RECOIL DISTANCE METHOD LIFETIME MEASUREMENTS VIA
GAMMA-RAY AND CHARGED-PARTICLE SPECTROSCOPY AT NSCL
By
Philip Jonathan Voss

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics
2011

ABSTRACT
RECOIL DISTANCE METHOD LIFETIME MEASUREMENTS VIA
GAMMA-RAY AND CHARGED-PARTICLE SPECTROSCOPY AT NSCL
By
Philip Jonathan Voss
The Recoil Distance Method (RDM) is a well-established technique for measuring lifetimes of electromagnetic transitions. Transition matrix elements derived from the lifetimes
provide valuable insight into nuclear structure. Recent RDM investigations at NSCL present
a powerful new model-independent tool for the spectroscopy of nuclei with extreme protonto-neutron ratios that exhibit surprising behavior.
Neutron-rich 18 C is one such example, where a small B(E2; 2+ → 0+ ) represented a
gs
1
dramatic shift from the expected inverse relationship between the B(E2) and 2+ excitation
1
energy. To shed light on the nature of this quadrupole excitation, the RDM lifetime technique
was applied with the Köln/NSCL plunger. States in 18 C were populated by the one-proton
knockout reaction of a 19 N secondary beam. De-excitation gamma rays were detected with
the Segmented Germanium Array in coincidence with reaction residues at the focal plane of
the S800 Magnetic Spectrometer. The deduced B(E2) and excitation energy were both well
described by ab initio no-core shell model calculations.
In addition, a novel extension of RDM lifetime measurements via charged-particle spectroscopy of exotic proton emitters has been investigated. Substituting the reaction residue
degrader of the Köln/NSCL plunger with a thin silicon detector permits the study of shortlived nuclei beyond the proton dripline. A proof of concept measurement of the mean lifetime of the two-proton emitter 19 Mg was conducted. The results indicated a sub-picosecond
lifetime, one order of magnitude smaller than the published results, and validate this new
technique for lifetime measurements of charged-particle emitters.

for laura

iii

ACKNOWLEDGMENTS
This work culminates many long years of learning, preparation, frustration, and triumph
and has been made possible with the assistance, guidance, and support of many individuals.
My humble gratitude extends far beyond this page. If I have completed anything, it is only
through Him who gives me strength.
My undergraduate experiences set me on this course. I’d like to thank the many wonderful
people in the Central Michigan University Physics Department and Honors Program for all
they’ve done. Fire up Chips! I then was fortunate to continue my education at Michigan
State University, where I had the opportunity to learn under the expert instruction of many
talented professors. Their commitment to education and excellence in research is inspiring.
I thank my guidance committee members for their collective role in shaping my graduate school career: Dr. B. Alex Brown, Dr. R. Sekhar Chivukula, Dr. David Morrissey, and
Dr. Michael Thoennessen. In addition, the listening ear, patient consultation, and critical
insight provided by Dr. Hironori Iwasaki and Dr. Thoennessen during my final months were
most appreciated. My deepest gratitude goes to my graduate advisor, Dr. Krzysztof Starosta.
His tutelage, guidance, and support were instrumental in spurring me forward.
My research would not have been possible without the critical support of many different
groups at NSCL. The assistance and dedication of the Computer Group, the Cyclotron
Operations Department, and the A1900 Fragment Separator Group was pivotal. Daniel Bazin
and Dirk Weisshaar warrant special acknowledgment for their tireless work on the S800 and
SeGA. I offer my many thanks to the members of the Gamma Group and collaborators from
IKP Köln and LBNL who aided in experimental setup, shift-taking, and analysis discussions.
To the many friends I made- may our common bond forged over late-night study sessions
and informal gatherings withstand both time and distance! My gratitude and admiration
belongs to Dr. David Miller, the elder half of Team Starosta. Andrew Ratkiewicz deserves

iv

much credit for his keen insight as both a draft editor and as one never afraid to offer his
opinion and counterpoint. Our discussions over coffee made me a better physicist. I must
also thank my fantastic officemates both past and present. “Save Kashy!”
I will always remember my comrades of various enjoyable diversionary ploys. Thanks to
the Tuesday Morning Basketball squad and Team Physics softball for slaking my competitive
thirst. I am grateful for the gathering of friends on Sunday evenings at our most beloved East
Lansing establishment. The Cyclotron DALMAC crew provided many blissful and spandexclad hours pedaling through the beautiful Michigan countryside. Remember fondly the image
of me rising unscathed from the ditch at least once per ride!
I am most indebted to my parents, Jonathan and Deborah Voss, for their constant love,
support, and encouragement to walk my own path. The greatest debt of gratitude is owed
to my beautiful wife, Laura Jean Voss. You were with me from the beginning of this long
process and still are now, even as I write these very words. Your love, immense patience, and
companionship helped bring me to this point. Yet our journey is just beginning...

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Nuclear Landscape . . . . . . . . . . . . . . . . . . . . . .
1.2 Nuclear Structure within the Shell Model . . . . . . . . . . . .
1.2.1 Nuclear Shell Structure . . . . . . . . . . . . . . . . . .
1.2.2 Ab Initio No-Core Shell Model . . . . . . . . . . . . . .
1.3 Nucleon Knockout from Radioactive Beams . . . . . . . . . .
1.4 Köln/NSCL Plunger Lifetime Measurements . . . . . . . . . .
1.4.1 Lifetime Measurements of Electromagnetic Transitions
1.4.2 Lifetime Measurements of Proton Radioactivity . . . .

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2 Experimental Methods . . . . . . . . . . . . . . . . .
2.1 Introduction and Overview . . . . . . . . . . . . . . . .
2.2 Radioactive Ion Beam Production . . . . . . . . . . . .
2.2.1 Primary Beam Ion Source . . . . . . . . . . . .
2.2.2 Primary Beam Acceleration and Fragmentation
2.2.3 Fragmentation Separation for Secondary Beams
2.3 Experimental Technique and Devices . . . . . . . . . .
2.3.1 The Recoil Distance Method . . . . . . . . . . .
2.3.2 The Köln/NSCL Plunger . . . . . . . . . . . . .
2.3.3 The S800 Spectrometer . . . . . . . . . . . . . .
2.3.4 The Segmented Germanium Array . . . . . . .
2.4 Analysis Software . . . . . . . . . . . . . . . . . . . . .
2.4.1 Data Histogramming with gf3 . . . . . . . . . .
2.4.2 The Geant4/ROOT Simulation Package . . . .

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3 Lifetime Measurements in 18 C . . . . . . . . . . . . .
3.1 Experimental Considerations . . . . . . . . . . . . . . .
3.2 Experimental Details . . . . . . . . . . . . . . . . . . .
3.2.1 Radioactive Secondary Beam . . . . . . . . . .

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4 Two-Proton Decay Lifetime of 19 Mg . . . . . . . . . . .
4.1 Experimental Considerations . . . . . . . . . . . . . . . . .
4.2 Experimental Details . . . . . . . . . . . . . . . . . . . . .
4.2.1 Radioactive Secondary Beam . . . . . . . . . . . .
4.2.2 Outgoing Reaction Residues . . . . . . . . . . . . .
4.2.3 Observed DSSD Energy Loss Spectra and Analysis
4.3 Lifetime Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Specifics of Geant4/ROOT Simulations . . . . . . .
4.3.2 Two-Proton Decay Lifetime of 19 Mg . . . . . . . .

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117

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 18 C Commentary . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Nuclear Structure of Neutron-Rich Carbon Isotopes
5.1.2 Comparison to ab initio NCSM . . . . . . . . . . .
5.1.3 Suggestions and Outlook . . . . . . . . . . . . . . .
5.2 19 Mg Commentary . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Comparison to Previous Work . . . . . . . . . . . .
5.2.2 Comments on Future Prospects . . . . . . . . . . .

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131
136
137
140

3.3

3.2.2 Outgoing Reaction Residues . . . . . . . . . . .
3.2.3 Observed Gamma-Ray Transitions and Analysis
Lifetime Analysis . . . . . . . . . . . . . . . . . . . . .
3.3.1 Specifics of Geant4/ROOT Simulations . . . . .
3.3.2 Electromagnetic Transition Rates in 18 C . . . .

6 Conclusions

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A Electromagnetic Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . 147
A.1 Parity Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.2 Angular Momentum Selection Rules . . . . . . . . . . . . . . . . . . . . . . . 148
B Relativistic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.1 Relativistic Kinematics Formulation . . . . . . . . . . . . . . . . . . . . . . . 150
B.2 Doppler Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
C Gamma-Ray Interactions with Matter
C.1 Interaction Mechanisms . . . . . . . .
C.1.1 Photoelectric Absorption . . . .
C.1.2 Compton Scattering . . . . . .
C.1.3 Pair Production . . . . . . . . .
C.2 Gamma-Ray Attenuation . . . . . . . .

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

vii

LIST OF TABLES

3.1

19 N secondary beam characteristics . . . . . . . . . . . . . . . . . . . . . . .

78

4.1

20 Mg secondary beam characteristics . . . . . . . . . . . . . . . . . . . . . .

104

4.2

19 Mg two-proton decay lifetime dependence on R and R
p
σ . . . . . . . . . .

118

5.1

Measured 18 C excited state properties

5.2

Ab initio no-core shell model calculations for 18 C . . . . . . . . . . . . . . . 131

5.3

19 Mg mean lifetime as a function of R . . . . . . . . . . . . . . . . . . . . .
p

5.4

Comparison of 19 Mg mean lifetime to previous work . . . . . . . . . . . . . . 138

viii

. . . . . . . . . . . . . . . . . . . . . 124

136

LIST OF FIGURES

1.1

Chart of the nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Energy considerations for two-proton radioactivity . . . . . . . . . . . . . . .

18

1.3

Diagram of partial rp-process pathway . . . . . . . . . . . . . . . . . . . . .

21

2.1

Diagram of radioactive beam production devices . . . . . . . . . . . . . . . .

35

2.2

Schematic of the Recoil Distance Method . . . . . . . . . . . . . . . . . . . .

36

2.3

Illustration of the Köln/NSCL plunger piezoelectric actuators . . . . . . . .

40

2.4

Köln/NSCL plunger distance calibration plot . . . . . . . . . . . . . . . . . .

41

2.5

The Köln/NSCL plunger mounted in the beam line . . . . . . . . . . . . . .

43

2.6

Gain-matched silicon double-sided strip detector spectra . . . . . . . . . . .

48

2.7

Diagram of the S800 Magnetic Spectrometer . . . . . . . . . . . . . . . . . .

49

2.8

Illustration of trajectory-corrected particle identification in S800 . . . . . . .

55

2.9

Diagram of a single Segmented Germanium Array crystal. . . . . . . . . . .

58

2.10 SeGA gamma-ray energy calibration plot . . . . . . . . . . . . . . . . . . . .

62

2.11 SeGA photopeak efficiency curves . . . . . . . . . . . . . . . . . . . . . . . .

64

2.12 Screenshot of the Geant4/ROOT simulation user interface . . . . . . . . . .

72

3.1

Identification of incoming 19 N secondary beam component . . . . . . . . . .

79

3.2

Identification of outgoing 18 C reaction residues . . . . . . . . . . . . . . . . .

81

3.3

Doppler reconstruction and particle gating effects on γ-ray spectra . . . . . .

82

3.4

Observed gamma-ray transitions in 18 C . . . . . . . . . . . . . . . . . . . . .

83

3.5

18 C level scheme

84

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ix

3.6

SeGA experimental and simulated photopeak efficiency curves . . . . . . . .

87

3.7

Experimental and simulated secondary beam target parameter fits . . . . . .

89

3.8

Best-fit simulated gamma-ray lineshapes and linear background . . . . . . .

91

3.9

χ2 minimization for the lifetime of the 2+ state in 18 C . . . . . . . . . . . .

92

3.10 Lineshape comparison of data and best-fit simulations . . . . . . . . . . . . .

93

3.11 18 C production yield ratio from fit to 2D χ2 hypersurface . . . . . . . . . . .

95

3.12 Best-fit simulations for 932 keV feeder transition . . . . . . . . . . . . . . . .

98

4.1

Identification of incoming 20 Mg secondary beam component . . . . . . . . . 104

4.2

Identification of outgoing 17 Ne reaction residues . . . . . . . . . . . . . . . . 106

4.3

Particle gating effects on DSSD energy loss spectra . . . . . . . . . . . . . . 108

4.4

Select experimental ∆E DSSD spectra . . . . . . . . . . . . . . . . . . . . . 109

4.5

Experimental and simulated secondary beam target parameter fits . . . . . . 112

4.6

Experimental and simulated unreacted beam DSSD energy deposits . . . . . 114

4.7

Fitting functions for high-energy ∆E DSSD background . . . . . . . . . . . 116

4.8

Reduced χ2 lifetime distributions . . . . . . . . . . . . . . . . . . . . . . . . 119

4.9

DSSD energy loss fits of data at several simulated lifetimes . . . . . . . . . . 120

4.10 Best-fit 19 Mg lifetimes as a function of Rp . . . . . . . . . . . . . . . . . . . 121
4.11 χ2 hypersurface as a function of τ and Rp . . . . . . . . . . . . . . . . . . . 122
ν
5.1

Observed B(E2; 2+ → 0+ ) trend for neutron-rich carbon isotopes . . . . . . 126
gs
1

5.2

Sample ab initio no-core shell model results for 18 C . . . . . . . . . . . . . . 130

5.3

Enhanced gamma-ray interaction position information with DDAS . . . . . . 135

5.4

DSSD energy loss fit with best-fit lifetime value . . . . . . . . . . . . . . . . 139

5.5

DSSD energy loss fit with literature lifetime value . . . . . . . . . . . . . . . 139

5.6

Silicon DSSD thickness comparisons . . . . . . . . . . . . . . . . . . . . . . . 141

B.1 Contributions to gamma-ray energy resolution . . . . . . . . . . . . . . . . . 157

x

Chapter 1
Introduction
The complexity of the natural world belies the fact that all tangible aspects of day-to-day life
are comprised of fewer than 300 fundamental building blocks. From this relatively small basis,
everything from a blade of grass to the tallest skyscraper can be constructed. These stable
building blocks, called atoms, consist of a densely packed nuclear core of positively charged
protons and chargeless neutrons surrounded by a cloud of negatively charged electrons. The
number of protons within a nucleus determines the chemical element; fewer than 100 occur
naturally on Earth.
Elements come in distinct varieties, or isotopes, that are differentiated from one another
based upon the number of neutrons within the nucleus. From all possible combinations of
neutrons and protons, less than 300 are found to be stable. Perhaps a more striking way to
phrase this is that of the roughly 7000 [1, 2] possible combinations of protons and neutrons
verified or thought to exist, all but approximately 300 undergo spontaneous radioactive decay
to achieve a more stable configuration.
The study of the processes responsible for the stability of some isotopes and the radioactivity of the remainder is the fundamental pursuit of nuclear physics. The challenge to
discern the properties and behavior of these microscopic objects is daunting. Typical atomic

1

diameters are on the order of angstroms (10−10 m), but over 99.9% of the atomic mass
resides within the nucleus with a diameter on the order of 10 femtometers (10−14 m). A
staggering 150 million million (1.5 × 1014 ) nuclei stacked one atop another are required to
reach the height of an average human. This roughly corresponds to the number of football
fields necessary to span the distance between our Sun and the edge of the Solar System.
Yet the current progress is equally impressive. New experimental facilities provide energetic beams of radioactive isotopes that can produce even more exotic nuclei which normally
exist only in the most violent of stellar explosions. To study the properties of these exotic
nuclei, state-of-the-art techniques and experimental devices have been developed. In addition, the advent of high-power computing has permitted detailed and complex theoretical
calculations with ever-increasing predictive power. A small, but meaningful contribution to
this progression of knowledge is presented in this work.

1.1

The Nuclear Landscape

The labeling of nuclei follows the notation A Z, where the number of protons Z determines
the element- denoted by a particular chemical symbol- and A is the total number of nucleons.
Thus, the number of neutrons follows immediately as N = A−Z. A conventional arrangement
of nuclei is illustrated in Fig. 1.1. Each isotope is plotted according to its unique combination
of neutrons and protons. The stable isotopes comprise the “line of stability”, beyond which
exist unstable, or radioactive, nuclei.
Studying the chart of nuclides depicted in Fig. 1.1 illuminates several key observations of
nuclear structure. First, as noted above, the overwhelming majority of isotopes are unstable.
In addition, the line of stability coincides with N = Z for the lightest nuclei, but tends
toward more neutron-rich (N > Z) isotopes for heavier systems. There, an excess of neutrons
is required to bind the nucleus against the increasing Coulomb repulsive force.

2

120

Proton number

100

80

60

r-process
rp-process

40

Observed nuclei

20

Stable nuclei

0
0

20

40

60

80

100

120

140

160

180

Neutron number
Figure 1.1: The chart of the nuclides with neutron number N along the abscissa and proton
number Z along the ordinate taken from [3]. Stable isotopes are indicated with black boxes,
observed radioactive nuclei in blue, and predicted unstable nuclei not yet observed in red.
The dashed lines indicate magic numbers (see text). Several astrophysical processes are
indicated as well. The rp-process shall be discussed in Subsect. 1.4.2. For interpretation of
the references to color in this and all other figures, the reader is referred to the electronic
version of this dissertation.
Most of the unstable isotopes to the left and right of the line of stability are bound systems
of protons and neutrons, meaning the nuclear binding energy (Eq. 1.20) is a positive quantity.
However, if the nucleus can become more bound by the transformation of a neutron into a
proton or vice versa, then β − or β + radioactivity may proceed via the weak force. These two
decay processes correspond to the emission of an electron or a positron, respectively. Other
decay processes such as the emission of a 4 He nucleus (α-decay) or the fission of a nucleus
into two lighter nuclei are also mechanisms by which binding energy is liberated and more
3

stable configurations are achieved. The production of radioactive isotopes beyond the proton
or neutron driplines at the very limits of nuclear existence result in nuclei where the final
nucleon is no longer energetically bound to the nucleus. This results in the direct emission of
that unbound nucleon. Together, the various modes of radioactive decay transmute unstable
nuclei back to the line of stability. Hence, in a sense, nuclear physicists practice modern-day
alchemy.
Finally, the vertical and horizontal dashed lines overlayed atop the chart of nuclides in
Fig. 1.1 indicate the canonical magic numbers 2, 8, 20, 28, 50 82, and 126, which occur when
major nuclear shells (see Subsect. 1.2.1) are filled. Nuclei with magic numbers of protons
and/or neutrons generally exhibit especially stable properties, such as increased binding and,
relative to neighboring even-even nuclei, an enhanced 2+ excitation energy and low transition
1
probability B(E2; 2+ → 0+ ). However, the study of exotic nuclei has demonstrated both
gs
1
the disappearance of magic numbers and shell gaps [4,5] and the emergence of new ones [6,7]
far from stability. The migration of magicity is but one clear indicator that nuclear structure
1
is not a constant; it can evolve dramatically as a function of isospin Tz = 2 (N − Z).

1.2

Nuclear Structure within the Shell Model

Describing how the various facets of nuclear structure, such as stability, energy levels, and
transition rates, fundamentally arise from the interactions of individual nucleons is a major
goal of nuclear physics. Since the nucleus is an inherently many-body quantum mechanical
system, a complete theoretical description for an A nucleon system is akin to the solution of
the A-body time-independent Schrödinger equation,

HΨ = EΨ.

4

(1.1)

The properties of the nucleus are governed by the interplay between the strong, weak, and
electromagnetic forces. The exact form of the Hamiltonian H is not known; instead, accurate
nucleon-nucleon and three-body interactions have been developed from both phenomenological and more fundamental approaches. The Argonne v18 [8] and CD-Bonn [9] potentials are
examples of the former approach and are derived from accurate reproduction of nucleonnucleon scattering data. Interactions derived from chiral effective field theory calculations
belong to the latter approach [10]. Nuclear theoretical models built upon these realistic twoand three-body interactions successfully calculate various nuclear properties. However, calculations quickly become computationally intractable and thus approximations and truncations
must be made.
In the standard shell model [11], the solution of Eq. 1.1 is simplified by splitting the
nucleus into an inert core and valence nucleons. The core, typically a nearby doubly-magic
nucleus, has fully-occupied single-particle levels and is treated as non-interacting. The implicit assumption is that all nuclear structure properties can be derived entirely with the
valence proton and neutron particles or holes. The allowable excitations of the valence nucleons must also be truncated to abet the calculations. Together, these two simplifications
preclude interactions between the core and valence particles and restrict the allowable excitations. Therefore an effective interaction must be derived from the realistic two- or three-body
interactions above to account for the truncated model space.
The no-core shell model [12] (NCSM) approach expands upon this, breaking the core and
including all nucleons within the calculation. Therefore, in principle, the full A-body problem
can be solved with this ab initio approach. However, truncations on allowable excitations
still must be made in the NCSM and therefore effective interactions are again required. To
obtain the full model space solution, extrapolation methods are used. Extending this method
to heavier nuclei is challenging. Accurate experimental data of exotic isotopes can provide
important benchmark tests. The lifetime measurements of electromagnetic transitions in 18 C
5

presented in Ch. 3 fulfill this role; a comparison between experiment and theory is provided
in Ch. 5. In preparation, a discussion of the ab initio NCSM is provided below, but first a
brief description of shell structure nomenclature is provided. A familiarization with terms
will be useful for a shell model discussion in Subsect. 5.1.1.

1.2.1

Nuclear Shell Structure

Nucleons have an intrinsic spin s = 1 and an orbital angular momentum
2

= 0, 1, 2, 3, . . .

that is labeled spectroscopically as s, p, d, f, . . . respectively. The spin and orbital angular
momentum couple to the total angular momentum j = ± s. Hence a complete description
of the single-particle shell model orbital for a nucleon is given by σn j , where σ denotes
the type of nucleon (π for protons and ν for neutrons) and n = 0, 1, 2, 3, . . . is the principle
quantum number. Nucleons obey Fermi-Dirac statistics; no two nucleons can occupy the
same orbital. Rather each orbital can accommodate 2j + 1 nucleons in different magnetic
substates. Pairs of nucleons in a given j state couple to spin 0 and thus even-even nuclei have
a ground state spin/parity 0+ . In contrast, odd-A nuclei have ground state spins typically
determined by the unpaired nucleon.

1.2.2

Ab Initio No-Core Shell Model

The goal of NCSM calculations is to solve the full A-body quantum mechanical problem
using realistic two- and three-body interactions such as those mentioned above [13]. A brief
primer to the method is given in Ref. [14]. The NCSM model space is typically spanned by
Slater determinants constructed with single-particle harmonic-oscillator (HO) basis states.
The spacing between energy levels is denoted by the HO frequency Ω. The model space
is restricted to a maximum number of harmonic-oscillator excitations Nmax such that all
Slater determinants with an energy less than Nmax Ω are included within the calculation.

6

Effective interactions are used to account for effects of configurations outside of this finite
model space [12].
The use of effective interactions reproduces a subset of eigenvalues from the original
two- or three-body interactions. Therefore as the model space approaches the full space
(Nmax → ∞), the approximate solution approaches the exact solution of the original full
interaction [13]. This convergence implies that calculations performed over several Nmax Ω
cutoffs will tend towards the full model space result and more importantly, calculations of
Nmax sequences for various Ω all must converge to the same result. The existence of a common solution for various pathways provides an asymptotic constraint for the extrapolation
to the full space result.
The ab initio NCSM therefore provides an approach by which exact solutions of an A-body
quantum mechanical problem using a realistic Hamiltonian can be obtained. The method
has been successfully applied to p-shell nuclei [14], however computational limits hinder
calculations for nuclei beyond A = 16 with sufficient Nmax to attain accurate extrapolations.
For instance, the current limit for 18 C calculations is Nmax = 6 [15] and that of 16 O is
Nmax = 8 [16]. In these cases, reliable convergence may not be computationally tractable.
To ameliorate this, a recent extension of NCSM calculations uses the importance truncation (IT) scheme [16,17]. With this approach, a measure of the importance of individual HO
basis states via perturbation theory precedes the calculation [17]. Many states contribute to
a calculation with vanishingly small amplitudes; the elimination of these unimportant basis
states truncates the model space while preserving the physically relevant basis states. This
reduces the computational intensity and thus permits calculations to higher Nmax , improving
the accuracy of the extrapolation to the full model space result [16]. For example, IT-NCSM
calculations have permitted calculations of the ground state energy of 16 O to Nmax = 22 [16]
and, as will be shown in Ch. 5, excited state properties of 18 C to Nmax = 8.

7

1.3

Nucleon Knockout from Radioactive Beams

The emerging importance of powerful theoretical approaches that derive nuclear properties
from the fundamental interactions of all nucleons, such as the ab initio NCSM, benefits from
comparison to precise experimental measurements at the extremes of isospin. Such data is
readily obtained with nucleon knockout reactions of radioactive beams. This technique has
provided the greatest reach from the line of stability; two examples of such measurements
shall be introduced in Ch. 3 and 4. Neither analyses are reaction-model dependent, therefore
a brief introduction to nucleon knockout reactions will suffice.
The first step involves the production of an energetic radioactive ion beam. Two main
techniques exist. The in-flight production and selection of radioactive beams is employed at
NSCL and shall be described in detail in Sect. 2.2. The isotope separation on-line method
is a complementary technique employed, for example, at HRIBF at ORNL and TRIUMF
in Canada. There, high-energy stable beams (such as protons) impinge upon a thick production target. The subsequent spallation or fission of the target nuclei produce a variety
of radioactive isotopes which upon extraction, ionization, and purification are accelerated to
the experimental station.
Delivery of radioactive ion beams to experimental targets and the subsequent singlenucleon knockout reaction is a powerful spectroscopy tool for nuclear structure studies. For
instance, the extraction of partial cross sections via gamma-ray tagging of final states can be
used to calculate spectroscopic factors which can help infer information about the occupation
of shell model single-particle orbitals [18]. For the present work, single-nucleon knockout
reactions provide a simple production mechanism by which exotic nuclei can be produced
and which can be well quantified within the framework of the analysis.
Single-nucleon knockout reactions are peripheral processes where a nucleon residing near
the nuclear surface is removed from the incident secondary beam (of mass A + 1) by a light

8

target, such as beryllium or carbon. The resulting reaction residue (of mass A and often in an
excited state) emerges from the target with nearly the same velocity as that of the incoming
beam [19]. Within the target, two distinct mechanisms are responsible for the removal of the
nucleon [18]. In stripping reactions the energy of the removed nucleon is absorbed by the
target and produces excited target states while the competing diffractive breakup involves
nucleon removal mediated by scattering off a target nucleus. If, as in the studies of Ch. 3 and
4, only the heavy-ion reaction residue is observed, the total single-particle knockout cross
section can be approximated by the sum of these two components. The relative contribution
of each mechanism is of little consequence for the present works.1
The energetic beams produced at NSCL are ideal for studies involving nucleon removal
from exotic beams. In particular, they permit the use of thick targets which enhance experimental luminosities despite low secondary beam rates and knockout cross sections. In
addition, the energetic residues are strongly kinematically forward-focused in the laboratory frame, preventing significant acceptance loss issues within tracking detectors [23]. This
ensures that event-by-event particle tagging can proceed to produce high signal-to-noise
spectra. These features were imperative for the lifetime measurements reported in this work.

1.4

Köln/NSCL Plunger Lifetime Measurements

The direct one-proton knockout reaction of an intermediate energy radioactive 19 N secondary beam at NSCL populated excited states in 18 C. After the reaction, the de-excitation
gamma radiation was observed with the Segmented Germanium Array. Via the Recoil Distance Method (RDM) with the Köln/NSCL plunger, the lifetime of the 2+ → 0+ electric
gs
1
quadrupole transition was measured. In addition, a lifetime upper limit was extracted for a
1 Lifetime measurements of electromagnetic transitions similar to that reported in Ch. 3

do not explicitly require nucleon knockout reactions for excited-state production; Coulomb
excitation on heavier targets has been effectively used as well [20–22].
9

higher-lying excited state feeding the 2+ state. The importance of lifetime studies such as
1
these to the basic understanding of the nucleus is introduced in Subsect. 1.4.1. The experimental technique and devices are introduced in Ch. 2, while a detailed discussion of the data
and analysis is found in Ch. 3. The results and comparisons to theoretical calculations are
presented in Ch. 5.
A modified Köln/NSCL plunger was utilized for the two-proton decay lifetime measurement of 19 Mg. This exotic proton emitter was produced by the one-neutron knockout reaction of a 20 Mg secondary beam on a carbon plunger target. A thin silicon detector mounted
downstream of the target recorded the energy loss signature of reaction and heavy-ion decay
residues. From these energy loss spectra, the 19 Mg ground state two-proton decay lifetime
was measured. The measurement served as a proof of concept for “particle plunger” RDM
lifetime measurements which may prove especially important for nuclear astrophysics calculations in the proton-rich region of the nuclear landscape. This importance is expanded upon
in Subsect. 1.4.2. The specific details of the technique are provided in Ch. 2 and the data
and analysis are presented in Ch. 4. The results are discussed and compared to the previous
measurement, and subjects for future investigation are given in Ch. 5.

1.4.1

Lifetime Measurements of Electromagnetic Transitions

RDM lifetime measurements provide a non-intrusive and model-independent probe by which
the detection of electromagnetic radiation elucidates information on the nuclear charge distribution. Such an electromagnetic probe is desirable as it only weakly perturbs the systemnucleon motion is governed by the strong force which dominates the electromagnetic force
at such short distances. Furthermore, the technique is versatile; any excited state population
mechanism can be used with RDM measurements to probe nuclear charge and current distributions without destroying the state of the system. This method is entirely complimentary
to hadron inelastic scattering probes which deduce information on nuclear matter distribu10

tions. Thus the study of electromagnetic transitions is key to understanding the nucleus and
can provide stringent tests of various and sometimes competing theoretical nuclear structure
calculations.
The nuclear electric and magnetic fields are produced by the distribution of charges and
currents within the nucleus. The multipole expansion of these fields separates the contributions of individual multipole moments based upon their characteristic radial dependence
(rx ). For instance, the electric monopole moment is just the 1/r2 field due to the net charge.
The 1/r3 and 1/r4 electric fields are the dipole and quadrupole moments, respectively. The
magnetic field multipoles behave in a similar fashion except that the monopole moment vanishes as no individual magnetic charges have been found in nature. The first non-vanishing
term is the 1/r3 magnetic dipole moment and arises from electric current moving in a circular
loop.
Therefore measuring the various electromagnetic moments of a nucleus tells us about the
distribution of the nuclear charge and current. In particular, the detection of electric quadrupole (E2) radiation in even-even nuclei probes the collective structure of nuclear excitations.
In a collective picture, the coherent behavior of all nucleons collectively influences nuclear
structure. The degree of this collective behavior is quantified by the reduced transition probability matrix element B(E2). This important experimental measurement is directly obtained
from lifetime measurements of E2 transitions.
The following discussion reflects, in a simplified manner, the spirit of the nuclear structure
texts of Ref. [24, 25]. The extent to which the charge distribution of a nuclear state |A
deviates from spherical symmetry is given by the static quadrupole moment

eQ = A| ME2 |A ,

(1.2)

where ME2 is the electric quadrupole operator. eQ thus represents the diagonal elements

11

of the matrix linking all possible initial and final quantum mechanical states via the more
general electric quadrupole tensor operator, ME2,µ . The non-diagonal elements represent E2
transitions between different states; studying these transitions sheds light on the quadrupole
deformation of the nuclear charge distribution.

Mathematical Formulation
Electromagnetic nuclear decays are spontaneous photon emissions via the interaction of a
nucleus with an external electromagnetic field and are mediated by the four vector (φ, A).
The scaler and vector potentials, φ and A, couple to the nuclear charge density ρ and current
density j, respectively. The expansion of the scaler field for a collection of A particles into a
summation of electric multipoles yields
A

φ(r) =

ej

j |r − r j |

=
λµ

4π
1
Y ∗ (Ω)MEλ,µ
λ+1 λµ
2λ + 1 r

(1.3)

where the electric multipole moment MEλ,µ is a spherical tensor of multipolarity λ and is
given by
A
λ
ej rj Yλµ (Ωj ).

MEλ,µ =

(1.4)

j

In a similar manner, the magnetic multipole moment of multipolarity λ is given by
A
s
gj sj +

MM λ,µ =
j

2 l
g l ·
λ+1 j j

λ
rj Yλµ (Ωj ) .

(1.5)

In Eq. 1.3, 1.4, and 1.5, Yλµ (Ω) are spherical harmonics of multipolarity λ with projection
s,l

µ = −λ, −λ + 1, . . . , λ − 1, λ and Ωj , ej , gj , and sj and lj represent the coordinates on the
unit sphere (θ, φ), the electric charge, the spin and orbital gyromagnetic ratios, and angular
momentum operators of the j th nucleon, respectively.

