QUADRUPOLE COLLECTIVITY IN NEUTRON-DEFICIENT SN NUCLEI: 104 SN
By
Vincent Maximilian Bader

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2014

ABSTRACT
QUADRUPOLE COLLECTIVITY IN NEUTRON-DEFICIENT SN NUCLEI: 104 SN
By
Vincent Maximilian Bader
This thesis is about the study of quadrupole collectivity in the neutron-deficient nucleus
104 Sn using intermediate-energy Coulomb excitation. The B(E2; 0+ → 2+ ) value for the
1
1

excitation of the first 2+ state in 104 Sn has been measured to be 0.180(37) e2 b2 relative
to the well-known B(E2 ↑) value of 102 Cd. This result disagrees by more than one sigma
with a recently published measurement [1]. The result indicates that the most modern manybody calculations remain unable to describe the enhanced collectivity below mid-shell in Tin
approaching N = Z = 50. The enhanced collectivity is attributed to proton particle-hole
configurations beyond the necessarily limited shell-model spaces and suggests the asymmetry
of the B(E2 ↑)-value trend around mid-shell to originate from enhanced proton excitations
across Z = 50 as N = Z is approached.

Dedicated to my parents Verena Bader and Dr. Raoul Bader.

iii

ACKNOWLEDGMENTS

First and foremost I would like to express my gratitude towards my adviser Alexandra Gade.
She was always there to discuss ideas, answer my questions and help me out whenever I
needed help. It was a great pleasure working with her. I would further like to thank my
guidance committee for their support: Alex Brown, Phillip Duxbury, Wayne Repko and
Michael Syphers. I would also like to give special thanks to Dirk Weisshaar without whom
this experiment would not have succeeded. Furthermore I highly appreciate the help of
Kathrin Wimmer who provided me with the root code for the analysis and Travis Baugher
who provided me with the simulation code. Both endured countless times I bothered them
with problems I encountered. There are many more people who have my thanks, all the
people in the Gamma group at the NSCL and of course the staff working here. Not to
forget my office mates Daniel Winklehner, Michael Jones and Michael Bennett who were a
welcomed source of knowledge and distraction.
Last but not least I would like to thank my family without who this stay at MSU would not
have been possible.

This work was in part funded by NSF and MSU.

iv

TABLE OF CONTENTS

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

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
KEY TO SYMBOLS AND ABBREVIATIONS
Chapter 1

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xv

Introduction and Motivation . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Physics of Nuclei . . . . . . . . . . . . . . . .
2.1 Atomic Nuclei . . . . . . . . . . . . . . . . . . . . . .
2.2 Nuclear Shell Model . . . . . . . . . . . . . . . . . .
2.3 Shell Model Calculations and Effective Charges
2.4 Excitations and Collective Excitations . . . . . .
2.5 Reduced Quadrupole Transition Strength . . . .
2.6 Physics of the Tin Isotopes: Leading to 104 Sn . .

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3
3
8
14
17
20
26

Chapter 3 Quadrupole Collectivity: Experimental Methods . . . . . . . .
3.1 Excited-state Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Intermediate-Energy Coulomb Excitation . . . . . . . . . . . . . . . .

30
30
31

Chapter 4 Experimental Setup . . . . . . . . . .
4.1 Overview of the Setup . . . . . . . . . . .
4.2 Radioactive Isotope Production . . . . .
4.2.1 Primary Beam Production . . . . .
4.2.2 Secondary Beam Production . . .
4.3 Radio Frequency Fragment Separator . .
4.4 CAESAR . . . . . . . . . . . . . . . . . . . .
4.4.1 Photoelectric Effect . . . . . . . . .
4.4.2 Compton Effect . . . . . . . . . . . .
4.4.3 Pair Production . . . . . . . . . . .
4.4.4 CAESAR, the Array . . . . . . . .
4.4.4.1 CAESAR, the Detectors .
4.4.5 Energy Resolution . . . . . . . . . .
4.4.6 Efficiency . . . . . . . . . . . . . . . .
4.5 Other Detectors . . . . . . . . . . . . . . .
4.5.1 P-I-N Detector . . . . . . . . . . . .
4.5.2 Parallel Plate Avalanche Counters
4.5.3 Phoswich . . . . . . . . . . . . . . . .

42
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44
46
48
52
55
57
58
59
60
62
63
64
65
66
68

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Chapter 5 Data Analysis . . . . . . . . . . . . . . . .
5.1 Calibrations and Corrections . . . . . . . . . .
5.1.1 Energy and Time Calibration . . . . . .
5.1.2 PPAC Calibration . . . . . . . . . . . . .
5.1.3 Corrections . . . . . . . . . . . . . . . . .
5.2 Particle Identification . . . . . . . . . . . . . . .
5.2.1 PID . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Trigger Modes and Contamination . . .
5.3 γ-ray Spectra . . . . . . . . . . . . . . . . . . . .
5.3.1 Velocity Determination . . . . . . . . . .
5.3.2 Gating . . . . . . . . . . . . . . . . . . . .
5.3.2.1 Prompt γ-ray Timing . . . . . .
5.3.2.2 Particle Scattering Angle Cuts
5.3.2.3 Multiplicities . . . . . . . . . . .
5.3.3 Spectra . . . . . . . . . . . . . . . . . . . .
5.4 Geant4 Simulations . . . . . . . . . . . . . . . .
5.5 Fits . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Relative Measurement Results . . . . . . . . .

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72
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109

Chapter 6

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 7

Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . 124

APPENDIX

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

BIBLIOGRAPHY

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vi

. . . . . . . . . . . . . . . . . . . 126
. . . . . . . . . . . . . . . . . 143

LIST OF TABLES

Table 5.1

Calibration sources and energies. . . . . . . . . . . . . . . . . . . . .

72

Table 5.2

Sources and energies used for σ(E) fits. . . . . . . . . . . . . . . . . 104

Table 5.3

Input settings used for the in-beam GEANT4 simulations. . . . . . . 107

Table 5.4

Important fit and calculation results. . . . . . . . . . . . . . . . . . 113

Table 6.1

B(E2; 0+
1
this work
Eq. 5.5. .

102 Cd and 104 Sn from literature and from
→ 2+
1 ) values for
(104 Sn); the results from [2] are used for normalization in
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vii

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Visualization of the nuclear landscape, plot of the chart of nuclei.
The number of protons (Z) is plotted versus the number of neutrons
(N ). Stable nuclei are shown as black boxes, observed unstable nuclei
as blue boxes, and unstable nuclei predicted to exist, but not yet
observed in red. Magic numbers (discussed in the text) are indicated
by dashed lines. Figure is taken and modified from [3]. . . . . . . .

5

Top panel: one-neutron separation energies for even-even nuclei with
N > Z as a function of neutrons. Bottom panel: difference between
one-neutron separation energies for the same set of nuclei. Solid lines
connect isotopic chains (same Z). The dashed lines indicate the magic
numbers, 20, 28, 50, 82 and 126. Figure is taken and modified from
[4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Calculated neutron single-particle energy levels for 208 Pb with a 3D
harmonic oscillator potential (a), a Woods-Saxon potential (b) and
a Woods-Saxon potential plus a spin-orbit coupling term (c). Gaps
in between energy levels are marked with big numbers and they represent the magic numbers for each potential. The levels in (c) are
also marked with their quantum numbers nℓj and are followed by
two additional numbers. The first one (in square brackets) shows the
maximum allowed number of nucleons for this level and the second
one counts the total number of nucleons that can be fit in all the
shells up to this one. Figure is taken and modified from [4]. . . . .

13

Energies of the first 2+ states in even-even nuclei. Dashed lines mark
the magic numbers. Figure is taken and modified from [5]. . . . . .

18

Schematic of the excitation of a nucleus (solid arrow) and the deexcitation via γ-ray emission (squiggly arrow). Figure is taken and
modified from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Measured energy of the first excited 2+ state for the even-even Tin
isotopes between the N = 50 and N = 82 shell closures. Data is
taken from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

viii

Figure 2.7

Measured B(E2 ↑) values for the chain of even-even Tin isotopes.
Data are taken from: Adopted values are published by the National
Nuclear Data Center [6], RIKEN [7], REX-ISOLDE [8,9], GSI-DSA [10],
GSI [1, 11, 12], NSCL [13], ORNL [14], IUAC [15], HRIBF [16]. The
dotted line reproduces the theoretical calculation shown in [11] with
gds (tπ = 4, see Sec. 6 for details). . . . . . . . . . . . . . . . . . . . 28

Figure 3.1

Schematic picture of the intermediate-energy Coulomb scattering process. Shown are the projectile and target nucleus as well as center-ofmass scattering angle θ and impact parameter b (see text for details).
The insert shows the excitation and de-excitation process in a level
scheme. The γ-ray energy is equal to the energy difference of the
nuclear levels. Figure is taken and modified from [3]. . . . . . . . .

34

Schematic overview of the beam production at the NSCL. Figure is
modified from [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Schematic overview of the experimental setup after the A1900 in the
S2 experimental vault. . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Schematic of the K500 cyclotron. The dees are red and the hills are
blue. Figure is taken and modified from [5]. . . . . . . . . . . . . . .

46

Simulation of the momentum distribution of the contaminants as
well as 104 Sn that are produced during the fragmentation process
for the settings used in this experiment. In red the 104 Sn is shown
and in black the contaminants. The area in between the two green
lines shows the momentum acceptance. The figure was made with
LISE++ [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Top: Drawing of the RFFS; Bottom: Photo of the RFFS. Taken and
modified from [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

LISE++ simulation for this setup. The yellow shaded area indicates
the beam blocker at the end of the RFFS. Everything in the yellow
area gets blocked out of the beam cocktail. . . . . . . . . . . . . . .

52

Technical drawing of the CAESAR setup. On the bottom is the
mounting table shown with which one can move the array horizontally.
On top of the table is CAESAR mounted. The mounting brackets
are clearly visible as well as the detectors’ PMTs and the encased
crystals (green and blue). . . . . . . . . . . . . . . . . . . . . . . . .

54

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

ix

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Schematic illustration of the arrangement of the CAESAR detectors.
Left: cross sectional view of the rings F and J perpendicular to the
beam axis. Right: Picture of the ten rings A (most upstream) to J
(most downstream) with the target position in red. The gray scale
corresponds to the position on the rings, as shown on the left. Picture
taken and modified from [20]. . . . . . . . . . . . . . . . . . . . . . .

55

Example of a γ-ray detector response to many mono-energetic γ rays.
The components of the spectrum are explained in the text. Picture
taken and modified from [3]. . . . . . . . . . . . . . . . . . . . . . .

56

Schematic illustration of the photoelectric absorption process. Picture taken and modified from [3]. . . . . . . . . . . . . . . . . . . . .

57

Schematic illustration of Compton scattering. Picture taken and modified from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Schematic illustration of pair production. Picture taken and modified
from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Schematic illustration of a detector. Picture taken and modified
from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Schematic illustration of the regions in a p-i-n detector. Picture taken
and modified from [21]. . . . . . . . . . . . . . . . . . . . . . . . . .

65

Figure 4.15

Schematic illustration of a PPAC detector. Left side is a cross section
and the right side shows one side. Picture taken and modified from [22]. 67

Figure 4.16

Schematic illustration of the resistor chain. Picture taken and modified from [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 4.17

Schematic illustration of the Phoswich detector. . . . . . . . . . . .

69

Figure 4.18

Schematic illustration of the Phoswich PMT signals and the integration ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Figure 4.19

Picture of the Phoswich detector.

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

71

Figure 5.1

Uncalibrated 88 Y source spectrum for one CAESAR detector. The
green lines show the two Gaussian fits used to determine the position
of the peaks in channel numbers. . . . . . . . . . . . . . . . . . . . .

74

x

Figure 5.2

Source peak energies vs channel numbers for one detector. The green
line is a polynomial of order 2 fit. . . . . . . . . . . . . . . . . . . .

75

Figure 5.3

Overview of all detectors and their calibrated energies for a 88 Y source. 76

Figure 5.4

Example of the time calibration fit, see text for details. . . . . . . .

77

Figure 5.5

Overview of all detectors and their calibrated timing information. . .

78

Figure 5.6

Prompt timing peak position for all 111 runs of the experiment for
γ-ray energies greater than 500 keV. . . . . . . . . . . . . . . . . . .

79

Uncalibrated mask run for the downstream PPAC. The positions of
the different features correspond to the positions of the holes and slits
in the mask placed in front of the PPAC during the calibration. . . .

80

Position data taken using the pilot beam and no target. Left column
has the x-positions and the corresponding angles. Right column has
the y-positions and the corresponding angles. The first row is at the
target position, the second at the downstream position and the third
at the upstream PPAC position. . . . . . . . . . . . . . . . . . . . .

81

Figure 5.9

X and Y projections of the beam spot at the target position. . . . .

81

Figure 5.10

Downstream PPAC beam spot position for 102 Cd over time. . . . . .

82

Figure 5.11

Upstream PPAC beam spot position for 102 Cd over time. . . . . . .

83

Figure 5.12

Corrections. The left side shows uncorrected data for all events that
were recorded during the experiment and the right side shows the
same data but with the run-by-run corrections applied as explained
in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Particle Identification spectrum for the neutron-deficient projectile
beam passing through the Au target. The energy loss measured in
the p-i-n detector is plotted versus the ion’s flight time. 104 Sn and
102 Cd can be clearly identified. The red lines show the incoming gates.
See text for details. . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

PID plot for downscaled single events with two dimensional Gaussian
fits (black lines). See text for more details. . . . . . . . . . . . . . .

87

Figure 5.7

Figure 5.8

Figure 5.13

Figure 5.14

xi

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Figure 5.24

Plot to find the right β for the Doppler correction of 104 Sn. The γ-ray
energy is at 1260 keV [6]. For details see the text. . . . . . . . . . .

89

Plot to find the right β for the Doppler correction of 102 Cd. The
γ-ray energy is at 777 keV [6]. For details see the text. . . . . . . . .

90

Plot to show the prompt γ-ray time gate (red). Only 1 hit events
in CAESAR are displayed. The two main features from prompt and
off-prompt γ rays are clearly visible as well as the two horizontal
background lines at 1461 keV and 2615 keV. For details see the text.

91

Scattering angle distribution for 104 Sn (left) and 102 Cd (right). Blue
line: “safe” angle which was used, green line: “safe” angle for beginning of the target, red line: “safe” angle for middle of the target. . .

93

Downstream PPAC gated on coincidence events, centered and gated
on 104 Sn (left) and 102 Cd (right). . . . . . . . . . . . . . . . . . . .

94

Effect of straggling. Pink is the measured distribution with downscaled singles trigger condition. Red is the result of the whole Monte
Carlo simulation. Blue are the events that fulfilled the scattering angle cut condition but started with angles bigger and green are the
events that did not fulfill the angle cut condition but started with
angles smaller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

CAESAR multiplicities for a single run. Black: all events; blue:
events with no CAESAR detector in overflow; orange: at least one
detector in overflow. . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Ring distribution of CAESAR detectors in overflow. Ring 1 (J) is the
most forward ring and 10 (A) the most backward. . . . . . . . . . .

98

Phoswich energies for different multiplicity ranges. On the left side
the different multiplicity ranges are shown in steps of 10 and on the
right side the corresponding Phoswich energies. All detected events
were used to generate these plots. The very top panel of both columns
has no gates applied, the full multiplicity range is used. . . . . . . .

99

Lab frame (upper panels) and Doppler corrected (projectile frame,
lower panels) energy spectra for 104 Sn and 102 Cd, respectively. Characteristic energies are indicated. For more details see text. . . . . . . 101

xii

Figure 5.25

Low-energy part for a high-activity 137 Cs spectrum for all detectors
in ring 5. In green is the half Gaussian fit shown. . . . . . . . . . . . 102

Figure 5.26

Comparison of the low-energy part of a rescaled high activity 137 Cs
(black) and a low-activity 60 Co (red) spectrum. . . . . . . . . . . . . 103

Figure 5.27

Development of σ(E) for all detectors in ring 5. . . . . . . . . . . . . 104

Figure 5.28

Comparison of simulated and measured source spectra. The black
histogram is the measured histogram, the red is the pure scaled simulation and the green is the simulation plus a scaled background. The
bottom right histogram shows the room background measurement. . 106

Figure 5.29

Shown are several simulated variables. Comparison with the input
values from Table 5.3 shows a good agreement. . . . . . . . . . . . . 108

Figure 5.30

Simulated γ-ray spectra for 104 Sn. On the left side are the spectra
Doppler corrected and on the right side not. The simulation is again
in good agreement with the input data and with the expectations of
the response of CAESAR. . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 5.31

Last test of the simulation. Two more variables that were simulated
are shown and again a comparison with the input values from Table
5.3 shows a good agreement. . . . . . . . . . . . . . . . . . . . . . . 110

Figure 5.32

Event-by-event Doppler reconstructed γ-ray spectra detected in coincidence with scattered 104 Sn (a) and scattered 102 Cd (b). γ-ray
transitions at 1260 keV and 777 keV can be clearly identified and are
attributed to the de-excitation γ rays from the first excited 2+ state
in 104 Sn and in 102 Cd, respectively. . . . . . . . . . . . . . . . . . . 111

Figure 5.33

Event-by-event Doppler reconstructed γ-ray spectra detected in coincidence with scattered 102 In (a) and scattered 100 Ag (b). Shown
are the 104 Sn background fit scaled to 102 In (a) and the 102 Cd background fit scaled to 100 Ag. In both spectra, no de-excitations are
observed in the sensitive region. . . . . . . . . . . . . . . . . . . . . 112

xiii

Figure 6.1

(a) Measured B(E2 ↑) values for the chain of Sn isotopes: Adopted [6],
Riken [7], REX-ISOLDE [8,9], GSI-DSA [10], GSI [1,11,12], NSCL [13],
ORNL [14], IUAC [15], HRIBF [16]. Solid line: Large-scale shellmodel (LSSM) calculations with NNLO interaction [23]. Dashed line:
LSSM with N3 LO interaction [24]. These two lines correspond to
calculations performed with an effective neutron charge set to 0.5e.
An increase to 1e would yield and increase by a factor of 4 for the
calculated B(E2 ↑) values. The dotted line reproduces the 0g1d2s
(tπ = 4) calculation shown in [11]. See text for details. (b) Energies
of the proton 2p − 1h intruder states in Sb nuclei [25]. . . . . . . . . 116

Figure 6.2

Shell model configuration with a closed 100 Sn core. The closed core is
marked in red, the model space for the valence neutrons is marked in
blue and the Fermi energy is marked in green. The visible numbers
are the same as in Fig. 2.3. Figure is taken and modified from [4].

117

Figure 6.3

Shell model configuration with a closed 90 Zr core. The closed core is
marked in red, the model space for the valence neutrons and up to 4
protons (tπ = 4) is marked in blue and the Fermi energy is marked
in green. The visible numbers are the same as in Fig. 2.3. Figure is
taken and modified from [4]. . . . . . . . . . . . . . . . . . . . . . 119

Figure 6.4

Calculated evolution of the proton ESPE with the neutron number
N along the chain of Sn isotopes. Figure is taken and modified from
[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 6.5

Schematic representation of the proton 2p − 2h excitations across the
Z = 50 shell gap in 102 Sn. The occupation of the same proton and
neutron orbitals above the Z = N = 50 shell leads to α-like correlations between the valence nucleons. Figure is taken and modified
from [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xiv

KEY TO SYMBOLS AND ABBREVIATIONS

A . . . . . . . . . . . . . . . . . . . . . . . Atomic mass number (sum of neutrons and protons in a nucleus)
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron number (number of neutrons in a nucleus)
Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proton number (number of protons in a nucleus)
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed of light in vacuum (≈ 3 · 108 m/s)
β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic factor beta (β = v/c)
√
γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic factor gamma (γ = 1/ 1 − β 2 )
NSCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Superconducting Cyclotron Laboratory
ECR ion source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Cyclotron Resonance ion source
ARTEMIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Room TEMperature Ion Source
K500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First cyclotron at the NSCL
K1200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second cyclotron at the NSCL
A1900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fragment separator at the NSCL
CAESAR . . . . . . . . . . . . . . High-efficiency CsI(Na) scintillator array; CAESium iodide ARray
RFFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radio Frequency Fragment Separator
p-i-n detector . . . . . . . . . . . . . . . . . . Semiconductor detector used for energy loss measurements
PPAC . . . . . . . . . . . . . . . . . .Parallel Plate Avalanche Counter; used for position measurements
Phoswich detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combination of two plastic scintillators
PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photomultiplier Tube

xv

Chapter 1
Introduction and Motivation
One of the overarching goals of nuclear physics is the development of a comprehensive model
of the atomic nucleus with predictive power across the nuclear chart. While the structure of
nuclei close to stability is fairly well understood, significant modifications compared to stable
nuclei have been observed for short-lived rare isotopes with unbalanced numbers of protons
and neutrons. The driving forces behind these structural changes are manifold, including
spin-isospin parts of the nuclear interaction [26] and various facets of many-body correlations [27]. Of particular importance for the development of nuclear models is experimental
data that consistently tracks the effect of isospin and changed nuclear binding energies, for
example. The chain of Sn isotopes has been a formidable testing ground for nuclear models
as some spectroscopic data is available from N = Z = 50 100 Sn [28] in the proximity of the
proton drip-line to 134 Sn [29], beyond the very neutron-rich doubly-magic nucleus (nucleus
with a magic number of neutrons and protons, see Sec. 2) 132 Sn [30]. In even-even nuclei
+
(even number of protons and neutrons), the electromagnetic B(E2 ↑) = B(E2; 0+
1 → 21 )

excitation strength is a measure of quadrupole collectivity (see Sec. 2.5), sensitive to the
presence of shell gaps, nuclear deformation, and nucleon-nucleon correlations, for example.
In the Sn isotopes, this transition strength has been reported from 104 Sn to 130 Sn, spanning
a chain of 14 even-even Sn isotopes. The trend is asymmetric with respect to mid-shell and
not even the largest-scale shell-model calculations have been able to describe the evolution of
transition strength across the isotopic chain without varying effective charges. In this work,
1

the determination of the 104 Sn B(E2 ↑) value from intermediate-energy Coulomb excitation
is reported. The value exceeds the recently published result [1] and – continuing the trend
below mid-shell – is found at variance with the largest-scale shell-model calculations. It is
though in agreement with the value from [7]. The drawn conclusion is very different from
[1] and explains the enhanced collectivity for neutron-deficient Sn nuclei by considering proton particle-hole intruder configurations observed in neighboring nuclei and α correlations
towards N = Z. It is suggested that the interplay of proton intruder configurations and
quadrupole collectivity is a common phenomenon along proton-magic isotope chains.

