PROBING PROTON CROSS-SHELL EXCITATIONS IN 70NI USING

NUCLEON KNOCKOUT REACTIONS

By

Brandon Alexander Elman

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics — Doctor of Philosophy

2019

ABSTRACT

PROBING PROTON CROSS-SHELL EXCITATIONS IN 70NI USING NUCLEON

KNOCKOUT REACTIONS

By

Brandon Alexander Elman

The neutron-rich Ni isotopes serve as an important benchmark for nuclear structure mod-

els because they exhibit interesting phenomena such as shape or configuration coexistence

driven by cross-shell excitations. Reported here are the excited-state population distribu-

tions following one-proton, one-neutron, and two-proton knockout from 71Cu, 71Ni, and

72Zn, respectively, into excited states of 70Ni. By comparing these final-state populations,

this work will attempt to determine the role of neutron and proton excitations separately

in the configurations of excited states. The specific aim is to identify those excited states

associated with configurations containing proton excitations across the Z = 28 shell gap.

These levels are suggested in the literature to be associated with prolate deformation at

relatively low excitation energy compared to neighboring isotopes in the nickel chain.

TABLE OF CONTENTS

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

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
1.1.2 Electromagnetic transitions in nuclei
1.2 Shape and configuration coexistence in exotic nuclei . . . . . . . . . . . . . .
1.3 Shape coexistence in the Ni isotopes
. . . . . . . . . . . . . . . . . . . . . .

v

vi

1
1
3
4
7
9

Chapter 2 Experimental details and detection systems

. . . . . . . . . . .
15
2.1 Beam production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2 The S800 magnetic spectrograph . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.1 Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.2 Trajectory reconstruction . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2.3
Isomer detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
In-beam γ-ray spectroscopy with GRETINA . . . . . . . . . . . . . . . . . .
24
2.3.1 The Gamma-Ray Energy Tracking In-Beam Nuclear Array (GRETINA) 26
27
2.3.2 Doppler reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Nearest-neighbor add-back . . . . . . . . . . . . . . . . . . . . . . . .
29
30
2.3.4
Simulating GRETINA’s peak response . . . . . . . . . . . . . . . . .
37
2.3.5 Lifetime effects on peak centroids . . . . . . . . . . . . . . . . . . . .
2.3.6 Energy calibration and efficiency determination . . . . . . . . . . . .
39

2.3

Chapter 3 Data analysis: calibrations and corrections . . . . . . . . . . . .
3.1 Particle identification in the entrance and exit channel for the one-proton
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Doppler reconstruction example for one-proton knockout setting . . . . . . .
3.3 Determining transition intensities . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Isomeric content of incoming beams . . . . . . . . . . . . . . . . . . . . . . .

knockout setting

40

40
45
52
55

Chapter 4 Results: γ-ray spectroscopy of 70Ni following nucleon knockout 60
61
75
80

4.1 One-proton knockout from 71Cu . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 One-neutron knockout from 71Ni
. . . . . . . . . . . . . . . . . . . . . . . .
4.3 Two-proton knockout from 72Zn . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 Comparison of reaction channels

. . . . . . . . . . . . . . . . . .
5.1 Comparison of γ-ray spectra . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Comparison of feeding-subtracted state populations . . . . . . . . . . . . . .

85
85
87

iii

Chapter 6

Summary and future work . . . . . . . . . . . . . . . . . . . . . .

90

93

94

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX A INTERMEDIATE-ENERGY COULOMB EXCITATION OF
88,90KR AND 86SE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX B EVENT-BY-EVENT CORRECTION FOR THE PARTICLE VE-

LOCITY USING DT A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

iv

LIST OF TABLES

Table 4.1: Measured intensities for all observed, placed 70Ni transitions relative to the
number of detected 70Ni residues in the specific reaction channel expressed
as a percentage. The error includes both statistical uncertainties and sys-
tematic contributions from varying the fit assumption for peak shapes and
backgrounds, and a 2.1% uncertainty from the efficiency determination.
Note that no feeding subtraction is attempted here.
. . . . . . . . . . . .

62

Table 4.2: Measured intensities for all observed, unplaced 70Ni transitions relative
to the number of detected 70Ni residues in the specific reaction channel
expressed as a percentage. See the caption to Table 4.1 for more information. 63

Table A.1: Measured cross sections, σ, and transition strengths, B(E2), for all isotopes
studied in this work, as well as transition strengths from the shell-model
calculation described in Section IV are shown. The uncertainty includes
both statistical uncertainties and systematic contributions from varying
the background model, as well as a 5.4% uncertainty from the efficiency
determination.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table A.2: Shell-model occupations of the proton and neutron orbitals for the observed

levels in 88,90Kr and 86Se. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

v

LIST OF FIGURES

Figure 1.1: Simplified picture for the single-particle configuration of the ground state
. . . . . . . . . . . . . . . . .

of 17O with the shell gaps at N, Z = 2, 8.

Figure 1.2: A schematic view of γ decay from the excited state of energy Ei to a state
at lower energy Ef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3: Common shapes of atomic nuclei. In panels (b) and (c) the magnitude of
. . .

the quadrupole deformation parameter was chosen to be |β| = 0.2.

Figure 1.4: Potential energy surface for 68Ni calculated using Hartree-Fock calcula-
tions constrained by the quadrupole moment. The three minima in the
potential energy surface are associated with spherical, oblate, and prolate
deformed 0+ states. Figure adapted from Ref. [1]. . . . . . . . . . . . . .

Figure 1.5: Potential energy surface for 70Ni calculated using Hartree-Fock calcula-
tions constrained by the quadrupole moment. There is a considerable
deepening of the prolate minimum in 70Ni compared to 68Ni, which is
shown in Fig. 1.4. Figure adapted from Ref. [2].
. . . . . . . . . . . . .

Figure 1.6: Results of shell-model calculations from Ref. [3] using the Monte Carlo
shell model (MCSM) [4] with the full proton and neutron f pg9/2d5/2
model space compared to results using traditional shell-model calculations
utilizing only the neutron f5/2pg9/2 model space and effective interactions:
jj44bpn [5], jj44pna [6], and JUN45 [7]. Only the 0+ and 2+ levels are
shown for comparison.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1: An overview of the layout for the Coupled Cyclotron Facility at Michigan
State University, including the K500-K1200 coupled cyclotrons and the
. . . . . . . . . . . . . . . . . . . . . . . . .
A1900 fragment separator.

Figure 2.2: The secondary cocktail beams were delivered to the S800 spectrograph,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pictured here.

Figure 2.3: Calibrated mask run for the downstream CRDC used to determine the
knockout residue’s position in the focal plane. Note that the non-
dispersive position is shifted so that it is centered on the knockout residue
of interest, rather than the center of the focal plane.
. . . . . . . . . . .

2

4

8

10

11

12

16

20

22

vi

Figure 2.4: Photograph of GRETINA surrounding the target position of the S800 in
front of the entrance quadrupole of the spectrograph. The beam pipe is
removed, and the beam direction would be into the page. Note that there
was one less detector at 90◦ utilized for this experiment as compared to
. . . . . . . . . . . .
the photograph, as the tenth was not yet available.

Figure 2.5: Fit with peak response determined from the GEANT4 simulation, assum-
ing a position resolution of σ = 2 mm to smear the simulated interaction
points. There are clear tails on the peak that are not reproduced by the
simulation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.6: Fit with peak response determined from the GEANT4 simulation, assum-
ing a position resolution of σ = 2 mm for the narrow component and a
position resolution of σ = 7 mm for the wide component, which are added
together in the ratio 80% − 20% as described in the text.
. . . . . . . .

Figure 2.7: Determination of the position resolution for the narrow component of the
. . . . . . . . . . . . . . . . . .

peak response, as described in the text.

Figure 2.8: Comparing the simulated full-energy peak for the narrow component of the
peak template and the one described in the text containing both a narrow
and wide component combined with the ratio 80%-20%, respectively. The
dashed lines show the region of integration to determine the peak areas in
this work, which accounts for 95% of the area of the spectrum containing
only the narrow component.
. . . . . . . . . . . . . . . . . . . . . . . .

25

33

34

35

36

Figure 2.9: The full energy peak for a 1260-keV transition simulated with various
different lifetimes. The peak is Doppler reconstructed using the expected
velocity for the 1260-keV transition in 70Ni, which has a known lifetime of
τ = 1.5 ps. There are clear shifts at relatively low lifetimes and the peaks
become increasingly wider towards lower energies as the lifetime increases. 38

Figure 3.1:

Incoming particle identification plot for the S800 setting centered on one-
proton knockout from 71Cu showing the xf p and obj time-of-flight distri-
butions discussed in more detail in Section 2.2.1. Outlined in red is the
contour around incoming 71Cu projectiles. Also contained in the contour
. . . . . . . . . .
are some 69Ni and 73Zn incoming beam contaminants.

Figure 3.2: Time-of-flight correction for position in the focal plane (top row) and an-
gle in the focal plane (bottom row), before and after corrections. This
corrects for the path-length dependence of the time-of-flight as discussed
in Section 2.2.1. Note this spectrum was already restricted to events con-
taining incoming 71Cu projectiles and outgoing 70Ni knockout residues for
illustrative purposes. A projection of the time-of-flight axis can quantify
the gain, which is very visible already in the two-dimensional plot.
. . .

41

42

vii

Figure 3.3: Outgoing particle identification plot presenting ∆E ∝ Z2 and T oF ∝ A
Z
gated on incoming 71Cu projectiles. The obj-E1 time of flight shown here
has been corrected for the angle and position of the knockout residue in
the focal plane. The presence of the hot spots to the right of the 70Ni
reaction products is due to incoming beam contaminants. . . . . . . . . .

Figure 3.4: Spectra used for removing outgoing beam contaminants, which are high-
lighted by the red arrows. It is clear that the contaminants are localized
at the edge of the focal plane based on the measured x position in the
CRDCs shown in panels (a-c). Gating on these hot spots leads to the dis-
tribution of dispersive angles in the focal plane in panel (d), where there
is a clear excess of counts related to the contamination. . . . . . . . . . .

Figure 3.5: Before (left) and after (right) the φ correction, which is the correction for
the trajectory of the particle at the target position. The x axis is the
angle between the reaction plane and the detection plane, while the y axis
is the detected γ-ray energy assuming the mid-target velocity.
. . . . . .

Figure 3.6: Before correction to remove correlation between γ-ray energy following
the φ and yta corrections. Each panel corresponds to a different crystal.
Crystals are ordered ring-by-ring, from smallest to largest polar angle, and
then within rings from smallest to largest azimuthal angle.
. . . . . . .

Figure 3.7: After correction to remove correlation between γ-ray energy following the
φ and yta corrections. Each panel corresponds to a different crystal. Crys-
tals are ordered ring-by-ring, from smallest to largest polar angle, and then
within rings from smallest to largest azimuthal angle.
. . . . . . . . . .

Figure 3.8: Before correction to remove correlation between γ-ray energy following the
φ and yta corrections and the particle’s kinetic energy after the target.
Each panel corresponds to a different crystal. Crystals are ordered ring-
by-ring, from smallest to largest polar angle, and then within rings from
. . . . . . . . . . . . . . . . . . . .
smallest to largest azimuthal angle.

Figure 3.9: After correction to remove correlation between γ-ray energy following the
φ and yta corrections and the kinetic energy after the target. Each panel
corresponds to a different crystal. Crystals are ordered ring-by-ring, from
smallest to largest polar angle, and then within rings from smallest to
largest azimuthal angle.
. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10: Spectra showing the gradual improvement of the resolution of the Doppler-
corrected 1260-keV γ ray in 70Ni following each of the different corrections
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
described in the text.

43

44

46

49

50

51

52

53

viii

Figure 3.11: Spectrum observed in coincidence with 70Ni knockout residues showing
. .

different components of the fit template in the region 1500-2000 kev.

54

Figure 3.12: Spectrum in coincidence with 71Cu projectiles implanted into an alu-
minum stopper at the center of GRETINA. The area of the peak at 133
keV, which corresponds to the de-excitation from the long-lived (19/2−)
state in 71Cu, was measured relative to the number of implants to estimate
the isomeric content in the beam.
. . . . . . . . . . . . . . . . . . . . . .

Figure 3.13: Spectrum in coincidence with 71Ni projectiles implanted into an aluminum
stopper at the center of GRETINA. The area of the peak at 454 keV,
which corresponds to the de-excitation from the long-lived (1/2−) state
in 71Cu to which the isomeric (1/2−), 499-keV level in 71Ni β-decays,
was measured relative to the number of implants to estimate the isomeric
content in the beam.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.1: The level scheme for 70Ni based on the transitions observed in each reac-
tion channel for this experiment. Transitions that are placed for the first
time into the level scheme are shown in red, while dashed lines indicate
tentative placements. The levels highlighted in orange are expected to
exhibit proton-particle hole configurations.
. . . . . . . . . . . . . . . .

Figure 4.2: An example of a fit used to extract intensities from the 70Ni spectrum
populated via one-proton knockout from 71Cu. The fit consisted of simu-
lated line shapes for all peaks observed in the spectrum as well as a back-
ground composed of a double exponential to model the beam-correlated
background in addition to stopped background lines from both inelastic
reactions of neutrons and other light particles with the aluminum beam
pipe or Ge detectors as well as annihilation radiation. Clearly present in
the spectrum are the 970-keV, 4+
1 transi-
tions in the yrast sequence along with many other transitions discussed
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
in the text.

1 and 1260-keV, 2+

1 → 2+

1 → 0+

Figure 4.3: Sample of coincidence spectra used for placing transitions into Fig. 4.1 for
the one-proton knockout reaction. Clear coincidences between the 1260-
970-1584 keV transitions are observed, making it possible to place the
1584 keV transition in the level scheme. Also shown are a weaker 480 keV
transition believed to feed into the newly placed 3814-keV state as well as
several of transitions in the suspected prolate structure at 279 keV, 609
keV, 640 keV, and 1249 keV. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: Coincidence spectra used for placing transitions in the suspected prolate
structure in the 71Cu one-proton-knockout reaction channel. Fig. 4.3 con-
tains the associated spectra in coincidence with transitions in the yrast
sequence.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

56

58

61

64

66

67

Figure 4.5: Coincidence spectra used for placing decays from the 4293-keV and ten-
tative 3662-keV levels in the level scheme of Fig. 4.1 for the 71Cu one-
proton-knockout reaction channel. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.6: Coincidence spectra used for placing decays from the 4017-keV and 2942-
keV levels in the level scheme of Fig. 4.1 for the 71Cu one-proton-knockout
reaction channel.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.7: Coincidence spectra used for placing decays the 1955-keV, 2701-keV, and
1321-keV transitions in the level scheme of Fig. 4.1 for the 71Cu one-
proton-knockout reaction channel. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: Spectra utilizing nearest-neighbor add-back from the one-proton-knockout
reaction where the different lines correspond to different detector multi-
plicities. Note that the multiplicity is the corrected multiplicity after
add-back; i.e., if an event only contains two hits and these are combined,
it is a detector multiplicity-1 event. The dominance of the 3846-keV tran-
sition in the detector multiplicity-1 spectrum suggests that it could be
a transition directly into the ground state, or some long-lived, low-spin
state, as discussed in the text.
. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.9: Spectra utilizing nearest-neighbor add-back from the one-proton-knockout
reaction where the different lines correspond to different detector multi-
plicities. Note that the multiplicity is the corrected multiplicity after
add-back; i.e., if an event only contains two hits and these are combined,
it is a detector multiplicity-1 event. The appearance of the 1260-keV
transition in each detector multiplicity spectrum is expected based on the
number of different ways it can be populated in the one-proton knockout
(see Fig. 4.1).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.10: The Doppler-corrected γ-ray spectrum observed in coincidence with the
70Ni residues after one-neutron knockout from 71Ni. Clearly present in the
spectrum are the 970-keV, 4+
1 transitions
in the yrast sequence. Also visible are the previously unplaced 385-keV
transition as well as the 682-keV γ ray associated with decay from the
(5−) state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 and 1260-keV, 2+

1 → 2+

1 → 0+

Figure 4.11: Coincidence spectra for the 385-keV, the 676-682 keV doublet, and the 234-
keV transitions from the sum of the one-proton and one-neutron knockout
reaction channels. Due to low statistics, it is only possible to tentatively
place the 385-keV transition. The displacement of the yrast energies in
panel (b), as indicated by the dashed lines at the expected yrast energies,
. . . . . . . . . . . . . . . . . . . . . . . . . . .
is discussed in the text.

x

69

70

72

73

74

76

77

Figure 4.12: Gamma-ray spectrum from the hodoscope taken in coincidence with out-
going 70Ni reaction residues following one-neutron knockout from 71Ni.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Figure 4.13: Comparison of experimental level scheme for states populated in the one-
neutron knockout reaction channel to results of a traditional shell-model
calculation only allowing neutron excitations using the jj44pna [6] effec-
tive interaction. For the positive-parity states, g9/2 shell-model spec-
troscopic factors C2S for one-neutron knockout from the 71Ni ground
state to individual 70Ni final states are listed when C2S(0g9/2) > 0.1.
For the negative-parity levels, the 0f5/2, 1p3/2, 1p1/2 shell-model spec-
troscopic factors are listed when C2S > 0.02 for states where the sum
C2S(0f5/2) + C2S(1p3/2) + C2S(1p1/2) > 0.1.
. . . . . . . . . . . . . .

Figure 4.14: The Doppler-corrected γ-ray spectrum observed in coincidence with the
70Ni residues after two-proton knockout from 72Zn. Clearly present in the
1 and 1260-keV, 2+
spectrum are the 970-keV, 4+
1 transitions
in the yrast sequence. Also visible are the 609-keV, 640-keV, and 1868-keV
. . . . . . . . . .
γ rays associated with the predicted prolate structure.

1 → 2+

1 → 0+

Figure 5.1: Comparison of Doppler-corrected γ-ray singles spectra in coincidence with
detection of 70Ni knockout residues. The spectra are scaled based on the
detected number of 70Ni residues in each setting relative to the number
detected in the two-proton knockout channel. The scaling factors are
listed in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.2: Comparison of feeding-subtracted state populations between the proton
knockout and neutron knockout. The populations were divide by the
number of 70Ni knockout residues detected in the channel under consider-
ation. Plotted is the difference over the sum of these relative populations
comparing both proton knockout reactions with neutron knockout.
. . .

Figure A.1: Adopted B(E2; 0+

1 → 2+

1 ) values from Ref. [8], except for the value for
98Zr and 86Se. In the case of the former, the value is based on a more re-
cent Coulomb excitation experiment [9] that determined the upper bound
for the strength, where previously only a lower limit had been reported.
In the case of the latter, the value is taken from a recoil-distance Doppler
shift measurement [10].
. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure A.2: Adopted energies from the NNDC database [11] for the 2+

1 level for the
. . . . . . . . . . . . . . . . . . . . . . . . .

discussed even-even nuclei.

83

84

86

88

96

97

Figure A.3: Particle identification plot for the setting centered on 88Kr35+ in the focal

plane. The beam components are cleanly separated and identifiable.

. . 101

xi

Figure A.4: Inelastic scattering distribution in the case of 88Kr. The red and blue
lines are Monte Carlo simulations of the inelastic scattering process based
on an input probability distribution from the Alder-Winther relativistic
Coulomb excitation model [12], either with (blue) or without (red) the
effects of angle emittances and angular straggling. The dashed green line
shows the chosen, more restrictive scattering-angle cut for the analysis of
88Kr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure A.5: (a)-(c) Doppler corrected γ-ray spectra including the fits to extract the
angle-integrated Coulomb excitation cross sections for each isotope. The
fit template based on the GEANT4 simulations is described in the text.
(d) The spectrum of the beam contaminant 90Rb with a double expo-
nential fit to model the prompt, beam-correlated background, using an
exclusion region in the fit around the peaks near 500 and 1400 keV. The
resulting double exponential adequately reproduces the behavior in the
region of interest, which is from 600 keV to 1.7 MeV.

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

Figure A.6: Spectrum coincident with the observation of the self-coincident 775 keV-
802 keV doublet of γ rays, including a fit to extract the intensity of the
feeders. A detector multiplicity-2 condition is applied as explained in the
text.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure A.7: Level scheme for transitions observed in coincidence with scattered 88Kr
projectiles. The given branching ratios are the adopted values from the
NNDC database [11]. The dashed line giving an upper limit for the
(2+
4 ) → 0+

1 branching ratio is based on the present work. . . . . . . . . . 108

Figure A.8: Level scheme for transitions observed in coincidence with scattered 90Kr
projectiles. The given branching ratios are the adopted values from the
NNDC database [11]. The dashed line giving an upper limit for the
1 branching ratio is based on the sensitivity limit at this en-
(2+) → 0+
ergy in the present work. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure A.9: Evolution of collectivity from N = 36 to N = 60 for the krypton and
selenium isotopes, including adopted values and the values measured here.
The adopted values are taken from Ref. [8], except for the case of 86Se [10].
Note the different units used here compared to Fig. A.1 (see Sec. A).

. . 114

Figure A.10: Comparison between measured B(E2) values and the results of a shell-

model calculation for 88,90Kr as a function of the energy of the 2+

n states. 116

xii

Chapter 1

Introduction

1.1 Nuclear physics

Nuclear physics is concerned with nuclei, strongly-interacting systems composed of neutrons

and protons. Neutrons and protons are fermions, particles of half-integer spin s = 1/2, and

are collectively known as nucleons. The number of protons and neutrons will henceforth be

referred to as Z and N, respectively. In the shell-model picture of nuclei, nucleons fill energy

orbitals with distinct quantum numbers based on the Pauli exclusion principle, which states

that indistinguishable fermions in the same quantum system cannot have the same quantum

numbers.