12

These electromagnetic multipoles act as quantum mechanical tensor operators on states
in a nucleus. Considering the simple case of a system containing a nucleus in an external
electromagnetic field, the initial state |i can be represented by the nucleus in its excited
state and the photon vacuum electromagnetic field. The final state |f is thus the nucleus in
its ground state with one photon of energy Eγ in the electromagnetic field. The transition
between these two states is described by one or more of the above multipoles with the
appropriate selection rules (discussed in Appendix A). The interaction between the nucleus
and electromagnetic field is weak and thus a perturbative treatment of the decay yields the
transition probability per unit time
σλµ

Tf i

Eγ 2λ+1
λ+1
2
f | Mσλ,µ |i
c
λ[(2λ + 1)!!]2
0
2
≡ κ f | Mσλ,µ |i .
=

2

(1.6)

Here, σ, λ, and µ denote the electric or magnetic character of the multipole, the multipolarity, and the projection, respectively. The initial and final states are characterized by their
respective quantum numbers such that for a state a, |a = |ξa , Ja , Ma where Ja and Ma
are the total angular momentum and angular momentum projection of the state and ξa are
the remaining quantum numbers necessary to describe the state.
For the present work, the initial and final magnetic substates are not measured. Thus
averaging over the initial substates Mi and summing over the final substates Mf and all
multipolarity projections µ leads to the new expression
σλ
Tf i =

1
2Ji + 1

σλµ

Tf i
Mi Mf µ

≡ κB(σλ; ξi Ji → ξf Jf ),

13

(1.7)

where B(σλ; ξi Ji → ξf Jf ) is the reduced transition probability given by
1
| ξ J || Mσλ ||ξi Ji |2 .
B(σλ; ξi Ji → ξf Jf ) ≡
2Ji + 1 f f

(1.8)

The reduced transition probability arises from the application of the Wigner-Eckart Theorem. The derivation is beyond the scope of this work; it is sufficient to note that the
theorem reduces the original matrix element to a form independent of the projection quantum numbers while retaining the physical information to be extracted from the observable.
The dependence of the matrix element on the magnetic projection numbers are concentrated
into Clebsch-Gordan coefficients.
For the 2+ → 0+ transition in 18 C, the E2 gamma ray is the experimental observable
gs
1
from which the lifetime is extracted. From this lifetime, the value of the reduced quadrupole
transition strength B(E2; 2+ → 0+ ) can be calculated by the inverse relationship between
gs
1
the lifetime and transition rate of Eq. 1.7
1
τ (2+ → 0+ ) =
gs
1
T2+ →0+
gs
1
=

1
75 0

−1
Eγ 5
+ → 0+ )
B(E2; 21
.
gs
c

(1.9)

Quantifying Nuclear Collectivity
Equation 1.9 provides the link between the experimentally measured lifetime and the reduced
quadrupole transition strength. This B(E2) value is a model-independent measure of the
degree of quadrupole collectivity in even-even nuclei. In particular, it can be associated with
nuclear shape deformation.
Following the liquid drop model, the quadrupole deformation of the nuclear radius about

14

the average value R0 is given by

R(Ω) = R0 1 +

α2,µ Y2,µ (Ω) ,

(1.10)

µ

where α2,µ are shape variables describing the deformation of the nucleus. If the laboratoryfixed coordinate system is arbitrarily rotated by some choice of angles Ω = (θ , φ ), the
intrinsic nuclear shape does not change. All that changes is how the shape is oriented with
respect to the new set of coordinates. In particular, the original spherical function Y2,µ (Ω)
will form new combinations of Y2,µ (Ω ) with new coefficients α2,µ . Since neither the shape
nor the experimental observable changes by this coordinate rotation, it is convenient to work
in a body-fixed coordinate system where the symmetry axis of the nucleus forms one of the
major coordinate axes. In this body-fixed frame, the quadrupole deformation of the nuclear
radius is expressed in a similar manner to Eq. 1.10,

R(Ω ) = R0 1 +

a2,µ Y2,µ (Ω ) ,

(1.11)

µ

where the new deformation shape variables a2,µ are linear combinations of the original variables α2,µ with coefficients reflecting the rotational angles from the laboratory-fixed to bodyfixed frames. Inserting the expressions for the λ = 2 spherical harmonics, the deformation
then becomes
R(Ω ) − R0
R0

5
4π

= a2,0

+
+

3
1
cos2 θ −
2
2

3
sin θ cos θ a2,−1 e−iφ − a2,1 eiφ
2
3 2
sin θ a2,2 e2iφ + a2,−2 e−2iφ .
8

15

(1.12)

Since the nuclear shape is invariant under coordinate rotations, it is reasonable to expect
the 5 deformation shape variables (a2,µ for µ = −2, −1, 0, 1, 2) can be reduced to 2 invariant
shape variables. Indeed, in the body-fixed frame the reflection about the equatorial plane
changes the polar angle from θ to π − θ while leaving φ unchanged. Thus cos θ switches
sign and to preserve the invariance of the deformation about this symmetry reflection,

a2,1 = a2,−1 = 0.

(1.13)

Furthermore, reflection about the vertical plane changes the azimuthal angle from φ to −φ
while leaving θ unchanged. Such a reflection will switch the a2,2 and a2,−2 terms in Eq. 1.12
and thus keeping the shape invariant under this symmetry reflection requires

a2,2 = a2,−2 .

(1.14)

Hence two invariant shape variables remain: a2,0 and a2,2 . These are commonly expressed
by the deformation parameter β and the non-axiality parameter γ (which quantifies the
departure from cylindrical symmetry) as
1
a2,2 = √ β sin γ.
2

a2,0 = β cos γ,

(1.15)

The deformation parameter β quantifies the departure from nuclear sphericity. For the
limiting case of a rotating ellipsoid (with axial symmetry), it can be directly related to the
measured B(E2) value with the additional assumptions that the deformation is both small
in magnitude and static (without vibration).
1
B(E2; 2+ → 0+ ) =
gs
1
5

16

2
3
2β .
ZeR0
4π

(1.16)

Final Remarks
The preceding discussion illuminates the important role of lifetime measurements in quantifying nuclear collectivity. The reduced quadrupole transition strength is inversely proportional
to the measured lifetime and from the B(E2) value, the degree of nuclear deformation can
be quantified. A customary way to indicate this is the ratio between the experimental B(E2)
value and the single-particle estimate,

B(E2)/BW (E2).

(1.17)

Here, the single-particle transition strength estimate BW (E2) is taken from Eq. A.3. A large
value for this ratio (∼10) indicates collective behavior while a value much closer to unity
suggests behavior associated with closed-shell nuclei, such as doubly-magic 16 O, 40 Ca, and
48 Ca which all have ratios of roughly 3 or smaller.

On a final note, an empirical approach within the liquid drop model to quantify global
B(E2) systematic trends for even-even nuclei [26] has found
E(2+ ) · B(E2; 2+ → 0+ ) ∝ Z 2 A−2/3 .
gs
1
1

(1.18)

Hence the general expectation is that the B(E2) will be inversely proportional to the 2+
1
excitation energy. For neutron-rich carbon isotopes, the 2+ excitation energy of the N = 8
1
nucleus 14 C drops dramatically from approximately 7.0 MeV to 1.8 and 1.6 MeV for 16 C
and 18 C, respectively. Accordingly, the B(E2) for 16,18 C is expected to be much larger than
the 14 C value of 3.74 e2 fm4 [27]. However, as reported in Ref. [28, 29] and later in Ch. 3, the
B(E2) values are observed to remain relatively constant with that of 14 C. This deviation
from systematic behavior may indicate a shift in quadrupole excitation mechanisms and
illustrates the particular importance of lifetime measurements for probing nuclear behavior

17

Figure 1.2: The pairing contribution to the nuclear binding energy lowers the ground state
energy of the two-proton emitter A = (Z, N ) with respect to that of the (Z − 1, N ) system.
Thus the nucleus decays via simultaneous 2p emission (solid arrow) directly to the ground
state of the (Z − 2, N ) daughter. Sequential 1p emissions (dashed arrows) via the unbound
intermediary (Z − 1, N ) daughter is possible from resonances within the continuum of the
parent (Z, N )∗ .
rather than relying solely upon energy systematics from in-beam gamma-ray spectroscopy.

1.4.2

Lifetime Measurements of Proton Radioactivity

The simultaneous emission of two protons is the most recently discovered nuclear decay
mechanism. This exotic decay mode was first discussed in the seminal work of Ref. [30]. Twoproton radioactivity only exists for even-Z nuclei where nuclear pairing of the valence protons
energetically prevents single-proton emission. Though the parent is bound to 1p decay, it
is unbound to 2p decay because of the symmetry energy and Coulomb interaction [31]. A
schematic of this scenario is presented in Fig. 1.2. True two-proton emission must also proceed
from narrow nuclear ground states; broad resonant states of neighboring nuclei beyond the
dripline can overlap, resulting in an ambiguity between simultaneous 2p decay and sequential
1p emissions.
Likely candidates for two-proton radioactivity were theoretically identified between 26 ≤
18

Z ≤ 36, where the Coulomb barrier would yield narrow ground states decaying with observable lifetimes [32, 33]. Two-proton radioactivity was later measured and confirmed for
45 Fe [34–36] and 54 Zn [37]. These measurements yielded lifetimes of several milliseconds,

well within the reach of implantation-type experiments [37] or the optical time projection
chamber method [36].
In contrast, the lower Coulomb and centrifugal barriers of lighter elements result in much
faster proton radioactivity. Indeed, a lifetime on the order of picoseconds was reported in
Ref. [38] for the two-proton decay of the 19 Mg ground state. The result was found to be
in agreement with quantum mechanical calculations treating the 2p-decay precursor as a
three-body composite of two independent protons and a 17 Ne core [39]. Such experimental
verification is important because simpler lifetime calculations, based upon a semi-classical
treatment of a tunneling diproton (2 He nucleus), predicted lifetimes much shorter than those
experimentally measured as discussed in Ref. [35, 40].
The fast proton radioactivity of 19 Mg provides an ideal testing ground for a new lifetime
measurement technique for proton emitters beyond the dripline. The technique is based
upon the well-established Recoil Distance Method; details are given within Ch. 2 while
the data and analysis are presented in Ch. 4. This proof of concept experiment yielded an
independent lifetime measurement of ground state proton emission in 19 Mg. The method
complements the other established techniques for lifetime studies at the proton dripline
[36–38,41] and affords sensitivity for precision lifetime measurements of both 1p and 2p decays
in the picoseconds range. These studies, interesting in their own right, are also of interest
for nuclear astrophysics. Proton decay is the inverse reaction of radiative proton capture;
such processes play a vital role in galactic proton-rich nucleosynthesis. Furthermore, as the
following discussion illustrates, the proton-decay lifetimes can provide important constraints
on mass models and proton-capture rates vital to the stellar isotopic synthesis calculations.

19

Importance for Nuclear Astrophysics
If the heaviest isotopes to exist in nature belonged to the iron group (A ≈ 56), our knowledge
of stellar nucleosynthesis would be nearly complete. Hydrogen core burning via the pp-chains
and CNO cycles produces an abundance of helium. After the hydrogen is exhausted, helium
burning commences via the triple alpha process. Once the helium is exhausted, the burning
proceeds more and more rapidly through carbon, neon, oxygen, and finally silicon. Here
nucleosynthesis ends as the resulting iron group nuclei are the most tightly bound in nature
and further fusion reactions are generally endothermic. References [42] and [43] provide a
detailed overview of these processes.
Yet the existence of the stable isotopes from iron to uranium has spurred the development
of a small arsenal of nucleosynthesis processes complimentary to stellar burning. Among these
are the slow and rapid neutron capture processes and the proton capture process. These three,
first postulated in the late 1950s [44], are responsible for the stellar production of almost all
stable isotopes beyond iron.
Several well-documented instances where nucleosynthesis calculations underproduce specific nuclei with respect to solar abundances may indicate the need for even more processes.
Such a suggestion was prompted by the underproduction of 92,94 Mo and 96,98 Ru [45] and
several isotopes of Sr, Y, and Zr [46]. An attractive mechanism to ameliorate this underproduction was the neutrino p-process [47]. Occurring in the neutrino-/antineutrino-rich environment of Type II core-collapse supernovae, the νp-process initially drives supernovae ejecta
proton rich by neutrino capture νe + n → p + e− . This facilitates the production of protonrich isotopes via (p, γ) reactions in a manner identical to the rp-process [48, 49]. In later
stages, the reaction flow stalls at waiting-point nuclei (such as 64 Ge, 68 Se, and 72 Kr) due to
a combination of long β + decay half-lives and small proton-capture rates. To produce heavier
nuclei, these waiting points must be bridged. This is accomplished with (n, p) reactions in the
νp-process, where antineutrino capture on the proton-rich ejecta, νe + p → n + e+ , increases
¯
20

Figure 1.3: Reaction flow of the rp-process for a portion of the nuclear chart during a Type
I x-ray burst. Reaction flows indicating proton captures and β + decays (gray arrows) were
taken from Fig. 1 of Ref. [51]. The red arrows indicate the possible two-proton capture of
68 Se which may bridge this long-lived waiting point.
the abundance of free neutrons. The waiting points can be bypassed in the rp-process via
(2p, γ) reactions which effectively jump over the unbound (Z + 1) isotone [49, 50]. Figure
1.3 illustrates a small portion of the rp-process reaction flow and the possible importance of
two-proton capture in bridging waiting-point nuclei.
The role of the νp-process in galactic chemical evolution and its uniqueness from the
rp-process in explosive environments ultimately depends upon the models’ success of reproducing observed abundances [52,53]. Both calculations involve proton-capture rates that are
sensitive to temperature, density, composition, and (most critically) masses [49, 53]. From
the masses, one- and two-proton separation energies (Sp and S2p ) can be calculated; their
importance in nucleosynthesis calculations is illustrated in the following brief derivation.

21

Mathematical Formulation
For simplicity, consider the one-proton capture reaction between two nuclei,

1 + p → 2 + γ.

(1.19)

The binding energy can be calculated directly if nuclear masses are known via
B(Z, N ) = Zmp + N mn − mnuc c2 .

(1.20)

After setting the speed of light to unity, the binding energy difference between the two nuclei
is given by

B(Z + 1, N ) − B(Z, N ) =

(Z + 1)mp + N mn − m2 − Zmp + N mn − m1

= mp + m1 − m2 .

(1.21)

This is simply the Q-value of the reaction and corresponds to the energy of the photon
in Eq. 1.19. If instead a photon of that same energy impinges upon nucleus 2, the reverse
reaction of Eq. 1.19 would occur. Thus the Q-value is equal in magnitude, but opposite in
sign, to the one-proton separation energy (Q = −Sp ).
The proton-capture cross section σ (in units of area) is given by

σ=

N /t
.
[Np /(t · A)]N1

(1.22)

The numerator is the number of reactions per time and the denominator contains the proton
current density jp = Np /(t · A) and number of target nuclei N1 . Rearrangement yields an

22

expression for the number of capture reactions per unit volume and time,
Np
N Np
N
= σN1
= σ(v)v 1
.
V ·t
V ·A·t
V V

(1.23)

Here, the velocity dependence of the cross section has been introduced. In the stellar environment, the protons and target nuclei have some distribution of relative velocities. This
distribution can be described by the probability function P (v) such that P (v)dv is the probability that the relative velocity is between v and v + dv. After absorbing the volume terms
into Ni , which should now be understood to be number densities, the reaction rate can be
expressed as
r ≡ N1 Np

∞
0

σ(v)vP (v)dv ≡ N1 Np σv .

(1.24)

The reaction rate per particle pair, σv , contains the nuclear physics information and in
most cases can be calculated assuming a Maxwell-Boltzmann velocity distribution.
With the help of Eq. 1.24, the destruction rate of the target nucleus can be written as
dN1
= −r = −N1 Np σv
dt
≡ −λp N1 .

(1.25)

The establishment of local equilibrium between the proton-capture (with decay constant λp )
and photodisintegration (with decay constant λγ ) implies
0 = r1+p→2+γ − r2+γ→1+p
= N1 Np σv − λγ N2 .

23

(1.26)

Therefore the Saha equation can be used to describe the equilibrium abundance ratios,
S1p
N2
σv
=
= αexp −
N1 Np
λγ
kT

(1.27)

.

The constant α contains terms related to the particles’ normalized partition functions,
masses, and spins. What is important here is the exponential dependence on the one-proton
separation energy. Further details are provided in Ref. [54].
With the machinery and definitions in hand, consider the sequential one-proton captures
from a waiting-point nucleus. One such example was illustrated with red arrows in Fig. 1.3
for 68 Se. The two relevant reactions are

1+p→2+γ
2 + p → 3 + γ.

(1.28)

If the first proton capture is in statistical equilibrium with photodisintegration, Eq. 1.26 and
1.27 can be used to express the equilibrium abundance of nucleus 2 as

N2 = N1 Np

S1p
σv
= αN1 Np exp −
λγ
kT

.

(1.29)

The rate of two 1p captures can be expressed in terms of the probabilities per unit time of
the one-proton capture from nucleus 2 or the sequential captures from nucleus 1 as

r1→3 = N2 λp = N1 λ2p ,

(1.30)

where Ni are the equilibrium abundances of nucleus i = 1, 2. Thus the probability of cap-

24

turing two protons per unit time λ2p can be written with the help of Eq. 1.29 as

λ2p =

S1p
N2
λp = αNp exp −
N1
kT

λp .

(1.31)

Using Eq. 1.25 to substitute the reaction rate per particle pair for λp , the final expression
for the destruction rate λ of the waiting-point nucleus is
2
λ = λβ + αNp exp −

S1p
kT

σv (Z+1,N ) .

(1.32)

The beta-decay rate has been included to illustrate the two competing mechanisms by which
the effective waiting-point lifetime is set.
From the exponential dependence on the proton separation energy evident in Eq. 1.32 it
is clear that nuclear masses play a vital role in characterizing nucleosynthesis pathways by
altering the effective waiting-point lifetimes. Correspondingly, much experimental effort has
been devoted to measuring relevant masses with high precision (see, for example, the works
of Ref. [55, 56]). Reliable nucleosynthesis network calculations require input parameters well
grounded upon experimentally measured quantities in order to accurately model observed
stellar abundances. In this manner, competing models can be vetted against one another.
Yet despite the utmost importance of accurate mass measurements, nuclei near or beyond the proton dripline suffer from both low production rates and short lifetimes [55]. For
these nuclei, proton separation energies are not directly tractable from mass measurements
and therefore calculations or extrapolations must be used. One such method calculates the
difference in binding energy (rather, the Coulomb displacement energy) between mirror-pair
nuclei. If the mass of the more neutron-rich nucleus is known, the mass of its proton-rich
mirror pair can be extracted from the binding energy difference calculation with an error of
roughly 100 keV [32, 33]. Proton separation energies along the dripline can also be obtained

25

from lifetime measurements of the proton emitters, as discussed below.

Application of Proton Decay Lifetime Measurements
Proton decay lifetime measurements are a valuable resource by which unknown masses (and
thus separation energies) can be experimentally constrained. Thus lifetime data is invaluable
to galactic nucleosynthesis calculations. The simplest, and correspondingly the most rudimentary, link between the measured proton decay lifetime and separation energy is given
within the framework of the semiclassical Wentzel, Kramers, and Brillouin (WKB) approximation. An example of constraining the proton separation energy from a one-proton decay
lifetime measurement is given in Ref. [57]. There, the non-observation of 69 Br set a halflife upper limit of 24 ns and constrained the nucleus to be unbound by at least 500 keV,
consistent with a recently measured one-proton separation energy of Sp = −785 keV [58].
A detailed derivation of the WKB approximation is beyond the scope of this work. Furthermore, the calculation for two-proton decay is based upon the emission of a diproton which
is certainly an oversimplification. As mentioned at the beginning of this section, a more appropriate quantum mechanical description is necessary for accurate predictions. Nonetheless,
a brief WKB discussion shall suffice to illuminate the point that decay lifetimes are intimately
linked to decay (separation) energies. Heuristically, this correlation is obvious- the proton,
despite having a positive Sp is trapped within the Coulomb and centrifugal barrier of the
nucleus and therefore takes a finite amount of time to tunnel outside.
The two-proton decay lifetime can be written approximately as
τ −1 = Γ2p ∝ exp −2

rout
rin

2µ m∗ (r)
Q2p − V (r)
2 m

dr.

(1.33)

Here, rin and rout are the classical turning points, µ is the reduced mass, m∗ (r)/r is the
proton effective mass inside the nucleus, and V (r) is some central potential. More details

26

are provided in Ref. [59]. A similar discussion is found in [33]. There, it is noted that the
largest uncertainty in estimating the lifetime is the two-proton Q-value. For instance, for
Q ∼ 500 keV, an uncertainty of ±100 keV results in a range of calculated lifetimes of nearly
six orders of magnitude. It should be noted that by Eq. 1.33, an accurate experimentally
measured lifetime can pinpoint the separation value. However, at such accuracy the uncertainties introduced by the simplified WKB diproton calculation and variations in both the
potential V (r) used and pre-factors not explicitly shown in the equation will dominate.
Though the WKB approximation provides a clear link between proton-decay lifetimes
and separation energies, it is too simplistic of a model. Besides only considering the emission
of a diproton, the calculations are only done at E = Q2p and therefore exclude the relative
motion of the protons after tunneling through the barrier. These shortcomings are noted
in Ref. [40, 60], where a more complex three body quantum mechanical calculation of the
two-proton decay is discussed. This more realistic approach yields narrower proton decay
widths than the diproton two-body WKB models, which leads to longer lifetimes in better
agreement with various experimental observations. Such calculations significantly narrow the
possible lifetimes within a given Q-value window (Ref. [39] provides an example).
Lifetime measurements are also useful for probing the validity of neutron-deficient nuclear
models. From a spectroscopic S-factor, defined as the ratio between calculated and measured
lifetimes, model deficiencies can be spotted. For example, large deviations of S from unity can
indicate the presence of nuclear structure effects not explicitly accounted for in the model,
such as deformation [61]. Indeed, deformation was suggested from the two-proton emission
study of 94m Ag, evidenced by a strong correlation between the emitted protons [62].
The study of proton emitters is therefore integral for the development of accurate elemental synthesis processes and reliable nuclear structure models along the proton dripline.
The body of work presented in Ch. 4 and 5 introduces a new measurement technique by
which the lifetime of such exotic decays can be studied.
27

Chapter 2
Experimental Methods
2.1

Introduction and Overview

Experiments conducted at the National Superconducting Cyclotron Laboratory (NSCL)
study the properties of exotic nuclei that exist for only fleeting moments before decaying.
They develop new production methods and measurement techniques necessary to explore
nuclear structure at the extremes of isospin. Shedding light on the behavior of the most rare
isotopes requires meticulous planning and the successful coupling of several state-of-the-art
experimental devices. A succinct overview of the experimental procedure for a typical unstable beam experiment at NSCL follows immediately; the individual experimental techniques
and devices utilized for the presented studies, along with the calibration routines and data
analysis tools are then discussed in greater detail. While it should be noted that some experiments make direct use of stable primary beams or use experimental equipment in novel ways
(see Ref. [63, 64] for examples), the following discussion is generalized for secondary beam
experiments involving the production of short-lived nuclear states via nucleon knockout or
Coulomb excitation and the subsequent study of the reaction products.
NSCL uses the fast-beam fragmentation method for isotope production and as such, ev-

28

ery experiment begins with stable and electrically neutral atoms which must be ionized and
accelerated to relativistic speeds near 0.5c, where c = 3 × 108 m/s is the speed of light in
vacuum. These high-energy projectiles (∼ 150 MeV/u) bombard a thick production target
where all isotopes lighter than the projectile beam are produced via fragmentation. The
production of isotopes can be treated with a variety of statistical models. Reference [65] and
references therein provide more details and also describe other radioactive ion beam production methods such as fusion, fusion-evaporation, spallation, and neutron-induced fission.
From the fragmentation remnants, a radioactive secondary beam is selected and delivered
to the experimental target where the isotope of interest (perhaps in an excited state) is produced via single-step nucleon knockout or Coulomb excitation reactions. In order to ascertain
the nature of the reaction in the target, both the incoming beam and the outgoing reaction
residue must be properly identified. The incoming beam components can be separated by
their time of flight between two scintillator detectors placed in the beam line upstream1 of the
experimental target. The reaction residues are identified on an event-by-event basis by their
time of flight and energy loss signatures obtained from a spectrometer placed downstream
of the experimental target.
For experiments such as the presented study of 18 C in Ch. 3, the excited nucleus emerges
from the target and de-excites via photon emission. The gamma-ray energy, angular distribution, and emission point all contain vital information about the nucleus in its excited state
and therefore photon detection is crucial. The Segmented Germanium Array (SeGA) [66],
with up to eighteen high-purity segmented germanium crystals, serves this purpose and provides acceptable detection efficiency and good energy resolution. The isotope of interest may
also decay by charged-particle emission if the excited state lies above the particle separation energy threshold. This is the case for the presented work on 19 Mg in Ch. 4, which lies
1 The use of “upstream” and “downstream” is a customary practice and denotes relative

position with respect to the propagation direction of the radioactive beam.

29

beyond the proton dripline and therefore decays by proton emission. In this scenario, the
proton decay can be studied from reaction- and decay-residue energy loss information in
charged-particle detectors.
In both cases, the desired experimental signal is masked by other events in the detectors
from the presence of other reaction channels or decays and ambient background sources of
radioactivity. Two actions reduce these sources of background in the experimental spectra.
First, the experimental electronics can be triggered only when the corresponding reaction
residue is detected downstream of the detector setup. This reduces background events from
decays occurring naturally in the experimental vault. In addition, software gates on the appropriate incoming and outgoing experimental channels can be applied during data analysis
to eliminate background events from unwanted contaminants and reaction residues. The reduction of experimental background improves the signal-to-noise ratio and is crucial for the
success of experiments involving weak secondary beam intensities, small experimental cross
sections, and low detection efficiencies.
It is therefore clear that experiments such as the presented studies of 18 C and 19 Mg
require many different experimental devices- from beam production to reaction residue
identification- to provide optimal beam current and purity, clean reaction residue separation, and high signal-to-noise experimental spectra. The following sections provide detailed
descriptions of each of these components in the experimental device chain.

2.2

Radioactive Ion Beam Production

Both experiments were performed at the Coupled Cyclotron Facility (CCF) [67] at the National Superconducting Cyclotron Laboratory located on the campus of Michigan State University. Bringing the experiments to successful completion at NSCL required the coupling
of several experimental components to produce and accelerate the primary beams and sub-

30

sequently select and purify the radioactive secondary beams. Figure 2.1 provides a succinct
summary of these components at the end of the section.

2.2.1

Primary Beam Ion Source

The primary beams accelerated by the CCF are comprised of stable ions. There are three ion
sources at NSCL: the Superconducting-Electron Cyclotron Resonance ion source (SC-ECR)
[68], the Advanced Room Temperature Ion Source (ARTEMIS) [69], and the Superconducting
Source for Ions (SuSI) [70, 71]. All three ion sources operate on the principle of electron
cyclotron resonance; a gas of the desired element is confined in a magnetic bottle and ionized
by bombardment with electrons accelerated in a microwave radiation field. After ionization,
the ions are extracted by a large electric field gradient of up to 30 kV and delivered to the
K500 cyclotron. In the simplest case, the desired stable beam is a naturally occurring gas
and can be directly injected into the magnetic bottle. If the desired element is a metal, an
oven is used to heat the metal and the resulting gaseous vapor is injected into the magnetic
bottle.
Both experiments utilized the SC-ECR ion source, from which high stable beam currents
at low charge states were extracted. The stable beam injected into the K500 cyclotron for the
18 C experiment was 22 Ne4+ and therefore neutral Ne gas was used. For the 19 Mg experiment,
24 Mg5+ was injected into the K500 cyclotron and therefore the source oven was used to

vaporize a sample of solid Mg.

2.2.2

Primary Beam Acceleration and Fragmentation

The K500 and K1200 cyclotrons of the CCF are responsible for the acceleration of the stable
primary beams for experiments at NSCL. Inside each cyclotron, a strong magnetic field
constrains the ion path to a circle whose radius increases with increasing ion velocity. The

31

cyclotrons are isochronous; their RF driving frequency remains constant while the magnetic
field strength increases with the ions’ orbital radius to account for relativistic effects. An
alternating electric field applied to three fan blade shaped “dees” provides the accelerating
potential at a fixed frequency near 23 MHz. The ions are extracted from each cyclotron once
their orbital radius can no longer be constrained internally. The maximum extraction energy
for a proton (in MeV) is designated by the “K” value of each cyclotron. For heavier ions, the
kinetic energy at extraction scales with Q2 /A as can be seen by equating the Lorentz and
centripetal forces in the non-relativistic limit and solving for the velocity.

Bqv =

mv 2
r

(2.1)

Here, q = Qe where Q is the net charge number of the ion and e is the unit of electric charge.
From Eq. 2.1 the Q2 /A dependence of the extraction energy is obtained.
1
E = mv 2 ∝ Q2 /A
2

(2.2)

Equation 2.2 suggests that to obtain the highest energy beams, only fully-stripped ions
should be accelerated. Indeed, the primary beams of 22 Ne and 24 Mg mentioned above are
light enough that they can be fully-stripped. However, the most intense beams from the
SC-ECR are produced at low charge states. To balance the need for primary beams of high
intensities and high energies, the K500 and K1200 cyclotrons are coupled. High intensity and
low charge state stable ions are vertically injected from the SC-ECR into the K500 cyclotron,
accelerated to approximately 0.15c, and then sent to the K1200 cyclotron. At the entrance
of the K1200, the beam passes through a thin carbon foil where the remaining electrons
are stripped allowing for maximum acceleration. For the presented experiments, the fullystripped primary beams of 24 Mg12+ and 22 Ne10+ were accelerated to 170 and 120 MeV/u,
corresponding to velocities of 0.53c and 0.46c, respectively.
32

After the production and acceleration of the primary beam, radioactive ions are produced
by fragmentation as the beam impinges upon a thick production target located at the object
position of the A1900 Fragment Separator [72]. For the 18 C and 19 Mg experiments, 1763 and
1081 mg/cm2 beryllium targets were used, respectively. Most primary beam particles pass
through the beryllium production target without undergoing any reactions. When reactions
do occur, the beam particle is fragmented into many pieces. These reaction byproducts
include every isotope lighter than the original primary beam. The radioactive secondary
beam must then be separated from this collection of fragments.

2.2.3

Fragmentation Separation for Secondary Beams

The A1900 Fragment Separator is responsible for magnetically selecting the desired secondary beam from the fragmentation residues. The A1900 consists of several large-bore
magnets: four 45◦ dipole bending magnets interspersed between a collection of 24 focusing
quadrupole magnets. The magnetic field strength of the dipole magnets can be individually
tuned such that the trajectory of the desired secondary beam follows the magnets’ radius
of curvature whereas unwanted fragmentation contaminants are either bent too much or too
little and are not accepted in the next section of beam line. Isotopes are deflected according
to their magnetic rigidity, Bρ, which from Eq. 2.1 is given as

Bρ =

Au
m
v=
v,
q
Qe

(2.3)

where the radius of curvature of the magnet r = ρ and u is the atomic mass unit.
The magnetic filtering still allows the transmission of beam contaminants with similar
momentum-to-charge ratios as that of the desired radioactive beam due to the large momentum acceptance of the A1900 (up to ∆p/p = 5.5%). Yet the beam components will be
spatially separated in the dispersive plane. To take advantage of this, three momentum aper33

tures are installed at each image position (Image 1, 2, and 3) between dipole magnet pairs.
A final mass aperture is installed at the A1900 focal plane. These adjustable slits restrict
the space available for the beam to pass through. Additional separation is provided by an
aluminum energy-degrading wedge installed between the second and third dipole magnets.
The wedge- actually a curved foil of uniform thickness- provides a varying effective thickness profile in the dispersive direction, thus preserving the spectrometer’s achromaticity [73].
Each beam component undergoes electronic stopping through the wedge and incurs a unique
energy loss, −dE/dx ∝ Q2 , given by the Bethe formula [74] as
−

dE
4πe4 Q2
2me v 2
=
N Z[ln
− β 2 ].
dx
me v 2
I(1 − β 2 )

(2.4)

In the above expression, v and eQ are the velocity and charge of the fully-stripped ion beam,
N and Z are the number density and the atomic number of the energy loss wedge, me is the
rest mass of the electron, I is the average excitation potential of the wedge material, and β is
the fraction of the beam velocity with respect to the speed of light, β = v/c. Appropriately
tuning the two dipole magnets downstream of the wedge provides further separation of beam
contaminants since their individual energy loss through the wedge results in contaminantspecific rigidities.
The final cocktail beam, comprised of the secondary radioactive beam of interest and any
remaining contaminants, then emerges from the A1900 and is delivered to the experimental
setup. Figure 2.1 illustrates the idealized situation where only the desired secondary beam
emerges from the A1900. Differentiation among the remaining beam components is provided
by time of flight measurements made by two thin plastic scintillators with respect to the
experimental trigger. One scintillator is installed at the A1900 focal plane and the other
is located near the experimental setup. For the presented studies, the cocktail beams were
delivered to the secondary fragmentation target of the Köln/NSCL plunger [75] located at the

34

ion sources

K500

coupling
line
K1200

A1900
image 1

focal plane

image 2

stripping
foil

image 3

production
target

Figure 2.1: The various radioactive secondary beam production devices. The stable ion primary beam (cyan) is injected into the K500 from the SC-ECR ion source, accelerated, and
sent to the K1200 (olive). The accelerated primary beam (magenta) impinges upon the production target and the fragments are separated according to their momentum-to-charge ratios
(purple, blue, and orange). Energy loss through the aluminum wedge at Image 2 changes the
rigidity of the beam (red, green, and blue) and only the desired secondary beam (green) is
transmitted after passing through an aperture at the focal plane.
target position of the S800 Magnetic Spectrometer [76] and the time of flight was measured
between the scintillators at the A1900 focal plane and S800 object position.