2

Chapter 2
Physics of Nuclei

2.1

Atomic Nuclei

The atomic nucleus is a many-body quantum system. It consist of fermions, namely two types
of nucleons, the protons and neutrons. Protons are positively charged and the neutrons are
electrically neutral. A nucleus is made up of N neutrons and Z protons, resulting in the
total number of A = N + Z nucleons. A nucleus is typically referred to in the notation
A X(Z), with X(Z) being the chemical symbol of the element that has Z protons. Nuclei

have typical radii of the order of 10 fm = 10−14 m. The radius for nuclei near stability is
usually approximated like this [31]:

1

R(A) = R0 A 3 ,

(2.1)

with R0 = 1.25 fm. Because the protons are positively charged, they are subject to the
Coulomb force. This force is repulsive between protons and is proportional to 12 and therer
fore, decreases fast with increasing radial distance. The neutrons are not affected by the
Coulomb force due to their electrical neutrality. If there was only the Coulomb force, nuclear systems would not exist. The nucleons are bound together by the attractive strong
force. This force has a typical range of the order of 10−14 -10−15 m which is of the order of
the size of a nucleon [32]. It is charge independent, acts on neutrons as well as protons, and

3

is about 100 times stronger than the Coulomb force on the length scale of a nucleus but it
is of negligible strength for longer distances. There are small difference in the strength of
the attraction depending on the composition of a nucleon pair. The nucleon-nucleon force is
charge symmetric, meaning that proton-proton and neutron-neutron interactions have the
same strength and it is nearly charge independent, meaning that the proton-neutron interaction is only slightly different from the proton-proton interaction [31].
More than 3000 nuclei have been observed so far but only about 250 are stable and the
unstable ones will eventually decay into stable nuclei. As a result of the Coulomb interaction, more neutron-rich than proton-rich nuclei exist. The boundary on the proton-rich side,
the proton drip line, is more well defined than the boundary on the neutron rich side, the
neutron drip line. This is because the proton drip line is closer to stability than the neutron
drip line and thus easier to reach experimentally. The Coulomb force makes it harder to
add protons than neutrons. A visualization of the nuclear landscape, the chart of nuclei, is
shown in Fig. 2.1. The number of protons is plotted versus the number of neutrons. Each
square represents a different nucleus. The black squares are the stable nuclei, the blue are
the unstable but known nuclei and the red squares represent the nuclei that are assumed to
exist but have not been observed yet. The so-called magic numbers (see below), 2, 8, 20, 28,
50, 82 and 126 are indicated as dashed lines.
There is plenty of evidence pointing to an internal structure of nuclei. The binding energy,
BE(N, Z) is the basis of four of those observables, the one-proton and -neutron separation energies, Sp (N, Z) and Sn (N, Z), and the difference in one-nucleon separation energies,
∆Sp (N, Z) and ∆Sn (N, Z). The binding energy of a nucleus is calculated by taking the
difference of the sum of the rest mass energies of the individual nucleons building a nucleus
and the measured rest mass energy of the bound nucleus itself [4]:
4

120
126

Proton number

100
80

82

60

82

50

40
20
20

28

r
Te

8

ra

ita
n
og
Inc
28

Observed nuclei
Stable nuclei
Unobserved nuclei

20

2
8

2

0
0

20

40

60

50

80
100
120
Neutron number

140

160

180

Figure 2.1 Visualization of the nuclear landscape, plot of the chart of nuclei. The number
of protons (Z) is plotted versus the number of neutrons (N ). Stable nuclei are shown as
black boxes, observed unstable nuclei as blue boxes, and unstable nuclei predicted to exist,
but not yet observed in red. Magic numbers (discussed in the text) are indicated by dashed
lines. Figure is taken and modified from [3].

(
)
BE(N, Z) = Zmp + N mn − m(N, Z) c2 .

(2.2)

M eV
In Equation 2.2, mp = 938.272046(21) M eV
2 is the proton mass, mn = 939.565379(21) 2
c

c

is the neutron mass [33], m(N, Z) is the mass of the nucleus with N neutrons and Z protons
and c = 299, 792, 458 m
s is the speed of light [34]. The one-proton and -neutron separation
energies, Sp (N, Z) and Sn (N, Z), are the energies needed to remove one proton or neutron
from the nuclei respectively. They are defined as the difference between the binding energies
of the neighboring nuclei:
5

Sn (N, Z) = BE(N, Z) − BE(N − 1, Z),

(2.3)

Sp (N, Z) = BE(N, Z) − BE(N, Z − 1).

(2.4)

The difference in one-nucleon separation energies, ∆Sp (N, Z) and ∆Sn (N, Z) are:

∆Sn (N, Z) = Sn (N, Z) − Sn (N + 1, Z),

(2.5)

∆Sp (N, Z) = Sp (N, Z) − Sp (N, Z + 1).

(2.6)

Plotting Sn (N, Z) and ∆Sn (N, Z) for nuclei with an even number of neutrons and protons
as a function of neutron number shows evidence of an internal structure. In Fig. 2.2 the
two observables are plotted for even-even nuclei with N > Z near stability. In the top
panel where Sn (N, Z) is plotted, the lines connect isotopic chains (same number of protons).
At the neutron numbers 20, 28, 50, 82 and 126 there is a sudden drop in the separation
energy for the next neutron, indicating that those numbers are special. The effect is even
more pronounced in the bottom part of the figure where ∆Sn (N, Z) is plotted. In this
representation there are clear peaks at the so-called magic numbers, again the solid lines
connect isotopic chains.
The features that can be observed come from the fact that for nuclei with a magic number
of neutrons an enhanced amount of energy is necessary to remove a neutron, which suggests
that those nuclei are more tightly bound than their neighbors. This points to an internal
structure of energy levels occupied by neutrons which have gaps at levels filled with a magic

6

15

Sn (MeV)

13
11
9
7
5

∆Sn (MeV)

3
6
4
2
0

0

50

100

150

Neutron Number
Figure 2.2 Top panel: one-neutron separation energies for even-even nuclei with N > Z as
a function of neutrons. Bottom panel: difference between one-neutron separation energies
for the same set of nuclei. Solid lines connect isotopic chains (same Z). The dashed lines
indicate the magic numbers, 20, 28, 50, 82 and 126. Figure is taken and modified from [4].
number of neutrons and the effects of those internal structures can be observed. The same
effects can be seen for plots of Sp (N, Z) and ∆Sp (N, Z) and result in the same magic numbers,
just for protons. Nuclei that have a magic number of either neutrons or protons are called
semi-magic or magic nuclei. Nuclei that have a magic number of both protons and neutrons
are called doubly magic. As shown, magic and doubly magic nuclei are more strongly bound
than their neighbors.
7

2.2

Nuclear Shell Model

The nuclear quantum system can be described by quantized energy levels that are available
to the nucleons, also called orbitals or shells. Gaps in the energy levels give rise to the
nuclear shell structure. These energy levels are characterized by a set of quantum numbers.
As for all fermions, the Pauli principle applies to nucleons, prohibiting two nucleons to
occupy a state with exactly the same set of quantum numbers. Protons are distinguishable
from neutrons and the Pauli principle only applies to indistinguishable fermions. Therefore,
they are considered separately and each can occupy their own energy levels. Both quantum
systems are very much alike with very similar energy levels and gaps between the same levels.
Nucleons have an intrinsic spin angular momentum quantum number s = ± 12 for the intrinsic
spin ⃗s, an orbital angular momentum quantum number ℓ = 0, 1, 2, 3, . . . for the angular
momentum ⃗ℓ and these two couple to the total angular momentum ⃗j = ⃗l + ⃗s with total
angular quantum number j = ℓ ± 21 . For each j exist 2j + 1 magnetic sub-states, with the
magnetic quantum number m which is the z-projection of ⃗j, running from −j to j in integer
steps.
Energy levels are labeled by several quantum numbers, namely nℓj . The quantum number
n = 0, 1, 2, 3, . . . is the principal quantum number, also called radial quantum number. It
counts the times the wavefunction of this state changes its sign and is directly related to the
harmonic oscillator model because the major quantum number of the harmonic oscillator
model is N = 2n + l (see Eq. 2.12 and 2.13). The states with ℓ = 0, 1, 2, 3, . . . are usually
labeled as ℓ = s, p, d, f, . . . which is called the spectroscopic notation. Because every orbital
j has 2j + 1 magnetic sub-states it can be occupied by 2j + 1 nucleons without two nucleons
having exactly the same set of quantum numbers nℓjm. Another characteristic of a state is

8

the parity π which describes the sign of a wavefunction when the following transformation
is done:

x → −x
y → −y

(2.7)

z → −z.
Parity can be π = ±1 and is determined by π = (−1)ℓ .

The total angular momentum of a nucleus is labeled J⃗ and is the sum of the ⃗j’s of the
nucleons:

J⃗ =

A
∑

⃗j

(2.8)

i=1

The discovery of the magic numbers led to the development of a theoretical nuclear model
to help understand and predict observable features in nuclei [35, 36]. This model is called
Single Particle Nuclear Shell Model. It treats the interaction of nucleons in a nucleus as the
motion of a single nucleon in a mean field potential generated by all the other nucleons in
the nucleus. In this model, the energies of the orbitals that can be occupied by the nucleons
are called single-particle energies.
The nucleons occupy orbitals/shells with certain energies and there are energy gaps between
certain orbitals. The nucleons that fill up all orbitals below a shell gap make up the so-called
inert core and the remaining nucleons which are not enough to fill up all orbitals between
this and and the next shell gap are called valence nucleons. If an orbital is filled with the

9

maximum number of nucleons allowed by the Pauli principle it is called a closed shell. Usually it is assumed that the inert core and the closed shells do not contribute to the total
angular momentum of the nucleus and only unpaired valence nucleons contribute to the total
angular momentum J⃗ of the ground state of the nucleus. This is especially true if there is
only one more or less nucleon than the inert core [32]. Of course, for more realistic shell
models, multiple nucleons outside the core have to be considered as well as mixing between
different nucleon configurations [37]. For nuclei with fewer than 12 nucleons in total it is
possible to include all nucleons in calculations and no cores are needed [38] (this is called
the no-core shell model).

The basis for the shell model is the solution of the Schr¨odinger equation. In general the
Hamiltonian of a system with A nucleons can be written like this [39]:
A
A
∑
∑
p2i
H=
Vik (ri − rk ) ≡ T + V ,
+
2mi
i=1

(2.9)

i>k=1

it describes the interaction between the different nucleons. The potential Vik is the potential
for the two-body interaction of two nucleons (3-body and higher-body forces are neglected
here), pi and mi are the momentum and mass, respectively, for each nucleon. The Coulomb
potential is so weak on the length scales of a nucleus that it may be neglected here. The exact
form of the Hamiltonian and of the involved potentials is not known. Therefore, assumptions
must be made. The first assumption, which simplifies the problem greatly, is that in first
order one nucleon moves in a central potential generated by the remaining A − 1 nucleons.
This is called the mean-field approximation. In this theory the nucleus is now a system of
A non-interacting fermions (quasi particles) in an external potential. One can arrive at this

10

form of the problem by adding and subtracting a summed single-particle potential energy
∑A

i=1 Ui (r), which is so far unspecified, to the Hamiltonian in 2.9:



[
]  A
A
A
∑



2
∑
∑
pi
H=
+ Ui (r)
+
Vik (ri − rk ) −
Ui (r)

 

2mi
i=1

i>k=1

i=1

(2.10)

≡ TM F + VM F + VRES ≡ HM F + VRES .
HM F is the mean-field Hamiltonian and VRES is the residual interaction which is much
weaker than the original interaction V [39]. The mean-field Hamiltonian describes the motion
of a single particle in the external mean-field potential, VM F , created by the A − 1 remaining
nucleons. VRES is a small perturbative potential describing the residual interactions. It is
easy to find a solution to the Schr¨odinger equation with just HM F because the problem is
now reduced to a one-particle problem. The contribution to the wavefunctions due to VRES
are dealt with in a perturbative way, see [39] for details.
The solutions for HM F depend on the choice of the mean-field potential, VM F = U (r). A
very common initial choice is the potential of a harmonic oscillator:

1
U (r) = µω 2 r2 ,
2

(2.11)

with µ being the reduced mass of the system and ω the oscillator frequency. The energy
levels of this system depend on the major quantum number N :
(
)
3
EN = N +
ω.
2
N depends on the radial quantum number n and orbital quantum number l:

11

(2.12)

N = 2n + l.

(2.13)

With this simple potential it is possible to reproduce the first three magic numbers, N or
Z = 2, 8 and 20 for shell closures but it cannot reproduce the higher numbers. In part
(a) of Fig. 2.3 are the energy levels shown for 208 Pb calculated with a harmonic oscillator
potential.
A more realistic choice for the potential is the Woods-Saxon potential:

U (r) =

−V0
r−R
1+e a

,

(2.14)

with V0 being the depth of the potential, R and a being the diffuseness and radius parameter
[39]. For details on the values of the parameters see [4]. This potential breaks the degeneracy
of the harmonic oscillator potential but still does not reproduce all magic numbers. In part
(b) of Fig. 2.3 are the energy levels shown for 208 Pb calculated with a Woods-Saxon potential.
In 1949 it was shown by Maria Goeppert-Mayer [36] and Otto Haxel, Hans Jensen and Hans
Suess [35] that all magic numbers observed can be reproduced by adding a strong, attractive
spin-orbit coupling term to the potential:

Uso (r, ⃗ℓ, ⃗s) = Uso (r)⃗ℓ · ⃗s.

(2.15)

In part (c) of Fig. 2.3 are the energy levels shown for 208 Pb calculated with a Woods-Saxon
potential plus the spin-orbit coupling term. All the magic numbers, 2, 8, 20, 28, 50, 82 and
128 are reproduced.
The levels that have their origin in the same harmonic oscillator quantum number N are

12

usually named together. For N = 0 it is the 0s orbital, for N = 1 they are the 0p orbitals,
for N = 2 they are the 1s0d (sd) orbitals, for N = 3 they are the 1p0f (pf ) orbitals and for
N = 4 they are the 0g1d2s (gds) orbitals.

[12] 82
[2] 70
[4] 68
[6] 64
[8] 58

[10] 50
[2] 40
[4] 38
[6] 34
[8] 28
[2] 20
[4] 18
[6] 14

[2]
[4]

8
6

[2]

2

Figure 2.3 Calculated neutron single-particle energy levels for 208 Pb with a 3D harmonic
oscillator potential (a), a Woods-Saxon potential (b) and a Woods-Saxon potential plus a
spin-orbit coupling term (c). Gaps in between energy levels are marked with big numbers
and they represent the magic numbers for each potential. The levels in (c) are also marked
with their quantum numbers nℓj and are followed by two additional numbers. The first one
(in square brackets) shows the maximum allowed number of nucleons for this level and the
second one counts the total number of nucleons that can be fit in all the shells up to this
one. Figure is taken and modified from [4].
13

2.3

Shell Model Calculations and Effective Charges

For systems with more than 12 nucleons (A > 12), the Schr¨odinger equation gets so complex
that it is too computationally intensive to solve it for every particle in the system. More
precisely, the Hamiltonian matrix gets too big to be diagonalized in a reasonable time. For
A < 12 it is feasible to solve the Schr¨odinger equation for all nucleons using the full Hamiltonian including 2-body interactions. For larger systems it is possible to find the single-particle
levels easily for the mean-field potential (VM F from Eq. 2.10) as shown in Fig. 2.3. This
creates roughly the right energy levels (assuming a more elaborate potential, like the WoodsSaxon potential plus spin-orbit term, is used) but to make them more realistic, one has to
include the residual interaction which includes VRES from Eq. 2.10 of the Hamiltonian and
takes into account all the two-body interactions which are not included in the mean-field
potential. VRES is usually dealt with in a perturbative way, for more details see [39].
Because of computational limits the problem needs to be simplified. This is done by choosing
an inert core of Acore nucleons that is mostly a nearby doubly magic nucleus (it has always
only closed shells). The remaining A − Acore nucleons are the so-called valence nucleons. To
further constrain the problem, a model space is chosen. This model space limits the singleparticle energy levels that can be occupied by the valence nucleons. This space is always
chosen to be as big as possible without reaching computational limits. The model space
available is usually divided into two parts. The so-called Fermi energy divides the active
hole orbitals (orbital energy lower than the Fermi energy) from the active particle orbitals
(orbital energy higher than the Fermi energy). The Fermi energy is usually chosen in a shell
gap between a completely filled shell and a not completely filled one. For Tin, the standard
choice is tho have the Fermi energy between the 0g9/2 and the 0g7/2 shells for protons and

14

neutrons (at the magic number 50). Excitations are labeled by the fact where the nucleon
is excited from. If the nucleon is excited from below the Fermi energy, leaving a hole, it is
a hole excitation. If the nucleon is excited from a level above the Fermi energy then it is
called particle excitation. In general, the excitations are labeled by the number of particle
and hole excitations (np-mh). For every created hole there must be a created particle, n
specifies how many nucleons are excited from levels above the Fermi energy and m specifies
how many nucleons are excited out of the orbitals below the Fermi energy into levels above.

A typical shell-model calculation determines the energy levels of a nucleus in the following
way. All possible multi-particle configurations of nucleons in the model space are calculated.
Next, the Hamiltonian matrix is built with those basis wavefunctions and its eigenfunctions
and eigenvalues are found. The lowest energy found (smallest eigenvalue) corresponds to
the ground state and the corresponding combination of wavefunctions belonging to that
eigenfunction determines the configuration of that state. The next higher energy and its
corresponding wavefunction combination describes the first exited state and so on. For more
details see [4, 39].

By restricting the calculations to a model space, effects the core or states outside of the
model space might have are not accounted for. To take these effects into account, for
example for the E2 transition operator (see Sec. 2.5 for details), effective charges, ep and
en , are introduced for protons and neutrons, respectively. The E2 transition rate between
nuclear states depends on the transition operator

15

ME2 =

∑

eri2 Y 2 ei ,

(2.16)

i

where e is the electric charge unit, ei is the effective charge of the ith nucleon, Y 2 are the
spherical harmonics of rank 2, r the radius and i is summing over all nucleons. See Section
2.5 for details. The effective charges enter Eq. 2.16 for ei in the sum. They take into
account the interactions of valence nucleons with the core which are not included in the
calculations [38]. These interactions can disturb the core and induce a quadrupole moment
to the core. The coupling of the single-particle movement of the valence nucleons with the
quadrupole vibrational modes of the core is taken into account with the effective charges [40].
The effective charges are defined as:

ep = (1 + δep )

(2.17)

en = (δen ),

(2.18)

with δep/n being polarization charges for the proton and neutron. For free nucleons, the clear
choice is ep = 1 and en = 0 which reproduces the standard values for the electric charge for
protons and neutrons. In shell-model calculations, the standard choice for the polarization
charges for valence nucleons is δep = 0.5 and δen = 0.5. Depending on the model space
and interaction, these values are phenomenologically adapted but typically assumed to be
constant. It is not clear if this assumption is correct for the whole model space in every
case [40–42].
The effect of the polarization charges can directly be seen in the shell-model calculations of

16

the reduced electric quadrupole transition probability (see Sec. 2.5):

1
B(E2; Ji → Jf ) =
(en An + ep Ap )2
2Ji + 1
1
=
|Mp |2 ,
2Ji + 1

(2.19)

as well as the proton and neutron transition matrix elements:

Mp = en An + ep Ap
Mn = ep An + en Ap ,

(2.20)

where An and Ap are the neutron and proton shell-model transition amplitudes, respectively.

2.4

Excitations and Collective Excitations

Nuclei with an even number of protons and neutrons (even-even nuclei) always have a ground
state with spin-parity J π = 0+ . Their first excited state is usually a J π = 2+ state. For
magic nuclei the energy gap between this state and the ground state is large (for example
1.3 MeV for 46 Ca and 1.2 MeV for 130 Sn). It is even larger for doubly magic nuclei (for
example 3.8 MeV for 48 Ca and 4.0 MeV for 132 Sn) and smaller for non magic nuclei as can
be seen in Fig. 2.4.
The single-particle shell model explains nuclear properties by the behavior of individual
nucleons. This can be very accurate near shell closures where only few nucleons contribute.

17

4
100

3.5

Proton Number

2.5
60

2
1.5

40

Energy [MeV]

3

80

1
20
0.5
20

40

60

80
100
120
Neutron Number

140

160

0

Figure 2.4 Energies of the first 2+ states in even-even nuclei. Dashed lines mark the magic
numbers. Figure is taken and modified from [5].
Away from closed shells, one can also observe collective behavior, many valence nucleons
contribute together to the excited state energies. Geometrical models have been developed to
describe the low-lying excitation energies due to the coherent motion of the valence nucleons.
Two models will be looked at closer here. One is a vibrational model and another one a
rotational model.
In the vibrational model, the nucleus is treated as an incompressible fluid which oscillates
around an average spherical shape. The energy levels of an even-even vibrational nucleus are
evenly spaced. They are ordered in the following way a ground state with spin-parity J π =
0+ , then a first excited state with J π = 2+ , ideally followed by an almost degenerate triplet
with J π = 0+ , 2+ , 4+ and then an almost degenerate quintet with J π = 0+ , 2+ , 3+ , 4+ , 6+
[31]. The first excited J π = 2+ state comes from a single phonon. A phonon, in the quantum
18

theory of mechanical vibrations, is a quantum of vibrational energy which is produced by
mechanical vibrations [31]. The triplet comes from the coupling of two phonons, resulting in
J = 0, 2, 4 and an energy twice the size of the single phonon. The coupling of three phonons
leads to the quintet with J = 0, 2, 3, 4, 6 and three times the energy of a single phonon.
The energy of the nth excited state can be approximated by:

Evib (n) = nE(2+ ).

(2.21)

In the simplest rotational model, the nucleus is treated as a rigid, deformed rotor. The
energy levels can be calculated like this [31]:

2

Erotor (J) =

2I

J(J + 1),

(2.22)

with I being the moment of inertia of the nucleus. The deformed nucleus rotates around
a axis perpendicular to the symmetry axis of the deformation. The most common form of
deformation is the quadrupole deformation. It can be described by the radius, R(θ, ϕ), of
the nucleus which is characterized by the quadrupole deformation parameter β:

R(θ, ϕ) = R0 (1 + βY2,0 (θ, ϕ)),

(2.23)

with R0 from Eq. 2.1 and Y2,0 (θ, ϕ) the spherical harmonic. The value of β determines the
shape of the nucleus. A β > 0 gives a prolate (elongated) shape, and a β < 0 describes an
oblate (flattened) shape.

Experimental evidence that may indicate that an even-even nucleus is collective, meaning

19

+
many nucleons participate in the excitation, is a large B(E2; 0+
gs → 21 ) excitation probabil-

ity.

2.5

Reduced Quadrupole Transition Strength

A nucleus that is in a bound excited state, |i⟩ (initial state), will most of the time decay via γray emission to a lower lying state, |f ⟩ (final state). The γ-ray energy of this electromagnetic
transition is equal to the energy difference of the initial and final state:

Eγ = Ei − Ef .

(2.24)

The energy that is lost due to the recoil of the nucleus can be neglected. The transitions
can be ranked by the parity, π, and the angular momentum, J, carried away by the photon.
The carried-away angular momentum and parity of the photon depends on the spin-parity
values of the initial, Jiπ , and final, Jfπ , state of the transition. A schematic of the transition
is shown in Fig. 2.5.