Orbitals are labeled by their quantum numbers nlj, where n = 0, 1, 2, . . . is the principal

quantum number and l = 0, 1, 2, . . . is the orbital angular momentum quantum number. The

orbital angular momentum and spin are coupled to a total angular momentum (cid:126)j = (cid:126)l + (cid:126)s,

and j = l ± 1/2 is the total angular momentum quantum number. Typically, the orbital
angular momentum is represented in spectroscopic notation, so that the different values l =

0, 1, 2, 3, 4, . . . are represented as l = s, p, d, f, g, . . . instead. For each j value, there are 2j +1

magnetic sub-states indicated by a magnetic quantum number m = −j,−j + 1, . . . , j − 1, j,
which is the z-projection of (cid:126)j. Every orbital labeled nlj can therefore be occupied by 2j + 1

nucleons, because no two fermions can occupy the same nljm state. Note that because

1

protons and neutrons are distinguishable, a proton and a neutron can share the same nljm

values, though two protons or two neutrons cannot. Normal-order configurations refer to the

distribution of the nucleons between the different single-particle orbitals in the case where

only the lowest-energy orbitals allowed by the Pauli exclusion principle are filled. Figure 1.1

presents a simple view of a normal-order configuration for the ground state of 17O.

Figure 1.1: Simplified picture for the single-particle configuration of the ground state of 17O
with the shell gaps at N, Z = 2, 8.

Parity (written as π) is a characteristic of nuclear states that describes the behavior of

the wavefunction under the parity transformation: x → −x, y → −y, z → −z. Parity can
either be even (π = +1) or odd (π = −1), and is determined by (−1)l. Energy levels in
nuclei are most-often presented in terms of the spin-parity, or J π, where (cid:126)J = (cid:80)A
(cid:126)j, A

being the number of nucleons.

i=1

2

1.1.1 Nuclear shell model

In the independent-particle shell model, nucleons are modeled as moving independently

in a mean-field created by all the other nucleons. There are relatively large energy gaps

between sets of orbitals at certain numbers of protons or neutrons known as magic numbers

(i.e., 2, 8, 20, 28, 50, 82, 126) analogous to the situation in the atomic shell model. A shell is

considered to be closed when all orbitals below it are filled by the allowed number of nucleons

based on the Pauli principle. So-called valence nucleons are the nucleons outside of these

closed shells. Nucleons tend to form pairs such that each pair has total spin 0 and even

parity, leaving the unpaired valence nucleons to determine the spin-parity of a given nuclear

state. For example, the unpaired nucleon in the 0d5/2 orbital in 17O (see Fig. 1.1) leads to
a ground-state spin-parity of 5/2+.

Shell gaps result in observable shell effects in nuclei that are singly-magic, wherein Z or

N is a magic number, or doubly-magic, wherein both are magic numbers, when compared

to neighboring nuclei. For example, even-even nuclei are nuclei with an even Z and N.

The ground state of even-even nuclei always has spin-parity J π = 0+, and the first excited

state has, in most cases, J π = 2+. For singly- and doubly-magic nuclei, the excitation

energy of the 2+ is generally larger than this same quantity in non-magic nuclei. Another

sign of magicity is a small electric quadrupole transition rate, written as B(E2; 0+

1 ),
1 → 2+
1 state in even-even nuclei, which is systematically smaller
when comparing magic nuclei to neighboring nuclei. This quantity will be introduced in the

between the ground state and the 2+

following section.

3

1.1.2 Electromagnetic transitions in nuclei

Decays between states of an atomic nucleus most commonly involve γ-ray emission. The

γ-ray energy is equal to the energy difference between the initial and final state, with a

negligible contribution from the recoil of the atomic nucleus. A schematic view of γ decay

is shown in Fig. 1.2

Figure 1.2: A schematic view of γ decay from the excited state of energy Ei to a state at
lower energy Ef .

Electromagnetic transitions are differentiated by the amount of angular momentum λ

carried away and the parity of the initial and final states. Consider an initial state of

πi
spin-parity J
i

πf
f . Because photons are bosons, they
must carry away at least one unit of angular momentum, meaning that γ-ray transitions

and final state with spin-parity J

between two states with Ji = Jf = 0 are forbidden. In this case, the decay is mediated

by emission of conversion electrons and/or internal-pair formation if the energy difference

exceeds 1022 keV. Because of conservation of angular momentum, λ for the photon can

take integer values between |Ji − Jf| ≤ λ ≤ Ji + Jf , except λ = 0 as noted above. This
carried-away angular momentum determines the multipolarity of the transition; i.e., λ = 1

is a dipole transition, λ = 2 is a quadrupole transition, and so on.

4

In terms of parity, the parity carried by photons is (−1)λ for those associated with the
electric multipoles or (−1)λ+1 for the magnetic multipoles. These are denoted by either E or
M, respectively. Therefore, due to parity conservation, if πi = πf M 1, E2, M 3, E4, . . . are

the possible transitions. The possible transitions are E1, M 2, E3, M 4, . . . otherwise. Because

photons cannot have λ = 0, E0 radiation denotes decays proceeding through conversion-

electron emission.

Let σ ∈ E, M represent whether the transition is electric or magnetic. The transition
rate T can be written [13] as the sum of matrix elements for the tensor operators Oσλµ of

rank λ, where µ is the projection of λ:

TMi,Mf ,µ =(cid:88)σλ (cid:18) 8π(λ + 1)

λ[(2λ + 1)!!]2(cid:19) 1

¯h(cid:18) Eγ
¯hc(cid:19)2λ+1

| < Jf Mf|O(σλ)µ|JiMi > |2

(1.1)

where Eγ is the γ-ray energy, ¯h is the reduced Planck constant, c is the speed of light,

and ¯hc = 197 MeV fm.

One can then sum over the Mf and µ values since they are not observed individually and

average over the initial states to rewrite TMi,Mf ,µ in terms of reduced matrix elements:

T =

1

2Ji + 1(cid:88)Mi (cid:88)Mf ,µ
=(cid:88)σλ (cid:18) 8π(λ + 1)
=(cid:88)σλ (cid:18) 8π(λ + 1)

λ[(2λ + 1)!!]2(cid:19) 1
λ[(2λ + 1)!!]2(cid:19) 1

TMi,Mf ,µ

¯h(cid:18) Eγ
¯hc(cid:19)2λ+1 | < Jf||O(σλ)||Ji > |2
¯h(cid:18) Eγ
¯hc(cid:19)2λ+1

B(σλ; Ji → Jf )

2Ji + 1

(1.2)

(1.3)

(1.4)

5

where B(σλ; Ji → Jf ) is the reduced transition probability.
Although λ can take all of the values allowed by angular momentum conservation, the

lowest-allowed multipolarity λ for either electric or magnetic transitions will have a transition

rate several orders of magnitude higher than transitions with the multipolarity λ+2; e.g., for

πf
πi
f = 3+, E2 and M 1 transitions are more than seven orders of magnitude
i = 4+ and J
J
stronger than E4 and M 3 transitions, which are also allowed. For transitions of mixed

multipolarity, such as the aforementioned 4+ → 3+ or 2+ → 2+, the multipole mixing ratio
δ is defined, where in the case of E2 − M 1 transitions:

δ2(cid:18) E2

M 1(cid:19) =

T (E2)
T (M 1)

The lifetime τ of an excited state is related to the transition rate:

τ =

1
T

For E2 transitions, the partial mean lifetime can be written as:

τp =

816
γB(E2)

E5

e2fm4MeV5ps,

(1.5)

(1.6)

(1.7)

where the B(E2) value is in e2fm4 and the energy Eγ is in MeV. The diagonal transition

matrix elements, i.e. the transition matrix elements where the initial and final states are the

same, are moments such as the electric quadrupole moment. These can quantify the shape

of a nucleus in a given state.

Shape coexistence is the peculiar nuclear phenomenon where, at low excitation energy

and within a narrow energy range, two or more states (or band structures of states) exist

6

which can be interpreted in terms of different intrinsic shapes.

1.2 Shape and configuration coexistence in exotic nuclei

On a microscopic level, shape coexistence in nuclei can be caused by the competition be-

tween normal-order configurations and configurations involving excitations across shell gaps,

referred to as cross-shell excitations, at relatively low excitation energy. These states result-

ing from cross-shell excitations often exhibit different electric quadrupole properties, such

as moments and transition rates, than those based on normal-order configurations. While

initially believed to be a rare occurrence on the nuclear chart, there is now evidence that

shape coexistence may occur in almost all nuclei [14].

The most common shapes for nuclear systems are spherical, prolate (cigar-like), and

oblate (pancake-like), which are shown in Fig. 1.3. More exotic nuclear shapes, such as

pear-like shapes due to octupole deformations, exist but are less prevalent in the regions of

the nuclear chart discussed here. Assuming neither triaxiality nor higher-order deformations

beyond quadrupole, Fig. 1.3 approximates the nuclear shape R(θ, φ) as:

R(θ, φ) = 1 + βY2,0(θ, φ),

(1.8)

where Y2,0(θ, φ) is a spherical harmonic, and β is the quadrupole deformation param-

eter, whose sign determines whether a nucleus is prolate or oblate. Prolate deformations

correspond to positive quadrupole deformation parameters, in contrast with oblate nuclei

which correspond to negative ones. It is possible for excited states of a nucleus to exhibit

dramatically different shapes compared to the ground state of the same nucleus.

On a microscopic level, the effect of cross-shell excitations across a shell gap are par-

7

Figure 1.3: Common shapes of atomic nuclei. In panels (b) and (c) the magnitude of the
quadrupole deformation parameter was chosen to be |β| = 0.2.

ticularly interesting in neutron-rich singly-magic nuclei. These cross-shell excitations into

so-called intruder orbitals outside of the normal-order shell fillings, become energetically

favored because of a variety of effects, such as quadrupole correlations [14] or the proton-

neutron tensor force [15].

Along the nickel isotopic chain, for example, the proton-neutron tensor force is one driver

responsible for changes to the nuclear structure. The effect of the proton-neutron tensor

interaction can be described in a simplified way by considering j> = l + 1/2 and j< = l− 1/2
orbitals separately. Consider the following example. The neutrons in the 0g9/2 (j>) orbital
interact with the proton 0f7/2 (j>) and 0f5/2 (j<) orbitals. The interaction between j>j>
In
and j<j< orbitals is repulsive;

i.e., the orbital being acted on becomes less bound.

contrast, the interaction between j>j< orbitals is attractive. Therefore, as the number of

neutrons in the 0g9/2 increases, the gap between the 0f5/2 and 0f7/2 decreases, and the
likelihood of excitations across the Z = 28 shell gap increases. This interaction is strongest

when the 0g9/2 orbital is half-filled; i.e., when it contains 2j+1

2 = 5 nucleons.

Effects like the aforementioned proton-neutron tensor force leads to states associated

8

(a) Spherical(b) Prolate(c) Oblatewith these excitations across shell gaps being at relatively low excitation energy compared

to states associated with similar configurations in stable nuclei. This work will focus on the

singly-magic (Z = 28) Ni isotopes, in particular on 70Ni, in which several states have been

associated with configurations based on such cross-shell excitations.

1.3 Shape coexistence in the Ni isotopes

Shape coexistence appears as a common feature in the neutron-rich Ni isotopes, as evidenced

by recent work on 66,68,70Ni [1, 3, 16–19] and predictions for 78Ni [20, 21]. In 78Ni, which is

doubly-magic, shell-model calculations [20] predicted that the first excited state is a prolate-

deformed, 2.65 MeV 0+ state. This 0+ state serves as the bandhead of a prolate-deformed

intruder band and signals an island of inversion centered on N = 50 analogous to those

at N = 8, 20, 28, and 40. An island of inversion [22, 23] is a region of the nuclear chart

where the traditional shell gaps seem to disappear, so that states associated with configura-

tions involving multiple particles being excited across shell gaps energetically compete with

normal-order configurations.

Proposed [19] as one of the lightest known systems to exhibit shape isomerism, 66Ni

is known to have four low-lying 0+ states. A hindered γ transition between the prolate-

deformed 0+

4 state and the yrast 2+

1 level has been reported in 66Ni based on the small
measured electric quadrupole transition strengths. Note that yrast refers to a state of a

nucleus with the lowest energy for a given spin-parity, such as the ground-state band (0+
1 ,
1 , . . . ). Such shape isomerism occurs when a large potential energy barrier exists

1 , 4+
2+
between the states associated with different deformations.

The situation is quite different for 68Ni, which appears to exhibit substantial mixing

9

between deformed and spherical configurations [17]. There are three known 0+ states in 68Ni

associated with different shapes [1, 16, 17]: the spherical ground state, an oblate deformed

level at 1604 keV, and a prolate deformed one at 2511 keV, where the shapes have been

inferred from shell-model calculations [1, 16]. A potential energy surface calculated [1, 16]

for 68Ni using Hartree-Fock constrained by the quadrupole moment determined from the

Monte Carlo shell model Hamiltonian is shown in Fig. 1.4. The minima in the potential

energy surface correspond to the different 0+ states associated with different shapes.

Figure 1.4: Potential energy surface for 68Ni calculated using Hartree-Fock calculations
constrained by the quadrupole moment. The three minima in the potential energy surface
are associated with spherical, oblate, and prolate deformed 0+ states. Figure adapted from
Ref. [1].

10

Figure 1.5: Potential energy surface for 70Ni calculated using Hartree-Fock calculations
constrained by the quadrupole moment. There is a considerable deepening of the prolate
minimum in 70Ni compared to 68Ni, which is shown in Fig. 1.4. Figure adapted from Ref. [2].

Considerably less is known concerning shape coexistence in 70Ni. Monte Carlo shell-

model calculations [16] predict both that the ground state of 70Ni is weakly oblate, and

that the prolate deformed level drops from 2.5 MeV in 68Ni to around 1.5 MeV in 70Ni,

highlighting the effect of shell evolution [16] when adding only a couple nucleons. Looking at

potential energy surfaces calculated using Hartree-Fock calculations [1, 2, 16] constrained by

the quadrupole moment determined from the Monte Carlo shell model Hamiltonian, there is

a clear deepening of the minima in the potential energy surface in 70Ni in Fig. 1.5 compared

11

to 68Ni in Fig. 1.4. Ref. [18] reported the observation of a (0+

2 ), 1567-keV level, possibly

corresponding to the predicted prolate 0+ state.

Figure 1.6: Results of shell-model calculations from Ref. [3] using the Monte Carlo shell
model (MCSM) [4] with the full proton and neutron f pg9/2d5/2 model space compared to
results using traditional shell-model calculations utilizing only the neutron f5/2pg9/2 model
space and effective interactions: jj44bpn [5], jj44pna [6], and JUN45 [7]. Only the 0+ and
2+ levels are shown for comparison.

A comparison between the predictions for the Monte Carlo shell model and traditional

shell-model calculations utilizing a number of different effective interactions is shown in

Fig. 1.6, which is based on the shell-model calculations discussed in Ref. [3]. The Monte

Carlo shell model is able to utilize the full proton and neutron f pg9/2d5/2 model space in
the calculation. For traditional shell-model calculations of 70Ni, including cross-shell proton

excitations requires diagonalizing intractably large matrices, and therefore the aforemen-

tioned calculations could only include neutron excitations to the f5/2pg9/2 orbitals. Unlike
the Monte Carlo shell model, these traditional calculations were unable to predict the sus-

pected prolate 0+ state, nor the 1868-keV, 2+

2 one, possibly suggesting that these states are

12

built on cross-shell proton excitations outside of the traditional model space. A predicted

oblate 0+ state at energies above 2 MeV from both traditional shell-model and Monte Carlo

shell-model calculations has not yet been observed. It is important to note that the prolate

shape of the structure in 70Ni is suspected solely from shell-model calculations and cannot

be currently inferred from any current experimental data.

Exploring the band structures, states of similar structure connected to each other by

strong transitions, built atop shape-coexisting 0+ states provides stringent tests for nuclear

models due to the demanding configuration spaces and the inclusion of cross-shell excitations.

This has been probed experimentally in 68Ni, for example, by observing the β decay of 68Co

and measuring the lifetimes and branching ratios of transitions from the three 0+ states

associated with different shapes. Another experimental technique, nucleon knockout, has

been used to probe cross-shell excitations in proton-magic nuclei in the past [24]. Here, the

role of excitations across Z = 28 is explored using complementary one- and two-nucleon

knockout reactions to determine the leading orbital configurations of excited states in 70Ni.

In the direct one- and two-nucleon knockout experiment performed in this work, a pro-

jectile of interest impinges on a 9Be target and one or two nucleons are removed, leaving

the projectile-like residue as a spectator to the sudden collision [25]. One- and two-nucleon

knockout provide a way of probing the single-particle configurations of excited states in nu-

clei. For example, by knocking out a proton from 71Cu, excited states in 70Ni associated

with configurations involving excitations across the Z = 28 shell closure can be selectively

populated, as the 70Ni nucleus is, with some probability related to wavefunction overlaps,

left in a configuration that corresponds to a one-particle one-hole proton excitation after

the reaction. The use of knockout as a direct reaction to probe spectroscopic factors was

not pursued here due to knockout from isomers that cannot be quantified event-by-event,

13

as well as to (1) mostly unknown or tentative final-state quantum numbers, to (2) complex

associated configurations, and to (3) expected high level densities (see for example [26]).

More details about these direct reactions can be found in Ref. [25]. Here, the focus is on the

population of excited states in 70Ni. By comparing the population of the different excited

states, as measured through their characteristic γ-ray de-excitations, to the resulting popu-

lations following one-neutron knockout from 71Ni, it is possible to determine which states are

associated primarily with proton or neutron excitations. The aim of the present work was

to probe the proton and neutron degrees of freedom in 70Ni with complementary reactions,

and more specifically to verify the role of cross-shell proton excitations in the configuration

of states proposed as belonging to the predicted prolate structure in 70Ni.

14

Chapter 2

Experimental details and detection

systems

The experiment was performed at the National Superconducting Cyclotron Laboratory

(NSCL) [27] at Michigan State University. An overview of the Coupled Cyclotron Facil-

ity at the NSCL, including the K500 and K1200 coupled cyclotrons as well as the A1900

fragment separator, is shown in Fig. 2.1.

2.1 Beam production

The production of the primary 76Ge beam proceeds as follows. A germanium vapor is created

by oven heating and then ionized in an electron cyclotron resonance (ECR) ion source, the

Advanced Room TEMperature Ion Source (ARTEMIS) [28]. The ions are then injected into

the K500 cyclotron. The primary 76Ge beam was ionized to 76Ge+12 before injection.

Cyclotrons take advantage of the circular motion of a particle moving perpendicular to

a uniform magnetic field. A particle with momentum p and charge q moving in a magnetic

field of strength B will trace out a circle of radius ρ, where:

ρ =

p
qB

15

(2.1)

Figure 2.1: An overview of the layout for the Coupled Cyclotron Facility at Michigan State
University, including the K500-K1200 coupled cyclotrons and the A1900 fragment separator.

A cyclotron is composed of pairs of "dees", copper electrodes surrounded by a supercon-

ducting electromagnet. A magnetic field serves to confine the particle along a circle of radius

ρ and a radio-frequency electric field accelerates the particle as it crosses a gap between the

dees. As the particle gains energy, its momentum p increases and it spirals outward from its

point of injection at the center of the cyclotron.

Through this process, the ions will eventually reach the maximum energy obtainable

from the K500 cyclotron and will be extracted and transported to the entrance point at the

center of the K1200 cyclotron. Following acceleration in the K500 cyclotron, the 76Ge+12

particles reached 11.6 MeV/nucleon. At the entrance of the K1200 cyclotron, the accelerated

16

ions strike a thin carbon "stripper" foil that further ionizes the particle; i.e., the charge

q becomes larger, which by Eqn. 2.1 results in higher achievable energies. After striking

the stripper foil, the 76Ge+30 particles are accelerated to 130 MeV/nucleon in the K1200

cyclotron, undergoing the same process as in the K500 cyclotron.

The 76Ge+30 primary beam then impinged on a thick 399 mg/cm2 9Be production target,

producing a cocktail of different fully-stripped ions including 71Cu, 71Ni, and 72Zn, referred

to as secondary beam, through projectile fragmentation. To select components of interest

from the secondary cocktail beam, the A1900 fragment separator was utilized [29].

The A1900 fragment separator is composed of 24 quadrupole magnets and four 45◦
dipoles. It separates the fragments using a Bρ − ∆E − Bρ technique, where Bρ = p
q is the
magnetic rigidity. The magnetic rigidity allows selecting the particles of interest, because

only those within a small range of momentum-to-charge ratio can pass through the dipole

magnets based on the fixed radius of curvature of the beam line element ρ and the selected

magnetic field strength B. Only particles with similar p

q = γmv
q

, where γ is the Lorentz

factor, m is the particle mass, and v is the velocity, will pass through the selection process.

Therefore, further beam purification is necessary to remove particles of similar momentum-

to-charge ratios. One way this is accomplished is through several slits located both in the

focal plane and the image points of the A1900, which can be partially closed to further limit

the range of accepted momenta. An aluminum wedge degrader with an areal density of

300 mg/cm2 located at the mid-acceptance position of the A1900 fragment separator was

also used to further purify the secondary beam cocktails. Ions striking this wedge will lose

differing amounts of energy based on the Bethe formula:

17

dE
dx

−

=

4πe4neZ2

meβ2

(cid:20)ln(cid:18)2meγ2β2

I

(cid:19) − β2(cid:21) .

