains toward N = 28 [25]. The valence space for the SDPF-MU

Hamiltonian includes the proton 0d5/2, 1s1/2, and 0d3/2 orbitals (sd shell) and the neutron

0f7/2, 1p3/2, 0f5/2, and 1p1/2 orbitals (pf shell). The interactions within the sd shell are
taken from the USD Hamiltonian [33, 34] except for the monopole interactions V T =0,1

0d3/2,0d5/2

which were taken from SDPF-M [35]. The interactions within the pf shell are taken from the

GXPF1B Hamiltonian [36] except for the monopole and quadrupole pairing matrix elements
(cid:104)0f7/20f7/2 |V | 0f7/20f7/2(cid:105)J=0,2 which were taken from KB3, another pf -shell Hamiltonian

[37]. Both the USD and GXPF1B effective interactions use the harmonic oscillator basis and

21

0d5/20d3/20f7/2protonneutronstart with a set of single-particle energies and two-body matrix elements derived from the

nucleon-nucleon interaction that are then allowed to vary to better reproduce experimental

data. The cross-shell interactions between the neutrons and protons are based on the VM U

interaction [38], which consists of a Gaussian central force, a two-body spin-orbit force taken

from the M3Y interaction [39], and a tensor force comprised of π and ρ meson exchanges.

Another shell-model interaction that has been successful in describing nuclei near N =

28 is the SDPF-U Hamiltonian [26]. SDPF-U also uses the USD interaction [33, 34] for

protons in the sd shell and a variant of the KB3 interaction [37] for neutrons in the pf

shell. The proton-neutron cross-shell matrix elements are taken from the Kahana-Lee-Scott

G matrix [40] with adjusted monopole terms to better fit experimental data. SDPF-U has
two formulations, one to be used for Z ≤ 14 and one for Z ≥ 15 with differences due to

consideration of proton excitations into the pf shell. The SDPF-U-MIX interaction extends

SDPF-U to include the sd shell for neutrons in addition to the f p shell [32].

1.7 Motivation for Studying the Neutron-Rich Sulfur

Isotopes

The breakdown of the magic number N = 28 discussed in the previous section is accompanied

by interesting changes in nuclear shell structure. For example, consider 43S, which has 16

protons and 27 neutrons. If the single-particle orbitals are filled in their normal order then

the one unpaired neutron will occupy the 0f7/2 orbital. Consequently, the ground state of 43S
should have a spin-parity of 7/2−. However, the ground state of 43S has been determined to
be 3/2− [41, 42]. In a simple shell-model picture, this corresponds to promoting the unpaired

neutron from the 0f7/2 orbital into the 1p3/2 orbital. Similarly, for even-even nuclei like 42Si

22

and 44S, the ground state would be expected to have a completely filled neutron 0f7/2

orbital. Instead, shell-model calculations suggest that the dominant configuration for the

ground state has two neutrons promoted to the 1p3/2 orbital, leaving two holes in the 0f7/2

orbital. Due to deformation, the energies of these so-called intruder configurations, which

normally correspond to excited states, are lower than the energies of the spherical normal-

order configurations. For this reason, the nuclei surrounding 42Si and 44S are said to belong

to the so-called N = 28 island of inversion [32]. Typically, the excited states with structure

dominated by the spherical normal-order configuration are at low excitation energy with

respect to the deformed ground state leading to a phenomenon known as shape coexistence.

The concept of an island of inversion is not restricted to the neutron-rich nuclei around the

conventional magic number N = 28. Historically, the first island of inversion investigated

was at N = 20 centered around 32Mg and it is now known that there are islands of shell

breaking for neutron-rich nuclei associated with normal-order shell closures and sub-shell

closures at N = 8, 14, 20, and 28 and the harmonic oscillator magic number N = 40 [43].

With 44S lying near the center of the island of inversion at N = 28, it is of interest to sys-

tematically study the structure of the sulfur isotopes with neutron numbers between 22 and

28 in order to gain understanding of the details of the mechanism driving shell evolution in

this region. Due to their importance, the even-even neutron-rich sulfur isotopes beyond the

heaviest stable isotope of sulfur at N = 20 have been studied using a variety of experimental

techniques. Levels in radioactive 38S have been studied using β decay, proton scattering,

two-neutron transfer, two-proton transfer, intermediate-energy Coulomb excitation, and in-

beam γ-ray spectroscopy following fragmentation of 48Ca [7]. States in 40S have been studied

using β decay, proton scattering, four-neutron transfer, intermediate-energy Coulomb exci-

tation, and in-beam γ-ray spectoscopy following fragmentation of 48Ca and 46Ar [7]. 42S

23

has been studied using β decay, intermediate-energy Coulomb excitation, and in-beam γ-ray

spectroscopy following fragmentation of 48Ca, fragmentation of 46Ar, and knockout from

44Cl [7]. Finally, 44S has been studied using intermediate-energy Coulomb excitation, in-

beam γ-ray spectroscopy following fragmentation of 48Ca, following two-proton knockout

from 46Ar, and following one-proton knockout from 45Cl [7]. In particular, lifetime measure-

ments in 44S [44, 45] and supporting theoretical calculations [46, 47] have established the

coexistence of levels dominated by zero-, one-, and two-neutron particle-hole configurations.

In a recent in-beam γ-ray spectroscopy measurement, an interesting systematic trend for

the 2+

2 → 0+

1 and 2+

2 → 2+

1 γ-ray branching ratios in the sulfur isotopic chain was observed

by Lunderberg et al. and discussed in comparison with shell-model calculations [10]. The

SDPF-MU interaction predicts that for 38S and 40S, the 2+

with 96.4% and 99.4% branching ratios, respectively.

1 transitions will dominate

2 → 2+
In 42S, the reverse occurs with a

prediction of 84.2% for the 2+

2 → 0+

1 transition and only 15.8% for the 2+

2 → 2+

1 transition.

These shell-model predictions were validated experimentally with the non-observation of the

1 branch in 38S [10, 48] and a measurement of 85(2)% for the tentative (2+

2 → 0+
2 ) → 0+
2+
branch in 42S [10]. Shell-model calculations link the abrupt change in decay pattern for 42S

1

with the difference in occupancies for the neutron 0f7/2 and 1p3/2 orbitals for the 2+

1 and
2+
2 states. The occupancies of the neutron 1p3/2 orbital for 0+ and 2+ levels up to 4.5 MeV
in excitation energy for 38,40,42,44S from SDPF-MU calculations are provided in Figure 1.8.

An increase in neutron 1p3/2 occupancy is strongly correlated with a decrease in occupancy

2 states differ the most for 42S. The 1p3/2 and 0f7/2 orbitals cannot be connected

for the neutron 0f7/2 orbital. As seen in Figure 1.8, the neutron 1p3/2 occupancies of the
2+
1 and 2+
by the magnetic dipole operator, leading to a strongly hindered B(M 1; 2+
strength in 42S of 0.14 × 10−3 µ2

1 ) transition
N in 38S. As a

2 → 2+

N compared to 0.19 µ2

N in 40S and 0.48 µ2

24

2 → 0+

result, the 2+

1 branch is favored in 42S. The dramatic increase in level density for
0+ and 2+ states below 4.5 MeV in 44S signals the sudden and massive gain in correlation

energy of the intruder configuration formed by moving two neutrons into the 1p3/2 orbital

compared to the closed 0f7/2 shell configuration, placing 44S inside of the N = 28 island

of inversion with 42S right at the boundary. Due to this particular interplay of the single-

particle configurations, a measurement of the B(E2) excitation strengths to higher-lying 2+

states provides a unique window into the breakdown of the N = 28 shell gap at Z = 16

and the underlying driving forces that are a cornerstone in explaining shell evolution in rare

isotopes.

As discussed above, one of the characteristics of the N = 28 island of inversion is the

coexistance of multiple shapes at low excitation energy. Studies of 44S have indicated that

the ground state is prolate deformed while the second 0+ state is spherical [42, 49]. One
neutron below 44S, the discovery of a long-lived (isomeric) 7/2− state at 320 keV provided
evidence for shape coexistence in 43S [41]. The half-life (T1/2 = ln(2) · τ ) of the isomer has

been reported as 415 ns [50, 51] and the structure of the isomeric state has been interpreted

as a spherical 0f7/2 single-neutron hole state while the intruder 3/2− ground state is prolate

deformed [50]. The spins and parities of other excited states have not been firmly established

[15, 52, 53], as will be discussed in later chapters. Furthermore, recent work has indicated

that proton-neutron correlations prevent the isomeric state from being regarded as purely

spherical and noted that it would be of great interest to experimentally identify the levels

belonging to the ground-state rotational band in order to give unambiguous evidence of

shape coexistence in 43S [51].

In this work, intermediate-energy Coulomb excitation is used to populate higher-lying

2+ states in the even-even S isotopes and to search for collective structures built on the 43S

25

Figure 1.8: SDPF-MU neutron 1p3/2 occupancies for 0+ (red) and 2+ states (blue) below
4.5 MeV in 38,40,42,44S. The increase in occupancy for the second 2+ state in 42S relative to
38S explains the observed change in γ-decay pattern. Figure modified from [10].

ground and isomeric states. Measurements of the experimental B(E2) strengths from the

0+ ground state to higher-lying 2+ states allow the onset of intruder configurations in the S

isotopic chain to be characterized. Shell-model calculations using the SDPF-MU interaction,

which predict a fragmentation of the strength over many different 2+ states in 44S are shown

in Figure 1.9. Testing the predictions of the most successful effective Hamiltonian in this

region is an important step toward confirming the underlying mechanism of shell evolution.

26

Ex (MeV)38S40S42Sp3/244S1.51.00.50.01.51.00.50.01.51.00.50.01.51.00.50.01.0occupancy22++122.03.04.01 → 2+) strengths in e2fm4 for levels in
Figure 1.9: SDPF-MU predictions for B(E2; 0+
38,40,42,44S. Arrow widths are proportional to transition strength. 2+ states above 4 MeV
with B(E2) values less than 10 e2fm4 are omitted in this figure. Effective proton and neutron
charges of 1.35 and 0.35, respectively, were used in the SDPF-MU calculations.

27

0500100015002000250030003500400045005000Energy(keV)38S0+21+22+23+160252640S0+21+22+23+3002.95442S0+21+22+25+370381644S0+21+22+23+24+25+45012110.62.7Chapter 2

Intermediate-Energy Coulomb

Excitation

Coulomb excitation is the inelastic excitation of a projectile (target) nucleus due to electro-

magnetic interaction with a target (projectile) nucleus and is a well-established experimental

probe in nuclear science [54–59]. With the high-intensity, low-velocity stable beams used in

the early days of this method, experiments were performed at energies below the Coulomb

barrier in order to ensure physical separation of the target and projectile, thus avoiding

nuclear contributions to the excitation mechanism [54, 55]. More recently, the in-flight pro-

duction of fast beams of radioactive nuclei has made it possible to study the structure of

nuclei far from the valley of stability, prompting the development of intermediate-energy

Coulomb excitation as an experimental tool [56, 57, 60, 61].

In intermediate-energy Coulomb excitation experiments, exotic nuclei at energies in ex-

cess of 30 MeV per nucleon are scattered off stable high-Z targets. The scattered nuclei

are detected in coincidence with the de-excitation γ rays and the Coulomb excitation cross

section is calculated from the efficiency-corrected number of detected γ rays. At these beam

energies, the analysis should be restricted to a maximum scattering angle, or equivalently

a minimum impact parameter, which is chosen to exceed the sum of the radii of the target

and projectile nuclei by several fm [57, 62]. The impact parameter b and center-of-mass

28

Figure 2.1: Schematic of intermediate-energy Coulomb excitation. The projectile (target)
nucleus is inelastically excited in the Coulomb field of the target (projectile) nucleus.
In
analysis, a maximum center-of-mass scattering angle θ (minimum impact parameter b) is
considered to ensure no nuclear contribution to the cross section. Figure adapted from [11].

scattering angle θ are related by:

(cid:18) θ

(cid:19)

,

2

b =

cot

a0
γ

where the Lorentz factor γ = 1/(cid:112)1 − (v/c)2 and a0 is the half-distance of closest approach:

(2.1)

a0 =

ZtZpe2
m0v2 ,

(2.2)

which depends on the proton numbers of the target and projectile, Zt and Zp, the reduced

29

γγγmass m0 of the system, and the velocity v of the projectile. The scattering angle restriction

ensures that the Coulomb interaction dominates over nuclear and nuclear-Coulomb interfer-

ence contributions to the excitation. A schematic illustrating intermediate-energy Coulomb

excitation is provided in Figure 2.1. From the angle-integrated cross section, the reduced
transition probability B(σλ; i → f ) from the initial state i of the projectile to the populated

excited state f can be calculated, for example, using the semi-classical theory of relativistic

Coulomb excitation developed by Alder and Winther [61].

2.1 Alder-Winther Formalism

In the Alder-Winther formalism [61], the relative motion of the projectile and the target

is described using the classical Rutherford trajectory while the single-step electromagnetic

excitation of the projectile is treated quantum mechanically using perturbation theory. Since

this theoretical framework assumes there is no overlap between the charge distributions of the

target and projectile, the excitation is pure Coulomb and the cross section can be expressed

in terms of the same electromagnetic multipole matrix elements as those in the expression

for the electromagnetic decay of nuclear states.

Assuming the relative motion takes place on the classical Rutherford trajectory, the

differential cross section for exciting the projectile from its initial nuclear state i to final state

f can be written as the Rutherford differential cross section multiplied by the probability of

excitation:

(cid:18) dσ

(cid:19)

(cid:18) dσ

(cid:19)

=

dΩ

i→f

dΩ

Rutherford

Pi→f .

(2.3)

The Coulomb excitation probability can be calculated using first-order time-dependent per-

30

turbation theory:

where

ai→f =

(cid:12)(cid:12)2 ,

Pi→f =(cid:12)(cid:12)ai→f
(cid:90) ∞
−∞ e

1
i

iωf it(cid:104)f |V (r(t))| i(cid:105)dt.

(2.4)

(2.5)

Here, ωf i = (Ef − Ei)/ = ∆E/ and V (r(t)) is the Coulomb potential which depends on

the relative positions of the projectile and target nucleus.

The transition amplitude ai→f can be expanded in terms of electric and magnetic mul-

tipole matrix elements and written as a product of two factors:

(cid:88)

λ

ai→f = i

χ

(λ)
i→f fλ(ξ),

with χ

(λ)
i→f being a measure of the strength of the interaction:

(cid:104)f(cid:12)(cid:12)O(σλ)µ

(cid:12)(cid:12) i(cid:105),

χ

(λ)

i→f ≈ Zte
cbλ

(2.6)

(2.7)

The function fλ(ξ) incorporates the dependence of the amplitude on the adiabaticity of the

reaction. The adiabatic parameter ξ is the ratio of the amount of time the projectile spends

in the vicinity of the target nucleus (called the collision time τcoll) to the timescale of internal

motion in the nucleus being excited τnuc. The collision time is expressed as:

(2.8)

τcoll =

b
γv

,

31

while the internal motion timescale is:

τnuc =

1
ωf i

=


∆E

.

Therefore, the adiabaticity parameter can be written as:

ξ =

τcoll
τnuc

=

∆E
γv

b.

(2.9)

(2.10)

For large values of ξ, corresponding to low projectile velocity, large impact parameter, or

high excitation energy, the reaction is referred to as adiabatic. In these cases, the projectile

is unlikely to be excited. Consequently, fλ(ξ) decreases exponentially as a function of ξ.