π

Ei

Ji i
Eγ = Ei - Ef

∆E

π

Jf f

Ef

Figure 2.5 Schematic of the excitation of a nucleus (solid arrow) and the de-excitation via
γ-ray emission (squiggly arrow). Figure is taken and modified from [5].
20

Because the photon is a spin-1 boson, it must carry away at least a total angular momentum
unit of 1 and can never be 0. The rules that determine the nature of this electromagnetic
transition are the following.
The carried-away angular momentum determines the multipolarity, λ, of the transition. As
mentioned, λ = 0 is not allowed because the photon must at least have an angular momentum
of 1. Transitions with λ = 1 are called dipole-, with λ = 2 quadrupole-, with λ = 3 octupoletransitions and so on (the order is equal to 2λ ). Because electromagnetic transitions conserve
angular momentum, the following restriction applies:

|Ji − Jf | ≤ λ ≤ Ji + Jf ,

(2.25)

λ ̸= 0.

Furthermore parity has to be conserved and hence this equation:

πi πphoton (−1)λ = πf ,

(2.26)

has to be fulfilled. Transitions can either be electric (E), πphoton = +1, or magnetic (M),
πphoton = −1. The rules that decide this are:

πi πf (−1)λ =





+1 electric



−1 magnetic.

(2.27)

Transitions are labeled as either electric transitions with multipolarity λ, Eλ or magnetic
transitions with multipolarity λ, M λ. A transition can have several possible electromagnetic

21

decays. All forms of electromagnetic radiation are allowed that fulfill Eq. 2.25 and from Eq.
2.27 it is clear that it has to be either a combination of

M 1, E2, M 3, E4, . . . for πi πf = (+1),

(2.28)

E1, M 2, E3, M 4, . . . for πi πf = (−1).

(2.29)

or

The transition probability, T , for one transition depending on π, λ and its projection µ, is
calculated by the ’golden rule’ of time-dependent perturbation theory and is [39]:

2
λ+1
πλµ
Tf i =
ϵ0 λ[(2λ + 1)!!]2

(

)
Eγ 2λ+1
|⟨Jf Mf |Mπλµ |Ji Mi ⟩|2 .
c

(2.30)

In this equation, Mπλµ is the nuclear electromagnetic transition operator associated with
the πλµ transition, Eγ is the energy of the transition, ϵ0 is the vacuum permittivity and

is

the reduced Planck constant. The magnetic sub-states are usually not observed individually,
therefore Eq. 2.30 is averaged over the initial sub-states (Mi ) and summed over the final
sub-states (Mf ) and projections µ [39]. This results in:

Tfπλ
i =

∑ ∑ πλµ
1
Tf i
2Ji + 1
Mi Mf ,µ

2

λ+1
=
ϵ0 λ[(2λ + 1)!!]2

(

)
Eγ 2λ+1
B(πλ; Ji → Jf ).
c

The reduced transition probability B(πλ; Ji → Jf ) is defined as:

22

(2.31)

(2.32)

1
B(πλ; Ji → Jf ) ≡
|⟨J ||Mπλ ||Ji ⟩|2 .
2Ji + 1 f

(2.33)

The double bar notation indicates a reduced matrix element in the sense of the Wigner-Eckart
theorem [39]. A normal matrix element of an operator Tλµ can be rewritten with the help of
Clebsch-Gordan coefficients into a product of a geometric factor containing the projection
quantum numbers Mf , λ and Mi , and a reduced matrix element which does not contain
them [39]:




λ Ji 
J −Mf  Jf
⟨Jf Mf |Tλµ |Ji Mi ⟩ = (−1) f
 ⟨Jf ||Tλ ||Ji ⟩.

−Mf µ Mi
The relation



(2.34)



∑

λ Ji 
J −Mf  Jf
(−1) f
=1

Mf ,µ,Mi
−Mf µ Mi

(2.35)

is the reason why in Eq. 2.32 the sum drops out.
The total rate of a transition from Ji to Jf is calculated by summing over all possible
transitions, meaning to sum over the allowed combinations of π and λ,
∑
π,λ

Tfπλ
i =

∑(

)
Tf i (E) + Tf i (M ) .

(2.36)

λ

For more details see [39].
The B(πλ) value depends on the direction of the transition. Assuming that a state b is
higher in energy than a state a, the notation B(πλ; ↑) ≡ B(πλ; a → b) is defined for an
excitation and B(πλ; ↓) ≡ B(πλ; b → a) for a decay. The relation between those two is [4]:

23

B(πλ; b → a) =

2Ja + 1
B(πλ; a → b).
2Jb + 1

(2.37)

The operators Mπλ are defined as follows. For electric transitions it is:

MEλ =

A
∑

rjλ Yλµ (θ, ϕ)ej e,

(2.38)

j=1

with the sum running over all nucleons in the nucleus. e is the electric charge unit and ej
the effective charge of the j th nucleon and Yλµ is the spherical harmonic.
For magnetic transitions it is:
]
A [
2 ℓ⃗
µN ∑
s
MM λ =
g ℓ + gj ⃗sj ∇j [rjλ Yλµ (θϕ)],
c
λ+1 j j

(2.39)

j

with gjs and gjℓ being the spin and orbital angular momentum g-factors, ⃗s and ⃗ℓ are the
spin and orbital angular momentum vectors, respectively. The values of the g-factors are
gps = 5.586, gns = −3.826, gpℓ = 1, gnℓ = 0 for free protons and neutrons, and

µN =

e
= 0.105 efm ,
2mp c

(2.40)

is the nuclear magneton.
From the factors in Eq. 2.32 (with Eγ ≈ 1 MeV) it is clear that the transition rates for
electric or magnetic transitions with multipolarity λ are about seven orders of magnitude
bigger than the rates for electric or magnetic transitions with multipolarity λ + 2 (also see
Sec. 3.1). Therefore, in most cases, only the lowest multipole transition is important for a
given type (electric or magnetic) of transition. If both electric and magnetic transitions are
allowed, then the λ + 1 transitions competes with the λ transition. This is not the case for
24

this experiment but for experiments where it is, it is important to define the multipole mixing
ratio. For the most common case, a mixing of E2 and M 1, the mixing ratio δ(E2/M 1), is:
(
δ

E2
M1

√

)
=

T (E2)
,
T (M 1)

(2.41)

with the transition rate T being the inverse partial lifetime, T = τ1p (see Sec. 3.1 for details).
With equation 3.4 for the E2 transition and Eq. 3.6 for the M 1 transition, the mixing ratio
is:
(
δ

E2
M1

√

)
= Eγ

B(E2)
µN
8.29 × 10−3
.
B(M 1)
efm2 MeV

(2.42)

To judge if transitions with different multipolarities are relatively weak or strong, the reduced
transition probabilities are expressed in the so-called single-particle units which are also called
Weisskopf units. This Weisskopf unit is an estimate of the B(πλ) value for a single-particle
(proton or neutron) and how it depends upon mass [4]. By convention it is defined by:
(
BW (Eλ) =

1
4π

)[

]
1
3
(1.2A 3 )2λ e2 f m2λ ,
3+λ

(2.43)

]
1
3
(1.2A 3 )2λ−2 µ2N f m2λ−2 .
3+λ

(2.44)

and
(
BW (M λ) =

10
π

)[

The most common ones are:

2

BW (E1) = 0.0645A 3 e2 f m2 ,

25

(2.45)

4

BW (E2) = 0.0594A 3 e2 f m4 ,

(2.46)

BW (M 1) = 1.790µ2N .

(2.47)

and

In general the unit in which B(E2 ↑) values are given is e2 f m4 . It can also be converted into
e2 b2 by dividing the e2 f m4 value by a factor of 1000. 1 b is one barn and equal to 10−28 m2 .

2.6

Physics of the Tin Isotopes: Leading to

104

Sn

There is a great theoretical and experimental interest in the study of nuclei far away from βstability. The Tin isotopes offer a unique laboratory for spectroscopic studies of semi-magic
isotopes (Z = 50) over a large range of neutron numbers. The Tin isotopes have a double
shell closure for 100 Sn (Z = 50 and N = 50) and for 132 Sn (Z = 50 and N = 82).

The comparison of theoretical descriptions of nuclear excited states with spectroscopic data
obtained for the Tin isotopes will help the current understanding of nuclear systems; consisting of valence neutrons beyond the doubly magic 100 Sn core or neutron holes relative to 132 Sn.

Two such properties of interest are the energy of the first excited 2+ state of the even-even
Tin isotopes and their reduced electric quadrupole transition strength from the ground state
to the first excited state, B(E2; 0+ → 2+ ). The selection rules listed in Sec. 2.5 show that
for a transition from a 2+ level to a 0+ ground state only E2 transitions are allowed.

26

The excitation energy E(2+ ) and the B(E2 ↑) strength for the light even-even Tin nuclei
provide information regarding the onset of collectivity with the addition of valence neutrons
to the doubly-magic 100 Sn core. The energies of the first excited 2+ state for the majority
of Sn isotopes between the N = 50 and N = 82 shell closures are well established and consistent with results from large-scale shell-model calculations. The energies of the first exited
2+ state for even-even Tin isotopes with N = 52 → 82 are shown in Fig. 2.6.

4500
4000

E(2+) [keV]

3500
3000
2500
2000
1500
1000
500
0
100

104

108

112

116

120

124

128

132

Mass number (A)
Figure 2.6 Measured energy of the first excited 2+ state for the even-even Tin isotopes
between the N = 50 and N = 82 shell closures. Data is taken from [6].

Prior to the current study, B(E2 ↑) data for even-even Tin nuclei existed only up to 106 Sn.
No data was available for the more neutron-deficient Tin nuclei. Measured B(E2 ↑) values
agree well with theoretical predictions of the shell model up to mid-shell (116 Sn). Below midshell, recent experiments have yielded B(E2 ↑) values consistently higher than predicted by
27

theory. The enhanced E2 transition strengths in the neutron-deficient Sn isotopes below
mid-shell are not reproduced by results from shell-model calculations. During the analysis
of this experiment, two B(E2 ↑) values were published for 104 Sn [1, 7] which disagree with
each other. In Fig. 2.7, an overview is shown of the known B(E2 ↑) values for even-even Tin
isotopes. Currently, no theoretical framework exists that describes the evolution of both the
energy of the first excited 2+ state and B(E2; 0+ → 2+ ) well for the Tin isotopes between

GSI-DSA
Adopted
Riken
REX-ISOLDE
GSI
NSCL
ORNL
IUAC
HRIBF

0.25

2 2

B(E2;0+1 2+1) [e b ]

N = 50 and N = 82.

0.2
0.15
0.1
0.05
50

55

60
65
70
Neutron number

75

80

Figure 2.7 Measured B(E2 ↑) values for the chain of even-even Tin isotopes. Data are taken
from: Adopted values are published by the National Nuclear Data Center [6], RIKEN [7],
REX-ISOLDE [8, 9], GSI-DSA [10], GSI [1, 11, 12], NSCL [13], ORNL [14], IUAC [15],
HRIBF [16]. The dotted line reproduces the theoretical calculation shown in [11] with
gds (tπ = 4, see Sec. 6 for details).

The goal of the current study is to measure the intermediate-energy Coulomb excitation
cross section for the population of the first excited 2+ state of 104 Sn and extract the reduced
transition strength B(E2; 0+ → 2+ ). Due to experimental restrictions (see Sec. 5.3.2.3),
the measurement was done relative to the well-established B(E2 ↑) value of 102 Cd [2, 43].
28

102 Cd was more intensely present in the cocktail beam than 104 Sn (see Sec. 5.2 and 5.5).

With that, another data point is added for the B(E2 ↑) value of 104 Sn. This will help
clarifying the disagreement between the current values [1] and [7]. The second goal is to
find a theoretical explanation for the behavior of the B(E2 ↑) values of the even-even Tin
isotopes below mid-shell.

29

Chapter 3
Quadrupole Collectivity:
Experimental Methods
In this chapter, two common techniques to measure the strength of the electric quadrupole
transition are described. Excited-state lifetimes are briefly discussed and the method of
intermediate-energy Coulomb excitation is explained in more detail.

3.1

Excited-state Lifetimes

The lifetime τif of an excited state is directly related to the transition probability per unit
πλ between an initial and final state (|i⟩ and |f ⟩) via
time Tif

τif =

1
.
Tif

(3.1)

The transition probability on the other hand is related to the excitation strength value B(πλ)
as follows (see Eq. 2.32):

λ+1
πλ = 2
Ti→f
ϵ0 λ[(2λ + 1)!!]2

(

)
Eγ 2λ+1
B(πλ; Ji → Jf ).
c

(3.2)

For the most common transitions, the partial mean lifetime τp and the transition probability
B(πλ ↓) are related by [4]:

30

B(E1) =

0.629 2 2
e f m M eV 3 f s,
Eγ3 τp

(3.3)

B(E2) =

816 2 4
e f m M eV 5 ps,
Eγ5 τp

(3.4)

B(E3) =

1760 2 6
e f m M eV 7 µs,
Eγ7 τp

(3.5)

for electric transitions and for magnetic transitions:

56.8 2
µN M eV 3 f s,
3
Eγ τp

(3.6)

B(M 2) =

74.1 2
µN f m2 M eV 5 ns,
5
Eγ τp

(3.7)

B(M 3) =

0.1585 2
µ f m4 M eV 7 s.
Eγ7 τp N

(3.8)

B(M 1) =

As shown, a lifetime measurement can be used to extract B(E2 ↑) values.

3.2

Intermediate-Energy Coulomb Excitation

The technique used in this experiment is called intermediate-energy Coulomb excitation. In
contrast to “safe” (energies below the Coulomb barrier) Coulomb excitation, the energy of
the projectile is higher than the Coulomb barrier energy. Typically projectile energies over 50
MeV/u are used together with high-Z targets like gold. Targets with high-Z values are used

31

to generate strong Coulomb fields. The projectile and/or target get excited in the Coulomb
field of one another, leading to excitations from an initial state |i⟩ to a final state |f ⟩. Because
this happens at energies higher than the Coulomb barrier, it can lead to situations where
projectile and target get so close that it is possible for nuclear interactions to contribute to
the excitation process. If this happens, the excitation process becomes complicated. For
such processes, the measured excitation cross section would not be easily translated into
the electric quadrupole transition strength. Only Coulomb interactions must contribute to
the cross section for the least model-dependent analysis. NSCL (National Superconducting
Cyclotron Laboratory) uses projectile fragmentation as an efficient production method for
fast rare-isotope beams (see Sec. 4.2). At high energies, events with small scattering angles,
corresponding to large impact parameters, have to be analyzed. Small impact parameters
can lead to nuclear interactions. If nuclear contributions were allowed, it would make the
extraction of the transition matrix element difficult and model dependent. Nuclear contributions are theoretically much more uncertain and these uncertainties would fold into the
uncertainty of the transition strength.

In the present experiment, the process of interest is where the projectile nuclei 104 Sn and
102 Cd start in the 0+ ground state (initial state), get excited to the first excited 2+ state

(final state) and de-excite again to the 0+ ground state (final-final state which is the initial
state). In all calculations the index i stands for the initial state from which the excitation
process starts, f stands for the final state where the excitation leads to and f f stands for
the final-final state, the state to where the final state (f ) de-excites to. In this experiment,
the initial (i) and final-final (f f ) state are the same.
Fig. 3.1 gives an overview of the excitation and de-excitation process. The minimum allowed
32

cm
impact parameter bmin is related to the maximum center-of-mass scattering angle θmax

by [44]:

a
bmin = 0 cot
γ
with γ = 1/

( cm )
θmax
,
2

(3.9)

√
1 − (v/c)2 and a0 is the half-distance of closest approach:

a0 =

Zt Zp e2
.
µv 2

(3.10)

Zt(p) is the atomic number of the target (projectile), e is the electric charge and µ the reduced
mass of the system. To ensure that only events without nuclear contributions are considered,
the analysis has to be restricted to impact parameters that are at least two Fermi larger
than the sum of the nuclear radii of the target-projectile system (“touching-sphere system”)
[44]:
)
(
1/3
1/3
+ 2 fm,
bmin = r0 At + Ap

(3.11)

where r0 = 1.25f m.
From here on it is assumed that only the Coulomb interaction contributed to the excitation
process. The following equations describe the Coulomb excitation of the projectile, to describe the excitation of the target all that needs to be done is to simply replace Zp and Zt
in the following equations.
The excitation strength is calculated from the cross section of the Coulomb excitation process.
The cross section for the intermediate-energy Coulomb de-excitation of the first exited 2+
state to the 0+ ground state of an even-even nucleus is deduced from the data like this:

33

Jf
γ-ray
Coulomb
excitation

γ-ray
decay
Ji=Jff

v/c

Θ

projectile

b
target

γ-ray
Figure 3.1 Schematic picture of the intermediate-energy Coulomb scattering process. Shown
are the projectile and target nucleus as well as center-of-mass scattering angle θ and impact
parameter b (see text for details). The insert shows the excitation and de-excitation process
in a level scheme. The γ-ray energy is equal to the energy difference of the nuclear levels.
Figure is taken and modified from [3].

σ2+ →0+ =
1

g.s.

Nγ
.
NBeam NT arget

(3.12)

It is the number of γ rays emitted (Nγ ) divided by the number of incident beam particles
(NBeam ) and the area number density of target nuclei (NT arget in #2 ).
cm

Alder and Winther developed a semi-classical theory that connects the cross sections of
intermediate-energy Coulomb excitation to the property of interest; the B(E2 ↑) value [45].

34

By assuming that the scattering process takes place along a classical trajectory, the total
differential excitation cross section can be described as the product of the inelastic scattering
Rutherford cross section [46] and the (comparably small) Coulomb excitation probability
Pi→f :
(

( )
)
dσ
dσ
=
P
.
dΩ i→f
dΩ Rutherford i→f

(3.13)

The excitation probability Pi→f is defined by the excitation amplitude ai→f :

Pi→f = ai→f

2

.

(3.14)

This amplitude can be evaluated via time-dependent perturbation theory if it is assumed
that the Coulomb potential VC (r(t)) is a small, time dependent perturbation:

1
ai→f =
i

∫ ∞
−∞

iω t
dt e f i ⟨f |VC (r(t))|i⟩ ,

(3.15)

(
)
with ωf i = Ef − Ei / = ∆E/ .
It is obvious now why it has to be ensured that the impact parameter is not too small. This
formalism assumes that only the Coulomb potential contributes to the perturbation and this
is only certain if nuclear contributions are excluded.
Now the potential VC (r(t)) can be expanded into its multipole components Vλµ (r). In turn,
these can be expressed in terms of electric and magnetic multipole matrix elements, M (πλµ),
where π = +1 ≡ E stands for electric transitions, π = −1 ≡ M for magnetic transitions, λ is
the multipolarity of the excitation and µ is its projection. The excitation amplitude written
in those terms is then:

35

ai→f = i

∑ (πλ)
χi→f fλ (ξ)

(3.16)

λ
(πλ)

with χi→f being a measure of the interaction strength,

(πλ)

χi→f ≈

Zt e ⟨f |M (πλµ)|i⟩
.
cbλ

(3.17)

(πλ)

Exactly which M (πλµ) contribute to χi→f and therefore to the sum in Eq. 3.16 depends
on the initial and final state of the transition. Only certain combinations are allowed. For
the transition from 0+ to 2+ only λ = 2 and π = E is allowed (see Sec. 2.5). The function
fλ (ξ) puts the cross section into connection with the adiabaticity of the reaction [45]. It is
formulated in terms of the adiabaticity parameter, ξ, which can be expressed as the ratio of
two time constants, namely the ratio of the collision time τcoll (time the projectile spends
in the vicinity of the target) and the timescale of the internal motion of the nucleus that is
undergoing the excitation, τnuc :

τ
ξ = coll .
τnuc

(3.18)

The two times can be calculated as follows:

τcoll =

b
,
γv

1
=
τnuc =
ωf i
∆E

(3.19)

(
)
with Ef − Ei = ∆E. Low velocities, large excitation energies or large impact parameters
result in a large ξ which means the reaction is adiabatic. This indicates the projectile moves
gently through the Coulomb field, following the slowly changing, time-dependent potential
36

and it is not very likely that it gets excited. For large ξ the function fλ (ξ) should fall of
exponentially as e−πξ . On the other hand, small values of ξ bring violent reactions and
excitation is more likely to occur. As ξ approaches zero (ξ → 0) fλ (ξ) approaches 1.
As mentioned before it is presumed that the trajectories are straight lines which is not
completely the case due to the Coulomb field repulsion. Alder and Winter showed that the
introduction of a rescaled impact parameter takes this first order deviation from a straight
line into account [45]. The distance of closest approach has to be increased by the following:

b→b+

π a0
= ba .
2 γ

(3.20)

This results in the adjusted adiabaticity parameter:

∆E
ξ=
γv

(
)
π a0
∆E
b+
=
ba .
2 γ
γv

(3.21)

Adiabatic cutoff sets in when ξ = 1 and this corresponds to an impact parameter:

ba =

vγ
,
∆E

(3.22)

which means that a maximum excitation energy of ∆E ≈ γ v/ba is possible.
The total cross section is obtained by summing up all contributions from all multipole matrix
elements:

σi→f =

∑

σπλ

(3.23)

πλ

The individual cross section for the different multipole elements can be approximated by
(πλ) 2

integrating the corresponding χi→f

from bmin to ba :
37

∫ ba
σπλ = 2π

bmin

b dbPi→f ≈ 2π

∫ ba

b db χ(πλ) (b)

2

(3.24)

bmin

resulting in:

(
σπλ ≈

Zt e2
c

)2

π

B (πλ; 0 → λ)
2λ−2

e2 bmin





(λ − 1)−1

for λ ≥ 2




2 ln (ba /bmin ) for λ = 1

,

(3.25)

where B(πλ; i → f ) is the reduced transition probability as defined in Eq. 2.33. For more
details see Sec. 2.5. The direct dependence on Zt , the target charge, is usually taken
advantage of by using high-Z targets in experiments.
Equation 3.25 shows that there is a proportionality between the B(E2 ↑) value and the
Coulomb excitation cross section. It is also important to note again that for transitions from
a 0+ state to a 2+ state only electric quadrupole transitions are allowed (see Sec. 2.5).
This means that the sum in Eq. 3.23 has only one contribution, namely σE2 and therefore there is a direct correlation between the total cross section σi→f and the sought after
B(E2; 0+ → 2+ ) value.
All calculations using cross sections to calculate excitation strengths and calculations of
angular distributions of γ rays are performed with a mathematica script described in [47].
For a more detailed calculation see [45, 47], the results are listed here. For completeness all
the math that is the basis of the matematica script is included even though not all of it is
used for this work.

