PROTON AND NEUTRON CONTRIBUTIONS FOR CROSS-SHELL EXCITATIONS NEAR
DOUBLY MAGIC 40CA STUDIED BY LIFETIME MEASUREMENTS OF MIRROR
TRANSITIONS

By

Andrew Sanchez

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2024

ABSTRACT

Under the assumption of isospin conservation in the strong interaction, mirror nuclei should have

similar level schemes and excitation properties; however, when deviations from expected trends

are observed in transition strengths of mirror nuclei, the opportunity may arise to learn about

physics mechanisms to account for the isospin symmetry breaking phenomena. It is known that the

isospin non-conserving force and Coulomb force can induce mirror energy differences. Nuclear

shape changes among mirror nuclei may also change the balance of the transition strengths among

mirror partners. The reduced transition strength 𝐵(E2) is proportional to the square of the proton

matrix element 𝑀𝑝 and, therefore, measurements of the transition strength provide valuable data

for comparison to shell-model calculations which utilize various effective interactions. Because

the proton matrix element 𝑀𝑝 for a transition in one nucleus is equivalent to the neutron matrix

element 𝑀𝑛 for the same transition in the mirror nucleus, measurements of mirror transitions

provide a fuller picture of neutron and proton configurations for excited states, especially since

the neutron matrix element is difficult to measure directly. Further, when measurements of mirror

transitions are conducted within the same experiment, and the data are analyzed with the same

methods, the resulting values of 𝑀𝑛 and 𝑀𝑝 may be compared more effectively due to reduction of

systematic uncertainty associated with comparing results from different experiments or by different

analysis methods.

In this work, we have utilized the recoil-distance method (RDM) to measure the lifetimes of the

11/2− states in mirror nuclei 39Ca and 39K by examining the analog 11/2− → 7/2− transitions in

both nuclei. The RDM lifetime measurements provide a model independent method of determining

the 𝐵(E2) and, hence, the matrix elements 𝑀𝑝 and 𝑀𝑛 for the analog transitions. The measurements

were performed utilizing the Coupled-Cyclotron Facility and A1900 fragment separator to produce

a 42Sc secondary beam which was directed to the TRIPLEX device where reactions on a 9Be target

produced 39Ca and 39K in excited states. The excited nuclei were identified in the S800 spectrograph

and gamma rays were collected with the GRETINA array. The analysis methods employed for the

lifetime measurement of the 11/2− state in 39Ca were validated by comparing the 39K results to

adopted values. Additionally, the data provide an improved lifetime measurement of the 9/2− state

in 39Ca.

From the lifetime measurements, the reduced transition strengths 𝐵(E2) are determined for the

11/2− → 7/2− mirror transitions. Using the matrix element decomposition, the 𝑀𝑝 and 𝑀𝑛 are

determined and the results are compared to shell-model calculations which utilize three effective

interactions common to this region of the nuclear chart. The comparison of the matrix elements to

shell-model calculations suggests an enhanced transition strength in 39Ca, suggesting both proton

and neutron contributions to core excitations across the 𝑍 = 𝑁 = 20 shell gaps.

ACKNOWLEDGEMENTS

Its been a long strange trip to get to this point and its required the support of many people, so where

do I start? Of course, at the beginning. Without my mother, Kathryn Gontarz, none of this would

have been possible and I must begin with acknowledgment of her unwavering support throughout

my life (good times, bad times, you know I’ve had my share).

I absolutely must thank Prof. Alan Wuosmaa from University of Connecticut for noticing the

potential in me and supporting each of my goals through undergrad and into grad school. Profs.

Artemis Spyrou and Jaideep Singh for providing the summer research opportunities that sparked

my interest in experimental nuclear physics. Of course my advisor and committee chair, Prof. Hiro

Iwasaki, who knew just how much to push me to attain my goals. I’d also like to thank my other

committee members Profs. Sean Liddick, Dean Lee, Stuart Tessmer, and Katharina Domnanich.

I must also thank MSU for gambling on me and giving me the opportunity to be a part of this

wonderful program.

There are a few more people that deserve extra acknowledgments for providing academic and

personal support above and beyond my expectations. First I must acknowledge my officemate Peter

Farris, who always found time to work through a problem with me and who I have had the pleasure

to watch grow into a great person and scientist. Roy Salinas, my other officemate, who has been

instrumental in my analysis by always taking time to work through issues in my code and being a

wonderful, generous person with a great sense of humor. I also must thank Cavan Maher for being

a good friend and for his emotional support through tough times.

Finally, my girlfriend Danielle Kohl. As the demands on my time became larger and larger,

Danielle was always there to take care of the things that needed to be done but that I just couldn’t

find the time or energy to do. She has been my biggest cheerleader throughout my final and most

demanding years in this program.

I’m so thankful that, by chance, our vessels passed, and we

finally met at last.

iv

TABLE OF CONTENTS

LIST OF TABLES .

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LIST OF FIGURES .

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vi

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CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
4
1.1 Nuclear Structure .
1.2 Systematic Trends in Matrix Element Values . . . . . . . . . . . . . . . . . . . 14
1.3 Analog Transitions in Mirror Nuclei 39Ca and 39K . . . . . . . . . . . . . . . . 19

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

.

CHAPTER 2

EXPERIMENT .

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

. 21
2.1 Primary Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Secondary Beam Development
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Particle Identification with the S800 Spectrograph . . . . . . . . . . . . . . . . 31
2.4 Gamma Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
. 49
2.5 RDM Lifetime Measurement with TRIPLEX . . . . . . . . . . . . . . . . .

CHAPTER 3

DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Experimental Data Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Simulation Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Analysis Overview .

CHAPTER 4

DISCUSSION OF RESULTS . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Experimental Data and Systematics . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Shell Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 Effective Interactions Employed in this Work . . . . . . . . . . . . . . . . . . 92

CHAPTER 5

CONCLUSION .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

BIBLIOGRAPHY . .

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. 107

v

LIST OF TABLES

Table 4.1 The present results of 𝐵(E2), proton (𝑀𝑝) and neutron (𝑀𝑛) matrix elements
for the 11/2− → 7/2− transitions in the 𝐴 = 39 system are compared to
adopted values for the 2+ → 0+ transitions in the 𝐴 = 38 system [43, 44].
The 𝑀𝑛 values and 𝑀𝑛/𝑀𝑝 ratios are presented only for 39Ca and 38Ca and
corresponding values in mirrors 39K and 38Ar, respectively, are omitted to
avoid redundancy. The table has been adapted from Ref. [88].

. . . . . . . . . . 90

Table 4.2 The present results of 𝐵(E2), proton (𝑀𝑝) and neutron (𝑀𝑛) matrix elements for
the 11/2− → 7/2− transitions in the 𝐴 = 39 system [88]. The 𝑀𝑛 values and
𝑀𝑛/𝑀𝑝 ratios are presented only for 39Ca and corresponding values in mirror
39K are omitted to avoid redundancy [88]. Reprinted table with permission
from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright
(2024) by the American Physical Society.

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

Table 4.3 The calculated values of the bare proton (𝐴𝑝) and neutron (𝐴𝑛) matrix elements
for the 11/2− → 7/2− transition of 39Ca from shell-model calculations with
the FSU [94], ZBM2 [95], and ZBM2m [96] effective interactions. The table
has been adapted from Ref. [88].

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

. 96

. 98

Table 4.4 The numbers of protons (Δ𝑝) and neutrons (Δ𝑛) excited from the (2𝑠1/2, 1𝑑3/2)
to (1 𝑓7/2, 2𝑝3/2) orbitals calculated with the ZBM2 [95] and ZBM2m [96]
interactions. The percentages (%) for different configurations are given for the
11/2− state of 39Ca (top) and for the 2+ state of 38Ca (bottom). The percentage
for each of the higher-order configurations is less than 6%, and only the sum
of these configurations is listed as “other” in the table. The table has been
adapted from Ref. [88].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Table 4.5 The present results of reduced transition strength 𝐵(E2), proton (𝑀𝑝) and
neutron (𝑀𝑛) matrix elements for the 2+ → 0+ transitions in the 𝐴 = 38 system
are compared to shell-model calculations with the FSU [94], ZBM2 [95], and
ZBM2m [96] effective interactions. The 𝑀𝑛 values and 𝑀𝑛/𝑀𝑝 ratios are
presented only for 38Ca and corresponding values in mirror 38Ar are omitted
to avoid redundancy [88]. Reprinted table with permission from A. Sanchez
et al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the
American Physical Society.

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

Table 4.6 The calculated values of the bare proton (𝐴𝑝) and neutron (𝐴𝑛) matrix elements
for the 2+ → 0+ transition of 38Ca (bottom) from shell-model calculations with
the FSU [94], ZBM2 [95], and ZBM2m [96] effective interactions. The table
has been adapted from Ref. [88].

. . . . . . . . . . . . . . . . . . . . . . . . . 101

vi

Table 4.7 The numbers of protons (Δ𝑝) and neutrons (Δ𝑛) excited from the (2𝑠1/2, 1𝑑3/2)
to (1 𝑓7/2, 2𝑝3/2) orbitals calculated with the ZBM2 [95] and ZBM2m [96]
interactions. The percentages (%) for different configurations are given for the
11/2− state of 39Ca (top) and for the 2+ state of 38Ca (bottom). The percentage
for each of the higher-order configurations is less than 6%, and only the sum
of these configurations is listed as “other” in the table. This table has been
adapted from Ref. [88].

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

vii

Figure 1.1

LIST OF FIGURES

(a) At the highest energies and shortest scales, nucleons are built from quark-
gluon interactions with many degrees of freedom at high computational costs.
(b) By using an effective model of quarks, hadrons can be modeled with less
degrees of freedom by integrating out the gluons.
(c) A further reduction
in degrees of freedom can be implemented by building nuclei from up from
inter-nucleon interactions utilizing a potential or pion-exchange. (d) A nu-
cleon in a mean-field potential allows for single-particle excitations up to the
nucleon separation energy. (e) Effective fields reduce the many-body problem
to critical building blocks such as proton and neutron densities and currents.
(f) Collective models can be implemented for large 𝐴 systems which are char-
acterized by vibrational and rotational modes. The figure has been adopted
from Ref. [1].

.

.

.

.

.

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

Figure 1.2 The potential energy for single particle proton states is presented as a function
of radius r for the 58Zn nucleus. The total potential (red) is displayed as the
sum of the relativistic mean-field (RMF) nuclear potential (black dash), the
Coulomb potential (green dash), and centrifugal potential (blue dash). Also
displayed are the single particle proton states below 0 MeV and continuum
states just above 0 MeV [28]. Used with permission of World Scientific
Publishing Co., Inc., from ‘’Study of neutron magic drip-line nuclei within
relativistic mean-field plus BCS approach”, G. Saxena et al., 22, 05 (2013);
permission conveyed through Copyright Clearance Center, Inc.

. . . . . . . . .

Figure 1.3

(a) The nuclear density 𝜌(𝑟) is understood to be relatively constant within
the the interior of the nucleus as the nucleon-nucleon forces tend to average
out, but decreases toward the nuclear skin. (b) The nuclear potential 𝑈 (𝑟) is
flat within the nuclear interior while tapering toward the nuclear skin. The
harmonic oscillator potential “H.O.” is used as a first order estimation for the
nuclear potential. (c) The nuclear shell model single-particle energy solutions
for various potentials. Harmonic oscillator solutions (left) could not explain
the experimentally observed magic numbers above 20. The development of
the Woods-Saxon potential (middle) split the degeneracy of the harmonic os-
cillator levels according to their orbital angular momentum 𝑙2. The inclusion
of the spin-orbit coupling term (cid:174)𝑙 · (cid:174)𝑠 (right) was necessary to replicate the
observed magic numbers above 20 [26]. Reprinted figure with permission
from T. Otsuka, A. Gade et al., Rev. Mod. Phys., vol. 92, p. 015002 (2020).
Copyright (2024) by the American Physical Society.

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

viii

2

5

7

Figure 1.4 The nuclear chart. Nuclei are arranged by atomic number 𝑍 and neutron
number 𝑁. The color scheme is representative of the ground-state half-life
with the valley of stability depicted in black, grey areas representing the
unbound nuclei, and nuclei with lifetimes ranging from single milliseconds
(light green) to years (dark blue). Vertical and horizontal rectangles outline
the traditional magic numbers. The inset depicts the area near doubly magic
40Ca with mirror nuclei 39Ca and 39K and the 𝑇 = 1 isobaric triplet of 38Ca,
38K, and 38Ar. The figure has been adapted from Ref. [33].

. . . . . . . . . . .

9

Figure 1.5 E2 transition matrix elements calculated with the USD, USDA, and USDB
Hamiltonians which utilize two-body matrix elements and single-particle
energies to fit 608 states over 77 𝑠𝑑-shell nuclei ranging from 𝐴 = 16 through
𝐴 = 40 [38]. Top: Experimental values of 𝑀𝑝(E2) compared to calculation
with the USDB effective interaction using free-nucleon charges 𝑒 𝑝 = 1.0
and 𝑒𝑛 = 0 (left) and effective charges 𝑒 𝑝 = 1.36 and 𝑒𝑛 = 0.45 (right).
Middle: Comparison of the bare matrix elements 𝐴𝑝 and 𝐴𝑛 calculated with
the USDA and USDB effective interactions. Bottom: Comparison of 𝐴𝑝 and
𝐴𝑛 calculated with the USD and USDB effective interactions [39]. Reprinted
figure with permission from W.A. Ritcher, B.A. Brown et al., Phys. Rev.
C, vol. 78, p. 064302 (2008). Copyright (2024) by the American Physical
Society.

.

.

.

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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 1.6 Proton matrix element values 𝑀𝑝 plotted as a function of isospin projection
𝑇𝑧 for the 2+ → 0+ E2 transition in six 𝑇 = 1 isobaric triplets [44]. Reprinted
figure with permission from F.M. Padros Estévez, A.M. Bruce et al., Phys.
Rev. C, vol. 75, p. 014309 (2007). Copyright (2024) by the American
Physical Society.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

.

Figure 1.7 The ratios 𝑀𝑛/𝑀𝑝 for the lowest 2+ → 0+ transitions in various single closed
shell (SCS) nuclei and doulby magic 48Ca and 208Pb. The data was obtained
from hardon scattering experiments and the mirror method described by
Eqn. 1.7. Measurements are compared to the homogeneous collective model
expectation (𝑁/𝑍), the no-free parameter schematic model (NPSM), and the
one-free parameter schematic model (OPSM) [46]. Reprinted from Physics
Letters B, Vol. 4, A.M. Bernstein et al. Neutron and proton transition matrix
elements and inelastic hadron scattering, Pages No. 255-258, Copyright
(1981), with permission from Elsevier.

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ix

Figure 1.8 The isoscalar 𝑀0 (black circles) and isovector 𝑀1 (red triangles) components
the of the 2+ → 0+ transitions for odd-odd 𝑇 = 1 nuclei across a wide, low-
mass region of the nuclear chart. The figure shows relatively no change in
the isovector components while isoscalar components begin a sharp increase
near 𝐴 = 40, where the 𝑠𝑑 shell becomes maximally occupied. Additionally,
the matrix elements derived from even-even 𝑇 = 0 nuclei along 𝑁 = 𝑍 are
provided by the solid line. The isoscalar components of the 𝑇 = 1 nuclei
appear to trend accordingly with the isoscalar values from the 𝑇 = 0 nuclei.
The figure has been adopted from Figure 4 of Ref. [51].

. . . . . . . . . . . . . 19

Figure 2.1 Beam line of the National Superconducting Cyclotron Laboratory (NSCL) [61].
The Superconducting Source for Ions (1) feeds ions to the Coupled Cyclotrons
(2) which accelerates the primary beam particles to their maximum energy.
The primary beam impinges on a production target (3) in which fragmentation
reactions create a beam of various species of nuclei. These species are sepa-
rated by the A1900 Fragment Separator (4) to produce the desired secondary
beam(s). The secondary beam is sent to the S3 vault where TRIPLEX and
GRETINA (sections 2.5.1 and 2.4.1, respectively) are located at the base of
the S800 Spectrometer (8). Nuclei in excited states are produced in reactions
on the TRIPLEX target, gamma rays are detected by GRETINA, and the
recoil nuclei are identified by the S800. The beam may also be directed to
other vaults where detectors such as the MONA (5) and the decay station (7)
may be utilized. The beam can also be stopped and reaccelerated by the ReA
facility (12) which directs a lower energy beam (single MeV/u) to a variety of
other detectors such as the ATTPC (14) and SOLARIS (not depicted). The
figure has been adopted from Ref. [61].

. . . . . . . . . . . . . . . . . . . . . 22

Figure 2.2 SUperconducting Source for Ions (SuSI) [54] at the NSCL. Inside the ion
source, a gaseous 58Ni is stripped of electrons with the Electron Cyclotron
Resonance (ECR) method. Ions in a variety of charge states are ejected from
the ion source and the beam is bent toward the Coupled Cyclotron Facility by
the red dipole magnet. Photograph adopted from Ref. [62].

. . . . . . . . . . . 24

Figure 2.3 Coupled Cyclotron Facility design concept logo. [65]. The figure depicts the
K500 and K1200 coupled cyclotrons and particles accelerated by the “dee”
electrodes of the K500 which are then injected into the K1200 where they are
accelerated to a higher energy, stripped of excess electrons, and then ejected
toward the production target. The figure has been adopted from Ref. [65].
Reproduced with permission from Springer Nature.

. . . . . . . . . . . . . . . 26

x

Figure 2.4 The A1900 Fragment Separator. The primary beam is directed from the
CCF to the production target (left) where reactions produce various species
of nuclei. Dipole magnets (green) bend the ions’ trajectories (red, blue, and
green lines) within the dispersive plane according to their magnetic rigidities.
Sets of quadrupole magnets (small rectangles and squares) focus the beam
after the production target, between dipoles, and before the focal plane slits
(right). The dipoles are tuned to allow species of the desired mass-charge
ratio 𝐴/𝑍 to follow a central trajectory in the dispersive plane while the
undesired ions are bent with larger or smaller radii 𝜌. An aluminum wedge
separates ions of similar 𝐴/𝑍 via differential energy loss thru the wedge-
shaped degrader according the square of the ions’ charge 𝑍 2. A second set
of two dipoles further separates the desired ions based on a different 𝐴/𝑍
ratio. Finally, slits at the focal plane cut out ions moving outside the central
trajectory, leaving the secondary beam to be transmitted to the experimental
vault. The figure has been adopted from Ref. [66].

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

. 28

Figure 2.5 The S800 Spectrograph. After exiting A1900 focal plane scintillator XFP
(left of diagram), the secondary beam components interact with the S800
object plane scintillator (OBJ). The time-of-flight (TOF) method is used to
determine the secondary beam components with the E1 scintillator at the end
of the focal plane acting as a reference and trigger. The focal plane and the
detector components that comprise it are presented in Fig. 2.6. The figure has
been adopted from Ref. [70].

. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.6 Focal plane of the S800 spectrometer. Detectors located at the focal plane
include a pair of CRDCs which provide each ion’s trajectory, an ion chamber
which provides energy loss information, and the E1 scintillator (labeled “plas-
tic scintillator”) which acts as a trigger for the S800 DAQ and provides timing
information from the XFP and OBJ scintillators. The IsoTagger hodoscope
provides information about the ion’s kinetic energy and charge state, and is
also utilized for identifying long-lived isomeric states but was not utilized in
this work. The figure has been adopted from Ref. [71].

. . . . . . . . . . . . . 33

Figure 2.7 Cathode Readout Drift Chambers (CRDCs) at the focal plane of the S800.
The two interaction points (X1,Y1) and (X2,Y2) determine the ion trajectory
in the focal plane. Beam ions and recoil nuclei ionize gas in the CRDC
chambers, the ionized electrons are collected by multiple anode wires, the
charge on the anodes induces an opposite charge on the cathode pads. The
ion’s position in the dispersive plane (X) is determined by a Gaussian fit to the
charge distribution across the cathode pads (inset) on an event-by-event basis.
The drift time of the ionized electrons to each anode wire is determined
with respect to the E1 trigger signal and provides the ion position in the
non-dispersive plane (Y). The figure has been adopted from Ref. [71].

. . . . . 36

xi

Figure 2.8 Unhoused view of the ion chamber at the focal plane of the S800. 16 parallel-
plate ion chambers are stacked perpendicular to the beam axis (left to right)
and cover an active area of ≈ 30×60 cm. When housed, the chamber contains
a mixture of 90% argon 10% methane, typically at a pressure of 300 torr. The
picture has been adopted from Ref. [71].

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

. 39

Figure 2.9 Energy loss in the S800 ion chamber as a function of time-of-flight between
the E1 and OBJ scintillators. The energy loss vs TOF method is used to distin-
guish recoil nuclei from reactions between our choice of incoming secondary
beam (in this case 42Sc) and outgoing reaction products.

. . . . . . . . . . . . 41

Figure 2.10 The Gamma-Ray Energy Tracking In-beam Nuclear Array with 12 modules
positioned for perpendicular and forward focused detection. The module
frame has been opened to allow connection of the TRIPLEX beam pipe to the
S800 and alignment of the target with the beam spot. The secondary beam
moves left to right and interacts with the TRIPLEX target which is located
13 cm upstream from the center of GRETINA. The beam then continues
through the S800 for particle identification.

. . . . . . . . . . . . . . . . . . . 44

Figure 2.11 Geant4 rendering of the GRETINA module configuration utilized for this
work. Each of the 12 GRETINA modules is comprised of a liquid nitrogen
cryo-module (grey), 4 hexagonal crystals (green and white representing type A
and B), with each crystal having 36-fold segmentation. Looking downstream
(left), the beryllium target (green) and the 90◦ ring of modules are rendered.
Looking upstream (right), the tantalum degrader (red) and the forward facing
modules are rendered. Not pictured for clarity are TRIPLEX and the beam
pipe which accounts for the uncovered area between 0 and 20 degrees with
respect to the beam axis (into and out of the page).

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

. 45

Figure 2.12 Diagram of a GRETINA “quad” (a) and a single crystal segmentation (b) [58].
Each GRETINA module contains a quad of four, closely-packed HPGe crys-
tals (a) with the two pairs of crystals each having unique tapered-hexagonal
shapes (A-type and B-type) and length of approximately 9 cm and maximum
width of 8 cm. The tapered-hexagonal shapes provide maximum detector
volume and solid-angle coverage. Each crystal contains a 36-fold segmen-
tation (b) with a central core contact to collect total energy deposited and
each segment with an individual outer-surface contact pad to determine in-
teraction location (36 pads per crystal). The segmentation of each crystal
allows for interaction position resolution of approximately 2 mm and track-
ing of gamma-ray interaction points throughout the detector volume [58].
Reprinted from Nuclear Instruments and Methods in Physics Research Sec-
tion A: Accelerators, Spectrometers, Detectors and Associated Equipment,
Vol. 709, S. Paschalis et al.,The performance of the gamma-ray energy track-
ing in-beam nuclear array GRETINA, Pages No. 44-55, Copyright (2013),
with permission from Elsevier.

. . . . . . . . . . . . . . . . . . . . . . . . . . 46

xii

Figure 2.13 The efficiency of GRETINA at the 152Eu gamma-ray energies observed in the
lab-frame. The efficiency of the simulation at a given energy is given in blue
while the efficiency of GRETINA is given in orange. . . . . . . . . . . . . . . . 47

Figure 2.14 Diagram of the TRIple PLunger for EXotic beams [57]. TRIPLEX is designed
to hold a production target and two degraders. In this diagram, the secondary
beam enters from the left then interacts with a 9Be target mounted on the target
cone which can be moved toward or away from the stationary first degrader
cone to allow measurement of lifetimes generally the 10s of picoseconds.
Although TRIPLEX can hold a second degrader in the secondary degrader
cone, the second degrader was not required for this experiment but the second
degrader cone held a stripper foil to increase the number of fully stripped ions.
Reprinted from Nuclear Instruments and Methods in Physics Research Section
A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol.
806, H. Iwasaki et al., The TRIple PLunger for EXotic beams TRIPLEX for
excited-state lifetime measurement studies on rare isotopes, Pages No.123-
131, Copyright (2016), with permission from Elsevier.

. . . . . . . . . . . . . 52

Figure 2.15 Full view of the TRIple PLunger for EXotic beams [57]. The secondary
beam enters TRIPLEX from the back (left) and interacts with the target and
degraders (right). The outer-most cylindrical housing protects motor control
electronics and the TESA precision distance measurement system. A target
motor is displayed on the left (small cylinder parallel with housing), this
motor moves the inner cylinder which connects to the target ring.

. . . . . . . . 53

Figure 2.16 The 9Be(42Sc, 39Ca*)X reaction produces nuclei in excited states (denoted by
“*”). Gamma rays may be emitted from “fast” nuclei of velocity 𝛽1 or “slow”
nuclei of 𝛽2, having been slowed by the degrader. The gamma-ray emittance
angle (y-axis) is plotted as a function of the Doppler-corrected gamma-ray
energy (x-axis) and the number of counts at a particular angle and energy
(z-axis).

. .

. .

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 3.1 Top: Angle 𝑎 𝑓 𝑝 in the focal plane as a function of the uncorrected time
between the E1 and OBJ scintillators. Bottom: Angle 𝑎 𝑓 𝑝 in the focal plane
as a function of the corrected time between the E1 and OBJ scintillators After
the timing correction, the tilted distributions (top) become aligned (bottom)
within narrow time distributions such that the secondary beam nuclei are well
separated in time. These plots serve as examples of the timing corrections
that were performed as the first step of the data calibration process.

. . . . . .

. 64

xiii

Figure 3.2 Secondary beam components separated by the TOF method where the time
differences between the E1 and XFP or OBJ scintillators are plotted on their
respective axes. The axes represent arbitrary units of time and the 𝑁 = 21
isotones have been labeled for the reader. This plot was created from a
run without the target installed in TRIPLEX and is used to characterize the
secondary beam properties.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 3.3 Secondary beam components separated by the TOF method for a target-only
data run. The axes represent arbitrary units of time and the 𝑁 = 21 isotones
have been labeled for the reader. This plot was created from a run with the
target installed in TRIPLEX. Gating on a particular secondary beam such as
42Sc for this work allows calibration and analysis of all reaction products from
that beam.

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

.

.

.

.

.

.

.

. 66

Figure 3.4 The corrected energy loss of 39Ca in the ion chamber (y-axis) is independent
of the ion’s position (x-axis) in the dispersive plane. Though this ion chamber
data correction was performed with a secondary beam of 42Sc and reaction
product 39Ca, the calibration holds for our mirror nucleus 39K as well.

. . . . . 67

Figure 3.5 The induced signal (y-axis) for a given CRDC pad (x-axis) will be relatively
constant after the pad gain-matching calibration. The pad readings from the
39K recoil nuclei are given as an example of a corrected spectrum for a single
CRDC. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

.

.

.

.

.

.

.

.

Figure 3.6 CRDC X and Y position calibration using the mask pattern. The aluminum
mask was inverted when placed in front of the first CRDC, but the position
calibration holds so long as each cluster of data points on the left is correctly
matched to the associated mask hole on the right. An outline of the “L”
pattern has been added to show correspondence between the data in the left
panel and the calibration pattern in the right panel. . . . . . . . . . . . . . . .

. 69

Figure 3.7 Trajectory distributions at the TRIPLEX target position for the unreacted 42Sc
beam. Experimental data for the angle in the dispersive plane 𝑎𝑡𝑎, angle in
the non-dispersive plane 𝑏𝑡𝑎, position in the non-dispersive plane 𝑦𝑡𝑎, and the
energy distribution 𝑑𝑡𝑎 are displayed in blue. The simulated secondary beam
trajectory information is displayed in red. . . . . . . . . . . . . . . . . . . . . . 71

Figure 3.8 Trajectory distributions at the TRIPLEX target position for the reaction prod-
uct 39Ca taken during a target-only run. Experimental data for the angle in
the dispersive plane 𝑎𝑡𝑎, angle in the non-dispersive plane 𝑏𝑡𝑎, position in the
non-dispersive plane 𝑦𝑡𝑎, and the energy distribution 𝑑𝑡𝑎 are displayed in blue
while the simulation is displayed in red.

. . . . . . . . . . . . . . . . . . . . . 72

xiv

Figure 3.9 Trajectory distributions at the TRIPLEX degrader position for the reaction
product 39Ca taken during a run with target and degrader mounted. Data for
the angle in the dispersive plane 𝑎𝑡𝑎, angle in the non-dispersive plane 𝑏𝑡𝑎,
position in the non-dispersive plane 𝑦𝑡𝑎, and the energy distribution 𝑑𝑡𝑎 are
displayed in blue while the simulation is displayed in red.

