SUBBARRIER COULOMB EXCITATION AT REA3 –
COMMISSIONING OF THE SEGA-JANUS EXPERIMENTAL SETUP
By
Eric Michael Lunderberg

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics — Doctor of Philosophy
2017

ABSTRACT
SUBBARRIER COULOMB EXCITATION AT REA3 – COMMISSIONING OF THE
SEGA-JANUS EXPERIMENTAL SETUP
By
Eric Michael Lunderberg
A new experimental setup for low-energy Coulomb excitation was developed at NSCL
and commissioned at the general purpose beam line of the ReA3 reaccelerator facility. The
so-called SeGA-JANUS (Joint Array for NUclear Structure) setup combines Îł-ray detection with the Segmented Ge Array (SeGA) with scattered particle detection using a pair
of segmented double-sided Si detectors. The low-energy Coulomb excitation program that
SeGA-JANUS will enable complements intermediate-energy Coulomb excitation studies that
have long been performed at NSCL by providing access to observables that quantify collectivity beyond the first excited state and that are sensitive to the sign and magnitude of
quadrupole moments. This opens up new research opportunities now at NSCL and in the
future at FRIB where the variety of available beams and their intensity will greatly increase.
The commissioning experiment used stable 78 Kr beam. With (Z,N)=(36,42), 78 Kr has
enough valence protons and neutrons to exhibit collective structures that are strongly excited
in Coulomb scattering. In addition to being a noble gas and well-studied, this made 78 Kr
an ideal choice for the SeGA-JANUS commissioning.
In this work, the new detector setup, including data acquisition and analysis will be
presented. The deduced measures of collectivity and shape are compared to literature values.
Due to delays in the ReA3 readiness, this thesis also includes the experimental study of
neutron-rich sulfur isotopes with mass numbers A=38-42. This work used the GRETINA
setup at the S800 spectrograph and resulted in a peer-reviewed publication in Physical

Review C that is reproduced in Appendix C.

Copyright by
ERIC MICHAEL LUNDERBERG
2017

To Elisira, my wife, who has been with me throughout graduate school, giving me
confidence and support.

v

ACKNOWLEDGMENTS

Throughout graduate school, I have enjoyed the continued support of friends, colleagues,
mentors and family. My list of acknowledgements is by no means exhaustive, and I am
thoroughly grateful to those named and unnamed for teaching, words of encouragement,
assistance, camaraderie and companionship that made this dissertation possible, productive
and enjoyable.
I thank all the members of the gamma group. My advisor, Professor Alexandra Gade,
has given me continuous support. Any time, day or night, I know that I can drop into her
office with a question about physics, and I will get an enthusiastic response. Often, I will not
even be back to my office before I have received emails with more information and papers to
read on the topic, thereby broadening my knowledge. I am grateful for this enthusiasm and
modeling of the thought processes of a scientist. To Dr. Dirk Weisshaar, who has forgotten
more about Îł-ray detectors than I will ever know, thank you. Your Socratic dialogues about
detectors do a marvelous job of emphasizing how simple detectors are in concept and yet
how complicated they can be in execution. To Peter Bender, whose knowledge of physics
and of ROOT are commendable, it was fun to talk about program design and deciding how
we would jointly design and write the GRUTinizer analysis package.
To my guidance committee, thank you. It has been a pleasure working with you, and I
appreciate all the advice that has led me through the PhD process.
To everyone who has taken the SeGA filling pager, thank you. It is a grave and solemn
duty to maintain the flow of liquid nitrogen to the detectors, being woken up in the middle of
the night to come to the lab and cool a warming detector; it is a duty better borne together.
I acknowledge Dr. Paul DeYoung, who I worked with in undergrad and whose guidance
vi

through research led me to the MoNA group, and then to NSCL as a graduate student. I
pay you to think, is my favorite PDY phrase, pushing students to understand, and never to
only work by rote action. Collaborators Ching-Yen Wu and Brian Bucher were instrumental
in the setup and analysis of this experiment, and I am most grateful for all the help. I thank
Shaofei Zhu, whose gracious sharing of GRETINA/CHICO2 spectra provided a marvelous
point of comparison for the SeGA-JANUS setup.
To my mom and my older brother Mark, who preceded me in their own journeys through
graduate school, thank you for your understanding and support. My first knowledge of the
word dissertation was as a four-year old, learning that it meant not to enter Moms study for a
while. Now dissertation means don’t enter my office and I will be home later than expected!
Your encouragement through the process of research and writing has been incredible.
To my Dungeons and Dragons group: who better to sit around a table in a dark basement
room and escape from the world of physics for a while? May the adventures we had, unlike
Harth and Garth, never be forgotten.
To anyone in the future reading this, I appreciate your dedication! Please know that this
work was a labor of love and I hope for your success in all of your academic endeavors. I
have a script watching for emails at thesis2017@lunderberg.com. Send an email that way,
and it will let you know how many people have made it as far as you have.
Last, I thank my wife, Elisira. She has been supportive and caring, listening when I
ramble about work, programming and physics, and still reminding me to nurture the other
aspects of my life. In my thesis experiment, we would joke that she was effectively widowed
and yet, through experiment and the remainder of the thesis process, I have known that
however much time the lab claims, I can come home. Thank you.

vii

TABLE OF CONTENTS

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

ix

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

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Coulomb Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Transition and Diagonal Matrix Elements . . . . . . . . . . . . . . . . . . .

1
1
8

Chapter 2 Experimental Setup . . . . . . . . . .
2.1 Particle Detection . . . . . . . . . . . . . . . .
2.1.1 Principles . . . . . . . . . . . . . . . .
2.1.2 JANUS Particle Detection . . . . . . .
2.2 Gamma-Ray Detection . . . . . . . . . . . . .
2.2.1 Principles . . . . . . . . . . . . . . . .
2.2.2 Doppler Shift of Îł Rays . . . . . . . .
2.2.3 SeGA . . . . . . . . . . . . . . . . . .
2.3 Data Acquisition . . . . . . . . . . . . . . . .
2.4 Testing of SeGA-JANUS with a 252 Cf Source

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

14
14
14
17
20
20
23
25
29
32

Chapter 3 Sub-barrier Coulomb Excitation of 78 Kr . .
3.1 Experimental Details . . . . . . . . . . . . . . . . . . .
3.1.1 Beam Production . . . . . . . . . . . . . . . . .
3.1.2 Target Details . . . . . . . . . . . . . . . . . . .
3.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Particle Identification and 78 Kr Reconstruction
3.2.2 Gamma-Particle Coincidences . . . . . . . . . .
3.2.3 Doppler Correction . . . . . . . . . . . . . . . .
3.2.4 Experimental Yields . . . . . . . . . . . . . . .
3.3 Matrix Element Extraction . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

36
36
37
39
40
40
44
47
52
57

Chapter 4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Chapter 5

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

72

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A GRUTinizer Development . . . . . . . . . . .
Appendix B Gosia Analysis Procedure in More Detail . . .
Appendix C Spectroscopy of Neutron-Rich Sulfur Isotopes

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

. 74
. 75
. 94
. 121

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

viii

LIST OF TABLES

Table 3.1: The experimental peak areas for each scattering angle region. Transitions
not observed in a given scattering angle region are left blank. . . . . . . .

57

Table 3.2: The lifetimes used as data points in the gosia χ2 minimization [52]. . . .

60

Table 3.3: The branching ratios used as data points in the gosia χ2 minimization [52]. 60
Table 3.4: The mixing ratios used as data points in the gosia χ2 minimization [52].
Signs of mixing ratios use the Krane sign convention [56]. . . . . . . . . .

60

Table 3.5: The matrix elements from [55] used as data points in the gosia χ2 minimization. Use of these transition matrix elements to so-called buffer states
is described in Ch. 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Table 4.1: The E2 matrix elements and uncertainties as extracted by gosia, compared
with values from NNDC [52] and Becker et al. [55]. Transitions below the
horizontal divider were not directly observed in this experiment, but were
included in the gosia minimization. Not shown are 3 ill-constrained matrix
+
elements, including transitions from the 8+
1 and 24 buffer states. . . . . .

65

Table 4.2: The M1 matrix elements and uncertainties as extracted by gosia, compared with values from NNDC [52] and Becker et al. [55]. (See text for a
discussion of the sensitivity to M1 transitions.) . . . . . . . . . . . . . . .

66

Table 4.3: The diagonal matrix elements and uncertainties as extracted by gosia,
compared with values from Becker et al. [55]. Results are quoted both as
diagonal matrix elements and as quadrupole moments. Not shown are 2
ill-constrained matrix elements. . . . . . . . . . . . . . . . . . . . . . . . .

66

Table C.1: Energies, intensities and coincidence relationships for Îł-ray decays observed
in 38 S. The 833(5) keV peak is significantly below the literature value of
849 keV, and has a peak shape indicative of a left tail. In comparison to
simulations, both may be explained by a lifetime of 100 ps < τ < 200 ps.
Transition energies placed in brackets indicate tentative identifications of
Îł-ray peaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Table C.2: Energies, intensities and coincidence relationships for Îł-ray decays in 39 S.

134

Table C.3: Energies, efficiency-corrected relative intensities, and coincidence relations
for Îł-ray decays observed in 40 S. As for 38 S, the transitions suspected to
form the even-spin yrast cascade are the most intense. . . . . . . . . . . . 137
ix

Table C.4: Energies, efficiency-corrected relative Îł-ray intensities, and coincidences for
41 S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Table C.5: Energies, efficiency-corrected relative Îł-ray intensities, and coincidences for
42 S. The 992 keV peak appears in coincidence with itself, suggesting that
a doublet cannot be excluded for this transition. . . . . . . . . . . . . . . 148

x

LIST OF FIGURES

Figure 1.1: Excitations and de-excitations in a simplified 3-state system. Each upward arrow represents a path for Coulomb excitation. Each downward
arrow represents de-excitation by Îł-ray emission. The excitations create
a population of excited states, which can then be tagged by spectroscopy
of the subsequent Îł-ray emission. . . . . . . . . . . . . . . . . . . . . . .

2

+
Figure 1.2: The B(E2; 0+
1 → 21 ) transition strength for isotopic chains, shown here
in Weisskopf units. Minima in the transition strength are clearly seen at
magic numbers N = 50, 82, 126. (Figure adapted from [4].) . . . . . . . .

3

Figure 1.3: The chart of the nuclides, with selected features labeled. Stable nuclei are
shown in black, observed unstable nuclei in blue, with the expected extent
of all possible nuclei shown in red. Figure adapted from [9]. . . . . . . .

4

Figure 1.4: The energy splitting caused by a strong electric field, as described in
Eq.1.14. The energy splitting is fairly large, for example on the order of
∟ 100 keV when using oxygen as the probe [20]. As oblate and prolate nuclei have quadrupole moments with opposite signs, their spin-degenerate
substates are split in the opposite direction. . . . . . . . . . . . . . . . .

11

Figure 2.1: An illustration of the experimental setup, showing a projectile scattering
off of a target nucleus, and subsequently emitting a Îł ray. The scattered
and recoiling nuclei are detected in the JANUS particle detectors, while
the emitted Îł rays are detected in the SeGA detectors. . . . . . . . . . .

15

Figure 2.2: The segmentation in rings on one surface and in sectors on the other of
a JANUS S3 detector. Shown in blue is a hypothetical energy deposition
into a ring and sector, with the corresponding pixel in which energy was
deposited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.3: The target position of the JANUS assembly. Shown here are the two
silicon detectors that compose JANUS, with the target wheel mounted
between them. From left to right, the target frame contains a 3 mm aperture, an empty frame, a 208 Pb target, and a 48 Ti target. The target
wheel is operated manually through a rod system that is not shown here.
The orange and green wires allow collected charge on the apertures to be
measured, for beam-tuning purposes. . . . . . . . . . . . . . . . . . . . .

19

Figure 2.4: Reaction probabilities of each interaction of a photon in a germanium
detector. Cross-section values taken from Ref. [41] . . . . . . . . . . . . .

21

xi

Figure 2.5: Idealized Îł-ray energy spectrum of a 3 MeV Îł-ray, showing characteristic
features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.6: Kinematics for the Doppler boost of a Îł ray emitted from a nucleus in
motion. All vectors shown are in the laboratory frame of reference. . . .

23

Figure 2.7: The effect of each experimental uncertainty on the Doppler reconstruction,
as a function of emission angle, based on equations from [11]. This plot was
generated using β = 8.5(5)%, σθ = 3◦ , and relative intrinsic resolution
σE
= 0.1%. In this plot, σE is used instead of full width at half
lab
maximum (FWHM), where FWHM = 2.355σE . . . . . . . . . . . . . .

24

Figure 2.8: The segmentation of a single SeGA crystal, as shown from front, side, and
diagonal perspectives. Figure adapted from [34]. . . . . . . . . . . . . . .

25

Figure 2.9: A single SeGA detector, outside of the experimental setup. The green
cylinder is the liquid nitrogen dewar, with the germanium crystal at the
bottom of the image. The preamplifiers are contained inside the metal
casing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

c
Figure 2.10: The SeGA array positioned around the target chamber. Photograph 
2016 NSCL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 2.11: The trigger logic for the JANUS setup, generating the raw trigger from
the individual channel triggers. . . . . . . . . . . . . . . . . . . . . . . .

31

Figure 2.12: The mass distribution of 252 Cf fission fragments. (Figure from [46].) . . .

33

Figure 2.13: The Îł-ray energies from fission fragments from 252 Cf plotted against the
angle between the lighter fission fragment and Îł ray. As an example, the
241 keV transition from 110 Ru, Doppler-shifted in the laboratory system,
is circled in red. Also visible is the 212 keV transition from 100 Zr. These
transitions are observed strongly, as they are near the peak of the lowmass fragment distribution, as shown in Fig. 2.12. Gamma rays emitted
from the heavier fission fragments are observed as well, appearing as faint
lines with orientation opposite that of the 212 keV and 241 keV transitions 34
Figure 2.14: The Îł-ray energy spectrum, with and without Doppler correction, for
fission fragments from 252 Cf. The resolution of the Doppler-corrected Îł
energy resolution is 2% FWHM. . . . . . . . . . . . . . . . . . . . . . . .

35

Figure 3.1: Diagram of the reaccelerator (ReA3) at NSCL.

37

xii

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

Figure 3.2: In black, the energy spectrum of a radioactive source containing the Îą
emitters 244 Cm, 241 Am, and 239 Pu. In red, the energy spectrum of the
same source, with the 208 Pb target put between source and detector. The
target thickness is deduced from the observed energy loss through the
target. In addition to energy loss, the spectrum also becomes wider due
to energy straggling through the target [39]. . . . . . . . . . . . . . . . .

40

Figure 3.3: The expected energies of scattered 78 Kr and recoiling 208 Pb, across the full
angular range, based on 3.9 MeV/u incoming beam energy. This calculation
assumes that the two particles scatter elastically. As the nuclear excitations are at maximum a few MeV, as compared against the ∟ 300 MeV
beam energy, this is a reasonable assumption. . . . . . . . . . . . . . . .

41

Figure 3.4: The angles of the scattered projectile and recoiling target nucleus, along
with relative positions of the two particle detectors. . . . . . . . . . . . .

42

Figure 3.5: Particle identification of scattered 78 Kr and recoiling 208 Pb in the forward
detector of JANUS. Kinematic lines for each particle show the energy when
scattering at a given angle. The clear separation between the energies
of the two isotopes allows for clear discrimination in analysis. Due to
electronics issues, the segment covering scattering angles 44-45◦ has poorer
resolution, but is still sufficient to discriminate between 78 Kr and 208 Pb.

43

Figure 3.6: Reconstruction of the scattered 78 Kr given the angle of the recoiling 208 Pb.
The gray region indicates the angular acceptance for recoiling 208 Pb, and
the corresponding angles of the reconstructed 78 Kr. . . . . . . . . . . . .

45

Figure 3.7: The correlation between lab-frame Îł-ray energy on the y-axis and the
time difference between Îł-ray detection and particle detection on the xaxis. The time-energy gate for determining coincidence between particles
and Îł rays is shown in red. The 1460 keV and 2614 keV room background
peaks from the decay of 40 K and 208 Tl. . . . . . . . . . . . . . . . . . . .

46

Figure 3.8: The lab-frame Îł-ray energy plotted against Îł-ray emission angle, for 78 Kr
detected in the downstream JANUS detector. This spectrum only shows
the dominant 455 keV transition in 78 Kr. . . . . . . . . . . . . . . . . . .

48

Figure 3.9: The lab-frame Îł-ray energy plotted against Îł-ray emission angle, for 78 Kr
events reconstructed from the measured trajectory of recoiling 208 Pb. This
spectrum only shows the dominant 455 keV transition in 78 Kr. Different
from Fig. 3.8, an unshifted component of the 455 keV Îł-ray energy is visible in addition to the Doppler-shifted component, caused by 78 Kr stopped
in the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

xiii

Figure 3.10: The lab-frame energy spectrum, compared against the Doppler-reconstructed
energy spectrum, for 78 Kr detected in the downstream JANUS detector,
using v/c = 0.085 in the Doppler reconstruction. This spectrum only
shows the dominant 455 keV transition in 78 Kr. . . . . . . . . . . . . . . 50
Figure 3.11: The velocity of the scattered 78 Kr as a function of scattering angle. The
three ranges of scattering angles of 78 Kr deduced from the silicon detector
coverage are indicated. The variation of v/c within a region is sufficiently
small for using a fixed Doppler-reconstruction v/c for each range. . . . .

51

Figure 3.12: The Doppler-corrected energy spectrum for each angular region of the
scattering 78 Kr. For performing Doppler correction, v/c = 0.085 was used
for the downstream detector, v/c = 0.06 for the reconstructed fragments,
and v/c = 0.04 for the upstream detector. . . . . . . . . . . . . . . . . .

53

Figure 3.13: The Doppler-corrected energy spectrum for each angular region of the
scattering 78 Kr, showing the energy region from 500 keV to 1000 keV. For
performing Doppler correction, v/c = 0.085 was used for the downstream
detector, v/c = 0.06 for the reconstructed fragments, and v/c = 0.04 for
the upstream detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Figure 3.14: The Doppler-corrected energy spectrum for each angular region of the
scattering 78 Kr, showing the energy region from 1000 keV to 2000 keV. For
performing Doppler correction, v/c = 0.085 was used for the downstream
detector, v/c = 0.06 for the reconstructed fragments, and v/c = 0.04 for
the upstream detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Figure 3.15: Measured SeGA efficiency curve using a 152 Eu source. The red line shows
a fit to the measured efficiencies, with functional form 1157 exp(−0.7511 log(E+
74.6)), where E is the energy in keV, and the resulting value is in percent. 56
Figure 3.16: The level scheme of 78 Kr, as used for fitting matrix elements to the observed data, with indications to show the use of literature values, contributing to the χ2 value. Levels and transitions in solid lines were directly observed in this experiment. Levels and transitions in dashed lines
are included in the coupled channels calculation, but do not have experimental yields contributing to the χ2 calculation. States marked with a τ
had their lifetimes used as components of the χ2 . States marked with a B
had branching ratios between all de-excitation Îł rays used as components
of the χ2 . Transitions marked with a δ had their E2/M1 mixing ratio used
as a component of the χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

59

Figure 4.1: The fitted experimental yields for each observed Îł ray transition. In black
are the experimental values, and in red are the fitted values. The experimental yield was extracted from the peak areas corrected by the Îł-ray
detection efficiency. On the left are the fits to each peak area, in efficiencycorrected counts. On the right are the z-scores of the comparison, the
number of standard deviations between the experimental values and the
fitted values, for each data point. Missing points for certain angle groups
indicate that a peak was not observed for that particular transition. . . .

63

Figure 4.2: A comparison between transition and diagonal matrix elements measured
in the present work, from Becker et al. [55], and from NNDC [52]. The
comparisons are discussed in the text. . . . . . . . . . . . . . . . . . . .

67


   + 
 
Figure 4.3: The χ2 curve of the 4+
1 E2 23 matrix element, with and without the
branching ratio constraint in the gosia fit. . . . . . . . . . . . . . . . . .

70

Figure A.1: The analysis pipeline of the simplest use of GRUTinizer . . . . . . . .

76

Figure A.2: Additional components that can be added to the GRUTinizer analysis
pipeline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure A.3: A screenshot of the GRUTinizer graphical user interface, exploring a
.root file containing histograms. . . . . . . . . . . . . . . . . . . . . . .

93

Figure C.1: Particle identification spectra for the reaction residues produced in 12 C(46 Ar,X)Y
(upper panel) and 12 C(48 Ca,X)Y (lower panel). The energy loss was measured with the ionization chamber of the S800 focal plane. The time-offlight was taken between plastic scintillators in the beam line and in the
back of the S800 focal plane. The S isotopes of interest are unambiguously
identified and separated. . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Figure C.2: Doppler-corrected (v/c = 0.357) Îł-ray spectrum taken with GRETINA in
coincidence with 38 S as identified with the S800 spectrograph. The inset
expands the energy range from 2 to 4 MeV. . . . . . . . . . . . . . . . . 127
Figure C.3: Background-subtracted γγ coincidence spectra for 38 S. Coincidence spectra for the 1282, 1515, and 1534 keV transitions are shown. The small
number of counts observed agrees with expectations based on the statistics
of the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure C.4: Proposed experimental level schemes for 38 S based on previous data and
coincidence relationships. The experimental level scheme is confronted
with shell-model calculations using the SDPF-MU Hamiltonian. . . . . 130

xv

Figure C.5: Doppler-reconstructed Îł-ray spectrum in coincidence with 39 S (v/c =
0.348). The inset expands the higher-energy region of the spectrum. . . 131
Figure C.6: Background-subtracted γγ coincidence spectra for 39 S. Software gates
on the 337, 466, and 533 keV transitions are displayed. The coincidence
spectra shown in the left and in the middle panel investigate the previously
claimed 337-466 keV coincidence [94], which seems plausible based on
our low-statistics data. The right panel provides weak evidence for a
coincidence between the newly observed 533 and 370 keV transitions. . 132
Figure C.7: Predicted level scheme and nanosecond lifetimes for 39 S from shell-model
calculations using the SDPF-MU effective interaction. . . . . . . . . . . 133
Figure C.8: Doppler-reconstructed Îł-ray spectrum in coincidence with 40 S (with v/c =
0.341 for 40 S from 48 Ca beam and v/c = 0.350 for 40 S from 46 Ar beam).

136

Figure C.9: Background-subtracted γγ coincidence spectra for 40 S. Spectra in coincidence with the strongest transitions at 902, 1350, and 1572 keV are
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Figure C.10: Proposed experimental level scheme for 40 S compared to the SDPF-MU
shell-model calculations. The level scheme is based on the γγ coincidence
spectra and intensities. The tentative placement of the 2057 keV transition
is indicated. The tentative identification of the low-lying even-spin yrast
sequence is based on comparison with the shell model and the population
pattern of excited states observed throughout this work. . . . . . . . . . 138
Figure C.11: Doppler-reconstructed Îł-ray spectrum in coincidence with 41 S (v/c =
0.342). The spectrum is expanded with changed binning beyond 500 keV
to highlight the high density of small peaks up to ∟3.2 MeV. . . . . . . 139
Figure C.12: Background-subtracted γγ coincidence spectra for 41 S. Spectra in coincidence with 451, 536, 1099, and 1633 keV are shown. . . . . . . . . . . . . 141
Figure C.13: Background-subtracted γγ coincidence spectra for 41 S. Spectra gated on
1611 and 1302 keV transitions in 41 S are shown. These transitions do
not appear in coincidence with 451 keV (see also Fig. C.12) or any other
transitions that they would feed. . . . . . . . . . . . . . . . . . . . . . . 141

xvi

Figure C.14: Proposed experimental level schemes for 41 S based on the observed coincidences, intensities, energy sums and comparison to the shell model
(SDPF-MU Hamiltonian). Solid lines and filled arrows indicate firm level
and transition assignments, the dashed line and unfilled arrows indicate a
tentative placement. Given that the 1611 and 1302 keV Îł-ray transitions
are strong and not in coincidence with 451 keV, we argue that they likely
populate the ground state directly. Comparison to shell-model energies,
decay branchings, and systematics was used to assign tentative J π values
(see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Figure C.15: Doppler-reconstructed Îł-ray spectrum in coincidence with 42 S (v/c =
0.335). The insets expand energy regions of the spectrum with weaker
intensity transitions. Transitions at 1143 and 2154 keV are tentative. . . 146
Figure C.16: Background-subtracted γγ coincidence spectra for 42 S. Spectra in coincidence with 902, 1787, 1820, and 2803 keV are shown. . . . . . . . . . . 147
Figure C.17: Background-subtracted γγ coincidence spectra for 42 S. Spectra in coincidence with 949, 992, and 2677 keV are shown. . . . . . . . . . . . . . . . 148
Figure C.18: Proposed experimental level scheme for 42 S based on the γγ coincidence
spectra, intensities, and energy sums. The experimental data is confronted
with shell-model calculations using the SDPF-MU effective interaction.
Comparison to shell-model energies and decay branchings was used to
assign tentative J π quantum numbers (see text). . . . . . . . . . . . . . 150
Figure C.19: Comparison of the excitation energy ratios of the first excited 2+ , 4+ , and
6+ states across the neutron-rich sulfur isotopes. For 36 S, the 6+
1 state
has not yet been identified in the literature [98], and is expected at high
excitation energy where the level density is significant. The collective 4+
and 6+ states of 44 S have not yet been observed either. For the calculated
44 S E(4+ )/E(2+ ) ratio, the shell-model energy of the second 4+ state
+
+
+
is used since the 6+
1 → 42 → 21 → 01 cascade is connected by the
strongest E2 transitions. However, the energy ratios would not change if
+
+
+
the 4+
1 or 62 energies were used instead since E(62 ) − E(61 ) = 56 keV
+
+
40,42 S
and E(4+
2 ) − E(41 ) = 134 keV. The tentative 61 assignments for
stem from the measurements presented here. . . . . . . . . . . . . . . . 153
Figure C.20: Shell-model (SDPF-MU) neutron p3/2 occupancy for the 0+ (red) and 2+
(blue) states of 38−44 S below 4.5 MeV. The rapid onset of neutron p3/2
occupancy together with the dramatic increase in the level density of 0+
and 2+ states in 44 S signals a sudden transition into the N = 28 “island
of inversion” in the S isotopic chain. The role of 42 S as a sensitive probe
for the neutron configurations is discussed in the text. . . . . . . . . . . 156
xvii

Chapter 1
Introduction

1.1

Coulomb Excitation

The nuclear excitation induced by the electromagnetic field acting between colliding nuclei
is referred to as Coulomb excitation. By restricting either the beam energy or the impact
parameter through particle scattering angles, the projectile and target can be kept separated
by more than the interaction distance of the short-range nuclear force. This technique enables
an analysis that does not rely on a model of the nuclear force, allowing consideration of only
excitations caused by the well-understood electromagnetic interactions [1].
The cross sections observed for the Coulomb excitation of individual final states in a
nucleus are related to the transition matrix elements that connect the excited states with
each other and the ground state. The cross sections can be deduced from the measured
intensity of, for example, the Îł-ray de-excitations of the level (see Fig. 1.1) relative to the
number of projectiles and number density of the target. It is noted that, in principle,
both the target and the projectile can be excited in the collision. Electric dipole (E1) and
electric quadrupole (E2) modes are predominantly excited. E3 excitations are observed as
well, while magnetic excitations are heavily suppressed at low beam energies [2] and will
be neglected here as an excitation mechanism. In even-even nuclei, Coulomb excitation is
mediated predominantly through E2 transition matrix elements that connect the low-lying

1

positive-parity yrast and yrare states.

I3
I2
I1
Figure 1.1: Excitations and de-excitations in a simplified 3-state system. Each upward arrow
represents a path for Coulomb excitation. Each downward arrow represents de-excitation
by Îł-ray emission. The excitations create a population of excited states, which can then be
tagged by spectroscopy of the subsequent Îł-ray emission.

Electromagnetic transition matrix elements deduced from Coulomb excitation cross sections can be translated into the reduced electromagnetic transition strength, B(E2; Ii → If ).
+
In even-even nuclei, the B(E2; 0+
1 → 21 ) reduced electric quadrupole excitation strength is

used extensively as an indicator of collective excitations, such as rotations or vibrations,
in which many nucleons outside of inert proton and neutron cores participate [3]. Near
shell closures, where the number of valence nucleons available for collective motion is small,
B(E2) values are small, while mid-shell, in between two magic numbers, the B(E2) strength
is maximized, as shown in Fig. 1.2.
Coulomb excitation was first used to study stable nuclei [5, 6]. The nucleus to be studied
was made into a target, which was then bombarded by 4 He nuclei or heavier ions such as 12 C
2

B(E2; 0+ to 2+) (weisskopf units)

N
+
Figure 1.2: The B(E2; 0+
1 → 21 ) transition strength for isotopic chains, shown here in
Weisskopf units. Minima in the transition strength are clearly seen at magic numbers N =
50, 82, 126. (Figure adapted from [4].)

at energies below the Coulomb barrier. Coulomb excitation would excite states with spins
as high as 30+ in the yrast band [1], quantified through the detection of Îł rays emitted by
the target nucleus. The energy of the Coulomb barrier is as follows, where ZP and ZT are
the proton number of the projectile and target, respectively, and AP and AT are the mass
number of the projectile and target:
1 ZP ZT e2
,
ECoulomb barrier =
4π0 rmin
h


i
1/3
1/3
rmin = 1.25 AP + AT
+ 5 fm.