The excitation amplitude is:

38

(c)
√
Zt e2 ∑
ai→f = −i
Gπλµ
(−1)µ Kµ (ξ (b)) k λ 2λ + 1
vγ
v
πλµ
⟨
⟩
Jf Mf M (πλ − µ) Ji Mi
. (3.26)
×
e

k = ∆E/ c and Kµ are modified Bessel functions. For electric transitions, the function
Gπλµ = GEλµ is defined as (µ ≥ 0):

GEλµ

(c)
v

√
(
)1 (
)− 1
2
16π
(λ − µ)! 2 ( c )2
λ+µ
=i
−1
λ (2λ + 1)!! (λ + µ)!
v
(
)
(λ + 1) (λ + µ) µ ( c ) λ (λ − µ + 1) µ ( c )
×
Pλ−1
−
Pλ+1
(3.27)
2λ + 1
v
2λ + 1
v

and for magnetic transitions, Gπλµ = GM λµ it is (µ ≥ 0):

GM λµ

(c)
v

√
)− 1
(
) 1 (( )
( )
2
2
2
(λ
−
µ)!
16π
c
µ c
λ+µ+1
=i
−1
,
µPλ
λ (2λ + 1)!! (λ + µ)!
v
v

(3.28)

µ

with Pλ being the associated Legendre polynomials. For negative µ apply the easy relations:
GEλ−µ = (−)µ GEλµ ,
GM λ−µ = (−)µ+1 GM λµ .
The full cross section ends up to be:

39

(3.29)

∫ ∞

∑
b db
2
ai→ f
bmin 2Ji + 1 M M
i f
(
)
(
)
2
(c) 2
∑
B
πλ;
J
→
J
Zt e2
i
f
=
k 2(λ−1)
G
gµ (ξ (bmin )) ,
πλµ
c
v
e2

σi→f = 2π

(3.30)

πλµ

with gµ (ξ (bmin )) being defined as

(
gµ (ξ (bmin )) = 2π

) ∫
ω 2 ∞
2
b db Kµ (ξ (b))
vγ
bmin

∫ ∞
= 2π

2

Kµ (x) x dx,

(3.31)

ξ

which can be rewritten using modified Bessel functions:

[
gµ (ξ (bmin )) = g−µ (ξ (bmin )) = πξ 2

Kµ+1 (ξ)

2

− Kµ (ξ)

2

]
2µ
K
(ξ) Kµ (ξ) .
−
ξ µ+1
(3.32)

The angular distribution of γ rays emitted during Coulomb excitation processes is not
isotropic due to the magnetic sub states not being equally populated [48]. The angular
distribution can be parametrized as:

W (θ) =

∑

ak Pk (cos θ)

(3.33)

k even

with k running from 0 to the smaller one of 2λ and 2Ji . Pk are the Legendre polynomials.
In [47] one can find that the coefficients ak are given by:
40

ak =

∑
µLL′

Gλµ

(c) 2
v





λ λ k 
gµ (ξ) (−1)µ 

µ −µ 0



Jf Jf k 
 (
)√
2k + 1 δL δL′ . (3.34)
×
Fk L, L′ , Jf f , Jf

λ λ J

i

(
)
The γ − γ correlation function Fk L, L′ , Jf , Ji is

(
)
J +J −1 √
Fk L, L′ , Jf , Ji = (−1) f i
(2k + 1) (2Ji + 1) (2L + 1) (2L′ + 1)





′
′


L L k  L L k
. (3.35)
×





Jf Ji Jf
1 −1 0

41

Chapter 4
Experimental Setup

4.1

Overview of the Setup

All measurements were performed at the National Superconducting Cyclotron Laboratory
(NSCL) at Michigan State University [49]. Due to the short lifetime of 104 Sn, namely 21
seconds [6], this experiment had to be conducted in inverse kinematics with the projectile
beam being excited, requiring the production of a radioactive 104 Sn beam. The radioactive
beam for this experiment was produced by fragmentation of stable 124 Xe ions, also called
primary beam. After the production of the primary beam ions in the Electron Cyclotron
Resonance (ECR) ion source of the NSCL, they were accelerated to 15% of the speed of light
in the K500 cyclotron. After further stripping (producing higher charge states) and acceleration in the K1200 cyclotron, the beam reached a velocity of roughly vc = 0.5. The primary
beam impinged upon a 240 mg/cm2 9 Be target to produce a wide variety of fragmentation
products, also called secondary beam. A first selection of the fragmentation products of
interest occurred within the A1900 fragment separator [17] (see Fig. 4.1 for details).
Due to high contamination with less exotic fragmentation products, the secondary beam
was sent to the Radio Frequency Fragment Separator (RFFS) [19] to increase the purity of
the isotopes of interest. Finally, the beam was sent to the experimental station consisting
of a 184 mg/cm2 197 Au target surrounded by the highly segmented CAEsium iodide ARray
(CAESAR) [20]. In addition to CAESAR, other detectors were used including a Silicon p-i-n
42

ion sources

K500
cyclotron

RT-ECR

20 ft

ST-ECR
10 m

K1200
cyclotron

focal
plane

A1900
image 2
stripper
foil

production
target

image 1
image 3
wedge

Figure 4.1 Schematic overview of the beam production at the NSCL. Figure is modified
from [17].
detector for energy loss measurements, two thin plastic timing scintillators for time-of-flight
(time it takes a projectile to travel a certain distance) measurements, one at the exit of the
A1900 (XFP position) and one at the exit of the RFFS (RF position), as well as a Phoswich
detector for particle tagging and two Parallel Plate Avalanche Counter (PPAC) for position
measurements (one in front and one after CAESAR). A layout drawing of the setup of the
experimental station is shown in Fig. 4.2
The experiment was performed at the end of 2011 over a period of 7 days. In total 111 data
runs with beam were recorded. This equals roughly 80 hours of beam data.

43

Beam
CAESAR
A1900
Downstream PPAC
RF ScinƟllator Upstream PPAC
RF
Fragment
Seperator

XFP
ScinƟllator

Θlab

300 µm Si p-i-n Detector

Fast-/SlowAu target Phoswich
Detector

197

Figure 4.2 Schematic overview of the experimental setup after the A1900 in the S2 experimental vault.

4.2
4.2.1

Radioactive Isotope Production
Primary Beam Production

The production of the stable beam starts at the ion source. Two different ion sources are
available at the NSCL which are both based on the Electron Cyclotron Resonance (ECR)
principle. The first one is the Advanced Room TEMperature Ion Source (ARTEMIS) [50] and
the second one is the Superconducting Source for Ions (SuSI) [51,52] which was used for this
experiment. The stable 124 Xe gas is injected into the source’s plasma chamber. To create
the needed plasma the few already existing free electrons are accelerated with a microwave
field, this starts the ionization process which creates more free electrons and in the end the
plasma. The microwave field is tuned to the electron cyclotron frequency ωc :

ωc =

eB
,
me

(4.1)

where e and me are the electron charge and mass, respectively, and B is the magnetic
field inside the plasma chamber. This magnetic field is created by a magnetic trap which is
44

needed to prevent the ions from escaping the source. The trap consists of six superconducting
solenoid magnets which create a magnetic mirror field.
To reach the highest possible energy the 124 Xe was stripped of 20 electrons in the plasma
and reached a charge state of 124 Xe20+ . At the end of the source the ions get extracted from
the trap by an extraction system and then accelerated and focused before they are injected
into the K500 cyclotron.
The cyclotrons use a magnetic field (between 3 - 5 Tesla) to confine the beam and a radio
frequency electric field for acceleration. The electric field is applied between the gaps of 3
“dees” and “hills” (see Fig. 4.3).
The Lorentz law gives the following dependence for the radius ρ of the motion [53]:

ρ=

p
γmv
=
,
Bq
Bq

(4.2)

where p is the momentum, B the magnetic field strength, q the charge, γ the relativistic
gamma factor, m the mass and v the velocity of the particle. The quantity Bρ is often
referred to as magnetic rigidity or just as rigidity. From Eq. 4.2 it is easy to see that Bρ is:

Bρ =

p
γmv
=
,
q
q

(4.3)

The radius of the circular motion of the beam depends on the velocity of the beam, as
the beam gets more energetic and therefore faster, the radius grows. When the radius gets
close to the radius of the cyclotron, the beam is extracted and gets transferred to the next
cyclotron, the K1200. The final energy after the K500 was 12.23 MeV/u for 124 Xe20+ . In
the middle of the K1200 is a carbon “stripper foil” which strips away more electrons and
the result is a 124 Xe48+ beam which helps to increase the beam energy even more. After
45

Dee
Injection
Gap

Extraction
Hill

K500 Cyclotron

Figure 4.3 Schematic of the K500 cyclotron. The dees are red and the hills are blue. Figure
is taken and modified from [5].
the K1200 the primary beam has reached a final energy of 140 MeV/u.

4.2.2

Secondary Beam Production

The beam of interest was produced by fragmentation of the primary beam on a stationary
9 Be target which is located after the K1200 cyclotron (see Fig. 4.1). A 240 mg/cm2 9 Be

46

production target was used. The primary beam was specifically chosen to achieve the highest
yield possible of 104 Sn, according to a LISE++ calculation [18]. The fragmentation can be
described as a two-step process. First there is a highly excited prefragment created by
removing some nucleons from the 124 Xe nucleus. In a second step, the prefragment decays
by statistical nucleon emission [54] (abrasion-ablation model). Of course not only are the
elements of interest produced but there are other, unwanted, fragmentation products, too.
These “contaminants” include a wide range of stable and radioactive nuclei, as well as a high
percentage of unreacted and now fully-stripped 124 Xe54+ . Therefore, it is very important to
filter out the isotope of interest, which is the fully-stripped 104 Sn50+ .
The first filtering is done with NSCL’s A1900 fragment separator [17, 55]. It is a three step
process. It employs the Bρ-∆E-Bρ technique. The first step is a magnetic filtering with
a 45◦ dipole magnet. Equation 4.3 shows that for a set Bρ value, meaning a magnetic
field with strength B and a path of radius ρ, only particles with a specific momentum to
charge ratio, pq , can pass through the dipole (p being the momentum of the particle and q its
electric charge). Particles that are not in the range of this set value will either be deflected
too much or too little and will run into either side of the magnet. Due to the size of the
magnet, several radii are possible for the particles to take and a range of pq values can make
it through this selection. Therefore, further purification is needed. Several slits are available
that can be opened and closed in the separator. They are positioned at the image positions
and at the focal plane of the A1900 (Fig. 4.1). Because of the difference in deflection in the
dipole magnets, slits are used to block certain particles which restricts the total momentum
spread. This means that contaminants that make it through the magnet but are deflected
more or less than the nuclei of interest are blocked out. Several more magnets for focusing
purposes are distributed along the beam line. The second step in the purification is an energy
47

loss filter with an achromatic Aluminum wedge degrader, consisting of a curved Aluminum
foil [56]. To calculate the energy loss, the Bethe formula can be used [21]:
[
(
4πe4 Zp2
dE
2m0 v 2
−
=
N Zt ln
− ln 1 −
dx
I
m0 v 2

v2
c2

)

]
v2
− 2 .
c

(4.4)

dE is the differential energy loss per differential path length for a charged particle in the
dx

degrader material, e is the electron charge, Zp and Zt are the atomic numbers of the projectile
and degrader respectively, m0 is the electron rest mass, v is the velocity of the charged
particle, N is the number density of atoms in the degrader and I stands for the average
excitation and ionization potential of the degrader. Clearly the energy loss is proportional
to Zp2 which means different elements lose different amounts of energy when they pass through
the wedge, depending on their atomic number. This helps with the selection of the particles
of interest, 104 Sn. In this experiment, the degrader used was a 150 mg/cm2 Al wedge. The
third and last step is again a Bρ filter. The slits along the beam line were so far closed that
the overall momentum acceptance was 0.41%.
This resulted in a 104 Sn purity of just 0.006%. The high levels of contamination originate
from low-momentum tails of higher rigidity fragmentation products that extend exponentially
and overlap with the momentum acceptance of the fragment separator [57]. Therefore,
NSCL’s Radio Frequency Fragment Separator (RFFS) was used for additional filtering.

4.3

Radio Frequency Fragment Separator

The main reason for the high level of contamination are the low-momentum tails of higher
rigidity fragments that extend exponentially and overlap with the momentum acceptance of
the fragment separator [57]. The process of projectile fragmentation, which is used to produce
48

the secondary beam, leads to the fact that the produced fragmentation residues exhibit
exponential-like tails on the lower end of the momentum distribution. This happens because
a large fraction of the projectile’s kinetic energy dissipates in the 9 Be target. Unfortunately,
neutron-deficient isotopes like the 104 Sn have a smaller Bρ compared to the more stable
isotopes and therefore fall under those exponential tails of the more abundant isotopes. In
Fig. 4.4 an example from a LISE++ [18] simulation for the production of 104 Sn is shown.

Yield [arb. units]

10 7
10 6
10 5
10 4
10 3
10

104

Sn

2

10 1
2.7

2.9

3.1

3.3
3.5
Bρ (Tm)

3.7

3.9

4.1

Figure 4.4 Simulation of the momentum distribution of the contaminants as well as 104 Sn
that are produced during the fragmentation process for the settings used in this experiment.
In red the 104 Sn is shown and in black the contaminants. The area in between the two green
lines shows the momentum acceptance. The figure was made with LISE++ [18].

It is evident that the order of magnitude of the amount of transmitted contaminants is many
orders higher than the one for the 104 Sn isotope. It also shows that the yield of more less
neutron-deficient contaminants increases exponentially with higher Bρ settings, whereas the
neutron-deficient isotopes show a steep fall off.
As the production and selection process within the A1900 produces isotopes with very different velocities, this can be used as an advantage. The velocity of an isotope for a fixed Bρ

49

Figure 4.5 Top: Drawing of the RFFS; Bottom: Photo of the RFFS. Taken and modified
from [19].

50

value is inversely proportional to its mass to charge ratio (see equation 4.3). This fact is
used for further purification with the Radio Frequency Fragment Separator (RFFS) which is
effectively a velocity filter.
A photo and drawing of the RFFS can be seen in Fig. 4.5. The cavity is 1.5 m long, has a
gap of 5 cm, a peak voltage of 100 kV and operates at frequencies between 19 and 27 MHz
(for this experiment it was tuned to match the cyclotron frequency of 23.1 MHz).
It employs a RF field perpendicular to the beam direction, synchronized with the RF of the
cyclotrons. This leads to periodic deflections varying in magnitude for the different beam
contaminants based on their velocities. The contaminants experience a different electric deflection than 104 Sn.

The transversal electric field leads to a phase dependent deflection of the different beam
contaminants. It was tuned to deliver a maximum positive deflection for 104 Sn and a blocker
was used to block out unwanted isotopes which were less deflected. In Fig. 4.6, a LISE++
simulation for this setup is shown, where the Tin gets deflected the most in the positive
direction and a majority of the contamination can be blocked out using a single-blade blocker.
This process increased the 104 Sn purity in the secondary beam cocktail by two orders of
magnitude (roughly a factor of 200). At the experimental end station downstream of the
RFFS, the resulting rare-isotope beam was composed of 1.3% 104 Sn and 2.9% 102 Cd at
average rates of 10 and 26 particles per second, for 104 Sn and 102 Cd, respectively.

51

20

Y-position (mm)

16

104

Sn

92

12

Zr

8
4
0
-4
-8
-12
-16

98

-20
455

465

Rh

475

485

495

505

515

Time of flight (arb. units)
Figure 4.6 LISE++ simulation for this setup. The yellow shaded area indicates the beam
blocker at the end of the RFFS. Everything in the yellow area gets blocked out of the beam
cocktail.

4.4

CAESAR

To measure the γ rays produced in the Coulomb excitation process, the CsI(Na) detector
array called CAESAR [20] was used. For the intermediate-energy Coulomb excitation experiment that was performed here, the incoming even-even 104 Sn and 102 Cd nuclei were
excited from the 0+ ground state to the first 2+ state in the Coulomb field of a gold target.
This was followed by a prompt (in the order of a few pico seconds) de-excitation of the 2+
state back to the 0+ ground state by emission of a γ ray. To calculate a B(E2 ↑) value it is
necessary to know how often this process happens compared to the total number of incoming
beam particles (see Eq. 3.12 and 3.25). CAESAR surrounded the gold target used in this
experiment. It was a 184 mg/cm2 197 Au target and placed in the center of CAESAR, about
52

12.8 m downstream from the RFFS. CAESAR consists of 192 detectors in total providing a
solid angle coverage of 95% of 4π. 48 of the detectors use 3 × 3 × 3 inch crystals and 144
were built with 2 × 2 × 4 inch crystals. To optimize angle coverage and spatial granularity,
a special geometry was applied, see Fig. 4.7 and 4.8 [20]. CEASAR is made out of 10 rings
as shown in the figures. The first and last ring (ring A and J) hold 10 detectors each, the
second and second to last ring (ring B and I) hold 14 detectors each and the remaining 6
rings (ring C-H) hold 24 detectors each. The high detection efficiency was needed due to the
low production rate of the secondary beam ions 104 Sn (purity of just 1.3%). With this setup
it is possible to reach a full-energy peak efficiency of 30% for 1 MeV γ rays. Because the
nuclei are traveling at a large fraction of the speed of light, the emitted γ-ray energies are
significantly Doppler shifted. This means the detected energy in the lab frame is connected
to the energy in the projectile frame by:
√
1 − β2
Eγ
=
,
(1 − β cos θ)
γ (1 − β cos θ)

Eγ
Eγlab =

(4.5)

with θ being the angle of emission of the γ ray with respect to the direction of the beam. If
β and θ are known, Eγlab can be Doppler corrected to obtain Eγ :

Eγ = Eγlab γ(1 − β cos θlab ).

(4.6)

As seen above, the Doppler corrected energy depends on the γ-ray emission angle. This
necessitates a good spatial resolution (of the array) in order to determine the emission angle
with acceptable precision (192 detectors).

53

Figure 4.7 Technical drawing of the CAESAR setup. On the bottom is the mounting table
shown with which one can move the array horizontally. On top of the table is CAESAR
mounted. The mounting brackets are clearly visible as well as the detectors’ PMTs and the
encased crystals (green and blue).
To explain how γ rays are actually detected it is necessary to explain how they interact with
matter and more precisely with the CsI(Na) crystals. There are three main processes which
describe how γ rays interact with matter. These are the photoelectric effect, Compton effect
and pair production. Each of these processes generates free electrons with a certain energy
in the crystal which then subsequently create electron-hole pairs in the crystal. When those
pairs recombine, the crystal emits light (a photon) with a wavelength in the visible spectrum
(around 430 nm [21]). This light is then collected in a Photomultiplier Tube (PMT) and

54

D E F G
C
B

H
I
J

A

beam

Figure 4.8 Schematic illustration of the arrangement of the CAESAR detectors. Left: cross
sectional view of the rings F and J perpendicular to the beam axis. Right: Picture of the
ten rings A (most upstream) to J (most downstream) with the target position in red. The
gray scale corresponds to the position on the rings, as shown on the left. Picture taken and
modified from [20].
transformed into a current that is proportional to the number of photons created in the
crystal, which itself is proportional to the incoming γ-ray energy. A typical spectrum for
mono-energetic γ rays is shown in Fig. 4.9. The different components of the spectrum are
explained in Sec. 4.4.1-4.4.3.

4.4.1

Photoelectric Effect

The typical energy range within which the photo electric effect occurs is between a few tens
of keV up to a few hundred keV. The γ ray is completely absorbed and an electron is ejected
from the atom (see Fig. 4.10). The electron has a remaining kinetic energy of [21]

Ee− = Eγ − Eb ,

55

(4.7)

full-energy peak
Compton edge

single-escape peak

double-escape peak

Counts

Compton continuum

E

γ

E

-2

x5 1
1k

γ

eV

-5

E
11

CE

E

γ

keV

Energy
Figure 4.9 Example of a γ-ray detector response to many mono-energetic γ rays. The components of the spectrum are explained in the text. Picture taken and modified from [3].
where Eγ is the energy of the photon and Eb the binding energy of the electron. The
probability per atom for this absorption to occur can be approximated by, Z n /Eγ3.5 , where
n slightly varies with the γ-ray energy and increases from about 4 at 100 keV to 4.6 at 3
MeV [58]. If the electron does not leave the detector volume then the full energy of the γ
ray is detected and contributes to the full-energy peak.

56

e-

E e - = E γ - E binding

Eγ
Nucleus

Atom

Figure 4.10 Schematic illustration of the photoelectric absorption process. Picture taken and
modified from [3].

4.4.2

Compton Effect

The typical energy range within which the Compton effect occurs is between hundreds of
keV and a few MeV. The photon scatters of an atom’s electron and transfers some of its
energy Eγ to the electron (see Fig. 4.11) (inelastic scattering). After the scattering process
it has the energy Eγ′ left. The partial energy transfer can be calculated as follows [59]:

(1 − cos θ)
1
1
=
−
,
′
Eγ
Eγ
me c 2

(4.8)

which can be rewritten as

Eγ′ =

(
1+

Eγ
)
.
(1 − cos θ)

Eγ
me c2

(4.9)

The transferred energy depends on the scattering angle θ. The upper and lower limits of the
energy transfer are easily calculated. It is 0 MeV and

Eγ′ (180◦ ) = ECE =

Eγ
1+

2Eγ
me c2

,

(4.10)

corresponding to the angles 0◦ and 180◦ respectively. This leads to the characteristic form

57

of the spectrum with the Compton edge and continuum. The broad feature in the spectrum
between 0 and ECE is called Compton continuum (see Fig. 4.9). If the scattered photon
does not escape the crystal and gets fully absorbed after the scattering process in the same
detector then this event contributes to the full-energy peak. This is because the full energy
of the incoming photon is deposited in one crystal. This process has an angular distribution
that follows the Klein-Nishina formula for the differential scattering cross section [21, 60]:

dσ
= r02 Z
dΩ

(

)
)2 (
)(
α2 (1 − cos θ)2
1
1 + cos2 θ
)
1+ (
,
1 + α (1 − cos θ)
2
1 + cos2 θ [1 + α (1 − cos θ)]
(4.11)

where α ≡ Eγ /m0 c2 and r0 is the classical electron radius:

r0 =

1
e2
≈ 2.82 × 10−15 m.
2
4πϵ0 me c

(4.12)

E '
E

Θ

e-

e-

Figure 4.11 Schematic illustration of Compton scattering.
from [3].

4.4.3

Picture taken and modified

Pair Production

If the incident γ-ray energy is larger than twice the rest mass of an electron (2m0 c2 =1.022
MeV) then pair production is possible. In the vicinity of a nucleus an electron-positron pair
is produced and the energy that remains is transferred to kinetic energy (shown in Fig. 4.12).
58

Ee− + Ee+ = Eγ − 2m0 c2 ,

(4.13)

The positron will annihilate in the scintillator material almost immediately and thereby
produce two back-to-back 511 keV γ rays. If one of those γ rays manages to escape the
detector then this will show as the characteristic single-escape peak line at the energy:

ESE = Eγ − 511keV,

(4.14)

and if both γ rays escape a double-escape peak is seen at:

EDE = Eγ − 1.022MeV,

(4.15)

see Fig. 4.9.

Figure 4.12 Schematic illustration of pair production. Picture taken and modified from [3].

4.4.4

CAESAR, the Array

Due to the low rates of very exotic beams, γ-ray detection efficiency plays a big role for
constructing a γ-ray detector. CAESAR was designed to especially address this point. As
mentioned, the array consists of 192 CsI(Na) detectors. It weighs roughly 300 kg in total.
59

The design and commissioning criteria for this array are discussed in [20]. The main goals
were to achieve an in-beam energy resolution of 10% FWHM (full width at half maximum)
or better for 1 MeV γ rays emitted from nuclei traveling at 40% of the speed of light.
The material CsI(Na) was chosen to get the best cross section for interactions with γ rays
which means a high-Z was important (see Eq. 4.4 and Sec. 4.4.1-4.4.3). The Sodiumdoped Caesium Iodide used here has a higher stopping power than other common choices for
scintillators, for example Thallium-doped Sodium Iodide (NaI(Tl)). Due to the arrangement
of the detectors (see Fig. 4.8) some of them are partially shielded by others. This (and the
size difference) results in the fact that not every detector detects the same number of γ rays
even if they are isotropically emitted from a source at its center and simulations have to be
used to model its in-beam response.