(2.2)

where e is the elementary charge constant, ne is the degrader’s electron number density,

me is the mass of the electron, β = v

c is the ratio of the ion velocity v to the speed of light
c, and I is the average excitation energy of the degrader material. This means that particles

with different atomic numbers Z will exit the degrader with significantly different energies

due to the Z2 term in Eqn. 2.2.

Following the wedge degrader, there is another Bρ selection process using the final two

dipoles of the A1900 fragment separator that functions in the same manner as the first stage.

The momentum acceptances for the different secondary beams of interest were limited by slits

to 1.0%, 2.0%, and 0.5% for the 71Cu, 71Ni, and 72Zn secondary-beam settings, respectively.

2.2 The S800 magnetic spectrograph

The separated fragments of interest were delivered from the A1900 through the transfer hall

to the analysis line of the high-resolution, high-acceptance S800 magnetic spectrograph [30],

shown in Fig. 2.2. The spectrograph is an optical system for charged particles that bends

and focuses ions in a way analogous to the way lenses and prisms act on light [31].

The S800 is composed of two main parts: the analysis line, which delivers the beam from

the transfer hall to the S800 target position, and the spectrograph itself. The analysis line

is composed of four dipoles, five sets of quadrupole triplets, one quadrupole doublet, and

four sextupoles [31]. The dipoles bend ion trajectories, the quadrupoles focus them, and the

sextupoles serve to correct higher-order aberrations. At the beginning of the analysis line is

the object position of the S800, where a thin plastic scintillator is located. This scintillator

18

is used for time-of-flight measurements.

At the target position of the spectrograph, fragments struck a thin 9Be reaction target

of thickness 100 mg/cm2 for the one-proton knockout reaction or 188 mg/cm2 for the one-

neutron and two-proton knockout reactions. The analysis beam line was operated in focus

mode [30].

At the entrance to the spectrograph is a quadrupole doublet that focuses the beam,

which is then directed into the S800 focal plane using two large dipole magnets. Similar

to the discussion of the A1900 fragment separator, the dipoles of the S800 spectrograph

are used to bend charged-particle trajectories into the focal plane based on their magnetic

rigidity. The magnetic field strength of the dipoles is adjusted to spatially center the reaction

products of interest in the focal plane. The detectors in the S800 focal plane [32] include:

two cathode-readout drift chambers (CRDCs) used to determine the position and trajectory

of the particles entering the focal plane, an ionization chamber used to measure energy loss

of the ions for particle Z identification, a thin plastic scintillator called the E1 scintillator

that enabled measurements of a particle’s time of flight and served as the particle trigger,

and a CsI(Na) array [33] that was used in this work to detect decays from long-lived states

of ions embedded into an aluminum plate directly in front of it.

The different detection systems in the S800 focal plane enabled trajectory reconstruction,

particle identification, and isomer detection, and will be discussed in more detail in the

following sections.

2.2.1 Particle identification

To identify components of the incoming cocktail beam, time of flight was measured between

scintillators placed at the object position of the S800 (obj) and at the focal plane of the A1900

19

Figure 2.2: The secondary cocktail beams were delivered to the S800 spectrograph, pictured
here.

(xf p) to the E1 scintillator in the S800 focal plane. Because ions entering the S800 focal

plane have been selected based on their magnetic rigidity Bρ = γmv
q

, the mass-to-charge

ratio m

q is inversely proportional to the velocity v assuming a fixed flight path through the
q . The

S800. Therefore, the time of flight is proportional to the mass-to-charge ratio m

time-of-flight difference xf p-obj is sensitive to the mass of the incoming beam.

When attempting to identify the outgoing reaction products, the time-of-flight difference

obj-E1 is utilized. It is important to correct for the correlation between the obj-E1 time-

of-flight difference and the angle and position in the S800 focal plane, as the time of flight

being proportional to the mass-to-charge ratio depends on the assumption of a fixed flight

path ρ through the spectrograph. In the case where the reaction products are fully stripped,

as is the case in this work, the mass-to-charge ratio m
q

is equivalent to A

Z , where A is

20

the mass number and Z is the atomic number of the ion. Using the time-of-flight difference

outgoing reaction products can be separated by A

Z , but this is not sufficient to identify specific
reaction products. Therefore, a gas-filled ionization chamber [32] is utilized to measure the

energy-loss ∆E of the reaction products in the S800 focal plane. The ionization chamber

is filled with a P10 gas and is partitioned into 16 1-inch anode segments. When the ions

enters the ionization chamber, it loses energy based on the Bethe formula (see Eqn. 2.2)

proportional to Z2, ionizing the gas along the particle track and causing the electrons to

drift towards the anodes. By plotting the energy deposited in the ion chamber ( ∝ Z2)
Z ), it is possible to discriminate between different
against the corrected time of flight (∝ A
outgoing reaction products. This is the standard ∆E − T oF technique.

2.2.2 Trajectory reconstruction

As noted in the previous section, the angles and positions of particles in the focal plane are

crucial for outgoing particle identification. Furthermore, based on this angle and position

information, the trajectory of a fragment can be reconstructed enabling the determination

of its dispersive (ata) and non-dispersive (bta) angles, non-dispersive position (yta), and

momentum (measured as the kinetic energy, dta) at the target position. While this momen-

tum information is essential in many experiments for obtaining momentum distributions [34],

here the reconstructed trajectories will play an important role in the Doppler-shift correction

discussed in detail in Chapter 3.

Trajectory reconstruction is accomplished using position information from two position-

sensitive cathode-readout drift chambers (CRDCs) [32] in the S800 focal plane. The CRDCs

are separated by about 1 m. The CRDCs provide information about the angle and position of

the particle in the focal plane of the S800. Particles passing through the CRDC ionize the gas,

21

causing electrons to drift towards the anode wire due to an applied voltage. The drift time

of the electrons to the anode determines the y position in each CRDC, while the distribution

of induced charge on the cathode pads determines the x position. Position information is

calibrated using a thick mask with holes and slits drilled into it at known positions, so

that particles can only pass through the holes and slits. The calibrated spectrum for the

downstream CRDC is shown in Fig. 2.3. By using two separate CRDCs, it is possible to

determine the dispersive (af p) and non-dispersive (bf p) angles in the focal plane from the

measured positions.

Figure 2.3: Calibrated mask run for the downstream CRDC used to determine the knockout
residue’s position in the focal plane. Note that the non-dispersive position is shifted so that
it is centered on the knockout residue of interest, rather than the center of the focal plane.

An inverse map generated with the ion optics code COSY Infinity [35], which takes into

account the magnet settings of the S800, is used to convert the quantities determined in

22

−300−200−1000100200300XPosition(mm)−40−30−20−1001020YPosition(mm)the focal plane (xf p, yf p, af p, bf p) to the previously mentioned ones at the target position

(ata, bta, yta, dta) through ray tracing. Note that the dta value is the difference between

the kinetic energy of the particle and the kinetic energy of a particle that would enter the

center of the focal plane. The determination of the beam trajectory at the target position is

important for Doppler reconstruction of γ rays emitted in-flight.

2.2.3

Isomer detection

Because the experiments require the detection of γ rays emitted from nuclei moving at

substantial fractions of the speed of light, in contrast to those that implant the nuclei of

interest in some material, the identification of long-lived excited states is difficult. These

so-called isomeric states arise from a number of different factors related to nuclear structure;

e.g., differences between the electric quadrupole properties of the excited states and the

ground states as in the case of shape coexistence or large angular-momentum differences

can lead to such behavior. Assuming a flight path through the spectrograph of about 15 m,

nuclei with velocities v ≈ 0.4c will have a flight time of roughly 125 ns. Therefore, it would
be necessary to measure de-excitations towards the end of the reaction product’s flight path

in the focal plane to identify the decays from states with lifetimes τ ≥ 125 ns.

To attempt the identification of such states, after the beam-like reaction residues pass

the E1 scintillator functioning as the particle trigger, they are embedded in a 6.35 mm thick

aluminum plate at a distance of 10.5 cm from a CsI(Na) hodoscope array [33], composed of

32 closely packed CsI(Na) crystals coupled to photomultiplier tubes. The energy resolution

typical of CsI(Na) scintillators is roughly 6% FWHM for a 137Cs source. The inclusion of

the aluminum plate for implanting the reaction products is known as the IsoTagger config-

uration [36].

23

The IsoTagger configuration enables the identification of decays from states of the knock-

out residues with lifetimes between 100 ns and several ms, such as the 2861-keV level

(τ = 335(1) ns [11]) in 70Ni.

In the range 50 keV to 1.9 MeV, the simulated full-energy

peak efficiency in this setup ranges between 6% and 16% based on the energy of the peak

assuming γ-ray emission from the center of the aluminum plate and an implantation profile

consistent with expectations for one nucleon removal reactions [36].

2.3 In-beam γ-ray spectroscopy with GRETINA

The purpose of this experiment is to measure the population of excited states in 70Ni. As

these states only decay by emitting a γ ray, this can be accomplished by measuring the in-

tensity of γ rays tagging the de-excitations. To this end, the Gamma-Ray Energy Tracking

In-beam Nuclear Array (GRETINA) [37], which in this experiment was composed of nine

cryostats each housing four 36-fold segmented high-purity germanium crystals, surrounded

the target position of the S800 magnetic spectrograph and was used for in-flight γ-ray de-

tection. The array was arranged in the NSCL standard configuration, which means the four

positions at 58◦ relative to the beam axis were populated, and the remaining detectors were

at 90◦. A photograph of GRETINA detectors arranged around the S800 target position in

front of the entrance quadrupole of the spectrograph is shown in Fig. 2.4. Note that there

are ten detectors shown here, one of which in the 90◦ ring was not available yet for this

experiment.

24

Figure 2.4: Photograph of GRETINA surrounding the target position of the S800 in front
of the entrance quadrupole of the spectrograph. The beam pipe is removed, and the beam
direction would be into the page. Note that there was one less detector at 90◦ utilized for
this experiment as compared to the photograph, as the tenth was not yet available.

25

2.3.1 The Gamma-Ray Energy Tracking In-Beam Nuclear Array

(GRETINA)

High-purity germanium detectors are semiconductor devices used for the detection of γ rays

that provide the best possible intrinsic energy resolution of about 0.2% FWHM at 1 MeV

for efficient γ-ray detection. Because γ rays are not charged, they cause no direct ionization

of a material. Therefore, it is necessary for the γ ray to go undergo some interaction with

the detection medium to produce energetic electrons, which are then detected and used to

reconstruct the energy of the initial γ ray. The three important types of interaction for γ

rays are: photoelectric absorption, Compton scattering, and pair production. As the cross

section for photoelectric absorption is small for γ rays with energies greater than a few 100

keV compared with Compton scattering, a γ ray undergoes multiple interactions before being

fully absorbed within a detector; e.g., multiple Compton scatters followed by photoelectric

absorption. Because of this, there are often multiple interaction points within a detector for

every γ ray that is detected as a full-energy-peak event in a spectrum.

The main aim of GRETINA is to measure these interaction points along with the energy

deposited at each point. The segmentation of the detectors makes it possible to determine

the sub-volume in a crystal where the γ-ray interactions occurred. By analyzing the detected

signal shapes, which are governed by the charge collection process, it is possible to achieve

sub-segment resolution. An interaction position in a segment is associated with a unique

set of signals observed on the hit segment contact, and via mirror charges on neighboring

segment electrodes as well. The unique set of signals is known as a signal basis. Because

there are several interaction points per event, a superposition of energy-scaled basis signals

are measured. To disentangle the different interaction points along with deposited energies at

26

each point, the measured signals from all 36 segments are fit against the signal basis, and up

to two interactions per segment are allowed. This process is known as signal decomposition,

and is accomplished online utilizing the GRETINA computer farm, which is capable of

processing 30,000 γ rays per second.

Measuring the interaction points and their associated deposited energies makes it possible

to perform γ-ray tracking. Using the kinematics of Compton scattering and the photoelec-

tric effect, γ-ray tracking makes it possible to determine whether a γ ray was completely

absorbed. This means it is feasible to eventually have a sphere providing full 4π coverage

with germanium without sacrificing the peak-to-total ratio, i.e. the ratio of counts in the

full-energy peak to the number of counts in the sum of the full-energy peak and the Compton

continuum. Although in the case of fast-beam experiments this is less of a concern due to

the relatively low γ-ray multiplicity for events, the number of GRETINA detectors provides

a large efficiency. In addition, the compactness of GRETINA detectors downstream of the

target is advantageous because of the forward focusing of γ rays emitted in flight at v

c > 30%

due to the Lorentz boost [38].

2.3.2 Doppler reconstruction

Because the γ rays are emitted from projectiles moving at substantial fractions of the speed

of light, they are detected in the laboratory frame with a significant Doppler shift. To

account for this, a Doppler reconstruction is performed to determine the energy of the γ ray

in the projectile frame (Ep) from the detected γ-ray energy in the lab frame Elab:

,

(2.3)

Ep = Elab

c cos θ

1 − v
(cid:114)1 − v2

c2

27

where v

c is the velocity of the projectile emitting the γ ray and θ is the γ-ray emission
angle. To properly determine the Doppler correction angle (θ in Eqn. 2.3) it is necessary to

determine the first interaction position in GRETINA, which is capable of providing interac-

tion points within σ ≈ 2 mm. To accomplish this, the assumption is that the first interaction
point is the main interaction point, which is the interaction point with the largest energy

deposition. The consequence of selecting the wrong first interaction point will be discussed

later.

The ability to reconstruct the energy of the γ ray is limited by knowledge of the emitting

particle’s velocity v and the detection angle θ from Eqn. 2.3. The reconstruction velocity

v is typically determined using the velocity at the mid-target position. The uncertainty

in the velocity determination is due to the thickness of the reaction target, because one

cannot determine the exact distance through the target at which the particle decays. The

dta value from the spectrograph only provides information on the kinetic energy after the

target. Therefore, as the decay can happen at any point in the target, there is an uncertainty

due to the energy loss in the target. Consider the knockout of one-neutron from 71Ni to 70Ni

using the 188 mg/cm2 9Be target. The energy after the target is measured based on the

magnetic rigidity of the spectrograph to be 66.8 MeV/nucleon. Therefore, the 71Ni decays

at some velocity between v/c = 0.394 and v/c = 0.3597, a 9% effect.

When a thinner target is used, such as the 100 mg/cm2 9Be target used for the one-proton

knockout setting, the largest contributor to the uncertainty in determining the energy of the

γ ray is the determination of the emission angle (θ in Eqn. 2.3). As mentioned previously,

the emission angle depends on the determination of the first interaction point, which is

assumed to be the interaction point with the largest energy deposition. Assuming a typical

distance between the target and GRETINA of 200 mm, the σ = 2 mm position resolution

28

of GRETINA results in an angular resolution of σ ≈ 10 mrad. This is much larger than the
few mrad uncertainty in the reconstruction of the particle trajectory using the S800. The

emission angle determination can be further refined by accounting for the non-dispersive

position of the particle at the target (yta) determined with the inverse map.

Finally, there is another relatively small effect due to the incoming beam being offset from

the optical beam axis. This can lead to the angle θ in Eqn. 2.3 being improperly shifted by

about 3 mrad in the worst case. [38]. The forward detectors are more sensitive to this effect

than those at 90◦, because the nearly-perpendicular emission angle for the 90◦ detectors are

not sensitive to small shifts in the xy-position of the beam on the target. All detectors,

however, would be sensitive to any beam angle offset from the optical axis. This is corrected

by systematically varying the beam spot offset (x0, y0) and beam angle offset (a0, b0) until

the measured centroids of individual crystals align. A discussion of corrections to manage

the aforementioned sources of uncertainty will be presented with examples in Chapter 3.

2.3.3 Nearest-neighbor add-back

It is common in GRETINA for neighboring crystals to detect interactions within the same

event, and this is likely caused by a single γ ray scattering from one crystal into the next.

This would lead to increased Compton background in a γ-ray singles spectrum, which is

simply the sum of the spectra from the central core of each individual crystal. The full

energy of the γ ray can be recovered by adding together the energy from the neighboring

crystals. This improves both the efficiency and the peak-to-total ratio as the two Compton

background γ rays are now part of the full-energy peak.

This process of recovering the full energy of γ rays scattered between crystals is known

as nearest-neighbor add-back. Two crystals are considered nearest neighbors if they share a

29

common boundary, whether they are in the same cryostat or in adjacent GRETINA modules.

In this work, add-back is accomplished by first sorting all interaction points in an event by

energy. Then, each interaction point is compared based on their distance and time difference.

If two interaction points lie within 80 mm and occurred within 440 ns, they are summed,

taking the position of the summed interaction points as that of the one associated with a

higher energy deposition.

Improving the peak-to-total ratio is particularly helpful for determining γγ coincidence

relationships, used in this work to place transitions into the level scheme, due to the decrease

in Compton scattered γ rays in the background when attempting to gate on individual tran-

sitions. For example, if one γ ray in a cascade scatters between crystals, it could lead to

misidentifying artifacts of this process as peaks in the γγ coincidence spectrum. A γγ co-

incidence spectrum for a specific γ ray is a spectrum containing all other γ rays observed

within the same events as that γ ray.

While nearest-neighbor add-back could be used to determine transition intensities, there

are number of difficulties; e.g., determining the absolute efficiency gain is subject to signifi-

cant systematic uncertainties, as add-back factors to scale the efficiency can only be reliably

determined for events with γ-ray multiplicity = 1 due to the possibility of summing two

coincident γ rays, particularly in this naive approach to add-back described above.

2.3.4 Simulating GRETINA’s peak response

The peak response was an important component of the fitting template used to extract the

peak areas from the spectra, particularly due to the presence of multiple, unresolvable peaks

in the spectra observed in this experiment. To this end, GEANT4 simulations using the

UCGretina framework [39] were utilized. The peak responses of both the observed γ rays

30

in the Doppler-corrected γ-ray spectra, and the stopped background lines, which can be

observed in the γ- spectra without a Doppler reconstruction, were simulated.

The simulation takes in to account: the geometry of GRETINA, including the segmen-

tation of the detectors, the cryostats, and the mounting hemispheres, the target nucleus

and target thickness, the aluminum beam pipe, the identity and energy of the incoming

projectiles, as well as the identity, excitation energy, and lifetime of the outgoing reaction

products. The inclusion of information concerning the target, incoming projectiles, and out-

going reaction products makes it possible to model the energy loss of the beam through

the target, and therefore the resulting uncertainty in the velocity for the Doppler correction

observed in the experiment due to the energy loss in the target. The reaction is modeled

as a single-step excitation, producing the reaction residue in an excited state with the given

excitation energy and lifetime. After the reaction, a new momentum vector is chosen for the

particle, and the distance it travels until decay is obtained from a Monte Carlo simulation

based on the lifetime of the excited state. The particle then continues to travel until it exits

the target, at which point the trajectory is recorded in the same format as from the S800

spectrograph; i.e., as ata, bta, dta, and yta.

The simulation process is complicated by the manner in which GRETINA obtains in-

teractions points, i.e., the signal decomposition process. While it is possible to determine

exactly the interaction points in the simulation, in reality multiple interactions within a sin-

gle segment are difficult for the decomposition process to distinguish. To approximate this in

the simulation, all interaction points within a segment are aggregated within a defined pack-

ing radius. In these simulations, the packing radius is chosen to be 6 mm, so all interaction

points determined in a single segment within 6 mm are combined into a single interaction

point with the energy as the sum of the energies of the different interaction points and with

31

position determined by an energy-weighted averaging. The resulting interaction points are

smeared based on the position resolution of GRETINA. In this analysis, the dependence of

the position resolution on the energy of the detected γ ray was explored, which has been

done previously for AGATA [40], the European γ-ray tracking detection array.

One important feature of the experimental data that is not directly reproduced in the

simulated peak response is the existence of tails extending both towards lower and higher

energies for peaks in the spectrum, a phenomena previously reported in Ref. [38]. This

is clearly visible in Fig. 2.5, where a simulation utilizing the nominal position resolution of

σ = 2 mm to smear the interaction points is unable to reproduce these tails when attempting

to fit a 1368-keV transition from 24Mg. This data comes from a measurement where the peak

shape is entirely dominated by the uncertainty in the detection angle due to the use of a thin

100 mg/cm2 target. It is clear that the simulated spectra is unable to reproduce the tails

on the side of the peak. This suggests that the γ rays are reconstructed using the wrong θ

in Eqn. 2.3. Some possible causes of this are selecting the wrong first interaction point or

the complex decomposition process returning incorrect interaction points for some fraction

of events. Reproducing the peak shapes was particularly important in this experiment as

there were a number of regions of the experimental spectra where multiple peaks could not

be resolved and intensity determination then relies on the modeling of the peaks within a

doublet.