(1.1)
(1.2)

Here, rmin is the minimum distance allowed in a Coulomb excitation analysis. Its phenomenological description is based on interaction radius studies [1].
When rare isotopes became available for experiments, changes to the well-known nuclear
shell structure [7, 8] were found and prompted experimental research programs world-wide.
Since these short-lived nuclei cannot be made into targets and are available as beams of ions
3

Figure 1.3: The chart of the nuclides, with selected features labeled. Stable nuclei are shown
in black, observed unstable nuclei in blue, with the expected extent of all possible nuclei
shown in red. Figure adapted from [9].
for experiments, well-established measurement techniques had to be adapted to deal with
“inverse kinematics,” a reaction scheme in which the nucleus of interest is the projectile. As
shown in Fig. 1.3, the vast majority of isotopes are unstable, including key isotopes relevant
for nuclear astrophysics along the r-process and rp-process paths.
The most exotic beams are created through projectile fragmentation at energies typically exceeding 100 MeV/u and subsequent in-flight separation [10]. After fragmentation
and separation, the fragments are typically available at velocities exceeding v/c ≈ 0.3. At
these velocities, the collision between projectile and target occurs above the Coulomb barrier
for all possible target-projectile combinations. Whereas earlier Coulomb excitation experiments could rely on sub-barrier energies to prevent nuclear interactions from occurring, these
intermediate-energy Coulomb excitation experiments [11] consider only events that scatter at
small scattering angles, which correspond to sufficiently large impact parameters. The exact

4

cm
relationship between impact parameter and scattering angle is given in Eq. 1.3, where θmax

is the maximum projectile scattering angle in the center of mass reference frame, β = v/c
is the beam velocity relative to the speed of light, Îł = q 1

1−β 2

is the Lorentz factor, ZP

and ZT are the proton numbers of projectile and target, respectively, and m0 is the reduced
mass of the projectile-target system [11]:

a
bmin = cot
Îł
a=

 cm 
θmax
2

ZP ZT e2
.
m0 c2 β 2

(1.3)
(1.4)

The higher beam energy also affects the range of states accessible in the Coulomb excitation process. The short interaction time of projectile and target at high collision velocities
suppresses multi-step excitations and restricts the accessible states largely to the ones that
can be excited from the ground state in a single step [12, 11]. This simplifies the analysis significantly, as each de-excitation Îł ray observed corresponds to a single excitation path
without feeding from higher-lying states reached in multiple steps. Using relativistic calculations developed by Alder and Winther, also in this regime, the observed Coulomb excitation
+
cross sections were related to the B(E2) transition strengths, such as B(E2; 0+
1 → 21 ) in

even-even nuclei [12]. In intermediate-energy Coulomb excitation, thick high-Z targets can
be used, increasing luminosity and allowing for experiments at a few ions per second [11].
Inverse kinematic Coulomb excitation, was first pioneered at RIKEN to quantify the
onset of collectivity as the neutron magic number N = 20 breaks down in exotic 32 Mg
[13], and subsequently implemented at the National Superconducting Cyclotron Laboratory
(NSCL) and other laboratories with beams in a similar energy regime. For example, using
the Segmented Germanium Array (SeGA) to detect Îł rays and the S800 spectrograph to
5

+
detect the scattered projectiles [14, 15, 16, 17, 18, 19], the B(E2; 0+
1 → 21 ) values for a wide

variety of rare isotopes were measured.
Extending low-energy Coulomb excitation to rare isotopes allows for more comprehensive
measurements of these nuclei, accessing collectivity beyond the first excited state through
multi-step excitations [1]. A particularly interesting case of a multi-step excitation can occur among the M -substates of a nuclear level. This so-called Coulomb nuclear reorientation
allows experimental measurements of the magnitude and sign of the diagonal matrix element hI||O(E2)||Ii, which is related to the quadrupole moment [20]. The magnitude of the
quadrupole moment quantifies the degree of deformation and its sign signals the shape as
prolate or oblate. In the regime of rare isotopes, shape coexistence has been observed where
magic numbers break down [21], and quantities that characterize shape become important.
At ISOL facilities, which complement facilities that use in-flight production and separation of rare isotopes, selected chemistry-dependent rare-isotope beams are available at a
few MeV/u, around the Coulomb barrier of typical projectile-target combinations [22]. ISOL
facilities produce rare isotopes by fragmentation of a heavy production target, for example
uranium carbide, with a high-intensity proton beam. The fragments produced then diffuse
out of the production target, are ionized, and are then accelerated to sub-barrier energies,
in order to perform Coulomb excitation. Some recent highlights are as follows. At the
TRIUMF-ISAC facility, studies of transitions in the radioactive nuclei 20,21 Na [23], 78,80 Zn
[24], and 11 Be [25] have been performed. At CERN-ISOLDE, measurements of 220 Rn and
224 Ra

have provided circumstantial evidence of octupole deformation [26], possibly relevant

for electric dipole moment (EDM) searches, and isomeric beams of 68,70 Cu and 106,108 In
have enabled the study of transition matrix elements based on isomeric states [27, 28]. A
different method of beam production is employed by CARIBU, located at Argonne National
6

Laboratory (ANL), where a 1 Ci 252 Cf source produces fission fragments, which are then
thermalized in a gas catcher, extracted, charge-bred, and accelerated to a few MeV [29].
CARIBU has been used to perform low-energy Coulomb excitation experiments, measuring
transition and diagonal matrix elements in 144,146 Ba [30, 31] and 110 Ru [32]. Such lowenergy experiments typically require projectile intensities exceeding 104 − 105 particles per
second, and are therefore limited to less exotic nuclei as compared to intermediate-energy
Coulomb excitation.
While ISOL facilities are limited to elements with a chemistry that allows extraction from
a thick production target and CARIBU to 252 Cf fission fragments, with NSCL’s new and
world-unique scheme of in-flight production and separation, combined with subsequent beam
thermalization in a gas cell, charge breeding, and re-acceleration [33], the element limitation
is largely removed and in principle all isotopes produced at sufficient intensities can be made
available as beams at Coulomb barrier energies for experiments. The variety of beams and
intensities available will increase at the upcoming Facility for Rare-Isotope Beams (FRIB),
extending the range of isotopes that can be studied with low-energy Coulomb excitation.
With this in mind, the development of a new experimental setup designed for low-energy
Coulomb excitation at the NSCL has been pursued and is the subject of this work, opening
new opportunities to comprehensively assess collectivity and shapes in nuclei not otherwise
available for such studies.
While Îł-ray tagging can still be used at low energies, the scattering-angle range of the
projectiles is essentially 180◦ , requiring a more complete angle coverage for the particle detection system, in contrast to the strongly forward-focused scattered projectiles at intermediate
beam energies. The Joint Array for NUclear Structure (JANUS), composed of two annular
double-sided segmented silicon detectors surrounded by the Segmented Germanium Array
7

(SeGA) [34] for Îł-ray detection, can detect particles scattering to large angles. With a solid
angle coverage of 29% of 4π, and an effective solid angle coverage after projectile reconstruction from the target recoil of 78% of 4π (see Sec. 3.2.1), the JANUS system presented here
is well-suited for low-energy Coulomb excitation.
The stable isotope 78 Kr was chosen as the in-beam commissioning beam, due to its high
collectivity and ease of use as a noble gas. It has proton and neutron numbers of Z = 36
and N = 42, far from the nearest magic numbers of 28 and 50. As a result, it exhibits large
B(E2) transition strengths connecting the low-lying positive-parity states. As it is stable
and gaseous, producing a beam of 78 Kr is a straightforward task. In addition, as 78 Kr has
previously been studied using low-energy Coulomb excitation, existing work will provide a
point of comparison, in order to verify the experimental scheme and data analysis.
In this work will be presented the commissioning experiment of the SeGA-JANUS experimental setup, performing low-energy Coulomb excitation of 78 Kr at the ReA3 facility at
NSCL.

1.2

Transition and Diagonal Matrix Elements

In inverse-kinematics Coulomb excitation experiments, the Coulomb field of the target nucleus provides a time-dependent perturbation on top of the nuclear Hamiltonian that describes the projectile. The Coulomb field felt by the projectile can be approximated using
the classical hyperbolic trajectory [35, 1]. By expanding the time-dependent wave function
in terms of the unperturbed eigenstates, one can determine the outgoing nuclear states in a
coupled channels calculation as shown below, where ψ(t) is the time-dependent wave function
of the projectile, H0 is the unperturbed nuclear Hamiltonian, En and φn are the eigenvalues

8

and eigenstates of H0 , an (t) are the excitation amplitudes, and V (t) is the time-dependent
potential generated by the target nucleus [1]:

ih̄ψ̇(t) = [H0 + V (t)] ψ(t)
ψ(t) =

X

(1.5)

an (t)φn

(1.6)

n

ih̄ȧn (t) =

X
m


hn|V (t)|mi exp

i(En − Em )t
h̄


am (t).

(1.7)

The electromagnetic interaction V (t) between the two extended nuclei is then expanded
in terms of electric and magnetic multipoles [1, 2]. After mathematical transformations that
can be found in [36], these coupled differential equations can be expressed in terms of the
reduced transition matrix elements hf ||O(Îť)||ii, where |ii is the initial state, hf | is the final
state, and O(Îť) is an operator for an electromagnetic transition with multipolarity Îť.
The operators for the electric transition of multipolarity Îť are given in Eq. 1.8, where
Y¾Ν are the spherical harmonics, etz are the charge of the particle being integrated over, e is
the elementary charge [4]:

O(Eλ) = rλ Yµλ (r̂)etz e,
 E
D 
 2

2
e.g. hf ||O(E2)||ii = f (r etz e)Yµ (r̂)i .

(1.8)
(1.9)

B(EÎť) transition strengths are a useful quantity derived from the transition matrix elements, as defined in Eq. 1.10. B(E2) values are often expressed in terms of Weisskopf units
(plotted in Fig. 1.2, which remove the dependency of the transition strength on the nucleon

9

number [4]:
hf ||O(E2)||ii2
,
2Ji + 1
 
2 
2Îť
1
3
1/3
Bw.u. (EÎť) =
1.2A
e2 fm2Îť ,
4π
3+Îť

B(E2; i → f ) =

Bw.u. (E2) = 0.0594A4/3 e2 fm4 .

(1.10)
(1.11)
(1.12)

While off-diagonal transition matrix elements characterize the transition between two
states, diagonal matrix elements hJ||O(EÎť)||Ji describe transitions between magnetic M
sub-states of the same J π state, and are related to the spectroscopic quadrupole moment
[37]:
r
Q=

16π hJ J 2 0|J Ji
√
hJ||O(E2)||Ji .
5
2J + 1

(1.13)

Inside a strong electric field gradient, there is a strong time-dependent interaction energy,
given in Eq. 1.14 [20]:

eQZ
E(t) = 3
r (t)



3M 2 − J(J + 1)
4J(2J − 1)


.

(1.14)

This interaction energy will cause spin-degenerate energies to split into the M sub-states,
as shown in Fig. 1.4. This splitting is sizable, for example typically on the order of ∟ 100 keV
when oxygen ions are used as probe [20]. Since the splitting depends on both the magnitude
and sign of Q, magnetic substates of prolate nuclei (Q < 0) and oblate nuclei (Q > 0) will
be split in opposite directions.
The obvious effect of this splitting is that some of the M states will be lower in energy,

10

0

Âą1

2+

Âą2

2+

Âą1

Âą2

0+

Jπ

Prolate

0

0

0+

M

Jπ

Oblate

0

M

Figure 1.4: The energy splitting caused by a strong electric field, as described in Eq.1.14.
The energy splitting is fairly large, for example on the order of ∟ 100 keV when using oxygen
as the probe [20]. As oblate and prolate nuclei have quadrupole moments with opposite signs,
their spin-degenerate substates are split in the opposite direction.
and therefore be preferentially populated in Coulomb excitation, causing a polarization effect for the Coulomb scattered particles [20]. For this reason, this effect is known as the
“reorientation effect.” The more prominent effect for the purpose of a low-energy Coulomb
excitation analysis is a change in the angle-dependent excitation probability. The excitation
probability depends strongly on the difference in energy between the initial and final states.
By altering the energy of some of the spin-degenerate substates, the reorientation effect will
cause changes to the excitation probabilities [20].
When two nuclei scatter off each other, the largest scattering angles correspond to the
smallest impact parameters. Since the magnitude of the quadrupole interaction energy
depends on 13 as shown in Eq. 1.14, events occurring at large scattering angles will have a
r
much larger splitting of spin-degenerate substates, and will be affected by the reorientation
11

effect more strongly. One can then quantify the reorientation effect and so access information
on the shape by measuring the Coulomb excitation cross section for a state in different
scattering angle regions.
For example, in the case of a projectile with spin 0+ beam impinged on a target with the
same spin, only the M = 0 substate of the 2+ state can be populated. In a prolate nucleus,
the M = 0 substate will be at higher energies when the projectile scatters at backward angles
than at forward angles, and so the 2+ state will have a lower excitation cross section. An
oblate nucleus, on the other hand, will have a decreased energy of the M = 0 substate at
large scattering angles, and will therefore have a larger excitation cross section to the 2+
state. By comparing the excitation cross section observed at high scattering angles relative
to that observed at low scattering angles, one obtains a clear experimental signature to
distinguish between prolate and oblate shapes.
Unlike intermediate-energy Coulomb excitation, which can only probe states accessible
in a single step from the ground state [11], low-energy Coulomb excitation can excite a
nucleus in multiple steps, probing higher excited states that form collective structures such
as rotational bands [1]. The prevalence of multi-step excitations in low-energy Coulomb
scattering, including the reorientation effect which is a two-step excitation into the magnetic
substates of a level, leads to an extensive network of coupled differential equations (see
Eq. 1.7). In practice, computer codes that model the semi-classical Coulomb excitation
theory are used to extract matrix elements from large sets of yields or cross sections measured
for different scattering angle ranges. The most modern code, gosia [36, 38, 1], takes as input
the efficiency-corrected Îł-ray yields that signal the direct and indirect population of excited
states and, in an involved χ2 minimization, provides a set of transition and diagonal matrix
elements that describe the measured data.
12

Gosia was designed and optimized to perform such high-dimensional χ2 minimization.
Gosia uses analytic approximations to increase the speed of the minimization, and can fit
up to 200 matrix elements simultaneously to measured yields of the directly and indirectly
populated states. In addition, previous measurements of lifetimes, multipole mixing ratios,
and branching ratios can be used as additional components of the χ2 calculation, helping to
constrain matrix elements that are not otherwise well-constrained by the measured yields,
without introducing model dependency.

13

Chapter 2
Experimental Setup
In an inverse kinematics low-energy Coulomb excitation experiment, a projectile nucleus
impinges on a stationary target, and is excited in the Coulomb field of the target nuclei.
Detection of de-excitation Îł rays emitted from the scattered projectile is used to tag and
quantify the population of states resulting from the Coulomb excitation process.
The SeGA-JANUS setup consists of two main parts: particle detectors, composed of
silicon semiconductors, to detect scattered projectiles or recoiling target nuclei, and Îł-ray
detectors, consisting of high-purity germanium, to detect the de-excitation Îł rays. Fig. 2.1
shows an illustration of the major parts of the setup.

2.1
2.1.1

Particle Detection
Principles

As charged particles pass through matter, they deposit energy into the surrounding material.
This occurs primarily through inelastic collisions with electrons in the material, with each
electron carrying away a small portion of the energy. These electron inelastic collisions
transfer energy to the electrons, resulting either in excitations of atomic levels or in ionization
of atoms in the surrounding material. Since the cross sections for these collisions are quite
high, on the order of 100 megabarns [39], a large number of collisions occur over a short

14

SeGA Îł-ray
Detector

JANUS Particle Detector

Scattered
Projectile

Incoming
Projectile

Target
Nucleus

Recoiling
Target Nucleus
Figure 2.1: An illustration of the experimental setup, showing a projectile scattering off of
a target nucleus, and subsequently emitting a Îł ray. The scattered and recoiling nuclei are
detected in the JANUS particle detectors, while the emitted Îł rays are detected in the SeGA
detectors.
distance, each removing a small amount of energy from the heavy ion, one can consider the
stopping power dE
dx of a material, rather than treating each collision individually.
Since each electron carries away only a small portion of the total energy of the passing
charged particle, a classical approximation of the energy loss can be performed by determining the energy imparted to an electron by a heavy particle of constant velocity [39]. The
energy loss per collision is then as follows, where Z is the charge of the heavy ion, e is the
electron charge, me is the electron mass, v is the velocity of the heavy ion, and b is the

15

impact parameter:

∆E =

2Z 2 e4
.
me v 2 b2

(2.1)

For a beam of 3.9 MeV/u 78 Kr, with reasonable values of the impact parameter b, this results
in energy losses per collision of a few eV up to a few keV. This validates the earlier assumption
that only a small amount of energy is transferred to an individual electron. Following this
classical approach, averaging over many such collisions in a solid, leads to the following
formula for the differential energy loss, where Ne is the density of electrons in the material,
Îł = q 1

1−v 2 /c2

, and ν̄ is the mean frequency of revolution of a bound electron around the

nucleus in the material:
4πZ 2 e4
dE
=−
Ne ln
dx
me v 2

 2

Îł me v 3
.
Ze2 ν̄

(2.2)

A fully quantum mechanical description of the inelastic collision between energetic heavy
ions and the electrons in a medium yields the Bethe-Bloch equation, where Na is Avagadro’s
number, re is the classical electron radius, me is the electron mass, c is the speed of light,
z and A are the atomic number and mass of the absorbing material, Z is the charge of the
heavy ion, β is the v/c of the heavy ion, γ = q 1

1−β 2

, I is the mean excitation potential of

the absorbing material, and Wmax is the maximum energy transfer to an electron in a single
collision [39]:
 

dE
z Z2
2me Îł 2 v 2 Wmax
2
2
2
= −2πNa re me c ρ
ln
− 2β
.
dx
A β2
I2

(2.3)

Modern calculations, such as those performed by SRIM [40], are based on the Bethe-Bloch
16

equation, with additional corrections applied.
In a semiconductor detector, such as a silicon detector used here for particle detection,
energy deposited in the material is converted into a proportional number of electron-hole
pairs in the band structure of the semiconductor. In the electric field of the applied bias
voltage, the electrons and holes travel to the corresponding contacts, and can be collected
and measured as a charge, usually with a charge-sensitive preamplifier. The amplitude of
the voltage signal delivered by the preamplifier is proportional to the energy deposited in the
detector. The contacts of the semiconductor detector can be segmented, and by identifying
the contact segments which collected the charges enables inference of the location where the
particle interacted within the detector, thus providing position sensitivity.

2.1.2

JANUS Particle Detection

The Joint Array for Nuclear Structure (JANUS) consists of two S3 detectors manufactured
by Micron Technology, Inc. These are annular double-sided silicon detectors, each with an
inner radius of 1.1 cm, and an outer radius of 3.5 cm. Each detector has at one surface a
24-fold segmentation in the radial direction, and on the other surface a 32-fold segmentation
in the φ direction. By recording in which ring and which sector energy was deposited, the
pixel of interaction can be determined, as shown in Fig. 2.2. In this geometry, the S3 detector
delivers 56 signals, but establishes a segmentation into 768 pixels of about 1 mm by 5 mm in
size each. At a typical distance of 3.0 cm from a particle source, this translates to an angular
resolution of about 1.5◦ in θ and 11.25◦ in φ.
The upstream detector had a thickness of 302 Âľm, and the downstream detector had a
thickness of 313 Âľm. Based on SRIM calculations [40], the 78 Kr ions will stop in under 50 Âľm
of silicon, so the full energy of the ions will be deposited and measured.
17

Front

Back

Effective Pixelation

Figure 2.2: The segmentation in rings on one surface and in sectors on the other of a JANUS
S3 detector. Shown in blue is a hypothetical energy deposition into a ring and sector, with
the corresponding pixel in which energy was deposited.
The detectors were placed perpendicular to the path of the beam, one at 3.0 cm upstream
of the target position, and one at 3.0 cm downstream of the target position. This gave angular
coverage at scattering angles of 20.1-49.4◦ in the downstream detector, and 130.6-159.9◦ in
the upstream detector.
The JANUS assembly is shown in Fig. 2.3, with both detectors, the target wheel, and
an upstream 5 mm aperture. The target assembly contains two thin-foil targets, one empty
frame, and one 3 mm aperture. A wire is connected to each aperture allowing beam current
deposited on the apertures to be measured for beam-tuning purposes. In addition, the target
assembly can be given a positive high voltage. This suppresses the emission of δ electrons1
from the target into the JANUS detector, which would otherwise become background in the
detector spectrum.
A bias voltage of 40 V was applied across each detector. As the bias voltage was increased,
the leakage current was observed to increase linearly at low voltages before leveling off. This
indicates that the detectors were fully biased, and were no longer in the ohmic region.
1 δ electrons are secondary electrons produced by the projectile passing through the target that have
sufficient energy to leave the target.

18

Figure 2.3: The target position of the JANUS assembly. Shown here are the two silicon
detectors that compose JANUS, with the target wheel mounted between them. From left
to right, the target frame contains a 3 mm aperture, an empty frame, a 208 Pb target, and a
48 Ti target. The target wheel is operated manually through a rod system that is not shown
here. The orange and green wires allow collected charge on the apertures to be measured,
for beam-tuning purposes.

19

The 112 signals from JANUS were pre-amplified using preamplifiers provided by Swan
Research, before being shaped using Shaper/Discriminator modules manufactured by PICO
Systems, which were controlled through a CAMAC interface. The shaped output of these
modules was sent to CAEN v785 analog-to-digital converters (ADCs) , while a secondary
output signal was sent to CAEN v775 time-to-digital converters (TDCs). The ADC and
TDC modules were read out through a VME interface. The output trigger signal from the
Shaper/Discriminator modules was used to generate the trigger condition for the complete
JANUS system.
The PICO shaper has an input range of 500 mV, while the silicon signal is expected
to have an amplitude of ∟ 1 V for energy deposition of 300 MeV, expected for projectiles
detected in typical in-beam experiments. Therefore, a simple attenuator consisting of a
100 Ω resistor was used to attenuate the output signal from the preamplifiers, prior to being
sent to the PICO shapers. This was necessary only for the downstream JANUS detector, as
discussed in Sec. 3.2.1.
To reduce triggering on noise, while still operating at a low particle energy threshold, a
coincidence between the two sides of a single detector was required to trigger the readout of
JANUS.

2.2
2.2.1

Gamma-Ray Detection
Principles

In order to detect a Îł ray, it must first interact with the bulk material of a detector to produce
a charged particle. There are three ways in which Îł rays can interact with the electrons of
the detector material: photoabsorption, Compton scattering, and pair production. The cross
20

section for these three interactions in bulk germanium as a function of Îł-ray energy is shown
in Fig. 2.4.

103

PhotoAbsorption
Compton Scattering
Pair Production

Cross Section (b)

102
101
100
10-1
10-2
10-3
10-4 0

500

1000

Energy (keV)

1500

2000

Figure 2.4: Reaction probabilities of each interaction of a photon in a germanium detector.
Cross-section values taken from Ref. [41]

First, the photon can undergo photoabsorption, transferring its entire energy to an electron. This results in the full energy of the photon being detected, which is the ideal situation.
This is the primary effect for low-energy Îł rays.
Second, the photon can scatter off an electron, a process known as Compton scattering.
If the photon Compton scatters, then only the energy imparted on the electron is detected.
This results in a continuous distribution of energy at less than the full energy of the Îł
ray. There is a maximum amount of energy that can be transferred in a single scattering,
occurring when the γ ray scatters at 180◦ , which is less than the full energy of the γ ray. This
results in a sharp cutoff, known as the Compton edge. The scattered Îł ray may interact with
the detector multiple times, losing energy with each scattering event. This brings the energy
21

of the photon down to the region where photoabsorption is dominant, at which point it may
be absorbed. In this way, a series of Compton scatters may conclude with a photoabsorption
event, contributing ultimately to the full energy peak.
Finally, the photon can undergo pair production, producing an electron/positron pair. In
this case, some amount of kinetic energy is transferred to the electron and positron, and is
deposited directly into the detector. The positron will later undergo pair annihilation with
an electron, producing two Îł rays at 511 keV. One or both of these Îł rays may escape from
the detector, producing the single- and double-escape peaks located at 511 keV and 1022 keV
below the initial energy of the photon. By conservation of energy, this process is limited to
photons with more than 2me c2 = 1022 keV of energy.
The idealized result of the three interactions is shown in Fig. 2.5 with the example of a
3 MeV Îł ray interacting with a germanium detector.

Relative counts

Full Energy Peak

0

Single Escape Peak
Compton Edge
Double Escape Peak
500

1000

1500

2000

Energy (keV)

2500

3000

3500

Figure 2.5: Idealized Îł-ray energy spectrum of a 3 MeV Îł-ray, showing characteristic features.

22

2.2.2

Doppler Shift of Îł Rays

In experiments using inverse kinematics, the excitation of the beam particle is of interest.
Since the scattered beam particle is moving at a significant fraction of the speed of light, the
Îł-ray energies signaling its de-excitation will be Doppler shifted when observed in the lab
frame. That is, while the energy of the Îł ray is well-defined in the rest frame of the excited
nucleus, the Îł-ray energy as detected in the lab frame depends on the emitter velocity v/c
and the angle between the direction of motion of the nucleus and the direction of the emitted
Îł ray, as shown schematically in Fig. 2.6.

θ

v/c

Figure 2.6: Kinematics for the Doppler boost of a Îł ray emitted from a nucleus in motion.
All vectors shown are in the laboratory frame of reference.

When the direction of the outgoing particle and Îł ray are both measured, the energy of
the γ ray in the rest frame of the nucleus can be reconstructed as follows. Here, β = v/c,
q
1 , and θ is the relative angle between the outgoing particle and the γ ray, as
Îł =
2
1−β

23

measured in the lab frame:

Erest = Elab γ(1 − β cos θ).

(2.4)

The uncertainty of the resulting reconstruction depends on the uncertainties of Elab , β,
and θ. Each term contributes differently, based on the angle of emission of the γ ray. An
uncertainty in β has the smallest effect close to 90◦ , while an uncertainty in θ has the smallest
effect close to 0◦ and 180◦ . The contributions to the uncertainties are shown in Fig. 2.7,
for conditions similar to those in the commissioning experiment, and can be derived from
Eq. 2.4 by means of finite-difference derivatives, and the result can be found for example in
Fig. [11].

0.008

Resolution Contributions of Doppler Correction

Total Uncertainty
Intrinsic Resolution
Angular Resolution
β Uncertainty

0.007
0.006

σE
E

0.005
0.004
0.003
0.002
0.001
0.0000

20

40

60

80

100 120 140 160 180

θ (deg)

Figure 2.7: The effect of each experimental uncertainty on the Doppler reconstruction, as
a function of emission angle, based on equations from [11]. This plot was generated using
β = 8.5(5)%, σθ = 3◦ , and relative intrinsic resolution σE
= 0.1%. In this plot, σE is
lab
used instead of full width at half maximum (FWHM), where FWHM = 2.355σE

24

2.2.3

SeGA

The Segmented Germanium Array (SeGA) consists of 16 32-fold segmented high-purity
germanium detectors. Each crystal is 80 mm long and 70 mm in diameter. The crystals
are segmented into 8 layers laterally, with each layer being divided into 4 quadrants, as
shown in Fig. 2.8.

Figure 2.8: The segmentation of a single SeGA crystal, as shown from front, side, and
diagonal perspectives. Figure adapted from [34].

While silicon’s larger band gap of 1.1 eV allows for operation of silicon detectors at
room temperature, the large leakage current that would result from thermal excitations
across germanium’s smaller band gap of 0.7 eV prevents operation of germanium detectors
at room temperature [42]. Liquid nitrogen is used to cool the SeGA detectors to operational
temperature of about 100 K, preventing thermal excitations of electrons across the band gap.
Each SeGA detector assembly consists of the germanium crystal, preamplifiers, and a liquid
nitrogen dewar, as shown in Fig. 2.9. The detectors are arranged around the target position
as shown in Fig. 2.10. This setup is known as the barrel configuration, and maximizes the
solid angle coverage of SeGA.
Electrons are collected at the single central contact, and holes at the segment contact.
25

Figure 2.9: A single SeGA detector, outside of the experimental setup. The green cylinder
is the liquid nitrogen dewar, with the germanium crystal at the bottom of the image. The
preamplifiers are contained inside the metal casing.
The central contact therefore delivers the total energy of the Îł-ray energy absorbed in the
detector, while the segment gives position-sensitive information about where within the
detector the Îł-ray interactions occurred.
The charges collected on the segment contacts are used for position determination. For
Doppler-shift reconstruction, the location of the first interaction of the Îł ray is required.
For that, the segment with the highest energy deposition is chosen. The detection position
is then used to determine the Îł-ray emission angle as part of the Doppler correction, as
described in Sec. 2.2.1. Each segment is 1 cm long, given an opening angle of 5.8◦ . This is
approximated in Fig. 2.7 with σθ = 2◦ .
Each detector assembly contains charge-sensitive preamplifiers. The preamplified signals
were then digitized using 100 MHz PIXIE-16 modules from the manufacturer XIA. Digitizerbased electronics such as the PIXIE-16 system have many benefits over analog electronics
26

c 2016
Figure 2.10: The SeGA array positioned around the target chamber. Photograph 
NSCL.
such as shaper modules or discriminators for the processing of detector signals. The overall
setup is simpler, as the digitizers take the role of a shaper, discriminator, analog-to-digital
converter, and time-to-digital converter in one module. In addition, the PIXIE-16 cards can
record the waveform of its input signal as a function of time. The waveforms captured can
be used to improve the position resolution of the SeGA detectors, determining the position
of the Îł-ray interaction within a segment. The radial position can be determined by the
shape of the waveform of the central contact, while the axial and azimuthal positions can be

27

determined by the size of transient signals observed on neighboring segments. The algorithms
necessary to determine the interaction location and their applicability to SeGA are discussed
in detail in the thesis of S.R. Stroberg [43]. Sub-segment resolution was not required for this
study of 78 Kr, and was not used in the present work.
Each central contact was allowed to generate a trigger on a leading-edge threshold. When
the central contact triggers, all segments in the associated detector are also read out. For
each channel, an energy, time, and trace were recorded at a 100 MHz sampling frequency.
The trace length for each central contact was 1 Âľs, and the trace length for each segment
was 2.5 Âľs. The total data collected for each Îł ray detected was approximately 16.7 kB.
Each detector contains 32 segments and the central contact, for a total of 33 channels.
Using the full array of 16 detectors, a single XIA crate2 could not contain sufficient PIXIE-16
modules to equip all channels. Three separate XIA crates were used to record all 33∗16 = 528
channels. The data acquisition system was run in a timestamped mode, where each channel
was registered along with the time at which it occurred, allowing offline reconstruction of
full events. A clock signal generated was distributed to all PIXIE-16 modules, as described
in [44]. No drift between the timestamps recorded in each crate was observed.
The system was limited to a Îł-ray detection rate of about 1.2 kHz. This limit is imposed primarily by the data throughput, that is the transfer of data from the XIA crates
to the computer. At the time of this experiment, the maximum data transfer rate was approximately 6-7 MB/s per crate, or 18-21 MB/s for the entire setup. With a per-event size of
16.7 kB, this corresponds to a system-wide limit of ≈ 1.2 kHz. The observed background rate
of 75 Hz per detector already exhausts this limit, before accounting for any beam-induced
2 The PIXIE-16 cards need a dedicated mainframe provided by the company XIA that are referred to as
XIA crates in this work.

28

Îł rays. In order to reduce the background rate, the system was gated on the beam spill of
the reaccelerator (see Sec. 3.1.1). A 30 ms gate around the 10 ms beam period allowed for a
reduction in the background rate by a factor of 5. The throughput limitation is primarily
exhausted due to the acquisition of trace data. If no trace data were recorded, each event
would be smaller by a factor of 30, resulting in a factor of 30 increase in the maximum Îł-ray
event rate.

2.3

Data Acquisition

The data was read out using the NSCLDAQ framework [45]. This framework allows for the
operation of multiple independent data acquisition systems, which are then merged into a
single output data stream. In this experiment, there were four separate computers, three for
the three XIA crates, and one for controlling the CAMAC and VME crates. The NSCLDAQ
framework provided a uniform way to start all four data acquisition systems, transfer the
recorded data to a single computer, merge all four data streams into a single output, and
write the resulting data to disk.
In order to correlate particle detections in JANUS and Îł-ray detections in SeGA, all
events were timestamped. The SeGA events, being recorded with PIXIE modules, have a
timestamp already built into the data structure. The JANUS timestamp was generated from
the 50 MHz clock signal, provided by the PIXIE modules, and recorded in a SIS3820 scaler
module. Unlike other scaler modules in this setup, which provide periodic diagnostics, this
scaler module was read out for each event. In addition, the module was set to the “deep
counter” mode, where it registers two 48-bit counters, rather than sixteen 32-bit counters.
This was necessary as a 50 MHz counter would overflow a 32-bit counter in 86 seconds,

29

whereas a 48-bit counter would require 65 days to overflow.
As the clock signal provided by the PIXIE modules starts somewhat before the PIXIE
modules themselves start counting, it is necessary to gate the clock signal based on whether
the PIXIE modules are in a running state. Otherwise, the JANUS electronics will start
counting earlier than the PIXIE modules, resulting in a significant timestamp offset, on the
order of hundreds of milliseconds.
Each SeGA central contact needs to be self-triggering, validated on the beam extraction
period of the EBIT (see Sec. 3.1.1), with the segments being read out whenever the corresponding central contact triggers. This is done primarily using the front panel diagnostics
of the PIXIE modules. To validate the central contact signals, the logical OR of all triggers
is taken from the front panel. The logical AND of this signal with the validation signal from
the beam spill is produced, with the resulting signal being sent back to the central contact
module as an external validation. Simply sending the validation signal directly into the
module will result in undesired behavior, as the PIXIE modules recognize transitions in the
external validation, rather than the absolute value. On each transition to a high voltage, the
PIXIE module opens a validation window, with a configurable length of up to a few Âľs. For
validation with the beam spill, which lasts tens of milliseconds, it is necessary to construct
the more elaborate validation signal, which gives a transition on each self-trigger, but only
if the beam spill validation is present.
To trigger the segment modules, the central contact self-triggers are first sent to the
backplane of the XIA crate, then a custom circuit board sends each central contact trigger
to an individual LEMO connector. Using analog logic modules, each trigger signal is gated
on the beam spill validation, and is then sent to the corresponding segment modules as an
external trigger.
30

The trigger for JANUS is done entirely through analog logic. Each JANUS detector has
32 sectors and 24 rings, and each PICO Shaper/Discriminator module is set up to process
either 16 sectors of a single detector, 16 rings of a single detector, or 8 rings of a single
detector. Each PICO module has as an output the logical OR of all of its channels. These
are used to generate the raw trigger from ring/sector coincidences for each detector, as shown
in Fig. 2.11. This raw trigger is then gated on the current status of the VME crate, and the
current run status of the XIA crate, in order to generate the live trigger.

Figure 2.11: The trigger logic for the JANUS setup, generating the raw trigger from the
individual channel triggers.

