SINGLE-PARTICLE STRUCTURE OF NEUTRON-RICH SILICON ISOTOPES AND
THE BREAKDOWN OF THE N=28 SHELL CLOSURE
By
Steven Ragnar Stroberg

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

ABSTRACT
SINGLE-PARTICLE STRUCTURE OF NEUTRON-RICH SILICON ISOTOPES AND
THE BREAKDOWN OF THE N=28 SHELL CLOSURE
By
Steven Ragnar Stroberg
The well-known shell structure of stable atomic nuclei has been observed to change in
systems with extreme proton-neutron ratios. Understanding this changing structure can
provide insight into the underlying forces between nucleons, and how they can lead to shell
structure. The N = 28 shell gap, which disappears in the region of the nuclear chart centered
on 42 Si, is an interesting case because this gap is the first gap which requires the assumption
of a strong spin-orbit splitting in the shell-model framework.
Previous experimental work on the breakdown of the N = 28 shell gap has generally utilized collective observables which have then been interpreted though theoretical calculations
in terms of changing shell structure. In this work, one-proton and one-neutron knockout
reactions are used to obtain a complimentary, single-particle picture of this region, focusing
on the neutron-rich silicon isotopes 36 Si, 38 Si, and 40 Si. This isotopic chain connects the
closed-shell nucleus 34 Si to 42 Si, which does not reflect a good shell closure. Level schemes
of knockout residues 35,37,39 Si and 35,37,39 Al are constructed, with new levels identified for
each nuclide. These results, in conjunction with large-scale shell model and eikonal reaction
model calculations, generally support the interpretation developed to explain the collective
observables. They further emphasize and illuminate the importance of excitations across
both the N = 20 and N = 28 shell gaps for describing nuclides in this region, and provide
additional guidance for shell model effective interactions in describing this region.
In addition, an empirical method is introduced to model the asymmetry observed in the

momentum distributions by utilizing inelastic scattering reactions to approximate the dissipative effects of target breakup, producing good agreement with the measured distributions.

To my dad, for encouraging my critical thinking and mistrust of authority (although this
can at times cause some trouble), and for toughing it out through two (now three) bouts with
cancer to see me graduate for the third time. And to my mom, for sharing her boundless
energy (often in the form of care packages sent regularly for nearly a decade throughout my
undergraduate and graduate school careers), and for keeping dad going though the darkest
times. You are the best people I know.

iv

ACKNOWLEDGMENTS

The list of those who have helped me along the way to this thesis is long, and while this section
is correspondingly long, I am sure I have omitted many people, and for this I apologize.
I would like to thank my teachers, especially my high school teacher Ken Umholtz who
gave me my first real taste of physics, and Professor Rick Norman, who took me into his
group at Berkeley and gave me the opportunity to go to my first conference and write my
first paper.
I thank the members of my committee, all of whom have done far more for me than
show up to meetings and sign my paper work. My advisor, Professor Alexandra Gade,
gave me freedom to explore my interests without letting me founder, and spoke to me as
an equal, despite her overwhelmingly superior knowledge of nuclear physics. She has been
an adamant promoter of my career throughout my five years here, encouraging me to seek
out opportunities and writing outstanding letters of recommendation. The idea for this
project, like many other ideas within this thesis, was originally hers. Professor Vladimir
Zelevinsky taught me both physics and humility in the many classes I took with him, as well
as during the summer after my first year, in which he was willing to provide a project for a
young experimentalist who was curious about theory. Professor Remco Zegers sat on shift
with me during my first experiment, shortly after I arrived in grad school, and patiently
explained much of the equipment that I would eventually use for my own thesis experiment.
He also spent countless hours teaching and developing a course on electronics that served
me well during my time here. Professor Sean Liddick generously gave his time, equipment,
and expertise to help me with the project that makes up the first appendix of this thesis.
Professor Phil Duxbury introduced me to computational physics (and to Fortran, although
v

I’m unsure if I should thank him for that part) and took me to the Netherlands to learn
about it.
Although he was not on my committee, Professor Alex Brown certainly received more
than his fair share of visits from me, asking all manner of questions about nuclear structure.
I am always amazed by his ability to catch up in thirty seconds to an idea that I had mulled
for weeks. Likewise, Professor Jeff Tostevin has proved an invaluable resource for reaction
theory, and devoted a substantial amount of time, thought, and energy to the ideas and
language contained in this thesis and the resulting publication. Professors Dick Furnstahl
and Achim Schwenk, who taught the 2013 TALENT school on nuclear forces, taught me a
semester’s worth of material in a month (some of which made it into the pages of this thesis),
and inspired me to pursue a post-doc in nuclear theory.
I am also indebted to the members of the Gamma Group. Dr. Dirk Weisshaar seems to
know absolutely everything about detecting γ rays, and is often happy to share his knowledge.
My favorite time in any Gamma Group meeting is always when Dirk starts out with “Ya, well
you see...” because I know that, without fail, I’m about to learn something. Without Dirk
working round the clock, the entire GRETINA campaign, including my thesis experiment,
would have been impossible. Dr. Andrew Ratkiewicz sat down in my office and talked to
me the week I arrived, and quickly became my first friend at the lab, introducing me to
the DALMAC and to Crunchy’s. He also taught me most of what I know about the SeGA
filling system, which is still far less than he knows. Dr. Travis Baugher became one of my
closest friends, and was always happy to pull my out-of-shape butt through the painful last
miles of a long bike ride, even when, for example, a blizzard appeared out of nowhere. Dr.
Vinny Bader helped me learn things I should have already known by asking questions that
I wouldn’t have thought of, or that I had been too embarrassed to ask. Professor Kathrin
vi

Wimmer and Eric Lunderberg developed the GrROOT analysis framework that I used to
analyze the experimental data for this thesis. Having previously analyzed an experiment
without this framework, I can confidently say that their work made my life much easier than
it otherwise would have been. I’m also thankful for the members of the Lifetime Group,
Chris Morse, Kenneth Whitmore, and Charles Loelius, who gamely took their turns with
the filling system pager and helped to lighten that oppressive burden.
Jon Bonofiglio, Renan Fontus, Dr. Thomas Baumann, Dr. Mathias Steiner, and the
rest of the cycling group made me feel like I was a part of the lab, rather than just a
grad student passing through. The Tuesday morning basketball crew helped to break up
the hours upon hours of sitting that is graduate school. My friends Samantha Keeney, Dr.
Abigail Lynch, Damien Sheppard, and James Vatter became my family in Lansing, and
kept me aware that there was a world outside the lab. Older grad students, especially Dr.
Rhiannon Meharchand, Dr. Andrew Ratkiewicz, Dr. Angelo Signoracci, Dr. Phil Voss,
and Dr. Josh Vredevoogd quickly made me feel like I belonged here and provided guidance,
either verbal or by example. Rachel Showalter has been a terrific friend and office-mate for
five years, especially in those not-so-infrequent cases in which she corrected my algebra on
the whiteboard.
I thank the Oak Ridge Associated Universities for giving me the opportunity to go to
the meeting of Nobel Laureates in Lindau, Germany, where I got to meet not only famous
old physicists but also brilliant young physicists.
Finally, I of course must thank my friend, partner, confidant, counselor, and inspiration,
Dr. Jenna Smith. Few people would have the patience to work on physics all day, only to
come home to me rattling on about even more physics, but she generally manages to take it
all with a smile and then point out where I’m wrong. Who could ask for more?
vii

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

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

x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Nuclear structure and the “magic numbers” . . . . . . . . . . . . . . . . . .
1.2 The breakdown of the N=28 shell gap . . . . . . . . . . . . . . . . . . . . . .

1
1
7

Chapter 2 Theoretical background . . . . . . . . . . . . .
2.1 Nuclear forces and their impact on structure . . . . . . .
2.1.1 One-pion exchange . . . . . . . . . . . . . . . . .
2.1.2 Other contributions . . . . . . . . . . . . . . . . .
2.1.3 The role of the tensor force . . . . . . . . . . . .
2.2 Solving the many-body problem . . . . . . . . . . . . . .
2.2.1 Many-body states . . . . . . . . . . . . . . . . . .
2.2.2 Hartree-Fock . . . . . . . . . . . . . . . . . . . .
2.2.3 The shell model (configuration interaction) . . . .
2.2.4 Effective interactions for shell-model calculations
2.2.5 Effective operators for γ decay . . . . . . . . . . .
2.3 Probing single-particle states with knockout reactions . .
2.3.1 Factorization of the cross section . . . . . . . . .
2.3.2 The eikonal approximation . . . . . . . . . . . . .
2.3.3 Generating the optical potential . . . . . . . . . .
2.3.4 Densities and form factors . . . . . . . . . . . . .
2.3.5 Reaction mechanisms and cross sections . . . . .
2.3.6 Momentum distributions . . . . . . . . . . . . . .

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13
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14
15
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20
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25
27
30
30
33
35
37
42
45

Chapter 3 Experimental method and analysis . . .
3.1 Radioactive beam production . . . . . . . . . . . .
3.2 The S800 spectrograph . . . . . . . . . . . . . . . .
3.2.1 Particle trajectories . . . . . . . . . . . . . .
3.2.2 Time-of flight . . . . . . . . . . . . . . . . .
3.2.3 Energy loss and particle-identification . . . .
3.3 The GRETINA array . . . . . . . . . . . . . . . . .
3.3.1 Doppler reconstruction . . . . . . . . . . . .
3.3.2 Sub-segment position resolution . . . . . . .
3.3.3 Origin of the exponential γ-ray background
3.3.4 GRETINA simulations . . . . . . . . . . . .
3.3.5 Energy and efficiency calibrations . . . . . .

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47
47
50
50
53
55
58
59
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62
70
73

viii

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3.4

3.3.6 Extracting lifetimes from peak shapes . . . . . . . . . . . . . . . . . .
Asymmetric momentum distributions . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Results and interpretation . . . . . . . .
4.1 Shell model calculations . . . . . . . . . . . . . .
4.2 Level schemes for the knockout residues . . . . . .
4.2.1 35 Si . . . . . . . . . . . . . . . . . . . . .
4.2.2 35 Al . . . . . . . . . . . . . . . . . . . . .
4.2.3 37 Si . . . . . . . . . . . . . . . . . . . . .
4.2.4 37 Al . . . . . . . . . . . . . . . . . . . . .
4.2.5 39 Si . . . . . . . . . . . . . . . . . . . . .
4.2.6 39 Al . . . . . . . . . . . . . . . . . . . . .
4.3 Knockout cross sections . . . . . . . . . . . . . .
4.3.1 Inclusive cross sections to all bound states
4.3.2 Exclusive cross sections to final states . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . .
Chapter 5

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75
77
82
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87
96
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105
105
112
114
114
119
123

Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 132

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Digital pulse-shape analysis for the localization of
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B Derivations of formulas . . . . . . . . . . . . . . .
Appendix C Parameters used in knockout reaction calculations

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γ-ray interaction
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135
136
147
164

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

ix

LIST OF TABLES

Table 2.1:

Table 2.2:

Table 4.1:

Table 4.2:

The monopole components of the SDPF-MU interaction between two
nucleons in orbits a and b, coupled to total isospin T , broken down
into central, spin-orbit, and tensor components. Energies are in MeV.
Some terms have been repeated for easier comparison. . . . . . . . .

19

Results for the two-body core-valence wave function for 35 Si+n, using
several prescriptions for determining the Woods-Saxon parameters:
Hartree-Fock (HF), well-depth (WD), changing the radius (∆R), and
changing the spin-orbit depth (SO). Primed prescription labels indicate a starting point of V0 =55, r0 =1.2, and VLS =6.0, instead of the
Hartree-Fock result. Energies are in MeV, radii are in fm, and cross
sections are in mb. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 35 Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 35 Al. Levels marked with an asterisk are tentative. . . . . . . . .

98

Table 4.3:

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 37 Si. Levels and transitions marked with an asterisk are tentative. 104

Table 4.4:

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 37 Al. Levels marked with an asterisk indicate a tentative assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table 4.5:

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 39 Si. Levels marked with an asterisk are tentative. . . . . . . . . 111

Table 4.6:

Gamma-ray energies, efficiency-corrected intensities, and coincidences
for 39 Al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Table 4.7:

Total and partial knockout cross sections to bound final states for
each of the knockout reactions studied. . . . . . . . . . . . . . . . . 124
Calculated spectroscopic factors to low-lying final states in 33,35,37,39 Si.125

Table 4.8:

x

Table 4.9:

Comparison of T = 1 (neutron-neutron) monopoles for the SDPF-U
and SDPF-MU interactions, evaluated for A = 42. The single-particle
energy gaps between the f7/2 and d3/2 are 4.325 and 3.147 MeV for
SDPF-U and SDPF-MU, respectively. . . . . . . . . . . . . . . . . . 131

Table B.1:

Coefficients of the Talmi integrals Ip for the calculation of the T = 1
monopole term for a tensor interaction Vt = X (2) · Y (2) f (r) for the
given orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Table C.1:

One-neutron and one-proton separation energies, in MeV, from [1]. . 165

Table C.2:

Calculated cross sections for a two-step 36 Si(0+ ) → 36 Si(2+ ) →35 Si(J π )
process. Energies and spectroscopic factors are from shell model calsp
culations with the SDPF-MU interaction. σ2step is the cross section
for excitation to 36 Si(2+ ), followed by a neutron knockout, assuming
a normalized neutron single-particle state. The last column gives the
summed cross section for a given final state in 35 Si. Only final states
which could decay via an E1 transition to the 7/2− ground state of
35 Si are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Table C.3:

Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 34 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 73.4 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 34 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 73.4 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 36 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 97.7 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 36 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 97.7 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 36 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 97.7 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .

Table C.4:

Table C.5:

Table C.6:

Table C.7:

xi

167

167

168

169

170

Table C.8:

Table C.9:

Table C.10:

Table C.11:

Table C.12:

Table C.13:

Table C.14:

Table C.15:

Table C.16:

Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 36 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 97.7 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 38 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 86.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 38 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 86.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 38 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 86.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 38 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 86.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 40 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 79.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for neutron knockout from 40 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 79.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 40 Si using level energies and
spectroscopic factors from the SDPF-MU interaction. The beam energy is 79.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . .
Parameters used in the calculation and the resulting theoretical cross
fi
sections σα for proton knockout from 40 Si using level energies and
spectroscopic factors from the SDPF-U interaction. The beam energy
is 79.0 MeV/u on a 9 Be target. . . . . . . . . . . . . . . . . . . . . .

xii

171

172

173

174

175

176

177

178

179

LIST OF FIGURES

Figure 1.1:

Figure 1.2:

Figure 1.3:

Figure 1.4:

Figure 2.1:

Figure 2.2:

The difference between experimental binding energies and binding
energies calculated within the liquid drop model, as a function of
neutron number N . Nuclides with Z ≥ 8 are shown. An excess of
binding energy can be clearly seen for neutron numbers 28, 50, 82,
and 126. Binding energy data taken from [1]. . . . . . . . . . . . . .

3

Eigenvalues for the Schr¨odinger equation for neutron orbits in 100 Sn.
In the spectrum labeled WS, the mean field potential is modeled as
a Woods-Saxon with parameters V0 = 50 MeV, r0 = 1.25 fm, and
a0 = 0.7 fm. The spectrum labeled WS+LS includes a spin orbit
term with strength VLS = 20 MeV. Note the (unlabeled) sub-shell
gap at 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Experimental energies of the first 2+ states as a function of neutron
number for the Ca, Ar, S, and Si isotopic chains. Data taken from
[2]. For the interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this
thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Ratio of the energy of the first 4+ state to the energy of the first 2+
state in silicon isotopes with 22 ≤ N ≤ 28. Experimental points from
Takeuchi et al. [3] are indicated with black dots, while shell model
calculations with two different effective interactions are indicated with
sold lines. The dashed lines indicate the simple vibrational (2.0) and
rotational (3.33) model predictions (see text). . . . . . . . . . . . . .

12

Diagrams for (a) a two-body interaction mediated by pion exchange,
and (b) a three-body interaction in which the middle nucleon is excited into a resonance – for example, the ∆(1232) resonance. . . . .

16

Value of the tensor operator S12 for some simple configurations of
two particles (indicated by circles) with spin (indicated by arrows). .

17

xiii

Figure 2.3:

Schematic representation of the effect of the tensor force when adding
neutrons to the f7/2 orbital. The monopole component of the tensor
force is attractive between the neutron f7/2 and proton d3/2 (indicated with the blue wavy line), while it is repulsive between the neutron f7/2 and proton d5/2 (indicated with the red wavy line). This
results in the reduction of the spin-orbit splitting in the proton d
orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

The 1 S0 channel of two phenomenological nucleon-nucleon potentials
[4, 5] which reproduce N N scattering data. . . . . . . . . . . . . . .

26

Schematic depicting a projectile nucleon scattering off of a single
target nucleon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Calculated proton and neutron densities for (a) a 35 Si core, calculated with Skyrme Hartree-Fock, and (b) a 9 Be target, from quantum
Monte Carlo calculations. The oscillations at small R for the QMC
calculation are statistical. The thin lines show the Gaussian densities
used for the target in this work. . . . . . . . . . . . . . . . . . . . .

38

Core-valence wave functions (dashed lines) for neutron orbits near
the Fermi surface in 36 Si, calculated with the well-depth prescription,
shown with their corresponding Woods-Saxon potentials (filled blue
curves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Results of three different prescriptions for obtaining the two-body
wave function (see text for details) for the 1p3/2 orbit in silicon isotopes 34−42 Si. (a) Rms relative radii (b) single particle cross sections
(c) the ratio between results for the well-depth prescription (WD)
and the potential radius prescription (∆R). . . . . . . . . . . . . . .

43

Illustrations of the three reaction mechanisms which contribute to the
knockout cross section. . . . . . . . . . . . . . . . . . . . . . . . . .

43

Figure 2.10: Momentum distributions calculated according to equation (2.45) for
neutron removal from orbitals near the Fermi surface in 36 Si. . . . .

46

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Figure 2.9:

Figure 3.1:

A schematic of the beam production at the Coupled Cyclotron Facility. 48

Figure 3.2:

A schematic of the S800 spectrograph. The secondary beam enters
from the left side and reacts at the target position. The reaction
residues are detected in the focal plane. . . . . . . . . . . . . . . . .

xiv

51

Figure 3.3:

Figure 3.4:

Figure 3.5:

Figure 3.6:

Figure 3.7:

A schematic of the S800 focal plane. Taken from [6], modified from
[7]. The CRDCs are separated by about one meter. . . . . . . . . .

51

Calibrated CRDC1 x and y positions for a mask run. The calibration
is performed using the known locations of the slits and pinholes in
the mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

(a) Time of flight parameter OBJ vs x position in the focal plane of
the S800 for the 1n knockout setting from 38 Si. (b) Corrected time of
flight parameter OBJC vs x. (c) OBJ vs angle afp in the dispersive
direction, and (d) OBJC vs afp. . . . . . . . . . . . . . . . . . . . .

54

Particle identification plots for each setting, with the gate on the
knockout residue. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Additional gates used to improve particle identification for 35 Si. (a)
The PID gated on incoming 36 Si shows poor separation between 35 Si
(the strongest blob) and 36 Si, lying to the left. (b) Dispersive angle
in the focal plane vs dispersive position in the focal plane gated on
the 35 Si PID blob, showing structures due to scattering of unreacted
beam. (c) PID resulting from cuts applied to (b) producing a much
cleaner separation. (d) The result of those cuts in angle vs position,
again gated on the 35 Si blob. . . . . . . . . . . . . . . . . . . . . . .

57

Figure 3.8:

The retracted northern (a) and southern (b) hemispheres of the GRETINA
array. When the array is in use, the two hemispheres are joined and
the detectors surround the target position. Images courtesy of S. Noji. 58

Figure 3.9:

A demonstration of the angular coverage of GRETINA. Four modules
were positioned in a ring at θ ≈ 55◦ , and the remaining three modules
were placed at θ ≈ 90◦ . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10: Gamma-ray energy spectra in coincidence with 35 Si obtained by summing all interactions within one crystal (blue filled histogram) and
by the add-back procedure described in the text (red histogram). . .

xv

60

62

Figure 3.11: Demonstration of the time structure of the γ-ray background. The
top panel shows the γ-ray spectrum in coincidence with 21 Ne detected
with LaBr3 detectors, with the time difference between the LaBr3
trigger and the particle trigger shown in the inset. The bottom figure
shows the γ-ray spectrum gated on the prompt peak in the timing
spectrum, indicated with the gray block. Most of the exponential
background is removed. The inset in the bottom figure shows the γ
spectrum gated off of the prompt peak, revealing an essentially pure
background spectrum. Reprinted with permission from [8]. . . . . .

64

Figure 3.12: (a) Parallel momentum distribution for outgoing 36 Si particles, with
two gates indicated. (b) The γ-ray spectrum for each of the two
momentum cuts indicated in (a), normalized to γ-rays per detected
ion. (c) The momentum distribution gated on no detected γ rays
(blue hatches), and γ rays with energy above 1 MeV (solid red). . .

66

Figure 3.13: Polar angle distribution, normalized to a 226 Ra source measurement,
for the 2+ → 0+ transition in 36 Si (blue triangles), the exponential γ-ray background above 1 MeV (magenta circles), and two other
source measurements (black diamonds and red squares) as a consistency check. The long-dashed line indicates an isotropic distribution
in the lab frame, and the short-dashed line indicates an isotropic distribution in a frame moving with beam velocity. The error bars are
statistical only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Figure 3.14: Distribution of the azimuthal angle φ between the direction of the
scattered beam and the direction of the detected γ ray. The red
circles show the distribution for γ rays with energy greater than 2
MeV, while the black diamonds show the distribution for low-energy
γ rays from atomic processes, which should be uncorrelated with the
beam direction. The dashed and dotted lines show fits with the two
lowest-order even Legendre polynomials. . . . . . . . . . . . . . . . .

69

Figure 3.15: Simulated γ ray detection efficiency compared to efficiency measured
with two sources. Note the discrepancy at low energy. . . . . . . . .

74

Figure 3.16: Simulated γ ray detection efficiency, with correction for losses at low
energy, compared to efficiency measured with two sources. The simulation now reproduces the measured efficiency from 100 keV to 3.5
MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

xvi

Figure 3.17: Experimental momentum distributions to specific final states, compared to eikonal model predictions folded with the incoming momentum distribution (blue dashed lines), or folded with the distribution
obtained from the inelastic setting (solid red curves, see text for details). 81
Figure 4.1:

Figure 4.2:

Figure 4.3:

Figure 4.4:

Figure 4.5:

Figure 4.6:

(a) The sdpf model space used in shell model calculations. The gray
boxes indicate excluded orbits. (b) The dependence of the calculated
energy spectra on the center-mass-parameter βCM (see text) for 35 Si
using the SDPF-MU interaction. The green and magenta lines indicate positive and negative parity states, respectively. . . . . . . . . .

83

A comparison of the matrix elements of the SDPF-MU interaction vs
the matrix elements of the SDPF-U interaction. If the interactions
were identical, all points would lie along the diagonal lines. Only the
T = 1 component of the f p interaction is shown, because protons are
not allowed into the f p shell in these calculations. . . . . . . . . . .

86

Doppler-reconstructed γ-ray spectrum detected in coincidence with
35 Si. The 908 keV transition is broadened by a lifetime effect. The inset figures show background-subtracted γγ coincidence spectra gated
on the 908 and 1134 keV transitions. The Doppler reconstruction was
performed with a velocity v/c = 0.426. . . . . . . . . . . . . . . . . .

88

Maximum likelihood fit of the lifetime of the state decaying by a 908
keV γ ray in 35 Si. The upper-left panel shows the negative log likelihood as a function of simulated lifetime, while the lower-left panel
demonstrates the effect of the lifetime uncertainty on the extracted
γ-ray intensity. The uncertainties shown in the figure are statistical
only. In the right panels, the magenta lines show the simulation with
the best fit lifetime for three different rings of GRETINA. The exponential background discussed in §3.3.3 is shown filled in dark gray,
and the lighter blue-filled curve shows the additional background due
to higher-energy transitions. . . . . . . . . . . . . . . . . . . . . . .

89

Parallel momentum distributions for the population of levels at 1688,
2042, 2164, and 2377 keV in 35 Si by neutron knockout from 36 Si. . .

90

Proposed level scheme for 35 Si from this work compared with shellmodel calculations (see text for details). The widths of the arrows
are proportional to the efficiency-corrected γ-ray intensity. . . . . . .

92

xvii

Figure 4.7:

The left panel shows the 3611 keV gamma ray peak (indicated with
an arrow) detected in coincidence with 35 Si, with an inset showing the
gates used on the outgoing parallel momentum distribution. The right
panel shows the γ spectra gated on the main peak in the momentum
distribution (blue hatches) and the low-momentum tail (solid red).
The 3611 keV peak appears to be associated with the tail of the
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Schematic showing the competition between γ decay and neutron
emission for the 3611 keV state in 35 Si. The 35 Si level scheme includes
a resonance at 5.5 MeV observed in [9]. The 34 Si level scheme includes
the recently-proposed first excited 0+ state [10]. The ground state of
34 Si appears to be the only final state for which neutron emission is
energetically favorable. . . . . . . . . . . . . . . . . . . . . . . . . .

94

Doppler-reconstructed γ-ray spectrum detected in coincidence with
35 Al. The section of the spectrum in the box labeled ×4 has been
rebinned by a factor 4. The blue dashed line shows the fitted background, suggesting a peak at 4275 keV. The inset in the upper-right
shows background-subtracted γγ coincidence matrices gated on the
1003 and 2237 keV transitions. The Doppler reconstruction was performed with a velocity v/c = 0.424 . . . . . . . . . . . . . . . . . . .

97

Figure 4.10: Parallel momentum distributions for the population of levels at 1972,
3243, and 4275 keV in 35 Al by proton knockout from 36 Si. . . . . .

98

Figure 4.11: Proposed level scheme for 35 Al from this work compared with shellmodel calculations (see text for details). The width of the arrows is
proportional to the efficiency-corrected γ-ray intensity. . . . . . . . .

99

Figure 4.8:

Figure 4.9:

Figure 4.12: Doppler-reconstructed γ-ray spectrum detected in coincidence with
37 Si. The section of the spectrum in the box labeled ×2 has been
rebinned and scaled by a factor of 2. The inset shows the backgroundsubtracted γγ coincidence matrix gated on the peaks at 156 and 903
keV. The Doppler reconstruction was performed with a velocity v/c =
0.403. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 4.13: Parallel momentum distributions for the population of the ground
state and excited states at 692 and 717 keV in 37 Si by neutron knockout from 38 Si. Note that the ground state distribution includes both
−
the 5/2−
1 and 7/21 states. . . . . . . . . . . . . . . . . . . . . . . . 101

xviii

Figure 4.14: Maximum likelihood fit of the lifetime of the state decaying by a 156
keV γ ray in 37 Si. The upper-left panel shows the negative log likelihood as a function of simulated lifetime, while the lower-left panel
demonstrates the effect of the lifetime uncertainty on the extracted
γ-ray intensity. The uncertainties shown in the figure are statistical
only. In the right panels, the magenta lines show the simulation with
the best fit lifetime for three different rings of GRETINA. The exponential background discussed in §3.3.3 is shown filled in dark gray,
and the lighter blue-filled curve shows the additional background due
to higher-energy transitions. . . . . . . . . . . . . . . . . . . . . . . 102
Figure 4.15: Level scheme for 37 Si from this work compared with shell-model calculations (see text for details). The width of the arrows is proportional
to the γ-ray intensity. Fine dashed lines indicate tentative levels and
transitions, while the thicker dashed line labeled Sn indicates the
neutron separation energy, with the gray shaded area indicating the
uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 4.16: Doppler-reconstructed γ-ray spectrum detected in coincidence with
37 Al. The section of the spectrum in the box labeled ×2 has been
rebinned and scaled by a factor of 2. The inset shows the backgroundsubtracted γγ coincidence matrix gated on the peak at 775 keV. The
Doppler reconstruction was performed with a velocity v/c = 0.400. . 106
Figure 4.17: Proposed level scheme for 37 Al from this work compared with shellmodel calculations (see text for details). The width of the arrows is
proportional to the efficiency-corrected γ-ray intensity. . . . . . . . . 107
Figure 4.18: Doppler-reconstructed γ-ray spectrum detected in coincidence with
39 Si. The Doppler reconstruction was performed with a velocity
v/c = 0.379. The inset shows the background-subtracted γγ coincidence matrix gated on the peaks at 172 and 879 keV, revealing no
strong coincidences with either peak. . . . . . . . . . . . . . . . . . 108
Figure 4.19: Maximum likelihood fit of the lifetime of the state decaying by a 172
keV γ ray in 39 Si. The upper-left panel shows the negative log likelihood as a function of simulated lifetime, while the lower-left panel
demonstrates the effect of the lifetime uncertainty on the extracted
γ-ray intensity. The uncertainties shown in the figure are statistical
only. The right panels show the best fit for three different rings of
GRETINA. The exponential background discussed in §3.3.3 is shown
filled in dark gray, and the lighter blue-filled curve shows the additional background due to higher-energy transitions. . . . . . . . . . 109

xix

Figure 4.20: Parallel momentum distributions gated on γ-ray transitions in 39 Si
(no feeding subtraction). . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 4.21: The left figure shows the proposed level scheme for 39 Si. The right figure is the level scheme proposed by Sohler et al. [11]. In the left figure,
the width of the arrows in proportional to the efficiency-corrected γray intensity. As described in the text, the levels could have an offset
if the lowest 3/2− state lies below the 5/2− . . . . . . . . . . . . . . 112
Figure 4.22: Doppler-reconstructed γ-ray spectrum detected in coincidence with
39 Al. The Doppler reconstruction was performed with a velocity
v/c = 0.388. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 4.23: A possible level scheme for 39 Al, although γγ cascades cannot be
ruled out due to low statistics. The center and right figures are shellmodel calculations (see text for details). . . . . . . . . . . . . . . . . 114
Figure 4.24: Knockout cross section to all bound states for each of the six settings,
calculated on a run-by-run basis. The error bars on each point are
statistical errors for counting the number of events inside the PID
gate. The dashed black line shows the error-weighted mean value of
all the runs, while the gray band indicates the uncertainty, which is
taken from the root mean squared deviation of the points from the
mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 4.25: Inclusive 1n and 1p knockout cross sections from silicon isotopes to
all bound states as a function of projectile neutron number, compared with theoretical prediction (solid red line) and the theoretical
prediction scaled by a systematic reduction factor R(∆S) [12]. The
theoretical error bars are generated by varying the neutron separation
energy by ±500 keV. The experimental value for 34 Si is taken from [13]118
Figure 4.26: One neutron knockout cross section to final states in 35 Si. Experimental data is shown in the top panel, and theoretical predictions are
shown in the bottom two panels. . . . . . . . . . . . . . . . . . . . . 120
Figure 4.27: One neutron knockout cross section to final states in 37 Si. Experimental data is shown in the top panel, and theoretical predictions are
shown in the bottom two panels. . . . . . . . . . . . . . . . . . . . . 120
Figure 4.28: One neutron knockout cross section to final states in 39 Si. Experimental data is shown in the top panel, and theoretical predictions are
shown in the bottom two panels. . . . . . . . . . . . . . . . . . . . . 121
xx

Figure 4.29: One proton knockout cross section to final states in 35 Al. Experimental data is shown in the top panel, and theoretical predictions
are shown in the bottom two panels. . . . . . . . . . . . . . . . . . . 121
Figure 4.30: One proton knockout cross section to final states in 37 Al. Experimental data is shown in the top panel, and theoretical predictions
are shown in the bottom two panels. . . . . . . . . . . . . . . . . . . 122
Figure 4.31: One proton knockout cross section to final states in 39 Al. Experimental data is shown in the top panel, and theoretical predictions
are shown in the bottom two panels. . . . . . . . . . . . . . . . . . . 122
Figure 4.32: Partial cross sections for the population of a bound final state with
J π = 7/2− and 3/2− in a one-neutron knockout reaction as a function
of mass number for the silicon projectile. Theoretical predictions
using the SDPF-MU (solid lines) and SDPF-U (dashed lines) effective
interactions are also shown. The dotted line shows the result for the
SDPF-MU interaction with the cross-shell tensor component set to
zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 4.33: Experimental cross sections to the lowest 5/2+ , 3/2+ , 1/2+ states
compared with the calculated cross sections using energies and spectroscopic factors from SDPF-MU (solid lines) and SDPF-U (dashed
lines). The data for 34 Si are taken from [13]. . . . . . . . . . . . . . 127
Figure 4.34: Energy difference between the J1π and 7/2−
1 levels in silicon isoπ
+
+
topes, where J = 3/2 , 1/2 . Experimental data are indicated
with points, while the shell model predictions are connected by lines.
See the text for how the 34 Si values were obtained. The shaded bars
indicate the SDPF-MU spectroscopic factors for one neutron removal
+
(the sum rule limit is 4 for the 3/2+
1 state and 2 for the 1/21 state). 130
Figure A.1:

An illustration of using induced charge to obtain position information.
(a) If the charge is collected on the left side of the center detector,
then the magnitude of the induced charge is larger in the left detector
than in the right detector. (b) Vice versa. . . . . . . . . . . . . . . 137

Figure A.2:

An illustration of using induced charge to obtain position information.
(a) If the charge is collected on the left side of the center detector,
then the magnitude of the induced charge is larger in the left detector
than in the right detector. (b) Vice versa. . . . . . . . . . . . . . . 139

xxi

Figure A.3:

A SeGA detector consists of a single HPGe crystal electronically divided into eight longitudinal slices and four azimuthal quarters, with
the cathode running down the center. . . . . . . . . . . . . . . . . . 140

Figure A.4:

(a) An illustration of the measurement taken, with three different
positions (indicated A, B, C) for the collimated 137 Cs source, each
aligned to illuminate a different region of the same segment. (b) An
example of the recorded signals in the segment with a net energy
deposited and two neighboring segments for an event with the source
at position A. (c) The resulting histograms of the linear interpolation
z parameter for each position, demonstrating subsegment resolution. 141

Figure A.5:

(a) Setup for the rough radial position measurement. (b) The two
alignments used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure A.6:

An example of the steepest slope algorithm for a single trace, showing the measured signal, a numerical derivative of that signal, and a
numerical second derivative, yielding the steepest slope time tss . In
(a), the parameters d = 1, w=0 are used, while in (b) the parameters
d = 3, w = 1 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Figure A.7:

The results of the rough radial position measurement showing (a) the
steepest slope parameter for each run (b) the azimuthal interpolation parameter for each run and (c) the combined (r, φ) coordinates
predicted by the pulse shape analysis. The black curves indicate the
geometry of the germanium crystal. . . . . . . . . . . . . . . . . . . 145

Figure A.8:

+
The Doppler-reconstructed γ peak corresponding to the 2+
1 → 01
transition in 16 C, at a beam velocity of β=0.36. The dashed black
line shows the peak obtained with the electronic segmentation of
SeGA (FWHM = 1.24%), while the solid red line shows the peak
obtained with subsegment resolution from digital signal processing
(FWHM=0.92%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Figure B.1:

Decomposition of two = 1 orbits in lab frame coordinates into rela2

tive and center-of-mass coordinates. For each component, Ym (θ, φ)
is shown. On the right side of the equation, the shapes represent the
relative wave function, while the center-of-mass motion is indicated
by a curved arrow. The straight arrows indicate the semi-classical
direction of the angular momentum vector. . . . . . . . . . . . . . . 157

xxii

Chapter 1
Introduction

All my life through, the new sights of Nature made me rejoice like a child.
(M. Curie)

1.1

Nuclear structure and the “magic numbers”

The atomic nucleus was proposed in 1911 by Rutherford [14], based on the experiments of
Geiger and Marsden [15], and a century later it remains the subject of vigorous experimental
and theoretical research. This is a testament to the depth and complexity of the problem of
understanding the nucleus.
At first consideration, one might think that attempting to describe the nucleus with
high precision would be a hopeless endeavor. The protons and neutrons (collectively called
nucleons) which make up the nucleus interact via all four known forces of nature 1 , and are
confined to a region of a few fermi (1 fm = 10−15 m) by the strong nuclear force. These
nucleons are bound by roughly 5-10 MeV, which, by the uncertainty principle, implies that
the spatial extent of their wave functions should be (2mE)−1/2 ∼ 1.5 fm, indicating that
this is very much a quantum system. Furthermore, because nucleons carry spin 1/2, they
obey Fermi-Dirac statistics. This requires that the wave functions of the individual nucleons
should be correlated with each other, in order to avoid having two nucleons simultaneously
1 Gravity

is, of course, irrelevant to the considerations of nuclear structure.