. . . . . . . . . . . . 73

Figure 3.10 Level diagrams for gamma rays observed in the target-only spectra for 39Ca
(left) and 39K (right). The energy information is from Ref.[45] and rounded
to keV. The relative intensities of the observed gamma-ray transitions, in-
dicated by the widths of the arrows, are normalized to the intensity of the
7/2− → 3/2+
𝑔.𝑠. transition for the respective nuclei. The energy information
for the (11/2−) state is highlighted in red, while those for the (9/2−) state are
highlighted in blue [88]. Reprinted figure with permission from A. Sanchez
et al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the
American Physical Society.

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

Figure 3.11 Doppler-corrected gamma-ray spectrum for 39Ca for the target-only run. Tran-
sitions from the (11/2−) state are labeled in red, from which the lifetime is
determined in the present work. Gamma rays from the (9/2−) state are labeled
in blue, for which we provide an improved lifetime result [88]. Reprinted fig-
ure with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p.
024322 (2024). Copyright (2024) by the American Physical Society.

. . . . .

. 74

. 76

Figure 3.12 Doppler-corrected gamma-ray spectrum for 39Ca taken with the 1-mm target-
degrader separation. The spectrum has been gated for the gamma-ray emis-
sion angle between 𝜃 = 29◦ and 𝜃 = 46◦ to improve sensitivity to Doppler
shifts. Transitions from the (11/2−) state are labeled in red, while transi-
tions from the (9/2−) state are labeled in blue. Laboratory-frame background
gamma rays, associated with neutron-induced reactions [89], are shown in
green and downscaled by a factor of 2 for clarity [88]. Reprinted figure with
permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024).
Copyright (2024) by the American Physical Society.

. . . . . . . . . . . . . . 79

Figure 3.13 Doppler-shift corrected gamma-ray spectrum for 39Ca taken with the 0-mm
target-degrader separation setting. The spectrum has been gated between
𝜃 = 29◦ and 𝜃 = 46◦ to improve sensitivity to Doppler shifts. Transitions
from the (11/2−) state are labeled in red, while transitions from the (9/2−)
state are labeled in blue. Laboratory-frame background gamma rays are
shown in green and downscaled by a factor of 2 [88]. Reprinted figure with
permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024).
Copyright (2024) by the American Physical Society.

. . . . . . . . . . . . . . 81

xv

Figure 3.14 Doppler-shift corrected gamma-ray spectrum for 39K from the target-only
measurement. Transitions from the (11/2−) state are labeled in red, while
transitions from the (9/2−) state are labeled in blue [88]. Reprinted figure
with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322
(2024). Copyright (2024) by the American Physical Society.

. . . . . . . . . . 85

Figure 3.15 Doppler-shift corrected gamma-ray spectrum for the 1-mm target-degrader
separation setting in 39K. The spectrum has been gated between 𝜃 = 29◦ and
𝜃 = 46◦ to improve sensitivity to Doppler shifts. Transitions from the (11/2−)
state are labeled in red, while transitions from the (9/2−) state are labeled
in blue. Laboratory-frame background gamma rays are shown in green and
downscaled by a factor of 2 [88]. Reprinted figure with permission from A.
Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright (2024)
by the American Physical Society.

. . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 3.16 Doppler-corrected gamma-ray spectrum for the 39K 0-mm target-degrader
separation. The spectrum has been gated between 𝜃 = 29◦ and 𝜃 = 46◦
to improve sensitivity to Doppler shifts. Transitions from the (11/2−) state
are labeled in red, while transitions from the (9/2−) state are labeled in
blue. Laboratory-frame background gamma rays are shown in green and
downscaled by a factor of 2 [88]. Reprinted figure with permission from A.
Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright (2024)
by the American Physical Society.

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

. 88

Figure 4.1 Ratios 𝑀𝑛/𝑀𝑝 for calcium isotopes near 𝑁 = 𝑍 compared to the homoge-
neous collective model (𝑁/𝑍) and the single-closed shell (SCS) esitmation
𝑒 𝑝/𝑒𝑛 = 3.0(4). The figure shows an enhanced 𝑀𝑝 when compared to the
SCS expectation. The 𝐴 = 36 ratio was obtained from Refs. [49, 48]. The
𝐴 = 38 ratio is from Refs. [43, 44]. The 𝐴 = 39 ratio is from this work [88].
And Ref. [40] provides the 𝐴 = 42 ratio.

. . . . . . . . . . . . . . . . . . . . . 91

Figure 4.2 Level energies of the lowest 7/2−, 9/2−, 11/2−, and 13/2− states of 39Ca from
experiment (Exp) compared to shell-model calculations with the FSU [94],
ZBM2 [95], and ZBM2m [96] effective interactions [88]. Reprinted figure
with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322
(2024). Copyright (2024) by the American Physical Society.

. . . . . . . . . . 94

xvi

CHAPTER 1

INTRODUCTION

The field of nuclear structure studies endeavors to explain how the nucleons within an atomic

nucleus arrange themselves under different conditions and how these configurations are expressed

in various measurable quantities such as reaction cross sections, transition strengths, and state

lifetimes. The nucleus offers a rich environment for physics research as it lies at the intersection

of three of the four fundamental forces:

the strong nuclear force, the Coulomb force, the weak

nuclear force, and the gravitational force, which is commonly ignored in this field due to its weak

interaction at small mass scales relative to the other three forces. While the interactions of these

forces provide many avenues for examination, even if one ignores the weak and gravitational

interactions, determining how nucleons arrange themselves quickly becomes a complex problem

even for some of the lightest nuclei.

In Fig. 1.1 , the problem of modeling the atomic nucleus is presented in terms of different scales

and degrees of freedom. Quantum chromodynamics attempts to build nuclei from fundamental

quark-gluon interactions [2], but the amount of computational power necessary to describe even

a helium nucleus can be daunting due to the number of degrees of freedom involved [3]. Some

ab-initio nuclear calculations such as Green’s function Monte Carlo (GFMC) [4, 5, 6], no-core shell

mode (NCSM) [7, 8, 9], variational Monte Carlo (VCM) [10], and coupled cluster (CC) [11, 12]

attempt to simplify the problem by restricting the degrees of freedom to the properties of protons

and neutrons; but even these simplified models of 2 and 3 body nucleon-nucleon interactions can

challenge technical capabilities for calculations involving light nuclei of 𝐴 < 20. Nuclear mean

field models which utilize effective interactions, such as Hartree-Fock based calculations [13],

can further reduce the degrees of freedom by modeling nuclei as a system of 𝐴 nucleons which

each move in an external field generated by the remaining 𝐴 − 1 nucleons [14] . Effective field

theories (EFTs) [15] utilize nucleon motion and density to reduce light systems down to manageable

problems while collective models can be used to describe large mass systems in terms of vibrational

and rotational modes, similar to the liquid-drop model.

1

Figure 1.1 (a) At the highest energies and shortest scales, nucleons are built from quark-gluon
interactions with many degrees of freedom at high computational costs. (b) By using an effective
model of quarks, hadrons can be modeled with less degrees of freedom by integrating out the
gluons.
(c) A further reduction in degrees of freedom can be implemented by building nuclei
from up from inter-nucleon interactions utilizing a potential or pion-exchange. (d) A nucleon in
a mean-field potential allows for single-particle excitations up to the nucleon separation energy.
(e) Effective fields reduce the many-body problem to critical building blocks such as proton and
(f) Collective models can be implemented for large 𝐴 systems
neutron densities and currents.
which are characterized by vibrational and rotational modes. The figure has been adopted from
Ref. [1].

2

Many of the challenges in structure study are a result of the complexity of the strong interaction,

which is strongly repulsive at nucleon separations of 𝑟 < 0.7 fm, attractive at 𝑟 ≈ 1 fm, and

diminishes rapidly as 𝑟 approaches 2 fm [16]. When one includes the electromagnetic effects of

protons, the dynamics of nucleon motion, and the effects of the angular momentum and intrinsic

spin of each nucleon, simplified models become increasingly necessary to describe heavier nuclei.

The problem of nuclear structure can be simplified when the concept of isospin is introduced. In

the isospin model, protons and neutrons are viewed as two states of the same particle, the nucleon,

with different isospin projections [17]. The study of mirror energy differences (MEDs); that is, the

difference in the energies of mirror states, is one common method of probing how well the isospin

model holds in mirror nuclei. Of course, the isospin model is an approximation as it ignores the

charge of the proton; but this approximation works remarkably well when one compares the spin

assignments and energies of level diagrams for mirror nuclei, with MEDs typically on the order of

10 − 100 keV [18]. Once isospin has been invoked, differences in mirror energy levels can then be

attributed to isospin non-conserving forces such as the Coulomb interaction.

Transition strengths between states is another common method of investigating nuclear struc-

ture. When significant discrepancies are observed in the transition strengths of mirror nuclei,

opportunities may arise to investigate the underlying mechanisms involved in isospin symmetry

breaking [19, 20]. The measurements of reduced transition strengths 𝐵(E2) between the first 2+ and

0+ ground state has led to the observation of unexpected behaviors such as collectivity [21], shape

coexistence [22, 23], intruder configurations [24], and shell evolution [25, 26] in exotic regions of

the nuclear chart such as the neutron-rich island of inversion [27].

This work endeavors to exploit isospin symmetry and reduced transition strengths through the

method of isospin decomposition and precision measurements to shed new light on cross-shell

excitations, even in close proximity to doubly magic 40Ca. From the reduced transition strengths

of mirror transitions, we can determine the neutron and proton matrix elements of a transition and

infer from the experimental data and theory calculations that significant contributions from both

proton and neutron cross-shell excitations must be invoked to describe low-lying excited states near

3

stability. The aim of the following discussion is to provide the background required for a qualitative

and quantitative understanding of the reduced transition strength 𝐵(E2) and the matrix elements

𝑀𝑝/𝑛, which serve as benchmarks for the testing of modern effective-interaction models and their

relevancy to modern nuclear structure studies.

1.1 Nuclear Structure

The mass of a nucleus can approximated as 𝐴u where 𝐴 = 𝑁 + 𝑍 is the sum of neutrons 𝑁

and protons 𝑍 and u is the atomic mass unit, defined as one-twelfth of the mass of a carbon-12

atom. If we neglect the relatively small amount of binding energy which contributes to the mass

of the nucleus, it allows for the reasonable approximation that the proton and neutron have equal

masses; indeed, the number 𝐴 is commonly referred to as the “mass number” of a nucleus. Given

these definitions, a common notation for describing a nucleus is 𝐴

𝑍 X𝑁 , where 𝑋 is the chemical

symbol of the element. A more concise and common notation, which we will employ in this work,

is given as 𝐴X since X implies the number of protons 𝑍 and the number of neutrons 𝑁 can be

deduced via 𝑁 = 𝐴 − 𝑍. A nucleus of 𝑍 protons may form a bound state with many different

numbers of neutrons 𝑁 and to these nuclei we assign the title “isotopes”; relevant to this work

are the calcium isotopes 38Ca and 39Ca which both have 𝑍 = 20 protons but contain 𝑁 = 18 and

𝑁 = 19 neutrons, respectively. Particularly relevant to this study are the subset of isobars (nuclei

of the same mass number 𝐴) which we call “mirror nuclei”, these are sets of nuclei whose proton

and neutron numbers are exchanged such as 39Ca with 𝑍 = 20 and 𝑁 = 19, and 39K with 𝑍 = 19

and 𝑁 = 20.

Protons carry a charge of +1 and tend to strongly repel one another via Coulomb repulsion

when protons are within a few femtometers of each other. Neutrons carry no charge but tend to

bind the nucleus together via the strong force, which attracts nucleons on the same distance scale

that the Coulomb force repels. Relevant to the following conversation is the concept of isospin

symmetry [17], which proposes that neutrons and protons be considered two states of the same

particle with different isospin projections 𝑇𝑧. Since both protons and neutrons have an intrinsic

spin of 1/2 and nearly equal masses, it is useful to model nucleons as an isospin doublet of isospin

4

𝑇 = 1/2 and projections 𝑇𝑧 = −1/2 for protons and 𝑇𝑧 = +1/2 for neutrons. When speaking of an

entire nucleus with many bound nucleons, we consider the total isospin projection of the nucleus

𝑇𝑧 = (𝑁 − 𝑍)/2 with 2𝑇 + 1 eigenvalues ranging from 𝑇𝑧 = −𝑇 up to 𝑇𝑧 = +𝑇 in integer steps. A

consequence of conservation of isospin is that one expects mirror symmetry from mirror nuclei;

that is, the energy level diagrams of mirror nuclei should appear very similar with differences

attributed to isospin non-conserving forces such as the Coulomb force.

Figure 1.2 The potential energy for single particle proton states is presented as a function of radius
r for the 58Zn nucleus. The total potential (red) is displayed as the sum of the relativistic mean-field
(RMF) nuclear potential (black dash), the Coulomb potential (green dash), and centrifugal potential
(blue dash). Also displayed are the single particle proton states below 0 MeV and continuum states
just above 0 MeV [28]. Used with permission of World Scientific Publishing Co., Inc., from ‘’Study
of neutron magic drip-line nuclei within relativistic mean-field plus BCS approach”, G. Saxena et
al., 22, 05 (2013); permission conveyed through Copyright Clearance Center, Inc.

Both protons and neutrons carry an angular momentum 𝑙 which, when large, provides additional

binding due to the centrifugal barrier. Fig. 1.2 provides a nuclear potential (in this case, for 58Zn)

in red which is the sum of the nuclear, Coulomb, and centrifugal components [28]. The figure

displays the energy as a function of distance 𝑟 and conveys the concepts of attraction and repulsion

5

at the nucleon scale via the various components that comprise the total potential. At distances less

than approximately 2.5 fm, where the total potential is positive, protons repel each other primarily

due to the Coulomb and centrifugal potentials. Between 2.5 and 5 fm, protons are bound within

the negative potential well and will occupy single particle states such as the 1𝑠1/2 through 1 𝑓7/2

states depicted in the negative energy region. One notes on Fig. 1.2, though the term "centrifugal

barrier" is used to describe the blue dashed line, the same term is also used to describe the humped

region of the total potential between 6 and 10 fm, though this hump arises from the sum of the

Coulomb and centrifugal components and acts as a barrier in that a nucleon within the potential

well requires a large kinetic energy to escape the well, and a nucleon outside the hump requires the

same energy to penetrate the potential well from outside. Of course, the above explanation relies

on a classical understanding of potential and kinetic energies, but an explanation more relevant to

this work requires a quantum mechanical understanding.

Once the potential of a nuclear system is modeled, such as in Fig. 1.2, the Schrödinger equation

𝐻Ψ = 𝐸Ψ can be solved for the energies 𝐸 of the states of the system where 𝐻 is the Hamiltonian

which contains information about the nucleon momentum (linear and angular), spin, and various

potential energy approximations, and Ψ is the wave function which contains information about

the orbital occupations of the nucleons. Generally, the Hamiltonian will tend to be a sum 𝐻 =

𝑉0 + 𝑉1 + 𝑉2 + ... of the various nuclear, and/or electromagnetic potentials 𝑉𝑛 that comprise the total

potential and is typically represented as a matrix with eigenvalue and eigenstate solutions calculated

from the Schrödinger equation for various wave function combinations. When the Hamiltonian has

been diagonalized with Ψ𝑛 wavefunctions (with 𝑛 = 1, 2, 3, ... distinguishing various single-particle

configurations), the diagonal matrix elements provide the level energies; therefore, level energies

are important, providing experimentally-measured values to which effective interactions can be fit

to and which model calculations are tested against.

In one of its simplest forms, a Hamiltonian can be modeled with a harmonic oscillator potential

which, when solved for, provides single-particle state energies in multiples of ℏ𝜔. Historically,

the variations of the 3-dimensional harmonic oscillator model, whose solutions are functions of

6

the principle quantum number 𝑛 and orbital angular momentum 𝑙, were the blocks upon which the

modern nuclear shell model was built. Experimental data accumulated in the mid 20th century had

shown that when nuclei contain certain numbers of neutrons or protons the binding energy of the

nucleus is greatly increased and the nucleus appears to exist in a more stable configuration, these

numbers would come to be described as “magic numbers” and found to be linked to the closed shell

orbitals; that is, orbitals whose shells are maximally occupied by nucleons.

Figure 1.3 (a) The nuclear density 𝜌(𝑟) is understood to be relatively constant within the the interior
of the nucleus as the nucleon-nucleon forces tend to average out, but decreases toward the nuclear
skin. (b) The nuclear potential 𝑈 (𝑟) is flat within the nuclear interior while tapering toward the
nuclear skin. The harmonic oscillator potential “H.O.” is used as a first order estimation for the
nuclear potential. (c) The nuclear shell model single-particle energy solutions for various potentials.
Harmonic oscillator solutions (left) could not explain the experimentally observed magic numbers
above 20. The development of the Woods-Saxon potential (middle) split the degeneracy of the
harmonic oscillator levels according to their orbital angular momentum 𝑙2. The inclusion of the
spin-orbit coupling term (cid:174)𝑙 · (cid:174)𝑠 (right) was necessary to replicate the observed magic numbers above
20 [26]. Reprinted figure with permission from T. Otsuka, A. Gade et al., Rev. Mod. Phys., vol.
92, p. 015002 (2020). Copyright (2024) by the American Physical Society.

Figure 1.3 shows that the harmonic oscillator model was able to reproduce the first few magic

7

numbers: 2, 8, and 20; but it predicted numbers above 20 that were not supported by the experimental

evidence. The development of the Woods-Saxon potential added a potential term that contained

information about the nuclear radius, surface diffuseness, and the potential well depth, which broke

the degeneracy of the harmonic oscillator solutions into levels of different 𝑙. A major breakthrough

in nuclear structure was the addition of the spin-orbit coupling term to the Woods-Saxon potential,

whose solutions are given on the right of Fig. 1.3. It was the work of Mayer and Jensen which

demonstrated that the addition of a spin-orbit term to the nuclear potential could replicate the magic

numbers above 20 [29, 30, 31]. By coupling the intrinsic spin 𝑠 and orbital angular momentum

𝑙, the degeneracy of the 𝑙 > 0 orbits is split into two levels of 𝑗 = 𝑙 ± 1

2. Today, energy states are
defined by the principle quantum number 𝑛, the angular momentum 𝑙 = 0, 1, 2, 3, .. = 𝑠, 𝑝, 𝑑, 𝑓 , ...,

and the total angular momentum 𝐽 = 𝐿 + 𝑆 such that every state in a shell structure can be described

with the 𝑛𝑙𝐽 notation and the state parity given by 𝜋 = (−1)𝑙. The introduction of spin-orbit

coupling led to a shared Nobel prize in 1963 [32] and served as a guidepost for nuclear research for

decades to follow.

Figure 1.4 depicts the currently observed and predicted nuclides with proton number on the

vertical axis and neutron number on the horizontal axis. The nuclei with traditional magic numbers

of closed shells are outlined with rectangular boxes and the intersections of vertical and horizontal

boxes correspond to the doubly magic nuclei such as 16O, 40,48Ca, 48,78Ni, etc... The chart has been

colored accordingly with the ground-state half-lives; unbound nuclei lie along the dripline and are

colored in grey, stable nuclei are colored black and form the valley of stability along the central

diagonal of the chart, the remaining nuclei have half-lives ranging from milliseconds (light green)

to years (dark blue). One should note that the chart does not follow a trend set by 𝑁 = 𝑍; the chart

favors neutron-rich nuclei as Coulomb repulsion between many protons tends to prevent binding of

nuclei with small 𝑁/𝑍 ratios. Also depicted in Fig. 1.4 is an inset depicting the region near doubly

magic 40Ca. The 𝐴 = 38 isobaric triplet of Ca, K, and Ar is shown, as is the 𝐴 = 39 doublet of Ca

and K, both of which constitute the focus of this work.

While the magic numbers from 8 through 126 guided structure studies in the era following

8

Figure 1.4 The nuclear chart. Nuclei are arranged by atomic number 𝑍 and neutron number 𝑁.
The color scheme is representative of the ground-state half-life with the valley of stability depicted
in black, grey areas representing the unbound nuclei, and nuclei with lifetimes ranging from single
milliseconds (light green) to years (dark blue). Vertical and horizontal rectangles outline the
traditional magic numbers. The inset depicts the area near doubly magic 40Ca with mirror nuclei
39Ca and 39K and the 𝑇 = 1 isobaric triplet of 38Ca, 38K, and 38Ar. The figure has been adapted
from Ref. [33].

their description by Mayer and Jensen, as research pushed further away from stability the idea of

immutable closed shells and permanent magic numbers was supplanted by the concept of shell

evolution. Traditionally, nuclei have been modeled as single particles bound by a mean field

potential, such as the one presented in Fig. 1.2. While this model simplifies the problem of n-

bodies each interacting with the other through the strong and Coulomb interactions, a problem

which quickly becomes unwieldy at even low 𝐴, it may struggle to describe sudden changes in

binding energy, nuclear radii, and transition strengths with respect to neighboring nuclei.

Some of the first indications of shell evolution were observed in what is now known as the

“island of inversion”, where the 𝑁 = 20 shell closure appears to decrease, resulting in first excited

9

states with unusually low energy [34]. One such famous example is the case of 32Mg, which lies

at the heart of the island of inversion and is characterized by a first 2+ state at 885 keV; one can

compare this to the doubly magic 40Ca with a first 2+ state residing at 3.9 MeV, or many non-magic

nuclei with first 2+ states typically near 2 MeV in this mass region. In the traditional spherical

shell-model picture, the ground state of 32Mg should be characterized by a fully occupied 1𝑑3/2

orbital; but due to a reduction in the 𝑁 = 20 shell gap the ground state is, in fact, described by a

2-particle 2-hole (2𝑝 − 2ℎ) configuration with two neutrons occupying the 1 𝑓7/2 or 2𝑝3/2 orbitals,

leaving the 1𝑑3/2 orbital unfilled. Similarly, the ground-state band 4+ state has been shown to

exhibit high collectivity with 𝐵(E2) strengths indicating significant contributions from 2𝑝 − 2ℎ and

4𝑝 − 4ℎ configurations [35].

The intruder configurations in the island of inversion may be difficult to explain with a traditional

spherical shell-model, but when one introduces nuclear deformation, which is closely tied to the

collective motion of nucleons, the evolution of shell gaps may be explained by the reduction or

even inversion of level energies and intruder configurations which become energetically favorable.

In fact, as the 1𝑑3/2 single-particle energy moves closer to the 1 𝑓7/2 orbital, one expects an increase

in contributions from 2𝑝 − 2ℎ and 4𝑝 − 4ℎ configurations as single-particle cross-shell excitations

into the 1 𝑓7/2 would result in negative parity states which are not observed [25]. Also, with the

reduction of the 𝑁 = 20 shell gap one would expect a shell gap to appear at 𝑁 = 16 arising from an

increased gap between the 1𝑑3/2 and 1𝑠1/2 orbitals. Indeed, such behavior has been observed near

the neutron drip line [36].

While it is now common to expect shell evolution in exotic nuclei with large 𝑁/𝑍 asymmetries,

there are still opportunities to uncover cross-shell excitations and loss of magicity near stability,

making the nuclei in close proximity to doubly magic 40Ca a region suitable for close examination

with highly sensitive experimental probes. This work is aims to reveal possible cross-shell excita-

tions in less exotic nuclei by examination of the electromagnetic transition which is governed by

the multipole operator O (𝜋𝜆), where 𝜋 is either electric 𝐸 or magnetic 𝑀 and 𝜆 is rank of the

operator as defined by the change in angular momentum 𝑙 from the initial and final states of the

10

electromagnetic transition.

When a photon is released by the transition of a nucleus from an initial state 𝐽𝑖 to a final

state 𝐽 𝑓 , the photon of spin 𝑆 = 1 will carry an amount of angular momentum 𝑙 such that

(cid:174)𝑙 = (cid:174)𝐽𝑖 − (cid:174)𝐽 𝑓 . The amount of angular momentum carried by the photon is given by the triangle

relation |𝐽𝑖 − 𝐽 𝑓 | ≤ 𝑙 ≤ 𝐽𝑖 + 𝐽 𝑓 , where 𝑙 is an integer. If the transition results in a change of parity,

the allowed 𝑙 are odd for electric transitions and even for magnetic transitions; e.g., transitions

such as E1, M2, E3, etc... If there is no change in parity between the initial and final states of

the transition, the allowed 𝑙 are odd for magnetic transitions and even for electric transitions; e.g.,

M1, E2, M3, etc... The multipolarity of the gamma radiation is defined by the angular momentum

carried by the photon where 𝑙 = 1 defines a dipole transition, 𝑙 = 2 a quadrupole transition, and

𝑙 = 3 is an octupole transition with higher-order transitions occurring in nuclei of large mass 𝐴.

This work is focused on the electric quadrupole (E2) transition and particularly the reduced

transition strength

𝐵(E2) =

| ⟨𝐽 𝑓 | |O (E2)| |𝐽𝑖⟩ |2
(2𝐽𝑖 + 1)

and the proton (𝑘 = 𝑝) and neutron (𝑘 = 𝑛) matrix elements

𝑀𝑘 = ⟨𝐽 𝑓 | |O (E2)| |𝐽𝑖⟩

where the one-body electric transition operator is given by

O (𝐸𝜆) = 𝑟𝜆

𝑘𝑌 𝜆 ( ˆ𝑟𝑘 )𝑒𝑘 𝑒

(1.1)

(1.2)

(1.3)

with 𝜆 = 2 for quadrupole transitions, 𝑟𝑘 giving the radial component of proton or neutron, 𝑌

representing the angular components as spherical harmonics , 𝑒𝑘 representing the charge of the

proton or neutron, and 𝑒 the fundamental charge [37]. The one-body operator acts on a single

particle (proton or neutron) and the overlap between the final and initial states represents the

transition of that particle between the states. Similarly, two-body operators may be introduced to

connect states with multiple particles.

Once the Hamiltonian has been solved for the eigen wave functions, the matrix element is

calculated from the overlap of final and initial states with O (𝐸2) according to Eqn. 1.2. The

11

matrix elements calculated for (𝐽𝑖 → 𝐽𝑖) provide the quadrupole moment of state 𝐽𝑖, while those

calculated for (𝐽𝑖 → 𝐽 𝑓 ) provide the reduced transition strength 𝐵(E2) (also known as the reduced

transition probability) between states 𝐽𝑖 and 𝐽 𝑓 , with “reduced” referring to the matrix element

being independent of the total angular momentum projection 𝑀𝐽. Experimentally, the 𝐵(E2) can

be calculated from measurements of the gamma-ray energy 𝐸𝛾, the half-life 𝑇1/2 of the initial state,

and branching ratio 𝑏 such that

𝐵(E2) =

564 · 𝑏
𝛾𝑇1/2
𝐸 5

𝑒2 𝑓 𝑚4

(1.4)

where the constant contains units of MeV5ps, making the reduced transition strength a valuable

measurement for testing nuclear structure models and fitting effective interactions.

For an effective interaction which can be used to predict 𝐵(E2) values in the framework of shell

models, the effective charges 𝑒𝑛 and 𝑒 𝑝 (for the neutron and proton, respectively) are employed

as a type of free parameter to account for properties of the model which may not be included or

accurately represented in the interaction such as core polarization, nucleon configurations within

the core, and orbitals that have been truncated from the model space but affect the accuracy of the

calculated 𝐵(E2) when compared to experimentally obtained values. If the free-nucleon charges

𝑒 𝑝 = 1.0 and 𝑒𝑛 = 0 are naively used for an effective interaction, the calculated results for the

proton matrix element 𝑀𝑝 tend to predict values less than the experimentally determined values,

as shown in the top-left panel of Fig. 1.5. To correct for these deficiencies, the bare proton 𝐴𝑝 and

bare neutron 𝐴𝑛 matrix elements are calculated from the effective interaction and then scaled by

the effective charges to determine the transition matrix element 𝑀𝑝 according to

𝑀𝑝 = 𝑒 𝑝 𝐴𝑝 + 𝑒𝑛 𝐴𝑛

(1.5)

These bare matrix elements allow 𝑀𝑝 to be described in terms of proton and neutron components

calculated with the free-nucleon charges and, as in the lower two panels of Fig. 1.5, may be used to

compare the results of effective interactions against each other. Similarly, under the assumption of

isospin symmetry, the neutron matrix element 𝑀𝑛, which may be difficult to precisely determine

12

Figure 1.5 E2 transition matrix elements calculated with the USD, USDA, and USDB Hamiltonians
which utilize two-body matrix elements and single-particle energies to fit 608 states over 77 𝑠𝑑-
shell nuclei ranging from 𝐴 = 16 through 𝐴 = 40 [38]. Top: Experimental values of 𝑀𝑝(E2)
compared to calculation with the USDB effective interaction using free-nucleon charges 𝑒 𝑝 = 1.0
and 𝑒𝑛 = 0 (left) and effective charges 𝑒 𝑝 = 1.36 and 𝑒𝑛 = 0.45 (right). Middle: Comparison of
the bare matrix elements 𝐴𝑝 and 𝐴𝑛 calculated with the USDA and USDB effective interactions.
Bottom: Comparison of 𝐴𝑝 and 𝐴𝑛 calculated with the USD and USDB effective interactions [39].
Reprinted figure with permission from W.A. Ritcher, B.A. Brown et al., Phys. Rev. C, vol. 78, p.
064302 (2008). Copyright (2024) by the American Physical Society.