Therefore, to reproduce the peak shapes with the aim of handling these tails when fitting

the full γ-ray spectrum to determine the peak areas, a strategy was devised where each peak

was split into two components: a narrow component corresponding to the expected peak

width, and a second, wider component, which would approximate the effect of the incorrectly

determined angle. The two components were summed with an 80%-20% scaling for the

32

Figure 2.5: Fit with peak response determined from the GEANT4 simulation, assuming a
position resolution of σ = 2 mm to smear the simulated interaction points. There are clear
tails on the peak that are not reproduced by the simulation.

narrow and wide component, respectively. Varying the widths of the narrow component in

the range 1.0 mm to 5.0 mm and the wide component in the range 5.1 mm to 7.5 mm,

the value resulting in the best fit for this 1368-keV transition as measured by the χ2-value

was determined to be σ = 2.0 mm and σ = 7.0 mm for the narrow and wide components,

respectively. This fit is shown in Fig. 2.6, and produced a factor of five improvement in

terms of χ2 compared to Fig. 2.5. Note that the fits were considerably more sensitive to the

narrow component than the wide component, and the wide component was not observed to

have an energy dependence.

33

13001400Energy(keV)01000200030004000Counts/keV24Mgv/c=0.369FitFigure 2.6: Fit with peak response determined from the GEANT4 simulation, assuming a
position resolution of σ = 2 mm for the narrow component and a position resolution of σ = 7
mm for the wide component, which are added together in the ratio 80% − 20% as described
in the text.

The energy dependence of the narrow component of the peak response was constrained

by fitting GRETINA data from light nuclei (21Ne, 23Na, 24Mg, 28Si, and 32S) on a 100

mg/cm2 9Be target with well-separated peaks at different energies and varying the simulated

position resolution of GRETINA, enabling the determination of the energy dependence of

the peak widths as shown in Fig. 2.7. The use of data from light nuclei on such a thin target

is advantageous, as it almost eliminates the uncertainty in the velocity determination and

therefore improves the sensitivity to the position resolution in GRETINA. Unfortunately,

34

13001400Energy(keV)01000200030004000Counts/keV24Mgv/c=0.369Fitthe fit in Fig. 2.7 is not well constrained in the region between 500-1200 keV; however,

because the beams used in this experiment are considerably higher-Z than those used for the

determination of the spatial resolution in Fig. 2.7, the greater energy loss in the target due

to the Z2 term in Eqn. 2.2 will cause the uncertainty in the velocity to be a more dominant

contribution to the width than the uncertainty in the position resolution of GRETINA. Note

that extrapolating to lower energies using this function does not work as the fit function

increases exponentially. Instead the optimal width of the narrow component is assumed to

remain constant below the 351-keV transition included in the fit.

Figure 2.7: Determination of the position resolution for the narrow component of the peak
response, as described in the text.

A caveat with this approach is that it causes one to overestimate the peak-to-total con-

35

0.00.51.01.52.02.53.0Energy(MeV)1.52.02.53.03.54.0Resolution(mm)y=2.1e−0.1x+60.0e−10.2xFittedCurveDataFigure 2.8: Comparing the simulated full-energy peak for the narrow component of the peak
template and the one described in the text containing both a narrow and wide component
combined with the ratio 80%-20%, respectively. The dashed lines show the region of inte-
gration to determine the peak areas in this work, which accounts for 95% of the area of the
spectrum containing only the narrow component.

tribution for individual peaks. This is because counts that would traditionally be placed in

the background are now able to be accounted for in the wide tails, as seen in Fig. 2.8, where

a 1260-keV full-energy peak is shown using only a narrow component (blue) and both a nar-

row and wide component (tan). To account for this, the peaks fit to experimental data were

integrated in a range around the centroid to avoid including background counts in the peak

areas. This region was chosen to be 2 FWHM from the peak centroid in either direction, as

shown by the dashes lines in Fig. 2.8, which recovers 95% of the intensity for the simulated

full-energy peak if only the narrow component was included. Uncertainties from including

36

1200125013001350Energy(keV)02004006008001000Counts/keVOnly Narrow Narrow and Widemore or less of the fitted peak were included in the uncertainty estimates for all measured

peak intensities discussed later in Chapter 4.

2.3.5 Lifetime effects on peak centroids

The lifetime of decaying states has a significant effect on the observed peak shapes and

centroids for γ rays emitted in flight due to the sensitivity of the Doppler correction to

the velocity and position of the emitter. While these effects are also important for prompt

transitions, the use of a single velocity to optimize spectral quality when performing the

Doppler reconstruction for spectra containing primarily prompt transitions means that the

long-lived states will be Doppler corrected with a velocity that is too high. Longer-lived

states decay at lower velocities than prompt transitions due to the on-average loss of more

energy through interaction with more of the target material. Because the emission point will

occur further along the trajectory for such states, the emission angle will be underestimated,

leading to an underestimation of the transition energy for γ decays from isomeric states.

Consider the 188 mg/cm2 9Be target, which is about 1 mm thick. Assuming a velocity of

c = 0.33, the beam takes 10 ps to traverse the target. In this lifetime range, the uncertainty
v
due to the lifetime is dominated by the uncertainty of the energy loss in the target. For

lifetimes in the range 10-30 ps, the decay happens mostly after target, leading to a well-

known velocity due to the dta measurement and an angle uncertainty comparable to the

σ ≈ 10 mrad uncertainty from the position resolution of GRETINA. For the range between
≈ 30 ps−1 ns, the angle plays a dominant role in the energy determination due to uncertainty
concerning the decay position. Finally, for lifetimes longer than approximately 2 ns, the decay

starts to happen in the S800 spectrograph.

For short lifetimes, on the order of τ < 100 ps, the net effect is a shift of the peak centroids

37

Figure 2.9: The full energy peak for a 1260-keV transition simulated with various different
lifetimes. The peak is Doppler reconstructed using the expected velocity for the 1260-keV
transition in 70Ni, which has a known lifetime of τ = 1.5 ps. There are clear shifts at
relatively low lifetimes and the peaks become increasingly wider towards lower energies as
the lifetime increases.

towards smaller values and a low-energy tail extending from the peak. Beyond τ ≈ 100 ps,
the peak shape widens substantially. This is shown in Fig. 2.9, where a 1260-keV decay was

simulated assuming a number of different lifetimes, and then Doppler-corrected as if it was

the same lifetime (τ = 1.5 ps [11]) as the 1260-keV state in 70Ni. This change in peak shape

and centroid position will be important to arguments later in Chapter 4.

38

90010001100120013001400Energy (keV)05001000150020002500300035004000Counts / keV1.5 ps75 ps400 ps400 ps200 ps2.3.6 Energy calibration and efficiency determination

To identify peaks in γ-ray spectra, it is important to perform an energy calibration. The

energy calibration for GRETINA in this work was performed prior to the experiment using

radioactive sources. It was set prior to the experiment for every segment and crystal, with

the signal decomposition outputting calibrated energies.

Ultimately, after identifying peaks in the spectra, it was important to determine peak

areas and then correct these for the detection efficiency to deduce the transition intensities.

The quality of the spectra produced by GRETINA at the low γ-ray multiplicity for fast-

beam experiments is sufficient to determine the peak areas for the detected γ rays. It is then

possible to determine the γ-ray transition intensities from the experimental efficiency curve

(see Figure 2 in Ref. [38]), which was determined using radioactive sources and was corrected

here for the Lorentz boost using simulation. The γ-ray transition intensities determined in

this work are all presented in Chapter 4.

39

Chapter 3

Data analysis: calibrations and

corrections

In the following chapter, examples of the necessary corrections and calibrations mentioned

in Chapter 2 will be discussed in more detail with specific examples from the analysis of the

setting centered on incoming 71Cu. The analysis proceeded in similar ways for the 9Be(71Ni,

70Ni+γ)X and 9Be(72Zn, 70Ni+γ)X settings.

3.1 Particle identification in the entrance and exit chan-

nel for the one-proton knockout setting

The following provides an example of the particle identification process for the one-proton

knockout from 71Cu to 70Ni. The 71Cu is delivered as part of a "cocktail" beam of 95% purity,

with primary contaminants being 69Ni and 73Zn, and first must be identified. The incoming

beam spectrum for the one-proton knockout reaction setting with a red contour drawn around

the 71Cu component for a subset of the experimental data is shown in Fig. 3.1. Plotting

the obj and xf p flight times against each other separates the different constituents of the

incoming beam by their velocity difference. Some of the 69Ni and 73Zn beam contaminants

remain in this contour.

40

Figure 3.1: Incoming particle identification plot for the S800 setting centered on one-proton
knockout from 71Cu showing the xf p and obj time-of-flight distributions discussed in more
detail in Section 2.2.1. Outlined in red is the contour around incoming 71Cu projectiles.
Also contained in the contour are some 69Ni and 73Zn incoming beam contaminants.

After identifying the incoming 71Cu projectiles, the goal was to discern the 70Ni knockout

residues exiting the target from the other reaction products that enter the large-acceptance

focal plane of the S800 spectrograph. To this end, the obj time of flight had to be corrected

for the angle and position in the focal plane for each particle to be able to cleanly separate the

outgoing beam components. This trajectory correction for events gated on 70Ni projectile-

like knockout residues and 71Cu incoming projectiles is shown in Fig. 3.2. In this figure, the

correlation between the obj-E1 time-of-flight difference and the position in the first CRDC

(top) and angle in the focal plane (bottom) of the beam-like reaction product is shown. This

correction accounts for the fact that particles that are, for example, on trajectories with

larger angles will have a longer flight time. The corrected time-of-flight difference provides

a measurement of A

Z for completely stripped fragments in the spectrograph.

41

OBJ Time of FlightXFP Time of FlightFigure 3.2: Time-of-flight correction for position in the focal plane (top row) and angle
in the focal plane (bottom row), before and after corrections. This corrects for the path-
length dependence of the time-of-flight as discussed in Section 2.2.1. Note this spectrum was
already restricted to events containing incoming 71Cu projectiles and outgoing 70Ni knockout
residues for illustrative purposes. A projection of the time-of-flight axis can quantify the gain,
which is very visible already in the two-dimensional plot.

After correcting the time of flight, the outgoing particles were identified using the stan-

dard ∆E − T oF technique to separate particles based on Z2, from the energy loss measured
Z , from
in the ionization chamber [32] in the focal plane of the S800 spectrograph, and A

42

OBJ(arb.units)−300−200−1000100200300CRDC1X(mm)Before−300−200−1000100200300CRDC1X(mm)AfterOBJ(arb.units)−0.10−0.050.000.050.10AFP(rad)Before−0.10−0.050.000.050.10AFP(rad)AfterCorrected OBJCorrected OBJthe aforementioned time-of-flight difference. The outgoing particle identification spectrum is

shown in Fig. 3.3, where a red contour is drawn around the 70Ni outgoing reaction products.

Z gated
Figure 3.3: Outgoing particle identification plot presenting ∆E ∝ Z2 and T oF ∝ A
on incoming 71Cu projectiles. The obj-E1 time of flight shown here has been corrected for
the angle and position of the knockout residue in the focal plane. The presence of the hot
spots to the right of the 70Ni reaction products is due to incoming beam contaminants.

A small amount of contamination from neighboring nuclei was present even after gating

on the projectile-like 70Ni knockout residues for each knockout setting. This contamination

is evident as increased intensity at the edge of the distributions of angle and position in the

focal plane, as in Fig. 3.4, where the position distribution in the focal plane as measured by

the CRDCs as a function of time of flight are shown in (a)-(c) and the angle distribution

measured between the CRDCs restricted to the contamination in (a)-(c) is shown in (d).

The hot spots signaling the contamination are highlighted with red arrows in each spectrum.

To remove this contamination, which is primarily localized at the edge of the focal plane,

gates on the position and angle in the focal plane were necessary. Events that satisfy a

43

70Ni71CuΔE (arb. unit)Time of flight (arb. unit)69Ni73Zncombination of cuts on the contaminants are removed from the analysis.

In this way, it

was possible to remove selectively the contaminants and to determine the amount of 70Ni

knockout residues detected. This process was repeated for both the 71Ni and 72Zn incoming

beams.

Figure 3.4: Spectra used for removing outgoing beam contaminants, which are highlighted
by the red arrows. It is clear that the contaminants are localized at the edge of the focal
plane based on the measured x position in the CRDCs shown in panels (a-c). Gating on
these hot spots leads to the distribution of dispersive angles in the focal plane in panel (d),
where there is a clear excess of counts related to the contamination.

44

OBJ (arb. units)−300−200−1000100200300CRDC 1 X (mm)OBJ (arb. units)−300−200−1000100200300CRDC 2 X (mm)XFP (arb. units)−300−200−1000100200300CRDC 1 X (mm)OBJ (arb. units)−0.050.000.05AFP (rad)(c)(a)(b)(d)After successfully identifying the 70Ni knockout residues, it is then possible to observe

the γ rays tagging de-excitation from excited states of 70Ni using GRETINA.

3.2 Doppler

reconstruction example for one-proton

knockout setting

As discussed in Section 2.3, significant Doppler shifts are observed for the γ rays emitted in-

flight at mid-target velocities of v/c = 0.38, v/c = 0.37, and v/c = 0.35 for the one-proton,

one-neutron, and two-proton knockout reactions, respectively. To measure the intensities

and identify peaks in the spectrum, it is necessary to perform a Doppler reconstruction.

The Doppler correction formula is given in Eqn. 2.3.

In the following, the Doppler

correction process is explored using the 70Ni spectrum from one-proton knockout. The

Doppler correction process starts by using the expected particle velocity v/c at the mid-target

position and the γ-ray emission angle θ determined event-by-event by the angle between the

main interaction point, the interaction point associated with the largest energy deposition,

and a vector pointing straight downstream from the target;

i.e., the optical beam axis.

The following corrections rely on the aforementioned reconstructed trajectory from COSY

Infinity. The corrections are described in order from largest effect on resolution to smallest.

The first correction that is applied takes into account the dispersive and non-dispersive

angles of the particle at the target position.

Instead of determining the emission angle

relative to a vector pointing along the optical beam axis, the angle is determined relative to

the reconstructed particle trajectory at the target. The emission angle θ is corrected event-

by-event based on the determined ata and bta, the dispersive and non-dispersive angles at the

target position (see Section 2.2.2). In practice, this means the emission angle is determined

45

Figure 3.5: Before (left) and after (right) the φ correction, which is the correction for the
trajectory of the particle at the target position. The x axis is the angle between the reaction
plane and the detection plane, while the y axis is the detected γ-ray energy assuming the
mid-target velocity.

relative to a vector:

(cid:126)u =(cid:18)sin(ata),− sin(bta),(cid:113)1 − sin2(ata) − sin2(bta)(cid:19) ,

(3.1)

rather than the initial choice of (cid:126)v = (0, 0, 1), where ˆz is chosen to go along the optical

beam axis. The negative sign on the second component is due to the change in coordinate

systems between the S800 focal plane and GRETINA.

To visualize this correction, one can look at the Doppler-corrected energy as a function

of the angle between the detection plane and the reaction plane, as shown in Fig. 3.5. The

detection plane is defined by the optical axis and a vector determined by the coordinate of

46

090180270PlaneAngle(deg)1200122012401260128013001320γ-rayEnergy(keV)Before090180270PlaneAngle(deg)Afterthe first interaction point in GRETINA (see Section 2.3.2). The reaction plane is defined by

the incoming particle trajectory, assumed to be the optical axis, and the outgoing particle

trajectory as determined with the S800 inverse map and Eqn. 3.1. This correction is often

referred to as the φ correction. This plane angle φP can be determined by determining the

azimuthal angle φT of the particle at the target position using the inverse map of the S800

and the azimuthal angle of the detector φD in which the γ ray was detected as follows:

φP = 2π − φT − φD.

(3.2)

Because the reconstructed trajectory from the S800 is accurate to the order a few mrad,

the dominant remaining uncertainty in determining the emission angle stems from the σ ≈ 10
mrad uncertainty from the position resolution of GRETINA.

So far, the point of γ ray emission is considered to be a fixed point for all events. In

reality, the beam spot size of the incoming beam on the target exhibits a sizable spread,

especially in the non-dispersive direction. The width of the yta distribution for the 71Cu

setting was σ ≈ 3 mm, which is considerable when compared to the σ = 2 mm spatial
resolution of GRETINA. This non-dispersive position is measured event-by-event using the

inverse map of the spectrograph as yta, which can be used to further refine the emission

angle determination for the Doppler shift correction.

Figure 3.6, in which each pane corresponds to an individual crystal, clearly shows the

correlation between the non-dispersive position on the target and the Doppler reconstructed

energy after the φ correction for the forward detectors. The panes are ordered ring-by-ring

in the GRETINA coordinate system from smallest polar angle to largest, and within rings

from smallest azimuthal angle to largest, where φ = 90◦ and φ = 270◦ correspond to the

47

non-dispersive direction. Note that this effect is most prominent on the forward detectors;

i.e., the first fifteen panels. This is because the detectors at 90◦ are not sensitive to the

non-dispersive position on the target, as shifts in the non-dispersive position will not have

a significant impact on the polar angle determining the Doppler shift for the 90◦ detectors.

The spectra after correcting for this correlation are shown in Fig. 3.7.

After this correction, the dta value is used to correct the velocity of the particle. As

noted previously, this is a measurement of the kinetic energy after the target, and therefore

cannot correct for the uncertainty in the velocity due to the energy loss through the target.

The dta value is dT

T , where T = E − mc2 is the kinetic energy. The velocity is corrected

event-by-event by an amount (see the derivation in Appendix B):

dβ =

dT
T

β

γ(γ + 1)

,

(3.3)

where β = v

c and γ is the Lorentz factor. Figure 3.8 and Figure 3.9 show before and after

the correction in Eqn. 3.3 is applied.

Finally, after the event-by-event corrections based on the inverse map are completed,

the resolution of the transitions in the spectrum can be further improved by a set of global

parameters to correct for the incoming beam not being perfectly aligned with the optical

beam axis; i.e., correcting for offsets in the xy position of the beam spot and the beam

angle offsets a0 and b0. These offsets are systematically varied until the correlation between

Doppler-corrected energy and crystal position is minimized. The magnitude of the offsets

were determined to be x0 = 2.5 mm, y0 = 0.1 mm, a0 = 6.8 mrad, and b0 = 2.3 mrad. The

spectra following the different steps of the Doppler correction process is shown in Fig. 3.10.

The final v

c used for reconstruction was chosen to optimize the resolution of the high-

48

Figure 3.6: Before correction to remove correlation between γ-ray energy following the φ
and yta corrections. Each panel corresponds to a different crystal. Crystals are ordered
ring-by-ring, from smallest to largest polar angle, and then within rings from smallest to
largest azimuthal angle.

intensity 1260-keV 2+

1 transition in 70Ni. Combining all of these corrections made
it possible to achieve energy resolutions of 0.9%, 1.1%, and 1.2% FWHM for the high-

1 → 0+

intensity 1260-keV peak in the one-proton, one-neutron, and two-proton knockout reactions,

respectively. The worse resolution for one-neutron and two-proton knockout was due to the

49

122512601295122512601295122512601295122512601295Energy(keV)122512601295122512601295-10010122512601295-10010-10010Non-dispersivePositiononTarget(mm)-10010-10010Figure 3.7: After correction to remove correlation between γ-ray energy following the φ and
yta corrections. Each panel corresponds to a different crystal. Crystals are ordered ring-
by-ring, from smallest to largest polar angle, and then within rings from smallest to largest
azimuthal angle.

use of a thicker 188 mg/cm2 9Be target. The use of the thicker target leads to increased

luminosity, but also increases the energy loss through the target, which is the largest source

of uncertainty in the determination of the velocity of the particle at the γ-ray emission

point. Now that the particles are identified and the Doppler reconstruction is complete, it

50

122512601295122512601295122512601295122512601295Energy(keV)122512601295122512601295-10010122512601295-10010-10010Non-dispersivePositiononTarget(mm)-10010-10010Figure 3.8: Before correction to remove correlation between γ-ray energy following the φ and
yta corrections and the particle’s kinetic energy after the target. Each panel corresponds to
a different crystal. Crystals are ordered ring-by-ring, from smallest to largest polar angle,
and then within rings from smallest to largest azimuthal angle.

is reasonable to start identifying more transitions and measuring their intensities.

51

122512601295122512601295122512601295122512601295Energy(keV)122512601295122512601295-0.0500.05122512601295-0.0500.05-0.0500.05DTA-0.0500.05-0.0500.05Figure 3.9: After correction to remove correlation between γ-ray energy following the φ and
yta corrections and the kinetic energy after the target. Each panel corresponds to a different
crystal. Crystals are ordered ring-by-ring, from smallest to largest polar angle, and then
within rings from smallest to largest azimuthal angle.

3.3 Determining transition intensities

To measure the transition intensities for 70Ni, it was necessary to fit the spectrum with a

fit template including the peak response for all observed transitions as well as a background

template, composed of both an exponential to model the beam-correlated background [34] as

52

122512601295122512601295122512601295122512601295Energy(keV)122512601295122512601295-0.0500.05122512601295-0.0500.05-0.0500.05DTA-0.0500.05-0.0500.05Figure 3.10: Spectra showing the gradual improvement of the resolution of the Doppler-
corrected 1260-keV γ ray in 70Ni following each of the different corrections described in the
text.

well as simulated stopped background lines, which are present due to the inelastic reactions

between neutrons and other light particles with both the germanium detectors and the

aluminum beam pipe. These stopped lines had a considerably smaller effect compared with

the exponential background and the peak response of observed transitions in 70Ni. The

scaling factor for each stopped line was fixed by fitting each individual transition in the non-

Doppler reconstructed spectrum, and the resulting sum of the spectra from each stopped

line was included as a fixed part of the background model.