2.3

Experimental Technique and Devices

The presented works involved detecting short-lived radioactive isotopes and their decays.
The experiments were designed to extract information on exotic nuclei existing far from the
line of stability; 18 C is only four neutrons removed from 22 C, the heaviest carbon isotope,
and 19 Mg lies beyond the proton dripline. For experiments involving such exotic nuclei, event
rates are quite low and therefore the capability to detect both decays and reaction residues
must be maximized. This involves using several detection systems concurrently to obtain
highly selective data. For example, SeGA experiments have low in-beam gamma-ray detection
efficiencies (on the order of a few percent) and it becomes vitally important to accurately

35

Counts

Target Degrader

Counts

Counts

Variable distance

Energy

Figure 2.2: Schematic of the Recoil Distance Method for gamma-ray spectroscopy lifetime
measurements taken from [3]. The intensity ratio of the two detected photopeaks varies as
the distance between the target and degrader is increased. See text for details.
ascertain the origin of each detected decay to cull useful experimental information from
the large swath of background signals. The successful event reconstruction in the presented
experiments was made possible with event-by-event particle identification and tracking in
the large-acceptance, high-efficiency S800 Magnetic Spectrometer, which facilitated the use
of particle-specific software gating in the analysis to produce high signal-to-noise spectra.

2.3.1

The Recoil Distance Method

The two experiments used variants of the Recoil Distance Method (RDM). This experimental
technique is described in general in Ref. [77]; the implementation optimized for intermediate
energy radioactive isotope beams at NSCL is discussed in Ref. [20]. For RDM lifetime measurements via gamma-ray spectroscopy (as depicted in Fig. 2.2), an excited nuclear state
36

emerges from the target (gray) and decays by gamma-ray emission either before (blue) or
after (green) passing through the reaction residue velocity degrader (red). Neglecting nuclear
recoil, the energy of the gamma ray in the rest frame2 , Erest , is simply the energy level spacing between the initial and final states. The resulting gamma-ray energy in the laboratory
frame, Elab , is Doppler shifted according to the velocity of the excited nucleus at the point
of gamma-ray emission, β = v/c, and the laboratory angle of emission relative to the beam
velocity vector, θ, and is given by

Elab = Erest

1 − β2
.
1 − β cos θ

(2.5)

The observed gamma-ray energies in the laboratory frame must be Doppler corrected
(or reconstructed) back to the rest frame to obtain an accurate description of the excited
level structure of the nucleus of interest. For the presented lifetime study of the 2+ → 0+
gs
1
transition in 18 C, the average beam velocity downstream of the plunger degrader, βslow , and
the angle between the beam axis and the line segment connecting the target position of the
S800 with the center of the SeGA segment in which the gamma-ray was detected, θlab , were
inserted into Eq. 2.5 solved for Erest . The Doppler correction therefore took the form

Erest = Elab

1 − βslow cos θlab
2
1 − βslow

.

(2.6)

Appendix B provides a detailed derivation of the relativistic Doppler effect of Eq. 2.5 and
a discussion of both the photopeak Doppler broadening and uncertainty of the Doppler
reconstructed gamma-ray energy due to experimental uncertainties in Eq. 2.6.
2 This work shall denote the two inertial reference frames as “rest” and “laboratory” frames.

In the rest frame, the emitted gamma radiation energies would be detected just as they would
if originating from a radioactive source at rest in the experimental setup. The laboratory
frame is the experimental frame of reference, where relativistic effects of beam velocities
around 0.3c are observed and must be accounted for.
37

After Doppler reconstruction, the gamma-ray energy spectrum exhibits two photopeaks
with different energies corresponding to decays upstream and downstream of the plunger
degrader. The intensity ratio of the two photopeaks will vary as a function of the lifetime, τ ,
and target-degrader distance, d. In the limits of very large or short distances compared to the
transit of the reaction residue in one lifetime, d >> γvτ or d << γvτ , only one photopeak will
be detected.3 From the experimental gamma-ray lineshapes at various distances, the excited
state lifetime is obtained by fitting simulated lineshapes and minimizing the resulting χ2 as
a function of lifetime.
For RDM lifetime measurements via charged-particle spectroscopy, the degrader is replaced with a silicon double-sided strip detector (DSSD) which provides Z identification of
the reaction and decay residues from energy loss measurements. To obtain the lifetime, the
intensity of the energy loss peak of the nucleus of interest is monitored as a function of
target-detector distance. To distinguish the experimental energy loss signal from contaminant isotopes with the same Z, and thus according to Eq. 2.4 the same energy loss, DSSD
spectra are collected in coincidence with reaction residues detected in the S800. For twoproton decay studies, software particle gates on the A−2 (Z − 2) daughter nucleus in the S800
therefore unambiguously identify the nucleus of interest A Z from isotopic contaminants in
the DSSD.
The essence of RDM lifetime measurements is to take advantage of the relationship
between distance, velocity, and time. Nuclear states with short lifetimes (∼ 1 ps) require
very fast clocks (∼ 1 THz) and excellent resolution for traditional lifetime measurements
and are prohibitively difficult. The RDM technique translates the difficult study of very fast
3 This neglects the impact of excited-state production in the plunger degrader. For very

large distances compared to the nuclear transit, all excited states produced in the plunger
target will decay upstream of the degrader. Thus the presence of a second photopeak at such
large distances is indicative of excited-state production in the degrader and the subsequent
gamma-ray decay further downstream. Determining this “production yield ratio” and its
impact on the lifetime analysis is discussed in Ch. 3.
38

decay events with clocks into the study of decay spectra differences over very short, yet
macroscopic, distances. For example, at typical NSCL secondary beam velocities of 0.3c, an
exotic isotope that decays 10 ps after its production will have traversed 1 mm. Therefore with
an accurate knowledge of the beam velocity and the precise control over distances provided by
the Köln/NSCL plunger, the lifetimes of very short-lived states can be accurately determined.

2.3.2

The Köln/NSCL Plunger

The Köln/NSCL plunger device was employed for the presented RDM measurements. The
device offers precise control of the distance between an experimental target and velocity
degrader or silicon DSSD. Such precise control over short distances, as mentioned above,
is required to probe picosecond-lived nuclear states. The distance is manipulated by a set
of plunger-mounted Exfo Burleigh Inchworm Motors. This motion system consists of three
piezoelectric transducer (PZT) ceramic actuators. PZT materials exhibit crystal expansion
along a linear axis in the direction of an applied electric field. A solid-state motion device
consisting of PZT materials is thus very practical due to the lack of rotational motion and the
absence of mechanical gears or hydraulic lines. Therefore, the system exhibits very precise
and uniform one dimensional motion without mechanical vibrations or drift. Figure 2.3
illustrates the simple concept of the plunger PZT motor system. Two outer components act
as clamps along the axis of motion while a central component acts as the linear expansion
or contraction motor. All three components are physically connected to one another, but
have individually applied bias voltages. By repeating the pattern of actions in Fig. 2.3, the
motor can in principle progress forward or backward indefinitely. In the plunger setup, the
degrader4 is mounted to a fixed position and the target is mounted to the PZT motor.
Therefore the motion of the target is restricted to a region less than 5 cm wide bounded by
4 Or DSSD- the information in the following two subsections applies equally to the plunger

for gamma-ray spectroscopy and the particle plunger for charged-particle spectroscopy.

39

(a)

(b)

(c)

(d)

Figure 2.3: An illustration of the four actions required to increment the position of the
plunger target. The rear PZT actuator clamps to the axis shaft (a), allowing the central
PZT actuator to expand (b). To retract back to its original length, the front actuator clamps
to the axis shaft while the rear shaft unclamps itself (c). Finally, the central PZT component
can retract to its original length (d). Repetition of this sequence advances the target.
the initial position of the motor shaft and the fixed degrader. Within this region, the target
can be moved in sub-micron steps in both directions.

Separation Distance Calibration
The distance between the plunger target and degrader is monitored and controlled by the
accompanying Inchworm Control System, which consists of computer hardware and software,
device controllers, and signal cables to connect the hardware to the PZT motors. The system
measures relative distances. To obtain absolute distance measurements, the hardware must
be calibrated to a zero-distance offset prior to plunger insertion into the beam line.
A proper zero-distance offset calibration first requires the parallel alignment of the target
and degrader. This alignment is typically done by eye, since the human eye can discern
separations of 50 µm or better. The plunger is placed on a large and clean white surface
and illuminated from above with a bright light. The target and degrader are brought very

40

Corrected Inverse Capacitance [1/nF]

0.01

0.008

0.006

0.004

0.002

0
40
60
80
100
120
140
Plunger Target-Degrader Separation Distance [µm]

Figure 2.4: The inverse capacitance corrected for a measured stray capacitance of 1 pF is
plotted against the relative distance between the plunger target and degrader. In this sample
measurement, the extrapolated linear behavior of 1/C vs D indicates an offset from true zero
of approximately 30 µm. Other measurements attained an offset closer to 10 µm.
close together and parallel alignment is achieved by adjusting the tension on a spring-screw
system which affixes the target and degrader mounts to the plunger’s support structure.
Done carefully, a very narrow and uniform region of white will emerge between the two foils.
Once the target and degrader are parallel, capacitance measurements are taken as a
function of distance to calibrate the zero-distance offset. Since the capacitance is inversely
proportional to the distance between two parallel plates, the inverse capacitance should linearly decrease to zero with decreasing target-degrader distance. However, mechanical contact
from imperfect parallel alignment or target/degrader surface imperfections is made prior to
achieving true zero and the linearity vanishes.
Figure 2.4 illustrates sample zero-distance offset calibration data. An extrapolation of the

41

data points beyond the breakdown in linearity observed at approximately 60 µm provides the
zero-distance offset. With care, the extrapolated distance offset calibration can be observed
closer to 10 µm. This suggests a maximum calibrated distance error on the order of 10 µm.
For the presented works, such a small error of the calibrated distance is insignificant. For
the 18 C study, data were collected at much larger distance increments (many hundreds of
microns) whereas only data from the shortest distance were used in the analysis of 19 Mg.
Furthermore, the thick targets used in both experiments result in an uncertainty in the
knockout reaction location of several hundred microns, providing the dominant contribution
to the uncertainty in the distance determination.

Plunger Alignment
After proper zero-distance plunger calibrations for both presented experiments, the device
was mounted in the plunger beam pipe, which has a diameter to match the remainder of the
S800 beam line, but is customized with positioning set screws and electrical feedthroughs
for remote manipulation of the target-degrader distance. The setup is illustrated in Fig. 2.5.
Care was taken to mount the plunger into the beam pipe so that the upstream face of the
degrader was aligned with the geometrical center of SeGA at the target position of the S800.5
Furthermore, the plunger’s cylindrical symmetry axis was fixed to coincide with that of the
beam pipe in order for the beam to pass unobstructed through the center of the plunger and
impinge upon the center of the target and degrader.
For proper positioning, a mock-up plunger was constructed with the same outer diameter
and length as the plunger and a thin aluminum cap imprinted with a crosshair reticle affixed
5 A Cartesian coordinate system for the experimental setup has its origin at the S800

target position. The positive z-axis corresponds to the direction of beam travel. The positive
x-axis corresponds to the dispersive direction of the beam (rather, the direction the beam is
bent by the S800 dipole magnets) which is pointing straight up (against gravity). The y-axis
is mutually orthogonal to the other two and therefore points perpendicular to the beam line
and parallel to the floor.
42

Figure 2.5: The Köln/NSCL plunger mounted in the plunger beam pipe. The beam passes
through the center of the plunger, from the lower right to upper left of the figure, and impinges
upon the target and degrader which are mounted by support rings on the downstream edge
of the device. The prominent cylinder on the plunger body is the PZT Inchworm motor
housing.
to its downstream edge. The mock-up was inserted into the beam pipe such that the aluminum cap was aligned at z = 0 mm and the distance from the lip of the beam pipe to the
upstream edge of the mock-up was noted. With the z position fixed, a surveyor’s telescope
placed between the two quadrupole magnets of the S800 analysis line was centrally aligned
with the beam axis using a crosshair reticle on the downstream edge of the final quadrupole
magnet. This reticle target was removed and the telescope was refocused onto the mock-up’s
crosshair reticle. Then, the plunger beam pipe positioning set screws were adjusted so that
the mock-up plunger was secure on all sides and aligned with the beam axis. With the final
alignment set, the top positioning set screws were loosened, the mock-up was retracted, and
the Köln/NSCL plunger was inserted and secured into its aligned position.
With the plunger installed and aligned, the PZT motor controls were connected through
the beam pipe feedthroughs to the external hardware system that was installed near the

43

beam line in the experimental vault. The Köln/NSCL plunger target-degrader distance was
manipulated remotely during the experiment via a coaxial cable link between the hardware
in the vault and a controller computer in the Data-U.

Specifics for Gamma-Ray Spectroscopy
The Köln/NSCL plunger was designed for RDM lifetime measurements at NSCL where the
state of interest is produced from the secondary beam on the plunger target via single-step
knockout reactions or Coulomb excitation [20–22, 78, 79]. For knockout reaction studies, carbon or beryllium foils are used for the experimental target and heavier metals, such as gold,
tantalum, or indium, are used for the degrader. For Coulomb excitation measurements, a
large target Coulomb field is necessary and therefore high-Z foils such as niobium or tungsten are used. Regardless of the reaction mechanism employed for the production of excited
states, the chosen foil thickness and degrader material depend upon several experimental
considerations.
The most important consideration is preserving the ability to resolve the two photopeaks
resulting from decays before and after the plunger degrader. The energy difference is due
to the Doppler effect (Eq. 2.5); for fixed SeGA angles, the choice of degrader material and
thickness dictates the change in beam velocity and thus the photopeak energy separation.
Another consideration is optimizing the target thickness. The target must be thin enough
that at the 0 µm distance the excited state can fully emerge from the target and slow in
the degrader before decaying. Conversely, the target must be thick enough for an acceptable
production yield of the state of interest. To balance these competing needs, proper experimental planning is crucial. This need has been met with a Monte-Carlo simulation package
that will be discussed in Subsect. 2.4.2.
As the experiment progresses, gamma-ray spectra are collected at several plunger targetdegrader distances. For the analysis, the most important piece of information that comes
44

directly from the plunger is this distance. If the zero-distance offset calibration was done
carefully, then the distance information is quite accurate since the Burleigh Inchworm PZT
motor system makes relative moves from one distance to the next with sub-micron accuracy.
The actual distance is therefore simply read as the relative distance minus the calibrated
offset. More importantly, there is no mechanical drift. Temperature fluctuations can have a
minimal impact on the target-degrader distance, and though this can be automatically monitored and corrected by the Burleigh Inchworm system, the effect was considered negligible.

Specifics for Charged-Particle Spectroscopy
The modified Köln/NSCL plunger was designed for RDM lifetime investigations of particledecaying isotopes by replacing the degrader with a silicon energy loss detector. The presented
19 Mg two-proton decay lifetime measurement therefore serves as a proof of concept for the

charged-particle RDM, or particle plunger, lifetime measurement technique. Despite the
various difficulties encountered in the preparation and execution of the experiment that are
mentioned below, the successful lifetime measurement as presented in Ch. 4 validates this
experimental method and warrants future work and improvements. The difficulties mentioned
serve as guidance for future particle plunger experiments.
For the 19 Mg two-proton decay lifetime measurement, two Micron Semiconductor silicon
DSSDs were purchased (thicknesses of 140 and 300 µm), each with an active area of 50 mm
x 50 mm and electronically segmented into 16 strips vertically on the upstream face and horizontally on the downstream face with a pitch of 3.125 mm. Manufacturing issues prevented
the delivery of the thinner DSSD and delayed the delivery of an operational thicker DSSD
until the day before the experiment. The DSSD silicon wafer was affixed to a ceramic mount
such that the upstream face of the wafer and the plunger target could be brought together
without the ceramic mount interfering.
The plunger PZT motor control signals are a source of electrical noise. Therefore the
45

DSSD was mounted to the plunger support with electrically insulating nylon screws and
spacers. Nonetheless, significant noise was observed in the detector when the external plunger
control hardware was connected to the beam pipe. Therefore all supporting plunger electronics were kept off except when the target-DSSD distance needed to be changed.
It is important to note that the exposed DSSD and ceramic mount are quite fragile and
therefore great care is necessary when aligning the plunger with the S800 target position and
ensuring its parallel alignment to the beam axis. The positioning set screws and a stopper
block meant to lock the z-axis position of the plunger protrude inside of the beam pipe. Sufficient care was not exercised after the experiment; upon extraction from the beam pipe, a
corner of the ceramic mount was fractured, severing several strip-to-pin connections and preventing post-experimental source calibrations from being made. Instead, the characterization
and calibration of the DSSD during the analysis was done with unreacted beam data.
The detector was biased to its depletion voltage of 80 V and the leakage current was
noted on a run-by-run basis to monitor beam damage. A small increase in leakage currentinsufficient to result in performance degradation- was observed. To bias the detector and extract the energy loss, strip identification, and timing information from the DSSD, a section of
beam pipe was modified with a vacuum-tight 34-pin ribbon cable feedthrough. The detector
signals were extracted via ribbon cable through the feedthrough and were passed to a simple
ground signal splitter board. This board replicated the two ground signals from the DSSD
(one from each face) into two sets of sixteen. From the splitter board, two ribbon cables each
transmitted 16 signal-ground input pairs to a MultiChannel Systems preamplifier. The two
preamplifier modules had inverted and non-inverted signal output options to allow the signal
from both DSSD faces to be passed with the same polarity.
From the preamplifiers, the detector signals were sent to a pair of 16-channel, double width
CAMAC shaper and discriminator modules from Pico Systems. From the discriminators, the
signals were first sent to ECL-to-NIM converters and then to Phillips Scientific 7186 TDCs to
46

extract the timing information. Timing troubles arose during the experiment; troubleshooting
was abandoned in an effort to collect energy loss data. The shaper signals were sent to a
pair of single-width CAMAC Phillips Scientific 7164 ADC modules where pulse amplitude
(energy) and channel (strip) information was converted to a digital signal.

DSSD Detector Characterization
The two-proton decay RDM lifetime measurement requires the identification and integration
of the energy loss (∆E) peak in the DSSD from the parent nucleus A Z in coincidence with
the detected heavy-ion decay residue A−2 (Z −2) as a function of target-detector distance. As
will be discussed in Ch. 4, the combined low statistics and short two-proton decay lifetime of
19 Mg prevented the analysis from proceeding in such a manner. Instead, lifetime information

was obtained by fitting simulated ∆E lineshapes to the experimental DSSD spectra at 0 µm
and minimizing the resulting χ2 as a function of lifetime.
The experimental ∆E DSSD spectra were generated by summing the signals from the
gain-matched upstream strips of the DSSD. 93% of the events consisted of energy deposits
in only one strip. Gain matching was done using unreacted beam data. The centroid energy
loss of each incoming beam component was extracted from a Gaussian fit to the peaks of
the strip spectra. These centroids were then used to extract strip-by-strip gains and offsets
from their linear fit to a representative strip with the smallest raw energy loss signal. Gainmatched unreacted beam ∆E spectra are presented in the first two columns of Fig. 2.6 for
select strips from the upstream DSSD face. The summed spectrum of all 16 gain-matched
strips is presented in the third column. The full-width at half-maximum energy resolution for
20 Mg was calculated to be 4.1%. Note that this resolution was dominated by the incoming

momentum spread of the beam, energy and angular straggling through the target, and similar
straggling through the dead layers of the detector; the intrinsic resolution of a silicon detector
is typically on the order of, or better than, 1% [80, 81].
47

60

7

2000

500

20
Counts/8 channels

40

Summed

10

750

16

20

O

Mg

250

0
300

8

200

750

11

300
200

100
0

1500

0

18

1000

100
9

0

17

12

75

500

F

500

50

250

Ne

25

0
750 1250 1750 2250

0
750 1250 1750 2250
Channel

0
750 1250 1750 2250

Figure 2.6: Silicon DSSD energy loss spectra for the unreacted beam components 16 O, 17 F,
18 Ne, and 20 Mg. Select vertical strips from the upstream DSSD face are plotted in the
first two columns. Panel labels denote the strip number; strips were sequentially numbered
from left to right when viewed upstream of the detector. The summed spectrum of all 16
gain-matched vertical strips is plotted in the third column.

2.3.3

The S800 Spectrometer

In the presented 18 C and 19 Mg studies, the reaction residues emerged from the Köln/NSCL
plunger target, decayed in flight after a distance governed by their velocity and half-life, and
were identified by the S800 Magnetic Spectrometer. The S800 is a large-acceptance, highefficiency magnetic spectrometer with high energy resolution and good angular resolution.
Like the A1900, it is an ion optical system- just as lenses and prisms are photon optical
systems- and operates on the principles of magnetic separation via the momentum-to-charge
ratio of ions as given in Eq. 2.3. Figure 2.7 illustrates the main features of the S800, including the two main components: the analysis beam line and the spectrometer. The analysis
line extends from the object to target positions and consists of four 22.5◦ bending dipole
48

Figure 2.7: An illustration of the S800 Magnetic Spectrometer. Radioactive ion beams are
delivered to the experimental target through the analysis line. Reaction residues emerge from
the target; those within the acceptance of the S800 spectrometer are delivered to the focal
plane, which consists of several detectors for position, timing, and energy measurements. See
text for more details.
magnets, five quadrupole focusing magnets, and several sextupole correcting magnets. The
spectrometer extends vertically from the target position to the focal plane as two large-bore
quadrupole magnets and two 75◦ dipole magnets. The spectrometer has a large solid angle
acceptance of Ω = 20 msr, which approximately corresponds to angles of 7◦ in the dispersive
and 10◦ in the non-dispersive directions.
The secondary beams selected by the A1900 Fragment Separator are delivered to the S800
and focused at the object position. Two main modes of S800 operation determine where the
final focus of the beam is located. In dispersion matching mode, both the analysis line and the
spectrometer are achromatically tuned such that the beam is focused at the focal plane and
momentum dispersed at the target position. This configuration allows for the best energy
resolution (∆E/E = 2 × 10−4 with a nominal 1 mm diameter beam spot at the object
position) since the intrinsic momentum spread of the beam at the object position is canceled
at the focal plane. However, the momentum dispersion at the target position restricts the
acceptance to ∆p/p = ±0.25% for a 5 cm square target. In focused mode, the analysis line
49

is achromatically tuned such that the beam is focused at the target position and momentum
dispersed at the focal plane. This allows for maximum momentum acceptance of the S800
(∆p/p = ±2.5%), as ions of all momenta accepted by the analysis line arrive together as a
small beam spot on the experimental target, but a lower energy resolution of ∆E/E = 10−3 .
Both experiments used the S800 in focused mode to track and detect outgoing reaction
residues emerging from the experimental target on an event-by-event basis. This facilitated
particle tagging of decay events detected in SeGA or the particle plunger DSSD and significantly reduced unrelated background signals in these detectors. Spectra demonstrating the
improvement in detector signal-to-noise ratios are provided in Ch. 3 and 4.
The following subsections detail the various detector systems of the S800 used for the
18 C and 19 Mg experiments. Several optional detectors, such as beam tracking parallel plate

avalanche counters at the intermediate image of the analysis line, were not utilized in these
works and therefore will not be discussed. The majority of the S800 detectors are located in
the focal plane [82], including the cathode readout drift chambers (CRDCs), the ionization
chamber (IC), and three plastic scintillators (E1, E2, and E3; 3 mm, 5 cm, and 10 cm thick,
respectively) . The cross sectional dimension of each focal plane detector is approximately
30 cm wide by 60 cm tall. An experimental “S800 singles” trigger is generated from the fast
timing pulse generated by an event in the E1 scintillator. This singles trigger can drive the
experimental data acquisition by initiating the electronic readout of the detector hardware.
This triggering scheme was utilized in the 19 Mg experiment, where the silicon DSSD energy
loss signals of all particles accepted by the S800 were recorded. Alternatively, a logical “AND”
between the S800 singles trigger and an external trigger generated by an auxiliary detector
system can drive the readout. This scheme was adopted for the 18 C experiment, where only
events corresponding to the detection of a reaction residue in the S800 and a gamma-ray
energy deposition in SeGA within a 600 ns coincidence window were recorded.

50

Time of Flight Scintillators
Thin organic plastic scintillators are used for time of flight measurements. The light output
generated by ionizing beam particles passing through the detectors is converted to electric
current in a photomultiplier tube attached to the scintillator by the photoelectric effect
and subsequent charge multiplication. In general, these detectors are not ideal for elemental
identification as they have relatively poor energy resolution. However they have very fast
decay times and therefore provide fast and reliable timing pulses.
A thin 1 mm plastic scintillator was installed at the object position of the S800 for both
of the presented experiments. Together with a plastic scintillator installed at the focus of the
A1900 (130 µm and 1 mm for the 19 Mg and 18 C experiments, respectively), the secondary
beam components were separated by their time of flight between the two. The time of flight
was computed from the timing difference between the two scintillator signals with respect to
the common S800 trigger and enabled precise selection of the incoming beam component in
the analysis. In addition, the thin E1 focal plane scintillator provided a timing signal which,
when compared to the scintillator signal at the object position, yielded the time of flight
through the S800.

Ionization Chamber
The ionization chamber is a Frisch gridded gas-filled ion chamber. The detector is segmented
into 16 one inch anodes and filled with 140 Torr of P10 gas (a mixture of 90% argon and 10%
methane). As the beam traverses the ion chamber perpendicular to the anode segmentation
and between the Frisch grid and cathode, ion pairs are created in the gas. The energy
deposited by the beam is measured by the collection of the electrons on the anode pads
and is correlated to the residues’ atomic number Z. This elemental identification together
with the time of flight measurements served to unambiguously identify the reaction residues
emerging from the plunger target.
51

To extract the average energy deposited for each event, the 16 anode channels were calibrated against one another. The procedure is outlined below. The calibrated energy readout
of each channel was assumed to sample a Gaussian energy loss distribution. Thus, each event
produced sixteen samples of this distribution, from which a centroid and standard deviation
were calculated. If a channel’s calibrated energy fell outside of one standard deviation from
this centroid value, it was neglected and the arithmetic average energy deposit for the event
was calculated from the remaining anode channels.

Cathode Readout Drift Chambers
The CRDCs are a pair of position sensitive drift detectors filled with a gas mixture of 80%
carbon tetrafluoride and 20% isobutane to a pressure of 50 Torr. Each detector has an active
depth of 1.5 cm. The high velocity reaction residues interact with the CRDC fill gas and, as
in the case of interactions in the ionization chamber, create ion pairs. The electrons from the
ionization of the CRDC gas drift through a Frisch grid and are amplified and collected on
an anode wire. The charge on the anode wire induces a signal on a nearby pair of cathodes
located on either side of the anode wire. Each cathode is segmented into 224 individual
readout pads forming a linear chain. To determine the dispersive (x) position of the beam
passing through the CRDC, the image charge distribution on the cathode pads is fit to an
assumed Gaussian peak shape and the centroid is extracted. The non-dispersive (y) position
of the beam passing through the CRDC is determined by the drift time of the electrons,
which is extracted from the timing difference between the charge pulse on the anode wire
and the S800 trigger from the E1 scintillator. The position resolution attained from these
measurements is approximately 0.5 mm in both the dispersive and non-dispersive directions.
It should be noted that for such large volume drift detectors, the charge collection process is
relatively slow (tens of microseconds) and therefore to avoid detector malfunction or efficiency
losses due to pile up, dead time, and other effects, the CRDC count rate is limited to 5 kHz.
52

The set of coordinates (x, y) determined by each CRDC were used to determine the
focal plane positions and angles of each beam particle. From the pair of coordinates and
the known distance of about 1 m between the two CRDCs, the angles at the focal plane in
the dispersive and non-dispersive directions (af p and bf p , respectively) were calculated. In
addition, CRDC2 is positioned at the S800 optical focus and therefore its extracted beam
particle coordinates (x, y) corresponded to the focal plane position in the dispersive and
non-dispersive directions (xf p and yf p , respectively). To extract proper position and angle
information about the beam, both CRDCs were calibrated by detecting beam ions with
well defined and precise spatial positions. These calibrations are discussed in the following
subsection.

Detector Calibrations
The position and angle information derived from the CRDCs were used to track the reaction
residues’ position, angle, and momentum spread as they emerged from the plunger target.
To provide accurate tracking information, the CRDCs were first calibrated such that the
detector response to an incident beam particle was correlated to a well defined (x, y) spatial
position. To accomplish this calibration, a pair of thick tungsten beam blocking masks were
used. A geometrical pattern of slits and holes are stamped through each mask at well defined
positions. Thus, only beam particles incident upon the slits or holes entered the detector and
the resulting electronic response was correlated to a precise spatial position. During a mask
calibration, unreacted beam was swept across the entire focal plane by varying the magnetic
field settings of the S800 to ensure the detector response over the entire face of the CRDC
was probed. To obtain a complete set of CRDC calibration parameters (gains and offsets
for CRDC1 and CRDC2), the experimental CRDC detector response to beam ions passing
through each slit and hole was linearly fit to the known slit/hole positions on the two masks.
Several mask calibration runs were taken during each experiment to properly characterize
53

the CRDCs as a function of time. This was necessary since the detector response is intimately
linked to the operating conditions, such as temperature, gas pressure, and radiation damage,
which affect the ion-pair production and electron mobility. The set of obtained gains and
offsets from each mask calibration must be appropriately implemented in the analysis. For
the analysis of the 18 C and 19 Mg data, run-by-run CRDC gains and offsets were obtained
by linear interpolation of the results from each mask calibrations as a function of time.
The 16 anode channels of the ionization chamber were also calibrated such that the energy
loss peak from each channel aligned with that of the other 15 channels. To complete this task,
data with two or more beam species present in the IC (corresponding to two or more distinct
energy loss peaks) was used; for the 19 Mg experiment, this was accomplished by delivering
the unreacted beam components to the S800 focal plane while for the 18 C experiment, the
reaction products from the 19 N secondary beam impinging upon the plunger target were
used. A Gaussian was then fit to each energy loss peak and the centroids were extracted
from each channel. From these centroids, gains and offsets were obtained from the linear fit
of each channel to a chosen representative channel with the smallest uncalibrated energy loss
signal.
The incoming secondary beam emittance, reaction kinematics, and straggling through
the plunger target and degrader all have an impact on the spread of trajectories through the
S800 for a given beam species. With properly calibrated CRDCs, the exact trajectory of each
beam ion in the S800 was deduced (discussed below). With this knowledge, a path length
correction was applied to both the calibrated ionization chamber energy loss and time of
flight data. This was done by constructing a two dimensional histogram plotting the energy
loss or time of flight information against the CRDC-determined focal plane position and
angle information. The position/angular dependence of the measured energy loss or time of
flight was manifest by a correlation in these plots. A simple constant of proportionality was
then determined by eye to remove the observed correlation. The resulting trajectory-corrected
54

20

1

Energy Loss [Arb.]