13

by experiment, can be calculated with effective charges and bare matrix elements [40] with

𝑀𝑛 = 𝑒 𝑝 𝐴𝑛 + 𝑒𝑛 𝐴𝑝.

(1.6)

The values of these effective charges may depend on the model space; where common values for

the 𝑓 𝑝 space may be 𝑒 𝑝 = 1.5 and 𝑒𝑛 = 0.5 [41], or 𝑒 𝑝 = 1.36 and 𝑒𝑛 = 0.45 for the 𝑠𝑑 space [42].

When effective charges are used, the 𝑀𝑝 calculated from the effective interactions are increased

and tend to agree with the experimental values, as can be seen in the top-right panel of Fig. 1.5.

The neutron matrix element 𝑀𝑛 for a transition may be experimentally obtained, though, from

the same transition in a mirror nucleus via the relation

𝑀𝑛 (−𝑇𝑧) = 𝑀𝑝 (+𝑇𝑧)

(1.7)

where the neutron matrix element from the nucleus with isospin projection −𝑇𝑧 is equivalent to

the proton matrix element from the nucleus with +𝑇𝑧 [40]. For this work, the proton matrix

element 𝑀𝑝 (38K) of the mirror 11/2− → 7/2− transition provides the neutron matrix element

𝑀𝑛 (38Ca). Using this isospin decomposition method for mirror transitions leads to a fuller picture

of the transition and provides the method employed in this work to compare matrix element trends

between isobars of different isospin projection 𝑇𝑧 by comparing the ratio of matrix elements 𝑀𝑛/𝑀𝑝.

1.2 Systematic Trends in Matrix Element Values

Previous work has formulated that the proton matrix elements 𝑀𝑝 for the 2+ → 0+ E2 transitions

(𝑀𝑝(E2)) of 𝑇 = 1 isobaric triplets should follow a linear trend as a function of isospin projection

𝑇𝑧 according to

1
2
with 𝑀0 and 𝑀1 representing the isoscalar and isovector components, respectively [40]. Generally,

(𝑀0 − 𝑀1𝑇𝑧)

𝑀𝑝 (𝑇𝑧) =

(1.8)

the isovector component 𝑀1 is positive in light nuclei and smaller than the isoscalar component 𝑀0.

This leads to many 𝑇 = 1 isobars having 2+ → 0+ transition matrix elements of ratios 𝑀𝑛/𝑀𝑝 < 1

for proton-rich 𝑇 = 1 nuclei, as observed in Fig. 1.6. While this trend holds over a wide mass range,

a notable deviation has been observed in the 𝐴 = 38 triplet of Ca, K, and Ar [43, 44], as observed

14

in the bottom-middle panel of Fig. 1.6. Cottle et al. [43] observed a significant enhancement in

the 38Ca matrix element, deviating from the trend set by the 38Ar and 38K matrix elements and

therefore hinting at symmetry breaking. Indeed, assuming perfect isospin symmetry and a single

closed-shell model, one would expect 𝑀𝑝 for the 2+ → 0+ transition in 38Ca to be near zero and

therefore expect a large ratio 𝑀𝑛/𝑀𝑝 >> 1.

Figure 1.6 Proton matrix element values 𝑀𝑝 plotted as a function of isospin projection 𝑇𝑧 for the
2+ → 0+ E2 transition in six 𝑇 = 1 isobaric triplets [44]. Reprinted figure with permission from
F.M. Padros Estévez, A.M. Bruce et al., Phys. Rev. C, vol. 75, p. 014309 (2007). Copyright
(2024) by the American Physical Society.

The expectation of a large 𝑀𝑛/𝑀𝑝 ratio for single closed-shell nuclei can be derived from some

naive shell-model assumptions starting with Eqn. 1.5. If we assume a single-closed proton shell

for calcium isotopes, then the matrix element will be comprised exclusively of neutron components

𝑀𝑝,38𝐶𝑎 = 𝑒𝑛 𝐴𝑛. Similarly, assuming a single-closed neutron shell for argon implies that its matrix

element will contain only proton terms 𝑀𝑝,38𝐴𝑟 = 𝑒 𝑝 𝐴𝑝. Making no assumptions about the sign of

15

the matrix elements, we square of the ratio and find that

(cid:19) 2

(cid:18) 𝑀𝑛,38𝐶𝑎
𝑀𝑝,38𝐶𝑎

=

(cid:19) 2

(cid:18) 𝑀𝑝,38𝐴𝑟
𝑀𝑝,38𝐶𝑎

=

(cid:18) 𝑒 𝑝 𝐴𝑝,38𝐴𝑟
𝑒𝑛 𝐴𝑛,38𝐶𝑎

(cid:19) 2

=

(cid:19) 2

(cid:18) 𝑒 𝑝
𝑒𝑛

(1.9)

where, due to mirror symmetry, we expect 𝐴𝑝,38𝐴𝑟 = 𝐴𝑛,38𝐶𝑎, which cancels the bare matrix element
𝑒 𝑝
𝑒𝑛 .

terms in the ratio and leaves a ratio of matrix elements equal to the ratio of effective charges

Using commonly accepted effective charge values from the USD, USDA, and USDB Hamiltonians

𝑒 𝑝 = 1.36(5) and 𝑒𝑛 = 0.45(5) [38] and the isospin decomposition equivalency in Eqn. 1.7, we

find an expected ratio of

𝑀𝑛,38𝐶𝑎
𝑀𝑝,38𝐶𝑎

=

𝑒 𝑝
𝑒𝑛

= 3.0(4)

(1.10)

which is much larger than the experimentally determined ratio of 𝑀𝑛,38𝐶𝑎/𝑀𝑝,38𝐶𝑎 = 1.1(1) for the

2+ → 0+ transition. This discrepancy reinforces the importance of the choice of effective charges

in nuclear shell-model calculations as well as the unrealistic nature of the closed-shell assumption.

The idea of a single-closed shell is naive as nuclear excited states are comprised of many different

configurations existing in a superposition of states with some configurations more probable than

others. And when effects from collective motion, deformation, and shell evolution are considered,

expectations may change drastically.

For the 𝐴 = 38 isobars, Cottle et al. called for a more precise measurement of the 𝑇𝑧 = 0

matrix element in 38K, suggesting that such a measurement would strengthen their argument for an

enhanced 38Ca proton matrix element. The call for a better measurement of the 𝑇 = 1, 2+ → 0+

transition in 38K would be answered by Estèvez et al. [44] who concluded that the 38Ca matrix

element was larger than predicted based on their measurement of 38K and shell-model calculations,

furthering interest in the 𝐴 = 38 anomaly. Since the publication of this finding, the error of the

adopted value of 𝑀𝑝 (𝐸2) for 38K has be increased, again introducing ambiguity to the 𝑇𝑧 = 0 data

point [45]. But for the purpose of this work, we will focus on the ratio of the 𝑇𝑧 = +1 and 𝑇𝑧 = −1

matrix elements 𝑀𝑛/𝑀𝑝, which is independent of the 𝑇𝑧 = 0 data point.

For a larger overview of 𝑀𝑛/𝑀𝑝 systematics, Fig. 1.7 has been adopted from Figure 2 of

Ref. [46] and plots 𝑀𝑛/𝑀𝑝 as a function of the mass number 𝐴 for single closed shell (SCS)

16

Figure 1.7 The ratios 𝑀𝑛/𝑀𝑝 for the lowest 2+ → 0+ transitions in various single closed shell (SCS)
nuclei and doulby magic 48Ca and 208Pb. The data was obtained from hardon scattering experiments
and the mirror method described by Eqn. 1.7. Measurements are compared to the homogeneous
collective model expectation (𝑁/𝑍), the no-free parameter schematic model (NPSM), and the one-
free parameter schematic model (OPSM) [46]. Reprinted from Physics Letters B, Vol. 4, A.M.
Bernstein et al. Neutron and proton transition matrix elements and inelastic hadron scattering,
Pages No. 255-258, Copyright (1981), with permission from Elsevier.

nuclei. The data represent the most recent measurements of lowest lying 2+ states at the time

of publication and consist mainly of data obtained from hadron scattering experiments along

with the mirror method described by Eqn. 1.7 for lighter nuclei. The data are compared to

the homogeneous collective model in which the ratio of neutron (𝑛) and proton (𝑝) transition

densities 𝜌𝑛/𝜌 𝑝 = 𝑀𝑛/𝑀𝑝 = 𝑁/𝑍 [46], the no-free parameter schematic model (NPSM), and the

one-free parameter schematic model (OPSM). One will notice that 𝑀𝑛/𝑀𝑝 > 𝑁/𝑍 for neutron-

valence nuclei (closed proton shells) while, for proton-valence nuclei (closed neutron shells), the

collective model over predicts the experimental data with 𝑀𝑛/𝑀𝑝 < 𝑁/𝑍. Both of these trends

can be attributed to core-polarization effects which tend to push the 𝑀𝑛/𝑀𝑝 ratios back toward

the collective model predictions. Subtantial deviations from the general trends in each panel are

observed near doubly magic 48Ca and 208Pb; these deviations are attributed to 0ℏ𝜔 transitions

in 48Ca on the neutron side only while 208Pb is characterized by both neutron and proton 0ℏ𝜔

transitions [47].

Over the past few years, there has been a surge of interest in proton excitation in the 0+ ground

17

and 2+ excited states of even-even, neutron-deficient calcium isotopes.

In fact, given isospin

conservation, another island of inversion should be located near unbound 32Ca, being the neutron

deficient island mirroring the neutron rich island near 32Mg [48]. Due to the 𝑍 = 20 shell closure,

the low-lying 2+ states are usually considered to be comprised of a large fraction of closed proton-

shell configurations, a reasonable assumption for a proton magic nucleus. But, recent examination

has demonstrated that the ground states of 36Ca and 38Ca require proton excitations into the 𝑓 𝑝

shells to explain the observed 𝐵(E2; 0+ → 2+) transition strengths [49] and the cross-section ratio

𝜎(2+

1 )/𝜎(0+) of 36Ca measured in the 2𝑛 removal reaction from 38Ca [50]. Dronchi et al. [49]
concluded that a fraction of proton closed-shell configurations as low as 50% best replicates the

𝐵(E2; 2+ → 0+) for both 36,38Ca, which is reinforced by the results of Beck et al. [50]. Naturally,

the enhancement of the proton matrix elements of 36,38Ca motivates the study of neighboring Ca

isotopes to determine if this enhancement persists. Furthermore, since mirror transitions can be

effectively used to study the isospin decomposition of transition strengths [40], this work will

examine 39Ca and 39K mirror nuclei to understand the relative importance of proton and neutron

contributions to core excitations in the vicinity of doubly magic 40Ca.

More recent systematic studies have been conducted [51] for the isoscalar 𝑀0 and isovector 𝑀1

components for 2+ → 0+ transitions in odd-odd nuclei in the lower mass region as shown in Fig. 1.8.

Most apparent are the flat trend of isovector values at approximately 0 efm2, suggesting that nuclei

on 𝑁 = 𝑍 up to 𝐴 = 74 are most significantly characterized by their isoscalar components, and the

sharp increase of isoscalar values starting near the closing of the 𝑠𝑑 shell near 𝐴 = 40, suggesting

increased collectivity in this region. Interestingly, the isoscalar components of the odd-odd 𝑇 = 1

isobars trends simultaneously with the isoscalar components of the even-even 𝑇 = 0 nuclei, as

shown by the solid line and uncertainty band. This similarity implies that the collectivity observed

in the 𝑇 = 0 and 𝑇 = 1 states may have a common origin and this implication is supported by work

which examines the pairing correlations in 𝑇 = 0 states and puts them on par with pairing in the

𝑇 = 1 states [52].

18

Figure 1.8 The isoscalar 𝑀0 (black circles) and isovector 𝑀1 (red triangles) components the of the
2+ → 0+ transitions for odd-odd 𝑇 = 1 nuclei across a wide, low-mass region of the nuclear chart.
The figure shows relatively no change in the isovector components while isoscalar components
begin a sharp increase near 𝐴 = 40, where the 𝑠𝑑 shell becomes maximally occupied. Additionally,
the matrix elements derived from even-even 𝑇 = 0 nuclei along 𝑁 = 𝑍 are provided by the solid
line. The isoscalar components of the 𝑇 = 1 nuclei appear to trend accordingly with the isoscalar
values from the 𝑇 = 0 nuclei. The figure has been adopted from Figure 4 of Ref. [51].

1.3 Analog Transitions in Mirror Nuclei 39Ca and 39K

As observed in Fig. 1.6, it has been suggested that the 2+ → 0+ transition matrix element 𝑀𝑝

in 38Ca is enhanced with respect to the linear trend set by 38Ar and 38K [43, 44]. In an effort to add

systematic context to this suggestion, we have examined an analog transition in 39Ca to determine

if this enhancement persists in neighboring nuclei. The (11/2−) → 7/2− transition in 39Ca is

expected to be structurally related to the 2+ → 0+ transition in 38Ca if we naively assume the

𝑍 = 20 shell closure and that the (11/2−) and 7/2− states in 39Ca can be modeled as an 𝑓7/2 neutron

coupled to a 38Ca 2+ or 0+ core, respectively. In a similar fashion, the 11/2− → 7/2− transition

in 39K can be related to the 2+ → 0+ transition in 38Ar if we assume that the 11/2− and 7/2−

states in 39K can be formed by an 𝑓7/2 proton coupled to an 38Ar core. However, the situation is

complicated by the fact that there can also be wave function components coming from the coupling

of a proton (39Ca) or neutron (39K) to states of a 𝑁 = 𝑍 = 19 38K core. Therefore, the present

study of the quadrupole transitions in 39Ca and its mirror 39K aims to examine the 𝑀𝑝 trend as a

19

function of the isospin projection. The data are also compared to shell-model calculations which

employ effective interactions available to date in this mass region to understand the role of proton

and neutron excitations near 𝑍 = 𝑁 = 20.

In the following sections we will discuss the experimental design, procedure, and the analysis

methods used to determine the reduced transition strengths 𝐵(E2) from the lifetime of the analog

11/2− states in 39Ca and 39K. We will also provide an improved lifetime measurement for the mixed

M1+E2 9/2− transition in 39Ca with a significantly decreased error from the previously obtained

measurement.

20

CHAPTER 2

EXPERIMENT

The measurements discussed in this dissertation are the results of an experiment performed in 2019

at the National Superconducting Cyclotron Laboratory (NSCL). An overview of the NSCL beam

line is provided in Fig. 2.1. The aim of this study was to determine the lifetime of the 11/2−

state in 39Ca via the Recoil Distance Method (RDM). From the lifetime of the state of interest, the

reduced transition strength B(E2) and proton matrix element 𝑀𝑝 would be determined, which would

enhance our understanding of matrix element trends along the calcium isotopic chain in the neutron

deficient region of the nuclear chart. Data were also available for the 11/2− state in the mirror

nucleus 39K; therefore, the opportunity was taken to utilize the matrix element decomposition,

where 𝑀𝑛(39Ca)=𝑀𝑝(39K), to provide a more detailed picture of the E2 transition [40]. Thus, the

lifetime of the analog 11/2− state in 39K was determined for two purposes: to validate the analysis

method used for 39Ca, and to determine the neutron matrix element 𝑀𝑛 for the state of interest.

Additionally, a measurement of the 9/2− state of 39Ca was obtained, being that the plunger distance

settings were sensitive to the lifetime of this state, which provided a more precise measurement

than was obtained from the only previous measurement [53].

In this chapter the relevant devices used to perform the experiment will be described in detail. We

will begin with the production of the stable 58Ni primary beam by the Superconducting Source for

Ions (SuSI) [54] and acceleration of the beam up to 160 MeV/u via the Coupled Cyclotrons [55].

Then we will move down the beam line to the A1900 Fragment Separator [56] where the 42Sc

secondary beam was selected and discuss the process of isotope separation via magnetic rigidity.

Finally we follow the secondary beam to the S3 vault where inverse-kinematics reactions on the

TRIPLEX [57] target produced 39Ca and 39K recoil nuclei in excited states. Gamma rays emitted

from the excited nuclei were detected by GRETINA [58, 59] and the recoil nuclei were identified

by the S800 Spectrograph [60]; both of these devices and their use in tandem will be described in

this chapter. From this powerful combination of devices we are able to measure lifetimes of excited

states which exist for only 10s of picoseconds and perform model independent determination

21

of reduced transition strengths where other techniques such as Coulomb excitation may not be

applicable.

Figure 2.1 Beam line of the National Superconducting Cyclotron Laboratory (NSCL) [61]. The
Superconducting Source for Ions (1) feeds ions to the Coupled Cyclotrons (2) which accelerates
the primary beam particles to their maximum energy. The primary beam impinges on a production
target (3) in which fragmentation reactions create a beam of various species of nuclei. These species
are separated by the A1900 Fragment Separator (4) to produce the desired secondary beam(s). The
secondary beam is sent to the S3 vault where TRIPLEX and GRETINA (sections 2.5.1 and 2.4.1,
respectively) are located at the base of the S800 Spectrometer (8). Nuclei in excited states are
produced in reactions on the TRIPLEX target, gamma rays are detected by GRETINA, and the
recoil nuclei are identified by the S800. The beam may also be directed to other vaults where
detectors such as the MONA (5) and the decay station (7) may be utilized. The beam can also
be stopped and reaccelerated by the ReA facility (12) which directs a lower energy beam (single
MeV/u) to a variety of other detectors such as the ATTPC (14) and SOLARIS (not depicted). The
figure has been adopted from Ref. [61].

2.1 Primary Beam Production

For this experiment, a primary beam of 58Ni was created by the combination of SuSI and the

CCF. This primary beam was accelerated up to an energy of 160 MeV/u and then impinged on a

22

9Be production target to produce fragments which could be separated by their magnetic rigidity

and selected for secondary beam development in the A1900 fragment separator.

In the following sections we will describe the design principles and operation of the Super-

conducting Source for Ions (SuSI) and the Coupled Cyclotron Facility (CCF) which were used to

create the 58Ni primary beam.

2.1.1 Superconducting Source for Ions (SuSI)

The first step of the rare isotope production begins with the Superconducting Source for Ions

(SuSI) [54]. A metallic source foil of 58Ni is heated within an oven until a phase change produces

58Ni in a gaseous state which is injected into the 100-mm diameter plasma chamber. Inside the

chamber, magnetic fields on the order of 100 T are created by tunable superconducting magnets

and free electrons are injected into the chamber. The nickel atoms are then ionized by microwave

radiation of approximately 18 GHz using the Electron Cyclotron Resonance (ECR) method. The

ECR method is achieved by using magnetic fields 𝐵 (created by multiple superconducting solenoids

of various strengths) which are tuned to induce oscillation of both the 58Ni ions and the free electrons

to their respective cyclotron frequencies 𝜔𝑐 which are characterized by their unique charge-mass

ratios 𝑞/𝑚 and described by the relation

𝜔𝑐 =

𝑞
𝑚

𝐵

(2.1)

By applying a resonance frequency (RF) field to the oscillating particles, the kinetic energy of the

free electrons is increased disproportionately compared to the more massive nickel ions. Colli-

sions between the highly-energetic free-electrons and the heavier nickel atoms induces ionization,

unbinding electrons from the nickel atoms and leaving the 58Ni ions in a positively charged state.

The nickel ions, existing in a variety of charge states, are then guided by electric fields to the

coupled cyclotrons where the ions are accelerated up to the desired energy and their charge states

are increased for more efficient acceleration, transmission, and separation by the A1900 Fragment

Separator.

23

Figure 2.2 SUperconducting Source for Ions (SuSI) [54] at the NSCL. Inside the ion source, a
gaseous 58Ni is stripped of electrons with the Electron Cyclotron Resonance (ECR) method. Ions
in a variety of charge states are ejected from the ion source and the beam is bent toward the Coupled
Cyclotron Facility by the red dipole magnet. Photograph adopted from Ref. [62].

24

2.1.2 Coupled Cyclotron Facility (CCF)

The 58Ni ions are transmitted to the Coupled Cyclotron Facility (CCF) [55, 63, 64] which is

composed of a pair of coupled-superconducting cyclotrons, the K500 and K1200, where the suffix

number represents the maximum kinetic energy (in MeV) that the cyclotron can accelerate a single

proton to; i.e., the K500 can accelerate a proton to a maximum energy of 500 MeV and K1200 to

and energy of 1200 MeV. The K500 cyclotron accelerates 58Ni ions to an energy of approximately

10 − 20 MeV/u. At this energy, the ions are suitable for injection into the coupled K1200 cyclotron

where the nickel ions are stripped of electrons to the desired charge state and accelerated to an

energy of approximately 160 MeV/u. The nickel ions can then be guided to the 9Be production

target upon which reactions occur to create a “cocktail” beam of radioactive nuclei, from which the

secondary beam can be isolated by the A1900 Fragment Separator.

Upon injection into the center of the K500 cyclotron (entering the cyclotron upward from the

bottom), the nickel ions are subject to a magnetic field created by a superconducting magnet. The

ions are bent by a Lorentz force acting perpendicular to the ion’s velocity, sending the ions into a

spiral trajectory emanating outward from the center of the cyclotron and ending at the ejection port

in the outer radius. The positively charged nickel ions are accelerated by oscillating electric fields

created by three “dee” electrodes, each separated by a gap as shown in Fig. 2.3. As the ions travel

along the spiral trajectory they experience an electric potential, created by a pair of neighboring

dees of opposite polarity, which accelerates the ions across the gap between each dee. The polarities

of pairs of neighboring dees are switched at a frequency in accordance with the particular ion’s

cyclotron frequency. This alternating change in polarity between neighboring dees is timed such

that the ions experience an acceleration as they cross the gaps between each of the dees. With each

crossing of a gap between dees, the ions are accelerated to higher energies. Since the period of

oscillation is independent of the ion velocity, the ion will traverse an increasingly longer path with

each period of oscillation as its tangential velocity increases with each oscillation. When the ions

reach their final period they are guided out of the K500 with a velocity of approximately 0.2c and

directed toward the K1200 cyclotron.

25

Figure 2.3 Coupled Cyclotron Facility design concept logo. [65]. The figure depicts the K500 and
K1200 coupled cyclotrons and particles accelerated by the “dee” electrodes of the K500 which
are then injected into the K1200 where they are accelerated to a higher energy, stripped of excess
electrons, and then ejected toward the production target. The figure has been adopted from Ref. [65].
Reproduced with permission from Springer Nature.

Along the coupling line between the K500 and K1200 cyclotrons a combination of magnets is

used to shape the beam profile to the specific matching conditions required for injection into the

K1200. If we consider the z-axis to be parallel to the beam trajectory we then can build a coordinate

system which includes the dispersive (x) and non-dispersive (y) axis. Upon ejection from the K500

the beam trajectory is bent 16◦ in the dispersive plane by a single dipole magnet. The beam then

passes through a series of quadrupole magnet sets that alternately focus the beam along the x and

y planes. Each quadrupole will focus the beam in one plane while dispersing it the other; thus, by

using sets of alternately aligned quadrupoles a beam can be focused into a profile which is suitable

for acceptance by the K1200. The coupling line utilizes a quadrupole singlet and four quadrupole

doublets to achieve the proper beam profile before injection into the K1200.

The K1200 cyclotron works on the same principle as the K500 but ejects a beam of higher

energy, higher intensity, and greater charge state. The placement of a carbon stripping foil in

the ion’s outward-spiraling path, approximately 30 cm from the center of the K1200, strips the

26

remaining electrons from the nickel ions. This stripper foil enables the K1200 to more efficiently

accelerate ions as the higher charge state is more susceptible to the Lorentz bending force and the

electric potential between the dees. Upon ejection from the K1200, the 58Ni ions exist in their

highest charge state and achieve a maximum energy of approximately 160 MeV/u. At this point

the primary beam of 58Ni has been produced and can be directed through another transmission line

toward a production target.

The primary beam is impinged upon a 9Be production target where a multitude of reactions

occur to create a radioactive “cocktail” beam of nuclei of various charges, masses, and charge-

mass ratios. To isolate a specific secondary beam or set of secondary beams the cocktail beam is

transmitted to the A1900 fragment separator where the secondary beam(s) can be selected by their

charge and charge-mass ratio.

2.2 Secondary Beam Development

The primary beam of 58Ni impinges upon a 9Be production target where interactions with the

target material produce a variety of reaction products which can then be isolated with the A1900

Fragment Separator [56] by means of their magnetic rigidities and charge. After the secondary

beam is separated by each species’ unique radial dependence within the magnetic field of the last

dipole, a set of slits at the focal plane of the A1900 rejects the undesired species, allowing only

the desired secondary beam to pass to the transmission line leading to the experimental vault. In

this experiment, the resultant secondary beam of rare isotopes was mainly composed of 𝑁 = 21

isotones ranging from potassium (𝑍 = 19) to vanadium (𝑍 = 23) which were identified by their

time-of-flight (TOF) as determined by two plastic scintillators along the beam line (section 2.3.1).

In this section we will discuss the principles of projectile fragment separation as they relate

to the A1900 Fragment Separator shown in Fig. 2.4. We begin the discussion at the production

target then move through the first two dipoles of the A1900 fragment separator which separates the

reaction products by their magnetic rigidities. We will then move through the aluminum wedge-

shaped velocity degrader, which further separates nuclei of similar 𝐴/𝑍 by differential energy

loss and charge squared (𝑍 2) dependence. A second pair of dipoles further separates the beam

27

Figure 2.4 The A1900 Fragment Separator. The primary beam is directed from the CCF to the
production target (left) where reactions produce various species of nuclei. Dipole magnets (green)
bend the ions’ trajectories (red, blue, and green lines) within the dispersive plane according to their
magnetic rigidities. Sets of quadrupole magnets (small rectangles and squares) focus the beam
after the production target, between dipoles, and before the focal plane slits (right). The dipoles
are tuned to allow species of the desired mass-charge ratio 𝐴/𝑍 to follow a central trajectory in
the dispersive plane while the undesired ions are bent with larger or smaller radii 𝜌. An aluminum
wedge separates ions of similar 𝐴/𝑍 via differential energy loss thru the wedge-shaped degrader
according the square of the ions’ charge 𝑍 2. A second set of two dipoles further separates the
desired ions based on a different 𝐴/𝑍 ratio. Finally, slits at the focal plane cut out ions moving
outside the central trajectory, leaving the secondary beam to be transmitted to the experimental
vault. The figure has been adopted from Ref. [66].

by a different rigidity dependence and sends the beam through the focal plane slits, which rejects

undesired species and isolates the desired secondary beam before transmission to the experimental

vault.

2.2.1 A1900 Fragment Separator

Upon ejection from the K1200 cyclotron the primary beam is directed toward a thick 9Be

production target. Stable beryllium-9 is a commonly preferred material for Rare Isotope Beam

(RIB) production due to its high ratio of nucleons to areal density (high target number), its stability

in atmosphere (unlike highly reactive lithium), and the tendency of the reaction products to remain

in a forward-focused beam with a narrow angular distribution and momentum spread due to the

minimal effect of Coulomb forces from the target, allowing easier transmission to the A1900

fragment separator.

28

Many models can be used to describe the reaction mechanism of projectile fragmentation but

the common and intuitive abrasion-ablation model will suffice for this description. In the abrasion

process, the projectile (beam) particles experience a shearing-off of nucleons by interaction with

the stationary target nuclei. The resultant RIB, consisting mostly of ions with mass numbers less

than the mass number of the incoming primary beam nuclei, tends to carry the forward momentum

of each reaction product with little angular dispersion and a narrow momentum distribution due

to minimal energy loss in the recoil fragments. Of course, there is some loss of kinetic energy

in this interaction, but generally the fragmented projectiles continue along their initial trajectory

with a velocity only slightly lower than the incoming projectile velocity. In the ablation process,

the projectile fragments are in excited states and radiate excess energy as they continue along the

beamline trajectory. The resultant RIB consists of a cocktail of dozens to hundreds of species of

nuclei, all traveling within a narrow momentum and angular distribution, making transmission to

the A1900 fragment separator by sets of focusing quadrupole magnets a process very similar to the

coupling line described in section 2.1.2.