31

2.4

Testing of SeGA-JANUS with a

252

Cf Source

The experimental setup was tested using a thin-walled 252 Cf source. 252 Cf undergoes spontaneous fission, producing fission fragments that can be in an excited state. These excited
fission fragments then deexcite, emitting Îł-rays while in-flight, typically at v/c of 3-4%. This
produces a Doppler shift based on the relative angle between the fission fragment and the
emitted Îł-ray.
Since the 252 Cf source provides correlated fission fragments and Îł rays, it can be used
to test the integration of the SeGA and JANUS systems, rather than testing each system
in isolation. This test can confirm that the two systems are triggering simultaneously, that
the timestamps between the two systems are synchronized, and that the relative position
between the two systems is well-known.
When 252 Cf undergoes spontaneous fission, it splits into two fragments. These fragments
are typically of unequal mass, as shown in Fig. 2.12, and are emitted directly opposite
from each other, according to conservation of momentum. This symmetry, along with the
symmetry of the upstream and downstream JANUS detectors, causes two fission fragments
to be detected simultaneously with a high probability. Since the relative angle between the
fission fragment and the Îł ray is necessary to perform Doppler reconstruction, one must
select which detected fission fragment emitted the Îł ray. This is done by selecting either the
fragment with higher or lower kinetic energy. In order to select the lower-mass fragments
associated with the most intense Îł-ray transitions, events with higher energy deposition in
the particle detectors are selected.
This selection of fission fragments is an additional complication as compared to the inbeam commissioning, where due to the kinematics and detector layout, when a scattering

32

Figure 2.12: The mass distribution of 252 Cf fission fragments. (Figure from [46].)
78 Kr

particle is detected the recoiling 208 Pb will be outside the acceptance of the particle

detectors, and vice versa.
In this source test, both JANUS detectors were used in conjunction with six SeGA
detectors. For logistic reasons, the full setup of 16 detectors was only used during the
experiment itself. Shown in Fig. 2.13 is the lab-frame energy of detected Îł rays, plotted
against the relative angle with the fission fragment. As expected, emissions at low angles,
traveling in the same direction as the fragment, are shifted to higher energies, while emissions
at high angles, traveling in the opposite direction as the fragment, are shifted to lower
energies.
The effect of the Doppler reconstruction is shown in Fig. 2.14. The resolution of the
241 keV transition from 110 Ru is 2% FWHM. The achieved FWHM of 2% at 241 keV

33

Figure 2.13: The Îł-ray energies from fission fragments from 252 Cf plotted against the angle
between the lighter fission fragment and Îł ray. As an example, the 241 keV transition from
110 Ru, Doppler-shifted in the laboratory system, is circled in red. Also visible is the 212 keV
transition from 100 Zr. These transitions are observed strongly, as they are near the peak
of the low-mass fragment distribution, as shown in Fig. 2.12. Gamma rays emitted from
the heavier fission fragments are observed as well, appearing as faint lines with orientation
opposite that of the 212 keV and 241 keV transitions
was benchmarked with a provided, reconstructed 252 Cf source spectrum measured at ANL
with the GRETINA/CHICO2 setup [30]. With GRETINA and CHICO2 having a superior
angular resolution compared with the setup discussed in this thesis, the same reconstructed
energy resolution was achieved, suggesting that the measured resolution of the data measured
with the SeGA-JANUS setup is entirely limited by the uncertainty of v/c for the Doppler
reconstruction. The successful Doppler reconstruction shows that both detectors are being
read out correctly, can be time-correlated together, and that their relative positions are
well-understood.

34

Figure 2.14: The Îł-ray energy spectrum, with and without Doppler correction, for fission
fragments from 252 Cf. The resolution of the Doppler-corrected Îł energy resolution is 2%
FWHM.

35

Chapter 3
Sub-barrier Coulomb Excitation of
78Kr

3.1

Experimental Details

The experiment ran from January 27-30, 2017 at the National Superconducting Cyclotron
Laboratory, located at Michigan State University, with the experimental setup arranged at
the multi-purpose beamline of the ReA3 reaccelerator facility. A total of 55 hours beam on
target was run in the production settings.
A beam of 3.9 MeV/u 78 Kr was impinged on a 1.09 mg/cm2 208 Pb target. Initially, the
beam rate was 1.6¡105 particles per second, which was later increased to ∟ 3 ¡ 105 particles
per second as the experiment progressed.
16 SeGA detectors were arranged in the barrel configuration, a close-packed geometry
that maximizes solid angle coverage and detection efficiency. During the run, one SeGA
detector exhibited a changing energy calibration every few minutes, and was excluded from
analysis. Three other detectors exhibited slow drifts in the energy calibration, which could
be corrected. Two other detectors exhibited sudden jumps in calibration, which could be
corrected.
During the experiment, the target frame was biased to 2 kV in order to prevent the

36

Su
pe

rco
nd

uc

EB

tin

IT

gL
IN

AC

General Purpose
Beamline

High Energy Beam Transport (HEBT)

Figure 3.1: Diagram of the reaccelerator (ReA3) at NSCL.
emission of δ electrons from the target.

3.1.1

Beam Production

A beam of 3.9 MeV/u 78 Kr was produced using the reaccelerator (ReA3) [47] at the National
Superconducting Cyclotron Laboratory (NSCL). Isotopically pure 78 Kr was bled from a gas
bottle into the electron-beam ion trap (EBIT) charge breeder [48], and was ionized toward
higher charge states. Ions at the 25+ charge state were selected and then accelerated to the
desired energy using the ReA3 linear accelerator, and directed at the experimental station,
located at the general purpose beamline. The layout of the accelerator is shown in Fig. 3.1.
The EBIT traps ions radially using the electric field of an electron beam, and axially
with an applied electric field [49]. While ions are in the trap, collisions with an electron
beam remove electrons from the captured ions, increasing the charge state. By removing the
external field, axial confinement is removed, allowing the charged ions to escape the trap
[48]. During this experiment, the ReA3 charge breeder ran at a frequency of 7 Hz, cycling
37

between injection of low charge state ions and extraction of high charge state ions. During
extraction, 78 Kr was released to the accelerator for a period of 10 ms. Outside of that period,
no beam was provided to the experimental setup. A logic signal indicating when the beam is
present is provided to the experimental area, and was used as part of the trigger condition of
the experimental setup, reducing Îł-ray events in SeGA stemming from natural background
radiation, and therefore staying within the total Îł-ray detection rate of the experimental
setup (see Sec. 2.3).
After the ions have been released from the EBIT, the desired charge state is selected
using a magnetic separation system. The ions selected are then accelerated, first with a
room-temperature radio-frequency quadrupole, then a superconducting linear accelerator
[33]. The ReA3 accelerator can accelerate U 238 to an energy of 3 MeV/u, or can accelerate
other isotopes to higher energies, with the maximum energy depending on the Q/A of the
isotope being accelerated [50].
With a beam of 78 Kr impacting a 208 Pb target, the Coulomb barrier is 4.2 MeV/u. The
selected beam energy of 3.9 MeV/u is therefore sufficient to stay below the Coulomb barrier.
This ensures that all excitations of the 78 Kr are solely due to electromagnetic interaction
with the 208 Pb target, rather than nuclear interactions (see Sec. 1.1).
The beam was tuned using a pilot beam of 36 Ar. For tuning purposes, the beam current
at two apertures aligned with respect to the optical beam axis were available in the setup.
The aperture at the target location had a 3 mm diameter, and the other aperture 6.7 cm
upstream of the target location had a 5 mm diameter. With the final tune of the pilot beam,
it was found that 100% passed through the upstream aperture and 90% through the aperture
at the target position, suggesting a beam spot size of approximately 3 mm.

38

3.1.2

Target Details

The target consisted of 1.09 mg/cm2 208 Pb deposited onto a carbon backing of 0.05 mg/cm2 .
With Z = 82, using a 208 Pb takes advantage of the increased Coulomb field at high values of Z [20]. In addition, as 208 Pb has closed proton and neutron shells, it has reduced
B(E2) transition strengths, as discussed in Sec. 1.1. This simplifies the analysis as the Îł-ray
spectrum only contains events from projectile excitations.
The thickness of the deposited 208 Pb was determined by measuring the energy loss of
Îą particles passing through the target. The Îą particles were produced from a collimated
source containing 244 Cm, 241 Am, and 239 Pu, which primarily emits Îą particles of 5.805 MeV,
5.486 MeV, and 5.157 MeV, respectively. The result of the energy loss measurement is shown
in Fig. 3.2.
The thickness was determined by using SRIM [40], varying the thickness of the 208 Pb
until the energy loss matched the energy loss observed, assuming the nominal thickness of
0.05 mg/cm2 for the carbon backing layer. When measuring without the target in place, the
three energies of the Îą particles provided the energy calibration.
The measurement was repeated at multiple locations of the target, to determine the
variability of the target thickness. Different areas had very similar energy loss, corresponding
to an inhomogeneity in target thickness of ∟ 1%.

39

Figure 3.2: In black, the energy spectrum of a radioactive source containing the Îą emitters
and 239 Pu. In red, the energy spectrum of the same source, with the 208 Pb
target put between source and detector. The target thickness is deduced from the observed
energy loss through the target. In addition to energy loss, the spectrum also becomes wider
due to energy straggling through the target [39].
244 Cm, 241 Am,

3.2
3.2.1

Data Analysis
Particle Identification and

78

Kr Reconstruction

By conservation of momentum, when a beam particle is scattered off of a target nucleus,
there is some momentum imparted to the target nucleus. If the recoiling target nucleus hits
a silicon detector, it can be detected. It is therefore necessary to distinguish between events
caused by scattered 78 Kr and events caused by recoiling 208 Pb.
The expected energy of the scattered 78 Kr and the recoiling 208 Pb are shown in Fig. 3.3,
based on an assumption of elastic scattering in relativistic kinematics. The angles of the
scattered projectile and recoiling target nucleus are shown in Fig. 3.4. As the total kinetic

40

350

Scattered 78 Kr
Recoiling 208 Pb

300

Energy (MeV)

250
200
150
100
50
00

20

40

60

80

100

120

Detection Angle (deg)

140

160

180

Figure 3.3: The expected energies of scattered 78 Kr and recoiling 208 Pb, across the full
angular range, based on 3.9 MeV/u incoming beam energy. This calculation assumes that the
two particles scatter elastically. As the nuclear excitations are at maximum a few MeV, as
compared against the ∟ 300 MeV beam energy, this is a reasonable assumption.
energy transferred to internal degrees of freedom is on the order of a few MeV, while the beam
energy is over 300 MeV, elastic scattering is a valid approximation. Relativistic kinematics
were used for availability of existing implementations. At projectile velocities of v/c <
0.1, the errors introduced by using classical kinematics are less than the sensitivity of the
detectors. The recoiling 208 Pb can only scatter at angles less than 90◦ in the lab frame.
In the upstream detector, at angles greater than 90◦ , no particle identification is necessary
to discriminate between recoiling 208 Pb and scattered 78 Kr. However, in the downstream
JANUS detector, located at lower scattering angles, both particles are present, and must
be separated. The experimental energies are shown in Fig. 3.5, with the two nuclei being
clearly distinguishable.
Using relativistic kinematics, the relationships between center of mass scattering angles,
41

Figure 3.4: The angles of the scattered projectile and recoiling target nucleus, along with
relative positions of the two particle detectors.
lab frame scattering angles, and kinetic energies are as follows, where Θ is the scattering
angle of the projectile in the center-of-mass frame of reference, ν is the scattering angle
of the projectile in the laboratory frame of reference, Îą is the angle of the recoiling target
nucleus, relative to the incoming projectile’s trajectory, in the laboratory frame of reference,
Îł = q 1

1−v 2 /c2

is the Lorentz factor of the center-of-mass frame, relative to the lab frame,

mp and mt are the rest masses of the projectile and target, respectively, M = mp + mt is
the total rest mass of the system, ρ = mp /mt is the mass ratio, ξp =

Tp
mp c2

is the kinetic

energy of the projectile in terms of the rest mass of the projectile, and Tp , Ts , and Tr are the
kinetic energy of the incoming projectile, scattered projectile, and recoiling target nucleus,

42

400

105

Scattered 78 Kr
Recoiling 208 Pb

350

104

250

103

Counts

Energy (MeV)

300

200

102

150
100

101

50
0

25

30

35

40

Detection Angle (deg)

45

50

100

Figure 3.5: Particle identification of scattered 78 Kr and recoiling 208 Pb in the forward detector of JANUS. Kinematic lines for each particle show the energy when scattering at a
given angle. The clear separation between the energies of the two isotopes allows for clear
discrimination in analysis. Due to electronics issues, the segment covering scattering angles
44-45◦ has poorer resolution, but is still sufficient to discriminate between 78 Kr and 208 Pb.
respectively [51]:

tan ν =

sin Θ
γ(cos Θ + ρg)

sin Θ
γ(1 − cos Θ)
Tp + M c2
Îł=
M c2
1 + ρ(1 + ξp )
g=
(1 + ξp ) + ρ

tan Îą =

(3.1)
(3.2)
(3.3)
(3.4)

2ρ(1 − εp /2)
Ts
=1−
(1 − cos Θ)
Tp
(1 + ρ)2 + 2ρξp

(3.5)

Tr = Tp − Ts .

(3.6)

43

The upstream detector does have some events occurring at energies higher than those
allowed by kinematics. These events are in coincidence with 78 Kr Îł rays emitted in the
laboratory frame without Doppler shift. These events can therefore be understood as 78 Kr
particles that scattered off an upstream beam element, as for example the 5 mm aperture,
and that were then excited while stopping in the JANUS detector. These only compose less
than 1 part in one thousand of the total data, and can be easily removed.
A measurement of the recoiling 208 Pb still contains useful information due to the twobody kinematics. Even though the 78 Kr is not detected, it may still be left in an excited
state, producing Îł rays that can be observed in SeGA. The observed scattering angle of
the detected 208 Pb along with an assumed interaction location at the center of the target is
sufficient to reconstruct the scattering angle of the excited 78 Kr, using Eq. 3.1 and Eq. 3.2,
as shown in Fig. 3.6. The angle between the reconstructed 78 Kr angle and the emitted Îł ray
can then be used to Doppler-correct the measured Îł-ray energy, as described in Sec. 3.2.3.
As a result of using the reconstructed 78 Kr trajectories, the effective solid angle of JANUS
is greatly increased. While the physical extent of JANUS covers angles only at 20-49◦ and at
131-160◦ , the reconstructed trajectories cover angles at 62-121◦ . The relationship between
the scattering angle of 78 Kr with the recoil angle of 208 Pb in the laboratory system is plotted
in Fig. 3.6, and allows inference of the angle for 78 Kr from 208 Pb events.

3.2.2

Gamma-Particle Coincidences

In order to remove room background Îł-ray events from the analysis, only Îł rays that occurred
in coincidence with a particle in the silicon detectors are considered. The gate being used
is reproduced in Fig. 3.7, which shows the Îł-ray energy plotted against the time difference
between the Îł-ray detection and the particle detection. Particle-Îł coincidences are seen as
44

180

78

Kr Scattering Angle (deg)

160
140
120
100
80
60
40
20
00

10

20

208

30

40

50

60

Pb Recoil Angle (deg)

70

80

90

Figure 3.6: Reconstruction of the scattered 78 Kr given the angle of the recoiling 208 Pb.
The gray region indicates the angular acceptance for recoiling 208 Pb, and the corresponding
angles of the reconstructed 78 Kr.
the vertical line in the spectrum. At low Îł-ray energies, the time difference changes, due to
walk in the Îł-ray trigger. Therefore, a two-dimensional gate is used to identify prompt Îł-ray
emission, with the gate being based both on relative time and Îł-ray energy.
Room background Îł rays can be observed in Fig. 3.7 as horizontal lines. The 1460 keV
peak from the decay of 40 K and the 2614 keV peak from the decay of 208 Tl into 208 Pb are
clearly visible. These backgrounds are effectively removed with the gate. As there is no
significant intensity near the 2614 keV region after the gate has been applied, it is verified
that occurrences of target excitations are minimal.

45

Figure 3.7: The correlation between lab-frame Îł-ray energy on the y-axis and the time
difference between Îł-ray detection and particle detection on the x-axis. The time-energy
gate for determining coincidence between particles and Îł rays is shown in red. The 1460 keV
and 2614 keV room background peaks from the decay of 40 K and 208 Tl.

46

3.2.3

Doppler Correction

Photons emitted from a moving source will be Doppler-shifted according to the relative
velocity between the source and the observer (see Sec. 2.2.2). As the incoming beam has a
velocity on the order of v/c ≈ 10%, the measured γ-ray energy in the laboratory frame of
reference will be Doppler-shifted relative to the emitted Îł-ray energy in the rest frame of the
78 Kr.

This is described in more detail in Sec. 2.2.1.

The relationship between observed Îł-ray energy and emission angle obtained from the
particle and Îł-ray detection is shown in Fig. 3.8. Gamma rays emitted at angles less than
90◦ are shifted to higher energies and γ rays emitted at angles greater than 90◦ are shifted
to lower energies, while γ rays emitted at about 90◦ are unshifted. The effect of the Doppler
reconstruction is shown in Fig. 3.10. The resulting Doppler-reconstructed peak has a FWHM
of 1.6%. This is better than the resolution obtained using the 252 Cf source as described in
Sec. 2.4, proving that the resolution in the 252 Cf is indeed limited by the velocity uncertainty. For the in-beam data, the limitation in the achievable Îł-ray energy resolution is given
primarily by the angular resolution of SeGA.
For 78 Kr events reconstructed from the measured trajectory of recoiling 208 Pb, the 78 Kr
may have scattered at angles close to 90◦ . In these cases, the 78 Kr travels a longer distance
before leaving the target, losing large amounts of kinetic energy to the target. This results
in Îł-ray emission at lower velocities, with smaller shifts due to the Doppler effect. This
can be seen in Fig. 3.9, which shows the relationship between Îł-ray emission angle and labframe Îł-ray energy for reconstructed 78 Kr events. Gamma rays emitted in flight produce a
Doppler-shifted energy curve, similar to that in Fig. 3.8, while Îł rays emitted at rest form a
vertical distribution along the 455 keV Îł-ray energy without a dependence on the measured

47

180

90
80

140

70

120

Counts/keV/deg

Îł-ray Emission Angle (deg)

160

60

100

50

80

40

60

30

40

20

20

10

0
350

400

450

500

Îł-ray Energy, Lab Frame (keV)

550

Figure 3.8: The lab-frame Îł-ray energy plotted against Îł-ray emission angle, for 78 Kr detected in the downstream JANUS detector. This spectrum only shows the dominant 455 keV
transition in 78 Kr.
emission angle. This effect contributes to worse Îł-ray energy resolution for reconstructed
78 Kr

events.

In Coulomb excitation, a significant fraction of the beam’s original kinetic energy can
be imparted onto the target nucleus. The greater the projectile scattering angle, the more
energy has been transferred to the target nucleus. By conservation of energy and momentum
in the two-body kinematics, the velocity of the outgoing beam particle can be determined by
the scattering angle of the outgoing beam, using Eq. 3.1 and Eq. 3.5, as shown in Fig. 3.11.
In order to account for the angle-dependent value of v/c, the 78 Kr events were split into
3 regions of scattering angle coverage, consisting of the downstream detector, the upstream
detector, and the 78 Kr events reconstructed from measured 208 Pb (see Sec. 3.2.1). In each
region, the value of v/c used to perform the Doppler reconstruction was varied to minimize

48

Figure 3.9: The lab-frame Îł-ray energy plotted against Îł-ray emission angle, for 78 Kr events
reconstructed from the measured trajectory of recoiling 208 Pb. This spectrum only shows
the dominant 455 keV transition in 78 Kr. Different from Fig. 3.8, an unshifted component
of the 455 keV Îł-ray energy is visible in addition to the Doppler-shifted component, caused
by 78 Kr stopped in the target.

49

7000

Dopper-corrected, v/c =0.085
Lab Frame

6000

Counts/keV

5000
4000
3000
2000
1000
0
400

420

440

460

480

500

Îł-ray Energy (keV)

520

540

Figure 3.10: The lab-frame energy spectrum, compared against the Doppler-reconstructed
energy spectrum, for 78 Kr detected in the downstream JANUS detector, using v/c = 0.085
in the Doppler reconstruction. This spectrum only shows the dominant 455 keV transition
in 78 Kr.

50

Outgoing 78 Kr Velocity (v/c)

0.10
0.08
0.06
0.04
0.02
0.000

20

40

60

80

100

120

Scattering Angle (deg)

140

160

180

Figure 3.11: The velocity of the scattered 78 Kr as a function of scattering angle. The
three ranges of scattering angles of 78 Kr deduced from the silicon detector coverage are
indicated. The variation of v/c within a region is sufficiently small for using a fixed Dopplerreconstruction v/c for each range.

51

the width of the 455 keV peak in 78 Kr. The resulting values of v/c were consistent with those
predicted by kinematics for each scattering angle region. The final Doppler-corrected Îł-ray
spectra are shown in Fig. 3.12. The region from 500 keV to 1000 keV is shown in greater
detail in Fig. 3.13, and the region from 1000 keV to 2000 keV is shown in greater detail in
Fig. 3.14.

3.2.4

Experimental Yields

The measured Îł-ray yields are extracted from the number of counts in the corresponding
peak in the Îł-ray spectrum, corrected for the detection efficiency of the SeGA array. The
energy-dependent detection efficiency was measured using a 152 Eu source of known intensity,
then interpolated to the energies of the observed 78 Kr Îł rays. The measured efficiencies and
the fit used for interpolation are shown in Fig. 3.15.
The measured yields must be combined into scattering angle ranges. Each region of
scattering angle corresponds to a different momentum transfer between the beam and target
nuclei. The four angle regions used in this analysis were the inner 12 rings of the downstream
detector, covering angles 20-37◦ , the outer 12 rings of the downstream detector, covering angles 37-49◦ , the reconstructed 78 Kr fragments, covering angles 62-121◦ , and the upstream
detector, covering angles 131-160◦ . These regions were selected to maximize angular granularity, while still ensuring that low-intensity peaks had a sufficient number of counts for
extracting peak intensities.
For the reconstructed 78 Kr, the presence of the Îł rays emitted at rest, described in
Sec. 3.2.3 must be accounted for. These Îł rays form a broad component on the sides of each
peak in the Doppler-shifted Îł-ray energy spectrum, as can be seen for the high statistics
455 keV peak in Fig. 3.12. In peaks with fewer counts, the broad component cannot be
52

859

1148

859

1148

693

664

1301
1301

1148

859

562

693

664

562

693

664

562

451
451
451

Figure 3.12: The Doppler-corrected energy spectrum for each angular region of the scattering
78 Kr. For performing Doppler correction, v/c = 0.085 was used for the downstream detector,
v/c = 0.06 for the reconstructed fragments, and v/c = 0.04 for the upstream detector.

53

859

693

664

859

693

664

859

693

664

562
562
562

Figure 3.13: The Doppler-corrected energy spectrum for each angular region of the scattering
78 Kr, showing the energy region from 500 keV to 1000 keV. For performing Doppler correction, v/c = 0.085 was used for the downstream detector, v/c = 0.06 for the reconstructed
fragments, and v/c = 0.04 for the upstream detector.

54

1148

1301

1148

1301

1148
Figure 3.14: The Doppler-corrected energy spectrum for each angular region of the scattering
78 Kr, showing the energy region from 1000 keV to 2000 keV. For performing Doppler correction, v/c = 0.085 was used for the downstream detector, v/c = 0.06 for the reconstructed
fragments, and v/c = 0.04 for the upstream detector.

55

SeGA Detection Efficiency (%)

22

Measured Efficiency Curve
Fitted Efficiency Curve

20
18
16
14
12
10
8
6
4
200

400

600

800

1000

Îł-ray Energy (keV)

1200

1400

1600

Figure 3.15: Measured SeGA efficiency curve using a 152 Eu source. The red line shows a fit
to the measured efficiencies, with functional form 1157 exp(−0.7511 log(E + 74.6)), where E
is the energy in keV, and the resulting value is in percent.
easily identified. Therefore, to account for the full peak intensity including both in-flight
and at-rest Îł-ray emission, a narrow integration region is used around each peak, and the
resulting integral is scaled by the proportion of total peak area to narrow peak area, as
determined from the 455 keV peak.
The 1148 keV peak in the reconstructed 78 Kr presents additional difficulties, as the choice
of background region has a very strong influence on the extracted peak area. Here, the known
2+ →2+

2
1 = 0.624(12) [52] was used to inform the choice of background
branching ratio of +
22 →0+
1

regions to reproduce the known value.
The measured Îł-ray peak areas for all scattering angle region are shown in Table 3.1.
Transitions not observed in a given scattering angle region are left blank. When determining
the experimental yield for each transition, a 3% systematic uncertainty was included.

56

Transition
+
2+
1 → 01
+
0+
2 → 21
+
4+
1 → 21
+
2+
2 → 21
+
61 → 4+
1
+
2+
2 → 01
+
2+
3 → 21

Energy (keV)
455
562
664
693
859
1148
1301

Scattering Angle Region
37-49◦
62-121◦
28510(170) 17826(136) 21146(154)
55(11)
541(64)
686(27)
1227(36)
4125(90)
124(12)
171(14)
935(60)
31(7)
43(10)
550(63)
57(8)
92(10)
410(30)
80(15)
20-37◦

131-160◦
1344(40)
106(14)
292(20)
100(15)
100(17)
39(12)
17(9)

Table 3.1: The experimental peak areas for each scattering angle region. Transitions not
observed in a given scattering angle region are left blank.

3.3

Matrix Element Extraction

In low-energy Coulomb excitation, multi-step excitations can occur, and so the population
of an excited state may result from direct excitation, excitation through an intermediate
state, or feeding from higher-lying states. In order to extract all transition matrix elements,
the coupled-channel Coulomb excitation analysis code gosia [38] was used, a high-level
description of which is present in the Introduction of this work.
Gosia takes as input the measured Îł-ray yields, the angular coverage of particle and
Îł-ray detectors, the spins, parities, and energies of the nuclear energy levels, transition
and diagonal matrix elements to be varied, incoming beam energy and energy loss through
the target, and internal conversion coefficients. The exact form of the inputs provided
to gosia and their interpretation are described more fully in App. B. gosia performs a
χ2 minimization, varying the matrix elements of the nucleus being studied, in order to
reproduce the measured yields. In addition, values for lifetimes, multipole mixing ratios,
branching ratios, and matrix elements can be added as additional data points, contributing
to the χ2 value. Gosia allows the absolute normalization of experimental yields to vary,
accounting for absolute detector efficiency, target thickness, and integrated beam on target.

57

The level scheme supplied to the gosia input is shown in Fig. 3.16. Levels and transitions
beyond those observed are included in the calculation as buffer states [36], described in Ch. 4.
As no experimental yields are present for these transitions, the predicted yields of these
transitions do not contribute to the total χ2 . Should the matrix elements extracted predict
a yield for these unobserved transitions that is above a user-defined threshold, gosia will
issue a warning, indicating that either the calculation should be rerun, or the data should
be re-examined for the transition. For all transitions included in the fit, E2 matrix elements
are included. In transitions with ∆J ≤ 1, M1 matrix elements are also included. As only
positive parity states are observed in the measurement, E1 and E3 matrix elements are not
relevant.
For the initial fit parameters, estimations of matrix elements were made assuming an
asymmetric rigid rotor model [53], which predicts the relative values of transition and diagonal matrix elements within a deformed nucleus. The asymmetry parameter Îł of the rigid
rotor determines both the energy levels and matrix elements of the nucleus being described.
The γ deformation parameter was chosen as 22.5◦ , because at that value, the asymmetric
+
rigid rotor model predicts that the energies of the 4+
1 and 22 states will be nearly equal

[53], as is the case in 78 Kr. The absolute value of initial parameters was determined by
scaling these relative values from the asymmetric rigid rotor model to match a previous
+
measurement of the E2 matrix element of the 2+
1 → 01 transition.

During initial fits, while determining the minimum number of degrees of freedom necessary to describe the data, most matrix elements were constrained to stay at the same value
+
78
relative to the 2+
1 → 01 transition. As Kr is not very well-described by either an asym-

metric rigid rotor or a vibrator [54], it was necessary to relax these constraints as the fit
settled into a local minimum. In order to avoid having many ill-constrained matrix elements,
58

78 Kr
2994
1016

81+

2444
1978
1873
1756
859

739
637
609 (δ)
1418
725
754

61+ (τ)
42+
23+ (B)

1148
1119
1017

1756

22+ (τ)
41++ (τ)
02 (τ)

21+ (τ)

455

455

664
562
693 (δ)
1301 (δ)
1148

Energy (keV)

688

(24+ )

0

01+

Figure 3.16: The level scheme of 78 Kr, as used for fitting matrix elements to the observed
data, with indications to show the use of literature values, contributing to the χ2 value.
Levels and transitions in solid lines were directly observed in this experiment. Levels and
transitions in dashed lines are included in the coupled channels calculation, but do not have
experimental yields contributing to the χ2 calculation. States marked with a τ had their
lifetimes used as components of the χ2 . States marked with a B had branching ratios between
all de-excitation γ rays used as components of the χ2 . Transitions marked with a δ had their
E2/M1 mixing ratio used as a component of the χ2 .

59

Level
2+
1
0+
2
4+
1
2+
2
6+
1

Mean Lifetime [ps]
31.2(10)
13.0(36)
3.63(17)
4.8(9)
0.94(10)

Table 3.2: The lifetimes used as data points in the gosia χ2 minimization [52].
2+
3
2+
3
2+
3
2+
3

Transitions
+
+
→ 0+
1 /23 → 21
+
+
→ 0+
2 /23 → 21
+
+
→ 41 /23 → 2+
1
+
+
→ 2+
2 /23 → 21

Branching ratio
0.254(8)
0.518(8)
0.119(4)
0.047(10)

Table 3.3: The branching ratios used as data points in the gosia χ2 minimization [52].
additional data from previous measurements were included in the χ2 fit. Additional values
used for the lifetimes, branching ratios, and mixing ratios are listed in Tables 3.2, 3.3, and
3.4, respectively, and were taken from NNDC [52]. In addition, higher-lying matrix elements
from [55] shown in Table 3.5, where these are ill-constrained from experimental yields in the
current work.
Transition
+
2+
2 → 21
+
2+
3 → 22
+
2+
3 → 21

E2/M1 mixing ratio
0.45(10)
4.0(35)
-1.32(55)

Table 3.4: The mixing ratios used as data points in the gosia χ2 minimization [52]. Signs
of mixing ratios use the Krane sign convention [56].

60

Matrix
Value
 + 

 +  Element




0.19(3)

81+ E261+ 
E22
4
0.093(12)

 2+   2+ 
 
0.040(5)
2 E241 

4+
+




42 E2 21
0.006(1)
Table 3.5: The matrix elements from [55] used as data points in the gosia χ2 minimization.
Use of these transition matrix elements to so-called buffer states is described in Ch. 4.

61

Chapter 4
Results
Using gosia, matrix elements were successfully extracted. The reproduction of the measured
yields is shown in Fig. 4.1. The left column of Fig. 4.1 shows how the fitted yield for each
observed Îł-ray transition compares to the measurements for the four scattering angle groups.
It is noted that the yields in Coulomb excitation vary by orders of magnitude depending on
the transition and the scattering angle region, underlining the sensitivity of the approach.
The right column of Fig. 4.1 shows the number of standard deviations between measured
and fitted yields. With only a few exceptions, the fitted and measured yields agree within
2 standard deviations or better, which is reasonable for quantities that vary over orders of
magnitude.
The fit contained 26 matrix elements and 4 normalization constants (see Sec. 3.3), being
fit to 25 γ-ray yields and 16 values from previous measurements. The reduced χ2 reported
from gosia resulting from this fit was 1.6. The 16 values from previous experiments are
listed in Tables 3.2, 3.3, 3.4, and 3.5. These values are incorporated into the calculation of
the χ2 , attempting to drive the multidimensional minimization towards a viable minimum.
Four of the 16 values, listed in Table 3.5, provide a χ2 contribution for so-called buffer
+
states. These are the 8+
1 and 42 , the first nonobserved states in the yrast and yrare bands,

respectively. In such Coulomb excitation calculations, these states account for possible highlying states that were unobserved [57]. For the third of such buffer states, the 2+
4 , no input

62

102
104

Scattering Angles 131-160°

4
2
0
2
4

1301 keV, 23+ → 21+

1148 keV, 22+ → 01+

859 keV, 61+ → 41+

693 keV, 22+ → 21+

664 keV, 41+ → 21+

562 keV, 02+ → 21+

102
101

455 keV, 21+ → 01+

103

Scattering Angles 131-160°

1301 keV, 23+ → 21+

103

Scattering Angles 62-121°

1148 keV, 22+ → 01+

104

4
2
0
2
4

Scattering Angles 37-49°

859 keV, 61+ → 41+

Scattering Angles 62-121°

4
2
0
2
4

693 keV, 22+ → 21+

Scattering Angles 37-49°

664 keV, 41+ → 21+

Fitted Yield
Experimental Yield

Scattering Angles 20-37°

562 keV, 02+ → 21+

105

4
2
0
2
4

455 keV, 21+ → 01+

Efficiency-Corrected Counts

106
105
104
103
102

Scattering Angles 20-37°

Z-score

106
105
104
103
102

Figure 4.1: The fitted experimental yields for each observed Îł ray transition. In black are the
experimental values, and in red are the fitted values. The experimental yield was extracted
from the peak areas corrected by the Îł-ray detection efficiency. On the left are the fits to each
peak area, in efficiency-corrected counts. On the right are the z-scores of the comparison,
the number of standard deviations between the experimental values and the fitted values,
for each data point. Missing points for certain angle groups indicate that a peak was not
observed for that particular transition.

