NUCLEAR STRUCTURE OF PROTON-RICH INTERMEDIATE MASS NUCLEI
STUDIED WITH ADVANCED LIFETIME MEASUREMENT TECHNIQUES
By
Christopher Scott Morse

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

ABSTRACT
NUCLEAR STRUCTURE OF PROTON-RICH INTERMEDIATE MASS
NUCLEI STUDIED WITH ADVANCED LIFETIME MEASUREMENT
TECHNIQUES
By
Christopher Scott Morse
The transition matrix elements between excited nuclear states provide valuable information about the structure of exotic nuclei. Lifetime measurements are a model-independent
way to deduce these matrix elements from experimental data. Two studies of proton-rich
nuclei near mass 70 are presented herein using advanced lifetime measurement techniques.
+
The first study is a measurement of the 8+
1 and 91 states in the odd-N , odd-Z nucleus
70 As.

The lifetimes of these states were determined by the application of the γ-ray lineshape

method to γ-γ coincidence data. The states were populated using with the 9 Be(78 Rb,70 As)
reaction and γ rays were detected with the Segmented Germanium Array in coincidence with
reaction products detected in the focal plane of the S800 Spectrograph. The B(E1; 8+ → 7− )
and B(M 1; 9+ → 8+ ) transition strengths were deduced and were found to support the
assignment of these states to a coupling of the odd proton and neutron in the g9/2 orbital.
74
The second study is a measurement of the 2+
1 state lifetime in the N = Z nucleus Rb.

A novel technique called the Differential Recoil Distance Method was used to extract the
lifetime from the γ-ray spectra. The next-generation γ-ray detector array GRETINA was
used in the experiment, again coupled to the S800 Spectrograph to detect residues from
9 Be(74 Kr,74 Rb)

+
charge exchange reactions. The B(E2; 2+
1 → 01 ) strength was calculated

and is consistent with the measured strength of the transition between the isobaric analogue
states in 74 Kr, which may be a signature of shape coexistence in 74 Rb.

Copyright by
CHRISTOPHER SCOTT MORSE
2015

AMDG

iv

ACKNOWLEDGMENTS

No work is done in a vacuum, especially when that work comprises several years of one’s
life. Many people have contributed to this work either directly or indirectly, and it is beyond
my capability to acknowledge everyone individually. However, know that if you have talked
with me, laughed with (or possibly at) me, sung, studied, worked, played, or otherwise been
a part of my life during these past five years, you have helped me reach this point and for
that I am grateful.
Special thanks must go first to my advisor, Dr. Hironori Iwasaki, for being a continual
source of help and encouragement. Although I knew nothing of nuclear physics at the start
of my graduate career, he has always been patient and willing to guide me when I became
lost. Being his student has been the opportunity of a lifetime. I also wish to thank the
members of my guidance committee, Drs. Alex Brown, Wade Fisher, Alexandra Gade, and
Carlo Piermarocchi, for their insightful comments and for always pushing me to become a
better scientist. I would also be remiss if I did not acknowledge my undergraduate physics
professors at Westmont College. They laid the foundation of my work in physics, and that
foundation has proven itself sound.
Many people at the NSCL have been a great help over the years. My fellow Lifetime group
members have always been there to listen and discuss, and in some instances to play. Many
members of the Gamma group have also acted as mentors to me, for which I am extremely
grateful. The staff at the NSCL also deserve no small amount of credit in bringing this
document to fruition, as without their tireless efforts there would be nothing to write.
Several individuals have played a pivotal role in bringing me to this place and also
deserve recognition. I am indebted to Antoine Lemasson for recognizing that, while I had
v

the enthusiasm, I lacked both the skill to be a competent programmer and the knowledge to
be a capable experimenter. Our impromptu discussions on these and many other subjects
were invaluable both for enabling me to be an effective researcher and kindling an interest in
programming of which I was only vaguely aware. Second, I wish to extend my heartfelt thanks
to Warren Rogers, who introduced me to the idea of studying nuclear physics. Without his
invitation to visit the NSCL, I surely would have taken a different path. Similarly, the fact
that I chose to study physics at all is due in large part to Chris Sullivan. If not for his
enthusiastic exclamation of “Join me!” upon hearing that I was considering a physics major,
I might still be stuck doing titrations.
Many friendships have been forged during this journey through graduate school. I have
enjoyed immensely the chance to make music with the members of the Grand Canonical
Ensemble. I will always remember with fondness the late-night board game sessions and
their attendant shenanigans (“MAUOOOOB!”). Likewise, the encounters shared with adventuring companions gathered around the DM screen have formed a staple of my time here
at MSU. For those who tolerated my brief stint running a D&D campaign, may our “frondships” endure forever. And for all the other friends I have made, even if our relationship
was as simple as shooting the breeze from time to time, thank you for making my time as a
graduate student that much more enjoyable.
Last, but certainly not least, my family deserves an immense amount of thanks for their
never-ending support. They have endured an inordinate amount of venting and lamentation
from me and provided in return a constant stream of optimism and confidence. Without
their encouragement, I could not have achieved what I have. I can only hope to be a son
and brother worthy of their faithfulness.

vi

TABLE OF CONTENTS

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

ix

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

x

Chapter 1 Introduction . . . . . . . .
1.1 The atomic nucleus . . . . . . . .
1.2 Nuclear structure models . . . . .
1.2.1 The nuclear shell model .
1.2.2 The Nilsson model . . . .
1.2.3 Collective nuclear models .
1.3 Lifetime measurements . . . . . .

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1
1
6
6
10
12
14

Chapter 2 Lifetime Measurement Techniques .
2.1 Overview of lifetime measurements . . . . . .
2.1.1 Electronic timing measurements . . . .
2.1.2 The Recoil Distance Method . . . . . .
2.1.3 The Doppler Shift Attenuation Method
2.2 The Lineshape Method . . . . . . . . . . . . .
2.3 The Differential Recoil Distance Method . . .

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20
20
22
26
31
33
36

Chapter 3 Experimental Methods . . . . . . . . . . . . . . .
3.1 Beam production . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Primary beam production . . . . . . . . . . . . . .
3.1.2 Secondary beam selection . . . . . . . . . . . . . .
3.2 Experimental devices and detector systems . . . . . . . . .
3.2.1 The S800 Spectrograph . . . . . . . . . . . . . . . .
3.2.1.1 S800 particle detection systems . . . . . .
3.2.1.1.1 Time of Flight Scintillators . . .
3.2.1.1.2 The Ionization Chamber . . . . .
3.2.1.1.3 Cathode Readout Drift Chambers
3.2.1.1.4 Trajectory Reconstruction . . . .
3.2.1.2 Detector calibrations . . . . . . . . . . . .
3.2.1.2.1 CRDC calibrations . . . . . . . .
3.2.1.2.2 Ionization chamber calibration .
3.2.1.2.3 Time of flight corrections . . . .
3.2.2 γ-ray detectors . . . . . . . . . . . . . . . . . . . .
3.2.2.1 γ-ray interactions with matter . . . . . . .
3.2.2.2 The SeGA detector array . . . . . . . . .
3.2.2.2.1 SeGA calibrations . . . . . . . .

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43
43
44
48
51
51
53
54
54
55
56
57
57
58
59
61
61
65
68

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vii

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3.2.2.3

3.3

The GRETINA detector array . . .
3.2.2.3.1 Calibration of GRETINA
3.2.3 The TRIPLEX plunger device . . . . . . . .
Geant4 simulations . . . . . . . . . . . . . . . . . .

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70
73
73
77

Chapter 4 Lifetime Measurements in 70 As
4.1 Physics motivation . . . . . . . . . . . .
4.2 Experimental details . . . . . . . . . . .
4.3 γ-ray spectra . . . . . . . . . . . . . . .
4.4 Lifetime analysis . . . . . . . . . . . . .
4.4.1 Geant4 simulations . . . . . . . .
4.4.2 Lifetime determination . . . . . .
4.5 Results and discussion . . . . . . . . . .

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. 82
. 83
. 86
. 92
. 99
. 100
. 104
. 112

Chapter 5 Lifetime Measurements in 74 Rb
5.1 Physics motivation . . . . . . . . . . . .
5.2 Experimental details . . . . . . . . . . .
5.3 γ-ray spectra . . . . . . . . . . . . . . .
5.4 Lifetime analysis . . . . . . . . . . . . .
5.4.1 Geant4 simulations . . . . . . . .
5.4.2 Lifetime determination . . . . . .
5.5 Results and discussion . . . . . . . . . .

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117
118
126
130
133
134
138
141

Chapter 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
BIBLIOGRAPHY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

viii

LIST OF TABLES

Table 5.1:

+
A summary of the values for the mixing of the 0+
gs and 02 states
for the isotopes of krypton shown in Fig. 5.3, taken from Ref. [95].
The energy difference between the observed states is given by ∆ ,
while the energy difference between the intrinsic states is given by ∆
and the mixing matrix element is V . The amplitude of the prolate
2
wavefunction in the 0+
2 state is given by b . . . . . . . . . . . . . . 123

ix

LIST OF FIGURES

Figure 1.1:

Figure 1.2:

Figure 1.3:

Figure 1.4:

A drawing of the chart of the nuclides. The number of protons is plotted on the vertical axis and the number of neutrons on the horizontal
axis. The black squares are the stable nuclei, and lie along the socalled “valley of stability.” The blue area indicates nuclei which have
been observed, while the red area represents nuclei which are predicted to exist but have not been seen experimentally. Finally, the
dashed lines indicate the locations of the canonical magic numbers.
For interpretation of references to color in this and all other figures,
the reader is referred to the electronic version of this dissertation. .

2

A schematic diagram showing a portion of the level schemes for several nuclei with mass A = 58, with 58 Ni bolded to indicate that it
is stable. The levels joined together by dashed lines indicate isobaric
analogue states with the indicated value of isospin T . The isobaric
analogue states have been drawn at exactly the same energy in this
figure to emphasize their relationship with each other, effectively removing the contribution of the Coulomb interaction which violates
isospin symmetry. Also shown for the Tz = 0 nucleus 58 Cu are several T = 0 states that have no analogue states and are thus isospin
singlet states. Figure is taken and modified from [5]. . . . . . . . .

5

The energies of the orbitals generated with several different interactions in the shell model, calculated for 208 Pb. In the left and center
columns are shown the harmonic oscillator and Woods-Saxon potentials, respectively, neither of which is able to reproduce the correct
degeneracy of levels that gives the magic numbers. Adding a spinorbit term to the Woods-Saxon potential, however, does reproduce
the magic numbers, as shown by the column on the right. Figure is
taken from [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

A Nilsson diagram showing the single-particle energy levels in the
Nilsson scheme, plotted against the deformation parameter along the
horizontal axis. For zero deformation, the shell model magic numbers
are obtained, but for other deformation parameters (prolate only in
this figure), the splitting of the shell model orbitals fills in the gaps
between magic numbers, giving rise to a more complicated series of
single-particle energy levels. Figure is taken from [14]. . . . . . . .

11

x

Figure 2.1:

Figure 2.2:

Figure 2.3:

Figure 2.4:

Figure 2.5:

A diagram showing the approximate ranges in which various lifetime
measurement techniques are effective. The methods labeled “direct”
measure the lifetime itself, whereas the “indirect methods” measure
the width Γ = /τ or cross-sections of electromagnetic transitions.
Figure is from [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

An illustration of the concept of the plunger setup for measuring the
lifetimes of excited nuclear states. A beam of nuclei comes from the
left and interacts with a target foil, producing the excited states to
be studied. The beam then recoils out of the target, emitting γ rays
as it travels toward a second foil. In this figure, the beam is stopped
in the second foil, where any nuclei which have not yet decayed will
do so. Both Doppler shifted and non-Doppler shifted photons are
detected, and the ratio of the shifted and unshifted photons gives the
sensitivity to the lifetime. Figure is from [25]. . . . . . . . . . . . .

27

The change in the spectral lineshape of a 500 keV γ-ray transition for
10 ps, 100 ps, and 1000 ps lifetimes, shown in blue, red, and green,
respectively. For the 10 ps lifetime, there is a significant high-energy
shoulder which is smeared out over a large range, plus a sharp peak.
This is due to some decays happening in the target where the velocity
is faster than assumed having a smeared out Doppler correction and
can be used as an indication of a short lifetime. The 100 ps lifetime
shows only a peak, but it is shifted somewhat compared to the sharp
portion of the 10 ps peak, and this shift is due to the lifetime. There
is also a small low-energy tail. Finally, the 1000 ps lifetime peak has
mostly a smeared out low-energy tail with only a small number of the
decays happening near the target. . . . . . . . . . . . . . . . . . . .

35

+
An example decay curve measured for the 2+
1 → 01 transition in
120 Xe with the DDCM. The function S plotted on the vertical axis
ij
is proportional to Rij . As can be seen, many data points are used to
determine the lifetime, which in turn requires a large amount of data
to be taken. Figure is taken from [33]. . . . . . . . . . . . . . . . .

38

A schematic drawing of a differential plunger used for the DRDM.
The beam enters from the right and excited states are created by
interaction with the target. The recoiling nuclei exit the target with
velocity v and travel a distance x before being slowed in the degrader.
After the degrader they travel a short distance ∆x at a reduced velocity v before being stopped in the stopper. As the excited nuclei
relax, they emit γ rays with different Doppler shifts at velocities v,
v , and at rest, producing the γ-ray spectrum shown at the bottom.
Figure is from [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

xi

Figure 3.1:

Figure 3.2:

Figure 3.3:

Figure 3.4:

Figure 3.5:

An illustration of the equipment used to create a beam of rare isotopes
at the NSCL. A beam of ions with a only a few electrons removed
is generated in one of the ion sources, then injected into the K500
cyclotron. After being accelerated, the beam is transported to the
K1200, where it is further stripped of most or all of its electrons
and accelerated again. The beam is reacted on the production target
and a single species is selected to study in the A1900 before being
transported to an experimental endstation. Figure is from [36]. . .

44

A diagram showing the layout of the S800 spectrograph. The beam
enters at the Object position and is bent down through an analyzing beamline towards the location labeled scattering chamber, where
experimental apparatus and target materials are typically located.
After passing this location, the beam is bent upward through the
two dipoles of the spectrograph and beam particles are identified in
a suite of detectors located inside the focal plane. Figure adapted
from [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

An illustration of the detectors located in the focal plane of the
S800 spectrograph. The beam first passes through the two CRDCs
which provide position and angle information. Next the beam passes
through the ionization chamber which measures the energy loss of
the beam as it passes through the detector. Next the beam passes
through the E1 scintillator, which serves as both a timing signal and a
trigger. In the figure there are two more plastic scintillators, but these
have been replaced with a 32 crystal CsI(Na) hodoscope for charge
state identification [49]. Figure is taken and modified from [47]. . .

53

The measured signals in the CRDCs when the masks are inserted into
the beamline. The first CRDC is shown on the left, and the second on
the right. By matching the pattern of slits and holes on the mask to
the pattern of signals generated in the CRDCs, the CRDC positions
can be calibrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

This figure shows the gain matching of the ionization chamber. On
the left, the raw energy loss signals (in arbitrary units) are shown for
each channel. On the right, the energy loss in each channel has been
gain matched to channel zero, which has the lowest raw energy loss
signal. As a result, the centroids of the energy loss in each channel
are the same, and this provides much cleaner elemental separation
when performing particle identification. . . . . . . . . . . . . . . .

59

xii

Figure 3.6:

The effect of correcting the time of flight measured for the RF signal
(top row), object scintillator (middle row), and extended focal plane
scintillator (bottom row). Uncorrected spectra are shown on the left,
corrected spectra on the right. . . . . . . . . . . . . . . . . . . . . .

60

An illustration of the photoelectric effect. An incident photon interacts with an atom and deposits its energy, causing the ejection of an
electron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

An illustration of the process of Compton scattering. An incident
photon with energy hν interacts with an electron in the medium and
imparts some of its energy to it. The scattered photon then propagates away at an angle θ with respect to its original trajectory and
with a lower energy hν , given by Eq. 3.10, while the electron recoils
away at some other angle φ and energy given by Eq. 3.11. . . . . .

63

An illustration of the process of pair production. If a photon has at
least 1.022 MeV of energy, it may be absorbed by a nucleus and an
electron-positron pair emitted in its place. The positron will annihilate with another electron in the medium and release two 511 keV
photons, one or both of which may escape the detector. . . . . . . .

65

Figure 3.10: An illustration of the segmentation scheme of the crystals of the SeGA
detectors, as well as the nomenclature used to identify each segment.
The crystals are divided into eight disks with a width of 10 mm each,
labeled A through H along the axis of the cylindrical crystal. Each
disk is further segmented into quarters numbered 1 through 4, with
quarters 2 and 3 closest to the beampipe for the setup described in
this work. Figure adapted from [52]. . . . . . . . . . . . . . . . . .

67

Figure 3.11: The response of several SeGA detectors to a 226 Ra source before calibration. The photopeaks from each detector are clearly misaligned,
which would result in a γ-ray spectrum for the entire array with poor
resolution. The detectors are therefore calibrated to match the raw
signal in ADC channels to the actual γ-decay energies, which are
well-known. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 3.12: An illustration of the segmentation of the GRETINA crystals, as well
as the modules as a whole. The crystals are segmented into layers
which are, in order from the layer labeled α to φ, of thickness 8, 14,
16, 18, 20, and 14 mm. On the far left is the arrangement of the two
different hexagonal geometries used for grouping the crystals into a
quad. Figure adapted from [55]. . . . . . . . . . . . . . . . . . . . .

71

Figure 3.7:

Figure 3.8:

Figure 3.9:

xiii

Figure 3.13: A schematic drawing of the TRIPLEX plunger device. The components are labeled as follows: (A) The outer casing of the TRIPLEX.
(B) One of the piezoelectric motors for positioning the targets. (C) The
outermost movable cylinder, to which the second degrader is attached.
(D) The central, immobile cylinder to which the first degrader is attached. (E) The innermost movable cylinder to which the target
is attached. (F) The target, positioned by the moving cylinder E.
(G) The first degrader, held at a fixed position. (H) The second degrader frame, on which the second degrader (obscured by the first
degrader) is mounted. . . . . . . . . . . . . . . . . . . . . . . . . .

74

Figure 3.14: An illustration of the operation of the piezoelectric stepper motors
used in the TRIPLEX plunger. The red and blue rectangles are the
piezoelectric elements while the gray bars are the runner and the wall
of the chamber containing the stepper motor. Starting with frame 1,
a potential is applied to one of the blue piezoelectric elements, causing
it to expand and clamp to the wall. Another potential is then applied
to the red element on the end of the blue element, causing it to expand
and move the stepper assembly to the right in frame 2. The potential
is then switched from one blue element to the other, switching which
one is clamped (frame 3), and then the red element on the right has a
potential applied to move the assembly to the right again, while the
potential on the red element on the left is reversed to prepare it for
the next step. This process is then repeated until the assembly has
moved the desired distance. . . . . . . . . . . . . . . . . . . . . . .

75

Figure 4.1:

The composition of the secondary beam after purification in the
A1900 fragment separator, distinguished by their time of flight between the A1900 extended focal plane (xfp) scintillator on the vertical
axis and the S800 object (obj) scintillator on the horizontal axis. 78 Rb
is the primary component, with smaller contributions from 77 Kr and
76 Br. These latter components were removed with software gates in
the analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

87

Figure 4.2:

Figure 4.3:

Figure 4.4:

Figure 4.5:

A plot of the energy loss ∆E vs. time of flight Tobj of each particle
used to identify the various nuclei produced in the reaction of the
secondary beam. The energy loss separates elements vertically according to Z 2 while the time of flight separates isotopes horizontally
according to A/Z (note that for this plot, the horizontal axis has been
reversed, so that increasing time of flight corresponds to decreasing
A/Z). This results in the slanted lines shown. Stepping right along a
“horizontal” line corresponds to removing one neutron per locus, as
shown by the labeled Rb isotopes. Moving vertically downward along
a “column” corresponds to removing one proton and one neutron per
locus. Thus, moving the four loci from 78 Rb to 70 As in the figure
corresponds to removing four protons and four neutrons. . . . . . .

88

A partial level scheme for 70 As, showing the levels which are relevant to this study. Level energies are shown on the right, while spins
and parities are shown on the left (parentheses indicate tentative assignments), and the arrows indicate transitions between levels of the
associated energy. Two transitions are enclosed in boxes, indicating that these were used as gating transitions in the γ-γ analysis,
described in detail in the text. Figure is from [64]. . . . . . . . . .

89

The γ-ray singles spectrum in coincidence with 70 As fragments detected in the S800, with panel (a) showing the γ rays detected in the
backward ring and (b) those detected in the forward ring. Several
prominent transitions are identified by arrows, including the 8+ → 7−
at 788.3 keV which has a noticeable asymmetry due to lifetime effects.
The strong peak at low energy in the backward ring is due to both
the 9+ → 8+ transition and the REC process described in Sec. 4.2,
which cannot be resolved in this experiment. As mentioned, REC can
have a very large cross section, which explains the large amplitude of
this peak. In each panel, the inset shows the same spectrum focused
around the 788.3-keV γ ray from the 8+ state as well as weakly populated γ rays from the 10+ and 11+ states, which were important for
choosing the coincidence gates. Figure is modified from [64]. . . . .

93

The coincidence γ-ray spectra for the lifetime measurement of the
9+ state in 70 As, Doppler corrected and split into (a) the backward
ring and (b) the forward ring. The gating condition is that there is
a photon within the energy range covered by the combined REC and
9+ → 8+ transition peak. Several decays are labeled that can come
in the coincidence gate but are not related to the decay of the 9+
state. As can be seen in the figure, due to poor statistics the shape of
the 8+ → 7− peak is not well-defined in the backward ring and was
not able to be fit reliably. For this reason, only the forward ring was
used for analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

xv

Figure 4.6:

Figure 4.7:

The coincidence spectra used to determine the 8+ state lifetime in
70 As, split by photons detected in (a) the backward ring detectors
and (b) the forward ring detectors. In contrast to the coincidence
spectra for the 9+ state lifetime, there are no other obvious transitions
observed. The one bin which goes beyond the y-axis range in panel
(a) is due to accidental coincidence from REC which is not important
in this figure and had a bin content of 451 counts, and so the axis
range was set to emphasize the rest of the spectrum. The 8+ →
7− transition is labeled, and as for the 9+ coincidence spectra, the
peak shape is only well-defined in the forward detectors, and so the
backward ring was not used in the lifetime analysis of the 8+ state
either. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

A comparison of the true coincidence rate with the background coincidence rate for (a) the 10+ → 8+ coincidence spectrum for the 8+
lifetime analysis and (b) the 9+ → 8+ coincidence spectrum for the
9+ state lifetime analysis. In both panels, the dots represent the true
coincidence spectrum and the crosses represent the coincidence with
background, and only data from the forward ring are shown. For both
cases, the true coincidences show a clear presence of the 8+ → 7−
transition while the accidental coincidence spectra do not exhibit a
signal from this transition. Figure is adapted from [64]. . . . . . . .

98

Figure 4.8:

Comparison of the simulated and measured efficiency of SeGA for
various γ-ray energies with a 152 Eu source in (a) the forward ring
and (b) the backward ring. In both panels, the simulation is shown
by the red triangles while the data is shown by the black dots. The
simulations reproduce well the efficiency of the real SeGA detectors,
especially in the ∼600-1000 keV energy range of interest to this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 4.9:

A comparison of the measured beam profile (black) with the simulated one (red). Panels (a) and (b) show the spread in the angular
divergence of the beam in the dispersive and non-dispersive directions, respectively. Panel (c) shows the spatial distribution in the
non-dispersive direction, and panel (d) shows the spread in the energy of the beam relative to the energy specified by the Bρ of the
S800. All the distributions show very good agreement with the exception of yta , where the simulation does not reproduce the tails of
the data. However, the position in the non-dispersive direction does
not strongly affect the γ-ray spectrum, and so this level of agreement
is sufficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xvi

Figure 4.10: A first attempt at fitting the γ-ray singles spectrum from the forward
ring. For this fit, no attempt was made to account for any feeding,
and a lifetime for the 8+ state of 150 ps was used. Panel (a) shows
the data (black points with error bars) with the fit overlaid as the
solid line in red. The assumed exponential background is shown by
the dashed line. The individual components of this fit can be seen
in panel (b). The legend identifies the various γ-ray energies, which
correspond to those listed in Fig. 4.3 except for the 944.9-keV γ ray,
which was included to explain the accumulation of counts at that
energy. Otherwise, the 944.9-keV γ ray is unrelated to the analysis,
as the state from which it originates lies below the 8+ state. The
vertical dashed lines denote the location of the gate used to generate
the coincidence spectra for determining the 8+ lifetime. Figure is
from [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 4.11: The results of fitting simulations of the 8+ state with various lifetimes
(red line) to the experimental data for the downstream ring in coincidence with the 10+ → 8+ transition (black points with error bars).
The lifetime determined from the fit is τ8+ = 80(28) ps, including
only statistical uncertainties. Also shown is a simulation with a lifetime of 0 ps for the 8+ state for comparison, which demonstrates the
sensitivity of the data to the lifetime. The inset shows the reduced
χ2 distribution with five degrees of freedom, the minimum of which
was used to determine the lifetime. Figure is from [64]. . . . . . . . 107
Figure 4.12: The results of fitting the coincidence spectra gated on the 9+ state
for 70 As. The simulation with the best fit lifetime of 85 ps is shown
by the solid red line drawn over the black data points. As with the
8+ lifetime analysis, a simulation with the 9+ lifetime set to 0 ps is
shown by the blue dashed line to show the sensitivity to the lifetime.
The inset shows the reduced χ2 distribution which was calculated
with four degrees of freedom, the minimum of which was used to
determine that the 9+ state lifetime is τ9+ = 85(30) ps, including
statistical uncertainties only. Figure is from [64]. . . . . . . . . . . 109

xvii

Figure 4.13: A fit to the 9+ coincidence data using simulations which do not include the feeding from the 9+ state. This fit should be poor if the
measured lifetime is truly due to the 9+ state. Inspecting the spectrum, it can be seen that the shape of the photopeak is not reproduced
by the simulations under these conditions, which indicates that the
9+ state lifetime has a significant effect on the 8+ lineshape. The
reduced χ2 distribution in the inset, which was calculated using four
degrees of freedom as in Fig. 4.12, does not show a sharp minimum as
in Fig. 4.12 nor is it consistent with the result for the 8+ state lifetime
in Fig. 4.11. This indicates that the 8+ state lifetime alone cannot explain the lineshape of the peak under these gating conditions. Figure
is from [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 4.14: The results of a fit in which both the 8+ and 9+ state lifetimes are
varied simultaneously. Panel (a) shows the resulting fit to the data,
while panel (b) shows the individual simulations involved in the fit.
Also shown in the inset to panel (b) is the 2-dimensional reduced χ2
surface along with its projection onto the axes for the 8+ and 9+
state lifetimes. The minimum value is χ2 /N ≈ 1.15 at τ8+ ≈ 100 ps
and τ9+ ≈ 80 ps, in agreement with the coincidence fits. The number
of degrees of freedom for this fit is 58, which reflects the fact the fit is
over a much wider energy range. For reference, the maximum value
given by the red color in this contour plot is χ2 /N = 2. Figure is
adapted from [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Figure 4.15: A diagram showing the distribution of transition strengths for several types of electromagnetic transitions, divided into several different
mass regions [77]. The central set of distributions is for E1 transitions, and unfilled histogram is the relevant one for 70 As. This
histogram includes transition strengths in the range from roughly
10−6 to 10−2 W.u. With a strength of B(E1; 8+ → 7− ) = 1.2(4) ×
10−5 W.u., it is clear that this transition in 70 As is relatively hindered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xviii

Figure 4.16: A plot of B(M 1; 9+ → 8+ ) of several nuclei close to (and including)
70 As, as well as the strength predicted by the particle-vibration coupling model expressed in Eq. 4.3. All of the isotopes shown in the
figure are believed to have a 9+ state belonging to the πg9/2 νg9/2
configuration. Most of the transition strengths in the figure are similar to that measured in 70 As in this work, which indicates that 70 As
also has a 9+ state which is a part of the πg9/2 νg9/2 multiplet. The
outlier is 66 Ga, which has a significantly smaller strength. However,
this point is tentative because it is not certain that the decay is of an
M 1 nature, and this may explain the discrepancy. Figure is from [64],
and data are taken from [82, 83, 84, 85]. . . . . . . . . . . . . . . . 116
Figure 5.1:

The single-particle energy levels for 80 Sr in the Nilsson model, calculated for neutrons but which will be similar for protons in N ≈ Z
nuclei. The horizontal axis shows the quadrupole deformation parameter β, while the single-particle energies are plotted along the vertical
axis. While the usual gaps at the magic numbers are present at β = 0,
new gaps arise at non-zero values of the deformation. In particular,
gaps at roughly the same energy appear at neutron number 36 and
38, but corresponding to opposite signs of the deformation. This can
be a signature that shape coexistence may occur in nuclei that have
proton and neutron numbers similar to these values. Figure is taken
from [93]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 5.2:

Potential energy surfaces generated for 72,74,76 Kr. As described in
Ch. 1, the β axis gives the quadrupole deformation. Values of γ = 0◦
or γ = 60◦ correspond to prolate and oblate deformation, respectively,
while intermediate values of γ indicate triaxiality. All three krypton
isotopes clearly exhibit multiple minima, with a fairly strong prolate
deformation as well as a weaker oblate minimum. 72 Kr also has some
slight triaxiality associated with its prolate minimum. Figure taken
from [94]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 5.3:

Several partial level schemes for even-even proton-rich krypton isotopes, showing the systematics of the low-lying levels. For each isotope, the set of levels on the left is associated with a prolate shape,
while the 0+ level on the right is an oblate or spherical shape. In addition to the levels themselves, the positions at which the 0+ states
would be located if there was no mixing between them are indicated
by the dashed horizontal lines. These unperturbed levels become
almost degenerate at 74 Kr, at which point the mixing has become
almost maximized. The observed 0+ states are therefore almost
even mixtures of the intrinsic prolate and oblate shapes. Figure is
from [95]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xix

Figure 5.4:

A partial level diagram of the low-lying T = 1 states in 74 Rb and
74 Kr. Level energies in keV are shown to the left of the states, while
the spin-parities are given on the right. The isobaric analogue states
can be identified based on the similar excitation energies up to spin
8+ . However, there is no known excited 0+ state in 74 Rb which would
be the analogue of the excited 0+ state in 74 Kr. Data for 74 Kr are
taken from [97], while data for 74 Rb are from [98]. . . . . . . . . . 124

Figure 5.5:

The particle identification plot for the 74 Rb experiment. The vertical
axis gives the timing difference between the A1900 xfp scintillator and
the S800 E1 scintillator while the horizontal axis is the time difference
between the radio frequency (RF) signal from the cyclotrons and the
E1 scintillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 5.6:

The outgoing particle identification plot for the 74 Rb experiment.
The vertical axis gives the energy loss in the S800 ionization chamber while the horizontal axis gives the time of flight between the
A1900 xfp scintillator and the S800 E1 scintillator. As in Fig. 4.2,
the horizontal axis is reversed so that increasing time of flight indicates decreasing A/Z. The unreacted 74 Kr beam as well as the 74 Rb
reaction products are labeled and indicated by the position of the
ellipses in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 5.7:

The γ-ray spectra obtained in coincidence with 74 Kr nuclei in the
S800 Spectrograph. Panel (a) shows data taken with a 10 mm separation between the target and first degrader, while panel (b) shows
data taken with only a 1 mm separation. In both panels, transitions
+
can be clearly identified which correspond to the 2+
1 → 01 transition,
+
+
+
the 4+
1 → 21 transition, and the 61 → 41 transition, and are labeled
accordingly. The insets to each panel show the same spectra focused
+
on the 2+
1 → 01 transition to make clear the three-peaked structure,
where the components of the decay from fast, reduced velocity, and
slow recoils are also labeled. . . . . . . . . . . . . . . . . . . . . . . 131

Figure 5.8:

The γ-ray spectra obtained in coincidence with 74 Rb detected in the
S800 focal plane, again divided into (a) data taken with 10 mm separation between the target and the first degrader and (b) 1 mm separation between target and degrader. The insets again focus on the
+
2+
1 → 01 transition to show the three-peak structure. However, unlike for 74 Kr, there is very little population of states above the 2+
1
+ transition can be seen at the position
state. A hint of the 4+
→
2
1
1
labeled in the figure, but no other transitions can be identified. . . 132

xx

Figure 5.9:

A comparison between the measured (blue) and simulated (red) beam
properties. Panel (a) shows the angle of the beam in the dispersive
direction, while panel (b) shows the angle in the non-dispersive direction, and both are reproduced well. As with 70 As, the position
of the beam in the non-dispersive direction shown in panel (c) is not
reproduced well because the beam is assumed to be circular in the
simulation, but in reality may have any spatial distribution. The
spread in the energy is shown in panel (d), and is reproduced well.

