‘

dissertation entitled

QUADRUPOLE COLLECTIVITY MEASUREMENTS
IN EVEN-EVEN, NEUTRON-RICH SILICON AND
SULFUR ISOTOPES APPROACHING N = 28

presented by

Christopher M. Campbell

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Ph sits and Astronom

 

T Altai

Majorl Professor’s Signature
§ » KB~ 07

Date

MSU is an affinnative-act‘ion, equal-opportunity employer

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

DATE DUE DATE DUE DATE DUE

 

QUADRUPOLE COLLECTIVITY MEASUREMENTS
IN EVEN-EVEN, NEUTRON-RICH SILICON AND
SULFUR ISOTOPES APPROACHING N = 28

By

Christopher M. Campbell

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2007

ABSTRACT

QUADRUPOLE COLLECTIVITY MEASUREMENTS IN EVEN-EVEN,
NEUTRON-RICH SILICON AND SULFUR ISOTOPES APPROACHING N = 28

By

Christopher M. Campbell

An inelastic proton scattering experiment was performed at the National Super-
conducting Cyclotron Laboratory to study quadrupole collectivity in the even-even
silicon and sulfur isotopes near N = 28. Experiments on neutron-rich sulfur isotopes
have found significant collectivity and have been interpreted as pointing to the col-
lapse of the N = 28 shell gap. Narrowing of a proton subshell gap in the sulfur
isotopes may, however, be responsible for the increased collectivity. This experiment
gives a quantitative measurement of the decrease in collectivity between 42S and 44S
showing that the N = 28 shell gap does not vanish at Z = 16. In the silicon isotopes,
the large, stable Z = 14 subshell gap directly ties collective trends to the strength of
the N = 28 shell closure. Quadrupole collectivity and 21+ excitation energies in the
isotopes 36’38’408i give clear evidence for the narrowing of the N z 28 shell gap in

the absence of strong proton collectivity.

to my wife Bethany

iii

ACKNOWLEDGMENTS

This thesis has benefited from the hard work of dozens of people to whom I am
most grateful. I will start by thanking my advisor Thomas Glasmacher for introducing
me to the field of rare isotopes when I was in the Research Experience for Under-
graduates program. I thank Thomas for being a supportive and challenging mentor.
I am grateful to Michigan State University and the National Science Foundation for
financial support during my graduate education.

I thank the members of the Gamma group for their insights, support, and friend-
ship. I thank Katie Yurkewicz for her help during my REU. I thank Heather Zwahlen
for helping a new student get up to speed and for all the fun we had with the filling
system. I will always remember installing the SeGA detectors for your experiment. I
thank Dan-Cristian Dinca for working with me on the SeGA pulse-shape acquisition
and analysis. I thank Russ Terry and Alexandra Cade for our discussions of nuclear
structure and Doppler correction of 'y-rays.

I am very grateful to my Japanese collaborators who provided, constructed, and
maintained the RIKEN-Kyushu-Rikkyo liquid hydrogen target used in my thesis ex-
periment. I especially thank Nori Aoi and Shoko Kanno for their helpful discussions.
I thank Ken Yoneda and Nori Aoi for their assistance in understanding y-ray angular
distribution calculations based on ECIS outputs.

I thank the many outside collaborators with whom I have had the pleasure of
working. In particular, I thank Lew Riley for many helpful discussion on proton scat-
tering, probe sensitivity, and ECIS. I also owe a debt of gratitude to my collaborators
from Florida State University, especially Kirby Kemper and Paul Cottle for focusing
attention on the question of changing proton collectivity in the neutron-rich sulfur
isotopes. I thank Mihai Horoi for shell model calculations and Hiroyuki Sagawa for
his help with effective charges.

This work was made possible by the NSCL staff, and I am grateful for all their

iv

help. I thank Len Morris for all his help in designing detector stands, beam line
components, and experimental setups for the liquid hydrogen target campaign. I
thank the operations and A1900 groups for good beams. I thank Daniel Bazin and
John Yurkon for helping me understand the S800 and the focal plane detectors —
and for rebuilding the focal plane detectors after a minor disaster during the start of
my experiment.

For their support and thoughtful questions, I thank my committee members: Si-
mon Billinge, Wayne Repko, Krzysztof Starosta, Vladimir Zelevinsky, and Thomas
Glasmacher. I also thank Alex Brown —— who served on my committee until his sab-
batical —- for his insights, his shell model calculations, and our many discussions on
nuclear structure.

Thank you all.

List of Tables

TABLE OF CONTENTS

List of Figures

1 Introduction
The nucleus ................................
Nuclear structure .............................

1.1
1.2

1.3

1.2.1
1.2.2

The nuclear shell model .....................
Collective models of the nucleus .................

Exploring new territory ..........................

2 Experimental Method
2.1 Proton scattering .............................

2.2

2.3

2.1.1
2.1.2

Scattering theory .........................
Deformation extraction ......................

2.1.3 Sensitivity of (p, p’) to neutrons and protons ..........

2.1.4

Additional reaction channels ...................

Exotic beam production .........................

2.2.1
2.2.2

Projectile fragmentation .....................
Fragment separation and delivery ................

Reaction target and detectors ......................

2.3.1
2.3.2
2.3.3
2.3.4

Thick-target, y-ray tagging method for (p, p’ ) .........
Liquid hydrogen target ......................
The S800 spectrograph ......................
The Segmented Germanium Array (SeGA) ...........

2.4 Systematic corrections and uncertainties ................

3 Analysis
3.1 Particle and reaction channel identification ...............

3.2

3.1.1

Event selection ..........................

Analysis of 7-ray spectra .........................

3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6

3631 ................................

vi

viii

ix

casual—Md

14
15
15
17
18
21
26
26
27
28
28
30
34
38
44

49
49
50
57

4 Results and Discussion 85

4.1 Spectroscopy of 408i ........................... 85
4.2 Quadrupole collectivity .......................... 90
4.2.1 Collectivity in silicon isotopes .................. 93

4.2.2 Collectivity in sulfur isotopes .................. 98

5 Conclusion 103
Bibliography 105

vii

2.1

2.2
2.3
2.4
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
4.1

4.2

4.3

4.4

LIST OF TABLES

Even mass silicon and sulfur isotopes, number and energy delivered on
target ...................................

LH2 cell thickness vs. pressure. .....................
A list of sources used for 7—ray efficiency calibration. .........
Systematic correction and uncertainty table for 368i ..........

Incoming particles, 7-ray yields for reactions populating 388i ......

Energies and cross sections for 7 rays produced in 12(36Si,36 Si + 7 p’.

Energies and cross sections for states populated in 19(368i,36 Si + 7 p’.

Energies and cross sections for 7 rays produced in p(38Si,38 Si + 7 p’.

Energies and cross sections for states populated in p(38Si,38 Si + 7)p’.

Energies, observed cross sections for 7 rays produced in p(42P,40 Si+7)X.

Energies and cross sections for 7 rays produced in p(408,40 S + 7 p’. .

Energies and cross sections for states populated in 19(408,40 S + 7 p’. .

Energies and cross sections for states populated in 19(428,42 S + 7)p’. .

Energies and cross sections for 7 rays produced in 19(448,44 S + 7)p’. .
I

Energies and cross sections for states populated in p(44S,44 S + 7 p . .

Inelastic proton—scattering cross sections, deformation parameters of

the current experiment (I [32 (p p/)|) are given for 36’38’408i and 40’42’448.

Deformation parameters of the current experiment (I 62 (p 11’)” and a
previous Coulomb excitation measurement (Iflgcl) [84], shell-model

matrix elements Ap and An, and experimental and shell-model (fl) / (-Izvv)

ratios are given for 36’38’40Si.

Three sets of polarization charges considered for analyzing ftp = 1;.

Deformation parameters of the current experiment (I32 (1, ml) and a
previous Coulomb excitation measurement (Iflggl) [84], shell-model

matrix elements Ap and An, and experimental and shell-model (ID?) / (1;.

ratios are given for 40"12’4‘4S ........................

viii

45
49
62
64
66
70
73
76

to

2

93

100

1.1
1.2
1.3
2.1

2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

LIST OF FIGURES

Chart of the nuclides ........................... 2
Isotopes studied in this thesis ...................... 10
Proton single-particle energy shifts near N = 28 ............ 12
Ratio of neutron-to—proton sensitivity, bn/bp, in (p, p’). ........ 20
Spectra of 7 rays in coincidence with single-neutron removal from 48Ca

impinging on a 9Be target and on a proton target ............ 24
Diagram of coupled cyclotrons and A1900 fragment separator . . . . 27
Schematic diagram of LH2 target system ................ 31
Effective LH2 target thickness ...................... 33
Diagram of 8800 ............................. 34
TOP correlation with momentum and corrected TOF ......... 37
SeGA energy residuals after calibration ................. 39
SeGA-LH2 geometry in GEANT ...................... 41
SeGA full—energy efficiency measured and simulated .......... 42
Ratio of simulated to measured SeGA full-energy efficiency ...... 43
Gamma-ray angular distribution from (p, p’) .............. 47
Calculated differential (p, p’) cross section and momentum gating losses 48
Energy loss after target vs. before target ................ 51
Incoming PID for both beam settings .................. 52
PID overlap checked by projection .................... 54
Momentum loss corrected by TOP, mass, and charge ......... 55
Reduced contamination by corrected momentum loss gating ..... 56
Full 7—ray spectrum of 368i ........................ 60
Spectrum of 7—rays in coincidence with 368i(p, p’); high energy . . . . 61
Spectrum of 7-rays in coincidence with 36Si(p, p’); low energy ..... 62

ix

3.9

3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
4.1

4.2

4.3
4.4
4.5
4.6

4.7

Level scheme of 368i ...........................
Full 7-ray spectrum of 388i ........................
Spectrum of 7-rays in coincidence with 38Si(p, p'); low energy .....
38Si 7 — 7 spectra .............................
Spectrum of 7-rays in coincidence with 39Si(p, X )38Si; low energy
Level scheme of 38Si ...........................
Spectrum of 7—rays in coincidence with 40Si(p, p’) ...........
Spectrum of 7-rays in coincidence with p(42P,40 Si + 7)X .......
Level scheme of 40Si ...........................
Full 7-ray spectrum of 408 ........................
Spectrum of 7-rays in coincidence with 40S(p, p'); high energy
Spectrum of 7—rays in coincidence with 40S(p, p'); low energy .....
Level scheme of 40S ............................
Spectrum of 7-rays in coincidence with 42S(p, p’) ............
Level scheme of 42S ............................
Full 7—ray spectrum of 44S ........................
Spectrum of 7—rays in coincidence with 448(1), p'); high energy
Spectrum of 7-rays in coincidence with 44S(p, p'); low energy .....
Level scheme of 44S ............................

Evolution of 2'; energy with N for even-even nuclei ..........

Measured and predicted silicon 2'1Jr energies ...............

Comparing silicons with 1820C

Ratios of excitation energies, E(4+)/E(2+) ...............
Deformation parameters vs. N for silicon isotopes ...........

B(E2T) values vs. N for silicon isotopes, shell-model calculations using
polarization charges are compared with data ..............

(%)/(%) vs. N for silicon isotopes ...................

65
66
67
70
71
72
73
74
75
76
77
78
80
80
81
82
83
84
86
87

89
91
94

96
97

4.8 Deformation parameters vs. N for sulfur isotopes ........... 99

4.9 B(E2T) values vs. N for sulfur isotopes, shell-model calculations using

polarization charges are compared with data .............. 101
M N .
4.10 (MED/(7) vs. N for sulfur lsotopes ................... 102

Images in this dissertation are presented in color.

xi

Chapter 1

Introduction

1 .1 The nucleus

The nucleus is a bound collection of protons and neutrons — typically found at the
center of an atom. Protons (positively charged) and neutrons (neutral) are collectively
called nucleons due to their shared properties. Both types of nucleons have an intrinsic
spin of 1 / 2, have masses near 940 MeV/c2, and are fermions. (Here, and throughout
this thesis, all spins and angular momenta are given in units of h.) The like-nucleon
interaction strength (71. — n or p — p) is independent of the nucleon type —- once the
Coulomb interaction in p — p has been removed. Interactions between nucleons arise
from the strong interaction, which also holds together quarks inside a nucleon.
Nuclei have a characteristic size on the order of 10‘15 m, or 1 fm (SI unit fem-
tometer, historically known as a fermi). This is far smaller than the characteristic
atomic scale of 10‘10 m. Nuclear excitations are typically of order 0.1 — 10 MeV
(1 MeV = 106 eV), while atomic excitations are typically on the eV scale. This sep-
aration of scales effectively decouples the nuclear and atomic realms in most cases.
Similar to the arrangement of elements in the periodic table, Figure 1.1 shows
the chart of the nuclides which plots number of neutrons (N) on the horizontal axis

and number of protons (Z) on the vertical axis. Nuclei having the same number of

 

Proton number
8
T

   

. Observed nuclei
I Stable nuclei

   

 

0 I." I 1 l I l n-

o 20 4o 60 80 100 120 140 160 180
Neutron number

 

 

Figure 1.1: Chart of the nuclides

protons (neutrons) are referred to as isotopes (isotones). Stable nuclei — shown by
black squares in figure 1.1 — lie along an are known as the valley of stability. Here,
stability refers to B—stability, i.e. these nuclei do not undergo fl—decay which convert
protons into neutrons, or vice—versa. The vast majority of our knowledge of the nucleus
has come from this stable subset of possible nuclei.

Nuclei which are fl-unstable may allow one to study the effects of changing proton
and neutron number independently and may allow access to nuclei along the various

nucleosynthesis pathways which produce the heavy elements.

1.2 Nuclear structure

Nuclear properties are determined by the number of protons and neutrons and their
arrangement, i.e. the relative positions and motions of all nucleons. Nuclear structure

is the study of nuclear properties and how they arise due to the motions of nucleons.

2

Like atoms and molecules, nuclei have ground states and may have a number of bound
excited states which arise from the interactions among their constituents. These states
are identified by their total angular momentum J, parity 7r, and sometimes order in
a level scheme 2' written as Ji". So, the first level having an angular momentum of 2
and a positive parity would be denoted 2?.

In the case of atoms and molecules, there are long—range, Coulomb potentials
created by the nucleus or nuclei which determine the motions of most of the electrons
involved. Only the outermost and least bound (valence) electrons actively participate
in changes between the ground and excited state. Furthermore, the nuclei themselves
are unchanged by rearrangements of these electrons.

Nuclei, on the other hand, have no such strong, unchanging, long—distance, central
potential. Instead, the nucleon-nucleon strong interaction has a short range and the
nucleons —— as a group — create the potential which binds them together. The average
potential experienced by a nucleon interacting with all other nucleons in the nucleus
is called the nuclear mean-field potential and may be approximated by a central po-
tential. Given a central potential, the constituents may be described as filling orbitals
whose properties, ordering, and spacing are determined by the underlying potential.

Shell structure in nuclei was discovered by studying isotopic abundances and the
number of stable isotones for a given element [1]. These observations pointed to par-
ticularly stable numbers of protons or neutrons called magic numbers. The nuclear
magic numbers are 2, 8, 20, 28, 50, 82, and 126.

Atomic systems having so-called magic numbers of electrons were also known to
display particular stability. In the atomic case, electrons —- which are also spin 1/2
fermions — may only occupy the same orbital if they have different spin orientations.
Electrons fill the orbitals pairwise from lowest to highest energy until all electrons
have been placed. The electron magic numbers correspond to the atomic numbers
of the noble gasses on the periodic table: 2, 10, 18, 36, 54,. ..These atomic magic

numbers result from a degeneracy in the eigenvalues of the Coulomb potential for a

3

system of spin 1 / 2 fermions.

In nuclei, a first-principles calculation of the energy levels is complicated by the
fact that the potential, which leads to orbitals is created by the particles filling the
orbitals. Instead, the nuclear magic numbers were initially used to learn about the
nuclear mean field. The three-dimensional harmonic oscillator potential yields the
sequence 2, 8, 20, 40, 70, . . . for nucleons filling harmonic oscillator shells. However,
the harmonic oscillator potential — a parabola extending to infinity —— is rather
unphysical, and terms were added to better approximate the nuclear potential. The
addition of an attractive 8 - 6 term is equivalent to deepening the potential at large
radii. This changed the magic numbers but failed to reproduce the data. Finally, a
strong spin-orbit term was introduced which depressed the j = E + 1 / 2 orbital with
the maximum orbital angular momentum (6) from each harmonic oscillator shell. This
lead to the nuclear magic numbers: 2, 8, 20, 28, 50, 82, and 126 [2,3].