Conversely, for violent reactions with ξ nearing 0, excitation is likely to occur and fλ(ξ)

approaches 1.

At relativistic velocities, the violent reactions that have nuclear contributions are avoided

by considering small scattering angles (large impact parameters). Furthermore, excitations

are predominantly single-step due to the limited amount of time the projectile is in the

vicinity of the Coulomb field of the target nucleus. Therefore, Winther and Alder have made

the approximation that the projectile travels on a straight-line trajectory in the equations

above. In order to account for the distortion of the trajectory due to the Coulomb interaction

in first order, the impact parameter is rescaled by:

b → b +

π
2

a0
γ

,

(2.11)

32

correcting the adiabaticity parameter to:

(cid:18)

(cid:19)

b +

π
2

a0
γ

=

∆E
γv

ba.

(2.12)

ξ =

∆E
γv

(cid:12)(cid:12)(cid:12)χ

(λ)
i→f

(cid:12)(cid:12)(cid:12)2

Neglecting fλ(ξ) for simplicity, the Coulomb excitation cross section can be approxi-

mated by integrating

over impact parameters from bmin to bmax. The lower limit

is the experimentally-chosen minimum impact parameter and the upper limit is the impact

parameter corresponding to ξ=1:

bmax = ba =

γv
∆E

.

(2.13)

Note that this also sets the maximum excitation energy to be ∆E = γv/bmax. The cross

section is then:

σλ = 2π

bdbPi→f ≈ 2π

(cid:90) bmax

bmin

(cid:90) bmax

bmin

(cid:12)(cid:12)(cid:12)χ

bdb

(λ)
i→f

(cid:12)(cid:12)(cid:12)2

.

(2.14)

Evaluating this expression results in the Coulomb excitation cross section in terms of the

reduced transition probability:

(cid:19)2

(cid:18) Zte2

c

σσλ ≈

π

e2b2λ−2

min

(λ − 1)−1

2 ln(bmax/bmin)

for λ ≥ 2

for λ = 1,

B(σλ; 0 → λ)

(2.15)

and the total cross section is the sum over all allowed electric and magnetic multipolarities:

(cid:88)

σi→f =

σσλ.

(2.16)

Here, σ is electric E or magnetic M in the sum and subscript. As seen in Equation 2.15,

σλ

33

the Coulomb cross section is directly proportional to the reduced transition strength B(σλ).

For the 0+

1 → 2+ excitations of interest in this work, the multipolarity is pure E2 and the

cross section is proportional to B(E2; 0+

1 → 2+) strength.

The exact expression for the Coulomb excitation cross section is derived in detail by

Winther and Alder in [61]. Their result for the excitation amplitude is:

ai→f = −i

Zte2
γv

(cid:16) c

(cid:17)

v

Gσλµ

(cid:88)

σλµ

(−1)µKµ (ξ(b))

(cid:18)∆E

c

(cid:19)λ √
× (cid:104)Jf Mf

2λ + 1

(cid:12)(cid:12)O(σλ)−µ

(cid:12)(cid:12) JiMi(cid:105)

,

(2.17)

e

where Kµ are the modified Bessel functions of the second kind. For electric transitions, the
function GEλµ is defined for µ ≥ 0 as:

(cid:16) c

(cid:17)

v

GEλµ

= iλ+µ

√

16π

(cid:18)(λ − µ)!
(cid:18)(λ + 1)(λ + µ)

(λ + µ)!

(cid:19)1/2(cid:18)(cid:16) c
(cid:19)−1/2
(cid:17)2 − 1
(cid:17) − λ(λ − µ + 1)
(cid:16) c

v

P µ
λ−1

2λ + 1

2λ + 1

v

×

λ(2λ + 1)!!

(cid:16) c

(cid:17)(cid:19)

v

P µ
λ+1

,

(2.18)

and for magnetic transitions, the function GM λµ is defined for µ ≥ 0 as:
(cid:19)−1/2

(cid:18)(λ − µ)!

(cid:19)1/2(cid:18)(cid:16) c

(cid:16) c

(cid:17)

16π

√

(cid:17)2 − 1

GM λµ

v

= iλ+µ+1

λ(2λ + 1)!!

(λ + µ)!

v

(cid:16) c

(cid:17)

v

.

(2.19)

µP µ
λ

In these expressions, P µ

λ are the associated Legendre polynomials. Note that for negative µ

there are the relations:

GEλ−µ

= (−1)µGEλµ

GM λ−µ

= (−1)µ+1GM λµ

,

(cid:17)

.

(2.20)

(2.21)

(cid:17)
(cid:16) c
(cid:16) c

v

v

(cid:17)
(cid:16) c
(cid:17)
(cid:16) c

v

v

34

The full Coulomb excitation cross section is then:

σi→f = 2π

(cid:90) ∞
(cid:19)2(cid:88)
(cid:18) Zte2

bmin

bdb

1

2Ji + 1

(cid:18)∆E

c

σλµ

=

c

(cid:12)(cid:12)2

(cid:12)(cid:12)ai→f

(cid:88)
(cid:19)2(λ−1) B(σλ; Ji → Jf )

MiMf

e2

(cid:12)(cid:12)(cid:12)Gσλµ

(cid:16) c

v

(cid:17)(cid:12)(cid:12)(cid:12)2

(2.22)

gµ (ξ(bmin)) ,

(2.23)

where the function gµ (ξ(bmin)) is:

gµ (ξ(bmin)) = 2π

= 2π

bdb(cid:12)(cid:12)Kµ (ξ(b))(cid:12)(cid:12)2

(cid:19)2(cid:90) ∞
(cid:18) ∆E
(cid:90) ∞
(cid:12)(cid:12)Kµ (x)(cid:12)(cid:12)2 xdx.

γv

bmin

ξ

(2.24)

(2.25)

This integral can be evaluated in terms of the modified Bessel functions of the second kind.
For µ ≥ 0:

gµ(ξ) = g−µ(ξ) = πξ2

Kµ+1 (ξ) Kµ (ξ)

.

(2.26)

(cid:20)(cid:12)(cid:12)Kµ+1 (ξ)(cid:12)(cid:12)2 −(cid:12)(cid:12)Kµ (ξ)(cid:12)(cid:12)2 − 2µ

ξ

(cid:21)

The full result of Equation 2.23 also shows a direct proportionality between the Coulomb

excitation cross section and the reduced transition strength. A Mathematica script has been

developed to perform the full Alder-Winther calculations and is used in this work to deter-

mine reduced transition strengths given the experimentally-measured Coulomb excitation

cross sections. The basis of the Mathematica script is provided in [63].

35

2.2 Experimental Considerations

As seen in Equation 2.23, the Coulomb excitation cross section is proportional to Z2

t . In this

work, the heaviest stable nuclide 209Bi (Z = 83) is utilized for the target material in order to
maximize Zt. 209Bi is one proton above doubly-magic 208Pb and the adopted B(E2; 9/2− →
7/2−) strength between the ground state and first excited state at 896.28(5) keV is only

26.1(16) e2fm4 [7]. Consequently, the Coulomb excitation of target nuclei due to the S

(Z = 16) projectiles is hindered and γ rays from the target do not significantly contribute

to the observed γ-ray spectra. The general expression for the experimental cross section is:

σi→f =

Nγ,f→g

Nbeamntarget

.

(2.27)

Here, Nγ,f→g is the efficiency-corrected number of γ-ray transitions from the populated state

to all levels g it γ decays to, Nbeam is the number of incoming beam particles, and ntarget is

the areal number density of the target (number of target nuclei per unit area). Cross sections
in nuclear physics are often given in units of millibarns (1 mb = 10−31 m2 = 0.1 fm2).

In the Coulomb excitation process, the magnetic substates µ are not populated equally

[64]. Therefore, the de-excitation γ rays are not emitted isotropically in the center-of-mass

frame. The γ-ray angular distribution can be parameterized in terms of Legendre polyno-

mials Pk:

(cid:88)

k even

W (φ) =

akPk(cos φ),

(2.28)

where φ is the center-of-mass angle of the γ ray with respect to the beam axis. The index k

runs from 0 to the smaller of 2λ or 2Ji. The angle measured in the center-of-mass frame φ

36

is related to the laboratory-frame angle φlab by:

cos φ =

cos φlab − β
1 − β cos φlab

.

(2.29)

The coefficients ak found using the Alder-Winther excitation amplitudes are [63]:

(cid:12)(cid:12)(cid:12)Gλµ

(cid:16) c

v

(cid:17)(cid:12)(cid:12)(cid:12)2

(cid:88)

µLL(cid:48)

ak =

gµ(ξ)(−1)µ

×

λ

k
µ −µ 0

λ
Jf Jf

λ


 Fk(L, L(cid:48), Jf f , Jf )

k

λ Ji

√
2k + 1δLδL(cid:48),

(2.30)

where Jf f is the spin of the state the γ decay populates and the γ-γ correlation function
Fk(L, L(cid:48), Jf , Ji) is:

Fk(L, L(cid:48), Jf , Ji) = (−1)

Jf +Ji−1(cid:112)(2k + 1)(2Ji + 1)(2L + 1)(2L(cid:48) + 1)
 L L(cid:48)


L L(cid:48)

k
1 −1 0

×

Jf Ji Jf

 .

(2.31)

k

The effect of the γ-ray angular distribution on the detection efficiency must be taken into

account when determining the efficiency-corrected number of γ decays if the γ-ray detection

array does not cover the full 4π solid angle.

The uncertainties associated with the approximations used in the semi-classical method

of Winther and Alder have been studied through fully quantal coupled-channels calcula-

tions. Due to nuclear effects, mult-step excitations, and unobserved feeding, the theoretical

uncertainty for the semi-classical model is approximately 13% compared to a theoretical

37

uncertainty of less than 7% for the coupled-channels analysis [65]. The results obtained for

reduced transition strengths from intermediate-energy Coulomb excitation experiments have

shown excellent agreement with the results found using other experimental probes [66].

2.3 Reduced Transition Probabilities and Lifetimes

The lifetime τ of a nuclear state is the inverse of the transition rate W introduced in Sec-

tion 1.4:

(cid:88)

σλ

1
τ

= W =

(cid:18) 8π(λ + 1)

λ[(2λ + 1)!!]2

(cid:19)2λ+1

(cid:19) 1



(cid:18) Eγ

c

B(σλ; Ji → Jf ).

(2.32)

Therefore, lifetime measurements can be performed to extract B(σλ) strengths for γ decay,

which are related to the B(σλ) strengths for Coulomb excitation by Equation 1.18. Plugging

in the values, the B(σλ) strengths for γ decay are related to the partial lifetimes (τp = τ /b

where b is the branching fraction) by [5]:

B(E1) =

B(E2) =

B(E3) =

0.629
E3
γτp

816
E5
γτp

1760
E7
γτp

e2fm2MeV3fs,

e2fm4MeV5ps,

e2fm6MeV7µs,

B(M 1) =

56.8
E3
γτp

µ2
N MeV3fs,

B(M 2) =

74.1
E5
γτp

µ2
N fm2MeV5ns,

B(M 3) =

0.1585
E7
γτp

N fm4MeV7s.
µ2

38

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

(2.38)

Chapter 3

Experimental Method

The neutron-rich isotopes 38,40,42,43,44S were studied using intermediate-energy Coulomb

excitation at the National Superconducting Cyclotron Laboratory (NSCL) [67] at Michigan

State University. In this experiment, each sulfur isotope was studied at a different time over

the period of a week. Roughly 12 hours of data were taken for 38,40S and about 24 hours

of data were taken for 42,43,44S. The five separate beams of radioactive 38,40,42,43,44S were

all produced by the Coupled Cyclotron Facility using the projectile fragmentation method

[68] and transported to the experimental hall where they were impinged on a 492 mg/cm2

209Bi target to electromagnetically excite the levels of interest. The de-excitation γ rays were

detected using the CsI(Na) scintillator array CAESAR [14] and the outgoing beam particles

and reaction products were identified on an event-by-event basis with the S800 magnetic

spectrograph [69]. The following sections describe these experimental systems in detail.

3.1 Beam Production and Purification

A schematic of the NSCL Coupled Cyclotron Facility is provided in Figure 3.1. In order

to produce the radioactive sulfur isotopes of interest in this work, a stable primary beam

of 48Ca was first developed. A solid block of metallic 48Ca was heated in an oven until

vaporization. Once in gaseous form, the 48Ca atoms were ionized via the electron cyclotron

resonance (ECR) technique in the Advanced Room TEMperature Ion Source (ARTEMIS)

39

Figure 3.1: Layout of the Coupled Cyclotron Facility at the National Superconducting Cy-
clotron Laboratory showing the ion sources, K500 and K1200 cyclotrons, and A1900 fragment
separator [12]. Figure adapted from [13].

[70]. In Figure 3.1, ARTEMIS is represented by RT-ECR standing for room temperature

ECR. In the ECR method, the gas is injected into the ion source and confined in a magnetic

trap. Electrons are accelerated using microwave radiation tuned to the resonant frequency

ω of the cavity:

ω =

eB
me

,

(3.1)

where e and me are the charge and mass of an electron and B is the strength of the magnetic

field. The accelerated electrons ionize the gas atoms by stripping electrons during collisions.

In this experiment, 48Ca was ionized in ARTEMIS to a charge state of 8+ meaning that

eight electrons were removed from the neutral atom.

Next, the 48Ca8+ ions were injected into the K500 cyclotron. According to the Lorentz

law, particles with charge q and momentum p perpendicular to a magnetic field B will follow

a radius of curvature ρ given by:

ρ =

p
qB

=

γmv
qB

,

40

(3.2)

SC-ECRRT-ECRK500K1200A1900productiontargetwedgefocal plane20 ft10 mstrippingfoilion sourcesdipole magnetfocusingmagnetsimage 1image 2image 3where m and v are the mass and velocity of the particle and γ is the relativistic factor. The

quantity Bρ is referred to as the magnetic rigidity:

Bρ =

p
q

=

γmv

q

.

(3.3)

The K500 and K1200 cyclotrons at the NSCL operate using this principle. The cyclotrons

consist of three “dees”, copper electrodes surrounded by a superconducting electromagnet,

which produce a radio-frequency (RF) electric field that accelerates the ions in the gaps

between the dees. The trajectories of the ions are confined by the applied magnetic field

from the superconducting electromagnets. As the energy of the ions increases, the radius of

curvature gets larger. Once the ions reach the maximum energy given the radius constraints

of the cyclotron, they are extracted at a port on the outer edge. For the 48Ca8+ ions in

this experiment, the primary beam energy was accelerated to 12.28 MeV per nucleon (12.28

MeV/u) after the K500 cyclotron.

After extraction from the K500 cyclotron, the ions are transported and injected into the

K1200 cyclotron. A carbon stripper foil is located at the center of the K1200 cyclotron to

increase the charge state of the ions in order to increase the maximum possible final energy.

In this work, the 48Ca ions were fully stripped (48Ca20+) and accelerated to 140 MeV/u.

The 48Ca primary beam was then impinged on a thick 9Be production target, producing

a cocktail secondary beam of different fully-stripped ions, including the rare neutron-rich

sulfur isotopes, through projectile fragmentation. To purify the secondary beam, the A1900
fragment separator [12] was utilized. The A1900 consists of four large 45◦ dipole bending

magnets and 24 quadrupole focusing magnets with additional magnets for higher-order cor-

rections and achieves isotopic separation using the Bρ-∆E-Bρ technique [12]. In the initial

41

Bρ stage, the secondary beam is purified by tuning the magnetic fields of the first two dipoles.