4.4.4.1

CAESAR, the Detectors

All detectors were build by ScintiTech. The housing is made of 1 mm thick Aluminum and
protects the hygroscopic crystals from moisture. In between the Aluminum housing and the
crystals there is a 1.5 mm thick layer of reflective material to improve the light collection.
The Aluminum housing surrounds the front and sides of the crystals. The back of the crystal
is attached to a round borosilicate window framed in Aluminum. This enables the connection
to a photomultiplier tube (PMT).
In Fig. 4.13 a schematic of a detector is shown. The photomultiplier tube is coupled to the
crystal. The PMT is necessary to convert the scintillation light into an electric current. As
mentioned before, the number of scintillation photons is proportional to the energy of the
detected γ ray. This results in the current being directly related and proportional to the
γ-ray energy. The photons produced in the crystal hit the photocathode of the PMT and
60

Focusing electrode

γ ray

Photocathode

Dynode

Anode

Light

Photomultiplier tube

Scintillator
Electrons
Borosilicate
window

Connectors for signal
and high voltage

Figure 4.13 Schematic illustration of a detector. Picture taken and modified from [5].
produce electrons via the photoelectric effect (see Sec. 4.4.1). These then get focused by the
focusing electrode and multiplied by a set of dynodes (see [21]). Between each dynode, a
bias voltage is applied which increases from dynode to dynode. This leads to an acceleration
of the electrons between each dynode. The material of the dynode is such that it produces
a multiplication of the electrons every time it gets hit by an electron with a high kinetic
energy. So this system leads to an amplification of the number of electrons at every dynode
stage. In the end all those electrons are collected at the anode, producing a current signal
which can be easily read out. It is proportional to the initial number of photoelectrons (and
therefore the γ-ray energy). To get the best light transmission, the glass of the tube, placed
in between the crystal and in front of the photocathode, was chosen to match the index of
refraction of the detector material and window. The photo cathode itself is at the face of the
PMT and is coupled to the crystal using optical grease. Two different PMTs are used for
the CAESAR detectors, both made by Hamamatsu Photonics and also enclosed within an
Aluminum casing. For the larger detectors the model R1307 (3 inch diameter) is used and
61

for the smaller detectors model R1306 (2 inch diameter). To allow for operation in magnetic
fringe fields, several layers of µ-metal were added between the PMT and the Aluminum
housing. Thanks to this shielding the detectors stay operable in fields up to 2-3 mT [20].

4.4.5

Energy Resolution

The energy resolution of the CAESAR detectors,

R=

∆Eγ
,
Eγ

(4.16)

where ∆Eγ is the FWHM of the peak at energy Eγ [21], has three main contributions for
in-flight data. The first part is the intrinsic energy resolution, ∆Eγintr , of the detectors for
γ rays emitted from a resting source. The intrinsic resolution is defined as [21]

Rintrinsic ∝ √

1

.

(4.17)

Eγintr

All CAESAR detectors have an intrinsic resolution that is better than 7 % for 1 MeV γ rays
emitted at rest [20]. The second contribution comes from the uncertainty in the projectile’s
velocity, ∆β, at the point of the γ-ray emission. The projectile looses energy (slows down)
while passing through the target and for excited states with less than ∼ 1 ps lifetimes the
γ-ray emission after Coulomb excitation can occur at any point in the target (∼ 92 µm thick)
and therefore results in this uncertainty. The last contribution comes from the uncertainty
in the angle between the emitted γ ray and the scattered projectile, ∆θ. This uncertainty
is dominated by the finite opening angles of the detectors. θ is determined for CAESAR
by taking the angle between the center axis of CAESAR and the center of the individual
detector that fired. In total, this results in the following energy resolution [61]:
62

(

(
)2
)
(
)2
)
(
∆Eγ 2
cos
θ
−
β
β sin θ
∆Eintr 2
2
2
)
=
(∆θ) + (
(∆β) +
.
Eγ
1 − β cos θ
Eγ
1 − β 2 (1 − β cos θ)
(4.18)

All quantities on the right hand side of Eq. 4.18 are in the lab frame. CAESAR has been
optimized so that the intrinsic contribution is matched by the contribution of ∆θ and ∆β
for 1 MeV γ rays for velocities between 30% and 40% of c, leading to a total in-beam energy
resolution of better than 10% FWHM [20].

4.4.6

Efficiency

To determine an absolute cross section for a specific γ-ray transition, it is important to know
the number of emitted γ rays with this specific energy, not just the number of total detected
γ rays. The number of observed γ rays, Nγobs is related to the number of emitted γ rays, Nγ
via the absolute detection efficiency, ϵ:

ϵ(Eγ ) =

Nγobs
# of detected γ rays
=
.
Nγ
# of emitted γ rays

(4.19)

The efficiency is energy dependent and decreases with increasing energy of the γ ray. If the
source of the emission is moving then the Doppler-shifted energy (see Eq. 4.5) has to be
taken. In the case of a moving source, the Lorentz boost has to be taken into account, too,
which effectively changes the solid angle coverage of the system. The angle of detection in
the lab frame is related to the angle of emission in the projectile frame for photons (β = 1
and E = pc) by [46]:

63

tan(θlab ) =

sin(θcm )
γ(cos(θcm ) + 1)

(4.20)

with θlab/cm the angles of emission in the lab/projectile frame. If the angular distribution
of the γ rays is isotropic in the projectile frame, then this leads to a forward focusing in the
lab frame. The result is that the efficiency for a 1 MeV γ ray emitted at rest is different
to one emitted at 30% to 40% of the speed of light. For spherical symmetric systems, the
difference can be taken into account for the total efficiency by folding the detection efficiency
with the boosted angular distribution of the emitted γ rays (W (Ω)) [48]. But CAESAR
is not spherically symmetric and therefore another approach is needed to obtain in-beam
efficiencies. For this reason, T. Baugher et al. developed a Geant4 [62] simulation [63] of
the CAESAR array to take into account the complicated geometry of CAESAR, the target
position, thickness and material, the aluminum housing, small gaps between detectors, the
beam pipe, velocity of the projectile at the point of emission of the γ ray as well as lifetime
effects and other experimental conditions. Measured source spectra and efficiencies agree
with the Geant4 simulations which were used to model the in-beam spectral response of
CAESAR [63] (30% full-energy peak efficiency at 1 MeV [20], see also Sec. 5.4). The
simulation takes into account detection efficiencies, angular distributions and the Lorentz
boost. For more details on the simulation see Sec. 5.4.

4.5

Other Detectors

Three more detectors were used in this experiment. They are explained in detail in Sections
4.5.1-4.5.3.

64

4.5.1

P-I-N Detector

A p-i-n detector was used for energy-loss measurements of the incoming beam. This measurement was necessary for the particle identification process. The different incoming beam
components can be distinguished by their difference in energy-loss and time-of-flight (see
Sec. 5.2). A p-i-n detector is a type of semiconductor detector. In this case it was made
out of Silicon and was 300 µm thick. It consists of three major layers. The main layer is
the intrinsic Silicon semiconductor. Intrinsic means the material is not doped and has the
highest possible purity. Due to the fact that every material has some impurities, the starting
point was a slight p-type (positively doped) Silicon crystal and the so-called Lithium ion
drift process [21, 64] was used to create an “intrinsic” Silicon crystal by compensating the
acceptor impurities. This process creates also a highly doped n+ layer which is used as an
electrical contact on one side of the crystal. On the other side, due to the process, a p-type
doped Silicon crystal was generated which is used as a contact, too, or a heavily doped p+
layer is added (Fig. 4.14).

Figure 4.14 Schematic illustration of the regions in a p-i-n detector. Picture taken and
modified from [21].
The crystal’s compensated region is completely depleted and almost no leakage current exists
65

thanks to the p+ and n+ contacts. Because the intrinsic region ideally has no net charge
and is completely depleted, every electron-hole pair created by beam passing through the
material is collected. Therefore, the current signal provides a good measure of the energy
loss of the ion passing through the detector. The name “p-i-n” comes from using the three
different layers of the Silicon semiconductor, p-layer, intrinsic and n-layer.

4.5.2

Parallel Plate Avalanche Counters

Parallel Plate Avalanche Counters (PPACs) were used for position measurements, for example for the scattered particles after the reaction in the target. The PPAC used for this
measurement was 583 mm downstream of the gold target. The PPACs were developed and
built at the NSCL. They provide a very small, uniform, energy loss (they do not influence
the projectile too much, energy loss is less than 0.5 MeV per nucleon) while still having a
good position resolution [65]. They consist of two parallel planar electrodes on either side
of a central biased electrode. The central anode and cathode are close together, namely 3
mm apart, and the whole setup is enclosed by a container filled with the proportional gas
isobutane at 5 to 7 Torr. A homogeneous electric field of approximately 600 V is applied
between the electrodes. The two electrodes are stripped readout foils connected to a resistive
divider chain, see Fig. 4.15.
When a charged particle passes through the detector, it creates two independent avalanches.
The four signals at the end of the two resistor chains can be used to obtain a position
measurement. This is done by comparing the strength of the two measured induced charges
at one chain (see Fig. 4.16), performing a relative measurement this way.

66

Q2
Q1

Pressure foil
X-cathode
Anode
Y-cathode
Pressure foil

Figure 4.15 Schematic illustration of a PPAC detector. Left side is a cross section and the
right side shows one side. Picture taken and modified from [22].

P osition ∝

Q1 − Q2
Q1 + Q2

(4.21)

Because the PPAC is stripped in x and y, a clear position measurement can be achieved after
it is calibrated with a “position mask” (see Sec. 5.1.2).

QTotal
R1
Q1

R2
Q2

Figure 4.16 Schematic illustration of the resistor chain. Picture taken and modified from [22].

For more details on the specifications of how this detector is constructed, see D. Swan et al.,
“A simple two-dimensional PPAC” [65].

67

4.5.3

Phoswich

A Phoswich detector is usually used for ∆E-E measurements and made out of two plastic
scintillators and a single PMT. For this experiment it was mainly used as a particle trigger.
CAESAR was read out in coincidence with it. The detector was located 0.96 m downstream
of the gold target. The two scintillators are optically coupled and have two very different
decay times. The first scintillator is a thin scintillator (1 mm) from ScintiTech with a very
fast decay time. The second scintillator, also from ScintiTech, was thick enough (2 inches)
to completely stop all particles in the beam cocktail and had a slow time constant. Both
had a diameter of 6 inches and were connected to the PMT via a 6 inch long light guide that
had a diameter of 6 inches on one side and 3 inches on the other. To prevent loosing energy
in a dead layer between the fast and slow scintillator a special optical glue was implemented
which was optimized for this purpose to connect all the parts. In Fig. 4.17 is a schematic
shown of the detector.
The two different decay times of the scintillators lead to a very specific shape of the output
pulse of the PMT. It is an overlap of the very fast signal and the slow signal (see Fig. 4.18).
Depending on the relative contribution of the scintillation light from the two scintillators,
the signal changes it shape [21, 66]. The two different decay times are needed so that it is
clear which part of the resulting signal stems from which part of the detector.
Being able to distinguish between the two components is important for the ability to use
the detector for ∆E-E measurements. To get the full energy deposited, it is necessary to
integrate over the full signal of the Phoswich (fast and slow part combined) and to get the the
energy loss information the same signal was integrated but with a constant time offset. The
time offset enables skipping the first part (the fast part of the signal, main source of energy

68

Beam

light guide
thin, “fast”
scintillator

PMT

thick, “slow”
scintillator

Figure 4.17 Schematic illustration of the Phoswich detector.
loss) and integration occurs only over the slow part [66] (see Fig. 4.18 for details). The
result is an indirect measure of ∆E because the main energy loss happens in the first thin
part of the Phoswich and integrating over the second part of the signal gives the information
for E-∆E.
Because of the limited resolution of the Phoswich detector for high-Z nuclei used here, it was
not used mainly for ∆E-E measurements but as a particle trigger. A photo of the detector
is shown in Fig. 4.19.

69

full signal, “fast” and “slow” part combined

energy loss integration range

“slow” part of the signal

“fast” part of the signal

full energy integration range
Figure 4.18 Schematic illustration of the Phoswich PMT signals and the integration ranges.

70

Figure 4.19 Picture of the Phoswich detector.

71

Chapter 5
Data Analysis

5.1

Calibrations and Corrections

The first step of the data analysis was to calibrate all detectors and also apply some corrections where necessary. In the following sections it is explained how the calibrations and
corrections were done as well as why they were needed.

5.1.1

Energy and Time Calibration

The first calibrations done were energy and time calibrations of all 192 CAESAR detectors.
The sources used for the energy calibrations are listed in Table 5.1.
Source

γ-ray energy 1 [keV] γ-ray energy 2 [keV]

22 Na

511

1275.5

137 Cs

661.7

–

88 Y

898.0

1836.1

133 Ba

302.8

356.0

60 Co

1173.2

1332.5

background (40 K)

1460.8

–

background (208 Tl)

2614.5

–

Table 5.1 Calibration sources and energies.

Every peak in the source spectra was fitted with a Gaussian on top of a linear background.
This was done for every single detector in the array. This way, the exact position of the
72

peaks in channel numbers was extracted. Fig. 5.1 shows the fit for a 88 Y source for one
detector, where the x-axis shows the channel numbers.
After obtaining this information for every energy listed in Table 5.1 for every detector, the
fitting for the energy calibration was done. To do the calibration fits, the γ-ray energy (in
keV) is plotted versus the channel number and a polynomial of order two is fitted to the
points for each detector. In Fig. 5.2 the calibration fit for one CAESAR detector is shown.
The obtained function was then used to convert channels to energy (in keV).
All detectors were calibrated like this. Fig. 5.3 shows that the calibration worked well. It
shows all 192 detectors on the x-axis and on the y-axis is the γ-ray energy (in keV) of a
88 Y source shown. For simplicity, 24 detector channels were plotted for all 10 rings but as

mentioned previously (see Sec. 4.4), the first and last ring only have 10 detectors and the
second and second to last ring only 14 instead of 24. This is why there are white bands
visible in the Fig. 5.3. Proof that the calibration worked well is that the two main peaks in
the 88 Y spectrum are clearly visible as horizontal bands and that they are aligned.
Next the timing information of every detector was calibrated. Every detector has its own
timing information with 250 ps per channel dispersion. To ensure that all detectors are
aligned and have the same time offset, the following was done. A part of the in-beam data
with enough statistics was used and the timing information of all the detector hits with lab
energies bigger than 1000 keV were plotted. This resulted in a strong peak due to prompt
γ rays and a second, weaker peak due to off-prompt γ rays (see Fig. 5.4). Prompt γ rays
come from reactions with the target and off-prompt γ rays come from reactions with material
downstream of the target (e.g. the Phoswich). Because the Phoswich was located 0.96 m
downstream of the gold target, the nuclei needed more time to reach it and therefore γ rays
coming from reactions with it have this delay. These spectra were fitted for every detector
73

300
88

Y
source
Counts per channel

250
200
150
100
50
0
0

200 400 600
Channel number

800

1000

Figure 5.1 Uncalibrated 88 Y source spectrum for one CAESAR detector. The green lines
show the two Gaussian fits used to determine the position of the peaks in channel numbers.

74

Energy [keV]

2500
2000
1500
1000
500
200

400

600

800

Channel number

Figure 5.2 Source peak energies vs channel numbers for one detector. The green line is a
polynomial of order 2 fit.
with a double Gaussian and the mean of the Gaussian corresponding to prompt γ rays (the
first peak) was used as offset to center the timing for every detector. In Fig. 5.4 an example
of the fit for one detector is shown. The blue line is the total fit, the green is the prompt
part and the pink the off-prompt part.
That this method worked well can be seen in Fig. 5.5. All detectors are shown on the x-axis
and the calibrated timing information on the y-axis. It is very clear that the times are very
well aligned after the calibration. The white vertical lines have the same origin as the ones
in the energy calibration (see Fig. 5.3).
To make sure that there was no shift during the experiment, the prompt position for all 111
runs for energies bigger than 500 keV were checked. The positions changed by less than
8 channels (equaling 2ns) which is a lot smaller than the width of the prompt timing peak
therefore no additional corrections were needed (see Fig. 5.6).
75

Energy [keV]

3000
2500
10

2

2000
1500
10
1000
500
1
0

0

40

80

120

160

200 240
detector #

Figure 5.3 Overview of all detectors and their calibrated energies for a 88 Y source.
The times needed to be aligned so that one universal time-cut could be applied to all CAESAR detectors simultaneously (see Sec. 5.3.2.1).

5.1.2

PPAC Calibration

The calibration of the PPAC was done with a position mask. In front of the downstream
PPAC was a mask which was used for the position calibration. The position mask was made
out of a piece of metal with very well defined small holes and openings (in the order of few
millimeters). If the position mask is inserted in front of the PPAC in the path of the beam,
then the ions can only pass through the well-known openings. There was no position mask
in front of the upstream PPAC but it was possible to do a coarse calibration by using the
calibration of the downstream PPAC and the known dimensions of the PPACs.
In Fig. 5.7 a calibration run for the downstream PPAC is shown. With the knowledge of how
76

Counts / 250 ps

500
400
300
200
100
0

0

100

200

300

400

500

Channels

Figure 5.4 Example of the time calibration fit, see text for details.
far apart the different holes in the mask are, the gain in the x and y direction was determined.
To determine the positions of the holes in channel numbers the mean values of the features
corresponding to the holes were used. This was done for all clearly visible holes and the gain
was calculated for every row and column of holes. As a final value for the gain in the x and y
direction, the mean value of the different gains for the different rows and columns was used
respectively. The uncertainty of the position measurement was σ = 1.5 mm. It was also
checked that there was no pillow effect (because the PPAC is filled with gas (see Sec. 4.5.2)
77

1000

Timing after calibration [channels]

800

600

600
500

400
200

400

0
300

-200
-400

200

-600
100
-800

-1000
0
0 20 40 60 80 100 120 140 160 180 200 220 240
Detector #
Figure 5.5 Overview of all detectors and their calibrated timing information.
the surface of the PPAC could have been bent outwards). The necessary offset in the x and
y direction was determined for every beam component individually. It was made sure that
the PPAC was centered for coincidence data which was used for the scattering angle cuts.
This was done because only one PPAC was available behind the target and therefore the
assumption was made that on average the beam was centered. See Sec. 5.3.2.2 for details.
The total difference in the offsets for 104 Sn and 102 Cd was 12 mm (7.1 mm in x and 9.6 mm
in y).
With this knowledge it was possible to do a coarse calibration of the upstream PPAC, too.
It was done by ensuring that the gains used for the upstream PPAC resulted in the fact that
the overall image size of the PPAC was the same as the downstream PPAC.

78

Time offset [channels]

10

5

0

-5

-10
0

20

40

60
80
100
120
Run # of the experiment

140

160

180

Figure 5.6 Prompt timing peak position for all 111 runs of the experiment for γ-ray energies
greater than 500 keV.
All events with a bad position measurement were omitted from the analysis. This means
events where only the left or right information or only one up or down information was
available and therefore it was not possible to get an x or y position (see Sec. 4.5.2).
During a dedicated pilot beam, tuning, no-target run it was made sure that the beam was
focused at the target position. A good focus was achieved during the available time. In
Fig. 5.8 the x versus corresponding angle and y versus corresponding angle plots are shown
at the target position, upstream PPAC and downstream PPAC position.
With this no-target run, the beam spot size at the target position was determined, too. Fig.
5.9 shows the projection onto the x-axis and y-axis in mm for the beam species 102 Cd. The
beam spot has a σ ≤ 5.5 mm in any direction and is the same for any beam species that is

79

7

y-position uncalibrated

0.6

6

0.4
5
0.2
4
0
3
-0.2
2
-0.4
1

-0.6
-0.6

-0.4

-0.2
0
0.2
x-position uncalibrated

0.4

0.6

0

Figure 5.7 Uncalibrated mask run for the downstream PPAC. The positions of the different
features correspond to the positions of the holes and slits in the mask placed in front of the
PPAC during the calibration.
present in the beam cocktail. Therefore, any position uncertainty is dominated by this value.
But this measurement was performed as a relative measurement (see Sec. 5.3.2.3) and these
systematic uncertainties will not play a role for the final result.
To ensure that the beam spot did not move during the experiment, the mean value, rms value
and skewness of the upstream and downstream beam spots were monitored for all runs. This
was done while gated on different beam components. As shown in Fig. 5.10 and Fig. 5.11
for 102 Cd the spots did not really move and stayed constant. The mean values moved by
less than 5 mm which is smaller than the beam spot size and the second and third moment
(rms and skewness) also do not show any unexpected jumps or shifts. The same behavior
was observed for the other beam components.
80

Target positon
0.04
0.02
0.00

Corresponding anle [mrad]

-0.02
-0.04
-40 -30 -20 -10 0
Downstream PPAC

10 20 30 40 50

50 0.04
40 0.02
30 0.00
20
-0.02
10
-0.04
0

Target positon

-40 -30 -20 -10 0
Downstream PPAC

10 20 30 40 50

0.04

0.04
0.02
0.00

40
0.02
30
0.00
20
-0.02
10
-0.04
0

1

-0.02
-0.04
-40 -30 -20 -10 0
Upstream PPAC

10 20 30 40 50

-40 -30 -20 -10 0
Upstream PPAC

10 20 30 40 50

60
50
40
30
20
10
0
60
50
40
30
20
10
0

0.04

0.04
0.02
0.00
-0.02
-0.04
-40 -30 -20 -10 0

10 20 30 40 50

40
0.02
30
0.00
20
-0.02
10
-0.04
0

60
40
20
-40 -30 -20 -10 0

X-position [mm]

10 20 30 40 50

0

Y-position [mm]

Figure 5.8 Position data taken using the pilot beam and no target. Left column has the
x-positions and the corresponding angles. Right column has the y-positions and the corresponding angles. The first row is at the target position, the second at the downstream
position and the third at the upstream PPAC position.
45
40
35
30
25
20
15
10
5
0
-60

60
50
40
30
20
10
-40

-20

0

20

40

0
-60

60

X-position [mm]

-40

-20

0

20

Y-position [mm]

40

60

Figure 5.9 X and Y projections of the beam spot at the target position.