63

was provided to gosia. The uniqueness and robustness of this result was tested by providing
randomized matrix elements as the initial parameters to the gosia minimization routine [36],
then examining the resulting minimized χ2 , to identify if a better minimum could be found.
Of 350 such minimizations, none of the random initial parameter sets resulted in a better
χ2 than the fit presented here.
In Tables 4.1, 4.2, and 4.3 are shown the transition E2 matrix elements, M1 matrix
elements, and diagonal quadrupole matrix elements, respectively. It is noted that sensitivity
to M1 transition matrix elements in low-energy Coulomb excitation is encoded in the Îł-ray
angular distributions of the de-exciting transitions with mixed E2/M1 multipolarity. The
present setup and level of statistics did not lend itself to a detailed study of Îł-ray angular
distributions and, therefore, the extracted M1 matrix elements are based on the literature
multipole mixing ratios that were part of the gosia minimization. For the same value of an
M1 and E2 matrix element, the cross section for the excitation of an M1 mode is suppressed
by (v/c)2 and so any M1 information in the measured cross sections is negligibly small [35].
Transitions included in the Coulomb excitation calculation but constrained according to
the asymmetric rigid rotor model (see Sec.3.3) are not included in the table. Measured results
are compared with values taken from the 2009 NNDC evaluated data set [52], and with a
low-energy Coulomb excitation measurement performed by Becker et al. [55], which used a
beam of 78 Kr impinged on targets of 208 Pb, 48 Ti, and 26 Mg at various energies below the
Coulomb barrier, in effect providing a data set that results from 6 different measurements.
Most matrix elements agree with the literature values, as shown in Fig. 4.2. The results for
each observed state and its transitions are discussed in detail below.
78 Kr, decays by emitting a single Îł
The 2+
1 state at 455 keV, the first excited state of

ray, transitioning to the ground state. This 455 keV transition has an extracted E2 matrix
64


If ||E2||Ii [eb]




NNDC If ||E2||Ii [eb]




Becker et al. If ||E2||Ii [eb]

0+
1

0.81+0.01
−0.01

0.82+0.01
−0.01

0.82+0.02
−0.02

2+
2

0+
1

0.159+0.005
−0.005

0.13+0.01
−0.01

0.157+0.003
−0.004

0+
2

2+
1

0.243+0.016
−0.007

0.31+0.01
−0.01

0.30+0.01
−0.01

4+
1

2+
1

1.30+0.02
−0.02

1.25+0.04
−0.04

1.27+0.05
−0.02

2+
2

2+
1

0.43+0.02
−0.03

0.24+0.05
−0.05

0.26+0.06
−0.05

6+
1

4+
1

1.63+0.07
−0.07

1.56+0.09
−0.09

1.61+0.06
−0.08

2+
3

2+
1

0.084+0.004
−0.005

0.068+0.004
−0.004

0.50+0.02
−0.05

2+
3

0+
1

+0.0007
0.0384−0.0007

0.020+0.001
−0.001

0.180+0.007
−0.008

4+
2

2+
1

0.074+0.005
−0.006

0.069+0.005
−0.005

0.073+0.002
−0.005

2+
3

0+
2

0.48+0.01
−0.01

0.25+0.01
−0.01

0.26+0.010
−0.01

2+
3

4+
1

0.330+0.006
−0.006

0.178+0.008
−0.008

0.22+0.20
−0.05

2+
3

2+
2

0.15+0.02
−0.37

0.12+0.02
−0.02

0.19+0.32
−0.05

4+
2

2+
2

0.89+0.05
−0.07

0.95+0.06
−0.06

0.91+0.06
−0.04

Ii

If

2+
1




Table 4.1: The E2 matrix elements and uncertainties as extracted by gosia, compared with
values from NNDC [52] and Becker et al. [55]. Transitions below the horizontal divider were
not directly observed in this experiment, but were included in the gosia minimization. Not
+
shown are 3 ill-constrained matrix elements, including transitions from the 8+
1 and 24 buffer
states.

65




NNDC If ||M 1||Ii [ÂľN ]




Becker et al. If ||M 1||Ii [ÂľN ]

+0.02
0.33−0.02

0.30+0.03
−0.03

0.38+0.01
−0.03

2+
1

−0.149+0.003
−0.002

0.057+0.004
−0.004

−0.41+0.12
−0.04

2+
2

+0.02
0.09−0.02

0.016+0.014
−0.008

Ii

If

2+
2

2+
1

2+
3
2+
3





If ||M 1||Ii [ÂľN ]

Table 4.2: The M1 matrix elements and uncertainties as extracted by gosia, compared with
values from NNDC [52] and Becker et al. [55]. (See text for a discussion of the sensitivity
to M1 transitions.)

I

hI||E2||Ii [eb]

Becker et al. hI||E2||Ii [eb]

Q [eb]

Becker et al. Q [eb]

2+
1

−0.96+0.11
−0.47

−0.80+0.04
−0.04

−0.73+0.08
−0.36

−0.61+0.03
−0.03

4+
1

−1.2+2.4
−1.0

−0.73+0.15
−0.14

−0.9+1.8
−0.7

−0.55+0.11
−0.10

2+
2

1.0+0.3
−0.7

0.58+0.04
−0.08

0.7+0.2
−0.6

0.44+0.03
−0.06

2+
3

1.6+1.8
−0.5

−0.22+0.09
−0.14

1.2+1.4
−0.4

−0.17+0.07
−0.10

6+
1

−0.8+2.2
−1.3

−0.87+0.16
−0.12

−0.6+1.5
−0.9

−0.61+0.11
−0.08

Table 4.3: The diagonal matrix elements and uncertainties as extracted by gosia, compared
with values from Becker et al. [55]. Results are quoted both as diagonal matrix elements
and as quadrupole moments. Not shown are 2 ill-constrained matrix elements.

66

ÂŽ

ÂŽ

61+ ||E2||41+

23+ ||E2||21+

ÂŽ

ÂŽ

22+ ||E2||21+

­

2

23+ ||E2||23+

ÂŽ

41+ ||E2||21+

­

1

Diagonal Matrix Elements
­

ÂŽ

02+ ||E2||21+

­

­

ÂŽ

22+ ||E2||01+

­

Transition Matrix Elements

­

ÂŽ
­

21+ ||E2||01+

0.2

ÂŽ

0.4

0

22+ ||E2||22+

0.6

1

­

0.8

ÂŽ

1.0

2

21+ ||E2||21+

1.2

0.0

Present Work
Becker et al.

3

1.4

E2 Matrix Element Value [eb]

E2 Matrix Element Value [eb]

1.6

4

Present Work
Becker et al.
NNDC

­

1.8

Figure 4.2: A comparison between transition and diagonal matrix elements measured in the
present work, from Becker et al. [55], and from NNDC [52]. The comparisons are discussed
in the text.

67

element that agrees within uncertainty both with the value reported in NNDC, and the value
from [55]. The lifetime of the 2+
1 state has been deduced before from various recoil-distance
Doppler shift measurements [58, 59, 60, 61, 62, 63], Doppler-shift attenuation measurements
[64, 65], and from the B(E2) value extracted in the earlier Coulomb excitation measurement
[55]. As typical in these massive minimizations with gosia [66], the lifetime of the 2+
1 state
was used as an input to the χ2 search in gosia.
+0.11
The diagonal E2 matrix element of the 2+
1 state was also extracted as −0.96−0.47 eb

corresponding to a quadrupole moment of −0.73+0.08
−0.36 eb, 1.4σ away from the value measured
by Becker et al. [55], but clearly indicating prolate deformation from its sign.
+
The second excited state, 0+
2 , at 1017 keV decays to the 21 by emitting a 561 keV Îł ray.

This transition is weak and not observed in the forwardmost 20-37◦ scattering angle region.
The extracted matrix element for this transition is similar to previously measured values,
though outside of the uncertainties of the measurements by 3.5σ. Previous measurements
of the lifetime of this state compiled in NNDC [52], derived from Doppler-shift attention
measurements [67] and Coulomb excitation [55], are used in the χ2 minimization. When

   + 
 
no previous measurements are used in the gosia minimization, the 2+
1 E2 02 extracted
matrix element does not change within the uncertainty reported here.
+
The 4+
1 state at 1119 keV is part of the yrast band, and only decays to the 21 state by

emitting a 664 keV Îł ray. The literature value of the lifetime of the 4+
1 state compiled in
NNDC [52] and derived from [58, 59, 60, 61, 62, 63, 68, 64, 65, 55] was used as an input to
the gosia minimization. The E2 matrix element reproduces previous measurements, both
from NNDC and from Becker et al.. The quadrupole moment is also reproduced, though
with a large uncertainty. In fact, the uncertainty is so large that the measurement carries
no sensitivity to the value or the sign.
68

+
The 2+
2 state at 1148 keV can decay either to the 21 state by emitting a 691 keV Îł ray, or

to the ground state by emitting a 1148 keV Îł ray. The previously measured lifetime of the 2+
2
was included in the gosia χ2 minimization from NNDC [52], derived from [55, 59, 62, 64].

   + 
 
The measured value of 0+
1 E2 22 agrees with the value obtained by Becker et al. to
within 0.3σ, though both measurements disagree with the value reported in NNDC [52].

   + 
+0.02
 
The 2+
1 E2 22 matrix element of 0.43−0.03 eb is larger than either previous value by
2.5σ. It is reminded that there is no sensitivity to the known M1 component in the decay,
and so the multipole mixing ratio of the 691 keV transition was input as δ = -0.45(10) [52].
The resulting M1 matrix element of 0.33+0.02
−0.02 µN agrees with the literature value from NNDC
[52]. The measurement of this M1 matrix element by Becker et al. as 0.38+0.01
−0.03 µN is outside
of the uncertainty both of the literature value from NNDC and the value in the present work.

   + 
 
Attempts were made to fit the experimental yields while keeping 2+
1 E2 22 fixed at the
+0.06 eb. This constraint increases the reduced χ2 from 1.6
previously measured value of 0.26−0.05

   + 

   
E20 to 1.71+0.06 eb and 2+ E22+
to 2.5, while also increasing the values of the 2+
2
1
−0.01
3
1

0.12+0.01
−0.01 eb, outside of the range of agreement with previous measurements.
+ + + +
+
The 2+
3 state at 1752 keV can de-excite to any of the 01 , 21 , 02 , 41 , or 22 states. Of these

transitions, only the strongest transition to the 2+
1 state was directly observed here. Other
transition matrix elements are included in the Coulomb excitation calculation, but do not
correspond to observed Îł-ray transitions, and are therefore poorly constrained in the gosia
χ2 minimization. Branching ratios from the evaluated NNDC database [52], themselves
derived from [69, 59, 70], of all transitions from the 2+
3 state are therefore included as
+
additional inputs the gosia (Table 3.3). These branching ratios and the observed 2+
3 → 21

transition are the primary source of the corresponding matrix elements. The remarkably low
uncertainty for the matrix elements extracted for such unobserved transitions is solely based
69

2.8

60

2.6

χ2red without constraint

χ2red with constraint

70

50
40
30
20
10
0
0.0

2.4
2.2
2.0
1.8
1.6

0.1

0.2

0.3

+
h4+
1 ||E2||23 i

0.4

[eb]

0.5

0.6

1.4
0.0

0.1

0.2

0.3

0.4

+
h4+
1 ||E2||23 i [eb]

0.5

0.6


   + 
 
Figure 4.3: The χ2 curve of the 4+
1 E2 23 matrix element, with and without the branching ratio constraint in the gosia fit.
on the branching ratios and their uncertainties. An extraction of such values is therefore
questionable. The agreement with Becker et al. is poor. One can explore the disagreement
and check for inconsistencies of the present data and literature values. For example, a

    + 
  
comparison of the χ2 curve when varying the 4+
1 E2 23 matrix element, with and without
the branching ratio constraint, is shown in Fig. 4.3. The large discrepancy between them
may be an indication of an inconsistency between previous measurements of the matrix
element and previous measurements of the branching ratio as they cannot be consolidated
in the present data set. The literature values of the E2/M1 multipole mixing ratios for the
+
+
+
2+
3 → 22 and 23 → 21 transitions (Table 3.4) were included as additional inputs to the

gosia χ2 .
The quadrupole moment of the 2+
3 state is significantly different than the previous measurement by [55]. Though the uncertainty in the current measurement is large, it does
exclude all negative values, suggesting that this state is oblate, rather than prolate, from

   
this work. Attempts to fit the measured yields here using the 2+ E22+ value of -0.22 eb,
3

3

allowing all other parameters to vary, resulted in a reduced χ2 of 1.8, as opposed to the mini-

70

mum reduced χ2 found of 1.6. Most matrix elements in the modified fit change only slightly,

   + 
+0.003
 
with the exception of the 2+
3 E2 21 matrix element, which changes to 0.131−0.004 eb,
far from previously measured values. This change, along with the increased χ2 , shows that
there is no obvious variation of parameters that can explain the measured yields using the

   + 
 
value of 2+
3 E2 23 from [55].
+
The highest state observed was the 6+
1 state at 1977 keV, which decays to the 41 state by

emitting a 857 keV γ ray. The known lifetime of this state was included in the χ2 calculation
(Table 3.2). The extracted transition matrix element agrees with [55] and NNDC.
In total, the agreement between the matrix elements extracted here and the literature
values is satisfactory, given that the 2-day measurement reported here was performed at a
single beam energy on a single target, whereas Becker et al. [55] used in effect 6 measurements
to extract their results. The inconsistences uncovered for some matrix elements of the 2+
2
and 2+
3 states require complementary data for a resolution. In particular, sensitivity to the
sign of the quadrupole moments is demonstrated for all 2+ states in this analysis.

71

Chapter 5
Summary and Outlook
The SeGA-JANUS experimental setup was successfully commissioned, and is ready to be
used for inverse-kinematics low-energy Coulomb excitation experiments with rare isotope
beams. The existing SeGA detectors were transitioned to a digital data acquisition system,
and coupled to an analog data acquisition system for the silicon detectors. The two systems
were read out using the NSCLDAQ framework, and analyzed using GRUTinizer. The
experimental setup was tested using a 252 Cf fission source, and was commissioned in-beam,
performing low-energy Coulomb excitation of 78 Kr. From this experience and work from
other low-energy facilities, it is concluded that rates of 104 -105 particles per second will
be sufficient for performing rare-isotope experiments, depending of course on the degree of
collectivity of the nucleus of interest and the length of the run.
Since the commissioning run, the silicon detectors used in JANUS were transitioned
from the analog data acquisition system described in this work to a digitizer-based system
similar to that used for SeGA. This greatly simplifies the electronics part of the experimental
setup data acquisition, and asynchronous reading of the processed events from the digitizer
modules allows for a near-zero deadtime of the particle detectors [44]. The modified data
acquisition system was successfully tested with a 252 Cf source, identical to the earlier test
described in Sec. 2.4.
The JANUS setup will be used to study collectivity in rare isotope beams at NSCL now,

72

and at FRIB in the future. Unlike for ISOL facilities, which have chemistry-dependent restrictions on beam production based on which elements can be efficiently extracted from a
thick production target, NSCL’s unique technique of beam production by in-beam fragmentation, gas stopping, and reacceleration allows for low-energy rare-isotope beams not limited
by chemical properties. As FRIB comes online, the variety of available beams and their
intensity will increase, enabling studies of collectivity far from the valley of β stability.
Of particular interest are nuclei not available at ISOL facilities, for example, the neutronrich sulfur isotopes, leading up to the N = 28 magic number. 44
16 S28 has previously been
found to have low-lying quadrupole collectivity [71, 72], and configuration [73, 74, 75, 76]
and shape coexistence [77, 78, 79], indicating that the N = 28 magic number breaks down
in this region and elucidating the mechanism behind this breakdown. Low-energy Coulomb
excitations of 42−44 S would enable — for the first time — to prove and quantify shape
coexistence from measurements of the sign and magnitude of the quadrupole moment, while
also tracking the collective structures to higher spin.
Preparatory research in this region is presented in Appendix C, which reports the detailed Îł-ray spectroscopy of the neutron-rich sulfur isotopes up to A = 42 using the Îł-ray
detector array GRETINA and the S800 spectrograph with NSCL’s fast beams. At present,
the beam rates available in this region are insufficient to perform low-energy Coulomb excitation. As the beam availability and intensity increases in the S chain and other regions of
shape coexistence, low-energy Coulomb excitation experiments will provide measurements
of collectivity beyond the first 2+ state and information on nuclear shapes, which will help
further the understanding of nuclear structure in exotic nuclei.

73

APPENDICES

74

Appendix A
GRUTinizer Development
GRUTinizer is a new analysis package designed to be configurable for use in many experiments, handle multiple input formats, and allow custom user-configurations for analysis
of any detector system. The project was developed in conjunction with the current work.
GRUTinizer is multi-threaded to best take advantage of modern multi-core computer architectures, allowing for quick sorting of raw data, either read from disk or streamed live
during an experiment. While the underlying sorting framework is original, many of the user
friendly analysis and display features included are based on the ROOT framework[80].

Pipeline
The primary goal of GRUTinizer is to transform raw binary data into human-readable
structures while minimizing the amount of programming knowledge required by the experimenter. on the part of the experimenter. GRUTinizer presents the user with an analysis
pipeline, each section of which performs a single step in the analysis process. These steps
represent high-level actions, such as reading raw binary data from a file or time-correlating
events, and can be customized according to one’s individual needs.
The overall analysis pipeline consists of many steps, with each physics event being processed by all steps, one at a time. In the most basic setting, raw data read from a source
undergoes four basic steps. First, the individual raw data packets are sorted by timestamp
75

Read
raw data

Raw data
packets

Time
correlate
raw data

Correlated
raw data
packets

Unpack
raw data

Physics
events

Write
.root file

Physics
events

Free
memory

Figure A.1: The analysis pipeline of the simplest use of GRUTinizer
and grouped into time-correlated event packets. These event packets may contain raw data
from multiple independent detector systems. Next, each raw data packet within an event is
sent to the appropriate unpacker for the detector system that produced it. The job of each
unpacker is to convert the raw data, as packed by its corresponding data acquisition(DAQ)
systems, into a readable structure usable for representing the specific detector system. These
unpacked data structures then have the option to either be written directly to disk in the
form of a ROOT tree or to have desired information extracted and histogrammed, discussed
in more detail in App. A. Finally, the in-memory representation of the data is freed. This
flow of data through the analysis pipeline is shown in Fig. A.1.
Each computation step described above is independent, and is performed on a different
thread. This allows, for example, the reading of data from disk to occur simultaneously
with unpacking, rather than each step being performed sequentially by one processor. In
addition, each processing step is interchangeable, requiring only that the appropriate inputs
be provided, and the outputs be consumed. This allows additional computational steps to
76

Read from
.root file

Physics
events

Physics
events

Filter
events

Fill
histograms

Physics
events

Physics
events

Physics
events

Figure A.2: Additional components that can be added to the GRUTinizer analysis pipeline.
be added to the pipeline, so long as they provide the appropriate outputs for later processing
steps.. Rather than reading from a raw data file, for example, GRUTinizer can read from
a previously generated .root file, reading the unpacked data structure directly from the
stored tree to further process the data. Additional computational steps commonly added to
an analysis pipeline are shown figuratively in Fig. A.2.
This flexibility allows for a wide variety to the analysis pipeline. For example, one could
read from a .root file, filter the values, histogram the results, then write the filtered events
to a .root file.
The read loop is capable of reading from multiple different file formats, containing binary data packets. The raw data may be read from a raw file, from a compressed file, or
directly from an NSCL ringbuffer. The data format can be either the NSCL’s internal EVT
format, the output of the Global Event Builder (GEB) used by GRETINA, or a ROOT tree.
GRUTinizer handles the unpacking of these data formats, presenting a uniform interface

77

for all unpacking steps, allowing for a large amount of code reuse. For example, S800 data
written either through the NSCL data packer or through the GRETINA data packer can be
unpacked with the same code.
While being read out, a timestamp is extracted for each data packet. These data packets
are sorted according to their timestamps, and are grouped into packets similar in time.
Correlated data packages originating from multiple detector systems, referred to earlier as
event packets,are then processed together throughout all subsequent pipeline steps. The
correlation is done using a FIFO-like queue with a user-definable depth and time window,
and alerts the user if the depth is insufficient to sort the data.
Unpacking is largely done with user-defined code specific to an individual detector. This
code takes a group of raw data packets and converts it into a user-friendly data structure
representing an entire detector system. These systems are usually composed of detectors
containing many individual hits, each of which is derived from a single raw data packet.
Implementation of additional unpackers is described in App. A.
After the unpacking step, many options are available for use. The user-friendly data
structures can be written to disk in the form of a ROOT tree. Alternatively, these data
structures can be re-packed into raw data and written to disk. This is particularly useful, as
the flexibility of the pipeline allows a filter to be applied before writing, pruning unwanted
from the data, for example from another reaction channel. In such a way, one can perform
future steps in the analysis including only the events most important to the overall physics
goal. Additionally, events can be histogrammed, then either saved to a ROOT data structure
or displayed on screen.
The ability to unpack, filter, histogram, and store sorted data allows for powerful analysis
using GRUTinizer. Some of the common usage will be explained in the following section,
78

which can be expanded on a case-by-case basis.
Having a common sorting framework and analysis tools has proven to be extremely useful.
This framework has allowed many detector systems to be analyzed in a common manner.
GRUTinizer handles the reading of input files, the parsing of timestamps, the grouping of
events that happen nearby in time, and the writing of the output file in a semi-automatic
fashion. These are steps that are common to all analyses, and need not be recreated for
each analysis. Such an approach has had the added side effect of providing a large user base,
which in turn helps gather all user contributions into a common code.

Usage
Please be aware, usage and implementation details may change in the future.

Installation
GRUTinizer requires a current installation of ROOT, version 6, along with Python2.7. Once
these are installed, to compile GRUTinizer, first clone the current version from github, then
compile using make.
git clone https :// github . com / pcbend / grutinizer
cd grutinizer
make
In order to load environment variables which allow path independent user-code compilation and running GRUTinizer, the included thisgrut.sh file can be sourced. The primary
effect of this file is adding the bin directory of GRUTinizer to the user’s PATH variable.

79

source / my / path / to / grutinizer / thisgrut . sh

Class Structure
Detector systems in the GRUTinizer frame work derive from two main classes, TDetector
and TDetectorHit. TDetector is intended as a full description of a detector system’s output
in a single event. TSega and TJanus are two examples of TDetector subclasses, representing
the Îł-ray and particle detections in the SeGA-JANUS setup. Each subclass must provide
a BuildHits() method, which unpacks raw binary data from the detector into the class
structure. This is required for any new system that is added to the GRUTinizer framework.
The TDetectorHit class is not required, but provides features that make it easier to
implement a new detector. It is intended to represent a single channel, or detecting element,
on a detector. Each TDetectorHit is intended to hold the uncalibrated time and energy
collected in a single event for a single channel. These are calibrated using the TChannel
tool, described in App. A as needed. Additional information specific to each detector can be
included by creating a subclass of TDetectorHit, such as the channels individual position.
Once a user has created their new subclass of TDetector, it needs to be integrated
into the unpacker, so that GRUTinizer can identify data originating from that detector
system. This is done by editing the GRUTinizer/include/TGRUTTypes.h file to include
a new entry in the kDetectorSystems enum, then editing the GRUTinizer/libraries/
TGRUTint/TGRUTTypes.cxx file to include references to the new system. TGRUTTypes.cxx
contains two maps that are used by the GRUTinizer unpacker. The first maps from
strings to enums, and must have a new entry made corresponding to the new entry in
kDetectorSystems. This is used in the parsing of the .env file described in App. A. The

80

second maps from enums to classes, and is used when determining which class needs to be
constructed. In both cases, edit the map to have a new entry for the new detector, following
the format of the existing entries.
Additional details on this process can be found at the GRUTinizer wiki, located at
https://github.com/pcbend/GRUTinizer/wiki.

Configuration Files
GRUTinizer is designed to handle many different data formats and detector systems.
While much of the configuration need to sort the data is assumed by the program, sometimes
additional input is needed. For GRETINA data, the event type of each data packet is used
to determine to which detector the data packet belongs. These event types correspond to a
static list maintained by the GRETINA project. For NSCL data, the source id plays a similar
role. However, the source id may change from experiment to experiment. For this reason,
a configuration file is used to specify which source ids correspond to which detectors. The
default configuration file is located at GRUTinizer/config/DetectorEnvironment.env, and
can be overridden by passing a replacement file to GRUTinizer on the command line. For
example, in the SeGA-JANUS commissioning experiment, source ids 1, 2, and 3 corresponded
to SeGA data, while source id 4 corresponded to JANUS data. The following configuration
file was therefore used.
Sega : 1 2 3
Janus : 4
While histogramming or filtering objects, it may be desirable to have some user-configurable
settings. In order to handle such variables, GRUTinizer has the concept of GValues, which

81

can be passed in from the command line and accessed from within the histogram or filtering
libraries. Additionally these parameters help in real-time analysis; for example, in making
Doppler reconstructed histograms with SeGA and JANUS systems, the value of β, was a
GValue to allow it to be determined at run time. The values are intended to be configuration
parameters relating to a detector system as a whole, such as relative positioning between
detector systems. These configuration parameters are globally available, and can easily be
loaded from a configuration file, or modified at runtime through the GRUTinizer GUI.
Additionally, the values are written in plain-text files with the file extension .val. An example of such a file is shown below. In a histogram or filter library, these values can be
accessed by calling GValue::Value("beta"), where the argument is the name of the value
being requested.
beta {
Value : 0.085
}
While the GValues are good for handling of high-level parameters, detectors can also
require a large number of calibration parameters. In this case, it would become unwieldy
to have an independent GValue for the slope and offset of each channel. For this purpose,
GRUTinizer has the concept of TChannels, which provide a few common uses. For example, below is the channel specifying that the SeGA detector located at address 0x01010201
has calibration energy = 0.086637*channel + 0.017.
SEG02AP00 {
Address : 0 x01010201
EnergyCoeff : 0.017 0.086637

82

}
On its own, the TChannel provides a useful parser, and functions to apply channelspecific corrections, such as time and energy calibrations. This utility is coupled with the
GRUTinizer data structures, in order to provide ease of use to users.

Histogramming and Filtering
Throughout an experiment, different histograms may be needed for diagnostic and analysis
purposes. These histograms depend on the detectors present in the given experiment, and
may change rapidly as the needs of analysis change. To accommodate this, GRUTinizer
makes it easy to create a histogram definition file detailing the histograms that should be
produced. These definition files consist of compiled C/C++ code, allowing full flexibility
for any calculations that need to be run. While the definition file fills the histograms,
all memory management and output files are managed by GRUTinizer, simplifying the
process significantly. Additionally, these histograms can be viewed and manipulated as they
are being filled, giving real-time feedback to the experiment.
In order to histogram data, a histogram library must be generated. Samples of these
are included in the histos subdirectory of the GRUTinizer source directory. Additional
histogram files created in this directory will be automatically compiled by the GRUTinizer
makefile into a shared library. A simple example of such a file is shown below.
# include " TRuntimeObjects . h "

extern " C "
void MakeHistograms ( TRuntimeObjects & obj ) {
83

TSega * sega = obj . GetDetector < TSega >();
TJanus * janus = obj . GetDetector < TJanus >();

if ( sega & & janus ) {
obj . FillHistogram ( " tdiff " ,
400 , -2000 , 2000 ,
sega - > Timestamp () - janus - > Timestamp ());
}
}
The entrance point is the MakeHistograms function. This will be called once for every
time-correlated event in the input file. The single argument, TRuntimeObjects& obj, is the
entrance point both for accessing data about the event, and for specifying histograms to be
generated.
In order to access data from the event, use the obj.GetDetector method. This is
a templated method that will return a pointer to that detector object if the current event
contains that detector type, or will return a null pointer if the current event does not contain
that detector.
To make a histogram, use the obj.FillHistogram method. This accepts the name of
the histogram to be filled, the number of bins, the low end of the histogram range, the high
end of the histogram range, and finally the value that should be filled into the histogram. By
combining these parameters into a single function call, it is possible to add new histograms
by editing only a single location in code.
Filters to remove unwanted data are generated in a similar manner. The main difference

84

is that the entrance function is named FilterCondition rather than MakeHistograms. The
FilterCondition function should return true if the event is to be kept, and false if the event
should not be kept. Below is an example filter that will keep only events in which both SeGA
and JANUS data are present.
# include " TRuntimeObjects . h "

extern " C "
bool FilterCondition ( TRuntimeObjects & obj ) {
TSega * sega = obj . GetDetector < TSega >();
TJanus * janus = obj . GetDetector < TJanus >();

if ( sega & & janus ) {
return true ;
} else {
return false ;
}
}
In both histogram libraries and filter conditions, it may be advantageous to use twodimensional gates defined previously. These gates can be saved using the TCutG class provided by ROOT or the reimplemented GCutG class, provided by the GRUTinizer to ease
S800 analysis. If TCutG objects are saved to a ROOT file with the file extension .cut and
passed to GRUTinizer, they will be available for use inside histogram and filter libraries.
They can be accessed by calling obj.GetCut("MyCutName");.

85

Program Usage
The simplest usage of GRUTinizer is as an interpreter, based on the default ROOT interpreter. To start the GRUTinizer interpreter, simply run the program without any
arguments.
grutinizer
Options accepted by GRUTinizer are listed by passing the --help option, the output
of which is shown below.
GRUT_RELEASE : 1.02.00
Release the Chipper Chartreuse Chipmunk !
Options :
arg

Input file ( s )

-o [ -- output ] arg

Root output file

-f [ -- filter - output ] arg Output file for raw filtered data
-- hist - output arg

Output file for histograms

-r [ -- ring ] arg

Input ring source ( host / ringname ) .

Requires -- format to be specified .
-l [ --no - logo ]

Inhibit the startup logo

-H [ -- histos ]

attempt to run events through

MakeHisto lib .
-n [ --no - sort ]

Load raw data files without

sorting
-m [ -- sort - multiple ]

If passed multiple raw data files ,

treat them as one file .
86

-s [ -- sort ]

Attempt to loop through root files

.
-t [ -- time - sort ]

Reorder raw events by time

-- time - sort - depth arg

Number of events to hold when time

sorting ; default value 100000
-- build - window arg

Build window , timestamp units

-- long - file - description

Show full path to file in status

messages
-- format arg

File format of raw data .

Allowed

options are " EVT " and " GEB ". If unspecified , will be
guessed from the filename .
-g [ -- start - gui ]

Start the GUI

-w [ -- gretina - waves ]

Extract wave forms to data class

when available .
-q [ -- quit ]

Run in batch mode

-h -? [ -- help ]

Show this help message

-v [ -- version ]

Show version information

These are described in more detail below.
• --output allows the user to specify the name of the ROOT file to contain the unpacked
data in a ROOT tree. If a filter condition has been passed, this output will be filtered. If
GRUTinizer is sorting over raw data and no output file has been explicitly specified,
a name will be automatically generated. To disable output of the ROOT tree, pass
/dev/null as the output file.
87

• --filter-output enables output of raw data to the specified file, with only events
that have passed the filter condition being written.
• --hist-output allows the user to specify the name of the output ROOT file to contain
generated histograms. If GRUTinizer is histogramming data and no output file has
been explicitly specified, a name will be automatically generated. To disable output
of the ROOT tree, pass /dev/null as the output file.
• --ring specifies a NSCL ringbuffer[45] from which to read data. If this option is
specified, the --format option must also be given.
• --no-logo suppresses the display of the GRUTinizer logo on program start.
• --histos enables histogram generation.
• --no-sort will disable sorting over raw data files. This is used to examine raw data
files manually.
• --sort-multiple causes multiple raw files passed in to be sorted concurrently. This
is intended for data acquisition systems that write to multiple files simultaneously. In
this mode, events from multiple files are read in, and are sorted in order in increasing
timestamp, regardless of which raw file contains each individual event. Without this
option, multiple raw files will be sorted sequentially, in the order they are passed in on
the command line.
• --sort will enable sorting over ROOT trees.
• --time-sort enables sorting of events by increasing timestamp.