Orbitals are typically labeled as nlfj where n is the number of nodes in the radial
wavefunction, K is the orbital angular momentum, and j = 6 :l: 1 / 2 is the total angular
momentum. The prefixes 1/ and 7r specify neutron and proton, respectively. Node
counting may start from zero or one depending on the author. In this work, n will
start at zero. Orbital angular momentum is given by s, p, d, f, . . . for 8 = 0, 1, 2, 3, . . .
As an example, the first neutron orbital having 6 = 3 has j = 3 + 1/2 = 7/2 due to

the spin—orbit force and is denoted V0 f7 /2.

1.2.1 The nuclear shell model

Given this shell structure, it is then natural to consider a separation of nucleons into
two categories: core and valence. Core nucleons are those filling orbitals up to a magic
number, and valence nucleons will fill orbitals above that magic number. Consider-
ing only valence nucleons restricted to a few orbitals, detailed calculations including
nucleon-nucleon couplings become possible. This is commonly known as a shell model

calculation —— but a better term may be interacting shell-model calculation. In this

4

way, the nuclear level energies and properties may be calculated.

Why — or when —— is such a separation reasonable? The energy spacing between
valence orbitals, and the highest-energy filled orbitals of the core is ~ lfiw in terms
of the harmonic oscillator potential used to describe the mean field. This energy
is approximately given as hw z 45A'1/3 — 251‘1—2/3 [4]. For 16O and 208Pb, the
estimates are 14 MeV and 7 MeV, respectively. For energy levels well below lhw, the
approximation appears valid. It is important to point out that even though core and
valence nucleons appear to have many opportunities to interact, the Pauli principle
prevents many such interactions [5]. That is, a valence nucleon may not scatter off
a core nucleon with a small energy-momentum transfer because all orbitals to which
the core nucleon would scatter are filled.

An introduction to the shell model is a book in itself, so only a brief discussion is

presented here. An interacting shell model calculation is defined by the selection of:

o a core,

a set of valence orbitals,

a set of single-particle energies,

a two-body interaction, and

the number of valence protons and neutrons.

The core is typically chosen as the largest doubly-magic nucleus having fewer neutrons
and protons than the nuclei being calculated. Valence orbitals may consist of all
orbitals in the major proton and neutron shells above the core or some subset thereof.
Single-particle energies (SPEs) are typically taken from nuclei having one nucleon
more or fewer than a doubly-magic nucleus; a SPE represents the binding energy of
a single particle due to the mean field produced by the core. A two-body interaction
describes the interactions between pairs of valence nucleons; this interaction is not

the free nucleon-nucleon interaction. Instead, the two-body interaction implicitly or
5

explicitly contains many-body physics. In practice, the two-body interaction — or
two-body matrix elements (TBME) — may be fit to well known nuclear levels [6].

The nucleus is a quantum system. The Hamiltonian describing nuclear levels re-
sults from a sum over single—particle energies and two-body interactions of valence
particles. An initial set of states may be defined as a sum over partitions of nucleons
into orbitals, which is further summed over spin couplings for each partition which
yield a given J“. This initial set of states is then diagonalized to find eigenvalues
(level energies) and eigenvectors (wave functions). Thus, shell-model states arise from
a mixing of many configurations.

These states then give level schemes, spectroscopic factors, multipole moments,
and transition probabilities (transition matrix elements). Spectroscopic factors are
found by taking the wavefunction overlap of a state in a nucleus of mass A and
the result of a particle creation operator acting on a state in a daughter nucleus
of mass A — 1. Experiments compare to the C23 values which includes an isospin
Clebsch—Gordan coefficient. Multipole moments are found by taking the expectation
value of a multipole operator O, i.e. < \IIIO|\II >. The electric quadrupole moment
of the ground state would then be found using O = r2Y02e and the ground state
wavefunction. Similarly, the proton transition matrix element for an E2 excitation of

the 2] from a 0] ground state is given by

+2
167r
Mp = ]/—5— Z < 2+; M = m Zr2Y3(k)e|0f >,
u=—2 k
where the sum is over all protons, k [5]. The reduced transition probability -— termed
the B(E2; 0f ——) 2]), or B(E2T) value — would then be given by B(E2 T) = [Mp|2.
Although the shell model has been quite successful in describing low-lying nu-
clear structure [7—9], it does not offer a complete description of nuclear states and
transitions. To make calculations tractable, the model space available to nucleons is

truncated to a few orbitals and the two-body interactions are modified to remove the

6

hard core of the n — n potential. Direct comparisons with spectroscopic studies and
measurements of transition rates reveal discrepancies. Theory and experiment show
that the valence particles of the shell model behave as independent particles bound
in the mean field of the nucleus ~ 2 / 3 of the time; the remaining time being spent in
a correlated state [10].

Similarly, mixing with configurations outside the model space is required to ex-
plain observe transitions between bound states of the nucleus. Specifically, quadrupole
transition strengths between low-lying states are larger than estimated by the shell
model. This is explained as a small admixture of Zhw components which arise from
excitation of the core [11]. Essentially, the core is not, in fact, inert and may be polar-
ized by the valence nucleons. The simplest analogy might be the tides experienced on
Earth due to the gravitational pull and motion of the Moon. However, in the nuclear
case, one must consider separately the effects of valence protons and neutrons on the
protons and neutrons of the core. In equations 1.1 and 1.2, the generalized effective
charge model [12] includes four core—valence couplings (5CmeValence) and describes
the effective matrix elements (Mp, Mn) in terms of the shell model — or valence ——

matrix elements (Ap, An).

1.2.2 Collective models of the nucleus

In contrast to the shell model, collective models of the nucleus do not explicitly con-
sider the individual motions and couplings of all valence nucleons. Instead, nuclei are
treated much like a rubber ball or a drop of liquid; this is termed a macroscopic model.
Such objects are constrained to have constant volume, but their surfaces may be dy-
namically or statically deformed [13]. Details of this deformation are then governed by

some energy functional depending on surface, volume, and correlation energies. These
7

correlation energies may then be related back to specific nucleon orbital overlaps in
a more microscopic model. Unpaired nucleons may then couple to the underlying
collective modes of the even—even core.

The two fundamental paradigms for collective motion are vibration and rotation.

Both may be viewed as a volume conserving modification of the nuclear surface:

R = R0[1 + Zita/\uyAuwi 45)] (1'3)

where /\ denotes the multipole order [14]. The strongest collective excitations are typi-
cally A = 2, i.e. quadrupole. Quadrupole parameters (12],, or simply a”, are simplified

by choice of quantization axis and converted into the more useful parameters ,8 and 7.

do = fic037 (1.4)
Orig =2 flsin7 (1.5)
ail = 0 (1.6)

The magnitude of a quadrupole deformation is given by 6 while 7 describes the degree
of axial symmetry. Axial symmetric deformation is found when 7 = 0°. This thesis
will treat all deformations as axially symmetric unless otherwise noted.

Vibration occurs when a nucleus may be deformed but energetically favors an
average spherical shape, and a harmonic oscillator potential in terms of 6 is often
used. The shape of the nuclear surface may then oscillate. Excitation involves the
additions of phonons to this system. In a rotational nucleus, a minimum in the po-
tential V(fl) occurs away from ,3 = 0, and the nucleus is statically deformed in its
ground state. Positive deformation is termed prolate; negative deformation is oblate.
Excitation energy goes into rotating this deformed nucleus about some axis other
than the deformation axis.

Doubly-magic nuclei are spherical and have excitations built from particle—hole

states. Nuclei having a single closed shell, or only a few valence nucleons, may be
vibrational and typically show only modest collectivity. Rotation is associated with
a coupling of proton and neutron degrees of freedom [15]. Thus, both neutron and
proton valence particles —— perhaps several of each —— are needed to form a statically

deformed state.

1.3 Exploring new territory

Exploration of fl—unstable nuclei provides additional tests of models developed near
stability. In these exotic nuclei — also called rare isotopes — one may also study
novel couplings of valence nucleons. Specifically, this thesis will probe the evolution
of quadrupole collectivity in neutron-rich silicon and sulfur nuclei near N = 28. Figure
1.2 shows the isotopes studied in this thesis, 36’38’40Si and 40’42’448.

The ordering of single-particle orbitals and the gaps in excitation between them
form the foundation of the shell model. These single-particle energies (SPEs) are
known to vary with mass, which changes the size and depth of the mean-field potential
[16]. These SPEs are also known to vary with the filling of particular orbitals due
to specific nucleon overlaps giving increased, or decreased, binding [14, 17,18]. In
exotic nuclei, these changes can be more extreme and even major shell gaps are
affected [19]. Nuclei in the island of inversion [20-22] —— canonically N = 20-22, Z =
10 — 12 —— have intruder configuration ground states (V(0f7/2)x (51012—33) Neutrons
are promoted across the N = 20 shell gap because this gap is smaller in these neutron—
rich nuclei and the intruder states have increased correlation energy. Increased binding
of the ground states [20] and strong collectivity in the low-lying 2;“ states of even-even
nuclei [23—25] were the first observables used to discover and study this region. Much
work is still left to do in this region. For example, the first quantitative measurement
of intruder contributions to the ground state wavefunctions of 28Ne was recently

performed [26].

20".4‘Ca‘. ‘ . “.4SCa".47Ca‘.

- 4oK . 42K 43K 44K 45K 46K 47K

 

I8 0 ‘ 39Ar ‘ 0 A 4IAr 42Ar 43Ar 44Ar 45Ar 46Ar
380 390 40c: , 41CI 42Cl 430 440 450

I6 0 375 385 395 405 415 425 435 445
35P 36p 37P 38p 39p 40P 41P 42p 43P

14] 34Si 355i 355i 375i 385i 39Si “Si “Si 425i

20 22 24 26 28
Neutron number

 

 

 

Proton number

 

 

 

 

 

 

 

 

 

 

 

Figure 1.2: An expanded view of the chart of the nuclides is given for the region studied
in this thesis. Isotopes shown as black squares are stable, and isotopes studied in this
thesis are shown as grey squares.

In the N = 28 isotones, the first experimental indications of a reduction in the
V(0f7/2 — 1p3/2) shell gap came from the fi-decay lifetime of 44S, which was used
to infer static deformation [27]. Coulomb excitation was then used to measure the
B(E2T) value, confirming significant collectivity in this N = 28 nucleus [28]. The trend
in B(E2T) values across the sulfur (Z = 16) isotopes showed quadrupole collectivity
rose from a minimum at N = 20 to a maximum at N = 26; the small decrease between
N = 26 and N = 28 was nearly the same as the error bar for such a comparison.

An alternate explanation was put forward by Dr. Cottle and Dr. Kemper [29].
They argued that shifts in the proton SPEs might be the cause of increase collectivity
in the neutron-rich sulfur isotopes. Using data taken collected in (d,3 He) reactions
on calcium isotopes [30], Cottle and Kemper traced the 7r131/2 and 7r0d5/2 single-
particle energies relative to the 7r0d3/2 orbital as a function of neutron number from
N = 20 — 28. (Note: these were energy centroids summed over a large part of the
single-particle strengths.) Figure 1.3 shows their results with the proton subshell gaps
labeled. As N increases from 20 to 28, the subshell gap at Z = 16 decreases. Instead
of filling the 131/2 subshell, the last two valence protons in 42’44s are free to couple
in a larger space of 131/20d3/2. Proton collectivity increases and coupling of proton
and neutron degrees of freedom can lead to a region static deformation centered on
423 [31,32].

Thus, the rise in collectivity approaching N = 28 in the sulfur isotopes could be

a sum of
o proton collectivity arising from the collapse of the Z = 16 subshell gap,
0 neutron collectivity resulting from a diminished N _—_ 28 shell gap, and
0 increased p — n correlation energy leading to static deformation.

Measurements of sulfur isotopes are then inconclusive with respect to changes in

N = 28. Silicon isotopes, on the other hand, have a rather large Z = 14 subshell gap

11

 

 

 

gfldalz ' gsp(MeV)
A

d5/2

O)
I

 

 

 

20 22 24 26 28 28
N

Figure 1.3: Proton single-particle energy shifts near N = 28 are shown for the 181/2
and 0d5/2 orbital relative 0d3/2 [29]. As the Z = 16 subshell gap narrows, proton
collectivity is expected to increase. The 7r(131/2 — 0d3/2) degeneracy is also observed

in neutron-rich 43P [33,34].

12

from N = 20 — 28 and are well suited to determine whether the N = 28 shell gap has
truly weakened.

Shell model calculations in this region are typically performed using the SDPF-
NR interaction [9] placing N — 20 neutrons in the pf shell and Z — 8 protons in the
3d shell —— or more succinctly, 7r(sd)Z “81/(p f)N ’20. The SDPF-NR interaction re-
sulted from combining the USD interaction between protons with the KB’ interaction
between neutrons and a proton-neutron cross-shell interaction taken from G-matrix
calculations then fit to data [31]. Initially, single particle energies were fixed by val-
ues measured at or near stability, but fi-decay measurements found that the 1/1p3 /2
state in 35Si was significantly lower in excitation than predicted [35]. The interac—
tion was modified to reproduce the [smaller V(0f7/2 - 1p3/2) splitting observed at
N = 21, Z = 14. These changes had little effect on the calculated spectra of neutron-
rich sulfur and argon isotopes, but the 2'] excitation energy of 42Si was reduced from
2.5 MeV to 1.5 MeV, indicating a strong sensitivity to the choice of interaction. This
modified interaction is denoted SDPF-NR [9] and is compared to the levels observed

in this thesis.

13

Chapter 2

Experimental Method

This experiment was designed to measure the collectivity of several fi-unstable neutron-
rich nuclei by extracting quadrupole deformation parameters from inelastic proton
scattering cross sections.

A beam of these exotic nuclei — traveling at N 40% of the speed of light —
was produced and directed onto a liquid hydrogen target. The hydrogen nuclei ~
protons -— were occasionally scattered off projectile nuclei, which were sometimes
(roughly 1 in 104) left in an excited state. After some very short time —— typically
much less than 100 ps —~ the bound excited nuclei would de—excite by emitting 7
rays. A fraction of these 7 rays deposited their full energy in one of several 7—ray
detectors arranged around the target. The beam of nuclei then passed through a
series of magnets and were detected. Data were collected and cross sections for the
population of specific excited states were extracted. These cross sections were then
used to determine deformation parameters, |fl2,(p,p/)|, in a collective model.

The first section of this chapter discusses proton scattering at intermediate beam
energies. It focuses on the theory and inputs needed to extract deformation parameters
from cross sections. Remaining sections describe the experimental setup, calibrations,

and uncertainties in the cross section measurement.

14

2.1 Proton scattering

Inelastic proton scattering provides a method to directly measure nuclear collectivity.
Specifically, the sensitivity of proton scattering to protons and neutrons in the nucleus
has been employed along with the selective proton-sensitivity of Coulomb excitation
to measure differences in the proton and neutron contributions to the excitation of
specific states [36—38]. More recently, inverse-kinematics proton scattering has been
used instead of Coulomb excitation to study collective states in the most exotic nuclei,
as proton scattering experiments on liquid or solid hydrogen targets are feasible with
beam rates of about a factor of ten less than is required for Coulomb excitation.
The present experiment was devised to measure quadrupole collectivity in 36’38’40Si
and 40’42’44S. Deformation parameters (6271900) were deduced from inelastic excita-
tion cross sections (0194/) using the DWBA code ECIS [39]. Collective form-factors
for vibrational and rotational excited states were employed in deducing deformation
parameters from the measured cross sections. A brief review of the underlying theory

which allows the Up I); —> [32 (p pl) deduction is presented in the next section.

2.1 . 1 Scattering theory

Scattering of a particle off a nucleus is a basic, but fundamental, problem in the field
of nuclear reactions. The present experiment is concerned only with the scattering of
~ 100 MeV protons on a nucleus. Furthermore, only the cross section for scattering
which populates bound, excited states of that nucleus is relevant. Finally, only low-
lying, collective states of even-even nuclei are analyzed herein.

The full derivation of scattering from a potential is covered in quantum mechanics
and nuclear reaction texts [40—43]. Only a general overview is given here.