136

Figure 5.10: A comparison of the simulated and measured beam properties for
74 Rb, shown in red and blue, respectively. As for the 74 Kr beam, the
dispersive and non-dispersive angular distributions shown in panels
(a) and (b), respectively, are reproduced well, while the non-dispersive
position has a different distribution in the data than is assumed in
the simulation. The spread in beam energies about the central value
shown in panel (d) is reproduced well. . . . . . . . . . . . . . . . . 137
Figure 5.11: The results of fitting the simulated γ-ray spectra to the 74 Kr γ-ray
spectrum. The data points are shown in black, while the simulation
is shown in red and the background is shown in blue. The slow and
reduced velocity components of the data are reproduced well by the
simulation, but the fast component is not. This is expected, as the
observed feeding from higher-lying states should produce a deficit of
counts in this peak relative to the simulations. The inset shows the
distribution of reduced χ2 values obtained by varying the lifetime in
the simulations, and the minimum of this distribution gives a lifetime
of τ = 30.6(20) ps, with 15 degrees of freedom in the fit. . . . . . . 139
Figure 5.12: The results of fitting the 74 Kr γ-ray spectrum with simulations that
+
+
include the effects of feeding from the 4+
1 and 61 states. The 22 state
was also included, but observed to contribute almost nothing to the
fit. In this figure, the lifetime was not varied, but the literature value
of τ = 33.8 ps [106] was used and all three peaks were included in the
fitting region. Now all three peaks are reproduced simultaneously, in
contrast to Fig. 5.11, which demonstrates that all three components
can be reproduced if feeding is taken into account. . . . . . . . . . 140
Figure 5.13: The results of the fit to the 74 Rb data. The data points are shown
in black, while the red line shows the best-fit simulation and the blue
line shows the assumption of the background. All three peaks are
reproduced in this fit even though only the slow and reduced velocity
components are fit, which shows that feeding is not a strong contributor in this analysis. The inset shows the reduced χ2 distribution,
where the number of degrees of freedom is 19, and the minimum of
the distribution gives the lifetime as τ = 26.7(60) ps. . . . . . . . . 141
xxi

+
Figure 5.14: Theoretical B(E2; 0+
1 → 21 ) values for several N = Z nuclei calculated using the shell model with the f5/2 p1/2 g9/2 d5/2 model space.
The value for 74 Rb is underpredicted by roughly a factor of three,
which the authors of the study note is due to the omission of the p3/2
orbital. Figure taken from [110]. . . . . . . . . . . . . . . . . . . . 145

Figure 5.15: Systematics of the measured B(E2) values for several N = Z nuclei, where the even-even nuclei are indicated by the filled symbols
and the odd-odd nuclei correspond to the open symbols. The red
point is the B(E2) value for 74 Rb measured in this work. These
values are compared to the predictions calculated from the results
of the macroscopic-microscopic model in [112] (solid line). Overall,
the trend is reproduced well, although the prediction for 70 Br differs
significantly from the measured value. Data are taken from [32, 60,
66, 114]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xxii

Chapter 1
Introduction

1.1

The atomic nucleus

At the center of every atom is an atomic nucleus. The nucleus carries the positive charge of
the atom in the form of electrically positive protons, which keep the electrons bound through
the Coulomb force. It also contains neutrons, which are electrically neutral. A given nucleus
is labeled in the notation A
Z XN , where X is the chemical symbol which specifies the number
of protons Z, N is the number of neutrons, and A = N + Z is the mass number. This label is
often abbreviated to A X, since the number of protons determines the element and hence Z,
and N can be determined from N = A − Z. The nucleus occupies only a tiny fraction of the
atomic volume: typical atomic radii are on the order of angstroms (10−10 m) while typical
nuclear radii are on the order of 10 femtometers (10−14 m). Despite this, nuclei contain
nearly all of the mass of the atom (over 99.9%), making them extremely dense. Since only
positive charges exist in this dense object, another force must be present which keeps the
protons and neutrons (collectively called nucleons) bound. This force is the strong nuclear
force, and understanding how it governs nuclear behavior is one of the primary goals of
nuclear science.
Just as atoms are arranged according to the number of electrons which they have, nuclei
are organized according to the number of protons and neutrons that they contain. This
gives rise to the so-called chart of the nuclides (sometimes called the Segr`e chart, after
1

120
126

Proton number

100
80

82

60

82

50

40
20

20

28

Te

n

50

28
Observed nuclei
Stable nuclei

2

8

2

0

g

20

8

0

a
rr

In

co

a
it

20

Unobserved nuclei

40

60

80

100

120

140

160

180

Neutron number
Figure 1.1: A drawing of the chart of the nuclides. The number of protons is plotted on the
vertical axis and the number of neutrons on the horizontal axis. The black squares are the
stable nuclei, and lie along the so-called “valley of stability.” The blue area indicates nuclei
which have been observed, while the red area represents nuclei which are predicted to exist
but have not been seen experimentally. Finally, the dashed lines indicate the locations of
the canonical magic numbers. For interpretation of references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation.
Emilio Segr`e), shown in Fig. 1.1. In this plot, the nuclei are arranged such that the number
of protons in a nucleus is plotted on the vertical axis and the number of neutrons is plotted
on the horizontal axis. Rows of nuclei with the same number of protons are called isotopes,
while columns of nuclei with the same number of neutrons are called isotones. Nuclei which
have the same mass but different numbers of protons and neutrons lie along diagonal lines
sloping downward and to the right, and are termed isobars. The black boxes in the figure
indicate the stable nuclei, while the blue area represents observed unstable nuclei. The
red portion of the figure indicates those nuclei predicted to exist but which have never been

2

observed. The borders of the shaded area are known as the driplines and represent the limits
of nuclear binding, beyond which additional nucleons will simply “fall off” the nucleus. The
upper left boundary is known as the proton dripline, beyond which no more protons may
be added to a nucleus. On the other side of the shaded area is the corresponding neutron
dripline, beyond which no more neutrons may be added to a nucleus.
The dashed lines in Fig. 1.1 represent the so-called magic numbers of 2, 8, 20, 28, 50,
82, and 126 [1]. Nuclei that have a magic number of either protons or neutrons are called
magic nuclei, while those that have a magic number of both protons and neutrons are
said to be doubly magic. Such nuclei exhibit certain unique behavior, such as an increased
amount of energy necessary to unbind them compared to neighboring nuclei, as well as higher
first excited state energies and smaller transition probabilities between the first excited and
ground states (more on this in Sec. 1.3) compared to their neighbors. Together, these traits
suggest that magic nuclei are especially stable, and this in turn suggests that some internal
organization of the constituent nucleons in a nucleus exists. Understanding and predicting
how this structure arises is a natural goal once this observation has been made.
Another noteworthy feature of Fig 1.1 is that the stable nuclei all fall along a relatively
narrow band known as the valley of stability. These stable nuclei represent only a small
fraction of the nuclei depicted on the chart: of the roughly 7000 nuclear systems predicted
to exist [2], only about 300 do not undergo spontaneous decay to more stable species. In
light systems, the stable nuclei tend to have the same number of protons and neutrons, but
for heavier nuclei the valley of stability trends towards having more neutrons than protons.
This is because, as more protons are added to the nucleus, a greater number of neutrons are
required to balance the increasing contribution of the repulsive Coulomb force. This also
explains why the majority of the nuclei that are expected to exist but have so far not been
3

observed lie on the neutron-rich side of the valley of stability. As protons are added to a
nucleus, the Coulomb potential tends to unbind the nucleus, whereas neutrons do not feel
the Coulomb potential and so more may be added before the nucleus becomes unbound. For
this reason, the location of the proton dripline is determined to a much higher mass than
the neutron dripline, which lies much further from stability.
An important feature of the nuclear force is that it is charge symmetric and, more generally, charge independent. That is to say, the nuclear force does not distinguish between
protons and neutrons. It is therefore convenient to introduce a formalism which capitalizes on this feature, and treat protons and neutrons as different states of the same particle,
i.e. the nucleon. This formalism is called isospin, and the charge independence of the nuclear force is the manner in which the isospin symmetry between protons and neutrons is
manifested [3, 4]. This symmetry is only approximate, however, since it is violated by the
Coulomb potential. In the isospin description of the nucleus, nucleons (both protons and
neutrons) have isospin T = 12 , but opposite isospin projections Tz . The convention in nuclear physics is to assign neutrons an isospin projection of + 12 and protons a projection of
− 12 . In this sense, the isospin states of the nucleon are analogous to (and follow the same
mathematical rules as) intrinsic spin, and hence the name isospin was given to it.
Just as with angular momentum, the isospin of individual nucleons can be coupled to
describe the overall isospin of the nuclear state they comprise. This leads to the concept
of isospin multiplets, i.e. groups of states with the same isospin T but different isospin
projections Tz . Such states are termed isobaric analogue states, and they can be identified
by examining the spectra of nuclei with the same number of nucleons but different numbers
of protons and neutrons. Figure 1.2 illustrates this concept for several nuclei with mass
A = 58. The energies of several excited states, for which the contribution of the Coulomb
4

Figure 1.2: A schematic diagram showing a portion of the level schemes for several nuclei
with mass A = 58, with 58 Ni bolded to indicate that it is stable. The levels joined together
by dashed lines indicate isobaric analogue states with the indicated value of isospin T .
The isobaric analogue states have been drawn at exactly the same energy in this figure to
emphasize their relationship with each other, effectively removing the contribution of the
Coulomb interaction which violates isospin symmetry. Also shown for the Tz = 0 nucleus
58 Cu are several T = 0 states that have no analogue states and are thus isospin singlet
states. Figure is taken and modified from [5].
interaction has been removed, are drawn. The isobaric analogue states are shown connected
by dashed lines. As can be seen, the Tz = 0 nucleus 58
29 Cu29 has several states without
analogues, which are isospin singlet states with T = 0. Several other states form triplets
in the Ni, Cu, and Zn nuclei with T = 1, and higher states form T = 2 quintets with all
five nuclei represented. In principle, states with isospin up to T = A/2 will be present, but
in practice such states cannot be observed as they must lie at extremely high excitation
energies. However, the identification of the low-lying isobaric analogue states is important
because, due to isospin symmetry, the nature of the analogue states should be the same in
all nuclei in a multiplet. Therefore, information obtained on one nuclear state in a multiplet

5

can provide predictions about the other states in the multiplet, thereby giving guidance to
experimental studies. This concept will be important for the work presented in Ch. 5.

1.2

Nuclear structure models

The realization that nuclei possess an internal structure leads naturally to a desire to understand and predict this structure. Given that nuclei are composed of quantum mechanical
particles, their behavior can, in principle, be predicted by solving the Schr¨odinger equation
for a system of A nucleons:
HΨ = EΨ,

(1.1)

with H = T + U , where T is the kinetic energy and U is the nuclear potential. Unfortunately, while various properties of the nuclear force are known, the exact form of the nuclear
potential U is not known. For this reason, a great deal of effort has gone and continues to
go into discovering ways to model the nucleus despite not knowing the nuclear potential. As
the results of the studies presented within this work are primarily applicable to the study
of nuclear structure, a few comments about nuclear structure models are germane to this
discussion.

1.2.1

The nuclear shell model

The nuclear shell model [6] is perhaps the best known example of a nuclear structure model,
and it has been very successful in describing various facets of nuclear structure [7, 8, 9].
In this model, nucleons are organized into groups of orbitals or shells. These orbitals are
characterized by the quantum numbers of the nucleons that occupy them: the principal (or
radial) quantum number n, which denotes the number of times the radial wave function
6

changes sign; the orbital angular momentum quantum number l, which is usually given in
spectroscopic notation such that l = 0, 1, 2, 3, 4, 5, ... is denoted as s, p, d, f, g, h, ...; and the
total angular momentum quantum number j = l ±s, where s = 21 is the spin of the nucleon.
Since nucleons are fermions, no more than one can have any given combination of quantum
numbers in a nucleus. This restricts the number of nucleons that can occupy an orbital to
the number of magnetic substates mj (of which there are 2j + 1) associated with a given
orbital. Orbitals for protons and neutrons are treated separately because these two kinds of
particles can be distinguished by their isospin, and are typically denoted with a π for protons
or a ν for neutrons. The full notation for a given proton orbital with, e.g. n = 1, l = 2, and
j = 52 , would be π1d5/2 .
Historically, it was the observation of the magic numbers that prompted the development
of the shell model. After all, a proper model of the nucleus should be able to reproduce the
signature of structure that the magic numbers provide. However, it took some effort to
get the model to this point. As shown in Fig. 1.3, the convenient choice of the harmonic
oscillator as the nuclear potential proves to be insufficient. The levels drawn on the left-hand
side of the figure show that, while the harmonic oscillator does result in certain numbers of
nucleons that correspond to filled shells, they do not correspond to the magic numbers. A
more realistic potential is the Woods-Saxon potential, which has the form

U (r) =

U0
,
1 + exp(r − R)/a

(1.2)

where U0 is the depth of the potential well, R is the nuclear radius and a is a parameter
that describes the diffuseness of the nuclear surface. The main effect of this potential is
to break the degeneracy of the orbitals with different l in the harmonic oscillator, but this

7

10
Neutron Single-Particle Energies

0

[56] 168 N=6

[ 2] 168 3s
[10] 166 2d
[18] 156 1g
[26] 138 0i

[42] 112 N=5

[ 4] 184 2d3
[16] 180 0j15
[ 8] 164 1g7
[ 2] 156 3s1
[ 6] 154 2d5
[12] 148 0i11
[10] 136 1g9

126

[ 6] 112 2p
[14] 106 1f

single-particle energy (MeV)

-10
[22] 92 0h
[30] 70 N=4

[ 2] 70 2s
[10] 68 1d
[18] 58 0g

-20

82

[10] 50 0g9
[ 2] 40 1p1
[ 4] 38 1p3
[ 6] 34 0f5

[ 6] 40 1p
[14] 34 0f

[ 2] 20 1s

-30

[12] 82 0h11
[ 2] 70 2s1
[ 4] 68 1d3
[ 6] 64 1d5
[ 8] 58 0g7

50

[20] 40 N=3

[12] 20 N=2

[ 2] 126 2p1
[ 6] 124 1f5
[ 4] 118 2p3
[14] 114 0i13
[10] 100 0h9
[ 8] 90 1f7

28
20

[ 8] 28 0f7
[ 2] 20 1s1
[ 4] 18 0d3

[10] 18 0d

[ 6] 14 0d5
[ 6] 8 N=1

8
[ 2] 8 0p1
[ 4] 6 0p3

[ 6] 8 0p

2
-40

[ 2] 2 N=0

Harmonic
Oscillator
Potential

[ 2] 2 0s

Woods-Saxon
Potential

[ 2] 2 0s1

Woods-Saxon
Plus Spin-Orbit
Potential

-50

Figure 1.3: The energies of the orbitals generated with several different interactions in the
shell model, calculated for 208 Pb. In the left and center columns are shown the harmonic
oscillator and Woods-Saxon potentials, respectively, neither of which is able to reproduce
the correct degeneracy of levels that gives the magic numbers. Adding a spin-orbit term to
the Woods-Saxon potential, however, does reproduce the magic numbers, as shown by the
column on the right. Figure is taken from [10]

8

potential still does not reproduce the magic numbers, as shown in the center column of
Fig. 1.3. The breakthrough that solved the puzzle came with the addition of a term that
couples the intrinsic spin and orbital angular momentum of a nucleon (the so-called spinorbit term [11, 12]): Uso = −Uso (r)l · s. This term breaks the degeneracy between orbitals
with intrinsic spin and orbital angular momentum aligned and antialigned. The right-hand
column of Fig. 1.3 shows these energy levels, along with the location of the shell gaps which
correspond to the observed magic numbers.
The shell model can be used to make predictions about nuclear structure, such as the
nuclear energy levels and transition probabilities between those levels. However, the complexity of the calculations increases very quickly with increasing numbers of nucleons. To
make the computational size of the problem tractable, the standard approach in the shell
model is to make the assumption that a subset of the nucleons in a nucleus form an inert core
which does not contribute to the calculation. A common choice is to choose the core to be a
doubly magic nucleus close to the nucleus for which the calculation is being performed. The
number of particles in the calculation is then reduced to those remaining outside the core,
called valence nucleons. The number of orbitals available to the valence nucleons (called
the valence space) is also typically restricted to only a few orbitals above the closed core to
further reduce the computational cost of the calculation. To compensate for this truncation
of the calculation, effective interactions must be developed which make up for the effects of
the missing degrees of freedom. The validity of a given effective interaction can be tested
by comparing its predictions against experiments. In turn, the results of experiments can
prompt the development of new effective interactions.

9

1.2.2

The Nilsson model

While the shell model has been very successful in describing nuclei which have a spherical
shape, a large number of nuclei exhibit non-spherical properties. To describe these nuclei,
it is important to have tools that can deal with deformed shapes. One example of such a
theoretical tool is the Nilsson model, which is an extension of the shell model to non-spherical
(but still axially symmetric) potentials [13]. The effect of the non-spherical potential breaks
the degeneracy of the states with different projections of the angular momentum, so that
each spherical shell model orbital gives rise to (2j + 1)/2 Nilsson orbitals. Since the orbitals
are now distinct, they are given new labels according the scheme Ω[N nz Λ], where Ω is the
projection of the particle angular momentum onto the symmetry axis of the potential, N
is the principal quantum number of the major harmonic oscillator shell (see Fig. 1.3), nz is
the number of nodes of the wavefunction along the symmetry axis direction, and Λ is the
component of the orbital angular momentum along the symmetry axis of the potential. The
splitting of these orbitals is dependent on both the sign of the deformation (i.e. whether
the nucleus is prolate or oblate) and also on its magnitude. For a prolate deformation, the
orbitals with the lowest projection of their angular momentum (the lowest Ω) are the lowest
in energy, while for an oblate deformation the highest projections are lowered the most.
A diagram showing the Nilsson orbitals for nuclei with N ≈ 82 − 126 is shown in Fig. 1.4
and demonstrates the effect of the lowering or raising of the orbitals with different angular
momentum projections with changing deformation, which is plotted along the horizontal axis.
Notice that for no deformation (i.e. a spherical nucleus) the Nilsson model reverts to the
standard shell model with the canonical magic numbers. However, for non-zero deformation,
the large gaps between the orbitals corresponding to magic numbers quickly disappear as

10

Figure 1.4: A Nilsson diagram showing the single-particle energy levels in the Nilsson scheme,
plotted against the deformation parameter along the horizontal axis. For zero deformation,
the shell model magic numbers are obtained, but for other deformation parameters (prolate
only in this figure), the splitting of the shell model orbitals fills in the gaps between magic
numbers, giving rise to a more complicated series of single-particle energy levels. Figure is
taken from [14].
the now non-degenerate orbitals quickly fill the space between the spherical single-particle
orbitals. However, at certain deformations, new gaps will sometimes open up between the
Nilsson orbitals, which can indicate that a nucleus with a certain number of nucleons may
experience a particular stability at that value of the deformation. This can be a way to
predict the shape of a deformed nucleus associated with a particular nucleon number.

11

1.2.3

Collective nuclear models

Beyond the microscopic calculations of the shell model and the Nilsson model, attempts
to model nuclei have also been made using a macroscopic geometrical description of the
nucleus as a whole [15]. This is especially useful for nuclei in which many nucleons are
actively determining the properties of the nucleus. Such models are called collective models
because they attempt to describe the nucleus from the standpoint of a collection of nucleons
moving coherently, rather than many independently acting particles. Two such descriptions
are especially well-known: the vibrational model and the rotational model. These models are
usually discussed for even-even nuclei, which always have 0+ ground states and usually have
2+ first excited states. It is also possible to extend the discussion to odd-A and odd-odd
nuclei by considering the coupling of one or a few nucleons to an even-even core.
The vibrational model describes nuclei which, while still spherical, experience oscillations
involving many of their constituent nucleons. Most typically, quadrupole vibrations around a
spherical shape are considered, although in principle any order vibration could be considered.
However, since a dipole term in leading order corresponds only to the overall translation of
the nucleus, the quadrupole vibration is typically considered first. The excited states in this
model are treated as arising from quadrupole phonon excitations in the nucleus. Starting
from the 0+ ground state, the creation of one phonon produces a 2+ first excited state
(denoted 2+
1 ). Creation of another phonon produces a triplet of nearly degenerate states
at twice the energy of the 2+
1 state. These three states arise from the coupling of the two
phonons together to gives states with spin-parities of 0+ , 2+ , and 4+ . Adding yet another
phonon creates a quintet of states with 0+ , 2+ , 3+ , 4+ , and 6+ from the coupling of three
phonons. As might be expected, these five states are also nearly degenerate with an energy of

12

three times that of the 2+
1 state. Generally, the excitation spectrum of a vibrational nucleus
has sets of evenly spaced states with energies given by

En = nE2+ ,

(1.3)

1

where n denotes the number of phonons. Identification of such a spectrum is experimental
evidence that a nucleus has a vibrational character.
In a rotational nucleus, the spherical symmetry is broken in the intrinsic frame of reference
of the nucleus, which takes on a static deformation. A quadrupole deformation is the most
common shape considered, although higher order deformations are also sometimes included.
For a quadrupole deformation, the radius of the nucleus is given by

αµ Y2µ (θ, φ) ,

R = R0 1 +

(1.4)

µ

where R0 is the undeformed radius, Y2µ are the second order spherical harmonics, and the
αµ are the coefficients of the expansion of the nuclear shape in spherical harmonics. As with
the vibrational nucleus, the center of mass motion is not of interest, and so the expansion
coefficients which correspond to it (α1 and α−1 ) are set to 0. The remaining coefficients
are then parameterized in terms of the quadrupole deformation β and the axial asymmetry
γ: α0 = β cos γ and α2 = α−2 = β sin γ. In this formalism, β denotes the deviation of
the nucleus from sphericity along an axis of cylindrical symmetry and is positive-definite.
This assumes that the parameter γ takes one of two limiting values: γ = 0◦ or γ = 60◦ . If
γ = 0◦ , then β > 0 corresponds to a nucleus with an elongated ellipsoidal or prolate shape.
Conversely, γ = 60◦ and β > 0 corresponds to a compressed or oblate shape. In both cases,

13

the nucleus remains rotationally symmetric about the axis of deformation. For 0◦ < γ < 60◦ ,
however, the nucleus becomes triaxially deformed and is no longer rotationally symmetric
about any axis. With the nucleus now statically deformed, it can have excited states based
upon its rotation. For axially symmetric nuclei (the majority of nuclei), the energy levels of
a rotational nucleus are given by
2

E(J) =

2I

J(J + 1),

(1.5)

where I is the moment of inertia for a nucleus of the appropriate shape. Such level schemes
are well-known in many nuclear systems [16] and serve as a reliable method of identifying
rotation of nuclei.

1.3

Lifetime measurements

Just as a wide variety of models exist to predict nuclear structure, a large number of techniques have been developed to measure it. This dissertation focuses on studies using one
particular category of these techniques: the measurement of the lifetimes of excited nuclear
states. This section will briefly describe the utility of lifetime measurements in general and
their connection to nuclear structure. An overview of some of the techniques used in lifetime
measurements, including those used in this work, will be presented in Ch. 2.
In studying nuclear structure, one often desires to get information about the construction of states of a given nucleus. Lifetime measurements provide a particularly transparent
approach for gaining such insights. The connection of the lifetime to the nuclear structure

14

is given by the equation
1
= Ti→f =
τi

kλπ Eγ2λ+1
λ,π

| Jf ||M(πλ)||Ji |2
,
2Ji + 1

(1.6)

8π(λ + 1)
kλπ =
.
λ[(2λ + 1)!!]2 ( c)2λ+1
In this expression, the mean lifetime τi of the initial state i is the inverse of the transition
rate Ti→j between the initial state i and final state f . The expression on the right gives
this transition rate in terms of a sum over all the allowed multipolarities λ allowed in the
transition as well as the character π of the decay, where π = E for an electric transition
and π = M for a magnetic transition. The matrix element on the right gives the overlap
of initial and final wavefunctions of the nuclear states connected by the electromagnetic
transition operator M(πλ). For electric type decays, this operator takes the form

ej rjλ Yλµ ,

M(Eλ) =

(1.7)

j

and for magnetic decays

gjs sj +

M(M λ) =
j

2 l
g l · ∇[rλ Yλµ ]µN .
λ+1 j j

(1.8)

In these expressions, Yλµ are the spherical harmonics, ej is the charge of the j th nucleon,
µN is the nuclear magneton, and gjs and gjl are the spin and orbital gyromagnetic ratios,
respectively. The operators sj and lj are the spin and orbital angular momentum operators
for the j th nucleons, respectively.
Certain selection rules govern the permissible multipolarities for a given transition. The

15

Yλµ can only connect states with spin Ji and Jf that differ by at most λ. This is often
expressed as the so-called triangle condition

|Ji − Jf | ≤ λ ≤ Ji + Jf .

(1.9)

Turning this statement around, a transition that connects states that have a difference in
angular momentum ∆J must have multipolarity λ ≥ ∆J. Since photons have an intrinsic
angular momentum of one unit, λ = 0 is forbidden and therefore Ji = 0 → Jf = 0 transitions
cannot proceed by γ decay. The parities of the states involved are also important and
determine the electric or magnetic character of the decay. When the parities π are such that
πi πf (−1)λ is even, the transition must be of the electric type, otherwise it is magnetic.
The last fraction on the right of Eq. 1.6 is the so-called reduced transition strength and
is commonly given the symbol B(πλ):

B(πλ) =

| Jf ||M(πλ)||Ji |2
.
2Ji + 1

(1.10)

This quantity is the one that contains the nuclear structure information. When discussing
nuclear structure, it is common to do so in terms not of the lifetime itself, but of the transition
strength which is derived directly from the lifetime:

B(πλ) =

kπλ
τ Eγ2λ+1

.

(1.11)

With this expression, the transition strength and therefore the nuclear structure can be
discussed so long as one knows the energy of the transition, the lifetime of the initial state, and
the electric/magnetic character and multipolarity of the transition. Typically, the transition
16

energy is one of the first experimental observables obtained for a given level, and the character
of the decay is often not far behind. The lifetime is usually measured last, and constitutes
the final piece of information necessary to calculate the transition strength.
Finally, having defined the transition strength and its relationship to lifetime measurements, a word about the insight that these quantities can give about nuclear structure is in
order. First, since the transition strength is directly related to the overlap of the wavefunctions of the states involved in the transition, the transition strength provides insight into
the configuration of the nuclear states. It also provides a method for direct comparison with
theoretical predictions of the transition strength made using wavefunctions calculated with
any appropriate nuclear model, including those discussed above. Agreement between theory
and experimental results then offers a way to test whether the calculated wavefunctions accurately describe the physical states. Even without theoretical calculations, however, transition
strengths are still a valuable tool for interpreting the results of experiments. The relative
strength (or weakness) of a transition can be judged by expressing it in so-called singleparticle units or Weisskopf units [17]. The Weisskopf unit is an estimate of the strength
that would be observed for a given transition assuming that it is due to only a single nucleon. The expressions for the Weisskopf estimate for electric and magnetic transitions are
given by [18]
B(Eλ) =

1
4π

10
B(M λ) =
π

2
3
(1.2A1/3 )2λ e2 fm2λ
λ+3
2
3
(1.2A1/3 )2λ−2 µ2N fm2λ−2 .
λ+3

(1.12)

If a transition is observed to have a strength which is comparable to the single-particle
estimates given here, it can be taken as an indication that single-particle degrees of freedom
may be appropriate to describe the nuclear behavior. However, if the measured strength

17

deviates from this estimate by an order of magnitude or more, then it can be an indication
that the structure of the nucleus is significantly different than that which a single-particle
picture would predict. In this way, the structure of nuclear states can be discussed even
when detailed theoretical information is missing.
The specific case of electric quadrupole (B(E2)) transition strengths is especially interesting from the perspective of studying nuclear structure. In particular, for an even-even
nucleus, the ground state always has a spin-parity of 0+ and frequently the first excited state
will be 2+ . Therefore, the lowest multipole transition that can connect these two states is
an electric quadrupole transition, and this can be used to quantify the collective nature
of the nucleus. For quadrupole collectivity as described in Sec. 1.2.3, transitions between
states involve many nucleons acting coherently, and so the transition strength is significantly
enhanced over the single-particle estimate. The observation of such an enhancement can
therefore be taken as evidence for the collective nature of the nucleus being studied. Furthermore, for a statically deformed nucleus, the B(E2) can be related to the quadrupole
deformation parameter β [19]:

1
+
B(E2; 2+
1 → 01 ) = 5

2
3
ZeR02 β ,
4π

(1.13)

where R0 is the undeformed nuclear radius. Because of this sensitivity to the deformation
+
and the collectivity in general, B(E2; 2+
1 → 01 ) values provide valuable insight into the

structure of nuclei.
The remainder of this work will focus on the application of lifetime measurements to
understand nuclear structure. Two studies will be presented: one on the nucleus 70 As and
one on 74 Rb. These proton-rich nuclei lie in a portion of the nuclear chart which is known

18

to be a site of many nuclear structure phenomena. Some nuclei can be described well using
the single-particle language of the shell model, while others necessitate the use of collective
descriptions. In addition, the transition between these behaviors can be abrupt, both as a
function of the nucleons that make up a nucleus as well as the excitation energy present in
the system. However, these nuclei also present a challenge to experiment and theory alike.
On the experimental side, the nuclei near mass A ≈ 70 near the proton dripline can be hard
to produce in sufficient quantities to study. In addition, the nuclei in this work have an
odd number of both protons and neutrons, and such nuclei usually exhibit very complicated
spectra which can hinder analyses. From a theoretical standpoint, the rapid evolution of
nuclear structure in this region makes it difficult to create a universal description which
can simultaneously reproduce the features of all of these nuclei. These systems also have
enough nucleons that they require quite large model spaces for microscopic calculations such
as the shell model, which in turn requires a great deal of effort to keep the computational
requirements tractable. The work detailed in this dissertation is intended to contribute
to the solution of this situation from the experimental side. The results of the analyses
provide important information about the structure of the nuclei studied herein and help to
improve the understanding of nuclear structure in this region. In addition, the methods
used to perform these studies constitute a contribution in their own right. Both are lifetime
measurements that are advanced versions of their more well-known counterparts intended
to solve specific problems frequently encountered in nuclear spectroscopy. The next chapter
will go into detail about lifetime measurements in general, and will also introduce these two
methods and their utility. Following that, the equipment necessary for these studies will be
reviewed, and then each of the studies presented in turn. Finally, some closing remarks will
be made about the results of these studies and the future prospects they present.
19

Chapter 2
Lifetime Measurement Techniques
The measurement of the lifetimes of nuclear states has led to the development of a wide
variety of experimental techniques. One of the driving factors of this proliferation of techniques is the rather large range of values that these lifetimes can take, such that multiple
methods have been necessary to cover as much of the spectrum of lifetimes as possible. In
addition, since a nucleus can decay through multiple channels, it has been necessary to develop methods which are tailored to specific kinds of nuclear decays in order to measure
their lifetimes. Figure 2.1 shows several techniques for measuring lifetimes along with the
approximate range of lifetimes which can be measured with that technique. This chapter
will introduce and give a brief summary of a few common methods for measuring excited
state lifetimes, focusing mostly on those involving the detection of γ rays. After that, the
two methods used in the experiments described in this work will be covered: the lineshape
method and the Differential Recoil Distance method.