From Equation 3.3, the chosen magnetic rigidity selects the momentum-to-charge ratio p/q

of the ions that will pass through the fixed-radius dipoles. However, particles with similar

p/q ratios to the projectiles of interest will also make it through this selection process. In

order to limit these contaminants, slits at the image points and focal plane of the A1900 (see

Figure 3.1) can be employed to only allow a certain range of momenta. However, particles

with the same p/q ratio cannot be unambiguously selected with Bρ alone. Therefore, in

the ∆E step, the beam is impinged on an achromatic aluminum wedge degrader (actually a

curved aluminum foil) located at the mid-acceptance position of the A1900. The differential

energy loss of the ions traversing the wedge per differential path length is described by the

Bethe formula [71]:

(cid:20)

ln

(cid:18)2meγ2(v/c)2

(cid:19)

I

(cid:21)

,

− (v/c)2

− dE
dx

=

4πe4neZ2
me(v/c)2

(3.4)

where e is the electron charge, ne is the electron number density of the degrader, Z is the

atomic number of the incoming ion, me is the electron mass, v/c is the ion velocity relative

to the speed of light, γ is the relativistic factor, and I is the mean excitation potential of the

degrader. From this equation, the energy loss is proportional to the square of the charge of

the ion. Consequently, particles starting with the same momentum but with different atomic

numbers Z will have different momenta after passing through the wedge. A 450 mg/cm2

Al wedge was used for all settings in this experiment. The final step in the purification

process is another Bρ selection using the magnetic rigidities of the final two dipole magnets.

In this experiment, the slits were adjusted to give a momentum acceptance dp/p of about

1% for 38,40,42,43S and 2% for 44S. The production targets used for each secondary beam

42

are provided in Table 3.1 along with the energies of the projectiles before the 209Bi reaction

target calculated using the program LISE++ [72].

Table 3.1: Production target, momentum acceptance dp/p, and energy as inferred from the
magnetic rigidity setting of the segment of the analysis line just before the 492 mg/cm2 209Bi
reaction target in front of the S800 spectrograph for each secondary beam setting.

Setting

9Be Production Target (mg/cm2)

dp/p Energy (MeV/u)

38S
40S
42S
43S
44S

1034
987
940
1034
1081

1.07%
1.07%
1.08%
1.07%
2.02%

81.85
82.18
86.48
83.03
79.40

3.2 S800 Magnetic Spectrograph

After separation in the A1900, the secondary beam passed through a thin plastic scintillator

at the extended focal plane of the A1900 (XFP scintillator) and was delivered through the

transfer hall to the analysis beam line where it was impinged on the 209Bi reaction target at

the target position of the high-resolution, high-acceptance, S800 magnetic spectrograph [69].

In this experiment, the analysis line, which consists of four dipoles, five sets of quadrupole

triplets, one quadrupole doublet, and four sextupole magnets [73], was operated in focus

mode, where the focal point for the beam is at the 209Bi reaction target and the beam is

dispersed at the S800 focal plane. Following the reaction target, there is a quadrupole magnet

that refocuses the beam and two large dipole magnets that bend the charged particles based

on their momentum-to-charge ratios. The magnetic rigidity of the dipoles is set to center

the reaction products of interest in the focal plane of the S800. The analysis line and S800

magnetic spectrograph are shown in Figure 3.2.

There are a number of detectors at the focal plane of the S800 that are used for particle

43

Figure 3.2: Illustration of the S800 magnetic spectrograph including the analysis line. The
target position was surrounded by the CsI(Na) array CAESAR. Figure taken from [11].

identification and trajectory reconstruction [74]. A diagram of the S800 focal plane is shown

in Figure 3.3. The reaction products first pass through two cathode readout drift chambers

(CRDCs). The upstream (CRDC1) and downstream (CRDC2) CRDCs are spaced one meter

apart and are used to determine the xy-positions and angles necessary for the reconstruction

of the trajectory for each ion. Next, the energy loss of the ions is measured in an ionization

chamber to determine the atomic number of the ion. Afterward, the ions traverse a thin

plastic scintillator (E1 scintillator) that is used for timing measurements and to trigger the

data acquisition system. As described in detail in the sections below, timing measurements

and the S800 focal plane detectors allow the incoming beam particles and the reaction

products to be identified event-by-event.

44

Figure 3.3: S800 focal plane diagram showing the cathode readout drift chambers (CRDCs),
ionization chamber, and E1 plastic scintillator. Figure modified from [13].

3.2.1 Particle Identification and Trajectory Reconstruction

Near the beginning of the analysis line, there is another thin plastic scintillator at the ob-

ject position of the S800 (OBJ scintillator). The XFP and OBJ scintillators are used in

conjunction with the E1 plastic scintillator at the S800 focal plane to perform time-of-flight

measurements. The scintillators are coupled to photomultiplier tubes that detect the fluo-

rescence light generated when ions pass through the plastic. The incoming secondary beam

ions have been selected by Bρ = γmv/q. Therefore, assuming a fixed flight path through

45

Ionization ChamberE1 ScintillatorCRDCsBeamDirectionthe analysis line and S800, the time of flight, which is inversely proportional to velocity,

is proportional to the mass-to-charge ratio. Since the ions are fully stripped, selecting on

the ratio m/q is equivalent to selecting on A/Z. Consequently, the time-of-flight difference

XFP-OBJ (both relative to E1) is sensitive to the A/Z of the incoming beam components.

Since the proportionality to A/Z depends on the assumption that the flight path is fixed,

correcting for the correlations between time of flight (for XFP-E1 and OBJ-E1) and the

dispersive angle and position in the S800 focal plane provides significant improvement in

timing resolution.

To unambiguously identify the reaction products, the time-of-flight difference between

the OBJ and E1 scintillators is used in conjunction with the energy loss in the ionization

chamber. The ionization chamber consists of 16 individual parallel plate chambers and is

filled with P10 gas (90% Ar and 10% methane). When the reaction products travel through

the ionization chamber, the P10 gas is ionized and the electrons drift toward the anode

providing a signal proportional to the energy deposition. The energy loss in the ionization

chamber is proportional to Z2 as seen in Equation 3.4. As discussed above, the time-of-flight

difference is proportional to A/Z. Therefore, reaction products are uniquely identified by

plotting the energy deposited in the ionization chamber against the time of flight through

the S800 spectrograph (so-called ∆E-ToF technique).

In intermediate-energy Coulomb excitation, analysis must be restricted to small scatter-

ing angles to limit nuclear contributions to the cross section (see Chapter 2). The position

information from the CRDCs in the S800 focal plane is used to track the trajectory of the

scattered particles back to the reaction target. As a result, the scattering angle for each

particle on an event-by-event basis can be determined.

Each CRDC has an active depth of 1.5 cm and an active area of about 60 cm in the

46

x (dispersive) direction and 30 cm in the y (non-dispersive) direction [74]. The CRDCs

are filled with 80% carbon tetrafluoride (CF4) and 20% isobutane (C4H10). A negative bias

voltage is applied across each CRDC in the y direction. When nuclei pass through the CRDC

the gas is ionized and the electrons drift toward the anode wire. Bordering the anode wire

are 224 cathode pads, each 2.54 mm wide [74]. The collection of charge in the anode induces

a positive charge on the cathode pads. The interaction point in the x (dispersive) direction

for the incoming particle is determined by fitting the measured charge on the pads with a

Gaussian distribution. The y (non-dispersive) position of the interaction is determined from

the drift time of the electrons to the anode measured with respect to the timing signal from

the E1 scintillator.

In order to calibrate the x and y positions of the CRDCs during the experiment, thick

masks with holes and slits drilled into them at known positions can be inserted in front of

each CRDC. An example of a CRDC calibration run using a mask is shown in Figure 3.4.

During the experiment, the drift time can vary due to changes in the composition of the

CRDC fill gas. Therefore, mask runs for each CRDC were taken at several times during

the experiment and the slope of the linear y position calibration was adjusted run-by-run to

align the centroids of the measured y positions for the sulfur isotopes of interest.

From the xy-positions measured in the CRDCs (referred to as xf p and yf p, respectively),

the dispersive (af p) and non-dispersive (bf p) angles at the S800 focal plane can be calculated.

The trajectory of the ions from the reaction target through the spectrograph is reconstructed

using an inverse map of the S800 magnetic field generated from the ion optics code COSY

infinity [75]. The positions and angles calculated at the focal plane are related to quantities

47

Figure 3.4: Typical calibrated mask run for CRDC1 (upstream) taken from the 38S setting.

at the reaction target through the inverse map (S−1) by:

(ata, yta, bta, dta) = S−1(xf p, af p, yf p, bf p).

(3.5)

Here, ata and bta are the dispersive and non-dispersive angles at the target, yta is the y

position at the target, and dta is the deviation in energy relative to a particle taking the
central path through the spectrograph (dta = (E − E0)/E0). The scattering angle of the

projectile in the laboratory frame can be written as:

sin2(ata) + sin2(bta)

48

(cid:18)(cid:113)

θ = arcsin

(cid:19)

.

(3.6)

x position (mm)100-50-050100y position (mm)40-30-20-10-0102030405060010203040506070The uncertainty in the scattering angle due to the position resolution of the CRDCs is about

2 mrad [74].

As mentioned earlier in this section, there are correlations between the time-of-flight

differences between plastic scintillators XFP-E1 and OBJ-E1 and the dispersive positions

and angles (xf p and af p) measured in the S800 focal plane. These correlations are corrected

for by removing the linear dependence on xf p and af p via:

XFPcorr = XFP + A1 × xf p + A2 × af p,

OBJcorr = OBJ + B1 × xf p + B2 × af p.

(3.7)

(3.8)

An example of the ∆E-ToF particle identification using the S800 ionization chamber and

the corrected OBJ-E1 time of flight for the 44S setting is shown in Figure 3.5. The vertical

lines spanning energy loss at a constant time of flight are due to pileup in the ionization

chamber. The total number of pileup counts at the time of flight corresponding to 44S are

less than 3% of the counts in the energy loss peak.

3.2.2 Isomer Tagging

The 43S secondary beam at the NSCL is known to be produced with some fraction of the
43S nuclei in the low-lying 7/2− excited state at 320 keV with a half-life of 415 ns [50, 51].

In order to infer if the Coulomb excitation occurred with this isomeric state as the initial

state, the CsI(Na) hodoscope at the S800 focal plane [76] was utilized. The hodoscope

consists of 32 closely-packed CsI(Na) scintillator crystals and is located downstream of the

E1 scintillator. A 6.35 mm aluminum plate is added 10.5 cm before the hodoscope to stop the

49

Figure 3.5: Particle identification plot utilizing energy loss and time of flight for the 44S
setting. The energy loss is measured in the S800 ionization chamber and is proportional to
Z2 and the time of flight is measured between the OBJ and E1 plastic scintillators and is
proportional to A/Z.

reaction products in the so-called IsoTagger configuration [77]. Suppose a short-lived level
above the 7/2− isomeric state is populated in Coulomb excitation and then decays back to

the isomer via γ-ray emission. The prompt γ ray can be measured by CAESAR surrounding

the target. Since the flight path through the S800 spectrograph is approximately 15 m, a

particle moving at about v=0.4c will take roughly 125 ns to traverse the S800 spectrograph.

Consequently, the isomer (415 ns half-life) will predominantly decay after reaching the S800

focal plane. In this experiment, the aluminum stopper plate and CsI(Na) array were used to

search for 320-keV γ rays from the decay of the isomeric state in coincidence with prompt γ

50

Time of Flight (arb. units)2200−2100−2000−1900−1800−1700−1600−Energy Loss (arb. units)60080010001200140016001800200011021031041051044S46Cl45Clrays from higher-lying states in 43S at the reaction target.

3.3

In-Beam γ-Ray Spectroscopy with CAESAR

To measure de-excitation γ rays following the Coulomb excitation of the neutron-rich sulfur

isotopes, the 492 mg/cm2 209Bi reaction target at the target position of the S800 spectro-

graph was surrounded by the high-efficiency CAESium-iodide scintillator ARray CAESAR

[14]. CAESAR consists of 192 total CsI(Na) detectors arranged into 10 rings and provides
about 95% solid angle coverage. 48 of the detectors are 3× 3× 3 inches while the remaining
144 crystals are 2 × 2 inches on the front face and 4 inches in depth. An illustration of the

arrangement of the 192 CsI(Na) detectors is shown in Figure 3.6. Rings A (most upstream)

and J (most downstream) have 10 detectors each, rings B and I have 14 detectors each, and

the middle six rings C-H each hold 24 detectors. The full-energy-peak efficiency of the array

exceeds 35% for 1 MeV γ rays emitted by a source at rest. The in-beam energy resolution

is approximately 9.2% full width at half maximum (FWHM) for 1 MeV γ rays emitted from

a nucleus traveling at 0.35c [14]. A photograph of CAESAR set up in front of the S800

spectrograph is provided in Figure 3.7.

3.3.1 Interaction of γ Rays with Matter

The CsI(Na) scintillator crystals in the CAESAR detectors are housed in an aluminum

casing and coupled to a photomultiplier tube (PMT). When γ rays interact with the detector

material, the atoms in the scintillator become excited or ionized resulting in the emission of

de-excitation photons. When this scintillation light strikes the photocathode in the PMT,

electrons are ejected via the photoelectric effect. The photoelectrons are accelerated due to

51

Figure 3.6: Illustration of the arrangement of CAESAR into 10 rings of detectors. Left:
cross-sectional view of rings F and J perpendicular to the beam axis. Right: the 3 × 3 × 3
inch detectors occupy the two most upstream (A,B) and two most downstream (I,J) rings
while the 2 × 2 × 4 inch detectors occupy the six middle rings (C-H). The target position is
shown in red. Figure taken from [14].

an applied voltage across the PMT and collide with a series of dynodes, which release many

additional electrons. The resulting signal from the electric current, which is proportional to

the energy of the initial γ ray, can then be read out. Since PMTs are affected by magnetic

fields, several layers of µ metal were added between the aluminum housing and the PMTs.

[14]. In addition, a large soft-iron plate (the green metal sheet at the back of Figure 3.7)

was installed between CAESAR and the S800 spectrograph to shield the detectors from the

S800 fringe field.

There are three main processes by which γ rays interact with matter: the photoelectric

effect, Compton scattering, and pair production. The photoelectric effect is the dominant

interaction channel for γ ray energies below a few hundred keV [71]. In photoelectric ab-

sorption, the γ ray transfers all of its energy Eγ to an electron, ejecting it from the atom.

52

IHGCJFEDBAbeamoccupied by S800 spectrographFigure 3.7: Photograph of the 192 CsI(Na) detectors composing CAESAR in front of the
S800 spectrograph.

The remaining energy of the electron Ee− is:

Ee− = Eγ − Eb,

(3.9)

where Eb is the typically small binding energy of the electron. In this case, assuming the

electron remains in the detector volume, the full energy of the initial γ ray is collected. In

this case, a count will be added to the full-energy peak in the spectrum shown in Figure 3.8.

53

Figure 3.8: Illustration of the response of a γ-ray detector to many monoenergetic γ rays
showing characteristic features including the full-energy peak, Compton edge, Compton con-
tinuum, and single and double-escape peaks. Figure taken from [11].