Mg

2

300
18

C
17

C

200
15

B

100
800

1000

1200

800

1000

1200

Time of Flight [Arb.]
Figure 2.8: S800 trajectory-corrected (Panel 1) and -uncorrected (Panel 2) identification of
the outgoing reaction residues from the incoming 19 N secondary beam on the plunger target
by time of flight and energy loss information from the S800 Spectrometer. Increasing time is
from the right to left along the abscissa.
time of flight and energy loss spectra exhibited increased resolution and the resulting particle
identification plot of energy loss in the IC versus time of flight through the S800 was much
improved. Figure 2.8 illustrates the striking improvement in resolution attained by these
trajectory corrections from the presented work in Ch. 3.

Inverse Map Trajectory Reconstruction
For the presented study of 18 C and 19 Mg, the tracking of the outgoing reaction residues from
the focal plane back to the S800 target position was of great importance for understanding
the kinematics introduced by the knockout reaction. Comprehensive simulations played an
important role in the analysis of both experiments. In particular, the spread of outgoing
reaction residue momenta had to be accurately reproduced in the simulation in order to
obtain realistic Doppler broadened SeGA gamma-ray energy or silicon DSSD energy loss
simulated spectra. Thus, trajectory reconstruction via an inverse transfer map was done on
55

an event-by-event basis.
The analytical method for event-by-event beam ion trajectory reconstructed in the analysis was based upon the ion optics code COSY Infinity [83] which calculates a transfer map
S of the spectrometer based upon its measured magnetic field. The inverse transfer map S −1
relates the measured focal plane parameters xf p , yf p , af p , and bf p to the target parameters yta , ata , bta , and dta (the y position, dispersive and non-dispersive angles, and energy
deviation from the central trajectory, respectively) according to


ata 






 yta 



 = S −1 


 b 

 ta 



dta


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

(2.7)

Note that the dispersive position at the target, xta , is assumed to be negligibly small and is
set to 0 in order to extract the energy at the target, dta . Thus the inverse transfer map results
in the reconstruction of the momentum vector for each reaction residue (defined by ata , bta ,
and dta ) and position in the non-dispersive direction at the target. Since the spectrometer
separates particles according to their momentum-to-charge ratios, an inverse transfer map
must be calculated for each reaction residue.

2.3.4

The Segmented Germanium Array

The Segmented Germanium Array is used extensively at NSCL for gamma-ray spectroscopy
experiments. Short-lived excited states below the particle separation energy threshold typically decay by gamma-ray emission. Gamma-ray spectroscopy with SeGA in coincidence
with reaction residue identification in the S800 Spectrometer yields final state information
from nuclear reactions. It is therefore a powerful tool for probing nuclear structure properties

56

far from stability such as transition strengths, nuclear deformation and collective behavior,
cross sections, and spectroscopic factors.
SeGA consists of up to 18 high-purity, n-type cylindrical coaxial germanium crystals. Each
crystal measures 80 mm in length and 70 mm in diameter and is electronically segmented into
8 slices transverse to the cylindrical symmetry axis and radially divided into four quarters.
A central core electrode along the symmetry axis is biased during operation to ∼ 4000 V
while the outer contacts of each of the 32 segments are grounded. The crystals are oriented
such that the symmetry axis is perpendicular to the line segment connecting the S800 target
position and crystal midplane. Figure 2.9 provides a schematic representation of a single
crystal’s segmentation scheme and the nomenclature used for identifying each segment. The
crystals are kept at a temperature around 100 K by thermal contact with a cold finger to a
2 L liquid nitrogen reservoir oriented at 45◦ with respect to the crystal cylindrical axis.
The 32-fold electrical segmentation of each SeGA detector provides excellent angular sensitivity without relinquishing the detection efficiency afforded by a large crystal. This design
enables both the accurate determination of the angle of the first interaction and the ability
to absorb the full gamma-ray energy. The mechanism by which gamma rays interact with
a detecting medium depends upon the photon energy. Section C.1 of Appendix C provides
more details on the interaction of gamma rays with matter. For gamma-ray energies of several
hundred keV to a few MeV, single or multiple Compton scattering followed by photoelectric
absorption are the most probable interaction mechanisms; full energy deposition in a single
SeGA segment is quite unlikely. Thus in order to properly Doppler reconstruct each event
via Eq. 2.6, the angle of interaction, θlab , is taken from the segment with the largest energy
deposit while the charge collected by the central electrode from the sum of all interactions
within the crystal is used to gather the full energy deposition for the event, Elab . For photon
energies Eγ > 500 keV (and this is the case for all measured Doppler shifted transitions in
the 18 C study) this algorithm has been determined to be robust [84].
57

Figure 2.9: An illustration of a single Segmented Germanium Array crystal. Each of the
eight slices transverse to the cylindrical symmetry axis are labeled with a letter A through
H, with slice A closest to the beam pipe in the setup for the 18 C experiment. The four radial
quarters are numbered 1 through 4, with quarters 2 and 3 closest to the beam pipe in the
experimental setup. Preamplifiers and other electronics are housed behind slice H.
The intrinsic energy resolution of germanium detectors (on the order of 0.1%) is vastly
superior to the commonly used alternative sodium iodide scintillator detector (between 6%
and 8%). It is therefore desirable to utilize this enhanced resolving power in experiments that
involve the detection of multiple gamma rays. However, germanium as a detection medium
is quite costly and large volume single crystals are difficult to grow; for SeGA the price
of this excellent energy resolution is small solid angle coverage and thus a low gamma-ray
detection efficiency (on the order of 1%). The efficiency can be increased by installing the
detectors closer to the experimental target. However, this increases the opening angle of a
given segment and therefore the angular uncertainty for Doppler reconstruction.
The ultimate trade-off between efficiency and angular resolution drives several standard SeGA configurations that are available for different experimental needs. Delta SeGA

58

(c.f. James “Russ” Terry’s thesis [85]) is configured to cover many different angles for angular
correlation measurements. Classic SeGA (c.f. David Miller’s thesis [86]) has been utilized for
gamma-ray polarization measurements. Barrel (Beta) SeGA (c.f. Joshua Stoker’s thesis [87])
has been used for maximum solid angle coverage.
The 18 C experiment utilized 15 detectors in the Plunger SeGA configuration, which
consists of 8 detectors in the θ = 140◦ backward ring and 7 detectors in the θ = 30◦
forward ring.6 The detectors are spaced at 45◦ intervals azimuthally symmetric to the beam
axis (aside for one hole in the forward ring to accommodate a beam line vacuum valve).
The forward and backward rings were placed approximately 30 cm and 23 cm radially,
respectively, from the z = 0 target position of the S800 (which corresponds to the upstream
face of the plunger degrader). The detector placement of SeGA is such that each slice is at
a different polar angle θ with respect to the beam axis. The chosen angles for the forward
and backward rings are the extrema achievable with the plunger beam pipe and therefore
provide the maximum Doppler shift between gamma rays detected in the two rings.

Calibration Routine
Like the detectors in the S800 focal plane, the SeGA detectors must be calibrated and
characterized in order to extract useful gamma-ray energy spectra. Proper calibrations for
both the central contacts and segments are vital for the detectors to function as a single
array since the signals in each ring of detectors were summed together. Furthermore, energy
calibrations are vital to correlate the charge pulse from a detected gamma ray with the true
photon energy (and thus the nuclear transition energy). Here, the procedure for calibrating
all 480 segments and 15 central electrodes is outlined.
The first step is to calibrate all 15 central electrode channels against one another. This was
6 The θ designation of each ring is the angle between the beam axis and the line segment

connecting the S800 target position to the crystal midplane. Thus, the cylindrical symmetry
axis of detectors in a θ = 90◦ ring would lie parallel to the beam axis.
59

done by matching the detector response to a 226 Ra source with the known energy transitions
from that source. The goal of the calibration was to align the detected photopeaks from each
detector with one another such that the peaks fell into the same histogram bins which
corresponded to the correct transition energies. The calibration of each central contact was
automated- the raw data histograms were read into the histogram and analysis software gf3
(Subsect. 2.4.1), a source file containing the known transition energies for 226 Ra was loaded,
the peaks in the uncalibrated histograms were marked by the analyzer, and the data were
fit to the source information by a χ2 minimization of a second order polynomial. From this,
three calibration parameters for the central contact of each detector were obtained.
Next, the individual segments must be calibrated to the calibrated central contact of each
detector. This can quickly become a very complicated procedure; the set of three calibration
coefficients are dependent upon the number of segments that participated in a given gammaray event (known as the multiplicity or fold for the event). Thus, a given segment will have
a set of three fold-1 calibration parameters, 31 sets of three fold-2 calibration parameters
(corresponding to the 31 possible segment combinations for fold-2 events), and so on. The
procedure has been largely automated [88] and is described in brief below for a single detector.
First, the fold-1 subset of data was selected. There should be a one-to-one correspondence
between the energy signal in the only active segment and calibrated central contact. This
correspondence is independent of whether the detector signal resulted from partial or full
gamma-ray absorption; data from the Compton continuum is as useful as that from the
photopeak. A χ2 minimization was used to extract 32 sets of fold-1 calibration parameters
from a second order polynomial fit (one set for each segment). Next, the fold-2 subset of data
was selected and in a similar manner, fold-2 calibration parameters were extracted by the
one-to-one correspondence between the summed calibrated energy signals of the two active
31
sets of calibration
segments and the central contact. Thus, for fold-2 events a total of 32 ×
2
parameters were extracted from the χ2 minimization of a pair of second order polynomials
60

(6 correction coefficients total per set, three per segment).
For higher-order folds, the number of parameter sets grows with the possible combinations and the number of parameters in a set grows linearly. Therefore, the calibration for
segments with folds larger than 2 were done not by fitting, but by applying a higher-order
fold correction as given by
f

Ef
αi,j

=

f

f

1
Ei +

αi,j +

i=1
i=1 j=i+1
2 − E 1 − E 1.
= Ei,j
i
j

(f − 1)(f − 2)
ξ
2
(2.8)

1
Here, E f is the calibrated energy for segments with fold f > 2, Ei is the calibrated fold-1
2
energy for segment i, Ei,j is the calibrated fold-2 energy for a segment pair i, j, and ξ is

a fitting parameter for each detector that was generally found to be between 2 and 3 keV.
The result of the calibrations is shown in Fig. 2.10. The degradation in energy resolution for
higher folds is to be expected as the correction of Eq. 2.8 becomes a less and less accurate
approximation. The prominent decrease in statistics for events of fold-3 or higher mitigates
the effect of worsening resolution.

Photopeak Efficiency Calibration
For the presented work on 18 C, the lifetime information is extracted from the ratio of counts
in the two photopeaks as a function of target-degrader distance. Since this is a relative measurement, determining the energy-dependent photopeak efficiency of the Segmented Germanium Array is not a vital task per se. However, the photopeak efficiency strongly depends
upon the characteristics of the experimental setup such as the intrinsic detection efficiency
of the germanium crystals, the solid angle coverage of the array, the photon energy, and the
presence of absorbing materials such as the beam pipe, the aluminum casing of the detectors,
and the plunger target and degrader support structures. Therefore, measuring the efficiency
61

260

0

1

2

3

4

5

0

1

2

3

4

5

Energy [keV]

300

340

380
580
610
640
0 4500 0

1800

0

1250 0
Counts

375

0

75

0

15

Figure 2.10: Subset of calibrated 226 Ra gamma-ray energy spectra from the central contact
(panels 0) and segments with increasing event fold data from fold-1 to fold-5 (panels 1-5)
of a single SeGA detector for the 295.2, 351.9, and 609.3 keV transitions. The calibrated
energies for folds greater than two (panels 3-5) are given by Eq. 2.8.
with a standard radioactive source can reveal weaknesses in the experimental setup and indicate where future design improvements must be made. More importantly, the experimentally
determined photopeak efficiency provides a stringent verification of the analysis simulation
software’s ability to reproduce the experimental materials and setup, geometry, and gammaray attenuation. The simulation software is introduced in Subsect. 2.4.2 and discussed in
detail in Ch. 3.
The placement of the Köln/NSCL plunger in the beam pipe introduces a loss of efficiency
since the target and degrader material, along with the aluminum supporting structures,
shadow the SeGA detectors. It is therefore expected that the efficiency measured with a
radioactive source placed in an empty beam pipe will be larger than that of the experiment,

62

where the plunger materials attenuate the gamma-ray flux. Section C.2 of Appendix C
discusses gamma-ray attenuation in greater detail. To accurately determine the gamma-ray
photopeak efficiency for the 18 C measurement, data were collected with standard gammaray sources inserted between the 1.1 mm plunger beryllium target and 1.2 mm tantalum
degrader. 152 Eu (measured activity of 8.46 µCi on May 1st , 1978) and 226 Ra (measured
activity of 9.93 µCi on August 1st , 1975) sources were used. The 226 Ra source emits gamma
rays in an energy range from approximately 186 keV to 2448 keV. The corresponding range
of energies from the 152 Eu source is 122 keV to 1408 keV.
The transition of interest in the 18 C experiment was Doppler shifted to over 2100 keV in
the forward ring SeGA detectors (see Eq. B.24 of Appendix B) and thus the 226 Ra source
was needed to probe the efficiency at such high energies. The alpha-decay half-life of 226 Ra
is 1600(7) years. This imprecision results in an uncertainty in the current radioactivity calculated from the manufacturer’s measured activity. In contrast, the beta-decay half-life of
the 152 Eu ground state is precisely known to be 13.537(6) years. Thus, to obtain an absolute
efficiency calibration from both sources, the relative efficiency measured with 226 Ra source
data was scaled by the efficiency ratio of the two sources calculated from transitions of similar energies near 240, 350, 770, and 1120 keV. The energy-dependent absolute photopeak
efficiency (in %), (E), is given by

(E) = 100 ·

N (E)
.
A · T · P (E)

(2.9)

Here, N (E) is intensity of the photopeak of energy E, A is the source disintegration rate (in
Becquerels, or decays per second), T is the live-time duration of the measurement (in seconds), and P (E) is the probability of a gamma-ray emission of energy E per decay (obtained
from [89]).
The forward 30◦ and backward 140◦ SeGA ring photopeak efficiency curves are presented

63

3.5

Absolute Efficiency [%]

3
2.5
2
1.5
1
0.5
0
0

500

1000
1500
Gamma-Ray Energy [keV]

2000

2500

Figure 2.11: SeGA photopeak efficiency curves for the 30◦ ring (triangles) and 140◦ ring
(squares) from 152 Eu (red) and 226 Ra (black) source data. The functional forms of the fits
(dashed lines) were chosen phenomenologically and used to extract the in-beam photopeak
efficiency for the transition of interest after correcting for the gamma-ray intensity asymmetry
introduced by the relativistic Lorentz boost (see Ch. 3).
in Fig. 2.11. The efficiency is markedly higher in the backward ring because of its larger solid
angle coverage afforded by having more detectors and being installed closer to the S800
target position, corresponding to the source location (8 detectors compared to 7 and 23.3 cm
compared to 30.3 cm for the backward and forward rings, respectively.) In the forward ring,
gamma-ray attenuation through the thick tantalum plunger degrader diminishes the steep
increase in efficiency with decreasing photon energy observed in the backward ring and is
responsible for the sudden inflection of the efficiency curve near 300 keV. From the plot, the
summed absolute photopeak efficiency of the Plunger SeGA experimental setup for a 1 MeV
photon emitted at rest is approximately 1.9%.

64

DDAS
The 18 C measurement marked the first experiment at NSCL to fully instrument SeGA with
a complete array of digital electronics. The Digital Data Acquisition System (DDAS) was
developed at NSCL to enable the waveform capture and pulse shape analysis of segmented
radiation detectors. A more in depth discussion of DDAS can be found in [88] and references
therein. Here, the more salient features are discussed.
The signal pulse arising from a detected gamma ray in SeGA contains vital information
about the energy, timing, and position properties of the interaction. In order to fully extract
all of this information on an event-by-event basis, it is necessary to capture and store the
detected gamma-ray waveforms. Since the timing and interaction position information is
extracted from the rising edge of the signal, it is crucial to sample the waveform on a very
fine scale (relative to the rise time) to preserve the smooth, continuous nature of charge
collection. DDAS was designed and built to capture waveforms of a user-defined length with
a 100 MHz sampling rate and to perform pulse shape analysis of the signals extracted from
all 594 electronic channels of the full 18-detector Segmented Germanium Array.
The Digital Data Acquisition System architecture is designed around the 16-channel
Pixie-16 Digital Gamma Finder module. Each Pixie-16 module contains on-board computing
systems to control synchronous waveform capturing and digital pulse processing for 16 input
channels as well as module-to-module communications on a single-width PXI card. A 12bit 100 MHz Flash Analog to Digital Converter is allocated to each input channel which
facilitates the simultaneous analysis of up to 16 independent signals on a single card. The
cards are housed inside four Compact PCI/PXI 14-slot crates. Each crate is controlled via
an optical fiber bridge by a dedicated Linux-based PC server and a fifth server is designated
for event building. All five PC servers are connected to a 10 TB data storage array over
a fast Gigabit network. Together, the Pixie-16 modules, PCI/PXI crates, PC servers, and
storage array constitute the full Digital Data Acquisition System.
65

There are a total of 39 Pixie-16 modules installed in DDAS to fully instrument the
central contacts and segment electrodes of all 18 SeGA detectors. Their 16-channel input
design drives a natural separation of the roles each card plays. A pair of cards (the Manager and Worker) are required to serve the 32 segment signals from a single detector. Six
Manager/Worker pairs, and thus 6 detectors, are housed in each of three PCI/PXI crates.
The fourth PCI/PXI crate houses the remaining three cards- a pair of Assistants which each
serve 9 central contact signals in channels 0-8 and any potential auxiliary detector signals
in channels 9-15, and the Director which is responsible for receiving and distributing trigger information. All 39 cards have identical hardware and firmware; their unique roles are
designated by a set of circuit board jumpers and configuration software parameters that are
loaded upon system initialization. This design grants the system maximum flexibility in the
event of a module failure (for Plunger SeGA, only 15 detectors are used and therefore there
are spare cards) or to accommodate different experimental needs.
If a gamma-ray interaction within a SeGA detector produces a charge pulse that rises
above a user-defined energy threshold, several conditions must be met and tasks completed
before the resulting waveform can be written to disk. The process begins with the abovethreshold signal on one of the Assistant module channels. The active Assistant then communicates the presence of the event via a logic pulse sent through the PCI/PXI crate backplane
to the Director. This pulse prompts the Director to open a coincidence window of userdefined length and compute the event multiplicity based upon how many logic pulses it
receives from the Assistants within the coincidence window. If the computed multiplicity
satisfies the user-preset condition, the Director performs the actions outlined below.
• First, a validation window of a user-defined length is opened. The Director must receive
an external validation pulse within this window for the gamma-ray event to be written
to disk. This validation window signal is sent to the S800 analog electronics setup via
an optical fiber link. If the timing difference between the validation pulse and a reaction
66

residue signal from the S800 E1 scintillator falls within the S800-controlled coincidence
window, a master coincidence trigger is issued by the S800 electronics. This coincidence
trigger initiates the writing of the S800 reaction residue analog data to disk and also
serves as the external validation pulse for DDAS. (So the master trigger for the 18 C
experiment required the coincident detection of one reaction residue and at least one
gamma ray.)
• While the validation window is still open, the Director sends out a global fast trigger
to all DDAS Pixie-16 modules. This trigger initiates synchronous waveform capturing
and latches a time stamp provided by the 100 MHz DDAS clock that is common to all
Pixie-16 modules.
• The fast trigger is immediately proceeded by a hit pattern that indicates which modules
are to store the captured waveform data if the external validation pulse is received by
the Director. This hit pattern is based upon which Assistant channels sent eventdetected logic pulses within the coincidence window. Both the fast trigger and hit
pattern are distributed from the Director module to all other modules via a system
of equal length CAT-6 cables and Low Voltage Differential Signaling (LVDS) fanout
modules. Figure 1 of Ref. [90] illustrates this trigger/hit pattern distribution system.
• Finally, the arrival of the external validation pulse from the S800 trigger electronics
prompts the Director to send out a record trigger signal. Upon receiving the record
trigger signal, those modules indicated by the hit pattern assemble a data file with a
header and energy, timing, and waveform information.
For a given event, up to 39 module event files can thus be written to the storage array.
In order to assemble these individual module files into a single analysis-ready data file,
two steps are taken. First, a single DDAS event is assembled from the individual module

67

events by merging together those events that share a common DDAS time stamp and event
number. This module event merging is done after the experimental run is completed on the
fifth PC server and takes less than a minute total. Next, the complete DDAS digital run
file is merged with the S800 analog run file. To do this, a copy of the digital data stream
must be reduced to an equivalent analog data stream by retaining detector, segment, energy,
and timing information, but removing stored waveforms. This reduced digital data stream is
then merged with the S800 analog data stream on an event-by-event basis by another time
stamping routine. For the presented 18 C experiment, a 48-bit counter was set up on the
Director module and the S800 electronics rack. A common 1 MHz pulser was then sent to
each counter; the counters were latched by the coincidence trigger (external validation pulse).
At the conclusion of an experimental run, the reduced digital and analog data streams were
merged based upon common time stamps.
This merging method proved to be reliable and robust. Merging failures occasionally
occurred and were often caused by missing data from the Assistant module; in all but a
small handful of cases this issue was rectified. It is worth mentioning that both the original
analog data and full digital data for each run are preserved, allowing remerging after the
experiment or after an upgrade to the merging software. Finally, since an analog-equivalent
data file necessary for real time experimental analysis was not available until the completion
of the run, several analysis programs were developed to examine the data on the level of
individual module events online, including displaying waveforms and creating and plotting
module-specific energy histograms.

2.4

Analysis Software

Many software analysis tools exist at NSCL from which polished final results are extracted
from raw experimental data. All of the run sorting, particle gating, and data reduction

68

for the analysis of the 18 C and 19 Mg experiments was done with a collection of analysis
code written in both C and C++. The code structure was similar from one task-specific
program to the next- read in the data file (either the entire raw event file or a reduced data
file), load and apply any pertinent calibration or correction parameters, apply any gating
requirements, and output the results as a reduced data file or sorted histogram. During
production runs, raw data was written to disk when a reaction residue was detected in the
focal plane of the S800 (as in the 19 Mg experiment) or when a coincidence between a gammaray event in SeGA and a reaction residue in the S800 focal plane was registered (as in the 18 C
experiment). Therefore, raw data contains events that originate from all of the secondary
beam components that impinge upon the target and produce a reaction residue which falls
within the momentum acceptance of the S800 spectrometer.
To take advantage of the opportunity to analyze data from several incoming beam components and outgoing reaction residues, the analysis code was designed to sort the initial
raw data into various packets based upon the identification of incoming and outgoing beam
components on an event-by-event basis. This data file segmentation also improves the signalto-noise ratio of experimental spectra by removing events from contaminant beams and reaction residues. For example, the final data file from which a 18 C gamma-ray spectrum is
obtained is greatly reduced from the original raw data file by first removing events arising
from incoming beam contaminants and then eliminating from the remaining events those
attributable to contaminant reaction residues in the S800 focal plane. Working with progressively smaller data files inherently speeds up the analysis. Furthermore, new calibration or
correction parameters can be applied to only a specific data packet; the entire raw data file
need not be resorted.
Typically, the sorting programs output calibration and correction parameters or raw
and calibrated spectrum files. The output spectra are read, manipulated, and saved for
display using either ROOT [91] or gf3 [92]. For the presented analyses, the final gamma69

ray and DSSD energy loss spectra were passed to a Geant4/ROOT Monte-Carlo simulation
software package built upon the ROOT and Geant4 [93] toolkits. Both of these programs
are well documented with extensive online support and implementation examples; therefore
the following two subsections are devoted to describing the gf3 histogramming software and
the Geant4/ROOT simulations in brief detail.

2.4.1

Data Histogramming with gf3

In the presented analyses, the gf3 histogramming program from the RadWare software package was used almost exclusively for interactive graphical analysis of one dimensional histograms. These histograms were used for the analysis of SeGA gamma-ray spectra, particle
plunger DSSD and S800 ionization chamber energy loss spectra, and CRDC x- and y-position
spectra, among others. The data from these spectra are stored as unformatted integers in
4096 channels with 4 Bytes per channel. gf3 has many analysis features including spectra
manipulations such as zooming, contracting, peak fitting, background parameterizing, and
energy calibrating. Various histograms can be overlayed on top of one another or stacked for
comparison. Manipulated histograms can be saved or printed as postscript files.
At its heart, gf3 is a peak-fitting program that allows for the simultaneous fitting of up to
15 peaks on a quadratic background. The main component of the fitted peak is a Gaussian
that can be convoluted with a skewed Gaussian to account for incomplete charge collection
in the detector, which results in a low energy exponential tail. A smooth step function on
the low energy side of the peak can also be added for fitting gamma-ray spectra to account
for the Compton scattering of events from one detector into another. The user has control
over the parameters used to describe the peak shape and background. Parameters may be
iteratively determined given initial estimates or simply fixed via a start-up script or command
line prompt. The number of iterations to achieve convergence can also be controlled.

70

2.4.2

The Geant4/ROOT Simulation Package

The lifetime determinations reported in the subsequent two chapters were done by matching
simulated detector lineshapes produced with various input lifetimes to the experimental spectra obtained at various plunger distances. In order to produce accurate detector lineshapes
that can be reliably fit to experimental data, the simulations must reconstruct all pertinent
experimental details within the Geant4 framework. The lifetime is then determined by the
best fit via χ2 analysis within ROOT. This method of analysis has been tested and described
in detail in Ref. [94]. Chapters 3 and 4 describe the specific steps which brought the simulation to full complexity for the lifetime analyses. What is important to note here is that the
simulated lineshapes are dependent upon the detector system and plunger geometries, the
spatial extent and energy spread of the secondary beam, the reaction kinematics, and the
acceptance settings of the S800. For both analyses, the geometry of the experimental setup,
the phase space of the incoming secondary beam, the knockout kinematics and subsequent
energy and angular straggling of the reaction residues, and the decay and detection processes
were all accurately modeled within Geant4. ROOT was used to carry out the χ2 analysis of
the simulations’ fit to data, display and manipulate histograms, and provide a convenient
tree structure to access various histogrammed results.
The Geant4 and ROOT toolkits were seamlessly combined within G4UIROOT, which
formed the backbone of the simulations. As its name implies, the G4UIROOT program
provided a graphical user interface (GUI), built with ROOT, for simulations performed
by Geant4. G4UIROOT allows the user to access and browse all Geant4 commands in a
conveniently organized file system structure. Figure 2.12 shows several of the various onscreen windows available to the user during a simulation: the main GUI where commands
are accessed or macro scripts can be executed, a tree viewer from which different result
histograms can be directly accessed, and error and output message windows. In GUI mode,
the terminal provides the ROOT command line which is useful for modifying the output
71

Figure 2.12: A screenshot of the Geant4/ROOT simulation user interface illustrates several
of the GUI features of the program. All experimental commands can be accessed from the
main GUI window. A tree viewer provides convenient access to many different histograms of
the simulation results. Not pictured are the simulation output and error message windows or
the visualization window which draws the experimental setup and provides tracking of each
event. This last capability is computationally intensive and was not used in the analysis.
histograms generated at the completion of a simulation. The program can also run silently
in batch mode in the background. This is extremely useful in the final stages of analysis,
where all the experimental parameters in the simulation are well constrained and the lifetime
determination requires running long simulations from a script which increments the input
lifetime and plunger distance.
Finally, as noted in Subsect. 2.3.2, this simulation package proved useful in the prepa72

ration of RDM lifetime experiments, especially regarding the choice of plunger degrader
material and thickness. The choice of target material and thickness are largely motivated
by considerations concerning the reaction mechanism and experimental yield and can be
predicted with LISE++ [95]. However, the choice of degrader material and thickness can be
investigated by generating simulated lineshapes using various combinations and examining
the resulting peak separation between the fast and slow gamma-ray decay photopeaks. The
lifetime of the state of interest should also be varied; very short lifetimes in thick target
materials will decay in target rather than in vacuum and therefore the energy resolution will
worsen from velocity uncertainties in the Doppler reconstruction. Such material and thickness considerations were investigated using Geant4/ROOT simulations while preparing the
18 C experimental proposal.

73

Chapter 3
Lifetime Measurements in 18C
The Köln/NSCL plunger lifetime measurement of the 2+ → 0+ transition in 18 C was part
gs
1
of an experimental campaign investigating the collectivity and nuclear shape evolution of
the neutron-rich carbon isotopes 16,18,20 C. As shown in Ch. 1, the reduced quadrupole transition matrix element B(E2; 2+ → 0+ ) is inversely proportional to the lifetime. This B(E2)
gs
1
transition strength is a measure of the degree of quadrupole collectivity of the nucleus and
thus lifetime measurements of electromagnetic transitions in radioactive nuclei are useful and
simple probes of nuclear structure. The lifetime investigation was carried out at the National
Superconducting Cyclotron Laboratory via the application of the Recoil Distance Method
lifetime technique applied to the 9 Be(19 N, 18 C∗ )X one-proton knockout reaction in inverse
kinematics. This technique has been successfully applied to lifetime measurements in many
different regions of the nuclear chart at NSCL [20–22, 78, 79] and has proven to be a reliable
experimental method requiring minimal theoretical model dependence during analysis.
The lifetime of the 2+ → 0+ 1585 keV transition in 18 C was determined to be τ =
gs
1
22.4 ± 0.9(stat)+2.5 (syst) ps, which is in agreement with the literature value of τ = 18.9 ±
−1.6
0.9(stat) ± 4.4(syst) ps [29]. This corresponds to a reduced quadrupole transition matrix
element value of B(E2) = 3.64+0.46 e2 fm4 using Eq. 1.9. A 932 keV transition was observed
−0.48
74

to feed the 2+ state from above. The statistics for this state with unassigned spin and parity
1
were sufficient to set a lifetime upper limit of 4.5 ps. This is the first quantitative lifetime
information measured for a state above the first 2+ for even-even carbon isotopes with
N > 8. The experimental technique and detector characterization were presented at length
in Ch. 2. The present chapter details the experimental measurement, the data analysis, and
the subsequent experimental results.

3.1

Experimental Considerations

The Geant4/ROOT simulation software described in Subsect. 2.4.2 proved to be a powerful
analytic tool for extracting the lifetime from the 18 C RDM measurement. It was able to
accurately simulate the experimental setup, the secondary beam emittance, the one-proton
knockout kinematics and subsequent energy and angular straggling of the 18 C nuclei, and
the gamma-ray decay and detection processes- including the Doppler broadening of the
detected photopeaks, the asymmetric Lorentz boosted gamma-ray intensity distribution in
the laboratory frame, and the gamma-ray detection efficiency. Such a complete description
of all facets of the experiment required many input parameters. Properly constraining all of
these parameters necessitated the collection of a unique combination of experimental data.
• First, standard gamma-ray radioactive source spectra were collected before and after
the experiment for the energy and photopeak efficiency calibrations of SeGA (discussed
within Subsect. 2.3.4). Comparing the simulated and experimental source photopeak
efficiencies verified the proper implementation of the SeGA geometry, gamma-ray attenuation and detection processes, and the detector response within the simulation.
• In addition to SeGA calibrations, several in-beam calibrations for the S800 CRDCs
and IC were performed. The calibration procedures were discussed in detail within
Subsect. 2.3.3. Three CRDC mask calibrations were performed over the course of the
75

experiment by sweeping the 19 N secondary beam across the entire focal plane. The
reaction residues within the momentum acceptance of the S800 were also used for
calibrations. Their energy loss signals in the ionization chamber were used to energy
calibrate its sixteen channels.
• Prior to the insertion of the Köln/NSCL plunger, the 19 N secondary beam was delivered
unimpeded to the S800 focal plane. The data constrained the simulation parameters
describing the secondary beam emittance at the target position of the S800. From
the calibrated angles and positions in the CRDCs and the S800 rigidity settings, the
experimental spectra of the four S800 target parameters (ata , bta , yta , and dta ) were
produced by inverse map trajectory reconstruction. The simulation parameters were
set by fitting the corresponding simulated spectra to these experimental counterparts.
• Next, the target-only plunger was inserted and the energy and angular straggling of the
19 N secondary beam transported through the beryllium target was studied. To preclude

contributions from nuclear reactions, the S800 was tuned to accept only unreacted 19 N
beam ions. As above, the simulated target-only S800 target parameter spectra were fit
to their experimental counterparts and the simulation parameters describing energy
and angular straggling were fixed to appropriate values.
• The S800 was then tuned to accept 18 C reaction residues and the data acquisition
trigger was set for particle-gamma (S800-SeGA) coincidences. This target-only reacted
beam data provided straightforward information on the relative population of different
excited states in 18 C. It also constrained the simulation parameters required to describe
the proton knockout reaction and its effect on the momentum distribution of 18 C by
comparing the simulated ata , bta , yta , and dta reacted target-only beam spectra to
their experimental counterparts.
• Finally, with the data taken to both calibrate the experimental equipment and facilitate
76

the accurate parameterization of the experimental conditions in the simulation code,
the tantalum degrader was installed and the full plunger was inserted. To account for
the energy loss through the degrader, the S800 was retuned to accept 18 C reaction
residues. The simulation parameters describing the spread in momentum and angular
straggling introduced by the degrader were constrained by fitting the simulated fullplunger S800 target parameter spectra to their experimental counterparts. Figure 3.7
within Subsect. 3.3.1 illustrates the quality of the fit between the experimental data
and the fully constructed reaction residue simulation.