Elastic scattering is treated by solving for the wave functions which satisfy the

Schrtidinger equation:

2
(—2%V2 + V) \I/ = E\II. (2.1)

15

The Dirac equation may be used instead of the Schrodinger equation to handle rela-
tivistic kinematics; this is, in fact, done in the code ECIS [39]. Scattering comes in as
a modification of an initial plane wave. The asymptotic (large R) wave functions are
a superposition of the original plane wave and an outgoing spherical wave modulated
by a scattering amplitude, f [42].

ikr

@301?) ~ (all: F+ fE(r~)e ) (2.2)

—o
a
O

 

T'

The probability for the scattered particle to pass through a specific point is an integral

over [\II|2, and the differential cross—section for elastic scattering is:

(10' A 2
d—Q—leU)

 

(2.3)

When elastic scattering is the dominant reaction, other channels are treated
through absorptive potentials which remove flux —— or as perturbations. Inelastic
scattering is treated as a perturbation. In calculating the scattering amplitudes, elas-
tic scattering wavefunctions are used along with some additional potential responsible
for the inelastic excitation.

The distorted wave born approximation (DWBA) is often employed in such cal-
culations. Instead of plane waves, distorted waves (Xi) are used which satisfy the
Schréidinger equation for the elastic potential. The Born approximation is applied by
using the scattering wave functions calculated using the elastic potential. Essentially,
the part of the potential responsible for the inelastic transition is weak compared
to the elastic potential. Thus, the total potential and scattering wave functions are
well approximated by the elastic potential and the distorted waves derived from that
elastic potential [43].

For inelastic scattering, we consider not only the relative position wave function

for the proton-nucleus system, but also the wave function describing the internal

structure of the nucleus. Scattering amplitudes, f, for inelastic scattering are then
16

integrals over products of an incoming distorted wave, an outgoing distorted wave,
and a term which couples the initial and final states via the transition potential [43].
To calculate the amplitudes, and thus a cross section, the following inputs are

required:
0 a potential describing elastic proton—nucleus scattering,
o the initial and final states of the nucleus, and

o a potential which couples these wavefunctions.

2.1.2 Deformation extraction

In practice, nucleon—nucleus elastic scattering at energies from a few keV to hundreds
of MeV may be described using an optical model potential [44]. The optical model
potential is typically a sum over complex Woods—Saxon potentials and derivatives
thereof with parameters ——— depth, radius, and diffuseness —— fit locally to elastic
scattering on the nuclei of interest. Fitting the potential is not possible in the present
experiment due to the use of the thick-target, 7-ray tagging method (see section 2.3.1
for further discussion). Thus, a global optical model parameterization was used [44].
Similar parameterizations compare well with measurements of proton scattering on
nuclei in this region [45].

Inelastic proton scattering on even—even nuclei populates low-lying collective states
strongly [46,47]. In particular, shape oscillations of the nuclear surface and rotational
states in statically deformed nuclei are typically observed. Because the excitation
probability is small, only single-step excitations are expected.

To calculate scattering amplitudes for these states, the collective models discussed
in section 1.2.2 are used. Those models described deformations of the nucleus and are
appropriate as models for nuclear density. The assumption that changes in the po—

tential should follow changes in density distribution leads to the deformed optical

17

model [41]. This model was chosen for the analysis of the present experiment. Cou-
pling between initial and final collective states is modeled by a term proportional to
flLdU/dR [39], where 6 L is the deformation parameter for the L multipole and dU/dR
is the radial derivative of the optical potential. Here, the form-factor is a derivative of
the original optical potential and is surface peaked. The scattering amplitude f from
Eq. 2.3 is proportional to an integral over the product of incoming and outgoing dis-
torted waves and the initial and final states of the nucleus coupled by this collective
transition potential. Pulling [3 L out of the integral implicit in Eq. 2.3, the integrated
cross section is proportional to {3%.

Several codes have been developed to handle such DWBA calculations. In this
work, the code ECIS97 [39] was used to calculate DWBA cross sections for the (p, p')
reactions considered in this thesis. Total inelastic cross-sections were calculated for
multiple deformation parameters, [3L]. The quadratic dependence of the calculated
cross section on the deformation parameter was used to infer [6L] values for the mea-
sured cross sections and their :Elo uncertainties. Optical model parameters (OMP)
— consisting of the radius, depth, and diffuseness for each Woods-Saxon or derivative
of Woods—Saxon potential making up the proton-nucleus optical potential — were cal-
culated for each isotope at its mid-target energy [44]. Because ions traveling through
the target lose energy continuously, one must consider the energy dependence of the
OMP used. Using the mid-target energy instead of integrating over the distribution

of energies within the target leads to a negligible 1% reduction in inferred ISL].

2.1.3 Sensitivity of (p, p’) to neutrons and protons

Because proton scattering is sensitive to both protons and neutrons within the nu—
cleus, the deduced deformation parameters will depend on both proton and neutron
contributions to the collective excitation. Coulomb excitation, on the other hand, is
sensitive only to the protons. By combining measurements of both types, one may ex-

tract the relative strengths of the proton and neutron components of a given collective

18

excitation [36,37]. These ratios may then be compared to model predictions.

In the homogeneous collective model, neutrons and protons share a common de—
formation, dp = (in, where 6 = BR [48]. Transition densities for neutrons and protons
have a common shape but are scaled by the number of each type of nucleon. Proton

and neutron matrix elements are then given by

 

oo

21% = 1152: 2 f0 p’(r)r)‘+2dr, (2.4)
where p(r) is the normalized ground state density. So, the ratio of neutron-to—proton
matrix elements is Mn/Mp = N /Z .

Adding in nuclear shell structure modifies the observed trend for closed shell
nuclei [48]. Specifically, a nucleus with a closed-shell structure for one type of nucleon
and valence nucleons of the other type will have a large contribution to collective
transitions from the valence nucleons. As an example, one would expect N = 20 nuclei
with valence protons to show stronger proton than neutron collectivity. Thus, neutron
closed shell nuclei are expected to have Mn/Mp < N /Z , while proton closed-shell
nuclei show the opposite. The nucleon-nucleon interaction serves to couple valence
nucleon motions to the nucleons of the core, leading to core polarization [49]. Even
though a closed shell of nucleons may be considered inert in a shell-model calculation,
the core plays a role in transition strengths.

To compare deformation parameters from various probes, Bernstein, Madsen, and
Brown included the intrinsic sensitivity of the probe to protons and neutrons in the
nucleus [36]. Equation 2.5 uses the sensitivity parameter, bn/bp, for the specific case
of comparing proton scattering —— which samples both protons and neutrons —— to

Coulomb excitation — which samples only protons.
Mn bp 50, pl) bn N
—— = — —-’— 1 —-— — 1 2.5
Mp bu] 50 +pr ( )
Canonical proton scattering bn/bp values are 3 at low proton energy (10-30 MeV)

19

 

 

 

 

. — CPK
4 --- FL
0 Average

 

 

 

 

 

o , , 1
4o 60 80 100 120

Proton energy (MeV)

Figure 2.1: Ratio of neutron-to—proton sensitivity, bn/bp, in (p, p’ ) calculated using
two different parameterizations of the nucleon-nucleon interaction. One calculation
was performed by Carr, Petrovich, and Kelly (CPK) [50], and the other uses the
Franey and Love (FL) interaction [51].

and 1 at high energy (~1 GeV). This experiment was carried out at intermediate
energies, N100 MeV/nucleon. Following the procedure outlined by Carr, Petrovich,
and Kelly [50], the ratio of free nucleon-nucleon amplitudes, [tnp/tppl was used as
an estimate of bn/bp. Inelastic proton scattering is surface dominated, and the free
nucleon-nucleon cross sections are, therefore, a reasonable approximation.

Figure 2.1 shows the ratio Itnp/tppl as a function of bombarding energy for two pa-
rameterizations of the nucleon-nucleon interaction. Carr, Petrovich, and Kelly (CPK)
[50] used an interaction based on the M3Y [52] in the region 50 MeV < Ep < 100 MeV.
The parameterized interaction of Franey and Love (FL) was also used [51]. The mid—
target energy for nuclei in this study varied from ~ 90 — 100 MeV. An average energy,
Ep 2 95 MeV was taken, and an average of the two sensitivities was taken. This yields
bn/bp = 1.45(20), where the difference in the two parameterizations was used as an

estimate of uncertainty. At lower energies, the use of the free nucleon-nucleon interac—

20

tion is a poor approximation, and an effective interaction may be needed to determine
bn/bp

An additional concern for neutron-rich nuclei is the difference in the radial form-
factors for the neutron and proton transition densities. The issue is that the method
described in section 2.1.2 is not sensitive to total collective strength but only to
the strength which lies outside some absorptive potential. In a calculation with the
absorptive potential, the nuclei studied had calculated cross-sections which were ~
44% of the cross sections calculated with the absorptive potential removed. Because
the neutron transition density is centered at a larger radius than the proton transition
density in these nuclei, a smaller part of the neutron collective strength is lost to
absorption. Thus, the method favors neutrons relative to protons.

A sharp cutoff model was developed for this additional nucleon sensitivity in
the method. Shell model transition densities were used for each nucleus. rIi'ansition
densities were summed in proportion to the sensitivity, bn/bp, of proton scattering.
This value was multiplied by 7‘4 (one r2 term comes from the integral over the volume
of a sphere and another r2 term comes from the quadrupole operator) integrated from
R to co and squared to obtain a value proportional to the cross section lying outside a
given absorptive radius. The radius outside which 44% of the integrated cross section
lies was chosen as Rabsorptive- Then, a ratio of the fractions of each nucleon transition
strength lying outside Rabsm‘ptive was taken as the additional nucleon sensitivity.

These additional nucleon sensitivity values ranged from 1.2 — 1.6. Due to the ad
hoc nature of this estimate, the uncertainty was estimated as 50% of the effect in
each case. The final bn/bp value used for a given nucleus was simply a product of the

intrinsic and model nucleon sensitivities.

2.1.4 Additional reaction channels

While the goal of this experiment was to measure inelastic excitation cross-sections

in (p, p’), a number of other reactions were also observed. These included single and

21

multiple nucleon removal reactions. Proton pickup and (p, n) exchange reactions could
not be observed because these reactions fell outside the momentum acceptance of the
spectrograph. The single neutron removal channel (——1n) was of particular interest as
it had a large cross section, could populate a different set of states than (p, p’ ), and
held the possibility for relating level population to spectroscopic strength calculated
in the shell model.

Single-nucleon removal which occurs as a direct reaction, without transfer, is com-
monly called single-nucleon knockout, or simply nucleon knockout [53]. Two distinct
groups of reactions are typically included. The first group consists of (e, e’ p), (p, 2p),
and (p, pn), which were used to study the single particle composition of stable nu-
clei in normal kinematics [54,55]. High bombarding energies and kinematic selection
were used to minimize any non-direct components. Extending the technique to short-
lived, exotic nuclei required a more efficient technique. Thus, inverse-kinematics nu-
cleon knockout on thick light-ion (9Be) targets was developed. Light ions minimize
Coulomb breakup, strong absorption confines single nucleon removal reactions to the
surface, and the knockout residues are collected.

Nucleon pickup, i.e. (p, d), is another direct single-nucleon removal reaction in
which a proton ”picks up” a previously bound neutron from a heavy ion [41]. In
either reaction, the removed neutron may come from one of a few valence orbitals,
and the A —- 1 daughter nucleus will be left with some probability in one of several
final states. Partial cross sections for populating those states are proportional to the
spectroscopic factors (C25) for neutrons in the ground state of the original nucleus.
One can find spectroscopic factors using measured cross sections to specific excited
states, or one can use the predicted spectroscopic factors with the measured cross
sections to infer the identity (J 7r) of the populated excited states.

Unfortunately, the present experiment observed only the neutron removal residues
after some interaction with protons. The setup was optimized for inelastic scatter-

ing, not knockout. A large fraction of neutron removal residues were collected for

22

many incoming isotopes, but the fraction which fell outside the focal plane accep-
tance varied. Furthermore, the neutron removal reaction observed in this experiment
includes any reaction channel which populated bound final states in the —1n residue.
The observed cross section is a sum over (p, d), direct (p, pn), multi-step (p, pn), and
’2X N(p, p’ >21fo 49—1 X1"{J_1 + n reactions. This experiment was not designed to
separate these contributions.

Experimental studies of inclusive neutron removal reactions due to ~ 100 MeV
protons typically do not track the final states populated in residues. Those papers
which discuss the states populated in such reactions were mostly performed in the
1950’s and 1960’s, i.e. before the exclusive reaction techniques became common. Work
from that era suggests that 50% of the (p, pn) cross section for masses and energies
considered in this thesis comes from direct reactions [56]. Most of the remaining (p, pn)
cross section was estimated to come from a two-step process of inelastic scattering
followed by nucleon emission. The (p, (1) component near 100 MeV proton beam energy
was determined to be negligible [57]. Giant resonances carry much of the inelastic
scattering strength, and in light nuclei A § 40, semi-direct neutron emission can give
cause the population of final states in the daughter nucleus to follow the spectroscopic
factors [58,59]. So, a significant portion of the inclusive neutron removal reaction
considered here can be expected to populate states in the —ln residue in proportion
to the spectroscopic factors.

Spectra of 7 rays in coincidence with neutron knockout on 9Be were compared to 7-
ray spectra measured in coincidence with neutron removal observed in this experiment
for selected isotopes. Ptom the 7-ray spectra, the two reactions were found to populate
final excited states with similar relative intensities. Figure 2.2 compares the spectra
resulting from single-neutron removal of 48Ca on a 9Be target [60] and on the proton
target in this experiment. Similar agreement was also observed for —-1n reactions on
38Si and 413 [61]. The similarity in excited state populations between 1n knockout on

9Be and In removal on a proton target indicates that the (p, X) _1,, reaction populates

23

 

 

 

 

 

 

 

a l l T l j I F T l
600 — — 9Be target
—- Proton target

500 — -

400 -
9
3'2
CD .
g 300 _
:2
8
u 200 ‘ -

100 - -

.I
0 l l l l l l l l l
200 400 600 800
y-ray energy (keV)

120 b I I m I I I I I I

100 — ] —
S
.9"
co
5
3
+2
3
c
U

 

 

 

l l l l l l l l l l

i 200 1 600 2000 2400 2800
y-ray energy (keV)

Figure 2.2: Spectra of 7 rays in coincidence with single-neutron removal from 480a

impinging on a 9Be target and on a proton target. The two panels show low and high
energy portions of the spectrum. Deviations below 200 keV are likely due to differing
levels of low-energy, beam-induced background and different threshold settings.

24

final states in rough proportion to their spectroscopic factors.

In this thesis, dependence of -— 1n cross section on spectroscopic factors is used only
to assign the second strongest 7-ray transition in even-even residues as the 4] ——> 2']
transition. Spectroscopic factors for —In removal from odd-N, even-Z nuclei filling
the l/(0f7/2) orbital favor population of 4+ and 6+ states in the even-even residue [62].

Decays predominantly flow through the 4]" state.

25

2.2 Exotic beam production

Exotic, or rare, nuclei are unstable to ,3 decay and typically have short lifetimes.
Isotopes studied in this thesis are neutron-rich and undergo fi— decay with lifetimes
ranging from tens to hundreds of milliseconds. So, the experiment must be performed
in inverse kinematics with continuous production of the rare isotopes being studied.

Currently, two methods account for most of the production of isotopes far from
stability: ISOL and in—flight production [63]. Both utilize nuclear reactions which
break apart stable nuclei and use separation techniques to select the isotope(s) of
interest. In-flight production may be further separated into projectile fragmentation
and projectile fission. The latter will not be discussed in this thesis.

The name ISOL (isotope separation on-line) alludes to the methods historical de-
velopment. Once research reactors and accelerators became available, nuclear physi-
cists and chemists began creating long-lived, unstable isotopes by bombarding stable
isotopes with neutrons and protons. The new isotopes were extracted off-line by re-
moving the material to a lab and performing one or more physical and/ or chemical
separations. ISOL is essentially the same process but faster with continuous bombard-
ment and extraction. The extracted ions are then separated, accelerated, and used in
experiments. In this way, isotopes having shorter lifetimes may be studied.

Advantages of this technique include high production rate for some isotopes, good
beam quality, and beam energies comparable to those used at stable beam facilities.
Disadvantages include chemistry dependent extraction and beam development, decay
losses due to extraction time, and thin secondary targets, which lead to lower reaction

rates.