88

• --time-sort-depth gives the maximum number of events to be kept in memory simultaneously when sorting events by timestamp. Should this number be insufficient,
a diagnostic will be issued to the user.
• --build-window gives the maximum separation between two raw data packets in a
time-correlated event packet. This value is in the same units as the timestamps in the
raw data file.
• --long-file-description enables a long description of the file in status messages.
• --format explicitly states the file format of the raw data source, and must be either
“GEB” or “EVT”. If unspecified, it is assumed that files ending with the extension
.evt are NSCL data files and files ending with the extension .dat are GRETINA
data files. When reading from a ring with the --ring option, the --format option is
mandatory.
• --start-gui initializes the GRUTinizer graphical user interface on program start.
• --gretina-waves enables the extraction of waveforms when reading GRETINA data
that contains waveforms. Otherwise, the waveforms are skipped.
• --quit quits GRUTinizer after handling all arguments. If not specified, GRUTinizer will instead display an interactive C++ prompt.
• --help will display the help message, then quit.
• --version will display the version information, then quit.
In addition, input files are recognized by their file extension. Known file extensions are
as follows.
89

• .evt: NSCL data files.
• .dat: GRETINA data files.
• .cal: Calibration files containing TChannel definitions.
• .val: Calibration files containing GValue definitions.
• .root: ROOT files. These are automatically opened on startup.
• .c, .C, .c+, .C+, .c++, or .C++: ROOT macro files. These are executed after opening
all ROOT files.
• .env: The detector environment file, which provides a mapping from source id to
detector system, as described in App. A.
• .so: A shared object library containing either histogram definitions or a filter condition. If the function MakeHistograms is present, the library is recognized as a histogram
library. If the function FilterCondition is present, the library is recognized as a filter
condition.
• .inv: An inverse map, used for the S800 spectrograph[81].
• .cuts: A ROOT file containing 2-dimensional gates, saved as TCutG objects. These
are available for use in filter and histogram libraries, as described in App. A.
In order to sort data, pass the data file in as an argument. GRUTinizer will recognize
file extensions .evt as the NSCL data format, and .dat as GRETINA data files. In addition,
if the file extension is .evt.gz or .dat.gz, it will be interpreted as a gzipped file in NSCL
format or GRETINA format, respectively, and will be unzipped accordingly. An output file
90

can be manually specified with the -o option, as shown below. If no such option is given,
the output root file’s name will be automatically generated, based on the run number in the
filename.
grutinizer run -1234 -00. evt -o run -1234 -00. root -q
In order to signal to grutinizer that it should generate histograms, the histogram shared
library must be passed in, and an output histogram file specified. The output filename can be
explicitly specified using the --hist-output argument, or can be automatically generated
using the -H argument. Examples of these two methods are shown below.
grutinizer run -1234 -00. evt hist_lib . so \
-- hist - output = hist -1234 -00. root -q
grutinizer run -1234 -00. evt hist_lib . so -H -q
When generating histograms from a raw file, it may be desirable to suppress the writing
of the root tree itself. While useful for analysis, writing a root tree is the slowest part of the
analysis pipeline. In order to prevent a root tree from being created, specify the output file
as -o /dev/null.

Graphical User Interface
As it’s most visible aspect, GRUTinizer contains a graphical user interface (GUI) for
interacting with ROOT objects, as well as giving the user a convenient way to interact with
many of the internal pieces of the GRUTinizer framework.
The graphical user interface can be started from the command line. This contains a
browser for displaying histograms, along with access to internal variables used by GRUTinizer. The GUI can be opened by passing the -g option, as shown below, or by running
91

the StartGUI() function in the interpreter. A screenshot of the GUI is shown in Fig. A.3.
grutinizer -g merged . root

Additional Tools
This framework contains many tools built on top of the capabilities already provided by
ROOT. These are intended to make analysis steps easier by automating many common
tasks in Îł-ray spectroscopy, while being integrated into the standard ROOT user interface
wherever possible. A few such tools are described here, with further described available on
the GRUTinizer wiki, located at https://github.com/pcbend/GRUTinizer/wiki.
• By clicking on a one-dimensional histogram, one can define a region. With a keyboard
shortcut, the area within the region is then fit with a Gaussian distribution.
• The TCalibrator tool is designed to automatically perform calibration of a germanium
Îł-ray detector, given the known energies of a radioactive source and a measured spectrum. This automation in particular has saved a large amount of time in calibration
of the many channels present in segmented germanium detectors.
• Automatic shortcuts for performing projection and background subtraction of a twodimensional γ-γ matrix.
• A shortcut to perform an integral of a histogram within a marked region.
As with most software, GRUTinizer is still in development, and improvements continue
to be made. Modifications are welcome, and can be submitted at https://github.com/
pcbend/GRUTinizer.

92

Figure A.3: A screenshot of the GRUTinizer graphical user interface, exploring a .root
file containing histograms.

93

Appendix B
Gosia Analysis Procedure in More
Detail
Gosia [38] is a program designed to analyze Coulomb excitation experiments, extracting
transition strengths and moments. It takes as input a description of the level scheme, along
with allowed E1 through E6, M1, and M2 transitions between levels, as shown in Fig. 3.16.
Experimental data are given as intensities and uncertainties for observed Îł-ray transitions,
gating on different regions in scattering angles, as shown in App. B.
The general process of running gosia involves four main steps. First, the experimental
yields are corrected for finite angular range of the particle detectors and finite projectile
energy range due to energy loss in the target. Use of these corrected yields allows subsequent
processing done by gosia to calculate the yield at a single value of scattering angle and
projectile energy, rather than integrating over a range of each, saving computation time [36].
Second, the matrix elements are varied to best reproduce the experimental yields, minimizing the χ2 . This χ2 is computed based on the difference between the calculated yields
and the experimental yields. In addition, differences between calculated and experimental
values of any additional information provided to the fit, such as lifetimes, multipole mixing
ratios, and branching ratios, contribute to the χ2 being minimized.
Finally, the uncertainties of the extracted matrix elements are calculated. For reasons

94

expressed in detail in [36], this is not done through the traditional method of finding the
matrix elements that result in a χ2 value equal to χ2min +1, where χ2min is the minimum value
of χ2 . Instead, an integral is performed over the likelihood of observing the data, relative
to the integral over all values of the parameter. This is done in two steps, first finding the
uncertainty of the matrix elements while keeping all other matrix fixed, known as the diagonal
errors. The second step performs a full error estimation, determining the uncertainties of
each parameter including corrections for correlations between matrix elements.
Performing each step requires making modifications to the input file, then re-running
gosia. The exact modifications needed are described after the example input file. This
section is concerned with when each step should be run, rather than how they are performed.
The minimization step is the most frequent step performed, and so it is the mode described with the unmodified input file below. In order to correct for finite detector size and
target thickness, see the section on lines 126-150, describing the OP,CORR option. In order
to compute diagonal errors, see lines 152-155, describing the OP,ERRO option. For the final
correlated error calculation, see lines 162-165, also using the OP,ERRO option.
When performing the minimization, one adjusts the initial parameters, potentially adding
values from previous measurements. These values can include matrix elements themselves,
but can also include other measurements such as lifetimes, mixing ratios, and branching
ratios. These are used as additional values contributing to the total χ2 .
Since the matrix elements themselves affect the corrections for finite detector size and
target thickness, these corrections should be re-run whenever the fitted matrix elements
change significantly during a minimization step. Afterwards, the minimization is repeated
again.
When performing the correlated uncertainty calculation, the diagonal uncertainty calcula95

tion must have been performed immediately prior. This is because the correlated uncertainty
calculation relies on intermediate files generated by the diagonal uncertainty calculation.
gosia calculations run very quickly on modern hardware. As gosia was originally written in the early 1980s, many options described in the manual enable optimizations in order
to speed up computation, but these are rarely necessary anymore.

Gosia Input File
Shown below is an example file. Gosia is run using the command gosia < input file.
After the input file is a description of each line of the input file. The line numbers on the
left of the input file are for reference only, and are not part of the input file itself. After the
input file, a line-by-line explanation will be given.
The input file listed below is configured to run the minimization step of gosia. From
this input file, removing lines 122-125, from OP,MAP to OP,EXIT, and editing line 101 from 4
to 3, runs the corrections for finite angle range and target thickness. Copying lines 152-155
to line 123, just above OP,MINI, runs the diagonal uncertainty calculation, and copying lines
162-165 to line 123, just above OP,MINI, runs the correlated uncertainty calculation.
1 OP , FILE
2 22 ,3 ,1
3 Kr78 . out
4 9 ,1 ,1
5 chico97 . gdt
6 3 ,1 ,1
7 Kr78 . yld

96

8 4 ,3 ,1
9 Kr78 . cor
10 7 ,3 ,1
11 Kr78 . map
12 2 ,3 ,1
13 Kr78 . bst
14 15 ,3 ,1
15 Kr78 . err
16 0 ,0 ,0
17 OP , TITL
18

Coulomb excitation of 78 Kr by 208 Pb targets

19 OP , GOSI
20 LEVE
21 1 ,1 ,0.0 ,0.
22 2 ,1 ,2.0 ,0.455
23 3 ,1 ,0.0 ,1.017
24 4 ,1 ,4.0 ,1.119
25 5 ,1 ,2.0 ,1.148
26 7 ,1 ,2.0 ,1.756
27 8 ,1 ,4.0 ,1.873
28 9 ,1 ,6.0 ,1.978
29 10 ,1 ,8.0 ,2.994
30 12 ,1 ,2.0 ,2.444

97

31 0 ,0 ,0 ,0
32 ME
33 2 ,0 ,0 ,0 ,0
34 1 ,2 ,0.832 ,0.6 ,1.0
35 1 ,5 ,0.165 ,0.1 ,0.3
36 1 ,7 ,0.045 ,0.01 ,0.1
37 2 ,2 , -0.905 , -4.0 ,+0.5
38 2 ,3 ,0.265 ,0.2 ,0.8
39 2 ,4 ,1.285 ,1.0 ,2.0
40 2 ,5 ,0.440 ,0.1 ,1.0
41 2 ,7 ,0.135 ,0.05 ,3.0
42 2 ,8 ,0.073 , -0.5 ,1.0
43 3 ,7 ,0.560 ,0.01 ,1.0
44 4 ,4 , -1.2 , -4.0 ,+4.0
45 4 ,7 ,0.390 ,0.01 ,1.0
46 4 ,8 , -0.6 , -2 ,2
47 4 ,9 ,1.660 ,0.5 ,3.0
48 5 ,5 ,0.977 , -2.0 ,4.0
49 5 ,7 ,0.218 , -1.0 ,1.0
50 5 ,8 ,0.910 , -4.0 ,4.0
51 7 ,7 ,1.541 , -4.0 ,8.0
52 7 ,12 ,0.1 ,1 ,2
53 8 , -8 , -0.832 ,1 ,2

98

54 9 ,9 , -0.821 , -8.0 ,8.0
55 9 ,10 ,1.8 ,0.05 ,2.0
56 10 , -10 , -0.853 ,1 ,2
57 7 ,0 ,0 ,0 ,0
58 2 ,5 ,0.337 ,0.1 ,0.5
59 2 ,7 , -0.144 , -0.2 , -0.02
60 5 ,7 ,0.023 , -0.5 ,0.5
61 0 ,0 ,0 ,0 ,0
62 EXPT
63 4 ,36 ,78
64 -82 ,208 ,295 ,29 ,4 ,0 ,0 ,0 ,360 ,0 ,1
65 -82 ,208 ,295 ,44 ,4 ,0 ,0 ,0 ,360 ,0 ,2
66 -82 ,208 ,295 ,92 ,4 ,0 ,0 ,0 ,360 ,0 ,3
67 -82 ,208 ,295 ,145 ,4 ,0 ,0 ,0 ,360 ,0 ,4
68 CONT
69 PRT ,0.
70 1 , -1
71 2 ,0
72 4 , -2
73 5 ,1
74 7 ,0
75 11 ,0
76 12 ,0

99

77 14 ,0
78 16 ,0
79 0 ,0
80 END ,
81
82 OP , YIEL
83 0
84 15 ,2
85 0.10 ,0.12 ,0.15 ,0.18 ,0.22 ,0.26 ,0.32 ,0.38 ,0.46 ,
86 0.56 ,0.68 ,0.82 ,1.00 ,1.20 ,1.50
87 2
88 0.925 ,0.471 ,0.207 ,0.106 ,0.051 ,0.0281 ,0.0136 ,0.0076 ,
89 0.0041 ,0.0023 ,0.0013 ,0.0008 ,0.0005 ,0.0003 ,0.0003
90 7
91 0.129 ,0.078 ,0.043 ,0.027 ,0.016 ,0.010 ,0.0062 ,0.0041 ,
92 0.0026 ,0.0016 ,0.0011 ,0.0007 ,0.0005 ,0.0003 ,0.0003
93 1 , -1 , -1 , -1
94 1
95 0
96 0
97 2 ,1
98 1
99 0.05

100

100 1
101 4 ! which file gets used as input for the yields
102 4 ,1.0
103 7 ,1 ,7 ,2 ,0.254 ,0.008
104 7 ,3 ,7 ,2 ,0.518 ,0.008
105 7 ,4 ,7 ,2 ,0.119 ,0.004
106 7 ,5 ,7 ,2 ,0.047 ,0.010
107 5 ,1.0
108 2 , 31.2 , 1.0
109 3 , 13.0 , 3.6
110 4 , 3.63 , 0.17
111 5 , 4.8 , 0.9
112 9 , 0.94 , 0.10
113 3 ,1.0
114 5 ,2 ,0.45 ,0.10
115 7 ,5 ,4.0 ,3.5
116 7 ,2 , -1.32 ,0.55
117 4 ,1.0
118 2 ,9 ,10 ,1.80 ,0.15
119 2 ,5 ,8 ,0.91 ,0.06
120 2 ,4 ,8 , -0.60 ,0.03
121 2 ,2 ,8 ,0.073 ,0.005
122 OP , MAP

101

123 OP , MINI
124 2100 , 1000 , 0.000001 , 0.000001 , 1 , 0 , 0 , 1 , 0 , 0.0001
125 OP , EXIT
126 OP , INTG
127 6 ,8 ,286 ,304 ,20 ,38
128 270 ,280 ,290 ,300 ,310 ,320
129 15 ,18 ,23 ,28 ,33 ,38 ,43 ,48
130 6 ,7 ,286 ,304 ,38 ,49
131 270 ,280 ,290 ,300 ,310 ,320
132 34 ,37 ,40 ,43 ,47 ,50 ,53
133 6 ,16 ,286 ,304 ,62 ,121
134 270 ,280 ,290 ,300 ,310 ,320
135 55 ,60 ,65 ,70 ,75 ,80 ,85 ,90 ,95 ,100 ,105 ,110 ,115 ,120 ,125 ,130
136 6 ,10 ,286 ,304 ,131 ,160
137 270 ,280 ,290 ,300 ,310 ,320
138 125 ,130 ,135 ,140 ,145 ,150 ,155 ,160 ,165 ,170
139 6
140 270 ,280 ,290 ,300 ,310 ,320
141 18 ,18 ,18 ,18 ,18 ,18
142 10 ,50
143 0
144 10 ,10
145 0

102

146 10 ,100
147 0
148 10 ,100
149 OP , CORR
150 OP , EXIT
151 ! Diagonal error minimization
152 OP , RE , A
153 OP , ERRO
154 0 ,0 ,0 ,0 ,0 ,1000000000000
155 OP , EXIT
156 ! Full error calculation ( only important correlations )
157 OP , RE , A
158 OP , ERRO
159 1 ,0 ,0 ,1 ,0 ,1000000000000
160 OP , EXIT
161 ! Full error calculation ( all correlations )
162 OP , RE , A
163 OP , ERRO
164 1 ,0 ,0 ,1 ,1 ,1000000000000
165 OP , EXIT

1-16. File listings. This section starts with OP,FILE and continues until it reaches a line with
0,0,0. This section specifies the names of all input and output files. These entries
are not necessary, but provide human-readable names for files, rather than the default
103

fort.15 and similar. For a full description of all input and output files used by gosia,
and their assigned numbers, please refer to the “File assignments” chapter of the gosia
manual [36].
The first parameter is the number identifying which file is being given a name. That
is, an identifier of 3 indicates the output that would be written to fort.3. These are
referred to as “file number” and “tape number” in the gosia manual.
The second parameter is the current state of the file. 1 indicates that it is solely an
input file. 2 indicates that the file does not currently exist, and is an output file. 3
indicates that the file may or may not exist, and can be used either as an input or an
output file, according to the options given later in the file.
The third parameter will be 1 for all of the files listed. This indicates that it is formatted
text input, not binary input.
For example, the first entry in this section, 22,3,1, indicates that the tape 22 may be
opened either as an input or an output, and should be read as formatted text input.
The following line, Kr78.out, gives the name of the output file.
17-18. A title. Something human readable that will be printed to the output file.
19-81. The section containing setup for many different options describing the experiment and
the structure of the nucleus being investigated. Each is described in more detail below.
20-31. The levels in the nucleus being analyzed. The starts with the LEVE option, and
ends with a single line containing 0,0,0,0.
Each line has four values. The first is an index which will be used to refer to
the level from now on. The second is the parity, either +1 for positive or -1 for
104

negative. The third parameter is the spin. The fourth parameter is the energy of
the state, in MeV.
32-61. The matrix elements present in the calculation. Each line can take a number of
different forms. All uses of an integer to refer to a matrix element refer to the
order they appear in this list, with the first element having index 1.
Each line has five values A,B,C,D,E.
If B is 0, then this line gives the multipolarity of the matrix elements to follow. A is
the multipolarity, where values 1-6 correspond to E1-E6 transitions, 7 corresponds
to M1, and 8 corresponds to M2. In the example, 2,0,1,0,0 indicates that the
next set of matrix elements are E2 matrix elements. 7,0,0,0,0 indicates that
the next set of matrix elements are M1 matrix elements.
If B is a positive value, indicates a free parameter in the fit. This is a matrix
element connected states A and B. It has initial value C, and is allowed to vary
from D to E.
A,-B,C,D,E, where there is a negative sign before B, indicates a matrix element
that is present in the simulation, but has a value coupled to an earlier defined matrix element. That is, if the free parameter increases by 10%, then this parameter
also increases by 10%. This helps to keep the dimensionality of the problem low.
This matrix element connects states A and B. It has initial value C. It is coupled
to the transition that connects states D and E.
If the value E is greater than 100, then it means that the matrix element is
coupled to an element with a different multipolarity. The hundreds place of E
indicates which multipolarity contains the independent matrix element is in. For

105

example, the line 1,-2, 0.5, 1, 203 would couple the (1,2) transition in the
current section to the (1,3) E2 transition.
The line 0,0,0,0,0 indicates that all matrix elements have been input.
62-67. The experiment section. This describes some of the experimental details. The
first line is the number of “experiments”, Z of the nucleus of interest, then the
A of the nucleus of interest. In gosia, “experiments” refers to the number of
different angle bins being used.
Each experiment line contains 11 values, A, B, C, D, E, F, G, H, I, J, K.
-82, 208, 295, 29, 4, 0, 0, 0, 360, 0, 1 is an example line. A,B gives the
charge and mass of the particle not being investigated. Here, -82,208 specifies
208 Pb.

The negative sign of A indicates that the nucleus of interest is the projectile.

C is the mid-target kinetic energy of the projectile, in MeV. D is the central
angle of the scattering angle region being used. E,F as 4,0 are the number of
magnetic substates used in the full and approximate calculations. Giving G as
0 indicates that the particle detector is axially symmetric. H,I as 0,360 gives
the azimuthal coverage of the particle detector in degrees. The value of J selects
which scattering angle solution to use if the conversion from laboratory angles to
center of mass angles allows for two solutions. Using 0 as J selects the backward
angle, while using 1 selects the forward angle. Finally, the values of K gives an
index of a normalization constant to be used, adjusting the absolute value of the
fitted yields for this experiment. These normalization constants account for the
absolute detector efficiency, target thickness, amount of beam on target.
68-79. The control block. The main element here is the PRT block, controlling which

106

parameters are written as output. This section is described in detail in the gosia
manual [36].
80-81. The end of the OP,GOSI block. Two things of note here. First, the comma in END,
is necessary. Second, the line after END, is not entirely blank, but must contain
an empty space.
82-121. The OP,YIEL section contains additional experimental details, along with values measured from previous measurements. As there are many values in this section, each will
be mentioned in detail.
83. Whether to use a correction for finite distance traveled by the excited nucleus
before de-excitation. 0 indicates that this correction is disabled, and 1 indicates
that this correction is disabled.
84-92. The internal conversion coefficients. These can be acquired from BrIcc [82] on
NNDC. First, the number of points, and the number of multipolarities that will
be present. In this case, 15 points will be entered, for 2 different multipolarities.
On lines 85-86 are the x-values, in MeV, of the internal conversion coefficients.
Line 87 identifies which multipolarity is to follow (E2), with its coefficients next.
Similarly, line 90 identifies M1 as the next, with coefficients on lines 90-91.
93. The number of gamma ray detectors in each experiment. A negative sign indicates
that it has the same parameters as the previous set. The line 1,-1,-1,-1 indicates
that there is a single Îł-ray integration region being used for all four scattering
angle regions, and they all of the same parameters.
94-96. Number of detectors, theta angles, and phi angles. The experimental yields given
here to gosia treat the SeGA array as a single integration region. The abso107

lute coverage does not matter, as the total efficiency is included in an overall
normalization constant used by gosia.
97. Which transition is used for normalization between different experiments when
displaying predicted yields. Here 2,1 is given as the normalizing transition, indi+
78
cating that the 2+
1 → 01 transition in Kr should be be normalized to 1. This

option is only applicable when predicting experimental yields using the OP,POIN
option, which is not used in this example.
98. Number of data sets for each experiment. This experiment has one Îł-ray integration region, so this is 1.
99. Upper limit for intensity of unobserved Îł-ray yields. Any transitions that are
above this threshold, but not reported, will be warned about. This value is
relative to the intensity of the normalizing transition, given on line 97.
100. Relative normalization constant for Îł-ray yields detected in each Îł-ray integration regions of a single experiment. Since this analysis used only a single Îł-ray
summing region, no changes to the normalization constants are needed, and so
the value 1 is given.
101. The input file to be used for reading in experimental Îł-ray yields. This value
will change frequently while performing a gosia calculation. While performing
corrections for finite detector size and target thickness, this needs to be 3, indicating that the uncorrected yields are used as input. While performing any other
calculation, this should be 4, indicating that the corrected yields are used as an
input.
102-106. Branching ratios used as an input. First is the number of branching ratios to be
108

given, and the relative weight that they should be given in the fit. 6 numbers are
given for each branching ratio, A,B,C,D,E,F, where A, B, C, and D are indices of
A→B as E ± F .
states. This gives the branching ratio C→D

107-112. Lifetimes used as an input. First is the number of lifetimes to be given, and the
relative weight that they should be given in the fit. 3 numbers are given for each
lifetime, A,B,C. This gives the lifetime of state A as B Âą C ps.
113-116. The mixing ratios between E2 and M1 transitions. Line 113 has the number of
mixing ratios given, along with their weight in the fit. Each line has four values,
A,B,C,D, which gives the δ mixing ratio from A to B as C ¹ D.
117-121. Known matrix elements as input. Line 117 has the number of matrix elements
given, along with their weight in the fit. Each line has five values, A,B,C,D,E,
which specify that the matrix element with multipolarity A between states B and
C is D Âą E. Values of A from 1-6 correspond to E1-E6 multipolarity, and values
of 7 and 8 correspond to M1 and M2, respectively.
122. The OP,MAP command generates temporary files necessary for minimization and error
calculation.
123-125. The command to perform the minimization, varying the matrix elements to best fit
the data given. The values on line 124 mainly describe the convergence criteria. For
example, the 1000 is the number of iterations to try before bailing out. The parameters
here should be good for general use, and are described more fully in the manual.
The OP,EXIT command on line 125 causes gosia to exit before finishing the file.
Running gosia occurs in several steps, each requiring edits to the input file. All gosia
calculations require the same or similar input up to line 121, the end of the OP,YIEL
109

section, but have different requirements after that. Having the OP,EXIT command is
the documented way to comment out part of the file by having gosia exit early.
126-150. Options to correct the gamma ray yields for finite target thickness and finite angular
bin size. These depend on the matrix elements themselves, and so this must be re-run if
matrix elements change significantly during. It depends only on the initial values, listed
in the ME section. In order to correct based on the minimized values, the minimized
values must be copied into the ME section.
When performing corrections lines 122-125 are removed, so that the OP,MAP, OP,MINI
and OP,EXIT commands are not present.
127-138. There are three lines for each experiment. The first line is A,B,C,D,E,F. The
energy in the target varies from C to D, with A spline points. The angular bin
coverage for the experiment varies from E to F, with B points.
The second line is the A points used for interpolating the energy, and the third
line is the B points used for interpolating the energy. Due to the interpolation
method used, two points both above and below the endpoints of the region should
be given.
139-142. The energy loss through the target, as a function of energy. The number of points
is on the first line, with the incoming energy in MeV on the second, and the energy
loss in MeV on the third line. The fourth line contains the number of angle and
energy integration points to use.
143-148. Entries for experiments 2-4, in the same format as are in lines 139-142. Each
consists of only two lines. The 0 on the first line indicates that the explicit

110

listing of energies and energy losses will be omitted, using the same values as
were specified for the previous experiment.
149. OP,CORR indicates that the yields are to be corrected for finite target thickness
and angular coverage. This outputs on file 4. File 4 is named fort.4 by default,
and can be renamed in the OP,FILE section described earlier. While performing
corrections, line 101 must be equal to 3, so that gosia reads from the uncorrected
yields.
152-155. Calculation of diagonal errors. This varies each parameter independently until the χ2
increases. To use this, copy-paste this section above line 123, just above the OP,MINI
section.
OP,RE,A releases the coupling of parameters, so that each parameter can be varied
independently. OP,ERRO then performs the uncertainty calculation. Full description of
the parameters to OP,ERRO, given on the following line, are in the gosia manual [36].
157-160. Calculation of correlated errors, using only the largest of the correlations. This option
is rarely necessary, because the availability of modern processing power enables fast
use of the complete calculation as described in lines 162-165. If this optimization is
needed, the following steps can be performed.
1. The line SEL, is added to the CONT block. Make sure that you have 4,-2 in the
PRT block. gosia is then run, performing the χ2 minimization, and producing
fort.18 as a side effect.
2. The SEL, is then removed, and OP,SELECT, then OP,EXIT is added just above the
OP,MINI line. gosia is run, which produces fort.10 using fort.18. This file is
111

renamed to fort.18, overwriting what was there.
3. The OP,SELECT and OP,EXIT lines are then replaced with the contents of this
section (lines 157-160), after which gosia is run a third time, which produces the
correlated errors in the output file.
On a computer using an Intel Celeron Processor 2955U, this procedure reduced the
calculation time from 6 sec to 3 sec, as compared to the next section. This optimization
may be necessary for future analyses of Coulomb excitation experiments, but was not
used in the present work.
162-165. Calculation of correlated errors, using correlations between all matrix elements. This
section is active when copy-pasted above the OP,MINI command.

112

Gosia Yield File
Below is shown an example yield file. There is a section for each experiment. Within each
experiment, the experimental yields for each transition observed are given, along with their
uncertainty. In gosia, each “experiment” refers to the an different angle cut in the scattering
angle of the projectile.
1

1

36

78

304.2

2

1

273900

4

2

8470

420

5

2

1570

160

9

4

460

5

1

1030

2

1

36

2

1

171200

3

2

610

4

2

15150

5

2

2170

9

4

630

5

1

1660

3

1

36

2

1

203100

3

2

5960

4

2

50900

1900

5

2

11850

840

5

1.0

6

1.0

7

1.0

8400

100
150
78

304.2
5300

120
640
190
150
190
78

304.2
6300
730

113

9

4

8080

960

5

1

7380

580

7

2

1570

300

4

1

36

2

1

12910

3

2

1170

160

4

2

3600

270

5

2

1270

190

9

4

1470

250

5

1

700

220

7

2

330

180

78

304.2

7

1.0

550

The first line in each section has 7 values, A,B,C,D,E,F,G. A is the number of the experiment, and should increment 1-N. These values must correspond with the same experiment
index in the main input file. B should be 1. C and D are the charge and mass of the nucleus
of interest, respectively. E is the bombarding energy, in MeV. F is the number of Îł rays to
follow. G is the weighting factor for the experiment, usually 1.0.
After the header line, there will be F lines, each for a single Îł ray observed. Each line
consists of four values, A,B,C,D. This indicates that the transition from state A to state B
has C Âą D counts.
This file must be attached as file 3 when running gosia. By default, file 3 is named
fort.3, but a different filename can be specified in the OP,FILE section of the input file.

114

Gosia Output File
During each stage of running gosia, the relevant information for that step will be saved
to file. By default, this is saved as fort.22. However, it can be renamed to something
more descriptive by editing the appropriate line in the OP,FILE section of the input file,
as described in App. B. These output files are fairly self-explanatory, with human-readable
output.
Shown below is a portion of the output file when running the correlated uncertainties,
created during the analysis of the SeGA-JANUS commissioning run. First are shown the
matrix elements, in units of eb. The INDEX column gives the index of the matrix element,
in the order that they are listed in the input file, as described in App. B. The NI and NF
columns list the initial and final state indices, in the order in which they appear in the input
file.
OVERALL
ESTIMATED ERRORS

INDEX

1

NI

NF

1

2

......
2

1
......

3

1

-1.4 ,
5
-2.9 ,
7

ME AND ERRORS

0.81441

( -0.01172 ,

0.01392)

( -0.00467 ,

0.00463)

( -0.00073 ,

0.00067)

1.7 PC
0.15940
2.9 PC
0.03845

115

......
4

2
......

5

2
......

6

2
......

7

2
......

8

2
......

9

2
......

10

3
......

11
......

......

......

-3.0 ,
4
-1.6 ,
5
-8.1 ,
7
-6.0 ,
8
-7.6 ,
7

4

7

8
-5.3 ,

4
......

3

-1.9 ,
4

14

-49.0 ,

-80.4 ,
4

13

2

-2.0 ,
4

12

-1.9 ,

9
-4.4 ,

1.7 PC
-0.95705

( -0.46895 ,

0.10638)

( -0.00726 ,

0.01616)

( -0.02092 ,

0.02073)

( -0.03488 ,

0.01622)

( -0.00502 ,

0.00406)

( -0.00557 ,

0.00520)

( -0.00972 ,

0.00776)

( -0.96916 ,

2.40116)

( -0.00622 ,

0.00631)

( -0.03198 ,

0.02861)

( -0.07236 ,

0.07143)

11.1 PC
0.24314
6.6 PC
1.30478
1.6 PC
0.42944
3.8 PC
0.08398
4.8 PC
0.07362
7.1 PC
0.47660
1.6 PC
-1.20537
199.2 PC
0.33050
1.9 PC
-0.60618
4.7 PC
1.63204
4.4 PC

116

15

5
......

16

5
-75.0 ,

5

7

...... -244.8 ,
17

5
......

18

-7.8 ,
7

......
19

8

7
-34.1 ,

7

12

...... -195.5 ,
20

8
......

21

8
-22.0 ,

9

9

...... -163.5 ,
22

9
......

23

-16.7 ,

10
......

24
......

5
-5.4 ,

2
......

26

10
-20.5 ,

2

25

10

7
-1.4 ,

5

7

0.97168

( -0.72857 ,

0.26494)

( -0.37226 ,

0.02178)

( -0.07003 ,

0.04637)

( -0.53161 ,

1.80646)

( -1.95478 ,

0.97488)

( -0.17921 ,

0.43589)

( -1.33887 ,

2.17384)

( -0.29264 ,

0.09578)

( -0.17111 ,

1.43445)

( -0.01803 ,

0.01799)

( -0.00205 ,

0.00267)

( -0.02221 ,

0.01760)

27.3 PC
0.15204
14.3 PC
0.89297
5.2 PC
1.56100
115.7 PC
1.00000
97.5 PC
-0.81441
53.5 PC
-0.81893
265.4 PC
1.75631
5.5 PC
-0.83497
171.8 PC
0.33257
5.4 PC
-0.14939
1.8 PC
0.08830

117

......