Upon entering the A1900 the RIB encounters a magnetic field created by the first dipole

magnet. The dipole magnet bends each ion’s trajectory by approximately 45◦ according to its

magnetic rigidity. The magnetic rigidity 𝐵𝜌, commonly cited in units of Tesla-meters (Tm), for a

projectile fragment moving at relativistic velocity is given by the relationship

𝐵𝜌 =

𝑚𝜈
𝑞

𝛾

(2.2)

where B is the strength of the magnetic field, 𝜌 is the fragment’s radius of curvature, 𝑚 and 𝑞 are

the mass and charge of the fragment, 𝜈 is the velocity, and Lorentz factor is given by 𝛾 =
where 𝛽 = 𝜈

1−𝛽2
𝑐 . Although fragment separation is dependent on the momentum-charge ratio 𝑚𝜈𝛾/𝑞,

1√

in a suitable approximation we can assume that the various projectile fragments move with a

common velocity 𝜈𝛾. Under this assumption, we find that the magnetic rigidity is associated with

a fragment’s mass-charge ratio

𝐵𝜌 ∝

𝑚
𝑞

∝

𝐴
𝑍

29

(2.3)

where 𝐴 is total number of nucleons and 𝑍 is the number protons. With this approximation, it
is clear that within a uniform magnetic field 𝐵, ions of identical mass-charge ratio 𝐴

𝑍 will have

trajectories which are bent in the dispersive plane with a common radius 𝜌, while ions with
different 𝐴

𝑍 ratios will follow trajectories of different radii. In this manner, the dipole magnets can

be tuned to allow ions of the desired 𝐴/𝑍 to follow a central trajectory through the A1900 while

the undesirable fragments follow trajectories of larger or smaller radii and can be bent away from

the central trajectory and separated from the desired secondary beam.

After exiting the field of the first dipole, ions which entered the dipole with nearly identical

initial positions have been dispersed by the magnetic field into separate trajectories according

to their rigidities. The dispersed fragments are refocused by two sets of quadrupole magnets

and transmitted to the second dipole where they are bent again by 45◦ in the opposite direction.

This second bending further separates fragments by their rigidity, placing them on parallel but

spatially-separated paths in the dispersive plane.

After passing through a third set of focusing quadrupoles, the spatial separation of the frag-

mented beam is then exploited by the wedge-shaped aluminum velocity degrader pictured in yellow

at image 2 in Fig. 2.4. Here, spatially separated fragments of similar 𝐴/𝑍 will interact with the

aluminum degrader and lose energy proportional to the square of their charge 𝑍 2. Aluminum is

a commonly preferred degrader material due to its high heat conductivity and the smaller mass

number. As charged particles pass through the degrader material, differential energy loss in the de-
grader can be described in full by the Bethe-Bloch formula [67, 68] or approximated as 𝑑𝐸

𝑑𝑥 ∝ −𝑍 2.

Due to the differential energy loss and variable thickness of the wedge-shaped degrader, the beam

exits the degrader with fragments of variable momentum which can then be further separated
according to a different mass-charge dependence 𝐴2.5

𝑍 1.5 [69]. The second pair of dipoles bends the

beam ions according to this second mass-charge dependence and upon exiting the last dipole the

beam fragments have maximal spatial separation in the dispersive plane.

The final secondary beam selection is accomplished by a set of slits placed at the focal plane

of the A1900. The focal plane slits are orientated so as to reject ions of undesirable rigidity by

30

narrowing the aperture of the ejection port. The slits are adjusted to an aperture which allows the

maximum number of ions aligned with the central trajectory to be transmitted while minimizing

transmission of ions whose trajectories have diverged from the central trajectory.

Upon ejection from the A1900 the maximally purified secondary beam is transmitted through

the A1900 focal plane scintillator (XFP), along the transmission line to the S800 object plane

scintillator (OBJ), and then to the experimental vault where the S800 spectrograph is utilized to

identify each particle on an event-by-event basis. The purpose of the XFP, OBJ, and E1 scintillators

will be discussed in the next section along with the other components of the S800 spectrograph

which are used to identify the secondary beam(s) and recoil nuclei resulting from reactions on the

TRIPLEX plunger target.

In this experiment, the secondary beam was primarily composed of 𝑁 = 21 isotones ranging

from potassium to vanadium. Suffice it to say, of these isotones, the 42Sc beam with an energy of

85.2 MeV/u and purity of 44% was selected for the present study as the 9Be(42Sc, 39Ca)X reaction

on the TRIPLEX target populated excited states of 39Ca in greater yields than reactions from the

other available secondary beams.

2.3 Particle Identification with the S800 Spectrograph

Though most of the primary beam will pass through the targets and degraders unreacted, the

multitude of reactions that do occur can produce dozens of secondary beam fragments. Subsequent

reactions between the secondary beam and the TRIPLEX target then produce dozens more species

which must be identified. Thus, it is essential that the wide range of reaction products be identified

on an event-by-event basis so that gamma rays detected by GRETINA may be associated with

the particular nuclei from which they were emitted. The process of particle identification is

accomplished by the S800 Spectrograph [60], a powerful machine capable of high resolution and

broad acceptance achieved by the combined use of scintillators, cathode readout drift chambers

(CRDCs), an ion chamber, and a hodoscope. The primary functions of the hodoscope are to identify

an ion’s kinetic energy and charge state as well as isomeric states which may only be detected by

radioactive emission after implantation at the end of the focal plane; but this device was not utilized

31

for this work and shall not be discussed further.

Figure 2.5 The S800 Spectrograph. After exiting A1900 focal plane scintillator XFP (left of
diagram), the secondary beam components interact with the S800 object plane scintillator (OBJ).
The time-of-flight (TOF) method is used to determine the secondary beam components with the E1
scintillator at the end of the focal plane acting as a reference and trigger. The focal plane and the
detector components that comprise it are presented in Fig. 2.6. The figure has been adopted from
Ref. [70].

In this section we will describe the process of particle identification by the various components

which comprise the S800 spectrograph and discuss the basic working principles of each component.

Beginning with XFP scintillator at the end of the A1900 we will discuss how the time-of-flight

(TOF) method between the E1 and XFP, and E1 and OBJ scintillators is utilized to identify the

secondary beam components. We will then describe how the CRDCs are used to determine ion

trajectory in the S800 focal plane, which is subsequently used to determine each recoil nucleus’s

trajectory at the TRIPLEX target position. Finally we will describe how energy loss through the

ion chamber is used in conjunction with the OBJ scintillator to identify recoil nuclei created by

reactions between the secondary beam and the TRIPLEX production target.

2.3.1 Timing Scintillators

Upon ejection from the A1900 fragment separator, the secondary beam encounters a series

of three plastic scintillators that act as simple counting mechanisms which record that an event

occurred at a particular time. Due to the nature of the scintillator system, each scintillator provides

32

Figure 2.6 Focal plane of the S800 spectrometer. Detectors located at the focal plane include a
pair of CRDCs which provide each ion’s trajectory, an ion chamber which provides energy loss
information, and the E1 scintillator (labeled “plastic scintillator”) which acts as a trigger for the
S800 DAQ and provides timing information from the XFP and OBJ scintillators. The IsoTagger
hodoscope provides information about the ion’s kinetic energy and charge state, and is also utilized
for identifying long-lived isomeric states but was not utilized in this work. The figure has been
adopted from Ref. [71].

a very fast timing resolution and can handle signal rates up to 1 MHz. These properties make the

E1, XFP, and OBJ scintillator stations essential for particle identification at high beam intensities.

The first scintillator is located at the A1900 focal plane (not displayed in Fig. 2.5) and is commonly

referred to as the “XFP” scintillator, the second is located at the object position of the S800 in

Fig. 2.5 and is referred to as the “OBJ” scintillator, the last scintillator is positioned at the end of the

focal plane in Fig. 2.5 and is referred to as “E1”. Here one should note that a pair of dipoles placed

after the target position (D1 and D2 in Fig. 2.5) bends the ion beam, imposing a spatial separation

between ions in the dispersive plane and a unique trajectory through the S800 defined by each ion’s

mass-charge ratio.

33

The scintillators are mainly composed of a polyvinyltoluene material which produces photons

when energetic particles excite the scintillator atomic electrons into higher orbitals. The atoms then

emit photons as the electrons deexcite to lower orbitals. The photons produced in the luminescence

process are guided to two photomultipliers positioned at opposite ends of the scintillator block.

When photons interact with the photocathode material, electrons are ejected from the cathode

atoms in accordance with the photoelectric effect. The initially small electric signal from the

ejected electrons is then multiplied up to a readable current, this signal can then be sent to the

electronics stack where it can be digitized and recorded by the NSCLDAQ [72].

The E1 scintillator in the S800 focal plane provides a trigger for the coupled components of

the S800 and GRETINA as well as providing a reference timing to which the XFP and OBJ timing

can be compared to. For identification of the secondary beam 𝑁 = 21 isotones, the time difference

between the E1 and XFP scintillators was compared to the time difference between the E1 and OBJ

scintillators (Fig. 3.2). To determine the relationship between an ion’s mass-charge ratio and flight

time we recall that, in the absence of an electric field, the Lorentz force F is given by

𝐹 = 𝑞𝜈 × 𝐵 = 𝑞𝜈 sin 𝜃𝐵 = 𝑞𝜈𝐵

(2.4)

where q is the charge of the particle, 𝜈 is the velocity of the particle, 𝐵 is the strength of the

magnetic field, and 𝜃 = 90◦ as the velocity of the particle is perpendicular to the magnetic field.

We can then model particle’s trajectory through the magnetic field as a uniform circular motion

with a centripetal force

𝐹 = 𝑚𝑎 =

𝑚𝜈2
𝜌

(2.5)

where 𝜌 again is the radius of curvature. By equating the Lorentz and centripetal forces (Eqns. 2.4

and 2.5), we find

𝑞𝜈𝐵 =

𝑚𝜈2
𝜌

(2.6)

Further, we rearrange Eqn. 2.6 and use the definition of average velocity 𝜈 = 𝑑/𝑡 (distance 𝑑 over

time 𝑡) to find that

𝐵𝜌
𝑑

𝑡 =

𝑚
𝑞

=

𝐴
𝑞

34

(2.7)

where the mass-charge ratio of a species with total nucleons 𝐴 = 𝑁 + 𝑍 and charge 𝑞 (not necessarily

equal to 𝑍 due to charge states from incomplete stripping of electrons) is directly proportional to

the time-of-flight (TOF) 𝑡 since, for practical purposes, the ratio

𝐵𝜌
𝑑 is assumed constant. Using

Eqn. 2.7 we exploit the E1-XFP and E1-OBJ time differences to resolve the isotones composing

the secondary beam by plotting the time differences between each scintillator and E1 in arbitrary

time units, as displayed in Fig. 3.2.

2.3.2 Cathode Readout Drift Chambers

While we have glossed over exactly how the ion trajectory is determined, in this section we

will explain this process as we move our discussion to the focal plane of the S800 as depicted in

Fig. 2.6. In the focal plane a set of two Cathode Readout Drift Chambers (CRDCs) are utilized to

determine each ion’s trajectory on an event-by-event basis. A transformation matrix is the applied

to each ion’s trajectory as determined by the CRDCs which maps the ion’s path back through the

two dipole magnets (D1 and D2 in Fig. 2.5) to the TRIPLEX target position. In this manner we

are able to determine the ion’s trajectory and position at the TRIPLEX target, which is essential

information for the Doppler-shift energy reconstruction as it is a function of the angle of gamma-ray

emittance with respect to the ion’s trajectory, which will also be covered in Sect. 2.4.

Located at the focal plane of the S800 spectrograph (Fig. 2.6) are a series of detector stations

which provide crucial information for the particle identification process. After passing through the

S800, both the unreacted secondary beam ions and the reaction products created from interactions

between the secondary beam and the TRIPLEX target are guided through the series of dipoles (D1

and D2 in Fig. 2.5) and are transmitted to the focal plane detectors pictured in Fig. 2.6. The first set

of detectors that the beam encounters are a pair of Cathode Readout Drift Chambers which provide

each ion’s trajectory on an event-by-event basis. The pair of CRDCs will hereafter be referred to

as “CRDC1” and “CRDC2”, where CRDC1 provides an ion’s first interaction point (X1,Y1) along

its trajectory in the focal plane while CRDC2 provides the second interaction point (X2,Y2), as

depicted in Fig. 2.7. Simply stated, by using these two interaction points we can determine an

ion’s vector in the focal plane. A transformation matrix referred to as the “inverse map” can then

35

Figure 2.7 Cathode Readout Drift Chambers (CRDCs) at the focal plane of the S800. The two
interaction points (X1,Y1) and (X2,Y2) determine the ion trajectory in the focal plane. Beam ions
and recoil nuclei ionize gas in the CRDC chambers, the ionized electrons are collected by multiple
anode wires, the charge on the anodes induces an opposite charge on the cathode pads. The ion’s
position in the dispersive plane (X) is determined by a Gaussian fit to the charge distribution across
the cathode pads (inset) on an event-by-event basis. The drift time of the ionized electrons to each
anode wire is determined with respect to the E1 trigger signal and provides the ion position in the
non-dispersive plane (Y). The figure has been adopted from Ref. [71].

be applied to each ion’s vector to determine the ion’s trajectory backward from the focal plane,

through the dipoles, and back to the TRIPLEX target location. Knowing the ion’s trajectory at the

TRIPLEX target position is essential for event-by-event Doppler-shift correction as the ion-frame

energy is a function of angle 𝜃 between the ion’s trajectory and the direction of the gamma-ray

emission as determined by the main interaction point in the GRETINA detector.

Each CRDC is composed of an anode frame upon which the anode wires are mounted. The

36

anode frame is immersed in a carbon tetrafluoride gas which acts as the ionizing medium. Each

anode provides an electric potential to which the ionized electrons will migrate accordingly. Along

their paths to the anode, the electric field becomes stronger and the ionized electrons move with a

greater velocity causing ionization of more gas molecules along the way and creating an avalanche

effect which produces enough electrons to create a measurable charge on each anode. When charge

is collected on an anode wire an image charge is induced on the cathode pads located in front of

and behind of each wire. The induced charge from each pad can then be collected and the charge

distribution across multiple pads (inset Fig. 2.7) can be fitted with a Gaussian function to find the

mean position of greatest charge collection. The mean of the induced-charge distribution on the

cathode pads provides the ion position (commonly referred to as 𝑥 𝑓 𝑝) in the dispersive plane (X).

Simultaneously, the drift time of ionized electrons to the anode wire is determined with respect to

the E1 trigger signal, this drift time provides the ion position in the non-dispersive plane (Y) and is

commonly referred to as 𝑦 𝑓 𝑝.

After determining an initial position (X1,Y1) from CRDC1, the ion then travels to CRDC2

where the same principles are used to determine a second position approximately 1 m further along

the focal plane. Having two position values spatially separated along the ion’s trajectory allows us

to calculate the ion’s vector with respect to the central beam axis (z). From the second interaction

point (X2,Y2) we can determine the ion’s angle in the dispersive plane 𝑎 𝑓 𝑝 and the angle in the

non-dispersive plane 𝑏 𝑓 𝑝, both of which are calculated relative to the z-axis.

Given the ion’s trajectory as determined by the pair of CRDCs, we can then use the ion’s identity

(e.g., 39Ca for the main focus of this experiment), the S800 dipole magnet settings in Tesla-meters,

and the TRIPLEX target position relative to the S800 pivot point to calculate the inverse map. The

inverse map is a transformation matrix that is applied to the ion’s trajectory vector in the S800 focal

plane which provides the ion’s trajectory at the TRIPLEX target position. The trajectory at the

target position is commonly defined by 𝑎𝑡𝑎 (the ion’s angle in the dispersive plane x with respect

to the ion-of-interest’s central trajectory 𝑧), 𝑏𝑡𝑎 (the angle in the non-dispersive plane relative to

𝑧), and 𝑦𝑡𝑎 (the position in the non-dispersive plane relative to 𝑧). Further, the kinetic energy and,

37

hence, the ion’s momentum, can also be determined by the magnet settings and the position of

the ion in the dispersive plane at the S800 focal plane. The energy difference with respect to the

central value set for the ion of interest based on the S800 rigidity provides a distribution commonly

referred to as 𝑑𝑡𝑎.

The combined information from 𝑎𝑡𝑎, 𝑏𝑡𝑎, 𝑦𝑡𝑎, and 𝑑𝑡𝑎 are used to characterize the beam at the

target location. This characterization of the beam provides crucial information for the G4Lifetime

simulation, which requires a realistic beam profile (trajectory and energy) to reproduce the reaction

mechanisms and excited reaction products that provide the simulated gamma-ray spectra.

Of course, the description of the S800 particle identification process has excluded a very

important process; that is, the process of identifying recoil nuclei produced in reactions on the

TRIPLEX target. In the next section, we will discuss identification of recoil nuclei produced in the

TRIPLEX target via a combined use of the ion chamber and OBJ scintillator.

2.3.3

Ion Chamber

To determine the identity of reaction products produced by interactions between the secondary

beam and the TRIPLEX target we now move on to a description of the S800 ion chamber and

how it is used in conjunction with the OBJ scintillator to identify the recoil nuclei from which the

gamma rays are emitted. The S800 ion chamber, located just before the E1 scintillator and depicted

in Fig. 2.6, serves to identify the atomic number 𝑍 of the recoil nuclei by measuring energy loss

through a gaseous medium composed of 90% argon and 10% methane.

Contrary to its name, the S800 ion chamber is actually a series of 16 parallel-plate ion chambers,

as depicted in Fig. 2.8. Each chamber is composed of a pair of anode and cathode plates with a

gas-filled gap between each plate. As heavy nuclei travel through the ion chamber medium the

heavier, more energetic nuclei interact with the ion chamber gas and transfer a portion of their

kinetic energy to the gas atoms, ionizing the gas and sending electrons and gas ions toward the

anodes and cathodes, respectively. When the electrons and gas ions reach the anodes and cathodes

a change in voltage is observed and recorded. The heavier beam nuclei lose energy proportional

to their atomic number squared (𝑍 2) as per the Bethe-Bloch relation discussed briefly in section

38

Figure 2.8 Unhoused view of the ion chamber at the focal plane of the S800. 16 parallel-plate
ion chambers are stacked perpendicular to the beam axis (left to right) and cover an active area of
≈ 30 × 60 cm. When housed, the chamber contains a mixture of 90% argon 10% methane, typically
at a pressure of 300 torr. The picture has been adopted from Ref. [71].

39

2.2.1. The use of multiple chambers reduces electronic noise by averaging the energy loss over the

16 chambers.

The ion chamber alone is not enough to identify the dozens of recoil nuclei produced in reactions

on the target because many isotopes of the same 𝑍 may be produced in these reactions. Therefore,

another channel of information must be utilized to distinguish isotopes of the same atomic number,

in this case the time difference between the E1 and OBJ scintillators is utilized. Though all the

unreacted secondary beam ions of the same species will travel through the S800 with similar time

signatures, once those secondary beam nuclei interact with the target and produce reaction products

with various 𝐴/𝑍 ratios the time differences from the OBJ scintillator to the E1 scintillator will be

changed. By gating on a particular secondary beam in the data analysis phase (our incoming beam

gate), we can observe all events associated with that beam as determined on an event-by-event

basis by the OBJ timestamp and energy loss through the ion chamber. Due to energy loss from

the reaction mechanism and interactions with the target medium, each reaction product will have a

unique trajectory through the S800 dipoles and, thus, a unique TOF through the S800. Therefore,

to identify reaction products from a specific secondary beam we plot energy loss in the ion chamber

as a function of time difference between the E1 and OBJ scintillators, as shown in Fig. 2.9.

By examining Fig. 2.9 we can discern most of the reaction products from general principles.

The central column of nuclei all have similar time signatures because 𝐴/𝑍 = 2 for all 𝑁 = 𝑍

nuclei. Isotopes of a given 𝑍 have different times-of-flight due the difference in their mass-charge

ratio, thus we observe that heaver isotopes have longer (more negative) TOF and lighter isotopes

having shorter (less negative) TOF, as can be seen by the two calcium isotopes labeled in the figure.

Using these principles and the knowledge of the intensity of the incoming secondary beam we

can identify one nucleus along 𝑁 = 𝑍 and then follow the pattern to identify all the isotopes of

that nuclei by their unique TOFs and isotones by their unique energy loss. Further confirmation

of particle identification can come from gating on a suspected known nucleus and examining the

gamma rays associated with that “blob”; if the gamma ray energies from decays of that particular

nuclei and excited state are known, the spectrum in question can be compared to known data for

40

Figure 2.9 Energy loss in the S800 ion chamber as a function of time-of-flight between the E1
and OBJ scintillators. The energy loss vs TOF method is used to distinguish recoil nuclei from
reactions between our choice of incoming secondary beam (in this case 42Sc) and outgoing reaction
products.

confirmation. In this work all of the above methods were used to confidently identify the reaction

products labeled in the particle identification plot in Fig. 2.9.

Now that we understand the principals involved in primary beam production, secondary beam

fragmentation, and particle identification we will move to the area of the experimental vault where

GRETINA detects gamma rays emitted by nuclei in excited states which have been produced in

reactions between the secondary beam and the TRIPLEX target.

2.4 Gamma Ray Spectroscopy

When secondary beam nuclei interact with target nuclei, kinetic energy is transferred to the

nucleons and places them in an energetic state. The nucleus sheds this energy by de-exciting from

a higher energy state to a lower state and emitting a gamma-ray photon. The energy of the emitted

gamma ray is equal to the difference in energy of the initial and final states of the nucleus.

To determine the energy of a gamma ray emitted by an in-flight nucleus, the TRIPLEX target

is surrounded by 12 GRETINA detector modules. A single detector module consists of 4 high-

purity germanium (HPGe) crystals which are semiconductors that produce electron-hole pairs

41

when an energetic particle interacts with the detector medium. A high voltage is applied across the

medium that provides an electric potential which guides the electron-hole pairs to the electrodes

for collection. The charge collected by the electrodes is sent to a preamplifier and converted to a

voltage pulse with an amplitude that is proportional to the energy of the gamma-ray. The amplified

analog signal is then sent to the GRETINA DAQ where it is digitized. The digital signal is then

stored by the NSCLDAQ as an event with a timestamp and an associated energy.

There are three common interactions between the gamma ray and the detector medium which can

produce the electric signal: photoelectric absorption, Compton scattering, and pair production [67,

68]. Only photoelectric absorption will generate a voltage signal of amplitude proportional to the

full gamma-ray energy. In the case of Compton scattering and pair production, if all the interactions

occur within the detector medium the partial signals can be summed in order to obtain a voltage

signal proportional to the full gamma-ray energy.

In the process of photoelectric absorption, the energy of the gamma ray is completely transferred

to an atom of the medium material. This transfer of energy results in the ionization of an atomic

electron which is ejected from the medium atom with an energy equal to the gamma ray energy

minus the binding energy of the atomic electron. The highly energetic free electron is then

collected by the electrode with a signal amplitude proportional to the original gamma-ray energy.

Photoelectric absorption is the most common interaction type for gamma rays of energy lower than

a couple hundred keV and the process can be enhanced by using a detector medium of high Z.

For gamma rays of energy ranging from a couple hundred keV to several MeV, Compton

scattering is the most typical interaction type leading to gamma ray detection. In the process of

Compton scattering, a gamma-ray photon interacts directly with an atomic electron and transfers

part of its energy to this recoil electron while continuing along a diverted trajectory. The energy

transferred to the recoil electron is a function of the scattering angle and, thus, the photon can

transfer a very small fraction or a large fraction of its energy to the electron depending on the

scattering angle. The scattered gamma ray can then transfer any remaining energy to other recoil

electrons in a series of scattering events and those electrons can be collected at the anode where

42

the voltage amplitude is again proportional to the gamma ray energy.

When the energy of a gamma ray exceeds 1.02 MeV (twice the electron rest mass) the process

of pair production becomes a possible interaction type with the probability of this interaction

increasing with increased gamma-ray energy beyond the stated threshold. In the process of pair

production, a sufficiently energetic gamma ray interacts with the Coulomb field of a nucleus and

is converted into an electron-positron pair where the energy exceeding the threshold is converted

to kinetic energy which is shared by the electron and positron. The electron from this process

will be transferred toward the anode. The positron will be directed toward the cathode but will

likely interact with an electron and induce an electron-positron annihilation which will result in the

creation of two more photons. These photons, having conserved the energy of the annihilation pair,

can then proceed to interact again via the three processes discussed here until all the energy of the

initial incoming gamma ray has been converted to electrons that have been collected by the anodes.

When the processes are not contained to the detector volume, and only a fraction of the gamma-

ray energy is collected by the detector before the photon escapes the detector volume, various

structures may appear in the gamma-ray spectrum such as the Compton continuum, back scatter

peaks, and escape peaks. Therefore, detectors with high Z and large volume are preferred to

increase the chances of full-energy deposit within the detector volume, resulting in a full-energy

peak. Other techniques such as the addback method can be used in the data analysis phase to gather

a full energy peak from gamma rays which were scattered through multiple crystals or crystal

segments.

The experiment discussed in this work utilized the Gamma Ray Energy Tracking In-beam

Nuclear Array (GRETINA), pictured in Fig. 2.10, a multi-segmented HPGe gamma-ray detector

array which surrounds the TRIPLEX target and collects gamma rays for analysis. In the following

section we will delve into the specifics of the GRETINA configuration used in this experiment.

2.4.1 Gamma-Ray Energy Tracking In-beam Nuclear Array

The Gamma Ray Energy Tracking In-beam Nuclear Array (GRETINA) [58] is a device com-

posed of HPGe semiconductor detectors used in this work to detect gamma rays and determine

43

Figure 2.10 The Gamma-Ray Energy Tracking In-beam Nuclear Array with 12 modules positioned
for perpendicular and forward focused detection. The module frame has been opened to allow
connection of the TRIPLEX beam pipe to the S800 and alignment of the target with the beam spot.
The secondary beam moves left to right and interacts with the TRIPLEX target which is located
13 cm upstream from the center of GRETINA. The beam then continues through the S800 for
particle identification.

their energy. GRETINA, for this experiment, consisted of 12 detector modules (Fig. 2.11), each

containing four hexagonal HPGe crystals, with each crystal consisting of 36 segments (Fig. 2.12).

Each crystal has 37 contacts; a central contact which measures the total energy collected from in-

teractions in all segments, and 36 outer contacts corresponding (to the 36 segments) which provide

position information with the aid of a signal decomposition algorithm. The signal decomposition

algorithm reads signals from the outer contacts and compares them to a library of simulated signals

from different locations within each segment to match the real signal to a position provided by com-

parison to the signal library. Each segment is electrically separated which allows reconstruction of

the gamma-ray interaction point(s) to within 2 mm.

Initiated by the E1 trigger, an “event” within GRETINA can be composed of multiple “hits”

corresponding to interactions within the detector volume which coincide with a defined time

window. Under the main interaction point assumption [59], the interaction point of the largest

44

Figure 2.11 Geant4 rendering of the GRETINA module configuration utilized for this work. Each
of the 12 GRETINA modules is comprised of a liquid nitrogen cryo-module (grey), 4 hexagonal
crystals (green and white representing type A and B), with each crystal having 36-fold segmentation.
Looking downstream (left), the beryllium target (green) and the 90◦ ring of modules are rendered.
Looking upstream (right), the tantalum degrader (red) and the forward facing modules are rendered.
Not pictured for clarity are TRIPLEX and the beam pipe which accounts for the uncovered area
between 0 and 20 degrees with respect to the beam axis (into and out of the page).

energy deposit is taken as the first interaction point when multiple hits are registered within

the time gate. The position resolution is high enough to allow gamma-ray tracking of multiple

interaction points through the detector volume, enabling the addback method to be utilized in

the data analysis phase which increases the number of counts in the full-energy peaks by adding

the energy of multiple interactions together to create a single full-energy event in the gamma-ray

spectrum.

For the current work, 8 quads were mounted in the 90◦ ring of the GRETINA frame and 4

quads were mounted in the forward focused rings, as shown in Fig. 2.11. The TRIPLEX plunger

held the target approximately 13 cm upstream from the center of GRETINA. This configuration of

target placement and forward mounted quads was selected to achieve balance between detection

efficiency to forward focusing of gamma rays and sensitivity to varying degrees of Doppler shifts

caused by different recoil velocities. A trigger signal from the E1 scintillator initiated information

collection from the S800 spectrograph detectors and GRETINA. Using the coincidence between

45

Figure 2.12 Diagram of a GRETINA “quad” (a) and a single crystal segmentation (b) [58]. Each
GRETINA module contains a quad of four, closely-packed HPGe crystals (a) with the two pairs
of crystals each having unique tapered-hexagonal shapes (A-type and B-type) and length of ap-
proximately 9 cm and maximum width of 8 cm. The tapered-hexagonal shapes provide maximum
detector volume and solid-angle coverage. Each crystal contains a 36-fold segmentation (b) with
a central core contact to collect total energy deposited and each segment with an individual outer-
surface contact pad to determine interaction location (36 pads per crystal). The segmentation
of each crystal allows for interaction position resolution of approximately 2 mm and tracking of
gamma-ray interaction points throughout the detector volume [58]. Reprinted from Nuclear In-
struments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
Associated Equipment, Vol. 709, S. Paschalis et al.,The performance of the gamma-ray energy
tracking in-beam nuclear array GRETINA, Pages No. 44-55, Copyright (2013), with permission
from Elsevier.

the S800 and GRETINA allows one to identify gamma ray hits and associate them with a particle

event in each of the S800 detectors.