2.2.1 Projectile fragmentation

In projectile fragmentation, a stable heavy-ion beam is accelerated and directed onto

a thick production target. A given projectile nucleus may strike a target nucleus while

26

RT-ECR K500 20 II

 

, at: _, 2 :
Ion sources c. , -' ~ .-' WW
SC-ECR _ ~ 10m
ac
a -, § A1900 3’ focal plane
. ' ‘ a. ‘ $ r ,
. . H production \‘ ’
stnppmg target ‘ 15‘; t ‘21!" /
foil \

wedge

Figure 2.3: Diagram of coupled cyclotrons and A1900 fragment separator

passing through the target. Projectile nucleons in the target-projectile overlap region
are removed, and the remaining projectile fragment travels forward with its velocity
nearly unchanged [64]. This projectile fragment may then decay by nucleon emission
until it has reached a particle stable configuration.

The present experiment was performed at the National Superconducting Cyclotron
Laboratory (NSCL) on the campus of Michigan State University (MSU). Figure 2.3
provides a schematic of the Coupled Cyclotron Facility (CCF) [65] and the A1900
fragment separator [66], where the exotic nuclei used in this experiment were produced
and selected. Stable 48Ca was extracted from an ion source as beam with 8+ charge
state and injected into the K500 cyclotron. There, the beam was accelerated to an
energy of 12 MeV/u, extracted, and injected into the K1200 cyclotron. Inside the
K1200, the beam passed through a stripper foil and a beam with 19+ charge state
was accelerated to a final energy of 140 MeV/ u. This beam was extracted and delivered

to the production target.

2.2.2 Fragment separation and delivery

The 140 MeV/ u beam of 48Ca impinged on a 987 mg/cm2 9Be target placed at mid-

acceptance position of the A1900 [66]. Projectile fragmentation occurred within this

27

 

target. Following projectile fragmentation, the outgoing beam is a cocktail composed
of unreacted 48Ca and hundreds of stable and exotic nuclei, all having nearly the
same velocity. Nuclei are separated in the A1900 fragment separator by the Bp —
AE — Bp method which utilizes magnetic fields, differential energy—loss, and physical
slits. Equation 2.6 defines 3,0 with m being the ion rest mass, '0 the velocity, 7 the
relativistic factor, and q the ion’s charge. For isotope beams in this thesis, one may

replace m/q with A/Z.

Bp = — (2.6)

To make optimal use of the beam, a thin wedge was used at the intermediate
image of the A1900. This allowed many nuclei to be delivered to the target in a
single setting of the fragment separator —— such beams are called cocktail beams. To
collect data on the six even-A silicon and sulfur isotopes studied in this experiment,
two cocktail beams were developed. These will be referred to as dataset 1 and 2
or cocktail beam 1 and 2 in this thesis. Table 2.1 gives the integrated number and
mid-target beam energy of even-A silicon and sulfur isotopes incident on the target.
Cocktail beam 1 had a momentum acceptance of 0.5%, and cocktail beam 2 had a
momentum acceptance of 4% — imposed by the 8800 analysis line. The second, more
neutron-rich setting was developed with a large momentum acceptance to maximize

rate for 40Si.

2.3 Reaction target and detectors

2.3.1 Thick-target, 7-ray tagging method for (p, 19’)

Starting in the 1950’s, proton scattering experiments have been used to study both
the elastic and inelastic channels by detecting low-energy protons scattered off thin

targets of stable isotopes. Excitation energies in these target were reconstructed from

28

Table 2.1: Even mass silicon and sulfur isotopes studied in this thesis are listed.
The integrated number of nuclei delivered on the LH2 target and beam energies at
the mid-point of the target are listed. The cocktail beam in which each isotope was
measured is listed.

 

 

Cocktail Number of nuclei Mid-target
Isotope beam (dataset) on target (x106) beam energy (MeV/ u)
368i 1 14 90.3
38Si 2 23 99.5
4081 2 1.6 90.3
408 1 97 96.0
428 2 11 103.6
44s 2 36 94.9

 

 

 

 

 

 

measured proton energies and angles. By detecting protons at a number of fixed
scattering angles, differential cross sections are measured for scattering to each final
state. Starting in 1994 [67], such experiments have also been performed in inverse-
kinematics with rare isotope beams [45,68,69]. Here, rare isotope beams impinged on
targets containing protons, and low-energy protons were detected in the laboratory
close to 90 degrees with respect to the incoming beam, which corresponds to very for-
ward center-of—mass scattering angles. In all experiments where the scattered proton
is detected, thin targets (~ 2 mg/cm2) are employed to minimize energy straggling.
Thin targets also require beam intensities of ~ 104 particles per second of rare isotope
of interest to perform inelastic scattering experiments in a few days. The thick-target
method employed here can achieve the same luminosity with beam rates of about 104
less.

To expand inelastic proton scattering studies to more neutron—rich nuclei, the
thick-target, 7-ray tagging method was developed [70]. Instead of measuring the scat-
tered proton and kinematically reconstructing the excitation energy, 7-rays are de-
tected from decays of states populated in inelastic scattering events. An integrated
cross section, defined by the detection of the recoil ion, is obtained. This allows the
use of ~ 100 — 200 mg/cm2 thick plastic [(C3H6)n] or liquid hydrogen [LHQ] targets.

In this case, target thickness is limited by the impact of energy loss of the ion travers-

29

ing the target on the desired energy resolution of Doppler-corrected 7—ray spectra.
The present experiment used the thick-target, 7-ray tagging method to study nuclei
with a lower rate limit of ~ 10 particles per second.

This method improves energy resolution (~ 3% versus 100’s of keV), gives addi-
tional information through 7-ray branching ratios, and allows for the measurement of
long-lived states through changes in the 7-ray response. On the other hand, measur-
ing excitation energy directly determines the state populated; 7-ray measurements
are subject to feeding corrections and higher-lying level energies must be deduced
from energy sums.

Elastic scattering angular distributions may also be collected but typically have
poor angular resolution because the heavy-ion scattering angle is so small in the lab
frame. Thus, the optical model cannot be determined locally. In this experiment, the
energy and isotope dependent global optical model parameterization of Koning and

Delaroche was used [44].

2.3.2 Liquid hydrogen target

The RIKEN-Kyushu—Rikkyo liquid hydrogen target [71] was used in this experiment
to maximize the number of proton nuclei in the target while minimizing energy loss
and the effect of contaminant atoms. In a plastic target, for example, carbon atoms
also cause excitation of the nuclei under investigation, and a background run is re-
quired to measure and subtract this effect. The aluminum windows of the liquid
hydrogen target cell used in this experiment account for 5.7(2)% of the atom frac-
tion of the LH2 target cell. Therefore, their contribution to measured excitation cross
sections is negligible. Figure 2.4 is a schematic representation of the target system.
Because liquid hydrogen has a low density (0.071 g/cm3), quite thick targets are
required for exotic beam studies. A comparable experiment by Riley et al. used a 191
mg/cm2 polypropylene [(C3H6)n] target (i.e. 1.64 x 1022 protons/cm2) to study 46Ar.

In the present work, Doppler-corrected 7—ray spectra were used to measure total cross

30

sections for exciting particular states by inelastic proton scattering. So, although an
increase in target thickness would result in a proportional increase in reaction rate,
the effect of target thickness on Doppler reconstruction must be taken into account.

Target thickness adds to the angular uncertainty in the Doppler correction. In a
standard solid target experiment using SeGA, the angular resolution is dominated
by the detector segmentation, which is 1 cm along the detectors central axis. The
nominal target cell thickness chosen for this experiment was 9.0 mm, causing only a
small impact on the resulting resolution of Doppler-corrected spectra.

A greater concern is the change in target thickness caused by differential pressure
deforming the cell windows. The aluminum entrance and exit windows for this target
cell were 0.190 and 0.212 mm thick respectively. During the data runs the target
cell was placed into the vacuum of the beam pipe and contained liquid hydrogen

at a pressure of 970(20) hPa, or just under 1 atmosphere. This pressure difference

  
  
       

Refrigerator

Bellows —>

 

Target Cell

Gas Cylinder

learn I l

Gas Reservoir

Heat Shield

Figure 2.4: Schematic diagram of the RIKEN-Kyushu-Rikkyo liquid hydrogen (LH2)
target system [71].

31

Table 2.2: Measured maximum thickness of LH2 cell when deformed by differential
pressure. Measurements were performed at room temperature using an inert gas to
fill the cell to the desired pressure [72]. The first entry is the nominal value before the
aluminum windows were permanently deformed.

 

 

Differential pressure Thickness
(hPa) (mm)
flat 9.00
0 10.88
950 1 1.13
990 11.23

 

 

 

 

significantly bowed the target windows during the experiment increasing the effective
target thickness. Following the experiment, the target thickness was measured with
and without differential pressure [72]. Target thickness was measured with calipers at
the center of the target windows, see table 2.2.

This measurement gives the maximum target thickness, but a cross-section mea-
surement requires the effective target thickness and an associated uncertainty. Three
factors were considered in the calculation of an effective thickness: (i) deformation
profile, (ii) beam spot size, and (iii) beam position uncertainty. The deformation
profile of a window due to pressure depends critically on the materials response to
stresses involved [73]. In the limit of elastic response, the deformation is spherical.
At the other extreme, a thick plate may become permanently deformed and both
the deformation and its derivative must vanish at the boundary. Equation 2.7 de-
fines the surface elevation of a deformed, clamped plate [73]. Figure 2.5 shows the
half-thicknesses of the target in this experiment versus the Y position on the target
for 4500 randomly chosen points on the target surface. The spherical model is shown
in red and the deformed plate model in blue. Because the target remained deformed
even after the differential pressure was removed, the deformed plate model has been

used.

 

5.7

Target half-thickness (mm)

 

 

 

Y position on target (mm)

Figure 2.5: Effective LH2 target thickness for two models of window deformation: the
spherical model (red) and the deformed plate model (blue).

The effect of the beam spot was to average over some region of the target. This
averaging was computed numerically by random selection of points on the surface of
the target, see figure 2.5. An effective thickness was calculated at each point. These
thicknesses were weighted according to a Gaussian distributed beam with a FWHM
of 1 cm, summed, and normalized by the total probability due to all points. For a
known beam position, this gives the effective target thickness.

However, beam position on the target may also vary with the specific fragment
cocktail and the alignment of the target with the beam axis. A Gaussian distribution
of 1 cm FWHM was taken for the beam position, and effective thickness was calculated
as described above for beams centered at varying radii from the true target center. An
expectation value for the effective target thickness was calculated to be 0.931 of the
maximum target thickness. The range (0.912 — 0.946) contains 69% of the probability,

thus these values set the 1—sigma uncertainty in target thickness.

33

 

Figure 2.6: The 8800 analysis line and spectrograph are shown. The locations of
detectors used in this experiment are noted.

Target pressure varied between 950 and 970 hPa during the data runs giving a
maximum target thicknesses of 11.18(5) mm. The product of effective target thick-
ness, Avogadro’s number, and mass density of liquid hydrogen divided by hydrogen’s
atomic mass (1.008 g/mol) gives the areal proton density of this target: 4.41:; x 1022
protons / cm2. Because the main target volume is quite large compared to the deforma-
tion of the windows, the uncertainty in effective thickness is only ~ 2%, comparable

to or smaller than typical solid foil uncertainties.

2.3.3 The 8800 spectrograph

The 8800 analysis line and spectrograph [74] are shown in figure 2.6. The analysis line
serves to transport secondary beams prepared by the A1900 fragment separator to
the 8800 target position where secondary reactions may occur. The spectrograph then
selects reaction residues in a particular range of magnetic rigidity and delivers them
to the focal plane [75]. Detectors at the object and intermediate image of the analysis
line and the spectrograph’s focal plane uniquely characterize the nuclei impinging and
exiting the reaction target. For the current study, the 8800 was used in focused optics
mode with a liquid hydrogen cell placed at the reaction target position.

Two optics modes are typically used in the 8800 for inelastic scattering and nucleon

34

knockout experiments with fast beams. Dispersion-matched mode gives the highest
energy resolution, but limits the incoming momentum acceptance to 0.5% and results
in a momentum-dispersed beam on the reaction target. Focused mode allows for an
incoming momentum acceptance of i2% and results in a focused beam at the target
position. Because the present study requires a large momentum acceptance to make
efficient use of the projectile fragmentation beam and since the LH2 target cell has
a diameter of 3 cm, focused mode was used. In focused mode, the dispersive angle
of a particles trajectory at the intermediate image of the analysis line is proportional
to the incoming momentum. The momentum after the target is proportional to the
dispersive position measured in the focal plane.

In this experiment, energy loss in a 300 um Si-PIN detector placed at the object
position was measured to determine the charge of each incoming projectile nucleus.
Rare isotopes in this study are selected in the A1900 as fully stripped after emerg—
ing from the production target. They remain fully stripped when passing through
thin transmission detectors and the secondary target at all times due to their high
velocity and low atomic number. A pair of X—Y sensitive high-rate Tracking Parallel
Plate Avalanche Counters (TPPACS) placed up- and down—stream of the intermediate
image determined both the position and angle of each nucleus in the dispersive and
non-dispersive planes. In the focal plane, a single Cathode Readout Drift Chamber
(CRDC) determined the position of each nucleus in the dispersive and non-dispersive
planes. [Notez usually two CRDCs are used to determine both positions and angles,
but detector malfunctions left only one working CRDC for this experiment] Energy
loss in an ion-chamber determined the charge of each outgoing nucleus. Finally, time-
of—flight was measured between a timing scintillator at the extended focal plane of
the A1900 and a timing scintillator in the 8800 focal plane for a total flight path of
about 58 111.

Of these measured particle identification (PID) parameters, only the angle deter-

mined by the TPPACs and the dispersive position measured by the CRDC were used

35

as quantities needing calibration. Fortunately, the calibration in each case was fixed
by the physical size of the readout elements and the detector placement.

Strong kinematic correlations among particle identification parameters offer an
opportunity to improve resolution by combining separate measurements on an event-
by-event basis. Energy loss parameters for the 8i—PIN detector and the ion chamber
were corrected for velocity differences of incoming projectiles to give a more uniform
response proportional to projectile charge, Z. The largest improvement came from
time—of—flight (TOF) correction by dispersive angle and position at the intermediate
image of the 8800. At that position — when the 8800 is operated in focused mode
— the dispersive angle is proportional to the relative momentum of a given particle.
Thus, for each ion, its angle is proportional to velocity. Because the incoming cocktail
beams had momentum widths of 0.5% and 4% Ap/ p, a significant reduction in TOF
width occurred. Figure 2.7 shows the dispersive angle versus the raw and corrected
time of flight and the one dimensional time-of-flight spectra for 42’4'3’448 from the
second dataset.

From the TPPACs and the CRDC, a measure of the momentum difference due to
target interactions was constructed. This longitudinal momentum change parameter
was then corrected by the incoming momentum to remove the differential energy loss
due to the incoming momentum spread. Transverse momentum is expected to be
small in the lab frame due to the forward focusing of reaction products but could not
be measured with a single functioning CRDC. Likewise, scattering angle could not
be measured. The maximum lab scattering angle kinematically allowed for projectiles
scattering off protons in this (p, p') study was ~ 1.5°. This falls well within the 3° x 5°
nominal angular acceptance of the S800 spectrograph, and no angular acceptance

correction is required.

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A a I I I I m I I I 1 b I I I I I I r I 50
'0 __ _.__—‘ ' _
E 25.0 1 40
‘E’ 125 - —- -
a, .
w - - 30
g“ 0.0 - -- -
.E. ‘ ‘ i 20
4-3 '12.5 _ __ _'
E _
-25.0 e _ -.—— — ‘0
l 1 l 1 i 1 . I. I 1 l l 1 l 1 .l . 1 1 m - O
240 280 320 360 400 240 280 320 360 400
Raw Corrected
Time of flight (a.u.)
30000 1 1 1 1 1 1 1 1 30000
LC__] I I I I i] I I I I
4 _ 445 -- _ 445 -
_ 43S — 43
20000 — — 42$ -- — 425 — 20000
§ 1
o
u
10000 - —- - 10000
I l 1 1 l l . 1 Jl \ A l

 

 

 

 

240 280 320 360 400 240 280 320 360 400
Raw Corrected

 

Time of flight (a.u.)