In Compton scattering, a γ ray with initial energy Eγ inelastically scatters off of an
atomic electron, transferring only some of its energy. The remaining energy of the γ ray E(cid:48)

γ

is related to the initial energy by:

E(cid:48)
γ =

1 +

(cid:16) Eγ

mec2

(cid:17)

Eγ

(1 − cos θ)

54

,

(3.10)

Energysingle-escape peakdouble-escape peakCompton continuumCompton edgefull-energy peakCountsEγECEEγ - 511 keVEγ - 2x511 keVwhere θ is the scattering angle and me is the mass of the electron. The minimum energy
transfer is 0 for scattering at 0◦ while the maximum energy transfer is for scattering at 180◦:

ECE =

2E2
γ

mec2 + 2Eγ

.

(3.11)

Here, the maximum energy transferred to the electron ECE is the energy of the Compton

edge. As a result, the energy collected in the detector from Compton scattering ranges from

0 to ECE leading to a feature in the energy spectrum known as the Compton continuum as

seen in Figure 3.8. The scattered γ ray can leave the detector volume or undergo multiple

Compton scatterings until it is absorbed via the photoelectric effect. Compton scattering is

the dominant interaction process for γ rays with energy between a few hundred keV and a

few MeV [71].

If the γ ray energy is above two times the rest mass of the electron, pair production

can occur through interaction with the electric field of one of the nuclei in the detector

material.

In this case, the γ ray is converted into an electron-positron pair and through

energy conservation:

Eγ = Ee− + Ee+ + 2mec2.

(3.12)

Here Ee− and Ee+ are the kinetic energies of the electron and positron and mec2 = 511 keV.

The positron will quickly annihilate with an electron in the detector material producing two

511-keV photons. These photons can undergo photoelectric absorption, Compton scattering,

or leave the detector. If one of the annihilation photons escapes the detector while the other
is fully absorbed, the total energy detected will be ESE = Eγ − 511 keV leading to a

single-escape peak in the spectrum. If both annihilation photons escape the detector, the
double-escape peak will be at EDE = Eγ − 1022 keV. The single and double-escape peaks

55

are shown in Figure 3.8. Pair production only becomes the dominant interaction process for

γ-ray energies of several MeV [71].

3.3.2 Doppler Reconstruction

In this experiment, the velocity of the sulfur isotopes at mid-target was about 0.38c on

average. Therefore, the γ rays are emitted from a source moving at a relativistic velocity

with respect to the detectors in the laboratory frame. As a result, the energy of the γ ray

measured in the laboratory frame Elab is Doppler shifted with respect to the energy of the

γ ray in the rest frame of the nucleus Erest according to:

Erest = Elabγ(1 − β cos θ),

(3.13)

where γ is the relativistic factor, β = v/c is the velocity of the nucleus relative to the speed

of light, and θ is the angle of emission of the γ ray with respect to the velocity vector of

the nucleus in the laboratory frame.

In order to precisely Doppler reconstruct the rest-

frame energy of the γ ray, the uncertainty in the detection angle θ should be kept as small as

possible. The high-granularity of CAESAR resulting from the use of 192 individual detectors

provides acceptable angular resolution. The total energy resolution of a detector is related

to the uncertainties in detection angle (∆θ) and beam velocity (∆β) by [56]:

(cid:18)∆Eγ

(cid:19)2

Eγ

(cid:18) β sin θ

(cid:19)2

=

1 − β cos θ

(cid:32)

(∆θ)2 +

(cid:0)1 − β2(cid:1) (1 − β cos θ)

cos θ − β

(cid:33)2

(∆β)2 +

(cid:18)∆Eintr

Eγ

(cid:19)2

,

(3.14)

where ∆Eγ is the FWHM of the peak at energy Eγ. The uncertainty in the velocity ∆β

arises from the fact that the γ decay can occur anywhere in the reaction target. As the

56

nucleus passes through the reaction target, it losses energy and its velocity is lowered. The

uncertainty in detection angle ∆θ is dominated by the finite angle coverage of the detector.

∆Eintr is the intrinsic energy resolution of the detector. For CAESAR, the detectors have

an intrinsic energy resolution of about 7% at 1 MeV [14]. The intrinsic energy resolution

characteristically goes as 1/(cid:112)Eγ [71].

3.3.3 Nearest-Neighbor Addback

In order to increase the full-energy peak efficiency, a technique known as nearest-neighbor

addback can be employed. At typical γ-ray energies, Compton scattering has a significant

contribution to the observed γ-ray spectra. Due to the selectivity of the states populated in

intermediate-energy Coulomb excitation, the average number of de-excitation γ rays for each

excitation (the γ-ray multiplicity) is low. Suppose that two neighboring crystals in CAESAR

register within a short time coincidence window. Since the average γ-ray multiplicity is low,

the probability that two coincident γ rays hit two neighboring crystals in CAESAR is lower

than the probability that one γ ray has Compton scattered from one detector into the

neighboring detector. Assuming that the Compton scatter has occurred, the full energy of

the γ ray can be recovered by summing the energies deposited in the neighboring detectors

together. This transfers two counts in the Compton continuum into one count in the full-

energy peak of Figure 3.8. For in-beam γ-ray spectroscopy with CAESAR, the detector with

the highest energy deposition is used to determine the emission angle of the γ ray for Doppler

reconstruction. In source measurements, utilizing nearest-neighbor addback has been shown

to recover more than 20% of the full-energy peak intensity for γ-ray energies above 1 MeV

and more than 30% of the full-energy peak intensity for γ-ray energies above 2 MeV [14].

In order to derive the results discussed in Chapter 5, both addback and non-addback

57

spectra were utilized. To determine Coulomb excitation cross sections from the de-excitation

γ-ray yields, no addback was applied to the singles spectra. Using addback would add an

additional systematic uncertainty to the γ-ray detection efficiency, which would in turn

increase the uncertainty on the cross section. Specifically, when using addback the energy

thresholds for each detector need to be considered. Furthermore, the efficacy of the GEANT4

simulation for handling the scatter between detectors, which is sensitive in reality to the

detector housing and insensitive layers of the scintillator crystals, would have to be studied

in great detail. Therefore, the additional uncertainty in the detection efficiency does not

justify the gain in full-energy-peak statistics provided by employing addback. The gain in

statistics afforded by addback was exploited in the multiplicity-two γ-γ plots used to identify

coincidences and in the multiplicity-one plots used to identify transitions to the ground state.

Here, using addback properly converts a multiplicity-two event where one γ ray Compton

scatters from one detector to another detector into a multiplicity-one event.

3.3.4 CAESAR Calibrations

The first step in the analysis of the data was to perform the necessary calibrations and

corrections for the detector systems used in the experiment. To begin, all 192 CAESAR

detectors were energy calibrated using γ-ray spectra obtained from a set of standard γ-

ray calibration sources including 22Na, 60Co, 88Y, 137Cs, 133Ba, and 207Bi.

In addition,

natural background lines from the decays of 40K and 208Tl were utilized. Energies ranging

from 356 keV to 2614 keV were used in the calibration. To account for drifts in energy

calibration over time, calibration runs were taken at the beginning and end of the experiment.

Furthermore, the calibrations were performed with the rigidities of the S800 dipoles set since

the detector PMTs are sensitive to magnetic fields. For all detectors individually, the channel

58

numbers for the full-energy-peak centroids from the source data were obtained through

Gaussian fits and mapped to the known γ-ray energies with a second order polynomial. The

results of the energy calibration for the 88Y source are shown in Figure 3.9.

Figure 3.9: Energies after calibration for the 192 CAESAR detectors for the 88Y source.
The 898-keV and 1836-keV transitions are aligned for all detectors. The 2614-keV transition
from room background is also visible.

The timing calibration was performed for all 192 detectors using a portion of the in-beam

data. The CAESAR timing is taken relative to the time of flight of the projectile from the

OBJ scintillator to the E1 scintillator at the S800 focal plane. However, the timing dispersion

for CAESAR is 250 ps/channel while the timing dispersion for the plastic scinitllators is 100

ps/channel. Therefore, the corrected CAESAR time is given by:

CAESARcorrtime = CAESARtime − E1time × (0.1/0.25) − OBJtime × (0.1/0.25).

(3.15)

59

Detector Number020406080100120140160180200Energy (keV)05001000150020002500300035004000110210310410For detector hits with laboratory energies greater than 500 keV, the corrected CAESAR

timing spectrum was generated for all detectors using one of the in-beam runs for the 38S

setting. One detector was then chosen as the reference and the centroids of the timing

spectra for all other detectors were aligned to the timing centroid for the reference detector.

The results of the timing calibration are shown in Figure 3.10.

Figure 3.10: Alignment of the corrected CAESAR timings for all 192 detectors after cali-
bration gated on γ-ray energies above 500 keV in the laboratory frame for one of the 38S
runs.

In order to avoid issues stemming from high count rates, the absolute full-energy peak γ-

ray detection efficiency of CAESAR was determined using a low-activity 207Bi source (around

3433 decays per second). The β+ decay of 207Bi results in γ-ray transitions with energies

60

Detector Number020406080100120140160180200Timing (channels)600-400-200-0200400600800100012000510152025303540at 570 keV, 1064 keV, and 1770 keV. In this measurement, the efficiency was determined

from the number of full-energy peak counts above background (modeled with an exponen-

tial) divided by the number of emitted γ rays at that energy by the source. Furthermore,

efficiency data points for 60Co, 88Y, and 207Bi sources were obtained using coincidences with

a LaBr3(Ce) scintillator placed in the beam pipe near the source. Consider, for example,

gating on the 1332-keV full-energy peak from a 60Co source measured in the LaBr3(Ce)

detector. Since the 1332-keV transition feeds the 1173-keV level, the number of coincident

1173-keV full-energy peak counts in CAESAR divided by the total number of 1332-keV

counts in the LaBr3(Ce) detector within the applied gate is the absolute full-energy peak

efficiency for CAESAR at 1173 keV. More details on the method are provided in [78].

A GEANT4 [79] simulation of CAESAR called UCCAESAR was developed to deter-

mine the in-beam efficiency of the array (for some details on the simulation see [11, 80]).

The simulation takes into account the Lorentz boost, energy loss of the incoming projec-

tiles in the reaction target, absorption of emitted γ rays in the target and beam pipe, the

angular distributions of the γ rays emitted following Coulomb excitation, and the lifetime

of the populated state. The efficiencies predicted from the simulation are compared with

the efficiencies extracted from source data in Figure 3.11. The vertical error bars for the

experimental data in the direct efficiency determination from the 207Bi source include sta-

tistical and fit uncertainties added in quadrature with a 3% systematic uncertainty in the

source activity. The vertical error bars for the experimental data from coincidences with

the LaBr3(Ce) detector include statistical and fit uncertainties added in quadrature with
uncertainties due to random coincidences, γ − γ angular correlations, and scattering out of

the LaBr3(Ce) scintillator (see also [78]). On average, the experimental source efficiencies

are 5% higher than the simulated source efficiencies. Accordingly, in the analysis of the

61

Figure 3.11: Full-energy peak efficiency of CAESAR as a function of energy as determined
from direct measurement of a low-activity 207Bi source and from coincidence measurements
with a LaBr3(Ce) detector for 60Co, 88Y, and 207Bi sources compared to predictions from
GEANT4 simulations. The efficiency curves here were produced treating the 192 detectors
of CAESAR independently (no addback applied).

neutron-rich sulfur isotopes the in-beam efficiency of CAESAR was taken as the GEANT4

efficiency times a scaling factor of 1.05 and a systematic uncertainty of 5% was adopted

following the method of [81].

The energy resolution of CAESAR was parameterized in the form:

where the resolution σ is related to the FWHM by FWHM = 2(cid:112)2 ln(2)σ. The fit parameters

σ(E) = aEb,

(3.16)

a and b for each ring of CAESAR were found from the widths of full-energy peaks from source

62

0500100015002000Energy (keV)00.10.20.30.40.50.6Full-Energy Peak EfficiencyExp (direct)Exp (LaBr)GEANT4data over the same range of energies used for the energy calibration. This parameterization

was then used as an input for the CAESAR GEANT4 simulation to set the laboratory-frame

resolution as a function of energy. The values of a and b (for energies in keV) are provided

in Table 3.2 for each ring.

Table 3.2: Energy resolution parameters for each of the 10 rings of CAESAR for energies in
keV.

Ring

a

b

A
B
C
D
E
F
G
H
I
J

0.6346
0.5376
0.7518
0.6151
0.6416
0.6099
0.6183
0.5899
0.6041
0.5005

0.5589
0.5826
0.5252
0.5507
0.5475
0.5511
0.5507
0.5600
0.5691
0.5930

63

Chapter 4

Data Analysis and Results

4.1 General Method

Reduced transition strengths for excited states in the neutron-rich sulfur isotopes

38,40,42,43,44S were determined by measuring the cross sections for the excitation of these

states in intermediate-energy Coulomb excitation. The experimental excitation cross section

can be written as:

σCE =

Ndecays

Nbeamntarget

,

(4.1)

where Ndecays is the total number of γ decays from the excited state to all final states, Nbeam

is the number of incoming beam particles incident on the reaction target, and ntarget is the

areal number density of the reaction target.

The areal number density of the 492(4) mg/cm2 209Bi target used in the experiment is

1.42(1)×10−6 nuclei/mb and can be calculated using:

ntarget =

ρANA

M

,

(4.2)

where ρA is the areal density of the target, NA is Avogadro’s number (6.0221409 × 1023

particles/mole), and M is the molar mass of the target.

64

The number of incoming beam particles is given by:

Nbeam =

NS · DSS800

LTS800

.

(4.3)

Here, NS is the number of sulfur projectiles of interest detected at the S800 focal plane in

the gate applied on the ∆E-ToF particle identification plot, DSS800 is the downscale factor

applied to S800 particle triggers, and LTS800 is the livetime of the S800 data acquisition

(DAQ) system, which is triggered by the E1 scintillator. Note that there are two trigger

conditions, one for particles that is downscaled and one for particle-γ coincidences that is

not downscaled. After a trigger initiates the readout cycle of the event, there is an amount of

time in which the DAQ cannot process another trigger known as the deadtime. During this

deadtime, any particle interacting with the E1 scintillator will not be read out by the DAQ.

The number of times the DAQ is read out is the number of S800 singles events (Nsingles).

This number is extracted from the data using information from the S800 trigger register

included with each event. The S800 livetime is found by dividing the number of S800 singles

by the total number of events meeting the S800 trigger condition recorded by the S800 scaler

module (S800.T riggers), which is not affected by the deadtime:

LTS800 =

Nsingles

S800.T riggers

.

(4.4)

A downscaler was applied to the S800 singles in order to reduce the deadtime of the system.

For all settings in this experiment, a downscale factor of 20 was used, meaning that only

every 1/20 S800 events meeting the trigger condition of the DAQ were read out.

The total number of γ decays is the number of counts in the full-energy peak detected by

65

CAESAR Nf ep divided by the corresponding energy-dependent full-energy peak efficiency


(Eγ) and corrected for the livetime for detecting a reaction product in the S800 focal plane

in coincidence with a γ ray in CAESAR

Ndecays =

Nf ep


(Eγ) · LTcoinc

.