Shown in Fig. 3.11 is the region between 1500 and 2000 keV, where there were a number

of γ rays observed in the one-proton knockout setting. This fit template included all the

53

1200125013001350Energy(keV)0500100015002000Counts/keVInitial Doppler CorrectionAfter MCorrectionAfter yta+dta CorrectionAfter final correctionBefore Doppler CorrectionFigure 3.11: Spectrum observed in coincidence with 70Ni knockout residues showing different
components of the fit template in the region 1500-2000 kev.

peaks shown and an exponential background. The scaling factor for each peak included in

the template as well as the scaling and decay parameters for the exponential were allowed

to vary. It was important to utilize multiple different fitting ranges when determining the

peak areas, because the Compton background of peaks at higher energies with significant

intensity contribute background to peaks at lower energies. By varying this fitting range

and including other transitions, one can determine a systematic fitting uncertainty for each

determined peak area by observing the change in peak areas.

After fitting the spectrum, the full-energy-peak response from simulation was scaled by

the scaling factor determined by the fit and integrated in a region of width 4 FWHM centered

54

DataFull FitExponential Background1584 keV1682 keV1787 keV1868 keV1955 keV15001600170018001900Energy(keV)050010001500Counts/keVon the peak, as described in Section 2.3.4. After determining this peak area, it was divided

by the efficiency, which is discussed in Section 2.3.6, to deduce the measured transition

intensity. The final-state populations can then be determined by summing up the γ-ray

transition intensities for all decay branches from a given state and then subtracting any

intensity feeding into that state from de-excitations of higher-lying levels.

To compare the results of the different knockout reactions, it is important to ensure that

the deduced populations are not affected by the existence of long-lived states in the incoming

beams. This is important because these isomeric states will likely exhibit different single-

particle configurations than the ground-state and this can lead to drastically different final-

state populations following nucleon knockout. Hence, the isomeric content of the incoming

beams needed to be measured.

3.4 Isomeric content of incoming beams

Because both 71Cu and 71Ni are known to exhibit isomers in their level schemes, it was

necessary to determine the amount of isomeric content for states with lifetimes exceeding a

few hundred nanoseconds in the incoming beams, which would reach the reaction target of

the S800 before decaying. This is necessary because knockout from the incoming projectile’s

excited states, which will have configurations different from the ground state, will lead to

different final-state populations in 70Ni, and may even access states that are not directly

accessible via nucleon knockout from the ground state. To determine the isomeric content, a

5.1 mm thick Al stopper was placed at the center of GRETINA, which was used to measure

the γ rays emitted from the long-lived states. GRETINA is free running during these

measurements, such that every γ ray in GRETINA is recorded and the coincidence between

55

implantation and the detected γ ray is accomplished by comparing event time stamps.

For these measurements, the particle trigger is switched from the E1 scintillator in the

S800 focal plane to the obj scintillator, so that each incoming particle triggers the data

acquisition system. The xf p-obj time-of-flight difference can be used to identify the incoming

particle of interest, and the number of incoming particles corrected for the beam purity within

this time-of-flight difference gate determines the number of implanted particles.

Figure 3.12: Spectrum in coincidence with 71Cu projectiles implanted into an aluminum
stopper at the center of GRETINA. The area of the peak at 133 keV, which corresponds to
the de-excitation from the long-lived (19/2−) state in 71Cu, was measured relative to the
number of implants to estimate the isomeric content in the beam.

In the case of 71Cu, there is a known (19/2−) state at 2756 keV with mean lifetime

56

100150Energy(keV)0100200300Counts/keVImplanted71Cuτ = 391(20) ns [11]. First, the number of particles was determined by the number of triggers

from the obj scintillator, corrected for the 95% beam purity for this setting. No time-of-flight

difference restriction was possible, because the 69Ni and 73Zn components of the incoming

cocktail beam largely overlapped in time of flight with the 71Cu.

The incoming beam rate was on the order of 1.1 kHz, meaning that the resulting low γ-

ray decay activity would compete with natural background radiation. To reduce the natural

background in order to measure the number of decays from the (19/2−) isomer, only γ-ray

events within 1 µs of implantation were considered. The intensity of the 133-keV γ ray

tagging the de-excitation from this state in Fig. 3.12 is determined by fitting the peak with

a simple Gaussian peak shape and a linear background to determine the peak area and then

correcting this for the detection efficiency of GRETINA.

Because the 133-keV γ ray was low in energy, the absorption inside the 5.1 mm aluminum

stopper needs to be taken into consideration. To account for this, a simulation was performed

with UCGretina (see Section 2.3.4) including the aluminum plate at the center of GRETINA.

The γ ray was assumed to be emitted at the center of the aluminum plate, which is the

position that would maximize absorption, to estimate the number of transitions that would

be absorbed and correct the measured transition intensity. The transition intensity corrected

for loss in the aluminum stopper is then divided by the number of implanted particles to

determine the isomeric content, which for incoming 71Cu was Ic = 0.47(7)%.

The isomeric content for the 71Ni cocktail beam is more difficult to measure, because the

long-lived (1/2−), 499-keV state, which has a mean lifetime of τ = 3.3(4) s [11], does not γ

decay. Instead, it β decays into either the 3/2(−) ground state or the (1/2−) excited state at

454 keV of 71Cu [41]. The intensity of the 454-keV γ ray tagging the de-excitation from this

state in Fig. 3.13 is determined by fitting the peak with a simple Gaussian peak shape and

57

a linear background to determine the peak area and then correcting this for the detection

efficiency of GRETINA.

The xf p-obj time difference was particularly important for this setting in determining

the number of 71Ni ions implanted, because there is a large amount of 72Cu in the incoming

cocktail beam. After applying a gate to the xf p-obj time-of-flight difference, the beam purity

satisfying the gate was determined to be 88%, compared to 63% without any time-of-flight

restriction. The number of particle triggers determined with the obj scintillator within this

gate was corrected for this beam purity.

Figure 3.13: Spectrum in coincidence with 71Ni projectiles implanted into an aluminum
stopper at the center of GRETINA. The area of the peak at 454 keV, which corresponds
to the de-excitation from the long-lived (1/2−) state in 71Cu to which the isomeric (1/2−),
499-keV level in 71Ni β-decays, was measured relative to the number of implants to estimate
the isomeric content in the beam.

Unlike the case of 71Cu, the long τ = 3.3(4) s lifetime of the isomeric state in 71Ni meant

that using a timing restriction to minimize natural background was impossible, so the 454-

keV peak had to be discriminated on top of the natural background. Analysis was limited

58

440460480500Energy(keV)010002000300040005000Counts/keVImplanted71Ni454 keVto events occurring after the first 6.9 s, three mean lifetimes of the decaying state, which

reduces the amount of background in the spectrum from other shorter-lived decays.

Because the adopted NNDC β-decay branching ratio to the excited (1/2−) state is an

upper limit of 40%, the lower limit of the isomeric content of the incoming 71Ni beam based

on the observed intensity of the 454-keV, (1/2−) → 3/2(−) transition is Ic > 6%. This value
takes into account the absorption of the γ ray in the thick aluminum stopper in the same

manner as described above for 71Cu.

59

Chapter 4

Results: γ-ray spectroscopy of 70Ni

following nucleon knockout

In the following chapter, the results for each knockout reaction will be presented, including

γγ coincidence spectra used for placing transitions into the level scheme and measured in-

tensities for both placed and unplaced transitions. The results for the measured transition

intensities relative to the number of detected 70Ni knockout residues for each reaction are

shown in Table 4.1 for placed transitions and Table 4.2 for transitions that remain unplaced,

respectively. All intensities are determined from γ-ray singles spectra unless otherwise noted.

The number of 70Ni was determined for each setting based on the number of triggers sat-

isfying the particle identification gates corrected for the dead time of the data acquisition

system.

All fits to γ-ray spectra used to extract transition intensities included the simulated peak

responses determined with GEANT4 [39] (see Sec. 2.3.4), along with an exponential used

to model the beam correlated background and simulated stopped background lines, which

are present due to the inelastic reactions between neutrons and other light particles with

both the germanium detectors and the aluminum beam pipe. The quoted uncertainties are

described in more detail in the caption for Table 4.1. A systematic energy uncertainty of 2.2

keV was determined by comparing known transition energies from the NNDC database [11]

60

Figure 4.1: The level scheme for 70Ni based on the transitions observed in each reaction
channel for this experiment. Transitions that are placed for the first time into the level scheme
are shown in red, while dashed lines indicate tentative placements. The levels highlighted in
orange are expected to exhibit proton-particle hole configurations.

to the peak energies observed in the experiment. The level scheme for 70Ni based on the

observed transitions in each reaction channel is given in Fig. 4.1.

To determine the coincidence relationships between different transitions in the spectra,

nearest-neighbor add-back was utilized. This is described in more detail in Section 2.3.3.

All γγ coincidence spectra presented in this chapter exploited nearest-neighbor addback to

improve the peak-to-total ratio and therefore make it possible to place more transitions.

4.1 One-proton knockout from 71Cu

In the one-proton knockout reaction, states in the yrast band and the suspected prolate

sequence are populated, along with several higher-lying states of undetermined spin-parity.

A total of 26 previously unplaced transitions were observed in this reaction channel, and

61

1075Newly PlacedTentativeStates with Proton Particle-Hole StructuresTable 4.1: Measured intensities for all observed, placed 70Ni transitions relative to the num-
ber of detected 70Ni residues in the specific reaction channel expressed as a percentage. The
error includes both statistical uncertainties and systematic contributions from varying the
fit assumption for peak shapes and backgrounds, and a 2.1% uncertainty from the efficiency
determination. Note that no feeding subtraction is attempted here.

State

Transition

Rel. Intensities (%) Rel. Intensities (%) Rel. Intensities (%)

Energy (keV) Energy (keV)

1p Knockout

1n Knockout

2p Knockout

21(1)
2.4(4)
2.5(5)
7.8(8)
0.5(2)
2.2(4)
1.0(6)
0.6(2)
1.6(5)
1.0(9)
0.7(5)
1.3(5)

0.16(5)

-

-
-
-
-
-

-
-
-
-
-
-

0.4(3)

1.2(6)

1260(2)
1868(2)

2230(3)
2508(4)

2912(4)

2942(4)
3215(4)
3297(5)
3551(5)
3588(6)
3662(4)

3814(3)
3846(5)
3961(4)
4017(4)

4293(4)

1260(2)
609(2)
1868(2)
970(2)
279(3) a
640(2)
1249(3) b
234(4) c
682(3) d
1682(3)
1955(3)
385(3)
1321(4)
676(4)
2402(3)
1432(4)
1584(2)
3846(5)
2701(3)
1075(3) e
1787(3)
2757(3)
479(3)
631(4)
2063(3)
3033(5)

44(2)
3.6(4)
3.5(6)
16.5(9)
0.5(2)
2.2(3)
2.4(4)
0.4(1)
1.1(3)
1.3(3)
2.1(2)
1.3(2)
0.7(1)
0.12(3)
0.8(2)
0.3(2)
4.4(4)
0.7(1)
1.6(3)
0.9(5)
1.8(6)
1.4(2)
0.7(2)
0.9(2)
0.75(16)
1.2(2)

14.5(8)
1.9(3)
1.3(5)
5.7(7)
0.3(2)
1.2(6)
1.7(3)
1.5(4)
3.9(6)

0.8(3)
2.6(7)

-

-
-
-
-
-
-
-

-
-
-
-
-
-

aThe expected intensity for the 279-keV transition is below the detection sensitivity in the one-neutron
and two-proton reaction channels, so the intensity for these channels was estimated based on the branching
ratio measured in one-proton knockout

bThe discrepancy between the different measurements of the 1249-keV transition intensity relative to the
640-keV one is due both to the difficulty of determining the intensity in a self-coincident doublet and to
possible lifetime effects as discussed in Section 4.2.
cThe intensities given for the 234-keV γ ray for the one-proton and two-proton knockout are determined

based on the NNDC adopted branching ratio [11].

dences for the one-proton knockout. This splitting was then used for the two-proton knockout as well.

dThe splitting between the self-coincident 676-682 keV doublet is determined based on add-back coinci-
eThe non-observation of the 1787- and 2757-keV transitions in the one-neutron and two-proton reactions

is understood based on the small, highly uncertain intensity observed for the 1075-keV γ ray.

62

Table 4.2: Measured intensities for all observed, unplaced 70Ni transitions relative to the
number of detected 70Ni residues in the specific reaction channel expressed as a percentage.
See the caption to Table 4.1 for more information.

Rel. Intensities (%) Rel. Intensities (%) Rel. Intensities (%)

1p Knockout

1n Knockout

2p Knockout

Transition

Energy (keV)

424(5)
660(3)
714(4)
915(5)
930(5)
958(5)
1212(5)
1225(5)
1440(3)
1467(3)
2026(3)
2114(3)
2342(5)
2980(4)

0.6(4)

0.9(3)
0.33(25)
0.6(3)

-

-

-
-

0.8(3)
0.5(2)
1.2(2)
0.8(2)
0.5(2)
0.4(3)

0.6(3)
1.2(11)

1.5(4)
0.9(3)
1.8(5)

1.7(4)

0.6(3)

-

-

-
-
-

-
-

0.7(3)
1.2(4)
1.4(9)

1.1(5)

2.2(5)

-
-
-

-
-
-

-
-
-

16 of these transitions were placed into the level scheme for 70Ni in Fig. 4.1. There was

a combination of insufficient statistics, resolution, or coincidence relationships to place the

remaining transitions. The fit used to extract the intensities of the transitions is presented

in Fig. 4.2. This fit was composed of simulated peak responses for all peaks in the spectrum

as well as a background composed of double-exponential component to model the beam-

correlated background and simulated stopped background lines, which are present due to

the inelastic reactions between neutrons and other light particles with both the germanium

detectors and the aluminum beam pipe.

The levels populated in the proton-knockout channel are presumably positive parity ones

based on the available orbitals from which protons can be removed from 71Cu. Population

of negative-parity states at low excitation energies in the proton knockout reaction would re-

quire indirect feeding from higher-lying positive parity ones. This is because negative-parity

63

Figure 4.2: An example of a fit used to extract intensities from the 70Ni spectrum populated
via one-proton knockout from 71Cu. The fit consisted of simulated line shapes for all peaks
observed in the spectrum as well as a background composed of a double exponential to model
the beam-correlated background in addition to stopped background lines from both inelastic
reactions of neutrons and other light particles with the aluminum beam pipe or Ge detectors
as well as annihilation radiation. Clearly present in the spectrum are the 970-keV, 4+
1 → 2+
1
1 transitions in the yrast sequence along with many other transitions
and 1260-keV, 2+
discussed in the text.

1 → 0+

levels would imply knockout from the sd shell for the proton knockout reaction channels.

The states populated via the knockout of sd-shell protons would be expected to appear at

energies above 6 MeV in the spectrum [42].

Within the suspected prolate structure, the majority of previously known transitions

were observed, with the notable exception of any transitions into or out of the (0+

2 ) isomeric

64

100020003000Energy(keV)1000300050007000Counts/8keV71Cu-1pv/c=0.376Fit20003000100400700state. Unfortunately, the measured lifetime [17] of the suspected 0+

2 level excluded any
1 , 307-keV transition, as it would decay after entering
the S800 entrance quadrupole but well before implanting within the aluminum plate in front

possibility of measuring the (0+

2 ) → 2+

of the hodoscope. In addition to the previously known transitions, two new transitions were

observed and placed in the possible prolate sequence: a (4+

1 , 1249-keV transition
1 , 279-keV transition. The coincidences between the yrast transitions and
these two new transitions in the prolate structure are present in the coincidence spectra in

2 ) → 2+

2 ) → 4+

and a (4+

Fig. 4.3(a) and (b), respectively. Also present in Fig. 4.3(a) are other previously reported

keV, (2+

2 ) → 2+

transitions within the prolate structure: the 640-keV, (4+

2 transition and the 609-
1 decay was also observed both in the γ-ray
2 transition
(Fig. 4.4(c)). The spectra in coincidence with the transitions in the prolate structure is

singles spectrum and in the spectrum of γ rays in coincidence with the (4+

1 one. The 1868-keV, (2+

2 ) → 2+

2 ) → 0+

2 ) → 2+

presented in Fig. 4.4. The 279-keV and 1249-keV transitions were not previously observed in

the two-neutron knockout [3] reaction that detected the 640-keV de-excitation from the same

state. This is possibly because, in the case of the 279-keV transition, there is a poor peak-

to-background ratio in this low-energy region of the spectrum. The 1249-keV de-excitation

is part of a self-coincident doublet with the high-intensity 1260-keV one, making it difficult

to observe without a large amount of statistics. While the 1260-keV transition appeared to

be self-coincident in Ref. [3] in the two-neutron knockout reaction channel, they were unable

to determine the exact energy of the self-coincident γ ray. No evidence of the 1256.8-keV

self-coincident transition observed in β decay from the low-spin isomer of 70Co in Ref. [43]

was observed in this experiment.

The majority of the new 16 transitions placed into the level scheme de-excited higher-

lying states of unknown spin-parity into states in the yrast sequence. The strongest of

65

Figure 4.3: Sample of coincidence spectra used for placing transitions into Fig. 4.1 for the
one-proton knockout reaction. Clear coincidences between the 1260-970-1584 keV transitions
are observed, making it possible to place the 1584 keV transition in the level scheme. Also
shown are a weaker 480 keV transition believed to feed into the newly placed 3814-keV state
as well as several of transitions in the suspected prolate structure at 279 keV, 609 keV, 640
keV, and 1249 keV.

these transitions was a 1584-keV one, which de-excited a newly-placed 3814-keV level. The

coincidence relationships used to place this transition are presented in Fig. 4.3. The spectrum

in coincidence with the 1260-keV, 2+

1 → 0+

1 ground-state transition in panel (a) exhibits a

66

0150300450(a)Gatedon126071Cu-1p609640970124915840150300Counts/8keV(b)Gatedon97027912601584050010001500Energy(keV)050100(c)Gatedon15844809701260Figure 4.4: Coincidence spectra used for placing transitions in the suspected prolate structure
in the 71Cu one-proton-knockout reaction channel. Fig. 4.3 contains the associated spectra
in coincidence with transitions in the yrast sequence.

67

255075(a)Gateon609keV71Cu-1p640126015304560(b)Gateon1868keV640153045Counts/8keV(c)Gateon640keV60912601868050010001500Energy(keV)153045(d)Gateon279keV1260970clear coincidence between this transition and the 1584-keV transition. The 970-keV, 4+

1 → 2+
is also clearly coincident with the 1584-keV transition based on panels (b) and (c). Placing

1

and 2+

1 → 0+

it did feed the 6+

this 1584-keV de-excitation as feeding into the long-lived 6+

the 1.51(4) ns lifetime [44] of this yrast state would make it impossible to observe the 4+

1 state can be ruled out because
1 → 2+
1
1 de-excitations below it in the cascade in coincidence with the 1584-keV one if
1 state, as seen
1 state and the available
orbitals from which protons can be removed, it is possible that the 3814-keV level is a (5+
1 )
particle-hole configuration. A weaker, 480-keV de-excitation was

1 level. The 3814-keV state appears to only decay to the 4+

in Fig. 4.1. Because of the large amount of direct feeding to the 4+

state with a π1p+1
3/2

0f−1
7/2

observed in coincidence with the 1584-keV transition (see Fig. 4.3(c)), and was placed into

the level scheme as being one of four possible decay branches from a newly placed 4293-keV

state.

The 4293-keV and 4017-keV states are two newly proposed levels observed in the one-

proton knockout. These exhibit considerably more branching to other states when compared

to the aforementioned 3814-keV one, including strong, direct transitions to the 2+

1 state.

These possible decay branchings suggest that these states have spin I ≤ 4.

The coincidence spectra for transitions from the 4293-keV and 3662-keV states are shown

in Fig. 4.5. The strongest decay branch from the 4293-keV state is to the 2+

1 level, with
1 level, and a 3662-keV one
of unknown spin-parity. This suggests that the 4293-keV state has possible spin-parities

three marginally weaker transitions to the (5+) level, the 4+

of (3+,4+). Based on similar arguments, the spin-parities of the 3662-keV and 4017-keV

levels could be (2+,3+,4+), because in this case no transition connecting the (5+) and these

states is present to rule out the possibility of a 2+ assignment. The spectra coincident with

transitions from the 4017-keV and 2942-keV levels are shown in Fig. 4.6. The 2942-keV level

68

Figure 4.5: Coincidence spectra used for placing decays from the 4293-keV and tentative
3662-keV levels in the level scheme of Fig. 4.1 for the 71Cu one-proton-knockout reaction
channel.

69

1020(a)Gateon3033keV71Cu-1p1260102030(b)Gateon631keV1260970143224025101520Counts/8keV(c)Gateon2402keV63112605101520(d)Gateon1432keV631126097051015(e)Gateon2063keV970050010001500Energy(keV)153045(f)Gateon479keV97012601584Figure 4.6: Coincidence spectra used for placing decays from the 4017-keV and 2942-keV
levels in the level scheme of Fig. 4.1 for the 71Cu one-proton-knockout reaction channel.

de-excites by a 1682-keV transition, which may correspond to a previously observed 1676-keV

transition in the β decay of the (1+, 2+) state of 70Co [18, 45]. The source of the discrepancy

70

1020(a)Gateon1787keV71Cu-1p97012601020(b)Gateon2757keV1260102030Counts/8keV(c)Gateon1075keV12601682050010001500Energy(keV)102030(d)Gateon1682keV12601075between the measured centroids is unclear. The 631-keV, 1075-keV, 1432-keV, and 2402-keV

transitions are admittedly tentative placements due to the small measured intensities and

weak coincidence relationships. Further complicating the placement of the 1075-keV γ ray

is its position near the Compton edge of the high-intensity 1260-keV line.