3.2

Experimental Details

The remaining experimental data were taken with the full plunger apparatus, a particlegamma coincidence trigger, and the S800 tuned to accept 18 C reaction residues. Gammaray energy spectra were collected at five target-degrader distances of 0.0, 600.0, 1500.0,
2200.0, and 3000.0 µm in coincidence with reaction residues in the S800. The details concerning the incoming secondary beam, outgoing reaction residues, and observed gamma-ray
de-excitations follow.

3.2.1

Radioactive Secondary Beam

The delivery of the secondary beam to the Köln/NSCL plunger target involved the development, acceleration, and fragmentation of a 22 Ne primary beam and the subsequent selection
and purification of the 19 N component. The method and devices employed were discussed in
detail in Sect. 2.2.
The Coupled Cyclotron Facility at NSCL accelerated a primary beam of 22 Ne+10 to
120 MeV/u (≈ 46% the speed of light, or β = v/c = 0.46) with an average intensity of roughly
150 pnA. Following acceleration, the beam was delivered to a 1763 mg/cm2 (≈ 9.5 mm
77

Table 3.1: 19 N secondary beam characteristics. Bρ and ∆p/p were taken from the experimental beam line savesets provided by the A1900 Fragment Separator Group. The beam energy
was calculated from Bρ3,4 . The 19 N average rate and transmission to the S800 were calculated from run-by-run scaler information while the purity was calculated from integrating
the peaks of Fig. 3.1.
Bρ1,2
3.59090 T · m
Bρ3,4
3.46591 T · m
Final Energy
75.46 MeV/u
∆p/p
0.7%
Average Rate
2000 pps/pnA
Purity
97.7%
Transmission to S800
82%
thick) 9 Be production target. From the resulting fragmentation reaction products, the 19 N
secondary beam was selected by the A1900 Fragment Separator. The settings of the A1900
were optimized for transmission and purity of 19 N. In particular, the momentum acceptance
was restricted to 0.7% with momentum apertures installed at Image 1 and 2 (refer to Fig. 2.1).
A 300 mg/cm2 Al energy degrading wedge at Image 2 was used for additional contaminant
separation. The desired 19 N radioactive secondary beam emerged from the A1900 with better
than 97% purity. The remaining beam contaminants, 17 C and 21 O, were distinguished from
the 19 N fragment of interest by their time of flight between 1 mm thin plastic scintillators at
the A1900 focal plane and the object position of the S800. The properties of the secondary
beam are listed in Tab. 3.1. There, the subscripts refer to the four A1900 dipole magnets.
The magnetic rigidities of the final two A1900 dipole magnets were set to deliver the 19 N
secondary beam at 75.46 MeV/u to the plunger at the S800 target position. The beam rates
emerging from the A1900 and entering the S800 were monitored during the experiment by
the plastic scintillators. Throughout the experiment, the secondary beam intensity at the
A1900 focal plane was above 3.1×105 pps and the transmission losses en route to the S800
were less than 20%. Figure 3.1 shows the time of flight of the secondary beam components
during a run where the unreacted secondary beam was delivered to the S800 focal plane;

78

19

N

Counts

103

102
17

C

21

O

10

1
2300

2400
2500
Time of Flight [Arb. Units]

2600

Figure 3.1: Identification of the secondary beam components by their time of flight between
two thin plastic scintillators at the A1900 focal plane and object position of the S800. Note
the logarithmic scale of the ordinate. Increasing time is from right to left along the abscissa.
the clear separation between 19 N and the 17 C and 21 O contaminants allowed for offline
separation during the analysis.

3.2.2

Outgoing Reaction Residues

The 19 N secondary beam was sent to the S800 Spectrometer. The experiment utilized the
S800 in focused mode; the beam was focused at the object position of the S800 and the
analysis line delivered a focused beamspot onto the 9 Be 196 mg/cm2 (≈ 1.1 mm thick)
Köln/NSCL plunger target. States in 18 C were populated by the one proton knockout reaction of 19 N on the beryllium target. The magnetic rigidity of the spectrograph was set
to Bρ = 2.83 T · m to maximize the acceptance of 18 C and reject unreacted 19 N and other
reaction contaminants.
For full-plunger production runs, the 18 C reaction residues emerged from the target with
79

a velocity distribution centered at βf ast = v/c = 0.3565(5). After a specified flight distance,
the 18 C nuclei were slowed in the nat Ta 2010 mg/cm2 (≈ 1.2 mm thick) degrader to a
final velocity distribution centered at βslow = 0.2920(5). For target-only production runs,
the 18 C reaction residues emerged from the target with a velocity distribution centered at
βf ast = 0.3610(5). The difference between βf ast and βf ast is strictly due to small variations
in the momentum acceptance from tuning discrepancies of the S800 rigidity for data taken
with and without the tantalum degrader. The true momentum distribution of 18 C as it
emerges from the target is identical regardless of whether the degrader is present or not;
the observed momentum distribution upstream of the degrader is affected by the acceptance
settings of the S800. The procedure for determining the beam velocity and its uncertainty is
mentioned in Sect. B.2 of Appendix B.
The knockout reaction residues within the acceptance of the S800 were identified on an
event-by-event basis by their energy loss through the ionization chamber and time of flight
between the object position and E1 focal plane scintillators. The clear separation between
the 19 N knockout reaction residues accepted by the S800 is illustrated in Fig. 3.2. The
subset of events plotted were selected by a run-by-run software gate on the 19 N peak in
the secondary beam time of flight spectra. In order to achieve the level of separation in
the particle identification plot, the ionization chamber channels were calibrated and flightpath corrections were applied to the energy loss and time of flight signals using focal plane
angle and position information determined from the calibrated CRDCs. These trajectory
corrections were described within Subsect. 2.3.3.

3.2.3

Observed Gamma-Ray Transitions and Analysis

Doppler shifted de-excitation gamma rays were measured with the Segmented Germanium
Array in coincidence with reaction residues in the S800. The experiment utilized the Plunger
SeGA configuration with 15 detectors surrounding the plunger in two rings at laboratory
80

Energy Loss [Arb. Units]

18

C

250

17

C

200

150

15

B

100

900

1000

1100

1200

Time of Flight [Arb. Units]
Figure 3.2: Identification of the outgoing reaction residues from the incoming 19 N secondary
beam by time of flight and energy loss information from the S800 Spectrometer. Increasing
time is from the right to left along the abscissa.
angles of 30◦ and 140◦ with respect to the beam axis. The electronics trigger was issued upon
the coincident detection of a reaction residue in the S800 E1 scintillator and a gamma ray in
SeGA within 600 ns of one another (discussed in detail within Subsect. 2.3.4). This coincidence requirement significantly reduced background contaminants in the gamma-ray spectra
resulting from ambient sources and gamma-ray transitions from fragmentation contaminants
that were not accepted by the S800. Further background suppression was achieved in the
analysis by incoming 19 N and outgoing 18 C software particle gates. These gates rejected
events that originated from secondary beam contaminants or undesired reaction channels
that had similar momenta to 18 C and therefore were transmitted to the S800 focal plane.
The enhancement in the signal-to-noise ratio (S:N ) of gamma-ray spectra provided by software gating is illustrated by Panel 3 of Fig. 3.3, where Doppler reconstructed gamma-ray
spectra are plotted with no software gates (black dashes), gates on only incoming 19 N (solid

81

120

250
1

100

3

80
200

60
20
0
0

1000

2000

3000

80

2
*

60

150

100

*
*

40

Counts/8 keV

Counts/8 keV

40

50

20
0
0

1000
2000
Energy [keV]

3000

800

1200 1600
Energy [keV]

Figure 3.3: Gamma-ray energy spectra for the forward ring of SeGA from target-only production data. Panels 1 and 2 demonstrate the improvement in energy resolution offered by
Doppler reconstruction; Panel 3 illustrates the improvement of S:N via software particle
gating. See text for details.
gray), and gates on both incoming 19 N and outgoing 18 C (solid black). The improvement in
the full-width at half-maximum energy resolution, Eres , achieved by Doppler reconstruction
based upon the mean velocity of the 18 C reaction residue and the angle of detection with
respect to the beam axis is illustrated in the incoming and outgoing particle-gated spectra
of Panels 1 (corrected, Eres ≈ 2%) and 2 (uncorrected, Eres ≈ 5%). The asterisks in Panel
2 mark three background contaminants corresponding to the 511 keV annihilation line and
neutron inelastic scattering in the detector at 609 keV and 844 keV.
Two transitions in 18 C were observed in the Doppler corrected gamma-ray spectra at
energies of 932(11) and 1585(19) keV as indicated in Panel 1 of Fig. 3.3. The energies are
in good agreement with the values of 919(10) and 1585(10) keV reported in Ref. [96]. These
two photopeaks are shown in greater detail in the target-only spectra of Fig. 3.4. The full

82

140°

100

1585 keV

60
Counts/8 keV

932 keV
20
30°

100
60
20
800

1000

1200
1400
Energy [keV]

1600

1800

Figure 3.4: Doppler reconstructed gamma-ray energy spectra for the forward (bottom) and
backward (top) rings of SeGA. The data were collected with the plunger degrader removed
and therefore each transition is represented by a single photopeak. Note the misalignment
of the 932 keV transition; the Doppler reconstruction was performed to align the 1585 keV
transition corresponding to decays downstream of the target.
width at half maximum energy resolution for the forward and backward rings is 2.27(9)%
and 1.89(7)% respectively, in excellent agreement with the estimated resolution of 2.21% and
1.96% as discussed in Appendix B.
After in-beam efficiency corrections at the Doppler shifted laboratory energies, 80% of
the observed photopeak intensity was in the 1585 keV photopeak which corresponds to the
2+ → 0+ transition of interest. The remaining photopeak intensity was in the 932 keV
gs
1
transition populating the 2+ state from a higher-lying state with undetermined J π . Their
1
placement in the 18 C level scheme is illustrated in Fig. 3.5. The in-beam photopeak efficiency
was extracted from source efficiency measurements by correcting for the asymmetric gammaray intensity distribution introduced by the Lorentz boost. Thus, assuming no additional
83

4180

Sn

4000

Excitation Energy (MeV)

4

2

3200

1585

2

1500

0

0

0

2415
2517
932

2
2

1585

0

0
Experiment

Ab Initio

Figure 3.5: Experimental level scheme of 18 C (left). The solid arrows indicate transitions
observed in this work; their relative thickness corresponds to the measured in-beam efficiencycorrected intensity. The 4 MeV level and an unobserved gamma-ray (dashed arrow) were
reported in Ref. [96]. The neutron separation energy was taken from Ref. [97]. The energy
levels from ab initio calculations [15] discussed in Ch. 5 are given on the right.
feeding contributions to either observed state, the proton knockout reaction was found to
preferentially populate the 2+ state over the higher-lying feeder state by a ratio of 4 : 1.
1
For the lifetime analysis of 2+ → 0+ transition in 18 C, the effect of the 932 keV feeder
gs
1
transition was determined to be negligible. This simplification is justified by the 4 : 1 ratio
of proton knockout reactions which directly populated the 2+ state. Furthermore, aligning
1
the 932 keV photopeaks in the two rings of SeGA in the Doppler reconstructed target-only
gamma-ray energy spectra required β = 0.3650(5). Thus the 932 keV transition must proceed
within the target and therefore has a very short lifetime of only a few picoseconds. This is
clear when compared to the β = 0.3610(5) necessary to align the 1585 keV photopeaks,

84

as these transitions proceed in vacuum downstream of the target. From these two pieces
of information, it was decided to neglect feeding effects in the lifetime analysis. The error
introduced from this simplification is negligible compared to the uncertainty of t0 (the time
from which “counting” begins for the lifetime), corresponding to where in the thick 1 mm
plunger target the 2+ state in 18 C was produced. Nonetheless, despite low statistics for the
1
932 keV transition and thick plunger foils optimized for larger lifetime measurements, an
upper limit for the lifetime of the feeder transition was extracted.

3.3

Lifetime Analysis

The 18 C particle-gated gamma-ray spectra collected at all five target-degrader distances
yielded gamma-ray lineshapes with components from fast, slowing, and fully-slowed decays
that occurred before, inside, and after the plunger degrader. Fitting these experimental
spectra with simulated lineshapes generated from various input lifetime values yielded the
measured lifetime result. Details of the simulations and lifetime analysis for the observed
transitions in 18 C are presented here.

3.3.1

Specifics of Geant4/ROOT Simulations

Monte-Carlo simulations based upon the Geant4/ROOT toolkits were used to determine the
lifetime of the observed transitions in 18 C. These simulations required the parameterization
of experimental details to produce accurate simulated gamma-ray lineshapes. Section 3.1 described the assorted experimental data collected to properly constrain the parameters. The
full simulation- describing the experimental geometry and detector response, beam properties, reaction kinematics, and gamma-ray decay and interaction processes- was methodically
constructed in small increments; each added complexity was vetted against experimental data
to ensure accuracy and consistency. The more salient steps in this process are presented here.
85

SeGA Geometry
The canonical Plunger SeGA ring angles of 30◦ and 140◦ are often just approximations of
the true angles. The exact values must be known to obtain proper Doppler reconstructed
gamma-ray spectra. Variations in the installation of the two SeGA rings and/or inaccurate
placement of the plunger1 result in forward and backward ring angles that deviate from
their canonical values. A laser tracking system can be employed to determine the exact
angles of the SeGA rings. In practice however, this is a difficult and time consuming set of
measurements to make in the confines of the experimental vault.
In the present analysis, the eight SeGA slice spectra in each ring from target-only production data were compared to their simulated counterparts. As the slice spectra are not Doppler
corrected, the photopeak centroid varies from one slice to the next according to that slice’s
angle with respect to the beam axis. This varying photopeak position is used to determine
the correct SeGA ring angles; by varying the input angles in the simulated slice spectra, the
best fit yields the exact angles of the forward and backward rings. From this procedure, the
forward ring was found to deviate by nearly 2◦ from the canonical value and was determined
to be at 31.9(2)◦ . The backward ring, at 140.3(2)◦ , was much closer to the canonical value of
140◦ . These angles were thus used to calculate the true segment angles used for the Doppler
reconstruction of both the experimental and simulated gamma-ray lineshapes.

Photopeak Efficiency
With the geometry fixed, the simulated photopeak efficiency for 152 Eu and 226 Ra gammaray sources was compared to the measured source photopeak efficiency (discussed within
Subsect. 2.3.4). Figure 3.6 demonstrates the agreement between the scaled simulated (red)
1 Recall that the upstream face of the degrader should correspond to the target position of

the S800. It is from this position that the laboratory angles are determined and therefore if
the plunger is shifted along the beam axis, the angles will differ from their canonical values.

86

4

Absolute Efficiency [%]

3.5
3
2.5
2
1.5
1
0.5
0
0

500

1000
1500
Gamma-Ray Energy [keV]

2000

2500

Figure 3.6: Absolute SeGA photopeak efficiency curves from measured 152 Eu and 226 Ra
source data (black) and scaled simulations (red) for the 30◦ ring (triangles) and 140◦ ring
(squares). See text for details.
and experimental (black) photopeak efficiency curves. The scaling factors of both SeGA
rings coincided within uncertainties and were roughly 0.8 and 0.6 for the 152 Eu and 226 Ra
sources, respectively. This scaling is inconsequential as the lifetime analysis incorporates a
scaling factor for the simulated lineshapes of each SeGA ring to attain the best match to
the data. What is important is the equivalent shapes of the efficiency curves for the forward
and backward rings after scaling- especially in the 1 to 2 MeV range which corresponds to
the Doppler shifted energies of the 1585 keV transition. The low-energy deviation observed
in the backward ring may reflect the absence of plunger target mounting structures in the
simulation. The plot suggests these features, which are present in the experiment but not built
into the simulation, may account for the additional low-energy photon attenuation in the data
with respect to the simulated efficiency curve. Overall, the agreement indicated by Fig. 3.6

87

validates the simulation’s implementation of gamma-ray attenuation in the experimental
materials (the plunger target and degrader, beam pipe, and SeGA detector casing), the
detector response, and the energy-dependent gamma-ray interactions with matter.

Beam Characteristics and Reaction Kinematics
The 19 N secondary beam emittance and straggling through the plunger target was parameterized by fitting the simulated ata , bta , yta , and dta S800 target spectra to their experimental
counterparts for target-only, unreacted beam production data. Angular straggling from the
one-proton knockout reaction and reaction residue transport through the plunger target and
degrader Coulomb fields was properly constrained by similar S800 target fits using fullplunger reacted beam production data. In addition, the effective thickness of the plunger
foils and knockout reaction longitudinal momentum transfer were fine-tuned by simultaneously requiring consistency between the simulated and measured beam velocity downstream
of the degrader, βslow , and a proper fit between simulated and experimental dta spectra.
The excellent agreement between the simulated S800 target spectra and their experimental
counterparts illustrated in Fig. 3.7 indicates an accurate parameterization of the incoming
secondary beam properties, plunger foil thicknesses, and knockout reaction dynamics.

Lineshape Generation
The generation of de-excitation gamma-ray spectra at all five plunger distances was the final
task in bringing the simulation to full complexity. Having constrained all the parameters
describing the experimental geometry, secondary beam emittance, reaction kinematics, and
reaction residue properties, the only additional parameters needed to obtain the lineshape
spectra were the level and transition energies, the decay lifetimes, and the 18 C production
yield ratio (Ntar /Ndeg ) of the number of one-proton knockout reactions on the target and
degrader. The first two pieces of information were well constrained by the experimental
88

Counts/5 mrad

Counts/5 mrad

250
200
150
100
50
0

-100

-50
0
50
Ata [mrad]

250
200
150
100

0

100

Counts/0.1%

Counts

400
300
200
100
0

-10

0

10
Yta [mm]

20

50

30

-100

-50
0
50
Bta [mrad]

100

60
40
20
0
-10

-5

0
Dta [%]

5

10

Figure 3.7: Fit of simulated (blue) and experimental (red) 18 C reaction residue S800 target
spectra. The angular distributions in the dispersive (ata ) and non-dispersive (bta ) directions,
spread in non-dispersive position (yta ), and kinetic energy deviation from that corresponding
to a central trajectory through the S800 (dta ) are obtained from inverse map trajectory
reconstruction. See discussion within Subsect. 2.3.3 for more details.
Doppler reconstructed gamma-ray spectra of Fig. 3.4. The excited state lifetimes were the
object of study and therefore were varied in the simulations to ascertain the best fit to
the experimental gamma-ray lineshapes. The target-degrader yield ratio is an important
parameter and will be described below. The simulation could also incorporate gamma-ray
background from target and degrader excitations, however 9 Be has no bound excited states
and no de-excitations from the nat Ta degrader were observed in the experiment.
The simulated 18 C Doppler reconstructed gamma-ray spectra included both of the observed transitions- the 932 keV transition feeding the 2+ state and the 1585 keV transition
1
depopulating this state to the 18 C ground state. The photons produced the spectral features

89

(photopeak and associated Compton background) of the βf ast and βslow decay components
with an intensity ratio dependent upon the input lifetimes, the target-degrader spacing and
yield ratio, and the reaction residue velocity. Decays in the degrader at an intermediate
velocity were included as well. This required the simulation of properly Doppler shifted photons with a Lorentz boosted intensity distribution (from an assumed isotropic distribution
in the rest frame), implementing the in-beam photopeak efficiency, and then correcting for
the Doppler shift exactly as done in the experiment- with an input beam velocity β and the
calculated SeGA segment angles.
Two simplifications were made for the lifetime fits of the simulated gamma-ray spectra to
the experimental lineshapes. First, as discussed in Subsect. 3.2.3, the effect of feeding the 2+
1
state from above with the 932 keV gamma ray was determined to be negligible. Therefore the
final lifetime simulations for the 2+ → 0+ transition did not incorporate the feeder state.
gs
1
Furthermore, in accord with previous Köln/NSCL plunger lifetime measurements utilizing
gamma-ray lineshape simulations (see Ref. [22, 78, 94]), only the most sensitive spectral
features to lifetime effects- namely the two photopeaks and their relative centroid amplitudeswere included in the lifetime χ2 analysis. Thus from the simulated gamma-ray spectra, only a
region localized around the transition of interest was utilized for the lifetime determination.
This simplification facilitated the use of a simple polynomial background function to be
added atop the simulated gamma-ray lineshapes.

3.3.2

Electromagnetic Transition Rates in

18

C

With simulated plunger gamma-ray lineshapes at each of the five target-degrader distances,
the lifetimes of the observed electromagnetic transitions in 18 C were extracted. To do this,
the χ2 of the fit between the simulated and experimental spectra at various input lifetimes
was calculated within ROOT. The excited state lifetimes were extracted from the minimum
in χ2 as a function of simulated lifetime.
90

30°

Counts/8 keV

100

50

140°

100

50

1500

1600
Energy [keV]

1700

Figure 3.8: Example fit of experimental gamma-ray spectra (red) and best-fit linear background (gray) and gamma-ray lineshapes (blue) for the 1585 keV transition in 18 C for the
forward (top) and backward (bottom) rings of SeGA at a target-degrader distance of 1.5 mm.
2+ → 0+ Lifetime
gs
1
To determine the electric quadrupole transition rate of the 2+ state in 18 C, simulated gamma1
ray spectra were produced over a coarse lifetime grid. A distribution of χ2 values was obtained
from the fit of these simulated lineshapes to the forward and backward ring experimental
data at each distance. To properly account for the unique gamma-ray background present
in each ring and at each target-degrader distance, the coefficients of a linear background
were allowed to vary freely in the fit. The optimal set of background parameters were taken
from the simulation nearest the extracted χ2 minimum of this broad lifetime scan. Figure 3.8
demonstrates the fit to data (red) acheived by the simulation (blue) with the optimal linear
background parameters (gray) and best-fit lifetime (see below) for both the forward and
backward rings. With the background parameters fixed, a final lifetime scan was performed
in 1 ps increments around the broad lifetime scan minimum. A variable normalization fac91

1

2

200

Summed χ

2

35

25

140

15
80
5
20
15

20

25

30

0
Lifetime [ps]

2

4

6

Figure 3.9: χ2 minimization for the lifetime of the 2+ → 0+ (Panel 1) and 2+ feeder (Panel
gs
1
1
2) transitions. The summed χ2 (black points) was obtained from the unweighted sums of the
individual χ2 values from the fits to the forward (blue points) and backward (red points)
ring spectra at each target-degrader distance. The cubic polynomial fits (black lines) were
used to determine the χ2 minimum from which the lifetime was extracted. Despite the lack
of lifetime sensitivity exhibited by the backward ring in Panel 1, a complete and independent
error analysis for each ring yields consistent lifetime results.
tor accounted for the different statistics obtained in each ring and at each target-degrader
distance. A lifetime of τ = 22.4 ± 0.9(stat) ps was extracted from the χ2 minimization of
the summed individual χ2 values from the fits to the forward and backward ring spectra at
all five target-degrader distances. Panel 1 of Fig. 3.9 indicates the quality of the summed χ2
minimization; the minimum was obtained from the cubic polynomial fit to the summed χ2
data. The statistical error was taken to be the average half-width of this cubic polynomial
at a value of χ2 = χ2 + 1 as prescribed in Ref. [98].
min
Figure 3.10 illustrates the best fit of simulated lineshapes to the experimental gamma-ray
spectra for all five plunger distances. The fits were generated with the background parameters
fixed to their optimal values as described in the previous paragraph and the lifetime extracted

92

30°

0.0 mm 140°

100
50

50

30°

0.6 mm 140°

100

0.6 mm

100

50

Counts/8 keV

0.0 mm

100

50

30°

1.5 mm 140°

100
50

1.5 mm

100

50

30°

2.2 mm 140°

100
50

2.2 mm

100

50

30°

3.0 mm 140°

100
50

3.0 mm

100

50

1550

1600

1650

1700 1450

1500

1550

1600

Energy [keV]
Figure 3.10: Experimental gamma-ray spectra with statistical uncertainties (red) and simulated lineshapes (blue) for the 1585 keV 2+ → 0+ transition in 18 C for the forward (left
gs
1
column) and backward (right column) rings of SeGA and all five plunger target-degrader distances. The simulated lineshapes were generated with τ = 22.4 ps. Bins with experimental
error bars indicate the lifetime-sensitive region of the lineshapes where the χ2 was calculated.
from the summed χ2 minimum (Panel 1 of Fig. 3.9). The figure indicates the narrow lifetimesensitive region over which the fit was performed.
The most dominant source of systematic error in the reported measurement was an
uncertainty in the 18 C production yield ratio of one-proton knockout reactions on the target
and degrader (Ntar /Ndeg ). For target-degrader distances several times that of the average
93

flight distance of the 2+ excited state (d = γvτ ≈ 2 mm), the expectation is that all decays
occur upstream of the degrader. Hence the observation of a slow component in the Doppler
reconstructed gamma-ray spectrum at these so-called “infinite distances” would elucidate the
18 C production yield via one-proton knockout reactions on the degrader. Production data

at this large distance were not obtained due to experimental time constraints. This led to a
large uncertainty in the target-degrader yield ratio.
To deduce the ratio, a two dimensional χ2 hypersurface was generated from the fit of
experimental spectra in the forward ring to the corresponding simulated lineshapes obtained
with various input lifetimes and target-degrader yield ratios. A bivariate quadratic function
f was fit to this χ2 surface and both the minimum fmin and the ellipse tracing the function
at a value of fmin + 1 were obtained. Figure 3.11 provides more details. Projecting the
minimum and this ellipse onto the 18 C production yield ratio axis generated the range of
permissible ratios 2.15 ± 0.74. An additional constraint restricted the lower limit: assuming
the production data obtained at the 3 mm distance corresponded to the infinite distance, a
ratio of 2.09 ± 0.28 was obtained. Thus the actual ratio must be larger than 1.81 since this
distance is insufficient to completely remove the target contribution to the slow component
of the Doppler reconstructed gamma-ray spectrum. Therefore the final 18 C production yield
ratio was determined to be 2.15+0.74 .
−0.34
To ascertain the uncertainty of the lifetime measurement introduced by the poorly constrained 18 C production yield ratio, lifetime scans were performed with the ratio set to the
extracted upper and lower limits. From the minima of the resulting summed χ2 of the fits,
+2.2
the systematic error introduced by the uncertainty in this ratio was found to be τ−1.1 ps.

Deviations between the experimental and simulated momentum distribution were also found
to be an important source systematic error. Uncertainties in the simulated 18 C momentum
distribution impact the Doppler shift and thereby alter the simulated Doppler reconstructed
gamma-ray spectra. It was determined that a scant 2% change in the degrader thickness
94

24
32

23

28

22

24

Lifetime [ps]

21

20

20

1

1.5

2

2.5

3

19
18
17
16
15
0

0.5

1

1.5

2

2.5

3

3.5

18

C Production Yield Ratio

Figure 3.11: Ellipses tracing the fmin + 1 (black solid), fmin + 2 (gray solid), and fmin + 3
(black dashed) cross sections of the bivariate quadratic function fit to the two dimensional
χ2 hypersurface described in text. From the global minimum (square) and dashed vertical
traces, the 18 C production yield ratio was determined to be 2.15 ± 0.74 from the forward ring
(main plot) and 2.16 ± 0.62 from the backward ring (inset). The 18 C production yield ratio
was taken from the forward ring fit data as it yielded a slightly wider range of permissible
values and therefore a more conservative estimate of the uncertainty in the ratio.
was tolerable before disrupting the fit of the experimental and simulated dta spectra. Thus
another pair of lifetime scans were performed at these extrema. The resulting χ2 minima
indicated a systematic uncertainty of τ ± 1.1 ps. Other sources of error were found to be
negligible and thus the root sum of squares was adopted for the final systematic error.
Including both statistical and systematic errors, the lifetime of the 2+ → 0+ transition
gs
1
+2.5
in 18 C was found to be τ = 22.4 ± 0.9(stat)−1.6 (syst) ps. This is in agreement with [29],

which reported a lifetime of 18.9 ± 0.9(stat) ± 4.4(syst) ps.

95

2+ Feeder Lifetime
1
The plunger target and degrader thicknesses (∼ 1 mm) were optimized for the lifetime
measurement of the 2+ → 0+ transition in 18 C. Thus the setup was not ideal for measuring
1
1
lifetimes of only a few picoseconds as most de-excitations would occur in the plunger foils
rather than in vacuum. Indeed, the beam velocity required to align the 932 keV transition in
the Doppler reconstructed spectra of each SeGA ring was larger than that used to align the
1585 keV transition, which corresponded to decays in vacuum downstream of the target. In
addition, the one-proton knockout from 19 N preferentially populated the 2+ excited state by
1
a ratio of 4 : 1. Nonetheless, a lifetime upper limit was established for the 932 keV transition
which populated the 2+ state.
1
The combination of thick plunger foils and a fast de-excitation resulted in the majority of
gamma-ray decays proceeding within the same foil as the knockout reaction. Very few decays
occurred in vacuum and therefore the Doppler reconstructed lineshapes were qualitatively
the same regardless of target-degrader distance. Thus the experimental gamma-ray spectra
from all five distances were summed and the resulting effective 0 mm distance spectrum was
fit with simulated lineshapes also generated at a 0 mm target-degrader distance. Note that
the summing does not improve the signal-to-noise ratio of the data, but rather has the effect
of smoothing out statistical fluctuations in the background. This improved the accuracy of
the background polynomial fit functions.
The lineshape simulations of the 932 keV transition were generated in much the same
manner as those for the 1585 keV transition. Simulated spectra were produced over a course
lifetime grid and the parameters of a cubic polynomial background fit were allowed to freely
vary. The optimal background parameters were obtained from the simulation nearest the
extracted χ2 minimum of this broad lifetime scan. A final lifetime scan was then performed
and as before, a variable normalization factor accounted for the different statistics obtained
in each SeGA ring. The input target-degrader yield ratio of 2.15 from the analysis of the
96

1585 keV transition was used. The lifetime was determined to be τ = 3.2 ± 0.7(stat) ps from
the minimum of the summed χ2 obtained from the individual χ2 of the fits to the forward and
backward ring experimental spectra. This minimum was obtained from the cubic polynomial
fit to the χ2 data (Panel 2 of Fig. 3.9) and the statistical error was taken from the half-width
of the polynomial at a value of χ2 = χ2 + 1.
min
Figure 3.12 illustrates the resulting best fit of simulated lineshapes generated with this
lifetime to the summed experimental gamma-ray spectra. The asymmetric peaks primarily
arise from excited-state production on the tantalum degrader and the subsequent decay
before emerging into vacuum. The black arrows indicate the location of these diminutive
degrader components after Doppler reconstruction with the improper velocity corresponding
to the more prevalent decays within the target. As before, bins with error bars indicate the
narrow region over which the χ2 fit was performed.
It should be noted that the 0 mm experimental 932 keV gamma-ray lineshape in the
forward ring could be fit with two Gaussian components. These corresponded to excitedstate production and subsequent decay entirely within the target or the degrader. Even
with the low statistics, a production yield ratio of 3.5 ± 1.9 was obtained for this 2+ feeder
1
state. This is consistent with the value of 2.15 used to help generate Fig. 3.12, but given
the large uncertainty and also the general lack of experimental sensitivity to such short
lifetimes, the conservative approach of only pursuing a lifetime upper limit was adopted.
This limit was extracted from the χ2 minimum of the lifetime scans using the upper limit on
the excited-state production yield ratio of 5.4. Degrader contributions were thus minimized
within the simulation, which then required longer lifetimes to fit the experimental spectra.
In this manner, the conservative lifetime upper limit of the 932 keV transition in 18 C was
found to be τ < 4.5 ps.
The reported results from the 2+ → 0+ lifetime measurement and that of the 932 keV
gs
1
feeder transition are discussed in Ch. 5. There, the nuclear structure implications of the
97

30°

Counts/10 keV

600

400
850
800 140°

900

950

600
400
900

950

Energy [keV]

1000

Figure 3.12: Doppler reconstructed experimental gamma-ray spectra with statistical uncertainties (red) and simulated lineshapes (blue) for the 932 keV transition feeding the 2+ state
1
in 18 C for the forward (top) and backward (bottom) rings of SeGA. The simulated lineshapes were generated with τ = 3.2 ps. The black arrows indicate the centroid positions of
gamma-rays from knockout reactions and decays within the degrader. See text for details.
lifetime measurements are examined both within the context of other recent measurements
and in comparison to the results obtained by state-of-the-art ab initio no-core shell model
(NCSM) theoretical calculations. This comparison provides a vital benchmark for the theoretical tools used, specifically the truncation of unimportant basis states in the NCSM
calculation for this A = 18 system.