1

in the same state. Finally, there exists no simple analytic form for the strong interaction
between nucleons, in contrast to the Coulomb potential of atomic physics.
Nonetheless, some simple properties do emerge in the structure of nuclei, one of the most
notable being the appearance of the so-called magic numbers. As a simple first approximation, we may model a nucleus as a drop of nuclear liquid, and calculate the binding energy
[16]. We take a term which is attractive in the bulk – proportional to the volume, which
for a liquid should be proportional to the number of nucleons A – and add terms for surface
tension (proportional to A2/3 ) and the Coulomb repulsion of the charged protons. Further,
since there are two kinds of nucleon, which each obey the Pauli exclusion principle, we would
expect that for a given A, nuclei with an equal number of protons Z and neutrons N , should
have a lower energy than asymmetric nuclei. Finally, since there is a noted staggering of
binding energies between even and odd numbers of protons or neutrons, we add a pairing
term which is attractive for even N and even Z and repulsive for odd N and odd Z.



aP A−1/2
N odd and Z odd




δ(N, Z) = −aP A−1/2 N even and Z even






0
N + Z odd

(1.1)

This results in the semi-empirical mass formula [16]:

BE = aV A − aS A2/3 − aC

Z2
A1/3

− aA

(N − Z)2
+ δ.
A

(1.2)

The parameters of this model can be fit to experimental data, and give a quite good
reproduction of the binding energies across most of the chart. Figure 1.1 shows the difference
between the experimental binding energies [1] and those calculated with (1.2) using the
2

Exp. Binding Energy - Liquid Drop Energy (MeV)

15

10

5

0

-5

-10
0

20

40

60
80
100
Neutron number N

120

140

160

Figure 1.1: The difference between experimental binding energies and binding energies calculated within the liquid drop model, as a function of neutron number N . Nuclides with
Z ≥ 8 are shown. An excess of binding energy can be clearly seen for neutron numbers 28,
50, 82, and 126. Binding energy data taken from [1].

3

parameters (in MeV) aV = 15.54, aS = 17.23, aC = 0.697, aA = 22.6, aP = 12.0 (cf . [17]).
The first feature we note is that the binding energies are reproduced to within 10 MeV for
almost all nuclei, while nuclear binding energies are of the order of 10 MeV per nucleon.
Second, it is clear that there is some additional structure to the binding energies. Nuclei
with 28, 50, 82, and 126 neutrons (as well as 2, 8 and 20, although these are more difficult
to see here) are particularly well-bound with respect to their neighbors. They also exhibit
high neutron separation energies, and high first excited states. The same effect is observed
for nuclei with the equivalent number of protons. These numbers, called magic numbers (a
term likely coined by Wigner [18]), suggest that some additional structure beyond the liquid
drop picture makes these nuclei particularly stable.
By the 1930’s the presence of these magic numbers had been remarked upon [19], and
some properties of nuclei had been interpreted in terms of a shell model [20], but a microscopic
picture which robustly predicted these numbers remained elusive. As an illustration, consider
a nucleon moving in a mean field potential, generated by all the other nucleons in the nucleus.
It is known that the force between nucleons is short-ranged, so the shape of the potential
should be similar to the shape of the density. A simple parameterization is given by the
Woods-Saxon form:
V (r) = −V0 fws (r)
fws (r) =

1
1 + e(r−R0 )/a0

(1.3)
.

(1.4)

Typical values for the parameters are V0 = 50 MeV, R0 = 1.25A1/3 fm, and a0 = 0.7 fm
[21]. Solving the Schr¨odinger equation for this potential, we obtain allowed energies which
are shown on the left side of Figure 1.2. This reproduces the magic numbers 2, 8, and 20,
but fails for numbers beyond.
4

1d3/2
0g
7/2
2s1/2
1d5/2

2s
1d
0g

50
0g

40

9/2

1p
1/2
0f5/2
1p

1p

3/2

28

0f

0f7/2

20

20
1s1/2
0d3/2

1s
0d

0d5/2

8

8
0p
1/2
0p

0p

3/2

2

2
0s1/2

0s

WS

WS+LS

Figure 1.2: Eigenvalues for the Schr¨odinger equation for neutron orbits in 100 Sn. In the
spectrum labeled WS, the mean field potential is modeled as a Woods-Saxon with parameters
V0 = 50 MeV, r0 = 1.25 fm, and a0 = 0.7 fm. The spectrum labeled WS+LS includes a spin
orbit term with strength VLS = 20 MeV. Note the (unlabeled) sub-shell gap at 14.

5

In 1949, it was suggested independently by Mayer [22], and Haxel, Jensen, and Suess
[23] that one should make a further refinement and assume a strong, attractive, one-body
spin-orbit potential, given by

VLS (r) = ( · s) VLS

d
fws (r).
dr

(1.5)

The radial form of the potential is chosen to make it surface-peaked. This is because in the
interior of the nucleus we would expect the spin-orbit forces from all directions to cancel
out, while at the surface there is no such cancellation. The addition of this term, with
VLS = 20 MeV, gives levels which are shown on the right side of Figure 1.2. With this
adjustment, all observed magic numbers (at stability) are reproduced. However, as observed
by Goeppert-Mayer:

There is no adequate theoretical reason for the large observed value of the spin
orbit coupling. The Thomas splitting has the right sign, but is utterly inadequate
in magnitude to account for the observed values. A proper type of meson potential can be made to predict splitting qualitatively similar to the Thomas splitting
and therefore qualitatively similar to the observed, but greater in magnitude than
the Thomas splitting although usually somewhat less than the observed value.
(M. G. Mayer [24])
(The Thomas term to which she refers is a relativistic effect acting on a vector, in this case
the spin, in non-uniform motion, which contributes to the spin-orbit term in atomic fine
structure splitting [25].)
Relativistic mean field (RMF) theories, initiated in the 1950s [26, 27] and developed
further in the 1970s [28], do in fact predict a strong spin-orbit splitting, relying on the
combination of strong scalar and vector potentials which add destructively for the central
potential, and constructively for the spin-orbit potential (see [29] for an overview). These
6

RMF theories are phenomenological (they often omit explicit pion fields) and so the magnitude of these potentials is fit to data. However, if the potentials are constrained by the
requirement of saturation, the empirical spin-orbit splitting is then reproduced without further adjustments (see, for example [30] and references therein).
In a non-relativistic framework, spin-orbit splitting arises from various sources, including
a two-body spin-orbit force between nucleons [9], tensor forces [31], and three-body forces
[32]. The exact degree to which each of these sources contributes, and the correspondence
between the sources in the relativistic and non-relativistic frameworks, remains an open
question. It has even been suggested that there may be cancellations between the various
sources of spin-orbit splitting [33], and that these cancellations may change for different situations. It is therefore interesting to investigate how the observed spin-orbit splitting evolves
with different parameters, such as proton-neutron asymmetry, in order to help elucidate the
various contributions.

1.2

The breakdown of the N=28 shell gap

Recent developments in the study of exotic nuclei have shown that far away from stability,
magic numbers can disappear, and new ones can emerge [34]. This changing shell structure
can provide insight into how the various components of the nuclear interaction – for example
the spin-isospin [35] or tensor [31] components – contribute to the observed structure.
I concentrate here on the magic number N = 28, as it is the first magic number that
arises in the mean field picture due to the empirical strong spin-orbit potential. It is hoped
that an understanding of how this shell gap changes will provide complimentary information
to that obtained by investigating the N = 8 and N = 20 gaps, which arise due to the mean

7

field alone.
Early suggestions of the disappearance of the N = 28 shell gap came from measurements
of the energy of the first J π = 2+ state and reduced quadrupole transition probability
+
44
B(E2; 2+
1 → 01 ) in the neutron-rich nucleus S [36]. I will briefly outline, in a qualitative

way, why these quantities are indicators of the presence of a shell gap.
In a simple collective model along the lines of the liquid drop picture, we can consider
a vibrational excitation of a spherical nucleus (see for example [37]). The lowest-energy
excitation should be a single quadrupole phonon excitation (a monopole corresponds to the
compression of the nuclear fluid, and a dipole corresponds to the translation of the center of
mass, which is not an intrinsic excitation). A quadrupole phonon has J π = 2+ and energy
ω, where ω is the vibrational frequency. If the drop is tightly bound, it will be more “rigid”
and vibrate at a higher frequency, and thus the 2+ one-phonon excitation will lie at a higher
energy.
If we now consider an electromagnetic transition from the excited 2+ state to the 0+
ground state, we see that this transition must be an electric quadrupole (E2) transition.
If the 2+ state is well described by a quadrupole phonon, i.e. it is a collective state, then
this oscillating charge distribution will readily emit an electric quadrupole photon 2 , and this
transition will proceed rapidly, with a large corresponding transition probability B(E2; 2+ →
0+ ). These collective states consist of many particles being coherently excited out of the
closed core, and so they are unfavored if there is a large energy gap between the core and
the next available state. For a good shell closure, the excited 2+
1 state is not well-described
by a collective vibration and so the decay to the 0+ ground state proceeds more slowly.
2A

photon has spin S = 1, so an electric quadrupole photon with J π = 2+ must additionally carry what is often interpreted as orbital angular momentum L = 1 or 3.

8

The result is that good shell closures should be associated with high 2+
1 energies and
low B(E2) strengths. 44 S, which has 28 neutrons and 16 protons, exhibits a relatively low
2+ and high B(E2), indicating that the N = 28 shell gap is not so large. However, it was
unclear whether this collectivity is mainly due to the protons, which do not have a good
shell closure. This prompted the investigation of 42 Si, which has a proton sub-shell3 gap at
Z = 14 [38]. A measurement of a very small two-proton knockout cross section for 44 S → 42 Si
[39, 40] was interpreted as evidence that the Z = 14 gap remained good at 42 Si and that
42 Si

might be doubly magic. The observation of a very low lying 2+ energy in 42 Si in a

subsequent measurement [41] provided convincing evidence that 42 Si is in fact not a good
doubly magic nucleus, suggesting that in this extreme region of the nuclear landscape, the
N = 28 shell gap vanishes. Figure 1.3 shows the experimental 2+ energies for the calcium,
argon, sulfur, and silicon isotopes. Large 2+ energies can be clearly seen in doubly-magic
calcium (Z = 20) isotopes at N = 20 and N = 28, indicating good shell closures. At
N = 20 silicon also displays closed-shell behavior, enhanced by its Z = 14 sub-shell closure.
However, at N = 28 the 2+ energy of 42 Si is even lower than its open-shell neighbor 40 Si.
More recently, the (tentative) measurements of the first excited 4+ state in these silicon
isotopes [3] reinforced the notion that N = 28 is no longer a good closure, and that 42 Si is
in fact well-deformed. In the simple phonon model described above, the next excited state
should be a two-phonon state, where the two phonons can couple to J π = 0+ , 2+ , 4+ , with
energy 2 ω. Thus for a vibrational nucleus, we would expect a triplet of states with an
excitation energy equal to twice that of the first excited 2+ state. On the other hand, if the
nucleus is not spherical but deformed, then there is the possibility of a rotational excitation
3 The

distinction between a sub-shell gap and a proper shell gap is not well defined. In
fact, many older works (e.g. [23, 37]) list 14 as a magic number.

9

with energy 2 J(J + 1)/2I, where I is the moment of inertia. For the ground-state band
of an even-even nucleus, J takes on only even values (see appendix B), and so the first
excited state has J π = 2+ and the second excited state is 4+ . Assuming a constant I, the
ratio of the 4+ energy to the 2+ energy is 4(4 + 1)/2(2 + 1) = 20/6 = 3.333. Figure 1.4
shows the experimental ratio E(4+ )/E(2+ ) for a range of silicon isotopes as a function of
neutron number N . The upward trend in the ratio with increasing N indicates the onset
of deformation, with the greatest deformation found at N = 28, in clear defiance of the
expected shell gap.
It is therefore established that 42 Si does not behave as a traditional doubly-magic nucleus. The next step is then to understand the mechanisms which lead to the disappearance
of the N = 28 shell gap. The most promising proposed explanation [41, 42] is that the
action of the tensor force (discussed in §2.1.3) leads to the simultaneous reduction of the
N = 28 and Z = 14 shell gaps at large proton-neutron asymmetry, enhancing quadrupole
deformation and collective behavior. This explanation is framed in a single-particle picture
of nuclear structure, while the experimental data so far are of a collective nature. In this
work, we use single-nucleon knockout reaction cross sections (very much a single-particle
picture observable) to shed more light on the underlying mechanism behind the evolving
shell structure in the vicinity of 42 Si.

10

4

Ca (Z=20)
Ar (Z=18)
S (Z=16)
Si (Z=14)

3.5

2.5

1

+

E(2 ) (MeV)

3

2

1.5
1
0.5
0

18

20

22
24
26
Neutron number N

28

30

Figure 1.3: Experimental energies of the first 2+ states as a function of neutron number for
the Ca, Ar, S, and Si isotopic chains. Data taken from [2]. For the interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this thesis.

11

3.5

36-42

Si

SDPF-U

1

E(4+) / E(2+)

3

SDPF-MU

2.5

1

experiment
2

1.5

22

24
26
Neutron number N

28

Figure 1.4: Ratio of the energy of the first 4+ state to the energy of the first 2+ state in
silicon isotopes with 22 ≤ N ≤ 28. Experimental points from Takeuchi et al . [3] are indicated
with black dots, while shell model calculations with two different effective interactions are
indicated with sold lines. The dashed lines indicate the simple vibrational (2.0) and rotational
(3.33) model predictions (see text).

12

Chapter 2
Theoretical background

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand
nor deserve.
(E. Wigner)
In this chapter, I discuss the connection between nuclear structure and the underlying
nuclear forces, with an emphasis on the role of the tensor force. I then outline the theoretical
techniques used in the analysis and interpretation of this work.

2.1

Nuclear forces and their impact on structure

The fundamental theory describing the nuclear force is quantum chromodynamics (QCD)
which describes the interaction between quarks and gluons. Thus far, no deviations from
the predictions of QCD have been observed and so, in principle, the whole of nuclear physics
should be derived from QCD. Unfortunately, unlike in quantum electrodynamics (QED) –
where the coupling α ≈ 1/137 is small and varies only weakly with momentum [43], thus
admitting a good perturbative expansion – the strong coupling αs increases with decreasing
momentum [44, 45] and becomes much greater than 1 for momenta near the QCD momentum
scale ΛQCD ≈ 250 MeV/c [43]. This means that at the energies of interest for nuclear physics
(the Fermi energy in a nucleus is ∼ 30 MeV [17]), QCD is highly non-perturbative, which is
to say that any calculation in perturbation theory requires an infinite number of infinitely

13

complicated Feynman diagrams. Non-perturbative QCD calculations can be performed with
lattice gauge theory [46] (called Lattice QCD), and while such calculations have had success
in predicting the properties of hadrons (2 and 3 quark systems), the calculations of multihadron systems are more difficult. Some progress has been made on systems of up to a
few nucleons [47], but the calculation of heavier nuclei using these methods will not likely
be achieved in the near future (see [48] for a summary of the status of this work). For
quantitative predictions about nuclear systems, we must rely on effective theories of the
nuclear force.
Fortunately, the interaction between nucleons is not quite so unwieldy as that of quarks
and gluons. This is because hadrons are color-neutral and at large distances (more than a few
fermi) the interaction between nucleons goes to zero, analogous to the interaction between
neutral atoms. At smaller distances, the interaction is due to fluctuations in color-charge
density, analogous to the van der Waals force, while at the smallest distances, there is a
strong repulsion due to the Pauli exclusion principle between the constituent quarks. At
distances greater than about 1 fm, the interaction may be modeled as due to the exchange
of pions.

2.1.1

One-pion exchange

Pions are pseudoscalar (J π = 0− ) isovector (T = 1) particles, and the potential between two
nucleons due to pion exchange is given by [49]

Vπ (r) =

e−mπ r
fπ2
(τ1 · τ2 )(σ1 · ∇1 )(σ2 · ∇2 )
,
4πmπ
r

14

(2.1)

where 1 and 2 label the two nucleons, r = |r1 − r2 |, σ and τ are the spin and isospin Pauli
matrices, ∇ is the spatial derivative, mπ ≈ 140 MeV is the pion mass, and fπ is the pion
coupling.
This may be decomposed into scalar and tensor parts:

Vπ (r) =

fπ2
(τ · τ ) (σ1 · σ2 )
12π 1 2

e−mπ r 4π 3
−
δ (r) + S12
r
3

1
3
3
+
+
r mπ r (mπ r)2

e−mπ r
r
(2.2)

with the tensor operator S12 defined as (see appendix B)
1
S12 ≡ (σ1 · rˆ)(σ2 · rˆ) − σ1 · σ2
3
8π
[σ , σ ](2) · Y (2) ,
=
15 1 2

(2.3)
(2.4)

where σ1 and σ2 have been coupled to form a rank-2 tensor, Y (2) is the rank-2 spherical
harmonic for the relative coordinate, and the dot indicates a scalar product. At finite
distance, the one-pion exchange potential has a central term that depends on the magnitude
of the distance r between the nucleons, and a tensor term that depends on a combination
of the relative positions and the orientation of the spins. This tensor part has a subtle but
important impact on nuclear structure which will be discussed in §2.1.3.

2.1.2

Other contributions

There are, of course, other mesons aside from the pion, and they will also contribute to the
interaction between nucleons. These heavier mesons, such as the ρ and ω, also produce a
−mr
potential proportional to e r and their range is correspondingly shorter than that of the

one-pion exchange potential. Additionally, because nucleons are composite particles, the

15

N

N

N

N

N
π

π

N

π

N

N

(a)

N∗

N
(b)

N

Figure 2.1: Diagrams for (a) a two-body interaction mediated by pion exchange, and (b) a
three-body interaction in which the middle nucleon is excited into a resonance – for example,
the ∆(1232) resonance.
presence of a third nucleon will modify the internal structure of nucleons 1 and 2 – shown
schematically in Figure 2.1b – and thus modify their interaction in a way that cannot be
described by a sum of pairwise interactions [50]. This gives rise to 3-body and higher-body
parts of the potential which – while weaker than the pairwise interactions – have a significant
impact on structure.

2.1.3

The role of the tensor force

In this section, I describe the effect of the tensor force on the evolution of nuclear structure
along the lines laid out in [31]. The term tensor force is commonly used in the literature
to refer to the component of the two-body interaction in which the spins of each nucleon
combine into a rank-2 tensor. This is in contrast to the central (scalar, rank-0) component,
which is proportional to σ1 · σ2 , and the spin-orbit (vector, rank-1) component which is
proportional to σ1 + σ2 . The tensor component is written as:

VT = (τ1 · τ2 ) S12 fT (r).
16

(2.5)

+1
+2

-2
-1

Figure 2.2: Value of the tensor operator S12 for some simple configurations of two particles
(indicated by circles) with spin (indicated by arrows).
Here, S12 is the tensor operator as defined in (2.3), and fT (r) some function of r which,
in the one-pion-exchange case, is positive for all r. This form is (aside from the isospin
dependence) identical to that of a dipole-dipole interaction encountered in electrostatics
[51]. An interaction of this form was originally proposed by Rarita and Schwinger [52] to
explain the non-zero electric quadrupole moment of the deuteron [53]. The value of S12 for
some simple illustrative cases are shown in Figure 2.2.
We wish to know the effect that an interaction of this type will have on nuclear structure.
Consider two nucleons orbiting a nucleus, in orbits a and b, with quantum numbers (na , a , ja )
and (nb , b , jb ). For (relative) simplicity, we consider only the monopole (i.e. angle-averaged)
component of the tensor force, which is defined as

T
V¯ab

=

ma mb

ja ma jb mb |V |ja ma jb mb T
1
ma mb

=

J

(2J + 1) ja jb |V |ja jb J,T
(2J + 1)

.

(2.6)

J

By averaging over all projections m (or, equivalently, all J couplings), the monopole term
gives a measure of the average effect that the nucleons in orbit a have on the nucleons in
orbit b.
For a given , an orbit may have one of two values for j: j> =

17

+ 12 and j< =

− 21 .

It is stated here without proof (although a motivation is given in appendix B) that the
tensor monopole is in general attractive for V¯j< ,j> and V¯j> ,j< , and repulsive for V¯j< ,j< and
V¯j> ,j> .
This behavior of the tensor force plays an important role for the evolution of shell structure away from stability. We may consider, for example, the evolution of the splitting between
the proton d5/2 and d3/2 orbits from N = 20 to N = 28, where, to a first approximation,
neutrons are added to the f7/2 orbital (see Figure 1.2). The tensor force monopole is repulsive between the neutron f7/2 and the proton d5/2 (j> j> ), while it is attractive between
the neutron f7/2 and proton d3/2 (j> j< ). Table 2.1 lists some monopole terms from the
SDPF-MU effective interaction to illustrate this. We would then expect that going from
N = 20 to N = 28 (adding neutrons to the neutron f7/2 ), the gap between the proton d5/2
and d3/2 should become smaller, as indicated in Figure 2.3. Likewise, beginning at 48 Ca
(with 20 protons and 28 neutrons) and removing protons from the d3/2 orbital will cause
the energy of the neutron f7/2 orbital to increase. The energy of the neutron 1p3/2 orbital
will also increase in this case, but the effect will be reduced because the 1p3/2 orbit has a
node in the radial wave function, reducing the radial overlap. Thus, going from 48 Ca to
44 S,

the tensor force should cause a reduction of the N = 28 gap, in good agreement with

experimental findings such as those described in §1.2.
One may further show that (see appendix B)

(2j< + 1)V¯jT ,j + (2j> + 1)V¯jT ,j = 0
<
>

(2.7)

for either T = 0 or T = 1, which indicates that if both j< and j> orbits are filled, there is
no net effect of the tensor force This may be numerically verified with the monopoles from

18

f7/2 (j>)

d 3/2 (j<)
d 5/2 (j>)

protons

neutrons

Figure 2.3: Schematic representation of the effect of the tensor force when adding neutrons
to the f7/2 orbital. The monopole component of the tensor force is attractive between
the neutron f7/2 and proton d3/2 (indicated with the blue wavy line), while it is repulsive
between the neutron f7/2 and proton d5/2 (indicated with the red wavy line). This results
in the reduction of the spin-orbit splitting in the proton d orbitals.

Table 2.1: The monopole components of the SDPF-MU interaction between two nucleons in
orbits a and b, coupled to total isospin T , broken down into central, spin-orbit, and tensor
components. Energies are in MeV. Some terms have been repeated for easier comparison.
T
0
1
0
1

a

b

f7/2
f7/2
f7/2
f7/2
d3/2
d3/2
d3/2
d3/2

d5/2
d3/2
d5/2
d3/2
f7/2
p3/2
f7/2
p3/2

Va,b (central) Va,b (spin-orbit) Va,b (tensor)
-2.21412
-0.01714
0.20831
-2.09208
-0.00640
-0.31247
-0.10499
-0.04497
0.06944
-0.10499
-0.06868
-0.10416
-2.09208
-0.00640
-0.31247
-1.46457
0.03143
-0.06835
-0.10499
-0.06868
-0.10416
0.10713
0.09216
-0.02278

19

Table 2.1; for example, taking j = f7/2 and j< , j> = d3/2 , d5/2 , we have

4 × (−0.31247) + 6 × (0.20831) ≈ 0.

The effect of the tensor force is consequently greatest when one of the spin-orbit partners
is filled and the other is empty. In the case of the silicon isotopes studied in this work, the
proton d5/2 orbit is approximately filled, while the proton d3/2 is approximately empty, and
the neutron f7/2 is being filled while the f5/2 remains empty. Therefore, the tensor force
should play an important role, in terms of comparison to the calcium isotopes, in which the
proton d3/2 is also filled, as well as evolution along the isotopic chain.

2.2

Solving the many-body problem

Once an interaction between nucleons has been obtained, the task remains to calculate
observable properties of nuclei containing multiple nucleons. This consists principally of
finding the eigenstates and eigenvalues of the Hamiltonian, as well as expectation values of
other operators.

2.2.1

Many-body states

An A-body fermionic wave function Φ(r1 , r2 , . . . , rA ) may be described by a product of onebody wave functions φi (ri ) that obey Fermi-Dirac statistics, which requires that the entire
wave function must be antisymmetric with respect to the exchange of any two particles. For

20

a two-particle wave function, this is satisfied by

1
Φ(r1 , r2 ) = √ (φ1 (r1 )φ2 (r2 ) − φ1 (r2 )φ2 (r1 )) .
2

(2.8)

In general, for A particles, this is achieved by using a Slater determinant:




 φ1 (r1 ) φ2 (r1 ) . . . φA (r1 ) 






φ
(r
)
φ
(r
)
.
.
.
φ
(r
)
1
2
2
2
2
A
1


√
det 
Φ(r1 , r2 , . . . , rA ) =


.. 
.
..
..
A!
 ..
.
. 
.




φ1 (rA ) φ2 (rA ) ... φA (rA )

(2.9)

Equivalently, one may use the language of second quantization [54], where the manybody wave function is represented by a Fock state, which is the result of creation operators
a† applied to the vacuum:
1
†
† †
ΦA = √ aA . . . a2 a1 |0
A!

(2.10)

Here, the antisymmetrization is accounted for by the anticommutation of the creation and
annihilation operators.

†

ai , aj

= δij ,

†

†

(2.11)

ai , aj = 0 = ai , aj

† †

A Fock state may be labeled simply by the index of its orbits: |ij ≡ ai aj |0 . The Fock state
and the Slater determinant represent the same object, and sometimes the labels are used
interchangibly when speaking in the abstract. The actual eigenstate of the Hamiltonian for
an A-body system is in general some linear combination of A-body Slater determinants.

21

2.2.2

Hartree-Fock

The lowest-energy single Slater determinant may be found variationally, using the HartreeFock or self-consistent mean-field method [54]. The main idea is to treat each nucleon as
moving independently in a mean field generated by all the other nucleons. One starts with an
ansatz for the ground state, in which the nucleons occupy orbits obeying the Pauli principle.
These orbits are then used to generate a mean field potential, and the one-body Schr¨odinger
equation is solved for each nucleon. These new orbits generate a new mean field, and the
procedure is iterated until the change in wave functions from one iteration to the next are
within some numerical tolerance.
The Hartree-Fock method is often used with phenomenological interactions, such as the
Skyrme interaction which parameterizes the nuclear interaction with various terms, including
density dependence [17]. Skyrme-Hartree-Fock calculations do a good job of reproducing
bulk trends and properties such as binding energies and root-mean-squared (rms) radii for
closed-shell nuclei, but they are unable to capture detailed structure because they neglect
many-body correlations beyond the mean field. Additionally, as it is variational, the HartreeFock method is not well-suited to calculating excited states of nuclei. In this work, Skyrme
Hartree-Fock calculations are used in the generation of optical potentials and form factors
for the knockout reaction described in §2.3.4.

22

2.2.3

The shell model (configuration interaction)

Another technique is to solve for the ground state and excited states by expressing the
interaction as a matrix in some basis (usually constructed from Slater determinants)

ˆ + Vˆ |Φj
Hij = Φi |K

(2.12)

ˆ is the kinetic energy and Vˆ is the interaction between nucleons).
and diagonalizing. (Here, K
The resulting eigenstates are then a linear combination of slater determinants:

Ψj =
i

cij |Φi .

(2.13)

With an infinite basis1 and a realistic interaction, this would be an exact method. However,
to make the calculation tractable, the interaction is generally limited to two-body terms and
the basis is truncated.
The eigenstates of the 3-dimensional harmonic oscillator potential provide an attractive
basis for shell model calculations because the relative and center-of-mass motion may be
separated analytically. This is useful because the interaction between nucleons depends only
on the relative coordinates. Additionally, the oscillator wave functions are a reasonable first
approximation to well-bound orbits2 . The eigenstates are labeled by the radial quantum
number n, the orbital angular momentum , the total angular momentum j =

± 21 and

projection mj . They have an energy (N + 32 ) ω where N = 2n+ [56], and ω is the oscillator
1 The

non-relativistic approximation, implicitly or explicitly, makes a high-momentum
cutoff and so the basis is in fact finite, albeit large [55].
2 It should be noted that wave functions in the oscillator basis die off asymptotically as
2
e−ar , while the actual wave function should go as e−ar .

23

frequency of the potential. The difference between the harmonic oscillator potential VˆHO
and the nuclear interaction VˆN is treated as a perturbation

ˆ =H
ˆ HO + VˆN − VˆHO ≡ H
ˆ HO + Vˆ .
H

(2.14)

ˆ HO = K
ˆ HO + VˆHO is diagonal in the oscillator basis, so
The harmonic oscillator term H
ˆ HO |i = Ni ωδij .
we may replace it with its eigenvalues (omitting the constant 23 ω): j|H
Neglecting 3-body and higher parts of the nuclear interaction, the potential is given in
terms of two body matrix elements (TBME) k |Vˆ |ij . The nuclear potential conserves total
angular momentum and isospin (although isospin is only an approximate symmetry), so we
may specify the TBME as ab|Vˆ |cd JT , indicating that nucleons in orbits a and b are coupled
to total angular momentum J and isospin T (and likewise for c and d).
As was mentioned earlier, in order to make the calculation tractable it is necessary to
truncate the infinite oscillator basis. For light systems (A

12), it is possible to cut off at

high N , usually of the order of 12-14, and extrapolate to N → ∞. This method is called the
no-core shell model (NCSM), to distinguish it from the standard shell-model approach used
for heavier systems for which the NCSM is computationally infeasible. In the standard shell
model, in addition to truncating at high N , the active Hilbert space, or model space is also
truncated at low N , essentially freezing a core of nucleons into a single Slater determinant3 .
A good choice for the core is a closed-shell or “doubly magic” nucleus, such as 16 O or 40 Ca,
as excitations out of these cores are suppressed by the large shell gaps. The remaining
orbits are collectively called the valence space. In this case, we may sum the one-body and
3 Sometimes,

truncations are made within a single oscillator shell if the inclusion of the
full shell is intractable.

24

two-body matrix elements over the core orbits to obtain a zero-body term

E0 =
a

ˆ HO |a +
a|H

1
2

δ a=

ab|Vˆ |ab ,

a,b∈core

ab|Vˆ |ab

(2.15)

and additional one-body terms

b∈core

(2.16)

which result in the single-particle energy (SPE) of state a: a = Na ω + δ a . Neglecting
three-body and higher-body forces, the interaction is then specified by the TBME and singleparticle energies (as well as the zero-body E0 term, which, like the constant 23 ω oscillator
term, is unimportant for calculating excited-state spectra).

2.2.4

Effective interactions for shell-model calculations

The interaction used in a shell-model calculation should ideally be derived from the interaction between free nucleons, obtained, for example, in nucleon-nucleon scattering experiments.
Nucleon-nucleon potentials have been developed [5, 4, 57, 58] which can reproduce nucleonnucleon (N N ) scattering data up to a few hundred MeV. In principle, all that remains is to
evaluate one of these potentials in the chosen basis and diagonalize. However, this method
will produce accurate results only if contributions from configurations outside the model
space are small, and this is not necessarily the case.
Two potentials which have been derived from N N scattering are plotted in Figure 2.4 for
the 1 S0 channel4 . The most striking feature of this figure is the very repulsive potential at
4 This

spectroscopic notation indicates two nucleons in a spin singlet, with relative orbital
angular momentum = 0 and total angular momentum J = 0
25

Argonne v

1

S0 nn

300

18

Reid 93
V(r) (MeV)

200

100
0

-100
0

0.5

1

1.5
r (fm)

2

2.5

3

Figure 2.4: The 1 S0 channel of two phenomenological nucleon-nucleon potentials [4, 5] which
reproduce N N scattering data.
short distances, often referred to as a hard core. This leads to short-range correlations, due
to the fact that no two nucleons should be in the same place. These short-range correlations
disrupt the picture of independent single-particle orbits in a mean field, and the result is
that the true eigenvectors |Ψ contain contributions from many different configurations |Φi ,
including those outside the model space, and thus the results of calculations using bare N N
interactions within a truncated model space do not accurately reproduce the data.
There are two main strategies which are used (in concert) to mitigate this problem. One
is to develop N N potentials which have a softer core, but still reproduce scattering data at
low energy. Historically, this was done with the Brueckner G-matrix formalism [59]. More
recently, the Vlowk [60] and similarity renormalization group (SRG) methods [61, 62] have
been developed to do this. The second strategy is to generate effective interactions for a

26

given model space which take into account the excluded configurations. This can be done
with many-body perturbation theory (MBPT) [63] or, very recently, within the in-medium
SRG (IM-SRG) [64, 65] and coupled cluster formalisms [66]. In all cases, approximations
and truncations must be made, and the results are still not exact.
An alternative approach is to abandon the direct connection to the underlying N N interactions and instead fit to existing data, treating the TBME (or alternatively Talmi integrals)
as free parameters [67]. The gold standard of this approach is the USD effective interaction
[68], which can reproduce known energy levels in 1s0d shell nuclei with a root-mean-squared
deviation of less than 200 keV, better than any microscopically-based interaction has been
able to achieve. This approach requires a large amount of existing data, and its accuracy
beyond the fitted region quickly deteriorates, making it less reliable for extrapolating into
unknown regions of the nuclear chart. Often, what is done is to start with the microscopic
approach, and then to phenomenologically adjust the TBME to better reproduce existing
data. This is the approach adopted for both of the shell model effective interactions employed
in this work.