(4.5)

No downscale factor was applied for particle-γ (S800-CAESAR) coincidence events. The

livetime for coincidences LTcoinc is given by:

LTcoinc =

Ncoinc

Coinc.T riggers

,

(4.6)

where Ncoinc is the number of coincidence events recorded by the S800 trigger register and

Coinc.T riggers is the total number of events meeting the coincidence trigger condition

recorded by the scaler module, which is not affected by the deadtime.

The average livetime for particle triggers in the S800 magnetic spectrograph LTS800

and the average livetime for particle-γ coincidences LTcoinc for each setting, along with the

number of beam particles for the sulfur isotopes of interest corrected for S800 livetime and

the downscaler factor of 20 (Nbeam), are provided in Table 4.1. The quoted uncertainties

are statistical. The efficiency of both CRDCs combined relative to the ionization chamber

was approximately 99% for all settings.

Each time a particle triggers the E1 scintillator in the S800 focal plane, a coincidence

window between particles in the S800 and γ rays in CAESAR is opened. In order to reduce

background from random coincidences measured in CAESAR in the Doppler-corrected γ-

ray spectra, prompt time-energy cuts were utilized. An example of the two-dimensional

66

Table 4.1: Corrected number of sulfur isotopes Nbeam and average livetimes for particle
triggers in the S800 LTS800 and S800-CAESAR coincidences LTcoinc for each of the settings.

Setting

38S
40S
42S
43S
44S

Nbeam

4.522(2) × 108
4.106(2) × 108
1.0463(3) × 109
7.343(3) × 108
2.934(2) × 108

LTS800
0.9159(3)
0.9151(4)
0.8662(2)
0.9237(3)
0.9557(4)

LTcoinc
0.8548(4)
0.8543(4)
0.8454(3)
0.8830(3)
0.9512(5)

prompt time-energy cut used for 40S is provided in Figure 4.1. Applying the time-energy cut

removes background counts without affecting the prompt γ rays from de-excitation following

Coulomb excitation.

The number of full-energy peak counts Nf ep was determined through the use of GEANT4

simulations. As discussed in Subsection 3.3.4, the UCCAESAR GEANT4 simulations take

into account the incoming beam energy, energy loss in the 492 mg/cm2 209Bi target, ab-

sorption of γ rays in the target and beam pipe, the angular distributions of the emitted γ

rays, and the state lifetime. The GEANT4 simulation outputs histograms as ROOT files [82]

that were converted into ROOT TF1 fit functions using the ParamFunctor class. Multiple

TF1s, including a double exponential background model, were then summed together with

individual scaling factors and energy shifts in order to perform a simultaneous fit over the

desired energy range of the spectrum. In addition, UCCAESAR was used to simulate the

effect of the background lines at 511 keV from electron-positron annihilation and 1460 keV

from the β decay of 40K (naturally-occurring background radiation) in the laboratory frame.

These spectra were Doppler-corrected in the same way as the prompt γ-ray simulations and

added to the fit function. In general, the fit functions were of the form:

f (E) = a1 exp(a2E) + a3 exp(a4E) + b1S511 + b2S1460 +

ciSi(E − di).

(4.7)

(cid:88)

i

67

Figure 4.1: Doppler-corrected energy versus corrected CAESAR timing gated on 40S. As
seen from the 904-keV peak, the prompt time-energy cut (red) removes random background
counts without affecting the prompt γ decays that occur after Coulomb excitation.

Here, the first two terms compose the double exponential background and b1 and b2 are

scaling factors for the Doppler-shifted 511 keV and 1460 keV background spectra (S511 and

S1460). The simulations of the observed prompt γ-ray transitions Si are scaled by the factors

ci and are allowed to shift in energy by the parameter di. In the fits, di was constrained to be

less than 10 keV. The values of the parameters are determined by fitting to the experimental

spectrum using the MINUIT fitting package [83] in ROOT. The experimental number of

full-energy peak counts Nf ep is taken as the scaling factor ci times the number of full-energy

peak counts in the GEANT4 simulation and is divided by the livetime for coincidences

LTcoinc and the efficiency to calculate the total number of γ decays Ndecays. After the

68

Timing (channels)0100200300400500600700Energy (keV)010002000300040005000600070008000110210310904 keVCoulomb excitation cross section was determined, the reduced transition strength B(E2)

was calculated using the relativistic Alder-Winther theory with a Mathematica script [63].

4.2 Results for Even-Even Neutron-Rich Sulfur Iso-

topes

The experimental data were analyzed using the GRUTinizer software package built on the

ROOT framework, which is available at [84] and also described in [85]. The results for the

intermediate-energy Coulomb excitation of the even-even neutron-rich isotopes 38,40,42,44S

are described in the following subsections. Extracted B(E2) values are compared to calcula-

tions using the SDPF-MU interaction [25] with effective proton and neutron charges of 1.35

and 0.35, respectively.

4.2.1 Intermediate-Energy Coulomb Excitation of 38S

The Doppler-corrected γ-ray spectrum for 38S measured with CAESAR is provided in Fig-

ure 4.2 with a laboratory-frame scattering angle cut of 40 mrad applied. The low-energy

region of the spectrum is dominated by bremsstrahlung, which is electromagnetic radiation

produced by the acceleration of electrons from the target atoms due to interactions with

the beam, and other beam-correlated background such as target breakup and reaction prod-

ucts interacting with the beampipe. The highest intensity peak corresponds to the known

1 → 0+
2+

1 transition in 38S at 1292.0(2) keV [7]. The energy of the transition measured in this

work is 1289(5) keV. The weak γ-ray peaks at 1513 keV and 2508 keV were included on the

basis of γ-γ coincidence data. Furthermore, Doppler-corrected background from 1460-keV γ

69

rays emitted in the laboratory frame is clearly visible and was included in the fit.

The two-dimensional matrix for events where exactly two γ rays were detected with

CAESAR within the coincidence time window is shown in Figure 4.3. In this plot, nearest-

neighbor addback was employed and no scattering angle cut was applied in order to max-

imize statistics. As seen in the matrix, the 1292-keV and 1513-keV γ decays are in coin-

cidence. In addition, events with both the 1292-keV and 2508-keV transitions are visible.

The background-subtracted projection of this matrix gated on the 1292-keV peak is shown

in Figure 4.4. The background subtraction was performed by gating on a projection of the

same width as the 1292-keV gate but at higher energy and subtracting the resulting spec-

trum from the 1292-keV projection. This plot provides further evidence for the coincidence

relationships inferred from the matrix. The 1513-keV peak has been observed previously and

tentatively assigned as the (2+

2 ) → 2+

1 transition [10, 48, 86–88]. The adopted energy for the

transition is 1513(2) keV [7] and the energy measured here is 1512(9) keV. Its observation

from intermediate-energy Coulomb excitation in this work confirms the 2+

2 assignment. The

uncertainty in energy was derived from the fit uncertainty added in quadrature with a sys-

tematic uncertainty of 4.5 keV [89]. The 2508(25)-keV transition from a level at 3800(25) keV

has not been observed in previous works. The SDPF-MU interaction predicts that the 2+
3

level in 38S is at 3553 keV in excitation energy and decays with a relative branching ratios

of 100% to the 2+

1 , 18% to the ground state and 18% to the 2+

2 level. This is consistent with

the 2508-1292 keV cascade from a 3800-keV level established here. Due to low statistics,

any additional γ-decay branches from the 3800-keV state were not observed.

The nominal maximum safe scattering angle is 66 mrad in the center-of-mass frame or

70

Figure 4.2: Top: Doppler-corrected energy spectrum for 38S gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the individual compo-
nents of the fit function derived from GEANT4 simulations along with a double exponential
background. The red curve is the total fit function. Bottom: Low-energy portion of the
Doppler-corrected energy spectrum for 38S. The large background at low energies is due to
bremsstrahlung and other beam-correlated background.

71

100015002000250030003500Energy (keV)0200400600800Counts / 8 keV0200400600Energy (keV)103104105106Counts / 8 keVv/c = 0.378 40 mrad lab38S Coulex12921513bg from 1460250838S Coulexv/c = 0.378 40 mrad labbremsstrahlungand other beam-correlatedbackgroundFigure 4.3: Doppler-corrected coincidence matrix for 38S events with exactly two γ rays
detected in CAESAR. Both the x and y axes have 16-keV wide bins. The matrix is filled
symmetrically. The 1292-keV and 1513-keV transitions are in coincidence and the 1292-keV
and 2508-keV transitions are in coincidence.

55 mrad in the laboratory frame. This corresponds to a minimum impact parameter of:

(cid:17)

fm = 13.16 fm,

(4.8)

(cid:16)

bmin =

1.2 · (209)1/3 + 1.2 · (38)1/3 + 2

using the touching spheres plus 2 fm approximation [57, 66]. The maximum center-of-mass

maximum scattering angle θmax was calculated from the minimum impact parameter bmin

with the rescaling factor of Equation 2.11:

bmin =

a0
γ

cot

(cid:18) θmax

(cid:19)

2

72

+

π
2

a0
γ

.

(4.9)

Energy (keV)1000150020002500300035004000Energy (keV)1000150020002500300035004000012345678910Figure 4.4: Background-subtracted, Doppler-corrected γ-ray spectrum for 38S counts in
coincidence with the 1292-keV transition. The 1513-keV and 2508-keV peaks are visible.

The maximum scattering angle was then converted from the center-of-mass frame to the

laboratory frame using the relativistic kinematics calculator in LISE++ [72].

It is known that the angular emittance of the beam and angular straggling in the target

can necessitate a more restrictive choice for the maximum scattering angle [81, 90]. Figure 4.5

shows the number of background-subtracted counts in the 1292-keV full-energy peak as a

function of scattering angle in the laboratory frame. As shown here, the angle-integrated

cross section would be underestimated for a scattering angle cut of 55 mrad. The angle-

integrated cross section for the 1292-keV transition as a function of scattering angle cut

is shown in the top panel Figure 4.6. Here, the cross section has been corrected for the

feeding contributions of the 1513-keV and 2508-keV transitions. Furthermore, GEANT4

73

1000200030004000Energy (keV)0204060Counts / 32 keVv/c = 0.378 38S Coulex15132508Gated on 1292 keVsimulations were used to take into account the small changes in detection efficiency with

choice of scattering angle cut due to the anisotropic angular distribution of the emitted

γ rays. The corresponding B(E2; 0+

1 → 2+

1 ) strengths calculated using the semiclassical

approach of Alder and Winther are shown in the bottom panel of Figure 4.6. Up until about

40 mrad, the extracted B(E2; 0+

1 → 2+

1 ) values are rather flat while for larger scattering

angle cuts, some cross section is lost consistent with Figure 4.5.

Figure 4.5: Counts in the 1292-keV peak as a function of scattering angle in the laboratory
frame. Using the nominal safe scattering angle of 55 mrad would lead to an underestimation
of the angle-integrated cross section. A maximum safe scattering angle of 40 mrad was
considered in the analysis.

Using a maximum scattering angle of 40 mrad in the laboratory frame, the B(E2; 0+

2+
1 ) strength was determined to be 229(19) e2fm4. Similarly, the B(E2; 0+
was found to be 21(8) e2fm4 and the B(E2; 0+

1 → 2+

1 →
2 ) strength
3 ) strength was found to be 11(8) e2fm4

1 → 2+

74

020406080100Laboratory Frame Scattering Angle (mrad)0200400600Counts / 2 mradGated on 1292 keV38S Coulex38S Coulexv/c = 0.378for the same maximum scattering angle. It should be noted that the 1513-keV 2+

2 → 2+
1 decays have contributions from both M 1 and E2 multipoles. Since

and 2508-keV 2+

3 → 2+

1

the multipole mixing ratio affects the angular distribution of the emitted γ rays, there

is an effect on the detection efficiency of the array. The quoted B(E2) strengths were

extracted using the predicted multipole mixing ratios from the SDPF-MU interaction. From

GEANT4 simulations of CAESAR for the 1513-keV 2+

2 → 2+

1 transition, the difference in

efficiency using the angular distribution predicted from the shell-model multipole mixing

ratio compared to assuming a pure E2 transition is about 0.1%. The SDPF-MU multipole

mixing ratios were used to determine the angular distributions for all mixed transitions in

this work.

The literature value for the B(E2; 0+

1 → 2+

1 ) strength from the previous intermediate-

energy Coulomb excitation experiment by Scheit et al. is 235(30) e2fm4 [28]. The 1513-keV

and 2508-keV feeding transitions were not observed by Scheit et al. due to limited statistics.

If feeding is neglected in this analysis, the extracted B(E2; 0+

1 → 2+

1 ) strength is 261 e2fm4,

which is also consistent with the previous value [28].

75

Figure 4.6: Top: feeding-subtracted angle-integrated cross section as a function of choice of
scattering angle cut derived from the experimental data for the 2+
1 state in 38S (black). The
1 → 2+
red curve is the expected cross section assuming a B(E2; 0+
1 ) strength of 229 e2fm4.
Bottom: B(E2; 0+
1 ) strength calculated using Alder-Winther theory for different
choices of maximum scattering angle.

1 → 2+

76

020406080Cross Section (mb)ExperimentB(E2) = 229 e2fm40102030405060Laboratory Frame Scattering Angle (mrad)150200250300350B(E2) (e2fm4)38S 1292 keV4.2.2 Intermediate-Energy Coulomb Excitation of 40S

The Doppler-corrected γ-ray spectrum for 40S measured with CAESAR is shown in Fig-

ure 4.7. Here, a laboratory-frame scattering angle cut of 40 mrad has been applied. The

known 2+

1 → 0+

1 transition in 40S at 903.68(9) keV [7] is clearly visible. The energy of the

transition measured in this work is 902(5) keV. In addition, there is a weak transition at

2452(23) keV. The 2452-keV transition is more clearly observed in the two-dimensional γ-γ

coincidence matrix shown in Figure 4.8. As for 38S, nearest-neighbor addback was utilized

and no scattering angle cut was applied for this plot to maximize statistics. Figure 4.9 shows

that the 2452-keV peak is also clearly seen in the background-subtracted projection of the

coincidence matrix gated on the 904-keV transition. From this coincidence relationship, the

2452(23)-keV transition is inferred to originate from a level at 3356(23) keV.

The angle-integrated cross section for the 904-keV transition, corrected for feeding by the

2452-keV transition, is shown in the top panel Figure 4.6 as a function of scattering angle cut.

The nominal maximum scattering angle in the laboratory frame from the touching spheres

plus 2 fm approximation is 51 mrad. The bottom panel shows the corresponding B(E2; 0+

1 →
1 ) strength adopted in this work is 284(26) e2fm4 using

2+
1 ) values. The B(E2; 0+

1 → 2+

the more conservative 30 mrad laboratory-frame scattering angle cut. Angle cuts beyond

30 mrad begin to yield smaller B(E2; 0+

1 → 2+

1 ) values due to angular emittance of the beam

and angular straggling in the reaction target. The value for the B(E2; 0+

1 → 2+

1 ) strength

reported by Scheit et al. from intermediate-energy Coulomb excitation experiment, in which

no feeding was observed due to limited statistics, is 334(36) e2fm4 [28]. If feeding from the

2452-keV transition is neglected in this work, the extracted B(E2; 0+

1 → 2+

1 ) strength is

305 e2fm4, which is consistent with the previous measurement within uncertainties.