Figure 2.7: Time of flight correlation with incoming dispersive angle is shown. Because
dispersive angle is proportional to momentum, it may be used to correct for the
velocity spread of the incoming beam. Raw and corrected time of flight Show the gain
in resolution.

37

2.3.4 The Segmented Germanium Array (SeGA)

The Segmented Germanium Array (SeGA) is composed of 18 32-fold segmented high-
purity germanium (HPGe) detectors [76]. SeGA was designed and built to study nuclei
by detecting de—excitation 7 rays emitted in-flight. Segmentation of electrical contacts
introduce position resolution which aids in Doppler correction of 7 rays emitted at
roughly 40% of the speed of light.

The detection of 7 rays is accomplished by collecting the charges resulting from
ionization events within the germanium crystal. Gamma rays may introduce high-
energy electrons into the germanium by undergoing the photo-electric effect, Compton
scattering, or pair production (which results in a high-energy positron). These lose
energy through interactions with electrons in the semiconductor material, and after
a short time, the energy deposited has been converted into lattice vibrations and
electron-hole pairs. Through proper surface doping, the germanium semiconductor is
made into a diode, which is depleted of free charges by reverse biasing with a high
voltage. When a 7 ray deposits energy creating electron-hole pairs, the electrons and
holes drift toward the biasing electrodes and are collected.

In SeGA, the germanium crystals are cut to form cylinders with a coaxial cylin-
drical well. The central contact (in the well) collects all electrons excited into the
conduction band. The outer contact (on the surface of the cylinder) is divided into
eight slices along the cylinders axis and four quadrants in the azimuthal direction.
Thus, one or more of the 32 outer segmented contacts may collect the holes.

The energy measurement from SeGA detector central contacts was calibrated us-
ing a number of known 7-ray transitions from radioactive sources ranging from 81 to
3253 keV. While this calibration is expected to be linear for a semiconductor, non-
linear effects arising from the amplification chain or the analog-to—digital conversion
are possible. Here a quadratic calibration was used; the energy residuals after cal-
ibration for 15 of the 16 detectors used were remarkably similar. (The detector in

SeGA electronics channel 3 required a three-part calibration with two linear sections
38

 

 

 

 

 

 

 

 

 

 

0.8
O Quadratic fitting
05 _ A Bi-Linear fitting
1r *
A 0.4 -' A q) 7-
>
s 1-
v 0.2 4
8
3.3
EL 0.0 ------ -------------------------------- [I -------------------- a -----
'O .11.
.9 ,1
g -0.2 - -1
< -Ii-
'0.4 "- .1. .1].
<9 --
-0.6 -
D
‘0-8 I I f I I 1
0 500 1000 1500 2000 2500 3000 3500

Accepted Energy (keV)

Figure 2.8: Energy residuals for 15 of 16 SeGA detectors following calibration. Results
are shown for the quadratic function currently used and a two-part piecewise linear
function.

and one quadratic section.) The averaged energy residuals for those 15 detectors are
shown in figure 2.8. A two-part piecewise linear calibration was also attempted, but
not used in the data analysis. To minimize free parameters, each detector was al-
lowed one slope and intercept for variation; global linear relations between the slopes
and offsets in the two linear pieces were varied; and a global cutoff between the two
linear pieces was varied. The global fit residuals for this two-part piecewise linear
calibration are substantially smaller, and more importantly, have little or no trend
with energy. Future versions of the SeGA calibration routines should include such a
bilinear calibration.

The efliciency for full-energy detection of 7 rays was measured as a function of
energy using calibrated sources between 81 and 1528 keV for absolute measurement.
To extend the energy range of the measured efficiency to 3253 keV, uncalibrated

56Co and 226Ra sources with multiple 7 rays of known branching ratios were used.

39

Table 2.3: A list of sources used for 7-ray efficiency calibration. Sources with docu-
mented absolute calibrations are listed with their relative uncertainties. The Mixed
source contained nine radionuclides; most had short half-lives and were not used due
to low peak intensity. Transitions arising from decays of 137Cs and 60Co in the Mixed
source were used.

 

Source ID Parent isotope Measurement conditions Calibration (rel. unc.)
H7141 Mixed beam pipe only absolute (2.1, 1.6%)
H7961 1338a beam pipe & in-cell absolute (3%)

Z2707 152Eu beam pipe only absolute (1.4%)
Z7044 226Ra beam pipe & in—cell relative to Z2707
F8U(1) 56Co beam pipe only relative to Z2707

 

 

 

In these cases, the source efficiencies were scaled in an energy region overlapping
the calibrated sources. The most precise source (152Eu, serial number Z2707) had a
relative uncertainty of 1.4%; uncalibrated sources were scaled relative to this source.
Since this experiment uses a liquid hydrogen (LHg) target enclosed in an aluminum
target cell, efficiency data was collected both in and out of the cell to verify correct
modeling of the target cell in the simulation. The size of the cell restricted the sources
which could be measured inside the target cell. Efficiency results are displayed in figure
2.10.

The high-velocity of the nuclei at the moment of 7 ray emission leads to a Doppler
shift in the measured energies (see below) and a Lorentz boost which effects the
angular distribution of 7 rays. Both of these relativistic effects and a number of
kinematic uncertainties are included in a GRANT [77] simulation of in—flight SeGA 7-
ray detection. This simulation includes the photon scattering and attenuation due
to beam pipe, target cell, and detectors. Figure 2.9 shows the geometry used in the
GEANT simulation.

Response functions generated in this simulation are used in fitting the Doppler-
corrected 7 ray spectra, and the scaling found in the fit is used to calculate cross-
sections. The ratio of the measured to the simulated efficiency must be applied as
a correction to the cross-section. Figure 2.10 shows the GRANT simulated efficiencies

for a source in the beam pipe and a source in the target cell superimposed on the

40

 

(a)

Target cell

 
   
   
 

 

 

 

 

 

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

 

 

 

            

   

   
  

Beam I

* 1'9 J
] gun .

Eff-1,533]

.
.

';I'I'1‘O'o'u‘
11.111111“
1031175131:

1121119111171

 

 
   

 

 
 

 

 

side view

 

 

 

 

 

Figure 2.9: The geometry used in GEANT [77] simulations of this experiment is shown.
from side and front viewpoints. The SeGA detectors surround the LH2 target cell in
two rings at 37° and 90°.
measurements. The ratio of measured to simulated efficiencies is shown in figure 2.11.
Above 600 keV, the simulation systematically underestimates the efliciency. Below
600 keV, a sharp transition to overestimation occurs. Above 600 keV, a constant scale
factor of 1.01(3) may be used. The scale factor uncertainty includes the scatter of scale
factors and uncertainty in absolute source efficiency. Because the LH2 was not present
during the efliciency measurements, and the inclusion of LHg led to unphysical effects
in the simulation, the small 7-ray attenuation due to passage through the LH2 was
included as a 7-ray energy dependent correction to the cross-section. This correction
was typically less than 1% and an uncertainty of 20% of the correction was included.
Nuclei produced by projectile fragmentation have velocities that are a significant
fraction of the speed of light (I) ~ 0.4c in this study). To minimize losses, these
exotic nuclei are often studied in-flight (i.e. without being stopped and reaccelerated)
in inverse kinematics (i.e. the beam is being studied and the target is probe). This
allows for the use of thick secondary targets, which in turn, maximizes reaction yields;
kinematic forward-focusing often leads to collection of a large fraction of reaction

products.

41

 

 

 

 

 

 

 

 

 

0.06 .1
O LH2 cell
A Beam pipe
005 _ — GEANT, LH2 cell
— GEANT, beam pipe
,2; 0.04 -
.9
0
LE1
0.03 -
E
E
0.02 -
0.01 -
0.00 I r l
0 1000 2000 3000 4000
y-ray Energy (keV)

- Figure 2.10: SeGA full-energy efficiency is plotted as a function of energy for calibrated
source measurements and GEANT simulations. Data are shown for sources placed in
the beam pipe (analogous to a standard SeGA measurement) and for sources placed
in an empty LH2 cell.

The method maximizes reaction rate, but introduces problems in determining
what reaction took place. In normal kinematics, stable isotope experiments, the reac-
tion rate is not a dominant concern, and thin targets may be used. Often, a complete
kinematic description of each reaction event could be determined by detecting both
the beam and target reaction products. In other cases, 7—ray detection was used to
detect decay transitions from excited nuclei stopped in the target. When the stopping
time is much shorter than the excited state lifetime, the 7 rays are emitted from rest
and are detected with an energy resolution determined solely by the detector.

In experiments using projectile fragmentation beams on thick targets, target thick-
ness precludes a kinematic reconstruction of the scattering event because the scattered

target either stops inside the target or its energy is poorly defined due to target strag-

gling. Since 7 rays may exit the target without energy loss, they are used to probe

42

which state was excited in a given scattering event. The 7—ray energy observed in the
lab is related to the energy of the emitted 7 ray in the projectile frame by the Doppler
shift (see Eq. 2.8). However, uncertainties in the velocity of the nucleus emitting the
7 ray and the angle of emission cause significant uncertainty in the deduced 7-ray

energy.

Eprojectz'le Z Elab X '7 i1 _ fiCOS (6)] (2-8)

The external contact of the HPGe diodes are segmented into eight slices and four
quadrants in the SeGA detectors. By detecting the net charge deposited in each of
these 32 segments, one may estimate the location of the first interaction of a 7-ray
scattering inside the detector. Currently, the favored algorithm is Hit-On—Max which
uses the geometric center of the segment with the largest energy deposited as the

location of the first interaction point. Knowing this first interaction point and the

l I

1.10-

7 I

;:::‘ 111,1 I Ii

 

1.15

Efficiency Ratio: Experiment/GEANT

 

 

 

0.90 —
0.85 -
0-80 I I I I I I
0 500 1000 1500 2000 2500 3000 3500

y—ray Energy (keV)

Figure 2.11: The ratio of measured SeGA full-energy efficiency to GEANT simulated
full-energy efficiency is plotted as a function of energy.

43

location of 7 ray emission (typically taken as the center of the reaction target), one
can determine the angle between the 7-ray emission vector and the beam particles
velocity vector. This gives the cos(f9) term in the Doppler correction formula, Eq. 2.8.
The beam velocity may be determined for a given isotope from its position in the
S800 focal plane and the Bp setting of the spectrograph.

The detectors are typically arranged at a fixed distance from a reaction target in
rings at specific angles with respect to the beam axis. In the present work, a ring of
seven detectors at 37° and another ring with nine detectors at 90° were placed with
the centers of the crystals approximately 24 cm from the center of the LH2 target.
Only 16 of the 18 detectors could be used due to space constraints. To minimize the
angular uncertainty, the segmentation axis is oriented to be perpendicular to a line
connecting the target and detector centers and is coplanar with the beam axis. Figure

2.9 illustrates the relative SeGA detector and target positions.

2.4 Systematic corrections and uncertainties

This section describes corrections and systematic uncertainties applied to cross sec—
tions computed in the next chapter. Uncertainties listed here are added in quadrature
to the statistical uncertainties obtained from fitting and feeding corrections. Table
2.4 presents the corrections and uncertainties which were applied to the 2f excita-
tion cross section of 36Si. Correction values were defined as the estimated change in
7-ray yield with respect to the baseline assumptions for a fixed cross section. The
product of all corrections gives the total correction. Because the cross sections are
calculated based on measured numbers 7-ray events, the cross section is multiplied
by the reciprocal of this correction. The determination of these correction values and
uncertainties are explained in the remainder of this section.

Target thickness was discussed in section 2.3.2. This correction is the ratio of

the average target thickness — integrated over some beam spot — to the maximum

44

Table 2.4: Systematic correction and uncertainty table for 3°81

 

 

 

 

Correction Uncertainty Description
0.931 0.022 target thickness
1.011 0.030 efficiency ratio: experiment / GEANT
0.993 0.001 7-ray attenuation in LH2
1.040 0.020 7-ray angular distribution
0.990 0.005 momentum loss cut
0.962 0.042 total
1.039 0.046 total~1 2 cross section correction

thickness. Measured and GEANT calculated 7-ray efficiencies were discussed in section
2.3.4 and plotted as a function of 7-ray energy in figure 2.10. All 7-ray transitions
observed in this thesis have energies above 600 keV. Therefore, a constant scaling
between experiment and simulation has been used. Attenuation of 7 rays due to
interactions with the liquid hydrogen in the target are computed using standard
photon attenuation coefficients [78] instead of being included in the GEANT simulation;
this correction depends on 7-ray energy.

Figure 2.12 shows the 7-ray angular distribution in the projectile frame, which was
calculated from statistical tensors output by ECIS [39,79]. The quantity W(0) - sini9
is shown due to the implementation of 7—ray angular distributions within the SeGA
GEANT code [80]. Because 7 rays were detected over a large range of reaction scattering
angles, the integrated effect is small. SeGA detects 7 rays at 37° and 90° in the lab
frame — giving emission angles near 45° and 135° in the projectile frame —— further
minimizing the effect of any angular distribution.

Although proton scattering events are not cut by the angular acceptance of the
S800 focal plane, the momentum loss gate which separates the inelastic and —In
reaction channels introduces a cut on center-of—mass scattering angle, see section
3.1.1 for gating details. Figure 2.13, panel (a) shows the differential cross section as
a function of center-of—mass scattering angle for inelastic excitation of the 2] state
of 408 by 100 MeV protons calculated by ECIS [39]. Panel (b) shows the same cross

section as but differential with respect to scattering angle, not solid angle. In the

45

lab frame, the scattering event involves a momentum transfer from the beam particle
to the target proton. The relationship between longitudinal momentum transfer ——
i.e. along the beam axis —— and center-of-mass scattering angle was calculated using
relativistic kinematics and is shown in figure 2.13, panel (c). A cut on longitudinal
momentum transfer is then equivalent to a cut on center-of-mass scattering angle
and requires an acceptance correction. Because each cocktail beam had a different
intrinsic momentum width, the gating varied a small amount between cocktails to
keep 1n contamination to a minimum. Figure 2.13, panel ((1) shows the fraction of
the total inelastic excitation cross-section lost due to the gates used in datasets 1 and

2. This calculation was used for all isotopes.

46

 

8000 r
6000 _

4000 ~ ,
_ — Isotropic

2000 — W(0./) calculated from
~ ECIS outputs

Counts

 

 

 

20 40 60 5 100 120 140 160 180

 

 

 

 

 

 

 

 

0
y-ray emission angle (OCM)

700 :_ b 600 *— .-..:-“'n- C

600 T 500 i "
‘3 50° 5 400 —
g 400 _— ~
U 300 — 30° 5

200 l 200 f

100 ' 100 j

0 40 60 80 0100 120 140

y-ray detection angle (GCM)

Figure 2.12: Gamma—ray angular distribution from (p, p’ ) (dashed line) is compared
with an isotropic angular distribution (solid line) in the projectile frame (a). The
quantity W(0) - sinf) is shown due to the SeGA GEANT code implementation. Panels
(b) and (0) show the projectile frame angular distribution of 7—rays detected in the
37° and 90° rings of SeGA.

47

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100_ a 4_ ]] b
1: "’4‘ ”'6
a s 3‘
g 10‘2- B -
v E
_3_ V 2—
% 10 c;
\ 'O ..
b .4_ 5
'o 10 1,
10'5-
I I I
0 45 90 135 180
9CM
5_1 c #3
Ci
1; I I“
.9 4— E
E ‘ 8
3 3— To
C ‘5
Q) '1 '—
E 2_ o
o 1 I15
2 . c
a? 1.... .9
4.0
_ U
9
LL
0 I I I I I I I
0 45 90 135 180 0.0 0.5 1.0 1.5 2.0 2.5
BCM % Momentum lost

Figure 2.13: Calculated differential (p, p’) cross sections for excitation of the 2‘] state
in 408 per unit solid angle (a) and per unit scattering angle (b). The relationship
between momentum lost along the beam axis by the scattered ion in the lab frame
and scattering angle in the center-of—mass frame is shown (c). The fraction of the
total cross section not observed due to momentum gating losses is calculated as a
function of momentum loss ((1). For each dataset, the momentum loss gate position
and effect on cross section are shown with error bars.

48

Chapter

Analysis

3

3.1 Particle and reaction channel identification

Particle and reaction channel identification were crucial to the extraction of inelastic

excitation cross sections (op 1,1) in this experiment. While such gating is always impor-

tant in intermediate-energy, thick-target experiments, a quick example makes clear

the necessity of proper gating in this experiment. Table 3.1 lists the four incoming

isotopes giving the highest absolute yields for the 2] —> 0'] transition in 38Si. These

isotopes were measured with a single incoming cocktail beam. The channel of interest

—- 13(388i,38 Si + 7)p' — gives the 3rd largest 7—ray yield. Thus, clean identification

of incoming particle and reaction channel are required.