To determine the efficiency of GRETINA, a 152Eu source run was conducted. Gamma rays

collected during the source run are matched to adopted energy and intensities 𝐼𝛾 for 152Eu decays.

Given the activity of the source at the last measurement 𝐴0, the amount of time since the last
𝑙𝑛(2)
𝑇1/2

, the activity 𝐴 of the source at the time of the

measurement 𝑡, and the decay constant 𝜆 =

46

Figure 2.13 The efficiency of GRETINA at the 152Eu gamma-ray energies observed in the lab-
frame. The efficiency of the simulation at a given energy is given in blue while the efficiency of
GRETINA is given in orange.

source run can be calculated from the radioactive decay equation. From the calculated activity

𝐴, the source run duration 𝑡𝑟𝑢𝑛, and the adopted intensities of each of the gamma rays detected,

the number of gamma rays emitted from the source is calculated according to 𝑁𝑒𝑚𝑖𝑡 = 𝐴𝐼𝛾𝑡𝑟𝑢𝑛.

By integrating fits to the observed 152Eu gamma-ray peaks, the number of gamma rays observed

𝑁𝑒𝑚𝑖𝑡 in the lab-frame (𝛽 = 0) is then calculated at each gamma-ray energy, as displayed

𝑁𝑜𝑏𝑠 by GRETINA within the run time is determined. The experimental efficiency of GRETINA
𝜖 𝑒𝑥 𝑝
𝛽=0 = 𝑁𝑜𝑏𝑠
by the orange data points in Fig. 2.13. When the experimental efficiency is compared to the
𝑁 𝑠𝑖𝑚
𝑜𝑏𝑠
𝑁 𝑠𝑖𝑚
𝑟𝑢𝑛 𝐼𝛾

(blue data points in Fig. 2.13), where 𝑁 𝑠𝑖𝑚

efficiency of a simulated source run 𝜖 𝑠𝑖𝑚

𝛽=0 =

𝑟𝑢𝑛 is

the number of simulated events, one observes that the simulation overestimates the efficiency of
the GRETINA array. Therefore, to determine the experimental ion-frame efficiency 𝜖 𝑒𝑥 𝑝

𝛽>0, we must

scale the simulated ion-frame efficiency by a factor of

𝜖 𝑒𝑥 𝑝
𝛽=0
𝜖 𝑠𝑖𝑚
𝛽=0

. For this experiment, the simulated

ion-frame efficiency was scaled by a factor of 0.84. With the efficiencies determined, the intensities

47

of each of the 39Ca peaks could be calculated from simulated fits.

GRETINA was utilized to collect gamma rays emitted from 39Ca and 39K recoil nuclei. Because

the gamma rays are emitted from ions moving at relativistic velocities (approximately 0.3c) the

energies that were determined by GRETINA detectors in the lab frame must be converted to

energies that coincide with the ions’ rest frame. In the next section we will discuss the Doppler-

shift correction on the gamma ray energies detected by GRETINA.

2.4.2 Doppler-Shift Correction

The energies detected by GRETINA must be corrected for the difference in reference frames

between the GRETINA detectors in the laboratory frame and gamma rays emitted in the ion frame

at relativistic energies. Gamma rays that are emitted from in-flight nuclei with a velocity moving

toward an observer will appear to have a shorter wavelength from the perspective of the lab-frame

observer while gamma rays emitted from nuclei moving away from the observer will appear to have

a longer wavelength. Since the wavelength 𝜆 of a photon is inversely proportional to the energy 𝐸
of the photon according to 𝐸 = ℎ𝑐

𝜆 , it is essential that the energy detected by GRETINA be Doppler

corrected to account for the difference in photon wavelength and, hence, the energy collected by

detectors at various angles.

Given the energy detected in the lab-frame by the GRETINA detectors 𝐸𝑙𝑎𝑏, we can Doppler

correct this value to the desired ion-frame energy 𝐸𝑖𝑜𝑛 via

𝐸𝑖𝑜𝑛 = 𝐸𝑙𝑎𝑏𝛾(1 − 𝛽 cos 𝜃)

(2.8)

where 𝜃 is the angle of gamma-ray emittance with respect to the trajectory of the in-flight nuclei,
and 𝛾 is a function of 𝛽 = 𝜈

𝑐 defined in section 2.2.1. Therefore, knowledge of the ion velocity 𝜈
with respect to the lab-frame and the angle of emittance 𝜃 is essential for determining the ion-frame

energy of the gamma ray.

As shown in Eqn. 2.2, the velocity 𝜈 can be obtained from the rigidity settings of the S800

dipoles and the angle of emittance 𝜃 can be obtained from the main interaction point within the

GRETINA crystal. In practice, the uncertainty associated with 𝜈 and 𝜃 result in an uncertainty

in the determination of the ion-frame energy known as Doppler broadening. This uncertainty

48

expresses itself in a broadening the full-energy gamma-ray peak in the ion-frame energy spectrum.

An example of a source of uncertainty in 𝜈 is the assumption of reaction location in the target,

which affects the amount of differential energy loss and hence the velocity of the outgoing particle

since a 42Sc secondary beam particle of 𝑍 = 21 loses more energy than a 𝑍 = 20 calcium

isotope; therefore, correcting data with a 𝛽 which assumes a reaction in the middle of the target

may introduce error to the energy measurement. Along similar lines, the location of gamma-ray

emission may also affect the ion-frame energy spectrum; if the gamma was emitted from inside the

target due to a short lifetime or was emitted further past the target than TRIPLEX was configured

for (due to a long lifetime) the gamma-ray peak may have a high or low energy tail in the ion-frame

energy spectrum, respectively. Sources of uncertainty in 𝜃 may arise from the position resolution

of GRETINA, incorrect determination of the main interaction point, and incomplete knowledge of

the ion trajectory at the target position.

Despite the uncertainties listed above, we are able to constrain the parameters to obtain a

spectrum with sufficient resolution to utilize the recoil-distance method (RDM) for lifetime mea-

surement. In the following section we will move to the target station of the S800 where TRIPLEX

holds the secondary reaction target and the velocity degrader. We will describe the TRIPLEX

configurations utilized in this experiment and see how the recoil distance method is utilized to

determine the lifetimes of excited states.

2.5 RDM Lifetime Measurement with TRIPLEX

At the target station of the S800 the TRIple PLunger for EXotic beams (TRIPLEX) is positioned

in the beam line. As the secondary beam approaches TRIPLEX it first encounters a 2-mm thick

beryllium-9 target. Reactions on this target produce nuclei in excited states which travel some

distance before decaying by gamma-ray emission according to the lifetime of the excited state. The

resultant recoil nuclei then encounter a 0.13 mm stable-tantalum degrader before entering the S800

dipole magnetic fields and continuing on to the focal plane for particle identification.

Depending on the lifetime of the excited state, the recoil nuclei may decay before or after losing

velocity in the tantalum degrader, thus gamma rays may be emitted from nuclei with a velocity

49

𝛽1 which is “fast”, having not been slowed by the degrader, or with a “slow” velocity 𝛽2, having

lost energy and velocity to the tantalum degrader. When we Doppler correct the data for a single

𝛽 value, the resultant spectrum will show two peaks for a single transition; one peak at the true

ion-frame energy of the gamma ray and one peak that has been over or under corrected by the choice

of 𝛽. The relative intensities of these two peaks as a function of target-degrader separation distance

provides the data necessary to measure the lifetime of the excited state. In this section we will

briefly describe the basic operating principles of TRIPLEX, the settings used in this experiment,

and the recoil distance method for lifetime measurements.

2.5.1 TRIPLEX Design and Operation

The differential plunger device is, in essence, a target and degrader or stopper mount with

a precision distance-measurement system. Differential plunger devices have been employed at

laboratories around the world for various measurements. The Differential-Plunger for lifetime

measurements of tagged Unbound Nuclear States (DPUNS) [73] was designed to measure lifetimes

of unbound states and deformations. The TIGRESS Integrated Plunger device (TIP) [74] has

been utilized at TRIUMF [75] to determine electromagnetic transition rates via lifetime and low-

energy Coulomb excitation measurements. The Triple-foil differential Plunger for Exotic Nuclei

(TPEN) [76] was developed to measure excited state lifetimes with small cross sections. The

charge plunger device [77] has been designed to work with the MARA mass separator [78] to

determine lifetimes by detection of charge-state distribution of recoil nuclei. The Cologne Compact

Differential Plunger (CoCoDiff) [79] has been compactly designed to work with many different

spectrometers and detectors and to be sensitive to two regions of lifetimes simultaneously. The

TRIple PLunger for EXotic beams has been used with both the GRETINA and SeGA [80] gamma-

ray detectors to measure excited state lifetimes and cross sections using fast beams of rare isotopes.

TRIPLEX allows multiple targets to be mounted at various distances to conduct recoil-distance

measurements for excited states with lifetimes ranging from hundreds to tens of picoseconds. The

cylindrical shape of TRIPLEX allows it to be mounted inside a custom beam pipe which provides

feed-through connections from the TRIPLEX motors and conductance wires to the electronics

50

stack outside the beam line and finally to a computer dedicated to TRIPLEX operation. The

main components of TRIPLEX are three coaxial tubes separated by bearings to allow motion with

minimal friction and the motors that move the inner and outer tubes. Each tube is attached to a ring

upon which the target, degrader, and second degrader cones are mounted, with the target, degrader,

second degrader being mounted on their respective cones. The central tube remains stationary

and holds the first degrader ring upon which the first degrader cone is mounted. The inner tube is

connected to a precision motor which can move the target toward and away from the first degrader.

The outer tube holds the second degrader and is also connected to a precision motor that can move

the second degrader toward or away from the first degrader.

The distance between the target and degrader is measured by capacitance between the two

materials as the capacitance is inversely proportional to the distance between the plates. Great

effort is expended to ensure that the target and degraders are mounted parallel to each other so as

to model the system as a parallel plate capacitor. To determine the separation distance, a pulsed

charge is sent to the first degrader by a pulse generator in the electronics crate; the charge on the

first degrader induces an image charge on the target or second degrader which creates an electric

field which can be measured as a voltage difference. The electronics crate amplifies and shapes the

induced voltage signal received from the plunger and sends the signal to a multi-channel analyzer

which, in turn, transmits the signal to a computer that displays motor position and voltage and

provides a control panel through which to send movement commands to the plunger. TRIPLEX

also benefits from a second measurement system called the TESA [81], which relies on a mechanism

similar to a ball and spring interaction to determine position differences down to a tenth of a micron.

The position of the target and second degrader relative to their start points can be displayed in

the TRIPLEX LabVIEW GUI, but the absolute separation distance must be calculated from the

electrical contact position between plates, the relative distance the target has been moved away

from the contact position, and the voltage at each target position. When the motor reading, TESA

reading, and the voltage measurements are all calibrated sufficiently the position of the target or

second degrader with respect to the stationary first degrader can be determined to within tens of

51

Figure 2.14 Diagram of the TRIple PLunger for EXotic beams [57]. TRIPLEX is designed to
hold a production target and two degraders. In this diagram, the secondary beam enters from the
left then interacts with a 9Be target mounted on the target cone which can be moved toward or
away from the stationary first degrader cone to allow measurement of lifetimes generally the 10s
of picoseconds. Although TRIPLEX can hold a second degrader in the secondary degrader cone,
the second degrader was not required for this experiment but the second degrader cone held a
stripper foil to increase the number of fully stripped ions. Reprinted from Nuclear Instruments and
Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated
Equipment, Vol. 806, H. Iwasaki et al., The TRIple PLunger for EXotic beams TRIPLEX for
excited-state lifetime measurement studies on rare isotopes, Pages No.123-131, Copyright (2016),
with permission from Elsevier.

microns. This precision reduces systematic error resulting from uncertainty in the target-degrader

separation distance.

Once the target position has been calibrated, the TRIPLEX plunger uses the voltage readings

to maintain a constant distance setting by utilization of the feedback system. Using distance

measurements and voltage readings, LabVIEW operational software plots this data and a fit of the

data points is used by the feedback system to convert the voltage reading at any given moment into

a distance reading. When TRIPLEX is used in feedback mode, the software will automatically

adjust and maintain a separation distance defined by the plunger operator. The feedback system is

52

Figure 2.15 Full view of the TRIple PLunger for EXotic beams [57]. The secondary beam
enters TRIPLEX from the back (left) and interacts with the target and degraders (right). The
outer-most cylindrical housing protects motor control electronics and the TESA precision distance
measurement system. A target motor is displayed on the left (small cylinder parallel with housing),
this motor moves the inner cylinder which connects to the target ring.

necessary to maintain continual confirmation of the target position and correct any deviance from

the user-defined location by automatically moving the target motors to maintain a constant voltage.

2.5.2 TRIPLEX Experimental Configurations

For gamma-ray energy and efficiency calibrations, a 2-mm thick 9Be production target was

mounted to the target cone of TRIPLEX and moved to the position where the stationary tantalum

degrader would normally be located. For energy calibrations, a 152Eu gamma-radiation source was

mounted on the downstream face of the target and the TRIPLEX plunger was inserted into the beam

pipe so that the target was positioned 13 cm upstream from the center of GRETINA, as discussed in

section 2.4.1. GRETINA data was recorded for 60 minutes with the europium source to determine

the experimental efficiency of the GRETINA detectors with consideration given to attenuation of

gamma rays which are absorbed by the target and aluminum crystal shielding and to account for

areas lacking in solid angle detector coverage.

53

After the source run was completed, the target was dismounted and the beam was sent to the

experimental vault so characterization of secondary beam could be performed. The secondary

beam rate was measured over the course of two short runs (3 and 5 minutes) where the two runs

correspond to two different incoming beam acceptances, 𝑑𝑝/𝑝 = 1% and 𝑑𝑝/𝑝 = 0.5%. These

two runs provided important beam properties such as secondary beam energy, ion momentum, and

ion trajectory which are essential for producing the most realistic simulation of the incoming beam.

The target was then mounted again, the secondary beam was directed to the experimental vault,

and the S800 was tuned for the target-only data runs. The S800 dipoles were tuned to a rigidity

of 𝐵𝜌 = 2.2188 Tm, which placed the unreacted secondary beam nuclei on a central trajectory

through the S800, and two runs were again performed for the two momentum acceptances. This

configuration of rigidity settings and beam on target aids in characterizing the target itself as the

density of the material cited by the manufacturer may not be uniform; thus, our simulation allows

for a scale factor that compensates for discrepancies between the target manufacturer’s quoted

density and the effective density. This scale factor can be determined by simulating the position

distributions (𝑎𝑡𝑎, 𝑏𝑡𝑎, 𝑦𝑡𝑎) and the energy distribution 𝑑𝑡𝑎 of the unreacted beam on target, fitting

simulations to the data, and adjusting the scale parameter accordingly to achieve reasonable fits.

For this target, a scale factor of approximately 0.93 was found to properly reproduce the effective

thickness of our target. Two mask calibration runs were performed to calibrate the position readings

from CRDCs 1 and 2, and a 10 minute background run was conducted to quantify the level and type

of environmental background radiation. With the secondary beam on target, the S800 magnets were

re-tuned to center on the reaction products with a rigidity setting of 𝐵𝜌 = 2.1296 Tm. Data was

then taken in the target-only configuration for approximately 18 hours with coincidence between

the S800 and GRETINA triggered by event detection in the E1 scintillator. The target-only runs

provided valuable information regarding which excited states were populated during reactions on

target and how often they were populated relative to the other states.

Upon completion of the target-only runs, the target was moved 1 mm upstream and the 0.13-mm

Ta degrader was mounted to the stationary first-degrader cone. The S800 was re-tuned to center on

54

the unreacted secondary beam with rigidity of 𝐵𝜌 = 1.9578 Tm and two short runs were taken with

the two incoming-beam acceptance settings. These unreacted beam runs provide the information

necessary to characterize the tantalum degrader in the same manner described above for the target.

The S800 dipoles were then tuned to 𝐵𝜌 = 1.8577 Tm to center on the reaction products and data

was taken with the target-degrader separation distance of 1 mm for approximately 2 days. This

two-foil data-collection period resulted in the gamma-ray spectrums shown in the Data Analysis

chapter and provide data essential for the recoil-distance lifetime measurement.

Finally, the target was moved downstream toward the first degrader to take the 0-mm separation

data. The S800 settings were maintained from the 1-mm configuration as the energy loss remains

unchanged when no other components are added to the system. The 0-mm separation was main-

tained by the feedback system and data was taken for approximately 2 days in this configuration.

The second configuration is essential for the recoil distance lifetime method as this method relies

on the relative change in peak intensities as a function of separation distance. The short separa-

tion distance also allows for measurement of faster lived states, though for this work, those states

primarily decayed within the thick production target.

Reactions on target for all target and target-degrader configurations produced 39Ca and 39K

in excited states. When these nuclei de-excite by gamma-ray emission the resultant gamma-rays

are detected by GRETINA. In the next section we will describe how the relative intensities of

the gamma-ray peaks in a spectrum are used to determine the lifetime of an excited state via the

recoil-distance method.

2.5.3 Recoil-Distance Method

The recoil-distance method for lifetime measurements was utilized for this experiment as the

target-degrader separation distances were sensitive to the lifetime of the states of interest. As with

other types of radioactive decay, the decay of a nucleus by gamma-ray emittance is a spontaneous

and probabilistic process which can be described by an exponential decay function. Given some

initial number 𝑁0 of a particular species of nuclei in a particular excited state, the number of nuclei

55

remaining in the excited state 𝑁 as a function of time 𝑡 is given by

𝑁 (𝑡) = 𝑁0𝑒− 𝑡

𝜏

(2.9)

where 𝜏 is the mean lifetime of the excited state. The mean lifetime of a state is the time necessary

for the number of excited state nuclei to be reduced by a factor of 𝑒 and is approximately 44%

longer than the half-life of a nuclei, a value that is commonly quoted in other fields which measure

properties of radioactive nuclei. Of course, the mean lifetime 𝜏 can be related to the half-life 𝑇1/2 of

an excited state by 𝑇1/2 = ln(2)𝜏, but mean lifetime is most commonly used in RDM experiments

and thus we will restrict our discussion to this value.

The recoil-distance method exploits the exponential nature of the decay process to determine

the mean lifetime of an excited state by using the ratio of peak intensities of gamma rays emitted

from fast nuclei 𝐼 𝑓 and slow nuclei 𝐼𝑠 according to

𝐼𝑠
𝐼𝑠 + 𝐼 𝑓

= 𝑒− 𝑥

𝜈 𝜏

(2.10)

where 𝑥 is the target-degrader separation distance and 𝜈 is the velocity associated with the slow

nuclei (i.e., nuclei which emit gamma rays after losing velocity to the degrader). While this relation

is sufficient to describe the decay of a single excited state, when a nucleus decays by a series of

emissions from a higher orbital to an orbital multiple levels below (commonly referred to as a

“cascade”), the relation becomes more complicated as each state has a unique lifetime which feeds

the observed lifetime of the lower states. Rather than repeat what has already been sufficiently

described, this author will direct readers to Ref. [82] for a rigorous derivation of the relevant

relations. However, Fig. 2.16 provides an illustration of the basic principles underpinning the RDM

lifetime measurement with a more intuitive visual description of the relation between fast and slow

peak intensities resulting from a single transition.

The 2D-plot in figure 2.16 depicts the relation between the Doppler-corrected energy of the

gamma rays (x-axis), the angle of emittance as determined by the main interaction point in

GRETINA (y-axis), and the number of gammas rays detected at a particular angle and energy

(z-axis). The diagram below the 2D spectrum depicts the relation between gamma-ray energy

56

and Doppler correction when a single velocity 𝛽2 is used to correct gamma-ray energies from fast

(blue) and slow (red) in-flight nuclei. When the secondary beam interacts with the target, a reaction

occurs which produces 39Ca* in an excited state (indicated by the asterisk). The recoil nuclei then

travel beyond the target, through the degrader, and into the S800 for particle identification. Upon

excitation in the target the radioactive decay process begins. Some nuclei will decay before the

degrader with a fast velocity 𝛽1, other nuclei will lose energy in the degrader and decay with a slow

velocity 𝛽2. Because the nuclei are randomly oriented, each gamma ray will be emitted with a dif-

ferent trajectory with respect to the trajectory of the in-flight nucleus; therefore, GRETINA detects

gamma rays from interactions with crystals at all angles covered by the detector configuration (see

Fig. 2.10 for a rendering of the solid angle coverage for this experiment).

The detection of gamma rays at a single energy by detectors at all angles can be observed

in the 2D plot in Fig. 2.16 as the vertical lines at approximately 1100 keV and 900 keV, these

are the gamma rays emitted from the slow nuclei and Doppler corrected with 𝛽2 according to

Eqn. 2.8. Meanwhile, gamma rays originating from decays of the same excited state, but emitted

before the degrader and Doppler corrected with 𝛽2 (the beta associated with velocity after the

degrader), result in counts with under-corrected energy, producing higher energy components at

low detector angles. These fast components gradually converge with the straight, properly-corrected

gammas at high-angle detectors since the Doppler shift is very small to an observer (or detector)

perpendicular to the ion-trajectory. This merging of gamma ray energies in the 2D spectrum at

high angles gives a characteristic “chopstick” pattern for gamma rays originating from the same

transition, with the same energy, but Doppler corrected with different 𝛽. Gamma rays emitted

in the lab-frame appear over corrected in the 2D spectrum, the diagonal lines with low energy at

low angles and higher energy at high angles are typically the result of laboratory-frame gamma

rays where recoil neutrons created by the fragmentation of the secondary beam excite aluminum

nuclei in the beam pipe or germanium nuclei in the detector crystals which then emit gamma rays;

these gamma rays are commonly referred to as “neutron-induced background” gamma rays. The

excitation of the surrounding beam pipe, TRIPLEX components, and germanium nuclei result in

57

Figure 2.16 The 9Be(42Sc, 39Ca*)X reaction produces nuclei in excited states (denoted by “*”).
Gamma rays may be emitted from “fast” nuclei of velocity 𝛽1 or “slow” nuclei of 𝛽2, having been
slowed by the degrader. The gamma-ray emittance angle (y-axis) is plotted as a function of the
Doppler-corrected gamma-ray energy (x-axis) and the number of counts at a particular angle and
energy (z-axis).

gamma rays emitted from stationary nuclei and produce the over-corrected diagonal lines when

Doppler corrected with a 𝛽 associated with in-flight nuclei.

Given the 2D spectrum in Fig. 2.16, we can “cut” the spectrum at a low angle (represented by

the dashed line) and project the spectrum onto the x-axis for all events below our cut angle; this

projection produces a 1D gamma-ray energy spectrum with counts on the y-axis and gamma-ray

energy on the x-axis, as shown in Fig. 3.13. When we make this low-angle projection onto the x-

58

axis we see a characteristic 2-peak structure for a single gamma ray transition; e.g., the true-energy

gamma peak at 1100 keV and the under-corrected peak at 1130 keV in Fig. 3.13. From the 1D

gamma-ray spectrum we can then calculate the intensities of the slow peak 𝐼𝑠 at 1100 keV and the

fast peak 𝐼 𝑓 at 1130 keV and use Eqn. 2.10 with the separation distance 𝑥 and the velocity 𝜈 to solve

for the lifetime 𝜏 of the excited state. But again, this simple calculation omits important details.

The number of counts observed in each peak includes gamma-ray counts that originated from

cascades which fed the state-of-interest, this makes the slow peak appear to have more counts than

it would if the all the decays were the result of direct population of the state-of-interest. A nucleus

may be excited to a higher-lying state in the target, travel some distance associated with the lifetime

of that higher-lying state, emit a gamma ray by de-excitation to the lower-lying state-of-interest,

then travel more distance associated with the lifetime of the state-of-interest before finally decaying

at some location along its trajectory. The cascade of decays from higher-lying states results in

an increase in the intensity of the slow peak, resulting in a calculation of an apparent lifetime

which is longer than the actual lifetime of the state-of-interest. Therefore, knowledge of the direct

population of all higher-lying states that feed the state-of-interest is crucial for proper utilization

of the recoil-distance method for lifetime measurements. This topic will be explored extensively

in the Data Analysis section of this work as the states-of-interest which are the focus of this work

are fed by two higher-lying states and significant effort was expended to account for feeding effects

that contribute to the systematic uncertainty in the lifetime measurements.

In practice, the relevant relations utilized in RDM lifetime measurements are coded into a

GEANT4 [83, 84] simulation package which utilizes parameters such as secondary beam charac-

teristics, target-degrader separation distance and material properties, direct population fractions

of the excited states in a cascade, and the mean lifetimes of excited states to produce a simulated

gamma-ray spectrum. Many of these parameters can be constrained with knowledge of the beam

profile, S800 dipole settings, and the position, angle, and energy distributions at the target. Once

these parameters are constrained one can focus on the two most important parameters for this work:

the lifetimes of the excited states, and the direct population fractions for excited states which feed the

59

states of interest. The simulation package and the relevant parameters utilized to fit the simulated

gamma-ray spectra to the experimental spectra will be discussed further in the following chapter.

60

CHAPTER 3

DATA ANALYSIS

This chapter describes the analysis of data taken with all plunger settings, including detailed

procedures of calibrations, simulations and determination of the lifetimes of interest. The analysis

is conducted in three main steps: calibration of the experimental data, the adjustment of parameter

sets used in the simulation, and determination of the lifetime of the state of interest by fitting

simulations to the experimental data.

In the case of the present work, each step was repeated

separately for the analysis of 39Ca and again for the analysis of the mirror nucleus 39K.

Within the lifetime analysis phase for each nucleus, the analysis of the both the 11/2− and 9/2−

states was performed. The analysis of the 11/2− state lifetime in 39Ca was the primary focus of

this work due to its structural similarity to the 2+ state in 38Ca, while the analysis of the 9/2− state

was conducted because the TRIPLEX configuration happened to be sensitive to the lifetime of the

9/2− state. The analysis of the 11/2− state in 39K was performed for two reasons; first, since the

lifetime of the state had been measured previously, the current analysis served as a verification

of the techniques used to study the lesser known 39Ca state. Secondly, the analysis of the mirror

11/2− → 7/2− transition in 39K provided the opportunity for a self-consistent application of the

matrix element decomposition being that the 39Ca analysis provided the proton matrix element

while the 39K provided the neutron matrix element.

In the following sections, we will discuss the relevant calibrations of the data and simulations

used in this work and then move into the lifetime analysis. Because the analysis spans two states

in two nuclei from data taken with three plunger settings, we will attempt to minimize repetitive

explanations and keep the majority of the analysis technique descriptions restricted to the 39Ca

analysis, unless there was reason for a significant departure in the analysis of the mirror nucleus, in

which case, the difference in techniques utilized will be explained when required.

3.1 Experimental Data Calibration

During the experiment, the data collection was monitored using the SpecTcl software pack-

age [85]. SpecTcl allows for online viewing of data and limited offline analysis tools. A rough

61

calibration of the relevant detector data was used during the experimental run time to view a sample

of the data taken and ensure that all the devices were working, the device settings were suitable for

the experiment, and to make necessary adjustments to the device settings during experimental run

time. But the SpecTcl online analysis tool was used only to display a small fraction of the actual

number of events that the NSCLDAQ is recording. For a proper analysis of the experimental data,

all events must be analyzed and the data must be calibrated to a higher standard to get the most

accurate representation of each event. Thus, after the experiment was over, the raw-event data was

unpacked and the data from specific devices was corrected for a complete analysis.

When experimental data is collected by the NSCLDAQ, the data is stored in run files with each

run file containing a number of events. The events coincide with the S800 triggers and the data

collected from all the coincident detectors that were triggered by each event, this data structure

allows for event-by-event detail and correction. A C-type library called GrROOT [86] is used to

unpack the raw event data, perform corrections to the data based on the device settings, and create

histograms from the events. The histograms are displayed in ROOT trees [87] which can then

be analyzed. Using the histograms created from raw data from each of the detector components

described in the previous section, we calibrated the data to correct for any mismatching of device

settings and irregularities to provide the most accurate representation of the secondary beam, the

recoil nuclei produced in reactions on the TRIPLEX target, and the gamma-ray events that provide

the basis for the lifetime analysis.