2.1

Overview of lifetime measurements

Despite the great variety of strategies for measuring lifetimes, all lifetime experiments seek
to answer the same question: Given that excited nuclear states follow the law of radioactive
decay,
N = N0 e−t/τ ,
20

(2.1)

Figure 2.1: A diagram showing the approximate ranges in which various lifetime measurement techniques are effective. The methods labeled “direct” measure the lifetime itself,
whereas the “indirect methods” measure the width Γ = /τ or cross-sections of electromagnetic transitions. Figure is from [20].
how can the lifetime τ be extracted from the experimental data? The answer to this question
is very much dependent on the details of the state being analyzed, including the level scheme
of the nucleus, the difficulty of populating the state, and the value of the lifetime itself. An
additional complication presents itself when using secondary beams of rare isotopes, as the
intensities of such beams are usually limited and the beam emittance (that is, the distribution
of the beam in position and momentum space) can be large. This section will briefly introduce
a few of the more common methods employed to measure lifetimes in nuclear physics. More
specifically, since this dissertation is focused on direct methods for lifetime measurements
with fast rare isotope beams, the indirect methods shown in Fig. 2.1 will not be discussed.

21

In addition, the blocking and X-ray coincidence techniques listed in Fig. 2.1 will also be
omitted, as they are suited for lifetime measurements orders of magnitude shorter than the
lifetimes presented herein. The interested reader is referred to [20] for a review of these
methods.

2.1.1

Electronic timing measurements

Perhaps the most conceptually straight-forward method of measuring the lifetime of a nuclear
state is by directly measuring the time difference between two signals, one which corresponds
to the population of the state and another to its depopulation. This kind of measurement
is called an electronic timing measurement. Broadly speaking, there are two methods used
within the category of electronic timing methods: the slope method, used for longer lifetimes
greater than about 1 ns, and the centroid shift method, for shorter lifetimes less than about
1 ns [20]. Of necessity, this discussion of electronic timing measurements will be brief, but
much more information can be found in Ref. [21], from which this discussion is largely drawn.
In general, the timing signals generated by any detector system are subject to some level
of fluctuation, due both to the intrinsic timing characteristics of the detector system and to
the electronics used with it. Therefore, even for two radiation signals emitted simultaneously,
there is some finite width associated with the distribution of timing signals recorded by the
radiation detectors that sense this radiation. This distribution is known as the prompt time
distribution (PTD) or prompt response function (PRF). This is typically characterized by
the full width at half maximum of the distribution, which may also be referred to as the
“time resolution,” and an exponential tail which has some characteristic decay constant
called the “apparent half-life (τ1/2 )” or the “slope” of the distribution. It is the slope of
the PTD that determines which of the methods listed in the previous paragraph will be
22

used. If the apparent half-life of the PTD is small compared to the half-life of the state
to be measured, then the slope method can be used. On the other hand, if this is not the
case, then the difference between the centroids of the PTD and the time distribution of
the delayed signal from the decaying state can be used to measure the lifetime. As stated
in [20], centroid shifts as small as 0.5% of the FWHM of the PTD may be measured in this
fashion. For reference, values for the FWHM of the PTD may be a few hundred picoseconds
for scintillator detectors, while for germanium detectors this may be several nanoseconds.
Typical values of the apparent half-life are tens to around 100 ps for scintillators, and several
hundred picoseconds for germanium detectors.
Mathematically, the problem can be stated as an analysis of the time spectrum of the
delayed coincidence events F (t), which is a convolution of the PTD P (t) and an exponential
function f (t) with lifetime τ :
f (t) = (1/τ ) exp(−t/τ ),

(2.2)

for t ≥ 0 and 0 for t < 0. The function F (t) can then be written as
∞

f (t )P (t − t )dt ,

F (t) =

(2.3)

0

as long as F (t), P (t), and f (t) have their areas normalized to unity. The time delay t − t
in Eq. 2.3, which is just due to the width of the PTD, can be eliminated by the substitution
y = t − t , and further substituting in f (t) gives
t

F (t) = 1/τ exp(−t/τ )

exp(y/τ )P (y)dy,
−∞

23

(2.4)

which when differentiated yields

dF (t)
P (t) − F (t)
=
.
dt
τ

(2.5)

Moving F (t) to the left side, it can be seen that

d ln F (t)
dF (t)/dt
1
=
=−
dt
F (t)
τ

1−

P (t)
F (t)

.

(2.6)

Equation 2.6 gives the condition for using either the slope method or the centroid shift
method. Specifically, as stated earlier, at those times when the prompt timing distribution
makes a negligible contribution to the overall decay curve, i.e. when F (t)

P (t), then the

slope method can be used. In this case, the lifetime is found simply as

1
d[ln F (t)]
=− ,
dt
τ

(2.7)

and can be obtained from the slope of the function F (t) on a logarithmic plot at points
sufficiently far from the prompt timing curve.
For cases when the lifetime to be measured is small compared to the decay time of the
prompt timing distribution, then the condition F (t)

P (t) is never fulfilled. In this case,

the centroid shift method can be used. With this technique, the lifetime is given by the
difference of the first moments (centroids) of the delayed time distribution F (t) and the
prompt time distribution P (t). This result is not particularly difficult to derive. The first
moment of Equation 2.3 is
M(1) [F (t)] =

∞

tF (t)dt.
−∞

24

(2.8)

From Eq. 2.5, this integral can be rewritten as

M(1) [F (t)] =

∞

t P (t) − τ
−∞

dF (t)
dt

dt.

(2.9)

The term on the left is the first moment of P (t), while the term on the right can be integrated
by parts:
M(1) [F (t)] = M(1) [P (t)] − τ

∞

∞

tF (t) −∞ −

F (t)dt .

(2.10)

−∞

Because F (t) is normalized to unit area, the last integral in this expression is unity. Since
f (t) = 0 for t < 0, the lower evaluation limit in the first term in brackets can be set to zero:

∞

M(1) [F (t)] = M(1) [P (t)] − τ tF (t) 0 − 1 .

(2.11)

Physically, the delayed time spectrum F (t) must approach zero for very long times t, since
the population of a radioactively decaying state must eventually approach zero. The first
term in the brackets therefore vanishes at both the upper and lower limits, leaving the desired
relation:
τ = M(1) [F (t)] − M(1) [P (t)].

(2.12)

This simple expression is attractive because all that is required to determine the lifetime is
to read off the centroids of the two distributions F (t) and P (t). However, care needs to be
taken in practice to avoid systematic errors when the centroids of the distributions become
very close to each other, as noted in [20, 21]. It has also been shown that higher moments of
F (t) and P (t) may be used in the analysis [22]. Under the right circumstances this can allow
the lifetime to be analyzed even when the PTD is not known very precisely, but systematic

25

errors may become large. More details can be found in [21].

2.1.2

The Recoil Distance Method

The Recoil Distance Method (RDM), originally described in Ref. [23] and in more detail in
Ref. [24], is applicable to the measurement of lifetimes in the range from a few picoseconds to
nanoseconds and is described in its modern incarnation in Ref. [25]. Although there is some
overlap with electronic timing methods in terms of the lifetimes which can be measured,
the RDM represents a very different measurement paradigm than the electronic methods
of the previous section. In particular, rather than attempting to directly measure the time
delay between the creation and destruction of an excited state, which is quite challenging
for picosecond lifetimes, the RDM transforms the problem into one of measuring distances.
This takes advantage of the fact that an excited nuclear state is often created “in-flight”
by reaction of a moving nucleus on a fixed target. Assuming the reacting nucleus is not
stopped in the target material, it will recoil out of the target into vacuum, where it will have
a well-defined velocity. If the lifetime and the velocity are large enough, then the average
distance this nucleus will travel before decaying becomes macroscopic and can be measured
accurately. The problem is therefore translated from evaluating challengingly short times to
determining measurable distances.
The challenge of measuring the distance that the decaying nuclei travel still remains.
To solve this problem, the concept of a device which is called a “plunger” was developed,
illustrated in Fig. 2.2. These devices are designed to mount two foils in the beamline and
position them very precisely. The first foil that the beam encounters is called the target and
is used to populate excited states of the nucleus to be studied. The function of the second
foil is to change the velocity of the recoiling nucleus. For a low-energy study, the beam
26

Figure 2.2: An illustration of the concept of the plunger setup for measuring the lifetimes
of excited nuclear states. A beam of nuclei comes from the left and interacts with a target
foil, producing the excited states to be studied. The beam then recoils out of the target,
emitting γ rays as it travels toward a second foil. In this figure, the beam is stopped in the
second foil, where any nuclei which have not yet decayed will do so. Both Doppler shifted
and non-Doppler shifted photons are detected, and the ratio of the shifted and unshifted
photons gives the sensitivity to the lifetime. Figure is from [25].
velocity is typically a few percent of the speed of light, in which case the recoiling nucleus
is brought to rest in the second foil. In this case, the second foil is called a “stopper.” In
a high-energy experiment, which is more typical of the experiments performed at the NSCL
where the beam velocity is closer to 30% of the speed of light, the recoiling nucleus will pass
through the second foil. For these experiments, the second foil is called a “degrader.” In
either case, the reaction of the beam in the target defines a time t = 0. The second foil is
placed at a precisely measured distance D from the target, and if the velocity of the nucleus
as it recoils out of the target is known, then the time t = D/v at which it reaches the second
27

target is well-determined. The number of excited states which decay before the nucleus is
slowed or stopped in the second foil is governed by the lifetime. By systematically changing
the distance between the two foils during the experiment, the number of decays observed as
a function of distance can be mapped out, converted to a function of time, and the lifetime
extracted.
In principle, any method of detecting the decay of an excited state in a nucleus can
be used with the RDM, so long as it is sensitive to whether the decay happens before or
after the nucleus encounters the degrader/stopper. In practice, this most often involves the
detection of γ rays that are emitted as the nucleus decays to lower energy states. However,
some experiments have detected other radiations for the purposes of measuring lifetimes, for
example, from a proton-emitting nucleus [26]. Whatever the radiation detected, however, it
is necessary to determine whether it is emitted before or after the second foil is reached. For
the most common case of detecting photons, this can be done by noting that those nuclei
which decay before reaching the second foil do so in-flight, and therefore the γ-rays they
emit will be Doppler shifted according to

Eγlab =

Eγcm
,
γ(1 − β cos θlab )

(2.13)

where Eγcm is the photon energy in the center-of-mass frame, θlab is the laboratory-frame
angle at which the photon is emitted relative to the velocity vector of the nucleus, β is the
velocity of the nucleus as a fraction of the speed of light, and γ = (1 − β 2 )−1/2 is the Lorentz
factor. On the other hand, those nuclei that decay after encountering the second foil will
exhibit either no Doppler shift if the second foil is a stopper or a reduced Doppler shift if
the second foil is a degrader. Therefore, in the γ-ray spectrum, there will be two photopeaks

28

associated with the decay of each excited state which is populated in the nucleus: the shifted
peak for those nuclei decaying in-flight and the unshifted or less shifted peak for those nuclei
that decay after reaching the second foil. The number of decays happening before and after
reaching the second foil (and hence the lifetime of the state being studied) can then be
determined from the areas of the two peaks.
As it was originally stated [23], the mathematical justification for the Recoil Distance
Method is very simple. If the total number of γ rays observed from an excited state is given
by I0 , then the number of counts in the Doppler shifted peak is given by

Is = I0 (1 − exp(−D/vτ ))

(2.14)

and the counts in the unshifted peak (assuming a stopper is used) is

Iu = I0 exp(−D/vτ ).

(2.15)

If these two quantities are measured for several different distances D, then the ratio

R(D) =

Iu
= exp(−D/vτ )
Is + Iu

(2.16)

can be constructed as a function of D and the lifetime can be extracted from this function.
However, this doesn’t take into account the possibility of feeding from higher-lying states.
As written in Ref. [25], a more general approach can be taken to find the correct lifetime. In
this case, if the level of interest is Li , the higher-lying levels are Lh , and the levels to which

29

Li decays are Lj , the population of the level Li is governed by the differential equation
dni (t)
= −λi ni (t) +
dt

N

λh nh (t)bhi ,

(2.17)

h=i+1

where N denotes the highest state considered, λk = 1/τk is the decay constant of the state
k, and bhi is the branching ratio which indicates the fraction of the population of the state
h which decays to state i. Each of the higher states is also governed by a similar differential
equation, so that a system of coupled differential equations must be solved in order to find
the lifetime of the state i. The solution to this system of equations is given by
N

Mhi [(λi /λh ) exp(−tλh ) − exp(−tλi )],

Ri (t) = Pi exp(−tλi ) +

(2.18)

h=i+1

where Pn is the direct population of the state n which is not due to feeding and the term on
the right describes the feeding to this state, with


1
b P − bhi
Mhi =
λi /λh − 1  hi h

N

h−1

Mmh +

Mhm bmi (λm /λh )
m=i+1

m=h+1




.

(2.19)



In this last equation, the first term quantifies the direct population of the feeding state h, the
second term accounts for even higher-lying states that feed h, and the last term describes any
feeding of i by h through any intermediate states. Practically speaking, what this means is
that the data from an experiment with significant feeding must be fit with many parameters,
so that the lifetime, populations, and branching ratios for each state are extracted from the
data if they are not already known.

30

2.1.3

The Doppler Shift Attenuation Method

For lifetimes of nuclear states that are shorter than about one picosecond, measurements
that rely on the excited nuclei recoiling into vacuum become difficult. This is due to the fact
that a significant number of the excited states will decay before the nucleus exits the target,
and so a different approach is necessary. The Doppler Shift Attenuation Method (DSAM)
is applicable in this circumstance and makes use of the fact that the recoils are decaying inmedium. As the excited nuclei pass through the target material, they will be slowed due to
interactions with the target, with the consequence that the γ rays that are emitted as they
relax will have a continuous distribution of Doppler shifts according to the characteristic
timescale of the decay, which is just the lifetime of the state. Likewise, if the beam velocity
as it enters the target is chosen appropriately, then it will be stopped inside the target with
its own characteristic time scale. Comparing this stopping timescale, which is known in
principle, to the timescale exhibited by the distribution of Doppler shifts of the photons
from the decaying nucleus allows the lifetime to be extracted.
Practically speaking, the DSAM is based upon the measurement of the average velocity
v with which the excited states being studied decay. This is then used to define a function
F = v/v0 , with v0 the initial velocity of the nucleus [27]. An experimental measurement of
this function can be made in one of two ways. In the first method, the centroid of the observed
Doppler shifted γ-ray spectrum may be measured and used to determine v according to
Eγcm

lab

Eγ =

γ(1 − β cos θ)

.

(2.20)

The measured value of F can then be compared with a theoretically calculated value of F
using stopping theory. In the second case, the lifetime can be determined by fitting the
31

lineshape of the γ-ray spectrum to a theoretical lineshape, again calculated using stopping
theory, and which may generated using Monte Carlo methods [28]. In either case, it is
important to understand the stopping theory (e.g. [29, 30]) used as well as the stopping
powers they predict so that any possible uncertainties from this theory can be quantified.
Some insight into the nature of the dependence of the DSAM on stopping theories can
be gained by considering the actual form of the function F , which is

F =

∞
v
1
=
v(t) exp(−t/τ ) dt.
v0
v0 τ 0

(2.21)

From this, it can be seen that to evaluate F , it is necessary to know the functional form of
the velocity as a function of time. As a first approximation, the energy loss in the material
can be assumed to be proportional to the velocity, dE/dx ∝ v, if the velocity is around 1%
the speed of light [27]. This leads to the result that the velocity as a function of time behaves
exponentially, v(t) = v0 exp(−t/α), where α in this expression represents the characteristic
slowing down time of the material. Inserting this into Eq. 2.21 and evaluating the integral
results in
F =

1
.
1 + τ /α

(2.22)

As stated in [27], most solids have a characteristic slowing down time around 3 × 10−13 to
∼ 10−12 seconds. If values of F in the range 0.1 to 0.9 are those which can be determined
with sufficient accuracy, then the lifetimes that lie within the measurable range within this
assumption are roughly 10−14 to ∼ 10−11 seconds. However, the assumption that the velocity change is exponential is not necessarily valid. It is valid for velocities of about 1%
of the speed of light, where the energy loss is chiefly due to electronic stopping. However,
for higher velocities the energy loss scales more slowly with velocity, and for lower veloci32

ties nuclear stopping comes into play. Unfortunately, the treatment of the nuclear stopping
necessarily involves making approximations, and therefore DSAM measurements are inherently dependent upon the choice of some model. Nevertheless, DSAM is an important tool
for measuring lifetimes of femtosecond (10−15 s) order since few other direct methods can
measure lifetimes of less than one picosecond, as shown by Fig. 2.1. For more details and
further discussion of the stopping models used, see Refs. [20, 27].

2.2

The Lineshape Method

As described in Ch. 4, one of the goals of the experiments covered by this work was to measure
the lifetime of some of the excited states in 70 As. In order to accomplish this goal, a lifetime
measurement technique known as the lineshape method [31, 32] was used and applied to rare
isotope beams. Depending upon the experimental conditions, this technique can be applied
to lifetimes in the range of a few tens of picoseconds to a few nanoseconds. This section will
describe the lineshape method in detail as it was used in the 70 As experiment.
As with many of the methods used to measure prompt γ rays, the lineshape method is
predicated upon the fact that photons emitted from a source moving at relativistic speeds
will be Doppler shifted, such that the energy of the γ rays emitted in the rest frame of the
nucleus is related to the energy of the photons in the lab frame by the expression

Eγcm = γ(1 − β cos θ)Eγlab ,

(2.23)

where β is the velocity of the emitting nucleus expressed as a fraction of the speed of light,
θ is the angle (in the lab frame) at which the photon is emitted relative to the trajectory

33

of the nucleus, and γ is the Lorentz factor. In a typical experimental setup, the radioactive
isotopes which are to be studied are produced by passing a beam of nuclei through a thin
piece of target material to induce nuclear reactions. If these reactions result in nuclei in
excited states, the nuclei will quickly relax to their ground state, mostly by the emission
of one or more photons. Since many of these photons will be emitted extremely quickly,
Eq. 2.23 is used to reconstruct the energy of the photons in the center of mass frame of
the nuclei assuming that the angle of emission θ can be measured from the target position.
For short-lived states this is a good assumption, but for sufficiently long-lived states the
nucleus may travel a significant distance from the target before it decays. In this case the
assumption of emission at the target is no longer valid and the measured emission angle no
longer corresponds to the true one.
While incorrect Doppler correction is normally considered a liability, the lineshape method
makes use of it to extract useful information. For a sufficiently long-lived state, the observed
and true emission angle will differ enough that an observable shift in the reconstructed energy will appear. In addition, where normally a sharp peak would be present for a properly
Doppler corrected γ ray, a long-lived state will produce a peak which is broadened in such
a way that it acquires a low-energy tail. This shift in the peak energy and the shape of the
tail are characteristic of the lifetime of the decay, and therefore can be used to deduce the
lifetime of the state. Figure 2.3 shows typical lineshapes for decays with 10 ps, 100 ps, and
1000 ps lifetimes for decaying nuclei with a velocity of about 30% of the speed of light. As
can be seen from the figure, while there is no low-energy tail for the 10 ps lifetime, the 100 ps
lifetime has a small tail, and the 1000 ps lifetime has a prominent tail. This demonstrates
the sensitivity of the γ-ray spectral shape to the lifetime.
For the purposes of this dissertation, however, the lineshape method was not sufficient
34

4500
4000
10 ps simulation

3500

Counts

100 ps simulation

3000

1000 ps simulation

2500
2000
1500
1000
500
0
400

420

440

460

480

500

520

540

560

580

600

Eγ (keV)

Figure 2.3: The change in the spectral lineshape of a 500 keV γ-ray transition for 10 ps,
100 ps, and 1000 ps lifetimes, shown in blue, red, and green, respectively. For the 10 ps
lifetime, there is a significant high-energy shoulder which is smeared out over a large range,
plus a sharp peak. This is due to some decays happening in the target where the velocity
is faster than assumed having a smeared out Doppler correction and can be used as an
indication of a short lifetime. The 100 ps lifetime shows only a peak, but it is shifted
somewhat compared to the sharp portion of the 10 ps peak, and this shift is due to the
lifetime. There is also a small low-energy tail. Finally, the 1000 ps lifetime peak has mostly
a smeared out low-energy tail with only a small number of the decays happening near the
target.
without some improvement. This is because 70 As is an odd-odd nucleus and has a very
complicated level scheme. The presence of many photopeaks located close together in the
γ-ray spectra made it difficult to be certain that the lineshape observed for a given state
was, in fact, due to the lifetime and not to the presence of another peak at roughly the same
place. In addition, when there are multiple decays happening sequentially, the lifetime of one
state will have an effect on the measured lifetime of all of the states that it “feeds.” If this
“feeding lifetime” is long or at least comparable to the lifetime of a lower-lying state under
study, then it can cause a lifetime to be measured which is too long. For this reason, in order
to achieve a sufficiently clean spectrum and to control feeding, it was necessary to use the

35

lineshape method with γ-γ coincidence data. This enforces the requirement that a specific
state is populated first before another state, and therefore the history of each observed γ-ray
transition is known. It is then only necessary to know the lifetime of the state which is
feeding the lower-lying state(s) in order to correct for this effect on the lifetime. In addition,
the number of observed transitions in such a case is often substantially reduced and so
understanding the spectrum is considerably simplified. However, the requirement that a γ
ray be present in coincidence with another of a specific energy causes a large reduction in the
γ-ray yields, possibly of an order of magnitude or more. Therefore, successfully implementing
this technique is also challenging, as the signals from γ-ray transitions may be reduced to
the point that they are difficult to identify above the background. Thus, care must be taken
to optimize the selectivity of the coincidence requirements while still maintaining sufficient
statistics to make a meaningful measurement.

2.3

The Differential Recoil Distance Method

As the name implies, the Differential Recoil Distance Method (DRDM) is a variant of the
Recoil Distance Method (RDM) described earlier in this chapter. As will be discussed in
74 Rb. Although the
Ch. 5, this technique was used to measure the 2+
1 state lifetime of

method was first proposed in 1989 [33], it was not possible to implement it previously due to
difficulties with experimental resolution [25]. Therefore, the use of this method in this work
constitutes the first successful implementation of the technique.
In order to show the value of the DRDM, it is convenient to first introduce a variant of
the classic RDM known as the Differential Decay Curve Method (DDCM), also described
in [33]. The motivation for the DRDM then arises naturally from the development of the

36

DDCM. Since it is related to the classic RDM, the DDCM begins with the same equation
(c.f. Eq. 2.17):
dni (t)
= −λi ni (t) +
dt

N

λh nh (t)bhi .

(2.24)

h=i+1

However, rather than seeking an integral solution to the differential equations, the DDCM
attempts an algebraic solution. By defining the quantities
∞

Ri (t) = λi

t

ni (t)dt
(2.25)

Rij = bij Ri ,
where bij is the branching fraction of the decay of state i to state j, Eq. 2.24 can be rewritten
in the form
dRij (t)
= −λi
dt

Rij (t) − bij

Rhi (t) .

(2.26)

h

The quantity Rij is the decay function of the level i as it decays to the level j, and is given by
the number of states that decay after the nucleus has been in flight for a time t. Equation 2.26
requires that this function as well as its derivative be known. This is accomplished by taking
data at many different flight times (that is, many different distances between the target and
degrader/stopper foils). If the time t is defined as the time it takes for a nucleus to travel
between the two plunger foils, then the number of states that decay after this time will just
be the number of observed γ rays in the peak corresponding to the degrader/stopper. The
function Rij can then be determined by fitting an appropriate analytical function to the set
of values of Rij measured at different distances, and the derivative of the function taken
to determine dRij /dt. If higher states h are present that feed the level i, then the number
of γ decays observed in the degrader/stopper peaks (the second term in the numerator of

37

+
120 Xe with
Figure 2.4: An example decay curve measured for the 2+
1 → 01 transition in
the DDCM. The function Sij plotted on the vertical axis is proportional to Rij . As can be
seen, many data points are used to determine the lifetime, which in turn requires a large
amount of data to be taken. Figure is taken from [33].

Eq. 2.26) are subtracted from the decays associated with the level i in order to remove the
perturbing influence of the feeding on the lifetime. Compared to the Eq. 2.18 for the RDM,
which can quickly become very complicated when multiple feeding levels are involved, the
lifetime analysis in the DDCM is considerably more transparent, as only a few parameters
need to be determined. This facilitates accurate determination of the lifetime and reduces
the reliance on very complicated fitting routines.
However, despite the advantages of the DDCM, a rather large drawback of the technique
is its requirement that many data points be obtained and fit in order to determine the decay
function Rij , as illustrated in Fig. 2.4. For those nuclei which can be produced easily this
does not pose a challenge, but for very exotic nuclei this can be a serious problem, as the
amount of beam time necessary to accurately determine the derivative of the function Rij
38

Figure 2.5: A schematic drawing of a differential plunger used for the DRDM. The beam
enters from the right and excited states are created by interaction with the target. The
recoiling nuclei exit the target with velocity v and travel a distance x before being slowed
in the degrader. After the degrader they travel a short distance ∆x at a reduced velocity
v before being stopped in the stopper. As the excited nuclei relax, they emit γ rays with
different Doppler shifts at velocities v, v , and at rest, producing the γ-ray spectrum shown
at the bottom. Figure is from [25].
may become prohibitively large. This is where the DRDM method comes into play. Instead
of determining the derivative of the function Rij by fitting many data points, the DRDM
instead extracts the derivative of the decay function directly from the data. In order to
do this, a modification of the plunger is necessary. As shown in Fig. 2.5, a third foil is
added to the classic plunger design between the target and the degrader/stopper so that the
photopeak from the decay of each state is divided into three Doppler-shifted components.
This foil is always a degrader, as it is necessary that the nucleus being studied reach the
third foil. With this change, it becomes possible to determine the lifetime of a state with a
measurement at only one distance setting, thereby extending the family of Recoil Distance

39

measurements to much more exotic nuclei.
In order to see how a measurement with only one distance setting can yield the lifetime
of a nuclear state, it is useful to return to Eq. 2.26 and begin rewriting it in terms of the
quantities shown in Fig. 2.5. For the purposes of this discussion, it will be assumed that the
last foil in the experiment is a stopper for consistency with Fig. 2.5. As in the DDCM, the
decay function Rij is given by the number of excited nuclei that remain after a given time
t. It is convenient to take this time to be the time at which the decaying nucleus reaches
the stopper foil (denoted ts ), since the number of decays observed in the unshifted peak
from the stopper foil directly correspond to the number of excited states remaining at ts and
s . Similarly for any feeding
therefore to Rij (ts ). This number of decays will be denoted Iij
s . Now, to determine the derivative of R , it is necessary to
states, Rhi (ts ) will be written Ihi
ij

make an approximation. If the flight time of the nucleus is short compared to the lifetime of
the state under study, then the derivative of the decay function for the state i can be found
by a difference approximation of the decay function as the nucleus passes the degrader at
the time td and as it encounters the stopper at time ts :
dRij (ts )
Rij (td ) − Rij (ts )
≈
.
dt
ts − td

(2.27)

d = Id + Is ,
By noting that ts − td = ∆t = ∆x/v in the nomenclature of Fig.2.5 and that Rij
ij
ij

Eq. 2.27 can be written as
d + Is ) − Is
(Iij
dRij (ts )
Rij (td ) − Rij (ts )
ij
ij
d v .
≈
=
= Iij
dt
ts − td
∆x/v
∆x

40

(2.28)

Plugging this into Eq. 2.26 finally gives
s
s
∆x Iij − bij h Ihi
τ =−
,
d
v
Iij

(2.29)

which is the expression for the lifetime in the DRDM.
It is worth discussing the details of the DRDM further in order to make clear the advantages of the method. As stated, the addition of a third foil makes it possible to extract
lifetimes from experimentally measured quantities with only one distance setting between
the foils, which is a considerable improvement over the DDCM. This allows measurements
to be made of nuclei which are very difficult to produce, as significantly fewer counts are
necessary to determine the lifetime. Of course, feeding is a concern, and while the expression for the lifetime in the DRDM does explicitly treat feeding contributions, the effects
of feeding also warrant comment. For a feeding lifetime similar to that of the state being
studied, a significant fraction of the decays observed for the level of interest will be created
a significant distance from the target, and the term subtracted in the numerator of Eq. 2.29
will be very important. On the other hand, if the feeding states have a lifetime which is
much shorter than that of the state being studied, the design of the DRDM is such that it
automatically minimizes the effects of feeding. This is because the target is removed some
distance from the degrader, and therefore a state which is short-lived compared to the transit time to the degrader will have almost entirely decayed before reaching it. Therefore, the
DRDM effectively isolates the state being studied from the effects of feeding so long as the
target-degrader distance is properly chosen. This is reflected in the fact that the expression
for the lifetime in the DRDM does not make reference to the number of decays detected
t ). Of course, the ideal case is when there is no feeding at all, and
after the target (i.e. Iij

41

then the expression for the lifetime becomes especially transparent:
s

∆x Iij
τ =−
.
d
v Iij

(2.30)

As will be seen in Ch. 5, this expression can be very powerful for quickly obtaining an
estimate of the lifetime of a state before performing a full analysis, as only four quantities
are necessary to calculate the lifetime.

42

Chapter 3
Experimental Methods
The nature of rare isotopes is that they exist only for a short time before decaying towards
more stable species, mainly through β decay. Therefore, in order to study these rare isotopes,
it is necessary to produce them in a laboratory setting before transporting them to an experimental area for investigation. Doing so successfully often requires the use of sophisticated
techniques and the simultaneous operation of multiple state-of-the-art devices. This chapter
presents the process of developing and delivering a beam of rare isotopes at the National
Superconducting Cyclotron Laboratory (NSCL), as well as the experimental equipment and
methods relevant to the analyses presented in this work.