5.1.3

Corrections

Because the experiment was running for over 7 days and had high beam intensities, it was
necessary to do several re-tunes of the beam and among other thing move the plastic timing

81

1

0

-1

-2

20

60

100

140

12
10
8
6
4
2
0

180
# run

downstream Skew value X

downstream RMS value X

downstream mean value X

2

20

60

100

140

0.6

0.4

0.2

0

-0.2

-0.4

180
# run

20

60

100

140

180
# run

20

60

100

140

180
# run

1
0.5
0
-0.5
-1

20

60

100

140

180
# run

10

downstream Skew value Y

downstream RMS value Y

downstream mean value Y

0.8
1.5

8

6

4

2

0

20

60

100

140

180
# run

0.6
0.4
0.2

0
-0.2
-0.4

Figure 5.10 Downstream PPAC beam spot position for 102 Cd over time.
scintillators (see Sec. 4.1) to use different spots after radiation damage. Due to radiation
damage, the plastic scintillators were not able to detected particles with a high efficiency
anymore (the detectors were experiencing roughly a rate of nuclei of a little more than 1 kHz).
But the affected areas were small and it was possible to move to “unused” spots on the plastic
scintillators. This was necessary to keep running with a high efficiency. This also made it
necessary to apply corrections in the data analysis to account for small differences due to
different spots on the plastic scintillator detectors. Also the re-tunes sometimes changed the
secondary beam composition slightly and in combination with the long run time and heavy
use of the p-i-n detector, the energy loss measurements of the secondary beam components
were also affected. Therefore, this had to be taken into account, too.
All changes to detectors and re-tunes occurred in-between runs so it was possible to look

82

6

0.5
0
-0.5

-1

20

60

100

140

5
4
3
2
1
0

180
# run

0.4

upstream Skew value X

upstream RMS value X

upstream mean value X

1

20

60

100

140

0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4

180
# run

20

60

100

140

180
# run

20

60

100

140

180
# run

1.2

0.8

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
20

60

100

140

180
# run

upstream Skew value Y

5

upstream RMS value Y

upstream mean value Y

1.4

4
3
2
1
0

20

60

100

140

180
# run

0.6
0.4
0.2
0
-0.2
-0.4
-0.6

Figure 5.11 Upstream PPAC beam spot position for 102 Cd over time.
at data run-by-run with no change in the experimental conditions during the runs. For the
corrections, PID plots (see Sec. 5.2) and gates on the strongest beam component (101 Cd)
were made for every individual run.
The next step was to gate the variables that needed to be corrected on this strongest beam
component. After that the mean values for every 250 entries of the gated variables were
retrieved. Those mean values were then used as offsets to correct the variables.
The corrected variables were the timing information from the plastic scintillators at the RF
and XFP position and the difference of both. (Only the timing information from the XFP
position is used in the analysis because it has the longest time-of-flight information and
therefore gave the best resolution for the Particle Identification, see Sec. 5.2). The same
correction was done for the energy information of the p-i-n detector. The effects of the
corrections are shown in Fig. 5.12.
83

Timing at xfp before correction

150
100

300
250
200

Timing at xfp after correction

50

103

103

0

150
100

-50

102

102

-100

50
0

-150

101

101

-200

-50
-100
0

10

20

Timing at rf before correction
-60
-80
-100
-120
-140
-160
-180
-200
-220
-240
0
10
20

30
Event # (10^6)

40

50

-250

100

100
80
60
40
20
0
-20
-40
-60
-80
-100

104
103
102
101
30
Event # (10^6)

40

50

100

Energy loss at p-i-n before correction

0

10

20

30
Event # (10^6)

40

50

100

Timing at rf after correction
104
103
102
101
0

10

20

30
Event # (10^6)

40

50

100

Energy loss at p-i-n after correction

2500

300

2400

200

2300

102

102

100

2200

0

2100

-100

101

2000

101

-200
1900
1800
0

10

20

30
Event # (10^6)

40

50

-300

100

0

10

20

30
Event # (10^6)

40

50

100

Figure 5.12 Corrections. The left side shows uncorrected data for all events that were
recorded during the experiment and the right side shows the same data but with the run-byrun corrections applied as explained in the text.

5.2

Particle Identification

For this experiment, it was of course important to identify the incoming nuclei. It was done
with an incoming Particle Identification (PID) plot. Identification of the different beam
components is achieved by looking at an energy loss vs. time-of-flight (tof) plot. The energy
loss is proportional to the Z 2 value of the isotope (see Eq. 4.4) and the tof is proportional
to the A
q value (mass over charge).

5.2.1

PID

The tof was measured between the XFP scintillator and the Phoswich detector. This was
used because it provided the longest tof and with that the best resolution. It would have
been preferable to use the difference between the RF and XFP timing information because it
84

would have eliminated the common Phoswich but the tof was too short and the resolution not
good enough. The energy loss was measured with the p-i-n detector. In Fig. 5.13 the PID
plot is shown. The different beam components are marked with the incoming gates in red.
The assignment of the different species was confirmed by the observation of characteristic
γ rays in CAESAR found at [6]. It is also clearly visible that the separation between the
different beam components is not perfect. This is taken into account in the analysis (e.g. for
the total number of particles detected, see two dimensional fits in Sec. 5.2.2).

Energy loss [arb. units]

300

104
103

200

104

In

Sn
103

100

102

Cd
102

0
102

-100

101
100

-200
-25

-20

-15

In
10

Cd

Ag
-10

-5

0

5

10

15

20

25

1

Time of flight [arb. units]
Figure 5.13 Particle Identification spectrum for the neutron-deficient projectile beam passing
through the Au target. The energy loss measured in the p-i-n detector is plotted versus the
ion’s flight time. 104 Sn and 102 Cd can be clearly identified. The red lines show the incoming
gates. See text for details.

85

5.2.2

Trigger Modes and Contamination

Three different trigger modes were used during the experiment. The first and most common
was a coincidence trigger. The coincidence condition required a detected particle in the
Phoswich and a detected γ ray in CAESAR within a time window of 600 ns. The second
one was a downscaled particle singles trigger. For very 50th event that detected a particle
(independent of whether or not it was in coincidence with a γ ray) a trigger was issued. This
condition was needed to calculate the total number of incoming particles which is needed
for the B(E2 ↑) calculation (see Sec. 5.6). The downscaler of 50 was necessary so that
the data acquisition would mainly record coincidence events which are the main part of
this experiment and far less common. The last trigger condition required that the first two
occurred at the same time. After a careful analysis, it was noticed that trigger condition
two (downscaled particle singles) caused a time offset in the tof compared to the other two
trigger conditions. It was an offset of 3.5 ns. This offset was applied as a correction to the
events with this trigger condition.
Something else that had to be taken into account is the contamination of neighboring beam
components entering into the individual incoming gates. To account for this, two dimensional
fits were done. The two dimensional PID plot was taken and all 6 beam components were
fitted at the same time (see Fig. 5.14). For every component a two dimensional Gaussian
was used. With the functions obtained from the fits it was possible to calculate how many
particles in an incoming gate came from neighboring beam components. The corrections for
the two beam components of interest, 104 Sn and 102 Cd, were smaller than 2.5%. They were
calculated with the functions obtained from the fits of the downscaled particle singles trigger
PID plot. This trigger condition had to be used because the number of incoming ions was

86

counted with this plot. All the entries in this plot in an incoming gate have to be multiplied
by 50 which gives the total number of incoming ions for this gate. In Fig. 5.14 the PID
plot just for downscaled particle singles is shown and the two dimensional Gaussian fits are
shown superimposed.

gy
Ener
[a
loss
nits]
rb. u

t [arb. units
h
g
li
f
f
o
e
Tim

]

Figure 5.14 PID plot for downscaled single events with two dimensional Gaussian fits (black
lines). See text for more details.

5.3

γ-ray Spectra

In this section, it will be described how the γ-ray spectra, on which the results of this experiment are based, were created. All γ-ray spectra are so-called singles spectra. That means
that no addback was used to create them. Every detector was taken by itself without adding
back events that scattered into neighboring detectors (Compton scattering for example). The
trigger condition was the particle-γ ray coincidence trigger condition.

87

5.3.1

Velocity Determination

It was important to find the right β = vc to do the Doppler correction for the γ-ray spectra.
This was done by dividing the array into a forward and backward part (5 forward rings
and 5 backward rings) for 104 Sn and for the higher statistics 102 Cd case into 2 forward
parts (2 and 3 rings) and one backward part. Then the β for the Doppler correction was
systematically changed while tracking the position of a γ-ray peak which is characteristic
for the incoming nucleus. The positions of the γ-ray peaks were tracked separately for the
forward and backward parts of the array. For the right β the positions of the γ-ray peaks of
the forward and backward parts have to line up. In Fig. 5.15 and Fig. 5.16 the results of this
analysis are shown. For 104 Sn the red points mark the γ-ray peak position for the forward
part of CAESAR, the blue the backward part and the pink is for all of CAESAR. The yellow
is the FWHM of the simple Gaussian fit for all of CAESAR. For 102 Cd the red points mark
the most forward part of CAESAR, the green the second forward part, the blue the backward
part and the pink and yellow is the same as for 104 Sn. It is clear that β = 0.345 is the right
β for 104 Sn and β = 0.335 the right β for 102 Cd, because for those β values the γ-ray peaks
of the forward and backward parts of the array line up the best while simultaneously having
a small FWHM. Those results are consistent with the theoretical energy loss calculations
performed with LISE++ [18] that gave a before, mid and after target β of 0.382, 0.361 and
0.336 for 104 Sn and 0.375, 0.354, 0.328 for 102 Cd.

5.3.2

Gating

Several gates are applied to the data to clean up the γ-ray spectra. The first gate applied
is a low-energy cut. It is necessary due to a lot of low-energy counts coming from atomic

88

1300
1300

120
100

1280
1280
60

1260
1260

40

1240
1240

Forward part

20

Full array

1220
1220

104

Sn

FWHM [keV]

Energy [keV]

80

0

Backward part

-20
1200
1200

0.33
0.33

0.335
0.34
0.335
0.34
β -value

0.345
0.345

0.35
0.35

Figure 5.15 Plot to find the right β for the Doppler correction of 104 Sn. The γ-ray energy is
at 1260 keV [6]. For details see the text.
background. This background consist mainly of four components [67]:
X K and L shell X-rays from ionized gold target atoms,
X radiative electron capture (REC) of the gold target electrons into the projectile K

and/or L shells,
X primary Bremsstrahlung (PB) from target electrons produced by the collisions with

the projectile,
X secondary Bremsstrahlung (SEB) from energetic knock-out electrons re-scattering in

the target and/or the surrounding material.

89

810
810

80

800
800

60

790
790

40

780
780

20

FWHM [keV]

Energy [keV]

820
820

770
770
760
760

102

Most forward part
Full array
Backward part
Second forward part

Cd

0.33
0.33

0.335
0.335

0.34
0.345
0.34
0.345
β -value

0.35
0.35

0
-20

Figure 5.16 Plot to find the right β for the Doppler correction of 102 Cd. The γ-ray energy
is at 777 keV [6]. For details see the text.
To avoid being dominated by this background all energies below 100 keV in the lab frame
were cut out. The other gates that were used are explained below.

5.3.2.1

Prompt γ-ray Timing

Another very important gate was the prompt γ-ray time-cut. This cut eliminated random
coincidences, especially of γ rays that were produced by nuclei hitting the Phoswich detector
that was placed 0.96m downstream of the target. Fig. 5.17 shows a plot of lab energy versus
timing for all events with only 1 hit in CAESAR (no incoming gate). Only the one hit
events are shown. The cutoff at 100 keV is visible and so are two main features. The main
feature at time zero is from γ rays produced at the target position and the second feature
with roughly a 10 ns delay (the x-axis has 250 ps per channel) is from delayed reactions
90

e.g. in the Phoswich. The plot also clearly shows the two horizontal background lines from
40 K with 1460.8 keV and the other one from 208 Tl with 2614.5 keV stemming from random

coincidences with room background. The red line indicates the gate region which was used
to only allow prompt γ rays in the analysis. In this plot it might look like there are more
off-prompt γ rays than prompt ones but that is because of the choice to only show events
with 1 γ ray detected in CAESAR. It should also be noticed that the random events a bit
further away from the prompt events are not evenly distributed. Therefore it was decided
not to perform a random background subtraction.

Energy in the lab frame [keV]

7000

104

6000
103

5000
4000

102

3000
2000

10

1000
0
-1000

-600

-200 0 200
600
Timing [250ps per channel]

1000

1

Figure 5.17 Plot to show the prompt γ-ray time gate (red). Only 1 hit events in CAESAR
are displayed. The two main features from prompt and off-prompt γ rays are clearly visible
as well as the two horizontal background lines at 1461 keV and 2615 keV. For details see the
text.

91

5.3.2.2

Particle Scattering Angle Cuts

A “safe” angle cut was employed, too (see Sec. 3.2). As described, this is necessary to make
sure there are no nuclear contributions to the Coulomb excitation cross section by focusing on
events with large impact parameters, corresponding to small scattering angles. Calculations
for the maximum safe angle were performed with the Eq. 3.9 using bmin from Eq. 3.11. In
SI units, this formula is expressed as follows (exactly the same as Eq. 3.9):




Zp Zt α c
(
)  cot
bmin = 
Ap At
2
2
λβ mnuc c A +A
p

( max )
θcm
,
2

(5.1)

t

with mnuc c2 = 931.494 MeV the average bound nucleon mass, α = 0.007297 the fine structure constant, c = 197.327 MeV, Zp/t the projectile/target proton number and Ap/t the
projectile/target mass number.
The maximum scattering angle was calculated in the center-of-mass frame for the two beam
species of interest, 104 Sn and 102 Cd, with the assumption that the reaction happened mid
target (β = 0.361, β = 0.354, respectively) as well as at the front of the target (β = 0.382,
β = 0.375). The conversion to lab frame angles was performed with LISE++ [18]. This
resulted in the following maximum scattering angles in the lab frame: 3.126◦ and 2.756◦
for 104 Sn as well as 3.206◦ and 2.821◦ for 102 Cd. To be conservative, it was decided to
restrict the data to an even “safer” (meaning larger impact parameter) maximum scattering
angle, namely 45 mrad which corresponds to 2.578◦ . This was done for 104 Sn and 102 Cd. In
Fig. 5.18, the distribution of all scattering angles for all events of our two species of interest
are plotted. The different scattering angle cutoffs are shown, too. The red line corresponds
to the mid-target β, the green line to the front of the target β and the blue line marks the
actual cut-off used.
92

120

50

100
Counts [103]

Counts [103]

40
30
20
10
0

80
60
40
20

0

1
2
3
4
Scattering angle in deg

0

5

0

1
2
3
4
Scattering angle in deg

5

Figure 5.18 Scattering angle distribution for 104 Sn (left) and 102 Cd (right). Blue line: “safe”
angle which was used, green line: “safe” angle for beginning of the target, red line: “safe”
angle for middle of the target.
The measured scattering angles were determined the following way. The distance between
the target and the downstream PPAC was measured to be 583 mm. As the distance and
the x and y position in the PPAC are known, this makes it easy to calculate the scattering
angle. With the PPAC position information d, the distance from the center of the PPAC to
where the nucleus hit the PPAC, can be calculated in millimeters:
√
d = x2 + y 2 .

(5.2)

The scattering angle θ has therefore the following dependence:
(
θ = atan

d
583

)
.

(5.3)

For every species, different offsets had to be used (see Sec. 5.1.2). The offsets were chosen

93

in a way that made sure that all coincidence events together have a mean x and y position
of zero on the PPAC. This lead to a difference in offsets between 104 Sn and 102 Cd of 12 mm
in total (7.1 mm in x and 9.6 mm in y). The centered downstream PPAC plots can be seen
in Fig. 5.19.
Mean x

y position [mm]

60

6.252e-08

Mean x -3.492e-05

30

60

40

25

40

50

20

20

20

40

0

15

0

30

-20

10 -20

20

-40

5

-40

10

0

-60
-60

-60
-60

Mean y -1.912e-06

-40

-20

0

20

40

60

x position [mm]

Mean y -1.126e-06

-40

-20

0

20

40

60

60

0

x position [mm]

Figure 5.19 Downstream PPAC gated on coincidence events, centered and gated on 104 Sn
(left) and 102 Cd (right).
LISE++ [18] was used to calculated the angular straggling and it is less than σ = 0.4◦ . This
was then used as an input for a Monte Carlo Simulation to estimate how big the effect of
straggling would be on the angle cuts. The straggling effect makes it appear that an event
has an acceptable angle (below cutoff) where in fact it does not (and vice verca). This was
done for both cases of interest for the singles scattering angle spectra but only the higher
statistics case 102 Cd is displayed here. The downscaled singles trigger condition was applied
to avoid a bias towards elastic or inelastic scattering processes. An input distribution was
chosen that would reproduce the measured angular distribution after folding it with the
94

straggling distribution. The straggling distribution is a Gaussian distribution with σ = 0.4◦
from which a random straggling angle was taken and then this angle was randomly added
in the full 3D space to the initial angle. For both 104 Sn and 102 Cd it was determined that
for all angles larger than 1.5◦ , about 0.7-1.5% of the counts originate from larger scattering
angles that fulfill the maximum angle cut since they straggled to smaller angles (blue) and
about 3-4% of the events that would have fulfilled the maximum angle cut without angular
straggling are lost (green). The results for 102 Cd can be seen in Fig. 5.20.

Counts [103]

10
9
8
7
6
5
4
3
2
1
0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Scatter angle in deg
Figure 5.20 Effect of straggling. Pink is the measured distribution with downscaled singles
trigger condition. Red is the result of the whole Monte Carlo simulation. Blue are the events
that fulfilled the scattering angle cut condition but started with angles bigger and green are
the events that did not fulfill the angle cut condition but started with angles smaller.

95

5.3.2.3

Multiplicities

Very early in the analysis the problem surfaced that even with a prompt γ-ray time gate
and incoming PID gate it was not possible to see any γ-ray peaks unless the spectrum
was restricted to low CAESAR detector multiplicities. Multiplicities are determined by the
number of detectors hit in CAESAR during one event. When a prompt γ-ray time gate
is applied then the multiplicity takes only γ rays into account that lie in the gate. One
problem with this setup was the PID resolution of the Phoswich. The resolution was not
good enough to use the detector for particle identification after the reactions at the target.
This is one of the main reasons why no peaks were visible unless the analysis was restricted to
low multiplicities. An investigation into this matter revealed that events were present with
huge multiplicities (up to 60 hit detectors in CAESAR per event). Those high multiplicity
events contribute hugely to the background and obscured any peak. The source of those
high-multiplicity events was determined. It is now clear that they come from very violent
reactions in the target such as fragmentation. During those violent reactions, a lot of γ-rays
and charged particles, interacting with surrounding matter, are produced and it is likely that
one of the light charged particles hits one of the CAESAR detectors, causing it to go into
overflow (the detected energy was higher than the sensitive range of the electronics).
Looking at the multiplicity distribution for a single run (see Fig. 5.21) clearly shows that
events with no detector in overflow (blue) dominate the low-multiplicity range and that the
high-multiplicity events are dominated by events where at least one detector went into overflow (orange). This confirms the assumption that the high multiplicity events are connected
to violent reactions and charged particles with high energies being detected in some of the
detectors.

96

105
Counts

104
103
102
10
1
0

40

80
120
Multiplicity

160

200

Figure 5.21 CAESAR multiplicities for a single run. Black: all events; blue: events with no
CAESAR detector in overflow; orange: at least one detector in overflow.
To confirm that it actually was fragmentation that occurred, two additional effects were
investigated. First, if there is an event with a detector going into overflow, then it mostly
happened in the forward rings (0, 1 and 2) of the array, see Fig. 5.22. This shows that those
events are heavily forward focused, supporting the hypothesis.
Secondly, the Phoswich detector was considered. Even with its rather modest energy (∆E
and E) resolution, it was possible to see trends. Fig. 5.23 shows a trend going to higher
multiplicities. On the x-axis is the full energy E displayed and on the y-axis the energy loss
E − ∆E. There is a shift to the left along the x-axis visible and almost no shift along the
y-axis. This can be explained by the following. On the x-axis is the full energy deposited
in the Phoswich (integration over the full PMT signal) and on the y-axis is E-∆E shown

97

J
Counts

102

I
H
G
F

10

E D
C B
A

0

2

4

6
8
Rings

10

Figure 5.22 Ring distribution of CAESAR detectors in overflow. Ring 1 (J) is the most
forward ring and 10 (A) the most backward.
(integration with delay over the slow part of the PMT signal) which is an indication for
the energy loss (for details see Sec. 4.5.3). So this shift means that the particles loose less
energy in the thin, first part and almost the same energy is deposited in the thick second
part. This confirms that the particles associated with higher multiplicity events have lost
protons because the energy loss is proportional to Z 2 . So this also supports the theory of
violent processes in the target where nucleons are removed from the projectile. Some light,
charged particles hit CAESAR and surroundings and some lighter, projectile-like residues
make it into the Phoswich. This is the reason why the analysis had to be restricted to small
multiplicities and the decision was made to only allow multiplicity 1 events.
This greatly improves the peak-to-background ratio. The majority of the counts belong
98

Full range

40

60

80

100

120

140

160

180

200

Phoswich energy loss [a.u.]

Counts log scale [a.u.]

0

Full range

0

40

80
120
Multiplicity

160

200
Phoswich full energy [a.u.]

Figure 5.23 Phoswich energies for different multiplicity ranges. On the left side the different
multiplicity ranges are shown in steps of 10 and on the right side the corresponding Phoswich
energies. All detected events were used to generate these plots. The very top panel of both
columns has no gates applied, the full multiplicity range is used.
to multiplicity 1 events. Restricting the analysis to multiplicity 1 events and because this
setup and data acquisition system are not characterized as well as in previous work at the
S800 spectrograph (see [68], for example), it was not possible to establish reliable absolute

99

in-beam efficiencies for all detector systems. Therefore, a relative B(E2 ↑) measurement was
done where most absolute efficiencies cancel out. The measurement was done relative to the
well-known B(E2 ↑) value of 102 Cd [2, 43]. 102 Cd was present in the beam cocktail with a
higher intensity than 104 Sn and provided good statistics for a relative measurement (see Sec.
5.2 and 5.5).

5.3.3

Spectra

With all the gates applied as described in Sec. 5.3.2, the in-beam γ-ray spectra were created.
As mentioned before, transitions in 104 Sn and 102 Cd form the base of the relative B(E2 ↑)
measurement. The two most important variables that need to be known are the number of γ
rays emitted by the nucleus of interest and the total number of incoming ions of this nucleus.
To convert the detected γ-ray energy in the lab frame into the Doppler corrected energy in
the projectile rest frame, Eq. 4.5 was used.
In Fig. 5.24 are the lab frame (upper panels) and Doppler corrected (lower panels) spectra
shown for 104 Sn and 102 Cd. In the lab frame spectra, the background energy peak at 1461
keV from 40 K and the 548 keV + 511 keV peak from Coulomb excitations of the gold target
and pair production are clearly visible. In the Doppler corrected spectra, those peaks do not
appear anymore but the characteristic energies for the E2 transitions in 104 Sn and 102 Cd
respectively [6] are clearly visible.

100

Counts / 20 keV

Counts / 25 keV

25

(a) 104Sn

20

v/c = 0
15

1461 keV
background

10
5
0

500

1000

1500

2000

Energy [keV]

(b) 102Cd
v/c = 0

150
100
1461 keV
background

50
500

1000

1500

2000

Energy [keV]

2500

250

(c)

20

1260 keV

104

Sn

Counts / 20 keV

Counts / 25 keV

548 keV
+511 keV

200

0

2500

25

v/c = 0.345

15
10
5
0

250

500

1000

1500

2000

2500

Energy [keV]

(d) 102Cd

200

v/c = 0.335
150

777 keV

100
50
0

500

1000

1500

2000

2500

Energy [keV]

Figure 5.24 Lab frame (upper panels) and Doppler corrected (projectile frame, lower panels)
energy spectra for 104 Sn and 102 Cd, respectively. Characteristic energies are indicated. For
more details see text.