In the following sections we will describe the experimental data calibration processes. We will

then move the discussion to calibration of our simulation in a similar fashion. Finally, we will

explain the lifetime analysis process and results for each of our nuclei of interest.

3.1.1 Scintillator Data Calibration

As described in section 2.3.1, the differences between the E1 trigger and XFP and OBJ scintilla-

tors are used to identify the secondary beam components. The timing information of the scintillators

must be individually corrected as the first step of the data calibration phase in order to produce

a secondary beam PID where each isotone is well separated. Since the time-of-flight (TOF) of a

62

nuclei is dependent on the nuclei’s trajectory through the S800 as determined by the CRDCs, we

must correct the TOF with respect to the angle in the S800 focal plane 𝑎 𝑓 𝑝 and the position in the

dispersive plane 𝑥 𝑓 𝑝. As a backup measure, the scintillators are connected to multiple time signal

converters to add redundancy to the data acquisition system. Time-to-analog converters (TACs)

and time-to-digital converters (TDCs) offer two avenues of signal conversion to record the timing

signals in the NSCLDAQ. For this work, corrections were made for both of the converters, but

the TDC was used for further data calibration as it offered superior resolution over the TAC after

calibration.

As shown in Fig. 3.1, a linear correction on the E1-OBJ time difference produces a spectrum

where each species of the secondary beam is well separated in time (OBJ) while the trajectory angle

in the dispersive plane 𝑎 𝑓 𝑝 forms a distribution across the CRDCs. Similar timing corrections were

performed with the E1-XFP scintillators and the position in the focal plane 𝑥 𝑓 𝑝. After the scintillator

data has been corrected, we plot E1-XFP against E1-OBJ to produce the incoming secondary beam

histogram. Figure 3.2 is produced from data taken while no target was installed in TRIPLEX

and the S800 rigidity set to center on the secondary beam with an incoming-beam momentum

acceptance 𝑑𝑝/𝑝 = 1%.

As one can see in Fig. 3.2, the 𝑁 = 21 isotones are separated due to a calibration of the timing

signal data from the XFP and OBJ scintillators. In addition to the labeled groups of events, we

also see another group of events at lower XFP and higher OBJ time differences that is due to other

isotone contaminants which are present in the incoming beams. Although it is not included in the

plot of Fig. 3.2, there is an additional group of events which is a ghost-type image resulting from

the bunched structure of the beam and the timing window imposed on the data collection. When

the data collection time window for an event is wider than the time associated with the cyclotron

frequency, data will be recorded from the next (or previous) bunch of nuclei that interact with the

scintillator, resulting in the second image of the secondary beam isotones.

In practice, the timing corrections were performed with a run where the target was installed in

TRIPLEX. It is necessary to use a target-only run for the target-only data calibration as the desired

63

Figure 3.1 Top: Angle 𝑎 𝑓 𝑝 in the focal plane as a function of the uncorrected time between the
E1 and OBJ scintillators. Bottom: Angle 𝑎 𝑓 𝑝 in the focal plane as a function of the corrected
time between the E1 and OBJ scintillators After the timing correction, the tilted distributions (top)
become aligned (bottom) within narrow time distributions such that the secondary beam nuclei are
well separated in time. These plots serve as examples of the timing corrections that were performed
as the first step of the data calibration process.

end result of our calibration process is a particle identification diagram (PID, Fig. 2.8) which

displays the reaction products associated with the chosen secondary beam. With the PID, we can

gate on a specific reaction product and see all gamma rays associated with that ion of interest. The

reaction product PID requires proper calibration of the OBJ scintillator data and the ion chamber

64

Figure 3.2 Secondary beam components separated by the TOF method where the time differences
between the E1 and XFP or OBJ scintillators are plotted on their respective axes. The axes represent
arbitrary units of time and the 𝑁 = 21 isotones have been labeled for the reader. This plot was
created from a run without the target installed in TRIPLEX and is used to characterize the secondary
beam properties.

data which we will describe in the next section.

3.1.2

Ion Chamber Data Calibration

Once the timing corrections have been applied to the scintillator data the next step is calibration

of the ion chamber data. We begin the process by plotting the XFP and OBJ scintillator time

differences from a target-only run as shown in Fig. 3.3. One may notice that the plot created from

the target-only run is a “smeared” version of the plot with no target installed (Fig. 3.2). When

we place the target in the beam line, the differences in differential energy loss of individual ions

within a given species becomes apparent because each ion is interacting with the target at different

locations and losing different amounts of energy than other ions of the same species. This “energy

straggling” of the secondary beam ions is attributed to each ion’s unique interaction with the

Coulomb forces inside the target, resulting in different energy losses for each event. In addition,

the S800 spectrograph is set to have a fixed magnetic rigidity and therefore a different velocity

component is selected for each isotope based on its 𝐴/𝑍 value. The result of energy straggling

65

OBJ (arb)1100-1000-900-800-700-600-500-XFP (arb)300032003400360038004000420001020304050607080V44Ti43Sc42Ca41within the target is that each ion leaves the target with a slightly different energy, and thus, a slightly

different velocity, leading to a distribution of time differences from both scintillators with respect

to E1.

Figure 3.3 Secondary beam components separated by the TOF method for a target-only data run.
The axes represent arbitrary units of time and the 𝑁 = 21 isotones have been labeled for the reader.
This plot was created from a run with the target installed in TRIPLEX. Gating on a particular
secondary beam such as 42Sc for this work allows calibration and analysis of all reaction products
from that beam.

Once we have identified the species in the secondary beam PID, as labeled in Fig. 3.3, a

graphical tool within ROOT is used to draw a “gate” around the particular isotone we are interested

in seeing reaction products from. Using this gate we can then create an uncorrected ion chamber

energy loss (dE) versus TOF plot and see all the reaction products associated with that incoming

beam. The uncorrected dE vs TOF plot will appear smeared and difficult to resolve but this gives

a basic plot to which the corrected version can be compared to. We can then identify a particular

reaction product and create a gate that allows us to calibrate the ion chamber data such that energy

loss in the ion chamber does not depend on the ion’s position in the dispersive plane. This is an

important calibration as energy straggling in the target may cause recoil nuclei of the same species

to follow slightly different trajectories through the S800 and be detected at different positions in

the dispersive plane. Figure 3.4 displays the corrected energy loss in the ion chamber as a function

66

of position in the dispersive plane (x) for our reaction product of interest, 39Ca. One will note that

once the ion-chamber data has been properly corrected, the energy loss of a particular ion traveling

through the ion chamber is independent of the recoil nuclei’s position in the dispersive plane.

Figure 3.4 The corrected energy loss of 39Ca in the ion chamber (y-axis) is independent of the ion’s
position (x-axis) in the dispersive plane. Though this ion chamber data correction was performed
with a secondary beam of 42Sc and reaction product 39Ca, the calibration holds for our mirror
nucleus 39K as well.

3.1.3 CRDC Pad Calibration

As stated in section 2.3.2, the CRDCs detect positions by fitting a Gaussian to the induced-

charge distribution over multiple CRDC pads when an ion passes through the chambers. In the data

calibration phase, the signals from each of the 224 pads on each CRDC must be gain matched such

that they provide consistent voltage information independent of the channel and electronics used to

read the signal. The CRDC pad calibration is an iterative process in which the calibration of one

pad affects the induced signal read by a neighboring pad; therefore, one pad must be calibrated and

then the next pad may be calibrated based on the corrected signal from the previous pad. After

several iterations of pad corrections, the induced signal across all pads should be relatively constant,

as we see in Fig. 3.5. Though not pictured, both CRDCs must be calibrated in a similar fashion to

get accurate gain matching across all channels.

67

Figure 3.5 The induced signal (y-axis) for a given CRDC pad (x-axis) will be relatively constant
after the pad gain-matching calibration. The pad readings from the 39K recoil nuclei are given as
an example of a corrected spectrum for a single CRDC.

3.1.4 CRDC Mask Calibration

The final calibration performed on the CRDC data is referred to as the “mask” calibration. As

stated in section 2.5.2, two mask-calibration runs were conducted during the experiment with the

target installed on TRIPLEX. During these runs, an aluminum plate is placed in front of a CRDC

and data is taken for approximately 10 minutes per run. The aluminum plate contains a pattern of

holes through which beam particles are allowed to pass. Particles that pass through the mask create

a pattern in the CRDC that corresponds to the hole pattern on the aluminum plate. The readings

from the CRDC mask runs are then used to calibrate the position readings from the CRDC pads.

Figure 3.6 shows the result of a mask run in the left panel and the mask pattern in the right

panel. During the mask calibration the hole pattern from the data run on the left is matched to the

corresponding mask pattern on the right. Multiple holes are used in the calibration and the position

information from the data is plotted as a function of the known hole-position information from the

pattern on the right. A linear function is then fit to the calibration-plot data which contains a slope

and offset. The experimental data is then re-run with the proper CRDC slope and offset information

to calibrate the CRDC position data. Here we will note that although the aluminum mask was

68

Figure 3.6 CRDC X and Y position calibration using the mask pattern. The aluminum mask was
inverted when placed in front of the first CRDC, but the position calibration holds so long as each
cluster of data points on the left is correctly matched to the associated mask hole on the right. An
outline of the “L” pattern has been added to show correspondence between the data in the left panel
and the calibration pattern in the right panel.

inverted when placed in front of the upstream CRDC, as shown by the “L” outlines provided in

Fig. 3.6, the calibration process is independent of the plate orientation so long as each cluster of

data points is associated with the proper hole pattern.

3.1.4.1 S800 Inverse Map

Once the CRDC position data has been corrected, we calculate the trajectory of ions at the target

location from the CRDC data taken in the focal plane. To obtain the ion trajectory angle in the

dispersive plane 𝑎𝑡𝑎, the angle in the non-dispersive plane 𝑏𝑡𝑎, the ion position in the non-dispersive

plane 𝑦𝑡𝑎, and the difference in ion energy 𝑑𝑡𝑎 with respect to the energy of an ion along the S800

central trajectory, we must use a transformation matrix that maps the trajectory detected by the

CRDCs back to the trajectory at the target location on an event-by-event basis.

The transformation matrix, commonly referred to as the “inverse map”, is calculated from the

magnet settings of the S800 for a particular nuclei of interest; specifically: two quadrupole current

settings (in Amps), the two S800 dipole current settings, the rigidity setting (in Tm) of the S800

during the specific run type (no target, target-only, or target+degrader), the mass (A) and charge

69

(Z) of the nuclei of interest, and the distance between the TRIPLEX target and the S800 pivot

point. From these values a transformation matrix can be produced which, when applied to the data

calibration, builds the 𝑎𝑡𝑎, 𝑏𝑡𝑎, 𝑦𝑡𝑎, and 𝑑𝑡𝑎 distributions on an event-by-event basis.

The distributions are Gaussian in nature but may not have a mean centered at the zero position

or angle due to inconsistencies in the CRDC readings which may cause the position reading to drift

over the course of hours or days. Therefore, once the inverse map is applied to the data, run-by-run

corrections are made to the trajectory distributions to center the mean at the zero position and

zero angle. Because our simulation does not drift over time, the run-by-run corrections to 𝑎𝑡𝑎,

𝑏𝑡𝑎, and 𝑦𝑡𝑎 are applied to the data so that the simulated distributions align with the experimental

distributions.

This concludes the discussion of the data calibration. In the next section, a description of the

simulation calibration processes will be discussed followed by the lifetime analysis of the 39Ca and

39K states covered in two sections which are associated with the particular nucleus of interest.

3.2 Simulation Calibration

Calibrating the G4Lifetime simulation to the experimental data starts with the characterization

of the incoming beam profile. As displayed in Fig. 3.2, the time-of-flight method is employed for

two runs without a target installed to aid in determining the secondary beam characteristics, which

are parameters necessary for the most accurate reproduction of the experimental data. From the

S800 rigidity settings and data runs with no target installed we can determine the secondary beam

energy. But when the target is installed, reactions in the target produce recoil nuclei and knocked

out nucleons which carry energy and provide information about the reaction. Therefore, we must

adjust the parameters in our simulation to match the experimental measurements.

Figure 3.7 provides the corrected trajectory distributions for the unreacted secondary beam

of 42Sc for a single, target-only run. The incoming secondary beam energy of 85.2 MeV/u was

calculated from the S800 rigidity setting 𝐵𝜌 = 2.2188 Tm for the run. By fitting the simulated 𝑑𝑡𝑎

distribution (red) to the data (blue) it was determined that a scale factor of 0.93 was required to

replicate the effective density of the 9Be target. In a similar fashion, another unreacted beam run

70

was used to determine that the effective density of the 181Ta degrader was best replicated with a

scale factor of 0.96.

Figure 3.7 Trajectory distributions at the TRIPLEX target position for the unreacted 42Sc beam.
Experimental data for the angle in the dispersive plane 𝑎𝑡𝑎, angle in the non-dispersive plane 𝑏𝑡𝑎,
position in the non-dispersive plane 𝑦𝑡𝑎, and the energy distribution 𝑑𝑡𝑎 are displayed in blue. The
simulated secondary beam trajectory information is displayed in red.

Once the incoming beam, target, and degrader properties were determined we began char-

acterization of the outgoing, reaction-product beam. This process again relies on the trajectory

distributions at the target position. For outgoing beam properties relevant to our nuclei of interest,

we gate on the particular recoil nuclei and use the trajectory distributions to determine the outgoing

beam properties. Fig. 3.8 displays the trajectory distributions at the target for the reaction product

39Ca during a target-only run. From the S800 rigidity setting 𝐵𝜌 = 2.1296 Tm, we calculated that

our recoil nuclei have an energy distribution centered at 55.9 MeV/u. We then adjusted parameters

associated with the energy centroid and distribution width of the knocked-out nucleon and the

difference in velocity between the incoming beam and outgoing beam to fit the simulated 𝑑𝑡𝑎 (red)

to the data (blue), as is displayed in Fig. 3.8. By fitting the 𝑑𝑡𝑎 spectrum, the necessary reaction

information is passed to the simulation such that the angle distributions 𝑎𝑡𝑎 and 𝑏𝑡𝑎, and position

distribution 𝑦𝑡𝑎, are replicated by the simulation.

71

Figure 3.8 Trajectory distributions at the TRIPLEX target position for the reaction product 39Ca
taken during a target-only run. Experimental data for the angle in the dispersive plane 𝑎𝑡𝑎, angle in
the non-dispersive plane 𝑏𝑡𝑎, position in the non-dispersive plane 𝑦𝑡𝑎, and the energy distribution
𝑑𝑡𝑎 are displayed in blue while the simulation is displayed in red.

Finally, for the runs with both target and degrader installed on TRIPLEX another set of trajectory

distributions was created to characterize the outgoing beam information after interaction with the

tantalum degrader. From the S800 rigidity setting 𝐵𝜌 = 1.8351 Tm, we calculated that our recoil

nuclei have an energy distribution centered at 41.9 MeV/u. The distributions associated with the

plunger data (target and degrader) are given in Fig. 3.9. Parameters associated with the energy

of the knocked-out nucleon were adjusted, as was done with the target-only distributions, and the

relevant parameters were passed to the simulation to produce the most accurate representations of

the gamma-ray spectra.

The same calibration process described above was also performed for the 39K simulations.

The 9Be(42Sc, 39K)X reaction relied on the same 42Sc beam properties and target/degrader prop-

erties, therefore, the energy distribution 𝑑𝑡𝑎 was fitted to obtain the unique parameters required to

characterize the outgoing 39K beam. With both the data and the simulations calibrated we could

then create the gamma-ray spectra and move onto the analysis of the data by comparison to the

72

Figure 3.9 Trajectory distributions at the TRIPLEX degrader position for the reaction product 39Ca
taken during a run with target and degrader mounted. Data for the angle in the dispersive plane
𝑎𝑡𝑎, angle in the non-dispersive plane 𝑏𝑡𝑎, position in the non-dispersive plane 𝑦𝑡𝑎, and the energy
distribution 𝑑𝑡𝑎 are displayed in blue while the simulation is displayed in red.

simulations for the purpose of determining the lifetime of our states of interest.

3.3 Analysis Overview

As mentioned in section 2.5.2, the data were taken for three TRIPLEX configurations: target-

only, target and degrader with 1 mm separation, and target and degrader with 0 mm separation.

The gamma-ray peaks from the target-only spectra were compared to the adopted values [45] of the

transition energies to determine which excited states were populated and which gamma-ray peaks

were associated with which transitions. The states which were populated in this experiment as well

as the transitions observed are depicted in Fig. 3.10. The target-only spectra were also used to

determine in what percentages each excited state was populated in both 39Ca and 39K.

The 1 and 0-mm spectra were used to determine the lifetimes of the excited states of interest;

that is, the 11/2− and 9/2− excited states. Application of the recoil-distance method for the lifetime

measurement of the 11/2− state provided a model independent determination of the reduced

transition strengths (𝐵(𝐸2)). Evaluation of the lifetime of the 9/2− state in 39Ca was opportunistic

73

Figure 3.10 Level diagrams for gamma rays observed in the target-only spectra for 39Ca (left) and
39K (right). The energy information is from Ref.[45] and rounded to keV. The relative intensities
of the observed gamma-ray transitions, indicated by the widths of the arrows, are normalized to
the intensity of the 7/2− → 3/2+
𝑔.𝑠. transition for the respective nuclei. The energy information for
the (11/2−) state is highlighted in red, while those for the (9/2−) state are highlighted in blue [88].
Reprinted figure with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024).
Copyright (2024) by the American Physical Society.

74

10301511252109527968447571774347783281411303/20(15/2)6432(13/2)5402(11/2)3891(9/2)36407/227963/2015/2647513/2571811/239449/235977/22814 39Ca 39Kas the plunger configurations were, by chance, sensitive to the 9/2− state as well as the 11/2− state

of interest, thus allowing for a more precise measurement of the 9/2− state lifetime, which had only

been measured once prior to this work [53]. Agreement between the known lifetimes of the 11/2−

and 9/2− states in 39K and the lifetimes determined in this work strengthened confidence in the

analysis methods employed for lesser known lifetimes of the analog states in 39Ca.

For both nuclei, a rigorous analysis of systematic errors which contributed to uncertainty in the

lifetime measurements was performed. The systematic error analysis accounted for the effects of

lifetime and population of the higher lying 15/2+ and 13/2− states on the lifetime measurement

of the lower lying 11/2− and 9/2− states. The observation of the short-lived 13/2− state provided

data that were used to constrain the ratio of reactions on target to reactions on degrader, which was

another parameter used in the simulation.

In the following sections we will delve into the analysis of each nucleus by examining the

spectra obtained from each TRIPLEX configuration. Since the analysis performed on 39Ca was

validated by using the same method on the 39K data, most of the detailed analysis methods will be

contained in the 39Ca sections. The particular analysis procedures for 39K which diverged from the

39Ca procedures will be examined in the 39K analysis sections.

3.3.1

39Ca Target-only Data

The gamma-ray spectrum for 39Ca measured with the target-only configuration is provided in

Fig. 3.11. This figure presents the energy spectrum for gamma rays measured by all GRETINA

detectors where Doppler-shift corrections were applied with the gamma-ray emission angle 𝜃 and

ion velocity at the downstream face of the target, 𝛽 = 0.33. Figure 3.11 includes background

contributions from neutron-induced reactions which were evaluated in the laboratory frame and

whose responses in the ion frame were simulated and included in the fit [89]. The neutron-

induced backgrounds are depicted in green and it should be noted that a low-angle cut was not

applied to the target-only data; therefore, the lab-frame backgrounds appear “smeared out” in the

Doppler-corrected spectra; but when the cut is applied, as it is in the 1 mm and 0 mm data, the

contributions from the lab-frame backgrounds are evident. Two exponential functions are assumed

75

Figure 3.11 Doppler-corrected gamma-ray spectrum for 39Ca for the target-only run. Transitions
from the (11/2−) state are labeled in red, from which the lifetime is determined in the present work.
Gamma rays from the (9/2−) state are labeled in blue, for which we provide an improved lifetime
result [88]. Reprinted figure with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p.
024322 (2024). Copyright (2024) by the American Physical Society.

for the background and the simulation was fit to the data using the log likelihood method. The

gamma-ray peaks associated with 39Ca are clearly visible at 252 keV, 844 keV, 1030 keV, 1095 keV,

1511 keV, and 2796 keV. By matching the experimental gamma-ray energy to available data for

39Ca, the spin-parities and energies of each transition were clearly identified. Gamma-gamma

coincidence was used to verify that the gamma-ray transitions observed in Fig. 3.11 belong to the

cascade shown in Fig. 3.10, with the 6432-keV (15/2+) state being the highest populated.

Once the coincidences had been verified, the order of the cascade was discernible from the

Nuclear Data Sheets [45] for 39Ca, available from the National Nuclear Data Center (NNDC) [33].

A reaction in the target could result in the direct population of the 15/2+ state at 6432 keV and a

subsequent decay by emission of a 1030 keV gamma-ray to the 13/2− state, which then proceeds

to decay to the 11/2− state by emission of a 1511 keV gamma-ray. When the 11/2− state was

populated by direct population from the reaction or by indirect population from decay of the 13/2−

state, a transition to the 9/2− state at 3640 keV or to the 7/2− state could ensue. The probability

76

Energy (keV)500100015002000250030003500Counts/2keV0200400600800100012001400160018002000-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+23fi-27Ca39Target-onlyData rays + BackgroundgSimulated -ray Simulationg raysgLab-frame Exponential Backgroundsof the 11/2− state decaying to the 9/2− or to the 7/2− state is the branching ratio of the state. An

interaction with the target could also directly populate the 7/2− state at 2796 keV or the state could

be indirectly populated by decay from the 9/2− or 11/2− states.

The spectrum was compared to a simulation which utilizes the GEANT4 [83, 84, 90] package

developed to reproduce RDM experiments utilizing TRIPLEX and GRETINA [89, 91, 90, 51].

The gamma-ray transitions indicated by arrows in Fig. 3.10 were included in the simulation. Once

properly calibrated, the G4Lifetime simulation requires four parameters to produce a gamma-ray

spectrum: the level energy, the gamma-ray transition energy, the direct population fraction of the

state, and the lifetime of the state. Since the level energies and the gamma-ray transition energies are

known quantities, the two main parameters that were adjusted in the simulations were the lifetimes

and direct populations of the states. A cascade can be implemented in the simulation when the

level energy of the lower state is equal to the level energy of the higher state minus the energy of the

emitted gamma ray. In this manner, we were able to simulate cascades of decays starting with the

15/2+ state and ending with a decay from the 7/2− state to the 3/2+ ground state. Though the order

of the cascade was evident, quantifying the probability that a particular state was directly populated

in a reaction required a more rigorous methodology using two simulations to reproduce the two

cascade paths created by the branch in the 11/2− state. Suffice it to say, the direct populations to

all excited states in the cascade from the 9Be(42Sc, 39Ca)X reactions were determined from the

yields of the gamma-ray peaks in the target-only spectrum, with considerations given to indirect

populations from the observed higher-lying states.

The results of the analysis of the direct population fractions for each of the observed states

were normalized such that their sum is 100% and direct population to the ground state was not

calculated, nor were populations to unobserved states, due to the absence of gamma-ray peaks to

analyze. Reactions on the target resulted in a weak direct population of 11(3)% to the 6432-keV

(15/2+) state, which then decays by emission of the 1030-keV gamma ray to the (13/2−) state. The

weakest direct population of 1(1)% was observed for the 5402-keV (13/2−) state, which decays

by emission of the 1511-keV gamma ray to the (11/2−) state. The strongest direct population of

77

37(8)% was observed for the 3891-keV (11/2−) state, which emits the 252-keV gamma ray decaying

to the (9/2−) state and the 1095-keV gamma ray decaying to the 7/2− state. A direct population of

17(4)% was determined for the 3640-keV (9/2−) state, which decays via the 844-keV gamma-ray

transition to the 2796-keV 7/2− state. Finally, a direct population of 34(10)% was observed for

the 7/2− state, which decays directly to the 3/2+ ground state. One may also note that we did not

observe the population to the 1/2+ state at 2467 keV in our target-only measurement, as was the

case for a recent study of 39Ca by Gade et al. [92].

The relative branching ratio of the 252-keV gamma ray 𝑏𝑟𝑒𝑙 = 43(3) with respect to the

1095-keV gamma-ray branch 𝑏𝑟𝑒𝑙 = 100(6) for the (11/2−) state was calculated from the ratio

of gamma-ray intensities at each energy. Within quoted uncertainties, our result agrees with the

adopted value of the relative branching ratio for the 252-keV branch, 𝑏𝑟𝑒𝑙 = 51(6) [45]. Based on

our relative branching ratio, we find an absolute branching ratio 𝑏𝑎𝑏𝑠 = 70(2)% for the 1095-keV

decay; this value is later used with the gamma-ray energy and the lifetime to calculate the reduced

transition strength 𝐵(E2, (11/2−) → 7/2−).

3.3.2

39Ca RDM Analysis

Having determined the populations of excited states from the target-only data, we move to the

analysis of the recoil-distance lifetime data. The measured spectrum for the 1-mm target-degrader

separation is presented in Fig. 3.13 where Doppler-shift correction was performed with 𝛽 = 0.29,

corresponding to the velocity of the ions on the downstream face of the degrader. The spectrum

measured with the 0-mm separation is presented in Fig. 3.14 and was corrected for Doppler shifts

with the same 𝛽 value.

As discussed in Sect. 2.5.3, excited states with lifetimes to which the target-degrader config-

uration is sensitive show a characteristic double-peak structure in the 1D gamma-ray spectrum.

These structures can be seen in the 1-mm spectrum for the (11/2−)→7/2− and (11/2−)→(9/2−)

transitions labeled in red, and the (9/2−)→7/2− transition in blue, in Fig. 3.12. The two compo-

nents of the gamma-ray double-peak structure are defined by their decay location along the ion

trajectory; gamma rays emitted before an ion reaches the degrader are denoted “fast” components

78

Figure 3.12 Doppler-corrected gamma-ray spectrum for 39Ca taken with the 1-mm target-degrader
separation. The spectrum has been gated for the gamma-ray emission angle between 𝜃 = 29◦ and
𝜃 = 46◦ to improve sensitivity to Doppler shifts. Transitions from the (11/2−) state are labeled
in red, while transitions from the (9/2−) state are labeled in blue. Laboratory-frame background
gamma rays, associated with neutron-induced reactions [89], are shown in green and downscaled
by a factor of 2 for clarity [88]. Reprinted figure with permission from A. Sanchez et al., Phys.
Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

while gamma rays emitted after an ion passes through the degrader are called “slow” components.

Both spectra were produced from an angle gate of a 2D histogram for detectors between 29◦ and

46◦. The low-angle gate improved sensitivity to Doppler shifts while separating the fast component

of the 15/2+ → 13/2− transition at 1060 keV from the slow component of the 11/2− → 7/2−

transition at 1095 keV.

In Figs. 3.12 and 3.13, we observe that the low-angle cut results in sharpened background peaks

resulting from lab-frame emissions which are simulated in green and labeled with black geometric

icons. Starting at the low-energy end of the spectra, we first observed the pair-production peak at

approximately 400 keV. As discussed in Sec. 2.4, pair production can occur when a gamma ray

interacts with the Coulomb field of a nucleus, producing an electron-positron pair in which the

positron will annihilate with another electron to produce another pair of gamma rays. If the pair-

79

Energy (keV)2004006008001000120014001600Counts/4keV020040060080010001200140016001800-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+ 3/2fi-7/2SlowFastCa391-mm separationData rays + BackgroundgSimulated -ray Simulationg rays 1/2 scalegLab-frame Exponential Backgrounds-e+eGe(n,n')74Fe(n,n')56Al(n,n'), 27Ge(n,n'), 72Al(n,n')27Ge(n,n'), 70production occurs outside the detector volume, say by a gamma ray interacting with the aluminum

beam pipe, a gamma ray of energy 511 keV may enter the detector volume and deposit its full

energy, thereby creating a peak at 511 keV in the lab-frame spectrum. After the spectrum has been

Doppler corrected to the ion frame, the Doppler corrected energy at low angles is over corrected

and the 511 keV peak will be displayed at a lower energy after a low-angle gate is applied to

the spectrum. Another common process which produces lab-frame peaks in the plunger spectra

is neutron-induced radiation; in this process, an energetic neutron is released from the beam-

target reaction and proceeds to interact with other materials in the detector vicinity such as the

aluminum beam pipe or the germanium detector volume. When the energetic neutron inelastically

scatters off another nucleus, the lab-frame nucleus becomes excited and releases a gamma ray by

deexcitation. The gamma rays from neutron-induced radiation are released from lab-frame nuclei

and are absorbed in the detector volume. These gamma rays can be identified as originating from a

specific material by their energy in the lab-frame spectra and, as with the pair-production peak, are

overcorrected at low angles in the Doppler correction process and therefore appear at lower energies

when a low-angle gate is applied to the ion-frame data. In both the 39Ca and 39K spectra we observe

peaks at approximately 500 keV which are the result of the 596 and 609-keV gamma rays released

by neutron excitation of 74Ge nuclei in the detector volume. At approximately 700 keV we observe

a single peak which is the combined result of an 834-keV gamma ray from the excitation of 72Ge,

an 844-keV gamma ray from the excitation of 27Al, and an 846-keV gamma ray due to the excitation

of 56Fe. Finally, at approximately 825 keV we observe a peak which is the combined result of a

1014-keV gamma ray released from excited 27Al and a 1040-keV gamma ray released from excited

70Ge. The above listed background contributions were identified in the lab-frame spectra and

their responses in the ion-frame were simulated and added to the exponential backgrounds and the

simulated 39Ca spectra to produce the overall fits shown in red.