3.1

Beam production

The NSCL uses the projectile fragmentation method to produce exotic nuclei. In this
method, a beam of stable nuclei is first produced by ionizing them in an ion source and
then accelerating them to relativistic speeds in the K500 and K1200 cyclotrons, together
known as the Coupled Cyclotron Facility (CCF) [34, 35]. This beam of stable nuclei is called
the primary beam, as it is the first and most intense beam of nuclei created by the facility.
Once it is produced, the primary beam is then directed onto a piece of material, usually
beryllium, which is known as the primary target or production target. As the primary beam
passes through the primary target, nuclear reactions may be induced which can produce

43

Figure 3.1: An illustration of the equipment used to create a beam of rare isotopes at
the NSCL. A beam of ions with a only a few electrons removed is generated in one of the
ion sources, then injected into the K500 cyclotron. After being accelerated, the beam is
transported to the K1200, where it is further stripped of most or all of its electrons and
accelerated again. The beam is reacted on the production target and a single species is
selected to study in the A1900 before being transported to an experimental endstation.
Figure is from [36].
many different species of nucleus. These species typically contain fewer nucleons than the
primary beam, giving rise to the name “fragmentation.” Once past the primary target, one
or a few of the species produced can be selected in the A1900 Fragment Separator [36] based
on their charge and momentum in order to create the secondary beam, which then can be
transported to one of the experimental areas for further processing or analysis. Figure 3.1
shows the arrangement of the equipment just mentioned, and details of this process are
presented in the following subsections.

3.1.1

Primary beam production

The creation of a beam of rare isotopes at the NSCL begins with the production of ionized
stable nuclei. There are two ion sources available to accomplish this task: the Superconducting Source for Ions (SuSI) [37, 38] and the Advanced Room TEMperature Ion Source
44

(ARTEMIS) (see Ref. [39] for a description of ARTEMIS-B, an identical copy of the ion
source used for primary beam production), both of which operate on the principle of electron cyclotron resonance. If the stable species that will be accelerated is gaseous, then it
may be used directly in the ion sources. However, if it is metallic, then the material must
be placed into an oven to produce a vapor that the ion sources can use. With the atoms
now prepared in a gaseous state, electron cyclotron resonance heating [40] is used to ionize
them. The atoms to be accelerated are injected into a plasma confined within a magnetic
field while they are being ionized by microwave radiation applied at the electron cyclotron
resonance frequency ωc , given by
ωc =

eB
,
me

(3.1)

where e is the charge of the electron, B is the magnetic field strength, and me is the mass of
the electron. This causes energy to be transferred to the electrons of the plasma, accelerating
them. Multiple ionization is achieved by the accelerated electrons repeatedly impacting the
species to be used to form the primary beam, ionizing the beam material step-by-step to
the desired charge state before it is extracted from the ion source by an applied electric field
gradient. For both of the experiments described in this work, SuSI was used to produce
78 Kr

ions with a charge state of 14+. For more details on electron cyclotron resonance ion

sources, see Ref. [40].
After extraction from the ion source, the two cyclotrons of the Coupled Cyclotron Facility [34, 35] are used to accelerate the ions up to the desired beam energy. The operating
principle of the cyclotrons is based upon the application of a radio-frequency electric field for
acceleration and a magnetic field for confinement of the ions. The electric field is produced
by three pairs of blade-shaped electrodes known as “dees,” to which a rapidly switched elec-

45

tric potential is applied. When the ions are injected into the cyclotrons, they experience an
acceleration whenever they pass through the space between each pair of dees. Meanwhile,
a magnetic field is applied transverse to the plane in which the dees lie. Because the ions
are in motion, the magnetic field causes them to experience a Lorentz force that produces a
circular trajectory in the cyclotrons. The radius ρ of this trajectory for a given velocity can
be found by equating the Lorentz force due to the magnetic field with the centripetal force
experienced by the ions in the non-relativistic limit:

qvB =

mv 2
,
ρ

(3.2)

where q, m, and v, are the charge, mass, and velocity magnitude of the ions, respectively,
and B is the magnitude of the applied magnetic field. In the absence of the accelerating
electric field, the radius of the circular orbit in the cyclotrons is then

ρ=

mv
.
qB

(3.3)

However, the acceleration from the electric field continually increases the velocity of the ions,
causing them to spiral outward towards the edge of the cyclotrons where they are extracted.
For a particular cyclotron with a given radius and magnetic field, the energy of the ions at
extraction is determined by their charge and mass and can be found by rearranging Eq. 3.3
for the velocity at the maximum radius:

1
1
E = mv 2 = m
2
2

qBρ 2
q2B 2
q2
=
∝ .
m
m
2mρ2

(3.4)

At the Coupled Cyclotron Facility, the NSCL operates two cyclotrons connected in series:
46

the K500 cyclotron and the K1200 cyclotron (see Ref. [41] for several schematic drawings
and photographs of the K500). The names correspond to the so-called “K-number” of the
devices, which corresponds to the maximum energy to which a beam of protons can be
accelerated, although in practice this is not typically done. In principle, however, the K500
can produce a beam of 500 MeV protons, while the K1200 can produce a beam of 1200 MeV
protons. Both cyclotrons are isochronous, such that the RF frequency of their accelerating
electric fields is constant at about 23 MHz. As a consequence, in order to compensate for
the relativistic effects of the ions accelerating in these fields, the magnetic fields increase in
strength from the center of the cyclotrons towards their outer radii. In order to accelerate
ions most efficiently, the cyclotrons are coupled to produce higher energy beams. According
to Eq. 3.4, the highest energy beams for a given species can be produced when the charge
of that species is maximized. However, in order to produce ions which have the maximum
number of electrons removed, a significant penalty in beam current must be incurred. To
compensate for this, partially stripped ions are extracted from the ions sources, which have
a much higher beam rate, and injected into the K500. These ions are accelerated to their
maximum velocity of about 0.15c in the K500 before being extracted and then directed
through a thin carbon stripper foil whose purpose is to remove the remaining electrons
from the ion species. The now fully- (or mostly-) stripped ions are then injected into the
K1200 where they are accelerated up to their final velocity of about 0.5c. For the works
described in this dissertation, the K500 produced beams of 78 Kr (Z = 36) with an energy of
13.02 MeV/nucleon and a charge state of 14+ which were subsequently sent to the K1200
to produce primary beams of 78 Kr with an energy 150 MeV/nucleon and a charge state of
34+.

47

3.1.2

Secondary beam selection

Although some experiments at the NSCL are performed using stable beams, the majority
of experiments make use of radioactive species to perform experiments. To produce these
radioisotopes, the technique of projectile fragmentation [42, 43] is used to produce the nucleus
to be used in the experiment. For this technique, the primary beam is impinged upon a piece
of material called the primary production target or simply the primary target. Beryllium is
usually chosen to be the target material because it has a high number density of target
atoms, a high melting point, and good thermal conductivity so that it can withstand and
release the significant heat deposition from the primary beam. Most of the primary beam
particles will pass through the primary target without interaction, but some fraction will
undergo a nuclear reaction which can produce many different species lighter than (or, in the
case of a charge exchange reaction like in Ch. 4, of an equal weight to) the primary beam.
At this point, it is necessary to select the desired isotope from all other isotopes produced by
the interaction of the primary beam with the primary target, as well as any beam particles
which did not react.
In order to select the desired nuclear species from all other species present in the nowcomposite beam, the A1900 Fragment Separator [36] is used. The A1900 is composed of four
superconducting 45◦ dipole bending magnets and 24 superconducting quadrupole focusing
magnets which are collectively used to eliminate unwanted reaction products as well as
unreacted primary beam particles from the beamline. Each dipole magnet is separated by
two triplets of quadrupole magnets, and between each pair of quadrupole triplets is an image
plane where a pair of slits with variable width is located. A wedge of material can be inserted
at the momentum dispersive focal plane (Image 2 in Fig. 3.1) position which serves to aid

48

in isotope separation [44], and finally a suite of detectors is located at the achromatic focal
plane at the end of the separator (the focal plane detectors, Image 3 in Fig. 3.1) which provide
information about the now purified secondary beam.
The A1900 achieves isotopic separation by taking advantage of the fact that each species
present in the beam will have a specific (although not necessarily unique) magnetic rigidity,
given by
Bρ =

mv
A
∝ v,
q
Z

(3.5)

where B and ρ are the magnetic field strength and radius of curvature of the dipole magnet
through which a particle is traveling, and m, v, and q are the mass, velocity, and charge of
the particle as defined in Eq. 3.3. This quantity (which is often simply called the “Bρ”) has
A of a given particle, independent of the device
the advantage that it identifies the ratio Z

used to produce the magnetic field. The velocity v is typically almost constant for species
produced in fragmentation, and so the velocity dependence of the Bρ does not interfere with
identifying nuclei. The first step in purifying the beam in the A1900 is to select in the dipole
magnets a specific Bρ which includes the isotope to be studied. Those nuclei which do not
match this Bρ will follow a path with a radius of curvature different than that of the magnets
and will fail to be accepted into the next section of the beamline. Because the beam is now
spread out in the dispersive direction (i.e. the direction in which the beam bends), further
selection is provided by the pairs of slits at the intermediate image locations. These slits
are positioned such that they block all but a narrow portion of the beam which selectively
contains the isotopes to be studied.
The separation of the beam based on the magnetic rigidity may still allow multiple species
with the same A/Z ratio as the desired nucleus to pass through the separator. In order to

49

remove such persistent contaminants from the beam, an aluminum “wedge” is inserted into
the beamline at the Image 2 location. This wedge (which actually may be a curved piece
of aluminum of uniform thickness if it is thin enough) results in the different components
of the beam experiencing a different effective thickness as they pass through the wedge.
The various components of the beam will experience a unique energy loss which is governed
mainly by their charge q and velocity v and is given according to the Bethe formula [45]:

−

4πe4 q 2
2me v 2
dE
=
N
Z
ln
− β2 .
dx
me v 2
I(1 − β 2 )

(3.6)

In this expression, N is the number density and Z the atomic number of the atoms in the
wedge, e is the unit of electric charge, I is a parameter specific to the stopping material (the
aluminum wedge in this case), and β is the beam velocity expressed as a fraction of the speed
of light c. This energy loss causes the different parts of the beam, which were degenerate
in Bρ, to now have a different magnetic rigidity. The desired isotope can then be selected
in the second half of the A1900 by a similar Bρ separation as in the first half, providing a
beam which is largely free of contaminants. This beam is then sent to the focal plane of the
A1900 where various detectors may be used for beam diagnostics and identification purposes.
The most important of these for the purposes of this work is the extended focal plane (xfp)
scintillator which, together with the object scintillator in the S800 (see Sec. 3.2.1), is used
for identifying the components of the beam in the experiment.

50

3.2

Experimental devices and detector systems

The previous section describes the radioactive ion beam production process that is common
to most experiments done at the NSCL. However, a wide array of other devices are available
at the laboratory. Depending on the physics goals of a given experiment, the secondary
beam may be sent to one of several different vaults that are dedicated to certain setups.
For the purposes of the experiments described in this work, it was necessary to select an
area where the products of secondary reactions induced in the beam could be detected along
with γ radiation being emitted by these reaction products. Therefore, in both experiments
presented in this work the secondary beam was transported to the S3 vault, where the
S800 spectrograph is located and can be coupled with γ-ray detection systems. This section
will discuss the experimental apparatus and detector systems used in these experiments,
including S800 and the two different γ-ray detector arrays used, as well as the TRIPLEX
plunger device, which was critical to the success of the lifetime measurement of 74 Rb.

3.2.1

The S800 Spectrograph

For in-beam experiments performed at the NSCL, further manipulation of the secondary
beam may be necessary in order to extract the desired physics. This may be the case, for
instance, for reaction studies in which it is necessary to measure the angular distribution
of reaction products or for experiments which are intended to study nuclear states created
in the reacted beam at the experimental station. In such cases where further reactions
of the secondary beam are necessary, the ability to again identify one reaction product
from the potentially many newly created species is necessary. For this purpose, the S800
spectrograph [46], shown in Fig. 3.2, may be used.

51

Figure 3.2: A diagram showing the layout of the S800 spectrograph. The beam enters at
the Object position and is bent down through an analyzing beamline towards the location
labeled scattering chamber, where experimental apparatus and target materials are typically
located. After passing this location, the beam is bent upward through the two dipoles of
the spectrograph and beam particles are identified in a suite of detectors located inside the
focal plane. Figure adapted from [46].
The S800 is composed of an analyzing beamline and the spectrograph itself. The beamline
is built of superconducting magnets which are used to direct and focus the beam, while the
spectrograph is comprised of two superconducting dipole bending magnets plus a suite of
particle detectors located just past these dipoles in the final focal plane [47]. The S800 can be
operated in two modes: focused mode and dispersion matching mode. In focused mode, the
beamline is achromatic such that the beam is focused at the target position but dispersed in
the focal plane. This mode allows for the widest momentum acceptance of roughly ±2.5%,
but the momentum resolution is then limited by the momentum spread of the incoming
beam. By contrast, in dispersion matching mode the entirety of the S800 is tuned to be
achromatic so that the beam is dispersed across the target but this dispersion is canceled
in the focal plane. For typical targets, the momentum acceptance is limited to ±0.25% in

52

Figure 3.3: An illustration of the detectors located in the focal plane of the S800 spectrograph. The beam first passes through the two CRDCs which provide position and angle
information. Next the beam passes through the ionization chamber which measures the
energy loss of the beam as it passes through the detector. Next the beam passes through
the E1 scintillator, which serves as both a timing signal and a trigger. In the figure there
are two more plastic scintillators, but these have been replaced with a 32 crystal CsI(Na)
hodoscope for charge state identification [49]. Figure is taken and modified from [47].
this mode [48], but greater precision in the momentum measurement of the beam can be
achieved. Both of the experiments described in this work were run in focused mode.

3.2.1.1

S800 particle detection systems

The S800 makes use of a variety of particle detectors for the purposes of beam diagnostics
and identification. The majority of these are housed in the focal plane, shown in Fig. 3.3,
and include two Cathode Readout Drift Chambers (CRDCs) for position and angle determination, an Ionization Chamber (IC) which measures energy loss, a plastic scintillator (called
the E1 scintillator) for timing and triggering information, and a hodoscope composed of
32 CsI(Na) crystals for charge state identification. Besides the focal plane detectors, there
is also a scintillator at the object position (known as the OBJ scintillator) which together
with the E1 scintillator gives the time of flight for the ions through the entire spectrograph.

53

Optionally, tracking detectors may be inserted at the intermediate object image of the spectrograph beamline to characterize the path of the beam before it interacts with the target.
However, these detectors were not used in the experiments described in this work, and so
will not be discussed further here.

3.2.1.1.1

Time of Flight Scintillators Thin plastic scintillators in the S800 serve to

identify the time of flight of the particles through the spectrograph. When beam particles
pass through these scintillators, light is emitted and collected in the photomultiplier tubes
attached to the ends of the scintillators. While they have poor energy resolution, the light
pulse that they emit has a very fast decay time, resulting in excellent time resolution (with a
FWHM of the timing signal on the order of 100 ps [47]) ideal for time of flight measurements.
The two scintillators used in the S800 are the E1 scintillator in the focal plane, and another
scintillator located at the object position of the spectrograph. In the focal plane, the E1
scintillator is located just before the hodoscope and is used as a trigger signal for the S800
readout as well as providing a timing signal. When this timing signal is combined with
that from the scintillator at the object position, the time of flight of a particle through the
spectrograph can be calculated.

3.2.1.1.2

The Ionization Chamber The ionization chamber is located in the S800

focal plane just after the CRDCs. As with the CRDCs, the ionization chamber has an active
area of roughly 30 cm by 60 cm, but has a depth of 16 inches. The volume of the detector is
filled with P10 gas (a mixture of 90% argon and 10% methane) and is divided into 16 1-inch
sections along the beam direction. Previously this segmentation was achieved by having
16 separate anodes paired with a cathode on the opposite side of the detector. However,
this design was replaced with a new detector in which there are 16 cathode-anode pairs
54

formed from aluminized mylar foils. As beam particles pass through the detector, energy
is deposited in the gas and a number of electron-ion pairs are created proportional to the
energy deposited. The electrons are collected on the anodes and the ions on the cathodes.
The charge collected in this way is determined by the number of pairs created and is therefore
proportional to the energy loss of the particle. Since the energy loss is in turn proportional
to the square of the charge of the particle as given by Eq. 3.6, this gives identification of the
element of the particle in the focal plane.

3.2.1.1.3

Cathode Readout Drift Chambers The CRDCs measure the position and

the angle of the beam as it traverses the focal plane of the S800. These detectors have
an active area of roughly 30 by 60 cm and a depth of 1.5 cm, and are separated by 1 m.
The CRDCs are filled with a gas mixture which is 80% CF4 and 20% C4 H10 , and when
a beam particle passes through this gas mixture electrons are dislodged from the gas and
are collected on an anode wire. The position in the dispersive direction (the x position) is
determined by measuring the induced signal on the neighboring cathodes which sandwich
the anode and are divided into 224 pads. The distribution of induced charges on the pads
is fit with a Gaussian function to determine the x-position. The y-position is determined by
the drift time of the electrons in the gas, where the time is measured between the collection
of the charge on the anode wire and the timing signal from the E1 scintillator. The first
CRDC is located at the optical focus of the S800 and so the dispersive and non-dispersive
coordinates of a beam particle in the focal plane (called xf p and yf p , respectively) are taken
from this detector. The angles in the dispersive and non-dispersive directions relative to a
central trajectory through the S800 focal plane (termed af p and bf p , respectively) at which
a particle travels is determined by the difference in the positions measured in each of the

55

CRDCs.

3.2.1.1.4

Trajectory Reconstruction While the detectors of the S800 provide the

ability to determine the position and momentum of beam particles in the focal plane, it
is often of greater interest to know the beam properties at the location of the target. For
this purpose, the S800 provides the capability to reconstruct the trajectory of the beam at
the target position on an event-by-event basis. This was critical for both of the experiments
described in this work, as both relied heavily on reproducing the experimental measurements
with simulations (see section 3.3 for more details). The simulations rely on the ability to
accurately reproduce the properties of the beam as it exits the target, and so the particle
tracking facility of the S800 is very important to these analyses.
The trajectory reconstruction is achieved by way of an analytical calculation in the code
COSY INFINITY [50]. Using the measured magnetic field maps of the magnets, this program
constructs a transfer map S which transforms the beam parameters at the target to those
at the focal plane. In practice, since the focal plane parameters are the ones measured, the
inverse transfer map S−1 is then calculated and used to determine the desired parameters
at the target location:
















ata 







yta 

−1
=S 



bta 




dta


xf p 


af p 

.

yf p 


bf p

(3.7)

Here the parameters with the subscript ta indicate beam parameters at the target while
f p indicates parameters at the focal plane. In either case, x and y indicate dispersive and
non-dispersive position, while a and b indicate angles in the dispersive and non-dispersive

56

directions, respectively. The parameter dta indicates the energy of the beam at the target.
For this transformation, the spread of the beam in the dispersive direction is assumed to be
negligible and so xta is set to zero, thereby allowing the energy dta to be calculated [46].

3.2.1.2

Detector calibrations

The detectors in the S800 must be properly calibrated to be of use in an experiment. As these
detectors were common to both of the experiments in this work, the calibration procedures
will be described here so that they do not need to be repeated in the chapters devoted to
the individual studies.

3.2.1.2.1

CRDC calibrations The CRDCs determine the (x, y) position of the beam

as it passes through these detectors, and this position needs to be calibrated so that any
offsets are removed. This is done during the experiment by inserting a thick metal plate
(called a “mask”) with a specific pattern of holes and slits drilled in it in front of each
CRDC. The beam is then swept across the face of the detector with the mask in front of
it, with the consequence that only the parts of the detector behind the holes in the mask
are illuminated. Figure 3.4 shows the signals measured in the CRDCs when the masks are
inserted. Since the locations of the holes are well-determined on the masks, the signals
recorded in the CRDC during a mask run correspond to known physical positions. The
positions recorded in the CRDC are then linearly scaled so that the pattern that is recorded
matches the physical dimensions of the pattern of holes on the mask. Several mask runs
are taken throughout an experiment, since the positions that are recorded in the CRDCs
are dependent upon experimental conditions such as the pressure of the fill gas and the
temperature. If there is any change in the calibration between mask runs, this can be

57

150

150

(b)

100

100

50

50

Y(mm)

Y (mm)

(a)

0

0

-50

-50

-100

-100

-150
-300

-200

-100

0

100

200

-150
-300

300

X (mm)

-200

-100

0

100

200

300

X (mm)

Figure 3.4: The measured signals in the CRDCs when the masks are inserted into the
beamline. The first CRDC is shown on the left, and the second on the right. By matching
the pattern of slits and holes on the mask to the pattern of signals generated in the CRDCs,
the CRDC positions can be calibrated.
taken into account by interpolating between the mask runs to correct for any changes in the
experimental conditions.

3.2.1.2.2

Ionization chamber calibration The ionization chamber has sixteen chan-

nels which are used to measure the energy loss of the beam as it passes through the detector.
Since the energy loss for the detector is determined by averaging the energy loss in each channel, the channels need to be gain matched so that no single channel has a greater influence
than the others. This is done by selecting at least two nuclear species and measuring their
energy loss as recorded in each channel. The energy loss peaks are then fit with Gaussians,
and the centroids are used to create a linear function that scales each channel to match that
of the channel which has the smallest unadjusted energy loss. Figure 3.5 shows the result of
this gain matching.

58

2200

3000

(a)

(b)

2000

2500

2200

3000

2000

2500

1800

1800
1600

∆E (arb. units)

∆E (arb. units)

1600
2000

1400
1200

1500
1000
800

1000

2000

1400
1200

1500
1000
800

1000

600

600

400

500

400

500

200
0
0

2

4

6

8

10

12

14

16

200

0

0
0

Channels

2

4

6

8

10

12

14

16

0

Channels

Figure 3.5: This figure shows the gain matching of the ionization chamber. On the left, the
raw energy loss signals (in arbitrary units) are shown for each channel. On the right, the
energy loss in each channel has been gain matched to channel zero, which has the lowest raw
energy loss signal. As a result, the centroids of the energy loss in each channel are the same,
and this provides much cleaner elemental separation when performing particle identification.
3.2.1.2.3

Time of flight corrections Due to the different paths that ions take through

the spectrograph, there is a correlation between the time of flight measured by the scintillators and the measured angle of the ion trajectories as they pass through the CRDCs.
Removing this correlation allows for better identification of nuclear species in the S800 focal
plane. The form of this correction is

Tscint,corrected = Tscint + maf p + nxf p ,

(3.8)

where af p and xf p are the dispersive angle and position in the focal plane, respectively,
and m and n are constants to be determined. Figure 3.6 shows the effect of this correction
for the extended focal plane and object scintillators, as well as for the RF signal from the
cyclotrons.

59

300

200
0

150
100

-50

-1500

-1000

TRF (arb. units)
50
0

-50

1300

1400

1500

1600

1700

200
0

150
100
50

0

-2000

-1000

100
50
0

50
0

-50

1300

1400

1500

1600

1700

350

afp (mrad)

250
200

0

150
100

-50

350

(f)

300

50

300

50

250
200

0

150
100

-50

50
1500

1600

1700

100
50
0

Tobj, cor (arb. units)

(e)

1400

0
400
350
300
250
200
150

(d)

Tobj (arb. units)

afp (mrad)

-1500

TRF, cor (arb. units)

400
350
300
250
200
150

(c)

250

50

-50

50
-2000

afp (mrad)

afp (mrad)

250

50

300

(b)

afp (mrad)

afp (mrad)

(a)

1800

50

0

1400

Txfp (arb. units)

1500

1600

1700

1800

0

Txfp, cor (arb. units)

Figure 3.6: The effect of correcting the time of flight measured for the RF signal (top
row), object scintillator (middle row), and extended focal plane scintillator (bottom row).
Uncorrected spectra are shown on the left, corrected spectra on the right.

60

3.2.2

γ-ray detectors

Both of the techniques for extracting lifetimes utilized in the experiments described in this
work are predicated on the detection of γ rays. The techniques themselves have already
been described in detail in Ch. 2, however, neither the actual process of detecting γ rays
nor the apparatus used to detect them has been discussed. This section and its subsections
therefore summarize the important aspects of the measurement of γ radiation as well as the
two different γ detector arrays used in each of the two analyses contained herein.

3.2.2.1

γ-ray interactions with matter

Unlike charged particles, which undergo a continuous process of slowing down in a material,
γ rays interact with matter in discrete steps. For the purposes of this work, this interaction
can be described by three processes which dominate at different energies: photoelectric absorption, Compton scattering, and pair production. While each of these processes is distinct
from the others, they are similar in that each involves the transfer of energy from the photon
to electrons present in the material. The remainder of this subsection will describe these
processes.
Photoelectric absorption occurs when a photon interacts with and transfers all of its
energy to an atomic nucleus, as illustrated in Fig. 3.7. This results in the ejection of one of
the atomic electrons as a photoelectron with a kinetic energy given by

Ee− = hν − Eb ,

(3.9)

where h is Planck’s constant, ν is the frequency of the incident photon, and Eb is the binding
energy of the photoelectron. This expression neglects the recoil velocity of the nucleus, which
61

hν

Figure 3.7: An illustration of the photoelectric effect. An incident photon interacts with an
atom and deposits its energy, causing the ejection of an electron.
is very small. Characteristic X-rays or Auger electrons are generated as the vacancy left by
the photoelectron is filled. If the energy of these secondary radiations is absorbed in the
detector, the result is the detection of the full energy of the incident γ ray. This is the
ideal case for γ-ray spectroscopy, as the identification of γ-ray energies corresponding to
specific transitions may be done by inspecting the spectrum of observed photon energies.
However, this process typically is dominant only at photon energies below about 140 keV
(see Fig. 12.17 in Ref. [45]), and for germanium detectors gives way to Compton scattering
for photon energies above roughly 200 keV (see Fig. 2.20 of Ref. [45]).
Compton scattering is another process by which a γ ray may interact with a detector and
is illustrated in Fig. 3.8. In this process, a photon scatters from an electron in the medium,
imparting to the electron some but not all of its energy. The electron will then recoil while
the lower-energy photon propagates away at some angle θ relative to its previous trajectory.
The energy of the scattered γ ray is given by

hν =

hν
,
1 + (hν/me c2 )(1 − cos θ)

(3.10)

where hν and hν denote the photon energy before and after scattering, respectively. The

62

hν

hν

θ
e−

φ

Figure 3.8: An illustration of the process of Compton scattering. An incident photon with
energy hν interacts with an electron in the medium and imparts some of its energy to it. The
scattered photon then propagates away at an angle θ with respect to its original trajectory
and with a lower energy hν , given by Eq. 3.10, while the electron recoils away at some other
angle φ and energy given by Eq. 3.11.
energy of the photoelectron, which is the amount of energy that will be registered in the
detector, is then the difference between the energy of the photon before and after scattering:

Ee− = hν − hν = hν

(hν/me c2 )(1 − cos θ)
1 + (hν/me c2 )(1 − cos θ)

.

(3.11)

For a given photon energy, the only parameter in this expression is the scattering angle θ,
and it is useful to examine the limiting cases of scattering at θ ≈ 0 and θ = π. In the θ ≈ 0
case, it is evident from Eq. 3.10 that hν ≈ hν and so the scattered photon deposits almost
no energy in the detector. In this case, the photoelectron has very low energy (as shown by
Eq. 3.11) and the event is essentially undetected. On the other hand, for the θ = π case, the
energy transferred to the photoelectron is maximized:

Ee− = hν

2hν/me c2
1 + 2hν/me c2

63

.

(3.12)

However, the photon still carries away a portion of its original energy. There is also a
probability for the photon to scatter into other angles, and the cross-section for scattering
at any given angle is predicted by the Klein-Nishina formula [51]:
dσ
=Zr02
dΩ

1
1 + (hν/me c2 )(1 − cos θ)

(hν/me c2 )2 (1 − cos θ)2
× 1+
(1 + cos2 θ)[1 + (hν/me c2 )(1 − cos θ)]

(3.13)
,

where r0 is the classical electron radius. For a given photon energy, this distribution leads
to a γ-ray spectrum that has a large continuum for Compton scattering, in contrast to the
single photopeak present for photoelectric absorption. Neglecting detector efficiency effects,
the continuum reaches from zero energy up to the maximum given by Eq. 3.12, with no
peak corresponding to the energy of the incident γ ray. However, if the scattered photon
undergoes subsequent interactions within the detector volume, it may deposit some or all of
its remaining energy and in this case the full-energy peak may be observed.
Finally, for sufficiently high-energy γ rays, pair production may occur. In this process,
a photon is converted into a positron-electron pair in the electric field near a nucleus in
the detection medium, as illustrated in Fig. 3.9. This process is only energetically allowed
if the incident photon has an energy of at least the mass of the newly created particles:
hν ≥ 2me c2 = 1.022 MeV. Any remaining energy from the photon goes into the kinetic
energy of the pair. Both particles quickly slow as they travel through the medium, but the
positron will eventually annihilate with another electron, creating two photons with energy
me c2 = 511 keV. One or both of these photons may then escape from the detector, giving
rise to a spectrum characterized by the full-energy peak for the photon, plus a single-escape
peak 511 keV below the full-energy peak corresponding to the escape of one photon from the

64

511 keV
e−
e+
511 keV
hν

e−
Figure 3.9: An illustration of the process of pair production. If a photon has at least
1.022 MeV of energy, it may be absorbed by a nucleus and an electron-positron pair emitted
in its place. The positron will annihilate with another electron in the medium and release
two 511 keV photons, one or both of which may escape the detector.
detector, and a double-escape peak 1.022 MeV below the full-energy peak corresponding to
the escape of both of the annihilation photons.

3.2.2.2

The SeGA detector array

One of the γ-ray detection devices used in the experiments described in this work is the
Segmented Germanium Array (SeGA) [52], specifically for the study on 70 As presented in
Ch. 4. As the name implies, SeGA is an array of 18 high-purity germanium detectors whose
crystals have been electrically segmented to provide sensitivity to the position at which γ
rays interact with the crystal. This position sensitivity is important because the γ rays
detected in the experiments in this work are Doppler shifted according to the relation

Eγcm = γ(1 − β cos θ)Eγlab ,

65

(3.14)

where the superscripts cm and lab indicate the photon energy in the center of mass frame
and in the laboratory frame, respectively, and θ is the emission angle of the photon in the
laboratory frame relative to the beam axis. Since the energy is measured in the laboratory
frame, the relation above must be used to find the γ-ray energy in the center of mass frame,
and therefore accurate knowledge of the emission angle (as well as the beam velocity) is
critical for good final energy resolution. The remainder of this section will describe the
characteristics of the SeGA detectors and their use in the present work.
Each of the detectors that comprise SeGA are composed of a highly-segmented germanium crystal. The germanium itself is of high-purity n-type material (sometimes called
ν-type to distinguish high-purity material with residual donor impurities from material which
is intentionally doped with donor atoms). The crystals are cylindrical with a diameter of
7 cm and a length of 8 cm. Electrical contacts are placed along the outside of the crystal that
divide the detector into 8 disks of 10 mm width along the axis of the cylinder, and each disk
is further divided into four quadrants to give a total of 32 segments which are individually
read out (see Fig. 3.10). In addition, there is a central contact in the center of the crystal
so that the total energy deposited in the crystal can be read out in addition to the energy
deposited in each segment. The crystal is kept cold at roughly 100 K by thermal contact
with a dewar of liquid nitrogen, mounted at a 45◦ angle to the crystal axis.
The operational characteristics of SeGA depend heavily on the arrangement of the detectors during an experiment. While many different considerations may go into choosing an
experimental configuration for the detectors, often the choice is centered around optimizing
the trade-off between γ-ray detection efficiency and energy resolution. Efficiency can be
increased by placing the detector crystals as close as possible to the γ-ray source in order to
maximize the solid angle which the crystals cover, thereby increasing the probability that a
66

Figure 3.10: An illustration of the segmentation scheme of the crystals of the SeGA detectors,
as well as the nomenclature used to identify each segment. The crystals are divided into
eight disks with a width of 10 mm each, labeled A through H along the axis of the cylindrical
crystal. Each disk is further segmented into quarters numbered 1 through 4, with quarters 2
and 3 closest to the beampipe for the setup described in this work. Figure adapted from [52].
photon will pass through the detectors’ active volume. However, doing so has implications
for the energy resolution of the array. As indicated by Eq. 3.14, there are three quantities
that are needed to compute the center of mass energy of a γ ray emitted from a nucleus
moving at relativistic speeds: the velocity as a fraction of the speed of light (β), the angle
of emission (θ), and the photon energy measured in the lab frame (Eγlab ). Each of these
parameters contributes to the resolution of the γ-ray energy:
∆Eγcm
Eγcm

2

=

2
β sin θ
(∆θ)2 +
1 − β cos θ

2
β − cos θ
(∆β)2 +
2
(1 − β )(1 − β cos θ)

∆Eγlab
Eγlab

2

.
(3.15)

The intrinsic resolution of the detectors (the final term in this expression) has been determined to be ∼ 0.2%, but the other two terms can be much larger, on the order of about 1%
(see text and Fig. 1 of [52]). In particular, the first term includes a factor in ∆θ, and as
the detectors are brought closer to the γ-ray source, the solid angle of the segments used to

67

determine the emission angle becomes wider and therefore causes this term to grow.
The trade-off between detector efficiency and energy resolution has caused several different standard configurations to be developed for SeGA, each designed to optimize the
performance of the array for specific science goals. For the experiment on 70 As, a configuration known as the “barrel” or “beta” configuration was chosen. In this configuration,
SeGA consists of fifteen detectors arranged in two rings of detectors with the crystal axes
aligned with and roughly 13 cm from the beam axis. One ring is composed of eight detectors
covering emission angles of roughly 95◦ to 125◦ relative to the target (the “backward” ring),
and the other ring of seven detectors covering roughly 50◦ to 80◦ with respect to the target
(the “forward” ring). The forward ring had one less detector to allow space for a gatevalve
in the beamline. This gives a detection efficiency for 1 MeV photons of roughly 7%, which
was important for the 70 As experiment, while still giving sufficient energy resolution. See
Ref. [53] for other possible configurations of SeGA.