98

Chapter 4
Two-Proton Decay Lifetime of 19Mg
The mean lifetime measurement of the exotic two-proton decay 19 Mg → p + p + 17 Ne
following the one-neutron knockout reaction nat C(20 Mg, 19 Mg)X in inverse kinematics was
conducted at the National Superconducting Cyclotron Laboratory. For the measurement, the
Köln/NSCL plunger was modified by replacing the velocity degrader with a silicon doublesided strip detector (DSSD). This enabled elemental identification of both reaction and decay
residues1 emerging from the carbon plunger target. In principle, the lifetime is determined
from the varying relative intensity of the 19 Mg and 17 Ne energy loss peaks in the silicon
DSSD as a function of target-DSSD distance. The unambiguous identification of 19 Mg from
the unreacted 20 Mg secondary beam was facilitated by a software particle gate on the twoproton decay residue 17 Ne detected in the S800 focal plane.
This study was the first of its kind and therefore serves as a proof of concept for the
particle plunger variant of the Recoil Distance Method for picosecond lifetime measurements
along the proton dripline. The low counting statistics in the 19 Mg energy loss peak at the
1 The signature reaction and decay residues in this study were 19 Mg and 17 Ne, respectively;

the lifetime analysis was not sensitive to the proton decay remnants. However, to avoid
confusion of terms, “reaction residues” in this chapter shall henceforth refer to both 19 Mg
and 17 Ne. A distinction will be made where additional clarity is necessary.

99

0 mm target-DSSD distance indicated a lifetime much shorter than the literature value of
τ = 5.8 ± 2.2 ps [38]. It also reduced the analysis to lineshape fitting at a single distance in a
manner similar to the lifetime analysis of the feeder transition in 18 C as presented in Ch. 3.
To properly fit the full energy loss lineshape, the contribution of the nonresonant breakup
20 Mg → p + p + n + 17 Ne to the prominent 17 Ne energy loss peak must be quantified.

However, the production mechanism ratio, given by
nat C(20 Mg, 17 Ne)X
Rp = nat 20
,
C( Mg, 19 Mg)X + nat C(20 Mg, 17 Ne)X

(4.1)

could not be constrained by the data, as the resulting 17 Ne DSSD energy loss signal is
indistinguishable regardless of origin.
Hence, the mean lifetime of the two-proton emitter 19 Mg was determined as a function
of the production mechanism ratio in the range Rp = [0.05, 0.95]. As Rp was increased, the
best fits were achieved with a monotonically increasing lifetime. In particular, within the
reasonable range Rp = [0.15, 0.85], the best-fit lifetimes fell in the range τ = [0.39, 2.44] ps.
Technical details and calibration procedures for the Köln/NSCL particle plunger and supporting experimental devices were discussed in depth in Ch. 2. The details regarding the
measurement, data analysis, and experimental results are presented here.

4.1

Experimental Considerations

Similar to the procedure reported in Ch. 3, several types of in-beam data were collected during the experiment to facilitate the proper characterization of the detector systems. These
data also helped constrain the many simulation parameters necessary to generate energy loss
DSSD lineshapes for the lifetime analysis. The experiment was not designed to be analyzed
via lineshape simulations; therefore without certain data some simulation parameters could

100

not be optimally constrained. For instance, without target-only production data, the effective target and detector thicknesses were fixed to the manufacturers’ specifications. However,
as indicated by the Bethe formula of Eq. 2.4, the energy deposited in the DSSD is inversely
proportional to the beam energy. Hence a good agreement between the simulated and measured reaction residue momentum distribution is vital for accurate energy loss simulations.
A description of the steps necessary to ensure this follows.
• The unreacted 20 Mg secondary beam was delivered through the full particle plunger
and swept across the focal plane to calibrate the S800 CRDCs. In total, 3 CRDC
mask calibrations were taken during the experiment. Gains and offsets were extracted
to correlate the raw detector signals with the well defined (x, y) coordinates of the
mask slits and holes. From the three sets of calibration parameters, a run-by-run set of
parameters was extracted as described within Subsect. 2.3.3. The resulting calibrated
CRDC positions and angles (xf p , yf p , af p , and bf p ) were used for the trajectory
reconstruction of the reaction residues back to the S800 target position.
• In addition, the unreacted beam data was used to calibrate the 16 channels of the
S800 ionization chamber (IC). The raw energy loss signals in each channel were gain
matched to a representative channel such that from the calibrated signals, an average
energy deposit could be calculated on an event-by-event basis.
• The data used for the IC calibration were also used to help constrain the simulation
parameters describing the secondary beam emittance and straggling through the target
and detector. This was done by constructing simulated S800 target parameter spectra
(ata , bta , yta , and dta ) and fitting them to their experimental counterparts obtained
from unreacted beam trajectory reconstruction.
• Finally, the S800 magnetic rigidity was tuned to center the two-proton decay remnant 17 Ne in the focal plane. From the calibrated focal plane positions and magnetic
101

field settings, the experimental S800 target spectra were used to constrain the oneneutron knockout reaction kinematics. Simulated S800 target spectra were fit to their
experimental counterparts and the extracted reaction parameters were used for both
reactions on the target and silicon detector, as well as both 17 Ne production mechanisms. The lack of target-only reacted beam production data necessitated the former
simplification. The latter was necessary from the indistinguishable origin of 17 Ne from
the breakup of 20 Mg or the two-proton decay of 19 Mg.
The quality of the fit between the experimental S800 target spectra and fully constructed
reaction residue simulation is illustrated in Fig. 4.5 within Subsect. 4.3.1. Despite an asymmetric beam profile in the non-dispersive direction, the excellent agreement between the
simulated and experimental momentum distribution spectra (dta ) was paramount to producing accurate simulated energy loss lineshapes.

4.2

Experimental Details

The remaining production data was devoted to collecting DSSD energy loss signals in coincidence with reaction residues in the S800 focal plane. The particle plunger Inchworm Control
System was kept off until the target-DSSD distance needed to be changed to reduce electronic noise in the DSSD. The S800 was operated in focused mode and was tuned to accept
the decay remnant 17 Ne. The data acquisition was triggered by the detection of a particle
(reaction residue or beam ion within the S800 momentum acceptance) in the E1 scintillator. Several other reaction residues with similar momentum-to-charge ratios were within the
acceptance cut. One of these, 18 Ne, proved particularly useful in the analysis as will be discussed in Sect. 4.3.1. A discussion of the properties of the radioactive 20 Mg secondary beam,
subsequent reaction residues, and observed DSSD energy loss signals follows immediately.

102

4.2.1

Radioactive Secondary Beam

A 24 Mg primary beam was accelerated by the Coupled Cyclotron Facility at NSCL and fragmented on a thick 9 Be production target. A 20 Mg radioactive secondary beam was selected,
purified, and delivered to the Köln/NSCL particle plunger target. The method and devices
used were discussed in Sect. 2.2.
The 24 Mg+12 primary beam was accelerated to 170 MeV/u (≈ 0.53% the speed of light)
by the coupled K500 and K1200 cyclotrons and delivered to a 1081 mg/cm2 (≈ 5.8 mm
thick) 9 Be production target. The settings of the A1900 were adjusted to provide an adequate balance between purity and intensity of the 20 Mg secondary beam. In particular, the
momentum acceptance was restricted to 0.5% with momentum apertures installed at Image
1 and 2. A 1050 mg/cm2 energy degrading Al wedge at Image 2 was used for additional isotopic separation. The desired 20 Mg secondary beam emerged from the A1900 with a purity
of roughly 36%. Three significant secondary beam contaminants- 16 O, 17 F, and 18 Ne- were
distinguished from the 20 Mg fragment by their time of flight between two thin plastic scintillators installed at the A1900 focal plane and the S800 object position. The time of flight of a
10 C contaminant could not be resolved from 20 Mg since they have the same mass-to-charge

ratio and thus the same time of flight. Instead, software particle gates on their unique energy
loss in the S800 ionization chamber provided the separation. The properties of the secondary
beam are listed in Tab. 4.1 where the subscripts refer to the four A1900 dipole magnets.
The desired 20 Mg radioactive secondary beam emerged from the A1900 with an energy
of 92.65 MeV/u alongside the four contaminants and was delivered to the particle plunger at
the S800 target position. The unreacted secondary beam time of flight from the A1900 focal
plane to the S800 object position is plotted in Fig. 4.1. Note the clean separation between the
16 O, 17 F, 18 Ne, and 20 Mg time of flight peaks which facilitated offline separation during the

analysis. After setting a software gate on the 20 Mg time of flight peak, the 10 C component
was removed by particle gating in the S800 (see Subsect. 4.2.2).
103

Table 4.1: 20 Mg secondary cocktail beam characteristics. Bρ and ∆p/p were taken from the
experimental beam line savesets provided by the A1900 Fragment Separator Group. The
beam energy was calculated from Bρ3,4 . The beam composition percentages were calculated
by integrating the appropriate time of flight peaks in Fig. 4.1. The 20 Mg average rate and
transmission to the S800 were calculated from run-by-run scaler information.
Bρ1,2
2.85500 T · m
Bρ3,4
2.36871 T · m
Final Energy
92.65 MeV/u
∆p/p
0.5%
10 C Contaminant
4.9%
16 O Contaminant
28.6%
17 F Contaminant
10.6%
18 Ne Contaminant
18.7%
20 Mg Total Purity
36.2%
Average Rate
2500 pps
Transmission to S800
65%

1600
1400

20

Counts

1200
1000

16

Mg

O

800
18

600
400

17

Ne

F
10

200
0

2400

2500
2600
2700
Time of Flight [Arb.]

C

2800

Figure 4.1: Identification of the secondary beam components by their time of flight between
two thin plastic scintillators at the A1900 focal plane and object position of the S800. To
resolve the 10 C (red) and 20 Mg (blue) components of the fourth peak, the time of flight data
was sorted from appropriately particle-gated events in the S800. Increasing time is from right
to left along the abscissa.
104

The plastic time of flight scintillators were also used to monitor the beam rates. The
secondary beam emerged from the A1900 with an average rate of 6800 pps as measured at
the A1900 focal plane. Transmission losses en route to the S800 were measured by the ratio
of beam rates at the A1900 focal plane and S800 object position to be nearly 35%.

4.2.2

Outgoing Reaction Residues

The S800 Magnetic Spectrometer provided event-by-event particle identification and data
acquisition triggering. The 20 Mg radioactive secondary beam was focused at the object position and then delivered via the analysis line as a focused beamspot onto the Köln/NSCL
particle plunger nat C 110 mg/cm2 (≈ 0.5 mm thick) target. States in 19 Mg were populated
by the one neutron knockout reaction and subsequently decayed by two-proton emission2
after a distance governed by the lifetime and beam velocity to available states in 17 Ne. A
similar one-neutron knockout reaction of a 20 Mg secondary beam on a thick beryllium target
was used in the 19 Mg lifetime measurement of Ref. [38]. The S800 magnetic rigidity was set
to Bρ = 2.29402 T · m in order to center the 17 Ne reaction residues in the focal plane and
reject reaction residues from the secondary beam contaminants.
Various reaction residues, the 17 Ne two-proton decay remnants, and the high-momentum
tail of unreacted 20 Mg fell within the momentum acceptance of the S800. Each component
was clearly resolved on an event-by-event basis by their energy loss through the S800 ionization chamber and time of flight between the object position and E1 focal plane scintillators.
2 There is no experimental differentiation between proton emission from the ground state

(via direct two-proton decay) and excited states (via sequential one-proton decays through
a resonance in the (Z − 1) intermediary 18 Na). Following Ref. [38], excited state decays will
have a lifetime far below the picosecond sensitivity we report here and thus these events
are indistinguishable from the nonresonant breakup of 20 Mg. Therefore the contribution of
these two processes to the energy loss data are lumped together within the 17 Ne production
mechanism ratio Rp . Hence, the term “two-proton decay” labels only ground state proton
emission of 19 Mg and variants of “direct production of 17 Ne” are taken to mean both the
breakup of 20 Mg and prompt 19 Mg excited state decay.
105

Energy Loss [Arb. Units]

1000

21

Mg

20

20

Mg

Na

800

18

Ne
17

Ne

600
14

O

400
10

C

200
2300

2350
2400
2450
Time of Flight [Arb. Units]

2500

Figure 4.2: Identification of the outgoing reaction residues and decay remnants from the
incoming 20 Mg secondary beam by time of flight and energy loss information from the S800
Spectrometer. Increasing time is from the right to left along the abscissa.
Figure 4.2 illustrates this particle identification spectrum. The quality of isotopic separation
indicated in the figure was achieved with flight-path corrected time of flight and calibrated IC
energy loss signals using the focal plane angles and positions determined from the calibrated
CRDCs (as discussed within Subsect. 2.3.3). The population of several reaction channels
close in mass to 20 Mg serves as indirect evidence that the one-neutron knockout reaction
producing 19 Mg likely occurred.

4.2.3

Observed DSSD Energy Loss Spectra and Analysis

The energy loss profile of each beam particle emerging from the particle plunger target was
detected in the 69 mg/cm2 (≈ 0.3 mm thick) silicon double-sided strip detector mounted

106

downstream of the target. Each face of the DSSD was electronically segmented into 16 strips;
the gain-matched front strip spectra were summed to produce run-by-run ∆E lineshapes.
More details were provided within Subsect. 2.3.2.
DSSD energy loss signals were recorded in coincidence with reaction residues detected
in the S800 E1 scintillator. This trigger requirement suppressed background energy loss signals from contaminant reaction residues. Further background reduction was achieved during
analysis by software particle gates on incoming 20 Mg and outgoing 17 Ne.3 The incremental
enhancement in the signal-to-noise (S:N ) ratio is demonstrated in Panel 1 of Fig. 4.3 where
∆E lineshapes are shown for ungated data (black dashes), data gated on incoming 20 Mg
(solid gray), and data gated on both incoming 20 Mg and outgoing 17 Ne (solid black). This
final set of gates removed all DSSD signals not directly related to either the two-proton decay
19 Mg → p + p + 17 Ne or the direct dissociation 20 Mg → p + p + n + 17 Ne.
17 Ne particle-gated DSSD energy loss spectra were collected at five target-DSSD distances

of 0.0, 100.0, 200.0, 500.0, and 1000.0 µm. The combination of reactions on the target and
detector for both direct production of 19 Mg and 17 Ne and also two-proton decays in the
target and detector obfuscated the origin of the ∆E components which together comprise
the full lineshape plotted in Panel 2 of Fig. 4.3. Events where only 17 Ne traversed the
detector- corresponding to the direct production of 17 Ne in the target or the production
and decay of 19 Mg in the target- generated the full energy 17 Ne peak. The small 19 Mg
peak originated from magnesium energy loss through the entire detector- corresponding to
19 Mg production in the target (or extreme front/back of the detector) and decay after the

detector. The broad plateau between the two peaks was generated by events with mixed
neon and magnesium energy deposits. Neither the experimental data or previous studies
addressed the relative importance of each of these components; thus the simulations discussed
3 Recall from Subsect. 4.2.1 that a time of flight gate on the incoming 20 Mg peak necessarily includes contributions from 10 C. The subsequent S800 particle gate on 17 Ne precludes
any contributions from 10 C in the final DSSD energy loss spectra.

107

600
1

10000

17

2
Ne

Counts/16 channels

500
1000

400

300

100

200
10
19

100

1
750

1500

2250

0
3000 1250
Channel

1750

Mg

2250

2750

Figure 4.3: Energy loss spectra constructed from summing the front (upstream) gain-matched
strips of the DSSD. The data were collected at the 0 mm target-DSSD distance. The improvement in S:N of the DSSD spectra via incoming 20 Mg particle gates (solid gray) and
incoming 20 Mg and outgoing 17 Ne particle gates (solid black) in the analysis over the ungated spectrum (black dashes) is demonstrated in Panel 1. Note the logarithmic scale of the
ordinate. Panel 2 replots the DSSD ∆E spectrum from the fully-gated data subset linearly.
below were constructed to probe the three dimensional parameter space of lifetime, 17 Ne
production mechanism ratio Rp , and target-detector yield ratio Rσ . This is the ratio of
reactions occurring in the carbon target compared to the silicon detector. The definition is
equivalent to the target-degrader yield ratio of Ch. 3, except here the reactions are both the
one neutron knockout producing 19 Mg and the direct dissociation yielding 17 Ne.

4.3

Lifetime Analysis

The 17 Ne particle-gated DSSD energy loss spectra for three of the five target-DSSD distances
are plotted in Fig. 4.4. The paucity of counts in the 19 Mg peak even at 0 mm strongly
108

600

0 mm

0 mm

0.1

0.5

1900

1900

1900

125
17

Ne

Counts/16 channels

450

100

75
300
50
150
19

Mg

25

0
1250

1750

2250
Channel

Figure 4.4: 17 Ne particle-gated silicon DSSD ∆E spectra. The full spectrum for data taken
at the 0 mm target-DSSD distance is plotted in the leftmost column. To the right, the region
surrounding the 19 Mg energy loss peak is examined for 0 mm (red), 0.1 mm (blue), and
0.5 mm (green) distances.
indicates a very short two-proton decay lifetime. This precluded the lifetime analysis via
counting 19 Mg as a function of target-detector distance. Instead, a modification of the MonteCarlo simulation software from the 18 C lifetime analysis was used.
It was decided to use only the 0 mm lineshape data for the lifetime analysis due to the
slight enhancement of the 19 Mg energy loss peak and the near equivalence of the energy loss
lineshapes for the remaining distances. This simplification also mitigated the effects of drifting DSSD calibrations due to radiation damage. Thus, 0 mm simulations were constructed
within the three dimensional parameter space of τ , Rp , and Rσ . At each step the χ2 of the
fit to the experimental DSSD spectrum was calculated by ROOT; the set of simulations at a
single Rp exhibit a clear minimum in the plot of χ2 against lifetime and minimal dependence

109

upon Rσ . Details of the lineshape simulations and the lifetime analysis are presented here.

4.3.1

Specifics of Geant4/ROOT Simulations

Similar to the simulation discussions in Ch. 3, the ∆E lineshape simulations required an
accurate parameterization of the experiment to properly describe the experimental geometry
and silicon DSSD response, the secondary beam emittance and knockout reaction kinematics,
and the two-proton decay process. The lack of certain production data, such as target-only
unreacted beam and detector-only reacted beam runs, prevented the data-based constraint
of several simulated parameters; examples were provided in Sect. 4.1. As noted there, the
production of realistic ∆E lineshape simulations hinges upon accurately reproducing the
reaction residue dta momentum distribution. Trajectory variations through the detector for
a given residue also impact the energy loss simulations. Yet this trajectory spread is restricted
by the S800 angular acceptance (7◦ and 10◦ in the dispersive and non-dispersive directions)
to deviations less than 0.4%. Thus, the ∆E depends much more strongly upon the reaction
residue momentum distribution. The steps towards obtaining accurate DSSD ∆E lineshapes
are outlined below.

Beam Characteristics and Reaction Kinematics
The simulated 20 Mg secondary beam emittance and S800 detector response was parameterized by fitting the simulated ata , bta , yta , and dta S800 target spectra to their experimental
counterparts from unreacted secondary beam data obtained prior to the installation of the
particle plunger in the beam line. The energy and angular straggling introduced by the
carbon target and silicon detector were constrained by similar fits to unreacted beam data
obtained after the particle plunger installation. The absence of target-only data prevented
the deconvolution of the straggling effects introduced separately by the target and detector.
Care was taken to accurately constrain the simulation parameters affecting the momen110

tum distribution. First, the plunger target and detector thicknesses were fixed to the manufacturers’ specifications. Next, the longitudinal momentum transfer from the one-neutron
knockout reaction was tuned to achieve agreement between the simulated and experimental dta spectra. Note that in principle, the competing reactions 19 Mg → p + p + 17 Ne and
20 Mg → p + p + n + 17 Ne on both the target and detector necessitate four separate parame-

terizations as they result in different 17 Ne momentum distributions and hence unique DSSD
energy deposits. Yet for reasons discussed within Sect. 4.1, only an effective target-detector
parameterization was implemented for each simulated Rp to properly fit the DSSD ∆E and
dta spectra simultaneously. An example of the fit quality between the experimental S800
reaction residue target spectra and their simulated counterparts at Rp = 0.55 is illustrated
in Fig. 4.5. Beam quality issues at the particle plunger target in the non-dispersive direction
(manifest in the bta and yta spectra) were not simulated as this had negligible impact on the
simulated DSSD lineshapes.

Silicon DSSD Response
Next, the simulated energy loss spectra (binned in MeV) were calibrated to the experimental
spectra (binned in channels, and thus indicating relative energy loss in the detector) with
unreacted beam data such that meaningful comparisons could be made. From a Gaussian
fit to each unreacted beam energy loss signature, the simulated and experimental energy
loss centroids were plotted against one another and from the least-squares linear fit, the
calibration gain and offset parameters a = 0.0315 MeV/channel and b = 2.11 MeV were
obtained, respectively.
For simplicity, the simulation regarded the DSSD as a passive silicon degrader. As such,
the simulated detector response was strictly a function of the momentum and trajectory
spread of the ionizing radiation. Therefore detector-specific sources of energy degradation
[99–101], such as incomplete charge collection due to charge carrier trapping or recombination
111

500
Counts/5 mrad

Counts/5 mrad

800
600
400
200
0

-100

-50
0
50
Ata [mrad]

200
100
0
-100 -50
140
120
100
80
60
40
20
0

0
50 100 150
Bta [mrad]

Counts/0.1%

400
Counts

300

100

500

300
200
100
0

400

0

10

20
Yta [mm]

30

40

-5

0
Dta [%]

5

Figure 4.5: Fit of simulated (blue) and experimental (red) 17 Ne particle-gated S800 reaction
residue target spectra. The angular distributions in the dispersive (ata ) and non-dispersive
(bta ) directions, spread in non-dispersive position (yta ), and kinetic energy deviation from
that corresponding to a central trajectory through the S800 (dta ) are obtained from inverse
map trajectory reconstruction. See discussion within Subsect. 2.3.3 for more details.
effects, energy straggling introduced in the detector dead layers, and electronic noise were
not explicitly accounted for. In addition, the energy loss lineshapes from reaction residues
exhibited a right-skewed asymmetry. The underlying cause of this is a matter for future
examination; it is likely to be a combination of correlated background events (discussed
below) and lifetime effects.
Thus both effects were accounted for to produce proper lineshape fits. A detector response function was incorporated into the simulation based upon a general peak shape from
the DAMM (Display, Analysis and Manipulation Module) software package [102]. On an
event-by-event basis, the simulated raw energy loss Eraw (based solely upon momentum and

112

trajectory variations through the detector) was taken as the centroid of a unique lineshape
probability distribution function P (x) of the form

 √ 1 e−(x−x0 )2 /2σ 2

 2πσ 2
P (x) =
(x−x0 )
−(x−x0 )2 / 2σ 2 1+

σ
 √1

e
2πσ 2

for x < x0 = Eraw
(4.2)
for x ≥ x0 = Eraw .

A random number was then generated based upon the above probability distribution function
and this value was passed as the simulated energy deposit for a single event. Both the
standard deviation σ and the strength of the asymmetric high-energy tail

were adjusted

to properly fit the experimental DSSD lineshape. Figure 4.6 demonstrates the fit using this
detector response implementation for unreacted beam energy loss data. Reacted beam ∆E
fits are provided below. It should be noted that for unreacted beam data, the strength of
the right-skewed parameter

was found to be very small compared to that required to fit

reacted beam data (≈ 5%) and the width σ was only 80% that necessary for reacted beam
data fits.

DSSD Lineshape Generation
Having constrained the 20 Mg secondary beam properties, parameterized the reaction kinematics, and modeled the silicon detector response to ionizing radiation, simulated reaction
residue energy loss spectra were generated. This required two functions to adequately describe
the experimental background shape and also the variation of the three previously mentioned
parameters: the 19 Mg two-proton decay mean lifetime, the 17 Ne production mechanism ratio
Rp , and the target-detector yield ratio Rσ . Each shall be discussed in turn.
There were two distinct types of background in the ∆E spectra. The low energy tails
extending from the full energy deposit peaks in Fig. 4.3, 4.4, and 4.6 are well understood
as a result of incomplete charge collection within the detector [103], variations in dead

113

Counts/0.252 MeV

1500

16

O

20

Mg

1000
18

Ne

17

F

500

0

30

40

50
60
Energy Loss [MeV]

70

Figure 4.6: Experimental (dashed) and simulated (solid) DSSD energy loss lineshapes for
the four dominant secondary beam components. Each labeled ∆E peak is from data gated
on both the incoming and outgoing labeled species; the resulting four individual histograms
were added together to produce a single spectrum.
layer thickness [99], and random background events. As the figures illustrate, this is a small
effect (< 2% for the 20 Mg peak in Fig. 4.6) and therefore a simple constant background
was employed to account for these events. Future improvements will certainly involve more
accurate reproductions of this shape; in the present analysis, more effort was dedicated to
modeling the detector response and the second background type.
The origin of the high energy background events (i.e. the events in channels beyond 2250
in Fig. 4.4) in the incoming 20 Mg and outgoing 17 Ne particle-gated DSSD energy loss spectra
is a subject for future analysis. The events cannot originate from Z > 12 contaminants- this
possibility is entirely excluded by the particle gating condition used to generate the spectra.
Nor can they be the coincident detection of the full set of decay daughters 17 Ne+p+p as the
114

proton energies are similar to the reaction residue energies (∼ 90 MeV) and thus the energy
deposited by each is insufficient (< 1 MeV). A possible candidate may be the simultaneous
detection of reaction residues in coincidence with dislodged target or detector nuclei.
For the present analysis, a phenomenological fit to these background events was performed. The functional form f of the background for channel x was chosen to be

f (x) = h · arctan

x − x0
a

exp −

x − x0
b

.

(4.3)

The arctangent function was chosen as it produces a rounded step function shape while
the decaying exponential tail fits the background trend at high energies. Four parameters
controlled the initial height h, the offset x0 , the roundedness of the step a, and the exponential
decay constant b to best fit the background.
The four background parameters were adjusted to fit the high energy background events
in the 0 mm DSSD energy loss spectrum. However, the 17 Ne particle-gated ∆E spectrum
includes the finite lifetime effects of 19 Mg. To exclude these effects from contaminating the
fit of the background parameters, a 18 Ne particle-gated DSSD energy loss spectrum was
used instead. The 18 Ne reaction channel was populated either directly from the two-proton
knockout of the secondary beam by 20 Mg → p + p + 18 Ne or from the one-proton decay of
19 Na which immediately proceeds from the one-proton knockout reaction 20 Mg → p + 19 Na.

The 19 Na ground state is unbound to proton emission by roughly 320 keV [104] and has a 1p
decay lifetime on the order of femtoseconds from Shell Model calculations of the proton decay
width [105]. Both production mechanisms result in the immediate production of 18 Ne and
the resulting DSSD spectrum exhibits no lifetime effect within the experimental sensitivity.
Thus the four high energy background parameters were fixed to fit the high energy events
in the 18 Ne particle-gated spectrum. It was decided to offset f (x) to the center of the 18 Ne
energy loss peak, although both this choice and that of the roundedness a are somewhat

115

25

300

Counts/0.504 MeV

20
15
10

200
5
0

75

80

85

100

0

50

60
70
Energy Loss [MeV]

80

Figure 4.7: 18 Ne particle-gated silicon DSSD ∆E spectrum at a target-detector distance of
0 mm (red) with statistical uncertainties. The simulated lineshape (blue) was constructed
assuming a 0 ps one-proton decay lifetime of 19 Na and includes contributions from the high
energy background function f (x) (dark gray) and the constant background (light gray). The
inset displays the high-energy background events, the constant background, and f (x) of
Eq. 4.3 with parameters h = 25, x0 = 48.4 MeV, a = 10 MeV, and b = 35 MeV.
arbitrary. The implementation of this background is illustrated in the inset of Fig. 4.7. For
the 19 Mg lifetime simulations, the roundedness and exponential decay constant parameters
(a and b) were held fixed while the offset x0 was adjusted to the center of the 17 Ne peak and
the height h was scaled by the ratio of counts in the 17 Ne and 18 Ne particle-gated spectra.
Next, the yield ratio Rσ of the one-neutron knockout reaction 20 Mg → n + 19 Mg on the
carbon target and silicon DSSD was constrained. Cross section ratio estimates within Lise++
using the fragmentation parameterizations EPAX 2.15 [106] and Abration Ablation [107,108]
gave Rσ = 2.9 and 3.3, respectively. In addition, cross section calculations at beam energies

116

of 80, 85, and 90 MeV/u produced a similar ratio of 3.2 [109]. Furthermore, both EPAX
2.15 and Abration Ablation calculations gave the exact same Rσ values as above for the
similar nucleon knockout reaction 20 Mg → p + 19 Na. Simulations of the 19 Na 1p decay
with τ = 0 ps were fit to the 18 Ne particle-gated ∆E lineshapes. The best fits were in the
range Rσ = [3.03, 3.43]. Therefore several target-detector yield ratios within this range, well
constrained by both calculations and data, were used for 19 Mg lifetime analysis.
The single largest uncertainty in the measurement was the 17 Ne production mechanism
ratio Rp of Eq. 4.1. Simple Lise++ calculations produced ratios of nearly 0.06 and 0.25
using Abration Ablation and EPAX 2.15 parameterizations, respectively. This ratio could
not be constrained with the experimental data or from previous work in the literature. It
was therefore decided to extract the two-proton decay lifetime as a function of Rp . A large
simulated production mechanism ratio, corresponding to the majority of 17 Ne production
from the dissociation of 20 Mg, required a longer 19 Mg lifetime to properly fit the energy loss
data. Conversely, a small ratio, corresponding to the majority of 17 Ne production from the
two-proton decay of 19 Mg, required a short lifetime to fit the data.

4.3.2

Two-Proton Decay Lifetime of

19

Mg

DSSD energy loss spectra were simulated with all the parameters modeling the secondary
beam emittance, reaction kinematics, silicon detector response to ionizing radiation, and
the high energy background function of Eq. 4.3 constrained as discussed in the previous
subsection. Lineshapes were generated for the 0 mm distance over a range of lifetimes for
each input production mechanism ratio and target-detector yield ratio. The reduced χ2 (χ2 )
ν
of the fit between the simulated and experimental energy loss spectra was calculated within
ROOT. For each Rp -Rσ pair, the mean lifetime of the two-proton emitter 19 Mg was extracted
from the minimum of a polynomial fit to the χ2 lifetime distribution.
ν
Energy loss lineshape simulations were performed over a range of 19 Mg lifetimes for
117

Table 4.2: 19 Mg lifetime (picoseconds) dependence on Rp and Rσ . See text for details.