-25.2 ,

19.9 PC

Also included in this output are shown the matrix elements converted to B(EÎť), B(MÎť),
or to the quadrupole moment, depending on what is appropriate for the matrix element
being displayed.
OVERALL
ESTIMATED ERRORS

INDEX

NI

NF

B (E , ML ) ( OR QUADRUPOLE MOMENT ) AND

2

1

0.13265

(

-0.00379 ,

1

0.00508

(

-0.00029 ,

1

0.00030

(

-0.00001 ,

2

-0.72538

(

-0.35544 ,

2

0.05912

(

-0.00348 ,

2

0.18916

(

-0.00602 ,

ERRORS

1

0.00457)
2

5
0.00030)

3

7
0.00001)

4

2
0.08063)

5

3
0.00812)

6

4
0.00606)

118

7

5

2

0.03688

(

-0.00575 ,

2

0.00141

(

-0.00016 ,

2

0.00060

(

-0.00009 ,

3

0.04543

(

-0.00183 ,

4

-0.90897

(

-0.73084 ,

4

0.02185

(

-0.00081 ,

4

0.04083

(

0.00442 ,

4

0.20489

(

-0.01777 ,

5

0.73647

(

-0.55221 ,

5

0.00462

(

-0.00462 ,

5

0.08860

(

-0.01335 ,

7

1.18314

(

-0.40293 ,

0.00284)
8

7
0.00014)

9

8
0.00009)

10

7
0.00149)

11

4
1.81071)

12

7
0.00084)

13

8
0.00442)

14

9
0.01833)

15

5
0.20081)

16

7
0.00508)

17

8
0.00944)

18

7

119

1.36917)
19

12

7

0.20000

(

-0.20000 ,

8

-0.61415

(

-0.13514 ,

9

-0.57095

(

-0.93346 ,

9

0.18145

(

-0.05543 ,

10

-0.53788

(

-0.11023 ,

2

0.02212

(

-0.00233 ,

2

0.00446

(

0.00012 ,

5

0.00156

(

-0.00069 ,

0.58003)
20

8
0.32870)

21

9
1.51559)

22

10
0.02033)

23

10
0.92407)

24

5
0.00246)

25

7
0.00012)

26

7
0.00068)

The extracted matrix elements are discussed in detail in Ch. 4.

120

Appendix C
Spectroscopy of Neutron-Rich Sulfur
Isotopes
In addition to my work with SeGA and JANUS, and in preparation for possible future work
on 44 S with the SeGA-JANUS setup, I also analyzed and published experimental data from
GRETINA and the S800 spectrograph, performing γ-ray spectroscopy of 38−42 S. Results of
this work have already motivated follow-up research on the even-even sulfur isotopes, with
a proposal accepted to perform intermediate-energy Coulomb excitation of 38,40,42,43,44 S at
the NSCL (part of the PhD thesis of B. Longfellow).
c
The material as follows has previously been published in Ref. [83], 
2016
American
Physical Society, used with permission.

Abstract
The low-energy excitation level schemes of the neutron-rich 38−42 S isotopes are investigated
via in-beam Îł-ray spectroscopy following the fragmentation of 48 Ca and 46 Ar projectiles on
a 12 C target at intermediate beam energies. Information on γγ coincidences complemented
by comparisons to shell-model calculations were used to construct level schemes for these
neutron-rich nuclei. The experimental data are discussed in the context of large-scale shellmodel calculations with the SDPF-MU effective interaction in the sd-pf shell. For the
121

even-mass S isotopes, the evolution of the yrast sequence is explored as well as a peculiar
change in decay pattern of the second 2+ states at N = 26. For the odd-mass 41 S, a level
scheme is presented that seems complete below 2.2 MeV and consistent with the predictions
by the SDPF-MU shell-model Hamiltonian; this is a remarkable benchmark given the rapid
shell and shape evolution at play in the S isotopes as the broken-down N = 28 magic number
is approached. Furthermore, the population of excited final states in projectile fragmentation
is discussed.

Introduction
Neutron-rich N = 28 isotones – comprising 48 Ca, 46 Ar, 44 S, and 42 Si – have provided
much insight into the changes of the structure of nuclei encountered in the regime of large
isospin. Evidence for a breakdown of the traditional N = 28 magic number resulted from
the pioneering observation of low-lying quadrupole collectivity in 44 S [71, 72] and fueled the
field of rare-isotope science in the quest to unravel the origin of shell and shape evolution in
exotic nuclei with experimental programs worldwide.
The structure of the neutron-rich sulfur isotopes displays a variety of phenomena that are
closely tied to shell evolution in exotic nuclei [84], with shape [77, 78, 79] and configuration
coexistence [73, 74, 75] driving the properties of 44 S (N = 28) at low excitation energy. It is
interesting to explore the evolution of the low-lying states as N = 28 is approached. It was
pointed out by Utsuno et al. [84] that tensor-driven shell evolution plays a critical role in the
rapid shape transitions that occur in the S and Si isotopic chains towards N = 28. These
effects are included in the SDPF-MU effective shell-model interaction introduced in [84]
and the resulting predictions for the 40,41,42 S level schemes will be tested in the present

122

work. The sulfur isotopes between N = 20 and N = 28 have been studied with a variety
of experimental techniques [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95], however, information
on the level schemes even at low excitation energy is still scarce. Beyond N = 28, very few
excited states have been reported in the S isotopic chain [96, 97].
Gamma-ray spectroscopy following, for example, β decay [94], intermediate-energy Coulomb
excitation [90], multinucleon transfer reactions [95, 93, 86], and projectile fragmentation [91,
92] provided a first, limited glimpse of the level structure of the neutron-rich S isotopes
approaching N = 28 [98]. Here, we report on the in-beam γ-ray spectroscopy of 38−42 S
following the fragmentation of 46 Ar and 48 Ca intermediate-energy projectile beams on a C
target in the center of the GRETINA Îł-ray spectrometer [99]. Complementing the comparisons by Wang et al. [93] of S level schemes to shell-model calculations with the SDPF-U
effective interaction [100], we compare our measurements with similar calculations based on
the SDPF-MU Hamiltonian, which was constructed to describe the shell and shape evolution
in the S and Si isotopic chains as N = 28 is approached [84].

Experiment
The measurements were performed at the Coupled Cyclotron Facility at the National Superconducting Cyclotron Laboratory [33] at Michigan State University.
The 46 Ar projectile beam was produced from a 48 Ca primary beam impinging upon a
1363 mg/cm2 9 Be production target and separated with a 240 mg/cm2 Al degrader in the
A1900 fragment separator [101]. The same production target was used to energy-degrade the
48 Ca

primary beam in a separate setting. The total momentum acceptance of the separator

was limited to ∆p/p = 0.25% for both projectile beams. In two separate runs, the projectile

123

beams impinged upon a 149 mg/cm2 glassy 12 C reaction target located at the pivot point of
the S800 spectrograph [81]. The 46 Ar and 48 Ca beams had mid-target energies of 67.0 MeV/u
and 66.7 MeV/u, respectively. The projectile-like reaction residues formed in the collision
with the target were identified event-by-event with the focal-plane detection system of the
S800 spectrograph and time-of-flight information involving plastic scintillators in the beam
lines upstream of the reaction target. The magnetic rigidity of the S800 spectrograph was set
to center the one-neutron pickup residues, 47 Ar [97] and 49 Ca [102], respectively. In the same
settings, due to the large acceptance of the spectrograph, 40−43 S and 38−40 S, respectively,
entered the S800 focal plane. The particle identification spectra correlating the energy loss
of the reaction residues measured in the S800 ionization chamber and their time of flight are
shown in Fig. C.1; the various S isotopes can be cleanly separated. The statistics for 43 S
were not sufficient to construct a level scheme and thus will not be discussed here. Most
transitions observed in 43 S can be associated with Îł rays previously reported in Ref. [88].
The reaction target was surrounded by the Gamma Ray Energy Tracking In-beam Nuclear Array (GRETINA) [99], consisting of seven detector modules, each containing four
high-purity, 36-fold segmented, Germanium crystals. The GRETINA detectors were arranged to cover forward angles, with four detector modules located at 58◦ and three at 90◦
with respect to the beam axis. The 3-dimensional coordinates of the Îł-ray interaction points
within the GRETINA crystals were determined from the signal decomposition of the digitized traces read out from each segment. The first interaction point, assumed to correspond
to the coordinate with the largest energy deposition, was used to deduce the Îł-ray emission
angle that is used in the event-by-event Doppler reconstruction of the Îł rays emitted by the
reaction products in flight. The spectra shown in this work employ addback, a procedure
recovering the Îł-ray energy of events scattered from one crystal into a neighbor [103].
124

Energy Loss (arb. units)

46

43

S

Ar beam

42

S

41

S

40

S

Time of Flight (arb. units)

Energy Loss (arb. units)

48

Ca beam

40

S

39

S

38

S

Time of Flight (arb. units)
Figure C.1: Particle identification spectra for the reaction residues produced in
12 C(46 Ar,X)Y (upper panel) and 12 C(48 Ca,X)Y (lower panel). The energy loss was measured with the ionization chamber of the S800 focal plane. The time-of-flight was taken
between plastic scintillators in the beam line and in the back of the S800 focal plane. The S
isotopes of interest are unambiguously identified and separated.

125

In-beam detection efficiencies, taking into account the Lorentz boost, were determined
with a GEANT4 simulation [104], with parameters adjusted to reproduce GRETINA’s
response to standard calibration sources at rest. These in-beam efficiencies were used to
obtain the relative Îł-ray intensities from recorded peak areas, as given in the tables in the
next section. To determine γγ coincidence relationships for placement of transitions in level
schemes, software cuts with appropriate background subtraction on γ-ray transitions in γγ
coincidence matrices were used.

Results
In the following, we present our results for each isotope separately. The proposed level
schemes are compared to large-scale shell-model calculations using the SDPF-MU [84] effective interaction for the sd-pf shell. The calculations adopted the full sd and f p model
space for protons and neutrons, respectively, and used effective proton and neutron charges
of eπ = 1.35e and eν = 0.35e [84] and standard spin and orbital proton and neutron g
factors. The calculations were carried out with the code NuShellX [105].

38

S

Figure C.2 shows the Doppler-reconstructed Îł-ray spectrum taken in coincidence with 38 S
reaction residues as produced in the fragmentation of the 48 Ca degraded primary beam.
Several Îł-ray transitions are present that will be discussed below.
We observe strong transitions at 1292(4), 1515(6), and 1534(5) keV that can be iden+
+
+
+
+
tified with the previously reported 2+
1 → 01 , (22 ) → 21 and 41 → 21 transitions, re-

spectively [106, 107, 95, 93, 108]. This is consistent with the coincidence spectra shown in

126

0

Counts/16 keV
Counts/16
keV

3000
Energy (keV)

0

[2344]

[768]
833

[380]
500

2000

20

1515
1534

1292

Counts/4 keV

S

80

40

[2344]

38

120

1000
1500
Energy (keV)

2000

2500

Figure C.2: Doppler-corrected (v/c = 0.357) Îł-ray spectrum taken with GRETINA in coincidence with 38 S as identified with the S800 spectrograph. The inset expands the energy
range from 2 to 4 MeV.

Fig. C.3, where the small number of counts observed agrees with expectations based on the
statistics in the singles spectrum and the detection efficiencies at the respective energies.
Within our limited statistics, the 1515/1534 keV doublet is coincident with the 1292 keV
+
2+
1 → 01 transition. No coincidence relationships could be established for the new, weaker

Îł-ray transitions.
+
The 833(5) keV transition in our spectrum is 16 keV lower than the (6+
1 ) → 41 transition

previously reported at 849 keV [95, 93]. Given the velocity of the S reaction residues,
v/c ∟ 0.35, and the C target thickness of 149 mg/cm2 , excited states with lifetimes of the
order of several tens to hundreds of picoseconds will predominantly decay downstream of the
target, signaled by a lowered peak energy and a left-tail in the Doppler-reconstructed Îł-ray
spectrum. The peak shape of the transition at 833 keV indeed seems to exhibit a left-tail in
addition to the down shift in energy. GEANT simulations for different lifetime values reveal

127

0

0

1292

gate on 1534
5

1292

Counts/8 keV

5
1515
1534

Counts/10 keV

5

gate on 1515
Counts/16 keV

gate on 1292

0

500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
Energy (keV)

Figure C.3: Background-subtracted γγ coincidence spectra for 38 S. Coincidence spectra for
the 1282, 1515, and 1534 keV transitions are shown. The small number of counts observed
agrees with expectations based on the statistics of the measurement.

that the position and shape of the 833 keV transition is consistent with the emission of a
849 keV γ ray from a state that has a mean lifetime, τ , between 100 and 200 ps. Shell-model
calculations with the SDPF-MU effective interaction, in fact, predict a lifetime for the 6+
1
state of ∟40 ps, which is an order of magnitude longer than the lifetime of the 2+
1 state [108]
but a factor of about 4 shorter than our estimate1 . The association of the 833 keV transition
+
reported here with the known 849 keV (6+
1 ) → 41 transition is plausible but would benefit

from more statistics for conclusive γγ coincidence and line-shape analyses.
Table C.1 lists the observed 38 S Îł-ray energies together with their relative intensities and
coincidence relationships. For the new weaker transitions reported here, coincidences could
not be established due to low statistics.
Figure C.4 compares the 38 S level scheme with the SDPF-MU shell-model calculations.
1 Using the measured transition energy instead of the one from the shell model only increases the lifetime
to 54 ps at constant B(E2) strength.

128

EÎł [keV]
[380(5)]
[768(5)]
833(5)
1292(4)
1515(6)
1534(5)
[2344(9)]

Rel. Intensities (%)
5(1)
7(1)
25(3)
100(10)
10(2)
29(4)
10(2)

Coinc.

1515, 1534
1292
1292

Table C.1: Energies, intensities and coincidence relationships for Îł-ray decays observed in
38 S. The 833(5) keV peak is significantly below the literature value of 849 keV, and has a
peak shape indicative of a left tail. In comparison to simulations, both may be explained by
a lifetime of 100 ps < τ < 200 ps. Transition energies placed in brackets indicate tentative
identifications of Îł-ray peaks.
The experimental scheme only contains the previously known transitions since the new Îł
rays reported here are too weak to be placed in the level scheme based on coincidence
relationships. The weak transition at 380 keV may correspond to the 383 keV transition
visible in the 38 S spectrum of Wang et al. [93]. In their work as well as here, this Îł ray
remains unplaced. We note that the association of the 2807 keV level with the 2+
2 state from
the shell model is supported by the decay branching ratio. It is predicted within the SDPF+
MU shell-model calculations that the 2+
2 → 21 transition is the dominant decay branch with

an intensity exceeding 96% of the total yield out of the 2+
2 state. No evidence for a 2807 keV
transition has been reported in any of the previous Îł-ray spectroscopy measurements that
observed the 1515 keV transition [106, 95] and there is no evidence for such a transition in
the present work (see Fig. C.2).
Consistent with previous studies, transitions from yrast states are the most prominent in
the Îł-ray spectra of reaction residues from secondary fragmentation reactions with several
nucleons removed from the projectile [109]. In the following, we will continue to explore this
population pattern and use it to argue possible level schemes for the more exotic S isotopes.

129

(6+)

1+
3+
6+

(2+)

0+

849

3000

4+

2+
4+
1515

2000

1534

Energy (keV)

4000

2+

2+

1292

1000

0

0+
38

0+

SDPF-MU

S

Figure C.4: Proposed experimental level schemes for 38 S based on previous data and coincidence relationships. The experimental level scheme is confronted with shell-model calculations using the SDPF-MU Hamiltonian.

130

39

S

0

500

1500
1000
Energy (keV)

20
2000

Counts/8 keV

20

40
1655
1728

1518

702

533

466
370
392

40

337

Counts/4 keV

60

Figure C.5: Doppler-reconstructed Îł-ray spectrum in coincidence with 39 S (v/c = 0.348).
The inset expands the higher-energy region of the spectrum.

39

S

Figure C.5 shows the Doppler-reconstructed Îł-ray spectrum taken in coincidence with 39 S
reaction residues produced in the projectile fragmentation of 48 Ca. The transitions at 337(4),
392(6), 466(4), 702(4), 1518(4), 1655(6), 1728(5) keV have been reported before from multinucleon transfer reactions [93, 110], β decay of 39 P [111], and 40 P βn emission [94]. We
identify the 392(6) keV line with the 398 keV transition reported in the references above
Coincidences of the 392, 337 and 466 keV transitions were reported from the β-decay
work [94]. In our intensity and peak-to-background regime at low energies, weak evidence
was seen only for the 337-466 keV coincidence (see Fig. C.6). The two new transitions
reported in this work, at 370(6) keV and 533(4) keV, appear to be in coincidence, with the
370 keV transition feeding the state that decays by emitting a 533 keV Îł ray, based on
intensity arguments.
The transition energies, intensities, and coincidence relationships are summarized in Ta131

gate on 466

gate on 337
Counts/8 keV

5

[466]

0

0

5

337

0

500

gate on 533
5

[370]

0

1000 0
500 1000 0
Energy (keV)

500 1000

Figure C.6: Background-subtracted γγ coincidence spectra for 39 S. Software gates on the
337, 466, and 533 keV transitions are displayed. The coincidence spectra shown in the left
and in the middle panel investigate the previously claimed 337-466 keV coincidence [94],
which seems plausible based on our low-statistics data. The right panel provides weak
evidence for a coincidence between the newly observed 533 and 370 keV transitions.

ble C.2. We confirm previously reported Îł-ray transitions and add two news ones at 370 and
533 keV that appear to be in coincidence.
From the present data on 39 S, it is hardly possible to propose a firm level scheme –
this is not just due to the lack of coincidences but also related to the expected structure at
low energies. The difficulty becomes apparent from the predicted level scheme displayed in
Fig. C.7. A triplet of states is expected within an energy range of ∟200 keV. Depending
on the exact excitation energies, the two lowest-lying excited states may be nanosecond
isomers, as predicted by the shell-model calculation. In-beam Îł-ray spectroscopy at our
beam velocities has limited sensitivity to nanosecond isomers. This makes it difficult to
construct a level scheme since transitions or cascades can feed the ground state or any of the
possible isomers.

132

4000

Energy ( keV)

3000

1/2-

2000

9/23/211/2-

1000

0

7/2- 4.3ns
5/2- 2.1ns
3/2-

SDPF- MU

Figure C.7: Predicted level scheme and nanosecond lifetimes for 39 S from shell-model calculations using the SDPF-MU effective interaction.

133

EÎł [keV]
337(4)
370(6)
392(6)
466(4)
533(4)
702(4)
1518(4)
1655(6)
1728(5)

Rel. Intensities (%)
28(5)
9(3)
42(7)
71(10)
38(7)
42(8)
100(15)
59(11)
43(9)

Coinc.
[466]

337
[370]

Table C.2: Energies, intensities and coincidence relationships for Îł-ray decays in 39 S.
Chapman et al. [110] propose a level scheme in comparison to shell-model calculations and
N = 23 isotones, with 398- and 339-keV transitions depopulating the (3/2−
1 ) excited state
−
to the (7/2− ) ground state and the (5/2−
1 ) first-excited level at 59 keV. The (3/2 ) is then

suggested to be fed by the 466-keV decay of the first (3/2+ ) cross-shell excitation. While this
is consistent with previously reported coincidence relationships, it would mean that, based
on our intensities Iγ (337) + Iγ (392) ≈ Iγ (466), there is no room for any significant direct
population or additional unobserved, discrete feeding of the (3/2− ) level. The transitions
reported here (Table C.2) are indeed indicative of positive-parity states, i.e. 3/2+ and
1/2+ , located in the gap from 300 keV to 1600 keV that separates the first two groups of
negative-parity states in 39 S (Fig. C.7). The higher-energy transitions are likely connecting
the second group of negative-parity states expected between 1.6 and 2 MeV to the first group
near the ground state. The observed energies of 1518, 1655 and 1728 keV fit this picture
well. Certainly, a firm level scheme for 39 S requires a measurement with sufficient statistics
for γγ coincidences, and sensitivity to low-energy γ-ray transitions and isomers.

134

40

S

Figure C.8 shows the Doppler-reconstructed Îł-ray spectrum taken in coincidence with the
40 S

reaction residues. 40 S was produced in the fragmentation of 48 Ca as well as from the

46 Ar

projectile beam (see Fig. C.1). The two data sets were added for the purpose of Îł-ray

spectroscopy. Previous information on the spectroscopy of 40 S stems from intermediateenergy Coulomb excitation [71], fragmentation [92], 40 P β decay [94], and most recently
multinucleon transfer [93].
Nine γ-ray transitions are apparent in our spectrum. Compared to the β-decay work,
the only common transitions are at 902 and 1350 keV [94]. This complementarity in the
population pattern can most likely be attributed to the suspected (2− , 3− ) ground state of
the β-decay parent and the resulting selective population of final states in the decay daughter.
This is in contrast to the observation that fragmentation reactions seem to populate low-lying
yrast states the strongest. Other overlapping transitions with previous work are 891(13) keV
from intermediate-energy Coulomb excitation [71], 909(5) and 1356(6) keV from projectile
fragmentation [92], and 904, 1352, and 1572 keV from multinucleon transfer [93].
In addition, γγ coincidence relations could be established for several transitions, as shown
in Fig. C.9. The coincidence spectra of the 902, 1350, and 1572 keV transitions show that
they are mutually coincident, consistent with decaying to each other in a cascade. Weak
evidence is visible in the spectrum gated on 1350 keV for a coincidence with the 2057 keV
transition.
The observed transition energies, intensities and coincidence relations are listed in Table C.3. It is clear from the coincidence spectra in Fig. C.9 that the statistics in the 1572 keV
line is just sufficient for a γγ coincidence analysis and, therefore, a placement of the weaker

135

40

1000
Energy (keV)

2057

[1850]

100
50

2000

Counts/8 keV

0

973
1102
1218

200

1572

1350

400

851

Counts/4 keV

902

S

0

500

1000
1500
Energy (keV)

gate on 902
gate on 1350
2057

1572

1350

0
5

1350

902

0
20

902

Counts/8 keV

20

1572

Figure C.8: Doppler-reconstructed Îł-ray spectrum in coincidence with 40 S (with v/c = 0.341
for 40 S from 48 Ca beam and v/c = 0.350 for 40 S from 46 Ar beam).

gate on 1572
2000

Figure C.9: Background-subtracted γγ coincidence spectra for 40 S. Spectra in coincidence
with the strongest transitions at 902, 1350, and 1572 keV are shown.

136

EÎł [keV]
851(4)
902(4)
973(4)
1102(6)
1218(4)
1350(4)
1572(4)
[1850(5)]
2057(6)

Rel. Intensity (%) Coinc.
5(1)
100(8) 1350, 1572
5(1)
9(1)
7(1)
76(6) 902, 1572, 2057
20(2) 902, 1350
4(1)
8(1)

Table C.3: Energies, efficiency-corrected relative intensities, and coincidence relations for
Îł-ray decays observed in 40 S. As for 38 S, the transitions suspected to form the even-spin
yrast cascade are the most intense.
transitions reported here in the level scheme was not possible.
Figure C.10 shows the experimental level scheme proposed in this work. Based on the
coincidences and the Îł-ray intensities reported here, see Fig. C.9 and Table C.3, we propose
+
+
+
the 1572 – 1350 – 902 keV cascade to correspond to the (6+
1 ) → (41 ) → 21 → 01 even-spin

yrast sequence, consistent with Wang et al. [93]. Also, the 902, 1350 and 1572 keV transitions
are the most intense in our spectrum, consistent with the population pattern reported in
Section III.A for 38 S where the strongest transitions were the decays within the ground-state
band up to the 6+ state. The 2057 keV transition is placed tentatively as feeding the (4+
1)
state based on the spectrum in coincidence with the 1350 keV line.
+
We note that Winger et al. attribute the 1350 keV transition to the (2+
2 ) → 21 decay.

This is at odds with our work and with the results from the multinucleon transfer [93] and the
earlier projectile fragmentation measurement [92], where the 902 and 1350 keV transitions
+
+
+
are attributed to the 2+
1 → 01 and (41 ) → 21 decays, respectively. We see no evidence for

the 1013 keV Îł ray that was tentatively proposed by Winger et al. to connect the yrast 4+
and 2+ states.
Shell-model calculations with the SDPF-MU effective interaction describe the even-spin
137

6000

5000

6+
4+
5+
6+
2+
0+

2057

3000

(6+)

1572

Energy (keV)

4000

4+
(4+)
4+
1350

2000

1000

2+
902

2+

0

0+

0+

40

SDPF- MU

S

Figure C.10: Proposed experimental level scheme for 40 S compared to the SDPF-MU shellmodel calculations. The level scheme is based on the γγ coincidence spectra and intensities.
The tentative placement of the 2057 keV transition is indicated. The tentative identification
of the low-lying even-spin yrast sequence is based on comparison with the shell model and
the population pattern of excited states observed throughout this work.

138

2000
Energy (keV)

3000

0

Counts/8 keV

2578

500
2338

1893
2099

1548
1611/1633

1276/1302

1099

451

1000

1000

3216

0

901

1000

502

2000

S

587 536

Counts/4 keV

3000

41

Figure C.11: Doppler-reconstructed Îł-ray spectrum in coincidence with 41 S (v/c = 0.342).
The spectrum is expanded with changed binning beyond 500 keV to highlight the high
density of small peaks up to ∟3.2 MeV.
yrast sequence of 40 S well as shown in the comparison in Fig. C.10. The level density in
40 S

is predicted to increase significantly at about 3 MeV. The many weak transitions not

placed within the level scheme will originate from the multitude of states in this excitation
energy region. Possible candidate states for the level established by the 2057 keV transition
are higher-lying 4+ or 6+ states or the first 5+ level (see Fig. C.10).

41

S

Figure C.11 shows the Doppler-reconstructed Îł-ray spectrum taken in coincidence with 41 S
reaction residues produced in the fragmentation of 46 Ar projectiles. Sixteen Îł-ray transitions
are visible in the complex spectrum. Of these, transitions that likely correspond to our 451,
902, and 1613 keV Îł-ray transitions have been previously observed in intermediate-energy
Coulomb excitation [90] (449 and 904 keV), in β decay from 41 P [111] (904, 1308 and
1613 keV) and in multinucleon transfer [86] (449 keV).

139

EÎł [keV]
451(4)
502(4)
536(4)
587(4)
901(4)
1099(4)
1276(4)
1302(4)
1548(4)
1611(4)
1633(4)
1893(4)
2099(5)
2338(6)
2578(5)
3216(8)

Rel. Intensity (%)
100(6)
1.0(2)
8.8(8)
2.5(2)
4.8(4)
41(3)
4.2(5)
13.4(1.2)
7.3(8)
11.5(1.1)
19(2)
7.4(8)
6.4(7)
5.0(6)
6.0(7)
6.7(8)

Coinc.
536, 1099, 1633
451, 1099

451, 536

451

Table C.4: Energies, efficiency-corrected relative Îł-ray intensities, and coincidences for 41 S.
Our level of statistics allowed for a γγ coincidence analysis, as shown in Fig. C.12, with
several conclusive relationships established. In a software gate on the 451 keV line, 536, 1099
and 1633 keV transitions are clearly visible. The 536 keV transition is in coincidence with
both 451 and 1099 keV and a gate on 1099 keV returns the 451 and 536 keV lines. The
1633 keV transition is cleanly observed only in coincidence with 451 keV. We note that the
peak structure at ∟1620 keV is a doublet of two peaks with centroids of 1611 and 1633 keV,
where a software gate on the right peak, mainly 1633 keV, returns the 451 keV while a
gate on the lower-energy side, narrowly on 1611 keV, does not (see Fig. C.13). Similarly,
the 1302 keV transition is comparably intense and no coincidence is apparent, as shown in
Fig. C.13. The transition energies, intensities, and coincidence relationships are summarized
in Table C.4.
The proposed level scheme is shown in Fig. C.14. The placement of the transitions is
based on γγ coincidences, energy sums, and intensities observed in the present work.
140

1633

1099

536

40

gate on 536

1099

451

20
0

gate on 1099

451
536

50
0
20
0

gate on 1633

451

Counts/10 keV

0

gate on 451

500

1000 1500 2000 2500
Energy (keV)

Figure C.12: Background-subtracted γγ coincidence spectra for 41 S. Spectra in coincidence
with 451, 536, 1099, and 1633 keV are shown.

451 keV

gate on 1302

0
10

451 keV

gate on 1611

Counts/10 keV

10

0
0

500

1000 1500 2000 2500 3000
Energy (keV)

Figure C.13: Background-subtracted γγ coincidence spectra for 41 S. Spectra gated on 1611
and 1302 keV transitions in 41 S are shown. These transitions do not appear in coincidence
with 451 keV (see also Fig. C.12) or any other transitions that they would feed.

141

4000

11/21/2-

0

1302

(3/2-,1/2-,3/2+)
1611

1548

1099

9/2-

(9/2-)

451

1000

3/2-

(3/2-,1/2-,3/2+)
1633

2000

3/2-

(11/2-)
536

Energy (keV)

3000

41

(7/2-)

7/2-

(5/2-)

5/2-

SDPF-MU

S

Figure C.14: Proposed experimental level schemes for 41 S based on the observed coincidences, intensities, energy sums and comparison to the shell model (SDPF-MU Hamiltonian). Solid lines and filled arrows indicate firm level and transition assignments, the dashed
line and unfilled arrows indicate a tentative placement. Given that the 1611 and 1302 keV
Îł-ray transitions are strong and not in coincidence with 451 keV, we argue that they likely
populate the ground state directly. Comparison to shell-model energies, decay branchings,
and systematics was used to assign tentative J π values (see text).

142

Based on comparison between shell-model calculations and observed decay patterns, spinparities of (5/2− ), (7/2− ), (9/2− ), (3/2− , 1/2− , 3/2+ ) and (11/2− ) are tentatively assigned
to the lowest-lying states in our experimental level scheme. These assignments provide
reasonable matches of measured and calculated excitation energies, and in addition, are
supported by comparison of the measured and calculated decay patterns. The 9/2− state is
predicted to have a branching ratio of 83% to the 7/2− state and 17% to the 5/2− ground
state. As listed in Table C.4, the branching ratio for (9/2− ) from our work is 85(2)% to
the (7/2− ) state and 15(2)% to the ground state. For the level that we tentatively identify
as the (11/2− ) state, the strongest decay leads to the (7/2− ) state with 68(3)% of the
total strength, and the remaining 32(3)% feeds the tentative (9/2− ) state. The predicted
branching ratios for these transitions are 69% and 31%, respectively, in good agreement
with the data. Our (3/2− , 1/2− , 3/2+ ) assignments are based on the fact that the 1302 and
1611 keV Îł rays are among the most intense transitions (see Table C.4) while not being in
coincidence with 451 keV or other strong transitions. We propose that both decay to the
ground state directly, forming excited states at 1302(4) and 1611(4) keV. Comparison to the
−
SDPF-MU shell-model calculations reveal the 3/2−
1 and 1/21 states as the closest in energy

with transitions to the ground state exceeding 97% of all de-excitations. The previous βdecay work offers support for this proposition. Winger et al. [111] report 1308 and 1614 keV γ
rays that likely correspond to the 1302 and 1611 keV transitions observed in the present work.
Our shell-model calculations with the SDPF-MU Hamiltonian suggest that the decay parent
41 P

has a ground-state spin-parity of 1/2+ and a first excited 3/2+ state at 274 keV. Either

of these possible J π values for the 41 P ground state could populate the 1/2− and 3/2− states
in 41 S, allowing their observation in [111]. If the 1/2− and 3/2− states were indeed at 1302
and 1611 keV, we would have observed all low-lying negative-parity states below 2.2 MeV
143

consistent with the systematics of excited states populated in fragmentation. However,
positive parity-states, corresponding to neutron cross-shell excitations across N = 20 as
discussed for the Si isotopic chain [112], may be found at low excitation energy as well. A
3/2+ level would be expected to decay to the (5/2− ) ground state and would have been
strongly populated in the β decay of the positive-parity ground state. Such a positive-parity
state is expected from systematics, but is based on cross-shell excitations and is therefore
outside of the shell-model space employed here.
We show the shell-model level scheme up to 4 MeV and it is clear that the multitude
of weaker, unplaced Îł-ray transitions likely depopulate the higher-lying states. It is noted
that our level scheme disagrees with the scheme proposed by Wang et al. [86] based on a
low-statistics Îł-ray singles spectrum obtained in multinucleon transfer. Wang et al. suggest
that the 904 keV Îł-ray transition reported in intermediate-energy Coulomb excitation [90],
although they did not observe it in their own work, corresponds to the decay of the 9/2−
state to the ground state. This contradicts the expected decay pattern for such a state that
would predominantly decay to the 7/2− state.
Since multistep processes are severely suppressed in intermediate-energy Coulomb excitation [12], the observed Îł rays in the work by Ibbotson et al. [90] were attributed to
the depopulation of states at 449 and 904 keV, respectively. Based on a particle-rotor approach, the ground state and the proposed 449 and 904 keV levels were assigned 7/2− ,
5/2− and 9/2− quantum numbers, respectively [90]. M 1 excitations are heavily suppressed
in Coulomb excitation and, in the absence of parity change, only E2 excitations have to
be considered. In intermediate-energy Coulomb excitation, the proportionality between the
excitation cross section and the B(Eλ; Jgs → Jf ) transition strength depends on the multipolarity, λ, but not explicitly on the spin values [12]. Therefore, we will refer to the E2
144

excitation strengths deduced by Ibbotson et al. as B(E2 ↑). Now, assuming the SDPFMU shell-model spin and parity assignments, the B(E2 ↑)449 keV = 167(65) e2 fm4 and
B(E2 ↑)904 keV = 232(56) e2 fm4 values from [90] have to be compared to B(E2; 5/2− →
7/2− ) = 147 e2 fm4 and B(E2; 5/2− → 9/2− ) = 59 e2 fm4 , respectively. While the measured
B(E2) strength to the first excited state agrees well with the shell-model picture, all other
calculated B(E2) excitation strengths, including the one to the 9/2− state, are expected to
−
−
be smaller by a factor of 4 (9/2−
1 ) or two orders of magnitude (3/21 and 1/2 ) than what is

reported for the B(E2 ↑)904 keV value in [90]. While a very weak γ-ray transition at 902 keV
is visible in our spectrum, it would be surprising if it corresponded to a low-lying state based
on the population pattern of excited states in projectile fragmentation that we have observed
so far. Ibbotson et al. explored the possibility of E1 excitations in their measurement and
concluded that the measured cross sections would be beyond the recommended upper limits
for E1 strength in the region but that this possibility of a parity-changing transition cannot
be fully excluded [90].
Wang et al. further report a Îł ray at 638 keV based on very low statistics and without
coincidence data and assign it to connect the 11/2− and the 7/2− states. We see no evidence
for a 638 keV transition in our 41 S spectrum.
The energies and Îł-ray branching ratios of our level scheme agree with the shell-model
calculation using the SDPF-MU effective interaction. The fact that we observe candidate
states matching all calculated levels below 2.2 MeV is consistent with a picture where, with
no discernible final-state selectivity, the lowest-lying states are the most prominent, likely
populated directly in the reaction and fed indirectly through a multitude of higher-lying
excited states that cascade toward the ground state.