77

SDPF-MU shell-model calculations for 40S predict that the 2+

2 level is at 3285 keV and

2 state in the calculation decays nearly 100% of

that the 2+

3 state is at 3579 keV. The 2+
1 level. Similarly, the 2+

the time to the 2+

2+
1 level and an 8% branch to the ground state. However, the predicted B(E2; 0+
strength is only 2.9 e2fm4 compared to 54 e2fm4 for the B(E2; 0+

3 state decays with a relative 100% branch to the
1 → 2+
2 )
3 ) strength. The

1 → 2+

experimental B(E2; 0+

1 → 2+

x ) strength for the 3356(23)-keV state calculated from the yield

of the 2452-keV transition in this work is 26(17) e2fm4 using the 30 mrad scattering angle

cut. No higher-lying 2+ states beyond the first 2+ are listed in the current data evaluation

[7].

Figure 4.7: Doppler-corrected energy spectrum for 40S gated on scattering angles smaller
than 40 mrad in the laboratory frame. The blue curves are the individual components of the
fit function derived from GEANT4 simulations along with a double exponential background.
The red curve is the total fit function.

78

500100015002000250030003500Energy (keV)0200400600800100012001400Counts / 8 keVv/c = 0.380 40 mrad lab40S Coulex904bg from 14602452Figure 4.8: Doppler-corrected coincidence matrix for 40S events with exactly two γ rays
detected in CAESAR. Both the x and y axes have 16-keV wide bins. The matrix is filled
symmetrically. The 904-keV and 2452-keV transitions are in coincidence.

79

Energy (keV)50010001500200025003000Energy (keV)50010001500200025003000024681012141618Figure 4.9: Background-subtracted, Doppler-corrected γ-ray spectrum for 40S counts in
coincidence with the 904-keV transition. A peak at 2452 keV is visible.

80

1000200030004000Energy (keV)010203040506070Counts / 32 keVv/c = 0.380 40S Coulex2452Gated on 904 keVFigure 4.10: Top: feeding-subtracted angle-integrated cross section as a function of choice of
scattering angle cut derived from the experimental data for the 2+
1 state in 40S (black). The
1 → 2+
red curve is the expected cross section assuming a B(E2; 0+
1 ) strength of 284 e2fm4.
Bottom: B(E2; 0+
1 ) strength calculated using Alder-Winther theory for different
choices of maximum scattering angle.

1 → 2+

81

020406080Cross Section (mb)ExperimentB(E2) = 284 e2fm401020304050Laboratory Frame Scattering Angle (mrad)200250300350400B(E2) (e2fm4)40S 904 keV4.2.3 Intermediate-Energy Coulomb Excitation of 42S

Figure 4.11 shows the non-addback Doppler-corrected γ-ray spectrum for 42S measured with

CAESAR using a scattering angle cut of 40 mrad in the laboratory frame. An intense peak

corresponding to the 2+

1 → 0+

1 transition reported at 903(5) keV [7] is clearly visible. The

energy of the transition extracted from the fit performed in this experiment is 904(5) keV.

Figure 4.11: Doppler-corrected energy spectrum for 42S gated on scattering angles smaller
than 40 mrad in the laboratory frame. The blue curves are the individual components of the
fit function derived from GEANT4 simulations along with a double exponential background.
The red curve is the total fit function.

Lunderberg et al. observed a 3002(4) keV γ ray in 42S and tentatively associated this

transition with the decay from the (2+

2 ) state to the ground state [10]. This weak transition

can be seen at 3005(13) keV in the nearest-neighbor addback spectrum gated on multiplicity

1 events shown in Figure 4.12. For an array like CAESAR covering nearly the full solid angle,

82

1000200030004000Energy (keV)0100020003000Counts / 8 keVv/c = 0.391 40 mrad lab42S Coulex903bg from 1460300241502100the relative enhancement of the 3002-keV peak for events where only one γ ray was detected

(a so-called multiplicity one event) compared to all events is indicative of a direct transition

to ground state. The observation of the 3002-keV transition in intermediate-energy Coulomb

excitation in this work confirms the assignment by Lunderberg et al. rather than the proposal

for the (2+

2 ) level lying at 2779 keV and decaying predominantly to the 2+

1 state put forth

by Sohler et al. [91].

Figure 4.12: Doppler-corrected energy spectrum for 42S for events where only one γ ray
was detected by CAESAR. Nearest-neighbor addback was utilized. The 3000-keV transition
from the 2+

2 state to the ground state is visible.

From [10], the 3002-keV level has relative decay branches of 100(2)% to the ground state

and 18(2)% to the 2+

1 state via a 2100(4)-keV γ-ray transition. The 2100-keV transition

can be seen in the two-dimensional γ-γ coincidence matrix shown in Figure 4.13, along with

83

1000200030004000Energy (keV)101001000Counts / 8 keVv/c = 0.391 Addback mult 142S Coulex903bg from 14603002a broad feature around 4150(110) keV. The projection of this matrix gated on the 903-

keV peak is provided in Figure 4.14, showing the same peaks. The energy of the 2100-keV

transition measured in this work is 2117(31) keV, consistent with the more precise value of

2100(4) keV from [10]. Due to the feature from Doppler-corrected 1460-keV background,

the relative intensities of the 3002-keV and 2100-keV transitions were fixed to the known

efficiency-corrected branching ratio from [10] in the fit of Figure 4.11 and in the similar fits

performed using different scattering angle cuts.

Figure 4.13: Doppler-corrected coincidence matrix for 42S events with exactly two γ rays
detected in CAESAR. Both the x and y axes have 16-keV wide bins. The matrix is filled
symmetrically. The 903-keV transition is in coincidence with peaks at 2100 keV (more clearly
observed in the projection of Figure 4.14) and 4150 keV.

Figure 4.15 shows the angle-integrated cross section for the 903-keV transition as a func-

tion of scattering angle cut. The cross sections have been corrected for feeding from the

84

Energy (keV)10002000300040005000Energy (keV)1000200030004000500005101520253035Figure 4.14: Background-subtracted, Doppler-corrected γ-ray spectrum for 42S counts in
coincidence with the 903-keV transition.

2100-keV and 4150-keV transitions. Unlike for 38S and 40S, this plot shows a clear and sig-

nificant dependence of extracted B(E2; 0+

1 → 2+

1 ) strength on laboratory-frame scattering

angle cut for all choices of maximum scattering angle. A method similar in principle to the

CRDC scaling method introduced in [92, 93] was developed to correct this dependence. The

shape of the theoretical cross section curve as a function of maximum scattering angle is

fixed by the A and Z of the target and projectile, multipolarity of the transition, energy of

the state being excited, and velocity of the beam at mid-target. Therefore, the experimental

cross section curve should have the same shape over the range of scattering angles smaller

than the scattering angle at which the angular straggling begins to affect the data, as seen

in Figure 4.6 for 38S and Figure 4.10 for 40S. The B(E2; 0+

1 → 2+

1 ) strength is propor-

85

10002000300040005000Energy (keV)0204060Counts / 32 keVv/c = 0.391 42S Coulex41502100Gated on 903 keVtional to the scaling factor of the curve. The theoretical cross section was parameterized as

atθ2 + btθ + ct where θ is the maximum scattering angle. Similarly, the experimental data

were parameterized as aeθ2 + beθ + ce for experimental laboratory-frame scattering angles

from 10 mrad to 35 mrad. A quadratic correction factor was then applied to the experimen-
tal scattering angles where the value of this factor was chosen to minimize (at/bt − ae/be)2.

Since the minimization is based on the ratios at/bt and ae/be, the shape of the experimental

curve is adjusted to match the shape of the theoretical curve without matching the absolute

scaling. Therefore, the B(E2; 0+

1 → 2+

1 ) strength is not assumed beforehand in the cor-

rection procedure. The best fit scaling factor for the theoretical curve after correcting the

experimental data corresponds to a B(E2; 0+

1 → 2+

1 ) strength of 326 e2fm4.

The 42S data were taken in two separate sets of beam time, one in the middle of the

experiment and one at the end of the experiment. The data from the first set and the

second set show the same correlation between B(E2; 0+

1 → 2+

2 ) strength and laboratory

scattering angle shown in Figure 4.15. For example, the cross sections found for the 40 mrad

laboratory-frame scattering angle cut from the first and second data sets are 48.4 mb and

48.2 mb, respectively. The experimental data for each of the neutron-rich sulfur isotopes was

taken in the following order: 38S, 40S, 42S (first half), 43S, 44S, and 42S (second half). As

seen in Section 4.3 and the following subsection, the 43S and 44S data do not clearly show

the same correlation between choice of scattering angle cut and B(E2; 0+

1 → 2+

2 ) strength

observed for 42S. Therefore, the issue did not appear at a certain time and persist for the

remainder of the experiment. Based on this, it is hypothesized that the problem is with the

scattering-angle reconstruction on an incoming beam basis rather than a scattering-angle-

dependent loss in γ-ray detection efficiency.

The B(E2; 0+

1 → 2+

1 ) in 42S measured in intermediate-energy Coulomb excitation by

86

Figure 4.15: Top: feeding-subtracted angle-integrated cross section as a function of choice of
scattering angle cut derived from the experimental data for the 2+
1 state in 42S (black). The
1 → 2+
red curve is the expected cross section assuming a B(E2; 0+
1 ) strength of 326 e2fm4.
The blue data points are the experimental data with the correction for scattering angle
explained in the text. Bottom: B(E2; 0+
1 ) strength calculated using Alder-Winther
theory for different choices of maximum scattering angle.

1 → 2+

87

020406080100120Cross Section (mb)ExperimentExperiment (corrected)B(E2) = 326 e2fm40102030405060Laboratory Frame Scattering Angle (mrad)150200250300350400450B(E2) (e2fm4)42S 903 keVScheit et al. was 397(63) e2fm4 [28]. The lifetime of the 2+

1 state in 42S has also been
measured using the recoil-distance method to be 20.6(15) ps [45] and 21.5+1.1−0.9 ps [53]. Using

Equation 2.34, Equation 1.18, and an excitation energy of 903(5) keV, the corresponding

1 → 2+

B(E2; 0+

1 ) strengths are 329(26) e2fm4 from [45] and 315+18−16 e2fm4 from [53]. The
value of 326 e2fm4 found after applying the correction to the scattering angle is in good
1 →
2+
1 ) strength was estimated by varying the number of experimental data points included in

agreement with these previous measurements. The uncertainty in the extracted B(E2; 0+

the fit and using both linear and quadratic correction factors. Including an additional 10%

systematic uncertainty, the adopted value is 326(48) e2fm4.

The Coulomb excitation cross section to the 3002-keV state was found using the 35 mrad

laboratory-frame scattering angle cut from the yields of the detected 2100-keV and 3002-keV

de-excitation γ rays. The B(E2; 0+

1 → 2+

2 ) strength was calculated using the same corrected

scattering angle used to extract the adopted result for the B(E2; 0+

1 → 2+

1 ) strength to be

27(14) e2fm4. SDPF-MU shell-model calculations predict a B(E2; 0+

1 → 2+

5 ) strength of

16 e2fm4 for the level at 5003 keV. The predicted reduced transition strengths to the 2+

3 and

2+
4 states are less than 0.001 e2fm4 and about 4 e2fm4, respectively. The 2+

4 level at 4924 keV

decays with a relative branching ratios of 100% to the ground state and 54% branching ratio

to the 2+

1 state (with smaller branching ratios to other states) while the 2+

5 level decays

with relative branching ratios of 100% to the 2+

1 state and 11% to the ground state in the

calculation. Employing the same method used for the 3002-keV level, the reduced transition

strength to the level at 5053(110) keV that decays via a 4150(110)-keV γ ray to the 903-keV

2+
1 state was calculated to be 11(9) e2fm4 assuming an E2 excitation from the data using

the 35 mrad scattering angle cut. Lunderberg et al. report γ-ray transitions at 4102(8) keV

and 4266(7) keV that were not placed in the level scheme [10]. Due to its large energy

88

uncertainty, the 4150(110)-keV feature may correspond to one or a combination of these

transitions.

4.2.4 Intermediate-Energy Coulomb Excitation of 44S

The Doppler-corrected γ-ray spectrum for 44S measured with CAESAR using a laboratory-

frame scattering angle cut of 40 mrad is shown in Figure 4.16. The previously-observed

1329.0(5)-keV γ ray corresponds to the 2+

1 → 0+

1 transition [49]. Other works report the en-

ergy of this transition as 1297(18) keV [29], 1350(10) keV [91], 1319(7) keV [44], 1321(10) keV

[94], and 1320(8) keV [95]. The energy of the transition measured in this work is 1324(6) keV.

The 949(5)-keV 2+

3 → 2+

1 transition reported in the literature [7] is more clearly observed

in the two-dimensional γ-γ coincidence matrix for events with exactly two detected γ rays

shown in Figure 4.17. The background-subtracted projection of this matrix gated on the

1329-keV peak is shown in Figure 4.18. The energy for the 2+

3 → 2+

1 transition measured

in this work is 941(19) keV. This transition has also been reported recently at an energy

of 954(4) keV [96]. SDPF-MU predicts that the 2+

3 state decays with relative branching

ratios of 100% to the 2+

1 level and 14% to the ground state. Due to low statistics, the

possible ground state branch was not observed in this experiment or in previous works.

The 2150(11)-keV transition reported in the literature is tentatively assigned as the γ de-

cay of the 2+

2 state to the ground state [44, 45]. Due to the Doppler-corrected 1460-keV

laboratory-frame background, the possible presence of this transition in this work is not well

constrained.

The angle-integrated cross section for the 1329-keV transition, corrected for feeding by

the 949-keV transition, is shown in the top panel Figure 4.19 as a function of the choice

of maximum scattering angle. As seen in the bottom panel of Figure 4.19, the correspond-

89

Figure 4.16: Doppler-corrected energy spectrum for 44S gated on scattering angles smaller
than 40 mrad in the laboratory frame. The blue curves are the individual components of the
fit function derived from GEANT4 simulations along with a double exponential background.
The red curve is the total fit function.

ing B(E2; 0+

1 → 2+

1 ) strength does not decrease with scattering angle cut with the same

large slope observed for 42S. The B(E2; 0+

1 → 2+

1 ) strength extracted using the 35 mrad

laboratory-frame scattering angle cut is 221(28) e2fm4. The nominal maximum scatter-

ing angle in the laboratory frame from the touching spheres plus 2 fm approximation is

48 mrad. Using the same angle cut and the yield of the 949-keV 2+

3 → 2+

1 transition, the

B(E2; 0+

1 → 2+
The B(E2; 0+

3 ) strength was determined to be 10(6) e2fm4.
1 → 2+

1 ) strength reported from intermediate-energy Coulomb excitation

by Glasmacher et al. is 314(88) e2fm4 [29] which overlaps with the value in this work of

221(28) e2fm4 within mutual uncertainties. No feeding transitions were observed in the

90

100020003000Energy (keV)0200400600800Counts / 8 keVv/c = 0.37440 mrad lab44S Coulex949bg from 1460bg from 5111329work of Glasmacher et al. [29]. If feeding is neglected in this work, the B(E2; 0+

1 → 2+
1 )

value increases to 231 e2fm4, slightly improving the agreement.

Figure 4.17: Doppler-corrected coincidence matrix for 44S events with exactly two γ rays
detected in CAESAR. Both the x and y axes have 16-keV wide bins. The matrix is filled
symmetrically. The 1329-keV transition is in coincidence with the 949-keV transition.

91

Energy (keV)50010001500200025003000Energy (keV)500100015002000250030000246810121416Figure 4.18: Background-subtracted, Doppler-corrected γ-ray spectrum for 44S counts in
coincidence with the 1329-keV transition.