Since the Doppler-shift correction was optimized for decays behind the degrader, the slow

components appear at the proper ion-frame energies for the transitions. On the other hand, the fast

components are under-corrected and therefore appear at higher energies. For instance, in Fig. 3.12,

80

Figure 3.13 Doppler-shift corrected gamma-ray spectrum for 39Ca taken with the 0-mm target-
degrader separation setting. The spectrum has been gated between 𝜃 = 29◦ and 𝜃 = 46◦ to improve
sensitivity to Doppler shifts. Transitions from the (11/2−) state are labeled in red, while transitions
from the (9/2−) state are labeled in blue. Laboratory-frame background gamma rays are shown in
green and downscaled by a factor of 2 [88]. Reprinted figure with permission from A. Sanchez et
al., Phys. Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

gamma rays from the (11/2−) state emitted after the degrader compose the true energy peak at

1095 keV, whereas those emitted before the degrader form the higher-energy fast-component peak

observed at 1150 keV, having been under-corrected by the choice of lower 𝛽.

Based on data available for the mirror states in 39K, the (15/2+) and (13/2−) states in 39Ca are

both expected to have lifetimes that are relatively short for this target thickness and foil separation

and, in Fig. 3.12, we observe only the fast components of the transitions at 1060 keV and 1570 keV,

respectively. The 7/2− peak at 2796 keV has been omitted from the analysis; since it is the lowest-

lying excited state, it does not feed any states of interest and therefore has no effect on the lifetime

measurements of the higher lying states. We also note from Fig. 3.13 that when the target-degrader

separation is 0 mm we observe a diminished fast component, which is consistent with more decays

occurring behind the degrader.

In Sect. 2.5, the basic methodology fo RDM measurements was introduced where the ratio of

81

Energy (keV)2004006008001000120014001600Counts/4keV020040060080010001200140016001800-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+ 3/2fi-7/2SlowFastCa390-mm separationData rays + BackgroundgSimulated -ray Simulationg rays 1/2 scalegLab-frame Exponential Backgrounds-e+eGe(n,n')74Fe(n,n')56Al(n,n'), 27Ge(n,n'), 72Al(n,n')27Ge(n,n'), 70fast and slow components provides a sensitivity to the lifetime. But because other experimental

conditions contribute to the yields of fast or slow components, the ratio must be measured at

multiple distance settings to quantify the evolution of the ratio due to the lifetime [93]. These

additional contributions can be constrained by taking additional data sets tailored to determine the

contributions, but usually it is not possible or practical to have such extensive data sets. Therefore,

data is taken such that the most significant contributions to peak ratios can be constrained.

For recoil-distance lifetime measurements with relativistic-energy beams, it is important to

understand and constrain contributions of reactions occurring within the degrader, which can act

as a second target to populate excited states of interest.

In practice, the ratio of reactions on

target to reactions on degrader should be quantified to obtain the best possible parameter set in

our simulations [35]. To quantify this ratio we used the 1-mm separation data and analyzed the

transitions from the (15/2+) and (13/2−) states, which are expected to be short-lived with respect

to flight time over the foil separation. As can be observed in Fig. 3.12, both transitions from

the (15/2+) and (13/2−) states are dominated by their fast components, with very little hint of

slow components for each transition. Since the lifetimes of these states are not known [45], we

estimated their lifetimes from the available data for analogue states in 39K by accounting for the

energy differences of the decays based on the 1/𝐸 2𝜆+1

𝛾

dependence; that is, the reduced transition

strength for an electromagnetic transition can be given by

𝐵(𝜋𝜆, ↓) =

𝐶
𝐸 2𝜆+1
𝛾

𝑇𝑝

(3.1)

where 𝜋 = 𝐸 for electric transitions, 𝜋 = 𝑀 for magnetic transitions, 𝐸𝛾 is the gamma-ray energy

for the transition, 𝜆 is the allowed change in angular momentum for the transition, 𝐶 is a constant

specific to each transition type (E1, E2, M1, M2, etc...), and 𝑇𝑝 is the partial half-life. If we assume

isospin symmetry, the ratio of transition strengths for a pair of mirror states could be equal to 1 and

the partial half-life of an unknown state 𝐽 𝜋∗ can be estimated from the ratio of gamma-ray energies

and the half-life of the known state 𝐽 𝜋 by

𝑇𝑝,𝐽 𝜋∗ =

𝐸 2𝜆+1
𝛾
𝐸 2𝜆+1
𝛾∗

𝑇𝑝,𝐽 𝜋

82

(3.2)

For this work, we assumed a dipole (𝜆 = 1) transition for the (15/2+) → (13/2−) case as

the quadrupole component with 𝛿(𝑀2/𝐸1) = 0.002 [45] is negligible, and we calculated a mean

lifetime of 𝜏 = 4.8 ps for the (15/2+) state of 39Ca. For the (13/2−) → (11/2−) transition in 39Ca

we also assumed a prominent dipole component as the evaluation of the (13/2−) state yields a 6%

quadrupole component (from 𝛿(𝐸2/𝑀1) = −0.25 [45]) based on data for analogue states in 39K,

from which we estimated a lifetime of 𝜏 = 0.4 ps. Therefore, both states in 39Ca are expected to be

relatively short-lived compared to our 1-mm separation distance. The gamma-ray spectra including

the (15/2+) → (13/2−) and (13/2−) → (11/2−) transitions were then simulated as a cascade to

account for feeding effects on the (13/2−) state. By adjusting the ratio of reactions on target to

degrader within our simulation we determined a lower limit of the ratio to be 15(1) : 1, indicating

that reactions on the target are dominant compared to those on the degrader in the 39Ca data.

Using the direct populations and the target-degrader reaction ratio calculated above, simulations

for the cascade in Fig. 3.10 were performed with a wide range of lifetimes for the states of interest.

The lifetime range was then narrowed down accordingly with fit results. The analysis of the (11/2−)

state was conducted utilizing both the 1095-keV and 252-keV branches, leading to the lifetime result

of 𝜏11/2− = 37(1) ps for the 1095-keV branch and 𝜏11/2− = 37(+1

−2) ps for the 252-keV branch, where

the error is statistical only.

To evaluate additional systematic errors, the direct population of the (15/2+) state was varied

and the effect on the lifetime of the (11/2−) state was examined. Since the direct population of the

(13/2−) state was consistent with zero within uncertainty, it was determined that variation of the

(15/2+) direct population was the dominant contributor to systematic uncertainties. Due to feeding

effects from the population of the (15/2+) and (13/2−) states, we determined a (+1

−4)𝑠𝑦𝑠𝑡 ps systematic
uncertainty for the mean lifetime of the (11/2−) state. Similarly, the lifetime of the (15/2+) state

was varied within our simulations and its effects on the observed lifetime of the (11/2−) state were

evaluated. From feeding effects due to the lifetimes of the (15/2+) and (13/2−) states we found an

additional systematic uncertainty of (+1

−3)𝑠𝑦𝑠𝑡 ps to our result.

The systematic uncertainties were added in quadrature with the statistical uncertainties and are

83

presented below as a single, combined error to our lifetime results. The analysis of the 1095-keV

branch resulted in a lifetime of 𝜏11/2− = 37(+2
−5

) ps. The analysis of the 252-keV branch yields

the same result for the 1095-keV branch with a lifetime of 𝜏11/2− = 37(+2
−5

) ps. The agreement

of the lifetimes from the evaluation of both the 252-keV and 1095-keV branches strengthened

confidence in the analysis. For the purpose of this study we adopt a mean lifetime 𝜏11/2− = 37(+2
−5
ps for the (11/2−) state in 39Ca. Using the lifetime above, the absolute branching ratio of the

)

(11/2−) → 7/2− transition 𝑏𝑎𝑏𝑠 = 70(2)%, and the gamma-ray energy 𝐸𝛾 = 1095(4) keV, we

determined the reduced transition strength 𝐵(E2;(11/2−) → 7/2−)=9.8(+14
−6

) e2fm4.

3.3.3

39K Target-only Data

The analysis method used for 39Ca was tested and validated by studying the reference data for

the mirror nucleus 39K, for which the 9Be(42Sc, 39K)X reaction was employed. The gamma-ray

spectrum for the target-only data is presented in Fig. 3.14 with the analogue cascade of transitions

originating from the 15/2+ state noted with arrows, as observed in 39Ca. The relevant level scheme

of 39K is provided in Fig. 3.10 and the analogue transitions in 39K that are the focus of this work

are highlighted in red and blue.

In Fig. 3.14 one may notice transition peaks near 900 keV and 1400 keV which are not labeled;

we have chosen to omit these transitions from our analysis because the associated states do not

significantly populate the 11/2− state, as explained below. The 7/2−

4 state at 5010 keV was

populated in our experiment and decays via two branches; the prominent branch emits a 883-keV

gamma ray to the 7/2−

2 state at 4127 keV which, in turn, decays by cascade to the 7/2−

1 state at

2814 keV via emission of a 1312-keV gamma ray. The minor branch of the 7/2−

4 state decays by

emission of a 1412-keV gamma ray to the 9/2−

1 state at 3597 keV with an absolute branching ratio

of 𝑏𝑎𝑏𝑠 = 23(4)%. Therefore, decays from the 7/2−

4 state bypass the 11/2−

1 state of interest and

have no effect on the lifetime measurement of the 11/2−

1 state. We also observed transitions from

the 9/2−

2 state at 4520 keV whose largest branch leads to the 9/2−

1 state at 3597 keV by emission

of a 923-keV gamma ray. The 9/2−

2 state also has a minor branch which feeds the 11/2−

1 state at

3944 keV by emission of a 576-keV gamma ray which we do not observe in our data. The absolute

84

Figure 3.14 Doppler-shift corrected gamma-ray spectrum for 39K from the target-only measurement.
Transitions from the (11/2−) state are labeled in red, while transitions from the (9/2−) state are
labeled in blue [88]. Reprinted figure with permission from A. Sanchez et al., Phys. Rev. C, vol.
110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

branching ratio for this 9/2−

has a short lifetime of 𝜏9/2−
1 state from the 9/2−

11/2−

2

1 transition is weak (𝑏𝑎𝑏𝑠 = 9.2(8)% [45]) and the state
2 → 11/2−
= 0.14(4) ps [45], from which we conclude that feeding effects to the

2 state are negligible, if any. Thus, we acknowledge the existence of these

transitions in our data while determining that their influence on our state of interest is negligible

and can therefore be excluded from the analysis, allowing us to focus on the mirror cascade in 39K

as shown in Fig. 3.10.

When comparing the 39Ca and 39K target-only spectra, the similarity of decay schemes is

apparent, but one should note a 300-keV increase in the energy of the 13/2− state in 39K. This

shift displaces the 15/2+ → 13/2− transition from 1030 keV in 39Ca to an energy of 757 keV in

39K, placing it in close proximity to the 783-keV peak from the 9/2− → 7/2− transition. As with

our observations of the fast components of the higher-lying transitions in 39Ca, the lifetime of the

13/2− state in 39K is short (𝜏13/2− = 0.27(7) ps [45]) and the direct population is small (2(2)%)

and therefore the 13/2− state has a negligible feeding effect on the 11/2− state.

85

Energy (keV)500100015002000250030003500Counts/2keV0500100015002000250030003500400045005000-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+23fi-27K39Target-onlyData rays + BackgroundgSimulated -ray Simulationg raysgLab-frame Exponential BackgroundsAs was explained for 39Ca, the intensities of the observed transitions in 39K were determined

from the 152Eu source data. Using the intensities, a relative branching ratio for the 11/2− state of

𝑏𝑟𝑒𝑙 = 52(4) was determined by the intensity of the 347-keV peak with respect to the intensity of

the major branch at 1130 keV with 𝑏𝑟𝑒𝑙 = 100(4). This result was found to agree with the literature

value of the relative branching ratio 𝑏𝑟𝑒𝑙 = 57.3(13) [45].

The analysis of the target-only data for 39K was accomplished by determining the direct pop-

ulation fractions of the populated states via the method explained for 39Ca. To the 15/2+ state

at 6475 keV, we assigned a population of 21(3)%. This state decays via emission of a 757-keV

gamma ray to the 13/2− state. The 13/2− state at 5718 keV has a direct population of 2(2)% and

decays to the 11/2− state via emission of a 1774-keV gamma ray. Again, we observed the strongest

population to the 11/2− state at 3944 keV, for which we calculated a direct population of 49(7)%.

This state decays by emission of either a 347-keV gamma ray to the 9/2− state or an 1130-keV

gamma ray to the 7/2− state. The 9/2− state at 3597 keV was observed to be much more weakly

populated in the 9Be(42Sc, 39K)X reaction with a direct population of 3(2)%, decaying by emission

of a 783-keV gamma ray to the 7/2− state. Finally, for the 2814-keV 7/2− state, we found a direct

population of 26(10)% with ambiguities due to feeding from states not included in this study.

3.3.4

39K RDM Analysis

The spectra for 39K taken with the 1 and 0-mm separation distance are provided in Fig. 3.15 and

Fig. 3.16, respectively. The decrease in the target-degrader ratio, compared to the case of 39Ca, is

evident in the 13/2− → 11/2− transition measured with the 1-mm distance as shown in Fig. 3.16,

where one can observe comparable yields for the fast and slow components. As was explained for

39Ca, the effective lifetime of the short-lived 13/2− state in 39K was used to determine the ratio

of reactions on target to reactions on degrader, which was found to be 2.3(4) : 1. By considering

the rigidity setting of the S800 spectrograph, the reduced ratio was attributed to an effect of the

momentum acceptance setting, which was sufficient to accept nearly all of the 39Ca recoil nuclei,

but was only able to cover the lower-momentum components of 39K. Since multi-nucleon (-2p-1n)

removal reactions of 42Sc are used to populate 39K, the final momentum of 39K depends largely

86

Figure 3.15 Doppler-shift corrected gamma-ray spectrum for the 1-mm target-degrader separation
setting in 39K. The spectrum has been gated between 𝜃 = 29◦ and 𝜃 = 46◦ to improve sensitivity
to Doppler shifts. Transitions from the (11/2−) state are labeled in red, while transitions from the
(9/2−) state are labeled in blue. Laboratory-frame background gamma rays are shown in green and
downscaled by a factor of 2 [88]. Reprinted figure with permission from A. Sanchez et al., Phys.
Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

on the reaction points, yielding low-momentum components when reactions occur in the degrader.

The rigidity setting used in this experiment therefore resulted in the decreased target-degrader ratio.

These features are incorporated in the simulation and the target-degrader ratio should be considered

as an effective parameter used for the lifetime analysis.

Lifetime analyses of the 39K states were performed using the adopted lifetime values for

higher-lying states [45] combined with the direct population of the excited states determined

from the target-only data. By varying the target-degrader ratio and observing the effect on the

lifetime of the 11/2− state we determined that a systematic error of (±2)𝑠𝑦𝑠𝑡 ps contributes to

the overall uncertainty of the lifetime measurement of the 11/2− state. As a result, a lifetime of

𝜏11/2− = 13(2) ps was determined for the 11/2− state at 3944 keV from fits to the 1130-keV and

347-keV peaks in both the 0- and 1-mm spectra. The statistical and systematic uncertainties were

added in quadrature for the result above and our result agrees with the adopted value of 12.3(14)

87

Energy (keV)200400600800100012001400160018002000Counts/4keV0500100015002000250030003500-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+ 3/2fi-7/2SlowFastK391-mm separationData rays + BackgroundgSimulated -ray Simulationg rays 1/2 scalegLab-frame Exponential Backgrounds-e+eGe(n,n')74Fe(n,n')56Al(n,n'), 27Ge(n,n'), 72Al(n,n')27Ge(n,n'), 70Figure 3.16 Doppler-corrected gamma-ray spectrum for the 39K 0-mm target-degrader separation.
The spectrum has been gated between 𝜃 = 29◦ and 𝜃 = 46◦ to improve sensitivity to Doppler shifts.
Transitions from the (11/2−) state are labeled in red, while transitions from the (9/2−) state are
labeled in blue. Laboratory-frame background gamma rays are shown in green and downscaled by
a factor of 2 [88]. Reprinted figure with permission from A. Sanchez et al., Phys. Rev. C, vol. 110,
p. 024322 (2024). Copyright (2024) by the American Physical Society.

ps. The experimental values for the absolute branching ratio of the 1130(4)-keV branch found from

the target-only data 𝑏𝑎𝑏𝑠 = 66(2)% and the lifetime above give the reduced transition strength of

𝐵(E2; 11/2− → 7/2−) = 22.4(35) e2fm4 as presented in the following section.

88

Energy (keV)200400600800100012001400160018002000Counts/4keV0500100015002000250030003500-29fi-211-27fi-29-213fi+215-27fi-211-211fi-213+ 3/2fi-7/2SlowFastK390-mm separationData rays + BackgroundgSimulated -ray Simulationg rays 1/2 scalegLab-frame Exponential Backgrounds-e+eGe(n,n')74Fe(n,n')56Al(n,n'), 27Ge(n,n'), 72Al(n,n')27Ge(n,n'), 70CHAPTER 4

DISCUSSION OF RESULTS

Having determined the reduced transition strengths for the 11/2− → 7/2− E2 transitions in the

𝐴 = 39 system we now discuss their relation to matrix elements 𝑀𝑝 and 𝑀𝑛 and compare the results

to the data systems and shell-model calculations. From comparison of the current results for the

𝐴 = 39 system to the adopted values for other neutron-deficient calcium isotopes, we observe an

increasing ratio 𝑀𝑛/𝑀𝑝 as we move from 𝐴 = 36 toward 𝐴 = 40, implying an increased degree

of neutron excitation. We then compare the results of this study to shell-model calculations for the

𝐴 = 38 and 𝐴 = 39 systems using three effective interactions common to this region of the nuclear

chart. The shell-model calculations also point toward the importance of both proton and neutron

cross-shell configurations for 𝐴 = 39 system, as opposed to the the 𝐴 = 38 system which appears

to be largely dominated by proton cross-shell excitations.

4.1 Experimental Data and Systematics

The experimental mean lifetimes 𝜏, the absolute branching ratios 𝑏𝑎𝑏𝑠 for the 11/2− states, and

the gamma-ray energies 𝐸𝛾 given in the previous sections were used to calculate the 𝐵(E2;11/2− →

7/2−) values for 39Ca and 39K as presented in Table 4.1. Also presented in Table 4.1 are the adopted

𝐵(E2) values and their related matrix elements for the 𝐴 = 38 system [43, 44]. The 𝐵(E2) values

were converted to the absolute value of proton matrix elements 𝑀𝑝 according to Eqn. 4.1 and are

presented in Table 4.1 for the 𝐴 = 38 system with 𝐽𝑖 = 2 and the 𝐴 = 39 system with 𝐽𝑖 = 11/2.

√︃

𝑀𝑝 =

𝐵(E2; 𝐽𝑖 → 𝐽 𝑓 ) × (2𝐽𝑖 + 1)

(4.1)

As described in Sect. 1.1, under the assumption of isospin conservation, the neutron matrix elements

𝑀𝑛 of 39Ca and 38Ca are obtained from the proton matrix elements of the mirror nuclei 39K and

38Ar, respectively, via Eqn. 1.7. For the present work, the equivalencies are 𝑀𝑛 (39Ca) = 𝑀𝑝 (39K)

and 𝑀𝑛 (38Ca) = 𝑀𝑝 (38Ar). Here we should note that, to avoid redundancies, the 𝑀𝑛 and 𝑀𝑛/𝑀𝑝

values have been omitted from the table for the 𝑇𝑧 ≥ 0 nuclei.

89

Table 4.1 The present results of 𝐵(E2), proton (𝑀𝑝) and neutron (𝑀𝑛) matrix elements for the
11/2− → 7/2− transitions in the 𝐴 = 39 system are compared to adopted values for the 2+ → 0+
transitions in the 𝐴 = 38 system [43, 44]. The 𝑀𝑛 values and 𝑀𝑛/𝑀𝑝 ratios are presented only for
39Ca and 38Ca and corresponding values in mirrors 39K and 38Ar, respectively, are omitted to avoid
redundancy. The table has been adapted from Ref. [88].

𝐵(E2)
(e2fm4)

𝑀𝑝
(efm2)

𝑀𝑛
(efm2)

𝑀𝑛/𝑀𝑝

9.8(+14
39Ca
)
−6
39K 22.4(35)

10.8(+8
)
−4
16.4(13)

16.4(13)
−

1.5(2)
−

As with the 𝐴 = 38 system, the 𝑀𝑝 (𝑇𝑧) values for the 𝐴 = 39 mirror nuclei increase as a

function of increasing 𝑇𝑧, which results in 𝑀𝑛 > 𝑀𝑝 for 39Ca, as is the case for the 𝐴 = 38 system.

This common feature of the 𝐴 = 38 and 𝐴 = 39 systems appears to go against the trends set by the

𝐴 = 22, 26, 30, 34, and 42 systems observed in Fig. 1.6 and implies that the 𝐴 = 39 and 𝐴 = 38

mirror nuclei have a negative isovector matrix element 𝑀1 [51, 40]. The ratios 𝑀𝑛/𝑀𝑝 observed

among neutron-deficient calcium isotopes are found to be 0.8(1) and 1.1(1) for the 2+ → 0+

transitions of 36Ca and 38Ca [49], respectively, and 1.5(2) for the 11/2− → 7/2− transition of 39Ca,

indicating an increased degree of neutron excitations toward 𝑁 = 𝑍.

In a broader view of the systematics, the 𝑀𝑛/𝑀𝑝 ratios for the 2+ → 0+ transitions in the

neutron deficient calcium isotopes near 𝑁 = 𝑍 are plotted in Fig. 4.1 along with the ratio for the

analog 11/2− → 7/2− transition for the 𝐴 = 39 system studied in this work. The 𝑁/𝑍 expectation

from the homogeneous collective model is also provided, as was similarly provided by Bernstein et

al. in Ref. [47] and displayed in this work as Fig. 1.7 in Sect. 1.2. Also plotted, is the expectation

for single-closed shell (SCS) mirror nuclei from the ratio of effective charges 𝑒 𝑝/𝑒𝑛 = 3.0(4), as

derived in Eqns. 1.9 and 1.10.

In Fig. 4.1 we observe agreement between the collective model assumption 𝑁/𝑍 and the 𝑀𝑛/𝑀𝑝

ratio for 𝐴 = 36 system of calcium and sulfur [49, 48], suggesting a large degree of collectivity

for this system and a positive isovector matrix element 𝑀1. As one adds neutrons to the system,

a divergence from the 𝑁/𝑍 assumption is observed as one approaches 40Ca. Recall that the ratio

90

Figure 4.1 Ratios 𝑀𝑛/𝑀𝑝 for calcium isotopes near 𝑁 = 𝑍 compared to the homogeneous collective
model (𝑁/𝑍) and the single-closed shell (SCS) esitmation 𝑒 𝑝/𝑒𝑛 = 3.0(4). The figure shows an
enhanced 𝑀𝑝 when compared to the SCS expectation. The 𝐴 = 36 ratio was obtained from
Refs. [49, 48]. The 𝐴 = 38 ratio is from Refs. [43, 44]. The 𝐴 = 39 ratio is from this work [88].
And Ref. [40] provides the 𝐴 = 42 ratio.

𝑀𝑛/𝑀𝑝 is independent of the 𝑇𝑧 = 0 data point; this is especially relevant for 𝐴 = 38 system [43, 44]

as recent evaluation has enlarged the error bars of the 38K data point. We observe in Fig. 4.1 that

the 𝑀𝑛/𝑀𝑝 ratio for the calcium isotopic chain begins to diverge from 𝑁/𝑍 at the 𝐴 = 38 system.

This work has further illustrated this divergence with the addition of the 𝐴 = 39 system [88]. Since

40Ca is self-conjugate (it is its own magic nucleus), we expect 𝑀𝑛/𝑀𝑝 = 1. However, from Fig. 1.7

in Sect. 1.2, we see that the ratios tend to increase drastically for doubly magic nuclei such as 48Ca

and 208Pb, diverging from the trend set by the single closed shell nuclei. This implies that 40Ca

similarly may have a ratio much larger than the 𝑁/𝑍 assumption, though the methods of probing

this neutron matrix element are outside the scope of this work. Upon filling of the 𝑠𝑑 shell, there

is a decrease in the 𝑀𝑛/𝑀𝑝 ratio as observed in the 𝐴 = 42 system [40]. Further, Fig. 4.1 suggests

that 39Ca displays more collectivity than one would expect when compared to the SCS expectation,

making its 𝑀𝑛/𝑀𝑝 ratio similar to the other neutron deficient Ca isotopes displayed in the figure.

91

4.2 Shell Model Calculations

In the following sections we will examine the effective interactions utilized to replicate the 𝐵(E2)

and proton matrix elements that are the focus of this work. We will begin with descriptions of the

shell-model effective interactions involved and provide the context in which these interactions were

developed. Following the brief description of each interaction, we will then consider the results of

the calculations for the energies of the negative parity states examined in this work. We then move

on to the results of the 𝐵(E2) and 𝑀𝑝 calculations from each interaction, as well as the predicted

ratios 𝑀𝑛/𝑀𝑝 for the𝐴 = 39 system, and in the last section we will discuss the same calculations

for the 𝐴 = 38 system.

4.3 Effective Interactions Employed in this Work

Three effective interactions common to this mass region were employed to replicate the ex-

perimental data: FSU [94], ZBM2 [95], and ZBM2m [96]. These interactions average the forces

acting between nucleons allowing for more manageable calculations when compared to ab initio

models. The choice of which interactions to use is highly dependent on the model space for orbitals

most likely to be occupied by the particle configurations for the nuclei in question. The parameters

within the effective interactions are often derived from fits to experimental data such as binding

energies, excitation energies, and other observables.

The FSU [94] effective interaction was developed from a modified version of the WBP [97]

interaction and adopted the WBP model space (𝑠𝑝𝑠𝑑𝑓 𝑝) as a starting point. This interaction

assumes isospin invarience while allowing Coulomb corrections to the binding energies. Lower

𝑠𝑝 orbital single-particle energies (SPEs) and two-body matrix elements (TBMEs) were taken

from the WBP interaction while the TBMEs for the 𝑠𝑑 shell were adopted from the USDB [38]

Hamiltonian. The 𝑠𝑑 − 𝑓 𝑝 cross-shell matrix elements required modification of the monopole and

isospin strengths while the remaining multipoles for the upper shells of the model space were fit

to data with the GXPF1A [98] Hamiltonian as a starting framework. Special attention was paid to

data from intruder states sensitive to the 𝑝 − 𝑠𝑑 shell gap, negative parity states sensitive to particle

excitation from 𝑠𝑑 to 𝑓 𝑝 shells, neutron-rich cross-shell nuclei with 𝑍 < 20 and 𝑁 > 20, and

92

𝑓 𝑝 shell nuclei of 𝑍 ≥ 20 and 𝑁 ≥ 21. Based on pure n-particle n-hole configurations of either

0ℏ𝜔, 1ℏ𝜔, and 2ℏ𝜔 excitations in the 𝑠𝑝𝑠𝑑𝑝 𝑓 model space, the FSU interaction provides accurate

predictions of level energies for nuclei in the 𝑠𝑑-shell region, including neutron-rich nuclei in the

island of inversion [94].