145

1000

2000

Counts/4 keV

[1143]

949
992

3000

4000

4592

1000

100
4102
4266

2000

200
0

3415

2011
2100
[2154]
2677
2803
3002
3150

902

3000

0

S

1787/1820

Counts/4 keV

4000

42

5000

Energy (keV)
Figure C.15: Doppler-reconstructed Îł-ray spectrum in coincidence with 42 S (v/c = 0.335).
The insets expand energy regions of the spectrum with weaker intensity transitions. Transitions at 1143 and 2154 keV are tentative.

42

S

Figure C.15 shows the Doppler-reconstructed Îł-ray spectrum taken in coincidence with 42 S
reaction residues resulting from the fragmentation of 46 Ar. More than 15 Îł-ray transitions
are identified in the spectrum. Of these transitions, only the 902 keV and 1820 keV Îł rays
have been reported before, in intermediate-energy Coulomb excitation (890(15) keV) [71]
and in the fragmentation of a 48 Ca primary beam (904 and 1821 keV) [92]. Two Îł-ray
transitions, at 1466(8) keV and 1875(9) keV, reported in [92] are not observed in the present
work.
In addition, γγ coincidences were observed between several transitions, as shown in
Figs. C.16 and C.17. First, the coincidence spectra for 902, 1787, and 1820 keV indicate
that all three transitions are in coincidence with each other, forming a cascade that can be
sorted by intensity. Furthermore, the 2100 keV transition is in coincidence with 902 keV

146

500

gate on 1787

Counts/10 keV

2100

2677

2803
2803

1820

902

0

gate on 902

gate on 182010

1787

0
10

1820

1820
902

0
100

1787

902

Counts/10 keV

50
0
20

gate on 2803

0

1000 1500 2000 2500 3000 3500
Energy (keV)

Figure C.16: Background-subtracted γγ coincidence spectra for 42 S. Spectra in coincidence
with 902, 1787, 1820, and 2803 keV are shown.

and the 2803 keV Îł-ray decay populates the state decaying by the 1820 keV transition.
An interesting structure emerges at high excitation energy. The background-subtracted
coincidence spectrum for the weak 949 keV transition (see inset of Fig. C.15) shows the
992 and 2677 keV transitions. A gate on the 992 keV line returns 902, 949 and 2677 keV
transitions and shows a 992 keV self-coincidence that may point to a doublet structure. In
coincidence with 2677 keV, all three transitions, 902, 949, and 992 keV, are visible.
The Îł-ray transition energies, intensities and coincidence relationships are listed in Table C.5.
Based on γγ coincidences, intensities and energy sums, the level scheme shown in Fig. C.18
is proposed. From the coincidence spectra of Fig. C.16 and the intensities listed in Table C.5
we propose the 1787 – 1820 – 902 keV cascade to correspond to the even-spin yrast sequence
(6+ ) → (4+ ) → 2+ → 0+ . This is in reasonable agreement with the shell-model calculation
where the biggest deviation is observed for the 6+ state with the calculation placing the

147

0

902/949/992

2677

902/949/992

gate on 949

2677

992

Counts/10 keV

5
0
10
0
20

gate on 992
gate on 2677

500 1000 1500 2000 2500 3000
Energy (keV)

Figure C.17: Background-subtracted γγ coincidence spectra for 42 S. Spectra in coincidence
with 949, 992, and 2677 keV are shown.

EÎł [keV]
902(4)
949(4)
992(6)
[1143(4)]
1787(4)
1820(4)
2011(4)
2100(4)
[2154(4)]
2677(4)
2803(4)
3002(4)
3150(4)
3415(9)
4102(8)
4266(7)
4592(7)

Rel. Intensity (%)
100(6)
1.2(1)
2.2(2)
1.6(2)
8.4(7)
33(2)
2.2(3)
1.8(2)
0.9(1)
10.6(9)
1.7(2)
10.1(9)
5.4(6)
5.1(5)
5.2(6)
3.1(4)
2.9(4)

Coinc.
1820, 2100, 2677
992, 2677
902, 949, 992, 2677
902, 1820
902, 1787

902, 949, 992
902, 1820

Table C.5: Energies, efficiency-corrected relative Îł-ray intensities, and coincidences for 42 S.
The 992 keV peak appears in coincidence with itself, suggesting that a doublet cannot be
excluded for this transition.

148

state about 400 keV higher than the suggestion from experiment.
Placing the 2100 keV transition on top of the 2+
1 state leads to a state at 3002(6) keV.
In fact, we observe a 3002 keV Îł-ray that then becomes a candidate to depopulate this
new state directly to the ground state. We associate this state tentatively with the second
2+ state of 42 S. The shell model predicts the 2+
2 level at 3072 keV with a 84% branch to
the ground state and the remaining 16% decaying to the 2+
1 state. From our intensities in
Table C.5 we obtain a decay branching of 85(2)% to the ground state and 15(2)% to the
2+
1 level. We note that our assignment is at odds with the level scheme proposed by Sohler
et al. [92]. We do not observe the 1875 keV transition that is attributed in their work to
+
depopulate the second 2+ state to the first 2+ state. Such a situation, where the 2+
2 → 01
+
transition is not observed while the 2+
2 → 21 is, would also be in contradiction to the shell+
model calculations that have 2+
2 → 01 as the strongest branch by a factor of five. We also

observe no evidence for the 1466(8) keV Îł-ray transition that establishes a 4245 keV state
in 42 S in the work by Sohler et al. [92].
The 2677 keV transition feeding the 2+
1 state leads to a state at 3579(6) keV that, based
on excitation energy alone, may be identified with the 3+
1 state from the shell model or with
+
+
a state from the group just above, comprising the 4+
2 , 32 and 02 states. From the decay
+
pattern, however, the 3+
1 and 42 levels are the only two with an essentially exclusive branch
+
to the first 2+ state. The 3+
2 and 02 states are expected to exhibit significant decays to

the second 2+ state. A 3− spin-parity assignment cannot be excluded and is outside of our
shell-model configuration space.
From Fig. C.17 and the intensities of Table C.5, we construct a cascade 949 – 992 –
2677 keV on top of the 2+
1 state. This leads to two new excited states, at 4571(7) keV and
5520(8) keV. The 2803 keV transition that was found in coincidence with the (4+
1 ) state now
149

949

6000

6+
1+

(6+)
2803

992

0+
3+

Energy (keV)

1787

4000

2+
4+

4+
3+

(2+)

2+

(4+)

2100

2677

3002

1820

2000

4+

2+

0+

0+

902

2+

0

42

SDPF-MU

S

Figure C.18: Proposed experimental level scheme for 42 S based on the γγ coincidence spectra, intensities, and energy sums. The experimental data is confronted with shell-model
calculations using the SDPF-MU effective interaction. Comparison to shell-model energies
and decay branchings was used to assign tentative J π quantum numbers (see text).

150

is a second branch of the new level at 5520 keV. Due to the high expected level density in this
excitation energy region, it is not possible to associate this structure with states and decays
of the SDPF-MU shell-model calculation. Many of the higher-lying 4+ and 6+ states, for
example, show decay patterns broadly consistent with the high-lying structure in our level
scheme.

Discussion
In Section III, we compare the 38,40,41,42 S level schemes from experiment to shell-model
calculations with the SDPF-MU Hamiltonian. The motivation for choosing this shell-model
effective interaction is rooted in its optimization to explain the complex structure of the
N = 28 isotones 42 Si and 44 S, comprising phenomena such as shape and configuration
coexistence, on a common footing [84]. Furthermore, SDPF-U level schemes are available
in the literature for 39 S [110], 40 S [93], and 41 S [86]. In contrast to SDPF-MU, the SDPFU effective interaction consists of two parts, one valid for Z ≤ 14 and one applicable to
Z ≥ 15 [100]. Earlier work benchmarked the performance of SDPF-MU in the chain of Si
leading up to N = 28 [112] and the present work extends this comparison to the S isotopic
chain. Below, (i) the character of the quadrupole collectivity of the even-mass S isotopes
is considered from E(4+ )/E(2+ ) and E(6+ )/E(2+ ) energy ratios, (ii) the transition into
the N = 28 “island of inversion” is characterized by an analysis of the decay properties of
the 2+
2 state, (iii) the odd-mass S isotopes are discussed, and the emerging pattern for the
population of excited states in fragmentation reactions is summarized.
For even-even nuclei, the ratios of yrast excitation energies have long been used to classify
collectivity in terms of vibrational, rotational, and transitional character. The chain of S

151

isotopes, however, is challenging as shape and configuration coexistence is at play. We use
+
+
+
E(4+
1 )/E(21 ) and E(61 )/E(21 ) energy ratios to compare the ground-state bands of our

proposed level schemes to the SDPF-MU shell-model calculations. Figure C.19 shows the
comparison of these ratios for the even-mass sulfur isotopes with N = 20 − 28. Assuming
the 6+
1 energies proposed in this work, good agreement is reached for the measured and
+
38,40,42 S. For 36,44 S, the 6+ state has not been identified
calculated E(6+
1 )/E(21 ) ratios in
1

in the literature. The systematics, which are not solely based on comparison with the shell
model but also the population pattern of excited states that has emerged in this work, lend
40,42 S. For the E(4+ )/E(2+ ) ratio, close
support to our new tentative 6+
1 assignments for
1
1

agreement is observed for 40,42 S while measurement and theory are only within ∟ 25% for
the semi-magic 36 S and neighboring 38 S. It is noted that the shell-model calculation is not
expected to work well for 36 S since the neutrons are restricted to the sd shell.
The case of 44 S is complex - a low-lying 4+ state has been observed [73] that, based
on two-proton knockout cross sections [73] and evidence for a long lifetime from a Îł-ray
line-shape analysis [73, 113], is suggested to correspond to a K = 4 isomer [73, 74]. This
+
+
state differs in configuration from the 2+
1 state, resulting in a strongly hindered 41 → 21

transition. The 4+ level of 44 S that is connected to the collective 2+
1 state [72] by a strong E2
decay has not yet been identified experimentally. With the intent of probing the collective
nature of states with a similar underlying structure, we use the energies of the 4+
2 shell-model
+
+
+
state for 44 S since the corresponding cascade 6+
1 → 42 → 21 → 01 is connected by the
+
strongest E2 transitions. Using the 4+
1 and 62 energies instead would not be noticeable

in Fig. C.19 as the energies of the first and second 4+ and 6+ states differ only by 56 and
134 keV, respectively. Future experiments will put the predictive power of the SDPF-MU
shell-model Hamiltonian to the test once the collective structures beyond the first 2+ state
152

E(6 )/E(2 )

+
+

Expt.
SDPF-MU

5.0
4.0
3.0
3.0

E(4 )/E(2 )

+
+

2.5
2.0
36

S

38

S

40

S

42

S

44

S

Figure C.19: Comparison of the excitation energy ratios of the first excited 2+ , 4+ , and
6+ states across the neutron-rich sulfur isotopes. For 36 S, the 6+
1 state has not yet been
identified in the literature [98], and is expected at high excitation energy where the level
density is significant. The collective 4+ and 6+ states of 44 S have not yet been observed
either. For the calculated 44 S E(4+ )/E(2+ ) ratio, the shell-model energy of the second 4+
+
+
+
state is used since the 6+
1 → 42 → 21 → 01 cascade is connected by the strongest E2
+
transitions. However, the energy ratios would not change if the 4+
1 or 62 energies were used
+
+
+
+
instead since E(6+
2 ) − E(61 ) = 56 keV and E(42 ) − E(41 ) = 134 keV. The tentative 61
assignments for 40,42 S stem from the measurements presented here.

are identified in the complex nucleus 44 S whose low-lying structure is sensitively determined
by shape and configuration coexistence.
An interesting systematic trend emerges for the 2+
2 states in the S isotopic chain. According to the shell-model calculations with the SDPF-MU Hamiltonian, the second 2+ state
+
+
+
in 42 S has a unique structure that is reflected in the 2+
2 → 01 and 22 → 21 branching ra+
tio. For 38 S and 40 S, the 2+
2 → 21 transitions are predicted to dominate with 96.4% and

99.4%, respectively. For 42 S, the branching is essentially reversed with 84% predicted for
+
+
+
the 2+
2 → 01 transition and only 16% for the 22 → 21 decay. The non-observation of

153

+
+
+
38
42
the 2+
2 → 01 branch in S here and in [95] and the 85(2)% branch for (22 ) → 01 in S

reported here are in remarkable agreement with this sudden structural change. We note that
in 40 S the 2+
2 level could not be identified — it is expected in a region of already high level
density — and, solely based on energy, the 1850 keV γ ray may be a candidate for the 2+
2
to 2+
1 transition.
42 S lies in
The reason for the abrupt change in the decay pattern of the 2+
2 state in
+
42
its neutron single-particle structure. The 2+
1 and 22 states in S differ in the occupancies

of the 0f7/2 and 1p3/2 neutron orbitals as detailed below. These two orbitals cannot be
connected by the M 1 magnetic dipole transition operator. Consequently, the B(M 1; 2+
2 →
−3 µ2 in 42 S
2+
1 ) ≡ B(M 1) transition strength is strongly hindered with B(M 1) = 0.1355·10
N
+
versus B(M 1) = 0.1924 Âľ2N in 40 S, disfavoring the 2+
2 → 21 branch at N = 26. Figure C.20

illustrates this by showing the occupancies of the neutron 1p3/2 orbital for the 0+ (red) and
2+ (blue) states up to 4.5 MeV from the calculations with the SDPF-MU Hamiltonian. Here,
an increase of the neutron 1p3/2 occupancy is correlated with a decrease of the neutron 0f7/2
occupancy. The 2+ state with the largest 1p3/2 occupancy is lowered in energy between 38 S
and 42 S due to a reduction in the 1p3/2 − 0f7/2 single-particle gap as the neutron number
increases. Up to 42 S, the configurations of the 0+ and 2+ states below 2 MeV are dominated
by the (a) = (0f7/2 )n configuration. The wave functions of the states above 2 MeV in 42 S
are dominated by the (b) = (0f7/2 )(n−2) (1p3/2 )2 configuration2 . Of all S isotopes shown,
+
42
the neutron p3/2 occupancy differs the most between the 2+
1 and 22 states in S, leading
+
to the hindrance of the corresponding 2+
2 → 21 M 1 transition and the resulting very small

B(M 1) value quoted above.
2 Due to the mixing of (a) and (b) and a small occupancy of the 0f
5/2 and 1p1/2 orbitals, the change in
the occupancy of the p3/2 orbital is not exactly 2 between the two groups of states.

154

Figure C.20 shows a dramatic change in the 0+ and 2+ level density below 4 MeV for 44 S
and ties this to the excitation of neutrons across the N = 28 shell gap into the p3/2 orbital.
In 44 S, the correlation energy of the shell-breaking (b) configuration now becomes larger
than that of the closed-shell configuration (a), putting 44 S inside the “island of inversion” at
+
N = 28. The sensitivity of the 2+
2 → 21 M 1 decay to the p3/2 neutron intruder occupancy

now provides a very stringent test for the shell evolution leading up to the N = 28 “island of
+
42
inversion”. Our observation of a small 2+
2 → 21 branch in S, in agreement with the SDPF-

MU calculations, indicates that this shell-model Hamiltonian indeed captures the changes in
the neutron single-particle structure in the S isotopic chain as N = 28 is approached. It also
illustrates how sudden the comparably simple structure of 42 S evolves into the complexity
encountered for 44 S as the N = 28 shell closure breaks down.
For the odd-mass isotope 39 S, the expected low-lying nanosecond isomers, to which the
present measurement is insensitive, prevent the construction of an experimental level scheme
based on energy sums in the absence of clear coincidences and knowledge of the energies of
the isomeric states. For 41 S on the other hand, the proposed experimental level scheme
seems complete below 2.2 MeV and agrees remarkably well with the shell-model predictions.
Given the complexity of the structure of the S isotopes, this agreement is noteworthy.
From all cases investigated here, a consistent picture emerges for the population of excited states in fragmentation reactions. Transitions from yrast states are the most prominent, visible even at low statistics (e.g. 38 S). For the higher statistics cases of 40,41,42 S,
the presence of a multitude of weaker transitions can be understood as resulting from connections between the regions of high level density, upward from 3-4 MeV excitation energy,
and the low-lying level scheme. While this may always have been the assumption behind
the population of excited states in fragmentation reactions, evidence is presented here for
155

1.5

44S

1.0
0.5
0.0
1.5

p 3/2 occupancy

1.0
0.5

42S

2 +2

2+1

0.0
1.5

40S

1.0
0.5
0.0
1.5

38S

1.0
0.5
0.0

1.0

2.0

3.0

4.0

5.0

E x (MeV )

Figure C.20: Shell-model (SDPF-MU) neutron p3/2 occupancy for the 0+ (red) and 2+
(blue) states of 38−44 S below 4.5 MeV. The rapid onset of neutron p3/2 occupancy together
with the dramatic increase in the level density of 0+ and 2+ states in 44 S signals a sudden
transition into the N = 28 “island of inversion” in the S isotopic chain. The role of 42 S as a
sensitive probe for the neutron configurations is discussed in the text.

156

the many feeding transitions that have remained unobserved in previous work discussing
fragmentation reactions specifically for S isotopes [92] or the population of excited states in
projectile-like fragmentation residues in general [109]. In the case of 41 S(42 S), all calculated
negative(positive) parity states below 2.2 MeV(3.5 MeV) have been matched to states in
our proposed level schemes, including off-yrast states, while many weaker transitions remain
unplaced. The prominence of yrast states can likely be attributed to their significant indirect
feeding from the regions of high level density in addition to their direct population in the
fragmentation reaction.

Summary
We have performed in-beam Îł-ray spectroscopy on neutron-rich sulfur isotopes populated
by fragmentation of intermediate-energy 48 Ca and 46 Ar projectile beams. New transitions
were identified in 39−42 S and new level schemes for 40−42 S are proposed from γγ coincidence
information, energy sums and comparison to the shell model. Shell-model calculations with
the SDPF-MU Hamiltonian provide remarkable agreement and consistency with the proposed
level schemes. For the even-mass S isotopes, the evolution of the yrast sequence is discussed
in terms of E(6+ )/E(2+ ) and E(4+ )/E(2+ ) energy ratios. For 42 S, a candidate for the
38,40 S. This is
2+
2 state is proposed that exhibits a unique decay pattern as compared to

rooted in its neutron single-particle structure and confirmed by the SDPF-MU shell-model
calculations. For the odd-mass 41 S, a level scheme is presented that appears complete below
2.2 MeV and consistent with the predictions by SDPF-MU shell-model Hamiltonian; this is
a remarkable benchmark given the rapid shell and shape evolution prevalent in this textbook
isotopic chain as the diminished N = 28 shell gap is approached.

157

BIBLIOGRAPHY

158

BIBLIOGRAPHY

[1] Douglas Cline. Nuclear shapes studied by Coulomb excitation. Annual Review of
Nuclear and Particle Science, 36(1):683–716, 1986.
[2] K. Alder, A. Winther, and John L. Gammel. Electromagnetic excitation: Theory of
Coulomb excitation with heavy ions. Physics Today, 29:63, 1976.
[3] A. Bohr and B.R. Mottelson. Nuclear Structure, volume 2 of Nuclear Structure. World
Scientific, 1998.
[4] B. Alex Brown. Lecture notes in nuclear structure physics. National Superconducting
Cyclotron Laboratory, 2017.
[5] G. M. Temmer and N. P. Heydenburg. Coulomb excitation of medium-weight nuclei.
Phys. Rev., 104:967–980, Nov 1956.
[6] P. H. Stelson and F. K. McGowan. Coulomb excitation of medium-weight even-even
nuclei. Phys. Rev., 110:489–506, Apr 1958.
[7] O. Sorlin and M.-G. Porquet. Nuclear magic numbers: New features far from stability.
Progress in Particle and Nuclear Physics, 61(2):602 – 673, 2008.
[8] Alexandra Gade and Thomas Glasmacher. In-beam nuclear spectroscopy of bound
states with fast exotic ion beams. Progress in Particle and Nuclear Physics, 60(1):161
– 224, 2008.
[9] Michigan State University. Cyclotron Laboratory. Isotope Science Facility at Michigan
State University: Upgrade of the NSCL Rare Isotope Research Capabilities. Michigan
State University, 2006.
[10] David J. Morrissey and Bradley M. Sherrill. Radioactive nuclear beam facilities based
on projectile fragmentation. Philosophical Transactions of the Royal Society of London
A: Mathematical, Physical and Engineering Sciences, 356(1744):1985–2006, 1998.
[11] Thomas Glasmacher. Coulomb excitation at intermediate energies. Annual Review of
Nuclear and Particle Science, 48(1):1–31, 1998.
[12] Aage Winther and Kurt Alder. Relativistic Coulomb excitation. Nuclear Physics A,
319(3):518 – 532, 1979.
[13] T. Motobayashi, Y. Ikeda, Y. Ando, K. Ieki, M. Inoue, N. Iwasa, T. Kikuchi,
M. Kurokawa, S. Moriya, S. Ogawa, H. Murakami, S. Shimoura, Y. Yanagiswaw,
159

T. Nakamura, Y. Watanabe, M. Ishihara, T. Teranishia, H. Okuno, and R.F. Casten.
Large deformation of the very neutron-rich nucleus 32 Mg from intermediate-energy
Coulomb excitation. Physics Letters B, 346(1):9–14, 1995.
[14] A. Gade, D. Bazin, C. M. Campbell, J. A. Church, D. C. Dinca, J. Enders, T. Glasmacher, Z. Hu, K. W. Kemper, W. F. Mueller, H. Olliver, B. C. Perry, L. A. Riley, B. T.
Roeder, B. M. Sherrill, and J. R. Terry. Detailed experimental study on intermediateenergy Coulomb excitation of 46 Ar. Phys. Rev. C, 68:014302, Jul 2003.
[15] K. L. Yurkewicz, D. Bazin, B. A. Brown, C. M. Campbell, J. A. Church, D.-C. Dinca,
A. Gade, T. Glasmacher, M. Honma, T. Mizusaki, W. F. Mueller, H. Olliver, T. Otsuka, L. A. Riley, and J. R. Terry. Intermediate-energy Coulomb excitation of 52 Fe.
Phys. Rev. C, 70:034301, Sep 2004.
[16] C. Vaman, C. Andreoiu, D. Bazin, A. Becerril, B. A. Brown, C. M. Campbell,
A. Chester, J. M. Cook, D. C. Dinca, A. Gade, D. Galaviz, T. Glasmacher, M. HjorthJensen, M. Horoi, D. Miller, V. Moeller, W. F. Mueller, A. Schiller, K. Starosta,
A. Stolz, J. R. Terry, A. Volya, V. Zelevinsky, and H. Zwahlen. Z = 50 shell gap near
100 Sn from intermediate-energy Coulomb excitations in even-mass 106−112 Sn isotopes.
Phys. Rev. Lett., 99:162501, Oct 2007.
[17] S. Ettenauer, H. Zwahlen, P. Adrich, D. Bazin, C. M. Campbell, J. M. Cook, A. D.
Davies, D.-C. Dinca, A. Gade, T. Glasmacher, J.-L. Lecouey, W. F. Mueller, T. Otsuka,
R. R. Reynolds, L. A. Riley, J. R. Terry, Y. Utsuno, and K. Yoneda. Intermediateenergy Coulomb excitation of 30 Na. Phys. Rev. C, 78:017302, Jul 2008.
[18] R. Winkler, A. Gade, T. Baugher, D. Bazin, B. A. Brown, T. Glasmacher, G. F.
Grinyer, R. Meharchand, S. McDaniel, A. Ratkiewicz, and D. Weisshaar. Quadrupole
collectivity beyond N = 28: Intermediate-energy Coulomb excitation of 47,48 Ar. Phys.
Rev. Lett., 108:182501, Apr 2012.
[19] T. Baugher, A. Gade, R. V. F. Janssens, S. M. Lenzi, D. Bazin, B. A. Brown, M. P.
Carpenter, A. N. Deacon, S. J. Freeman, T. Glasmacher, G. F. Grinyer, F. G. Kondev,
S. McDaniel, A. Poves, A. Ratkiewicz, E. A. McCutchan, D. K. Sharp, I. Stefanescu,
K. A. Walsh, D. Weisshaar, and S. Zhu. Intermediate-energy Coulomb excitation of
58,60,62 Cr: The onset of collectivity toward N = 40. Phys. Rev. C, 86:011305, Jul
2012.
[20] D.C. Kean. Measurement of quadrupole moments through Coulomb excitation. pages
80–94. Springer, 1976.
[21] A. Gade and S.N. Liddick. Shape coexistence in neutron-rich nuclei. Journal of Physics
G: Nuclear and Particle Physics, 43(2):024001, 2016.

160

[22] A.C.C. Villari. The accelerated ISOL technique and the SPIRAL project. Nuclear
Physics A, 693(1):465 – 476, 2001.
[23] M. A. Schumaker, D. Cline, G. Hackman, C. Pearson, C. E. Svensson, C. Y. Wu,
A. Andreyev, R. A. E. Austin, G. C. Ball, D. Bandyopadhyay, J. A. Becker, A. J.
Boston, H. C. Boston, L. Buchmann, R. Churchman, F. Cifarelli, R. J. Cooper, D. S.
Cross, D. Dashdorj, G. A. Demand, M. R. Dimmock, T. E. Drake, P. Finlay, A. T.
Gallant, P. E. Garrett, K. L. Green, A. N. Grint, G. F. Grinyer, L. J. Harkness, A. B.
Hayes, R. Kanungo, A. F. Lisetskiy, K. G. Leach, G. Lee, R. Maharaj, J-P. Martin,
F. Moisan, A. C. Morton, S. Mythili, L. Nelson, O. Newman, P. J. Nolan, J. N. Orce,
E. Padilla-Rodal, A. A. Phillips, M. Porter-Peden, J. J. Ressler, R. Roy, C. Ruiz,
F. Sarazin, D. P. Scraggs, J. C. Waddington, J. M. Wan, A. Whitbeck, S. J. Williams,
and J. Wong. Coulomb excitation of radioactive 20,21 Na. The European Physical
Journal A, 42(3):477, Jun 2009.
[24] J. Van de Walle, F. Aksouh, F. Ames, T. Behrens, V. Bildstein, A. Blazhev, J. Cederkäll, E. Clément, T. E. Cocolios, T. Davinson, P. Delahaye, J. Eberth, A. Ekström,
D. V. Fedorov, V. N. Fedosseev, L. M. Fraile, S. Franchoo, R. Gernhauser, G. Georgiev,
D. Habs, K. Heyde, G. Huber, M. Huyse, F. Ibrahim, O. Ivanov, J. Iwanicki, J. Jolie,
O. Kester, U. Köster, T. Kröll, R. Krücken, M. Lauer, A. F. Lisetskiy, R. Lutter, B. A.
Marsh, P. Mayet, O. Niedermaier, T. Nilsson, M. Pantea, O. Perru, R. Raabe, P. Reiter, M. Sawicka, H. Scheit, G. Schrieder, D. Schwalm, M. D. Seliverstov, T. Sieber,
G. Sletten, N. Smirnova, M. Stanoiu, I. Stefanescu, J.-C. Thomas, J. J. ValienteDobón, P. Van Duppen, D. Verney, D. Voulot, N. Warr, D. Weisshaar, F. Wenander,
B. H. Wolf, and M. Zielińska. Coulomb excitation of neutron-rich Zn isotopes: First
80
observation of the 2+
1 state in Zn. Phys. Rev. Lett., 99:142501, Oct 2007.
[25] E. Kwan, C.Y. Wu, N.C. Summers, G. Hackman, T.E. Drake, C. Andreoiu, R. Ashley,
G.C. Ball, P.C. Bender, A.J. Boston, H.C. Boston, A. Chester, A. Close, D. Cline,
D.S. Cross, R. Dunlop, A. Finley, A. Garnsworthy, A.B. Hayes, A. Laffoley, T. Nano,
P. Voss, S.J. Williams, and Z.M. Wang. Precision measurements of the B(E1) strengths
in 11 Be. Technical report, Lawrence Livermore National Laboratory (LLNL), Livermore, CA, 2012.
[26] L. P. Gaffney, P. A. Butler, M. Scheck, A. B. Hayes, F. Wenander, M. Albers, B. Bastin,
C. Bauer, A. Blazhev, S. Bonig, N. Bree, J. Cederkall, T. Chupp, D. Cline, T. E.
Cocolios, T. Davinson, H. De Witte, J. Diriken, T. Grahn, A. Herzan, M. Huyse,
D. G. Jenkins, D. T. Joss, N. Kesteloot, J. Konki, M. Kowalczyk, Th Kroll, E. Kwan,
R. Lutter, K. Moschner, P. Napiorkowski, J. Pakarinen, M. Pfeiffer, D. Radeck, P. Reiter, K. Reynders, S. V. Rigby, L. M. Robledo, M. Rudigier, S. Sambi, M. Seidlitz,
B. Siebeck, T. Stora, P. Thoele, P. Van Duppen, M. J. Vermeulen, M. von Schmid,
D. Voulot, N. Warr, K. Wimmer, K. Wrzosek-Lipska, C. Y. Wu, and M. Zielinska. Studies of pear-shaped nuclei using accelerated radioactive beams. Nature, 497(7448):199–
204, May 2013.