Rσ

0.05 0.15
3.03
3.23
3.43

0.25

Rp
0.35 0.45

0.55

0.65

0.75

0.85

0.95

0.32 0.35
0.35 0.39
0.39 0.42

0.41
0.46
0.50

0.43 0.54 0.68
0.49 0.60 0.76
0.54 0.66 0.82

0.88
0.96
1.04

1.24
1.35
1.45

2.32
2.44
2.59

7.52
7.26
7.37

ten 17 Ne production mechanism ratios from 0.05 to 0.95 in increments of 0.1. This process
was repeated three times at target-detector yield ratios of 3.03, 3.23, and 3.43. A variable
normalization factor was implemented to account for statistical differences between the simulation and the data. Each simulation used the same functions to describe the experimental
background and detector response, but as mentioned previously, each Rp required slightly
different parameterizations of the longitudinal momentum distribution to simultaneously fit
the dta and ∆E spectra. Distributions of χ2 values from the fits were constructed for all
ν
thirty Rp -Rσ pairs and fit with cubic polynomials. The best-fit lifetimes were then extracted
from the minima. Table 4.2 summarizes the results. With the exception of the final column,
smaller yield ratios, which correspond to more reactions on the silicon detector, result in
shorter best-fit lifetimes as expected. Note that for a given Rp , the spread of lifetimes from
the three target-detector yield ratios is insignificant compared to the overall uncertainty
introduced by the unconstrained 17 Ne production mechanism ratio.
The three panels of Fig. 4.8 indicate the sensitivity of nine sample fits with production
mechanism ratios of Rp = 0.35, 0.55, and 0.75 and yield ratios of Rσ = 3.03, 3.23, and
3.43. The lifetime sensitivity is enhanced for small values of Rp as expected; with limited
contributions from the breakup 20 Mg → p + p + n + 17 Ne, the intensity ratio of the small
19 Mg and prominent 17 Ne energy loss peaks is heavily dependent upon the two-proton decay

lifetime. Statistical errors were taken from the cubic polynomials fit to the χ2 values in the
ν
same manner as in Ch. 3. The main contribution to the systematic error was taken to be

118

6
0.35

0.55

0.75

Reduced χ

2

5

4

3

2

1
1

2

1
Lifetime [ps]

2

1

2

Figure 4.8: χ2 from the fit of simulated ∆E lineshapes to 0 mm experimental data for 17 Ne
ν
production mechanism ratios of 0.35, 0.55, and 0.75 and target-detector yield ratios of 3.03
(red), 3.23 (black), and 3.43 (blue). Table 4.2 lists the best-fit 19 Mg lifetimes, extracted from
the minima of the cubic polynomial fits.
the average spread of best-fit lifetimes from simulations at the upper (3.43) and lower (3.03)
limits of Rσ . Other sources of systematic error are discussed within Sect. 5.2 of Ch. 5.
The four plots in Fig. 4.9 illustrate the sensitivity of the simulated ∆E lineshape features
to small changes in lifetime. The χ2 was calculated by ROOT from the fit over the entire
ν
range of the simulated lineshape (blue). The superior fit of τ = 0.76 ps simulation is clear;
this lifetime value was extracted from the minimum of the black curve in the middle panel
of Fig. 4.8.
Figure 4.10 plots the Rσ = 3.23 data from Tab. 4.2 to illuminate the general lifetime
trend as a function of an increasing Rp (red points). The sum of the statistical and total
systematic uncertainties discussed above was taken as the error in the best-fit lifetime at each

119

0.1 ps

0.5 ps

400

400

300

Counts/0.504 MeV

500

500

300

200

200

100

100

0.76 ps

1.25 ps

500

500

400

400

300

300

200

200

100

100

40

50

60

70

40

50

60

70

Energy [MeV]
Figure 4.9: 17 Ne particle-gated silicon energy loss spectra at a target-detector distance of
0 mm. The fit between experimental data (red, with statistical uncertainties) and simulated
lineshapes (blue) produced with Rp = 0.55 and Rσ = 3.23 is plotted for four simulated
lifetimes of τ = 0.1, 0.5, 0.76, and 1.25 ps.
Rp and is included in the figure. No minimum was obtained for the χ2 fits of the Rp = 0.95
ν
lifetime scan. Rather, the distribution flattened out. For Rσ = 3.23 this occurred at 7.26 ps
and therefore this value has been taken as a lifetime lower limit at Rp = 0.95. Within the
reasonable range of 17 Ne production mechanism ratios 0.15 to 0.85 (see discussion in Ch. 5),
the extracted lifetimes span the range from 0.39(9) ps to 2.44(54) ps.
The 19 Mg two-proton decay lifetime results of Ref. [38] are also plotted in Fig. 4.10 for
comparison (black points). The quoted value of τ = 5.8 ± 2.2 ps at Rp = 0.25 is given, along
with lifetimes of 4.5 ± 1.5 ps and 9+3 ps derived at Rp = 0.0 and 0.65, respectively, from the
−6
same work. (The Rp = 0.0 data has been shifted off the ordinate for clarity.) The obvious
disparity between the two measurements can only be resolved at the two-sigma level for the
120

12

Best Fit Lifetime [ps]

10
8
6
4
2
0
0

0.2

0.4

0.6

0.8

1

17

Ne Production Mechanism Ratio

Figure 4.10: Best-fit 19 Mg ground state two-proton decay lifetime as a function of increasing
17 Ne production via the dissociation 20 Mg → p + p + n + 17 Ne (red points). The error bars
indicate the sum of the statistical and systematic uncertainties. The literature values [38]
are plotted for comparison (black points). See text for details.
Rp = 0.65 result or if the two experimental techniques probed different 17 Ne production
ratios.
Plotting the χ2 of the lineshape fits to the experimental energy loss spectrum for all
ν
simulated Rp and lifetime values results in a two dimensional χ2 hypersurface. Figure 4.11
ν
illustrates this hypersurface from scans with Rσ = 3.23. The plot indicates the χ2 valley,
ν
representing the extracted lifetimes given in the second row of Tab. 4.2, trends towards larger
lifetimes with increasing production of 17 Ne via the direct breakup of 20 Mg. The same trend
was depicted in Fig. 4.10; here, the lack of lifetime sensitivity at Rp = 0.95 is clear.
The lifetime results presented here for the two-proton emitter 19 Mg are discussed further

121

Reduced χ2

5
4
3
2
1
0
16.0

1
0.8

8.0
4.5
Life 2.5
tim
e [p1.25
0.1
s]
0.01

0.6
0.4
0.2

Rp

0

Figure 4.11: χ2 of the energy loss lineshape fits to the 0 mm experimental data from various
ν
input τ and Rp values at a fixed target-detector yield ratio of 3.23. For display purposes,
the lifetime axis is non-linear. The bins- each corresponding to a single lifetime simulation
point- are of equal width regardless of the spacing between consecutive simulated points.
in Ch. 5. The permissible range of lifetimes determined by this work is inconsistent with
the result reported in Ref. [38], but lends credence to this proof of concept study. Within
this context, a brief prospectus outlining opportunities for future study and improvements
of charged-particle lifetime measurements at NSCL with the particle plunger shall be given.

122

Chapter 5
Discussion
5.1

18

C Commentary

The lifetime measurement of the 2+ → 0+ transition in 18 C via the Recoil Distance Method
gs
1
was presented in Ch. 3. The mean lifetime of 22.4 ± 0.9(stat)+2.5 (syst) ps is in agreement
−1.6
with the literature value 18.9 ± 0.9(stat) ± 4.4(syst) ps reported in [29]. The lifetime is
inversely proportional to the electromagnetic transition rate and thus the reduced quadrupole
transition strength B(E2; 2+ → 0+ ) was determined via Eq. 1.9 to be 3.64+0.46 e2 fm4 . A
gs
1
−0.48
932 keV gamma-ray transition was observed to feed the 2+ level from above. Though this
1
higher-lying state was populated with an intensity only 25% that of the first 2+ state, a
lifetime upper limit τ < 4.5 ps was extracted. This quantitative lifetime measurement is the
first of its kind for states above the first 2+ in even-even carbon isotopes with N > 8.
1
The observed excited state properties are summarized in Tab. 5.1. There, the assumed
J π assignment of 2+ for the observed 2+ feeder level shall prove convenient for comparison
1
2
with ab initio theoretical calculations in Subsect. 5.1.2.1 In addition, shell model calculations
1 Experimental spin and parity assignments for excited states populated by nucleon knock-

out reactions from gamma-ray spectroscopy with SeGA are daunting (c.f. Ref. [86,110]) and
not feasible for the measurement presented here.
123

Table 5.1: Measured electromagnetic transition properties of 18 C excited states.
Observable
Experiment Unit
E(2+ → 0+ )
gs
1
E(2+ → 2+ )
2
1
τ (2+ )
1
τ (2+ )
2
B(E2; 2+ → 0+ )
gs
1

1585(19)
932(11)
22.4+3.4
−2.5
< 4.5
3.64+0.46
−0.48

keV
keV
ps
ps
2 fm4
e

with the WBP interaction [111] also predict the second excited state to be 2+ [5]. From the
lifetime upper limit of this state, a reference lower limit B(E2; 2+ → 2+ ) = 8.1 e2 fm4 is
2
1
calculated. This is just an estimate; a mixing ratio (δ ∝ Eγ

B(E2)/B(M 1)) of −0.18(15)

was adopted from the 2+ → 2+ transition in 16 C [112] and 100% branching was assumed
2
1
from the non-observation of a 2517 keV transition to the ground state. If instead a pure E2
transition is assumed, the lower limit B(E2; 2+ → 2+ ) increases thirty fold to 257 e2 fm4 .
2
1
Note that the selection rules from Appendix A also permit an M 3 or E4 transition, but both
can be excluded as lower multipoles dominate higher ones.
With the aid of Eq. A.3, the single-particle Weisskopf estimate of B(E2; 2+ → 0+ ) is
gs
1
found to be BW (E2) = 2.8 e2 fm4 . The ratio of the experimental reduced quadrupole matrix
element to the single-particle value (Eq. 1.17) indicates the degree of collectivity for the
excited state. The deduced ratio of Bexp /BW = 1.3 is near unity and suggests the excitation
should be well described within the framework of the single-particle shell model. This will
be demonstrated below.
The experimental results tabulated in Tab. 5.1 serve as useful probes of the nuclear structure of 18 C. In particular, the electric quadrupole transition strength is a powerful indicator
of changes in shell structure [23]. The discussion provided immediately below illustrates this
fact; there the near constancy of B(E2; 2+ → 0+ ) values for neutron-rich carbon isotopes
gs
1
is explored. The measurements are also valuable benchmarks which can hone the predictive
power of various nuclear models. In turn, accurate models can help interpret data and iden124

tify interesting objects for future studies. RDM lifetime measurements are certain to play
an important role as this mutual feedback loop advances the forefront of the field; thus the
section concludes with a few general suggestions and comments regarding future Köln/NSCL
plunger measurements.

5.1.1

Nuclear Structure of Neutron-Rich Carbon Isotopes

The final remarks surrounding Eq. 1.18 in Ch. 1 alluded to the marked deviation of 16,18 C
from the systematic liquid drop behavior B(E2) ∝ E(2+ )−1 with respect to 14 C as evidence
for a very different excitation mechanism. Indeed, the hindered B(E2) observed in 16,18 C
with respect to other light open-shell nuclei [28, 29] has been reported by some to indicate
exotic proton and neutron decoupling modes [113]. The divergence from expected behavior is
demonstrated in Fig. 5.1, where the B(E2) values (main plot) for 14,16,18 C remain relatively
constant while the E(2+ ) (inset) drops significantly from nearly 7.0 MeV for 14 C to less
1
than 2.0 MeV for 16,18 C. The present 18 C results, together with previous measurements of
16,18,20 C (Ref. [28,29,79]), can systematically deduce the degree of quadrupole collectivity for

the 2+ excitations of these neutron-rich carbon isotopes. It will be shown below that simple
1
shell model calculations suffice to describe the observed behavior of neutron-rich carbon
isotopes without evoking exotic decoupling modes.
The four valence protons of 18 C completely fill the π0p3/2 orbital. Shell model calculations
discussed in Ref. [29] for 16,18 C indicate a large energy gap between the π0p3/2 and π0p1/2
orbitals. It can therefore be expected that the 2+ excited state is dominated by neutron
1
contributions. However, the quadrupole transition strength is strictly an electromagnetic
phenomenon and therefore neutrons cannot contribute directly. Instead, the valence neutrons
(all in the ν0d5/2 shell for 16−20 C, but with a strong admixture of ν1s1/2 contributions due
to the disappearance of the N = 14 subshell closure for carbon isotopes [5] which induces
a near degeneracy between ν0d5/2 and ν1s1/2 orbitals) couple to the proton core. This
125

12

6

10

B(E2) [e2fm4]

4
8

6

2

E(2+) [MeV]
1

8

0
14

15

16

17

18

4

2

0
14

15

16
17
18
A [Carbon Isotopes]

19

20

Figure 5.1: Observed B(E2; 2+ → 0+ ) trend for even-even 14 ≤ A ≤ 20 carbon isotopes
gs
1
(main plot). The squares indicate measured B(E2) data for 14 C (black) [27], 16 C (red) [28],
16,18 C (gray) [29], and 20 C (blue) [79]. The green circle is taken from the present work
discussed in Ch. 3. B(E2) shell model calculations for 16,18,20 C are given by the black line
and are discussed in Ref. [79]. See text for more details. The relatively constant B(E2) for
14,16,18 C together with the E(2+ ) trend (inset) demonstrates the marked deviation from
1
even-even isotope global systematics of Ref. [26].
collective behavior induces polarization of the proton core and the B(E2) transition strength
is indicative of the extent of this core polarization. The polarization effects are weaker at
large isospin (small neutron binding energies) [114], as the spatially extended neutrons with
increasing ν1s1/2 admixtures induce polarization less efficiently.2
Therefore a relatively hindered core polarization and suppression of proton excitations
leads to the small B(E2) values observed in 16 C [28, 29] and 18 C (as presented in Ch. 3 and
Ref. [29]). In addition, the transition strength should increase for 20 C due to the monopole
2 It is interesting to note that this spatially extended ν1s
1/2 orbit also plays a vital role
19 C [115, 116].
in the known halo configuration of

126

interaction of the tensor force [117], where the ν0d5/2 orbital is filled and therefore the
attractive ν0d5/2 − π0p3/2 and repulsive ν0d5/2 − π0p1/2 interactions reduce the shell gap
between the two proton 0p orbitals. Thus the B(E2) strength contribution from proton
excitations is enhanced, as has been observed [79].
The observed behavior of the B(E2; 2+ → 0+ ) values for neutron-rich carbon isotopes
gs
1
can be reproduced with shell model calculations discussed in Ref. [79]. This is illustrated in
Fig. 5.1 where B(E2) data and calculations are shown as a function of neutron number. The
calculations were constructed from
1
2
B(E2; 2+ → 0+ ) =
Mp ep + Mn en ,
gs
1
2Ji + 1

(5.1)

where Ji is the spin of the initial state and Mp (Mn ) are the shell model proton(neutron)
matrix elements coupling the two states of the 2+ → 0+ transition. The ep and en are the
gs
1
isospin-dependent proton and neutron effective charges as parameterized in Ref. [118]. The
effective charges are introduced to account for contributions to the quadrupole transition
strength from excitations outside of the truncated shell model space. They are useful here as
they quantify the core polarization; ep and en will approach the canonical charge values of
+1e and 0, respectively, as the core polarization disappears. The numerical values for both
the matrix elements and effective charges were extracted from Ref. [79].
The agreement between the calculations and data presented in Fig. 5.1 presents clear evidence that the collective behavior of neutron-rich carbon isotopes can be described within
the shell model utilizing isospin-dependent effective charges. Strong proton-neutron decoupling did not need to be invoked, but rather a lessening of neutron-proton core polarization
due to the spatial extent of ν1s1/2 valence neutrons. More immediately important is the
excellent agreement between the B(E2) derived from the current lifetime measurement and
that of the previous study [29].

127

5.1.2

Comparison to ab initio NCSM

The 18 C excited state lifetime measurements presented in Ch. 3 and discussed above provide
a unique opportunity to test the accuracy of ab initio calculations, such as the no-core shell
model (NCSM) [12]. This fundamental theoretical approach begins with realistic two- and
three-nucleon interactions and the full set of nucleon degrees of freedom. A brief description
of the ab initio NCSM and several pertinent definitions were provided in Sect. 1.2. Here, a
set of NCSM calculations performed by Ref. [15] are compared to the 18 C electromagnetic
transition data.
Besides the extreme neutron-to-proton ratio of 18 C, which as shown above can result in
interesting quadrupole excitation behavior, this A = 18 system is slightly beyond current
computational limits for full NCSM calculations. For example, extracting stable excitation
energies for low-lying states requires a very large model space and prohibits results for
A > 16 [119]. To extend the applicability of ab initio NCSM to either heavier nuclei or larger
model spaces, approximations such as the importance-truncation (IT) scheme [16, 17] must
be applied.
In the IT-NCSM approach, unimportant basis states are first identified with many-body
perturbation theory and then removed prior to the full matrix eigenvalue calculation. A
substantial number of basis states are irrelevant for the description of low-lying excited
states and therefore removing these inconsequential states truncates the model space without
compromising predictive power. Using this reduced model space extends ab initio NCSM
predictions to heavier nuclei and higher harmonic-oscillator excitations. The approach has
been tested with calculations of properties of the stable nuclei 4 He, 16 O, and 40 Ca [17], as
well as light radioisotopes of helium [16]; the present discussion marks a vital benchmark
test for the scheme’s applicability to heavier short-lived radioactive nuclei.
Large-scale ab initio NCSM calculations were performed for the low-lying excited states
of 18 C using two different realistic nucleon-nucleon interactions: CD-Bonn 2000 [9] and
128

INOY [120]. The former provides a charge-dependent potential, built upon fits to global
proton-proton and neutron-proton scattering data with an excellent χ2 /datum ∼ 1, while
the latter introduces non-local interactions to account for three-body effects without explicitly including them. For brevity, only the CD-Bonn based calculations will be compared with
the experimental data. A more complete comparison will be provided in a forthcoming paper.
From the two-body realistic potentials, effective interactions were derived to accelerate convergence issues introduced by strong short-range nucleon-nucleon correlations. No adjustable
parameters or effective charges were used in the NCSM calculation. The calculations were
performed using harmonic-oscillator (HO) basis states constrained by a total HO energy
cutoff characterized by a maximum number of HO excitations Nmax above the ground state.
The results exhibited a dependence upon both Nmax and the HO frequency Ω that is
expected to disappear as the model space is increased [14]. This implies that Nmax sequences
obtained at different Ω values should all eventually converge to the same result. Thus the
fits of multiple sequences can be constrained to converge at the same result [13]. To this
end, the calculations employed as large an Nmax basis as feasible over a wide range of Ω
and extrapolated the calculated observables to infinite space. As the computational limit for
full space 18 C calculations is Nmax = 6, the importance-truncation scheme was employed to
extend the calculations to Nmax = 8, thereby improving the reliability of the infinite space
extrapolation. Figure 5.2 presents the NCSM full space (closed squares) and importancetruncated (open squares) results as well as the extrapolations for two of the observables
considered.
Table 5.2 presents a comparison of the extrapolated NCSM results to the corresponding
18 C experimental measurements. In particular, the 2+ excitation energy and the calculated
1
B(E2; 2+ → 0+ ) = 4.3(5) e2 fm4 are in excellent agreement with the experimental results.
gs
1

The calculated quadrupole moments for the two 2+ states are Q(2+ ) = +3.8(1) efm2 and
1
Q(2+ ) = -3.8(2) efm2 . An infinite space extrapolation for the 2+ excited state would permit
2
2
129

Figure 5.2: Ab initio NCSM results from Ref. [15] for electromagnetic observables of the
2+ excited state in 18 C illustrating the Nmax dependence of Q(2+ ) and B(E2; 2+ → 0+ )
gs
1
1
1
calculated with the CD-Bonn 2000 interaction. The filled (open) squares correspond to full
(importance-truncated) model space results. Each extrapolation curve corresponds to the
sequence of results obtained at a fixed HO frequency. The quoted theoretical results in
Tab. 5.2 are derived from the calculated convergence of the extrapolation curves.

130

Table 5.2: Ab initio no-core shell model calculations of low-lying 18 C electromagnetic transition properties [15] compared to the experimental results tabulated in Tab. 5.1. The theoretical E(2+ ) value is from a preliminary Nmax = 2 calculation; the full extrapolated value
2
was not calculated.
Observable
Experiment Theory Unit
E(2+ )
1
B(E2; 2+ → 0+ )
gs
1
+)
Q(21

1.585(19)
3.64+0.46
−0.48
—

1.5(4)
4.3(5)
+3.8(1)

MeV
e2 fm4
efm2

E(2+ )
2
Q(2+ )
2

2.517(22)
—

3.2(4)
−3.8(2)

MeV
efm2

a meaningful comparison to the data, but only some of the properties of the two lowest 18 C
excited states (2+ and 2+ ) were converged due to the complexity of the calculation. However,
1
2
in a much smaller model space calculation (Nmax = 2) the properties of many states were
calculated, including the 2+ excitation energy E(2+ ) = 3.2(4) MeV.
2
2

5.1.3

Suggestions and Outlook

The successful lifetime measurement presented in Ch. 3 and discussed here serves to further
validate the Köln/NSCL plunger RDM technique with fast radioactive beams. This work
has been unique; as part of a larger campaign it has aided in deciphering the evolution
and degree of quadrupole collective behavior in the neutron-rich 16,18,20 C isotopic chain.
Together, the results are well described within the shell model (Subsect. 5.1.1) using isospindependent effective charges. Alone, the current 18 C results agree well with the literature
value [29], but are reported with greater precision. In addition, the observation and analysis
of two electromagnetic transitions in this exotic isotope with a 2:1 neutron-to-proton ratio has
provided a benchmark for reliable ab initio no-core shell model calculations far from stability.
As shown in Subsect. 5.1.2, the E2 transition properties of the first excited state are well
reproduced; more work must be done to achieve agreement for the higher-lying state. In light

131

of the successes reported here and in past NSCL plunger lifetime measurements [21,22,78,79],
the Köln/NSCL plunger technique will continue to play an important role in elucidating the
nuclear structure of exotic radioactive isotopes. In the spirit of continual improvement, several
practical suggestions are offered below and a brief outlook is presented.
The dominant source of systematic error in the 18 C analysis was an uncertainty in the
ratio of excited-state production on the target and degrader. A satisfactory constraint of
this ratio was obtained from a two dimensional χ2 minimization by varying both the ratio
and the lifetime and fitting the resulting simulated lineshapes to the gamma-ray energy
spectra. However the construction of the χ2 hypersurface is computationally intensive. A
much simpler solution is to collect gamma-ray data at the “infinite distance” (Subsect. 3.3.2)
such that any observed radiation emitted downstream of the degrader is directly attributable
to reactions on the degrader. This data simultaneously reduces systematic uncertainties and
accelerates the analysis.
In addition, the employed Geant4 Monte-Carlo simulations were shown to accurately reproduce all of the key experimental features that impact the gamma-ray energy lineshapes
including the experimental geometry, the reaction residue momentum distribution, and the
photon detection efficiencies. Future plunger data will undoubtedly be analyzed within a
similar framework. It would therefore be advantageous to add the following to the simulations: a more complete description of the plunger geometry and the capability to generate
an anisotropic gamma-ray angular distribution within the rest frame. These augmentations
would enhance the realistic description of experiment.
The first suggestion entails the addition of the plunger foil mounting hardware- including the aluminum cones supporting the foils and the rigid structures affixing the cones to
the plunger body- into the simulation. The attenuation of low energy gamma rays within
these structures may influence the intensity ratio of photopeaks from decays upstream and
downstream of the degrader. This is not accounted for in the simulation and may skew the
132

lifetime analysis. Furthermore, it is difficult to simply quantify this effect with gamma-ray
source measurements as the effect depends upon the emission point z (and thus the lifetime)
of the nucleus.
The second suggestion would also improve the simulation’s ability to accurately reconstruct all pertinent experimental parameters. The present implementation uses an isotropic
angular distribution in the rest frame which is then Lorentz boosted into the laboratory
frame. The boosted angular distribution integrated over θ is different for the forward and
backward rings. More importantly, the relativistic boost is velocity-dependent and hence it
differs for decays occurring upstream or downstream of the degrader. Therefore the photopeak intensity ratios are unique to each ring. The simulations fully reproduce this behavior
with the assumption of an isotropic distribution in the rest frame. Experimental deviations
from this assumption may impact the analysis, but a preliminary investigation found no
effect on the result using various anisotropic distributions. A more detailed study may be
prudent.
Finally, several experimental advances on the horizon will play an important role in
improving lifetime measurements with the Köln/NSCL plunger. As discussed within Subsect. 2.3.2, preserving the ability to resolve the photopeaks from decays upstream and downstream of the plunger degrader is essential for a successful RDM lifetime measurement. The
present resolution is limited by uncertainties in the Doppler reconstruction of gamma-ray
energies in the laboratory frame from the spread of both the detection angles (∆θ) and reaction residue velocities (∆β). As illustrated in Fig. B.1, the most dominant contribution to
the energy resolution comes from this angular uncertainty. The advent of a new generation
of gamma-ray detector arrays such as AGATA [121], GRETA [122], and the more immediate
GRETINA [123] will usher in a new paradigm for gamma-ray spectroscopy where gammaray tracking [124] will provide an accurate determination of the first interaction point. The
measured position resolution of GRETINA has been demonstrated to be roughly 2 mm [125]
133

whereas a single SeGA segment is 1 cm wide. This improved position resolution translates
directly to improved energy resolution of Doppler reconstructed energy spectra.
It should be noted that sub-segment resolution has already been demonstrated in principle for SeGA instrumented with the Digital Data Acquisition System (DDAS) [88]; Fig. 5.3
illustrates the effect. The synchronous capture of waveforms in all 32 segments of a SeGA detector provides information about the gamma-ray interaction position on an event-by-event
basis. The caption of Fig. 5.3 provides more details. The decomposition of these signals
to provide precise position information is non-trivial and beyond the scope of this work.
Nonetheless, the figure illustrates that the improvement of SeGA angular resolution is possible with DDAS and may play an important role in future studies where GRETINA (or
GRETA) and the associated digital electronics are unavailable.
Currently, the plunger degrader thickness is chosen to provide adequate separation between the two photopeaks. Thus the improvement in gamma-ray energy resolution afforded
by gamma-ray tracking and waveform capturing will permit the use of thinner degraders. The
most immediate consequence of this improvement is the ability to probe shorter lifetimesthinner degraders equate to shorter transit times and thus more decays downstream of the
degrader. A secondary benefit of thinner degrader foils will be the reduction of contaminant
gamma-rays from the de-excitation of states produced by reactions on the degrader.
Aside from thinner degraders, another interesting prospect for future plunger lifetime
measurements at NSCL involves the use of multiple degraders. The so-called “double plunger”
is currently in the design phase [126] and will provide multiple configurations for different
lifetime measurement requirements. In the simplest approach, a second degrader is added
to provide a third photopeak corresponding to decays downstream of the second and final
degrader. Thus a single plunger setting can provide two different distances; in principle data
can be taken at more distances, though the photopeak intensity reduction from splitting two
decay regions into three requires consideration. Another configuration utilizing one degrader
134

Amplitude

A4

600

400
0.5

B4
600

C4
600

D4
600

E4
600

F4
600

G4
600

400

600

400

400

400

400

400

400

B3
600

C3
600

D3
600

E3
600

F3
600

G3
600

1500

E2

1000

1

A3

600

H4

600

H3

500
0.5

1

1000 CC
400

400
0.5

400

400

400

400

400

B2
600

C2
600

D2
600

E2
600

F2
600

G2
600

800

1

A2

600

400

600

H2

600
0.5

400

400

400

400

400

400

400

1

400

3000 Ref.
0.5

600

400

1

A1

600

B1
600

C1
600

D1
600

E1
600

F1
600

G1
600

400

400

400

400

400

400

400

0.5 1.0

0.5 1.0

0.5 1.0

0.5 1.0

0.5 1.0

0.5 1.0

0.5 1.0

H1

2000
1000

0.5 1.0

0.5

1.0

Time [us]

Figure 5.3: Digital waveforms for several events in all 32 channels of a single SeGA detector
corresponding to the full absorption of a 1332 keV gamma-ray from 60 Co in a single segment
(E2) in coincidence with a 1173 keV gamma-ray detected in a fast CsF scintillator. This
scintillator was used as a timing reference (“Ref.”, bottom right). The solid black signals are
the average of many such signals and should be contrasted to two individual events (dashed
and dotted lines). The observable rise time differences are taken as evidence of sub-segment
position resolution. Figure from Ref. [88]
close to the production target (∼ 1 mm) and another much further away (∼ 1 cm) can
simultaneously collect two sets of data in very different lifetime regimes. Note that the
contaminant events from production of excited states on the multiple degraders will have to
be characterized for both implementations.

135

Table 5.3: 19 Mg two-proton decay lifetime (picoseconds) as a function of Rp .
17 Ne Production Mechanism Ratio

0.05
τ
∆τ

5.2

19

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

0.35
0.09

0.39
0.09

0.46
0.12

0.49
0.14

0.60
0.15

0.76
0.18

0.96
0.22

1.35
0.31

2.44
0.54

7.26
–

Mg Commentary

The 19 Mg mean two-proton decay lifetime study was presented in Ch. 4. The extracted
lifetime depended strongly upon the 17 Ne production mechanism ratio Rp . This ratio was
not well constrained by either the data or from previous studies. There was also a dependence,
albeit a much weaker one, upon the target-detector reaction yield ratio Rσ . Hence the bestfit lifetime depended upon the combination of Rp and Rσ ; the results were presented in
Tab. 4.2.
As discussed within Subsect. 4.3.1, Rσ was well constrained between 3.03 and 3.43. The
best-fit lifetime from the central value Rσ = 3.23 has been taken as the unique lifetime
result at each Rp . The best-fit lifetimes corresponding to the upper and lower Rσ limits
were taken as the main source of systematic uncertainty. Accurate simulated DSSD ∆E lineshapes required excellent agreement between the simulated and experimental dta spectra (as
illustrated in Fig. 4.5). The contribution of small, but tolerable, variations in the simulated
reaction residue momentum distribution to the systematic error was explored and found to
contribute half as much as the uncertainties in Rσ . All other sources of systematic error
were found to be negligible, including variations in the two-proton decay Q-value. The root
sum of squares of these two systematic errors was adopted for the final results, which are
presented in Tab. 5.3. There, the overall uncertainty is given as ∆τ = ∆τstat + ∆τsyst .
It is prudent to compared these results with those of the 19 Mg lifetime measurement
reported in Ref. [38]. This exercise is presented in the following subsection. There, the stark

136

difference between the measurements shall be discussed. The present results indicate a lifetime one order of magnitude smaller than the literature value of τ = 5.8±2.2 ps. In addition,
the demonstrated sensitivity to picosecond lifetimes in this work validates the Köln/NSCL
particle plunger technique for lifetime measurements of charged-particle emitters.

5.2.1

Comparison to Previous Work

The method used for the first measurement of the two-proton emitter 19 Mg is presented
in detail in Ref. [38, 127, 128]. There, the energy and angles of the three-body decay remnants of 19 Mg were measured independently. The decay vertex was reconstructed from the
deduced trajectories’ point of closest approach. Plotting the observed decay vertices against
the production target profile yielded a distribution from which the lifetime was determined.
The authors noted the indistinguishability between the vertex distribution from the nonresonant breakup 20 Mg → p + p + n + 17 Ne and the prompt proton decay of 19 Mg excited
states with lifetimes well below the experimental sensitivity. By comparing the observed
17 Ne-proton angular correlations to simulations, the contribution of these prompt events to

the ground state 2p decay data was determined to be 25%. The quoted lifetime result was
derived using this percentage. Factoring in the uncertainties of the shape of these background
contributions yielded an upper limit contribution of 65%. A lifetime was given for this case
and also the 0% lower limit case where no background contribution was considered.
Note that these prompt contribution percentages to the 17 Ne yield are entirely equivalent
to the 17 Ne production mechanism ratio Rp introduced for the analysis presented in Eq. 4.1
of Ch. 4. Thus, their contribution to the decay vertex distribution of the delayed two-proton
emission from the 19 Mg ground state is synonymous with the impact of the simulated Rp on
the ∆E lineshape simulations. For this reason, the range of reasonable values for the 17 Ne
production mechanism ratio was set to 0.15 ≤ Rp ≤ 0.85 to fully encompass the findings
reported in [38].
137

Table 5.4: Comparison of 19 Mg lifetime to the previous study [38] as a function of Rp . For
ease of comparison, the literature value at Rp = 0.00 is approximated by Rp = 0.05. See text
for details.
17 Ne Production Mechanism Ratio
0.05
0.25
0.65
Current Work 0.35 ± 0.09 0.46 ± 0.12 0.96 ± 0.22
Ref. [38]
4.5 ± 1.5
5.8 ± 2.2
9+3
−6
This fortuitous equivalence allows the two measurements to be compared on equal footing.
Table 5.4 presents the lifetimes for three Rp values. There, the lifetime derived from the 0%
prompt 17 Ne contribution scenario is used as an approximate value for comparison with the
Rp = 0.05 result from the analysis of Ch. 4. The table demonstrates the results from the two
measurements differ by one order of magnitude.
Figures 5.4 and 5.5 provide a direct contrast between the two measurements at Rp =
0.25. In both figures, the data for 0 mm 17 Ne particle-gated energy losses are plotted with
statistical uncertainties against simulated lineshapes. The former is generated with the bestfit lifetime τ = 0.46 ps from Tab. 5.4 and the latter with the literature reported lifetime
τ = 5.8 ps. Both simulations incorporated the exact same parameters to model the incoming
20 Mg secondary beam, knockout reaction kinematics, detector response, and background

functions. Thus the only difference between the two, save for an overall scaling factor to
normalize the simulated 17 Ne peak height to the data, is the input lifetimes. The discrepancy
between the two results is clear; the presented data does not support such a long lifetime.
The demonstrated picosecond lifetime sensitivity to the ground state two-proton emission
of 19 Mg indicates that lifetime measurements of exotic proton emitters with the Köln/NSCL
particle plunger technique merit further consideration. This proof of concept measurement
opens the door for future studies of nuclei beyond the proton dripline. The region surrounding
the N = Z rp-process waiting-point nuclei is of immediate interest, as the introduction
given in Subsect. 1.4.2 indicated. In particular, the bridging of 68 Se by sequential one-proton
138

Counts/0.504 MeV

500
400
300
200
100
0

40

50
60
Energy Loss [MeV]

70

Figure 5.4: 17 Ne particle-gated silicon energy loss spectra at 0 mm with statistical uncertainties (red). The simulated lineshape (blue) was generated from the best-fit lifetime τ = 0.46 ps
from the lifetime scans with Rp = 0.25 and Rσ = 3.23.
600

Counts/0.504 MeV

500
400
300
200
100
0

40

50

60

70

Energy Loss [MeV]

Figure 5.5: Same as in Fig. 5.4, but with the simulated lineshape generated with the lifetime
τ = 5.8 ps from Ref. [38].