The 1955-keV transition only appears to exhibit a coincidence relationship (see

Fig. 4.7(a)) with the 1260-keV ground-state transition. This decay appears in all reaction

channels, though it is strongest in the one-proton knockout reaction. It has been previously

observed in a β decay measurement [45] following decay from the (1+, 2+) state of 70Co

and from two-neutron knockout [3]. Similar to the 1955-keV transition, the 2701-keV γ ray

appears only in coincidence with the 1260-keV transition, as seen in Fig. 4.7(b). The 1321-

keV transition exhibits a weak coincidence relationship with both the 970-keV and 1260-keV

transition, as shown in Fig. 4.7(c). The 1321-keV and 2701-keV transitions had not been

previously observed.

The 3846-keV transition was previously reported following the β decay of the low-spin

isomer in 70Cu, and was tentatively placed as feeding into the suspected prolate, 1868-keV,

2 state [46]. No coincidence relationships with this 3846-keV state were observed in this
2+
experiment. In fact, this transition was observed almost exclusively in events with detector

multiplicity-1. Fig. 4.8 shows the spectra for events with detector multiplicity-1, -2, and -3 or

higher. This absence of coincidence information and dominance in the detector multiplicity-1

spectrum supports the placement of this state as feeding into the ground state; however, it

is possible that this state could decay into the isomeric (0+

2 ) state at 1567 keV. It is unlikely
1 yrast
levels, as the 3846-keV transition was observed in the β decay of the (1+, 2+) state of 70Co.

that it would decay into the other known, higher-spin long-lived states, the 6+

1 and 8+

For comparison, presented in Fig. 4.9 is the 1260-keV transition with this same breakdown by

71

Figure 4.7: Coincidence spectra used for placing decays the 1955-keV, 2701-keV, and 1321-
keV transitions in the level scheme of Fig. 4.1 for the 71Cu one-proton-knockout reaction
channel.

72

1020(a)Gateon1955keV71Cu-1p12601020(b)Gateon2701keV1260050010001500Energy(keV)1020(c)Gateon1321keV9701260Figure 4.8: Spectra utilizing nearest-neighbor add-back from the one-proton-knockout re-
action where the different lines correspond to different detector multiplicities. Note that
the multiplicity is the corrected multiplicity after add-back; i.e., if an event only contains
two hits and these are combined, it is a detector multiplicity-1 event. The dominance of
the 3846-keV transition in the detector multiplicity-1 spectrum suggests that it could be a
transition directly into the ground state, or some long-lived, low-spin state, as discussed in
the text.

detector multiplicity. Because the 1260-keV transition is expected to appear in each possible

detector multiplicity here, it is unsurprising that it is observed clearly in Fig. 4.9 in each.

As mentioned previously, all the transitions discussed so far for the one-proton knockout

have been assumed to be from positive parity states, as negative-parity states are expected

to be populated at much higher energies in this reaction channel. Surprisingly, a transition

near 682 keV corresponding to the decay from the (5−), 2912-keV state was observed in

the proton-knockout reaction channel. As discussed in Section 4.2, this is possible due to

73

34003600380040004200Energy (keV)050100150200250300350Counts / 16 keVMult. = 1Mult. = 2Mult.  > 2Figure 4.9: Spectra utilizing nearest-neighbor add-back from the one-proton-knockout reac-
tion where the different lines correspond to different detector multiplicities. Note that the
multiplicity is the corrected multiplicity after add-back; i.e., if an event only contains two
hits and these are combined, it is a detector multiplicity-1 event. The appearance of the
1260-keV transition in each detector multiplicity spectrum is expected based on the number
of different ways it can be populated in the one-proton knockout (see Fig. 4.1).

feeding from higher-lying states. This explanation is consistent with the aforementioned

expectation that negative-states populated in the proton-knockout reaction at excitation en-

ergies below 6 MeV must be fed by positive-parity levels. The 682-keV transition appears

to be a self-coincident doublet with a 676-keV transition. This was previously observed in

the β decay of the high-spin (6−, 7−) state in 70Co [18]. Only the sum of the counts in

each peak could be determined by fitting the γ-ray singles spectrum because of insufficient

resolution to resolve the peaks. Instead, the number of coincident transitions between the

74

1200125013001350Energy(keV)0100020003000400050006000Counts/4keVMult. = 1Mult. = 2Mult.  > 2two is determined utilizing nearest-neighbor addback by fitting the spectrum in coincidence

with the two transitions and correcting for the NNDC adopted branching ratio [11] from

the (5−) state. The add-back factor was determined by simulating the 683-keV transition

and determining the efficiency gain in nearest-neighbor addback compared to γ-ray singles.

Because of the poor peak-to-background in the region, the intensity of the 234-keV decay

branch could not be measured directly in the proton-knockout reactions. Instead, the afore-

mentioned branching ratio was used to estimate the intensity of the 234-keV γ ray based on

the measured 682-keV transition intensity. It was necessary to determine this intensity to

determine the feeding-subtracted state populations used later in Section 5.1 to compare the

different knockout reaction channels.

As noted in the previous paragraphs, the one-proton-knockout reaction can only populate

negative-parity states at energies below 6 MeV, such as the 2912-keV (5−) level, through

indirect feeding from positive-parity states. Assuming the 676-keV transition and the 385-

keV transition, which will be discussed in the following section, de-excite positive parity

states, the population of the (5−) level can be entirely attributed to indirect feeding in the

proton-knockout reaction channels.

4.2 One-neutron knockout from 71Ni

The one-neutron knockout reaction channel exhibited a different distribution of strength

among final states than the proton-knockout one. The most interesting differences relative

to the one-proton knockout are the decrease in the relative intensity of decays associated

with the prolate structure, which is clear from Table 4.1, and increase both in the feeding-

subtracted state population of the previously discussed (5−) level and in the relative intensity

75

Figure 4.10: The Doppler-corrected γ-ray spectrum observed in coincidence with the 70Ni
residues after one-neutron knockout from 71Ni. Clearly present in the spectrum are the
970-keV, 4+
1 transitions in the yrast sequence. Also visible
are the previously unplaced 385-keV transition as well as the 682-keV γ ray associated with
decay from the (5−) state.

1 and 1260-keV, 2+

1 → 2+

1 → 0+

of the previously-unplaced 385-keV transition. These feeding-subtracted state populations

are discussed in more detail in Section 5.1. The Doppler-corrected γ-ray spectrum observed

for the one-neutron knockout reaction is shown in Fig. 4.10.

The 385-keV transition is the strongest unplaced peak in the one-neutron knockout re-

action.

It was present in both proton-knockout reaction channels, albeit weakly in the

two-proton knockout one, and in the two-neutron knockout channel in a previous measure-

76

500100015002000Energy(keV)20070012001700Counts/8keV71Ni-1nv/c=0.365Figure 4.11: Coincidence spectra for the 385-keV, the 676-682 keV doublet, and the 234-keV
transitions from the sum of the one-proton and one-neutron knockout reaction channels.
Due to low statistics, it is only possible to tentatively place the 385-keV transition. The
displacement of the yrast energies in panel (b), as indicated by the dashed lines at the
expected yrast energies, is discussed in the text.

ment [3]. The 385-keV transition is placed based on the coincidence spectra in Fig. 4.11,

where the coincident spectra from the one-proton and one-neutron knockout reactions are

added together. The two-proton knockout channel was not included in the sum as the 385-

keV intensity was considerably smaller for that reaction. In Fig. 4.11(a), the spectrum in

77

0204060(a)Gateon385keV1pKO+1nKO2346820204060Counts/8keV(b)Gateon676+682keV385676+682970126004008001200Energy(keV)04080120(c)Gateon234keV385coincidence with the 385-keV transition appears to be coincident with the 234-keV and 682-

keV transitions from the (5−) state. The 676-keV transition, which is part of a self-coincident

doublet with the 682-keV transition in the one-proton knockout reaction, is not observed in

the one-neutron knockout channel.

The limited statistics in the coincidence spectra in Fig. 4.11 as well as the position of the

957-keV and 1247-keV peaks in Fig. 4.11(b), which are suspected to be the 970-keV and 1260-

keV transitions in the yrast sequence, complicated the placement of the 385-keV transition.

This shift of roughly 13 keV for these suspected yrast transitions from their expected energies

is shown by the dashed lines at the expected transition energies in Fig. 4.11(b). Offsets in

energy for Doppler-corrected γ-ray spectra can be caused by lifetime effects, as discussed

1 and 2+

in 70Ni through the 4+

in Section 2.3.5. To explore this possibility, a simulation of a cascade from the (5−) state
1 levels to the ground state was performed. The lifetime of
the (5−) state was varied and the resulting γ-ray spectrum was observed. To account for
such an energy shift by a lifetime effect would require τ (5−) ≥ 75 ps. These offsets can
account for some of the intensity of unplaced transitions such as the 957-keV transition in

Table 4.2 as well as part of the intensity of the 1249-keV transition, which was placed into the

prolate structure based on the self-coincidence in the one-proton knockout reaction channel

(see Fig. 4.3). The statistics in the one-neutron reaction channel was insufficient to observe

the expected coincidence relationship between the 1249-keV and the 1260-keV transitions.

Due to the presence of isomeric states in the level scheme for 70Ni, some coincidence

relationships cannot be observed. This is likely the reason some of the unplaced transitions

in Table 4.2 remain unplaced. For example, the 915-keV de-excitation observed is the same

energy as a known transition into the isomeric 6+

statistics to identify any coincidence relationships. This is because the 2678-keV, 6+

1 state; however, there are insufficient
1 (τ =

78

1.51(4) ns [44]) state decays within the bounds of GRETINA, but considerably after the

target. As discussed in Section 2.3.5, this leads to a deformation of the Doppler-corrected γ-

ray spectrum due to incorrect determination of velocity and emission angle. For states with

even longer lifetimes, like the 2861-keV, 8+

1 (τ = 335(1) ns) [11] state, the reaction residue
would not decay before entering the S800 focal plane, making detection with GRETINA

of the γ ray tagging the de-excitation impossible.

Instead, it would be necessary to rely

on gating on the detection of the 183-keV γ ray that signals the de-excitation from the

isomer in the hodoscope (see Sec. 2.2.3). The hodoscope spectrum taken for the one-neutron

knockout reaction channel is shown in Fig. 4.12. Unfortunately, although the 183-keV γ

ray and subsequent cascade from the aforementioned isomeric state are observed, there were

insufficient statistics to measure any coincidences between decays from the long-lived state

in the hodoscope and prompt decays in GRETINA indicating feeding into this state.

A shell-model calculation was performed using the jj44pna [6] interaction for 70Ni with

a model space allowing only neutron excitations. The model space included the neutron

0f5/2, 1p3/2, 1p1/2, and 0g9/2 orbitals with the restriction that there were a minimum of
two neutrons in the 0g9/2 orbital. The comparison between the experimental level scheme
in Fig. 4.1 restricted to states observed in the one-neutron knockout and the results of

the calculation are shown in Fig. 4.13. Listed to the right of the states is the shell-model

spectroscopic factor for the knockout of a neutron from 71Ni for all states with energies less

than 4.5 MeV satisfying the restrictions on C2S noted in the figure caption. The absence of

the predicted prolate structure in the calculation is expected due to the absence of proton

cross-shell excitations in the model space. The spectroscopic strength is highly fragmented to

higher-lying, negative-parity levels, with minimal strength to positive parity states above the

1 yrast level. Due to this fragmentation of higher-lying strength primarily to negative-parity
8+

79

Figure 4.12: Gamma-ray spectrum from the hodoscope taken in coincidence with outgoing
70Ni reaction residues following one-neutron knockout from 71Ni.

states, it is possible that the unplaced transitions observed in the one-neutron knockout

channel are de-excitations from negative-parity levels.

4.3 Two-proton knockout from 72Zn

Because of the existence of isomeric states in the level schemes of the 71Cu and 71Ni beams

that would affect the comparison between the final-state populations, the two-proton knock-

out reaction channel from 72Zn, which exhibits no known long-lived level, was utilized. The

80

50010001500Energy(keV)050100150200250300350Counts/32keV1834489701260population distribution was similar to the one-proton knockout, as primarily decays from

the yrast sequence and the suspected prolate levels were observed. The Doppler-corrected

γ-ray spectrum from this reaction channel is shown in Fig. 4.14.

The 682-keV and 676-keV transitions cannot be separated in this reaction channel, similar

to the situation in one-proton knockout (see Section 4.1). Furthermore, due to the consider-

able decrease in statistics relative to the one-proton knockout channel, the splitting between

the two transitions cannot be determined in this setting using the coincidence relationship

between the two decays. Instead, the splitting observed in the one-proton knockout channel

is utilized.

It is possible that this estimate of the intensity splitting is incorrect for this

reaction channel. Using this splitting between the two decays, the expected intensity of the

676-keV transition in coincidence with the 682-keV transition would be below the experimen-

tal sensitivity for this reaction. Assuming there were no 676-keV decays in this setting would

only increase the intensity for the 682-keV transition by 10%. Therefore, results reported

in Table 4.1 utilize the assumption that the splitting between the intensities of the 676-keV

and 682-keV is the same as that from the one-proton reaction.

A difficulty unique to the two-proton reaction channel in determining the intensity of

the 682-keV transition is the presence of a structure with a large excess of counts near 700

keV, which is visible in Fig. 4.14. Based on the line shape, it is possible that this structure

corresponds to either a 714-keV decay from a state with a significantly long lifetime or a

number of unresolved transitions. Assuming this structure is a de-excitation from a state

with a 75-ps lifetime leads to a 21% decrease in the measured intensity of the 682-keV

transition compared to assuming that the 714-keV peak is a prompt transition. This large

uncertainty is reflected in the measured intensities in Table 4.1 and Table 4.2 of the 682-

and 714-keV transitions, as well as the that of the 676- and 234-keV lines, which are derived

81

from the intensity of the 682-keV de-excitation.

82

Figure 4.13: Comparison of experimental level scheme for states populated in the one-neutron
knockout reaction channel to results of a traditional shell-model calculation only allowing
neutron excitations using the jj44pna [6] effective interaction. For the positive-parity states,
g9/2 shell-model spectroscopic factors C2S for one-neutron knockout from the 71Ni ground
state to individual 70Ni final states are listed when C2S(0g9/2) > 0.1. For the negative-parity
levels, the 0f5/2, 1p3/2, 1p1/2 shell-model spectroscopic factors are listed when C2S > 0.02
for states where the sum C2S(0f5/2) + C2S(1p3/2) + C2S(1p1/2) > 0.1.

83

Energy (keV)ExperimentTheory J+Theory J-0+0+2+2+4+6+8+4+6+8+(4+)2+(5-)5-71Ni-1n0+2+4-3-3-4-4+0.04 (0f5/2 ), 0.88 (1p1/2 )0.04 (0f5/2 ), 0.88 (1p1/2 )0.29 (0f5/2 ), 0.23 (1p3/2 )0.29 (0f5/2 ), 0.23 (1p3/2 )0.07 (0f5/2 ), 0.04 (1p3/2 )0.07 (0f5/2 ), 0.04 (1p3/2 )0.66 (1p1/2 )0.66 (1p1/2 )0.30 (0f5/2 ) 0.30 (0f5/2 ) 0.70 (0f5/2 ) 0.70 (0f5/2 ) 0.47 (0f5/2 ), 0.04 (1p1/2 )   0.47 (0f5/2 ), 0.04 (1p1/2 )   0.90 (0f5/2 ) 0.90 (0f5/2 ) 0.45 (0f5/2 ) 0.45 (0f5/2 ) 0.11 (0f5/2 ) 0.11 (0f5/2 ) 0.30 (0f5/2 ) 0.30 (0f5/2 ) 0.08 (0f5/2 ), 0.06 (1p3/2 ) 0.08 (0f5/2 ), 0.06 (1p3/2 ) 0.11 (0f5/2 ) 0.11 (0f5/2 ) 0.27 (1p3/2 ) 0.27 (1p3/2 ) 0.19 (1p3/2 ) 0.19 (1p3/2 ) 0.18 (1p3/2 ) 0.18 (1p3/2 ) C2S (0g9/2) 1.060.640.630.240.65Figure 4.14: The Doppler-corrected γ-ray spectrum observed in coincidence with the 70Ni
residues after two-proton knockout from 72Zn. Clearly present in the spectrum are the 970-
keV, 4+
1 transitions in the yrast sequence. Also visible are
the 609-keV, 640-keV, and 1868-keV γ rays associated with the predicted prolate structure.

1 and 1260-keV, 2+

1 → 2+

1 → 0+

84

500100015002000Energy(keV)100200300400500Counts/8keV72Zn-2pv/c=0.352Chapter 5

Comparison of reaction channels

The primary aim of this work was to probe the underlying single-particle configurations for

excited states in 70Ni by comparing the final-state population distributions following different

reaction mechanisms. To this end, the γ-ray spectra and final-state, feeding-subtracted state

populations will be compared in the following sections.

5.1 Comparison of γ-ray spectra

The Doppler-corrected γ-ray singles spectra detected in coincidence with 70Ni beam-like

knockout residues are compared in Fig. 5.1, where the Doppler-corrected singles spectra are

presented normalized by the ratio of reaction residues in each setting to that of the two-proton

knockout. There are clear differences between the spectra following the different knockout

reactions, signaling differences in the in the population of excited final states following the

different reactions. There is a much stronger relative intensity of the 385-keV transition in

the one-neutron knockout channel compared to the proton-knockout ones. Multiple peaks,

such as the 479- and 1584-keV γ rays, are only present in the spectrum from the one-proton

knockout. The region between 600-700 keV exhibits structures with considerably different

intensity distributions between the different reaction channels. While the behavior near 200

keV is dramatically different in the one-proton knockout, it was not possible to determine

any coincidence relationships with the structure here due to the poor peak-to-background

85

Figure 5.1: Comparison of Doppler-corrected γ-ray singles spectra in coincidence with de-
tection of 70Ni knockout residues. The spectra are scaled based on the detected number of
70Ni residues in each setting relative to the number detected in the two-proton knockout
channel. The scaling factors are listed in the legend.

86

30080013001800Energy(keV)100200300400500600Counts/4keV1p*0.0821n*0.3542p385Multiplet1584480ratio at this energy. The peak-like structure lines up with the energy expected to be in

coincidence with the Compton edge of the strong 1260-keV transition.

5.2 Comparison of feeding-subtracted state populations

To probe the differences in the single-particle configurations of the states associated with the

prolate structure, the direct population of states is explored in Fig. 5.2. In this figure, the

intensities were first corrected for observed feeding from higher-lying levels. Then, these were

turned into relative intensities by dividing by the number of 70Ni reaction residues detected

in the reaction channel being considered. Finally, the differences in these feeding-subtracted

relative intensities divided by their sum was used to compare the one-proton and one-neutron

channels (circles) and the two-proton and one-neutron ones (squares). In this way, points

above (below) the line correspond to states which exhibited more direct feeding from proton

(neutron) knockout. Note that only levels that were populated in each reaction channel

are shown in Fig. 5.2. One notable exclusion is the 3214-keV level, which was dominantly

populated in the one-proton knockout.

Figure 5.2 suggests that the prolate states were populated more strongly in the proton

knockout channels, which is consistent with the picture that the prolate levels are associated

with cross-shell proton excitations. Although states in the yrast sequence also appear to have

a larger direct population following proton knockout, this can also be explained by either

incomplete feeding subtraction due to unplaced transitions or neutron knockout populating

both long-lived positive parity states, such as the 6+

1 levels, and higher-lying
negative-parity states which decay into these isomeric ones. Furthermore, the populations

1 and the 8+

are consistent with the expectation that the negative-parity states, such as the (5−) one, can

87

Figure 5.2: Comparison of feeding-subtracted state populations between the proton knockout
and neutron knockout. The populations were divide by the number of 70Ni knockout residues
detected in the channel under consideration. Plotted is the difference over the sum of these
relative populations comparing both proton knockout reactions with neutron knockout.

88

123ExcitationEnergy(MeV)−1.0−0.50.00.51.0(Rp−Rn)/(Rp+Rn)2+12+24+1(cid:0)4+2(cid:1)(cid:0)5−1(cid:1)J(+)1p-1n2p-1nProlateYrastNegativeParityUnknownJπonly be directly populated in the neutron knockout reaction.

Marked as the J (+) state in Fig. 5.2, the newly-placed, 3297-keV level, which de-excites

by the 385-keV transition, appears to be populated primarily in the neutron-knockout reac-

tion. This transition, as well as a 676-keV one, were previously speculated in Ref. [3] to be

candidates for the 2+

2 transition because these γ rays were observed in the two-neutron
knockout reaction channel but absent in the deep-inelastic scattering data. The placement

2 → 0+

of the 385-keV transition as feeding the (5−) state contradicts this idea. Furthermore, the

676-keV reconstructed transition energy observed in Ref. [3] is compatible with a 682-keV γ

ray emitted from a long-lived (τ ≥ 75 ps) state. Reference [18] placed the the (0+
at 1567 keV, resulting in an expected energy of 301 keV for the 2+

2 ) state
2 transition. These
observations along with the observed coincidence relationships with the 682-keV, (5−) → 4+
transition makes the association with the proposed prolate structure unlikely. Unfortunately,

2 → 0+

1

at this level of statistics, the suspected small branching ratio due to competition with the

higher-energy 1868-keV and 609-keV transitions precluded any detection of the 2+

2 → 0+

2

one in this measurement.