161

[27] I. Stefanescu, G. Georgiev, F. Ames, J. Äystö, D. L. Balabanski, G. Bollen, P. A.
Butler, J. Cederkäll, N. Champault, T. Davinson, A. De Maesschalck, P. Delahaye,
J. Eberth, D Fedorov, V. N. Fedosseev, L. M. Fraile, S. Franchoo, K. Gladnishki,
D. Habs, K. Heyde, M. Huyse, O. Ivanov, J. Iwanicki, J. Jolie, B. Jonson, Th. Kröll,
R. Krücken, O. Kester, U Köster, A. Lagoyannis, L. Liljeby, G. Lo Bianco, B. A.
Marsh, O. Niedermaier, T. Nilsson, M. Oinonen, G. Pascovici, P. Reiter, A. Saltarelli,
H. Scheit, D. Schwalm, T. Sieber, N. Smirnova, J. Van De Walle, P. Van Duppen,
S. Zemlyanoi, N. Warr, D. Weisshaar, and F. Wenander. Coulomb excitation of 68,70 Cu:
First use of postaccelerated isomeric beams. Phys. Rev. Lett., 98:122701, Mar 2007.
[28] A. Ekström, J. Cederkäll, C. Fahlander, M. Hjorth-Jensen, T. Engeland, A. Blazhev,
P. A. Butler, T. Davinson, J. Eberth, F. Finke, A. Görgen, M. Górska, A. M. Hurst,
O. Ivanov, J. Iwanicki, U. Köster, B. A. Marsh, J. Mierzejewski, P. Reiter, S. Siem,
G. Sletten, I. Stefanescu, G. M. Tveten, J. Van de Walle, D. Voulot, N. Warr, D. Weisshaar, F. Wenander, and M. Zielińska. Coulomb excitation of the odd-odd isotopes
106,108 In. The European Physical Journal A, 44(3):355–361, Jun 2010.
[29] G. Savard, R. C. Pardo, S. Baker, C. N. Davids, A. Levand, D. Peterson, D. G. Phillips,
T. Sun, R. Vondrasek, B. J. Zabransky, and G. P. Zinkann. CARIBU: a new facility for
the study of neutron-rich isotopes. Hyperfine Interactions, 199(1):301–309, Jul 2011.
[30] B. Bucher, S. Zhu, C. Y. Wu, R. V. F. Janssens, D. Cline, A. B. Hayes, M. Albers, A. D.
Ayangeakaa, P. A. Butler, C. M. Campbell, M. P. Carpenter, C. J. Chiara, J. A. Clark,
H. L. Crawford, M. Cromaz, H. M. David, C. Dickerson, E. T. Gregor, J. Harker, C. R.
Hoffman, B. P. Kay, F. G. Kondev, A. Korichi, T. Lauritsen, A. O. Macchiavelli, R. C.
Pardo, A. Richard, M. A. Riley, G. Savard, M. Scheck, D. Seweryniak, M. K. Smith,
R. Vondrasek, and A. Wiens. Direct evidence of octupole deformation in neutron-rich
144 Ba. Phys. Rev. Lett., 116:112503, Mar 2016.
[31] B. Bucher, S. Zhu, C. Y. Wu, R. V. F. Janssens, R. N. Bernard, L. M. Robledo, T. R.
RodrĹ́guez, D. Cline, A. B. Hayes, A. D. Ayangeakaa, M. Q. Buckner, C. M. Campbell,
M. P. Carpenter, J. A. Clark, H. L. Crawford, H. M. David, C. Dickerson, J. Harker,
C. R. Hoffman, B. P. Kay, F. G. Kondev, T. Lauritsen, A. O. Macchiavelli, R. C. Pardo,
G. Savard, D. Seweryniak, and R. Vondrasek. Direct evidence for octupole deformation
in 146 Ba and the origin of large E1 moment variations in reflection-asymmetric nuclei.
Phys. Rev. Lett., 118:152504, Apr 2017.
[32] D.T. Doherty, J.M. Allmond, R.V.F. Janssens, W. Korten, S. Zhu, M. Zieliska, D.C.
Radford, A.D. Ayangeakaa, B. Bucher, J.C. Batchelder, C.W. Beausang, C. Campbell,
M.P. Carpenter, D. Cline, H.L. Crawford, H.M. David, J.P. Delaroche, C. Dickerson,
P. Fallon, A. Galindo-Uribarri, F.G. Kondev, J.L. Harker, A.B. Hayes, M. Hendricks,
P. Humby, M. Girod, C.J. Gross, M. Klintefjord, K. Kolos, G.J. Lane, T. Lauritsen,
J. Libert, A.O. Macchiavelli, P.J. Napiorkowski, E. Padilla-Rodal, R.C. Pardo, W. Reviol, D.G. Sarantites, G. Savard, D. Seweryniak, J. Srebrny, R. Varner, R. Vondrasek,

162

A. Wiens, E. Wilson, J.L. Wood, and C.Y. Wu. Triaxiality near the 110 Ru ground
state from Coulomb excitation. Physics Letters B, 766:334 – 338, 2017.
[33] A. Gade and B.M. Sherrill. NSCL and FRIB at Michigan State University: Nuclear
science at the limits of stability. Physica Scripta, 91(5):053003, 2016.
[34] W.F. Mueller, J.A. Church, T. Glasmacher, D. Gutknecht, G. Hackman, P.G. Hansen,
Z. Hu, K.L. Miller, and P. Quirin. Thirty-two-fold segmented germanium detectors
to identify Îł-rays from intermediate-energy exotic beams. Nuclear Instruments and
Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment, 466(3):492 – 498, 2001.
[35] Kurt Alder, Aage Bohr, Torben Huus, Ben Mottelson, and Aage Winther. Study
of nuclear structure by electromagnetic excitation with accelerated ions. Reviews of
Modern Physics, 28(4):432, 1956.
[36] D. Cline, T. Czosnyka, A. Hayes, P. Napiorkowski, N. Warr, and C.Y. Wu. GOSIA
user manual for simulation and analysis of Coulomb excitation experiments. Gosia
Steering Committee, 18:19–60, 2012.
[37] Kris Heyde. Basic ideas and concepts in nuclear physics: an introductory approach.
CRC Press, 2004.
[38] T. Czosnyka, D. Cline, L. Hasselgren, C.Y. Wu, R.M. Diamond, H. Kluge, C. Roulet,
E.K. Hulet, R.W. Lougheed, and C. Baktash. E2 properties of the ground state band
in 248 Cm. Nuclear Physics A, 458(1):123 – 136, 1986.
[39] William R. Leo. Techniques for nuclear and particle physics experiments: a how-to
approach. Springer Science & Business Media, 2012.
[40] James F. Ziegler, M.D. Ziegler, and J.P. Biersack. SRIM — the stopping and range of
ions in matter (2010). Nuclear Instruments and Methods in Physics Research Section
B: Beam Interactions with Materials and Atoms, 268(1112):1818 – 1823, 2010. 19th
International Conference on Ion Beam Analysis.
[41] M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S.
Zucker, and K. Olsen. XCOM: Photon Cross Sections Database (version 1.5). National
Institute of Standards and Technology, Gaithersburg, MD, 2010.
[42] Glenn F. Knoll. Radiation detection and measurement. John Wiley & Sons, 2010.
[43] Steven Ragnar Stroberg. Single-particle structure of neutron-rich silicon isotopes and
the breakdown of the N=28 shell closure. Michigan State University, 2014.

163

[44] C.J. Prokop, S.N. Liddick, B.L. Abromeit, A.T. Chemey, N.R. Larson, S. Suchyta,
and J.R. Tompkins. Digital data acquisition system implementation at the national
superconducting cyclotron laboratory. Nuclear Instruments and Methods in Physics
Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment,
741:163 – 168, 2014.
[45] NSCL data acquisition documentation. http://docs.nscl.msu.edu/daq/newsite/
home.php. Accessed: 2017-07-25.
[46] J.S. Fraser, J.C.D. Milton, H.R. Bowman, and S.G. Thompson. Precise kinetic energy
measurements and fine structure in the spontaneous fission of Cf252 . Canadian Journal
of Physics, 41(12):2080–2097, 1963.
[47] A.C.C. Villari, D. Alt, G. Bollen, D.B. Crisp, M. Ikegami, S.W. Krause, A. Lapierre,
S.M. Lidia, D.J. Morrissey, S. Nash, R.J. Rencsok, R. Ringle, S. Schwarz, R. Shane,
C. Sumithrarachchi, S.J. Williams, and Qiang Zhao. Commissioning and first accelerated beams in the reaccelerator (ReA3) of the National Superconducting Cyclotron
Laboratory, MSU. International Particle Accelerator Conference, pages 1287–1290,
2016.
[48] A. Lapierre, S. Schwarz, T. M. Baumann, K. Cooper, K. Kittimanapun, A. J. Rodriguez, C. Sumithrarachchi, S. J. Williams, W. Wittmer, D. Leitner, and G. Bollen.
First charge breeding of a rare-isotope beam with the electron-beam ion trap of the
rea post-accelerator at the national superconducting cyclotron laboratory. Review of
Scientific Instruments, 85(2):02B701, 2014.
[49] Morton A. Levine, R.E. Marrs, J.R. Henderson, D.A. Knapp, and Marilyn B. Schneider. The electron beam ion trap: A new instrument for atomic physics measurements.
Physica Scripta, 1988(T22):157, 1988.
[50] A. Gade and H. Iwasaki. ReA energy upgrade. 2016.
[51] Herbert Goldstein, Charles Poole, and John Safko. Classical mechanics, 2002.
[52] Ameenah R. Farhan and Balraj Singh. Nuclear data sheets for A = 78. Nuclear Data
Sheets, 110(9):1917 – 2080, 2009.
[53] H. Toki and Amand Faessler. Extended VMI model for asymmetric deformed nuclei.
Zeitschrift für Physik A Atoms and Nuclei, 276(1):35–43, Mar 1976.
[54] R. L. Robinson, H. J. Kim, R. O. Sayer, W. T. Milner, R. B. Piercey, J. H. Hamilton,
A. V. Ramayya, J. C. Wells, and A. J. Caffrey. High-spin levels and band structure in
78 Kr. Phys. Rev. C, 21:603–613, Feb 1980.

164

[55] F. Becker, A. Petrovici, J. Iwanicki, N. Amzal, W. Korten, K. Hauschild, A. Hurstel, Ch. Theisen, P.A. Butler, R.A. Cunningham, T. Czosnyka, G. de France,
J. Gerl, P. Greenlees, K. Helariutta, R.-D. Herzberg, P. Jones, R. Julin, S. Juutinen, H. Kankaanp, M. Muikku, P. Nieminen, O. Radu, P. Rahkila, and Ch. Schlegel.
Coulomb excitation of 78 Kr. Nuclear Physics A, 770(3):107 – 125, 2006.
E2 multipole mixing ratios of
[56] K. S. Krane and R. M. Steffen. Determination of the M
1
the gamma transitions in Cd110 . Phys. Rev. C, 2:724–734, Aug 1970.

[57] L. P. Gaffney, A. P. Robinson, D. G. Jenkins, A. N. Andreyev, M. Bender, A. Blazhev,
N. Bree, B. Bruyneel, P. A. Butler, T. E. Cocolios, T. Davinson, A. N. Deacon,
H. De Witte, D. DiJulio, J. Diriken, A. Ekström, Ch. Fransen, S. J. Freeman, K. Geibel,
T. Grahn, B. Hadinia, M. Hass, P.-H. Heenen, H. Hess, M. Huyse, U. Jakobsson,
N. Kesteloot, J. Konki, Th. Kröll, V. Kumar, O. Ivanov, S. Martin-Haugh, D. Mücher,
R. Orlandi, J. Pakarinen, A. Petts, P. Peura, P. Rahkila, P. Reiter, M. Scheck, M. Seidlitz, K. Singh, J. F. Smith, J. Van de Walle, P. Van Duppen, D. Voulot, R. Wadsworth,
N. Warr, F. Wenander, K. Wimmer, K. Wrzosek-Lipska, and M. Zielińska. Collectivity in the light radon nuclei measured directly via Coulomb excitation. Phys. Rev. C,
91:064313, Jun 2015.
[58] E. Nolte, Y. Shida, W. Kutschera, R. Prestele, and H. Morinaga. Investigation of
neutron deficient nuclei in the region 28 < N, Z < 50 with the help of heavy ion
compound reactions. Z.Phys., 268:267, 1974.
[59] D. S. Andreev, K. I. Erokhina, I. Kh. Lemberg, A. A. Pasternak, L. A. Rassadin, and
I. N. Chugunov. Lifetime measurements of excited states of 78 Kr. Izv.Akad.Nauk.
SSSR, Ser.Fiz., 46:30 – 44, 1982.
[60] G. Winter, F. Dubbers, J. Doring, L. Funke, P. Kemnitz, E. Will, D.S. Andreev, K.I.
Erochina, I.Kh. Lemberg, A.A. Pasternak, L.A. Rassadin, and I.N. Chugunov. On the
irregularities in the band structures of 78 Kr. Journal of Physics G: Nuclear Physics,
11(2):277, 1985.
[61] G. Garca Bermdez, M.A. Cardona, and A. Filevich. A simple plunger device and
error analysis of nuclear lifetime measurements. Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated
Equipment, 292(2):367 – 372, 1990.
[62] H.P. Hellmeister, J. Keinonen, K.P. Lieb, U. Kaup, R. Rascher, R. Ballini, J. Delaunay,
and H. Dumont. Study of high spin states in 78 Kr. Nuclear Physics A, 332(1):241 –
268, 1979.
[63] P.K. Joshi, H.C. Jain, R. Palit, G. Mukherjee, and S. Nagaraj. Shape evolution of
yrast-band in 78 Kr. Nuclear Physics A, 700(1):59 – 69, 2002.

165

[64] R. L. Robinson, H. J. Kim, R. O. Sayer, W. T. Milner, R. B. Piercey, J. H. Hamilton,
A. V. Ramayya, J. C. Wells, and A. J. Caffrey. High-spin levels and band structure in
78 Kr. Phys. Rev. C, 21:603–613, Feb 1980.
[65] T. J. Mertzimekis, N. Benczer-Koller, J. Holden, G. Jakob, G. Kumbartzki, K.-H.
Speidel, R. Ernst, A. Macchiavelli, M. McMahan, L. Phair, P. Maier-Komor, A. Pakou,
S. Vincent, and W. Korten. First measurements of g factors in the even Kr isotopes.
Phys. Rev. C, 64:024314, Jul 2001.
[66] M. Zielińska, L.P. Gaffney, Kasia Wrzosek-Lipska, E. Clément, Tuomas Grahn, Nele
Kesteloot, P. Napiorkowski, Janne Pakarinen, Piet Van Duppen, and N. Warr. Analysis
methods of safe Coulomb-excitation experiments with radioactive ion beams using the
GOSIA code. The European Physical Journal A, 52(4):99, 2016.
[67] A. Giannatiempo, A. Nannini, A. Perego, P. Sona, M. J. G. Borge, and O. Tengblad.
78
Decay properties of the 0+
2 state and spin-parity assignments in Kr. Phys. Rev. C,
52:2444–2447, Nov 1995.
[68] J. Billowes, F. Cristancho, H. Grawe, C. J. Gross, J. Heese, A. W. Mountford, and
M. Weiszflog. Neutron alignment at the first crossing in 78 Kr. Phys. Rev. C, 47:R917–
R921, Mar 1993.
[69] G. K. Bavaria, J. E. Crawford, S. Calamawy, and J. E. Kitching. The decay of 78g Rb
and 78m Rb to levels in 78 Kr. Z.Phys., A302:329, 1981.
[70] N. Yoshikawa, Y. Shida, O. Hashimoto, M. Sakai, and T. Numao. Levels in 74 Se,
78,80 Kr and 84 Sr and systematics of quasi-γ bands. Nuclear Physics A, 327(2):477 –
489, 1979.
[71] H. Scheit, T. Glasmacher, B. A. Brown, J. A. Brown, P. D. Cottle, P. G. Hansen,
R. Harkewicz, M. Hellström, R. W. Ibbotson, J. K. Jewell, K. W. Kemper, D. J.
Morrissey, M. Steiner, P. Thirolf, and M. Thoennessen. New region of deformation:
The neutron-rich sulfur isotopes. Phys. Rev. Lett., 77:3967–3970, Nov 1996.
[72] T. Glasmacher, B.A. Brown, M.J. Chromik, P.D. Cottle, M. Fauerbach, R.W. Ibbotson, K.W. Kemper, D.J. Morrissey, H. Scheit, D.W. Sklenicka, and M. Steiner.
Collectivity in 44 S. Physics Letters B, 395(3):163 – 168, 1997.
[73] D. Santiago-Gonzalez, I. Wiedenhöver, V. Abramkina, M. L. Avila, T. Baugher,
D. Bazin, B. A. Brown, P. D. Cottle, A. Gade, T. Glasmacher, K. W. Kemper, S. McDaniel, A. Rojas, A. Ratkiewicz, R. Meharchand, E. C. Simpson, J. A. Tostevin,
A. Volya, and D. Weisshaar. Triple configuration coexistence in 44 S. Phys. Rev. C,
83:061305, Jun 2011.

166

[74] Yutaka Utsuno, Noritaka Shimizu, Takaharu Otsuka, Tooru Yoshida, and Yusuke
Tsunoda. Nature of isomerism in exotic sulfur isotopes. Phys. Rev. Lett., 114:032501,
Jan 2015.
[75] J. Luis Egido, Marta Borrajo, and Tomás R. RodrĹ́guez. Collective and single-particle
motion in beyond mean field approaches. Phys. Rev. Lett., 116:052502, Feb 2016.
[76] J. J. Parker, I. Wiedenhöver, P. D. Cottle, J. Baker, D. McPherson, M. A. Riley,
D. Santiago-Gonzalez, A. Volya, V. M. Bader, T. Baugher, D. Bazin, A. Gade, T. Ginter, H. Iwasaki, C. Loelius, C. Morse, F. Recchia, D. Smalley, S. R. Stroberg, K. Whitmore, D. Weisshaar, A. Lemasson, H. L. Crawford, A. O. Macchiavelli, and K. Wimmer. Isomeric character of the lowest observed 4+ state in 44 S. Phys. Rev. Lett.,
118:052501, Jan 2017.
[77] C. Force, S. Grévy, L. Gaudefroy, O. Sorlin, L. Cáceres, F. Rotaru, J. Mrazek, N. L.
Achouri, J. C. Angélique, F. Azaiez, B. Bastin, R. Borcea, A. Buta, J. M. Daugas,
Z. Dlouhy, Zs. Dombrádi, F. De Oliveira, F. Negoita, Y. Penionzhkevich, M. G. SaintLaurent, D. Sohler, M. Stanoiu, I. Stefan, C. Stodel, and F. Nowacki. Prolate-spherical
shape coexistence at N = 28 in 44 S. Phys. Rev. Lett., 105:102501, Sep 2010.
[78] T.R. Werner, J.A. Sheikh, W. Nazarewicz, M.R. Strayer, A.S. Umar, and M. Misu.
Shape coexistence around 44
16 S28 : the deformed N = 28 region. Physics Letters B,
335(3):259 – 265, 1994.
[79] M. Kimura, Y. Taniguchi, Y. Kanada-En’yo, H. Horiuchi, and K. Ikeda. Prolate,
oblate, and triaxial shape coexistence, and the lost magicity of N = 28 in 43 S. Phys.
Rev. C, 87:011301, Jan 2013.
[80] Rene Brun and Fons Rademakers. Root – an object oriented data analysis framework. Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment, 389(1):81 – 86, 1997.
[81] D. Bazin, J.A. Caggiano, B.M. Sherrill, J. Yurkon, and A. Zeller. The S800 spectrograph. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 204:629 – 633, 2003. 14th International Conference
on Electromagnetic Isotope Separators and Techniques Related to their Applications.
[82] T. Kibdi, T.W. Burrows, M.B. Trzhaskovskaya, P.M. Davidson, and C.W. Nestor.
Evaluation of theoretical conversion coefficients using BrIcc. Nuclear Instruments and
Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment, 589(2):202 – 229, 2008.
[83] E. Lunderberg, A. Gade, V. Bader, T. Baugher, D. Bazin, J. S. Berryman, B. A.
Brown, D. J. Hartley, F. Recchia, S. R. Stroberg, D. Weisshaar, and K. Wimmer.
In-beam γ-ray spectroscopy of 38−42 S. Phys. Rev. C, 94:064327, Dec 2016.
167

[84] Yutaka Utsuno, Takaharu Otsuka, B. Alex Brown, Michio Honma, Takahiro Mizusaki,
and Noritaka Shimizu. Shape transitions in exotic Si and S isotopes and tensor-forcedriven Jahn-Teller effect. Phys. Rev. C, 86:051301, Nov 2012.
[85] R. Chevrier, J. M. Daugas, L. Gaudefroy, Y. Ichikawa, H. Ueno, M. Hass, H. Haas,
S. Cottenier, N. Aoi, K. Asahi, D. L. Balabanski, N. Fukuda, T. Furukawa, G. Georgiev,
H. Hayashi, H. Iijima, N. Inabe, T. Inoue, M. Ishihara, Y. Ishii, D. Kameda, T. Kubo,
T. Nanao, G. Neyens, T. Ohnishi, M. M. Rajabali, K. Suzuki, H. Takeda, M. Tsuchiya,
43
N. Vermeulen, H. Watanabe, and A. Yoshimi. Is the 7/2−
1 isomer state of S spherical?
Phys. Rev. Lett., 108:162501, Apr 2012.
[86] Z. M. Wang, R. Chapman, F. Haas, X. Liang, F. Azaiez, B. R. Behera, M. Burns,
L. Corradi, D. Curien, A. N. Deacon, Zs. Dombrádi, E. Farnea, E. Fioretto, A. Gadea,
A. Hodsdon, F. Ibrahim, A. Jungclaus, K. Keyes, V. Kumar, A. Latina, N. Mărginean,
G. Montagnoli, D. R. Napoli, J. Ollier, D. O’Donnell, A. Papenberg, G. Pollarolo, M.D. Salsac, F. Scarlassara, J. F. Smith, K. M. Spohr, M. Stanoiu, A. M. Stefanini,
S. Szilner, M. Trotta, and D. Verney. Collectivity in 41 S. Phys. Rev. C, 83:061304,
Jun 2011.
[87] L. Gaudefroy, J. M. Daugas, M. Hass, S. Grévy, Ch. Stodel, J. C. Thomas, L. Perrot,
M. Girod, B. Rossé, J. C. Angélique, D. L. Balabanski, E. Fiori, C. Force, G. Georgiev,
D. Kameda, V. Kumar, R. L. Lozeva, I. Matea, V. Méot, P. Morel, B. S. Nara Singh,
F. Nowacki, and G. Simpson. Shell erosion and shape coexistence in 43
16 S27 . Phys. Rev.
Lett., 102:092501, Mar 2009.
[88] L. A. Riley, P. Adrich, T. R. Baugher, D. Bazin, B. A. Brown, J. M. Cook, P. D.
Cottle, C. Aa. Diget, A. Gade, D. A. Garland, T. Glasmacher, K. E. Hosier, K. W.
Kemper, A. Ratkiewicz, K. P. Siwek, J. A. Tostevin, and D. Weisshaar. Rotational
and neutron-hole states in 43 S via the neutron knockout and fragmentation reactions.
Phys. Rev. C, 80:037305, Sep 2009.
[89] R. Ringle, C. Bachelet, M. Block, G. Bollen, M. Facina, C. M. Folden, C. Guénaut,
A. A. Kwiatkowski, D. J. Morrissey, G. K. Pang, A. M. Prinke, J. Savory, P. Schury,
S. Schwarz, and C. S. Sumithrarachchi. High-precision Penning trap mass measurements of neutron-rich sulfur isotopes at the N = 28 shell closure. Phys. Rev. C,
80:064321, Dec 2009.
[90] R. W. Ibbotson, T. Glasmacher, P. F. Mantica, and H. Scheit. Coulomb excitation of
odd-A neutron-rich π(s − d) and ν(f − p) shell nuclei. Phys. Rev. C, 59:642–647, Feb
1999.
[91] F. Azaiez, M. Belleguic, D. Sohler, M. Stanoiu, Zs. Dombrádi, O. Sorlin, J. Timár,
F. Amorini, D. Baiborodin, A. Bauchet, F. Becker, C. Borcea, C. Bourgeois, Z. Dlouhy,
C. Donzaud, J. Duprat, D. Guillemaud-Mueller, F. Ibrahim, M.J. Lopez, R. Lucas,
S.M. Lukyanov, V. Maslov, J. Mrazek, C. Moore, F. Nowacki, B.M. Nyakó, Yu.-E.
168

Penionzhkevich, M.G. Saint-Laurent, F. Sarazin, J.A. Scarpaci, G. Sletten, C. Stodel,
M. Taylor, C. Theisen, and G. Voltolini. Probing shell structure in neutron-rich nuclei
with in-beam Îł-spectroscopy. The European Physical Journal A - Hadrons and Nuclei,
15(1):93–97, 2002.
[92] D. Sohler, Zs. Dombrádi, J. Timár, O. Sorlin, F. Azaiez, F. Amorini, M. Belleguic,
C. Bourgeois, C. Donzaud, J. Duprat, D. Guillemaud-Mueller, F. Ibrahim, J. A.
Scarpaci, M. Stanoiu, M. J. Lopez, M. G. Saint-Laurent, F. Becker, F. Sarazin,
C. Stodel, G. Voltolini, S. M. Lukyanov, V. Maslov, Yu.-E. Penionzhkevich, M. Girod,
S. Péru, F. Nowacki, G. Sletten, R. Lucas, C. Theisen, D. Baiborodin, Z. Dlouhy,
J. Mrazek, C. Borcea, A. Bauchet, C. J. Moore, and M. J. Taylor. Shape evolution in
heavy sulfur isotopes and erosion of the N = 28 shell closure. Phys. Rev. C, 66:054302,
Nov 2002.
[93] Z. M. Wang, R. Chapman, X. Liang, F. Haas, F. Azaiez, B. R. Behera, M. Burns,
E. Caurier, L. Corradi, D. Curien, A. N. Deacon, Zs. Dombrádi, E. Farnea, E. Fioretto,
A. Gadea, A. Hodsdon, F. Ibrahim, A. Jungclaus, K. Keyes, V. Kumar, A. Latina,
S. Lunardi, N. Mărginean, G. Montagnoli, D. R. Napoli, F. Nowacki, J. Ollier,
D. O’Donnell, A. Papenberg, G. Pollarolo, M.-D. Salsac, F. Scarlassara, J. F. Smith,
K. M. Spohr, M. Stanoiu, A. M. Stefanini, S. Szilner, M. Trotta, and D. Verney. Îł-ray
spectroscopy of neutron-rich 40 S. Phys. Rev. C, 81:054305, May 2010.
[94] J. A. Winger, P. F. Mantica, R. M. Ronningen, and M. A. Caprio. Low-energy structure
of 40 S through 40 P β decay. Phys. Rev. C, 64:064318, Nov 2001.
[95] B. Fornal, R. H. Mayer, I. G. Bearden, Ph. Benet, R. Broda, P. J. Daly, Z. W.
Grabowski, I. Ahmad, M. P. Carpenter, P. B. Fernandez, R. V. F. Janssens, T. L.
Khoo, T. Lauritsen, E. F. Moore, and M. Drigert. Gamma ray studies of neutron-rich
sdf shell nuclei produced in heavy ion collisions. Phys. Rev. C, 49:2413–2418, May
1994.
[96] A. Gade, P. Adrich, D. Bazin, B. A. Brown, J. M. Cook, C. Aa. Diget, T. Glasmacher, S. McDaniel, A. Ratkiewicz, K. Siwek, and D. Weisshaar. In-beam Îł-ray
spectroscopy of very neutron-rich nuclei: Excited states in 46 S and 48 Ar. Phys. Rev.
Lett., 102:182502, May 2009.
[97] A. Gade, J. A. Tostevin, V. Bader, T. Baugher, D. Bazin, J. S. Berryman, B. A.
Brown, C. Aa. Diget, T. Glasmacher, D. J. Hartley, E. Lunderberg, S. R. Stroberg,
F. Recchia, A. Ratkiewicz, D. Weisshaar, and K. Wimmer. Single-particle structure at
N = 29: The structure of 47 Ar and first spectroscopy of 45 S. Phys. Rev. C, 93:054315,
May 2016.
[98] Evaluated nuclear structure database. http://www.nndc.bnl.gov/ensdf/.

169

[99] S. Paschalis, I.Y. Lee, A.O. Macchiavelli, C.M. Campbell, M. Cromaz, S. Gros, J. Pavan, J. Qian, R.M. Clark, H.L. Crawford, D. Doering, P. Fallon, C. Lionberger,
T. Loew, M. Petri, T. Stezelberger, S. Zimmermann, D.C. Radford, K. Lagergren,
D. Weisshaar, R. Winkler, T. Glasmacher, J.T. Anderson, and C.W. Beausang. The
performance of the gamma-ray energy tracking in-beam nuclear array GRETINA. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 709:44 – 55, 2013.
[100] F. Nowacki and A. Poves. New effective interaction for 0h̄ω shell-model calculations
in the sd −pf valence space. Phys. Rev. C, 79:014310, Jan 2009.
[101] D.J. Morrissey, B.M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhoever. Commissioning the A1900 projectile fragment separator. Nuclear Instruments and Methods in
Physics Research Section B: Beam Interactions with Materials and Atoms, 204:90 –
96, 2003. 14th International Conference on Electromagnetic Isotope Separators and
Techniques Related to their Applications.
[102] A. Gade, J. A. Tostevin, V. Bader, T. Baugher, D. Bazin, J. S. Berryman, B. A. Brown,
D. J. Hartley, E. Lunderberg, F. Recchia, S. R. Stroberg, Y. Utsuno, D. Weisshaar,
and K. Wimmer. One-neutron pickup into 49 Ca: Bound neutron g9/2 spectroscopic
strength at N = 29. Phys. Rev. C, 93:031601, Mar 2016.
[103] D. Weisshaar, D. Bazin, P.C. Bender, C.M. Campbell, F. Recchia, V. Bader,
T. Baugher, J. Belarge, M.P. Carpenter, H.L. Crawford, M. Cromaz, B. Elman,
P. Fallon, A. Forney, A. Gade, J. Harker, N. Kobayashi, C. Langer, T. Lauritsen,
I.Y. Lee, A. Lemasson, B. Longfellow, E. Lunderberg, A.O. Macchiavelli, K. Miki,
S. Momiyama, S. Noji, D.C. Radford, M. Scott, J. Sethi, S.R. Stroberg, C. Sullivan,
R. Titus, A. Wiens, S. Williams, K. Wimmer, and S. Zhu. The performance of the
Îł-ray tracking array GRETINA for Îł-ray spectroscopy with fast beams of rare isotopes. Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment, 847:187 – 198, 2017.
[104] L.A. Riley. UCGretina GEANT4. Ursinus College (unpublished).
[105] NuShellX. http://www.garsington.eclipse.co.uk/.
[106] J. W. Olness, E. K. Warburton, J. A. Becker, D. J. Decman, E. A. Henry, L. G. Mann,
and L. Ussery. 36 S(t, pγ)38 S reaction. Phys. Rev. C, 34:2049–2054, Dec 1986.
[107] E. K. Warburton, D. E. Alburger, and G. Wang. Confirmation of the 2805-keV level
of 38 S. Phys. Rev. C, 36:429–430, Jul 1987.
[108] John A. Cameron and Balraj Singh. Nuclear data sheets for A = 38. Nuclear Data
Sheets, 109(1):1 – 170, 2008.

170

[109] A. Obertelli, A. Gade, D. Bazin, C. M. Campbell, J. M. Cook, P. D. Cottle, A. D.
Davies, D.-C. Dinca, T. Glasmacher, P. G. Hansen, T. Hoagland, K. W. Kemper, J.-L.
Lecouey, W. F. Mueller, R. R. Reynolds, B. T. Roeder, J. R. Terry, J. A. Tostevin,
K. Yoneda, and H. Zwahlen. Population of bound excited states in intermediate-energy
fragmentation reactions. Phys. Rev. C, 73:044605, Apr 2006.
[110] R. Chapman, Z. M. Wang, M. Bouhelal, F. Haas, X. Liang, F. Azaiez, B. R. Behera,
M. Burns, E. Caurier, L. Corradi, D. Curien, A. N. Deacon, Zs. Dombrádi, E. Farnea,
E. Fioretto, A. Gadea, A. Hodsdon, F. Ibrahim, A. Jungclaus, K. Keyes, V. Kumar, S. Lunardi, N. Mărginean, G. Montagnoli, D. R. Napoli, F. Nowacki, J. Ollier,
D. O’Donnell, A. Papenberg, G. Pollarolo, M.-D. Salsac, F. Scarlassara, J. F. Smith,
K. M. Spohr, M. Stanoiu, A. M. Stefanini, S. Szilner, M. Trotta, and D. Verney. First
in-beam Îł-ray study of the level structure of neutron-rich 39 S. Phys. Rev. C, 94:024325,
Aug 2016.
[111] J. A. Winger, H. H. Yousif, W. C. Ma, V. Ravikumar, W. Lui, S. K. Phillips, R. B.
Piercey, P. F. Mantica, B. Pritychenko, R. M. Ronningen, and M. Steiner. Low-energy
structure of neutron-rich S, Cl and Ar nuclides through β decay. AIP Conference
Proceedings, 455(1):606–609, 1998.
[112] S. R. Stroberg, A. Gade, J. A. Tostevin, V. M. Bader, T. Baugher, D. Bazin, J. S.
Berryman, B. A. Brown, C. M. Campbell, K. W. Kemper, C. Langer, E. Lunderberg,
A. Lemasson, S. Noji, T. Otsuka, F. Recchia, C. Walz, D. Weisshaar, and S. Williams.
Neutron single-particle strength in silicon isotopes: Constraining the driving forces of
shell evolution. Phys. Rev. C, 91:041302, Apr 2015.
[113] L. A. Riley, P. Adrich, N. Ahsan, T. R. Baugher, D. Bazin, B. A. Brown, J. M.
Cook, P. D. Cottle, C. Aa. Diget, A. Gade, T. Glasmacher, K. E. Hosier, K. W.
Kemper, A. Ratkiewicz, K. P. Siwek, J. A. Tostevin, A. Volya, and D. Weisshaar.
Îł-ray spectroscopy of one-proton knockout from 45 Cl. Phys. Rev. C, 86:047301, Oct
2012.

171