3.2.2.2.1

SeGA calibrations As SeGA consists of many separate germanium crystals

which operate independently of one another, each detector will exhibit a different response
to γ rays of the same energy, as shown in Fig. 3.11. Therefore, in order to make use of the
array in an experiment, it is necessary to calibrate SeGA against a source of γ rays with
known energies in order to ensure that each detector will give the same response. For this
purpose, in the 70 As experiment data were taken with several different radioactive check
sources, including 56 Co, 133 Ba, 152 Eu, and 226 Ra.
Calibrating the SeGA detectors first requires determining the centroids of the peak positions for a radioactive source with well-determined γ-ray transition energies. These peak
positions are then matched to the known peak positions and a fit is performed using a second

68

1600

Counts / channel

1400
1200
1000
800
600
400
200
0

300

400

500

600

700

800

Eγ (channels)
Figure 3.11: The response of several SeGA detectors to a 226 Ra source before calibration.
The photopeaks from each detector are clearly misaligned, which would result in a γ-ray
spectrum for the entire array with poor resolution. The detectors are therefore calibrated to
match the raw signal in ADC channels to the actual γ-decay energies, which are well-known.
order polynomial to map the uncalibrated energies to the calibrated energies for each crystal.
A χ2 minimization is used to determine the parameters of the fit, and these parameters are
then used to align the peaks in each crystal so that when the spectra from all the detectors
are summed the photopeaks are aligned. However, the segments of each crystal also provide
energy information and must be calibrated as well. This is a much more complicated problem, as there is cross-talk between the segments that causes the calibration to depend on
the number of segments which register an interaction in an event (the “fold” of the event).
For single-fold events, the energy deposited in the segment must be the same as that for
the central contact, and so the second-order polynomial fit can be performed to match the

69

segment calibration to the central contact calibration. Similarly, a χ2 fit is performed to
match each pair of calibrated segments to the central contact energy in 2-fold events, which
results in 32 × 31/2 sets of calibration parameters, showing how quickly the number of parameters grows with the fold of the event. Higher-fold events are therefore calibrated using
a correction derived from the single-fold and two-fold events. This process is described in
Ref. [54].

3.2.2.3

The GRETINA detector array

While SeGA provides good energy resolution, the next generation of γ-ray detectors aims
for significantly better resolution. To this end, the development of a new detector array
called the Gamma-Ray Energy Tracking In-beam Nuclear Array (GRETINA) [55] has been
undertaken. This is part of a project to construct a larger array, the Gamma-Ray Energy
Tracking Array (GRETA) [56], which will cover a full 4π of solid angle. The experiment on
74 Rb

described in this work was performed using GRETINA during its first experimental

campaign at the NSCL. This subsection will therefore describe the characteristics of the
array as it was during this first campaign.
During the experiment on 74 Rb described in Ch. 5, GRETINA was composed of seven
detector modules (called “quads”), each of which contains four hexagonally-shaped highpurity germanium crystals. There are two different irregular hexagonal geometries of crystal
(type A and type B), and each quad contains two of each type. Similarly to SeGA, each
crystal is electronically segmented; however, the GRETINA crystals are divided into six slices
of width 8 mm, 14 mm, 16 mm, 18 mm, 20 mm, and 14 mm in order from the face to the
back of the crystal. Each slice is further divided into six segments, for a total of 36 segments
per crystal. The electronics for reading out the photon signals are housed just behind the
70

Figure 3.12: An illustration of the segmentation of the GRETINA crystals, as well as the
modules as a whole. The crystals are segmented into layers which are, in order from the layer
labeled α to φ, of thickness 8, 14, 16, 18, 20, and 14 mm. On the far left is the arrangement
of the two different hexagonal geometries used for grouping the crystals into a quad. Figure
adapted from [55].
crystals, and a dewar of liquid nitrogen keeps the crystals cold. For an illustration of the
GRETINA detector modules, as well as the segmentation of the crystals, see Fig. 3.12.
Although both SeGA and GRETINA make use of segmented germanium crystals, the
design of GRETINA actually represents a significant advancement in γ-ray detection technology. This is due to two features of GRETINA: signal decomposition and γ-ray tracking.
The signal decomposition is one of the most important aspects of GRETINA, in that it
allows the γ-ray interaction position to be determined with subsegment resolution. In order
to do so, a set of simulated detector responses to the deposition of a unit of charge at various locations in the crystal are generated. This set of simulated responses is called a basis.
The basis signals are generated for a grid of non-uniform spacing with an average distance
between points of 1 mm. When a γ ray interacts with the crystal, it will generate a signal
in the segment it hits as well as an image charge in the surrounding segments. By fitting
the basis signals to the real and image charge signals in the detector, the position of the
interaction can be determined. The fit can also be performed for multiple-scattering by the
71

same photon in the crystal, in which case the basis signals are fit to the linear combination
of signals generated by the multiple hits from multiple-scattering events. During testing, the
resulting position resolution of the detectors was found to be roughly 2 mm [55], significantly
better than assuming that the hit occurred at the center of the segment.
GRETINA is also designed to have the ability to do γ-ray tracking. This is the capability
to examine an event in which the array experienced multiple photon scattering events in the
crystals and reconstruct the path(s) of the photon(s) that interacted with the detectors. This
is done by grouping the interaction points according to the likelihood that they originate
from the same photon. Next, for each group of points a scattering sequence is generated and
compared to the Compton scattering formula (Eq. 3.11) and a figure of merit is generated
according to how well the deposited energies and angles of the scattering sequence agree
with the Compton formula. The sequence is permuted for all possible orderings. A detector
that had perfect energy and position resolution would have a figure of merit of zero for
a fully absorbed photon. In practice, the figure of merit will not be exactly zero for a
correctly reconstructed scattering sequence, but the figure of merit should be large for partial
absorption or incorrect reconstruction. Finally, other possible clustering arrangements of the
interaction points may be iterated to seek an optimum solution. Placing a condition that
only events in which the figure of merit is above a certain threshold then eliminates events
that can be identified as only partial deposition of the γ-ray energy, resulting in a spectrum
with an increased ratio of events in the full-energy peak compared to those that populate
the Compton continuum.
For the 74 Rb experiment, GRETINA was set up in a “standard” configuration, with four
quads positioned to cover angles of roughly 20◦ to 50◦ and the remaining three quads placed
around 70◦ with respect to the beam axis. This provided the necessary sensitivity to the
72

lifetime of the state being investigated (see Ch. 5 for more details). The signal decomposition
is done automatically, as it is key to the superior energy resolution of GRETINA, and is
available on-line during experiment. However, it was found that γ-ray tracking was not
necessary for this particular experiment, and so this capability was not used in the present
work. Finally, the software GrROOT [57], which was developed for the GRETINA campaign
at the NSCL, was used for unpacking the data from the NSCL event format into ROOT [58]
objects.

3.2.2.3.1

Calibration of GRETINA In order to operate effectively, the GRETINA

detectors must be calibrated. Similarly to SeGA, this may be done using radioactive check
sources in order to map the measured energy in ADC channels to the actual γ-ray energies
emitted by a given source. The experiment on 74 Rb presented in this work was performed
as part of a year-long campaign of experiments with GRETINA at the NSCL, and so the
detectors remained in a calibrated state between experiments. This calibration was checked
before each experiment to ensure that the calibration parameters had not changed. For the
data presented in this work, this check was done by comparing the energies recorded for a
152 Eu

source in each crystal against the known source energies. A linear fit was performed

to extract “recalibration” parameters for each crystal and these parameters were used in the
data analysis to account for any drift in the calibration. For the γ-ray energies of interest
for the 74 Rb study (∼450 keV), this resulted in a typical correction of less than 1 keV.

3.2.3

The TRIPLEX plunger device

The TRIple PLunger for EXotic beams (TRIPLEX) [59, 60] is a device used to precisely
position various experimental targets along the axis of the beamline. The TRIPLEX is

73

Figure 3.13: A schematic drawing of the TRIPLEX plunger device. The components are
labeled as follows: (A) The outer casing of the TRIPLEX. (B) One of the piezoelectric motors for positioning the targets. (C) The outermost movable cylinder, to which the second
degrader is attached. (D) The central, immobile cylinder to which the first degrader is attached. (E) The innermost movable cylinder to which the target is attached. (F) The target,
positioned by the moving cylinder E. (G) The first degrader, held at a fixed position. (H)
The second degrader frame, on which the second degrader (obscured by the first degrader)
is mounted.
in the class of devices known as “plungers” commonly used in Recoil Distance Method
lifetime measurements. Unlike most plungers, however, which typically support mounting
at most two foils, the TRIPLEX is designed to support three foils in the path of the beam.
This addition is critical for enabling the use of the Differential Recoil Distance Method
(section 2.3), among other possible uses.
The design of the TRIPLEX is shown in Fig. 3.13, highlighting its various components.
The three foils are mounted on frames which are attached to the plunger by screws. The
middle foil is stationary, but the other two may be moved up to 25 mm away from the
center foil. This movement is achieved by moving the cylindrical tubes on which the foils are
mounted relative to the one on which the central foil is mounted. The tubes themselves are
moved by two piezoelectric stepping motors. Each motor contains several pairs of piezoelec-

74

Figure 3.14: An illustration of the operation of the piezoelectric stepper motors used in
the TRIPLEX plunger. The red and blue rectangles are the piezoelectric elements while the
gray bars are the runner and the wall of the chamber containing the stepper motor. Starting
with frame 1, a potential is applied to one of the blue piezoelectric elements, causing it to
expand and clamp to the wall. Another potential is then applied to the red element on the
end of the blue element, causing it to expand and move the stepper assembly to the right in
frame 2. The potential is then switched from one blue element to the other, switching which
one is clamped (frame 3), and then the red element on the right has a potential applied to
move the assembly to the right again, while the potential on the red element on the left is
reversed to prepare it for the next step. This process is then repeated until the assembly has
moved the desired distance.
tric elements attached to each other as well as to a runner, as shown in Fig. 3.14. Whenever
an electric potential is applied to the elements, they expand along the direction of the applied electric field. To achieve motion, a potential is applied to the elements attached to the
runner so that they expand and clamp against another surface. Then, another potential is
applied so that the piezoelectric elements deform along the axis of the runner, causing it to
move. The remaining elements are then clamped while the original elements are allowed to
unclamp and the deforming potential is reversed so that the motion is continued in the same
direction. By performing this process (shown in Fig. 3.14) repeatedly, uniform motion can

75

be achieved.
There are three methods for verifying the position of the three foils relative to one another.
First, the motors internally track how far they have moved and report a position relative
to their starting position when they are initially given power. Testing has shown that these
reported positions are generally reliable, but may begin to show some drift when the motors
are operated near the extremes of their range. A second method involves two micrometers,
one each for the position of the first foil relative to the second, and the third to the second.
These micrometers are extremely precise and can be used as a check against the positions
reported by the motors. However, they have a limited range which is less than the range of
motion of the motors. They are also fragile and may be damaged if too much force is applied
to them. Finally, the three mounting locations of the plunger are electrically isolated from
each other, which allows capacitive measurements of the distance to be used. Since the foils
are usually flat pieces of metal, they resemble parallel plate capacitors, and therefore the
TRIPLEX is designed to allow an electrical potential to be applied to the middle foil and the
induced signal read out from the other two foils. If properly calibrated, the control software
of the TRIPLEX is able to translate this capacitance into a position. Under conditions in
which the foils are thin enough that they may deform under the application of the beam,
such as low-energy experiments with beam energies less than about 10 MeV/nucleon, the
software can even use this capacitance measurement to correct the separation of the foils
in real time via a feedback mechanism. However, as this was not necessary for the present
work, this feedback mechanism was not used. Instead, the capacitance reading was used to
note the precise moment when contact was achieved between the foils, and this was used to
define the “zero distance” for the motor and micrometer readings.

76

3.3

Geant4 simulations

Much of the work to extract lifetimes from the experimental γ-ray spectra is performed with
the aid of the Geant4 simulation package [61]. These simulations need to accurately reproduce all of the important aspects of the experimental setup as well as handle the appropriate
physics processes. Geant4 itself is designed to take care of the physics processes, including
the physics of the beam propagating through target materials and photon physics. For the
specific needs of the experiments described in this work, a simulation package previously
developed and used at the NSCL [62] was employed. This package has been updated so that
it incorporates both the SeGA and GRETINA detector systems in the geometries used in
this work. This section will briefly describe the operation of the simulation software and the
manner in which it is used to analyze lifetime data.
The simulation software accepts many parameters in order to accurately describe an
experiment. Many of these are determined by the equipment used in an experiment. As
the simulation is designed to be used for lifetime measurements at NSCL, it accepts input
which tells it whether to use a single target, a classic two-foil plunger, or the TRIPLEX
plunger. It can also accept values for the material type, thickness, position, and a factor by
which to scale the density of each target in order to reproduce the measured energy loss in
the experiment. There is also the possibility to translate the plunger away from its default
position within the simulation. With respect to the detector systems involved, the simulation
is able to virtually construct either SeGA or GRETINA and use it as a detector system for
γ rays. For SeGA, this includes the capability to translate and rotate each SeGA detector
individually to precisely reproduce the experimental setup. The description of the SeGA
detectors is also very complete, including both the active volumes of the detectors as well as

77

the crystal housings and dewars, and includes the ability to simulate the individual segments
of SeGA. For GRETINA, the individual quads are described and can be positioned precisely
in a similar manner to SeGA. However, instead of individual segments, the simulation mimics
the position sensitivity of GRETINA by taking the actual interaction point of a photon in
the simulated detector volume and shifting it according to a three dimensional gaussian
distribution with a variable width parameter. In this way, the observed position resolution
can be matched in the simulation.
In addition to the parameters which describe the experimental setup, the simulation also
accepts many values which describe the experimental conditions specific to a given nucleus.
These include the position, spatial extent, energy, momentum spread, and angular spread
of the incoming beam; the change in charge and mass number undergone in a reaction in a
foil, the probability that the reaction will happen in a given foil as opposed to the others,
and the change in momentum of the reacted beam due to the reaction; and the charge state
of the outgoing beam, a reference energy that the outgoing beam is expected to have, and
the acceptance of the S800. Finally, the simulation accepts parameters which describe one
or more excited states in the nucleus that results from the reaction on the target, including
the excitation energy of the states, the energy of the γ rays coming from these states, the
fraction of the nuclei produced which populate each state, and the lifetime of each state. In
this way it is possible to describe the beam itself and reproduce a potentially complicated
level scheme involving many excited states which decay to each other.
With all the relevant parameters set, the simulation program can be executed. The
parameters can be set at runtime in interactive mode or a set of macros can be defined to
automate the process. Once executed, the program will use Monte Carlo methods to generate
incoming beam particles in a distribution defined by the given incoming beam parameters.
78

The simulation will then propagate the beam through the various foils, simulating the energy
loss that the beam particles experience as they pass through the material. According to the
parameters entered, the simulation will select one of the foils to be the one in which the
beam undergoes a reaction, producing the outgoing nucleus in one of the states defined
in the input file. The parameters that determine the outgoing beam momentum include
a fraction of the momentum that is lost in the reaction, the width of a gaussian profile
that is sampled to create a spread in the outgoing momentum, and finally the strength of a
kick in a random direction. If the outgoing nucleus is in an excited state, the program will
sample the decay curve of the excited state based on its entered lifetime and determine the
location at which it will decay, propagating the nucleus to that point and emitting a γ-ray
according to the given decay properties of that state. The photon emission distribution is
isotropic, and if it encounters part of the detector volume its interaction will be determined
by the included Geant4 physics processes. The response of the detector is then reproduced,
including the various possible photon-material interactions, and the energy deposited in the
germanium crystals recorded. This continues until the nucleus has deexcited to its ground
state. This process is then repeated many times until a given number of beam particles have
been simulated.
Once the program has finished simulating the experiment, it will output a set of histograms that can be compared to experimental data. Internally, it uses the ROOT analysis
package [58] to store data during the simulation process, then generates a predefined set of
histograms from this data. These histograms are then written to a ROOT file which can be
opened and manipulated by the user. The tree structure which stores the data internally
can also be written to file, although this is atypical as the files are very large. Since the
experimental data in this work was also analyzed using the ROOT package, it is then possi79

ble to directly compare the simulations of data with those from the simulation for analysis
purposes.
In this work, the simulations are fit to the data using a χ2 minimization procedure. A
short overview of this procedure is given here, which represents a summary of the information
found in Ref. [63]. If the data is represented by a histogram with N bins denoted xi , where
i is the index of a bin, and the simulation by a histogram with the same number of bins
denoted yi , then the quantity χ2 is calculated by
N

χ2

=

(xi − yi )2
,
2
σ
i
i=1

(3.16)

where σi is the uncertainty in the number of entries in bin i of the data. If the simulation
reproduces the spectral shape of the experimental spectrum well, then the χ2 will be small,
as the differences between the data x and the simulation y will be small. However, if the
simulation doesn’t reproduce the data (if, e.g., the lifetime input to the simulations is not
the true lifetime), then the χ2 will be large. This provides a metric by which to gauge the
goodness of a particular fit. By fitting many simulations with different lifetimes to the data,
a distribution of χ2 can be generated, and the minimum taken to be the “best fit.” If this
distribution of χ2 is quadratic in the lifetime, then the χ2 for a value of the lifetime which
is one standard deviation away from the minimum χ2 is given by

χ2 = χ2min + 1.

(3.17)

This provides a useful way of determining the uncertainty in the lifetime. However, it is also
important to judge whether the obtained χ2 indicates a good fit or not. This can be done,

80

assuming that the individual bins of the data are independent and the simulation gives the
correct spectral shape, by noting that the minimum value of the χ2 should be distributed
according to the χ2 distribution:

f (z; n) =

1
2n/2 Γ(n/2)

z n/2−1 e−z/2 ,

(3.18)

where z is a continuous variable (the lifetime, in this case), n is the number of degrees of
freedom of the fit, and Γ is the gamma function defined by
∞

Γ(x) =

e−t tx−1 dt.

(3.19)

0

The expectation value of the χ2 distribution is the number of degrees of freedom, which
in the present case is the number of bins less the number of fit parameters fixed from the
data. For a good fit, then, it is expected that χ2 /n = 1. This condition can therefore be
used to judge the quality of the fit, and in conjunction with the procedure for finding the
uncertainty, is used to determine the lifetime in the analyses presented in this work.

81

Chapter 4
Lifetime Measurements in 70As
The first of the lifetime analyses to be discussed in this work was done on 70
33 As37 [64]. This
+
experiment used the lineshape method to determine the lifetimes of the 8+
1 and 91 states of

this nucleus. More specifically, the analysis applied the lineshape method to γ-γ coincidence
data, where it was required that a γ ray populating the state of interest be observed in the
same event as a γ ray depopulating the state. This presents an experimental challenge, as the
γ-ray yield is substantially reduced in a coincidence measurement and therefore maintaining
sufficient statistics for an accurate measurement becomes difficult. However, as 70 As is an
odd-odd nucleus (i.e. it has an odd number of both protons and neutrons), it has a high
level density of excited states which are interconnected. This makes it difficult to ensure
that the shape of the γ-ray spectrum being studied is due to just one transition and not
multiple transitions superimposed on one another. In addition, with significant feeding, it
is possible that a given state may appear to have a longer lifetime than it actually does due
to the influence of the lifetime of a feeding level. The use of γ-γ coincidence was therefore
necessary for this analysis, and also demonstrates an important tool for the determination of
lifetimes in similarly complicated nuclear systems where the analysis of γ-ray singles spectra
may prove insufficient.

82

4.1

Physics motivation

The motivation for studying 70 As was to understand the structure of this nucleus. 70 As
is located in a region of the nuclear chart which is known to be a location where nuclear
structure is rapidly evolving [16, 32]. For instance, nuclear systems of roughly mass 70 near
the N = Z line have been shown to rapidly change their shapes as a function of the number
of protons and neutrons they have [60, 65, 66]. In addition, as more nucleons are added
to nuclei in this mass region, they exhibit greater levels of collectivity and this behavior
is attributed to increasing occupation of the g9/2 orbital, which is understood to drive the
collectivity in this region [67]. However, for nuclei which have A

70, it can be expected

that their low-lying states can be described by nucleons that occupy primarily the p and
f shell-model orbitals. Still, in higher-lying excited states these nuclei may show evidence
of the g9/2 orbital [68] as nucleons are promoted out of the p and f orbitals. Studying the
excited states of 70 As and looking for evidence of collective behavior can therefore provide
information about the structure of these states.
Another aspect of 70 As that needs to be considered is that it is an odd-odd nucleus.
Without collectivity, an odd-odd nucleus can be described primarily by the coupling between
the last proton and last neutron which are then coupled to the rest of the nucleons in the
form of an even-even core. In such an arrangement, the excited states of the nucleus are
built primarily upon the interactions of the two odd nucleons, and so the transitions between
these excited states can be expected to display a strongly single-particle character. On the
other hand, for states that are at sufficiently high excitation energy, the odd particles may
be promoted into the g9/2 orbital, which might be expected to cause the nucleus to evolve
collective characteristics as mentioned above. Therefore, it is also of interest to understand

83

how nuclei evolve with increasing excitation energy, as well as changing nucleon number.
There have been a number of previous studies performed on 70 As that have provided
information about its nuclear structure. In addition to identifying many excited states and
assigning spin and parity values to them, some studies have sought to interpret the structure
of the nuclear states in terms of the excitations of the two odd particles. It was found that
the ground state can be described by placing both the odd proton and the odd neutron into
the f5/2 orbital, and several excited states were described in terms of the coupling of the
odd proton and odd neutron in the f5/2 , p3/2 , p1/2 , and g9/2 orbital space [69, 70]. This
would lend support to the premise that 70 As does not exhibit strong collective features.
Indeed, in Ref. [70] it was found that the low-lying states are well-described by a spherical
shape and in Ref. [69] there was no evidence found for collective behavior when the odd
nucleons are not both in the g9/2 orbital. However, a later study found that for the 11+
1
and 13+
1 states there is evidence of strong collectivity based on the lifetime measurements of
these states [71]. This would indicate that at some point the structure of 70 As changes from
non-collective to collective. The purpose of the study presented in this chapter is to provide
further information about this change in structure.
The two states under investigation in the analysis presented here are the yrast (that
is, lowest-lying) 8+ and 9+ states in 70 As. In both Refs. [69] and [70] the 9+ state was
suggested to belong to the configuration where both the odd proton and the odd neutron
are in the g9/2 orbital. This claim was based upon the observation that the level energies
of the 29

+

states in the neighboring odd-A nuclei sum to roughly the energy of the 9+

+
+
state in 70 As (i.e. E(69 Ge, 92 ) + E(69 As, 92 ) = 397.9 keV + 1306.7 keV = 1704.6 keV [72],

compared to E(70 As,9+ ) = 1752.2 keV [73]). This would indicate that the 9+ state behaves
similarly to the lower-lying states which arise from the coupling of the valence nucleons,
84

in that it behaves as an excitation of the two odd nucleons to the g9/2 orbital rather than
collectively as the higher-lying states do. The 8+ level was proposed to also belong to this
configuration (referred to as πg9/2 νg9/2 hereafter) based on the fact that a strong M 1 branch
was observed to connect the 9+ and the 8+ states, which could suggest that the two states
have a similar structure since the overlap of their wavefunctions would be large. However,
more experimental information can help to set these assignments on firmer ground.
In order to investigate whether the 8+ and 9+ states do belong to the πg9/2 νg9/2 configuration, the lifetimes of these two states were measured. From this, the strengths of the
transitions B(M 1; 9+ → 8+ ) and B(E1; 8+ → 7− ) can be extracted in order to discuss the
structure of these states. As a prediction, the strength of the 9+ → 8+ transition should be
on the order of unity in single-particle units if the assignment of these states to the πg9/2 νg9/2
configuration is correct, since only the last odd nucleons should be participating in the transition. On the other hand, to make a prediction about the strength of the 8+ → 7− transition,
some knowledge of the 7− state is necessary. The stretched configuration πf5/2 νg9/2 can
account for the spin-parity of the state, and the measured magnetic moment of the state
agrees well with theoretical calculations [70, 74]. Therefore, the assignment of this state to
the πf5/2 νg9/2 configuration can be taken with some confidence. However, this means that
the 8+ → 7− transition, which has a dipole character, must connect two states in which the
proton makes an angular momentum change of 2 between the f5/2 and g9/2 orbitals, which
is forbidden. Given that the transition occurs, it means that if the odd-particle configuration assignments of these two states are accurate then the transition is strongly hindered
and proceeds because of small admixtures of other configurations. A similar situation for the
starting and ending configurations occurs in the neighboring odd-odd isotope 72 As, where
the strength of the 8+ → 7− transition is known to be 1.22(16) × 10−5 W.u. since the
85

lifetime [75] and the E1 character of the γ ray [76] are known. This is a strongly hindered
decay [77], consistent with the proton having to make a ∆j = 2 transition via a forbidden
dipole decay. Therefore, if the 8+ → 7− decay has a strength which is similarly hindered, it
will support the assignment of the 8+ state to the πg9/2 νg9/2 configuration.

4.2

Experimental details

The experiment that produced the data for this analysis was run in December of 2010 at the
NSCL. The primary beam from the cyclotrons was a standard 78 Kr beam with an energy of
150 MeV/nucleon and an intensity of 25 particle nano-Amperes (pnA). This primary beam
was impinged upon a 9 Be target to produce 78 Rb via a nuclear charge exchange reaction,
rather than the typical fragmentation reactions used at the facility, which was necessary to
fulfill other science goals in the experiment. The 78 Rb was selected in the A1900 fragment
separator, using an aluminum wedge of thickness 240 mg/cm2 and a longitudinal momentum
acceptance of 0.5%. This resulted in a secondary beam that was approximately 70% 78 Rb,
with an energy of 101.6 MeV/nucleon and a rate of about 1 × 105 particles per second.
Figure 4.1 shows the different species present in the secondary beam after purification in the
A1900, identified by their time of flight between the extended focal plane (xfp) scintillator
in the A1900 focal plane box and the object scintillator in the S800. Software gates were
used to remove the remaining components besides 78 Rb in the offline analysis. This beam
was then transported to the S800 vault for the measurement.
Once the beam reached the experimental station in the S800 vault, further reactions were
necessary to produce the 70 As nuclei in the appropriate excited states. For this purpose, a
second 9 Be reaction target of thickness 376 mg/cm2 was placed at the S800 target position

86

1200

Txfp (arb. units)

1100

78

104

Rb

1000
3

77

900
800

76

10

Kr
102

Br

700
10
600
500
-600

-500

-400

-300

-200

-100

0

1

Tobj (arb. units)
Figure 4.1: The composition of the secondary beam after purification in the A1900 fragment
separator, distinguished by their time of flight between the A1900 extended focal plane (xfp)
scintillator on the vertical axis and the S800 object (obj) scintillator on the horizontal axis.
78 Rb is the primary component, with smaller contributions from 77 Kr and 76 Br. These latter
components were removed with software gates in the analysis.
to induce further reactions in the secondary beam. The beam particles that were within the
acceptance of the spectrograph (which was operated in focused mode for this experiment,
resulting in a large momentum acceptance of about ±2.5%) were then bent through the
analyzing dipole magnets to the S800 focal plane. In general, many species of particle will
be present in the focal plane detectors. They can be identified by correlating their time of
flight from one of the scintillators in the beamline (the obj scintillator in this case) with
their energy loss in the S800 ionization chamber. As shown by Eq. 3.6, the energy loss of
an ion in a material is proportional to the square of the charge of the ion, Z 2 , which gives

87

3

2300

10
78

2200

Rb
76

∆E (arb. units)

2100

Rb
102

2000
1900
1800

70

As

10

1700
1600
1500

1400

1450

1500

1550

1600

1

Tobj (arb. units)
Figure 4.2: A plot of the energy loss ∆E vs. time of flight Tobj of each particle used to identify
the various nuclei produced in the reaction of the secondary beam. The energy loss separates
elements vertically according to Z 2 while the time of flight separates isotopes horizontally
according to A/Z (note that for this plot, the horizontal axis has been reversed, so that
increasing time of flight corresponds to decreasing A/Z). This results in the slanted lines
shown. Stepping right along a “horizontal” line corresponds to removing one neutron per
locus, as shown by the labeled Rb isotopes. Moving vertically downward along a “column”
corresponds to removing one proton and one neutron per locus. Thus, moving the four loci
from 78 Rb to 70 As in the figure corresponds to removing four protons and four neutrons.
elemental separation. Meanwhile, the time of flight Tobj is related to the ratio of the mass
to the charge:
Bρ =

m
A
A L
v≈ v=
,
q
Z
Z Tobj

(4.1)

A , which gives isotopic separation. Correlating the energy loss with the time of
or Tobj ∝ Z

flight therefore uniquely identifies the nuclei that enter the S800 focal plane. Figure 4.2 shows
the particle identification for this experiment with several nuclear species labeled, including
70 As.

Software gates were again used to select only the nucleus under study.