2.2.5

Effective operators for γ decay

Once the eigenvalue problem has been solved, yielding energies and wave functions, other
observables may be calculated using those wave functions. Here, I concentrate on the calculation of γ decay rates. The rate for a γ transition from an initial state with spin parity
Jiπ to a final state with Jfπ is given by the reduced transition probability B(σλ; Jiπ → Jfπ )
multiplied by a phase-space factor which depends on the parity and multipolarity of the

27

emitted photon (see appendix B). The reduced transition probability is defined as

B(σλ; Jiπ

→

Jfπ )

=

Jf Mσλ Ji
2Ji + 1

2

(2.17)

where σ can indicate either electric or magnetic transition, λ indicates the multipolarity of
the transition, Mσλ is the appropriate multipole operator, and the double bar indicates a
reduced matrix element.
The free-space operators for an electric transition are [17]

ME
λ =

riλ Y (λ) (ˆ
ri )ei ,

(2.18)

i

and the free space operators for a magnetic transition are

MM
λ =

i

2gi
+ gis si ∇ rλ Y (λ) ,
2λ + 1 i

(2.19)

where i sums over all nucleons. The quantities ei in (2.18) are the charges of each nucleon,
which in free space are eπ = 1 for the proton and eν = 0 for the neutron. The quantities
g and g s in (2.19) are the spin and orbital g factors, respectively, with free-space values
gπ = 1, gπs = 5.586 for the proton and gν = 0, gνs = −3.826 for the neutron.
When evaluating these operators between shell-model wave functions, one must keep
in mind that the shell-model calculation was performed with truncations which cut out
some degrees of freedom. Just as an effective Hamiltonian is required to obtain accurate
energies in the truncated space, so too is an effective transition operator required to obtain
accurate electromagnetic observables. Formally, there should be a unitary transformation
Meff = U MU † , where U is the same transformation that was used to obtain the effective
28

Hamiltonian. However, it is often the case (e.g. with phenomenological Hamiltonians) that
U is unknown. In that case, the electromagnetic operator must also be phenomenologically
adjusted, typically using the parameters ei or gi and gis .
The most common application of this is to E2 transitions, in which protons and neutrons
are given an effective charge, typically eπ = 1.5, eν = 0.5 [21]. The fact that the neutrons
acquire a charge can be made not quite so alarming by considering the case of a single neutron
orbiting a closed-shell doubly-magic core. As the neutron moves, it attracts the nucleons
in the core, producing an instantaneous non-spherical charge distribution that follows the
motion of the neutron. One can account for this approximately by assuming the core remains
spherical and associating an effective charge with the neutron. Various terms used to describe
this phenomenon include saying that the neutron has induced a core polarization, or that the
neutron is a dressed quasiparticle [69]. This prescription has in fact been shown to reasonably
represent the effective E2 operator in cases in which the unitary transformation is known
[70].
Likewise, corrections to the M 1 operator must be made to account for the model space
truncation. It is further expected that corrections should be made for meson exchange
currents, delta isobar excitations, relativistic effects, and processes beyond the impulse approximation which are not included in the shell model calculation [71, 72]. Regardless of
relative importance of each of these effects, it is clear that the M 1 operator used in shell
model calculations should differ from the free one. In addition to effective g and g s factors,
another term is sometimes added [71], proportional to the tensor product σY (2)
dle cases in which the

(1)

to han-

and s operators cannot connect the initial and final state, such as a

nucleon transitioning from the 1s1/2 orbit to the 0d3/2 orbit (see, for example [73]).

29

2.3

Probing single-particle states with knockout reactions

Direct reactions which involve the addition or removal of a single nucleon provide a useful experimental probe of the single-particle structure of nuclei [21]. Historically, electron-induced
proton knockout, or (e, e p), experiments were used (and still are) to identify single-particlelike states in stable nuclei. Two drawbacks to (e, e p) experiments are that (i) they are only
able to probe proton orbits, and (ii) they can currently only be performed for a nuclide which
is long-lived enough to allow the production of a target. Knockout reactions on a hadronic
target at intermediate energies solve both of these problems [74, 75]. In a knockout reaction,
the nuclide of interest is the projectile and the target is most often beryllium or carbon.
A single nucleon is removed from the A-body projectile in a reaction with the target, and
the remaining A − 1-body residue is detected. Since the target is hadronic, either a proton
or a neutron may be removed. Further, since the nuclide of interest is the projectile, the
often-difficult process of target fabrication is not required and so short-lived nuclides may be
studied in experiments using exotic beams. In this section, I describe the formalism used in
the description of these knockout reactions at energies on the order of 100 MeV per nucleon,
based on developments in [74, 76, 12].

2.3.1

Factorization of the cross section

ko such that the
We consider a knockout reaction with some associated transition operator Oα
fi

reaction amplitude Aα for removing a nucleon with quantum numbers α = (n, , s, j, m, tz )

30

and populating a final state ΨA−1
in the residue is given by
from an initial state ΨA
i
f
fi

ko ΨA .
Aα = ΨA−1
Oα
i
f

(2.20)

This operator also should depend on the target nucleus and the incident beam energy. The
fi 2

fi

reaction cross section is given by σα = Aα

. If the reaction is fast compared to the

internal motion of the nucleus, then we can make the sudden approximation and the knockout
operator becomes proportional to the annihilation operator aα 5 .

fi

fi

Aα ⇒ Cα ΨA−1
aα ΨA
i ,
f

(2.21)

fi

where Cα is a complex number. In most experiments, the polarizations of the incoming and
outgoing particles are not measured, and so the cross section is obtained by summing over
final projections and averaging over initial projections. We use k to denote all the quantum
numbers α, except for m (by angular momentum conservation, m = Mi − Mf ), and obtain
fi

σk =

1
2Ji + 1

fi 2

Mi M f

Ck

ΨA−1
ak,m ΨA
i
f

2

.

(2.22)

fi

We now approximate Ck by taking its average value over Mi , Mf . This approximation,
essentially the assumption of a spherical projectile, is exact if either Ji or Jf are zero, as is
the case for the projectiles studied in this work. The remaining sum is evaluated using the
5 Technically,

aα is not a tensor operator and we should instead use a
˜k,m = (−1)j+m ak,−m
[17]. For clarity, we neglect this additional notation. In any rate, this quantity is squared in
the cross section, so the sign is irrelevant.

31

Wigner-Eckart theorem, yielding

fi

fi

σk = Ck
=

The term

sp
σk

=

fi
Ck

2

1
2Ji + 1

ΨA−1
ak ΨA
i
f

2

(2.23)

sp f i
σk Sk .

2

is called the single-particle cross section, and from (2.21) we see

that it is the cross section if we assume ΨA
i

†

fi

= ak ΨA−1
. The term Sk is called the
f

spectroscopic factor and only depends on the structure of the final and initial states. Often,
fi

the spectroscopic factor is calculated in the isospin formalism, yielding Sk (T ) [17]. This
fi

quantity is related to the above Sk by an isospin Clebsch-Gordan coefficient, and so the
fi

spectroscopic factor in the proton-neutron formalism is often written as C 2 Sk (T ) or even
simply C 2 S, with implied dependence on the final and initial states. A small center-ofmass correction should be added, which for spectroscopic factors calculated in a harmonic
oscillator basis takes the form [77]

C 2S →

N
A
C 2 S,
A−1

(2.24)

where N = 2n + is the principle quantum number of the oscillator shell.
We are interested in the nuclear structure which is contained in the spectroscopic factor.
The extent to which we may compare this quantity with experimentally measured cross
sections, is determined by the extent to which we can reliably calculate the single-particle
cross section.
I will note here that the spectroscopic factors themselves are not experimental observables
[78]. One may change the spectroscopic factor while simultaneously changing the single-

32

particle cross section in order to produce the exact same observable quantity, i.e. the cross
section. However, if we keep in mind that the spectroscopic factors are scheme-dependent,
we may still obtain insight into the structure within that scheme. Further, it may be the case
that this scheme dependence is relatively weak among sensible formulations of the reaction
theory. The remainder of this chapter is concerned with the calculation of the single-particle
cross section and momentum distribution.

2.3.2

The eikonal approximation
sp

The single-particle cross section σk of (2.23) is calculated with a Glauber (eikonal) model
[79], assuming the impulse approximation described in §2.3.1 and treating the core as a
spectator – that is, internal reconfigurations of the core during the reaction are neglected.
The wave function of a projectile nucleus can be separated into relative and center-of-mass
components
Ψ(r1 , r2 , . . . , rA ) = ψCM (RCM ) ⊗ ψrel (r1 , r2 , . . . , rA ).

(2.25)

The relative part is the subject of the nuclear structure calculations of §2.2. The center-ofmass motion is characterized by the total momentum of the system k. In free space, the
center-of-mass contribution to the wave function may be approximated by a plane wave with
wave vector k:
ψCM (r) = eik·r .

(2.26)

In the vicinity of the target, this plane wave will be distorted by an optical potential U (r),
which we take to be local and constant in time, i.e. we neglect target recoil and breakup.
For a high-energy projectile, where the rate of change in the reduced Compton wavelength
λ
λ
¯ = k −1 is small d¯
dr

1, we can make a WKB approximation [80] and express the wave
33

number k as a slowly-varying function of position:

k(r) =

(E − U (r))2 − m2

k∞ U (r)
≈ k∞ −
,
2Ek

(2.27)

where E is the total energy at r → ∞, Ek = E − m is the kinetic energy, and k∞ =
√
E 2 − m2 . For the second line I have taken U (r)
Ek
m, which is reasonable for the
values U (r)

50 MeV [21], Ek ∼ 100 MeV/u, m ∼ 1 GeV/u.

If we assume that the optical potential can be approximated by a Woods-Saxon shape,
U

0
with depth U0 and diffuseness a0 , we can estimate dU
dr ≤ 4a0 and

d¯
λ
dr

1 U0
.
U )2 2E 4a
(1 − 2E
0
k
λ
¯∞

(2.28)

k

Using typical values of U0 = 50 MeV and a0 =0.7 fm, with a beam energy of 100 MeV/u,
this amounts to k∞

30 MeV for a single nucleon. At 100 MeV/u, k∞ ∼ 400 MeV, so the

WKB approximation is reasonable.
With the approximation (2.27), and assuming an azimuthally symmetric potential, the
center-of-mass wave function in (2.25) becomes

ψCM (R) = eik∞ z eiχ(b,z) ,

(2.29)

where the eikonal phase χ(b, z) is defined as

χ(b, z) = −

k∞ z
U (b, z)dz.
2Ek −∞

34

(2.30)

{
Figure 2.5: Schematic depicting a projectile nucleon scattering off of a single target nucleon.
With this we identify the elastic S-matrix (not to be confused with the spectroscopic factor
of the previous section) in the eikonal limit:

S(b) = eiχ(b,+∞) .

(2.31)

For a real potential, |S(b)|2 = 1 and integrating over all impact parameters gives the far-field
angular distribution for elastic scattering (see for example [81]). For a complex potential,
|S(b)|2 may be less than 1, representing the loss of flux to other channels, such as nucleon
transfer, inelastic excitation, or breakup.

2.3.3

Generating the optical potential

It is desirable to be able to generate the optical potential U (r) for an arbitrary core-targetvalence combination. In this section, I make a heuristic derivation of the potential which
reproduces the first-order result from a more rigorous consideration [82]. First, as sketched
in Figure 2.5, we consider a projectile nucleon incident on a nucleus, which is treated as
a collection of scattering centers (i.e. other nucleons) with density ρt . The probability
dP (z) that the nucleon with impact parameter b = (b, φ) will interact and scatter within a

35

infinitesimal path length between z and z + dz is given by the probability (1 − P (z)) that
it has not scattered up until z, multiplied by the nucleon-nucleon scattering cross section
σN N , times the areal density of scattering centers ρt (b, z)dz:

dP (b, z) = 1 − P (b, z) σN N ρt (b, z)dz.

(2.32)

Assuming P (z = −∞) = 0, we integrate (2.32) to get
z

1 − P (b, z) = exp −σN N

−∞

ρt (b, z)dz .

(2.33)

The quantity 1 − P (b, z), the probability for the nucleon to survive up to z, may be identified
2

as the modulus squared of the elastic S-matrix S(b, z) . Consulting (2.31), the imaginary
part of the eikonal phase is then
∞
1
dzρt (b, z),
Im{χ(b)} = σN N
2
−∞

(2.34)

which suggests a corresponding imaginary part of the nucleon-nucleus optical potential

Im{U (b, z)} =

−Ek σN N
ρt (b, z).
k∞

(2.35)

This may be extended to nucleus-nucleus scattering by integrating this potential over the
projectile nucleon density ρp , yielding the imaginary part of the folded nucleus-nucleus optical
potential:
Im{U (b, z)} =

−Ek σN N
k∞

36

d3 rρt (r)ρp (r − R(b, z))

(2.36)

where R(b, z) is the center-to-center separation of the target and projectile. This result is
identical to the first-order optical potential obtained in multiple-scattering theory [82], often
referred to as the tρρ approximation.
For the systems considered in this work, the real part of the optical potential has a
relatively small effect on the resulting cross section, and we parameterize it following Ray
[83] as UN N = (i − α)Im {UN N }, where αpn = 1.0 and αpp = αnn = 1.87 have been
extracted from fits to nucleon-nucleus reaction cross section data. We also add a term to
approximate the finite range of the nucleon-nucleon interaction with a Gaussian (this also
has a small effect the cross section) and the full optical potential becomes:

UN N (b, z) = (i − α)

−ik∞ σrN N
2E∞

d3 rp d3 rt ρt (rt )ρp (rp ) exp −

(rt − rp − R(b, z))2
.
γ2
(2.37)

We take γnp = γnn = γpp = 0.5 fm, as in [84].

2.3.4

Densities and form factors

The next ingredient is to calculate the target and core densities (the valence density is a
delta function, since nucleons are treated as point particles). We calculate the core density
using Skyrme Hartree-Fock, which is described in §2.2.2. An example is shown for 35 Si in
Figure 2.6a, calculated using the Skx Skyrme parameter set [85]. For the 9 Be target, we
assume a Gaussian density with rms radius 2.36 fm for all calculations, for consistency with
previous studies. Using instead the results of a quantum Monte Carlo calculation [86] or
Hartree-Fock results in a change in the relative importance in the stripping and diffraction
mechanisms at the order of 5-10%, but the overall single-particle cross sections only change
on the order of 1-2%.
37

0.12

35

Si

9

neutrons
protons

0.1

Be

0.06

neutrons
protons

ρ (fm-3)

ρ (fm-3)

0.05
0.08
0.06
0.04

0.03
0.02

0.02
0
0

0.04

0.01
1

2

3

4

5

0
0

6

1

R (fm)
(a) core density

2

3

4

5

6

R (fm)
(b) target density

Figure 2.6: Calculated proton and neutron densities for (a) a 35 Si core, calculated with
Skyrme Hartree-Fock, and (b) a 9 Be target, from quantum Monte Carlo calculations. The
oscillations at small R for the QMC calculation are statistical. The thin lines show the
Gaussian densities used for the target in this work.
Finally, we need the core-valence two-body wave function, or form factor φ(r). It was
shown in [12] that the calculated cross section depends strongly on the rms (root-meansquared) radius of the two-body wave function, and so it is important to have a consistent
way to determine this component of the calculation. The adopted method, proposed in [12],
utilizes the so-called well-depth prescription. The wave function is take to be the solution to
the Schr¨odinger equation for a particle in a real Woods-Saxon potential of the form

d
V (r) = −V0 fws (r) + ( · s) VLS fws (r)
dr

fws (r) =

1
1 + e(r−r0 )/a0

.

(2.38)

(2.39)

It then remains to choose the parameters V0 , VLS , r0 , a0 . It was shown in [12] that the
resulting cross section is relatively insensitive to the diffuseness a0 , so this is fixed at 0.7
fm. For the same reason, the spin-orbit depth VLS is fixed to 6 MeV. We are left with
two parameters, the depth V0 and radius r0 , which are fixed in a two-step procedure. We
38

first adjust both parameters to simultaneously reproduce the binding energy and rms radius
of the single-particle orbit obtained with a Skyrme Hartree-Fock calculation. Then, as
the asymptotic form of the wave function must be an exponential with the decay constant
determined by the binding energy, we adjust V0 with r0 held fixed to match the experimental
binding energy Ef − Ei . The results of this procedure for neutron orbits in 36 Si are shown in
Figure 2.7. Note that for the fully-occupied orbits 1s1/2 , 0d3/2 , shown in 2.7a and 2.7b, the
Woods-Saxon potential is similar to optical potentials used in this mass region [21]. For the
partially occupied f7/2 orbit (Figure 2.7c), and especially for the weakly-occupied p3/2 orbit
(Figure 2.7d), which lies above the Fermi surface in the Hartree-Fock calculation, the depth of
the potential is significantly greater. This reflects the fact that, for these orbits, correlations
beyond the mean field play a significant role in the binding, and therefore an unrealistic
mean field is required to reproduce the experimental binding energy. The proper treatment
of the form factor for such orbits is an old problem which, as yet, has no satisfactory solution
[87, 88, 89].
As a measure of the impact of this issue on the resulting cross sections, we consider
the effect of using different prescriptions. Instead of adjusting the depth of the potential,
V0 , to reproduce the experimental binding energy, we adjust either the radius parameter r0
(referred to here as the ∆R prescription) or the spin-orbit depth VLS (SO prescription). The
results of these calculations are given in Table 2.2, for the f7/2 and p3/2 neutron orbits in
36 Si.

Noting that the p3/2 orbit requires an unnaturally large well depth to reproduce the

Hartree-Fock calculation, we also perform the calculations starting with the more typical
parameters V0 =55 MeV, r0 =1.2 fm, and VLS =6.0 MeV, and adjust one of the parameters
to reproduce the experimental binding energy. These calculations are indicated in Table 2.2
with a primed label.
39

40

35

φ(r)

Si + n
1s1/2

20

V(r) (MeV)

V(r) (MeV)

0

-20
V(r)

-40

-20

-60

-60

-80

-80
2

4

6

8

10

V(r)

-40

0

2

4

r (fm)

8

10

(b)
35

φ(r)

40

6
r (fm)

(a)

Si + n

40

0f7/2

20

35

φ(r)

Si + n
1p

20

3/2

0
V(r) (MeV)

0
V(r) (MeV)

Si + n
0d3/2

20

0

0

35

φ(r)

40

-20
-40

-20
-40

V(r)
-60

-60

-80

-80

V(r)

0

2

4

6

8

10

r (fm)

0

2

4

6

8

10

r (fm)

(c)

(d)

Figure 2.7: Core-valence wave functions (dashed lines) for neutron orbits near the Fermi
surface in 36 Si, calculated with the well-depth prescription, shown with their corresponding
Woods-Saxon potentials (filled blue curves).
40

Table 2.2: Results for the two-body core-valence wave function for 35 Si+n, using several
prescriptions for determining the Woods-Saxon parameters: Hartree-Fock (HF), well-depth
(WD), changing the radius (∆R), and changing the spin-orbit depth (SO). Primed prescription labels indicate a starting point of V0 =55, r0 =1.2, and VLS =6.0, instead of the
Hartree-Fock result. Energies are in MeV, radii are in fm, and cross sections are in mb.
orbit
f7/2

p3/2

prescription
HF
WD
∆R
SO
HF
WD
∆R
SO
WD
∆R
SO

V0
54.2
59.4
54.2
54.2
73.9
92.1
73.9
73.9
64.2
55.0
55.0

2 >1/2
r0
VLS
BE < rrel
1.191 6.0 3.348
4.06
1.191 6.0 6.129
3.87
1.256 6.0 6.129
4.02
1.191 12.6 6.129
3.94
0.943 6.0 1.023
5.45
0.943 6.0 7.029
3.81
1.093 6.0 7.029
4.02
0.943 63.0 7.029
3.87
1.200 6.0 7.029
4.18
1.331 6.0 7.029
4.37
1.200 52.3 7.029
4.24

σsp
19.4
15.5
17.9
16.5
61.5
17.7
22.2
18.9
26.0
31.2
28.0

The results for the f7/2 orbit, which is below the Fermi surface in the Hartree-Fock
calculation, are only weakly sensitive to the method of perturbing the potential. The p3/2
results, on the other hand, show a variation of greater than 50%. This indicates that the
calculations of the absolute cross sections for removal from an orbit above the Fermi surface
have a substantial uncertainty, owing to there not being a clear reason to believe one of
these prescriptions more than another. For consistency in comparing with other work, we
choose the well-depth prescription, constrained by the Hartree-Fock calculation. The hope
is that while the absolute value of the calculated single-particle cross sections may have
a large uncertainty for these orbits, we may still obtain insight from the evolution of the
cross section along the isotopic chain. As can be seen in Figure 2.8c, the shift in the cross
section between the well-depth procedure and the ∆R procedure remains constant within
approximately 20% between 34 Si and 42 Si, and within approximately 10% for the isotopes
studied in this work.

41

Obviously, these weakly occupied orbits have a relatively small impact on the inclusive
cross sections. For example, the removal a neutron from any of the 1p3/2 , 1p1/2 , or 0f5/2
orbits of 36 Si is predicted to contribute a combined total of less than 5% of the inclusive cross
section to bound states. It should be noted that these weakly occupied orbitals also carry a
large uncertainty in the shell-model spectroscopic factor. One might be tempted to consider
if this effect could contribute to the observed systematic reduction of knockout cross sections
relative the theoretical predictions [12]. However, aside from the fact that this only becomes
important when the spectroscopic factor is small, the behavior is qualitatively in the wrong
direction: the deficient species should be deeply bound and the different prescriptions should
have a relatively small impact on the radius whereas experimentally the cross section for the
deficient species is strongly quenched.

2.3.5

Reaction mechanisms and cross sections

Following [74], the cross section is divided into two reaction mechanisms. The first – and
more important in the removal of well-bound nucleons – is the so-called stripping mechanism.
In a stripping reaction, illustrated in Figure 2.9a, the valence nucleon is absorbed by the
imaginary part of the optical potential (with probability 1-|Sv |2 ), while the core survives
(with probability |Sc |2 ). The cross section for this process is given by

σstr =

1
2J + 1 m

2
2
d2 b φm
0 |Sc | 1 − |Sv |

φm
0 ,

(2.40)

where J is the projectile angular momentum, b is the core-target impact parameter, φm
0
is the core-valence form factor of §2.3.4, and Sc and Sv are evaluated at their respective
positions.
42

6.5

90
HF

HF

80

WD
70

∆R

5.5

1.4

WD
∆R

radius
σsp

1.3

5
4.5

∆R / WD

60
σsp (mb)

relative rms radius (fm)

6

1.5

50
40

1.2
1.1

30
4

1
20

3.5
3

0.9

10
34

36

38
A

40

(a)

42

0

34

36

38
A

40

(b)

42

0.8

34

36

38
A

40

42

(c)

Figure 2.8: Results of three different prescriptions for obtaining the two-body wave function
(see text for details) for the 1p3/2 orbit in silicon isotopes 34−42 Si. (a) Rms relative radii
(b) single particle cross sections (c) the ratio between results for the well-depth prescription
(WD) and the potential radius prescription (∆R).

(a) stripping

(b) diffraction

(c) refraction

Figure 2.9: Illustrations of the three reaction mechanisms which contribute to the knockout
cross section.

43

The second mechanism is actually composed of two processes: diffraction due to the
absorptive imaginary part of the potential, shown in Figure 2.9b, and refraction due to the
real part of the potential, shown in Figure 2.9c. These mechanisms add coherently and are
both contained in the expression

σdiff =

1
2J + 1

d2 b

φm |Sc Sv | φm
0

d3 k

k

mm

2

.

(2.41)

This gives the probability that in the outgoing channel, due to the presence of the potential,
the core and valence are in a state with relative momentum k between them, i.e. they have
been excited into a continuum state. In practice, rather than integrating over all relative
momentum states we use a closure relation [90].

d3 k φm
k

m

φm = 1 −

φm
i

k

φm
i

(2.42)

mi

where i labels bound states. Inserting this into (2.41) gives

σdiff =

1
2J + 1

d2 b
mm

2 m
φm
0 |Sc Sv | φ0 −

i

2
m
φm
i Sc Sv φ0

.

(2.43)

In this form, the cross section is given by the probability that both the core and target
survive, minus the probability that they end up in a bound final state. A further approximation is made that the ground state is the only bound final state or, more generally, that
m
φm
i Sc Sv φ0 ≈ 0 for i = 0, which is often a reasonable assumption [76].

The two reaction mechanisms are in principle distinguishable; diffraction leaves the target
in its ground state, while stripping excites the target to another state (unbound in the case
of 9 Be). This means that they add incoherently (there is no interference) and we may assign
44

the single-particle cross section to be

σko = σstr + σdiff .

2.3.6

(2.44)

Momentum distributions

When the valence nucleon is removed, the knockout residue recoils with some momentum
k. The probability for a particular k is given by the momentum-space core-valence wave
function, which is just the Fourier transform of the position-space wave function. Note,
however, that the reaction is surface-peaked, and so the valence nucleon is not removed from
all parts of the position-space wave function with equal probability. We must instead weight
each point in position space by the knockout probability at that point. The momentum in
the transverse direction (perpendicular to the incoming beam direction) is complicated by
Coulomb effects, so we instead consider the longitudinal (parallel) momentum distribution.
For the case of a stripping reaction, this is given by [74]

1
dσstr
=
dkzc
2J + 1 m

c

d2 b

v

2

d2 ρ |Sc |2

1 − |Sv |

2

e−ikz z
dz
φ(ρ, z) ,
2π

(2.45)

where bv is the impact parameter of the valence nucleon, (ρ, z) is the vector between the
core and valence nucleon, and the integration is over all space.
It is observed that the distributions for diffraction, which is also surface-peaked, have
a nearly identical shape [91]. The result of applying (2.45) to neutron orbits near the
Fermi surface in 36 Si is shown in Figure 2.10. These calculated distributions may then be
compared with experimental distributions to infer the orbital angular momentum

of the

removed nucleon. This procedure is analogous to the angular distributions used in classical
45

1s1/2 ( = 0)

0.12

dσ/dP

0.1

1p3/2 ( = 1)

0.08
0.06

0d3/2 ( = 2)

0.04

0f7/2 ( = 3)

0.02
0

-400

-200

0
∆P (M eV /c)

200

400

Figure 2.10: Momentum distributions calculated according to equation (2.45) for neutron
removal from orbitals near the Fermi surface in 36 Si.
light-ion transfer reactions to infer the

of the transferred nucleon [90].

46

Chapter 3
Experimental method and analysis

If your experiment needs statistics, you ought to have done a better experiment.
(E. Rutherford)
For the current experiment, three different secondary beams (36 Si, 38 Si, 40 Si) were delivered by the Coupled Cyclotron Facility and impinged upon a metallic beryllium target at
the target position of the S800 spectrograph [92]. For each of these secondary beams, the
rigidity of the S800 was centered on both one-neutron and one-proton knockout residues,
as well as inelastically scattered beam particles. De-excitation γ rays were detected in the
GRETINA array of high-purity germanium detectors [93], surrounding the target position.
These systems will be detailed in the following sections.

3.1

Radioactive beam production

The silicon isotopes 36,38,40 Si are unstable to β decay, all having mean lifetimes of less than
1 second [94, 95, 96]. The production of beams of these nuclides was achieved by in-flight
fragmentation of a stable beam of 48 Ca. The process begins with a block of metallic 48 Ca
which is heated in an oven to produce a vapor. These neutral atoms are then ionized via
electron cyclotron resonance (ECR), in which electrons – confined by a magnetic field and
driven by a radio-frequency (RF) electric field – collide with the atoms and ionize them.
The singly or double ionized atoms are injected into the K500 cyclotron (see the schematic
47

Figure 3.1: A schematic of the beam production at the Coupled Cyclotron Facility.
in Figure 3.1).
A particle with charge q, mass m, and momentum p moving in a uniform magnetic field
B will trace out a circle of radius ρ at frequency ωc given by

p
ρ = qB
,

qB ,
ωc = γm

(3.1)

where γ is the relativistic factor (see appendix B). A cyclotron confines a charged particle
with a uniform magnetic field and accelerates it with an RF electric field tuned to the
cyclotron frequency ωc . As the particle’s momentum increases, so does its radius of orbit,
according to (3.1), until it is extracted at the outer edge of the cyclotron.
Ions are accelerated up to approximately 15% of the speed of light in the K500 cyclotron
and then extracted and transported to the K1200 cyclotron1 where they are accelerated
further. Note that, according to (3.1), the momentum of a particle leaving a cyclotron
is p = qBρ, so we may increase the final energy by increasing the charge of the particle,
1 The

K500 and K1200 cyclotrons are so named because they can accelerate a proton up
to 500 MeV and 1200 MeV, respectively.
48

increasing the strength of the magnetic field, or increasing the size of the cyclotron. The size
and magnetic field of the cyclotron are not easily increased, but we may increase the charge
q by further ionizing the particle. A carbon stripper foil is placed in the center of the K1200
cyclotron to do this (in the case of 48 Ca, the beam becomes fully ionized), and the primary
beam leaves the K1200 at approximately 50% of the speed of light.
The beam is then impinged on a thick beryllium target, producing a cocktail of secondary
beam particles by fragmentation. The secondary beam of interest is selected from the rest
of the fragmentation products by the so-called Bρ − ∆E − Bρ method using the A1900
fragment separator [97].
The quantity Bρ = pq is called the magnetic rigidity, and is useful for particle separation,
as it depends on the (presumably) fixed radius of curvature of a beam line element and the
adjustable magnetic field at that element. Fragmentation reactions at this energy approximately conserve energy (and momentum) per nucleon, and so – assuming the fragmentation
products are fully-stripped, which is generally the case for light and medium-mass nuclei –
A . This selection is achieved with slits which
selecting on pq is equivalent to selecting on Z

only allow ions with a small range of radii of curvature ρ to pass. Next the beam is passed
through an aluminum foil (called a wedge2 ) which causes the beam to lose energy from collisions with atomic electrons. The energy loss of a charged particle in matter due to collisions
with electrons is given by the Bethe formula [98]
dE
4πe4 ne Z 2
−
=
log
dx
me β 2

2me γ 2 β 2
I

− β2 ,

(3.2)

where e is the electron charge, ne is the target electron density, Z is the atomic number of
2 The

“wedge” is actually a flat piece of aluminum which is bent so as to achieve the
desired effective thickness as a function of transverse position.
49

the projectile, me is the electron mass, β is the projectile velocity, γ is the relativistic factor,
and I is the ionization potential of the target material. Assuming a constant beam velocity
β, the energy loss for a given degrader material depends only on Z, and the outgoing beam
is left with a momentum p(Z) which depends on the atomic number. By passing the beam
through another Bρ filter, a particular A,Z combination may be selected and delivered to
the experiment. For more details on radioactive beam production, see [99].

3.2

The S800 spectrograph

The beam delivered from the A1900 was then impinged on a thinner 9 Be reaction target
located at the target position of the S800 spectrograph (see the schematic in Figure 3.2),
and this is where the reactions of interest took place. Projectile-like reaction residues leaving
the target were bent by the two dipole magnets of the S800 and were detected in the focal
plane. The focal plane, shown in Figure 3.3, consists of two cathode-readout drift counters
(CRDCs) to detect the trajectory of the particle, a gas-filled ionization chamber to measure
energy loss for particle identification, and a scintillator used to trigger the system. Two
additional scintillators, located at the extended focal plane (XFP) of the A1900 and at the
object (OBJ) position of the S800 analysis line are used for time-of-flight (TOF) information.
These detector systems and their analysis are discussed in the following sections.

3.2.1

Particle trajectories

The S800 focal plane uses two CRDCs to measure the trajectory of the reaction residue. The
CRDCs are gas-filled position-sensitive detectors separated by approximately 1 m. Knowledge of the particle’s coordinates (x, y) at both of the CRDCs determines the position

50

Figure 3.2: A schematic of the S800 spectrograph. The secondary beam enters from the left
side and reacts at the target position. The reaction residues are detected in the focal plane.

ion chamber

CRDC1
beam direction
E1 scintillator
CRDC2

Figure 3.3: A schematic of the S800 focal plane. Taken from [6], modified from [7]. The
CRDCs are separated by about one meter.

51

y position (mm)

50

0

-50
-200

0
x position (mm)

200

Figure 3.4: Calibrated CRDC1 x and y positions for a mask run. The calibration is performed
using the known locations of the slits and pinholes in the mask.
(xf p, yf p) and angle (af p, bf p) at the focal plane. Here, af p is the angle in the dispersive (x) direction, and bf p is the angle in the non-dispersive (y) direction. The angle and
energy at the target position can be obtained by the use of an inverse map which describes
the trajectory of a charged-particle passing through the magnetic field of the S800. The
CRDC x position is obtained by the cathode segment which collects the most charge following ionization of the fill gas, and requires trivial calibration. The y position is obtained
by the drift time of that charge, and is calibrated with a mask which has holes at known
positions. An example of one such mask calibration is shown in Figure 3.4.
An inverse-map generated with COSY Infinity [100] allows the conversion of the focalplane parameters (xf p, yf p, af p, bf p) to the target parameters (dta, yta, ata, bta) – the x
position at the target is assumed to be zero. The parameter dta is the difference between the
energy of the particle and the energy of a particle that would arrive at the center of the focal

52

plane. The direction of the incoming beam may not perfectly coincide with the zˆ direction
in the S800 coordinates, so a correction is applied to ata and bta such that the distributions
are centered at zero. With the four target parameters and the rigidity Bρ of the S800, we
may calculate the energy-momentum 4-vector of the outgoing particle (see appendix B).