Table 3.1: Incoming particles and 2] —> 0] 7-ray yields for reactions populating 38Si.

 

 

 

 

 

Incoming isotope Number of beam particles (x106) Area of 1077 keV 388i peak
41p 77 500
39s; 7 490
3881 23 400
401:) 58 I40

 

49

 

3. 1 . 1 Event selection

The final set of gates was the result of an iterative process; only the final gating
solution is discussed here. To extract a cross section without physical bias, one must
ensure that neither the gates applied nor the experimental setup itself introduces any
physical bias. To that end, gating for the two datasets was developed in parallel.
When applicable, gates were constucted with uniform geometries and positioned in
a consistent way relative to peak centroids in the relevant parameter space. 80, all
incoming isotopes were treated consistently, and trends and parameter correlations
were treated simultaneously for all gates. All particle and reaction channel gating
were performed using the code SpecTcl [81].

Specifically, the inelastic channel gating requires that the unreacted beam and the
inelastically scattered beam be treated equally. Equal treatment must be confirmed
because a change in longitudinal momentum occurs in the scattering process, see
section 2.4. Detection efficiencies cancel in the cross section calculation as long as
their is no preference with respect to momentum.

The inelastic / -—1n gates for a given isotope is the AND of the following gates:
0 Valid particle

— TPPAC —— Intermediate image center
— PlN-IC — both detectors have reasonable energy loss

— CRDCl single hit — exactly one hit recorded in multi-hit TDC

PIN-IC — select the change initial & final proton number

PIN—TOF — select incoming isotope

IC-TOF —— select outgoing isotopes

Momentum Loss—TOF — select the final mass

50

 

I104 I I I I I I I I I I I I

 

 

 

 

 

 

b .
3 41;
1:. . zou,=16 _
Q) __ _
.Q
E
(U
.C
U ..
c ._ _.
2 - _
Lu _ _ -
< _-0p -2p Zin"16 -

O I I I l I I I l I I I I I

100 200 300 400
AE Si-PIN (a.u.)

Figure 3.1: Energy loss after the target AEionchamber versus AESF p I N is plotted
for the second dataset. Gates select charge (Z) of ions before and after reactions on
the LH2 target.

Valid particle selects events in which all required particle detectors had reasonable
values. Figure 3.1 shows energy loss in a detector after the target (A — E ion chamber)
plotted versus energy loss in a detector before the target (A — E Si-PIN). Here the
incoming and outgoing projectile charges are selected.

Choosing events with the same number of protons (Z) before and after the target
selects the —0pa:n channels, i.e. the (p, p'), —1n, and —2n channels. Incoming isotopic
selection was made in the A — E Si-PIN versus time of flight (TOF) spectra for each
dataset, shown in figure 3.2. Here each horizontal line is one isotopic chain with mass
increasing from right to left. The outgoing —0pxn channel was again selected in the
A - E ion chamber versus time of flight (TOF) spectra. Figure 3.2 shows incoming
isotopes are especially well—separated in the dataset containing 408. This is because
Bp selection of a projectile fragmentation beam causes the TOF parameter to vary as
.M / Q. For the fully—stripped ions studied here, this is simply A/Z. The first dataset
selects isotopes near 408, thus A / Z ~ 2.5. This give optimal PID separation by placing
the TOF centroids of the Z :t 1 nuclei exactly between the TOF centroids of a given
isotopic chain.

The separation is less than optimal in the second dataset, figure 3.2 panel (d). The

51

 

 

IIIIIIT‘II’II’II

ITIIIIIIIIIII

 

 

    

a b .
— 40 r ‘2 - - 44 ,1 " ’” -
; 5?. ..i ; ; S i. ‘ 2.3425;
- 36 - : : 40 1 _
_ SI 111' - _ SI (‘5 (“1.3339 _
‘I If 4hl‘iflitflililif
C q To d I I I Z
400 — ,. -- _ 400
3: : : . :
m I ._ .'\ l .—
Z 300 _ ..31 .7. _ i 300
n- _ .'. "u — I... '1:
w 200 — 3.1:- —, 5 f — 200
Q l' I; __1 .. “1"- 1, " -

It" . i ‘--':I .1

1OOIIIIILLlllllIlI OIIIIIIIIJIIALIIA1OO
I 00 200 300 400 1 00 200 300 400

<—————

Time of flight (a.u.)

 

 

 

 

 

 

 

Figure 3.2: Incoming particle identification (PID) plots for both incoming cocktail
beams. Panels (a) and (b) show the locations of isotopes studied in this thesis for the
first and second cocktail beams, respectively. Panels (c) and ((1) show the data for
the respective settings with the isotope locations as marked in panels (a) and (b). A
logarithmic color scale is used.

52

worst observed separation occurs for diagonal neighbors in the A — E ion chamber
versus time of flight (TOF) spectrum for this second dataset. Figure 3.3 presents a
magnified View of this spectrum. Possible contamination was investigated by selecting
a diagonal grouping of isotopes, including 448 and 38Si. Panels (b) and (c) of figure
3.3 Show the selected region and a projection onto a diagonal line running parallel
to the group. The gaussian estimate for particle misidentification is N 9 x 10‘4 of
the given peak area. Contamination of particle identification gates, therefore, gives a
negligble contribution to the cross-section uncertainty.

Finally, the (p,p'), —1n, and —2n channels were separated on the basis of mo-
mentum lost while passing through the target. The momentum loss parameter is
described in section 2.3.3 and is sensitive to differences in the logitudinal momentum
change. Figure 3.4 shows the momentum loss versus time-of-flight spectrum gated for
—0p:rn reactions arising from incoming 428. A residual correlation in the parameters
is present because all TOF corrections were based on detectors before the target. Pro-
ton scattering, —1n, and —2n channels were initially separated by diagonal cuts in
such 2-D spectra. These cuts were made individually as the momentum loss parameter
was not calibrated for incoming mass and charge. To remove the —ln contamination
from the (p, p’) 7—ray spectra and to standardize the momentum loss cut, a time-of-
flight, incoming-isotope corrected momentum loss parameter was created. Panel (b)
of figure 3.4 shows a corrected momentum loss spectrum for the final inelastic gat—
ing compared with the —1n events which had contaminated the inelastic spectrum
and the —1n—gated events. Figure 3.5 (a) shows the corrected momentum loss versus
the Doppler-corrected 7-ray energy of the original inelastic gating. Above momentum
loss channel 320, the spectrum changes. Figure 3.5 (b) presents the original, final,

and contaminant-gated Doppler—corrected 7-ray spectra.

53

AE Ion chamber(a.u.)

 

 

 

 

 

 

 

 

 

Time of flight (a.u.)

105 C I I I :

g ‘03 E 3851 445 E
g E E
U 102 E— 1:
101 E a;

1 : l l I :

100 150 200 250

Projection axis(a.u.)

Figure 3.3: PID overlap was checked in the AEIonchamber vs. time of flight spectrum
for the second beam setting. A diagonal selection of isotopes was made and projected
on to the axis shown in panel (b). Overlap of neighboring isotopes in panel (c) is
below 10*3 of total number of counts for a given isotope, which is negligible in this
analysis.

54

 

 

 

 

 

 

 

 

 

I I I 40 250 ' | I I I
450 a / _— 3 b — Removed
/ -1n : -:_‘ ' - Eiznalgaffid (p.p’)‘
I. . =2 iii:— _ — S(p,X) Sgated
400 finelastic 1'..- -_ 32 200
i} 350 - _ _ ' 24 1.1 150 -
'c 1 ‘ - ' E
1% 300 F i 8 —
H— — __ =' -.- U
8 _ 4:. 16 100 —
§ - _
l- 250 — - '
1‘ .‘H.
_ '. 8 50 -
200 ~ :13: = r
L . I _ .' _' 0 o . -
340 360 380 200 300 400 500
Momentum loss(a.u.) Corrected momentum loss(a.u.)

Figure 3.4: Initial momentum loss gating is shown in panel (a) for the 42S -0p:cn
channel. After correcting for time of flight, incoming mass, and incoming charge,
panel (b) shows the corrected momentum loss spectrum.

55

 

filTrI—I_IIJ—I I I'IT—TTE—Ill_l I I I I I I [T
_ —— :

    
        

 

 

 

 

’5

2 1

‘_6 350 1L

E i

a I

5 300

E

O

E

'0 250

w ‘ —-

U — = _ r _

cu I —- _
52001L"|""IllIIIIIJ—LIIIIII1111I1
U 0 250 500 750 1000 1250 1500

y—ray energy (keV)

 

— Original gate
—- Corrected gate 1
— Contaminant‘l‘S

Counts

 

 

‘1-':E‘ul..1. '11. .
1250 1500
y-ray energy (keV)
Figure 3.5: Corrected momentum loss is plotted versus Doppler corrected 7-ray energy
for the original inelastic gating shown in figure 3.4 (a). Above momentum loss channel
320, the spectrum clearly changes. The original and final gated 7—ray spectra are
superimposed in panel (b). Also shown is the spectrum corresponding to —In channel

7 rays.

56

3.2 Analysis of 7-ray spectra

After events were selected according to incoming beam particle and reaction channel,
gated 7-ray spectra were produced for the nuclei of interest. The following sections
will focus on the analysis of the 7-ray spectra of 36,38,403; and 401421443 measured in
coincidence with (in)elastic scattering in the LH2 target. The gating and analysis for
each isotope was handled in a consistent fashion and is outlined here.

The incoming nuclei and (in)elastic scattering were selected by taking an AND of
the gates described in section 3.1. Spectra are Doppler corrected on an event-by-event
basis according to Eq. 2.8. The beam velocity, 6, is determined for a nucleus from the
focal plane dispersive position and the magnetic rigidity of the S800 spectrograph.
The 7-ray emission angle is calculated from measured segment positions and a fixed
target center.

To reduce random coincidences for the inelastic-scattering 7—ray spectra, a timing
cut was added to select only 7—rays in a prompt time window of 60 us. The timing
parameter has been corrected for the time of flight between the target and the S800
focal plane. Due to an energy dependence in the triggering time of the SeGA detec-
tors, 7-rays depositing S, 600keV may be prompt, but fall outside this timing cut.
While a linearization of the time response could be performed, a strong, low-energy 7
background in the inelastic channel would still affect this energy region. Spectra were
investigated with and without this cut, and no low-energy peaks were observed for the
inelastic-gated spectra of the nuclei currently under study. A study of efliciency loss
due to the timing cut will be necessary to extract inelastic excitation cross sections
from the neighboring odd-mass and odd-odd nuclei.

The remaining background in each inelastic-gated spectrum was due primarily
to beam interactions with the aluminum windows of the LH2 target cell. This back-
ground was measured by raising the target cell temperature above the boiling point

of hydrogen. The magnetic rigidity of the S800 spectrograph was then increased to

57

compensate for the reduced energy loss of ions traversing the cell. This data was
then analyzed to extract the 7-ray spectrum in coincidence with events which would
satisfy an (in)elastic scattering gate. The total number beam particles contributing
to this spectrum was computed from the number of down-scaled events satisfying the
gate multiplied by the downscale factor. In the following sections, this collection of
7—ray events has been Doppler corrected using the flp’rojectile appropriate for each
nucleus and sealed in proportion to the number of incoming beam particles to form
the beam-related background for each inelastic spectrum.

The sum of all individual SeGA detector spectra is used for fitting. Detectors
were also summed by rings to check full-energy peaks for energy shifts relating to
the lifetime of a particular state. Other than the previously measured lifetimes of
the 2? —> 0:“ transitions, no lifetime shift large enough to affect the analysis was
observed. Spectra have been fit using response functions generated by a GRANT [77]
simulation of SeGA added on top of this scaled background spectrum. The scale and
position of each response were fit to find the number of 7-rays emitted and the energy
centroid of the peak. Fits were performed in PAW [82] using the negative log-likelihood
option of Minuit [83].

The SeGA GRANT simulation was written by Dr. W. F. Mueller to address the
variation of Doppler-corrected peak shape with experimental conditions [80]. Exper-
imental uncertainties including the finite segment size of SeGA, projectile scattering
angle, energy loss through the target, physical thickness of the target, and lifetime
of a decaying state may change the width — and sometimes position —— of the re-
constructed 7-ray peak. The simulation accepts information on the target, beam, and
emitted 7-ray, then applies the appropriate Lorentz transformation on the 7—ray emit-
ted in the projectile frame to find its lab-frame properties, and simulates the 7-ray
interactions with SeGA. Then, the 7—ray is reconstructed as in the experiment, using
an average velocity and the angle of the center of the segment receiving the most

energy.

58

Cross sections for 7-ray production were calculated according to the definition of

cross section.

Nreactions
= 3.1
N beam ° (N/A)target ( )

The terms Nreactz’ons and Nbeam denote the total numbers of 7 rays emitted due

 

0’

to reactions and incoming beam particles, respectively. (N /A)target is the number of
target nuclei per unit area, also called the areal number density of the target. Note
that this is not a level population cross section. The level population cross section
is given by adding the cross sections for 7-rays which decay from a given level and

subtracting the cross sections of those 7-rays which decay to that same level.

3.2.1 3681

Prior to this experiment, the only published 7-ray information on 36Si was the
intermediate-energy Coulomb excitation work of Ibbotson et al. [84]. They observed a
single peak at 1399(25) keV and labeled it as the 2? —+ 0:5 transition. Figures 3.6 - 3.8
show three views of the 7-ray spectrum observed in coincidence with p(3°Si,36 8i+7)p’ .
The dominant peak in the full spectum is clearly near 1400 keV, but the spectrum is
significantly above background at high energies.

Because 7—ray responses feature a peak with some low-energy continuum, fitting
often proceeds from high energies to low energies. Often the full-energy peak (also
known as the photopeak) will be fit; then its intensity will be fixed and used to fix its
contribution to lower-energy background due to any escape peaks and the Compton
continuum. However, these Doppler-corrected peaks are broadened compared to tra-
ditional forward—kinematics, or lab-frame, peaks. Thus for Doppler-corrected spectra,
the relative importance of the full-energy peak is diminished, and the Compton edge
and escape peaks may significantly decrease the uncertainty in a fit.

Fitting of the 3681 spectrum was split into two energy regions, 1700-5000 keV

59

 

 
 

 

Counts / (32 keV)

-. I .
II I.
'II -l‘]‘1".1']
. ‘ . . . .  _ -. 11111111111111.1110
0 1 000 2000 3000 4000 5000
y—ray Energy (keV)

Figure 3.6: Full 7-ray spectrum of 36Si. The background (grey-filled) and fit (blue)
are superimposed on the full spectrum (black).
and 1000-2100 keV, shown in Figs. 3.7 and 3.8, respectively. First, the high-energy
fit was performed stepwise from high to low energy. Each peak was added, fit in a
narrow region about the full-energy peak, and fixed. Once all peaks were fit, the fit
was performed once again with all peak parameters allowed to vary. Figure 3.7 shows
the resulting peak fit. The solid line histograms are the measured and fitted spectra;
grey fills the top of the fit spectrum histogram. Filled histograms of various colors
Show the fitted response functions. These are stacked, to Show their sum, with the
window background as grey as the top member of the stack. The window background
is jagged compared to the response functions and was placed at the top to preserve
the visual clarity of this figure. Smoothing was not performed on the background
because several distinct feature arise from lab—frame 7 rays being Doppler corrected
to the projectile frame.

The step-wise fitting procedure was then repeated in the low-energy region with
the high-energy peaks fixed. The high-energy responses contributed a nearly flat con-
tinuum component in this region. The peak near 1900 keV was allowed to vary to

take advantage of its Compton continuum. Figure 3.8 shows the resulting peak fit.

60

 

 

Counts / (32 keV)

 

 

2000 2500 3000 3500 4000 6600
y—ray Energy (keV)

Figure 3.7: Spectrum of 7-rays in coincidence with 36Si(p,p’) shown from 1700-
5000 keV. Response functions for fitted 7—ray transitions (colors) are stacked with
the scaled background (grey) at the top. Data is shown as an unfilled histogram

(black line).