88

(13+ )

1342.7

4075.6 keV

11+
(10+ )
791.3

827.7

1752.21 keV
1676.11 keV
1454.3 keV
743.0

221.8

788.3

9+
8(+)

II
76.1

980.7

I

903.8

2732.9 keV
2579.9 keV
2467.5 keV

933.16 keV
887.76 keV

7(−)

Figure 4.3: A partial level scheme for 70 As, showing the levels which are relevant to this
study. Level energies are shown on the right, while spins and parities are shown on the left
(parentheses indicate tentative assignments), and the arrows indicate transitions between
levels of the associated energy. Two transitions are enclosed in boxes, indicating that these
were used as gating transitions in the γ-γ analysis, described in detail in the text. Figure is
from [64].
The deexcitation γ rays coming from the excited nuclei produced by reactions in the
target were detected by SeGA in its barrel configuration (see Sec. 3.2.2.2). The lineshape
method was used to extract the lifetimes of the states of interest in 70 As. To do so, γ-γ
coincidence data were used, despite the large reduction in statistics that this imposes on
the data. The reason for this is that the level scheme of 70 As is both dense and highly
interconnected as illustrated by Fig. 4.3, which shows the portion of the excitation scheme of
70 As

relevant to this study. As stated earlier, the two quantities of interest to this study are

the 8+ and 9+ state lifetimes. However, inspection of Fig. 4.3 shows that multiple higherlying states decay to both of these levels. This presents a problem because only some of

89

these higher-lying states have known lifetimes. If any of these states should have a lifetime
which is comparable to or longer than the ones they feed, then the lifetimes of the 8+ and 9+
states measured here would be systematically higher than the true values. Therefore, gates
were placed on the γ-ray spectra such that only photons coincident with a particular state
with a sufficiently short lifetime were present in the spectra, thereby ensuring that feeding
from long-lived states is removed.
To measure the 8+ state lifetime, the 8+ → 7− transition was analyzed in coincidence
with a γ ray from the 10+ → 8+ transition. This transition was chosen because it was
observed to be present in the experiment and, more importantly, there is good reason to
expect that the 10+ state has a short lifetime. Although the 10+ state lifetime in 70 As is
unknown, it is reasonable to expect that the 10+ → 8+ transition strength will be similar
to that in the neighboring arsenic isotopes. Therefore, the 10+ state lifetime in 70 As can
be estimated based on the strength of the 10+ → 8+ transition in 72 As, which is known to
be B(E2; 10+ → 8+ ) = 67 W.u [78]. Under the assumption that the strength of this decay
is identical in 70,72 As, this leads to an estimated lifetime of τ10+ = 0.73 ps, which is well
below the sensitivity of this measurement and can be neglected. The possible impact of this
estimation is discussed further in the discussion of the results of this measurement, Sec. 4.5.
Therefore, analyzing the lineshape of the 8+ → 7− transition when the 8+ state is fed by
the 10+ state should ensure that the lifetime determined from this measurement is free from
feeding effects.
The 9+ state was analyzed in a similar fashion to the 8+ state by analyzing the lifetime
of the 9+ state in coincidence with a γ ray from another state. However, several of the
experimental conditions made a direct measurement of the 9+ → 8+ decay lineshape difficult.
As can be seen in the spectra in Sec. 4.3, the threshold of the SeGA detectors was set such
90

that the transition was just on the edge of the range of energies that could be detected, and
below the threshold for the forward detectors. Since a portion of the photopeak then lies
over the part of the spectrum which is rapidly decreasing at the threshold, the shape of the
peak is no longer reliable because the background shape cannot be accurately characterized.
It was also observed during the experiment that there was a strong background component
at roughly the same energy as the 76.1-keV transition depopulating the 9+ state. This
corresponds to a photon from radiative electron capture (REC) [79, 80], which is the inverse
process of the photoelectric effect in which the projectile nucleus captures an electron from
the target and emits a photon. The photon energy in the beam frame is the sum of the
electron K-shell binding energy Eb in the projectile and the kinetic energy of the electron in
the beam frame:
EREC =

1
1 − β2

− 1 me c2 + Eb .

(4.2)

Since REC can have a cross section as high as tens of barns (see Fig. 1 of Ref. [79]), there is a
significant chance for this process to be due to the 78 Rb beam before it reacts to form 70 As,
or to the reacted beam itself. Therefore, the upper bound of REC energies corresponds to
78 Rb

as it enters the target (traveling at β = 0.43) and the lower bound to 70 As as it leaves

the target (β = 0.34). Eb can be approximated by assuming that the binding energy is the
same as a hydrogen-like atom of the appropriate species, so that the range of energies in
which an REC photon can be emitted is between 47 keV and 74 keV in the projectile frame.
Considering the 9+ → 8+ photon has an energy of 76.1 keV, it will not likely be possible to
distinguish it from the REC photopeak. Therefore, instead of measuring the shape of the
9+ → 8+ transition, it was decided to use the photon depopulating the 9+ state as a gating
condition and evaluate its effect on the lineshape of another transition which the 9+ state

91

feeds. The 8+ → 7− transition was the natural choice since it had already been measured
in this work. With a sufficiently long lifetime, the feeding from the 9+ state would cause
the 8+ state to appear to have a longer lifetime than that measured in coincidence with the
10+ → 8+ transition. With the 8+ state lifetime already measured, any remaining effect
would then be attributable to the 9+ state and allow its lifetime to be determined.

4.3

γ-ray spectra

With the identification in the previous section of the particles entering and exiting the
target, the γ rays that were detected in SeGA could be filtered to select only those that
were detected in coincidence with 78 Rb particles incident on the target and 70 As fragments
exiting the target. This insures that only the γ rays coming from excited states in 70 As are
present in the analysis, so that transitions from other nuclei do not influence the lifetime
measurement. These γ-ray spectra will be discussed in this section, both the singles spectra
(those on which the only condition placed is coincidence with the incoming and outgoing
beams) and the coincidence spectra (those which are gated on the presence of a specific γ
ray).
The singles γ-ray spectra for 70 As are shown in Fig. 4.4, split into forward and backward
rings and corrected for Doppler shifting. While the singles spectra may be problematic for
measuring lifetimes in the present case (for the reasons discussed in Sec. 4.2), they have the
highest statistics and are therefore useful for the purpose of identifying the various γ rays
present. It is also important to inspect the singles spectra carefully before deciding upon
the conditions for generating the coincidence spectra, as it is important to choose a gating
condition which maximizes the number of true events in the gate while excluding as many

92

Counts / 4 keV

(a)

70

As backward

4000

+

1000
800

+

REC, 9 → 8

11+ → 9+

600

3000
-

400

+

21 → 11

2000

800

3+4 → 2+2

1000
0
0
(b)

200
70

400

+

600

800

Eγ (keV)

As forward

1000

-

800

+

21 → 11

2000
1500

600
+

1000

-

8 →7

2500

Counts / 4 keV

-

8+ → 7
10+ → 8+

1000

1200

1400

-

8+ → 7
10+ → 8+
11+ → 9+

400

+

34 → 22

+

-

8 →7

1000

800

1000

500
0
0

200

400

600

800

Eγ (keV)

1000

1200

1400

Figure 4.4: The γ-ray singles spectrum in coincidence with 70 As fragments detected in the
S800, with panel (a) showing the γ rays detected in the backward ring and (b) those detected
in the forward ring. Several prominent transitions are identified by arrows, including the
8+ → 7− at 788.3 keV which has a noticeable asymmetry due to lifetime effects. The strong
peak at low energy in the backward ring is due to both the 9+ → 8+ transition and the REC
process described in Sec. 4.2, which cannot be resolved in this experiment. As mentioned,
REC can have a very large cross section, which explains the large amplitude of this peak.
In each panel, the inset shows the same spectrum focused around the 788.3-keV γ ray from
the 8+ state as well as weakly populated γ rays from the 10+ and 11+ states, which were
important for choosing the coincidence gates. Figure is modified from [64].

93

accidental coincidences as possible. Inspection of Fig. 4.4 shows several places where clear
peaks are present in the γ-ray spectrum, including the 8+ → 7− peak and the peak which
is a combination of the REC process mentioned earlier and the 9+ → 8+ transition. The
insets to the figures show the peak from the 8+ state along with two transitions from the
10+ and 11+ states, which are only weakly populated. These states feed the 8+ and 9+
states, which demonstrates the necessity of using γ-γ coincidence.
The coincidence spectra were generated by requiring that a γ ray within a given energy
range corresponding to a particular transition be present and then plotting all other γ rays
that were detected in the same event. The determination of the coincidence condition for the
9+ state lifetime measurement was straightforward, as the peak from the 9+ state lay within
the large peak at low energy in the backward ring singles spectrum. Therefore, the condition
for plotting γ rays for the coincidence spectrum was that a photon was detected in the range
covered by this combination of the 9+ → 8+ and REC peaks in the backward ring. This
gating condition can include events where REC satisfies the gating condition and creates
false coincidence. However, since REC is an atomic process and not related to the nuclear
states, it is completely uncorrelated with any nuclear transition and should contribute only a
random background. The resulting γ-ray spectra are shown in Fig. 4.5, where the upstream
and downstream rings have once again been plotted separately. Several transitions besides
the 8+ → 7− have been tentatively identified in this figure, but they are not related to the
transition in question. In addition, it can be seen that the backward ring of detectors suffers
from poor statistics and the shape of the 8+ → 7− peak is not well-defined. Because of this,
the backward ring was not able to be fit reliably and was excluded from the analysis and the
lifetime was extracted from the forward detectors only.
The placement of the gate for generating the spectra for measuring the 8+ state lifetime
94

70

200 (a)

Counts / 16 keV

+

+

As backward, 9 coincidence

+

31→21

(-)

-

6 →51
(5) →51

150

1

+

-

8 →7

100

50

0
0

200

400

600

800

1000

1200

1400

Eγ (keV)
70

200 (b)

Counts / 16 keV

(-)

+

As forward, 9 coincidence

-

6 →51
(5) →51

150

+

-

8 →7

1

100

50

0
0

200

400

600

800

1000

1200

1400

Eγ (keV)
Figure 4.5: The coincidence γ-ray spectra for the lifetime measurement of the 9+ state
in 70 As, Doppler corrected and split into (a) the backward ring and (b) the forward ring.
The gating condition is that there is a photon within the energy range covered by the
combined REC and 9+ → 8+ transition peak. Several decays are labeled that can come in
the coincidence gate but are not related to the decay of the 9+ state. As can be seen in
the figure, due to poor statistics the shape of the 8+ → 7− peak is not well-defined in the
backward ring and was not able to be fit reliably. For this reason, only the forward ring was
used for analysis.

95

was more difficult to determine due to the much lower statistics in the 10+ → 8+ transition
at 903.8 keV. In the end, the placement of the gates was chosen to require either a photon
with an energy between 878 keV and 938 keV in the forward ring or a photon with an
energy between 886 keV and 920 keV in the backward ring, as this was observed to give
the best signal-to-noise ratio for the 8+ → 7− transition. This choice was also validated
later by comparing to the simulated peak shape for the 10+ → 8+ transition (see Sec. 4.5).
The resulting γ-ray spectra are shown in Fig. 4.6, again split into forward and backward
detectors. As with the coincidence spectra for the 9+ state, the backward ring spectrum
has very low statistics for the 8+ → 7− transition, which does not have a clear shape for
the peak. Therefore, as for the analysis of the 9+ state lifetime, only the forward ring of
detectors were used to analyze the lifetime of the 8+ state.
Finally, given that the signal-to-noise ratio is not very high in the singles spectra, it
can be asked whether the signal for the 8+ → 7− transition observed in the coincidence
spectra is due to true coincidence or accidental coincidence with background events. To
evaluate the false coincidence rate, spectra were generated with gates beside the true gates
that were specifically placed in locations where no coincidence with the 788.3-keV γ ray
from the 8+ state is expected. If a signal is observed in these spectra at 788.3 keV, then
this will correspond to the accidental coincidence rate and can be used to determine whether
it will be significant. These false-coincidence spectra are shown in Fig. 4.7, with the dots
corresponding to the true coincidence data and the cross marks corresponding to the false
coincidence data. In panel (a), the coincidence spectra for the 8+ lifetime measurement are
shown, and it is clear that while the background in both spectra are comparable, there is
only a signal present in true coincidence data. The background spectra were generated by
putting a gate on the energy spectrum around 1400 keV, with the same width as the gate on
96

Counts / 16 keV

(a)

70

+

As backward, 8 coincidence

200
150
100

-

8+→7

50
0
0

200

400

600

800

1000

1200

1400

Eγ (keV)

Counts / 16 keV

(b)

70

+

As forward, 8 coincidence

200
150
-

8+→7
100
50
0
0

200

400

600

800

1000

1200

1400

Eγ (keV)
Figure 4.6: The coincidence spectra used to determine the 8+ state lifetime in 70 As, split
by photons detected in (a) the backward ring detectors and (b) the forward ring detectors.
In contrast to the coincidence spectra for the 9+ state lifetime, there are no other obvious
transitions observed. The one bin which goes beyond the y-axis range in panel (a) is due to
accidental coincidence from REC which is not important in this figure and had a bin content
of 451 counts, and so the axis range was set to emphasize the rest of the spectrum. The
8+ → 7− transition is labeled, and as for the 9+ coincidence spectra, the peak shape is only
well-defined in the forward detectors, and so the backward ring was not used in the lifetime
analysis of the 8+ state either.

97

Counts / 16 keV

(a)

Gated on 903 keV

200

Gated on background

150
-

8+→7
100
50
0
0

200

400

600

800

1000

1200

Eγ (keV)
(-)

Counts / 16 keV

(b)

-

6 →51

Gated on 76 keV

(5) →51
1

200

Gated on background
-

8+→7

150
100
50
0
0

200

400

600

800

1000

1200

Eγ (keV)
Figure 4.7: A comparison of the true coincidence rate with the background coincidence rate
for (a) the 10+ → 8+ coincidence spectrum for the 8+ lifetime analysis and (b) the 9+ → 8+
coincidence spectrum for the 9+ state lifetime analysis. In both panels, the dots represent
the true coincidence spectrum and the crosses represent the coincidence with background,
and only data from the forward ring are shown. For both cases, the true coincidences show
a clear presence of the 8+ → 7− transition while the accidental coincidence spectra do not
exhibit a signal from this transition. Figure is adapted from [64].

98

the true coincidence spectrum. A gate closer to the actual energy would have been used, but
several transitions that feed the 8+ state surround the 10+ → 8+ transition (see Fig. 4.3),
and so a gate at higher energy had to be chosen to evaluate the accidental coincidences.
Panel (b) shows a similar comparison for the 9+ state lifetime analysis, and once again only
the true coincidence spectrum shows a signal from the 8+ → 7− transition, as well as a peak
from other transitions as in Fig. 4.5. This false coincidence gate was set just above the true
coincidence gate to try to use background as similar as possible to the true coincidence gate,
and also shows no signal near the 788.3-keV γ ray from the 8+ state. This shows that the
gates used for this analysis do in fact select events which are in true coincidence with the
8+ → 7− transition, as opposed to being in coincidence with the background. Normally, the
background spectra would be subtracted from the true coincidence spectra in order to have
a cleaner signal, however, it is usual to subtract a background spectrum which is an average
of background gated just above and just below the transition of interest to try to minimize
the impact of choosing any particular location to characterize the background. In this case,
since the background in panel (a) was taken at significantly higher energies, and that in
panel (b) could not be taken below the peak due to the threshold settings, no background
subtraction was performed, and the background spectra are presented to demonstrate the
quality of the gates.

4.4

Lifetime analysis

With the γ-ray spectra generated, the analysis of the lifetime of the 8+ and 9+ states could
be performed. This was done by using Geant4 simulations (see Sec. 3.3) to reproduce the
experimental γ-ray spectrum under the assumption of a range of different lifetimes for the

99

state being studied, and then fitting these simulated spectra to the real ones. In doing so,
a reduced χ2 distribution was generated as a function of the lifetime, and the minimum of
this function was taken to be the lifetime. For this dataset, since coincidence data were
used to determine the lifetimes, simulated coincidence spectra were also generated for use in
the fitting procedure. The remainder of this section will describe the preparation of these
simulations and the data analysis itself.

4.4.1

Geant4 simulations

In order to accurately reproduce experimental γ-ray spectra, it is important for the Geant4
simulations to mimic as much as possible the conditions under which the experiment was
performed. The first step towards reproducing the experiment is to ensure that the placement of the SeGA detectors in the simulation corresponds to the placement of the physical
detectors. While the geometry of the barrel configuration of SeGA is largely set by the
detector mounting frame, there is some room for adjustment of the detectors. To determine the detector locations as accurately as possible, a device known as a “ROMER arm”
is used. This device is an armature with a pressure-sensitive probe attached to the end.
As the armature is moved, the device tracks the location of the probe in space and records
the coordinates of the probe each time it is pressed against a surface. Known points on the
SeGA detectors are then touched by the probe to record their coordinates, and the location
of the ROMER arm is determined against well-known calibration locations in the lab by
laser survey. These coordinates are then used to generate the simulated SeGA detectors so
that their locations are the same as those used in the experiment.
With the geometry fixed, the next thing to be validated is the γ-ray detection efficiency
of the simulation as a function of photon energy. This is important because the detector
100

20

18

16

16

14

14

Efficiency (%)

Efficiency (%)

18

20

(a) Forward ring

12
10
8
6

12
10
8
6

4

4

2

2

0
0

0
0

200 400 600 800 1000 1200 1400

Eγ (keV)

(b) Backward ring

200 400 600 800 1000 1200 1400

Eγ (keV)

Figure 4.8: Comparison of the simulated and measured efficiency of SeGA for various γ-ray
energies with a 152 Eu source in (a) the forward ring and (b) the backward ring. In both
panels, the simulation is shown by the red triangles while the data is shown by the black
dots. The simulations reproduce well the efficiency of the real SeGA detectors, especially in
the ∼600-1000 keV energy range of interest to this experiment.
efficiency is intimately connected with the photopeak shape. If the simulation were to have
a detection efficiency with a much greater slope as a function of energy than the real detectors, it would lead to an exaggerated low-energy tail and thus produce erroneous lifetime
results. The efficiency can be checked by taking data with a standard radioactive calibration
source with known activity and γ-ray branching ratios and comparing it to the corresponding
spectrum generated by the simulation. The efficiency curve for a 152 Eu source is shown in
Fig. 4.8, divided into the forward and backward rings. The red triangles represent the data
while the black dots represent the simulations. The overall shape of the efficiency curve for
the data is reproduced very well by the simulations, especially for the energy range between
about 600 keV and 1000 keV which is relevant for this analysis. While some small deviation
between the data and simulations may be present, this is ultimately inconsequential, as the
simulations are scaled to match the data in the analysis and this will effectively normalize

101

the simulated efficiency. What is important is that the relative efficiency at various energies
remains constant between the simulation and the data. This figure aptly demonstrates that
this is the case.
With the performance and placement of the SeGA detectors determined, the properties
of the beam in the experiment need to be input into the simulation. In particular, it is
important to fix the physical size and direction of the beam as well as its energy, because
the simulation attempts to apply the same Doppler correction to photons that it generates
as is applied to the γ rays detected in the experiment. Since the real beam is tracked in
the S800 and its reconstructed trajectory at the target used for the Doppler correction, this
same condition needs to be recreated in the simulation as well. This was done by providing
the focus point of the incoming 78 Rb beam as well as its spatial and angular spread in the
dispersive and non-dispersive directions. The energy of the incoming beam and the energy
spread as measured in the experiment are also provided. The reaction mechanism is input
as the change in the number of protons and neutrons, and the momentum change of the
beam is described by a loss of longitudinal momentum according to a Gaussian distribution
with an adjustable width, plus a kick of adjustable strength in a random direction. The
target thickness and density are also provided, and the simulation uses this information to
calculate the energy loss of the ion through the target (see Sec. 3.3). The kinetic energy of the
beam corresponding to the Bρ setting of the S800 is provided as well as the acceptance of the
spectrograph so that the acceptance of the simulation is similar to what is actually observed.
In the end, the simulation outputs the distributions of the angles in the dispersive and nondispersive direction (ata and bta , respectively), the non-dispersive position (yta ), and the
spread in the energy ∆E/E (dta ). Figure 4.9 shows a comparison between the experimental
(black) and simulated (red) properties of the beam. All of the simulated spectra reproduce
102

(b)
Counts (arb. units)

Counts (arb. units)

(a)

-60 -40 -20

0

20 40 60

-60 -40 -20

ata (mrad)

0

20 40 60

bta (mrad)

(c)
Counts (arb. units)

Counts (arb. units)

(d)

-5

0

5

10

15

y (mm)

-10 -8 -6 -4 -2 0

2 4

6 8 10

dta (%)

ta

Figure 4.9: A comparison of the measured beam profile (black) with the simulated one (red).
Panels (a) and (b) show the spread in the angular divergence of the beam in the dispersive
and non-dispersive directions, respectively. Panel (c) shows the spatial distribution in the
non-dispersive direction, and panel (d) shows the spread in the energy of the beam relative to
the energy specified by the Bρ of the S800. All the distributions show very good agreement
with the exception of yta , where the simulation does not reproduce the tails of the data.
However, the position in the non-dispersive direction does not strongly affect the γ-ray
spectrum, and so this level of agreement is sufficient.

103

the data very well with the exception of the yta , which shows a marked deviation from
the tails of the data. This is because the simulation assumes a uniform ellipsoidal beam,
but in reality the beam may have any shape. However, the distribution of the beam in
the non-dispersive direction does not have a strong effect on the γ-ray spectrum, and so
instead of attempting to modify the simulation code to accommodate every experiment, the
approximation of an ellipsoid is used and adjusted to the data as much as possible. The
position in the dispersive direction is not measured, but is usually assumed to be zero in
order to be consistent with the assumption made for the inverse tracking in the S800.
Finally, with the detector and beam properties constrained, the last inputs necessary to
generate the simulated γ-ray spectra are the level scheme, decay energies, and lifetimes. For
this work, the level and γ-decay energies were already determined in previous studies, and
so these parameters were taken from the literature. The lifetime is to be determined here,
and so this final parameter was varied in the simulations. The simulated spectra and the
lifetimes determined from fitting them to the data are presented in the next section.

4.4.2

Lifetime determination

With the Geant4 simulations prepared, the lifetimes of the 8+ and 9+ states were determined
by fitting the simulated γ-ray lineshapes to those observed. First, a preliminary fit with no
feeding was performed on the singles spectrum in order to verify which transitions could be
observed in the spectrum. Then, fits to the coincidence data were performed to determine
the 8+ and 9+ state lifetimes. Finally, a fit to the singles data was performed with full
feeding where both the 8+ and 9+ state lifetimes were varied simultaneously, and this was
used to verify the results of the coincidence fit.
The first step in understanding the γ-ray spectra was to determine which transitions
104

could be observed and to have an estimate of their intensities. To do this, a preliminary
investigation of the γ-ray singles spectrum in the range between roughly 600-1000 keV was
performed by fitting γ-ray spectra for all of the transitions shown in Fig. 4.3 on top of an
exponential background (excluding the 1342.7-keV, 76.1-keV, and 221.8-keV transitions, as
these were outside of the initial fitting range chosen). A transition at 944.9 keV (not shown
in Fig. 4.3) was also added in order to explain the counts observed at that energy, however, it
is otherwise unrelated to the analysis. The results of this fit are shown in Fig. 4.10, where the
upper panel shows the simulated γ-ray spectrum (red) fit to the data (black) in the forward
ring and the lower panel shows the simulation broken up into its individual components.
For this first fit, a lifetime of 150 ps for the 8+ state was observed to give a reasonable
reproduction of the experimental spectrum. However, it should be noted that feeding was
ignored for this first attempt, and so this value should not be taken to be representative of
the true 8+ lifetime. What is important to see from this fit is that there are many γ rays
present in the spectrum and that many of them are significantly overlapping each other. In
particular, the 791.3-keV γ ray is completely unresolved from the 788.3 keV γ ray from the
8+ state, which severely complicates the determination of the lifetime of the 8+ state in the
singles spectrum. The other point to notice about this fit is that it shows that the 903.8-keV
transition from the 10+ state is well-separated from the 827.7-keV and 980.7-keV transitions
that feed the 8+ state through the 9+ state, which makes it ideal as a gating condition. It
does have significant overlap with the 944.9-keV transition mentioned earlier, but as this is
located below the 8+ state and does not feed any of the states of interest to this analysis, it is
inconsequential. This provides confirmation that the placement of the gate used to generate
the 8+ state coincidence spectrum is valid. The dashed vertical lines indicate the placement
of the gate.
105

Counts / 8 keV

(a)

Simulation
Background

1500

1000

500

0

600

700

800

(b)

900

1000

Eγ = 788.3, 743.0 keV

Counts / 8 keV

Eγ = 791.3 keV
Eγ = 903.8, 827.7 keV
Eγ = 944.9 keV

400

Eγ = 980.7 keV

200

0

600

700

800

900

1000

Eγ (keV)
Figure 4.10: A first attempt at fitting the γ-ray singles spectrum from the forward ring. For
this fit, no attempt was made to account for any feeding, and a lifetime for the 8+ state of
150 ps was used. Panel (a) shows the data (black points with error bars) with the fit overlaid
as the solid line in red. The assumed exponential background is shown by the dashed line.
The individual components of this fit can be seen in panel (b). The legend identifies the
various γ-ray energies, which correspond to those listed in Fig. 4.3 except for the 944.9-keV
γ ray, which was included to explain the accumulation of counts at that energy. Otherwise,
the 944.9-keV γ ray is unrelated to the analysis, as the state from which it originates lies
below the 8+ state. The vertical dashed lines denote the location of the gate used to generate
the coincidence spectra for determining the 8+ lifetime. Figure is from [64].

106

Simulation varying τ8+

Counts / 16 keV

200

Simulation with τ8+ = 0 ps
Background

χ2/ndf

Data gated on 903.8 keV γ

2

1
0

100

50

100 150 200 250

τ8+ (ps)

0

400

500

600

700

800

900

1000

1100

Eγ (keV)

Figure 4.11: The results of fitting simulations of the 8+ state with various lifetimes (red
line) to the experimental data for the downstream ring in coincidence with the 10+ → 8+
transition (black points with error bars). The lifetime determined from the fit is τ8+ =
80(28) ps, including only statistical uncertainties. Also shown is a simulation with a lifetime
of 0 ps for the 8+ state for comparison, which demonstrates the sensitivity of the data to
the lifetime. The inset shows the reduced χ2 distribution with five degrees of freedom, the
minimum of which was used to determine the lifetime. Figure is from [64].
In order to determine the lifetime of the 8+ state, simulations were fit to the coincidence
spectrum for the forward detectors shown in Fig. 4.6(b). The simulations were generated
with the lifetime varied in 10 ps steps, and each of these was fit to the data on top of
an exponential background. The background was constrained by fitting on a region that
included the 8+ → 7− peak plus background on either side, then fixing the parameters of the
exponential. The simulations were then fit to the data over a smaller region which included
only the peak area, the results of which are shown in Fig. 4.11. For each simulation, the
χ2 value per degree of freedom was calculated and saved. The minimum of the distribution
of reduced χ2 values (shown in the inset of the figure) was then used to determine the 8+
lifetime as τ8+ = 80(28) ps. In the figure, the data are shown by the black points, the
solid red line shows the simulation generated with a lifetime of 80 ps, and the blue dashed

107

line is a simulation with an 8+ lifetime of 0 ps to show the sensitivity of the peak shape
to the lifetime. The uncertainty of this measurement given in parentheses is only due to
the statistical uncertainty of the fit. Systematic uncertainties were also evaluated and were
mostly due to the assumptions associated with the background, including its shape and
the region over which it was fit, which when varied produced a sensitivity of 6.4% of the
measured lifetime value. Varying the position of the detectors relative to the target in the
simulations produced a 3% variation in the results, consistent with the findings in [32] where
the same setup was used. Finally, the assumption that the 10+ state lifetime is 0.73 ps
also introduces some uncertainty. To evaluate this effect, the lifetime of the 8+ state was
also evaluated under the assumption that the 10+ state lifetime is five times longer (i.e.
3.65 ps, which would already indicate a substantial change in the structure of the 10+ states
between 70,72 As and can therefore be considered unlikely). In this case, the lifetime of the
8+ state in 70 As is reduced by 3.4%. Added in quadrature and combined with the statistical
uncertainty, this gives a final result for the 8+ state lifetime of τ8+ = 80(29) ps.
A similar approach was taken to determine the 9+ state lifetime using the coincidence
spectrum shown in Fig. 4.5(b). Simulations were generated with the 9+ state decaying into
the 8+ state, and the 8+ in turn decaying into the 7− state. The lifetime of the 8+ state
was fixed to 80 ps as measured in this experiment, while the 9+ state lifetime was varied
in 10 ps increments. Since the 827.7-keV γ ray from the 10+ state and the 980.7-keV γ
ray from the 11+ state can both be in coincidence with the 9+ → 8+ decay, simulations
were also generated for these states and included in the fit. As with the 8+ state lifetime
analysis, the background as well as the amplitude of the peaks from the 10+ and 11+ states
were fixed by fitting over a wide region including the 8+ peak, then the simulations were
fit to a region including only the peak itself and a reduced χ2 distribution was generated.
108

Data gated on 76.1 keV γ

Counts / 16 keV

200

Simulation with τ9+ = 0 ps
Background

3

χ2/ndf

Simulation varying τ9+

1
0

100

0

2

50

100 150 200 250

τ9+ (ps)

400

500

600

700

800

900

1000

1100

Eγ (keV)

Figure 4.12: The results of fitting the coincidence spectra gated on the 9+ state for 70 As.
The simulation with the best fit lifetime of 85 ps is shown by the solid red line drawn over
the black data points. As with the 8+ lifetime analysis, a simulation with the 9+ lifetime
set to 0 ps is shown by the blue dashed line to show the sensitivity to the lifetime. The
inset shows the reduced χ2 distribution which was calculated with four degrees of freedom,
the minimum of which was used to determine that the 9+ state lifetime is τ9+ = 85(30) ps,
including statistical uncertainties only. Figure is from [64].
The results of the fit are shown in Fig. 4.12. From the χ2 distribution, the lifetime was
found to be τ9+ = 85(30) ps, where the uncertainty is again only statistical. The sources
of systematic uncertainty were similar to that for the 8+ state lifetime analysis, with the
background constraint contributing 9.4% to the uncertainty, the same 3% uncertainty due
to detector placement, and the assumption of the 10+ state lifetime gives an additional
uncertainty of 4.4%. Since the lifetime of the 8+ state was used in the determination of the
9+ state, its uncertainty was also propagated to the 9+ state lifetime result. Combined in
quadrature, these extra sources of uncertainty give a final result for the 9+ state lifetime of
τ9+ = 85(43) ps.
As a check that the lifetime measured for the 9+ state is not from differences in the
lineshape due to other features of the spectrum, for example the shape of the background, a
109

5

Simulation varying τ8+

4

χ2/ndf

Counts / 16 keV

200

Data gated on 76.1 keV γ

Simulation with τ8+ = 0 ps
Background

3
2
1
0

50

100

0

100

150

200

250

τ8+ (ps)

400

500

600

700

800

900

1000

1100

Eγ (keV)

Figure 4.13: A fit to the 9+ coincidence data using simulations which do not include the
feeding from the 9+ state. This fit should be poor if the measured lifetime is truly due to
the 9+ state. Inspecting the spectrum, it can be seen that the shape of the photopeak is
not reproduced by the simulations under these conditions, which indicates that the 9+ state
lifetime has a significant effect on the 8+ lineshape. The reduced χ2 distribution in the
inset, which was calculated using four degrees of freedom as in Fig. 4.12, does not show a
sharp minimum as in Fig. 4.12 nor is it consistent with the result for the 8+ state lifetime
in Fig. 4.11. This indicates that the 8+ state lifetime alone cannot explain the lineshape of
the peak under these gating conditions. Figure is from [64].
set of simulations without feeding from the 9+ state was generated and fit to the coincidence
data gated on the 9+ state. The results of this fit are shown in Fig. 4.13. Under these
conditions, the shape of the 8+ → 7− transition cannot be adequately reproduced for any
lifetime of the 8+ state. In addition, the reduced χ2 value is considerably worse than the
fit including feeding from the 9+ state. This indicates that the measured lifetime is in fact
attributable to the 9+ state.
As a final check for consistency, the 8+ and 9+ lifetimes were analyzed by varying them
simultaneously in the simulations and fitting the resulting spectra to the γ-ray singles spectrum. The full feeding of every level to the 8+ and 9+ levels was implemented for this fit
to ensure full consistency. The result of this fitting procedure is shown in Fig. 4.14, where
110

Counts / 8 keV

2000

(a)

1500
1000
500
0

600

700

800

900

1000

1100

Eγ (keV)
800

Eγ = 788.3, 743.0 keV

τ9+ (ps)

Counts / 8 keV

(b)

Eγ = 76.1 keV
Eγ = 791.3 keV
Eγ = 903.8, 827.7 keV

600

150
100
50

Eγ = 944.9 keV

0
1 1.2
χ2/N 1.2
1
0

Eγ = 980.7 keV

400

Total fit

200
0

600

700

800

900

1000

50

100
τ8+ (ps)

150

1100

Eγ (keV)

Figure 4.14: The results of a fit in which both the 8+ and 9+ state lifetimes are varied
simultaneously. Panel (a) shows the resulting fit to the data, while panel (b) shows the
individual simulations involved in the fit. Also shown in the inset to panel (b) is the 2dimensional reduced χ2 surface along with its projection onto the axes for the 8+ and 9+
state lifetimes. The minimum value is χ2 /N ≈ 1.15 at τ8+ ≈ 100 ps and τ9+ ≈ 80 ps,
in agreement with the coincidence fits. The number of degrees of freedom for this fit is 58,
which reflects the fact the fit is over a much wider energy range. For reference, the maximum
value given by the red color in this contour plot is χ2 /N = 2. Figure is adapted from [64].
the top panel shows the summed simulations (red line) overlayed on the data points for the
downstream ring of detectors as well as the exponential background (dashed line). As shown
in the inset to panel (b), a 2-dimensional χ2 surface was generated from these fits and the
projection of this surface onto the 8+ and 9+ χ2 axes are also shown. Two minima can be
seen in the inset, one at τ8+ ≈ 0 ps and τ9+ ≈ 160 ps, and another at τ8+ ≈ 100 ps and
τ9+ ≈ 80 ps. The coincidence fits clearly exclude the possibility of τ8+ = 0 ps, but the other
minimum is fully consistent with the conclusions drawn from the coincidence fits and serves
111

as a validation of these results.