3.2.2

Time-of flight

Particles which arrive in the focal plane of the S800 have been selected for rigidity Bρ = pq ,
and so for a given q, the mass M of the fragment is inversely proportional to its velocity,
and thus proportional to its time of flight, assuming a fixed path length. The experimental
setup consists of three timing scintillators, the XFP scintillator at the end of the A1900, the
OBJ scintillator at the object position of the S800 analysis line, and the E1 scintillator at
the back of the S800 focal plane, which is used to trigger the readout of the S800 detector
systems. This setup yields two time differences, which we take relative to the E1 scintillator.
The difference between the XFP time and the OBJ time is proportional to the mass of
the incoming particle, while the OBJ-E1 time difference is sensitive to the mass of the
outgoing particle. The flight path through the S800 depends on the angle and momentum
of the particle, so the OBJ time of flight must be corrected for these quantities, yielding
a corrected time of flight OBJC. In practice, this correlation is corrected by removing the
linear dependence of OBJ on the x position and angle at the focal plane of the S800.

OBJC = OBJ + C1 × xf p + C2 × af p

This correction is demonstrated in Figure 3.5.

53

(3.3)

103

2360

2360

10

2300

102
2320

2300

2280

10

2280
-200

0
xfp (mm)

1

200

-200

(a)

0
xfp (mm)

1

200

(b)
103

103

2360

2360

2340

2340

10

2300

Counts

2320

OBJC

102
Counts

102
OBJ

Counts

OBJ

2320

OBJC

2340
102

Counts

2340

103

2320
10

2300

2280

2280
-50

0
afp (mrad)

50

100

1

-50

(c)

0
afp (mrad)

50

100

1

(d)

Figure 3.5: (a) Time of flight parameter OBJ vs x position in the focal plane of the S800 for
the 1n knockout setting from 38 Si. (b) Corrected time of flight parameter OBJC vs x. (c)
OBJ vs angle afp in the dispersive direction, and (d) OBJC vs afp.

54

3.2.3

Energy loss and particle-identification

As discussed in §3.1, the energy loss through matter is given by (3.2) and depends quadratically on the atomic number Zp of the projectile. For two different nuclides in a beam, the
only parameters which are different are Zp and β. Assuming β is very similar for similar
mass projectiles, we may discriminate Zp by the energy deposited in the ionization chamber. The energy loss through the ionization chamber also depends on the path length of the
projectile, and so the energy loss is usually corrected for the position x in the focal plane of
the S800. However, in the current experiment with Z ∼ 14, this correction is unnecessary to
obtain clear separation in Z. Combining the Z discrimination with mass discrimination from
time-of-flight, we can obtain a particle identification (PID) plot. The particle identification
plots used for the current work are shown in Figure 3.6.
Some additional gates were used in the case of 35 Si and 37 Si. In 35 Si, there is a significant
contamination from scattered unreacted beam which makes separation of 35 Si more difficult
(see Figure 3.7a). This scattered beam originates in a source close to the focal plane and so
its x position and angle in the focal plane are strongly correlated, as can be seen in Figure
3.7b. These structures in the x vs angle plot are gated out (see Figure 3.7d), yielding a
cleaner separation, shown in Figure 3.7c. Similar cuts are made for 37 Si, requiring x > −190
in the focal plane to cut out some of the unreacted beam, and af p − 0.18xf p < 50 to remove
scattered beam particles. These cuts result in a change of less than 1% in the total counts
inside the PID gate, but they allow a clearer separation, as demonstrated in Figure 3.7.

55

36

Si

35

Si

102

34

Si

700

33

Al

32

Al

600

10
30

Mg

500
2360

2380
OBJC

2400

2420

1

Si

700

36

Al

102

35

Al

34

Al

600

33

Mg

32

Mg

31

Mg

500

400
2380

2400

38

37

36

Si

10

Counts

Si

800
35

Al

700

34

Al
1

600

2300

2320
OBJC

Si

10

800
38

37

Al

700

Al

36

1

Al

34

Mg

600

10-1

2340

2300

2320
OBJC

2340

(d) 38 Si one proton knockout setting.

1200

1000

40

1100

103

42

1000

40

38

Si

Si

102

39

Si

900

Counts

P

10
800

-950

-900

-850
OBJC

-800

Ion Chamber dE (arbitrary units)

Si

900

39

Al

38

8

Al

800
6
700

4

600

500

1

10

2

2220

2240

2260

2280

0

OBJC

(e) 40 Si one neutron knockout setting.

(f) 40 Si one proton knockout setting.

Figure 3.6: Particle identification plots for each setting, with the gate on the knockout
residue.

56

Counts

Si

38

Counts

900

Ion Chamber dE (arbitrary units)

900

(c) 38 Si one neutron knockout setting.

Ion Chamber dE (arbitrary units)

1

2420

(b) 36 Si one proton knockout setting.

102

700

10

OBJC

1000
Ion Chamber dE (arbitrary units)

36

2360

(a) 36 Si one neutron knockout setting.

500

800

Counts

800

Ion Chamber dE (arbitrary units)

103

Counts

Ion Chamber dE (arbitrary units)

900

102
700

600

10

500
2380
OBJC

2400

2420

50

103

0

102

-100

1

10

-50

-200

(a)

102
700

600

10

500
2400

2420

x angle at focal plane (mrad)

800

Counts

Ion Chamber dE (Arbitrary Units)

100

103

2380
OBJC

1

(b)

900

2360

0
200
x position at focal plane (mm)

0

(c)

10

-50

-100

1

102

50

1
-200

0
200
x position at focal plane (mm)

(d)

Figure 3.7: Additional gates used to improve particle identification for 35 Si. (a) The PID
gated on incoming 36 Si shows poor separation between 35 Si (the strongest blob) and 36 Si,
lying to the left. (b) Dispersive angle in the focal plane vs dispersive position in the focal
plane gated on the 35 Si PID blob, showing structures due to scattering of unreacted beam.
(c) PID resulting from cuts applied to (b) producing a much cleaner separation. (d) The
result of those cuts in angle vs position, again gated on the 35 Si blob.

57

Counts

2360

x angle at focal plane (mrad)

800

Counts

100

103

Counts

Ion Chamber dE (Arbitrary Units)

900

(a)

(b)

Figure 3.8: The retracted northern (a) and southern (b) hemispheres of the GRETINA array.
When the array is in use, the two hemispheres are joined and the detectors surround the
target position. Images courtesy of S. Noji.

3.3

The GRETINA array

Prompt γ-rays were detected in the Gamma Ray Energy Tracking In-beam Nuclear Array
(GRETINA) [93], which in this experiment consisted of 7 modules, each containing 4 high
purity germanium (HPGe) crystals. The configuration of these modules is shown in Figure
3.9. The GRETINA array and the experimental details of γ-ray detection for this experiment
are detailed in the following sections.

58

3.3.1

Doppler reconstruction

Prompt γ rays are emitted in flight by the projectile, which is moving at a velocity v/c ≈ 0.4.
This causes the energy of the γ rays in the lab frame to be Doppler shifted as a function of
angle (see appendix B):
ELAB =

1 − β2
E
1 − β cos θLAB CM

(3.4)

The physical quantity of interest is the energy of the transition in the rest frame of the
projectile. In order to extract this from the measured energy, we must also know the velocity
of the beam and the angle of the emitted γ ray with respect to the beam’s trajectory. For
each reaction setting, the centroid reconstruction velocity β was chosen to align the strongest
peaks at forward and backward angles. The velocity was then corrected event-by-event using
the (after-target) beam velocity measured by the S800 spectrograph. The angle of emission
can be reconstructed if we know the origin (assumed to be the center of the target) and
the point of the first interaction. In this experiment, the GRETINA array consisted of
seven modules containing four HPGe crystals each. The crystals are further electronically
segmented into 36 segments. In order to maximize the position resolution of the array, digital
pulse-shape analysis is employed to obtain sub-segment resolution.

3.3.2

Sub-segment position resolution

When a γ ray interacts with a germanium detector – by Compton scattering, photoabsorption, or pair production – electron-hole pairs are created and drift in opposite directions
under the influence of the high voltage [98]. The moving charges induce a current on the detector electrodes, and this is read out as the signal. If the charge is collected on an electrode,
there will be a net current measured on the electrode. However, if the charge is collected on
59

South

Down

North

Gretina θ (degrees)

150

100

50

0

-150

-100

-50
0
50
Gretina φ (degrees)

100

150

Figure 3.9: A demonstration of the angular coverage of GRETINA. Four modules were
positioned in a ring at θ ≈ 55◦ , and the remaining three modules were placed at θ ≈ 90◦ .
the electrode of a nearby segment, there is no net current. The magnitude of the induced
charge depends on the proximity of the electrode to the drifting charge, so by comparing
the magnitude of the induced charge in two segments adjacent to a segment with net charge
collection we may obtain information about the spacial origin of the charge within the segment. Simple algorithms for doing so are detailed in Appendix A, with application to the
Segmented Germanium Array (SeGA) [101].
The GRETINA array employs a more sophisticated method to obtain sub-segment position resolution. Instead of the simple interpolation algorithms described in Appendix A,
each event is compared to a stored basis of signals, which correspond to a mesh of single
interaction points. These basis signals, unique to each interaction point, are calculated from
electrostatics using the detector geometry. A χ2 minimization procedure is used to find the
best-fit interaction point. This procedure, called decomposition, also allows for the possibil-

60

ity of more than one interaction within a single segment. In the case that only one segment
fires, up to three interactions are allowed to be placed in a single segment. Otherwise, only
two interactions are allowed in a single segment. The output of the decomposition is a list
of energies and interaction points, which are then stored with the event.
In order to obtain the emitted γ-ray energy ECM of 3.4, one needs to know the angle
of emission, which can be obtained from the point of first interaction, assuming that the γ
ray originated from the target. One of the main features of GRETINA – and the future full
4π GRETA – is the ability to use all the interaction points provided by the decomposition
algorithm to determine the most likely trajectory for the γ-ray using the Compton formula,

Eγ =

Eγ
1 + Eγ /m(1 − cos θ)

(3.5)

(see appendix B). In the current experiment, small inaccuracies in the basis signals used for
decomposition, combined with the non-ideal detector geometry make this procedure challenging, and the benefits are outweighed by the added complexity in obtaining an absolute
detection efficiency. For all the Doppler reconstructions in this work, it is simply assumed
that the first interaction point is the one with the largest energy deposition. This is a reasonable assumption because, even if the first interaction deposits a small amount of energy,
by (3.5), the scattering angle must be small as well, and so the error incurred by assuming
the subsequent interaction point was the first one is relatively small.
All interaction points within a single crystal are assumed to originate from the same γ
ray, which is also a reasonable assumption, considering the low γ-ray multiplicities seen in
this work. For an improved peak-to-total ratio, an add-back procedure is used in which all
adjacent crystals with an interaction are grouped into a cluster and assumed to originate

61

single crystal

Counts × 100 / (4 keV)

15

add back

10

5

0
0

200

400

600
800
Energy (keV)

1000

1200

Figure 3.10: Gamma-ray energy spectra in coincidence with 35 Si obtained by summing
all interactions within one crystal (blue filled histogram) and by the add-back procedure
described in the text (red histogram).
from the same γ-ray. As an example of the difference in results obtained by the two methods,
Figure 3.10 shows part of the γ-ray energy spectrum in coincidence with 35 Si obtained with
each procedure. In this work, add-back is used for peak identification and γγ coincidences
because of its superior spectral quality, while the single-crystal spectra are used to obtain
the absolute yields of γ-rays because of the more accurate efficiency determination.

3.3.3

Origin of the exponential γ-ray background

Gamma-ray spectra taken in coincidence with knockout reactions at intermediate energies
(approximately 100 MeV/u) inevitably exhibit a pronounced continuous background which

62

decays exponentially with increasing energy. The origin of this background has not been
clearly established, although brief suggestions exist in the literature. A similar background
is seen in other reactions at intermediate energy such as fragmentation, inelastic excitation,
and Coulomb excitation 3 , but not in pickup reactions (see, for example, [102]), which
are particular in their minimal momentum transfer. This ubiquity is evidence that the
background does not depend on the structure of the studied nucleus and so in most cases,
the origin of the background is unimportant; simply fitting it with an exponential or doubleexponential curve allows the extraction of the desired information. However, in the present
work we encounter two peaks of interest that lie just above the γ-ray detection threshold,
where the shape of the background is no longer a simple exponential. The approach taken
here is to attempt to understand the origin of the background sufficiently to simulate it and
extrapolate from the background at higher energy. We use three observables to illuminate
the source of this background: timing information, correlation with the momentum of the
detected outgoing particle, and the angular distribution of the γ rays.
Timing information on the background comes from an experiment using lanthanum bromide (LaBr3 ) scintillator γ-ray detectors, which have superior timing resolution to highpurity germanium (HPGe) detectors. Figure 3.11, taken from [8], demonstrates that the
exponential background is delayed by a few nanoseconds relative to the prompt de-excitation
γ rays. Further, a gate on later times reveals the 884 keV line in 27 Al (not shown in Figure
3.11), indicating inelastic reactions on the aluminum beam pipe.
The relationship between the γ-ray energy and the momentum of the outgoing particle is
shown in Figure 3.12. Panel (a) shows the parallel momentum distribution of outgoing 36 Si
3 Coulomb

excitation and inelastic scattering also have a significant contribution to the
background at low energy from atomic processes.

63

Figure 3.11: Demonstration of the time structure of the γ-ray background. The top panel
shows the γ-ray spectrum in coincidence with 21 Ne detected with LaBr3 detectors, with the
time difference between the LaBr3 trigger and the particle trigger shown in the inset. The
bottom figure shows the γ-ray spectrum gated on the prompt peak in the timing spectrum,
indicated with the gray block. Most of the exponential background is removed. The inset in
the bottom figure shows the γ spectrum gated off of the prompt peak, revealing an essentially
pure background spectrum. Reprinted with permission from [8].

64

particles, with two cuts indicated, one on the main peak and one on the low-momentum tail.
Panel (b) shows the γ-ray spectra for each of the indicated cuts, normalized by the number of
detected ions in each cut. It is clear that the high-energy exponential background originates
from events in which a significant amount of momentum is removed from the projectile,
indicating a nuclear collision with energy transfer to the target. The γ-rays associated
with the main peak are much more concentrated at low energies, which indicates that they
originate from atomic processes such as bremsstrahlung of scattered target electrons. The
main panel, labeled (c), shows the parallel momentum distribution in coincidence with no
detected γ-rays (blue hatched curve) and with γ-rays with energy above 1 MeV (solid red
curve), indicating a complementary relationship to that shown in panel (b). The blue curve,
with no γ-rays in coincidence, is essentially the incoming beam profile, while the red curve
shows a significant tail to low energies. 4 This relationship will be used in §3.4. If the blue
curve and the high-energy portion of the red curve are fit with Gaussians, the difference in
their centroids corresponds to an energy loss of approximately 5 MeV.
The angular distributions of the background γ-rays are shown in Figures 3.13 (polar angle
θ) and 3.14 (azimuthal angle φ). In obtaining these distributions, it was necessary to remove
the effect of the non-trivial geometry of the GRETINA detectors.
For the polar angle distribution, the measured distribution is normalized to the distribution from a 226 Ra calibration source, which is assumed to be isotropic, located at the target
4 The

γ-ray detection efficiency of the GRETINA array is, of course, not 100% and so
there are some counts in the blue curve which correspond to events with undetected γ-rays.
As a result, one might expect a low momentum tail on the blue curve. The tail is present,
but it is so overwhelmed by true no-γ events that it is invisible on a linear scale.

65

Counts

3

2

(c)

(a)

14.4
14.6
14.8
15
15.2
Parallel Momentum (GeV/c)

Counts / ion

Scaled Counts × 104

4

107
6
10
5
10
104
3
10
102
10

-3

10

(b)

10-4
-5

10

1

0

102
103
Lab Frame γ Energy (keV)

14.6

104

14.7
14.8
14.9
15
Parallel Momentum (GeV/c)

15.1

Figure 3.12: (a) Parallel momentum distribution for outgoing 36 Si particles, with two gates
indicated. (b) The γ-ray spectrum for each of the two momentum cuts indicated in (a),
normalized to γ-rays per detected ion. (c) The momentum distribution gated on no detected
γ rays (blue hatches), and γ rays with energy above 1 MeV (solid red).

66

position. In Figure 3.13 I show the weighting function W (θ) defined by

1
dN (θ) = N W (θ) sin θdθ
2

(3.6)

such that an isotropic distribution has W (θ) = 1. The distributions for two additional
calibration sources, 56 Co and 60 Co, are shown and are consistent with an isotropic distri+
36
bution. Also shown is the distribution for the 2+
1 → 01 transition in Si, emitted in-flight

with a velocity of β ≈ 0.41 after inelastic scattering on a 9 Be target. A calculated curve
shows the expected distribution assuming an isotropic distribution in the rest frame of the
emitting 36 Si particle. Finally, the angular distribution for all γ-rays with energy above 1
MeV in coincidence with a detected 36 Si particle is shown. This distribution appears to
indicate a source moving at a velocity greater than beam velocity, which is unreasonable.
However, such a distribution could also be generated by a source located approximately 10
cm downstream, consistent with fragments from the target drifting down stream for a few
nanoseconds before interacting with the beam pipe.
For the azimuthal distribution, the angle φ is taken relative to the scattering plane of the
reaction, obtained with the S800. Due to the non-uniform angular coverage by GRETINA,
and the non-uniform distribution of beam angles ata and bta, there could be some artificial
correlation in the angle of the detected γ-ray and the angle of the scattered particle. For
example, the GRETINA detectors are more concentrated along the y direction (see Figure
3.9), and if there were significantly more spread in the incoming bta (angle in the y direction)
relative to ata, then even totally unrelated events would show a preference for the γ ray to
be detected in the scattering plane. To remove this, a baseline correlation Irandom (∆φ)
was generated by constructing a distribution of φ in which the γ and scattered particle

67

2

36

1.8
1.6
1.4

W(θLAB)

+

Si 2 → 0
36
Si Eγ > 1MeV
60
Co source > 1 MeV
56
Co source > 1 MeV
β = 0.41
β = 0.00
+

1.2
1

0.8
0.6
0.4
0.2

30

40

50

60

70

80

θLAB (deg)

90

100

110

120

Figure 3.13: Polar angle distribution, normalized to a 226 Ra source measurement, for the
2+ → 0+ transition in 36 Si (blue triangles), the exponential γ-ray background above 1 MeV
(magenta circles), and two other source measurements (black diamonds and red squares)
as a consistency check. The long-dashed line indicates an isotropic distribution in the lab
frame, and the short-dashed line indicates an isotropic distribution in a frame moving with
beam velocity. The error bars are statistical only.

68

1.8

Eγ > 2 MeV

I(∆φ) / Irandom(∆φ)

1.6

Eγ < 0.2 MeV

1.4
1.2
1
0.8
0.6
0.4
-150

-100

-50

φγ - φ

0

ion

50

100

150

(deg)

Figure 3.14: Distribution of the azimuthal angle φ between the direction of the scattered
beam and the direction of the detected γ ray. The red circles show the distribution for
γ rays with energy greater than 2 MeV, while the black diamonds show the distribution
for low-energy γ rays from atomic processes, which should be uncorrelated with the beam
direction. The dashed and dotted lines show fits with the two lowest-order even Legendre
polynomials.
are taken from separate (uncorrelated) events. The actual distribution I(∆φ) was then
divided by the random distribution to remove artificial correlations. Figure 3.14 shows the
resulting distribution for two cuts in the γ-ray energy spectrum in coincidence with outgoing
36 Si.

The low-energy cut Eγ < 0.2 MeV selects primarily photons originating in electronic

bremsstrahlung, which should have no significant correlation with the scattering angle of
the outgoing particle, and indeed the distribution is flat. The high-energy cut Eγ > 2 MeV
selects the background of interest. A clear anti-correlation is evident, indicating that the
γ-rays are preferentially detected in the direction of the recoiling target.
69

Taken together, this evidence strongly suggests that the exponential γ-ray background is
produced by a collision with the target, which breaks up into nucleons and light ions. These
particles then interact with the aluminum beam pipe to produce the observed γ rays, either
by bremsstrahlung or by statistical γ cascades from highly excited states. In experiments
with a liquid hydrogen target [103, 104], the beam-correlated background may be reproduced
by the γ spectrum obtained with an empty target, scaled by the number of detected particles.
This suggests that protons (and, we may infer, light ions) play a small role in the production
of this background and that neutrons are the primary culprit 5 . Indeed, (n, n ) induced
γ-rays from germanium and aluminum confirm that neutrons are produced in abundance in
these reactions. Finally, we may also understand that the more delicate pickup reactions are
much less likely to occur in violent collisions which produce neutrons, and hence the absence
of this background in those reactions (see, for example, [102, 105, 106]).

3.3.4

GRETINA simulations

The response of the GRETINA array was modeled using the GEANT4 simulation package [107]. The simulation includes the fully segmented detectors with crysostats and two
mounting hemispheres, as well as the aluminum beam pipe and the target material. The behavior of the photons was modeled including Compton scattering, Rayleigh scattering, pair
production, and the photoelectric effect. For in-beam simulations, the incoming beam characteristics were matched to data from an unreacted setting, while the reaction is modeled in
a single step which produces the reaction residue in an excited state. A new momentum direction is selected, based on parameters which reproduce the measured outgoing momentum
5 This

is, of course, still speculative. A more definitive statement would require the
measurement of charged particles after neutron knockout or inelastic scattering.

70

distribution, and the ion proceeds until it decays based on the lifetime of the excited state.
The ion then continues until it has left the target, at which point its trajectory is recorded in
S800 format. The energy and position of all interactions in GRETINA are associated with
that event, and the simulated data may be passed through analysis identical to that applied
to the experimental data.
One additional layer of complexity is the fact that GRETINA obtains the interaction
points through a fairly complicated decomposition process, which has some associated error.
Further, multiple interactions within a single segment are very difficult for the decomposition
algorithm to distinguish. To treat this, all interactions in the same segment within a “packing
radius”, taken to be 8 mm, are combined into a single interaction with an energy equal to the
sum of the individual energies and a position given by the energy-weighted average position.
Each interaction point is then shifted in a random direction to generate the effective position
resolution of 2 mm (root-mean-squared) in x, y, and z.
The simulated γ-rays fall into three categories: prompt in-flight γ’s, neutron-induced
lab-frame γ’s, and the continuous exponential background described in §3.3.3. In principle,
γ’s from atomic processes should also be included, but this is found to contribute negligibly
to the γ-ray spectrum for any reaction in which the outgoing fragment is different from the
incoming beam. This may be understood by considering the relative likelihood of various
processes: In the unreacted setting, the vast majority of outgoing particles will not have
undergone a nuclear reaction in the target, and so no nuclear γ-rays are produced. There is,
however, a small probability for producing and detecting a photon from an atomic process,
and this small probability competes with the small probability of a nuclear reaction which
produces a γ ray. In the reacted setting, by definition all outgoing particles have undergone
a nuclear reaction, and the probability of producing a γ-ray is relatively large, while the
71

probability of producing a photon in atomic processes remains small.
Neutron-induced lines in germanium have a characteristic high-energy tail which comes
from the recoil energy of the excited germanium nucleus [98]. This was modeled by, instead
of generating γ rays with exactly the transition energy, selecting them from a distribution
of energies which reproduces the observed peak shape.
The background described in §3.3.3 is generated by selecting γ-ray energies from a
Maxwell-Boltzmann distribution

P (E) = E k e−E/E0

(3.7)

and emitting these isotropically in the rest frame. The parameter E0 is taken to be 2 MeV,
to reproduce the observed high-energy behavior of the background, and we take k = 0.2.
As this background is understood to be due to neutrons (and possibly light ions) from the
fragmented target interacting with the aluminum beam pipe, the origin of the γ rays is
taken to be a point on the beam pipe with z position selected from a Gaussian distribution
centered 10 cm downstream of the target.
At this point, all components of the simulated γ-ray spectrum have a fixed shape, and the
only remaining variables are their overall scaling factors. These scaling factors are obtained
by fitting to the experimental spectrum using the MINUIT fitting package [108] included in
the ROOT analysis framework [109]. To better constrain the lab-frame peaks, the spectra
are fit simultaneously in the lab and Doppler-reconstructed frame. For low statistics data, it
was found that a log-likelihood fit assuming Poisson-distributed errors produced a less biased
fit than the default χ2 fit, consistent with findings in [110]. However, the χ2 fit was found to
be significantly less sensitive to the initial parameter values. As a result, a two-step fit was

72

performed: first a χ2 fit to get reasonable initial parameters, followed by a log-likelihood
fit. The intensities of the γ-rays are then obtained by multiplying the fit parameters by the
number of γ-rays simulated.

3.3.5

Energy and efficiency calibrations

Energy and efficiency calibrations for GRETINA were performed before and after the experiment using radioactive sources. The energy calibration was set at the beginning of the
experiment for each segment and crystal, and the decomposition algorithm output calibrated
energies. As any uncertainty in these energies is small compared to the uncertainties due to
the Doppler reconstruction, no further energy calibration was performed6 .
All absolute γ-ray intensities from in-beam data were extracted by fitting with simulated
peaks and comparing to the absolute number of γ-rays generated in the simulation. It
is therefore critical to confirm that the simulation accurately reproduces the efficiency of
GRETINA.
The absolute efficiency was obtained by D. Weisshaar [111] by first starting with a multichannel analyzer (MCA) with a calibrated dead time estimation connected to a single crystal
of GRETINA. A spectrum was taken of a 152 Eu source with a known activity of 53.7(7) kBq,
and integrated to get the total number of counts within a fixed energy range spanning most of
the strong lines in 152 Eu. Next, the same crystal was run connected to the GRETINA data
acquisition system (DAQ) in free running mode (that is, self-triggering with no external
validation), and the same count rate was confirmed. The crystal was then connected to
the S800 DAQ to generate an external trigger. The same count rate was again confirmed.
6 As

described in [93], there is an additional adjustment made to correct for differential
nonlinearities in the digitizer modules. No further treatment was made for the current
experiment.
73

152

Eu measured efficiency

0.14
56

Co measured efficiency

Absolute Efficiency

0.12

Simulated efficiency

0.1
0.08
0.06
0.04
0.02
0
0

500

1000

1500
2000
2500
Energy (keV)

3000

3500

4000

Figure 3.15: Simulated γ ray detection efficiency compared to efficiency measured with two
sources. Note the discrepancy at low energy.
Finally, all detectors were connected and the full GRETINA efficiency was obtained for each
prominent line from 152 Eu, with branching ratios taken from the IAEA [112].
This efficiency was independently confirmed by using a 60 Co source which emits two
coincident γ rays, at 1173 and 1332 keV. A lanthanum bromide (LaBr) detector was placed
near the array and used as the trigger. For events in which a 1332 keV γ ray was seen in
the LaBr detector, exactly one γ ray of energy 1173 keV was emitted and the efficiency was
extracted by measuring the observed counts in the 1173 keV peak. The same procedure was
followed for a 88 Y source, which has two coincident lines at 898 keV and 1836 keV. The
efficiencies of these measurements were consistent with the efficiency curve obtained with
the 152 Eu source.
For calibration at higher energy, a 56 Co source was used, although its activity was not well
known. Its intensity was therefore scaled to match the efficiency of the 152 Eu source in the

74

range 1.0-1.5 MeV. The resulting efficiency curve is shown in Figure 3.15. A corresponding
simulated efficiency curve was generated by integrating the full-energy peaks in simulated
spectra and dividing by the number of γ rays emitted in the simulation. This curve is
also shown in Figure 3.15. The agreement is excellent above 500 keV, but the simulation
overpredicts the efficiency at lower energies, particularly below 200 keV. Tests indicated
that this discrepancy could not be due to inaccuracies in the simulation’s treatment of γray absorption at low energies, and the addition of absorptive material was limited by the
known detector geometry. Another potential source of the discrepancy is an inefficiency in
the decomposition procedure at these energies. Low-energy γ rays produce weak signals in
the segments of GRETINA, which may not reach the threshold, causing an event without
any segment information, which will fail in the decomposition. Even if the segment does
trigger, the weak pulse will be relatively noisy and difficult for the decomposition algorithm
to handle, leading to poor fit results. In both cases, and possibly others, these events may not
make it through the data stream to be recorded. This additional inefficiency was accounted
for by adding a modulating function into the simulation which throws away a small fraction
of low energy events. The resulting efficiency curve, shown in Figure 3.16, reproduces the
measured efficiency across the measured range, from 100 keV to 3.5 MeV.

3.3.6

Extracting lifetimes from peak shapes

If an excited state produced in the reaction target has a lifetime in the range of ∼1 ps to
∼10 ns, then the excited nucleus will travel some distance through the target, losing energy
according to (3.2), before decaying. The γ decay will occur at a slower velocity and greater
z position than that assumed in the Doppler reconstruction, resulting in a low-energy tail
on the peak, according to (3.4). The low-energy tail becomes more pronounced for longer
75

152

Eu measured efficiency

0.14
56

Co measured efficiency

Absolute Efficiency

0.12

Simulated efficiency

0.1
0.08
0.06
0.04
0.02
0
0

500

1000

1500
2000
2500
Energy (keV)

3000

3500

4000

Figure 3.16: Simulated γ ray detection efficiency, with correction for losses at low energy,
compared to efficiency measured with two sources. The simulation now reproduces the
measured efficiency from 100 keV to 3.5 MeV.
lifetimes, and so if the lifetime of the state is unknown, it may be extracted by the γ-ray
line shape. This is achieved in this work by simulating the entire process of excitation of
the nucleus, the traveling of the excited nucleus according to its lifetime, followed by the
γ decay and detection. By simulating this process with many different assumed lifetimes,
one may extract the lifetime as the one which produces the best fit to the experimental line
shape. A maximum likelihood fit is performed and the best-fit likelihood L for each simulated
lifetime (actually, the negative logarithm of the likelihood L ≡ − log L) is recorded. Near
the maximum overall likelihood (minimum negative log likelihood), the dependence of the
likelihood on the lifetime may be parameterized by a quadratic function of the lifetime. In
fact, because the relative change in the line shape for a given change in lifetime is greater at
shorter lifetimes, the dependence is parameterized as quadratic function of the logarithm of

76

the lifetime.
L(τ ) = C [log(τ /τ0 )]2 + L0

(3.8)

The extracted lifetime is taken to be τ0 , and the uncertainty ∆τ is estimated by [113]
L(τ ± ∆τ ) = L0 + 12 + δ where δ is the rms deviation for the quadratic fit to L(τ ).
As the excited nucleus moves in the z direction, the geometric efficiency for γ-ray detection changes, resulting in a dependence of the extracted peak intensity on the assumed
lifetime. To estimate the contribution of the lifetime effect to the uncertainty in the peak intensity, the best-fit peak intensity is also recorded for each simulated lifetime. The intensities
associated with τ ± ∆τ are taken to represent this uncertainty.
The effect of the lifetime on the γ peak line shape is dependent on the position of the
detector. Detectors at backward angles (∼ 90◦ ) have a much less pronounced tail, while
detectors at intermediate angles are “shadowed” by the forward-most detectors, leading
to a reduced efficiency for longer lifetimes. In order to maximize sensitivity, the peak is
simultaneously fit for forward, backward and intermediate angles. The results of fits to
lifetime-broadened peaks are shown in §4.2.1, §4.2.3, and §4.2.5.

3.4

Asymmetric momentum distributions

The sensitivity of knockout reactions to the orbital angular momentum

of the removed

nucleon (see §2.3.6) provides an essential tool in identifying the orbit from which the nucleon
was removed and, for a 0+ projectile, identifying the J π of the state populated in the
knockout residue. Extracting

from the parallel momentum distribution of the residue is

dependent on a theoretical prediction that can reproduce the data to sufficient precision to
differentiate between the possible values of . Often in knockout experiments, an asymmetric
77

momentum distribution with a tail to the low-momentum side is seen [114, 115, 116, 117,
118], whereas the results of the eikonal calculation (2.45) are by construction symmetric.
This is because in the eikonal formalism, the target is modeled as a static potential, and
so no target recoil or breakup is included in the calculation. By energy and momentum
conservation, any energy transferred to the target should result in a lowering of the parallel
momentum of the residue, and so it is assumed that the low-momentum tail is due to
the recoil and inelastic excitation of the target. Indeed, proper treatment of energy and
momentum conservation, using a coupled-channels formalism, reproduces the low momentum
tails convincingly for light, loosely-bound projectiles [76]. However, the knockout reaction for
these processes proceeds predominantly through diffraction (see §2.3.5), in which the target
is left in its ground state and the reaction is relatively simple to treat with coupled-channels.
For stripping reactions, where the target is left in an excited state, the coupled-channels
calculation requires a proper treatment of the target breakup.
Treating this theoretically is a tricky task, as 9 Be is only bound by 1.7 MeV, with no
bound excited states. No existing theory can describe accurately how 9 Be will break up in
a knockout reaction. As the stripping mechanism dominates for the cases studied here, we
therefore turn to the possibility of empirically obtaining the effects of the target breakup,
which we can then use to improve the eikonal description of the momentum distribution.
In order to isolate the effect of the target breakup from that due to the recoil from the
removed nucleon, we investigate events in which the outgoing nucleus is the same as the
incoming projectile. As we saw in §3.3.3 there is a clear correlation between the γ spectrum
and the momentum distribution for such events, namely that high-energy γ rays accompany
the more violent collisions. However, the requirement that the projectile survives constrains
these events to be peripheral. These are exactly the collisions we wish to select, because
78

knockout reactions are surface-peaked, owing to the requirement that the core survive.
It is to be expected then that – to some level of approximation – the momentum distribution for knockout should be given by the distribution for these surface-peaked inelastically
scattered events, folded with the additional kick from the knockout, given by the eikonal
calculation. Indeed, as shown in Figure 3.17, including the core-target interaction through
the inelastic scattering data clearly improves the agreement with the measured distributions.
In more detail, we denote the measured inelastic momentum distribution Finel (p), where
p is in the lab frame. In the knockout reaction, we approximately conserve momentum
per nucleon, so this distribution should be scaled by (A − 1)/A. For the measurements of
36 Si

and 38 Si, a different target thickness was used for the knockout and inelastic scattering

experiments. The extra energy loss in the thicker target was estimated by the difference
in centroid momenta between the inelastic scattering run (with no gate on γ-rays) and the
unreacted beam particles in the knockout setting:

∆tgt = P unreacted − P inel .