Again, the unfilled solid histogram is the measured spectrum, while a filled solid line
histogram is the fitted spectrum. Fitted responses are stacked, filled histograms of
various colors. At the base of the stack, in black, is the contribution of all responses
coming from 7 rays with energies above 2000 keV. The window background is at the
top of the stack in grey.

In this second fit, a doublet is shown between 1400 and 1450 keV. Hints of this
doublet stucture were observed in this experiment —— larger than expected peak width,
a shoulder visible in one ring spectrum —— but no firm conclusions could be drawn.
Even if a doublet was found, fitting the positions and intensities would be impossible
with the resolution of this Doppler-corrected spectrum. Fortunately, this doublet was
observed in a deep inelastic collision experiment [85] published during this analysis.
That experiment observed transitions at 1408(1), 1442(3), and 842(1) keV, which they
identify as the 2? —> 0+, 4? —+ 2+, and 6? -—> 4'] transitions respectively. An energy
spacing of 34 keV was used to fix the relative positions of the doublet members, and

the positions and intensities of the doublet were sucessfully fit.

61

 

60

      

 

 

9 50 l

8 -

\ 30 ~

9 _

g 20 —

C

o 10

-__ ' - ..._- 1' 11.5.1.1. -.
9000 1200 1400 1600 1800 2000

y—ray Energy (keV)

Figure 3.8: Spectrum of 7-rays in coincidence with 36Si(p,p') shown from 1000-
2100 keV. Response functions for fitted 7-ray transitions (colors) are stacked with
the scaled background (grey) at the top. The sum of all responses due to higher en-
ergy 7 rays is shown as a black filled histogram at the base of the stack. Data is shown
as an unfilled histogram (black line).

Table 3.2: Energies and cross sections for 7 rays produced in p(3°Si,3° 8i + 7)p'.

 

 

 

 

 

 

7—ray energy (keV) unc. Cross section (mb) unc.
1179 3 2.1 0.6
1412.0 1.6 18.0 1.0
1446.0 1.6 3.8 0.8
1501 5 2.1 0.4
1916 4 3.4 0.4
2472 5 2.3 0.4
2613 15 1.1 0.4
2928 16 1.5 0.5
3299 18 1.2 0.3
3925 17 1.0 0.2

 

 

Cross sections to produce each 7 ray were calculated using Eq. 3.1, and energy
uncertainties were calculated by adding the position uncertainty from the fit to the
calibration uncertainty in quadrature. Results are shown in table 3.2. For the 1412
and 1446 keV peaks, two additional uncertainties were also added in quadrature:
uncertainty due to lifetime of the 2] state (0.5 keV) and uncertainty in the doublet

energy spacing (0.7 keV).

62

 

 

 

 

(3') 4.028(4)
(22:: 2.910(4)
(4*) 2.850(3)

 

 

 

2+ 1 #1 1.408(1)

 

 

 

0" J 6 0
Experiment

3651

Figure 3.9: Level scheme of 36Si deduced in this experiment is compared with low-

0+

2+

4+

2+

4.045

3.623

3.100

1 .826

0

SDPF—NR

lying levels calculated in the shell model. Energies are given in MeV.

These 7 rays were then used to build a level scheme based on the energy sums
and the known 41' —+ 2? —> 0? cascade. Because the doublet was well separated in

the deep inelastic experiment, those energies will be used for the 2'1," and 41+ levels.

The following energy sums were found.

1408(1) + 1502(5) = 2909(5) e: 2928(16)

1442(3) + 1179(3) = 2621(4) a 2613(15)

The resulting level scheme is shown in figure 3.9. All level schemes in this thesis were

drawn using the LevelScheme package [86].

63

Table 3.3: Energies and cross sections for states populated in p(3°Si,36 8i + 7)p'.

 

 

Level energy (keV) unc. Cross section (mb) unc.
1408 I 7.4 2.4
2850 3 1.8 1.0
2910 4 4.0 0.7
4028 4 3.6 0.7

 

 

 

 

 

 

Feeding corrections were made according to this level scheme. Unplaced 7 rays
could decay to these levels causing further reductions to the population cross sections.
Decays to the ground state and first excited state are the most likely, and decays to
higher-lying states will not affect the population cross section of the 2?" state. Thus,
the effect of the unplaced 7 rays will be considered only for the 2? state. Feeding from
unplaced 7 rays is estimated as 50(25)% of the summed cross section of all unplaced
transitions. Corrections as discussed in section 2.4 were applied. Table 3.3 gives the
level energies, population cross-sections, and associated uncertainties.

Tentative J7r assignments are given in figure 3.9. The first two spin-parity assign-
ments are taken from the previously mentioned Coulomb excitation and deep inelastic
experiments [84,85]. The tentative 3" assignment is based on the decay to both 4+
and 2+ states and the observation that (p, p’) typically populates 2+ and 3“ states
in even-even nuclei. The tentative 2+ assignment of the 2910 keV level is based on
the decay to both 2Jr and 0+ states and the typical population of 2+ and 3" states

in even—even nuclei by inelastic proton scattering.

3.2.2 38Si

Published spectroscopy of 38Si prior to this experiment was confined to the Coulomb
excitation measurement of Ibbotson et al. [84]. The reported 2:” energy was 1084(20) keV.
Figures 3.10 and 3.11 Show the full 7—ray spectrum for the reaction p(388i,38 Si +7)p’
and the region from 800 to 1400 keV in detail. The 2'] —> 0;“ transition is dominant

with two smaller peaks near 1160 and 1300 keV. Around 2000 keV, the spectrum is

64

 

 

Counts / (8 keV)

 

ILIIIII 1 l
[TIMI I"l'l' 'I' II'
0 1 000 2000 3000 4000 5000

y—ray Energy (keV)

Figure 3.10: Full 7—ray spectrum of 38Si. The background (grey-filled) and fit (blue)
are superimposed on the full spectrum (black).

clearly above background, but no peak is visible.

As before, the fitting proceeded from high to low energy. In this case, the statistics
were too poor for a definative peak fit. Indeed, it is unclear how many peaks may be
contributing. The spectrum returns to the level of background near 2500 keV. The
intensity of a hypothetical 2400 keV peak was fit to provide some measure of the
intensity which may feed other peaks. This single peak fit gave a good match to the
data, but shifts in the starting position or two—peak fits would give similar results.
The Compton continuum from such a low-intensity, high-energy peak is negligible in
the region of the other peaks.

The region from 800 to 1400 keV was fit using three response functions and the
window background. Figure 3.11 shows the resulting fit. Again, the unfilled, solid line
histogram is the measured data; the grey—filled, solid line histogram is the sum of
all responses and background. The 7-ray response functions are stacked, in various
colors, with the window background at the top of the stack in grey. Table 3.4 gives
the cross sections for production of each 7 ray.

Unlike the above 36Si inelastic analysis, there were neither energy sums nor re-

65

 

    

 

 

$120 —
g 100 F
93 80 j
.72 50 f
S 40 —
O

0 20

0800 900 1000 1100 1200 1300 1400

y-ray Energy (keV)

Figure 3.11: Spectrum of 7-rays in coincidence with 38Si(p, p’) shown from 800-
1400 keV. Response functions for fitted 7—ray transitions (colors) are stacked with
the scaled background (grey) at the top. Data is shown as an unfilled histogram
(black line).

cently published level schemes to enable feeding correction. Systematics of even-even
nuclei would reject either the 1167 or 1303 keV 7 rays as decays to the ground state.
Whether these transitions should be parallel or form a cascade was unclear. However,
data from other reaction channels leading to excited states in 38Si is present in this
data set. As explained in section 2.3.3, the large momentum acceptance of the 8800
spectrograph allowed data on reaction residues from neutron removal reactions to be
collected simultaneously with the inelastic reaction. 80, the 7-ray spectrum recorded
in coincidence with p(398i,38 Si + 7)X was examined.

Figure 3.13 shows the 7-ray spectrum from 800 to 1400 keV for neutron removal

reactions populating bound states of 38Si. There are no definitive peak structures

Table 3.4: Energies and cross sections for 7 rays produced in p(3881,38 Si + 7)p'.

 

 

 

7-ray energy (keV) unc. Cross section (mb) unc.
1077 2 18.5 0.5
1167 3 2.5 0.3
1303 3 2.1 0.3
(2400) - 2.7 0.6

 

 

 

 

 

66

15

1O

 

600 750 1000 1250 1500 1750 2000 2250 2500

Figure 3.12: 38Si 7 — 7 spectra taken from a sum of several reaction channels. In
each panel, the energy gates listed were used to create the coincident (black) and
background (grey) spectra. Transitions in coincidence with the each gate are labeled

 

 

II

(a) Gate: 1050-1090 keV
Bkg: 980-1020 keV

11.1-1111... .. 1.1. .. .

I I

IIIIIEI

 

 

(b) Gate: 1135-1175 keV
Bkg: 1200-1240 keV

rIIITTIII

 

II

(0) Gate: 1280-1300 keV
Bkg: 1200-1240 keV

IIIIIIIITIIII

 

 

 

y-ray Energy (keV)

by arrows.

other than the three visible in this figure. However, the relative intensities of the
1167 and 1303 keV transitions precluded the cascade 1167 —> 1303 —+ 1070. Also, the
strong population of the 1167 —> 1070 cascade allowed the first observation of 7 — 7
coincidences in this nucleus. The statistics were meagre. To check for other coincidence
relations, all reaction channels populating bound excited states in 3881 were added to

form a single 7 — 7 matrix. Figure 3.12 shows 7-ray spectra resulting from gates on

67

the three full-energy peaks. A backgound spectrum gated on a nearby energy region
is superimposed as a grey histogram in each case. From this, the following 7 — 7

coincidences were obtained:

1077 — 1167
1077 — 1303
1167 — 1303

Thus, the cascade 1303 —> 1167 —> 1070 is affirmed, and two new levels have been
observed in 38Si.

Before quoting final level energies, the neutron removal spectrum was fit with three
7-ray responses and a linear background. The goal was to verify the peak eneries, and
possibly improve the energy determination for the two new transitions. Detailed fitting
of the 19(398i,38 Si + 7)X spectrum showed that although the peaks closely resemble
those of the inelastic spectrum, significant shifts in peak position have occurred. Al-
though the total uncertainty in peak energy is at the 2 keV level, the uncertainties
due to level lifetime, Doppler correction parameters, and energy calibration are com-
mon for peaks in these two spectra. Thus, only the fitting uncertainty is used when
comparing them. The 1077 keV peak is shifted by -3.0(6) keV in the neutron removal
spectrum, and the 1167 keV peak is shifted by -10.5(22) keV. These are 50 shifts,
which must be explained. Large error bars on the 1303 keV peak preclude a definitive
statement on the peak shift.

Though the addition of a peak would solve one of these shifts, it is unlikely that
both peaks would have doublets and that they would both be at lower energies than
the peaks observed in the inelastic spectrum. An error in the Doppler correction is also
unlikely. Such an error would cause a common fractional displacement in the energies,
i.e. the observed shifts would be directly proportional to the observed peak energy.
After normalizing the peak shifts listed above by their respective peak positions, the
fractional shifts are different by > 30. The remaining cause of peak shifts is the effect

of lifetime.

68

For a lifetime to cause a 7—ray peak to shift between two Doppler corrected spectra
of nearly the same projectile velocity, the state emitting the shifted 7-ray must be fed
by one or more higher-lying states having non-negligible lifetime(s). Taking relative
intensities and peak shifts into account, feeding of the level at 2244 keV gives the
simplest explaination of the shifts of both peaks.

The p(398i,38 Si + 7)X spectrum was fit under the assumption that the level at
2244 keV was fed by state(s) with lifetime(s) such that the resulting 1077 and 1167
keV peaks were shifted in the Doppler-corrected spectrum. Direct population of the
2244 keV level and population through fast decays was also included. The 1077 and
1167 keV response shifts were fixed at the values given by the inelastic fit; the 1303
keV response was free to shift. A new response formed by summing the 1077 and 1167
keV responses was free to shift; this represents the cascade decay of the 2244 keV level
when fed by a delayed 7-ray transition. All response scaling was free to vary in the
fit. Figure 3.13 shows the spectrum, fit, and responses as before. The color—coding of
peaks is common between Figs. 3.11 and 3.13. The additional delayed cascade decay
of the 2244 keV level is shown in cyan as a two—peak response. Uncertainties in the
7-ray energies from the inelastic fit were accounted for by repeating the fit with using
the 21:10 limits for each fixed peak.

The resulting energy shift of -14(2) keV for the 1167 keV shifted component trans-
lates to a feeder lifetime of 63(10) ps. Though this shifted component has a large
fraction of the excited state population, no feeding 7 ray was observed in the 7 —- 7
coincidences, nor was any transition other than the three previously mentioned ob-
served in the singles spectrum. There is, in fact, no reason to conclude that the feeding
of the 2244 keV level proceeds through a single transition. Because the energy shifting
of peaks is nearly linear with lifetime, this feeder lifetime may be thought of as the
average of a collection of feeding 7-ray transitions. Furthermore, 7 — 7 coincidences
resulting from Compton scattering off one SeGA detector into another is highly prob-

able. Thus, the region ,S 1000 keV has a physical background that makes it difficult

69

 

  
         

 

 

0300 900 10.00 1100 1200 13.00 14.00
y—ray Energy (keV)

Figure 3.13: Spectrum of 7-rays in coincidence with 3981(1), X )388i shown from 800—
1400 keV. Energy axis and colors correspond to figure 3.11. An added response (cyan)
corresponds to a delayed 7 cascade from the 2244 keV level.

Table 3.5: Energies and cross sections for states populated in p(3BSi,38 Si + 7)p’.

 

 

Level energy (keV) unc. Cross section (mb) unc.
1077 2 15.4 2.6
2244 4 0.4 0.4
3545 6 2.3 0.4
Unplaced estimate — 2.7 0.6

 

 

 

 

 

 

to locate new coincidences.

Though the source of the delayed feeding is unknown, it is clear that these are
the same peaks observed in the inelastic spectrum. The prompt 1167 keV transition
comes with more intensity than the 1296(4) keV transition, which is associated with
the 1303(2) keV transition found in the inelastic spectrum. (A weighted average yields
a transition energy of 1301(5) keV.) So, the cascade ordering made above remains
intact.

The deduced level scheme and feeding-corrected 10(388i,38 Si+7)p' excitation cross
sections are given in figure 3.14 and table 3.5, respectively. Coulomb excitation es-
tablished the 2f assignment for the 1077 keV level. Strong population of the level at

3545 keV suggests a 3‘ assignment. The excitation cross section of the 2244 keV level

70

(3‘) 3.545(6)

 

2+ 2.838
4+ 2.621
(4+)_! 2.244(4) 0+ 2.260
2+ 1.543

2+4 1.0772(19)

 

0* J 0 0+ 0

Experiment SDPF—NR
3831

Figure 3.14: Level scheme of 38Si deduced in this experiment is compared with low-
lying levels calculated in the shell model. Energies are given in MeV.

71

 

     

 

 
     

 

 

14 f-
9 12 f ]
50¢) 10 :-
33“ 8 r
a 6 5 I
1: -
4.1.1. lull,“ ~
0 II' 'I Ill:1;1!.l.!.I.L-].l].1..!.I.!.!.!.!.| ].i.|.!.l._ll!|- .-.I |._I _. .ll .
600 800 1000 1200 1400 1600 1800 2000
y-ray Energy (keV)

Figure 3.15: Spectrum of 7-rays in coincidence with 40Si(p,p') shown from 500—
2000 keV. The 2? —> 0;“ transition’s response function (blue) is stacked on the scaled

background (grey) and a small linear background term (green). Data is shown as an
unfilled histogram (black line).
is consistent with no direct population. Feeding from a tentative 3— and decays to

the established 21" suggest either 2+ or 4+ for the 2244 keV level. Strong population

in the neutron removal channel favors a 4+ assignment, see section 2.1.4.

3.2.3 40Si

This experiment was the first to observe 7-ray transitions in 40Si. Figure 3.15 shows
the 7-ray spectrum for the reaction p(408i,40 Si + 7)pI from 500 to 2000 keV. A single
peak at 986 keV dominates the spectrum. Superimposed on this spectrum are the
fitted peak response, the window background, and a small linear background function
added to improve the fit. These components are stacked in order linear background,
window background, and peak response, from bottom-to—top in figure 3.15.