4.5

Results and discussion

In order to interpret the results of the measurements reported in this chapter, the lifetimes of
the 8+ and 9+ states should be converted into reduced transition strengths. The transition
strengths contain the nuclear structure information of the states involved in a transition and
therefore they are useful for understanding the results of a measurement and comparing to
the structure of nearby nuclei. For the 8+ state, the decay that was measured has a primarily
E1 character with a small M 2 admixture (δ = 0.017(13)) [73]. Knowing this, the transition
strength can be calculated to be B(E1) = 1.3(5) × 10−5 e2 fm2 . In single-particle units, this
corresponds to 1.2(4)×10−5 W.u. The 9+ → 8+ transition, on the other hand, has an M 1
character with a small E2 admixture (δ = 0.01(3)) [73], and so the transition strength is
B(M 1) = 1.5(8) µ2N , or 0.85(42) W.u. These results can now be used to provide insight
into the nature of the 8+ and 9+ states, and specifically whether they can be described by
a πg9/2 νg9/2 configuration.
As discussed in Sec. 4.1, if the 8+ state has a πg9/2 νg9/2 configuration, then it is to be
expected that the 8+ → 7− transition is strongly hindered. As explained, there is good
reason to believe that the 7− state has a πf5/2 νg9/2 main configuration, and so these two
states cannot be connected by an electric dipole transition since the proton must change its
angular momentum by 2 between the f5/2 and g9/2 orbitals. Therefore, if the transition
proceeds it must be through small admixtures of other minor configurations (e.g. excitations
of the core) which are allowed, and therefore the overall strength must be very hindered.
To gauge the hindrance of this decay, it is helpful to compare its strength to that of other

112

Figure 4.15: A diagram showing the distribution of transition strengths for several types
of electromagnetic transitions, divided into several different mass regions [77]. The central
set of distributions is for E1 transitions, and unfilled histogram is the relevant one for 70 As.
This histogram includes transition strengths in the range from roughly 10−6 to 10−2 W.u.
With a strength of B(E1; 8+ → 7− ) = 1.2(4) × 10−5 W.u., it is clear that this transition in
70 As is relatively hindered.
E1 transitions in this mass region. Figure 4.15 shows a figure taken from Ref. [77] which
contains several histograms for different transitions in different areas of the nuclear chart. Of
interest for the present discussion is the unfilled area of the central collection of histograms in
this figure, which corresponds to E1 transitions in the mass A = 45 − 90 region. Inspection
of this distribution shows that the B(E1; 8+ → 7− ) value measured in this work is indeed
on low end of this distribution, indicating that it is a hindered transition. This supports the
assignment of the 8+ state to the πg9/2 νg9/2 configuration.
A more direct method of evaluating the configuration of the 8+ state in 70 As was also
mentioned in Sec. 4.1, which was comparison with the strength of the 8+ → 7− transition

113

in the neighboring isotope 72 As. In this nucleus, it has been established already that the
8+ state is a member of the πg9/2 νg9/2 configuration, and this state also has a decay to
a 7− level with a πf5/2 νg9/2 configuration, just as is believed to be the case in 70 As. As
mentioned at the beginning of this chapter, this decay has a known strength of B(E1) =
1.22(16) × 10−5 W.u. The measurement of the 8+ → 7− strength in 70 As in this work of
B(E1) = 1.2(4) × 10−5 W.u. is fully consistent with that measured in 72 As. Since it can
be expected that neighboring isotopes have similar structure, this is another indication that
the 8+ state in 70 As does in fact belong to the πg9/2 νg9/2 multiplet of states.
Evaluating the structure of the 9+ state relied upon interpreting the M 1 strength of the
transition between the 9+ and 8+ states. In Sec. 4.1, it was predicted that if the 9+ state
belongs to the πg9/2 νg9/2 configuration (along with the 8+ state), then the strength of the
transition should be about 1 W.u. in terms of order of magnitude because only the last odd
nucleons should be participating in the transition. Indeed, the strength that was measured
was 0.85(42) W.u., which is consistent with this interpretation. However, a more quantitative prediction can be made by employing a simple model calculation for the M 1 strength.
For this purpose, a model was chosen that has been used to describe 70 As previously [70],
specifically its lower-lying states. The model itself describes the nucleus as composed of
two quasiparticles (a quasiproton and a quasineutron) coupled to an even-even core which
has a vibrational character [81]. This model is a convenient choice, because it allows the
M 1 strength of a transition between two states in the same quasiparticle multiplet to be
calculated analytically with the equation [81]

B(M 1; Ii → If ) =

3(jp + jn + Ii + 1)(jp − jn + Ii )(−jp + jn + Ii )(jp + jn − Ii + 1)
4πIi (2Ii + 1)
×

µp
µn 2
−
,
2jp 2jn
114

(4.3)

where in this expression Ik represents the spin of the nuclear state k, jp,n the angular
momentum of the proton and neutron, respectively, and µp,n the magnetic moment of the
proton and neutron, respectively. It is important to note that this expression assumes that
the initial state is the higher spin state, and if it is not then all Ii must become If in
the numerator of Eq. 4.3. In Ref. [81], the term in the magnetic moments is evaluated by
assuming a quenching of the spin-g factor for free nucleons of 0.6, such that

µp
µn
1
−
=
2jp 2jn
2

1+

2.35
2.29
+
2lp + 1 2ln + 1

µN .

(4.4)

With this, and inserting jp = jn = 92 and Ii = 9, this model gives a predicted M 1 strength
for the 9+ → 8+ transition of B(M 1) = 1.2 µ2N , or equivalently 0.69 W.u. This is very
much in line with the original prediction of ∼1 W.u. transition strength, and also consistent
with the measured strength of 0.85(42) W.u. This model therefore supports the assignment
of the 9+ state to the πg9/2 νg9/2 multiplet.
The consistency of the strength of the 9+ → 8+ transition in 70 As with other nuclei in the
region can also be evaluated, similar to the manner done with the 8+ state. While there is
no 9+ → 8+ transition with measured strength in the neighboring arsenic isotopes, isotopes
of other nearby elements do have such transitions with measured strengths, and these can
be used for comparison with 70 As. The 9+ → 8+ strengths of transitions in several nuclei
including 70 As are shown in Fig. 4.16. These nuclei were chosen because they have a known
B(M 1; 9+ → 8+ ) and also because they all have the same isospin as 70 As, in order to try
to make as direct a comparison as possible. They also all originate from a 9+ state found
to be a member of the πg9/2 νg9/2 multiplet. As can be seen, all of the nuclei shown in
Fig. 4.16 have a very similar strength as that measured in 70 As except for 66 Ga. This data

115

N

B(M1;9+ → 8+) (µ2 )

10

70

As

1
74

-1

10

82

Br

78

Rb

Y

10-2
10-3
10-4

66

Ga

-5

10 64

66

68

70

72

74

76

78

80

82

84

A

Figure 4.16: A plot of B(M 1; 9+ → 8+ ) of several nuclei close to (and including) 70 As, as
well as the strength predicted by the particle-vibration coupling model expressed in Eq. 4.3.
All of the isotopes shown in the figure are believed to have a 9+ state belonging to the
πg9/2 νg9/2 configuration. Most of the transition strengths in the figure are similar to that
measured in 70 As in this work, which indicates that 70 As also has a 9+ state which is a
part of the πg9/2 νg9/2 multiplet. The outlier is 66 Ga, which has a significantly smaller
strength. However, this point is tentative because it is not certain that the decay is of an
M 1 nature, and this may explain the discrepancy. Figure is from [64], and data are taken
from [82, 83, 84, 85].
point is, however, tentative, in that it is known that it is a dipole transition [86] but not
necessarily that it is of M 1 character. The apparent discrepancy of this point with the others
in Fig. 4.16 may therefore lie in the fact that it could be an E1 transition. It could also be
that the transition is of M 1 character, but that the 9+ state of 66 Ga is not well-described
by a πg9/2 νg9/2 configuration. More information would be necessary to make a statement
about this situation. The remaining points in Fig. 4.16, however, show comparable strengths
to that measured in 70 As, which indicates that it shares with them a similar structure. The
dashed line indicates the strength calculated from the particle-vibration coupling model
discussed earlier for comparison, which also demonstrates that it does a reasonable job of
reproducing the strengths of these nuclei. Overall, this comparison further supports that the
9+ state of 70 As belongs to the πg9/2 νg9/2 configuration.

116

Chapter 5
Lifetime Measurements in 74Rb
The second lifetime study presented in this work utilizes the Differential Recoil Distance
Method (DRDM). As mentioned in Sec. 2.3, this is the first time that the DRDM has been
successfully implemented, due in large part to the superior energy resolution of the nextgeneration γ-ray detector array GRETINA (Sec. 3.2.2.3). The physics goal of the study was
74
to measure the lifetime of the 2+
1 state of 37 Rb37 , thereby making this the heaviest odd-odd

N = Z nucleus with a measured B(E2). This nucleus can be produced by a unique reaction
path involving charge exchange reactions at intermediate energies [32, 87] and therefore
has a small production cross-section. Thus, it was anticipated that the statistics for this
measurement would be low. However, as discussed in Sec. 2.3, the DRDM is well-suited to
such measurements as it requires fewer counts to make a lifetime measurement compared to
the well-known (non-differential) Recoil Distance method. The 74 Rb 2+
1 lifetime therefore
provides an excellent case to demonstrate the utility of the DRDM. However, unlike the γ-γ
coincidence lineshape analysis performed in Ch. 4, the DRDM produces several Dopplershifted components for each deexcitation γ ray, and therefore requires a relatively clean
spectrum with a wide separation between γ-ray decay energies compared to the dense level
scheme of 70 As. It is also sensitive to lifetimes considerably shorter (a few picoseconds)
than would normally be accessible to the lineshape method (several tens of picoseconds)
under the typical fast beam conditions at the NSCL. Therefore, the DRDM is a technique
complementary to the lineshape method, rather than a competing alternative.
117

5.1

Physics motivation

The physical motivation for undertaking this study was to measure the lifetime of the 2+
1
state in 74 Rb. As this state decays to the 0+ ground state of 74 Rb, the γ-ray decay is of an E2
+
+
nature, and so the lifetime of the 2+
1 state can be used to calculate the B(E2; 2 → 0 ) of the

transition. As discussed in Ch. 1, B(E2) transition strengths are often used when discussing
the collectivity of a nucleus and can be used to infer information about its shape. However,
in the case of 74 Rb, there is a complication involved which makes such a discussion less clear.
This complication is a phenomenon known as shape coexistence [88, 89], and determining
whether it is present in 74 Rb forms the primary basis for making the measurement reported
herein. This is of interest for understanding the structure of the 74 Rb nucleus itself, but also
has implications for other physics interests as well. For instance, 74 Rb is one of a few nuclei
that have very precisely measured superallowed Fermi beta decay strengths (also referred to
as f t-values) used to test the conserved vector current (CVC) hypothesis and the unitarity
of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [90]. Such tests rely critically on nuclear
structure corrections for their analyses, and so it is important to provide accurate nuclear
structure inputs for these nuclei, including information about the wavefunctions themselves.
A full discussion of the CVC hypothesis and the unitarity of the CKM matrix is beyond
the scope of this work, but more information can be found in Refs. [91, 92]. It is important
to note, however, that the data provided by this work can also provide information about
the wavefunctions and therefore the structure of the 74 Rb nucleus, which is important for
properly correcting for the nuclear structure effects in these tests of fundamental physics.
Shape coexistence occurs when a nucleus has available to it two or more different configurations with distinct shapes which are separated by only a small difference in energy

118

(typically about 1 MeV or less). Because of their close proximity, these two configurations
are said to “coexist” with each other. However, the coexistence of these two different configurations has a consequence, which is that the wavefunctions of the states that belong to the
two different configurations can mix via some interaction between them. Then, the states
which are observed physically have wavefunctions which are superpositions of the wavefunctions of the “intrinsic” states which belong purely to one configuration or the other. The
mixing causes one state to be lowered in energy while the other is raised.
Shape coexistence may be predicted in a number of ways. One is an examination of
the Nilsson orbits calculated in a given region. Figure 5.1 shows the neutron single-particle
orbitals calculated for the self-conjugate nucleus 80 Sr in the Nilsson scheme [93]. Inspection
of these orbitals shows that, as expected, when the quadrupole deformation parameter β
is zero, there are energy gaps between the single-particle energies at the canonical magic
numbers. However, when the nucleus takes on some non-zero deformation, these gaps begin
to close and new ones to appear at different particle numbers. Of particular interest to
this discussion is the occurence of gaps at almost equal energy but opposite sign of the
deformation parameter at nucleon numbers 36 and 38. This suggests that in nuclei which
possess N ≈ Z ≈ 36 − 38, there may be competition between these two gaps and this can
lead to shape coexistence. Another very useful tool for predicting the occurence of shape
coexistence is the potential energy surface of a nucleus as a function of the quadrupole
deformation parameter β and the triaxiality parameter γ. As discussed in Sec. 1.2.3, these
calculations can be used to predict the shape of a nucleus. It can also happen that there
may be multiple minima in these potential energy surfaces, which serves as another strong
indicator of shape coexistence. Figure 5.2 shows such potential energy surfaces for 72,74,76 Kr,
where the numbers along the contours give the depth of the potential in MeV. All three nuclei
119

Figure 5.1: The single-particle energy levels for 80 Sr in the Nilsson model, calculated for
neutrons but which will be similar for protons in N ≈ Z nuclei. The horizontal axis shows
the quadrupole deformation parameter β, while the single-particle energies are plotted along
the vertical axis. While the usual gaps at the magic numbers are present at β = 0, new
gaps arise at non-zero values of the deformation. In particular, gaps at roughly the same
energy appear at neutron number 36 and 38, but corresponding to opposite signs of the
deformation. This can be a signature that shape coexistence may occur in nuclei that have
proton and neutron numbers similar to these values. Figure is taken from [93].
clearly have multiple minima, one of which is prolate and the other oblate, while in 72 Kr
the prolate minimum also has a degree of triaxiality with γ ≈ 15◦ . This suggests that these
krypton isotopes exhibit shape coexistence, in agreement with the argument based on the
gaps in the Nilsson orbitals in Fig. 5.1.
Experimentally, often a signature of shape coexistence is the observation of low-lying
excited 0+ states above the ground 0+ state (in even-even nuclei). These excited 0+ states
can be considered as a kind of “alternative ground state” which might have become yrast had
the energy of the various configurations been different. These 0+ states typically also have

120

Figure 5.2: Potential energy surfaces generated for 72,74,76 Kr. As described in Ch. 1, the β
axis gives the quadrupole deformation. Values of γ = 0◦ or γ = 60◦ correspond to prolate and
oblate deformation, respectively, while intermediate values of γ indicate triaxiality. All three
krypton isotopes clearly exhibit multiple minima, with a fairly strong prolate deformation
as well as a weaker oblate minimum. 72 Kr also has some slight triaxiality associated with
its prolate minimum. Figure taken from [94].
excited state bands built upon them. However, if the ground band mixes with an excited
band, then it is usually the 0+ states which mix most strongly, as the J(J + 1) energy
dependence of the excited states of the two bands (assuming they are axially deformed)
causes progressively greater energy separation between pairs of levels of the same J π and
hence less mixing. This causes an interruption of the regular J(J + 1) spacing at low-spin,
and can be taken as an indication of mixing. However, the regular spacing at higher energies
can be used to extrapolate down to find the locations of the bandheads if there was no
mixing, and from this both the mixing matrix element and the amplitude of each of the pure
configurations in the mixed states can be deduced. Figure 5.3 shows the results of such a
calculation for the even-even krypton isotopes from mass A = 72 to mass A = 78 [95, 96].
For each nucleus, a few low-lying states are drawn divided into those belonging to prolate
shapes on the left and oblate or spherical on the right. Also drawn is the location at which
the 0+ states associated with each shape would be found if there was no mixing. Starting
with 78 Kr, the intrinsic states start far apart, and so there is only a small amount of mixing
and repulsion. However, the intrinsic states become closer together in the lighter isotopes

121

4+

612

4

+

+

558

2
+
0

0

611

+

0

+

644 562

346
+

2

∆

4+

4+

52

+

72

Kr

+

2

+

508(1)

456

0

0

2

710
671(2)

+

+

424

455

+

0

+

0
74

0
76

Kr

1017

770

Kr

78

Kr

Figure 5.3: Several partial level schemes for even-even proton-rich krypton isotopes, showing
the systematics of the low-lying levels. For each isotope, the set of levels on the left is
associated with a prolate shape, while the 0+ level on the right is an oblate or spherical
shape. In addition to the levels themselves, the positions at which the 0+ states would be
located if there was no mixing between them are indicated by the dashed horizontal lines.
These unperturbed levels become almost degenerate at 74 Kr, at which point the mixing has
become almost maximized. The observed 0+ states are therefore almost even mixtures of
the intrinsic prolate and oblate shapes. Figure is from [95].
and even switch places in 72 Kr, so that the oblate state becomes the ground state. Table 5.1,
which is taken from [95], summarizes the mixing of the intrinsic states in these nuclei. In
74 Kr,

the two intrinsic states become almost degenerate, and this corresponds to almost the

maximum possible mixing of the two intrinsic configurations, with half of the wavefunction
amplitude coming from the prolate shape and half from the oblate shape.
Turning now to the question concerning the work presented in this chapter, namely that
of shape coexistence in 74 Rb, the krypton isotopes just discussed offer a convenient entry
point. In Ch. 1, the concept of isospin was introduced. The nuclei 74 Rb and 74 Kr are isobars
with Tz = 0 and Tz = 1, respectively. Since this allows both nuclei to have states with T = 1,
122

Nuclide

∆ [MeV]

∆ [MeV]

V [MeV]

b2

78 Kr

1.01718(3)

0.80(1)

0.31(1)

0.11(2)

76 Kr

0.7700(2)

0.36(1)

0.34(1)

0.27(1)

74 Kr

0.508(1)

-0.02(1)

0.25(1)

0.52(1)

72 Kr

0.671(1)

-0.54(1)

0.20(1)

0.90(1)

+
Table 5.1: A summary of the values for the mixing of the 0+
gs and 02 states for the isotopes of
krypton shown in Fig. 5.3, taken from Ref. [95]. The energy difference between the observed
states is given by ∆ , while the energy difference between the intrinsic states is given by ∆
and the mixing matrix element is V . The amplitude of the prolate wavefunction in the 0+
2
state is given by b2 .

there should be a subset of levels in both nuclei which have similar properties. In Fig. 5.4,
which shows a few of the low-lying levels of 74 Kr and 74 Rb, it can be seen that several of
these levels follow a similar pattern of excitation energy up to spin 8+ , and these have been
proposed to be T = 1 isobaric analogue states in these nuclei. These analogue states should
all share similar nuclear structure properties under the assumption of isospin symmetry,
and therefore the fact that there is shape coexistence in 74 Kr means that 74 Rb ought to
manifest shape coexistence as well. However, as can be seen in Fig. 5.4, there is no known
excited 0+ state in 74 Rb, despite numerous searches [98, 99, 100, 101, 102, 103]. As has been
previously suggested, this most likely means that the level exists close to an excitation energy
of 500 keV in the vicinity of or even below the 2+
1 state and decays primarily or exclusively by
the emission of internal conversion electrons, to which the previous studies were not sensitive.
However, the non-observation of this 0+ state makes it difficult to investigate to what extent
it may manifest shape coexistence by the extrapolation method used in the krypton nuclei.
Another method is therefore required to explore this matter further.
To solve this problem, the study presented in this chapter was designed to measure the
74
lifetime of the 2+
1 state in Rb. This is a useful measurement to make because the transitions

123

(8+ ) 2747.9

8+

6+

1781.4

6+

1052.9

4+

1203.2
1013.3

(2+ )
4+

477.8

2+

509
455.6

0+

0

2812

1836.0

0

74

Rb

0+
2+
74

Kr

0+

Figure 5.4: A partial level diagram of the low-lying T = 1 states in 74 Rb and 74 Kr. Level
energies in keV are shown to the left of the states, while the spin-parities are given on the
right. The isobaric analogue states can be identified based on the similar excitation energies
up to spin 8+ . However, there is no known excited 0+ state in 74 Rb which would be the
analogue of the excited 0+ state in 74 Kr. Data for 74 Kr are taken from [97], while data for
74 Rb are from [98].
between the isobaric analogue states should share similar properties. In particular, the
+
quadrupole transition strength of the analogue 2+
1 → 01 decays should be very similar due

to isospin symmetry, and the lifetime can be used to calculate the strength. However, there
is a caveat to this last statement about the equivalence of the transition strengths. If the
matrix element M governing the transition is expressed in terms of isospin, then it is due to
a sum of two terms [104]:
1
M (Tz ) = [M0 − Tz M1 ],
2

124

(5.1)

where M0 is called the isoscalar matrix element and M1 is called the isovector matrix
element. In the Tz = 0 nucleus of an isobaric triplet (T = 1), the matrix element arises from
the isoscalar matrix element only, but the isovector component also contributes in the Tz =
±1 cases. Therefore, the statement that the transition strength B(E2) = |M (Tz )|2 /(2Ji + 1)
should be roughly equivalent between the analogue states of the Tz = 0 and Tz = 1 members
of an isobaric triplet is only true as long as the isovector matrix element is small compared
to the isoscalar matrix element. It has been observed in other isobaric triplets, however,
that the isovector matrix element is typically on the order of 10% of the isoscalar matrix
element [105], and so this statement may be reasonably expected to be true in the case of
the A = 74 nuclei as well.
As discussed in Ch. 1, the transition strength B(E2) is related to the overlap of the
initial and final states of the transition. This makes this quantity sensitive to the structure
of the states and can be used to infer information about the occurence of shape coexistence.
In 74 Kr, the lifetime of the 2+
1 state has already been measured [106] and so the B(E2)
is known. Since isospin symmetry (and a small isovector contribution) would predict that
+
the B(E2; 2+
1 → 01 ) is nearly identical in these two nuclei, this can be used to predict
74
the lifetime of the 2+
1 state in Rb. For this electric quadrupole transition, the transition

strength is given by
B(E2) =

816 2 4
e fm ,
Eγ5 τ

(5.2)

where Eγ is given in units of MeV and τ is given in picoseconds, which yields a predicted
74 Rb of about 26 picoseconds. However, if there is no shape
lifetime for the 2+
1 state in
+
coexistence, then the 0+
1 and 21 states should have a much larger overlap because there is

only one configuration involved, and so the B(E2) should be larger (by a factor of ∼ 2) and

125

the lifetime would then be about 13 picoseconds. These predictions therefore offer a metric
by which to judge whether shape coexistence persists between these two nuclei, and motivate
the measurement of the 2+
1 state lifetime.

5.2

Experimental details

As with the data on 70 As, the data for this analysis was obtained from an experiment
run at the NSCL. The primary beam was again 78 Kr at 150 MeV/nucleon and 25 pnA. The
secondary beam was produced by fragmentation on a 9 Be target and 74 Kr was selected in the
A1900 with the aid of an aluminum wedge and 0.5% momentum acceptance. The resulting
beam had an intensity of roughly 1 × 105 particles per second and a purity of about 40%.
For this experiment, the timing signals from the object scintillator were not sufficiently wellseparated to resolve the beam components, and so the radio frequency (RF) timing signal
from the cyclotrons was used instead of the object scintillator to identify the components
of the secondary beam. Figure 5.5 shows the timing signal from the A1900 extended focal
plane (xfp) scintillator plotted against the RF signal from the cyclotrons. Software gates
were used to isolate the 74 Kr from the contaminants in the beam during offline analysis. The
beam was then directed to the S800 vault where the lifetime measurement was performed.
Once in the S800 vault, the secondary beam was caused to undergo further reactions in
order to produce 74 Rb. The beam was directed to the target location of the S800 just in
front of the large analyzing dipole magnets where the TRIPLEX plunger (c.f. Sec. 3.2.3)
was installed in the beam line. The TRIPLEX held three metallic foils in this experiment: a
138-mg/cm2 9 Be target used to induce reactions, a 208-mg/cm2 nat Ta degrader to reduce the
beam velocity, and a 166-mg/cm2 second nat Ta second degrader to slow the beam further.

126

74
73

-900

Txfp (arb. units)

72

75

Kr

104

Rb
74

Br
73

Se

-1000

72

Kr

Br

3

10

Se

71

As

-1100
70

102

Ge

-1200
10
-1300

-1500 -1400 -1300 -1200 -1100 -1000

-900

1

Trf (arb. units)
Figure 5.5: The particle identification plot for the 74 Rb experiment. The vertical axis gives
the timing difference between the A1900 xfp scintillator and the S800 E1 scintillator while
the horizontal axis is the time difference between the radio frequency (RF) signal from the
cyclotrons and the E1 scintillator.
Behind the second degrader, a 7-mg/cm2 thick foil made of polyethylene was installed in
order to improve the charge-state distribution of reaction products. The beam then entered
the analyzing dipoles in the S800 and were directed into the focal plane where the reaction
residues were identified. Figure 5.6 shows the fragments identified by their energy loss in the
S800 ionization chamber (∆E) plotted against their time of flight between the E1 scintillator
and the A1900 xfp scintillator (Txfp ). The energy loss resolution in this experiment was not
as high as in the 70 As experiment, however, it was still sufficient to identify the fragments.
The location of the unreacted 74 Kr beam as well as 74 Rb reaction products are indicated by
the ellipses in the figure, and software gates were used to select these species in the analysis.
The γ rays emitted from the beam after interactions with the target were detected with

127

1600

3

74

∆E (arb. units)

1500

Kr

74

Rb

1400

10

102

1300
1200
10

1100
1000
2640

2660

2680

2700

2720

2740

1

Txfp (arb. units)
Figure 5.6: The outgoing particle identification plot for the 74 Rb experiment. The vertical
axis gives the energy loss in the S800 ionization chamber while the horizontal axis gives the
time of flight between the A1900 xfp scintillator and the S800 E1 scintillator. As in Fig. 4.2,
the horizontal axis is reversed so that increasing time of flight indicates decreasing A/Z. The
unreacted 74 Kr beam as well as the 74 Rb reaction products are labeled and indicated by the
position of the ellipses in the figure.
the GRETINA detector array (Sec. 3.2.2.3). The seven detector modules of GRETINA were
configured with four detectors in the most forward positions and the remaining three around
90◦ with respect to the beam axis. The TRIPLEX plunger was moved upstream from the
nominal target position at the center of GRETINA in order to shift the angular coverage
of GRETINA towards more forward angles, since these will show the most sensitivity to
Doppler shifting at different velocities. With this plunger placement, the forward detectors
covered roughly 20◦ − 50◦ with respect to the beam axis, and the remaining three detectors
were centered around 70◦ . In addition, thin sheets of lead and copper were installed on the
faces of the GRETINA modules in order to attenuate X-rays from the tantalum degraders,

128

which were observed with high intensity during the experiment. Data were taken for γ rays
in coincidence with 74 Rb detected in the S800 focal plane as well as for 74 Kr. The 74 Kr was
used as a reference measurement to demonstrate the validity of the DRDM, since it had not
been implemented previously. For each nucleus, data were taken with a separation of 1 mm
between the target and the first degrader and between the two degraders. A short time was
also spent taking data for both nuclei with the separation between the target and the first
degrader increased to 10 mm in order to quantify how many reactions are occurring in the
degraders compared to the target (see Sec. 5.4.1 for a detailed discussion).
74 Rb arises
A possible concern regarding the lifetime measurement of the 2+
1 state in

from the potential feeding of this state by the predicted 0+
2 state. Assuming that it lies
near about 500 keV as the analogue state in 74 Kr does, any γ decay that it experiences to
74 Rb would have an energy of ∼ 20 keV, which would be stopped in the
the 2+
1 state of

absorbers placed over the GRETINA modules even if it is not highly converted into electron
emissions. This means that the possible effect of feeding from this state to the 2+
1 state
74
cannot be observed. However, this is not a large concern, as the analogue 0+
2 state in Kr

has a lifetime of 20.5(14) ns [107]. Under the assumption of isospin symmetry, the 0+
2 state
in 74 Rb should have a similar lifetime. For the present experiment, it takes only about 20 ps
for the beam to travel from the target position to the second degrader position, and so the
majority of the possible population of the 0+
2 state should decay far downstream from the
TRIPLEX. As a consequence, the γ-ray spectrum should not be strongly sensitive to this
lifetime and so it should not have a large effect on the 2+
1 state lifetime measurement.