The ZBM2 [95] effective interaction employs a smaller model space of the 2𝑠1/2, 1𝑑3/2, 1 𝑓7/2,

and 2𝑝3/2 orbitals while allowing all configurations within this space. The ZBM2 interaction was

developed to reproduce the odd-even charge radii staggering in the 40Ca to 48Ca isotopic chain

while simultaneously replicating the parabolic dependence on mass 𝐴 of the low-lying excited-state

energies [95]. The TBMEs from Ref. [99], which are defined with respect to a 16O core, were

adopted as a starting point. The SPEs were modified to reproduce the 29Si spectrum and the 𝑠𝑑 − 𝑓 𝑝

monopoles from Ref. [99] were adjusted to replicate the masses of the neutron rich isotopic chains

then returned to reproduce the 40Ca shell gap and the spectra of 39K and 41Ca [95]. Because the

ZBM2 interaction does not rely on pure particle-hole states we are able to get a more detailed

picture of the mixed configurations underlying the states being studied.

In an attempt to have theoretical calculations which are in closer agreement to data, we also used

a modified ZBM2 interaction [96], denoted as ZBM2m. This interaction was developed to better

describe the charge radius of the 38K𝑚 isomer and reproduce the correct order of the 3+ ground

state and 0+ isomeric state of 38K [96]. While the ZBM2 interaction provided a fair description of

the low lying 0+ and 2+ states in 40Ca, it could not properly replicate the ordering of the ground and

isomeric states in 38K mentioned above and predicted a 3+ excited state at nearly 500 keV instead

of the 0 keV ground state which has been observed. For the ZBM2m interaction, the 𝑇 = 1 and

𝑇 = 0 strengths were adjusted within the monopole Hamiltonian, with one coefficient fixing the

poistion of the configurations of a given isospin 𝑇 and another coefficient fixing the the position of

configurations dependent on the particle number involved [96]. As with the ZBM2 interaction, the

ZBM2m interaction utilizes the 𝑠𝑑𝑓 𝑝 model space while allowing configuration mixing within the

space.

93

4.3.1 Shell Model Predictions of 39Ca Level Energies

The calculated energy level diagrams for the negative parity states in 39Ca which are the subject

of this work are provided in Fig. 4.2. The results from the three effective interaction calculations

(FSU, ZBM2, and ZBM2m) are compared to the experimental energies (Exp) observed in the

gamma-ray spectra presented in Sect. 3.3.

Figure 4.2 Level energies of the lowest 7/2−, 9/2−, 11/2−, and 13/2− states of 39Ca from experiment
(Exp) compared to shell-model calculations with the FSU [94], ZBM2 [95], and ZBM2m [96]
effective interactions [88]. Reprinted figure with permission from A. Sanchez et al., Phys. Rev. C,
vol. 110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

One will notice that the FSU calculations for all the observed levels in the 39Ca spectra are in

good agreement with the experimental level scheme, having replicated the correct 𝐽 𝜋 ordering of

each state correctly and predicting the energies of each state accurately to within approximately

200 keV at most, and often much closer. Most importantly for this work, the 11/2− state is

predicted accurately to within 30 keV of the experimental data while the 7/2− state shows a

200 keV discrepancy from the data. The FSU calculation predicts an energy difference between the

11/2− and 7/2− states of 984 keV which is in reasonable agreement with the gamma-ray energy of

94

1095 keV observed in the 39Ca spectra.

The ZBM2 calculations do not reproduce the levels in 39Ca as well as the FSU calculation.

Working from the ground state up, we first notice a 7/2− state that is approximately 1.2 MeV lower

than the experimental data. Generally, level energy predictions that agree with data to within a

couple hundred keV are considered “good” calculations; if this is an acceptable margin of error

then the ZBM2 prediction of the 7/2− state with energy 1.2 MeV below the data does not fall

into that category. Moving up in energy we next encounter the 11/2− state at an energy 800 keV

below the experimental data, this significant decrease in the 11/2− energy with respect to the

data results in an inversion of the 9/2− and 11/2− states which is not supported by observation.

The ZBM2 calculation predicts the energy of the 9/2− state best compared to other levels with a

difference between data and calculation under 500 keV. The 13/2− state, whose energy value is not

of particular importance to this work, does also suffer from a 800 keV descrepency with the data.

Overall, the ZMB2 calculation appears to under-predict excitation energies for every state while

predicting a gamma-ray energy of 1490 keV, which is not representative of the 1095 keV gamma

ray associated with the 11/2− → 7/2− transition that is the focus of this work.

For the ZBM2m calculation we again begin with the lowest excited state and work up the level

diagram. While the 7/2− state prediction is approximately 600 keV lower than the experimental

data, the difference between the 11/2− and 7/2− states is 1194 keV which, like the FSU calculation,

is in reasonable agreement with the data. As with the ZBM2 interaction, the ZBM2m interaction

also under predicts the energy of the 11/2− state by approximately 600 keV, again resulting in

the inversion of the 9/2− and 11/2− states. The 9/2− state is well replicated, as it is with each

interaction used in this work, predicting an energy about 100 keV lower than the experimental

data. Finally, the 13/2− state is calculated to have an energy of approximately 1 MeV lower than

the data, but as was stated for the ZBM2 calculation, this level energy is not tied closely to the

11/2− → 7/2− E2 transition which is the focus of this work.

Overall, while the FSU interaction best reproduces the absolute level energies, both the FSU and

ZBM2m interactions well replicate the 11/2− → 7/2− transition energy. While the FSU interaction

95

obtains close agreement for this transition energy by predicting the absolute energies of the two

states within suitable margins, the ZBM2m obtains similar agreement through a down scaling of

both the 11/2− and 7/2− energies resulting in a transition energy that is in good agreement by the

200 keV metric discussed above.

4.3.2 𝐵(E2) and 𝑀𝑝 for the 𝐴 = 39 system

We now employ Eqn. 1.4 to calculate the reduced transition strengths 𝐵(E2;11/2− → 7/2−)

for 39Ca and 39K based on our determination of the experimental mean lifetimes 𝜏, the absolute

branching ratios 𝑏𝑎𝑏𝑠 for the 11/2− states, and the gamma-ray energies 𝐸𝛾 provided in Sect. 3.3.

The 𝐵(E2) determined from this work are presented in Table 4.2 in the row labeled “Exp”. The

FSU, ZBM2, and ZBM2m effective interactions were also used to predict the 𝐵(E2) values for the

E2 transitions in both 39Ca and 39K and the values are presented in Table 4.2 accordingly with the

interaction name.

Table 4.2 The present results of 𝐵(E2), proton (𝑀𝑝) and neutron (𝑀𝑛) matrix elements for the
11/2− → 7/2− transitions in the 𝐴 = 39 system [88]. The 𝑀𝑛 values and 𝑀𝑛/𝑀𝑝 ratios are pre-
sented only for 39Ca and corresponding values in mirror 39K are omitted to avoid redundancy [88].
Reprinted table with permission from A. Sanchez et al., Phys. Rev. C, vol. 110, p. 024322 (2024).
Copyright (2024) by the American Physical Society.

𝐵(E2)
(e2fm4)

𝑀𝑝
(efm2)

𝑀𝑛
(efm2)

𝑀𝑛/𝑀𝑝

39Ca

39K

Exp
FSU
ZBM2
ZBM2m
Exp
FSU
ZBM2
ZBM2m

)

9.8(+14
−6
3.4
20.6
13.7
22.4(35)
10.8
27.0
18.5

)

10.8(+8
−4
6.4
15.7
12.8
16.4(13)
11.4
18.0
14.9

16.4(13)
11.4
18.0
14.9
−
−
−
−

1.5(2)
1.8
1.1
1.2
−
−
−
−

Upon inspection, we find that the FSU calculation tends to under predict the 𝐵(E2) for 39Ca

and 39K. For the E2 transition in 39Ca, the FSU calculation predicts a 𝐵(E2)= 3.4 e2fm4, which

is approximately 35% of the experimental result determined by this work. For 39K, the FSU

interaction predicts a value that is closer to the experimental value with 𝐵(E2)= 10.8 e2fm4 for

96

the analog 11/2− → 7/2− transition, which is approximately 50% of the value determined in this

work.

The ZBM2 prediction for the 39Ca transition is 𝐵(E2)= 20.6 e2fm4 which is over twice the

experimental value, making a stark contrast between its utility for this work compared to the FSU

and ZBM2m interactions. For 39K, the ZMB2 calculation of 𝐵(E2)= 27.0 e2fm4 is a much more

accurate prediction than what was calculated for 39Ca being only 20% larger than the experimental

value. This is unsurprising considering that the parameters which were adjusted from WBP to

make ZBM2 were modified to fit the 39K spectra, as described in Sect. 4.2.

Out of the three effective interactions, the ZBM2m most accurately reproduces the 39Ca E2

transition strength with 𝐵(E2)= 13.7 e2fm4 which is approximately 40% larger than the experimen-

tal value but is the best prediction out of the three interactions chosen for this work. The ZBM2m

also best replicates the analog transition in 39K with 𝐵(E2)= 18.5 e2fm4, 17% lower than the

experimental result. This is a bit unexpected considering that the ZBM2m interaction was tailored

to replicate 38K𝑚, as discussed in Sect. 4.2.

For a more detailed description of the origin of the differences among the interactions used for

this work, the bare matrix elements 𝐴𝑝 and 𝐴𝑛 used in the calculations of 𝑀𝑝 are examined below.

The effective interactions use Eqn. 1.5 to calculate 𝑀𝑝 using the bare matrix elements and the

effective charges. For each calculation, the effective charges of 𝑒 𝑝 = 1.36 and 𝑒𝑛 = 0.45 [42] were

used as Dronchi et al. [49] used for the ZBM2 calculations of 36,38Ca. One should also note that

the effective charges employed here are comparable to the charges 𝑒 𝑝 = 1.31 and 𝑒𝑛 = 0.46 which

were used with the SDPF-U-MIX Hamiltonian for calculations of 36Ca and 36S [48]. Therefore,

the calculations presently used should be on equal footing with respect to the choice of effective

charges for calculations of other neutron-deficient Ca isotopes.

The calculated values of 𝐴𝑝 and 𝐴𝑛 used in the shell-model calculations for 39Ca are provided

in Table 4.3; as discussed in Sect. 1.1, these correspond to the matrix elements calculated with the

free-nucleon charges 𝑒𝑛 = 0 and 𝑒 𝑝 = 1. The bare matrix elements 𝐴𝑝 and 𝐴𝑛 are scaled by the

effective charges mentioned above and summed to calculate the proton matrix elements provided

97

in Table 4.2. By examining 𝐴𝑝 and 𝐴𝑛 for the ZBM2 and ZBM2m interactions, we notice that

both values are larger than those calculated by the FSU interaction and both predict relatively even

distributions of proton and neutron contributions. Conversely, the FSU interaction predicts a much

larger neutron contribution to the 𝑀𝑝.

As described in Sect. 1.1, under the assumption of isospin conservation, the neutron matrix

elements 𝑀𝑛 of 39Ca are obtained from the proton matrix elements of the mirror nuclei 39K via

Eqn. 1.7. For the 𝐴 = 39 system, the equivalency is 𝑀𝑛 (39Ca) = 𝑀𝑝 (39K). Here we should note

that, to avoid redundancies, the 𝑀𝑛 and 𝑀𝑛/𝑀𝑝 values for the 𝑇𝑧 > 0 nucleus have been omitted

from Table 4.2.

Table 4.3 The calculated values of the bare proton (𝐴𝑝) and neutron (𝐴𝑛) matrix elements for the
11/2− → 7/2− transition of 39Ca from shell-model calculations with the FSU [94], ZBM2 [95],
and ZBM2m [96] effective interactions. The table has been adapted from Ref. [88].

39Ca (11/2− → 7/2−)

𝐴𝑝 (efm2) 𝐴𝑛 (efm2)

FSU
ZBM2
ZBM2m

2.14
8.06
6.51

7.65
10.57
8.81

In examining the predicted matrix elements for the 𝐴 = 39 system we note that all the 𝑀𝑝 (𝑇𝑧)

calculations increase as a function of increasing 𝑇𝑧, which replicates the in 𝑀𝑛/𝑀𝑝 > 1 feature

observed in the 39Ca data. Referring back to Table 4.2, the FSU calculation replicates the 𝑀𝑝 value

of 39Ca to within 40% of the 𝑀𝑝 derived from the data. Similarly, the FSU interaction replicates

the 𝑀𝑝 of 39K to within 30% of the experimental value. The ZBM2 calculation for the 𝑀𝑝 of

39Ca is approximately 40% larger than the experimental value with a calculated 𝑀𝑝 = 15.7 efm2.

For 39K, the ZMB2 interaction provides the best prediction of the proton matrix element with

𝑀𝑝 = 18.0 efm2 which is within 10% of the experimental value. Again, the ZBM2m interaction

appears to best replicate the results of the 39Ca values with 𝑀𝑝 = 12.8 efm2, predicting the proton

matrix element value to within 19% of the experimental value. And for 39K, the ZBM2m interaction

similarly reproduces 𝑀𝑝 to within 10% Therefore, when we examine the ratio 𝑀𝑛/𝑀𝑝 and see that

both the FSU and ZBM2m calculations are just outside the experimental uncertainty, one must also

98

take into account that the ZBM2m calculations predicted the individual 𝑀𝑝 values more accurately

than the FSU calculation.

Table 4.4 The numbers of protons (Δ𝑝) and neutrons (Δ𝑛) excited from the (2𝑠1/2, 1𝑑3/2) to (1 𝑓7/2,
2𝑝3/2) orbitals calculated with the ZBM2 [95] and ZBM2m [96] interactions. The percentages (%)
for different configurations are given for the 11/2− state of 39Ca (top) and for the 2+ state of 38Ca
(bottom). The percentage for each of the higher-order configurations is less than 6%, and only
the sum of these configurations is listed as “other” in the table. The table has been adapted from
Ref. [88].

39Ca (11/2−)

Δ𝑝 Δ𝑛 ZBM2 (%) ZBM2m (%)
0
1
1
0
1
2
1
2
other

38
18
20
9
15

26
21
23
10
20

While the FSU interaction describes the 11/2− state as a pure 1ℏ𝜔 configuration with ap-

proximately 60% proton and 40% neutron excitations into the 𝑓 𝑝 shell, the ZBM2 and ZBM2m

interactions provide a more robust picture of the state. Because the ZBM2 and ZBM2m interactions

do not rely on pure particle-hole configurations, we are able to estimate the probabilities of various

configurations which contribute the 11/2− state. Table 4.4 provides the percentages of different

configurations which compose the 11/2− state in 39Ca in terms of the number of protons (Δ𝑝) and

neutrons (Δ𝑛) excited from the (2𝑠1/2, 1𝑑3/2) to the (1 𝑓7/2, 2𝑝3/2) orbitals. In this notation, the

ZBM2 calculation predicts lowest-order excitations with (Δ𝑝, Δ𝑛) values of (1,0) and (0,1) and

probabilities of 26% and 21% respectively, which are followed by higher-order excitations of (Δ𝑝,

Δ𝑛)=(2, 1) with 23% and (1, 2) with 10%. Excitation configurations with low probabilities are

summed and grouped as “other” in the table and for the ZBM2 interaction these configurations

account for 20% of the configurations of the 11/2− state in 39Ca. The ZBM2m calculation predicts

a larger (Δ𝑝, Δ𝑛)=(1, 0) component that the ZBM2 calculation, but a further increase of 𝐴𝑛 may be

favored given that the ZBM2m interaction underestimates the 𝑀𝑛/𝑀𝑝 ratio. We see from Table 4.4

that both ZBM2 and ZBM2m calculations predict non-negligible proton and neutron contributions

to the cross-shell excitations, providing an enhancement of 𝑀𝑝 accompanied by an enhanced 𝑀𝑛

99

value for 39Ca. The present work supports the ZBM2m calculations for both 𝑀𝑝 and 𝑀𝑛 in 39Ca,

emphasizing the important role of cross-shell excitations of both protons and neutrons across the

𝑍 = 𝑁 = 20 shell gap [88].

4.3.3 𝐵(E2) and 𝑀𝑝 for the 𝐴 = 38 system

Presented in Table 4.5 are the adopted 𝐵(E2,2+ → 0+) values and their related matrix elements

for the 𝐴 = 38 system [43, 44]. We begin by comparing the adopted 𝐵(E2) values (Exp) to the

values calculated with the three effective interactions. We will then examine the effective interaction

predictions for 𝑀𝑝 in the 38Ca and 38Ar and compare these matrix elements to those calculated

from the adopted 𝐵(E2) values.

Table 4.5 The present results of reduced transition strength 𝐵(E2), proton (𝑀𝑝) and neutron (𝑀𝑛)
matrix elements for the 2+ → 0+ transitions in the 𝐴 = 38 system are compared to shell-model
calculations with the FSU [94], ZBM2 [95], and ZBM2m [96] effective interactions. The 𝑀𝑛
values and 𝑀𝑛/𝑀𝑝 ratios are presented only for 38Ca and corresponding values in mirror 38Ar are
omitted to avoid redundancy [88]. Reprinted table with permission from A. Sanchez et al., Phys.
Rev. C, vol. 110, p. 024322 (2024). Copyright (2024) by the American Physical Society.

𝐵(E2)
(e2fm4)

𝑀𝑝
(efm2)

𝑀𝑛
(efm2)

𝑀𝑛/𝑀𝑝

20.2(22)
2.8
22.0
22.5

10.0(5)
3.7
10.5
10.6

11.3(2)
11.3
13.4
13.4

1.1(1)
3.0
1.3
1.3

)

13(+10
−6
11.3
28.5
28.8

25.6(8)
25.5
35.8
35.9

)

8(+3
−2
7.5
11.9
12.0

11.3(2)
11.3
13.4
13.4

−
−
−
−

−
−
−
−

−
−
−
−

−
−
−
−

38Ca

38K

38Ar

Exp
FSU
ZBM2
ZBM2m

Exp
FSU
ZBM2
ZBM2m

Exp
FSU
ZBM2
ZBM2m

Perhaps the first and most obvious observation is that the FSU calculation dramatically under

predicts the 𝐵(E2,2+ → 0+) for 38Ca with a value just 10% of the experimental value. The FSU

interaction does predict the 𝐵(E2) for 38Ar quite accurately with a value within the experimental

100

uncertainty. In fact, no other interaction gets closer to the adopted 𝐵(E2) for 38Ar than the FSU

interaction.

Both the ZBM2 and ZB2m interactions quite accurately predict the 𝐵(E2) values for 38Ca.

While the ZBM2 interaction does predict the 𝐵(E2) within the uncertainty of the adopted value,

the ZBM2m prediction is only a negligibly outside the adopted uncertainty. For the cases of 38Ar

and 38K, both the ZBM2 and ZBM2m interactions significantly over predict the 𝐵(E2) values.

While the near 40% divergence from the adopted value for 38Ar is significant, both the ZBM2 and

ZBM2m intreacions predict a 38K transition strength over twice the experimental value.

As was explained for the 39Ca calculations, the proton matrix elements for the 𝐴 = 38 system

were calculated with the bare matrix elements 𝐴𝑛 and 𝐴𝑝 and the same effective charges used for

39Ca calculations. The bare matrix elements calculated by the three effective intreacions for the

2+ → 0+ transition in 38Ca are presented in Table 4.6. Here we see the similarity between the FSU

calculation and assumptions made for the single-single closed shell prediction derived in Eqn. 1.10.

The ZBM2 and ZBM2m calculations make near identical predictions for both 𝐴𝑝 and 𝐴𝑛, showing

the importance of both valence proton and neutron contributions to the matrix element 𝑀𝑝.

Table 4.6 The calculated values of the bare proton (𝐴𝑝) and neutron (𝐴𝑛) matrix elements for the
2+ → 0+ transition of 38Ca (bottom) from shell-model calculations with the FSU [94], ZBM2 [95],
and ZBM2m [96] effective interactions. The table has been adapted from Ref. [88].

38Ca (2+ → 0+)

𝐴𝑝 (efm2) 𝐴𝑛 (efm2)

FSU
ZBM2
ZBM2m

0.00
5.00
5.10

8.31
8.19
8.16

In examining the proton matrix element predictions in Table 4.5, we see that the trend observed

in the 𝐵(E2) calculations is propagated through to the 𝑀𝑝 calculations. The FSU interaction again

severely under predicts 𝑀𝑝 with a value of approximately 35% of the value calculated from the

adopted 𝐵(E2). The FSU calculation does predict the 𝑀𝑝 for 38Ar to within the experimental

uncertainty, as it did with the 𝐵(E2) value for the same nucleus. On the other hand, both the ZBM2

and ZBM2m interactions quite accurately predict the 𝑀𝑝 values for 38Ca with no discrepancy

101

between the 38Ca predictions and adopted values and a 20% over prediction for 38Ar.

In examining the ratio 𝑀𝑛/𝑀𝑝 for the 𝐴 = 38 system, we again see the similarity between

the FSU calculation and the single-closed shell model prediction from Eqn. 1.10; that is, the

FSU interaction corresponds to the prediction made by the ratio of effective charges 𝑒 𝑝/𝑒𝑛 = 3.0.

But as was stated during the derivation of Eqn. 1.1, the assumptions are naive as most nuclear

configurations exist as a superposition of states which makes the single-closed shell model an

unrealistic picture of an excited state. The ZBM2 and ZBM2m predictions for 𝑀𝑛/𝑀𝑝 ratio agree

with each other and with the experimental data only slightly over-predicting the ratio extracted from

the adopted 𝐵(E2) values. These calculations show that the FSU interaction under predicts the role

of proton contributions while the ZBM2 and ZBM2 interactions appear to attribute sizable proton

contributions to the 2+ state.

We have used the ZBM2 and ZBM2m interactions to decompose the cross-shell excitations into

percentage probabilities of proton (Δ𝑝) and neutron (Δ𝑛) cross-shell configurations in Table 4.7, as

was done with 39Ca. The table illustrates the importance of proton cross-shell contributions to the

38Ca 2+ state with nearly 30% of the state being composed of 2-proton excitations into the 𝑓 𝑝 shell.

Table 4.7 The numbers of protons (Δ𝑝) and neutrons (Δ𝑛) excited from the (2𝑠1/2, 1𝑑3/2) to (1 𝑓7/2,
2𝑝3/2) orbitals calculated with the ZBM2 [95] and ZBM2m [96] interactions. The percentages (%)
for different configurations are given for the 11/2− state of 39Ca (top) and for the 2+ state of 38Ca
(bottom). The percentage for each of the higher-order configurations is less than 6%, and only
the sum of these configurations is listed as “other” in the table. This table has been adapted from
Ref. [88].

38Ca (2+)

Δ𝑝 Δ𝑛 ZBM2 (%) ZBM2m (%)
0
0
0
2
1
1
other

36
30
11
23

39
32
10
19

Finally, we note that, recent evaluation of the systematic error on the 38K data point suggests

that the error bars are larger than those provided in Fig. 1.6. The most recent evaluated uncertainties

were used in Table 4.5 and more accurately represent the current state of the 𝐴 = 38 system and

particularly the 38K data point. We should note that, given the current error, the linearity of the

102

𝑀𝑝 trend in the 𝐴 = 38 system is not completely ruled out. However, the enhanced collectively

in 38Ca is still evident as indicated by 𝑀𝑛/𝑀𝑝 = 1.1(1), which suggests sizable proton excitations

contribute to the 2+ state.

103

CHAPTER 5

CONCLUSION

Given isospin conservation in the strong force, protons and neutrons can be modeled as different

states of a nucleon. Treating protons and neutrons as a single particle implies that mirror nuclei

should have similar excitation energies with differences attributed to isospin non-conserving forces

such as the Coulomb interaction, which acts only between protons. The similarity in spin assignment

and level energies between the level diagrams of mirror nuclei is a testament to the utility of this

model.

The concept of isospin formalism can be applied and tested using the information of the reduced

transition strengths 𝐵(E2) obtained in mirror nuclei. The set of data can also allow for the isospin

decomposition of the of the transition strengths, clarifying the relative contributions of proton and

neutron excitations. Lifetime measurements of mirror states, determined by analysis of gamma-ray

spectra of mirror transitions, provide model-independent determination of the reduced transition

strengths 𝐵(E2). The proton matrix elements 𝑀𝑝 are calculated from the 𝐵(E2) values and can

be compared to matrix elements calculated with shell-model effective interactions which utilize

effective charges.

Much progress has been made in determining the 𝑀𝑝 values for the 2+ → 0+ transitions in even

𝐴 isobars such as those presented in Fig. 1.6. The linear relation between 𝑀𝑝 of isobaric triplets

has been observed across a wide mass range, but an apparent enhancement of the 𝑀𝑝 for 38Ca

deviates from the trend set by 38K and 38Ar, making this anomaly a point of interest. By studying

a structurally similar (11/2−) → 7/2− quadrupole transition in the mirror nuclei 39Ca and 39K,

and by comparing the ratios 𝑀𝑛/𝑀𝑝, we observe that the apparent enhancement of 𝑀𝑝 observed

in 38Ca is similarly observed in 39Ca.

The lifetimes of the 11/2− states in mirror nuclei 39Ca and 39K were determined by simultaneous

RDM measurements. By constraining the direct populations for each of the states in the cascades,

quantifying the target-to-degrader reaction ratios, and estimating the lifetimes of the higher-lying

states from available data, the value 𝜏11/2− = 37(+2
−5

) ps has been obtained from analysis of the

104

39Ca gamma-ray spectra. While previous measurements were able to determine an upper bound

on the lifetime of this state, this is the first work to provide a precise lifetime measurement with

ample consideration given to feeding from higher states. Using the lifetime stated above, the

gamma-ray energy, and the branching ratio we have determined a reduced transition strength

𝐵(E2;(11/2−) → 7/2−)=9.8(+14
−6

) e2fm4. Similarly for 39K, a lifetime of 𝜏11/2− = 13(2) ps was

determined utilizing the same analysis methods used for 39Ca, which provides a reduced transition

strength 𝐵(E2; 11/2− → 7/2−) = 22.4(35) e2fm4. The lifetime values determined for the 39K

states agree with the adopted value [45] found by previous measurements and, therefore, strengthens

confidence in the analysis method utilized for 39Ca. Additionally, the lifetime of the 9/2− state in

39Ca was determined from the 9/2− → 7/2− mixed M1+E2 transition in the same experiment and

found to be 𝜏9/2− = 22(+6

−7) ps. This measurement agrees with the only previous measurement of

this state while providing a nearly 50% reduction in the error.

The 𝐵(E2) for the 11/2− → 7/2− transition in 39Ca was used to extract the proton matrix

element 𝑀𝑝 (E2) = 10.8(+8
−4

) efm2 while the result from the mirror transition in 39K was used to

determine the neutron matrix element 𝑀𝑛 = 16.4(13) efm2 via the isospin decomposition for mirror

transitions method developed by Bernstein et al. This work demonstrates the value of simultaneous

lifetime measurements for mirror transitions by providing a self-consistent, model-independent

determination of the reduced transition strengths and determination of both the proton and neutron

matrix elements, providing a more detailed and robust picture of the transition. The present data

were compared to shell-model calculations within different model spaces. Our results support

calculations which allow a large percentage of cross-shell excitations, indicating that the magicity

of 40Ca is not robust in the transitions studied for 39Ca and 39K.

As an outlook for future studies, we can extend lifetime measurements in mirror nuclei in

the vicinities of doubly magic nuclei in heavier mass regions. While the lower-lying states of

the mirror pair 55Ni and 55Co near 56Ni have been examined and found to display a mirror

asymmetry in partial cross sections for the lowest two 3/2− states and an additional asymmetry in

the 1/2+ → 3/2−

1,2 E1 transitions [100], there remains an opportunity to examine similar mirror

105

E2 transitions from the 11/2− states. Specifically, the lifetimes of the (11/2−) states in both 55Ni

and 55Co remain undetermined, providing an opportunity to employ the methods developed in

this work to understand neutron and proton cross-shell excitations near 𝑁 = 𝑍 = 28. Given the

proper experimental conditions, similar opportunities may arise near the doubly magic 𝑁 = 𝑍 = 50

nucleus 100Sn, where much remains unknown about excited states of the 𝐴 = 99 isobars Sn and In.

This thesis work demonstrated the power and utility of precision lifetime measurements with

rare isotope beams for determining transition strengths and matrix elements from mirror transitions.

By using matrix element decomposition to determine matrix element ratios for analog transitions,

such as the 11/2− → 7/2− E2 transition in the 𝐴 = 39 system, we can compare transition strengths

of odd 𝐴 nuclei to transitions in even-even and odd-odd nuclei where the 2+ → 0+ transition has

been a staple measurement of nuclear structure studies. The data provided in this work have been

used to test three effective interactions commonly used in the N=Z=20 region of the nuclear chart.

Measurements of mirror transitions near 𝑁 = 𝑍 = 50 may provide data useful for testing models

near the proton dripline. Indeed, the methods developed in this work may be applied in any area

of the nuclear chart where mirror transitions can be examined and provide alternate avenues of

exploration for future structure studies.

106

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