5.4

Geant4 Simulations

To determine the total number of emitted γ rays, a simulation was used to take into account
geometry, in-beam efficiencies, absorptions, angular distributions and other experimental circumstances. A Geant4 simulation written by T. Baugher and G.F. Grinyer was implemented
to do exactly this (see Sec. 7 and [63]).
For the simulations to work properly, two files had to be prepared. The first was a description
of the low-energy threshold for every detector and the second one contained energy resolution
functions for each detector.

101

To obtain a description of the low-energy threshold, the low-energy part of a 137 Cs highactivity source spectrum was fitted with half a Gaussian curve. This was done for every
detector in CAESAR separately. The received fitting parameters sigma (σ) and mean (µ)
were used to model the low-energy part of the simulated spectra. The results of the fits for
ring 5 are shown in Fig. 5.25.
120

0

0

450

Counts / 1 keV

0

450

140

0

0

0

0

400

0

0

350

0

0

450

0

0

0

400

0

0

400

0

0

500

0

0

0

400

0

0

350

0

0

500

0

0

0

300

0

0

350

0

0

400

0

0

0

350

0

0

450

120

400

120

450

0

120

120

120

350

0

120

120

120

120

400

0
120

120

120

500

0

300

120

120

120

120

0

0

120

120

0

120

120

0

0

400

120

400

0

0

400

Energy [keV]

Figure 5.25 Low-energy part for a high-activity 137 Cs spectrum for all detectors in ring 5.
In green is the half Gaussian fit shown.

To ensure that the low-energy part is not dependent on the energy of the full-energy peak or
on the source activity, the spectrum of the high activity 137 Cs source was compared to a low
activity 60 Co source spectrum. After rescaling, there was no visible difference in shape. This
confirms that the original 137 Cs fits were good and could be used. To see the comparison

102

for all the detectors in ring 5, see Fig. 5.26.
40

0

0

50

500

Counts / 1 keV

0

500

0

500

35

0

0

500

0

0

0

0

500

0

0

0

0

0

45

500
406.

500

40

500

0

0

500

0

0

0

0

500

35

0

0

500

0

0

0

0

0

0

500

45

500

0

0

500

0

0

500

0

0

0

500

0

0

500

40

500

0

0

500

50

50

500

0

30

50

35

500

50

50

45

40

40

500

0

35

45

50

0

0

35

35

0

0

40

500

0

0

500

Energy [keV]

Figure 5.26 Comparison of the low-energy part of a rescaled high activity 137 Cs (black) and
a low-activity 60 Co (red) spectrum.

In a next step, several γ-ray energies from different sources were fitted to gain the function
that describes how the full width at half maximum (FWHM) changes with energy. The sigma
values of the different Gaussian fits at the different energies were fitted with the following
formula [21]:

σ(E) = a × E b .

(5.4)

Again this was done for every detector and the parameters a and b were used to model the
FWHM behavior in the simulation. Fig.5.27 shows the fits for the detectors in ring 5. The

103

used sources and their γ-ray energies are listed in Table 5.2
Source γ-ray energy 1 [keV] γ-ray energy 2 [keV]
22 Na

511

1275.5

137 Cs

661.7

–

88 Y

898.0

1836.1

133 Ba

356.0

–

Table 5.2 Sources and energies used for σ(E) fits.

45

15
1800

40

400

1800

1800

15

15
400

1800

400

1800

40

40

400

1800

40

15

400

1800

1800

15

400

1800

15
1800

400

1800

400

1800

1800

15

400

1800

400

1800

40

15
400

1800

400

1800

15

1800

400

1800

400

1800

15

15
1800

400
40

40

400

1800

15

15
400

40

35

15

15
400

15

15

40

35

35

35

1800

45

40

400

400

400
35

20

20

15

35

15
400

40

40

40

40

15

15
400

σ (E) [keV]

40

40

15
400

1800

Energy [keV]

Figure 5.27 Development of σ(E) for all detectors in ring 5.

The first test of the GEANT4 simulation was to simulate source spectra and compare them
with recorded source runs. However, during the calibration runs, all the unused sources were
stored in a source locker roughly 7m away from CAESAR but the locker was not shielded

104

perfectly. Therefore, some of the γ rays from the strongest sources in the source locker were
detected in CAESAR and slightly contaminated every run. Nevertheless, it was possible to
confirm that the simulations reproduce the source spectra well. But because this problem
depends on the activity of the source used, we were not able to correct for this effect in a
consistent way.
For each source, the total number of γ rays that were expected to be emitted was calculated.
First, the activity of the source at the point of the measurement was calculated and then this
activity was multiplied with the time the detector system was live (ready for data) during
the measurement. This gives the total number of decays that happened during the time
of the measurement. Next the simulated spectrum (one energy at a time) was divided by
the number of simulated events (normalized), multiplied with the γ ray’s specific emission
probability (see [6]) and then multiplied with the total number of decays. If necessary
multiple energies that belong to the same source were simulated and added together. The
last step was to add a scaled background histogram to the simulation. The background
histogram was scaled with its data acquisition live time and was needed because the room
background is not included in the simulations. The recorded source spectra, on the other
hand, have only one gate applied and that is a prompt γ-ray time gate. In Fig.5.28 the results
of the comparison are shown. The black histogram is the measured source histogram, the
red is the pure scaled simulation and the green is the simulation plus the scaled background.

105

70k

0

60

0

1800

300k
Counts / 1 keV

Co 100k

22

0

1800
133

0

90k

Ba

700

2200
137

Cs
high act

0

0
30k

0

0

14k

137

Cs
low act

00

Y
high act

0

Na 200k

0

0
200k

0

88

1000 0 0

1000
88

Y
low act
2200
bg

7000

Energy [keV]
Figure 5.28 Comparison of simulated and measured source spectra. The black histogram is
the measured histogram, the red is the pure scaled simulation and the green is the simulation plus a scaled background. The bottom right histogram shows the room background
measurement.

5.5

Fits

For the in-beam simulation more information is needed. The kinetic energy of the beam
before the target needs to be known (this information was obtained by using recorded Bρ
settings and calculating energy losses with LISE++ [18]) in addition to the momentum

106

spread of the beam (also obtained from recorded settings), the angular distribution of the
γ rays which was calculated with the previously mentioned mathematica script [47] and the
angular spread of the beam after the target deduced from the data. Ata is the scattering
angle in x-direction and Bta the scattering angle in y-direction. Also the thickness, position
and dimensions of the target were required. If known, then the lifetime of the excited state
was added, too, although the effect is negligible because CAESAR is not sensitive enough to
detect changes in γ-ray peak shapes for short lifetimes.
The target used was a gold target of 184 mg/cm2 thickness and 5 × 5 cm area, it was placed
in the center of CAESAR about 12.8 m downstream from the RFFS.
The other settings are displayed in the Table 5.3.
104 Sn
Setting
Incoming kinetic energy [MeV/nucleon] 76.60
Incoming momentum spread dp/p
0.41 %
Simulated γ-ray energy [keV]
1260.2
Lifetime [ps]
–
Angular distribution a0
1.0
Angular distribution a2
-0.4690
Angular distribution a4
-0.1137
Sigma Ata [rad]
0.01710
Sigma Bta [rad]
0.01292

102 Cd

73.37
0.41 %
776.6
5.12
1.0
-0.5710
-0.1740
0.01676
0.01239

Table 5.3 Input settings used for the in-beam GEANT4 simulations.
The simulation was tested and in Fig. 5.29, 5.30 and 5.31 it is shown that the input settings
for 104 Sn are reproduced by the simulation. The simulation may be improved by getting the
energy loss closer to the energy loss calculated by LISE++. The energy loss calculations
done by LISE++ are based on a model that was specifically tuned for the energy region
of the nuclei that are produced at the NSCL. GEANT4 on the other hand uses a different
model that is not necessarily optimized for this energy region.
With the in-beam simulation tested, it can be used to fit the 104 Sn and 102 Cd spectra to
107

extract the number of detected γ rays.
beta before target
mean: 0.383

30
20
10
0
0.3818

30
10
β

0.3832

Counts

detection angle
in the lab frame

0

0.34

0.385
β

z position
of the reaction
m

beta after target

2000
0
0

30
20

20

80 mean: 0.343
60
40
20
0
0.34
β

6000
4000

beta at decay
mean: 0.364

beta at reaction
mean: 0.364

0.346

40 mean: 0.0 mm
30
20
10
0
-0.04
z [mm]

0.04

Scattering angle
Position: -0.0003
Sigma: 0.0173(3)

120
80
40
0
-0.2
detection angle 180
lab frame [deg]

10
0
0.34
β

0.385

z position
of the decay
m

40 mean: 0.0 mm
30
20
10
0
-0.04
0.04
z [mm]

Scattering angle
Position: -0.0002
Sigma: 0.0175(3)

200

x scattering
angle [rad]

0.1

100
0
-0.2

y scattering
angle [rad]

0.2

Figure 5.29 Shown are several simulated variables. Comparison with the input values from
Table 5.3 shows a good agreement.

108

Full Array
Position 1261.06 +- 0.12
FWHM 102.29 = 8.11 %
12000
v/c = 0.364
8000
4000
0
0
400
800
1200
Ring A
Position 1262.24 +- 0.90
FWHM 103.22 = 8.18 %

Full Array
6000

18000

Counts / 1 keV

16000

v/c = 0

4000
2000
1600

0

0

500

1000

1500

2000

Ring A

350
300

v/c = 0.364

200
100
0

0
400
800
1200
Ring J
Position 1255.83 +- 0.33
FWHM 85.42 = 6.80 %

1600

1600
1400

v/c = 0.364

800
400
0

0

400

800

1200

1600

400
300
200
100
0

0
Ring J
1000
800
600
400
200
0
0

Energy [keV]

v/c = 0

500

1000

1500

2000

v/c = 0

500

1000

1500

2000

Energy [keV]

Figure 5.30 Simulated γ-ray spectra for 104 Sn. On the left side are the spectra Doppler
corrected and on the right side not. The simulation is again in good agreement with the
input data and with the expectations of the response of CAESAR.

5.6

Relative Measurement Results

In previous experiments, the usual procedure was to determine the absolute angle-integrated
max ), and translate it into absolute B(πλ)
Coulomb excitation cross section, σ(θlab ≤ θlab

excitation strengths using the Winther-Alder description of intermediate-energy Coulomb
excitation [45]. However, this was not possible in the current work as the absolute efficiencies
of this new setup and data acquisition system were not as well characterized compared to
previous works (e.g. using the S800 spectrograph - see [68] for example). For more details see
104 Sn was
Sec. 5.3.2.3. Therefore, the B(E2 ↑) = B(E2; 0+ → 2+
1 ) excitation strength for

109

Counts

Kin energy per nuclean before target in MeV
mean: 76.595

Momentum distribution before target in GeV/c
mean: 40.086
momentum spread (dp/p): 0.41 %

35

35

30

30

25

25

20

20

15

15

10

10

5

5

0
7.93

7.95

7.97

0

7.99

Energy [GeV]

40

40.04 40.08 40.12 40.16

Energy [GeV]

Figure 5.31 Last test of the simulation. Two more variables that were simulated are shown
and again a comparison with the input values from Table 5.3 shows a good agreement.
determined relative to the well known B(E2 ↑) value of 102 Cd [2], which Coulomb excitation
yield was measured in this experiment at the very same time, under identical conditions and
with good statistics. The relative determination of the excitation strength was performed
this way:

B(E2 ↑)Sn = B(E2 ↑)Cd ·
Sn,Cd

with Nγ

Sn,Cd

and Nproj

Cd AW (Cd)
NγSn Nproj
Sn AW (Sn)
NγCd Nproj

,

(5.5)

104 Sn and
the number of 2+
1 de-excitation γ rays and projectiles for

102 Cd, respectively. The term AW (Sn, Cd) is the Alder-Winther angle-integrated Coulomb

excitation cross section, σ(θlab ≤ 2.578◦ ) per unit B(E2) value, for 104 Sn and 102 Cd, respectively, taking into account the proper kinematics and atomic number of the projectiles. The

110

error was calculated as follows:
(
∆B(E2 ↑)Sn = B(E2 ↑)Cd · 

∆NγSn

)2
+

NγSn
(
+

(

Cd
∆Nproj
Cd
Nproj

∆NγCd

+

NγCd
)2

(
+

(

)2

Sn
∆Nproj

)2

Sn
Nproj

)2  21
Cd
∆B(E2 ↑)
 .
B(E2 ↑)Cd

(5.6)

The simulated Geant4 response of CAESAR was scaled to the 104 Sn and 102 Cd data to determine the number of 2+
1 de-excitations, Nγ . The simulations takes into account CAESAR’s
γ-ray detection efficiency and the absorption in the gold target and surrounding materials
and include the calculated γ-ray angular distribution in Coulomb excitation [45, 48]. The
simulated response functions fitted on top of a double-exponential smooth background are
shown in Fig. 5.32.

1260 keV

15
10

(a) 104Sn

777 keV

(b) 102Cd

v/c = 0.345

v/c = 0.335

total fit
double exp BG
Geant4 sim

total fit
double exp BG
Geant4 sim

150
100
50

5

Counts / 20 keV

Counts / 25 keV

20

0

0
1000

1500
2000
Energy [keV]

500

1000
1500
2000
Energy [keV]

Figure 5.32 Event-by-event Doppler reconstructed γ-ray spectra detected in coincidence
with scattered 104 Sn (a) and scattered 102 Cd (b). γ-ray transitions at 1260 keV and 777 keV
can be clearly identified and are attributed to the de-excitation γ rays from the first excited
2+ state in 104 Sn and in 102 Cd, respectively.

The shape of the double exponential background was confirmed by rescaling it and comparing
it to a γ-ray spectrum of a neighboring nucleus in the cocktail beam which was created under
111

identical circumstances. To confirm the shape for 104 Sn, the spectrum of 102 In was chosen
and for 102 Cd it was 100 Ag (see Fig. 5.13). In both cases, the difference between each pair of
of the nuclei is one less proton and neutron so they have very similar properties. As shown
in Fig. 5.33, the shape of the background is confirmed by the spectra which have no γ-ray
transition in the region of interest. Of course, only the main features of the shape can be

Counts / 25 keV

30

(a) 102In

25

(a) 100Ag

v/c = 0.345
double exp BG
from 104Sn,
rescaled

20
15
10

v/c = 0.335
double exp BG
from 102Cd,
rescaled

5
0

800

1200 1600 2000
Energy [keV]

600

1000 1400 1600 2000
Energy [keV]

400
350
300
250
200
150
100
50
0

Counts / 20 keV

confirmed and no exact match is expected due to the slight difference in A and Z.

Figure 5.33 Event-by-event Doppler reconstructed γ-ray spectra detected in coincidence
with scattered 102 In (a) and scattered 100 Ag (b). Shown are the 104 Sn background fit scaled
to 102 In (a) and the 102 Cd background fit scaled to 100 Ag. In both spectra, no de-excitations
are observed in the sensitive region.

The number of projectiles, Nproj , was determined from the particle identification spectrum
shown in Fig. 5.14 which was gated on downscaled particle singles trigger. Two dimensional
Gaussian fits, also shown in the figure, were used to estimate the contamination from neighboring, highly intense constituents of the cocktail beam (see Sec. 5.2.2). These corrections
to the number of projectiles were below 3% for both 104 Sn and 102 Cd.
The results of the fits and otherwise necessary information are listed in table 5.4.

112

Property

104 Sn

102 Cd

AW coefficient [mb/(100 e2 f m4 )]

15.30

14.92

Corrected number of incoming ions 2.347(10)E+06
number of 2+
1 de-excitation γ rays

262(42)

6.022(17)E+06
1017(75)

Table 5.4 Important fit and calculation results.

113

Chapter 6
Results and Discussion
With the known value of B(E2 ↑) = 0.28(3) e2 b2 for 102 Cd [2], a B(E2 ↑) value of
B(E2 ↑) = 0.180(37) e2 b2 is deduced for 104 Sn. The uncertainty includes all statistical uncertainties of Nproj , uncertainties from the fits of the response functions used to derive Nγ
and the uncertainty of the 102 Cd B(E2 ↑) value. The aforementioned normalization eliminates systematic uncertainties stemming from the angle-determination with the PPAC and
absolute efficiencies in general. This result compares well with the work by Doornenbal [7]
and disagrees with the recently published value from a relativistic Coulomb excitation measurement performed at GSI [1]. In [7], the impact of unobserved feeding is discussed but for
the relative measurement performed here it does not play a role. 102 Cd and 104 Sn have rather
comparable proton separation energies, Sp = 5.614 MeV and Sp = 4.286 MeV, respectively,
with a similar potential of unobserved feeding from higher-lying 3− and 2+
n states. Nuclear
contributions are minimized in this work by very conservative minimum impact parameters
that the analysis is restricted to, avoiding model-dependent estimates of the nuclear contribution (see Sec. 5.3.2.2). The result of this work and the literature values are summarized
in Table 6.1. An overview of the measured B(E2 ↑) values along the Sn isotopic chain is
given in Fig. 6.1(a).
In order to understand the trend in the B(E2 ↑) strength toward 100 Sn, several large-scale
shell-model (LSSM) calculations were performed. A recently parametrized nucleon-nucleon
force based on chiral perturbation theory [23] (at next-to-next leading order, NNLO) was used
114

Isotope
102 Cd
104 Sn

B(E2 ↑)literature
(e2 b2 )
0.28(3) a
0.281(45)
0.10(4)
0.163(26)

Reference
[2]
[43]
[1]
[7]

B(E2 ↑)
(e2 b2 )

0.180(37)

+
102 Cd and 104 Sn from literature and from this work
Table 6.1 B(E2; 0+
1 → 21 ) values for
(104 Sn); the results from [2] are used for normalization in Eq. 5.5.
a The

error includes statistical and systematic uncertainties.

as well as the N3 LO nucleon-nucleon interaction of [24]. The influence of three-body forces
following Ref. [69] was also studied. 100 Sn was employed as a closed-shell core, which leaves
no valence protons and only 4 valence neutrons for 104 Sn. The model space for the neutrons
was defined by the quantum numbers of the 0h11/2 , 2s1/2 , 1d and 0g7/2 single-neutron states.
In Fig. 6.2 is this configuration shown. All LSSM calculations were performed by Morten
Hjorth-Jensen.
One major problem with this model space is that the spacing of the single-neutron states in
the model space is not well known. This is because the single-particle energies of 101 Sn are
+
not known experimentally, except for the spacing between the 7/2+
1 and the 5/21 states of

170 keV [70, 71]. The effective neutron charge was set to 0.5e for all calculations.
All of these calculations result in rather similar behaviors for the B(E2 ↑) transitions in
the neutron-deficient Tin isotopes. In Fig. 6.1 are therefore only the results presented for
the newly optimized NNLO interaction [23], together with those obtained with the N3 LO
interaction [24]. The latter interaction also gives the overall best reproduction of the excited
states and binding energies. Thus, unless one adopts a phenomenological adjustment of the
effective neutron charges, see for example Ref. [72], theory based on an inert proton core
fails to describe the B(E2 ↑) strengths in Tin.

115

2 2

B(E2;0+1 2+1) (e b )

0.25

This Work
GSI-DSA
Adopted
Riken
REX-ISOLDE
GSI
NSCL
ORNL
IUAC
HRIBF

(a)

0.2
0.15
0.1

E x( M e V )

0.05
4

Sb isotopes
2p-1h

2

(b)
0

50

55

60
65
70
Neutron number

75

80

Figure 6.1 (a) Measured B(E2 ↑) values for the chain of Sn isotopes: Adopted [6], Riken [7],
REX-ISOLDE [8, 9], GSI-DSA [10], GSI [1, 11, 12], NSCL [13], ORNL [14], IUAC [15],
HRIBF [16]. Solid line: Large-scale shell-model (LSSM) calculations with NNLO interaction [23]. Dashed line: LSSM with N3 LO interaction [24]. These two lines correspond
to calculations performed with an effective neutron charge set to 0.5e. An increase to 1e
would yield and increase by a factor of 4 for the calculated B(E2 ↑) values. The dotted line
reproduces the 0g1d2s (tπ = 4) calculation shown in [11]. See text for details. (b) Energies
of the proton 2p − 1h intruder states in Sb nuclei [25].
For further references, proton particle-hole excitations (np-nh) will be described by a single
letter, tπ = n. The number of particles and holes for protons has to be the same (both n)
because there are only valence protons below the Fermi energy.
The results of the here performed calculations are similar to the tπ = 0 results shown in
Fig. 3 of [11]. If the neutron effective charge is increased to 1.0e, then the B(E2 ↑) values
are increased by a factor of 4 and are in better agreement with the data. With that increase

116

Model space
[12] 82
[2] 70
[4] 68
[6] 64
[8] 58

Fermi energy
100

Sn core

[10] 50
[2] 40
[4] 38
[6] 34

[10] 50
[2] 40
[4] 38
[6] 34

[8] 28

[8] 28

[2] 20
[4] 18

[2] 20
[4] 18

[6] 14

[6] 14

[2]
[4]

8
6

[2]
[4]

8
6

[2]

2

[2]

2

Protons

Neutrons

Figure 6.2 Shell model configuration with a closed 100 Sn core. The closed core is marked
in red, the model space for the valence neutrons is marked in blue and the Fermi energy
is marked in green. The visible numbers are the same as in Fig. 2.3. Figure is taken and
modified from [4].

117

the solid and dashed line in Fig. 6.1 would be very similar to the dotted line. But they are
still too small for the neutron-deficient Tin isotopes.
In [11], the model space was increased to allow up to four protons to be excited from the
0g9/2 orbital to 1d, 2s1/2 and 0g7/2 , (tπ = 4). The inert core used for these calculations was
a 90 Zr core. This configuration is shown in Fig. 6.3.
The results from this calculation, with effective charges 1.5e for protons and 0.5e for neutrons,
are also shown in Fig. 6.1 as the dotted line. Overall, the data is much better described,
except that the extended calculation is symmetric around the middle, whereas experiment
shows an asymmetry with an enhancement at the neutron-deficient end.
With the help of B. A. Brown the following interpretation was developed. In [13], the
comparison to data and theory for the Nickel isotopes was discussed. 56 Ni (28 protons and
neutrons) and 100 Sn (50 protons and neutrons) are similar with regard to their structure
and shell gaps. At the closed-shell limit, they are both jj closed shells with the 0f7/2 and
0g9/2 orbitals filled, respectively. There are low-lying particle-hole (np − nh) excitations
across the shell gap that give a moderately large calculated B(E2 ↑) strength in 56 Ni (it was
experimentally observed in [73]) and 100 Sn (not yet experimentally observed but calculated
for Fig. 4 of [1]). These np − nh proton excitations couple coherently to the neutron
configurations. The resulting increase in the B(E2 ↑) values could be interpreted in terms
of an additional neutron core-polarization charge for a model space that involves only the
valence neutrons.
The 100 Sn effective single-particle energy (ESPE) shell gap for the calculations given was
about 7.5 MeV (see Fig. 6.4 taken from [11]). In comparison, the ESPE gap for the calculations used for the Nickel isotopes in Fig. 4 of [13] is 7.1 MeV. The correlated gap obtained
from the full 1p0f -shell calculation and from experimental ground-state binding energies of
118

Model space
[2]
[4]
[6]
[8]

70
68
64
58

[12] 82
[2] 70
[4] 68
[6] 64
[8] 58

Fermi
energy

[10] 50
[2] 40
[4] 38
[6] 34

[10] 50
[2] 40
[4] 38
[6] 34

90

Zr
core

[8] 28

[8] 28

[2] 20
[4] 18

[2] 20
[4] 18

[6] 14

[6] 14

[2]
[4]

8
6

[2]
[4]

8
6

[2]

2

[2]

2

Protons

Neutrons

Figure 6.3 Shell model configuration with a closed 90 Zr core. The closed core is marked in
red, the model space for the valence neutrons and up to 4 protons (tπ = 4) is marked in blue
and the Fermi energy is marked in green. The visible numbers are the same as in Fig. 2.3.
Figure is taken and modified from [4].