(3.9)

Further, the mass A and mass (A − 1) systems have different energy loss through the
target, particularly for proton knockout. This difference in outgoing momentum is estimated
by the parameter δp, which is the difference in outgoing momenta after the full target
thickness for the mass A and mass A − 1 systems, estimated with LISE++ [119]. The
reaction takes place at any depth in the target with approximately equal probability, so the
effect of the different energy loss is handled with Heaviside functions

h(p, δp) = |Θ(p − δp) − Θ(p)| .
79

(3.10)

We therefore obtain a corrected inelastic distribution

F(p) = Finel

A
p − ∆tgt ∗ h(p, δp)
A−1

(3.11)

where ∗ indicates a convolution. The distribution F(p) accounts for all effects which are not
due to the recoil of the removed nucleon.
The recoil is handled by the eikonal calculation, which produces a distribution Geik (pcm )
in the center of mass frame, corresponding to the curves shown in Figure 2.10. The final
theoretical curve K(p) which corresponds to the red curves in Figure 3.17 is obtained by
K(p) = F(p) ∗ Geik (p/γ)

(3.12)

where we have divided by the relativistic factor γ in the eikonal distribution to account for
the broadening from the Lorentz boost to the lab frame. The blue curves in Figure 3.17 are
obtained by replacing the inelastic scattering distribution with the unreacted distribution
Finel (p) → Funreacted (p) in (3.11) and by taking ∆tgt = 0.
The amplitude of the distribution K(p) is then scaled so that its integral is equal to
the total number of particles in the experimental momentum distribution, leaving no free
parameters for fitting. This procedure is used for all theoretical momentum distribution
curves shown in this work.
I note here that for momentum distributions for populating the ground state, as shown
in Figure 3.17(a) and (c), the distribution is obtained by the subtraction of momentum
distributions for all feeding transitions (scaled for γ detection efficiency) from the total
momentum distribution.

80

Counts / bin(×103 ) Counts / bin(×103 )

20

35

10

0
20

10

0

Si
gs
=3

(a)

35

Si
908 keV
=1

(b)

10

5

35

Al
gs
=2

(c)

14.5
15
Parallel Momentum (GeV/c)

37

Si
1596 keV
=0

(d)

14.6
14.8
15
Parallel Momentum (GeV/c)

0

2

0

Figure 3.17: Experimental momentum distributions to specific final states, compared to
eikonal model predictions folded with the incoming momentum distribution (blue dashed
lines), or folded with the distribution obtained from the inelastic setting (solid red curves,
see text for details).
The finding that this relatively straightforward folding procedure is so successful at reproducing the measured distributions suggests that the tails observed in the knockout reaction
are primarily due to the core-target interaction, and that higher-order effects are suppressed.
Further investigation of this method, using different projectiles and targets, would help to
confirm this suspicion.

81

Chapter 4
Results and interpretation

There are two possible outcomes: if the result confirms the hypothesis, then you’ve
made a measurement. If the result is contrary to the hypothesis, then you’ve made
a discovery.
(E. Fermi)
The desired observables from this experiment are partial knockout cross sections to individual final states. This requires the knowledge of the available final states, which in the
case of most of the knockout residues in this experiment are not previously known. Therefore, I must first establish level schemes for the six knockout residues studied. These levels
are constructed with the aid of γ spectra, γγ coincidence spectra, momentum distributions,
and comparison with shell model calculations. I begin in §4.1 by outlining the shell model
calculations used here, before giving the extracted level schemes in §4.2. I then proceed with
experimental knockout cross sections in §4.3, and a discussion in §4.4.

4.1

Shell model calculations

Shell model calculations in this work are carried out in the 1s0d1p0f model space, with the
restriction that protons are kept in the sd shell, and we limit neutrons to 1p-1h excitations
across the N = 20 shell gap, in order to allow for intruder parity states (i.e. those with
parity different from that of the independent particle model ground state). This model
space is illustrated in Figure 4.1. Allowing cross-shell excitations introduces an admixture

82

0g9/2
0f 5/2
1p1/2
1p3/2

4

28

0f 7/2

20

0d3/2
1s1/2
0d5/2

2

1

8

0p1/2
0p3/2

0s1/2

1p-1h

Energy (MeV)

3

2

0

protons

0

neutrons

10

20 30
β

40

50

CM

(a)

(b)

Figure 4.1: (a) The sdpf model space used in shell model calculations. The gray boxes
indicate excluded orbits. (b) The dependence of the calculated energy spectra on the centermass-parameter βCM (see text) for 35 Si using the SDPF-MU interaction. The green and
magenta lines indicate positive and negative parity states, respectively.

83

of spurious center-of-mass excitations, which are removed by adding a large positive centerof-mass term to the Hamiltonian [120].

H = H + βCM HCM

(4.1)

Strictly speaking, as pointed out in [121] and elsewhere, this prescription is only applicable
in cases where all possible 1 ω excitations are included in the calculation. For the present
model space, this would require including proton 0p → 0d1s and 0d1s → 0f 1p excitations,
as well as neutron 0f 1p → 0g1d2s excitations, in addition to the neutron 0d1s → 0f 1p
excitations considered in these calculations. However, (4.1) is approximately valid so long
as the other 1 ω excitations contribute a negligible amount to the low-lying intruder states,
which is a reasonable assumption for the nuclides studied here.
The effect of this change may be seen by performing a calculation with a range of values
for βCM . If the spurious components are removed, changing βCM should not change the
resulting low-lying energies. Such a calculation is shown in Figure 4.1, demonstrating that
above βCM ≈ 30, the spurious components have been removed. For all calculations in this
work, we take βCM = 50.
For comparison, we use two different phenomenological interactions developed specifically
for this model space: SDPF-U [122] and SDPF-MU [42]. SDPF-U consists of the USD [123]
interaction for the sd shell matrix elements and a modification of the KB interaction [124]
for the f p shell matrix elements. The sd-f p cross-shell matrix elements, which are the least
constrained by existing data, are taken from the Kahana-Lee-Scott G-matrix interaction
[125]. The cross-shell interaction is then altered by adjusting monopole terms in order
to better reproduce experimental data. Finally, the SDPF-U interaction is split into two

84

interactions, one for Z ≤ 14 and one for Z > 14, with the difference being that for Z ≤ 14
a schematic pairing interaction is subtracted from the f p part of the interaction. This
is justified with the claim that proton excitations into the f p shell – which are excluded
from the model space – should be accounted for by renormalizing the f p shell interaction.
Excitations from the proton d5/2 orbit (the last filled orbit for Z ≤ 14) should be suppressed
relative to excitations from the s1/2 and d3/2 orbits, due to the larger gap in single-particle
energies, and thus the corresponding renormalization should be less.
SDPF-MU also uses the USD interaction for the sd part, although with a small change to
the V¯d

5/2 d3/2

monopole. The f p matrix elements are a mixture of the GXPF1B interaction

and the same modified KB interaction used in SDPF-U. The cross-shell terms are based on
the VM U potential, which is a schematic type potential consisting of a single-range Gaussian
central interaction and a tensor interaction taken from π + ρ meson exchange. The central
interaction is adjusted to reproduce the monopole behavior of other phenomenological shell
model interactions. To this is added a two-body spin-orbit component taken from the M3Y
[126] interaction. There is no adjustment made for Z ≤ 14.
We may visually compare the interactions by plotting their two-body matrix elements
against one another, shown in Figure 4.2. As might be expected, the cross-shell matrix
elements – least constrained by data – have the largest deviation.
Reduced matrix elements for γ decays B(M 1) and B(E2) are also calculated using the
results of the shell model calculations with effective operators (see §2.2.5). The effective M 1
operator is constructed using gs = 0.75gsfree , g π = 1.1, and g ν = 0.1 [127], with no tensor
correction. The effective E2 operator is constructed using effective charges eπ = 1.35 and
eν = 0.35 [42].
In §4.4, the effect of the tensor component of the interactions is investigated. This tensor
85

6

4

4

2

2

0

6

fp
(T=1)

2

0

0

-2

-2

-2

-4

-4

-4

-6
-6

-4

-2

0

2

4

SDPF-MU

6

-6
-6

cross-shell

4

SDPF-U

sd

SDPF-U

SDPF-U

6

-4

-2

0

2

4

6

-6
-6

-4

-2

SDPF-MU

0

2

4

6

SDPF-MU

Figure 4.2: A comparison of the matrix elements of the SDPF-MU interaction vs the matrix
elements of the SDPF-U interaction. If the interactions were identical, all points would lie
along the diagonal lines. Only the T = 1 component of the f p interaction is shown, because
protons are not allowed into the f p shell in these calculations.
component is extracted from an interaction via spin-tensor decomposition as described in
[128], using the somewhat daunting formula:




 a sa ja   c


J
(−1) (2k + 1)


ab|Vk |cd JT =
 b sb jb   d


(1 + δab )(1 + δcd)

LL 
SS
L S J
L




L S J 
J
×
(−1) (2J + 1)


S L k 
J



 a sa ja   c sc jc 






×
 b sb j   d sd j  (1 + δa b
b 
d



ja jb 
L S J
L S J
jc j


sc jc  


 L S
sd jd 

 S L
S J



J

k

)(1 + δc d ) a b V c d J T

d

(4.2)
where k indicates the spin-tensor rank, the square brackets indicate a normalized 9-j symbol
[17], and the curly braces indicate a 6-j symbol, and |cd is shorthand for | c sc jc d sd jd .

86

4.2

Level schemes for the knockout residues

Prior to this work, few states were known in the odd-A silicon isotopes, and almost nothing
was known about the excited states in the odd-A aluminum isotopes. It is therefore necessary
to construct a level scheme for each of the knockout residues in order to identify which
final states were populated. This was achieved using γ spectra, γγ coincidences, relative
intensities, and comparison to shell model calculations. Spin-parity assignment of the levels
is aided by the parallel momentum distribution gated on de-excitation γ rays.
In §4.2.1-4.2.6 below, I present these ingredients, as well as a proposed level scheme. For
each fragment, the information is summarized in a table.
The γ ray intensities are obtained by a fit to simulation, as discussed in §3.3.4. The
uncertainties in the intensities are due to uncertainty in the fit (as given by the fit routine),
√
statistical ( N ) uncertainty in the peak area, uncertainty in the correspondence between
simulated and actual γ detection efficiencies, and uncertainty in the lifetime of long-lived
(τ ∼ 1 ns) states. The efficiency uncertainty is estimated to be 5%, except at low energies
(below 300 keV), where the need for a correction factor suggests a greater uncertainty,
estimated to be 10%. The uncertainty due to extracted lifetimes is assigned based on the
negative log likelihood minimization of the lifetime fit, as demonstrated in Figures 4.4, 4.14,
and 4.19.

4.2.1

35

Si

A previous β decay measurement [129] found three excited states in 35 Si: a 3/2− state
at 910 keV, a 3/2+ state at 974 keV, and a 5/2+ state at 2168 keV. The 974 keV state
decays predominately by a 64 keV γ-ray transition with a life time of τ =8.5(9) ns – too

87

long to be observed in-flight in the current experiment – and so its population could not be
detected here. The ground state was assumed to be 7/2− , from systematics and comparison
with shell model calculations, consistent with the long lifetime for the 3/2+ state. A recent
(d, p) reaction study [9] found an additional state at 2044(7) keV, and assigned it to have
spin-parity 1/2− .

1400

0
Cts/16 keV

Cts. / 16 (keV)

0

715

1000
800

1000

1000

2000

1000
2000
Energy (keV)

1970

1134

500

Gate 1134 keV

0

200
0
0

2000

E γ (keV)

10
10

0

400

1000

908

0

600

Gate 908 keV

50
50

1200

Counts / (4 keV)

780

1500
2000
Energy (keV)

E γ (keV)

2164
2275
2377

Cts/4 keV

Si

Cts. / 4 (keV)

35

780
908

1600

2500

3000

Figure 4.3: Doppler-reconstructed γ-ray spectrum detected in coincidence with 35 Si. The
908 keV transition is broadened by a lifetime effect. The inset figures show backgroundsubtracted γγ coincidence spectra gated on the 908 and 1134 keV transitions. The Doppler
reconstruction was performed with a velocity v/c = 0.426.
The γ-ray spectrum in coincidence with outgoing 35 Si ions in the current experiment is
shown in Figure 4.3. Two strong peaks at 780 and 908 keV are observed. The latter peak,
which we associate with the 910 keV transition from β decay is broadened, presumably due
to a lifetime effect. The lifetime extracted from a maximum likelihood test of simulated
88

line shapes is τ =80(20) ps (see Figure 4.4). This corresponds to B(E2; 3/2− → 7/2− ) =
2 4
17+4
−5 e fm (see appendix B), which compares well with the shell model value of 14 and 16

e2 fm4 calculated with SDPF-U and SDPF-MU, respectively.

Si 908 keV

400

Counts

τ = 83+7
-6 ns
400

300
200

0
400

Counts

300

200

Decays / 100 ions

Front ring

100

28

Middle ring

300
200
100
0
400

26

Iγ = 25.9(0.2)

Counts

Negative Log Likelihood

35

500

Backward rings

300
200
100

24
50

100

0
800

150

850

900

950

1000

Energy (keV)

Simulated Lifetime (ns)

Figure 4.4: Maximum likelihood fit of the lifetime of the state decaying by a 908 keV γ ray
in 35 Si. The upper-left panel shows the negative log likelihood as a function of simulated
lifetime, while the lower-left panel demonstrates the effect of the lifetime uncertainty on
the extracted γ-ray intensity. The uncertainties shown in the figure are statistical only.
In the right panels, the magenta lines show the simulation with the best fit lifetime for
three different rings of GRETINA. The exponential background discussed in §3.3.3 is shown
filled in dark gray, and the lighter blue-filled curve shows the additional background due to
higher-energy transitions.

The 780 keV transition is clearly seen in the inset to Figure 4.3 to be in coincidence with
89

Counts / bin(×103 ) Counts / bin(×103 )

10

5

(a) 1688 keV
=0
=1
=2
=3

(b) 2042 keV

(c) 2164 keV

(d) 2377 keV

(×6)

0
5

0
14.5
15
Parallel Momentum (GeV/c)

14.5
15
Parallel Momentum (GeV/c)

Figure 4.5: Parallel momentum distributions for the population of levels at 1688, 2042, 2164,
and 2377 keV in 35 Si by neutron knockout from 36 Si.
the 908 keV line, depopulating a state at 1688 keV. From the parallel momentum distribution
in coincidence with the 780 keV peak, we assign =0,1 for the removed nucleon. In this
knockout reaction, we expect to strongly populate hole states with J π =7/2− , 3/2+ , 1/2+
and so we assign this state to be 1/2+ . While we do not observe the 974 keV 3/2+ → 7/2−
transition observed in β decay due to its lifetime, a line at 715 keV is visible, which could
connect the 1688 keV and 974 keV states.
The relatively weak transition at 1134 keV likely corresponds to the unplaced 1130 keV
transition seen in [129] and the 1134(6) keV transition seen in [9], which was proposed to
depopulate a state at 2044 keV. The lower inset of Figure 4.3 reveals a coincidence with the
908 keV transition, confirming the state at 2044 keV. The momentum distribution gated on

90

the 1134 keV peak suggests

= 0 or 1, consistent with the 1/2− assignment.

The 2164 keV transition seen in β decay is also present in this measurement, and the
coincident momentum distribution suggests = 2 or 3, consistent with the 5/2+ assignment.
We do not see the reported 1194 keV transition connecting that state with the 974 keV
state, although the reported 30% branching ratio would put that transition at the limit of
our sensitivity.
Due to the low neutron separation energy of 2.48(4) MeV [1] in 35 Si and the absence
of any excited states below 900 keV, it is assumed that the three other transitions near 2
MeV directly populate the ground state, indicating levels at 1970, 2275, and 2377 keV. The
resulting level scheme is shown in Figure 4.6, and the observed transitions are given in Table
4.1.
Table 4.1: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 35 Si.
Eγ [keV]
715(4)
780(4)
908(4)
1134(5)
1970(6)
2164(6)
2275(6)
2377(7)
3611(8)

Yield/100 ions
1.9(2)
13(1)
25(2)
1.5(2)
1.4(2)
1.4(2)
2.0(3)
2.5(3)
1.0(2)

Coincident γ rays
908
780
908

Level [keV]
1688(5)
1688(5)
908(4)
2042(6)
1970(6)
2164(6)
2275(6)
2377(7)
3611(8)

Surprisingly, Figure 4.7 shows that we observe one transition at 3611 keV, which must
depopulate a state that is unbound to neutron emission by at least 1.1 MeV. Gamma decays
which compete with proton emission are seen occasionally, as the proton decay may be
hindered by the Coulomb barrier (see, for example, [130]). However, γ decays which compete
with neutron emission are far less common. In general, a strong angular momentum barrier
≥ 4 is required to hinder the neutron decay, but it is unlikely that an
91

≥ 4 state is

35
4.0

52
72
92

Si

5
7
7
1
9

2
2
2
2
2

52

52
72
72
12
92
32
52
11 2

32
32
52
52
12

32
32

0.0

12
32

908

1.0

72

12

52
72

715

12

12

32

72

32
72

Experiment

12

32

1970
2164
2275
2377
3611

2.0

32

52

Sn

1134
780

Energy MeV

3.0

32
72
52

SDPF U

72

SDPF MU

Figure 4.6: Proposed level scheme for 35 Si from this work compared with shell-model calculations (see text for details). The widths of the arrows are proportional to the efficiencycorrected γ-ray intensity.
populated directly in a knockout reaction, due to the expected very low occupation of the
neutron g9/2 and higher orbitals.
One possible explanation for this situation is the combination of a low neutron separation
energy in the 35 Si parent nucleus (2.48 MeV) and a doubly-magic 34 Si daughter nucleus with
a high first excited state (2.719 MeV [10]). While a 3.6 MeV state in 35 Si is unbound to

92

Sn = 2.474 MeV

(b)

Counts / bin

Counts / (16 keV)

700
600

(a)

60
40

(a)

500

20

400

14.4 14.6 14.8 15 15.2
Parallel Momentum (GeV/c)

150
(b)

300

100

200
100

1

50

2

3

4

5

Energy (MeV)

3

4

Energy (MeV)

5

Figure 4.7: The left panel shows the 3611 keV gamma ray peak (indicated with an arrow)
detected in coincidence with 35 Si, with an inset showing the gates used on the outgoing
parallel momentum distribution. The right panel shows the γ spectra gated on the main
peak in the momentum distribution (blue hatches) and the low-momentum tail (solid red).
The 3611 keV peak appears to be associated with the tail of the distribution.
neutron emission, the ground state of 34 Si is the only energetically allowed final state, and
so spectroscopic overlaps may significantly hinder the decay. The SDPF-MU (SDPF-U)
shell model calculations, combined with eikonal reaction theory, predict knockout to a 5/2+
3
state at 3.76 (4.07) MeV with a cross section of 16.8 (13.3) mb. This 5/2+
3 state would
necessarily contain a neutron hole in the sd shell, and could only decay by the emission of a
d5/2 neutron, resulting in a 2p-2h configuration dominated by two holes in the deeply-bound
neutron d5/2 orbit. Such a configuration should have a small overlap with the ground state in
34 Si

– which has a large contribution of 0p-0h configurations [10] – leading to a significantly

hindered neutron decay. The γ decay, on the other hand, would be a fast 5/2+ → 7/2− E1
decay, and could potentially compete with the neutron emission. Comparing the calculated
and measured cross sections, based on the intensity of the 3611 keV γ-ray (see Table 4.7),

93

the γ branching ratio would be on the order of 5-10%.

2
0

5.0

n

3.0

Sn

2.0
1.0
0.0

0
34

3611

Energy MeV

4.0

35

Si

Si

Figure 4.8: Schematic showing the competition between γ decay and neutron emission for the
3611 keV state in 35 Si. The 35 Si level scheme includes a resonance at 5.5 MeV observed in [9].
The 34 Si level scheme includes the recently-proposed first excited 0+ state [10]. The ground
state of 34 Si appears to be the only final state for which neutron emission is energetically
favorable.

A second explanation could be that this state is weakly populated in a more complicated
multi-step reaction pathway. To estimate the yield from such a higher-order removal mechanism, we assume a two-step reaction in which the incident 36 Si is first excited to its first
2+ state, and a neutron is then removed. Such a process is plausible given the (overlapping)
surface localization of the two reaction mechanisms. The joint probability for this two-step
reaction as a function of the projectile’s impact parameter, b, is computed from the deduced
impact parameter dependences of the cross sections (reaction probabilities) dσ/db, for the
0+ → 2+ inelastic excitation and the neutron removal from the 2+
1 state. The former is
deduced from a conventional inelastic scattering calculation made using the direct reactions
code fresco [131] and the latter is extracted from the eikonal model calculation of the
94

single-particle cross section. The resulting joint probability, integrated over impact parameters, is then multiplied by the SDPF-MU shell-model spectroscopic factor for knockout to
the candidate 3611 keV final states of 35 Si.
With our assumption of neutron knockout from the 36 Si(2+
1 ) state, the most likely candidates for the 3611 keV state, based on the shell model spectroscopic factors and knockout
+
cross sections, are the 7/2+
1 and 5/22 states, see Figure 4.6. Based on the two-step model

above, the estimated two-step cross sections are 0.29 and 0.21 mb, respectively. Essentially
identical results are obtained if, instead, one assumes the probability of projectile excitation
to the 2+
1 state to be the ratio of the total inelastic and elastic cross sections, with the latter
computed from the eikonal model S-matrix for 36 Si + 9 Be scattering using the methodology
discussed here. These two-step estimates are of the same order as the experimental 3611
keV state cross section of 0.8(2) mb. We note that in both shell model calculations the first
of these states lies below the neutron threshold and the second lies above, with energies near
3 MeV. So, if either candidate is the 3611 keV state, these energies suggest a significant
discrepancy between the shell model and experiment.
34 Si ground state by an
The 7/2+
1 state would need to decay to the

= 4 neutron

emission, whose spectroscopic factor would be very small, although it cannot be calculated
in the shell model, as the g7/2 orbit is well outside the model space. The 5/2+
2 state would
decay by

= 2 neutron emission and, as described above for the directly-populated 5/2+

state, would be hindered. In both cases the γ decay to the 7/2− ground state would be E1
and could compete with the hindered neutron decay.
An additional piece of information is given by the momentum distribution associated
with the population of this state. The statistics are insufficient to obtain a backgroundsubtracted momentum distribution for the 3611 keV γ-ray line, as was done for the states in
95

Figure 4.5. However, when the γ-ray spectrum is gated on high- and low-momentum regions
of the outgoing momentum distribution, as shown in the right panel of Figure 4.7, it is clear
that the 3611 keV line results predominantly from events with residue momenta in the tail of
the momentum distribution. As was discussed in §3.3.3 and §3.4, the low-momentum tail is
associated with more inelastic reactions, which would be consistent with a multi-step process.
However, the direct knockout of a more deeply-bound d5/2 neutron could potentially also
produce a distribution with a significant tail.
+
Clearly, if the 7/2+
1 and 5/22 states could be identified independently and their energies

found to be different from 3.6 MeV, then these observations would favor the direct process.

4.2.2

35

Al

The nuclide 35 Al was previously studied by Coulomb excitation, yielding one transition at
1020(8) keV [132]. The ground state is assumed to be 5/2+ from shell model calculations
and systematics, and so the spin of the Coulomb-excited state could be anything from 1/2
to 9/2.
In this work we observe many transitions and coincidences, shown in Figure 4.9 and
tabulated in Table 4.2. The two strongest transitions, at 802 and 1003 keV, are not in
coincidence and are assumed to feed the ground state. Transitions at 859, 968, and 2237
keV are clearly seen to be in coincidence with the 1005 keV transition, shown in the inset of
Figure 4.9, and we place them depopulating levels at 1864, 1972, and 3243 keV, respectively.
Transitions at 1064, 1174, and 2440 keV each differ from the previously mentioned transitions
by approximately 200 keV, and although a coincidence is not observed we tentatively assign
these transitions to connect the levels at 1862, 1972 and 3243 keV to the level at 802 keV.
The transition at 1932 keV is in coincidence with the 802 keV transition, as shown in the
96

800

859
968

20
20

Counts/(8 keV)

Cts/8 keV

Al

1003

35

802

900

Gate 1003 keV
2237

10
10

0
0

0

1000

0

1000

2000

2000

1000
2000
Energy (keV)
Eγ (keV)

4275

3060
3250

×4

2440

1972

100
0
0

0

2237

200

5

0

1932

300

802

5

0

1473

400

2000

Gate 1932 keV

Counts/(8 keV)

Cts/8 keV

600
500

1000
Eγ (keV)

968 859
1064 1174

Counts/(4 keV)

700

3000
Energy (keV)

4000

5000

Figure 4.9: Doppler-reconstructed γ-ray spectrum detected in coincidence with 35 Al. The
section of the spectrum in the box labeled ×4 has been rebinned by a factor 4. The blue
dashed line shows the fitted background, suggesting a peak at 4275 keV. The inset in the
upper-right shows background-subtracted γγ coincidence matrices gated on the 1003 and
2237 keV transitions. The Doppler reconstruction was performed with a velocity v/c = 0.424
lower inset of Figure 4.9, and it is placed depopulating a state at 2734 keV. The transition
at 1972 keV is not observed to have any coincident transitions, and we place it connecting
the level at 1972 keV to the ground state. Likewise, we place the transition at 3250 keV
connecting the level at 3243 keV to the ground state. The neutron separation energy for 35 Al
is 5244(92) keV [1], so the transition at 4275 keV, shown in Figure 4.12 above an exponential
background, could populate the ground state or one of the lowest two excited states. In the
absence of any observed coincidences, we tentatively place the 4275 keV transition feeding
the ground state.

97

Counts / bin(×103 )

10

(a)1972 keV
=0
=2

(b)3243 keV

(c)4275 keV

(×2)

5

0
14.5

15

14.5
15
14.5
Parallel Momentum (GeV/c)

15

Figure 4.10: Parallel momentum distributions for the population of levels at 1972, 3243, and
4275 keV in 35 Al by proton knockout from 36 Si.
Momentum distributions in coincidence with decays from the proposed levels at 1972,
3243, and 4275 keV are shown in Figure 4.10, and the momentum distribution for the
population of the ground state is shown in Figure 3.17(c). All of these distributions are
consistent with the removal of an

= 2 proton.

Table 4.2: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 35 Al.
Levels marked with an asterisk are tentative.
Eγ [keV]
802(4)
859(4)
968(4)
1003(4)
1064(4)
1174(5)
1473(5)
1932(6)
1972(6)
2237(6)
2440(7)
3060(8)
3250(8)
4275(9)

Yield/100 ions
10(1)
3.6(3)
4.4(3)
19(1)
0.8(2)
2.8(3)
1.1(2)
2.5(3)
7.5(5)
7.8(6)
1.4(2)
1.6(4)
3.3(4)
3(1)

Coincident γ rays
1923
1003
1003
859, 968, 2237

802
1003

Level [keV]
802(4)
1864(5)
1972(4)
1003(4)
1864(5)
1972(4)
2734(7)
1972(4)
3243(5)
3243(5)
3243(5)
4275(9)∗

98

35

Al

4.0

52

52

3.0

52

11 2
32
12
52

52

11 2

0.0

11 2
52
32

52
12
32

12
92

92

52
72
52
72
52
11 2
12
32

72

4275

72
3250

1972

1003

802

1.0

1932
2237
2440

2.0
859
1064
968
1174

Energy MeV

52

52

Experiment

52

52

SDPF U

SDPF MU

Figure 4.11: Proposed level scheme for 35 Al from this work compared with shell-model
calculations (see text for details). The width of the arrows is proportional to the efficiencycorrected γ-ray intensity.

4.2.3

37

Si

The structure of 37 Si has been previously studied by Coulomb excitation [132], in which the
authors tentatively proposed a level at 1437(27) keV, and also by the β decay of 37 Al [133],
in which levels at 68, 156, 717, and 1270 keV were proposed. The authors mention in [132]
that the 1437 keV level is uncertain and could be due to a stripping reaction, rather than
inelastic scattering. As no transition near 1437 is seen in the present work or the β decay,

99

+
36
and the 2+
1 → 01 transition in Si lies at 1408 keV [134], we do not include this level in the
37 Si

level scheme. The levels from β decay at 717 and 1270 keV were tentatively assigned

Si

15
15

Cts/4 keV

300

37
156

350

538
562

to 3/2+ and 5/2+ , respectively, based on log(f t) values.

10 156

Gate 903 keV

538

10

692

5
5

0
0

250

500

20

Cts/4 keV

538

10

0
0

-10
0
-10

1000

1000
Energy (keV)

500

1000
1500
Energy (keV)

2000

2323

50
0
0

1500

×2
2068
2115

1442

500

1279

746

100

1500

Gate 156 keV

10

0

150

1000

562

1750

200

903

692

20

716

Counts/(4 keV)

0

2500

Figure 4.12: Doppler-reconstructed γ-ray spectrum detected in coincidence with 37 Si. The
section of the spectrum in the box labeled ×2 has been rebinned and scaled by a factor of
2. The inset shows the background-subtracted γγ coincidence matrix gated on the peaks at
156 and 903 keV. The Doppler reconstruction was performed with a velocity v/c = 0.403.
Unlike in 35 Si, where the 7/2− ground state is well separated from the first excited
states, the shell model predicts low-lying 3/2− and 5/2− states, which correspond primarily
to seniority v = 3 (f7/2 )3 configurations, with the 5/2− predicted to be the ground state.
This suggests that, whatever the spin-parity of the ground state, the other two low-lying
states may be isomers. Additionally, these states allow the 3/2+ intruder state to decay by
a fast E1, making the population of the 3/2+ state detectable.
100

Counts / bin(×103 )

10 (a)gs
=0
=1
=2
=3
5

(b)692 keV

(c) 717 keV
(×5)

(×4)

0
14.5

15
14.5
15
14.5
Parallel Momentum (GeV/c)

15

Figure 4.13: Parallel momentum distributions for the population of the ground state and
excited states at 692 and 717 keV in 37 Si by neutron knockout from 38 Si. Note that the
−
ground state distribution includes both the 5/2−
1 and 7/21 states.
The γ-ray spectrum for the current measurement is shown in Figure 4.12. We observe
transitions at 562 and 716 keV, consistent with the β decay findings. The parallel momentum
distribution in coincidence with a decay from the 716 keV level (Figure 4.13(c)) suggests a
removal of an =2 neutron, consistent with the 3/2+ assignment. The transition at 156
keV is also seen, although it is broadened by a lifetime effect. From simulation, we estimate
the lifetime of the 156 keV state to be τ =4.4(10) ns. Assuming a pure M1 transition, this
−
+1.0
−3 µ2 , which may be compared with
corresponds to B(M 1; 3/2−
1 → 5/21 ) = 3.4−0.7 × 10
N

shell model values of 1 × 10−3 and 1.5 × 10−2 µ2N for SDPF-U and SDPF-MU, respectively.
In addition, we observe two new transitions at 538 and 692 keV, which are also separated
by approximately 156 keV. We place these transitions depopulating a state at 692 keV.
From the parallel momentum distribution shown in Figure 4.13(b) and comparison to shell
model calculations, we assign this level to be 3/2− . We observe that this 3/2− level is
nearly degenerate with the lowest 3/2+ level, as in 35 Si and as predicted by SDPF-MU.
3
In the calculations, the lowest-lying 3/2−
1 state is generated primarily by a neutron (f7/2 )

101

183

Si 156 keV

Front ring

150

Counts

τ = 4.4+0.8
-0.7 ns

182

100

50

181
0

Middle ring

180

Counts

Negative Log Likelihood

37

100

50

0

40

Backward rings
100

35

Counts

Decays / 100 ions

45

Iγ = 36+5
-4

30

50

25
3

4

5

0

6

100

200

300

400

Energy (keV)

Simulated Lifetime (ns)

Figure 4.14: Maximum likelihood fit of the lifetime of the state decaying by a 156 keV γ ray
in 37 Si. The upper-left panel shows the negative log likelihood as a function of simulated
lifetime, while the lower-left panel demonstrates the effect of the lifetime uncertainty on
the extracted γ-ray intensity. The uncertainties shown in the figure are statistical only.
In the right panels, the magenta lines show the simulation with the best fit lifetime for
three different rings of GRETINA. The exponential background discussed in §3.3.3 is shown
filled in dark gray, and the lighter blue-filled curve shows the additional background due to
higher-energy transitions.
2
configuration, while the 3/2−
2 state is generated primarily by an (f7/2 ) p3/2 configuration.

A transition at 903 keV is observed to be in coincidence with the 692 and 538 keV
transitions (see the inset in Figure 4.12), and it is placed connecting a level at 1596 keV to
the level at 692 keV. The corresponding parallel momentum distribution in Figure 3.17(d)
102

suggests = 0 or 1. Considering its strong population and comparing to the shell model and
the 35 Si level scheme, we assign this state to be 1/2+ .

37

2.5

Si

Sn

1.5

12

92

32

72
11 2
52
12
12

72
92
52
11 2

92

32
52

12

12

32

32

1.0

746
1279
1442

12
903

Energy MeV

2.0

52
92

0.0

716
562

0.5
32
52

538
692

32
32
32
32
72
52

156
Experiment

SDPF U

32
72
52
SDPF MU

Figure 4.15: Level scheme for 37 Si from this work compared with shell-model calculations
(see text for details). The width of the arrows is proportional to the γ-ray intensity. Fine
dashed lines indicate tentative levels and transitions, while the thicker dashed line labeled Sn
indicates the neutron separation energy, with the gray shaded area indicating the uncertainty.
A possible transition at 1279 keV could be associated with the 5/2+ level from β decay.
In that work, the 1270 keV level fed the ground state and excited states at 68 and 156 keV
with roughly equal branching ratios, indicating that we should see transitions at 1202 and
103

1115 keV, which we do not. As an alternative, we note that the difference between the
1279 and 746 keV transitions is 533 keV, which agrees within uncertainty with the energy
difference between the 692 and 156 keV levels, suggesting a level at 1438 keV. Indeed, we
also see a possible very weak transition at 1442 keV, which would connect this level to the
ground state. This state would be a candidate for the 1/2−
1 level predicted in both shell
model calculations in Figure 4.15, which would decay by M 1 to both 3/2− states and by
E2 to the 5/2− ground state. Decay to the 3/2+
1 intruder state would be hindered because
the predominant single-particle configurations, (d3/2 )−1 (f7/2 )4 and (f7/2 )2 p1/2 cannot be
connected with a one-body operator. We therefore tentatively place the 1/2− state in the
level scheme at 1438 keV.
Table 4.3: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 37 Si.
Levels and transitions marked with an asterisk are tentative.
Eγ [keV]
156(3)
538(4)
562(4)
692(4)
716(4)
746(4)
903(4)
1279(5)∗
1442(5)∗
1750(6)
2068(6)
2115(6)
2323(6)∗

Yield/100 ions
40(5)
11(1)
13(1)
6.6(9)
5.3(8)
0.8(6)
10(1)
1.4(5)
0.6(5)
2.3(7)
1.9(6)
1.8(6)
0.8(5)

Coincident γ rays
538, 562
156, 903
156
903

538, 692

Level [keV]
156(3)
693(4)
717(4)
693(4)
717(4)
1438(6)∗
1596(5)
1438(6)∗
1438(6)∗

The additional observed transitions at 1750, 2068, 2115 keV and a possible transition at
2323 keV are not placed in the level scheme, although due to the neutron separation energy
at 2250(100) keV, they will very likely only feed the ground state or other members of the
low-lying multiplet. The observed γ-rays are tabulated in Table 4.3 and the proposed level
104

scheme is shown in Figure 4.15.