129

5.3

γ-ray spectra

74
74
The γ-ray spectra used to determine the lifetimes of the 2+
1 states in Rb and Kr will be

discussed in this section. While the placement of the GRETINA modules allowed γ rays to
be observed at angles up to nearly 90◦ with respect to the beam axis, those γ rays emitted
nearly perpendicular to that axis will have a Doppler shift which is least sensitive to the
velocity at which the nucleus is moving. Since the DRDM relies critically on this velocity
difference, these γ rays are not useful for the present analysis. Therefore, all of the spectra
shown in this section include only those γ rays which are detected at an angle relative to the
beam axis of less than 40◦ . This allows sufficiently large separation between the photopeaks
originating from emission after each of the foils mounted in the TRIPLEX.
Figure 5.7 shows the γ-ray spectrum obtained in coincidence with 74 Kr nuclei detected
in the S800 focal plane. Transitions can be seen in these spectra which correspond to the
+
+
+
+
+
6+
1 → 41 decay, the 41 → 21 decay, and the 21 → 01 decay. In addition, the three-peaked

structure necessary for the DRDM can be clearly seen in the inset to panel (b) of the figure,
although for the 10 mm data in panel (a) most of the nuclei decay before reaching the
degraders and so only the fast peak is obvious. These spectra show that 74 Kr is well-suited
for demonstrating the validity of the DRDM, as each of the transitions shown in Fig. 5.7
originates from a state that has a known lifetime. The lifetime of the 2+
1 state can therefore
+
serve as an excellent test case, both by analyzing the 2+
1 → 01 transition alone as well as

analyzing the effects of feeding on this state.
The γ rays from 74 Rb are shown in Fig. 5.8. As with 74 Kr, the spectra have been
divided into data which were taken with a separation of 10 mm between the target and first
degrader in panel (a) and data which were taken with a separation of 1 mm between the

130

Counts / 1 keV

60

60 (a)
74
Kr, 10mm data
50

+

fast

+

2 →0

40
reduced

40

slow

20
30

0

20

+

420

+

440

460

480

4 →2

6+ → 4+

10
0

200

400

600

800

1000

1200

Eγ (keV)

Counts / 1 keV

140

150

(b)
74

slow

Kr, 1mm data

100

120

reduced

100
+

80
60

+

+

0

420

4 →2

40

+

440

460

480

+

6 →4

20
0

fast

50

+

2 →0

200

400

600

800

1000

1200

Eγ (keV)
Figure 5.7: The γ-ray spectra obtained in coincidence with 74 Kr nuclei in the S800 Spectrograph. Panel (a) shows data taken with a 10 mm separation between the target and
first degrader, while panel (b) shows data taken with only a 1 mm separation. In both
+
panels, transitions can be clearly identified which correspond to the 2+
1 → 01 transition, the
+
+
+
4+
1 → 21 transition, and the 61 → 41 transition, and are labeled accordingly. The insets
+
to each panel show the same spectra focused on the 2+
1 → 01 transition to make clear the
three-peaked structure, where the components of the decay from fast, reduced velocity, and
slow recoils are also labeled.

131

(a)

Counts / 2 keV

20

74

fast

20

Rb, 10mm data

reduced

15
15

10

2+ → 0+
4+ → 2+

10

slow

5
0

440

460

480

500

5
0

200

400

600

800

1000

1200

Eγ (keV)

Counts / 2 keV

40

(b)
74

40

Rb, 1mm data
+

reduced

30
+

2 →0

30

slow

fast

20
10

20
+

+

4 →2

0

440

460

480

500

10
0

200

400

600

800

1000

1200

Eγ (keV)
Figure 5.8: The γ-ray spectra obtained in coincidence with 74 Rb detected in the S800 focal
plane, again divided into (a) data taken with 10 mm separation between the target and the
first degrader and (b) 1 mm separation between target and degrader. The insets again focus
+
74
on the 2+
1 → 01 transition to show the three-peak structure. However, unlike for Kr, there
+
+
is very little population of states above the 2+
1 state. A hint of the 41 → 21 transition can
be seen at the position labeled in the figure, but no other transitions can be identified.

132

+
target and first degrader in panel (b), with the insets showing the 2+
1 → 01 transition in

greater detail. A striking difference between these data and those for 74 Kr is that the 74 Rb
data have much lower statistics. As noted in Sec. 2.3, the DRDM is well-suited for such
low-production rate experiments compared to the classic RDM method because it does not
require multiple measurements in order to extract a lifetime. Another difference between
the 74 Rb and 74 Kr data is that the 74 Rb spectra do not show strong population of excited
+
states above the 2+
1 state. There is some evidence for population of the 41 state, but it is

very small and considerably less than in 74 Kr. This means that the analysis of the 74 Rb
data will be simplified since feeding will not have a major impact on the 2+
1 state lifetime
measurement.

5.4

Lifetime analysis

Unlike the 70 As measurement presented earlier, once the γ-ray spectra were obtained very
little additional processing was necessary before proceeding to the analysis. This is partially
due to the relative simplicity of the low-energy level schemes of 74 Kr and 74 Rb, but is also
a feature of the DRDM. To show the transparency of the method, a rough estimate of the
74
74
lifetime of the 2+
1 states in Kr and Rb can be obtained from Eq. 2.30. From the insets

to Fig. 5.7(b) and Fig. 5.8(b), the ratio of the yield of the slow peak to that of the reduced
velocity peak for both nuclei was Is /Ir ≈ 3. The velocity of the recoils between the two
degraders was v ≈ 0.35c = 0.1 mm/ps, and the separation between the two degraders was
∆x = 1 mm. Following Eq. 2.30, the lifetime of the 2+
1 states can then be estimated to be:
s

τ=

1 mm
∆x Iij
≈
× 3 = 30 ps.
d
v Iij
0.1 mm/ps
133

(5.3)

This value is roughly consistent with the known value of τ = 33.8(6) ps [106], giving a
reasonable estimate for the unknown lifetime of 74 Rb. For a more thorough analysis, the
extraction of the 2+
1 state lifetimes from the γ-ray spectra was performed using Geant4
simulations. Although the method as it is described in Sec. 2.3 does not require the use
of simulations to perform a lifetime measurement, the use of simulations does offer certain
advantages. In particular, the DRDM hinges critically upon the ability to measure the
lifetime of a state using only the two photopeaks coming from decays after the first and second
degraders. However, this is motivated by the approximation that the slope of the decay
curve is linear in the space between the two degraders, while in actuality it is exponential.
By simulating the transport of the ions through the TRIPLEX plunger as they decay, this
approximation is in effect folded into the simulated spectra. Using simulations to determine
the lifetime therefore corrects for the assumptions inherent in the method and allows the
lifetime to be determined in a self-consistent way. It also takes into account the experimental
lineshape of the peaks, possible decays occurring inside the foils, and other similar small
effects which can affect the lifetime measurement. Feeding effects, if present, can also be
included. The material in this section will show the details of the analysis, including the
setup of the simulations and the fitting of the simulated γ-ray spectra to the data.

5.4.1

Geant4 simulations

As with the measurements performed on 70 As, the first step in the analysis is to set up the
simulations. This setup proceeded very much as was described already for 70 As in Sec. 4.4.1,
and so will only be summarized here. The GRETINA detectors were installed at the NSCL
as part of a campaign of experiments and so the location was determined by the laboratory
staff and provided to experimenters. This information and the dimensions of GRETINA
134

were used to fix the location of the detectors in the simulation. The beam properties were
determined in exactly the same fashion as for 70 As, and the agreement between the simulated
and measured beam properties are shown in Fig. 5.9 for 74 Kr and Fig. 5.10 for 74 Rb.
The chief difference in preparing the simulations between the analysis presented in this
chapter and that in Ch. 4 is that with the DRDM it is possible that some of the incoming
beam particles react not with the target as desired but instead react with the degraders.
Therefore, it is necessary to constrain the relative number of reactions occurring on the
target, the first degrader, and the second degrader. This is typically done by moving the foils
mounted on the plunger far apart, so that there is ample time after each foil for the excited
states to decay completely before the next foil is reached. Any counts observed in a given
peak can then be assigned unambiguously to reactions happening on the corresponding foil.
This information is provided to the simulation software as the fraction of reactions occurring
on the target, and the fraction of the remaining reactions occurring on the first degrader. To
fix these values for this experiment, the spectra in Fig. 5.7(a) and Fig. 5.8(a) were analyzed.
The number of counts observed at a Doppler shift corresponding to decays after the target
compared to those after the two degraders was used to determine the fraction of the reactions
happening in the target. This was determined to be about 90% for the 74 Kr data and about
80% for the 74 Rb data. Due to a mechanical failure in the drive controlling the position of
the second degrader, data with a large separation between the two degraders could not be
taken during the experiment. Therefore, the ratio of the reactions happening on the first
degrader compared to the second was fixed to the ratio of the thickness of the two degraders
for both 74 Kr and 74 Rb, which was about 55%.

135

(b)
Counts (arb. units)

Counts (arb. units)

(a)

-60 -40 -20

0

20 40 60

-60 -40 -20

ata (mrad)

0

20 40 60

bta (mrad)
Counts (arb. units)

(d)

Counts (arb. units)

(c)

-10

-5

0

5

10

-10 -8 -6 -4 -2 0

y (mm)

2 4

6 8 10

dta (%)

ta

Figure 5.9: A comparison between the measured (blue) and simulated (red) beam properties.
Panel (a) shows the angle of the beam in the dispersive direction, while panel (b) shows the
angle in the non-dispersive direction, and both are reproduced well. As with 70 As, the
position of the beam in the non-dispersive direction shown in panel (c) is not reproduced
well because the beam is assumed to be circular in the simulation, but in reality may have
any spatial distribution. The spread in the energy is shown in panel (d), and is reproduced
well.

136

(a)
Counts (arb. units)

Counts (arb. units)

(b)

-60 -40 -20

0

20 40 60

-60 -40 -20

ata (mrad)

0

20 40 60

bta (mrad)

(c)
Counts (arb. units)

Counts (arb. units)

(d)

-10

-5

0

5

10

-10 -8 -6 -4 -2 0

y (mm)

2 4

6 8 10

dta (%)

ta

Figure 5.10: A comparison of the simulated and measured beam properties for 74 Rb, shown
in red and blue, respectively. As for the 74 Kr beam, the dispersive and non-dispersive angular
distributions shown in panels (a) and (b), respectively, are reproduced well, while the nondispersive position has a different distribution in the data than is assumed in the simulation.
The spread in beam energies about the central value shown in panel (d) is reproduced well.

137

5.4.2

Lifetime determination

74
74
After preparing the simulations, determining the lifetimes of the 2+
1 states in Kr and Rb

was accomplished by generating a set of simulated γ-ray spectra for each nucleus with the
lifetime varying in a range from 0 ps to 50 ps. These simulations were then fit to the data
along with a background which was assumed to be exponential. In accordance with the
DRDM, only the peaks from the reduced and slow velocity recoils were fit, while the fast
component was excluded from the fitting region. For each simulation with a given lifetime,
a χ2 value was extracted, then the distribution of χ2 values was plotted and the minimum
used to extract the lifetime. The following shows the results of this analysis for each of the
two nuclei in question.
The analysis of 74 Kr was performed first in order to validate the DRDM. The 2+
1 state
has a lifetime of 33.8(6) ps [106], and so should lie within the range of lifetimes that was
simulated. As noted previously, a significant amount of feeding is present in the spectrum
obtained for 74 Kr, and so this data can also test how accurate the method is under these
conditions. First, the data was fit using only simulations that included the 2+
1 state, and did
not account for feeding from higher-lying states. An advantage of using the DRDM is that
this should give a result that is close to the true value. However, because there is significant
feeding, there should be an excess of counts in the target component in the simulations
compared to the data. This is because the lifetime of higher-lying states will delay the
population of a portion of the 2+
1 state, so that fewer decays from this state happen before
the first degrader. Figure 5.11 shows the results of this fit. As can be seen in the figure, the
slow and reduced velocity components are reproduced very well in this fit. However, the fast
component is overestimated considerably in the simulations, as expected given the observed

138

160

χ2/ndf

4

Counts / 1 keV

140
120

3
2

100

1
80

20 25 30 35 40 45 50

60

τ (ps)

40
20
0

380

400

420

440

460

480

500

520

540

Eγ (keV)

Figure 5.11: The results of fitting the simulated γ-ray spectra to the 74 Kr γ-ray spectrum.
The data points are shown in black, while the simulation is shown in red and the background
is shown in blue. The slow and reduced velocity components of the data are reproduced
well by the simulation, but the fast component is not. This is expected, as the observed
feeding from higher-lying states should produce a deficit of counts in this peak relative to
the simulations. The inset shows the distribution of reduced χ2 values obtained by varying
the lifetime in the simulations, and the minimum of this distribution gives a lifetime of
τ = 30.6(20) ps, with 15 degrees of freedom in the fit.
feeding. The inset in the figure shows the distribution of reduced χ2 values which has a
clear minimum. The lifetime extracted from this distribution is τ = 30.6(20) ps. Compared
to the accepted literature value of τ = 33.8(6) ps [106], this result is consistent within two
standard deviations, and demonstrates that the DRDM is giving accurate results.
To verify that it is actually the effects of feeding that are causing the overproduction of
the fast component of the simulation shown in Fig. 5.11, the higher-lying states can also be
included in the simulations. Doing so should reduce the contribution of the fast component
relative to that of the other two peaks and result in a simultaneous reproduction of all three
components. Figure 5.12 shows the result of fitting the 74 Kr data with simulations that
+ +
+
+
include the 2+
1 , 41 , 61 , and 22 states as well as the feeding among them. The 22 state was

139

160

Counts / 1 keV

140
120
100
80
60
40
20
0

380

400

420

440

460

480

500

520

540

Eγ (keV)

Figure 5.12: The results of fitting the 74 Kr γ-ray spectrum with simulations that include the
+
+
effects of feeding from the 4+
1 and 61 states. The 22 state was also included, but observed
to contribute almost nothing to the fit. In this figure, the lifetime was not varied, but the
literature value of τ = 33.8 ps [106] was used and all three peaks were included in the fitting
region. Now all three peaks are reproduced simultaneously, in contrast to Fig. 5.11, which
demonstrates that all three components can be reproduced if feeding is taken into account.
included to account for any possible population of this state, but is extremely small. For
this fit, instead of searching for a best-fit spectrum by varying lifetimes, the literature value
of τ = 33.8 ps was used for the 2+
1 state lifetime to generate the simulations in this figure.
Also, the fitting region was extended to include all three peaks. It is clear from looking at
this spectrum that all three peaks are reproduced well when the feeding from higher-lying
states is included, and confirms that the DRDM is performing as expected.
74 Rb was analyzed. Since
With the method validated, the lifetime of the 2+
1 state of

no significant feeding was observed in the data, it can be expected that even though only
the slow and reduced velocity peaks are included in the fitting range, the fast peak should
automatically be reproduced by the simulations if the lifetime is correctly determined. The
simulations were fit to the data and the reduced χ2 values plotted and used to determine

140

50

χ2/ndf

1.3

Counts / 2 keV

40

1.2

30

1.1

20

20

25

30

35

40

τ (ps)

10

0

420

440

460

480

500

520

540

560

Eγ (keV)

Figure 5.13: The results of the fit to the 74 Rb data. The data points are shown in black,
while the red line shows the best-fit simulation and the blue line shows the assumption of
the background. All three peaks are reproduced in this fit even though only the slow and
reduced velocity components are fit, which shows that feeding is not a strong contributor in
this analysis. The inset shows the reduced χ2 distribution, where the number of degrees of
freedom is 19, and the minimum of the distribution gives the lifetime as τ = 26.7(60) ps.
the lifetime. Figure 5.13 shows the results this analysis, with the data shown by the black
points and the simulation shown by the red solid line. The blue curve shows the assumed
exponential background. As expected, all three peaks are correctly reproduced, confirming
that there is no significant impact on the 2+
1 state lifetime from feeding. The distribution of
reduced χ2 values is shown in the inset, and the minimum gives a lifetime of τ = 26.7(60) ps.

5.5

Results and discussion

74 Rb measured, it is possible to discuss the physical
With the lifetime of the 2+
1 state of

meaning of the result. As mentioned in Sec. 5.1, isospin symmetry requires that the charac74
74
teristics of the 2+
1 states of Rb and Kr be similar. This led to the prediction that, based
+
74
on the previously measured 2+
1 → 01 transition strength measured in Kr, the lifetime of

141

74
the 2+
1 state in Rb should be about 26 ps. This prediction is born out by the results of this

study. This suggests that, similarly to 74 Kr, 74 Rb exhibits shape coexistence in its ground
state, and likely there is a similar amount of mixing between its ground state and the excited
0+ state that is hypothesized to exist at about 500 keV.
In order to have a quantitative discussion of the results of this study, it is useful to
+
74 Rb instead of the lifetime. From Eq. 5.2,
discuss the 2+
1 → 01 transition strength in
+
2 4
the transition strength can be calculated to be B(E2; 2+
1 → 01 ) = 1227(276) e fm . The

two-state mixing model described earlier in this chapter can be used to investigate to what
extent this result indicates mixing of intrinsic shapes. This can be done by formulating the
transition strength in terms of the overlap of the wavefunctions of the intrinsic states. For
the intrinsic wavefunctions of the 0+ and 2+ states of the different configurations, this would
be written as
+
|0+ = cos θ0 |0+
p + sin θ0 |0o

|2+

=

cos θ2 |0+
p

+ sin θ2 |0+
o

(5.4)
,

where θ0,2 is the mixing angle between the 0+ or 2+ states, respectively, and |0+
p denotes the
intrinsic wavefunction of the 0+ state belonging to the prolate configuration while |0+
o rep+
resents the same for the oblate configuration. Then, the B(E2; 2+
1 → 01 ) can be constructed

by writing the overlap of these wavefunctions through the E2 transition operator:
| 0+ ||M(E2)||2+ |2
2Ji + 1
1
+
+
+ 2
= | cos θ0 cos θ2 0+
p ||M(E2)||2p + sin θ0 sin θ2 0o ||M(E2)||2o | ,
5

+
B(E2; 2+
1 → 01 ) =

(5.5)
where it has been assumed that the oblate and prolate intrinsic configurations are orthogonal

142

+
and therefore all terms such as 0+
o ||M(E2)||2p = 0. From this expression the mixing am-

plitudes cos2 θ0 and cos2 θ2 can be extracted if the transition strengths between the intrinsic
configurations are known.
Unfortunately, a lifetime measurement of the 2+
1 state alone is not enough to deduce
the individual matrix elements on the right hand side of Eq. 5.5. However, an analysis
+
was performed in Ref. [108] on 74 Kr in which the matrix elements 0+
p ||M(E2)||2p and
+
0+
o ||M(E2)||2o were determined from the experimental data, as well as the mixing am-

plitudes cos2 θ0,2 . The technique used in the study is known as Coulomb excitation [109]
(more precisely, multiple Coulomb excitation), in which the nucleus is excited via the electromagnetic interaction with a high-Z material. This technique is sensitive to the shape of
the nucleus through the so-called “reorientation effect,” in which transitions occur between
magnetic substates of an excited state which gives access to the quadrupole moment of the
nucleus in that state. In order to discuss 74 Rb further, the assumption can be made that
these matrix elements have the same values in 74 Rb as in 74 Kr. While this is a strong
assumption, it is motivated by the expectation that isospin symmetry should preserve the
structure between the isobaric analogues states of these nuclei. There is also indirect evi+
dence to support this assumption, in that the B(E2; 2+
1 → 01 ) values in these two nuclei

are very similar, which in itself suggests that the structure of these states in these two nuclei
remains the same. With this justification, the values of the matrix elements from [108] can
+
2
+
+
2
be adopted here: 0+
p ||M(E2)||2p = 114(15) efm and 0o ||M(E2)||2o = −21(27) efm .

Since the oblate term is both small and consistent with zero, it can be neglected in Eq. 5.5,
which becomes
+ ) = 1 cos2 θ cos2 θ (114efm)2 .
B(E2; 2+
→
0
0
2
1
1
5

143

(5.6)

In Ref. [108], the term cos2 θ2 = 0.82(20), and therefore it is reasonable to set this value to
unity in Eq. 5.6. With the experimental value of the B(E2) measured in this work, the only
undetermined quantity left is cos2 θ0 , which is the amplitude of the prolate configuration in
the 0+
1 state. Solving for this quantity, it is found that
cos2 θ0 = 0.47(16)
sin2 θ0

(5.7)

= 0.53(16),

where the equation in sin2 θ0 is the amplitude of the oblate configuration. These values
74 Rb is nearly an equal mixture between intrinsic prolate
indicate that the 0+
1 state in

and oblate states. This is not a surprising result, as the assumptions made along the way
essentially necessitate this conclusion. However, what is important to note is the uncertainty
associated with these values, which indicates that even at the edge of the 1σ uncertainties,
there is still at least a 30%-70% mixture of the intrinsic shapes in 74 Rb. Therefore, within the
assumptions that have been made herein, this study provides good evidence for significant
shape coexistence and configuration mixing in 74 Rb.
Unfortunately, relatively few theoretical predictions which have published a B(E2; 2+
1 →
74
0+
1 ) value exist for Rb due to difficulties with performing detailed calculations in this mass

region. One spherical shell model study was published that used a truncated model space of
f5/2 p1/2 g9/2 d5/2 (i.e. the p3/2 orbital was omitted) and presented results for several N = Z
+
nuclei in this mass region, including 74 Rb [110]. The B(E2; 0+
1 → 21 ) values from this

study are shown in Fig. 5.14, which is taken from the paper. While a numerical value for
+
the B(E2) value for 74 Rb was not provided, from the figure the B(E2; 0+
1 → 21 ) value can
+
+
1
be estimated to approximately 0.22 e2 b2 . Noting that B(E2; 2+
1 → 01 ) = 5 B(E2; 01 →

144

+
Figure 5.14: Theoretical B(E2; 0+
1 → 21 ) values for several N = Z nuclei calculated using
the shell model with the f5/2 p1/2 g9/2 d5/2 model space. The value for 74 Rb is underpredicted
by roughly a factor of three, which the authors of the study note is due to the omission of
the p3/2 orbital. Figure taken from [110].
+
+
2 4
2+
1 ), this corresponds to a calculated transition strength of B(E2; 21 → 01 ) ≈ 440 e fm .

This is considerably smaller than the experimentally measured value of 1227(276) e2 fm4 .
However, the authors of the study note that their calculation underpredicts the other B(E2)
strengths in the figure as well, and attribute this to the omission of the p3/2 orbital from
the calculations. They make reference to another paper [111] in which the pf5/2 g9/2 model
space was used (i.e. the d5/2 orbital was not included but the p3/2 was) and note that
the magnitude of the B(E2) value calculated for 68 Se therein is increased by a factor of
three compared to the present calculation without the p3/2 orbital. Applying the same
factor of three increase to the calculated strength estimated from Fig. 5.14 for 74 Rb gives
+
2 4
B(E2; 2+
1 → 01 ) ≈ 1320 e fm , which is consistent with the value measured within this

work. It would be informative to see the results of a calculation performed for 74 Rb with
the p3/2 orbital included to see whether this factor of three increase is justified.
A different approach to predict the B(E2) strength can be taken by predicting the deformation parameter β of the nucleus and using Eq. 1.13 to convert this into a transition

145

strength. This procedure relies on the assumption that the nucleus is axially symmetric,
which can justified if the characteristic J(J + 1) energy dependence is observed in the energy
spectra. It then remains to find a suitable way to calculate the deformation parameter.
Using a macroscopic-microscopic model, these deformation parameters have been calculated
for an extremely wide range of nuclei in Ref. [112]. The macroscopic portion of the model
used was the “finite range droplet model,” which is essentially the well-known liquid-drop
model [113] that models the nuclear binding energy with terms such as the volume, surface area, and charge of the nucleus, just as if it were a drop of charged liquid, but with
refinements to account for the nuclear deformation. The microscopic portion of the model
provides corrections to the macroscopic portion from shell effects and nucleon pairing and
is based on a folded-Yukawa potential. A full discussion of these models, which is beyond
the scope of this work, can be found in [112]. The result of this calculation, however, is a
predicted deformation parameter of β = 0.381 for the ground state of 74 Rb. Putting this into
Eq. 1.13 and using the typical choice that the nuclear radius is given by R = 1.2A1/3 fm,
+
2 4
this predicts a value of B(E2; 2+
1 → 01 ) = 1459 e fm , which is in agreement with the

experimental value of 1227(276) e2 fm4 measured here. Values predicted by this model for
several neighboring N = Z nuclei are plotted in Fig. 5.15. For the most part, this model
does well in reproducing the B(E2) transition strengths for these nuclei, although it fails
in the case of 70 Br. However, it should be noted that this macroscopic-microscopic model
only calculates the ground-state deformation, and so it will not have any sensitivity to the
higher-lying states and possible mixing between them and the ground state. Therefore, the
results of this model in the particular case of 74 Rb should be interpreted with caution.
As a final remark about the comparison between the two models referenced above, while
the details of the calculations may not be exactly correct, it is important to note that
146

B(E2;2+1→0+1) (e2fm4)

2500
2000
1500
1000
500
0
67

68

69

70

71

72

73

74

75

76

77

A
Figure 5.15: Systematics of the measured B(E2) values for several N = Z nuclei, where the
even-even nuclei are indicated by the filled symbols and the odd-odd nuclei correspond to the
open symbols. The red point is the B(E2) value for 74 Rb measured in this work. These values
are compared to the predictions calculated from the results of the macroscopic-microscopic
model in [112] (solid line). Overall, the trend is reproduced well, although the prediction for
70 Br differs significantly from the measured value. Data are taken from [32, 60, 66, 114].
both models do correctly reproduce the overall trend of the B(E2) values in this region.
Specifically, for both the shell model and the macroscopic-microscopic model, there is a rapid
rise in collectivity that begins around 72 Kr and continues through 76 Sr which is reflected
in the increasing B(E2) values predicted by both models (see Figs. 5.14 and 5.15). The
macroscopic-microscopic model does predict that the increase in collectivity begins slightly
earlier in 70 Br, at variance with the data, but the overall agreement is still fairly good. It
will be very interesting to compare the results of these calculations with future theoretical
predictions, e.g. that can account for the full f p model space while including the g9/2 d5/2
orbitals as well.

147

Chapter 6
Concluding Remarks
In this work, two studies have been presented which used lifetime measurements to study
the structure of rare isotopes, namely 70 As and 74 Rb. In both cases the data presented
challenges to the analyses which required the use of advanced techniques in order to extract
results from the experimental spectra. A summary of these studies is given here as well as
a few words on the future prospects generated by this work before concluding.
For the 70 As experiment detailed in Ch. 4, data were taken with the intention of using
the lineshape method to extract lifetimes based on the asymmetric photopeaks produced by
long-lived states moving at relativistic speeds. However, the very high density of excited
states in this odd-odd nucleus necessitated a more sophisticated analysis, as the presence of
many photopeaks in close proximity introduced too much uncertainty into the determination
of the lineshape. To remedy this situation, the lineshape method was extended to use
γ-γ coincidence data in order to isolate the levels of interest, both from the perturbing
effect of feeding and to remove the presence of any possible background peaks in the singles
spectrum. While this improved the purity of the spectra greatly, as a consequence it reduced
the statistics available for the analysis considerably. However, with careful analysis, it was
possible to determine the lifetimes of the yrast 8+ and 9+ states. This study therefore
constitutes an important tool in the arsenal of nuclear spectroscopy for measuring excited
state lifetimes, particularly for complex spectra as is typical of odd-odd nuclei.
The physics goal of this study on 70 As was to provide a firmer experimental determination
148

of the configuration of the 8+ and 9+ states. As mentioned in Ch. 4, previous studies on
this nucleus had suggested that these states arise from a coupling of both the last proton
and the last neutron in g9/2 orbitals. This was based on the energy systematics of levels in
neighboring nuclei, however, and the lifetime measurements in this work were intended to
provide an independent verification of this assignment. The results of these measurements
support the assignment of the 8+ and 9+ states to the πg9/2 νg9/2 configuration, based
upon both the systematics of the reduced transition strengths calculated from the measured
lifetimes, and upon the agreement with the prediction of a particle-vibration coupling model
calculation.
The analysis of 74 Rb presented in Ch. 5 also presented an experimental challenge in
that it is difficult to produce this nucleus in large quantities with the current generation
of experimental facilities. To address this problem, this experiment implemented for the
first time the Differential Recoil Distance Method (DRDM). The analysis of 74 Rb is wellsuited to the use of the DRDM, as the technique is able to measure the lifetime of nuclear
states with considerably lower γ-ray yields than more conventional Recoil Distance methods.
74
The method was demonstrated to reproduce the measured lifetime of the 2+
1 state of Kr,

which has been determined very precisely by previous measurements, and then successfully
74
applied to the 2+
1 state of Rb. This analysis provides a way to retain the precision of the

Recoil Distance Method while simultaneously increasing the economy of the measurement
by reducing the number of counts necessary to determine the lifetime. This result therefore
paves the way to access very exotic nuclei which previously could not be produced in sufficient
quantities for lifetime measurements.
In addition to providing an opportunity to demonstrate the DRDM, the choice to measure
the lifetime of 74 Rb offered the chance to obtain important physics information as well. In
149

the isospin partner nucleus 74 Kr, it is well known that shape coexistence is present and that
this causes the two low-lying configurations that are present in this nucleus to experience
strong mixing. Because of isospin symmetry, states that arise from the same structure should
also be present in 74 Rb and their transitions should exhibit similar spectroscopic properties.
As discussed in Ch. 5, these isobaric analogue states in 74 Rb had been identified, and the
transition strength calculated from the lifetime measurement of the 2+
1 state in this work
agrees very well with the measured value in 74 Kr. This can be taken as an experimental
indication that shape coexistence persists across the isospin triplet at mass 74. By making
the assumption that the intrinsic shapes of the two coexisting configurations are the same,
the degree of mixing in 74 Rb was also quantified and found to be consistent with the roughly
equal mixing that occurs in 74 Kr, even with the relatively large uncertainty of ∼ 20%.
Finally, a few comments are offered about the future prospects that are raised by these
two studies. The preceding paragraphs have already mentioned how the techniques presented in this work can facilitate the analysis of challenging data. However, it is also worth
discussing the future physics interests that can be conceived based on this work. For 70 As,
the origin of the 8+ and 9+ states in the πg9/2 νg9/2 configuration was the sole focus, and
little consideration was given to the possible shape that the nucleus may adopt. However,
while it was mentioned that previous studies did not find strong evidence for deformation in
the lower-lying states, with both the odd nucleons populating the deformation driving g9/2
orbital a non-spherical shape might be expected. This can provide the motivation for future
studies which are designed specifically to quantify the degree of variation from sphericity in
the high-spin states of 70 As. Likewise, as was discussed in Ch. 5, the lifetime measurement
on 74 Rb presented in this work can be used to infer the presence of shape coexistence, but
cannot be used to directly determine its existence. However, future measurements specifi150

cally designed to detect the decay of the analogue of the excited 0+ state in 74 Kr, specifically
decays that proceed through the emission of conversion electrons, would provide more direct
evidence of shape coexistence. In addition, a multiple Coulomb excitation experiment similar
to the one used to determine the shapes and mixing of the intrinsic configurations similar to
that reported in [108] for 74 Kr should provide clear evidence for shape coexistence. Future
advances in experimental facilities may enable the successful use of these techniques in 74 Rb.
Lastly, as mentioned at the end of Ch. 5, there are relatively few theoretical treatments of
74 Rb,

+
and therefore future calculations that can reproduce the B(E2; 2+
1 → 01 ) value for

this nucleus are desirable.
In closing, lifetime measurements provide a sensitive probe of the structure and behavior
of exotic nuclei. They can provide a measure of the collectivity of a nucleus and are sensitive
to the wavefunctions of nuclear states. They are also model-independent and can serve as a
valuable tool for benchmarking the results of nuclear theory as well as evaluating systematic
trends in the data. The studies presented in this work demonstrate two advanced techniques
for measuring excited state lifetimes which are suitable for challenging systems with complicated level schemes or low production cross sections. The methods were developed and
74
used to study the 8+ and 9+ states in 70 As and the 2+
1 state in Rb, and the results used

to understand the structure of these exotic nuclei.

151

BIBLIOGRAPHY

152

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