119

Figure 6.4 Calculated evolution of the proton ESPE with the neutron number N along the
chain of Sn isotopes. Figure is taken and modified from [11].
55,56,57 Ni is 6.4 MeV. The correlated gap for 100 Sn is not known experimentally and the

theoretical value is not provided in [11]. The shell gaps obtained with some Skyrme energydensity functionals for 100 Sn are 5.2 MeV (Skx) [74], 5.9 MeV (SLy4) [75] and 5.8 MeV
(SkM*) [76], and for 56 Ni are 4.1 MeV (Skx), 4.8 MeV (SLy4) and 4.7 MeV (SkM*). Thus,
the shell gaps for 56 Ni and 100 Sn are similar.
The energy of the two-proton excitation is much lower than twice the shell gap energy due
to pairing and α correlations. For 102 Sn, the two excited protons go into the same deformed
quantum states that are occupied by the two valence neutrons (see Fig. 6.5 taken from [13]).
This lowers the energy due to the α-correlation energy and increases mixing with the neutron
120

configurations near N = Z.

Figure 6.5 Schematic representation of the proton 2p − 2h excitations across the Z = 50 shell
gap in 102 Sn. The occupation of the same proton and neutron orbitals above the Z = N = 50
shell leads to α-like correlations between the valence nucleons. Figure is taken and modified
from [13].
The α correlation persists for 104 Sn and gradually decreases for increasing number of neutrons
since the configurations for the added neutrons differ from the proton configurations. Thus, α
correlations provides a mechanism to generate an asymmetry in the B(E2 ↑) values between
N = 50 and N = 82. Part of the α correlations could be missed in the (tπ = 4) calculations
due to the truncation that had to be made for Tin. In addition, there is an asymmetryinducing contribution to the B(E2 ↑) values beyond (tπ = 4) that is obtained in the Nickel
calculations.
In contrast to the (tπ = 4) calculation for Tin isotopes that is symmetric around mid-shell,
the equivalent calculation for Nickel (Fig. 4 in [13]) shows an asymmetry with larger B(E2 ↑)
values at the neutron-deficient end. For 60 Ni, a calculation with a 56 Ni closed shell (4 valence
neutrons) yields a B(E2 ↑) value of only 0.010 e2 b2 whereas opening up the full 1p0f model
space results in a 10-fold increase to 0.105 e2 b2 . This large increase is due to the proton

121

excitations and their coherent contribution to the E2 transition matrix elements. In addition
to the increase, an asymmetry is introduced as a function of the number of valence neutrons
beyond 56 Ni such that the increase is larger at the neutron-deficient end. The energy of
the proton core excitations near N = Z is low due to α correlations for the protons and
neutrons in the orbitals above f7/2 . It is likely that the origin of the difference between the
calculations for Nickel and Tin comes from the truncations that must be made for Tin.
The energy of the low-lying intruder states with mp − nh configurations is also sensitive to
these correlations. The 2p − 2h state in 60 Ni is observed at 3.32 MeV from the 56 Fe(6 Li,d)
reaction [77]. The calculated energy of this state in the full 1p0f shell model is in agreement
with experiment (see Fig. 20 of [78]). In Sn, the proton core-excitations are connected to
the proton particle-hole intruder states observed in Sb isotopes. In the 100 Sn region, 2p − 1h
states have been observed down to N = 56 in the Sb chain as shown in Fig. 6.1(b). The
energies for the 2p − 1h 9/2+ levels have been identified and described in [79] for 113−119 Sb
nuclei and extended to 107,109 Sb in [80]. The trend, shown in Fig. 6.1(b) for Sb [25], may be
attributed in addition to the α correlation in part also to shell evolution driven by the robust
monopole parts of the proton-neutron tensor force [81]. The strongly attractive tensor force
between the νg7/2 and πg9/2 orbitals weakens as the g7/2 neutron occupation decreases from
108 Sn to 100 Sn, making the πg
9/2 orbital less bound and effectively narrowing the Z = 50

shell gap towards N = 50. On the neutron-rich end of the Sn chain, it is known that the
Z = 50 shell gap is large at N = 82 [82–84]. This is consistent with the energy trend of the
proton 2p − 1h excitations in Sb shown in Fig. 6.1(b), where the excitations across Z = 50
become increasingly more expensive towards N = 82, making the shell-model assumption
of an inert proton core more valid. The role of the protons can be quantified further by
comparison to 102,126 Cd with 4 neutron particles/holes relative to N = 50/82. The ratio of
122

B(E2 ↑, 126 Cd)/B(E2 ↑, 128 Sn) = 3 [16,85] indicates that with two protons added to 126 Cd,
the collectivity at 128 Sn drops by a factors of 3, owing to the absence of proton contributions
in the heavier Sn. For B(E2 ↑, 102 Cd)/B(E2 ↑, 104 Sn), the addition of two protons results
in a loss of only a factor of 1.6 in collectivity, consistent with more proton contributions to
the 2+ wave function in 104 Sn.
For nuclei closer to 100 Sn nothing is known yet about proton intruder excitations. Observation of these states and a comparison to the predictions of the models that include proton
excitations are required to complete the understanding of the asymmetry. This prevalence
of proton core excitations near N = Z is also important for understanding the B(E2 ↑)
values in 18 O, 42 Ca and 44 Ca [86,87] – making the situation encountered in the Tin isotopes
consistent with observations along proton-magic chains across the nuclear chart.

123

Chapter 7
Summary and Future Work
104 Sn was determined from intermediate-energy
In summary, B(E2; 0+ → 2+
1 ) strength for

Coulomb excitation. The result is at variance with a recently published measurement [1]. Unlike the conclusion of [1], the departure from large-scale shell-model calculations persists for
the neutron-deficient Sn isotopes approaching 100 Sn. This deviation from shell model seems
to originate from the interplay of proton particle-hole configurations beyond the necessarily
limited shell-model spaces. Those configurations and their effect on quadrupole collectivity
is a common phenomenon along proton-magic isotope chains.

In the future, experiments measuring the B(E2 ↑) values of 102 Sn and 100 Sn are a major
goal and will help the understanding of the theory in this region. Especially doubly magic
100 Sn is a very interesting case because not even the energy of the first excited 2+ state is

yet known experimentally. With the construction of FRIB at Michigan State, the feasibility
of producing 102 Sn and 100 Sn with a reasonable rate and purity increases greatly.
Following the arguments given in this thesis, one can predict the B(E2 ↑) value of 102 Sn.
Guidance can be given by the Nickel isotopes. In particular, 58 Ni is comparabel in its
structure to 102 Sn. Both isotopes have 2 neutrons in addition to a jj closed N = Z core,
namely 56 Ni and 100 Sn (see Sec. 6). For both isotopes calculations with up to 4 proton
particle-hole excitations (tπ = 4) were done as shown in Sec. 6. For 58 Ni the experimental
B(E2 ↑) value is known, too. Therefore, one can compare the experimental value for 58 Ni to
124

the shell model calculation and extract a scaling factor. This factor is estimated to be roughly
1.6 in comparison to the tπ = 4 LSSM calculation (estimated from Fig. 4 in [13]). Applying
this factor to the tπ = 4 calculation of 102 Sn gives an estimated value of B(E2 ↑) ≈ 0.08 e2 b2 .
With the development of more powerful computers it would also be very interesting to lift the
so far necessary restrictions of the LSSM calculations and open the proton 0g9/2 orbital up
to allow excitations for all 10 protons. This will most likely confirm the hypothesis that the
asymmetry in the experimentally observed B(E2 ↑) values stems from proton particle-hole
excitations and α correlations and can be reproduced by LSSM with a large enough model
space.

125

APPENDIX

126

CAESAR Simulations
Travis Baugher and Geoffrey Grinyer wrote the GEANT4 simulations [63] that were used in
this experiment to fit the data. In the following is a summary of how to use the simulation
which is taken from [3] with additions. It also entails a description of an analysis program,
ions2.C and instructions on how to use the fitting programs. The simulations have been used
in two publications so far [88, 89].

User’s Guide for CAESAR Simulations
Installation
To use the simulation and sorting code, Geant41 and ROOT2 need to be installed. Two
versions of the simulation exist, one that can do only in-beam data but can separate the
Compton edge from the full energy peak (fep) if wanted and one that cannot do the separation
but can do source simulations. The simulation tarball is available from
https://groups.nscl.msu.edu/gamma/wiki/doku.php?id=computers:computers
it should be extracted into a directory of choice (all folder references are found in this
directory) and compiled using the following commands.
1 http://geant4.web.cern.ch/geant4/
2 http://root.cern.ch/drupal/

127

On the NSCL network first run:
export G4WORKDIR=$PWD
and then
source $GEANT4_PATH/share/Geant4-9.6.2/geant4make/geant4make.sh
This sets the environment variables and then one can run
make clean
make

Running the Simulation
UCCAESAR run.mac
will start UCCAESAR and execute the commands in run.mac. See examples below.

Sorting
The program caesarsort is used to sort the output file into histograms. To compile, one
has to go into the folder caesarSort and then execute
make clean
make
After that one has to go back into the main folder.
Do caesarsort -h for usage instructions.

Depending on which version is used (the in-beam separation version or the source version),
128

there are different caesarsorts.
For the source version one can just go ahead and run it like this:
./caesarsort run.out -o run.out.root

For the in-beam version there are three options, first:
./caesarsort run.out -o run.out.root
which does not separate the Compton edge and the fep, second:
./caesarsort -fep run.out -o run fep.out.root
which gives one only the fep and lastly,
./caesarsort -compton run.out -o run compton.out.root
which gives one only the Compton scattered contribution.

Spectrum naming: The naming convention used by caesarsort is as follows.
X n0 means that each event in that spectrum did not have any neighboring detectors fire.
X n1 means that exactly one of the neighboring detectors fired and add-back was done

into the detector that registered the higher energy of the two.
X n0n1 is n0 and n1 added together.
X ng means that more than two neighboring detectors fired so we do not know what to

do (g for garbage).
X cal means not Doppler corrected
X dop means Doppler corrected.

129

Detector numbering: The most upstream ring is called ring A, most downstream is J.
Within each ring, detectors are numbered starting from 1 with the upper left corner going
clockwise if facing downstream (user is looking downstream).
Important text files:

caesarsort reads these files at run time. They must be in the

same directory as the executable.
X neighbors.txt has the neighbor relationships between the detectors for addback. Do

not edit this unless one really wants to.
X detectorPositions.txt contains the positions of the detectors in space relative to

the center of the array. This is another file one probably does not want to edit. This
file is also read by the simulation program to set up the geometry.
X omitDets.txt one lists in here numbers of the detectors that should be left out of the

sort. One might want to do this if there are missing detector signals, for example. Use
the command -x to activate.
X threshParams.txt Model the energy threshold of the CFD for each individual detector.

First column is the mean of a Gaussian, second column is FWHM
X widthParams.txt Parametrizes the energy resolution of each CAESAR detector as

σ(E) = aE b
X Doppler.txt Contains the β parameter for the Doppler reconstruction.
X caesarSort/Def.h Contains the parameter DEFAULT DIM which defines how many

bins are used for the sorting, the default is 8192.

130

Example Input Files
The following sections contain example macro files typically used to analyze an experiment.
There are separate inputs for simulation of sources, in-beam experiments, target excitations,
and ion tracking modes.
Before one does any of this it needs to be checked in src/Beam Tube.cc that the parameter
of the used beam pipe are the same as the one used in the experiment.

Source Simulations
Run source simulations to compare simulated and measured efficiency, determine resolution
and threshold parameters.
# set up a simulation of a source
/Experiment/RunSource
/Experiment/Source/Set simple
# $\gamma$-ray energy to simulate
/Experiment/Source/setEnergy 662. keV
# target material and thickness: ’G4_Galactic’ is vacuum, i.e., no target
/Target/Material G4_Galactic
/Target/Thickness 0.1 mm
# position of target and source emission point
/Target/SetPosition_Z 2.55 cm
/Experiment/Source/setZ 2.55 cm
# filename to save the output and number of $\gamma$ rays to simulate
/Output/Filename Cs137.out
131

/run/beamOn 1000000

In-beam Simulations
To simulate a γ-ray spectroscopy run, use:
# set up a simulation of an in-beam experiment
/Experiment/Reaction/On
# target material
/Target/Material Au
# thickness of target
/Target/Thickness 332.5 um
# size of target
/Target/X_length 5.0 cm
/Target/Y_length 5.0 cm
# position of target relative to the center of {\it CAESAR}
# positive values are downstream of center
/Target/SetPosition_Z 2.55 cm
# define the beam, this example is 74Ni
/BeamIn/A 74
/BeamIn/Z 28
# energy (per nucleon) of the incoming beam
/BeamIn/KEu 94.5 MeV
# fractional momentum spread of the incoming beam, 0.5\% = 0.005
/BeamIn/Dpp 0.005

132

# nucleons removed in the reaction, this is inelastic scattering
/BeamOut/DA 0
/BeamOut/DZ 0
#

excited state energy to simulate

/BeamOut/ProjectileExcitation 1024. keV
#excited state lifetime (not half life)
/BeamOut/tau 0 ps
# angular distribution coefficients for Coulomb excitation, calculated
for a certain "safe" angle cut (here it is isotropic which is not
realistic)
/BeamOut/seta0 1.0
/BeamOut/seta2 0.0
/BeamOut/seta4 0.0
# outgoing beam angular distribution in the dispersive and non-dispersive
/BeamOut/AngDistSigmaA 0.012 rad
/BeamOut/AngDistSigmaB 0.012 rad
# print information about the simulation parameters
/BeamIn/Report
/BeamOut/Report
/Target/Report
/ExpHall/Report
/CAESAR_Array/Report
# print detailed information during simulation steps (diagnostic
purposes and usually commented out because it generates a huge output)
133

#/IonPrint/Track_Set
# file to save the output
/Output/Filename 74Ni.out
# number of events to simulate
/run/beamOn 1000000

Ion Tracking
Ion tracking mode is useful for checking the beam parameters, such as kinetic energy, transverse and parallel momentum distributions, velocity, position and angles at several points
during the simulation. To make sure that the set up of the simulation is correct the following
steps need to be done:
X Make one simulation with the command /IonPrint/Track Set active (usually com-

mented out) in the .mac file and use 2000 particles.
X When one starts the simulation it has to be made sure that the output is written into

a .log file like this: UCCAESAR test.mac > test.log
X Do a second simulation with the print command commented out: #/IonPrint/Track Set

and use at least 100000 particles
X Sort the last simulation with caesarsort to create a root file
X Start root and Load ions2.C with .L ions2.C.
X Type in the command Do("test.log","real sim.root",A). The first input is the

.log file which was created with only 2000 particles, the second input is the sorted

134

root file created with at least 100000 particles and the third input is the mass number
A of the simulated beam.
This creates a ROOT file out of the .log file and also plots some overview spectra to check the
simulations. One only has to run this once, if the .log.root file already exits and one wants
to plot the overview spectra then one can use Plot("test.log","real sim.root",A).

Target Excitations
To simulate excitations of the target, 197 Au, for example, use:
# set up a simulation of an in-beam experiment
/Experiment/Reaction/On
# target material
/Target/Material Au
# thickness of target
/Target/Thickness 332.5 um
# size of target
/Target/X_length 5.0 cm
/Target/Y_length 5.0 cm
# position of target relative to the center of caesar
# positive values are downstream of center
/Target/SetPosition_Z 2.55 cm
# Produce only target excitations
/BeamOut/TargetExcitationFraction 1
#

excited state energy to simulate
135

/BeamOut/TargetExcitation 547.5 keV
# angular distribution coefficients for Coulomb excitation
# of the target
/BeamOut/setTargeta0 1.0
/BeamOut/setTargeta2 0.0
/BeamOut/setTargeta4 0.0
# print information about the simulation parameters
/BeamIn/Report
/BeamOut/Report
/Target/Report
/ExpHall/Report
/CAESAR_Array/Report
# file to save the output
/Output/Filename target_excitation.out
# number of events to simulate
/run/beamOn 1000000

Converting SpecTcl Spectra into Root Spectra
To use the fitting routines one needs to have all spectra in a root format. If one wants to
convert SpecTcl spectra into root spectra follow these steps:
1. Make sure that the following programs are in the .bashrc file:
module load root
module load tv
136

2. Create a new directory, lets call it NewDir for now.
3. Copy the spectra one wants to transform into NewDir.
4. Also copy the programs spc2root.sh and tvtxt2root.C into NewDir.
5. Edit the spc2root.sh file so that the file path points to the local copy of tvtxt2root.C
6. Run the spc2root.sh with the file one wants to convert as an argument:
./spc2root.sh Example histo.spc
This creates the root spectrum!

Creating the right Spectra for the Fitting Procedures
To use the fitting routines the right format of the created simulation spectra is needed, e.g.
the forward (ring F-J), backward (ring A-E) and full CAESAR (ring A-J) spectra are wanted.
For this there are two little scripts called ForwardBackward Simulation.C and
MakeFOrBackHistos Simulation.C.
ForwardBackward Simulation.C is the actual program that creates the spectra. It has
a function called plot that needs two input variables:
plot(char* inFile, char* Histogram)
The first is the file name of the created simulation root file but without the .root ending and
the second is the histogram type that is supposed to be used. It creates a new file that has
the same name as the input file with the difference that it ads Forward Backward.root
to the new file name. One can simply use it by starting root, loading it:
.L ForwardBackward Simulation.C
and then typing in the command
137

plot("test.out","cal")
which needs the file test.out.root created with caesarsort and then it creates all kinds of
groupings for the histograms called ”cal”.

To make life easier there is also the program MakeFOrBackHistos Simulation.C which
simply executes the ForwardBackward Simulation.C script several times in root. A
typical command list looks like this:
{
char buffer[1000];
sprintf(buffer,".L ForwardBackward_Simulation.C");
gROOT->ProcessLine(buffer);

plot("56Ni_2700.out","cal");
plot("56Ni_2700.out","dop");
plot("56Ni_2700.out","n0_cal");
plot("56Ni_2700.out","n0_dop");

plot("56Ni_2700_fep.out","cal");
plot("56Ni_2700_fep.out","dop");
plot("56Ni_2700_fep.out","n0_cal");
plot("56Ni_2700_fep.out","n0_dop");

plot("56Ni_2700_compton.out","cal");
plot("56Ni_2700_compton.out","dop");
138

plot("56Ni_2700_compton.out","n0_cal");
plot("56Ni_2700_compton.out","n0_dop");
}
And it can be executed by starting root and then using the command:
.x MakeFOrBackHistos Simulation.C

Fitting Scripts
A script was was written which fits the simulated CAESAR spectra to the data. It can do
this with as many simulated peaks as one wants and there are two versions. One version fits
the Compton part and the fep part separately with a scaling factor between the two as free
parameter. The other one fits the whole simulation to it. In both cases one can choose if
the fit should include all rings of CAESAR or only the forward or backward rings. Another
option is to choose what simulation should be fitted to the data, either all or n0 and so on.
In general the fits can either just fit the simulations to the data or they can include a double
exponential background. For the separate version the fitting routine has one free parameter
for every Compton-fep pair, one free parameter for an overall ratio between the Compton
and fep part and if wanted 4 more free parameters for the double exponential background.
The version that does the full simulation fit just has one free parameter for each simulated
peak and if wanted 4 more free parameters for the double exponential background. The first
script is called Fit seperate const.C and the second one Fit.C.

How the script for the fits with the separate Compton and fep part, Fit seperate const.C,
139

works is shown at hand of the Ni56 case. Before one can use the script some settings need
to be changed:
int comp = 2;

//number of total simulations one wants to fit

int comp2 = 1;

//number of simulation pairs (Compton+fep) one wants to

fit (should be comp/2)
int rebin = 16;

//rebinning for the simulations

int rebin1 = 16; //rebinning for the data
(in the end data and sim must have same bin size)
int simPart = 10000000; //number of simulated particles
and if one wants to use the double exponential background
//bg fit for initial parameters excludes everything between exclude1 and exclude2
double exclude1 = 1600;
double exclude2 = 3200;
Then one has to tell the program where the simulations are. The input here have to be the
files that were created with the MakeFOrBackHistos Simulation.C program. One has
to always give them as pairs so the fep part has to be 0 and the Compton part 1 and then
the next simulated energy has its fep part assigned to 2 and Compton to 3 and so on.
TFile *fsim[comp];
fsim[0] = new TFile("56Ni_2700_fep.out_Forward_Backward.root");
fsim[1] = new TFile("56Ni_2700_compton.out_Forward_Backward.root");
Next it needs to be specified what simulated histogram should be used for the fitting, all of
CAESAR or just the forward or backward part and if the full spectrum or for example n0
should be used. Only the spectrum that is desired should be commented in:
140

hsim[i] = (TH1F*)fsim[i]->Get("dop");
//hsim[i] = (TH1F*)fsim[i]->Get("dop_forwardrings");
//hsim[i] = (TH1F*)fsim[i]->Get("n0_dop");
//hsim[i] = (TH1F*)fsim[i]->Get("n0_dop_forwardrings");
The last change in the script is the naming of the histograms, make sure that all of them
are named individually:
hsim[0]->SetName("2700_fep_dop");
hsim[1]->SetName("2700_compton_dop");
To use the fitting routine start root and compile the script by doing: .L Fit seperate const.C+
The script has a function called dofit:
dofit(const char* file, char* Histo, double from, double to, double from2, double
to2, int Ex);
The first input is the root data file, the second one is the name of the histogram that is
supposed to be fitted. If one created the histogram by converting it from SpecTcl to root
then it is always called hist. The next two numbers define the total fit range and the following two numbers define the range where the program gets its initial parameters for the
double exponential background (it excludes the range one specified in the script). Not to get
confused this is only to get good starting parameters for the final fit. The last number has
to be either 0 or 1. It turns the background off or on in the final fit (0=off, 1=on). The fit
can be run like this:
dofit("a-j_40mrad_m01.root","hist",1700,4500,800,4500,0);
It creates a nice little plot with all important information one needs. There are two possible
warnings. One is about the exponential background, if this comes up it means that the
141

background is not monotone decreasing with increasing energy. The other error comes up if
the ratio Compton to fep is negative. On the plot also show up 2 green and 4 red lines. the
2 green lines indicate the fitting range for the total fit. In between the first two red lines
and in between the third and fourth is indicated the range that was used to get starting
parameters for the double exponential background.

142

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