92

100020003000Energy (keV)01530Counts / 16 keVv/c = 0.37444S Coulex949Gated on 1329 keVFigure 4.19: Top: feeding-subtracted angle-integrated cross section as a function of choice of
scattering angle cut derived from the experimental data for the 2+
1 state in 44S (black). The
1 → 2+
red curve is the expected cross section assuming a B(E2; 0+
1 ) strength of 221 e2fm4.
Bottom: B(E2; 0+
1 ) strength calculated using Alder-Winther theory for different
choices of maximum scattering angle.

1 → 2+

93

020406080Cross Section (mb)ExperimentB(E2) = 221 e2fm401020304050Laboratory Frame Scattering Angle (mrad)150200250300350400B(E2) (e2fm4)44S 1329 keV4.3 Results for 43S

4.3.1 43S Incoming Beam Isomeric Content

The production of the 43S secondary beam at the NSCL is known to leave a fraction of 43S
nuclei in the isomeric 7/2− state at 320 keV. Since this level has a half-life of 415 ns [50, 51],

some 43S nuclei will still be in this excited when they reach the 209Bi reaction target at

the center of CAESAR. Two methods were used to determine the isomeric content of the

incoming 43S secondary beam. In the first method, the 43S beam was stopped at the center

of CAESAR with a 5.1-mm thick Al stopper. For this measurement, the trigger for the

data acquisition system was switched from the E1 scintillator in the S800 focal plane to the

object scintillator in the beamline, which counts the number of incoming beam particles.

The proportion of 43S projectiles in the cocktail incoming beam was determined from the

particle identification plots in subsequent runs taken under the typical experimental settings.

The efficiency of CAESAR for detecting 320-keV γ rays from the decay of the isomeric

state in 43S was interpolated using the 303-keV and 356-keV lines from a 133Ba source. In

addition, GEANT4 simulations were performed to estimate the reduction in efficiency due

to absorption in the 5.1-mm Al stopper. A reduction factor of 0.82 was calculated using the

ratio of the CAESAR GEANT4 efficiency at 320 keV assuming the γ decay occurs in the

center of the Al stopper to the CAESAR GEANT4 efficiency at 320 keV without a stopper.

From the efficiency-corrected number of 320-keV γ rays from the decay of the isomer in

43S divided by the number of incoming 43S projectiles, the proportion of 43S nuclei in the

isomeric state at the center of CAESAR was found to be 14(3)%.

In the second method, the CsI(Na) hodoscope at the back of the S800 focal plane was uti-

lized to determine the isomeric content of the incoming 43S beam. In this setup, no reaction

94

target or stopper was installed at the center of CAESAR. Since no reactions occur without

the target, the composition of the incoming beam was determined using particle identifi-

cation from the energy loss in the S800 ionization chamber and the time-of-flight from the

object scintillator to the E1 scintillator. The efficiency of the hodoscope was estimated using

a GEANT4 simulation of the IsoTagger configuration which includes the Al stopping plate

before the CsI(Na) detectors and the surrounding stainless steel vacuum chamber [77]. The

extracted efficiency at 320 keV of 19(3)% is similar to the value measured by Momiyama

et al. from 43S produced from one-neutron knockout [15]. Dividing the efficiency-corrected

number of 320-keV γ rays measured in the hodoscope by the number of incoming 43S pro-

jectiles gives the isomeric content at the Al stopper of IsoTagger. In order to determine the

isomeric content at the reaction target, the half-life of the isomeric state must be considered.

For the no reaction target setting, the average velocity of the 43S nuclei is 0.396c. Therefore,

using a half-life of 415 ns and a distance of 15 m between the reaction target and the Al

stopper of IsoTagger, 81% of the 43S nuclei survive the flight from the reaction target to the

Al stopper in the S800 focal plane. Taking this into account gives an isomeric content of

18(4)% for 43S at the reaction target.

The average of the isomeric content measurements is 16(5)% where the difference in the

results using the two different methods has been adopted as the systematic uncertainty.

The distance from the 9Be production target to the A1900 focal plane is about 35.5 m

[97] and the distance from the A1900 focal plane to the S800 focal plane is about 58 m

[98]. Using approximately 15 m for the distance between the reaction target and the S800

focal plane, the distance from the production target to the reaction target is about 78.5 m.

Assuming a half-life of 415 ns, a velocity of 0.396c, and 16% of 43S in the isomeric state at

the reaction target, the initial isomeric ratio at the production target is about 0.5. Therfore,

95

the fragmentation of the 48Ca primary beam leaves 43S in the isomeric state or states that

feed the isomeric state about half of the time. For comparison, the fraction of the cross
sections from one-neutron knockout to the isomeric 7/2− state in 43S and to the state at

1852 keV that γ decays to the isomer to the inclusive cross section is also about 0.5 [15].

4.3.2 Intermediate-Energy Coulomb Excitation of 43S

The γ-ray spectrum from CAESAR for 43S after Doppler reconstruction is shown in Fig-

ure 4.20 using a laboratory-frame scattering angle cut of 40 mrad. The higher-energy region

of the spectrum is shown in the top panel of Figure 4.20 while the lower-energy region is pro-

vided in the bottom panel. The highest-statistics peak observed in this work is at 979(6) keV.

In previous experiments, this transition has been reported at 971(6) keV [52], 977(9) keV

[53], and 977(3) keV [15]. In [53], this transition was inferred to feed a level at 184(2) keV

from lifetime measurements. However, it was noted that the observed intensities of the

184-keV and 977-keV γ rays were equal within uncertainties [53], as was the case in [52].

In the recent one-neutron knockout work, the efficiency-corrected intensity of the 977-keV

transition was found to be larger than the intensity of the 185(1)-keV peak, establishing the

977-keV γ decay as a ground state transition and the 185-keV γ decay as originating from a

state at 1162 keV [15]. As seen in the bottom panel of Figure 4.20, the 185-keV transition

is not observed in this work, providing clear evidence for the level scheme proposed in [15].

For comparison, the Doppler-corrected γ-ray spectrum for 43S produced from one-neutron

removal measured in this experiment during the 44S setting is shown in Figure 4.21. In this

plot, the 185-keV transition, which is of similar efficiency-corrected intensity as the 977-keV

transition, is clearly visible. This figure shows events where either one or two γ rays were

detected in CAESAR after nearest-neighbor addback. The bottom panel of Figure 4.20 also

96

shows a wide feature resulting from the in-flight decay of the 320-keV isomer. Since the

γ decay of the isomer can occur at any point along the beam line, Doppler reconstruction

assuming the decay occurs at the middle of the reaction target does not result in a peak at

320 keV.

The top panel of Figure 4.20 also shows weaker peaks at 1168(27) keV, 1219(22) keV, and

1525(10) keV. The energies of these transitions were measured at 1154(7) keV, 1203(7) keV,

and 1529(9) keV in [52], at 1159(9) keV, 1213(10) keV, and 1543(13) keV in [53], and at

1155(3) keV, 1209(3) keV, and 1532(4) keV in [15]. The 1155-keV γ ray has been placed in

the level scheme for 43S as a transition to the ground state [15, 52, 53], as has the 1209-keV

transition [15].

In this work, no γ-γ coincidence relationships were observed in the data

collected with CAESAR, in agreement with the previous placements of transitions. The

prompt 1532-keV transition has been measured in coincidence with the isomeric 320-keV

decay in the hodoscope [15]. In this work, additional evidence consistent with the 1532-keV

transition feeding the 320-keV isomer is seen in Figure 4.22, which shows the CAESAR

spectrum for multiplicity one events after nearest-neighbor addback in coincidence with the

320-keV peak measured in the hodoscope with background subtraction applied.

The angle-integrated cross section for the 977-keV transition is shown in the top panel

Figure 4.23 as a function of maximum scattering angle. As for 38,40,44S, the extracted B(E2)

strength does not decrease with scattering angle cut in the same manner observed for 42S.

The nominal safe scattering angle in the laboratory frame from the touching spheres plus

2 fm approximation is about 47 mrad. Using the more restrictive choice of 40 mrad, the
B(E2) strength from the 3/2− 43S ground state to the level at 977 keV is 91(18) e2fm4. With

the same choice for the maximum scattering angle, the B(E2) strengths to the 1155-keV and

1209-keV states are 9(7) e2fm4 and 23(11) e2fm4, respectively. Similarly, the B(E2) strength

97

Figure 4.20: Top: Doppler-corrected energy spectrum for 43S gated on scattering angles
smaller than 40 mrad in the laboratory frame. The blue curves are the individual compo-
nents of the fit function derived from GEANT4 simulations along with a double exponential
background. The red curve is the total fit function. Bottom: Low-energy portion of the
Doppler-corrected energy spectrum for 43S. The wide feature present in the 43S data that is
not present for 38S is due to the continuous decay of the 320-keV isomer at different points
along the beam line. The large background at low energies is due to bremsstrahlung and
other beam-correlated background.

98

100015002000250030003500Energy (keV)020040060080010001200Counts / 8 keV0200400600Energy (keV)103104105106Counts / 8 keVv/c = 0.382 40 mrad lab43S Coulex9771155bg from 1460153243S Coulexv/c = 0.382 40 mrad lab38S Coulex scaledin-flight isomer decay1209Figure 4.21: Doppler-corrected energy spectrum for 43S produced from one-neutron removal
for addback events with exactly one or two detected γ rays. The labeled peaks were also
observed in the high-resolution 9Be(44S,43S)X data presented in [15].

from the 7/2− isomeric state at 320 keV to the level at 1852 keV that decays via a 1532-keV

γ-ray transition was found to be 101(42) e2fm4. When calculating these electric quadrupole

transition strengths, the numbers of incoming 43S nuclei in the ground and isomeric states

were utilized based on the proportion calculated in the previous subsection.

B(E2) strengths for low-lying states in 43S predicted using the SDPF-MU interaction

are shown in Figure 4.24 in comparison with the results found in this work. Based on the

intensity of the 977-keV transition observed experimentally, the 977-keV level is assigned
as the 7/2−
2 state, establishing it as a member of the rotational band built on top of the
3/2− ground state. Furthermore, the 1155-keV and 1209-keV states are most similar to the

99

0100020003000Energy (keV)020406080100120140Counts / 8 keVv/c = 0.380209Bi(44S,43S)X115526001209977625185Figure 4.22: Background-subtracted, Doppler-corrected energy spectrum for 43S measured
with CAESAR in coincidence with 320-keV γ rays detected on the hodoscope at the S800
focal plane from the decay of the isomer. The CAESAR spectrum is for addback multiplicity
one.

shell-model 3/2−

2 and 5/2−

1 levels. Note that SDPF-MU predicts that the 3/2−

2 and 5/2−

1

states decay predominantly to the ground state (100%) with only 20% and 22% relative

branching ratios to all other states, respectively. The second largest brancing ratios are to
the low-lying 1/2− state. Experimentally, this state is tentatively reported at 228 keV [15].

Since these other decay branches would result in γ rays at 927 keV and 981 keV, a portion

of the B(E2) strength to the 1155-keV and 1209-keV states may be misattributed to the

977-keV state. The 228-keV transition was not observed in this experiment. Finally, the
1852-keV state is assigned as the 9/2−

1 level built on top of the isomeric 7/2−

1 state.

In

the shell-model calculations, the B(E2) strength is 137 e2fm4, in good agreement with the

100

500100015002000250030003500Energy (keV)020406080Counts / 16 keVv/c = 0.38243S Coulex1532CAESARAddback mult 1Gated on 320 keV    in HodoscopeFigure 4.23: Top: feeding-subtracted angle-integrated cross section as a function of choice of
scattering angle cut derived from the experimental data for the 977-keV state in 43S (black).
The red curve is the expected cross section assuming a B(E2) strength of 91 e2fm4 from the
ground state to the 977-keV level. Bottom: B(E2) strength calculated using Alder-Winther
theory for different choices of maximum scattering angle.

101

010203040Cross Section (mb)ExperimentB(E2) = 91 e2fm401020304050Laboratory Frame Scattering Angle (mrad)50100150B(E2) (e2fm4)43S 977 keVmeasured value of 101(42) e2fm4. In [15], a peak at 1865(6) keV was observed with 17%

intensity relative to the 1532-keV transition and it was suggested that this γ ray could be

a transition from the 1852-keV state to the ground state based solely on its energy. Due
to the large spin difference between the 3/2− ground state and the 9/2− level at 1852 keV

established in the present work, this 1865-keV transition is strongly suggested to originate

from a different level.

Figure 4.24: SDPF-MU predictions for B(E2) strengths in e2fm4 for low-lying levels in 43S
compared with experimental results. Arrow widths are proportional to transition strength.

In a previous intermediate-energy Coulomb excitation experiment, the B(E2) strength

to a level around 940 keV in 43S was measured to be 175(69) e2fm4 [99] which is almost a

factor of two larger than the value of 91(18) e2fm4 reported to the later-observed level at

977(3) keV [15], measured in this work at 979(6) keV. It was noted by Ibbotson et al. that

the reported 940-keV peak may consist of multiple γ ray transitions within the 8% energy

resolution of the detector array utilized [99]. In this work and in the other in-beam γ-ray

spectroscopy experiments [15, 52, 53], no multiplets in this energy range were observed. The

102

0500100015002000Energy(keV)43SSDPF-MU3/21-1/21-7/21-3/22-7/22-5/21-5/22-1/22-9/21-67171815140.213743SExp3/21-7/21-7/22-(3/22-,5/21-)(5/21-,3/22-)9/21-91(18)9(7)23(11)101(42)T1/2 = 415 nsreported energy for the observed transition in 45Cl was 929(17) keV in [99] which is in very

good agreement with the literature value of 928(6) keV [7]. Suppose the 43S peak reported

at 940 keV in [99] has two components, the 977-keV transition reported in this work and in

the other in-beam γ-ray spectroscopy experiments [15, 52, 53] and a 903-keV transition from

the decay of the 2+

1 state in 42S. The number of incoming 42S particles was reported to be
nearly a factor of 7 larger than the number of incoming 43S particles in [99]. In addition, this

contaminant transition may be present due to one-neutron removal from 43S. The energy

of the 940-keV peak could then be explained as the centroid of the 977-keV and 903-keV

transitions with each transition contributing about half of the counts. Since the B(E2)

strength to the 903-keV state in 42S is larger than the B(E2) strength to the 977-keV state

in 43S, the average number of 903-keV γ rays per incoming particle is higher. Consequently,

the B(E2) strength to the 977-keV level would be overestimated. For a simple estimate of

the scale of this effect, suppose the experimental details for 42S and 43S are identical and

neglect any contribution from one-neutron removal so that the following relation holds:

N903 ∝ B(E2)903 × N42S

= B(E2)977 × N43S

∝ N977.

(4.10)

is the number of incoming 42S projectiles and N43S

Here, N42S
43S projectiles. From above, the number of 903 keV and 977 keV γ rays, N903 and N977,

is the number of incoming

are approximately equal. Using a B(E2) strength to the 903-keV transition of 326 e2fm4,

the measured value for the combined B(E2) strength of 175 e2fm4 [99] is then the weighted

average:

175 =

326 × N42S

+ B(E2)977 × N43S
N42S

+ N43S

.

(4.11)

Solving these relations simultaneously gives a purity of N43S

= 2.73N42S

and a B(E2)977

103

strength of 120 e2fm4, which is much closer to the value extracted in this experiment than

175 e2fm4.