4.2.4

37

Al

No previous data exist on excited states of 37 Al. In this work, we observe a strong transition
at 775 keV. This transition is seen to be coincident with the transition at 2558 keV and with
itself. Since the coincidence spectrum is background-subtracted, this indicates that there is
another transition with a nearly identical energy. In fact, the 775 keV peak has a shoulder
at low energy, suggesting a possible second peak. A fit with two Gaussians gives a best
description with peaks at 775 and 755 keV, suggesting levels at 775 and 1530 keV, as shown
in Figure 4.17. The transition at 3321 keV agrees within uncertainty with the sum of 775
and 2558, and so we place that transition connecting a level at 3327 keV with the ground
state. The transitions at 1255 and 1393 keV are not observed in coincidence with either
peak in the doublet at 775 keV, and they are not expected to feed the state at 3327, due to
the neutron separation energy at 4210(160) keV [1]. We therefore tentatively place the 1253
and 1393 keV transitions directly populating the ground state. Observed transitions at 861,
1612, and 2722 keV are not placed in the level scheme. We note that the intensities of these
three unplaced transitions could account for most of the remaining population of the state
at 775 keV.

4.2.5

39

Si

The excited states of 39 Si have been studied previously by fragmentation of 40,41 P and 42,43 S
[11]. Sohler et al . observed prominent peaks at 163(12) and 397(14) keV, as well as several
other more weakly-populated transitions, and their high efficiency allowed them to establish

105

775

37

Al

Counts/(16 keV)

160

2558

5
5

0

140

0

0

1000

0

3321

2722

40

×2

1612

861

60

3000

2558

80

2000

2000
Energy (keV)
E γ (keV)

1393

100

1255

120
755

Counts/(8 keV)

Gate 775 keV

755
Cts/16 keV

180

20
0
0

500

1000

1500
2000
Energy (keV)

2500

3000

3500

Figure 4.16: Doppler-reconstructed γ-ray spectrum detected in coincidence with 37 Al. The
section of the spectrum in the box labeled ×2 has been rebinned and scaled by a factor of
2. The inset shows the background-subtracted γγ coincidence matrix gated on the peak at
775 keV. The Doppler reconstruction was performed with a velocity v/c = 0.400.
Table 4.4: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 37 Al.
Levels marked with an asterisk indicate a tentative assignment.
Eγ [keV]
755(6)
775(4)
861(4)
1255(5)
1393(5)
1612(5)
2558(7)
2722(7)
3321(8)

Yield/100 ions
4.5(12)
23(2)
2.6(10)
7.0(13)
6.7(13)
3.2(11)
7.3(19)
3.8(16)
4.7(15)

Coincident γ rays
Level
775
1530(7)
755
775(4)
1255(5)∗
1393(5)∗
775

3327(8)
3327(8)

106

37

92

Al

3321

7
9
5
11

1393

1255

775

1.0

2
2
2
2

32
12
92

755

2.0

0.0

52
11 2

2558

Energy MeV

3.0

7
1
5
9
3

2
2
2
2
2

32
12
92

72

72

52

52
Experiment

SDPF U

52
SDPF MU

Figure 4.17: Proposed level scheme for 37 Al from this work compared with shell-model
calculations (see text for details). The width of the arrows is proportional to the efficiencycorrected γ-ray intensity.
several γγ coincidences. As in 37 Si, the shell model predicts two low-lying v = 3 states for
39 Si,

suggesting the presence of two isomers. As described before, these states allow the

prompt decay of the 3/2+ intruder state.
In the present work, we observe a strong peak at 172 keV, likely broadened by a lifetime
effect (see Figure 4.19). From simulation we estimate the lifetime of the 172 keV transition

107

39

Si

Cts/8 keV

172

160
140

Gate 172 keV
5
5

0
0

0

Cts/8 keV

40

2
2

1500

0
0

500

879

0

644

60

1000

Gate 879 keV

0

485

80

500

1000

1500

1000
Energy (keV)

928
974

100

290
350
379

Counts/(4 keV)

120

20
0
0

200

400

600
800
1000
Energy (keV)

1200

1400

1600

Figure 4.18: Doppler-reconstructed γ-ray spectrum detected in coincidence with 39 Si. The
Doppler reconstruction was performed with a velocity v/c = 0.379. The inset shows the
background-subtracted γγ coincidence matrix gated on the peaks at 172 and 879 keV, revealing no strong coincidences with either peak.
to be τ =1.4(2) ns. Assuming a pure M1 transition 7/2− → 5/2− , as discussed below, this
−
−3 µ2 , compared to shell model predictions
corresponds to B(M 1; 7/2−
1 → 5/21 ) = 8(1)×10
N

of 3 × 10−3 and 6 × 10−3 µ2N for SDPF-U and SDPF-MU, respectively.
In [11], the γ-ray spectrum resulting from proton knockout from 40 P was dominated
by the 163 keV transition, which we associate with our 172 keV transition due to the fact
that the lifetime effect – imperceptible with their energy resolution – shifts the centroid of
the peak to a slightly lower energy. Using the SDPF-MU spectroscopic factors and singleparticle cross sections from eikonal theory with a quenching factor R=0.4, we estimate the
cross sections for one proton knockout from 40 P to the lowest-lying states in 39 Si to be 5/2−
1:
108

Si 172 keV

Front ring

80

τ = 1.36+0.13
-0.12 ns
99

Counts

100

60
40
20

98
0

Middle ring

60

97

Counts

Negative Log Likelihood

39

45

40

0

Backward rings

60

40

Iγ =

39.3+1.5
-1.4

Counts

Decays / 100 ions

20

40

20

35
1.2

1.4

0

1.6

50

100

150

200

250

Energy (keV)

Simulated Lifetime (ns)

Figure 4.19: Maximum likelihood fit of the lifetime of the state decaying by a 172 keV γ ray
in 39 Si. The upper-left panel shows the negative log likelihood as a function of simulated
lifetime, while the lower-left panel demonstrates the effect of the lifetime uncertainty on the
extracted γ-ray intensity. The uncertainties shown in the figure are statistical only. The right
panels show the best fit for three different rings of GRETINA. The exponential background
discussed in §3.3.3 is shown filled in dark gray, and the lighter blue-filled curve shows the
additional background due to higher-energy transitions.
−
−
−
−
1.8 mb, 7/2−
1 : 1.1 mb, 3/21 : 0.1 mb, 1/21 : 0.06 mb, 5/22 : 0.4 mb, 3/22 : 0.2 mb. (Here we

assume a 2− ground state, although the results are similar if we assume a 3− ground state.)
−
This suggests that their 163 keV transition depopulates either the 5/2−
1 or 7/21 . Since we

also populate this transition strongly in one neutron knockout, it is most likely that this

109

transition depopulates the 7/2−
1 state. The parallel momentum distribution in coincidence
with this transition shown in Figure 4.20(a) suggests

= 2 or 3 for the removed nucleon, in

support of this assignment. Due to the low energy of the transition, it most likely proceeds
by E1 or M 1. From comparison with the shell model level schemes, the most likely candidate
for the final state is 5/2− . The shell-model calculations also predict a low-lying 3/2− state
which, within the uncertainty of the calculations, could lie below the 5/2− , making it the
ground state. If that were the case, the levels proposed here would be shifted up in energy

Counts / bin(×103 ) Counts / bin(×103 )

by the gap between the 3/2− and the 5/2− .

2

1

(a)172 keV
=0
=1
=2
=3

(b)379 keV

(c)879 keV

(d)926 keV

(×2)

0
(×3)
1

0
14.5
15
Parallel Momentum (GeV/c)

14.5
15
Parallel Momentum (GeV/c)

Figure 4.20: Parallel momentum distributions gated on γ-ray transitions in 39 Si (no feeding
subtraction).
We also expect to preferentially populate 3/2+ , 1/2+ and 3/2− states with the knockout
reaction. The next-strongest transition at 879 keV has a corresponding parallel momentum
110

distribution shown in Figure 4.20(c) that suggests

= 2 or 3, so we tentatively assign that

transition to depopulate a 3/2+ level. By comparison with the shell model, the most probable
final state for the decay of the 3/2+ state would be the 5/2− ground state, as the population
of the 7/2− state would require a slower M2 transition, and the population of the 3/2− and
1/2− states would be hindered by their strong admixtures of (f7/2 )4 p3/2 and (f7/2 )4 p1/2 ,
respectively. We therefore tentatively place the 3/2+ level at 879 keV, connecting to the 5/2−
ground state. The momentum distribution in Figure 4.20(b) in coincidence with the 379 keV
transition suggests

= 0 or 1 but without additional information from γγ coincidences, the

placement in the level scheme is unclear.
Table 4.5: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 39 Si.
Levels marked with an asterisk are tentative.
Eγ [keV]
172(5)
290(3)
350(3)
379(3)
485(4)
644(4)
879(4)
928(4)
974(4)

Yield/100 ions
36(6)
4(1)
6(2)
10(2)
2.2(8)
2.5(10)
12(2)
5.3(11)
2.3(8)

Coincident γ rays

Level [keV]
172(5)

879(5)∗

Additional lines at 290, 350, 485, 928, and 974 keV, as well as a broad structure around
644 keV, which likely corresponds to an unresolved multiplet, are likewise not placed in the
level scheme. The inset panel in Figure 4.18 shows the γγ coincidence matrix gated on the
two strongest peaks at 172 and 879 keV, revealing no clear coincidences. The most prominent
structure in the 172 keV coincidence spectrum is centered around 750 keV, which does not
correspond to any transition observed in singles. This supports the interpretation that these
transitions depopulate states which are directly populated in the knockout reaction.
111

2.0

39

Si

52
92

Sn

11 2
11 2
92
72
12

72
92

72
52
12

12

72
0.0

52

172

879

0.5

Experiment

32

52
12
1
7
3
5

2
2
2
2

32
72
52

SDPF U

SDPF MU

397

32
52

32

163

32

303
906

1.0

657
1143
1551

Energy MeV

1.5

Sohler

Figure 4.21: The left figure shows the proposed level scheme for 39 Si. The right figure is
the level scheme proposed by Sohler et al . [11]. In the left figure, the width of the arrows in
proportional to the efficiency-corrected γ-ray intensity. As described in the text, the levels
could have an offset if the lowest 3/2− state lies below the 5/2− .

4.2.6

39

Al

No previous measurements of the excited states of 39 Al exist. In the current measurement we
observe three transitions, at 764, 800, and 883 keV, and a possible peak at 995 keV. The low
statistics in this setting preclude any γγ coincidence measurement. The 800 keV transition

112

is the most intense, and we assign it populating the ground state. The other transitions are
not placed in the level scheme.
39

Al

764
800

22
20

16

883

14
12
10

995

Counts/(8 keV)

18

8
6
4
2
0
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Energy (keV)
Figure 4.22: Doppler-reconstructed γ-ray spectrum detected in coincidence with 39 Al. The
Doppler reconstruction was performed with a velocity v/c = 0.388.
Table 4.6: Gamma-ray energies, efficiency-corrected intensities, and coincidences for 39 Al.
Eγ [keV]
764(8)
800(8)
883(8)
995(8)

Yield/100 ions
10(4)
12(4)
9(4)
1(2)

Coincident γ rays

Level [keV]
800(8)

113

2.0
39

11
9
5
3
11
3
1
9
7
5

Al

Energy MeV

1.5

1.0

2
2
2
2
2
2
2
2
2
2

92

0.0

2
2
2
2

92

72
32
12

72
32
12

800

0.5

9
1
7
5

52

52

Experiment

SDPF U

52

SDPF MU

Figure 4.23: A possible level scheme for 39 Al, although γγ cascades cannot be ruled out
due to low statistics. The center and right figures are shell-model calculations (see text for
details).

4.3
4.3.1

Knockout cross sections
Inclusive cross sections to all bound states

Knockout cross sections are obtained by the equation

σko =

Nr
Nb nt

114

(4.3)

where Nr is the number of knockout residues detected, Nb is the number of incident beam
particles, nt is the areal target number density, and

is the efficiency for detecting the

knockout residues. In all knockout settings, the full momentum distribution was within the
acceptance of the S800, and so I take = 1 within the other uncertainties of the measurement.
The number of detected knockout residues is obtained by integrating the total number of
counts within the PID gates shown in Figure 3.6. The uncertainty in this number is given by
the statistical counting uncertainty

√

Nr and by the systematic error due to the placement

of the PID gate. The number of desired events excluded by the PID gate is estimated by
fitting the gated PID distribution with at two-dimensional Gaussian and evaluating what
fraction of the Gaussian lies outside the gate. In all cases, this amounts to less than 1% of
the total number of counts inside the gate.
The number of incident particles was inferred by monitoring the rate of the object scintillator and scaling that rate to a calibration measurement in which the rigidity of the S800
was centered on the unreacted beam. This accounts for fluctuations in beam rate, so the
main source of error in this method is fluctuations in the composition of the beam. The
uncertainty due to these fluctuations is estimated by calculating the cross section for each
one-hour run (shown in Figure 4.24) and taking the root-mean-squared deviation. This is
the dominant source of uncertainty for the inclusive cross section measurement, and is of the
order of a few millibarns.
The beryllium target used for knockout from 36 Si and 38 Si had an areal mass density of
287(3) mg/cm2 and the target used for knockout from 40 Si as well as inelastic scattering for
all three projectiles had a density of 376(4) mg/cm2 .
All these uncertainties are uncorrelated and so the total uncertainty in the inclusive cross

115

Cross section to bound states (mb)

Cross section to bound states (mb)

36

Si -1n

82
81.5
81
80.5
80
276

278

280
282
Run number

284

23.5

36

Si -1p

23
22.5
22
21.5
21

286

288

290

292
294
296
Run number

300

(b)

38

110

Cross section to bound states (mb)

Cross section to bound states (mb)

(a)

298

Si -1n

108
106
104
102
100

38

Si -1p

15
14
13
12
11

98
194

195

196
197
198
Run number

199

200

205

210

Cross section to bound states (mb)

40

125

Si-1n

120
115
110
105
100
240

250

260
Run Number

225

230

(d)

Cross section to bound states (mb)

Cross section to bound states (mb)

(c)

215
220
Run number

405 410 415

40

40

Si -1p

35
30
25
20
15
10
5
0
300

(e)

320

340
360
Run number

380

(f)

Figure 4.24: Knockout cross section to all bound states for each of the six settings, calculated
on a run-by-run basis. The error bars on each point are statistical errors for counting the
number of events inside the PID gate. The dashed black line shows the error-weighted mean
value of all the runs, while the gray band indicates the uncertainty, which is taken from the
root mean squared deviation of the points from the mean.
116

section ∆ko is given by
∆ko 2
=
σko

∆r 2
+
Nr

∆t 2
+
nt

∆b 2
Nb

(4.4)

The resulting inclusive one-proton and one-neutron knockout cross sections are shown in
Figure 4.25, compared to theoretical predictions using shell-model energies and spectroscopic
factors. In order to account for the uncertainty in both the neutron separation energies and
the shell model energies, the neutron separation energy was varied by ±500 keV to produce
the error bars shown on the theoretical curves. For calculations performed with SDPF-U,
the two lowest even-parity 1p-1h states were considered bound, regardless of their calculated
energies. Also shown in Figure 4.25 is the theoretical cross section scaled by an empirical
reduction factor R(∆S) ≈ 0.61 − 0.016∆S, taken from a fit to the systematics in [12]. Here
∆S is defined to be the difference in separation energy between the removed species and the
spectator species, e.g. for neutron knockout ∆S = Sn − Sp .
One notable feature in the trend in neutron knockout cross sections is the decrease from
34 Si

to 36 Si. In the shell model calculations, this is due to the fact that in 33 Si, which has a

neutron hole in the sd shell, the relevant energy for a d5/2 hole state is the d3/2 − d5/2 gap,
which is small enough for there to be bound 5/2+ states with substantial d5/2 hole strength.
At 35 Si, the neutron sd is filled and so the relevant energy becomes the f7/2 − d5/2 gap,
pushing d5/2 hole states up in energy, above the low neutron separation energy in 35 Si. As
a result, removal of a d5/2 neutron does not substantially contribute to the cross section to
bound states in 36 Si.

117

34-40

140

Si

SDPF-U

120

σko (mb)

100

-1n

80
60
40

-1p

20
0

SDPF-MU

140
120
-1n

σko (mb)

100
80
60
40

Experiment
Theory
Theory × R(∆S)

20
0

20

-1p

22
24
Neutron number

26

Figure 4.25: Inclusive 1n and 1p knockout cross sections from silicon isotopes to all bound
states as a function of projectile neutron number, compared with theoretical prediction (solid
red line) and the theoretical prediction scaled by a systematic reduction factor R(∆S) [12].
The theoretical error bars are generated by varying the neutron separation energy by ±500
keV. The experimental value for 34 Si is taken from [13]

118

4.3.2

Exclusive cross sections to final states

The knockout cross section to a specific final state i is given by the inclusive cross section
multiplied by the probability Pi of populating that state.

σi = σko Pi

(4.5)

This population is in turn calculated by summing the yield of all γ rays Di that de-excite
state i and subtracting the yield of all γ rays Fi that feed state i

Pi =
j∈Di

Yj −

Yj

(4.6)

j∈Fi

The population of the ground state is taken to be the remaining strength after subtracting
all populated excited states:
Pgs = 1 −

Pi

(4.7)

i

The uncertainty ∆ in the population probability is given by

∆Pi

2

=
j∈Di +Fi

∆Yj

2

(4.8)

In the case of unplaced γ transitions, the ground state cross section is taken to be the average
of the maximal and minimal feeding scenarios, and the cross sections for these unplaced
transitions are added in quadrature to the ground state uncertainty.
The resulting exclusive cross sections are shown in Figures 4.26-4.31, and tabulated in Table 4.7. Transitions which are not placed in the level scheme are assumed to feed the ground
state for plotting purposes, in order to better demonstrate the distribution of strength.
119

-

60

(7/2 + 3/2+)

36

Si -1n

This work

40

σpart (mb)

3/2

60
40

1/2+

-

20

SDPF-MU × 0.83

3/2+
-

7/2

1/2+

20

1/2+

60
40

5/2+

SDPF-U × 0.83
+

3/2
1/2+

-

7/2

20

0

1

2
Energy (MeV)

+

3/2+ 1/2 5/2
+

5/2+

3

4

Figure 4.26: One neutron knockout cross section to final states in 35 Si. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
60

-

38

7/2

40

-

3/2+

3/2

20

-

σpart (mb)

3/2
60
40

Si -1n

This work

1/2+

SDPF-MU × 0.86

-

7/2

3/2+
1/2+

20

5/2+

-

3/2
60
40

SDPF-U × 0.86

-

7/2

20

3/2+ +
1/2
7/2

-

3/2

0

1

2
Energy (MeV)

5/2+

3

4

Figure 4.27: One neutron knockout cross section to final states in 37 Si. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
Special care is required in evaluating partial cross sections for neutron knockout from
40 Si

because there are so many γ transitions with are not placed in the level scheme of 39 Si,

and which could be potential feeders. The 7/2−
1 state at 172 keV should be fed negligibly by

120

-

60
40

40

7/2
-

3/2

σpart (mb)

This work

+

[3/2 ]

20

60

Si -1n

-

SDPF-MU × 0.91

7/2

3/2+

40
-

3/2

20

1/2+ 7/2

+
3/2+ 3/2

-

60

7/2

SDPF-U × 0.91

40
20

-

3/2

7/2

-

3/2

0

0.5

3/2+ +
1/2

-

1
1.5
2
Energy (MeV)

2.5

3

Figure 4.28: One neutron knockout cross section to final states in 39 Si. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
20
15

36

5/2+

Si -1p

This work

10
5

σpart (mb)

20

5/2+

SDPF-MU × 0.39

15
10

5/2+

5
20

5/2+

SDPF-U × 0.39

15
10
5

+

5/2+

1/2

0

1

2
Energy (MeV)

3

4

Figure 4.29: One proton knockout cross section to final states in 35 Al. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
the 3/2+ and 1/2+ intruder states, due to the required M 2 and E3 transitions, respectively.
Low-lying 3/2− states could feed the 7/2−
1 state by an E2 transition. However, the γγ
coincidence spectrum gated on the 172 keV transition reveals no coincidences, and shell

121

10

38

5/2+

Si -1p

This work

σpart (mb)

5

5/2+

SDPF-MU × 0.36

10

5/2+

5

10

5/2+

SDPF-U × 0.36
5/2+
1/2+

5

0

1

2
Energy (MeV)

3

4

Figure 4.30: One proton knockout cross section to final states in 37 Al. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
8

40

5/2+

Si -1p

This work

6
4

σpart (mb)

2

8
6

SDPF-MU × 0.31

5/2+

4
2
8
6

1/2+

3/2+

SDPF-U × 0.31

5/2+

4

1/2+

2

0

0.5

+
5/2+ 3/2

1
1.5
Energy (MeV)

2

2.5

Figure 4.31: One proton knockout cross section to final states in 39 Al. Experimental data
is shown in the top panel, and theoretical predictions are shown in the bottom two panels.
model calculations using the SDPF-MU interaction predict a ∼1% branching ratio from
either of the 3/2− states to the 7/2−
1 state. Therefore, uncertainty in the cross section to
the 7/2−
1 state due to feeding should be small compared to the uncertainty due to extracting

122

the peak area.
The cross section to the 3/2−
1 state – which is obtained by subtraction because no deexcitation γ-rays are observed – is a more difficult case. This state should be fed by higherlying 7/2− , 5/2− , 3/2− , 1/2− , 3/2+ , and 1/2+ states and, furthermore, γ decays from
these states to the 5/2−
1 state should also be subtracted. Clearly, a detailed knowledge of
the level scheme is required to properly perform the subtraction. In the absence of such
knowledge, upper and lower limits on the cross section are obtained by considering minimal
and maximal feeding scenarios, constrained by requiring no direct feeding of the 172 and
879 keV transitions, and by the neutron separation energy of 1580(110) keV. The maximal
feeding case is obtained by assuming all observed γ transitions directly populate the ground
state (or an isomer), yielding the minimum cross section σ3/2− = 19(14) mb. The minimal
feeding case is obtained by assuming that only the 172, 350, 379, and 879 keV transitions
feed the ground state or an isomer. The remaining transitions are placed depopulating
higher states, with the 485 and 640 keV transitions feeding the 350 keV level, and the 664,
928, and 974 keV transitions feeding the 379 keV level. This results in the maximum cross
section σ3/2− = 40(9) mb. The cross section is taken to be the average of the maximal- and
minimal-feeding results, and the uncertainty from the feeding is added in quadrature to the
other uncertainties. The result is given in Table 4.7.

4.4

Discussion

The evolution of neutron single-particle strength along the silicon isotope chain may be
visualized by plotting the cross sections for populating the low-lying 5/2+ , 3/2+ , 1/2+ ,
7/2− , and 3/2− states, which gives an indication of the occupancy of orbits near the Fermi

123

Table 4.7: Total and partial knockout cross sections to bound
final states for each of the knockout reactions studied.

36 Si→35 Si

σko [mb] Efinal [keV]
81(2)
0
908
974
1688
1979
2042
2164
2275
2377
3611

π
Jfinal

[7/2− ]
[3/2− ]
[3/2+ ]
1/2+
[1/2− ]
[5/2+ ]

σpart [mb]
52(4)a
8(3)
b

13(1)
1.1(2)
1.3(2)
1.1(2)
1.6(2)
2.1(2)
0.8(2)

38 Si→37 Si

104(3)

?c
156
692
717
1438
1595

[7/2− ]d
[3/2− ]
[3/2− ]
[3/2+ ]
[1/2− ]
[1/2+ ]

47(9)
9(7)
7(3)
19(2)
3(1)
10(1)

40 Si→39 Si

116(5)

?c
172

[3/2− ]d
[7/2− ]

29(20)
49(7)

36 Si→35 Al

22(1)

0
802
1003
1862
1972
2731
3243
4275

[5/2+ ]

13(2)
1.0(7)
0.8(9)
1.0(2)
3.2(5)
0.5(1)
2.6(3)
0.5(1)

38 Si→37 Al

13(1)

0
775
1253
1393
3327

[5/2+ ]

7(2)
1.5(10)
0.9(2)
0.9(2)
1.6(6)

40 Si→39 Al

8(2)

0
800

[5/2+ ]

5(2)
0.9(4)

Includes population of the 3/2+
1 isomer.
b Not seen due to long lifetime (see text).
c Low lying level (energy is unknown).
d Includes (presumably weak) population of the 5/2− state.
1

a

124

Table 4.8: Calculated spectroscopic factors to low-lying final states in 33,35,37,39 Si.
interaction

7/2−
1

3/2−
1

3/2−
2

3/2+
1

1/2+
1

5/2+
1

3.48
3.61

1.38
1.55

2.42
3.69

2.61
3.07

1.00
0.96

0.26
0.09

34 Si

SDPF-U
SDPF-MU

36 Si

SDPF-U
SDPF-MU

1.73
1.71

0.09
0.13

38 Si

SDPF-U
SDPF-MU

2.48
2.81

0.02
0.11

0.26
0.27

2.19
2.79

0.97
0.80

0.05
0.03

40 Si

SDPF-U
SDPF-MU

3.33
3.19

0.51
0.90

0.17
0.08

1.69
2.31

0.72
0.53

0.04
0.02

surface. Figure 4.32 shows the experimental cross sections to directly populate the lowest
7/2− state, and the cross section to populate the lowest two 3/2− states (if they are bound),
compared with theoretical predictions. The lowest two 3/2− states are used because there
x and f x−1 p
configurations in 37 Si and 39 Si. In addition
should be mixing between the f7/2
7/2 3/2

to the curves predicted by the shell model calculations (shown in solid lines), results are
shown for calculations in which the cross-shell (sd-f p) tensor component of the SDPF-MU
interaction has been set to zero, shown in dotted lines. The results for SDPF-U with the
tensor removed are similar. All theoretical curves have been scaled by a reduction factor
R(∆S), obtained from a fit to the systematics in [12].
+
+
Figure 4.33 shows the cross sections to the 5/2+
1 , 3/21 and 1/21 states, corresponding to

the removal of an sd shell neutron, compared with theoretical predictions. The cross sections
for knockout from 34 Si are taken from a previous experiment [13]. As in Figure 4.32, the
theoretical cross sections have been scaled with the systematic reduction factor. Calculated
spectroscopic factors are given in Table 4.8 for reference.
A number of observations may be made regarding these plots. First, it appears that
both interactions reproduce the strength to the 7/2−
1 state quite well. From the calculations

125

Cross section to bound states (mb)

-

7/2

80

-

3/2

SDPF-MU
SDPF-U

60

SDPF-MU*
-

40

7/2

20
-

3/2
0

22

24
26
Projectile neutron number N

28

Figure 4.32: Partial cross sections for the population of a bound final state with J π = 7/2−
and 3/2− in a one-neutron knockout reaction as a function of mass number for the silicon
projectile. Theoretical predictions using the SDPF-MU (solid lines) and SDPF-U (dashed
lines) effective interactions are also shown. The dotted line shows the result for the SDPFMU interaction with the cross-shell tensor component set to zero.

126

Cross section to bound states (mb)

+

5/2

80

+

3/2

+

1/2

SDPF-MU

60

SDPF-U

3/2+

40

20
+

5/2
0

34

1/2+
36
38
Projectile mass number

40

Figure 4.33: Experimental cross sections to the lowest 5/2+ , 3/2+ , 1/2+ states compared
with the calculated cross sections using energies and spectroscopic factors from SDPF-MU
(solid lines) and SDPF-U (dashed lines). The data for 34 Si are taken from [13].
it is clear that the effect of the cross-shell tensor interaction becomes important at 40 Si for
reproducing the 7/2− strength. Second, the strength to 3/2− states is significantly underpredicted. These findings are consistent with findings in neutron knockout from 30,32 Mg [135]
and from 33 Mg [136], as well as a 30 Mg(t,p)32 Mg study [137], all of which found evidence
for an excess of p3/2 strength relative to shell model predictions.
Turning to the positive-parity states in Figure 4.33, two features are apparent: (i) the
knockout strength to positive-party states in 36,38,40 Si is over-predicted by the shell-model
calculations, and (ii) there is a large drop in the experimental knockout strengths from 34 Si
to 36 Si which is not predicted by theory. The observation (i) is perhaps not so surprising,

127

considering that the calculations were carried out in a model space restricted to 1p-1h neutron
excitations out of the sd shell. For the 0+ ground states of the even-A projectiles, this
is effectively a 0p-0h truncation, which artificially increases the occupation of neutron sd
orbitals. Additionally, the exclusion of higher np-nh excitations in the odd-A residues reduces
the fragmentation of the single-particle strength to some degree. Both of these effects lead
to overprediction of the knockout cross section to positive-parity states.
Observation (ii) is somewhat more surprising. The extracted spectroscopic factors for
+
33
knockout to the 3/2+
1 and 1/21 states in Si are at (or above, when one includes empirical

quenching [12]) their closed-shell sum-rule values of 2J + 1. As the shell-model predictions
are already restricted to 0p-0h excitations across the N = 20 gap for the initial and final
states considered, the only way to sufficiently increase the theoretical spectroscopic factors
is to freeze out proton excitations across the Z = 14 sub-shell gap, which is unrealistic.
Part of this discrepancy might be explained by considering effects that could lead to
artificially high experimental cross sections. While 34 Si has good proton and neutron shell
closures, there should still be some admixture of np-nh neutron excitations to the f p shell
in the ground state. The removal of a neutron from the f p shell would lead predominantly
to population of the lowest 7/2− and 3/2− states in 33 Si. The 7/2− state is a 15 ns isomer,
so it would be undetected by prompt, in-flight γ-ray spectroscopy and included in the 3/2+
ground state cross section. The 3/2− state decays to the 1/2+ state by emitting a 971 keV
γ-ray, which would have been difficult to distinguish from the 1010 keV γ-ray depopulating
the 1/2+ level, due to the γ-ray energy resolution of the scintillator array used in that
experiment. In this case, each population of the 3/2− state would have been counted twice
in the population of the 1/2+ state, and subtracted one time too many from the population
of the ground state.
128

In a rough estimate, assuming the ground state of 34 Si to consist of 80% 0p-0h neutron
configurations and 20% 2p-2h neutron configurations (probably an over-estimate of the 2p-2h
content [138]), with equal occupation of the 0f7/2 and 1p3/2 orbits, this would correspond to
a ∼3 mb cross section to each of the 7/2− and 3/2− states, with a corresponding reduction of
the 1/2+ cross section by 6 mb. In this case, the 1/2+ cross section would agree with the shell
model prediction within error, if the empirical quenching of [12] is neglected. Regardless,
it would be enlightening to have an independent measurement of the 34 Si cross sections in
order to ascertain the significance of the discrepancy.
Two additional observations may be made regarding the systematics of the excited state
spectra in 35,37,39 Si. The significant cross section to a state below the 172 keV 7/2−
1 state
in 39 Si, which is interpreted as a 3/2−
1 state, indicates that the N = 28 shell gap is already
quite small at 39 Si. This is consistent with the findings in 27 Ne [139] and 29,31 Mg [135,
140], that at large proton-neutron asymmetry the energy of the 3/2−
1 state drops below the
39
5
7/2−
1 state. However, in Si this picture is complicated because of the presence of (f7/2 )

configurations, which can mix with single-particle excitations to the p3/2 orbital and thwart
a simple interpretation in terms of single-particle orbits. Despite this complication, it is clear
that the structure of 39 Si does not reflect a large shell gap at N = 28. The significant cross
37
section to the 156 keV 3/2−
1 state in Si indicates that the reduction of the N = 28 gap is

not abrupt, but is instead relatively smooth as a function of neutron number.
+
−
+
Finally, Figure 4.34 shows the E(7/2+
1 ) − E(3/21 ) and E(7/21 ) − E(1/21 ) energy dif-

ferences for odd silicon isotopes, which gives an indication of the f7/2 -d3/2 and f7/2 - s1/2
single-particle gaps. For 33 Si, the ground state has spin-parity 3/2+ , and so special care
must be taken for comparison with the other silicon isotopes. The f7/2 -d3/2 gap is taken
−
to be E(3/2+
1 ) − E(7/21 ), while the f7/2 -s1/2 gap is taken to be the sum of the d3/2 -

129

π

+

π

+

4

J =3/2
J =1/2

SDPF-MU

0

3

SDPF-U

-1

2

-2
1
-3
3+ 1+
-4 2 2

19

3+ 1+
2 2

3+ 1+
2 2

21

23

SM spectroscopic factor

1

-

1

E(7/2 ) - E(Jπ) (MeV)

1

3+ 1+
2 2 0

25

Neutron number N

Figure 4.34: Energy difference between the J1π and 7/2−
1 levels in silicon isotopes, where
π
+
+
J = 3/2 , 1/2 . Experimental data are indicated with points, while the shell model
predictions are connected by lines. See the text for how the 34 Si values were obtained. The
shaded bars indicate the SDPF-MU spectroscopic factors for one neutron removal (the sum
+
rule limit is 4 for the 3/2+
1 state and 2 for the 1/21 state).