89

Chapter 6

Summary and future work

In agreement with expectations, the population of the observed levels in 70Ni following

one- and two-nucleon knockout reactions was found to depend on the reaction channel. By

observing the difference in final-state populations following the different nucleon knockout

reactions, it was possible to determine which states exhibited configurations dominated by

either proton or neutron excitations. As previously suggested [3], the proposed prolate

structure appears to be dominantly populated through cross-shell proton excitations across

the Z = 28 shell gap. Based on its relatively high intensity feeding the 4+

1 state and the
available orbitals from which protons can be removed, the level at 3813 keV also appears

to exhibit proton particle-hole character, and has been tentatively assigned a spin-parity of

(5+).

The previously observed [3] (5−) level as well as the newly-proposed tentative 3297-keV

one that decays through the 385-keV γ ray are populated more strongly in the one-neutron

knockout, suggesting that these are associated with neutron excitations. Two γ rays, the 676-

and 385-keV transitions, which were previously speculated [3] as possible candidates for the

1 decay based on their appearance in two-neutron knockout and absence in deep-
(0+
2 ) → 2+
inelastic scattering, appear to be unrelated to the suspected prolate structure based on the

results of this work. There were 14 observed yet unplaced transitions, which may feed into

the several isomeric states of 70Ni. In total, 16 transitions, including two transitions within

90

this proposed prolate structure, and 7 levels were added to the level scheme of 70Ni. Due to

the high level densities involved, the presence of several isomers in the level scheme of 70Ni,

the mostly unknown or tentative final-state quantum numbers, and the complex associated

configurations, it was unfortunately not feasible to probe the spectroscopic factors in this

measurement.

For the determination of the isomeric content in the incoming 71Ni beam, the precision

of the result will be improved when more precise results on the β-decay branching ratio

from the isomeric state of 71Ni are available. Because the current values from the NNDC

database [11] are only upper limits on this branching ratio, it is only possible to determine

the lower limit of the isomeric content of the incoming 71Ni secondary beam (see Section 3.4).

A more precise measurement would make it possible to definitively determine the amount of

isomeric content, and therefore determine the amount that long-lived states in the incoming

71Ni beam can affect the final-state populations.

The prolate nature of the previously-mentioned states is inferred solely from shell-model

calculations [16], and therefore one avenue for future inquiries would be a direct, experimental

determination of the quadrupole properties of the suspected prolate states; e.g., using sub-

barrier Coulomb excitation. Sub-barrier Coulomb excitation, which involves inducing nuclear

excitations through the electromagnetic field acting between nuclei, makes it possible to

measure the quadrupole moment. The magnitude of the quadrupole moment quantifies the

amount of deformation and its sign determines whether a state is oblate or prolate. This

measurement will require next-generation facilities like the Facility for Rare Isotope Beams

(FRIB), as reaccelerated 70Ni can currently only be produced at a rate of a few particles per

second, many orders of magnitude less than for the fast-beam experiments described in this

work.

91

Another complementary approach for probing the structure of the low-spin states in 70Ni

would be to use the two-neutron transfer reaction (t,p) in inverse kinematics as was recently

done for 68Ni [47]. It would be possible to determine the angular momentum transfer of the

reaction based on the observed angular distribution for the protons. Therefore, because the

excitation energy of 70Ni would be accessible based on the energy and detection angle of

the recoiling proton, the underlying configurations of different excited states in 70Ni could

be probed. In particular, it may be possible to observe population of the 0+

3 state near 2.5
MeV, which was predicted with traditional shell-model calculations [3] with a model space

only permitting neutron excitations. Such an experiment will likely only be possible at FRIB

due to the required beam intensity of reaccelerated 68Ni.

Determining the properties of nuclei with more extreme neutron-proton asymmetry will

be a major goal of nuclear physics moving forward. Approaching the limits of nuclear stabil-

ity, the doubly-magic nucleus 78Ni will be one major focus of future investigations. Already,

it is predicted that 78Ni exhibits shape coexistence, with a first excited state of spin-parity

0+, and acts as a "portal" to a new island of inversion at N = 50 [20]. These predictions have

been supported by experimental results from (p, 2p) and (p, 3p) reactions [21]. Performing

experiments to probe the properties of nuclei beyond 78Ni at the extremes of the nuclear

chart will become possible with FRIB.

92

APPENDICES

93

APPENDIX A

INTERMEDIATE-ENERGY

COULOMB EXCITATION OF 88,90KR

AND 86SE

Using nucleon knockout, it was possible to probe shape coexistence and the configurations

associated with the excited states in 70Ni using the single-particle degree of freedom. A

complementary approach to this type of problem is Coulomb excitation, which looks at

the collective degree of freedom directly. Collectivity refers to the participation of multiple

nucleons to excitation modes in nuclei, with the most common examples of collective motion

in nuclei being vibrational or rotational. Two indicators of quadrupole collectivity for even-

even nuclei are the energy of the first excited 2+ state and, more importantly, the electric

quadrupole transition strength from the ground state to the first excited 2+ state;

i.e.,

B(E2; 0+

1 → 2+

1 state energy or a large B(E2) value
compared to neighboring nuclei along an isotopic chain signals the onset of deformation. In

1 ), introduced in Section 1.1.2. A low 2+

an effort to probe collectivity beyond the N = 50 shell closure in neutron-rich krypton and

selenium isotopes, an intermediate-energy Coulomb excitation was performed to measure the

B(E2; 0+

1 → 2+

n ) values in 88,90Kr and 86Se. The results of this measurement were published

in Physical Review C [48].

94

This measurement resulted in considerably reduced uncertainties for the quadrupole tran-

sition strength in 90Kr and 86Se, as well as the first published result for this value in 88Kr.

There was considerable fragmentation of the strength to higher-lying 2+ levels observed for

the krypton isotopes. This fragmentation confirmed the results of a shell-model calculation

in the π(f pg) − ν(sdgh) shell with an only minimally-tuned effective interaction based on a
nucleon-nucleon interaction derived from effective field theory and effective charges adjusted

to reproduce the B(E2) value in singly-magic 86Kr.

Experimental motivation

Neutron-rich nuclei in the Se, Kr, Sr, and Zr isotopic chains all exhibit an onset of defor-

mation when charting their properties from N = 50 − 60; however, this onset manifests
differently for the different chains. For 38Sr and 40Zr, there is a rapid increase in collectivity

at N = 60, which is evident in Fig. A.1 by the large increase in the transition strength and

in Fig. A.2 by the noticeable decrease in the 2+

1 level energy. In the context of 100Zr, the
phenomena has been referred to as a quantum phase transition [49–51]. This rapid onset of

deformation can be explained by the downsloping πg9/2 intruder orbit, which becomes more
bound due to the proton-neutron tensor force as neutrons are added above the N = 50 shell

closure.

In comparison, the 34Se and 36Kr isotopic chains were expected to exhibit a more grad-

ual evolution of collectivity; however, in the case of 90Kr, the large uncertainties for the

measured values precluded making definitive statements concerning collective behavior. For

88Kr, the adopted value suggested a slight decrease in collectivity with two neutrons added

above the shell closure, which is a surprising result. Comparably little is known concerning

95

Figure A.1: Adopted B(E2; 0+
1 ) values from Ref. [8], except for the value for 98Zr
and 86Se. In the case of the former, the value is based on a more recent Coulomb excitation
experiment [9] that determined the upper bound for the strength, where previously only a
lower limit had been reported. In the case of the latter, the value is taken from a recoil-
distance Doppler shift measurement [10].

1 → 2+

the collective properties of neutron-rich Se beyond N = 50, with information limited to

excitation energies in 86−94Se [52–54], a measurement of the B(E2 ↑) value in 84Se using
Coulomb excitation [55], and lifetimes of excited states in 84,86Se measured using the recoil-

distance Doppler shift technique [10]. To determine B(E2) values with considerably lower

uncertainties compared to previously measured values and investigate the possible decrease

of collectivity in 88Kr, we used intermediate-energy Coulomb excitation. Before discussing

this, we will do a quick aside on the units and conventions used for reporting B(E2) values.

96

3436384042444648505254565860Neutron Number0.511.5B(E2; 0+→ 2+) (e2 b2)40Zr38Sr36Kr 34SeFigure A.2: Adopted energies from the NNDC database [11] for the 2+
even-even nuclei.

1 level for the discussed

Units and conventions for B(E2) values

It is important to note that electromagnetic transition strengths B(πλ), where π ∈ E, M
and λ = 0, 1, 2, 3... and which are also known as reduced transition probabilities, are often

reported using a variety of different units and conventions. Focusing on B(E2) values, which

are the most relevant quantities when discussing intermediate-energy Coulomb excitation of

even-even nuclei, these will be covered briefly here. For more detail, see Ref. [13].

First, it is important to note that the strengths can be written in terms of either excitation

97

343638404244464850525456586062Neutron Number500100015002000250021+ Energy (keV)40Zr38Sr36Kr34Se(B(E2 ↑)) or decay (B(E2 ↓)), and the relationship is:

B(J1 → J2) =

2J2 + 1
2J1 + 1

B(J2 → J1)

Therefore,

B(E2; 0+

1 → 2+

1 ) = 4B(E2; 2+

1 → 0+
1 )

(A.1)

(A.2)

In this work, all transition strengths are reported as excitations;

i.e. B(E2) =

B(E2; 0+

1 ).
1 → 2+

In terms of units, two types will be used in this work:

in the case of

Fig. A.1, the units used are e2b2, which can be converted to other commonly used unit

e2f m4 by noting that 1 barn = 100 fm2, and therefore 1 b2 = 10000 f m4. Weisskopf units

(W.u.), also called single-particle units, are commonly used units that are an estimate of

the mass-dependent decay strength for a single neutron or proton. Although it is at best

an approximation, one can think about the number of Weisskopf units as the number of

nucleons contributing to a decay. So a large number of Weisskopf units suggests a highly

collective decay, while a smaller number would suggest single-particle behavior. One B(E2)

Weisskopf unit for a nucleus of mass A is defined as:

Bs.p.(E2) =

9

100π(cid:16)1.2A1/3(cid:17)4

e2f m4

(A.3)

So, to determine the number of Weisskopf units for a given measured B(E2 ↓), simply
calculate B(E2 ↓)/Bs.p.(E2). Note that these units are only used for decays, not excitations.

98

Intermediate-energy Coulomb excitation

Intermediate-energy Coulomb excitation [56–58] involves sending a projectile of interest at

a high-Z target. The projectile is excited in the strong electric field of the target, and

then de-excites by emitting a γ ray. By measuring the intensity of the γ rays tagging the

electromagnetic interaction, the angle-integrated Coulomb excitation cross section can be

determined. The electric quadrupole transition strength is then determined using the Alder-

Winther theory of relativistic Coulomb excitation [12], which relates B(E2) values to the

cross section.

It is important to note that the energies used in intermediate-energy Coulomb excitation

are above the Coulomb barrier. Because of this, analysis must be restricted to peripheral

collisions to avoid nuclear contributions to the cross section. This is accomplished experi-

mentally by restricting analysis to events with a sufficiently small scattering angle; i.e., a

large impact parameter. The choice of the touching-spheres approximation + 2 fm has been

shown to be sufficient to ensure that the nuclear contribution is minimized [58–60]. The

touching spheres approximation is given by:

bmin > R0(A1/3

targ + A1/3
proj)

(A.4)

where R0 = 1.25 fm is the nuclear radius factor. Through kinematics [12], the minimum

impact parameter bmin is translated into a maximum scattering angle and used to restrict

the data.

At the relativistic beam energies used for intermediate-energy Coulomb excitation, mul-

tistep excitations are greatly hindered, unlike in classic low-energy Coulomb excitation. This

results in only 2+ states connected to the ground state by a sufficiently strong branch being

99

populated in the experiment. Typically, this means only the 2+

1 level is populated; however,
in this experiment, we observed significant strength to higher-lying 2+ states in 88,90Kr. We

determine the angle-integrated Coulomb excitation cross sections for each state based on

the γ-ray intensities of the de-exciting transitions relative to the number of projectiles and

number density of the target. This complicates the typical intermediate-energy Coulomb

excitation analysis, in which only the first 2+ state is observed, because the feeding through

1 transitions has to be subtracted to determine the cross section to the 2+

n → 2+
2+
branching ratios for the decays to the ground state and to the 2+

1 one, and
n levels

1 level from the 2+

have to be taken into account.

Experimental details and devices

The experiment was performed at the Coupled Cyclotron Facility in the National Super-

conducting Cyclotron Laboratory [27] at Michigan State University. A stable 96Zr primary

beam was accelerated to 120 MeV/nucleon by the K500 and K1200 coupled cyclotrons. The

primary beam impinged on a 364, 374, or 255 mg/cm2 9Be production target to produce

88Kr at 67 MeV/nucleon, 90Kr at 68 MeV/nucleon, and 86Se at 79 MeV/nucleon, respec-

tively, through projectile fragmentation. An aluminum wedge degrader with areal density

200 mg/cm2 located at the mid-acceptance position of the A1900 fragment separator [29]

was used to select the fragments of interest from the secondary beam cocktails.

The S800 spectrograph

The selected fragments were delivered to the analysis line of the S800 spectrograph [30].

At the target position of the spectrograph, a 246 mg/cm2 209Bi foil was used to induce

100

Figure A.3: Particle identification plot for the setting centered on 88Kr35+ in the focal plane.
The beam components are cleanly separated and identifiable.

projectile Coulomb excitation. Projectile identification and trajectory reconstruction were

accomplished using the focal plane detectors of the S800 [32]. Particle identification was

accomplished using the standard ∆E − T oF technique. Trajectory reconstruction was par-
ticularly important to measure the scattering angle of the projectiles, which is necessary as

noted above to ensure there is no nuclear contribution to measured cross sections. The S800

focal plane detectors are described in more detail in Section 2.2. The outgoing scattered

particle identification plot for the 88Kr secondary beam is shown in Fig. A.3, where the

energy loss is plotted versus the flight time from the obj scintillator. It was possible to select

the 88Kr outgoing projectiles separately from the other components of the cocktail beam,

namely 86Br and 90Rb. The scattered ions assumed a charge-state distribution following

passage through the bismuth target. In the case of the Kr isotopes, the H-like charge state

was the most intense and was centered in the S800 focal plane, in contrast to 86Se, in which

case the fully-stripped charge state was strongest and centered in the spectrograph. The

101

Time of flight (arb. unit)ΔE (arb. unit)86Br88Kr90Rbtypical S800 focal-plane rates were 1350 particles per second (pps) for 88Kr35+, 350 pps for

90Kr35+, and 20 pps for the fully stripped 86Se particles, with incoming A1900 momentum

acceptances of 0.5%, 1%, and 1%, respectively.

Refining the maximum scattering angle cut

The scattering angle was determined event-by-event using the position information in the

focal plane of the S800 and an inverse map generated with COSY Infinity [35]. A signifi-

cant, asymmetric angle emittance was observed in this experiment, and it was necessary to

account for this in the determination of the maximum-scattering angle. For each isotope,

the maximum scattering angle determined from the touching sphere + 2 fm minimum im-

pact parameter served as a starting point. These initial maximum scattering angles were

θmax = 56, 53, and 45 mrad in the laboratory frame for 88Kr, 90Kr, and 86Se, respectively.

We performed Monte Carlo simulation of the angular distribution for inelastic scattering

by using the Alder-Winther [12] formalism to generate the input probability distribution over

the range 0 ≤ θ ≤ θmax with the assumption of a sharp angle cutoff in a simplistic black-
disk model; i.e., all events with angles beyond θmax are assumed to be absorbed. Figure A.4

shows the simulated angular distribution assuming either no angular emittance (red curve)

or the experimental angle emittance and straggling determined with LISE [61] from at-most

elastically-scattered projectiles passing through the target. Close to the initial maximum-

scattering-angle cutoff, it is clear that one loses a fraction of counts corresponding to the

difference between the red and blue curves.

Comparing the simulated results with the experimental data (black histogram), which

was restricted to events in which only a 2+

1 decay was observed, the simulation
including the realistic beam emittance and straggling shows good agreement. To avoid

1 → 0+

102

Figure A.4: Inelastic scattering distribution in the case of 88Kr. The red and blue lines are
Monte Carlo simulations of the inelastic scattering process based on an input probability
distribution from the Alder-Winther relativistic Coulomb excitation model [12], either with
(blue) or without (red) the effects of angle emittances and angular straggling. The dashed
green line shows the chosen, more restrictive scattering-angle cut for the analysis of 88Kr.

potentially underestimating the angle-integrated cross sections, and the resulting B(E2)

value, a more conservative maximum-scattering-angle cut, indicated by the dashed green

line, was employed. This lead to maximum-scattering angles of 42 mrad, 35 mrad, and 30

mrad in the laboratory frame for 88Kr, 90Kr, and 86Se, respectively. This was similar to a

previous analysis [62], where it was shown that, away from the maximum-scattering angle

103

1020304050607080Scattering angle (mrad)255075100125Counts / 1.6 mradcorresponding to impact parameters where flux was removed from the (in)elastic channel,

the effect of angle emittance and straggling was minor due to an approximate balance of

events that scattered into and out of the angle cut applied to the data.

CAESAR: the CsI(Na) scintillator array

Surrounding the 209Bi target was CAESAR [63], the high-efficiency CsI(Na) scintillator

array. CAESAR is composed of 192 closely packed scintillation crystals that cover almost

4π in a cube-like geometry. Because of its high granularity, the array allowed for an event-

by-event Doppler reconstruction of the γ rays emitted in flight by the scattered projectiles.

The γ ray emission angle for the Doppler reconstruction is deduced using the position of the

CsI(Na) crystal that registered the largest energy deposition.

Compared to GRETINA, which is discussed in Section 2.3, CAESAR has a much higher

in-beam detection efficiency, exceeding 35%, at the cost of a considerably worse energy

resolution of approximately 10% FWHM for 1 MeV γ rays emitted in flight at v/c = 40% [63].

The corrections utilizing the inverse map beyond the φ correction described in Section 3.2

were not utilized, as they would have a negligible effect due to the thickness of the target.

The energy-dependent photopeak efficiency of the setup was determined using 88Y, 22Na,

60Co, and 137Cs standard calibration sources. A GEANT4 simulation was used to model the

in-beam response of the array, including the Lorentz boost of the emitted γ-ray distribution.

The GEANT4 efficiency curve was scaled to the measured efficiencies. Taking into account

the uncertainties in the activity of each sources, an overall efficiency uncertainty of 5.4% was

determined.

104

Isotope-by-isotope results

Figure A.5: (a)-(c) Doppler corrected γ-ray spectra including the fits to extract the angle-
integrated Coulomb excitation cross sections for each isotope. The fit template based on the
GEANT4 simulations is described in the text. (d) The spectrum of the beam contaminant
90Rb with a double exponential fit to model the prompt, beam-correlated background, using
an exclusion region in the fit around the peaks near 500 and 1400 keV. The resulting double
exponential adequately reproduces the behavior in the region of interest, which is from 600
keV to 1.7 MeV.

The Doppler-corrected γ-ray spectra observed with CAESAR are presented in Fig. A.5.

The final results for the measured angle-integrated Coulomb excitation cross sections and

B(E2) strengths for all isotopes and transitions observed in this work are presented in

Table A.1. In the following sections, the results are discussed isotope-by-isotope, including

105

80012001600Energy (keV)75150225Counts / 32 keV(c) 86SeTotal FitDouble Exp BGGeant4 Simulation150300450(b) 90Kr400800120080012001600300600900(a) 88Kr v/c = 0.372v/c = 0.345v/c = 0.341(d) 90Rbthe observed level schemes and assumptions that underlie the fitting process.

Table A.1: Measured cross sections, σ, and transition strengths, B(E2), for all isotopes stud-
ied in this work, as well as transition strengths from the shell-model calculation described in
Section IV are shown. The uncertainty includes both statistical uncertainties and systematic
contributions from varying the background model, as well as a 5.4% uncertainty from the
efficiency determination.

AZ

88Kr

90Kr

86Se

Ji

0+
1
0+
1
2+
2
0+
1
2+
3
0+
1
0+
1
0+
1
2+
2
0+
1
0+
1
0+
1
0+
1

Jf

2+
1
2+
2
2+
1
2+
3
2+
1
(2+
4 )
2+
1
2+
2
2+
1
(2+
3 )
2+
1
2+
2
2+
3

σ (mb)

210(30)
14(3)

-

50(8)

-

11(3)
180(30)
40(10)

-

17(13)
230(30)

-
-

B(E2; Ji → Jf ) (e2fm4)
shell model
measured

1310(190)

90(20)

≤ 2010(450)

320(50)
≤ 71 1
70(20)
1500(230)
330(90)
360(110)
150(110)
2110(320) 2

-
-

1450
80
440
450
69
170
1740
580
280
230
1910
150
340

88Kr

The primary fit template for 88Kr was composed of two simulated peak responses: the 775

keV transition from the 2+

level at 2216 keV to the 2+

1 state to the ground state and the 1440-keV transition from the 2+
3
1 one, both of which are visible in Fig. A.5(a). Two more feeding
1 decay shown in
Fig. A.6: one at 802 keV and another at 1644 keV. The relevant level scheme for transitions

transitions were observed in the spectrum coincident with the 2+

1 → 0+

observed in coincidence with the detection of scattered 88Kr projectiles is shown in Fig. A.7.