For 45Cl, Ibbotson et al. report the B(E2) strength to the 929(17)-keV state as

87(24) e2fm4 [99]. In this work, incoming 45Cl nuclei were present in the 44S setting. The

Doppler-corrected γ-ray energy spectrum for 45Cl projectiles identified from time-of-flight

and energy loss is shown in Figure 4.25. Here, the energy of the peak was measured as

928(9) keV. The resulting B(E2) strength using a 40 mrad labortory-frame scattering an-

gle cut is 82(29) e2fm4 which is in good agreement with the previous intermediate-energy

Coulomb excitation measurement. The touching spheres plus 2 fm approximation gives a

nominal laboratory-frame safe scattering angle cut of about 47 mrad.

The lifetimes of the 977-keV and 184-keV transitions were measured by Mijatovic et

al. using the recoil-distance method with a plunger device [53]. The best-fit results were

obtained assuming the 977-keV transition feeds the 184-keV transition which is shown in

this work and the work of Momiyama et al. to be incorrect [15]. Under the assumption

that the 184-keV transition feeds the 977-keV transition, a lower-quality fit was obtained

and the lifetime of the 1161-keV state was found to be 15(2) ps while the lifetime of the
977-keV state was inferred to be very short [53]. Assuming the 977-keV state is the 7/2−
level, the decay to the 3/2− ground state will be predominantly E2 and the limit of zero

2

lifetime corresponds to a B(E2) value approaching infinity. Interestingly, if the lifetime of

the 977-keV level is 15 ps instead of the 1161-keV level, the corresponding B(E2) strength
from the 3/2− ground state to the 7/2− 977-keV state would be 122 e2fm4. The lifetime
for the 7/2− → 3/2− γ decay calculated from the B(E2; 3/2− → 7/2−) value of 91 e2fm4

measured in this work is 20 ps.

104

Figure 4.25: Doppler-corrected energy spectrum for 45Cl gated on scattering angles smaller
than 40 mrad in the laboratory frame. The blue curves are the individual components of the
fit function derived from GEANT4 simulations along with a double exponential background.
The red curve is the total fit function.

105

50010001500200025003000Energy (keV)050100150200Counts / 16 keVv/c = 0.386 40 mrad lab45Cl Coulex928bg from 1460bg from511Chapter 5

Discussion

The B(E2) strengths determined for the even-even sulfur isotopes in this work using

intermediate-energy Coulomb excitaion are provided in Table 5.1 along with the values from

the previous intermediate-energy Coulomb excitation experiments [28, 29] and the strengths

predicted by the SDPF-MU interaction. In general, the B(E2; 0+

1 → 2+

1 ) strengths extracted

in this work are slightly lower than those published by Scheit et al. [28] and Glasmacher et

al. [29]. The effects of transitions feeding the 2+

1 states on the reduced transition strengths

were taken into account in this experiment while feeding transitions were not observed in

the previous works due to limited statistics and thus could not be subtracted. Further-

1 → 2+

more, the B(E2; 0+

1 ) strengths in 42S calculated from recent lifetime measurements of
329(26) e2fm4 [45] and 315+18−16 e2fm4[53] are slightly below the value of 397(63) e2fm4 from
Scheit et al. and in good agreement with the value of 326(48) e2fm4 found in the present

work. Figure 5.1 shows a visual comparison of the measured B(E2) strengths and the B(E2)

strengths predicted using the SDPF-MU interaction. Overall, there is very good agreement

between experiment and theory up to and including N = 26.

In 38S, shell-model calculations using the SDPF-MU interaction yield a B(E2; 0+

1 → 2+
1 )
strength of 160 e2fm4, which is slightly lower than the experimental value of 229(19) e2fm4.
1 →
2 ) strength was measured as

2 ) strength of 25 e2fm4 and a B(E2; 0+

In addition, SDPF-MU predicts a B(E2; 0+

2+
3 ) strength of 26 e2fm4.

In this work, the B(E2; 0+

1 → 2+

1 → 2+

106

Figure 5.1: B(E2; 0+
measured in this work compared to predictions from the SDPF-MU Hamiltonian for the first
five 2+ states.

1 → 2+) strengths in the even-even sulfur isotopes for the 2+ states

107

0100200B(E2) (e2fm4)ExpSDPF-MU0150300B(E2) (e2fm4)012345Energy (MeV)0200400B(E2) (e2fm4)0200400B(E2) (e2fm4)38S40S42S44STable 5.1: B(E2) strengths for the neutron-rich sulfur isotopes studied in this work com-
pared to the results of previous intermediate-energy Coulomb excitation experiments and
theoretical calculations using the SDPF-MU Hamiltonian with effective proton and neutron
charges of 1.35 and 0.35, respectively.

Nucleus Excitation This Work Previous Works Theory (SDPF-MU)

e2fm4

235(30) [28]

e2fm4
160

25

26

300

3

54

370

38

16

450

12

11

38S

40S

42S

44S

1

3

1

2

2

1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+
1 → 2+
0+

2

5

3

1

1

2

3

e2fm4
229(19)

21(8)

11(8)

284(26)

334(36) [28]

26(17)

326(48)

397(63) [28]

27(14)

11(9)

221(28)

314(88) [29]

10(6)

21(8) e2fm4 from the 1513-keV 2+

2 → 2+

1 transition while the reduced transition strength

from the ground state to the newly-observed level at 3800(25) keV was found to be

11(8) e2fm4 from the 2508-keV transition to the 2+

1 state. Based on the theoretical cal-

culations, the 3800-keV level is inferred to have structure most similar to the shell-model

2+
3 state. The 2+

2 state at 2805 keV was observed to γ decay to the 2+

1 level but not to the

ground state, consistent with the previous measurements [10, 48] and with the dominance

of the 2+

2 → 2+

1 branch predicted by SDPF-MU.

For 40S, the SDPF-MU Hamiltonian predicts a B(E2; 0+

1 ) strength of 300 e2fm4
compared to the measured value of 284(26) e2fm4. Moreover, the shell-model calculations

1 → 2+

predict a 54 e2fm4 B(E2; 0+

1 → 2+

3 ) strength while the B(E2; 0+

1 → 2+

2 ) strength is only

108

2.9 e2fm4.

In this work, the reduced transition probability from the ground state to the

presumed 2+ level at 3356(23) keV was found to be 26(17) e2fm4. Based on its observation

in this work, the 3356(23)-keV level corresponds more closely to the collective shell-model 2+
3

state. The predicted relative branching ratios for the 2+

8% to the ground state. In this experiment, only the 2+

3 state are 100% to the 2+
3 → 2+

1 branch of the newly-reported

1 level and

3356(23)-keV state was observed, consistent with the theoretical predictions given the level

of statistics. In the single-particle picture discussed in [10] and reproduced in Figure 1.8, the

neutron 1p3/2 occupancy for the ground state is very similar to the neutron 1p3/2 occupancies

for the 2+

1 and 2+

3 states, both of which states were observed in the present work. The SDPF-

MU neutron 1p3/2 occupancy of the unobserved 2+

2 level is only slightly higher but is nearly

identical to the neutron 1p3/2 occupancy of the 0+

2 state, which is predicted to be at nearly

the same excitation energy.

Shell-model calculations for 42S give a B(E2; 0+

1 → 2+

1 ) strength of 370 e2fm4 which

is slightly above the experimental result from this work of 326(48) e2fm4.

In addition,

SDPF-MU predicts a B(E2; 0+

1 → 2+

2 ) strength of 38 e2fm4 compared to the measured

value of 27(14) e2fm4. The shell-model calculations also predict a B(E2; 0+

1 → 2+

5 ) strength

of 16 e2fm4. Assuming an E2 excitation, the reduced transition strength to the level at

5053(110) keV is 11(9) e2fm4. However, the 4150(110)-keV transition from the 5053(110)-

keV state observed in this work may correspond to multiple unresolved transitions, such as

the 4102-keV and 4266-keV transitions reported in [10]. In addition, the branching ratios of

the tentatively assigned 2+

2 level at 3002-keV to the 2+

1 state and the ground state measured

by Lunderberg et al. were in remarkable agreement with the predictions from SDPF-MU

[10]. In this work, the 3002-keV state was confirmed as the 2+

2 level. Overall, the SDPF-

MU Hamiltonian predictions for the energies of 2+ states and E2 strengths in 42S are in

109

very good agreement with the experimental results, as seen in Figure 5.1, supporting the

single-particle evolution sketched with the neutron 1p3/2 occupancy in Figure 1.8.

The SDPF-MU interaction predicts a B(E2; 0+
1 → 2+

In this work, the measured B(E2; 0+

44S.

1 ) strength of 450 e2fm4 for N = 28

1 → 2+
1 ) strength is roughly half of this value

at 221(28) e2fm4. The B(E2; 0+

1 → 2+

2 ) and B(E2; 0+

1 → 2+

3 ) strengths from the shell-

model calculation are 12 e2fm4 and 11 e2fm4, respectively. The experimental value for

the B(E2; 0+

1 → 2+

3 ) strength from the 949-keV transition of the 2278-keV state in this

work is 10(6) e2fm4. The yield of possible 2150-keV transitions from the decay of the

2+
2 level to the ground state was not well constrained due to Doppler-corrected 1460-keV

laboratory-frame background. A similar overprediction of the B(E2; 0+

1 → 2+

1 ) strength by

shell-model calculations has been reported for N = 28 46Ar, just two protons above 44S.

While a lifetime measurement obtained a value of 570+335−160 e2fm4 [100], intermediate-energy
Coulomb excitation experiments have resulted in consistent strengths of 196(39) e2fm4 [28],

218(31) e2fm4 [62], and 225(29) e2fm4 (absoulte) and 234(19) e2fm4 (relative to 44Ca) [101].

Shell-model interactions including SDPF-MU predict a B(E2; 0+

1 → 2+

1 ) strength roughly a

factor of two larger at around 500 e2fm4 [101].

From the experimental results in the present work, the B(E2; 0+

1 → 2+

1 ) strength peaks

at N = 26 for the sulfur isotopes between N = 22 and N = 28. SDPF-MU predictions,

however, put the maximum collectivity at N = 28 rather than N = 26 over this range

1 → 2+

of neutron numbers. For comparison, the SDPF-MU B(E2; 0+

and 46S are nearly the same at about 450 e2fm4 while the SDPF-MU B(E2; 0+

1 ) strengths for 44S
1 → 2+
1 )
strength for 48S is lower at around 330 e2fm4. For the argon isotopic chain, there is a similar
1 →
relationship between the experimental results and shell-model calculations. The B(E2; 0+
1 ) strength in N = 28 46Ar [101] as discussed above and the B(E2) strength from the 3/2−
2+

110

model calculations by roughly a factor of two. At N = 30, the reported B(E2; 0+

ground state to low-lying states for N = 29 47Ar [102] are significantly overpredicted by shell-
1 → 2+
1 )
strength of 346(55) e2fm4 is in better agreement with theoretical calculations but is still

about 1.5 to 2 standard deviations lower than predicted [102]. Consequently, reproducing

the observed B(E2) strengths near N = 28 for argon and sulfur may serve as important

benchmarks for future shell-model interactions.

In 43S, collective structures built on top of the intruder 3/2− ground state and the normal-
order 7/2− isomeric state were characterized. The B(E2) strength to the 977-keV state was

measured to be 91(18) e2fm4 while the largest B(E2) strength predicted by the SDPF-MU
Hamiltonian from the ground state is 181 e2fm4 to the 7/2−

a factor of two is similar for the B(E2; 0+

2 level. The overprediction by
1 ) strengths in 44S and 46Ar discussed

1 → 2+

above. A previous intermediate-energy Coulomb excitation experiment measured a B(E2)

strength of 175(69) e2fm4 using a peak at appropriately 940 keV [99] which may include

some contamination from the 903-keV level in 42S. It would be interesting to revisit the

lifetime measurements performed for states in 43S [53]. In [53], the 184-keV transition was

assumed to be fed by the 977-keV transition and tentatively assigned as originating from the
low-lying 1/2− state. In both this work and [15], the 977-keV transition was found to be to

the ground state. Furthermore, the newly-observed 228-keV level is tentatively assigned as
the 1/2− state in [15]. Levels at 1155 keV and 1209 keV were also Coulomb excited from the

ground state of 43S with B(E2) values of 9(7) e2fm4 and 23(11) e2fm4, respectively. The
SDPF-MU B(E2) strengths from the ground state to the 3/2−

1 states are 17 e2fm4
and 51 e2fm4, respectively. The 1532-keV transition was found to be in coincidence with

2 and 5/2−

the 320-keV isomer in agreement with results from one-neutron knockout [15]. The B(E2)

strength from the 320-keV isomer present in the incoming beam to the 1852-keV level was

111

found to be 101(42) e2fm4. The SDPF-MU Hamiltonian predicts a B(E2) strength from
the 7/2− isomer to the 9/2− state of 137 e2fm4. As discussed in [51], the normal-order
configuration provides the largest contribution to the wavefunction of the 7/2− isomer but

intruder configurations also play a role. For example, the measured spectroscopic quadrupole

moment of 23(3) efm2 is larger than expected for a single-hole state and well-reproduced by
shell-model calculations [51]. Here, the measured B(E2; 7/2−

1 ) strength is also

1 → 9/2−

in agreement with the shell-model prediction, supporting the picture of shape coexistence

painted by the SDPF-MU interaction.

112

Chapter 6

Summary and Outlook

Reduced electromagnetic

transition strengths

in the neutron-rich sulfur

isotopes

38,40,42,43,44S were measured using the technique of intermediate-energy Coulomb excita-

tion with the CsI(Na) scintillator array CAESAR and the S800 magnetic spectrograph at

the National Superconducting Cyclotron Laboratory. For the even-even neutron-rich sulfur

isotopes the B(E2) strengths from the ground states to multiple low-lying 2+ states were

investigated. For 43S, the isomeric content of the incoming beam was exploited in order

to measure quadrupole collectivity from both the ground state and the isomeric state at

320 keV.

Looking forward, the Facility for Rare Isotope Beams (FRIB) is scheduled to begin op-

erations in 2022. With the new linear accelerator replacing the coupled cyclotrons, FRIB

will provide rare isotope beams at intensities roughly three orders of magnitude greater than

the NSCL [67]. With the increased beam intensities available, it may be possible to per-

form low-energy Coulomb excitation of the neutron-rich sulfur isotopes studied in this work.

For beam energies below the Coulomb barrier, multi-step excitations can occur, allowing

population of excited states that are high in energy. With semi-classical coupled-channels

Coulomb excitation codes such as GOSIA [103], the yields of the de-excitation γ rays de-

tected experimentally as a function of the scattering angle of the radioactive nucleus can

be used to determine electromagnetic transition matrix elements in a model-independent

113

fashion. The diagonal matrix elements are related to the static electromagnetic moments

and characterize the shape and degree of deformation of the nucleus. Furthermore, with the

increased beam intensities, it should be possible to perform intermediate-energy Coulomb

excitation of sulfur (Z = 16) isotopes beyond N = 28 toward the limit of nuclear existence.

It would also be interesting to probe the E2 strength distributions in the neutron-rich sili-

con (Z = 14) and magnesium (Z = 12) isotopes near N = 28 through intermediate-energy

Coulomb excitation in order to benchmark the performance of state-of-the-art shell-model

interactions throughout this island of inversion.

114

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