130

Table 4.9: Comparison of T = 1 (neutron-neutron) monopoles for the SDPF-U and SDPFMU interactions, evaluated for A = 42. The single-particle energy gaps between the f7/2
and d3/2 are 4.325 and 3.147 MeV for SDPF-U and SDPF-MU, respectively.
ja
d3/2
d3/2
s1/2
s1/2

jb
f7/2
p3/2
f7/2
p3/2

SDPF-U
−0.700
−0.262
−0.257
−0.875

SDPF-MU
−0.278
0.177
−0.107
0.202

+
+
−
s1/2 and f7/2 -d3/2 gaps, i.e. E(3/2+
1 ) − E(1/21 ) + E(3/21 ) − E(7/21 ). The evolution of

these intruder-configuration energies provides a constraint on the T = 1 (neutron-neutron)
component of the cross-shell sd − f p interaction. Without knowledge of the hole states in
this region, there was an ambiguity between sd-f p monopoles and f p shell single-particle
energies, since both simply contribute an energy proportional to the occupation of a given
orbit. For example, it is unclear how much binding for the f7/2 orbit comes from the interaction with the 16 O core versus interaction with the sd shell neutrons. As an illustration
of this point, cross-shell T = 1 monopole terms are given in Table 4.9 for the SDPF-U and
SDPF-MU interactions.

While both interactions produce similar results for non-intruder

0 ω states, their allocation of the f7/2 binding between core and sd-shell neutrons differ
significantly. The more attractive sd-f p monopoles in SDPF-U cause the neutron d3/2 and
s1/2 orbits to be pulled down in energy as neutrons are added to the f p shell, and predict a
+
37,39 Si.
correspondingly too-high energy for the 1p-1h 3/2+
1 and 1/21 states in

131

Chapter 5
Summary and outlook

A scientific person will never understand why he should believe opinions only
because they are written in a certain book. [Furthermore], he will never believe
that the results of his own attempts are final.
(A. Einstein [141])
This work investigated the evolution of nuclear structure along the chain of exotic silicon
isotopes between 34 Si and 42 Si in an attempt to understand the observed breakdown of the
N = 28 shell closure in the vicinity of 42 Si. The method used for this purpose was one-proton
and one-neutron knockout reactions, which provide insight into the single-particle structure
of these nuclides.
In the course of this study, previously unknown states have been identified in all six
reaction residues – 35,37,39 Si and 35,37,39 Al. Prior to this work, only one state in 35 Al was
known, and none in 37,39 Al. An unusual γ decay from a neutron-unbound state in 35 Si
was observed and interpreted in terms of a d5/2 hole state or, alternatively, as the result
of a two-step reaction mechanism. New spin-parity assignments were made for states in
35,37,39 Si,

using arguments from the selectivity of the knockout reaction, comparison of the

longitudinal momentum distributions to the predictions of eikonal theory, and the expected
multipolarities of γ decays.
An empirical method was developed to reproduce the asymmetric longitudinal momentum distributions of the outgoing knockout residue, resulting in good agreement with nearly
all measured distributions. The success of this method lends additional support to the
132

validity of the eikonal model for the reaction.
The measured trends in knockout cross sections are generally in good agreement with
the shell model predictions, providing strong support for the interpretation of the open-shell
behavior around 42 Si in terms of tensor-force driven evolution of single-particle structure.
The one-proton knockout cross sections of §4.3.2 did not reveal any significant spectroscopic
strength to orbits above the d5/2 , supporting the claim that the Z = 14 gap remains relatively
closed. One-neutron knockout indicated spectroscopic strength to the p3/2 orbit well above
the theoretical prediction. This was interpreted in terms of excitations across both the
N = 20 and the N = 28 shell gaps. Additionally, the persistence of the intruder 3/2+ and
1/2+ states was used to resolve an ambiguity in the T = 1 cross-shell matrix elements of
shell model effective interactions developed for this region.
Experimentally, the obvious next steps in this investigation would be to (i) remeasure
one-neutron knockout of 34 Si with improved γ-ray energy resolution in order to identify any
population of intruder states, and to clarify the discrepancy in the cross sections between
34 Si

and 36 Si; and (ii) measure one-neutron knockout in 42 Si, where the SDPF-MU shell

model calculation predicts the cross section to 3/2− states to become greater than that
to 7/2− states. In both cases, the presence of isomers would add to the experimental
challenge. Additionally, one-neutron knockout from 37 Si and 39 Si could potentially provide
clear ground state spin-parity assignments for these nuclei. It would be interesting as well
to perform further comparisons of momentum distributions from knockout and inelastic
scattering reactions with different projectile-target combinations with high statistics. This
would allow a better understanding of the range of validity of the method developed in this
work for reproducing the experimental momentum distributions.
From a theoretical point of view, it is clear that a proper treatment of the region studied
133

in this work requires the inclusion of many-particle many-hole neutron excitations across the
N = 20 gap. Shell model calculations in such a large model space, while computationally
demanding, are certainly feasible, and some calculations have already been carried out with
a new interaction based on SDPF-U [142]. New experimental information, such as that in
the current work, could provide guidance for effective shell model interactions in this large
space, while recent developments in the derivation of ab initio shell model interactions [65, 66]
provide hope that highly predictive interactions may be constructed without extensive prior
knowledge of the spectroscopy of a given region.

134

APPENDICES

135

Appendix A
Digital pulse-shape analysis for the
localization of γ-ray interaction points
When a γ-ray is emitted from a source moving at relativistic velocity, its energy is Doppler
shifted as a function of its angle of emission. In fast-beam γ spectroscopy it is therefore
crucial to ascertain the angle of emission in order to reconstruct the rest-frame energy. This
requires knowledge of the position of the first interaction point of the γ ray with the detector.
In GRETINA and the Segmented Germanium Array (SeGA), this is achieved to first order by
electronic segmentation of the germanium crystals. This position resolution can be further
improved by using the time structure of the electronic pulses to resolve where within a
given segment the interaction took place. Here, I discuss the basic ideas of using digital
pulse-shape analysis to obtain sub-segment position resolution, and give proof-of-principle
demonstrations both with source data and in-beam data, building on the previous work in
[143].

A.1

Principles

A charge moving in the vicinity of a grounded conductor induces an image current on that
conductor, the magnitude of which is given by the weighting potential – or, equivalently,
the weighting field – calculated by the Shockley-Ramo theorem [98]. The magnitude of this

136

+q

E

+q

(a)

E

(b)

Figure A.1: An illustration of using induced charge to obtain position information. (a) If the
charge is collected on the left side of the center detector, then the magnitude of the induced
charge is larger in the left detector than in the right detector. (b) Vice versa.
induced current generally decreases with increasing distance between the charge and the
conductor, and so measuring the induced current we may obtain information about the path
of the moving charge. This is illustrated in Figure A.1 for the simple case of three parallel
plate capacitors arranged side by side. We can clearly tell which “detector” collected the
charge, because there is a net charge in the center detector. However, for two different
positions within the center detector, we see different induced charges on the neighboring
detectors. With such a configuration, we may construct the quantity

x = x0 +

L AR − AL
,
2 AR + AL

(A.1)

where x is the position of the charge in the direction perpendicular to the field direction,
L is the size of the detector in that dimension, x0 is the x position of the center of the
detector, and AL , AR are the maximum amplitudes of the induced charges in the left and
right detectors, respectively. It is expected that this quantity should correlate with the actual
x position of the charge.

137

For a configuration in which there are no neighboring electrodes, such a method may not
be used. Such a situation arises with both the SeGA and GRETINA detectors, which are
not segmented in the radial direction. In this case, we may use the steepest slope algorithm.
To illustrate the steepest slope algorithm, we consider the analytically solvable case of an
infinitely long coaxial detector, shown in Figure A.2, with the cathode held at potential +V0
and the anode grounded. Much of this discussion follows that given in chapter 12 of [98]. In
a semiconductor detector, rather than a single charge, electron-hole pairs are created. For
simplicity, we consider a single electron-hole pair. In this case, the electric field is E(r) = βr rˆ,
V

and the potential is V (r) = β log rr , where β ≡ log(r 0/r ) . The weighting field and potential
2
1 2
for the cathode are the same, with V0 → 1. For sufficiently high V0 , the charges quickly
reach their maximum drift velocities, and so their positions are given by


r0 − ve t t < r0 −r1
ve
re (t) =
,

 r
otherwise
1



r0 + vh t t < r2 −r0
vh
rh (t) =

 r
otherwise

(A.2)

2

since once either the electron or hole is collected, its position is re (t) = r1 for electrons and
rh (t) = r2 for holes. The charged induced on the central cathode as a function of time is
then
Q(t) =

q
log
log(r1/r2)

rh (t)
re (t)

.

(A.3)

This is plotted in the top plot of Figure A.2b, for three different starting positions r0 .
Notably, for each position there is a visible kink in the Q(t) curve at the point where the
electrons are collected on the cathode, and the times of the kinks differentiate the different
starting positions. This can be seen more clearly if we look at the derivative dQ
dt , shown
in the lower plot of Figure A.2b, where a discontinuity occurs at the time at which the

138

r2

Q(t)/q

anode

0.8

r0 ~ r2

r1<r0<r2

1
r0 ~ r1

0.6
0.4
0.2

r0
cathode

_q

+V0

0

+q

-

collect e

-

collect e

-

collect e

time

r1
(1/q) (dQ/dt)

1
0.8
0.6
0.4
0.2
0
time

(a)

(b)

Figure A.2: An illustration of using induced charge to obtain position information. (a) If the
charge is collected on the left side of the center detector, then the magnitude of the induced
charge is larger in the left detector than in the right detector. (b) Vice versa.
electrons are collected. In practice there are multiple charges, and these are not collected
at exactly the same time, so the discontinuity becomes a steep slope. Identifying the time
of the steepest slope in the current I(t) = dQ
dt – achieved by finding the minimum of the
2
second derivative d Q
– gives the electron collection time which is strongly correlated with
dt2

the original position r0 . This steepest slope method, and the linear interpolation method of
(A.1) are used in the following two sections.

A.2

Sub-segment position resolution with SeGA

To test the efficacy of the above pulse-shape analysis, tests were performed using a single
SeGA detector and a collimated 137 Cs source. A SeGA detector, as shown in Figure A.3, consists of a single cylindrical HPGe crystal electronically divided in to 32 segments consisting

139

Figure A.3: A SeGA detector consists of a single HPGe crystal electronically divided into
eight longitudinal slices and four azimuthal quarters, with the cathode running down the
center.
of eight 1 cm slices in the longitudinal (“z”) direction and four azimuthal quarters.
A scan was performed in the z direction using three steps within a single segment in
order to determine if the pulse shape algorithm could distinguish between the runs. For each
event in which the full energy of the 660 keV γ ray from 137 Cs was detected, the parameter
zn was calculated as
zn =

An+1 − An−1
,
An+1 − An−1

(A.4)

where An−1 is the amplitude of the induced charge in the neighboring segment. The results
are shown in Figure A.4, and demonstrate a clear sensitivity to the source position, indicating
that using pulse shape analysis can effectively double the segmentation in the longitudinal
direction.
The scanning table used for the longitudinal direction scans was not designed for the
SeGA detectors, and so scans in the radial and azimuthal directions were not feasible due to
the detector geometry. Instead, a rough test was performed by pointing the collimated source
in the longitudinal direction of the detector along two different axes, one near the center,
and one near the outer edge, as shown in Figure A.5. The two positions also lay at different
azimuthal angles, allowing a simultaneous check of the linear interpolation algorithm for the

140

Left segment
1050

A

B

Signal amplitude

1000

C

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

60

80

100

120

140

160

180

200

Hit segment
1600
1400
1200
0

20

40

Right segment
1000

950

0

0.5

1

Time (µs)

(a)

1.5

2

(b)

400
B

Counts

300

C

A

200

100

0
-1

-0.5

0
Z parameter

0.5

1

(c)
Figure A.4: (a) An illustration of the measurement taken, with three different positions
(indicated A, B, C) for the collimated 137 Cs source, each aligned to illuminate a different
region of the same segment. (b) An example of the recorded signals in the segment with a
net energy deposited and two neighboring segments for an event with the source at position
A. (c) The resulting histograms of the linear interpolation z parameter for each position,
demonstrating subsegment resolution.

141

run B

(a)

run A

(b)

Figure A.5: (a) Setup for the rough radial position measurement. (b) The two alignments
used.
azimuthal direction.
For the implementation of the steepest slope algorithm, we need to perform numerical
derivatives on the digitized signal. For the first derivative at the ith time bin, we could take
a simple difference, for example
Q˙ i = Qi+1 − Qi .

(A.5)

However, this will amplify the high-frequency noise in the signal and lead to inaccuracies in
the second derivative. Instead we integrate over a small window of width w and differentiate
over a larger time step d to obtain the low-frequency part of the signal
w

Q˙ i =
j=−w

Qi+d+j − Qi−d+i .

(A.6)

1 , but since in the end
Note that the derivative in (A.6) should in principle contain a factor wd

we are only interested in finding the minimum of the second derivative, this normalization
is irrelevant. If we take the parameters w and d to be too large, then we will wash out
the signal and lose sensitivity. As shown in Figure A.6, the parameters d = 3, w = 1 are

142

1

Q

Q

1

0.5

0
0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.5

0

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

20

40

60

80

100

120

140

160

180

200

dQ/dt

0.05

d2Q/dt2

0
0.02

0.2
0

0

20

40

60

80

100

120

140

160

180

200

0.05

0.01

d2Q/dt2

dQ/dt

0.4

0

-0.01

-0.05

-0.02

0

0

0.5

tss

1

1.5

Time (µs)

2

(a)

0

0.5

tss

1

Time (µs)

1.5

2

(b)

Figure A.6: An example of the steepest slope algorithm for a single trace, showing the
measured signal, a numerical derivative of that signal, and a numerical second derivative,
yielding the steepest slope time tss . In (a), the parameters d = 1, w=0 are used, while in
(b) the parameters d = 3, w = 1 are used.
sufficient to remove the high-frequency noise.
The second derivative is obtained by applying (A.6) a second time
w

w

¨i =
Q
j=−w k=−w

Qi+2d+j+k + Qi−2d+j+k − 2Qi+j+k .

(A.7)

Since the parameters j and k only appear in the combination j +k, we may reduce the double
sum to a single sum by properly counting the ways j and k can sum to a given number m
2w

¨i =
Q
m=−2w

(2w + 1 − |m|) (Qi+2d+m + Qi−2d+m − 2Qi+m ) .

(A.8)

Applying (A.8) to the traces obtained in the radial position measurements and finding the
minimum of the second derivative yields the histogram shown in Figure A.7.a. The azimuthal
position is obtained in an analogous way to the longitudinal position, using (A.4) where the
neighboring segments are now the adjacent quarters, yielding an azimuthal parameter φn ,

143

which is plotted in Figure A.7.b. Clearly, both parameters are sensitive to the interaction
position within the segment. Taking the azimuthal position to be

φ = φ0 +

π
φn ,
4

(A.9)

where φ0 is the azimuthal angle of the center of the segment and taking the radial position
to be
r = rmin +

rmax − rmin
tmax − tmin

tss ,

(A.10)

where rmin and rmax are the inner and outer radii of the crystal and tmin , tmax are parameters taken to be 0.6 µs and 0.8 µs, respectively, we obtain the distribution of interaction
points shown in Figure A.7.c. It is therefore clear from the source measurements that digital
pulse-shape analysis is capable of improving the position resolution of SeGA by at least a
factor of two in each dimension. In Figure A.5, some points are localized outside of the
detector, reflecting the simplicity of the algorithm. This data was taken from the front-most
slice of the SeGA detector, which has a more complicated geometry than the inner slices,
due to the fact that the central contact ends partway through the slice. Charge deposited in
the front slice may therefore have a path length which is longer than the nominal distance
between the inner and outer electrodes. In practice, a more sophisticated algorithm should
be used for the front slice, or at the very least all events placed outside the crystal should
be moved to the edge of the crystal.

144

4

(a)

B

Counts

60

A

(c)

A

B

40

3

0
0.4

0.6

(b)

tss (µs)

0.8

1

y (cm)

20

2

B

A

Counts

60

1

40

20

0
-1

-0.5

0

φ parameter

0.5

0
0

1

1

2

3

4

x (cm)

Figure A.7: The results of the rough radial position measurement showing (a) the steepest
slope parameter for each run (b) the azimuthal interpolation parameter for each run and
(c) the combined (r, φ) coordinates predicted by the pulse shape analysis. The black curves
indicate the geometry of the germanium crystal.

A.3

Application to in-beam data with SeGA

As a proof of principle of the application of the sub-segment position resolution to in-beam
data, the above methods were applied to experimental data from NSCL experiment e07023,
which studied the structure of 16 C. The experiment used a plunger setup to measure the
16 C, populated by a one neutron knockout from 17 C on a 500
lifetime of the 2+
1 state in

mg/cm2 beryllium target. However, for this proof-of-principle, the target-only runs (no
plunger degraders) were used, to obtain the optimal beam energy resolution and isolate the
contribution of the γ-ray interaction point determination to the γ-ray energy resolution.
The results are shown in Figure A.8, comparing the γ-ray energy resolution obtained
with the standard segmentation and that obtained with the sub-segment positions. There is
an improvement in the resolution from a full width at half maximum (FWHM) of 1.24% to
0.92%. This marginal improvement is due to the fact that the classic SeGA configuration was

145

500
segments (1.24%)
subsegments (0.92%)

Counts per bin

400
300
200
100

1700

1750
1800
Doppler-corrected energy (keV)

+
Figure A.8: The Doppler-reconstructed γ peak corresponding to the 2+
1 → 01 transition in
16 C, at a beam velocity of β=0.36. The dashed black line shows the peak obtained with the
electronic segmentation of SeGA (FWHM = 1.24%), while the solid red line shows the peak
obtained with subsegment resolution from digital signal processing (FWHM=0.92%).

used for this experiment. In this configuration, the detectors are positioned approximately
30 cm from the target and the uncertainty in the angle of the emitted γ-ray due to the
uncertainty in the position of the interaction point within a given segment is relatively
small, and so the corresponding contribution to the resolution is small. It is hoped that for
the barrel configuration, in which the detectors are positioned closer to the target for higher
solid angle coverage, the improvement will be more substantial.

146

Appendix B
Derivations of formulas

B.1

Allowed J values for a rotational nucleus

Consider a deformed nucleus rotating about the zˆ axis (which is not the symmetry axis of
the deformation). We may take the symmetry axis to instantaneously be xˆ. Due to the
symmetry of the intrinsic shape, a rotation about the zˆ axis by an angle π is equivalent to
a parity transformation r → −r:
Rz (π)|Ψ = P |Ψ

(B.1)

where Rn (θ) is the operator for rotation about the n
ˆ axis by an angle θ. Angular momentum
is the generator of rotations, and we may write the rotation operator as Rn (θ) = e−iθJ·ˆn .
So we have
e−iπJz |Ψ = P |Ψ

(B.2)

For a positive-parity state P |Ψ = |Ψ , this means even-parity states are restricted to even
values of Jz , while odd-parity states are restricted to odd values of Jz . In all even-even
nuclei, the ground state has spin-parity 0+ , which requires J = Jz for rotational states,
since there is no other angular momentum to couple to, and we have J π = 0+ , 2+ , 4+ . . .
The energy of these states is taken in analogy to the classical picture where the energy

147

of an object with moment of inertia I rotating at an angular frequency ω
J2
1
,
Erot = Iω 2 =
2
2I

(B.3)

where J = Iω is the angular momentum. Moving from the classical world to the quantum
world, the quantity J 2 is replaced by the operator Jˆ2 , with eigenvalues 2 J(J + 1). Thus
the energy for a quantum mechanical rotor is given by
2 J(J

Erot =

B.2

+ 1)

2I

.

(B.4)

Reduced matrix elements for γ decays

In classical electrodynamics, the power radiated by an oscillating multipole distribution with
multipole rank λ, z-projection µ, and frequency ω, is given by [37, 17]

P (σλµ) =

8π
[(2λ + 1)!!]

2

λ+1
λ

2

σ
.
ω (2λ+2) Mλµ

(B.5)

σ is the amplitude of the
Where σ indicates either an electric or magnetic multipole and Mλµ

oscillation.
We may obtain the quantum mechanical version of this equation by interpreting the
σ with
power as the number of photons of energy ω emitted per second, and replacing Mλµ

the expectation value of the multipole operator Mσλµ between an initial state |Ji Mi and
final state Jf Mf . The number of photons per second is then given by P (σλ)/ω

Γ(σλµ) =

8π
[(2λ + 1)!!]2

λ+1
λ

ω (2λ+1)

148

Jf Mf Mσλµ Ji Mi

2

.

(B.6)

The total rate is obtained by averaging over initial projections Mi and summing over final
projections µ, Mf

Γ(σλ) =

λ+1
λ

8π
[(2λ + 1)!!]

2

ω (2λ+1) B(σλ; Ji → Jf ),

(B.7)

where the reduced transition probability B(σλ; Ji → Jf ) contains all the information on the
structure of the initial and final states and is defined as

B(σλ; Ji → Jf ) =

Jf M(σλ) Ji
(2Ji + 1)

2

.

(B.8)

The mean lifetime of a transition is related to the transition rate by τ = 1/Γ. This means
that measuring the lifetime of a transition is equivalent to measuring the reduced transition
probability, assuming one knows the energy and multipolarity of the transition. In the case
that more than one mode of decay is possible, the total decay rate is the sum of the individual
decay rates. The partial lifetime τp (σλ; i → f ) is defined for convenience as the inverse of the
decay rate for a particular decay channel. Note however, that the actual time dependence is
governed solely by the total lifetime τ , which is the reciprocal of the total decay rate.
The three most frequently used cases of (B.8), substituting ω = Eγ , are:

B(E1) =

0.62885 2 2
e fm MeV3 fs
Eγ3 τp

(B.9)

56.871 2
µ MeV2 fs
Eγ3 τp N

(B.10)

816.20 2 4
e fm MeV5 ps,
Eγ5 τp

(B.11)

B(M 1) =

B(E2) =

e is the Bohr magneton.
where µN = 2m
p

149

B.3
B.3.1

Relativistic formulas
Doppler shift

A photon is detected with energy E L in the lab frame at some angle θL . We know that the
photon was emitted by a particle moving with velocity β in the zˆ direction. We know that
the energy-momentum 4-vector for a photon P µ = (E, p), noting that the photon is massless
so E = |p|. To find the energy ECM of the photon in the projectile-photon center-of-mass
frame we use a Lorentz-transformation

ECM = γ (ELAB − βpzLAB )
= γ (ELAB − βELAB cos θLAB )

(B.12)

= γ (1 − β cos θLAB ) ELAB ,
where γ is the relativistic factor 1/

B.3.2

1 − β 2.

Compton formula

Consider a photon with 4-momentum K = (ω, k) scattering of an electron at rest, with
4-momentum P = (E, p), resulting in final momenta K , P , where the photon has been
scattered to an angle θ. We are interested in what the final energy ω will be. From
conservation of energy and momentum K −K = P −P . Squaring both sides and evaluating

150

in the lab frame gives

−2ωω + 2k · k = 2m2 − 2mE
ωω (1 − cos θ) = −m2 + m(m + E)
ω [m + ω(1 − cos θ)] = mω
where I have used k · k = ωω cos θ, E = E + m, and E = ω − ω . Solving for ω results in
the Compton formula:
ω =

B.3.3

ω
.
1 + ω/m(1 − cos θ)

(B.13)

Cyclotron frequency for a relativistic particle

I follow here the derivation given in [144]. A particle with charge q moving with velocity
β in the yˆ direction, traversing a magnetic field B in the zˆ direction experiences a Lorentz
force
F =

dp
= qBβ xˆ.
dt

(B.14)

This force is perpendicular to the particle’s motion and so, for an infinitesimal time interval
dt, does not alter the magnitude of β, only its direction. The direction is shifted by an angle

dθ =

dp
qBβdt
=
.
p
p

(B.15)

The particle has traversed a path length ds = βdt. In the next infinitesimal time step dt, the
force is again perpendicular to the motion, and again there is no change in the magnitude
ds , and substituting into (B.15)
of β. The particle then traces out a circle with radius ρ = dθ

151

we have
p
Bρ = ,
q

(B.16)

which is identical to the non-relativistic result. The frequency of the orbit, called the cyclotron frequency, is given by
ωc =

B.3.4

β
βqB
qB
=
=
.
ρ
p
γm

(B.17)

Energy and momentum from the inverse map

Given the rigidity Bρ of the S800 and the parameters dta, yta, ata, bta, and we wish to extract
the more physically relevant 4-vector P µ . First we convert Bρ = Pq into the useful quantity
γβ:
γβ =

BρZe
Z
Bρ
P
=
≈ ·
.
M
M
A 3.10715 Tm

(B.18)

For the last term, I have made the approximation M ≈ A × 931.502 MeV. The relativistic
parameter γ0 corresponding to the rigidity setting of the S800 is given by

γ0 =

1 + (γ0 β0 )2 =

1+

BρZe 2
.
M

(B.19)

The parameter dta is defined as the difference between the kinetic energy of the outgoing
particle (E − M ) and the kinetic energy (E0 − M ) corresponding to the rigidity setting of
the S800, expressed as a fraction of the central kinetic energy

dta =

E − E0
γ − γ0
=
.
E0 − M
γ0 − 1

152

(B.20)

Therefore,
γ = γ0 + (γ0 − 1)dta

(B.21)

Ekin = [γ0 + (γ0 − 1)dta − 1] M.

(B.22)

and

The magnitude of the momentum may be obtained, using P = γβM =

γ 2 − 1M ,

and γ 2 ≈ γ02 + 2 × dta × γ0 (γ0 − 1):
P
=
P0

γ2 − 1
γ02 − 1

≈ 1 + dta

(B.23)

γ0
γ0 + 1

Finally, the direction of the momentum vector is given by

Pˆ =

sin θ cos φ, sin θ sin φ, cos θ

(B.24)

where
sin θ =

sin2 ata + sin2 bta

tan φ =

sin bta
sin ata

153

(B.25)

(B.26)

B.4
B.4.1

The tensor force
Representations for the tensor operator S12

A rank-1 tensor is equivalent to a vector. For a vector a = (ax , ay , az ), the components of a
rank-1 tensor are
1
a+ = − √ (ax + iay )
2
1
a− = √ (ax − iay )
2

(B.27)

a0 = az .
The scalar product of two rank-1 tensors is given by

(a · b) = −a+ b− − a− b+ + a0 b0 .

(B.28)

Two rank-1 tensors may be combined to form a rank-2 tensor

(2)

Tµ

= a(1) ⊗ b(1)

(2)

=
µ1 ,µ2

1µ1 , 1µ2 |2µ aµ1 bµ2 .

(B.29)

The components of T are then given by
(2)

T±2 = a± b±
1
(2)
T±1 = √ (a± b0 + a0 b± )
2
1
(2)
T0 = √ (a+ b− + a− b+ + 2a0 b0 ).
6

154

(B.30)

The scalar product of two rank-two tensors is given by

T (2) · U (2) =

√

(2) (2)

1µ1 , 1 − µ1 |00 Tµ1 U−µ
1

5
µ1

(2) (2)

= T−2 U2

(2) (2)

− T−1 U1

(2) (2)

+ T0 U0

(B.31)
(2) (2)

(2) (2)

− T1 U−1 + T2 U−2 .

This may be written in terms of the constituent rank-1 tensors by inserting (B.30) into (B.31)

T (2) (a, b) · U (2) (c, d) = (a− b− )(c+ d+ )
1
− (a− b0 + a0 b− )(c+ d0 + c0 d+ )
2
1
+ (a+ b− + a− b+ + 2a0 b0 )(c+ d− + c− d+ + 2c0 d0 )
6
1
− (a+ b0 + a0 b+ )(c− d0 + c0 d− )
2

(B.32)

+(a+ b+ )(c− d− ).
With some reshuffling of terms, this becomes
1
(−a+ c− − a− c+ + a0 c0 )(−b+ d− − b− d+ + b0 d0 )
2
1
+ (−a+ d− − a− d+ + a0 d0 )(−b+ c− − b− c+ + b0 c0 )
2
1
− (−a+ b− − a− b+ + a0 b0 )(−c+ d− − c− d+ + c0 d0 ).
3

T (2) (a, b) · U (2) (c, d) =

(B.33)

Comparing this form with (B.28), this may be rewritten as

1
1
1
T (2) (a, b) · U (2) (c, d) = (a · c)(b · d) + (a · d)(b · c) − (a · b)(c · d).
2
2
3

155

(B.34)

The normalized rank-2 spherical harmonic is given by

Y (2) =

15
[ˆ
r, rˆ](2) .
8π

(B.35)

Inserting this into (B.34) and using rˆ · rˆ = 1, we obtain a relation for the tensor product of
Y (2) with a rank-2 tensor constructed from the spin operators:

[σ1 , σ2 ](2) · Y (2) =

B.4.2

15
8π

1
(σ1 · rˆ)(σ2 · rˆ) − (σ1 · σ2 ) .
3

(B.36)

The tensor force in the jj coupling scheme

The tensor force involves both the orbital and spin degrees of freedom and is most easily
dealt with in the LS coupling scheme. However, because the shell model is often treated in
the jj coupling scheme, it is convenient to have an idea of how the tensor force behaves in
that scheme. As a preface, we note that the direct term vanishes in the monopole term [31],
leaving only the exchange term, so the isospin dependence is simply given by (τ1 · τ2 )(−1)T ,
which is negative for both T = 0 and T = 1. Therefore, if the remaining part of the tensor
operator (the spin-coordinate part) produces a positive monopole, then the interaction will
be attractive; otherwise the interaction is repulsive.
The greatest impediment to an intuitive understanding of the tensor force in the jj
scheme is that the orbital part of the operator Y (2) acts on the relative coordinates of the
two particles, while the shell-model orbits are in the lab frame. We may transform between
lab frame and relative/center-of-mass coordinates by

|n1 1 n2 2 , LM =

N Λnλ

N Λnλ, L|n1 1 n2 2 , L |N Λnλ, LM .
156

(B.37)

Figure B.1: Decomposition of two

= 1 orbits in lab frame coordinates into relative and
2

center-of-mass coordinates. For each component, Ym (θ, φ) is shown. On the right side of
the equation, the shapes represent the relative wave function, while the center-of-mass motion
is indicated by a curved arrow. The straight arrows indicate the semi-classical direction of
the angular momentum vector.
In a harmonic oscillator basis, the coefficients N Λnλ, L|n1 1 n1 2 , L – called Moshinsky
brackets – have an analytic, although somewhat intimidating, formula [145], which will not
be presented here.
For some special cases, the results of the transformation may be understood somewhat
intuitively, and I consider here the simplest non-trivial pair made from the 0p-shell orbits.
For the j> , j> interaction (that is, 0p3/2 , 0p3/2 ), consider first the case mj = j, mj = j in
the uncoupled basis, which implies ms = ms = + 21 and m = , m = . These maximallyprojected orbits will both lie predominantly in the x-y plane, and so their relative separation
will be greater in the x and y directions than in the z direction, leading to an oblate shape
for the relative wave function.
Indeed, we have 0200, 2|0101, 2 = √1 and 0002, 2|0101, 2 = − √1 . This is displayed
2

2

in Figure B.1, which shows the decomposition graphically. The wave function consists of a
mixture of two cases: (i) two particles with relative angular momentum λ = 2 and centerof-mass angular momentum Λ = 0, and (ii) λ = 0 and Λ = 2. The average shape is, as

157

expected, oblate, and combining with both spin projections being +1, this leads to essentially
the configuration labeled “-1” in Figure 2.2, producing a repulsive interaction.
If the spin of one of the particles were to be flipped, this would result in predominantly
a j> , j< pair. The shape of the relative wave function would be the same, but now the
situation would be more like the configuration labeled “+1” in Figure 2.2. Combining this
with the isospin dependence, this gives an attractive interaction. For other projections m ,
the situation becomes more complicated. However, the property holds that the j> , j> and
j< , j< pairs are repulsive and the j> , j< pairs are attractive.

B.4.3

Sum rule for the tensor force monopole

I prove here the relation

(2j< + 1)V¯j ,j + (2j> + 1)V¯j ,j = 0
<
>

(B.38)

for the monopole component of the tensor force where j = j< and j = j> . Beginning with
the definition of the monopole (2.6)
jmj m V jmj m
V¯j,j = mm

(B.39)

(2j + 1)(2j + 1)

we sum over all possible j for a given , which amounts to two possibilities: j> and j< . We
may therefore write

(2j + 1) (2j> + 1)V¯j ,j + (2j< + 1)V¯j ,j
>
<

158

=

jmj m V jmj m
jm m

(B.40)

The sum over jm may be converted into a sum over m ms , leaving

(2j + 1) (2j> + 1)V¯j ,j + (2j< + 1)V¯j ,j
<
>

m sms j m V m sms j m

=
m ms m

(B.41)

For a given m and m , taking ms → −ms is equivalent to taking S12 → −S12 , as my be
seen from 2.3. Summing over ms = ± 12 then gives zero. The entire sum on the right side of
(B.41) becomes zero, and since (2j + 1) = 0, this proves (B.38). Note that if n, = n ,
then some of the ms projections in (B.41) will be Pauli blocked and (B.38) will not hold
exactly.

B.4.4

Explicit calculation of tensor monopole terms

Here we explicitly calculate a few terms to demonstrate some properties of the tensor
monopole. In order to evaluate the matrix element of an operator which is the scalar product of commuting tensor operators of rank k, Y (k) and X (k) , which operate on the L and S
coordinates, respectively, we use

LSJ Y (k) · X (k) L S J

= δJJ (−1)S+J+L



L

S

S



J

L


k

L Y (k) L

S X (k) S .
(B.42)

In order to transform from the jj coupling scheme to the LS coupling scheme, we use




 a sa ja 




| a sa ja , b sb jb , J =
 b sb jb  | a b L, sa sb S, J .


L,S 

L S J

159

(B.43)

To convert from lab coordinates to relative and center-of-mass coordinates, we use TalmiMoshinsky brackets [146]

|na a nb b L =

N Λnλ

na a nb b L|N Λnλ |N ΛnλL .

(B.44)

To evaluate the matrix element in relative and center-of-mass coordinates, we use

nλN ΛL Y (k) f (r) n λ N ΛL

= (−1)λ+Λ+L

(2L + 1)(2L + 1)

× λ Y (k) λ



L

λ



Λ


λ

L


k  (B.45)

nλ[ f (r) ]n λ .