In Fig. A.6, the spectrum of γ rays detected in coincidence with the 775-keV transitions

are shown with a "multiplicity-2" condition applied; i.e., exactly two of the 192 CAESAR

106

Figure A.6: Spectrum coincident with the observation of the self-coincident 775 keV-802
keV doublet of γ rays, including a fit to extract the intensity of the feeders. A detector
multiplicity-2 condition is applied as explained in the text.

detectors registered an event within a prompt time gate, with one of these being the afore-

mentioned 775-keV transition. Using this multiplicity condition, we selective reduce the

beam-correlated background. Adding events with higher detector multiplicities primarily

adds background to the coincidence matrix. It is clear that there is a self-coincident doublet

composed of the 775-keV, 2+

1 transitions. Due to the modest
energy resolution of CAESAR, it was not possible to observe the 802-keV decay from the

1 → 0+

1 and 802-keV, 2+

2 → 2+

1577-keV, 2+

2 level in the γ singles spectrum. Furthermore, there is a γ ray at 1644 keV, which
is consistent with a decay from the the 2419-keV, (3−1 ) level known in the literature [11]. It

107

80012001600Energy (keV)20406080Counts / 32 keV88Krv/c = 0.341Total FitDouble Exp BGGeant4 Simulation.775/802 keV1440 keV1644 keVFigure A.7: Level scheme for transitions observed in coincidence with scattered 88Kr projec-
tiles. The given branching ratios are the adopted values from the NNDC database [11]. The
1 branching ratio is based on the present
dashed line giving an upper limit for the (2+
work.

4 ) → 0+

108

was not possible to observe this transition in γ singles due to insufficient statistics.

Because the 2+

1 decay
to be present with a 26.34(24)% branching ratio [11]. This line is not visible in Fig. A.5(a)

1 transition was observed, one expects the 1577-keV, 2+

2 → 0+

2 → 2+

as it is below the sensitivity limit of the spectrum, but we use the known branching ratio

quoted above to determine the cross section σ(0+

2 ) based on the observed intensity
of the 802-keV γ ray. Both the 802-keV and 1644-keV transition intensities were extracted

1 → 2+

from the fit in Fig. A.6. Because the 775-keV and 802-keV peaks form a doublet within

our resolution, both were included in this fit with the constraint that each would occur with

equal intensity, as every 802-keV transition in multiplicity 2 is necessarily in coincidence with

a 775-keV transition. These intensities, along with the intensity of the 1440.5-keV γ ray were

subtracted from the measured intensity of the 775-keV line. In this subtraction, the 802-keV

transition had to be accounted for twice: as it is included in the peak area of the 775-keV

transition, and it is indistinguishable in the fit of Fig. A.5. The ratio of the feeding-transition

intensities to the total measured 775-keV one was determined to be 25(4)%.

The measured cross sections and deduced transition strengths for the transitions in 88Kr

are summarized in Table A.1. We note that a sub-barrier Coulomb excitation measurement

reported in conference proceedings [65] and a thesis [66] proposed a B(E2; 0+

1 → 2+

1 ) =

900(90) e2f m4, which is approximately 2σ lower than our measured value.

Using the B(E2; 0+

1 → 2+

2 ) value measured here for the first time and the known branch-

ing ratio, we could extract the transition strength to the 2+

1 ) =
δ2
1+δ2 2010(450) e2f m4, where δ is the unknown E2/M 1 multipole mixing ratio. Table A.1
quotes the upper limit of the strength as B(E2) = 2010(450) e2f m4.

1 state as B(E2; 2+

2 → 2+

Only the strong 2+

tion cross section, σ0+

3 → 2+
1 → 2+

1 branch was observed from the 2+
3 level. The Coulomb excita-
3 was deduced using the known branching ratio of 17.1(12)% [11]

109

relative to the 2+

3 → 2+

1 → 2+

1 transition. The resulting absolute measurement of B(E2; 0+

1 →
3 ) = 320(50) e2f m4 is consistent with the rather uncertain previous measurement of
2+
3 ) = 400(200) e2f m4 from relativistic Coulomb excitation, which is quoted
B(E2; 0+
s tate using the earlier-mentioned result
−0.05 [64]
N and

published solely in proceedings [65]. The E2/M 1 multipole mixing ratio δ = 0.08+0.09

and the known branching ratio [11], we determined B(M 1; 2+

in Ref. [64] relative to the excitation of the first 2+

3 → 2+

1 ) = 0.46(8)µ2

B(E2; 2+

3 → 2+

1 ≤ 71 e2f m4.

If the 2419-keV level has spin-parity 3− as suggested tentatively in the literature [11], the
1 γ ray would require an unphysically large excitation
1 → (3−)) > 204600 e3f m6, corresponding to B(E3 ↓) > 63 W.u. for
the de-excitation. Therefore we tentatively assign J π = (2+) for this state instead. No

measured intensity of the (3−) → 2+
strength of B(E3; 0+

previously observed ground state transition from this 2419-keV level has been observed, and

it was not observed in this experiment either, but we assign an upper limit on the branching

ratio of 21% based on the experimental sensitivity at this energy in the spectrum. In the

calculation of the cross section and the B(E2; 0+

1 → (2+)) in Table A.1, a branching ratio

of 10.5(105)%– exhausting the upper limit of the branching ratio–was used.

In addition to statistical and efficiency uncertainty, the quoted transition strengths and

cross sections involved uncertainties from each measured peak intensity, which were deter-

mined by systematically varying the fit template and background model and determining

the effect on the extracted peak area. The determined systematic uncertainty for the 775,

802, 1440, and 1644-keV transitions were 8%, 7.7%, 12%, and 9.1%, respectively. The error

quoted for the extracted values for the 775-keV decay of course include the error propagated

through the feeding subtraction.

110

90Kr

For 90Kr, the fit template used in Fig. A.5(b) included the four transitions represented by

solid black arrows in Fig. A.8. The 655-keV, 2+

1 transition was included despite forming
1 one by constraining its intensity through the known
branching ratio of 68(4)% [11]. All observed transitions in Fig. A.8 have been reported in the

a doublet with the 707-keV, 2+

1 → 0+

2 → 2+

literature at energies of 707.05(6), 655.17(6), 1362.32(10), and 1542.20(13) keV [11], except

for the 2249-keV ground state transition, which must exist due to the population of 2249-keV

level in this experiment.

In determining the σ(0+

it is necessary to subtract the feed-

1 → 2+

1 ) cross section,
1 and (2+) → 2+

ing intensities of the 2+

1 γ rays. This amounted to subtracting
a total of 11(5)% from the measured intensity of the 707-keV transition. The extracted

2 → 2+

value of B(E2; 0+

1 ) = 1500(230) is consistent with the only previously measured
value [67], but with greatly reduced uncertainties. Using the known E2/M 1 mixing ratio

1 → 2+

δ = 0.507+0.020

−0.022 [68], the B(E2; 2+
1 ).
1 → 0+

addition to the B(E2; 2+

2 → 2+

1 ) strength quoted in Table A.1 was determined, in

Although the level at 2249 keV has no J π assignment in the literature, we argue that this

is a higher-lying 2+ state based on its population in this experiment, similar to the argument

made for the 2420-keV level in 88Kr discussed in the previous section. An upper limit of 23%

for the ground-state decay branch was determined based on the sensitivity limit posed by

our spectrum. To determine the B(E2; 0+

1 → (2+)) value shown in Table A.1, a branching

ratio of 11.5(115)% was assumed, encompassing the upper limit.

The final determined uncertainties followed a similar process to 88Kr. By systematically

varying the fit template and background model, systematic uncertainties of 30.4%, 10.4%,

111

Figure A.8: Level scheme for transitions observed in coincidence with scattered 90Kr pro-
jectiles. The given branching ratios are the adopted values from the NNDC database [11].
1 branching ratio is based on the
The dashed line giving an upper limit for the (2+) → 0+
sensitivity limit at this energy in the present work.

and 62% were determined for the 655-, 707-, and 1542-keV peak intensities, respectively.

The final quoted errors includes these as well as statistical and efficiency uncertainties, and

are propagated through the feeding subtraction for the B(E2; 0+

1 → 2+

1 ) value.

112

86Se

The fit template for 86Se was considerably simpler than the krypton isotopes, as only the

expected 2+

1 decay was observed in Fig. A.5(c). There was, however, a low-energy
shoulder that may correspond to a peak near 613 keV. This transition energy has been re-

1 → 0+

ported previously [69], and was tentatively suggested to directly feed the 2+

1 state. This
would lead to a low-lying second-excited state of unknown nature. No coincidence relation-

ship was observed between the 613- and 704-keV γ rays in this work, and this tentative

613-keV decay is considered unplaced in the evaluated data [11]. To gauge the effect of in-

cluding this low-energy shoulder as peak, two fitting templates were used: one that includes

a 613-keV peak and one that does not. Figure A.5(c) shows the result of including this

613-keV peak. Removing it significantly worsens the χ2 of the fit, but only increases the

cross section and B(E2; 0+

1 → 2+

1 ) value from 227(34) mb and 2110(320) e2f m4 to 233(35)

mb and 2170(330) e2f m4. In Table A.1, the first set of values is quoted.

The measured B(E2) = 2110(320) e2f m4 agrees at considerably reduced uncertainty

with the literature value from the excited-state lifetime measurement [10] of B(E2) =
−855 e2f m4. The quoted uncertainty for our measured B(E2) value includes a sys-
tematic uncertainty of 12.4% determined by varying the fit template and background model,

2190+1295

which was then added in quadrature to the efficiency and statistical uncertainties.

Discussion

The final results for the evolution of collectivity along the selenium and krypton isotopic

chains, including the newly measured values discussed in the previous sections, are shown

in Fig. A.9. The newly measured values are consistent with the picture of a gradual onset

113

of deformation beyond N = 50 for Z = 34, 36. For 88Kr, the result shows the expected

behavior of a slight increase in collectivity relative to the singly-magic 86Kr, in contrast with

the previously adopted value [65, 66] with which it differs by more than 1σ. This increase

in collectivity is expected for even-even neighbors of singly-magic nuclei along an isotopic

chain. In the case of 90Kr and 86Se, the newly measured values are consistent with previous

measurements [10, 67], but with greatly increased precision. With these new more precise

values, the evolution of B(E2) strength is now clearly consistent with the predicted smooth

onset of deformation [70, 71].

Figure A.9: Evolution of collectivity from N = 36 to N = 60 for the krypton and selenium
isotopes, including adopted values and the values measured here. The adopted values are
taken from Ref. [8], except for the case of 86Se [10]. Note the different units used here
compared to Fig. A.1 (see Sec. A).

One unique result of this experiment was the observation of several higher-lying 2+

states for the krypton isotopes, signaling significant excitation strength. To understand

114

3436384042444648505254565860Neutron Number25005000750036Kr 34Se36Kr (NSCL)34Se (NSCL)B(E2; 0+→2+) (e2 fm4)11these E2 strength distributions, we compare to large-scale shell-model calculation using

the code NUSHELLX@MSU [72] with an effective two-body interaction derived from the

Idaho N 3LO nucleon-nucleon interaction [73]. The effective interaction was obtained us-

ing many-body perturbation theory to third order in the interaction following the recipe

outlined in Refs.

[62, 74]. A 78Ni core was used with single-particle energies taken

from either Ref. [75] or determined using Skyrme-Hartree-Fock calculations where miss-

ing. The model space included the proton 1f5/2, 2p3/2, 2p1/2, 1g9/2 orbitals and the neutron
1g7/2, 2d5/2, 2d3/2, 3s1/2, 1h11/2 orbitals, without sub-shell restrictions and with the energy
of the πg9/2 adjusted to reproduce the 9/2+ state in 87Rb. The software used to derive
the effective Hamiltonian, and the resulting effective interaction and single-particle energies

used in the shell-model calculations are available on GitHub [76].

It was necessary to adjust the effective charges that enter the calculation of B(E2) values

in the shell model calculation. To accomplish this, the transition strengths of 86Kr were

calculated in our shell-model setup, and the proton effective charge was adjusted to ep = 1.68

to reproduce the known B(E2; 0+

1 ) = 1056(95)e2f m4 [77]. Assuming a simple picture
of isoscalar polarization charges, where ep = 1 + δpol and en = δpol, the neutron effective

1 → 2+

charge is then en = 0.68. These choices of the effective charges are supported by the use of

similar choices of effective charges in other calculations that utilize a similar model space for

this region [78].

The results of the shell-model calculation for the krypton isotopes are shown both in

Table A.1 and in Fig. A.10, where the calculated (red) and measured (black) B(E2; 0+

are shown as a function of the excitation energy of the 2+

1 → 2+
n )
n state being considered. There

is good agreement between the measured and calculated values for both the energies and

transition strengths, which is somewhat remarkable considering the fact that only the single-

115

Figure A.10: Comparison between measured B(E2) values and the results of a shell-model
calculation for 88,90Kr as a function of the energy of the 2+

n states.

particle energies involved in the shell-model calculation are tuned in any way. Furthermore,

looking at Table A.1, the calculated B(E2; 2+

2 → 2+
comparison is more difficult for 88Kr, as the B(E2; 2+

1 ) for 90Kr agrees with experiment. The
1 ) reported in Table A.1 is only
2 → 2+
an upper limit due to the unknown E2/M 1 multipole mixing ratio. Agreement would be

exact assuming a multipole mixing ratio of |δ| = 0.53, which is similar to the known mixing
ratio of δ = 0.507+0.020

−0.022 for the corresponding transition in 90Kr.

The 2+

3 level of 88Kr has been proposed by Ref. [64] as a candidate for a proton-neutron
mixed symmetry state [79]. This is indicated by a collective M 1 transition to the first 2+ and

a weakly collective E2 decay to the ground state. Using the adopted branching ratio from the

116

50010001500(a) 88Kr5001000150020002500Energy (keV)50010001500(b) 90KrShell ModelMeasuredB(E2; 0+→2+) (e2 fm4)NNDC database [11], and the E2/M 1 multipole mixing ratio from Ref. [64], we determined

B(M 1; 2+

3 → 2+

1 ) = 64(10) e2f m4 ≡ 2.8(4)W.u.. These
values are consistent with the aforementioned expectations of a mixed symmetry state, and

N and B(E2; 2+

1 ) = 0.46(8)µ2

3 → 0+

are in good agreement with the shell-model calculation results of B(M 1; 2+

3 → 2+

1 ) = 0.51µ2
N

and B(E2; 2+

3 → 0+

1 ) = 90 e2f m4.

To quantify the amount of expected higher-lying strength in 86Se, which we did not

observe experimentally, we used the shell-model calculation and determined the peak areas

expected based on the calculated B(E2; 0+

n ) for the tentatively assigned 1399-keV,
3 ) levels from the literature [11]. We utilized the adopted branching
ratios from the NNDC database. Based on the calculated transition strengths, we’d expect

2 ) and 2208-keV, (2+

1 → 2+

(2+

6(2) and 22(5) counts to be observed for the 1399-keV, (2+

2 ) → 2+
decays. In terms of the 2208-keV state, we’d expect to observe 43(11) counts at 1504 keV for

1 and 695-keV, (2+

2 ) → 0+

1

the (2+

3 ) → 2+

3 ) → 0+
one. Because all of these expected peak areas are below the sensitivity limit of the measure-

1 transition and less than 2 counts at 2208-keV for the 2208-keV, (2+

1

ment, the observed 86Se data is consistent with the predicted E2 strength in the shell-model

calculations. Assuming the intensities deduced from the calculated shell-model strength and

branching ratios, the quoted B(E2; 0+

1 ) in Table A.1 would be reduced by 23%. Be-
cause we do not observe these transitions, this value serves as an upper limit on the feeding

1 → 2+

from known states.

Presented in Table A.2 are the proton and neutron orbital occupancies from the shell-

model calculation described above. Note that these are consistent with those from another

shell-model calculation for 90Kr [68] in terms of the proton occupations. For the neutron

occupations, there is a slight difference as the g7/2 and h11/2 orbitals have essentially 0
occupancy here, while in Ref. [68] they are substantial at 0.15 and 0.18, respectively, for the

117

Table A.2: Shell-model occupations of the proton and neutron orbitals for the observed levels
in 88,90Kr and 86Se.

Protons
p3/2
p1/2
0.62
2.74
0.65
2.74
2.84
0.72
0.79
2.69
0.55
2.64
2.96
0.79
0.80
2.99
1.15
2.80
0.87
3.10
1.84
0.41
0.41
1.82

f5/2
4.29
4.29
4.21
4.28
4.60
3.96
3.93
3.85
3.81
3.50
3.55

g9/2
0.35
0.33
0.23
0.24
0.21
0.29
0.27
0.20
0.22
0.24
0.21

88Kr

90Kr

86Se

0+
1
2+
1
2+
2
2+
3
2+
4
0+
1
2+
1
2+
2
2+
3
0+
1
2+
1

Neutrons

g7/2
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.01

d5/2
1.88
1.88
1.83
1.86
1.84
3.68
3.66
3.57
3.57
1.80
1.76

d3/2
0.04
0.03
0.03
0.05
0.04
0.16
0.14
0.19
0.11
0.06
0.05

s1/2 h11/2
0.02
0.06
0.01
0.08
0.12
0.01
0.01
0.08
0.02
0.09
0.13
0.03
0.02
0.18
0.02
0.21
0.02
0.29
0.12
0.01
0.01
0.17

ground state of 90Kr.

Summary

We were able to determine the electric quadrupole transition strengths for 88,90Kr and 86Se

with considerably increased precision compared to previous measurements. Furthermore, we

provide measurements of the strength to higher-lying 2+ states for the krypton isotopes, in

most cases for the first time. This fragmentation of strength to higher-lying states is remark-

ably consistent with the results of large-scale shell-model calculations with a minimally-tuned

single-particle basis and an effective interaction based on chiral effective field theory. In clos-

ing, our results definitively complete the picture of a gradual onset of collectivity for the

neutron-rich krypton isotopes beyond N = 50, and provide a good starting point for future

explorations beyond N = 52 for the selenium isotopic chain.

Recently, the measured B(E2; 2+

1 → 0+

1 ) from the measurement presented in this work

118

were confirmed, albeit with larger uncertainties, in a recoil-distance Doppler shift measure-

ment [80], which determines the B(E2) values by measuring the lifetimes of the corresponding

states.

For the krypton isotopic chain,the first spectroscopy [71] of 98,100Kr using (p, 2p) from

99,101Rb, respectively, was performed at the Radioactive Isotope Beam Factory (RIBF) at

the RIKEN Nishina Center. A picture of shape coexistence emerges in 98Kr, where a low

lying (0+

2 ,2+

2 ) state was reported.

In another experiment [54] utilizing nucleon removal reactions at RIBF, low-lying excited

states in 88,90,92,94Se were populated. The observed trend of the gradually-decreasing E(2+
1 )
energies along the isotopic chain presents a picture of gradually increasing collectivity when

approaching N = 60 for the selenium isotopes, a situation that is quite similar to that of the

krypton isotopic chain.

Future measurements utilizing intermediate-energy Coulomb excitation to measure the

B(E2) values of these short-lived isotopes will play an important role in determining the

evolution of collectivity at the extremes of neutron-proton asymmetry. The production of

these nuclei approaching the limits of nuclear stability at RIBF, and the opportunity to

produce these and even more exotic nuclei at FRIB with high-intensity beams, will enable

such measurements to be performed, and greatly improve our knowledge of the properties

of nuclei at the limits of existence.

119

APPENDIX B

EVENT-BY-EVENT CORRECTION

FOR THE PARTICLE VELOCITY

USING DT A

This appendix will provide a brief derivation of the correction for the particle velocity used for

Doppler reconstruction using the dta value described in Chapter 2. The correction proceeds

as follows. The dta value is dT

T , where T = E − mc2 is the kinetic energy. Let β = v

c and

γ = 1

1−β2 . The following equation makes use of the relation E = γmc2.

dT
T

=

=

d(E − mc2)
E − mc2
mc2(γ − 1)

dE

(B.1)

(B.2)

To determine dE, recall Eβ = pc for momentum p, and consider:

120

d(E2) = 2EdE = d((pc)2 + (mc2)2)

= 2pc2dp

dE =

pc2dp

E

= βcdp

(B.3)

(B.4)

(B.5)

Now plugging dE into the expression for dT

T and recalling the relations (γβ)2 = γ2 − 1,

and p = γβmc:

dT
T

=

=

=

=

=

βcdp

mc2(γ − 1)
(γβ)2cdp

γ2βmc2(γ − 1)
(γ2 − 1)cdp

γ(γβmc)c(γ − 1)
(γ + 1)(γ − 1)dp

γp(γ − 1)

γ + 1

γ

dp
p

(B.6)

(B.7)

(B.8)

(B.9)

(B.10)

Now we want an expression connecting dp

p and dβ

β . Recall that dγ

dβ = βγ3, and again

using (γβ)2 = γ2 − 1:

121

dp
p

=

=

=

=

d(γβmc)

γβmc
1

dβ

γβmc(cid:18) dγ
βmc + γmc(cid:19) dβ
γβmc(cid:16)β2γ3mc + γmc(cid:17) dβ

(β2γ2 + 1)dβ

1

β

= γ2 dβ
β

Finally, plugging this into the expression for dT
T :

dT
T

=

γ + 1

γ

= γ(γ + 1)

γ2 dβ
β
dβ
β

(B.11)

(B.12)

(B.13)

(B.14)

(B.15)

(B.16)

(B.17)

So, we correct the reconstruction velocity for each event by the determined dta = dT
T

with:

dβ =

dT
T

β

γ(γ + 1)

(B.18)

122

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