139

captures as depicted in Fig. 1.3 depends sensitively upon the 1p separation energy of the
proton unbound nucleus 69 Br via Eq. 1.32. A recent experiment determined the separation
energy to be Sp = −785 keV and therefore 69 Br is sufficiently unbound such that the proton
capture mechanism cannot bridge the 68 Se waiting point [58]. However, the analysis suffered
from the lack of a measured one-proton decay lifetime for 69 Br. An upper limit of 24 ns has
been set [57]; a more rigorous lifetime measurement for this nucleus (and others like it) can
aid in experimental analysis and provide a vital check for accurate mass models.

5.2.2

Comments on Future Prospects

The successful proof of concept measurement was not without experimental difficulties that
must be improved upon. Some suggestions, ranging from the very practical to the more complicated, are proffered below and may prove useful for future applications of the technique.
This discussion is framed within the scope of the current work so that the terminology and
context is familiar.
The significant contribution of reactions on the silicon DSSD to the ∆E lineshape was
discussed in Subsect. 4.2.3. The simplest remedy to remove these contributions is gathering
17 Ne particle-gated lineshapes from detector-only data. This would provide a quantitative

picture of the characteristic energy loss in the detector from the direct dissociation 20 Mg →
p + p + n + 17 Ne and the one neutron knockout and subsequent two-proton decay 19 Mg →
p+p+ 17 Ne processes. The detector contribution could then be subtracted from the data, but
care must be taken to ensure the data can be properly normalized to account for fluctuating
beam currents. The ∆E lineshape shift introduced by the difference in 20 Mg secondary beam
energies with and without the target installed must also be accounted for.
The use of a thinner silicon DSSD would be advantageous, as it would mitigate the effect
of contaminant reactions on the silicon and subsequently improve the 19 Mg peak-to-plateau
intensity ratio. Furthermore, for short-lived nuclei a thinner detector would permit a greater
140

50 µm 100

80

80

60

Counts/0.504 MeV

100

60

40

40

20

20

0

5

10

100

100 µm

0

15
200 µm

15

20

100

80

60

40

40

20

300 µm

80

60

25

20

0

30

40

50

0

40

50

60

70

Energy [MeV]
Figure 5.6: Simulated silicon detector energy loss profiles for four different detector thickness.
The simulation parameters were taken from the experimental analysis with Rσ = 3.23,
Rp = 0.55, and the corresponding best-fit lifetime τ = 0.76 ps.
fraction of 1n knockout residues to fully traverse the detector before decaying. This would
also improve the 19 Mg peak-to-plateau ratio and may open the possibility of investigating
the 17 Ne/19 Mg energy loss peak intensity ratios at various target-detector distances. Figure
5.6 illustrates the energy loss lineshapes from four different detector thicknesses. Note the
more prominent 19 Mg peak for detectors with thicknesses between 100 and 200 µm compared
to the 300 µm detector used in the present work. Too thin of a detector is also undesirable;
with smaller energy deposits the resolution between the 17 Ne and 19 Mg peaks is lost. The
plots suggest an optimal detector thickness of approximately 150 µm.
Constraining the uncertainty in the 17 Ne production mechanism ratio is perhaps the
most crucial task for future particle plunger studies. As discussed in Ch. 4 and Sect. 5.2,

141

the inability to distinguish between 17 Ne reaction residues from the two-proton decay of
19 Mg or the breakup of 20 Mg in the target prevents the extraction of a single two-proton

decay lifetime. Care should be taken during experimental planning to determine the optimal secondary beam reaction mechanism that minimizes such a contaminant production
mechanism without compromising the production of the nucleus of interest. Along the proton dripline, proton knockout reactions may not be a viable alternative to the one-neutron
knockout utilized in the present experiment; 2n and 1n1p knockouts should also be investigated. However, a cursory investigation with Lise++ using both Abration Ablation [107,108]
and EPAX 2.15 [106] parameterizations reveals these two to be inferior to 1n knockout when
both production rate and 17 Ne contaminant suppression are considered.
For proton decay studies along the dripline, it is likely that cross section measurements
or reliable calculations will be lacking beforehand. Therefore the greatest improvement in
constraining Rp will come from modifications to the particle plunger technique such that
either the ratio can be determined alongside the ground state proton emission lifetime or
the contaminant signals can be rejected. A possible implementation of the first modification
type would be to utilize the particle plunger with a separate charged-particle array such
that the angle between the 17 Ne decay residue and protons can be measured [126]. The
High Resolution Array [129] at NSCL would be one choice to measure the protons from the
decay, however the large scattering chamber would have to be modified to mount the particle
plunger. The effort may prove worthwhile; measuring 17 Ne-p angular correlations can help
to distinguish between ground state two-proton emission and prompt production of 17 Ne as
demonstrated in [38].
A less elegant modification would entail the addition of a thin silicon veto detector installed immediately downstream of the secondary target. This could generate a veto, or
anti-coincidence, flag in the analysis for events where 17 Ne emerged from the target. These
rejected events would originate from the nonresonant breakup of 20 Mg, the prompt proton
142

emission from excited state decays of 19 Mg, and ground state two-proton decays occurring
within the target. The addition of a thin veto detector would further suppress the experimental sensitivity to very short lifetimes and also have to be accounted for in the simulations,
although contaminant reactions would be minimal in such a thin detector. However, the loss
of resolution between the 17 Ne peak and other ∆E features evident in thin detectors (see
Fig. 5.6) would have to be addressed. The technical details are less complicated than the
above modification; if the resolution issues can be overcome, the contribution of prompt 17 Ne
contaminants would be eliminated.
Finally, the analysis of future particle-plunger measurements will likely incorporate similar Geant4 simulations. Including the DSSD dead layers into the simulation software should
be a simple improvement to implement. The dead layer on both sides is comprised of the
aluminum electrical contacts, plus the p+ -type junction contact on one side and the n+ -type
ohmic contact on the other. The current simulation implementation neglected these layers,
as it was estimated to be less than 1 µm on each side [130] and thus roughly only 0.5% of the
total detector thickness. If thinner detectors are used, this contribution will increase. Simply
decreasing the simulated detector width with respect to the manufacturer’s specifications
will not account for the dead layers; they represent regions where the beam can deposit energy or scatter. These experimental effects should be incorporated to improve the accuracy
of simulated ∆E lineshapes.

143

Chapter 6
Conclusions
Lifetime measurements with the Köln/NSCL plunger are sensitive probes of the behavior and
structure of exotic nuclei at the extremes of isospin. Picosecond-ordered decay processes are
studied in a model independent fashion. The lifetime is derived from changes in gamma-ray
energy spectra or charged-particle energy loss lineshapes elicited by micron-precision adjustments of the target-degrader (or detector) distance. The versatility of this Recoil Distance
Method technique with intermediate energy radioactive beams and inverse kinematics has
been demonstrated in this work.
The lifetime of the electromagnetic quadrupole 2+ → 0+ transition in 18 C was measured.
gs
1
The state was populated by the one-proton knockout reaction of a 19 N secondary beam
on the beryllium plunger target. The Doppler-shift corrected gamma-ray energy spectrum
exhibited two photopeaks corresponding to de-excitations upstream or downstream of the
tantalum plunger degrader. The lifetime was determined by fitting these lineshapes at five
target-degrader distances with simulated spectra. From the lifetime, the reduced quadrupole
transition strength B(E2) was extracted, which provides a straightforward means to quantify
the degree of collectivity in 18 C. The results agree well with a previous measurement [29]
and the fractional uncertainty has been reduced by a factor of two. The excitation energy

144

and B(E2) calculated with the ab initio no-core shell model [15] exhibit excellent agreement
with the data, providing an important test for ab initio calculations of heavier and more
exotic nuclei utilizing the importance truncation [16] approach. In addition, a weak transition
feeding the 2+ state was observed and a conservative lifetime upper limit was measured.
1
This represents the first quantitative lifetime information for a state above the first 2+ for
neutron-rich, even-even carbon isotopes.
A modified Köln/NSCL plunger was constructed with a thin silicon detector in place of
the degrader. The lifetime of the exotic two-proton emitter 19 Mg was measured as a proof
of concept of this device for lifetime measurements of charged-particle emitters beyond the
proton dripline. States were populated by the one-neutron knockout reaction of a 20 Mg
secondary beam on the carbon plunger target. A very short lifetime for the ground state
two-proton radioactivity was deduced by fitting the energy loss signatures of 19 Mg and
the two-proton decay residue 17 Ne in the silicon detector with simulated lineshapes. The
contribution of contaminant 17 Ne from both the immediate breakup of 20 Mg and the prompt
sequential one-proton emissions via an intermediary unbound 18 Na ground state could not be
quantified. Thus the lifetime was determined as a function of this prompt 17 Ne contribution.
Regardless of the extent of this contaminant, the results indicate a lifetime one order of
magnitude smaller than that of a previous measurement [38]. This work has demonstrated
the potential of this new experimental technique, especially along the proton dripline where
lifetime information provides important constraints on mass models and proton-capture rates
invaluable for the accurate characterization of nucleosynthesis pathways.

145

APPENDICES

146

Appendix A
Electromagnetic Selection Rules
A.1

Parity Selection Rules

Knowledge of the character and multipolarity of an electromagnetic transition linking two
nuclear states is paramount to interpreting the experimental results in terms of nuclear
properties. For instance, without knowing the transition observed in the 18 C experiment was
E2 in nature, the lifetime result cannot be interpreted as a gauge of nuclear quadrupole
collectivity. 18 C has an even number of protons and neutrons. A general property of all such
even-even nuclei is that the ground state is always 0+ due to the pairing of nucleons in a
particular l-orbital into spin up-spin down pairs. Furthermore, the first excited state is in
all but a very few instances 2+ , as noted in Ref. [131]. Thus from systematics, the spins and
parities J π of the initial excited and final ground state in 18 C are known, and from this the
character and multipolarity of the gamma ray transition can be deduced.
The conservation of parity for electromagnetic operators follows from the parity rules
implicit in Eq. 1.4 and 1.5. The parity of a spherical harmonic Yλµ is (−1)λ and that of
rλ is +1. Therefore the total parity of an electric λ-pole is +1 · (−1)λ = (−1)λ . For the
magnetic λ-pole, l and s are axial vectors and thus have a parity of +1. The total parity is

147

thus determined by the gradient vector (with a parity of −1) acting on rλ Yλµ (with a parity
of (−1)λ ) and is −1 · (−1)λ = (−1)λ−1 . The parity conservation selection rule can thus be


 (−1)λ−1 for Mλ
πi π f =

 (−1)λ
for Eλ.

stated as

A.2

(A.1)

Angular Momentum Selection Rules

The angular momentum selection rules follow the triangle condition, where the possible
multipolarities λ of the photon can be determined from the spins of the initial and final
states according to
Ji − Jf ≤ λ ≤ Ji + Jf .

(A.2)

From Eq. A.2 it is apparent that the 2+ → 0+ transition in 18 C proceeds with the emission
gs
1
of a λ = 2 photon that carries away two units of angular momentum. Furthermore, since
πi πf = +1 and λ was determined to be even, the transition must be an electric quadrupole
(E2) according to Eq. A.1.
As a brief aside, consider the case where the initial and final spins are such that several
3
electromagnetic λ-poles could contribute. For example, the 2

+

1+
→ 2 transition allows λ

values of 1 and 2 while the change in parity is πi πf = +1. Thus the emitted radiation would
be an admixture of M 1 and E2 components; to determine which component dominates in
order to relate the measured lifetime to the appropriate matrix element (B(E2) or B(M 1)),
Weisskopf single-particle transition probability estimates (BW (E2) or BW (M 1), for example) can be computed. The derivation of these estimates proceeds from the solution of the
reduced matrix element of Eq. 1.8 in Ch. 1 utilizing some simplifying assumptions and is

148

beyond the scope of this work. The results for electric and magnetic transitions are

BW (Eλ) =

BW (M λ) =

1.22λ
4π

10 (2λ−2)
1.2
π

2
3
A2λ/3 e2 f m2λ
3+λ

2
3
A(2λ−2)/3 (µN /c)2 f m2λ−2 .
3+λ

(A.3)

(A.4)

These equations can then be substituted into Eq. 1.7 and the transition probabilities can be
compared. For instance, assuming A = 100 and E = 1 MeV, the transition probability ratio
would be T (M 1) : T (E2) ≈ 103 and therefore the M 1 contribution would be expected to
dominate.
Calculations such as these using Weisskopf estimates provide a good starting point, but
are too simple as they only take single-particle excitations into account. For collective nuclear
states, the motion of many nucleons must be taken into account to describe the transition.
In these cases, the BW (E2) can underestimate the experimentally determined B(E2) by an
order of magnitude or more.

149

Appendix B
Relativistic Considerations
B.1

Relativistic Kinematics Formulation

As noted in Ch. 3 and 4, the reaction residues emerged from the plunger target at the S800
target position with velocities β = v/c ≈ 0.4. It is therefore necessary to account for time
dilation and the relativistic Doppler effect in the analysis. Here, the equations required for
the proper relativistic treatment are derived.
Consider the beam traveling along the z-axis with velocity v = βc. Let the position and
time coordinates in the laboratory frame be represented by unprimed coordinates (ct, x, y, z)
and those in the rest frame of the beam by primed coordinates (ct , x , y , z ). The four Lorentz
transformation equations that relate the coordinates in the laboratory frame to those in the
rest frame follow.

ct = γ ct + βz
x = x
y = y
z = γ z + βct
150

(B.1)

Here the relativistic factor γ is defined by
1

γ=

1 − β2

.

(B.2)

It is convenient to utilize the concept of 4-vectors, xα = (x0 , x1 , x2 , x3 ).1 The inner
product is a Lorentz invariant 4-scalar; therefore it is the same in all inertial reference
frames, and is given by
|x|2 = xα xα = (x0 )2 − (x1 )2 − (x2 )2 − (x3 )2 = invariant.

(B.3)

Here, x0 is the timelike component while x1,2,3 are the spacelike components. For uniform
motion along the z-axis, the components of the 4-vector xα undergo the Lorentz transformation from one inertial frame to another in a similar manner as those in Eq. B.1.

x0 = γ x0 + βx3
x1 = x1
x2 = x2
x3 = γ x3 + βx0

(B.4)

We can use the properties defined in Eq. B.3 and B.4 along with the universality of the speed
of light in all reference frames to derive several useful relationships.
Consider the position of a beam particle in four dimensional Minkowski space,

rα = (ct, r) = (ct, x, y, z).

(B.5)

1 In the literature, there is a notational distinction between x and xα as being covariant
α

and contravariant 4-vectors, respectively. In this brief mathematical formulation all 4-vectors
are contravariant, however the designation xα is used for ease of reading.

151

Suppose this particle decays after an elapsed time in the laboratory frame t12 = t2 − t1
and a change in laboratory position of r12 =

(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 =

2
2
x2 + y12 + z12 . In the beam rest frame a different time interval elapses, t12 , but there is
12

no change in position such that r 12 = 0. Since rα rα is an invariant quantity, the relationship
between the decay event coordinates in the laboratory and rest frames from Eq. B.3 is
2

2
2
c2 t 12 = c2 t2 − x2 − y12 − z12 .
12
12

(B.6)

Dividing both sides of the equation by c2 and taking the square root yields

t12 =

1
2
2
t2 − 2 (x2 + y12 + z12 ) = t12
12
c 12

1
1− 2
c

2
2
x2 + y12 + z12
12

t2
12

.

(B.7)

But the second term under the square root is simply the speed squared of the beam particle
as measured in the laboratory frame and therefore with the help of Eq. B.2, the final result
is
t12 = t12

t
v2
1 − 2 = 12 .
γ
c

(B.8)

Thus the lifetime of a particle measured in the laboratory frame, τlab = t12 , is related to the
lifetime of the particle in the rest reference frame, τrest = t12 , by
τlab = γτrest .

(B.9)

The Lorentz transformations of a generalized 4-vector in Eq. B.4 prove useful for determining the relationships between the gamma-ray energy and angle of emission as measured
in the laboratory reference frame and those in the rest frame. Consider the 4-vector

kα =

ω
ω
,k =
, kx , ky , kz
c
c
152

(B.10)

where for a photon, the wave vector, k, and frequency, ω = 2πν, are related by

(B.11)

ω=c k .

The invariant inner product of the wave 4-vector is
ω2
ω2
kα k α = 2 − k · k = 2 − k 2 = 0.
c
c

(B.12)

For a beam propagating along the z-axis, the kz component of the wave vector and
frequency ω in the laboratory frame are related to kz and ω in the rest frame by Eq. B.4.
ω
kz = γ kz + β
c

(B.13)

ω
=γ
c

(B.14)

ω
+ βkz
c

In addition, the kz component of the wave vector in the laboratory frame is given by
ω
cos θ,
c

kz = k cos θ =

(B.15)

where θ is the angle of photon propagation with respect to the z-axis. A similar expression
can be written in the rest frame for the wave vector k and angle of propagation θ . With
Eq. B.15 and its rest frame analog in hand, Eq. B.13 and B.14 can be rewritten.

kz =

ω
cos θ = γ
c
ω
=γ
c

k cos θ + β

ω
+ β k cos θ
c

153

ω
c

=γ

=γ

ω
cos θ + β
c

ω
(1 + β cos θ )
c

(B.16)

(B.17)

Rearranging the previous two equations yields
cos θ + β
ω
=γ
ω
cos θ

(B.18)

ω
= γ(1 + β cos θ )
ω

(B.19)

which when combined produce the relationship between the angle in the rest and laboratory
reference frames,
cos θ =

cos θ − β
.
1 − β cos θ

(B.20)

From this expression, the Lorentz boost of the solid angle from the rest frame to laboratory
reference frame can be extracted. For uniform motion along the z-axis, the azimuthal angle
is not boosted. Therefore from the expression for the solid angle

dΩ = sin θdθdφ = −d(cos θ)dφ,

(B.21)

the derivative with respect to cos θ of Eq. B.20 is taken to yield the result
dΩ
d(cos θ )
1 − β2
=
=
.
dΩ
d(cos θ)
(1 − β cos θ)2

B.2

(B.22)

Doppler Reconstruction

The Doppler shift of gamma-ray energies proceeds from Eq. B.19. Replacing ω with the
familiar relationship ω = E/ and substituting Eq. B.20 for cos θ yields
cos θ − β
E
=γ 1+β
E
1 − β cos θ

154

=

1 − β2
1 − β cos θ

(B.23)

where β is the beam velocity and θ is the laboratory angle of gamma-ray emission with
respect to the beam velocity vector.
As an example from the 18 C analysis, consider a photon with energy E = 1585 keV
in the beam rest frame emitted from a reaction residue with velocity β = 0.3565 and at
a laboratory angle of 30◦ . This photon can be detected in the forward ring of SeGA with
energy

√
1 − β2
1 − 0.35652
E=E
= (1585 keV)
= 2142 keV.
1 − β cos θ
1 − 0.3565 · cos 30◦

(B.24)

Similarly, if the photon is emitted at a laboratory angle of 140◦ , it can be detected in the
backward ring with energy
√
1 − β2
1 − 0.35652
E=E
= (1585 keV)
= 1163 keV.
1 − β cos θ
1 − 0.3565 · cos 140◦

(B.25)

Thus relativistic effects shift the measured gamma-ray energy by more than 25%.
The Doppler reconstruction of gamma rays detected at their shifted energies in the laboratory frame proceeds from the inverse of Eq. B.23,

E =E

1 − β cos θ
1 − β2

.

(B.26)

Here, the unmeasured angle of gamma-ray emission from Eq. B.23 has been replaced with the
laboratory angle θ of the SeGA segment in which the first gamma-ray interaction occurred
(see Subsect. 2.3.1 for more details). This introduces an uncertainty in the angle for Doppler
reconstruction, ∆θ, related to the finite opening angle of the segment. In addition, the
incoming momentum spread of the secondary beam, the energy straggling in the plunger foils,
and the one-proton knockout reaction all introduce a spread of reaction residue velocities,
∆β, on an event-by-event basis. Since most decays occur after the reaction residues emerge
from the plunger target, there are two narrow groupings of beam velocities at which the

155

nuclei decay- corresponding to decays in vacuum either upstream or downstream from the
plunger degrader- with centroid values of βf ast and βslow . The values were obtained by
first estimating the velocity from the magnetic rigidity of the S800 spectrometer in Lise++
(and for βf ast , then back tracking through the degrader) and then fine tuning the estimates
to properly align the fast or slow components of the 1585 keV transition in the forward
and backward rings of SeGA. Both ∆β and ∆θ increase the photopeak full-width at halfmaximum (FWHM) in the Doppler corrected gamma-ray energy spectrum.
The contribution from the uncertainties in the velocity and detection angles to the overall
energy resolution can be quantified. An expression for the energy resolution of the Doppler
reconstructed spectrum as a function of ∆β, ∆θ, and ∆E is given by
∆E 2 =

∂E
∂E

2

∆E 2 +

∂E
∂β

2

∆β 2 +

∂E
∂θ

2

∆θ2 .

(B.27)

The first term can be safely neglected as the contribution from the intrinsic resolution of
germanium detectors (approximately 0.2% for a 1 MeV photon) is very small. Taking the
appropriate partial derivatives of Eq. B.26 and rearranging terms yields
∆E
E

2

=

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

2
β sin θ
∆θ2 .
1 − β cos θ

(B.28)

Panel 1 of Fig. B.1 plots the resulting total gamma-ray energy resolution (blue) as well
as the individual ∆β (black) and ∆θ (red) components as a function of laboratory angle. An
input beam velocity of β = 0.361 was taken from the plunger target-only 18 C experimental
data. The experimental uncertainty in the velocity was estimated by the spread of β values
that align the 1585 keV transition photopeak at each plunger distance to be 0.15%. To err on
the side of caution, ∆β was set to twice this. ∆θ was fixed to 2.46◦ ; this corresponds to the
angle subtended by a single backward ring segment at a distance of approximately 23.3 cm
156

Gamma-Ray Energy Resolution FWHM [%]

4.5

1

3.5
2.5
1.5
0.5
4.5

2

3.5
2.5
1.5
0.5
0

20

40

60
80
100
120
Laboratory Angle [deg]

140

160

180

Figure B.1: Contributions to the total energy resolution FWHM in the Doppler reconstructed
gamma-ray spectra as a function of laboratory angle. The negligible contribution from the
intrinsic gamma-ray energy resolution of germanium is omitted. In Panel 1, the ∆θ (red)
and ∆β (black) components contribute to the total energy resolution (blue) via Eq. B.28.
Panel 2 plots the total energy resolution from different ∆θ contributions corresponding to
the detector-target distances of the forward (red) and backward (blue) rings of SeGA. See
text for details.
from the S800 target position. Panel 2 of Fig. B.1 compares the total gamma-ray energy
resolution at two detector-target distances corresponding to the backward ring (23.3 cm and
replotted in blue) and the forward ring (30.3 cm and plotted in red). Both curves have the
same ∆β component, but the forward ring detectors are further from the target position;
thus a single segment subtends an angle of 1.89◦ and ∆θ is reduced.
The dashed traces in the bottom panel of Fig. B.1 at θ = 31.85◦ and θ = 140.25◦
(corresponding to the true SeGA ring angles as discussed within Subsect. 3.3.1) indicate
that asymmetric placement of the rings with respect to the S800 target position results
in a better energy resolution in the backward ring than the forward ring even though the
157

opening angle of a single segment is larger in the backward ring. Using the estimates for ∆θ
and ∆β above, the energy resolution at FWHM in the forward and backward ring detectors
are calculated to be 2.21% and 1.96%, respectively (cf. Subsect. 3.2.3).
The deviation of the measured gamma-ray energy from the true energy in the rest frame
can also be quantified from Eq. B.27 with the first term instead corresponding to the uncertainty in the calibrated energy measurement of Doppler shifted gamma rays in the laboratory
frame, ∆E. As before, this term can be safely neglected; the calibrated energies from 152 Eu
and 226 Ra source data agree with the tabulated energies to better than 0.1%. Thus Eq. B.28
is used, but interpreted differently as ∆E /E is now taken to be the fractional uncertainty
in the Doppler reconstructed gamma-ray energy as opposed to the energy resolution. Using
∆θ and ∆β from above, the fractional uncertainties in the forward and backward rings are
found to be 1.2% and 0.8%, respectively. The value of 1.2% has been adopted for quoting the
uncertainty in Doppler reconstructed gamma-ray energies as the more conservative estimate.

158

Appendix C
Gamma-Ray Interactions with Matter
C.1

Interaction Mechanisms

There are three primary processes, hierarchically arranged by energy, by which a gamma ray
interacts within a detector. For gamma-ray detection with high-purity germanium crystals,
the end result of all three processes is a flow of charge carriers (electrons and holes) along
an externally applied electric field that forms the output current pulse. The processes differ
from one another in how the initial gamma-ray energy is transferred to the germanium semiconductor, promoting electrons from the valence band to the conduction band and creating
electron-hole pairs. The three processes listed in order of importance as a function of increasing incident gamma-ray energy are: photoelectric absorption, Compton scattering, and pair
production. To simplify the ensuing discussion, imagine all gamma-ray interactions occur in
the center of a very large germanium detector such that all secondary photons are eventually
detected. Hence, with the exception of electron binding energies, the total energy deposited
in the detector is equal to the incident photon energy.1
1 In reality, the finite detector size and/or interactions near the detector surface lead to

gamma-ray spectrum features indicative of partial energy deposition, such as the continuum
and edge of Compton scattering or the single and double escape peaks of pair production
159

C.1.1

Photoelectric Absorption

Photoelectric absorption is the process by which a low energy gamma ray (a few hundred
keV) disappears in the vicinity of an atom. In its place, an atomic electron from a deeply
bound orbital is ejected with energy

Ee− = Eγ − Eb .

(C.1)

Here, the electron energy is slightly less than the initial photon energy by the binding energy
of the electron’s orbital shell. The kinetic energy of the ejected electron is then absorbed by
the germanium detector by promoting valence electrons to the conduction band, resulting
in a detectable current flow. The probability of photoelectric absorption is enhanced for
absorber materials with high atomic numbers, as shown by the approximation adopted from
Ref. [133]
P ∝

Z 4.5
.
3.5
Eγ

(C.2)

This probability decreases sharply as a function of increasing gamma-ray energy. The superiority of sodium iodide (ZI = 53) over germanium (ZGe = 32) for the photoelectric
absorption process is evident from this equation.2
In addition, the orbital shell vacancy of the photoelectron is quickly filled by another electron which radiates its excess energy via a characteristic X-ray. This X-ray usually undergoes
photoelectric absorption as well, producing a second photoelectron such that, together with
the original photoelectron, their summed kinetic energy closely approximates the incident
[132].
2 For RDM lifetime measurements, the advantage of using germanium detectors rather
than sodium iodide scintillators is their superior energy resolution for resolving the fast
and slow photopeak components. With respect to the incident energy required to produce
one information carrier- 3 eV per electron-hole pair in Ge and 100 eV per photoelectron in
NaI(Tl)- the statistical fluctuation of charge collection is greatly suppressed in high-purity
germanium crystals.
160

gamma-ray energy. Note that this process of X-ray emission can continue iteratively or that
instead of an X-ray, an Auger electron can be emitted.

C.1.2

Compton Scattering

For gamma-ray energies ranging from several hundred keV to several MeV, Compton scattering is the most probable interaction between a gamma ray and detector. For the presented
spectroscopy of 18 C in Ch. 3, Compton scattering was the predominant effect. In this process, a gamma ray elastically scatters at an angle θ off an orbital electron in the detector
and deposits a portion of its energy to the recoiling electron. Neglecting the electron binding energy and assuming the electron to be initially at rest with a rest mass energy of
m0 c2 = 0.511 MeV, the new photon energy, Eγ , is given by

Eγ =

Eγ
.
Eγ
1+
(1 − cos(θ))
m0 c 2

(C.3)

The probability of Compton scattering increases with an increase in the number of available
scattering electrons, but decreases with increasing incident gamma-ray energy according to
the approximation [134]
P ∝

Z
.
Eγ

(C.4)

The Compton scattering process reduces the incident photon energy by transferring kinetic energy to the recoiling electron. This recoil electron then dissipates its energy in the
germanium crystal lattice in the same manner as a photoelectron does. The scattered photon energy is either absorbed via photoelectric absorption or subsequent Compton scattering
followed by photoelectric absorption.

161

C.1.3

Pair Production

Pair production can only occur in the Coulomb electric field of an atomic nucleus. In this
process, the gamma ray disappears and in its place an electron-positron pair are produced.
From energy considerations, the incident gamma-ray energy must be at minimum twice
the rest mass of the electron, or 1.022 MeV. Pragmatically speaking, the probability for
pair production is quite low until gamma-ray energies exceed several MeV. The gamma-ray
energy in excess of 1.022 MeV is converted to the kinetic energy of the electron-positron
pair. The electron loses its energy within the germanium crystal lattice as described for the
photoelectric absorption process. The positron does as well; however as it slows, it eventually
annihilates with an electron, producing a pair of 511 keV gamma rays. These gamma rays
will either Compton scatter or undergo photoelectric absorption, and thus the full incident
photon energy will be deposited within the detector. The probability of pair production
increases dramatically with increasing gamma-ray energy and the square of the absorber
atomic number.

C.2

Gamma-Ray Attenuation

Gamma-ray flux attenuation has a strong impact on the detection efficiency of an experimental setup. It occurs via absorption in experimental materials, such as the Köln/NSCL
plunger target and degrader, the beam pipe, and the aluminum housing for the germanium
crystals. The probability per unit length, σ, that a photon will be absorbed or scattered due
to photoelectric absorption, Compton scattering, or pair production is simply the sum of the
probability per unit length that any of those three processes will occur (σpe , σcs , σpp ) and is
given by
σ = σpe + σcs + σpp .

162

(C.5)

From the descriptions in Sect. C.1, the linear attenuation of gamma rays is clearly dependent
upon the photon energy and the density of the absorbing material. It is therefore more
common to characterize the probability of absorption per unit length by the mass attenuation
coefficient, which is simply σ scaled by the absorber density ρ.
For a collimated beam of monoenergetic photons impinging upon an absorber en route
to a detector, the rate of gamma-ray detection as a function of absorber thickness t, N (t),
will decrease exponentially according to
σ

− ρt
N (t) = N0 e−σt = N0 e ρ

(C.6)

where N0 is the gamma-ray detection rate in the absence of the absorber and the mass
thickness, ρt, has been introduced. This is a convenient parameter; indeed materials used in
the lab are often characterized by their mass thickness mg/cm2 .
Gamma ray attenuation is an important effect to consider in experiments where the integrated photopeak area is used to derive an experimental result such as a partial cross section.
Hence, many tables and calculations exist that document the mass attenuation coefficient
σ/ρ for pure elemental materials and common compounds over a wide range of photon energies (for example, Ref. [135] and references therein). Attenuation effects are also present in
RDM lifetime measurements with the Köln/NSCL plunger. Since the analysis depends upon
fitting the experimental photopeak intensity ratios with simulated lineshapes, it is imperative for the Geant4/ROOT simulation package to appropriately account for attenuation.
This was verified by comparing the measured source photopeak efficiency curves to their
simulated counterparts (see Fig. 2.11).

163

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