The strange bracket on the last term indicates an integration over the radial coordinates
only, which may be evaluated with Talmi integrals [147]:

nλ[ f (r) ]n λ

B(n, λ, n , λ , p)Ip ,

=

(B.46)

p

where
Ip =

∞
2
2p e−r2 f (r)r 2 dr.
r
Γ(p + 32 ) 0

(B.47)

When evaluating the reduced matrix element of a rank-2 harmonic oscillator, we use

λ Y (2) λ

=

5√
2λ + 1 λ020 λ 0 .
4π

(B.48)

Combining all of this, we have for the tensor force Vt = X (2) · Y (2) f (r), with X (2) acting

160

on the spins and Y (2) acting on the relative coordinates:

ab|Vt |cd J =

p

abcd I
CJp
p

(B.49)

abcd defined as
with the coefficient CJp








 a sa ja   c sc jc  


 L S



abcd =
CJp
(−1)S+J+L  b sb jb   d sd jd 





 S L
LL
L S J
L S J
×



J

k

na a nb b L|N Λnλ nc c nd d L N Λn λ

(B.50)

nλN Λλ

× (−1)λ+Λ+L
×

√

(2L + 1)(2L + 1)



L

λ



Λ


λ

L


k

2λ + 1 λ0k0 λ 0 B(n, λ, n , λ , p)

Here k = 2, S = 1 and I have absorbed the positive constants S X (2) S

and

5
4π

into

f (r). The monopole component may be written as
ab + E ab
DJp
Jp

Ip J

V ab =
p

(2J + 1)

(B.51)

J

where the direct and exchange terms

ja+jb−J−T
abba
ab ≡ (2J+1) C abab , and E ab ≡ (2J+1)(−1)
DJp
CJp
Jp
1+δab Jp
1+δab

have been defined for convenience.

161

(B.52)

The results of (B.50) are given for a few cases in Table B.1. It is clear that, regardless of
the form of the radial dependence f (r), the direct term vanishes in the monopole and that
(B.38) holds for j = j . We may further see that if we assume a Gaussian form for the radial
2

dependence f (r) = e−ar , then the Talmi integral in (B.46) is simply Ip = (1 + a)−(p+3/2) .
The tensor monopole is then attractive for j> , j< type configurations, while it is repulsive
for j> , j> and j< , j< configurations, for all positive values of a.

162

Table B.1: Coefficients of the Talmi integrals Ip for the calculation of the T = 1 monopole
term for a tensor interaction Vt = X (2) · Y (2) f (r) for the given orbits.
a
0d3/2

0d3/2

0p3/2

0p3/2

b
0p1/2

0p3/2

0p1/2

0p3/2

p
1
1
2
2
3
3
0
1
1
1
1
2
2
2
2
3
3
3
3
1
1
2
2
1
1
1
1
2
2
2
2

J
1
2
1
2
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
1
2
1
2
0
1
2
3
0
1
2
3

ab
DJp
0.0218
-0.0131
0.0355
-0.0213
0.0178
-0.0107
0.0000
-0.3165
0.1684
-0.0633
0.0183
0.3956
-0.2698
0.0791
0.0026
-0.2130
0.0568
-0.0426
0.0365
-0.0051
0.0030
-0.0761
0.0456
0.0304
-0.0203
0.0061
0.0000
0.0000
0.0304
0.0000
-0.0130

ab
EJp
0.1030
0.0113
-0.0862
-0.0213
0.0178
-0.0107
-0.0000
-0.1947
-0.0101
-0.0536
0.0183
0.2130
0.1075
-0.0304
-0.0026
-0.2130
0.0568
-0.0426
0.0365
-0.0862
0.0030
0.0761
-0.0456
0.0304
0.0203
0.0061
-0.0000
0.0000
-0.0304
0.0000
0.0130

163

J

ab
DJp

ab
J EJp

-0.0000

0.3651

-0.0000

-0.3651

0.0000
0.0000

0.0000
0.0000

0.0000

-0.3651

-0.0000

0.3651

-0.0000

0.0000

-0.0000

-0.2434

0.0000

-0.0000

0.0000

0.1217

-0.0000

0.0000

Appendix C
Parameters used in knockout reaction
calculations
π
Tabulated below are the parameters used in the eikonal model reaction calculations. Ji,f

and Ei,f are the spin-parities and excited state energies of the initial (projectile) and final
(residue) states. The orbit of the removed nucleon is indicated by ψα . V0 and R0 give the
depth and radius parameters for the Woods-Saxon potential used in the core-valence twobody wave function, or form factor. In all cases, a fixed diffuseness parameter a = 0.7 fm
and spin-orbit depth VLS = 6.0 MeV are used. The effective separation energy to each final
fi

state is given by Sn,p = Sn,p + Ef − Ei , where Sn,p is the neutron or proton ground-state
to ground-state separation energy from Table C.1. The resulting relative wave function
rms radius is given in the column marked rrms . The target density is approximated as a
Gaussian with rms radius 2.36 fm, and the incident beam energies were taken to be 73.4
[13], 97.7, 86.0, and 79.0 MeV/u for 34 Si,36 Si,38 Si and 40 Si, respectively. The calculated
stripping and diffraction cross sections, and their sum are indicated by σstr , σdif f , and
sp

σα , respectively. The shell-model spectroscopic factor is labeled C 2 S, and the resulting
theoretical cross section (including the [A/(A − 1)]N center-of-mass correction, see equation
fi

(1) in the text) is labeled σα . Table C.2 gives the results of the calculation of the twostep process for 36 Si described in section IV.A of the text. Parameters and cross sections

164

for neutron knockout from 34 Si using the SDPF-MU and SDPF-U interactions are given
in Tables C.3 and C.4. Parameters and cross sections for neutron and proton knockout
with each interaction are given in Tables C.5-C.8 (36 Si), Tables C.9-C.12 (38 Si), and Tables
C.13-C.16 (40 Si). All energies are in MeV, lengths are in fm, and cross sections are in mb.
Table C.1: One-neutron and one-proton separation energies, in MeV, from [1].
Sn
33 Si
34 Si
35 Si
36 Si
37 Si
38 Si
39 Si
40 Si
35 Al
37 Al
39 Al

Sp

4.508(08)
7.514(14)
2.470(40)
6.100(80)
2.270(110)
5.650(110)
1.580(110)
4.960(250)
5.220(100)
4.210(160)
3.290(560)

165

18.780(80)
19.460(100)
21.270(140)
22.860(550)

Table C.2: Calculated cross sections for a two-step 36 Si(0+ ) → 36 Si(2+ ) →35 Si(J π ) process.
Energies and spectroscopic factors are from shell model calculations with the SDPF-MU
sp
interaction. σ2step is the cross section for excitation to 36 Si(2+ ), followed by a neutron
knockout, assuming a normalized neutron single-particle state. The last column gives the
summed cross section for a given final state in 35 Si. Only final states which could decay via
an E1 transition to the 7/2− ground state of 35 Si are shown.
35 Si(J π )

7/2+
1
5/2+
2
9/2+
2

Ef [MeV]
1.733
3.015

4.447

sp

orbit σ2step (mb)
d5/2
0.26
d3/2
0.22
d5/2
0.26
d3/2
0.23
s1/2
0.36
d5/2
0.26

166

C 2S
0.0086
1.2997
0.0063
0.0006
0.5820
0.0531

σ2step
0.29

0.21
0.01

fi

Table C.3: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 34 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 73.4 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000

Jfπ
3/2+
1/2+
5/2+
3/2+

Ef
ψα
0.000 0d3/2
0.976 1s1/2
3.934 0d5/2
4.944 0d3/2

V0
45.706
49.446
46.257
53.087

R0
1.267
1.181
1.255
1.267

rrms
3.692
3.783
3.623
3.475

σstr
11.65
15.01
10.52
8.96

σdif f
3.72
5.50
3.05
2.48

sp

σα
15.37
20.51
13.58
11.45

C 2S
3.609
1.546
3.694
0.066

fi

σα
58.90
33.65
53.24
0.81

fi

Table C.4: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 34 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 73.4 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
3/2+
1/2+
5/2+
3/2+
3/2+
1/2+

Ef
ψα
0.000 0d3/2
0.725 1s1/2
3.931 0d5/2
4.030 0d3/2
4.312 0d3/2
4.650 1s1/2

V0
45.706
48.992
46.253
51.758
52.170
55.792

R0
1.267
1.181
1.255
1.267
1.267
1.181

167

rrms
3.692
3.805
3.623
3.507
3.497
3.534

σstr
11.65
15.33
10.53
9.34
9.22
11.62

σdif f
3.72
5.67
3.05
2.64
2.59
3.74

sp

σα
15.37
21.01
13.58
11.98
11.81
15.35

C 2S
3.478
1.382
2.422
0.152
0.000
0.475

fi

σα
56.76
30.82
34.92
1.93
0.00
7.75

fi

Table C.5: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 36 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 97.7 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
7/2−
3/2+
3/2−
1/2+
5/2+
1/2−
3/2+
5/2+
3/2+
1/2+
5/2+
5/2−
1/2+

Ef
0.000
0.499
1.222
1.240
2.298
2.340
2.600
3.014
3.268
3.297
3.755
4.289
4.420

ψα
0f7/2
0d3/2
1p3/2
1s1/2
0d5/2
1p1/2
0d3/2
0d5/2
0d3/2
1s1/2
0d5/2
0f5/2
1s1/2

V0
59.389
44.098
91.845
45.642
40.897
100.221
47.345
42.048
48.353
49.296
43.224
73.551
51.211

R0
1.191
1.245
0.949
1.188
1.241
0.950
1.245
1.241
1.245
1.188
1.241
1.224
1.188

168

rrms
3.876
3.751
3.794
3.929
3.788
3.680
3.631
3.749
3.600
3.747
3.712
3.620
3.668

σstr
12.25
11.37
13.11
15.61
11.55
11.67
9.93
11.08
9.57
13.18
10.63
9.32
12.20

σdif f
3.43
3.36
4.17
5.45
3.28
3.51
2.71
3.07
2.56
4.22
2.88
2.27
3.75

sp

σα
15.68
14.72
17.28
21.06
14.83
15.18
12.64
14.15
12.13
17.41
13.51
11.59
15.95

C 2S
1.709
3.070
0.132
0.957
0.086
0.022
0.184
0.096
0.162
0.552
1.176
0.057
0.000

fi

σα
29.16
47.83
2.48
21.31
1.36
0.36
2.46
1.44
2.08
10.17
16.82
0.72
0.00

fi

Table C.6: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 36 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 97.7 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
7/2−
3/2−
3/2+
1/2+
1/2−
5/2+
5/2+
3/2+
3/2+
1/2+
7/2−
5/2+
5/2−
5/2−
1/2+
7/2−
3/2−

Ef
0.000
0.932
0.955
1.234
2.144
2.518
2.802
2.925
3.317
3.738
4.055
4.071
4.162
4.725
4.752
4.886
4.929

ψα
0f7/2
1p3/2
0d3/2
1s1/2
1p1/2
0d5/2
0d5/2
0d3/2
0d3/2
1s1/2
0f7/2
0d5/2
0f5/2
0f5/2
1s1/2
0f7/2
1p3/2

V0
59.389
91.133
44.814
45.631
99.761
41.253
41.709
47.837
48.427
50.054
66.458
43.720
73.353
74.230
51.768
67.852
100.406

R0
1.191
0.949
1.245
1.188
0.950
1.241
1.241
1.245
1.245
1.188
1.191
1.241
1.224
1.224
1.188
1.191
0.949

169

rrms
3.876
3.822
3.722
3.930
3.696
3.775
3.760
3.616
3.597
3.715
3.701
3.697
3.625
3.607
3.647
3.673
3.528

σstr
12.25
13.46
11.01
15.62
11.86
11.40
11.21
9.75
9.55
12.77
10.10
10.46
9.36
9.17
11.94
9.77
9.95

σdif f
3.43
4.34
3.19
5.46
3.59
3.21
3.13
2.63
2.55
4.03
2.51
2.81
2.29
2.21
3.63
2.38
2.73

sp

σα
15.68
17.81
14.20
21.08
15.45
14.61
14.34
12.39
12.09
16.80
12.61
13.26
11.65
11.38
15.57
12.16
12.68

C 2S
1.727
0.090
2.609
0.998
0.019
0.260
0.163
0.163
0.495
0.446
0.025
0.950
0.045
0.032
0.010
0.012
0.000

fi

σα
29.46
1.75
39.20
22.25
0.33
4.02
2.47
2.14
6.34
7.93
0.35
13.33
0.58
0.40
0.17
0.16
0.00

fi

Table C.7: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 36 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 97.7 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
3/2+
1/2+
5/2+
5/2+
3/2+
1/2+
3/2+
5/2+
5/2+
1/2+
3/2+
3/2+
5/2+
5/2+
3/2+
3/2+
1/2+
5/2+

Ef
0.000
1.889
1.945
2.176
3.148
3.278
3.404
3.535
4.073
4.225
4.380
4.382
4.869
5.069
5.127
5.349
5.548
5.650
5.769

ψα
0d5/2
0d3/2
1s1/2
0d5/2
0d5/2
0d3/2
1s1/2
0d3/2
0d5/2
0d5/2
1s1/2
0d3/2
0d3/2
0d5/2
0d5/2
0d3/2
0d3/2
1s1/2
0d5/2

V0
59.899
65.943
70.227
62.798
64.082
67.716
72.221
68.042
65.298
65.497
73.545
69.117
69.733
66.600
66.676
70.340
70.591
75.257
67.512

R0
1.327
1.342
1.276
1.327
1.327
1.342
1.276
1.342
1.327
1.327
1.276
1.342
1.342
1.327
1.327
1.342
1.342
1.276
1.327

170

rrms
3.491
3.373
3.273
3.448
3.430
3.350
3.242
3.346
3.414
3.412
3.223
3.333
3.326
3.397
3.396
3.319
3.316
3.199
3.386

σstr
8.49
7.42
8.33
8.06
7.88
7.21
8.02
7.17
7.73
7.70
7.83
7.05
6.99
7.57
7.56
6.92
6.90
7.60
7.46

σdif f
1.85
1.55
1.96
1.71
1.66
1.49
1.85
1.47
1.61
1.60
1.78
1.44
1.42
1.56
1.56
1.40
1.39
1.70
1.53

sp

σα
10.34
8.98
10.29
9.77
9.54
8.70
9.87
8.65
9.34
9.30
9.61
8.49
8.40
9.13
9.11
8.32
8.29
9.30
8.99

C 2S
3.936
0.050
0.159
0.780
0.079
0.001
0.005
0.103
0.000
0.032
0.022
0.055
0.006
0.004
0.067
0.005
0.008
0.003
0.044

fi

σα
43.05
0.48
1.74
8.07
0.79
0.00
0.05
0.94
0.00
0.32
0.22
0.49
0.05
0.04
0.65
0.05
0.07
0.03
0.42

fi

Table C.8: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 36 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 97.7 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
1/2+
3/2+
5/2+
3/2+
1/2+
5/2+
3/2+
5/2+
5/2+
3/2+
1/2+
3/2+
5/2+
5/2+
3/2+
5/2+
3/2+
1/2+
3/2+
1/2+
1/2+

Ef
0.000
1.614
1.896
2.154
2.761
2.783
2.831
3.336
3.372
3.919
4.114
4.154
4.441
4.709
4.846
4.850
4.987
5.142
5.151
5.365
5.530
5.682

ψα
0d5/2
1s1/2
0d3/2
0d5/2
0d3/2
1s1/2
0d5/2
0d3/2
0d5/2
0d5/2
0d3/2
1s1/2
0d3/2
0d5/2
0d5/2
0d3/2
0d5/2
0d3/2
1s1/2
0d3/2
1s1/2
1s1/2

V0
59.899
69.772
65.952
62.769
67.057
71.374
63.664
67.789
64.377
65.096
68.778
73.239
69.192
66.130
66.309
69.709
66.493
70.078
74.585
70.360
75.095
75.300

R0
1.327
1.276
1.342
1.327
1.342
1.276
1.327
1.342
1.327
1.327
1.342
1.276
1.342
1.327
1.327
1.342
1.327
1.342
1.276
1.342
1.276
1.276

171

rrms
3.491
3.281
3.373
3.449
3.359
3.255
3.436
3.350
3.426
3.417
3.337
3.227
3.332
3.403
3.401
3.326
3.399
3.322
3.208
3.319
3.201
3.198

σstr
8.49
8.41
7.42
8.06
7.29
8.15
7.94
7.20
7.84
7.75
7.09
7.88
7.04
7.62
7.60
6.99
7.58
6.95
7.69
6.92
7.62
7.60

σdif f
1.85
1.99
1.55
1.72
1.51
1.89
1.68
1.48
1.65
1.62
1.45
1.80
1.44
1.58
1.57
1.42
1.56
1.41
1.73
1.40
1.71
1.70

sp

σα
10.34
10.39
8.97
9.78
8.80
10.05
9.61
8.69
9.49
9.37
8.54
9.67
8.48
9.20
9.17
8.41
9.14
8.36
9.42
8.32
9.33
9.30

C 2S
3.699
0.223
0.044
1.118
0.015
0.000
0.044
0.077
0.020
0.025
0.099
0.022
0.015
0.003
0.016
0.000
0.062
0.002
0.008
0.002
0.002
0.007

fi

σα
40.46
2.46
0.42
11.56
0.14
0.00
0.45
0.71
0.20
0.25
0.89
0.22
0.13
0.03
0.16
0.00
0.60
0.02
0.08
0.02
0.01
0.07

fi

Table C.9: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 38 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 86.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2−
7/2−
3/2−
3/2−
3/2+
1/2+
1/2−
5/2+
5/2−
3/2+
7/2−
7/2−
5/2−
5/2+
3/2−
1/2−
3/2+
1/2+
5/2+
1/2+
1/2−

Ef
0.000
0.091
0.248
0.751
0.763
1.464
1.509
1.773
2.055
2.127
2.381
2.774
2.805
2.827
2.828
3.036
3.100
3.237
3.519
3.534
4.619

ψα
0f5/2
0f7/2
1p3/2
1p3/2
0d3/2
1s1/2
1p1/2
0d5/2
0f5/2
0d3/2
0f7/2
0f7/2
0f5/2
0d5/2
1p3/2
1p1/2
0d3/2
1s1/2
0d5/2
1s1/2
1p1/2

V0
64.812
57.304
83.584
84.830
43.657
43.674
91.263
38.797
68.165
45.784
61.349
62.027
69.363
40.510
89.727
94.756
47.270
46.796
41.616
47.304
98.212

R0
1.212
1.185
0.966
0.966
1.224
1.195
0.971
1.227
1.212
1.224
1.185
1.185
1.212
1.227
0.966
0.971
1.224
1.195
1.227
1.195
0.971

172

rrms
3.833
3.930
3.992
3.934
3.766
3.990
3.842
3.865
3.738
3.682
3.817
3.800
3.708
3.801
3.743
3.713
3.631
3.823
3.764
3.800
3.605

σstr
10.68
11.68
14.57
13.79
10.45
15.32
12.57
11.34
9.53
9.47
10.27
10.07
9.19
10.56
11.37
10.99
8.90
13.08
10.12
12.78
9.76

σdif f
3.15
3.50
5.27
4.85
3.27
5.69
4.24
3.51
2.62
2.81
2.85
2.76
2.48
3.14
3.63
3.46
2.55
4.50
2.94
4.34
2.89

sp

σα
13.83
15.18
19.84
18.65
13.73
21.01
16.81
14.85
12.15
12.27
13.12
12.84
11.66
13.70
15.00
14.45
11.44
17.58
13.06
17.12
12.65

C 2S
0.150
2.814
0.109
0.270
2.793
0.799
0.062
0.029
0.067
0.032
0.113
0.008
0.000
0.120
0.010
0.006
0.043
0.027
0.341
0.091
0.001

fi

σα
2.25
46.28
2.34
5.45
40.45
17.71
1.13
0.46
0.88
0.42
1.61
0.11
0.00
1.74
0.17
0.10
0.52
0.50
4.69
1.65
0.01

fi

Table C.10: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 38 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 86.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2−
7/2−
3/2−
3/2−
1/2−
5/2−
7/2−
3/2+
1/2+
5/2−
7/2−
5/2+
3/2−
3/2+
1/2−
5/2+
3/2+
1/2+
5/2+
1/2−
1/2+

Ef
0.000
0.170
0.236
0.424
1.307
1.818
1.957
2.125
2.361
2.481
2.762
2.860
2.895
3.045
3.302
3.456
3.704
3.862
4.160
4.205
4.638

ψα
0f5/2
0f7/2
1p3/2
1p3/2
1p1/2
0f5/2
0f7/2
0d3/2
1s1/2
0f5/2
0f7/2
0d5/2
1p3/2
0d3/2
1p1/2
0d5/2
0d3/2
1s1/2
0d5/2
1p1/2
1s1/2

V0
64.812
57.447
83.554
84.023
90.787
67.784
60.614
45.781
45.274
68.847
62.006
40.563
89.880
47.186
95.347
41.516
48.180
47.861
42.627
97.322
49.162

R0
1.212
1.185
0.966
0.966
0.971
1.212
1.185
1.224
1.195
1.212
1.185
1.227
0.966
1.224
0.971
1.227
1.224
1.195
1.227
0.971
1.195

173

rrms
3.833
3.926
3.993
3.971
3.862
3.747
3.836
3.682
3.899
3.720
3.801
3.800
3.738
3.634
3.693
3.767
3.603
3.775
3.732
3.631
3.720

σstr
10.68
11.63
14.59
14.29
12.82
9.64
10.50
9.47
14.08
9.33
10.08
10.54
11.31
8.93
10.76
10.15
8.58
12.46
9.74
10.05
11.78

σdif f
3.15
3.47
5.28
5.12
4.37
2.67
2.95
2.81
5.02
2.54
2.77
3.13
3.60
2.56
3.35
2.96
2.41
4.18
2.78
3.02
3.84

sp

σα
13.83
15.10
19.87
19.41
17.19
12.32
13.45
12.27
19.10
11.87
12.85
13.67
14.91
11.49
14.11
13.11
10.99
16.64
12.52
13.07
15.62

C 2S
0.065
2.848
0.018
0.266
0.050
0.048
0.217
2.191
0.968
0.007
0.000
0.052
0.002
0.010
0.007
0.271
0.070
0.058
0.111
0.000
0.011

fi

σα
0.98
46.57
0.39
5.59
0.93
0.64
3.17
28.36
19.50
0.10
0.00
0.75
0.03
0.12
0.11
3.75
0.81
1.01
1.47
0.01
0.19

fi

Table C.11: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 38 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 86.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
1/2+
3/2+
5/2+
5/2+
3/2+
1/2+
3/2+
1/2+
3/2+
5/2+
1/2+
3/2+
5/2+
3/2+
1/2+
3/2+
1/2+
1/2+
1/2+

Ef
0.000
1.648
1.759
2.354
2.558
2.582
2.609
2.810
2.832
2.985
3.021
3.682
3.703
4.033
4.046
4.382
4.577
4.990
5.034
5.161

ψα
0d5/2
1s1/2
0d3/2
0d5/2
0d5/2
0d3/2
1s1/2
0d3/2
1s1/2
0d3/2
0d5/2
1s1/2
0d3/2
0d5/2
0d3/2
1s1/2
0d3/2
1s1/2
1s1/2
1s1/2

V0
60.568
70.124
66.424
63.638
63.902
67.459
71.409
67.745
71.707
67.965
64.501
72.837
68.863
65.805
69.292
73.764
69.954
74.566
74.624
74.791

R0
1.334
1.286
1.344
1.334
1.334
1.344
1.286
1.344
1.286
1.344
1.334
1.286
1.344
1.334
1.344
1.286
1.344
1.286
1.286
1.286

174

rrms
3.514
3.287
3.392
3.471
3.467
3.380
3.268
3.376
3.264
3.373
3.459
3.247
3.363
3.443
3.358
3.235
3.350
3.224
3.223
3.221

σstr
7.59
7.52
6.57
7.18
7.15
6.46
7.33
6.42
7.29
6.40
7.08
7.14
6.31
6.92
6.27
7.02
6.20
6.93
6.92
6.90

σdif f
1.73
1.83
1.44
1.59
1.58
1.40
1.76
1.39
1.75
1.39
1.56
1.69
1.36
1.51
1.34
1.65
1.32
1.62
1.61
1.61

sp

σα
9.31
9.35
8.01
8.78
8.73
7.86
9.10
7.82
9.04
7.79
8.64
8.83
7.67
8.44
7.61
8.67
7.53
8.54
8.53
8.50

C 2S
2.704
0.056
0.008
0.007
1.381
0.107
0.030
0.006
0.102
0.002
0.022
0.012
0.000
0.040
0.000
0.000
0.001
0.013
0.003
0.004

fi

σα
26.57
0.55
0.07
0.07
12.72
0.88
0.29
0.05
0.98
0.02
0.20
0.12
0.00
0.35
0.00
0.00
0.00
0.12
0.03
0.04

fi

Table C.12: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 38 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 86.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
1/2+
3/2+
5/2+
1/2+
5/2+
3/2+
5/2+
3/2+
1/2+
3/2+
1/2+

Ef
0.000
1.097
1.251
1.654
2.095
2.183
2.227
2.400
2.425
2.516
2.568
3.306

ψα
0d5/2
1s1/2
0d3/2
0d5/2
1s1/2
0d5/2
0d3/2
0d5/2
0d3/2
1s1/2
0d3/2
1s1/2

V0
60.568
69.384
65.783
62.729
70.723
63.416
67.013
63.697
67.262
71.285
67.441
72.338

R0
1.334
1.286
1.344
1.334
1.286
1.334
1.344
1.334
1.344
1.286
1.344
1.286

175

rrms
3.514
3.299
3.401
3.483
3.278
3.474
3.385
3.470
3.382
3.270
3.380
3.254

σstr
7.59
7.63
6.64
7.30
7.43
7.21
6.50
7.17
6.48
7.35
6.46
7.21

σdif f
1.73
1.87
1.46
1.63
1.80
1.60
1.42
1.59
1.41
1.77
1.40
1.72

sp

σα
9.31
9.50
8.10
8.93
9.23
8.81
7.92
8.77
7.89
9.12
7.86
8.92

C 2S
2.632
0.076
0.016
0.021
0.015
0.873
0.073
0.605
0.006
0.149
0.061
0.009

fi

σα
25.86
0.76
0.13
0.20
0.15
8.11
0.61
5.60
0.05
1.44
0.51
0.08

fi

Table C.13: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 40 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 79.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2−
7/2−
3/2−
1/2−
3/2+
5/2−
3/2−
1/2−
1/2+
7/2−
5/2+
5/2−
3/2+
7/2−
3/2−
7/2−
5/2−
5/2+
3/2−
1/2−
3/2+
1/2−
5/2+
3/2+
5/2+

Ef
0.000
0.044
0.098
0.378
0.635
0.654
0.775
1.134
1.279
1.461
1.499
1.777
1.821
1.837
1.918
1.992
2.237
2.383
2.383
2.605
2.775
2.805
2.841
3.093
3.341

ψα
0f5/2
0f7/2
1p3/2
1p1/2
0d3/2
0f5/2
1p3/2
1p1/2
1s1/2
0f7/2
0d5/2
0f5/2
0d3/2
0f7/2
1p3/2
0f7/2
0f5/2
0d5/2
1p3/2
1p1/2
0d3/2
1p1/2
0d5/2
0d3/2
0d5/2

V0
62.503
54.620
77.319
81.582
42.203
63.590
79.001
83.395
40.688
57.164
36.684
65.426
44.086
57.826
81.727
58.097
66.168
38.147
82.801
86.764
45.567
87.208
38.893
46.055
39.700

R0
1.202
1.179
0.981
0.993
1.204
1.202
0.981
0.993
1.202
1.179
1.216
1.202
1.204
1.179
0.981
1.179
1.202
1.216
0.981
0.993
1.204
0.993
1.216
1.204
1.216
176

rrms
3.894
4.008
4.150
4.103
3.829
3.858
4.058
4.008
4.129
3.927
3.952
3.802
3.745
3.907
3.930
3.900
3.781
3.890
3.885
3.858
3.687
3.840
3.861
3.670
3.831

σstr
10.34
11.51
15.64
14.94
10.21
9.89
14.36
13.64
16.18
10.48
11.32
9.23
9.22
10.24
12.65
10.15
8.99
10.55
12.08
11.71
8.58
11.50
10.20
8.39
9.85

σdif f
3.26
3.69
6.18
5.79
3.44
3.03
5.45
5.06
6.50
3.18
3.78
2.72
2.95
3.07
4.52
3.03
2.61
3.40
4.22
4.04
2.64
3.93
3.23
2.55
3.06

sp

σα
13.59
15.20
21.81
20.73
13.65
12.92
19.81
18.71
22.68
13.66
15.10
11.95
12.16
13.31
17.18
13.17
11.60
13.95
16.30
15.76
11.22
15.43
13.43
10.94
12.91

C 2S
0.095
3.191
0.903
0.031
2.312
0.153
0.084
0.131
0.526
0.305
0.016
0.065
0.026
0.114
0.001
0.131
0.003
0.182
0.011
0.000
0.262
0.004
0.132
0.340
0.005

fi

σα
1.39
52.33
21.26
0.68
33.20
2.13
1.79
2.64
12.55
4.50
0.26
0.84
0.34
1.63
0.01
1.86
0.03
2.67
0.20
0.00
3.10
0.07
1.87
3.91
0.07

fi

Table C.14: Parameters used in the calculation and the resulting theoretical cross sections σα for neutron knockout from 40 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 79.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2−
3/2−
7/2−
1/2−
5/2−
3/2−
1/2−
7/2−
5/2−
7/2−
3/2−
1/2−
3/2+
1/2+
5/2+

Ef
ψα
0.000 0f5/2
0.044 1p3/2
0.193 0f7/2
0.208 1p1/2
0.482 0f5/2
0.754 1p3/2
1.279 1p1/2
1.312 0f7/2
1.577 0f5/2
1.587 0f7/2
1.779 1p3/2
2.764 1p1/2
2.810 0d3/2
3.039 1s1/2
3.379 0d5/2

V0
62.503
77.182
54.892
81.165
63.305
78.949
83.736
56.900
65.101
57.386
81.402
87.117
45.621
43.818
39.761

R0
1.202
0.981
1.179
0.993
1.202
0.981
0.993
1.179
1.202
1.179
0.981
0.993
1.204
1.202
1.216

177

rrms
3.894
4.158
3.999
4.126
3.867
4.061
3.991
3.935
3.811
3.920
3.944
3.844
3.685
3.936
3.829

σstr
10.34
15.75
11.39
15.27
10.00
14.39
13.42
10.57
9.34
10.39
12.83
11.54
8.56
13.50
9.82

σdif f
3.26
6.24
3.63
5.98
3.09
5.47
4.94
3.23
2.77
3.14
4.62
3.95
2.63
4.99
3.05

sp

σα
13.59
21.99
15.02
21.25
13.09
19.86
18.36
13.80
12.11
13.54
17.46
15.50
11.19
18.49
12.87

C 2S
0.043
0.513
3.328
0.018
0.065
0.174
0.085
0.634
0.032
0.089
0.001
0.001
1.688
0.723
0.037

fi

σα
0.62
12.18
53.93
0.40
0.91
3.74
1.68
9.43
0.42
1.31
0.02
0.02
19.87
14.07
0.50

fi

Table C.15: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 40 Si
using level energies and spectroscopic factors from the SDPF-MU interaction. The beam energy is 79.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
1/2+
3/2+
5/2+
1/2+
3/2+
3/2+
5/2+
5/2+
3/2+
1/2+
1/2+

Ef
0.000
0.836
0.855
1.610
1.784
1.867
2.334
2.507
2.672
3.070
3.238
3.548

ψα
0d5/2
1s1/2
0d3/2
0d5/2
1s1/2
0d3/2
0d3/2
0d5/2
0d5/2
0d3/2
1s1/2
1s1/2

V0
61.131
69.344
65.810
63.198
70.592
67.069
67.648
64.342
64.552
68.560
72.495
72.899

R0
1.339
1.293
1.345
1.339
1.293
1.345
1.345
1.339
1.339
1.345
1.293
1.293

178

rrms
3.538
3.312
3.425
3.510
3.294
3.410
3.403
3.495
3.492
3.393
3.268
3.262

σstr
6.83
6.91
5.96
6.58
6.74
5.83
5.78
6.45
6.43
5.69
6.51
6.46

σdif f
1.60
1.72
1.36
1.52
1.66
1.32
1.30
1.47
1.47
1.27
1.58
1.56

sp

σα
8.43
8.63
7.32
8.10
8.41
7.15
7.08
7.93
7.89
6.97
8.08
8.02

C 2S
2.450
0.186
0.000
0.122
0.003
0.152
0.024
0.000
0.096
0.006
0.003
0.001

fi

σα
21.73
1.69
0.00
1.04
0.02
1.14
0.18
0.00
0.80
0.04
0.02
0.01

fi

Table C.16: Parameters used in the calculation and the resulting theoretical cross sections σα for proton knockout from 40 Si
using level energies and spectroscopic factors from the SDPF-U interaction. The beam energy is 79.0 MeV/u on a 9 Be target.
Jiπ
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+
0+

Ei
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Jfπ
5/2+
1/2+
3/2+
5/2+
1/2+
3/2+
3/2+
5/2+
5/2+
1/2+
3/2+
1/2+

Ef
0.000
0.496
0.559
1.147
1.324
1.431
1.517
1.625
1.790
2.353
2.482
2.602

ψα
0d5/2
1s1/2
0d3/2
0d5/2
1s1/2
0d3/2
0d3/2
0d5/2
0d5/2
1s1/2
0d3/2
1s1/2

V0
61.131
68.895
65.441
62.605
69.987
66.527
66.634
63.217
63.428
71.339
67.832
71.665

R0
1.339
1.293
1.345
1.339
1.293
1.345
1.345
1.339
1.339
1.293
1.345
1.293

179

rrms
3.538
3.319
3.430
3.518
3.303
3.417
3.415
3.510
3.507
3.283
3.401
3.279

σstr
6.83
6.97
6.00
6.65
6.82
5.89
5.88
6.58
6.55
6.65
5.76
6.61

σdif f
1.60
1.75
1.37
1.54
1.69
1.33
1.33
1.52
1.51
1.63
1.29
1.61

sp

σα
8.43
8.72
7.37
8.19
8.52
7.22
7.21
8.10
8.06
8.28
7.05
8.22

C 2S
2.231
0.184
0.018
0.130
0.005
0.159
0.026
0.103
0.006
0.002
0.016
0.003

fi

σα
19.79
1.69
0.14
1.12
0.05
1.21
0.20
0.88
0.05
0.01
0.12
0.03

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