Since the 7 ray at 990(5) keV is the only transition observed in p(4°Si,40 Si +7)p’,
it was placed as the 2? —> 0’] transition. As this is the only peak observed in the
spectrum, no feeding correction was applied, but an asymmetric uncertainty of -

20% was added in quadrature to the population cross—section uncertainty of the first

72

 

 

Counts / (8 keV)

 

1, 4.111111...)

600 300 1000 1200 14001600 1800 2000
y-ray Energy (keV)

 

 

 

 

 

 

 

 

 

 

 

 

O N «P- 05 (D O
I

Figure 3.16: Spectrum of 7-rays in coincidence with p(42P,4‘0 Si + 7)X shown from
500—2000 keV. Response functions for fitted 7-ray transitions (colors) are stacked on
a linear background (grey). Data is shown as an unfilled histogram (black line).

Table 3.6: Energies and observed cross sections for 7 rays produced in p(42P,40 Si +
7)X. Total observed reaction cross section is also shown.

 

 

 

 

7—ray energy (keV) unc. Cross section (mb) unc.
638 5 1.1 0.2

845 6 0.7 0.2

982 5 2.0 0.3

Total cross-section 3.7 1.0

 

 

 

 

 

excited state. Twenty percent is the final feeding estimate used in 3881 and is roughly
the size of a single transition which might be missed in this low-statistics spectrum.
This gives an inelastic excitation cross section, 00:42.], of 201:3 mb.

In addition to inelastic scattering, excited states in 40Si were observed in the
reaction p(42P,40 Si + 7)X. Figure 3.16 shows the 7-ray spectrum recorded in coin-
cidence with this reaction. Table 3.6 gives the observed cross sections for producing
each 7-ray transition. The total observed cross section was calculated using the to—
tal number of downscaled particles observed in this PID-reaction channel gate; this
value is also presented in table 3.6. Combining the both reaction channels gives the

2ir ——+ 0? transition an energy of 986(5) keV, where an uncertainty of 3 keV has been

73

4+ 2.92

 

 

 

 

 

0* 2.26
U
24%
........ 1.8.2.0).
----1.620). 2. ,_45
0.986(5)
0+ 0 0+ 0
Experiment SDPF—NR
4°51

Figure 3.17: Level scheme of 40Si deduced in this experiment is compared with low-
lying levels calculated in the shell model. Energies are given in MeV.

added to account for the unknown lifetime of the 2? state. Transitions at 991(10) and
624(10) keV have also been observed in the secondary fragmentation work of Grevy
et al. [87].

Though 7 — 7 coincidences and energy sums are impossible in this case, the place-
ment of at least one of the two new transitions as directly feeding the 2? level is quite
probable. In the cases of 36I38Si and 40S, the strongest observed 7—ray is from the
2'] —> 01+ transition, and the second strongest transition directly feeds the 2? level in
each case. This suggests at least one of the two new transitions directly feeds the 21"
level, and a second excited state of 40Si lies at either 1624(7) or 1838(8) keV. Figure

3.17 shows the deduced level scheme and the shell model prediction.

3.2.4 403

Prior to this experiment, the spectroscopy of 408 had been studied by Coulomb ex-

citation, fl—delayed 7 emission, and projectile fragmentation [88—90]. Figures 3.18,

74

Counts / (25 keV)

 

 

 

f I
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
y—ray Energy (keV)

Figure 3.18: Full 7—ray spectrum of 40S. The background (grey-filled) and fit (blue)
are superimposed on the full spectrum (black).

3.19, and 3.20 show full, high—energy, and low—energy portions of the 7-ray spectrum
collected in prompt coincidence with p(408,408 + 7)p’. The 21+ —> 0? transition at
906 keV clearly dominates the spectrum, but a strong peak near 1350 keV and smaller
peaks above 2 MeV indicate significant feeding. As before, solid-line, white-filled his-
tograms represent the data and fit components are stacked, color-filled histograms
superimposed on the data.

Fitting was performed first on the energy range 1400 to 5000 keV, as shown
in figure 3.19. A significant background in addition to the window background was
observed. A linear background was added and nicely accounted for these excess counts.
The linear background is shown as a black histogram on which the peak responses are
stacked in figure 3.19. After fitting the high—energy region, a fit to the region 500 to
1500 keV was performed. Additional fitting indicates ~ 1.4 mb may lie in other peaks
in this region. Figure 3.20 shows the fit with the linear background and the fixed
contributions from all higher-energy peaks shown as a narrow black-filled histogram.
Table 3.7 gives the 7—ray production cross sections for each of the fitted peaks.

The 905.9(11) keV transition is associated with the 21* —+ 0'] transition in 40S

75

 

 

1500 2000 2500 3000 3500 4000 4500 5000
y—ray Energy (keV)

Figure 3.19: Spectrum of 7—rays in coincidence with 40S(p,p’) shown from 1400-
5000 keV. Response functions for fitted 7-ray transitions (colors) are stacked with

the scaled background (grey) at the top. Data is shown as an unfilled histogram
(black line).

Table 3.7: Energies and cross sections for 7 rays produced in 17(408,4'0 S + 7)p’.

 

 

 

7—ray energy (keV) unc. Cross section (mb) unc.
905.9 1.1 27.7 0.3
1351.4 1.4 5.5 0.2
2457 5 1.7 0.2
3044 6 2.0 0.2
3957 9 1.6 0.1

 

 

 

 

 

which has been previously observed in fl—decay at an energy of 903.69(7) keV [89].
Similarly, the transition at 1351.4(14) keV is associated with the previously observed

1351.10(12) keV transition from fi—decay. This transition is known to decay to the 2:"

level.

A single energy sum was found:
3044(1) + 905.9(11) = 3952(5) 2’ 3957(9)

Taking a weighted average, the resulting level energy is 3952(5) keV. A level at

3947.0(3) keV was observed in fi—decay, but that level decayed by the emission of

76

 

700
600
500
400
300
200
1 00

  
  

Counts / (5 keV)

,a‘

1 400

   

   

   

600 800 1000 1200
y—ray Energy (keV)

 

 

Figure 3.20: Spectrum of 7—rays in coincidence with 40S(p,p’) shown from 500—
1500 keV. Response functions for fitted 7-ray transitions (colors) are stacked with
the scaled background (grey) at the top. The sum of all responses due to higher en-
ergy 7 rays is shown as a black filled histogram at the base of the stack. Data is shown
as an unfilled histogram (black line).

1692.6(9) and 3043.2(4) keV 7-rays with relative intensities of 1.1(4) and 4.4(5), re-
spectively. No 1693 keV transition was observed, so fitting was tried with a 1693
response function whose intensity relative to the 3044 response function was allowed
to vary in :tlo and i2o ranges. Both fits converged with the relative intensity pa-
rameter at the lower limit and significantly increased the X2 of the fit. Furthermore,
the B—decay experiment did not observed a transition near 3947 keV. Therefore, the
level seen here and the 3947 keV level observed in fi-decay are not the same.

The 2457(5) keV transition observed here was similarly checked for correspondence
with the 2469.8(2) keV transition observed in fi—decay [89]. If these transitions were
the same, a 2808-1013 keV cascade would be expected with an intensity relative to
the 2469.8 keV of 86(26)%. Fits at the :tla and i2o range for this relative intensity
failed to converge. Allowed to fit freely, the relative intensity of the 2808—1013 keV
cascade was 12(21)and consistent with zero intensity. Therefore, the transition seen
here and the 2469.8 keV transition observed in fl-decay are different.

Although the 408 has the highest statistics of any spectrum in this thesis, 7 — 7
77

(3‘) 3.952(5)

 

 

 

 

 

 

 

2+——3.7248

2,—3.4572

0 — .2802

4* 3.1829

(4") 2257403) 4+—2.2565

2+ 0.9059(11) 2+——0.9798
0+ 0 o+—o
Experiment SDPF—NR

408

Figure 3.21: Level scheme of 40S deduced in this experiment is compared with low-
lying levels calculated in the shell model. Energies are given in MeV.
coincidences were not helpful in placing transitions into a level scheme. Only the 906-
1351 keV coincidence was clear and unambiguous. In the remaining cases, the high
7-ray energy causes the intensity to be spread over the full-energy peak, Compton
edge, and escape peaks. Only in the lab frame are the Compton edge and escape
peaks at fixed energies relative to the full-energy peak. Doppler reconstruction over
many angles blurs the distinctions, and in the end, the response function for a 7 ray
above 2000 keV becomes a full energy peak with a mound of superimposed peaks
at lower energy. With three high-energy peaks, each having a similar low-intensity,
the 7-ray spectrum in coincidence with the 906 keV peak shows coincidences above
2000 keV, but no clear assignment is possible.

Figure 3.21 shows the level scheme as observed in p(4°S,408 + 7)pI along with

the shell model predictions for low-lying even parity states. Feeding of the 2? level

78

Table 3.8: Energies and cross sections for states populated in 13(4‘08,40 S + 7)p’.

 

 

 

Level energy (keV) unc. Cross section (mb) unc.
905.9 1.1 19.8 1.2
2257.4 1.8 6.0 0.4
3952 5 3.9 0.3

 

 

 

 

 

is taken as the summed intensity of the 1351 and 3044 keV transitions plus 50(25)%
of the unplaced, fitted 7-ray intensity. No feeding correction has been made for the
2257 and 3952 keV levels. Table 3.8 gives the observed level energies and excitation
cross sections; corrections discussed in section 2.4 have been applied to excitation
cross sections.

3.2.5 428

Prior to this experiment, the spectroscopy of 428 had been studied by Coulomb ex-
citation, and projectile fragmentation [88,90]. Figure 3.22 shows the 7-ray spectrum
collected in prompt coincidence with p(428,42 S + 7)p' from 500 to 3000 keV. In ad-
dition to the strong 2? —+ 0;” transition at 903.1(14) keV, a much weaker peak has
been fit at 1838(9) keV. Fit components are colored, stacked with grey-filled window
background at the top, and superimposed on the white-filled, solid-line histogram of
the data.

The 1838(9) keV transition observed here corresponds to the 1821(8) keV transi-
tion observed in projectile fragmentation [90]. As in the previous work, this transition
is placed as directly feeding the 2? level. Table 3.9 gives the feeding corrected exita-

tion cross sections for the two levels observed in p(4‘28,42 S + 7)p'.

Table 3.9: Energies and cross sections for states populated in p(428,42 S + 7)p’.

 

 

 

Level energy (keV) unc. Cross section (mb) unc.
903.1 1.4 16.3 1.0
2741 9 1.6 0.4

 

 

 

 

79

 

 

80
60
40

 

     

Counts / (10 keV)

 

20
0 "1'47““; .. ...I."4.1. .-.L -1-c 3. ,. ..1....
500 10 15 0 2000 2500 3000

y—ray Energy (keV)

Figure 3.22: Spectrum of 7-rays in coincidence with 428(1), p’) shown from 500-
3000 keV. Response functions for fitted 7-ray transitions (colors) are stacked with
the scaled background (grey) at the top. Data is shown as an unfilled histogram
(black line).

 

 

 

 

 

 

2+ 2.9817
(2". 4*) 2741(9) 4+ 2.952
2+_@.9031(14) 2 1'0233
0L! 0 0+ 0
Experiment SDPF-NR

423

Figure 3.23: Level scheme of 42S deduced in this experiment is compared with low-
lying levels calculated in the shell model. Energies are given in MeV.

80

 

 

 

Counts / (25 keV)

‘ . - . I 11.101191 .1. III]-
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
y-ray Energy (keV)

Figure 3.24: F1111 7-ray spectrum of 44S. The background (grey-filled) and fit (blue)
are superimposed on the full spectrum (black).

3.2.6 4:48

Prior to this experiment, the spectroscopy of 448 had been studied by Coulomb ex-
citation, projectile fragmentation, proton knockout, and isomer decay [34,88,90,91].
Figures 3.24, 3.25, and 3.26 show full, high-energy, and low-energy portions of the
7-ray spectrum collected in prompt coincidence with p(448,44 S + 7)p'. The 2? —> 0?
transition at 1328 keV is the strongest peak. Prominant full-energy peaks at both
higher and lower energies open the possibility of significant 7-ray feeding of 'the 2']
state. Solid-line, white-filled histograms represent the data and fit components are
stacked, color-filled histograms superimposed on the data.

Fitting was done in two steps. First, the energy range 1500-5000 keV was fit, see
figure 3.25, adding peak responses from high to low energy. Then, the energy range
800-1500 keV was fit, see figure 3.26, holding fixed the scaling and shifts of the high-
energy peaks. Table 3.10 gives the 7-ray energies, 7-ray production cross sections,
and corresponding uncertainties for all fitted transitions.

Low statistics precluded 7 — 7 coincidences, but two energy-sum correspondences

81

were observed.
1154(9) + 1331.0(14) = 2485(9) g 2491(13)

1331.0(14) + 954.8(15) + 2719(9) = 5004(9)
1900(3) + 3100(4) = 5000(5)
5004(9) % 5000(5)

Level energies were then calculated as weighted averages of these sums. Ordering of
the 955—2719 cascade was determined from '7-ray intensities. The 1900 keV transition
is marginally more intense than the 3100 keV transition, but finding a state at such
low excitation energy relative to the 2:" state which decays primarily to the ground
state is unlikely. Thus, a level is proposed at 3100 keV. Figure 3.27 shows the deduced
level scheme and the low-lying, positive parity states calculated in the shell model.

Tentative J 7’ assignments for 44‘8 were assigned by comparing with the shell-model

calculated decay scheme and the expected population of collective 2+ and 3‘ states.

Feeding corrections were applied according to this decay scheme. An additional cor-

 

  

 

  

HtflLI'I 111"

'Iiuid'uhII '1'!
Iqll-LIILJ-fi. L IléJ'.IJI_.1L..—L..!_L1J uni-Kl; 1 .11
1500 2000 2500 3000 3500 4000 4500 5000
y—ray Energy (keV)

Counts / (25 keV)

 

 

1O
0

Figure 3.25: Spectrum of 'y-rays in coincidence with 4“I*S(1o,p’) shown from 1500-
5000 keV. Response functions for fitted *y-ray transitions (colors) are stacked with
the scaled background (grey) at the top. Data is shown as an unfilled histogram
(black line).

82

 

 

 

 

600 800 1 000 1 200 1400
y—ray Energy (keV)

Figure 3.26: Spectrum of 7—rays in coincidence with 44S(p,p’) shown from 800—
1500 keV. Response functions for fitted 7-ray transitions (colors) are stacked with
the scaled background (grey) at the top. The sum of all responses due to higher en-
ergy 7 rays is shown as a black filled histogram at the base of the stack. Data is shown
as an unfilled histogram (black line).

Table 3.10: Energies and cross sections for 7 rays produced in p(44S,44 S + 7)p’.

 

 

 

 

 

 

7-ray energy (keV) unc. Cross section (mb) unc.
954.8 1.5 3.7 0.3
1154 9 0.6 0.3
1331.0 1.4 17.0 0.4
1599 7 0.9 0.2
1900 3 2.2 0.2
2328 14 0.5 0.2
2491 13 1.0 0.3
2719 9 1.2 0.3
3100 4 1.9 0.2

 

 

rection of 50(25)% of the summed intensity of all unplaced transitions was subtracted
from the 2? cross section. Level energies and excitation cross sections for the reaction

19(448,4‘4 S + 701), are presented in table 3.11.

83

(3‘) 5.001 (4)

 

4+ __3.3931

1411

(2*)-i__3.100(4)

 

 

 

(2*) 2.487(8)

 

 

 

 

 

 

 

 

(4+) - 228612) E _3:§§18
2+ 1 .3310(14) 0+ 1.2636
2+ 1.2182
0*m__o 0+ _0
Experiment SDPF—NR
445

Figure 3.27: Level scheme of 44S deduced in this experiment is compared with low-
lying levels calculated in the shell model. Energies are given in MeV.

Table 3.11: Energies and cross sections for states populated in 19(4‘48,44 S + 7)p'.

 

 

Level energy (keV) unc. Cross section (mb) unc.
1331.0 1.4 12.5 0.9
2286 2 2.7 0.4
2487 8 1.7 0.4
3100 4 -0.3 0.3
5001 4 3.6 0.4

 

 

 

 

 

 

84

Chapter 4

Results and Discussion

4.1 Spectroscopy of 408i

As discussed in section 1.3, the N = 28 shell gap has been predicted to weaken at
or below Z = 16. Observation of enhanced collectivity in the neutron-rich sulfur
isotopes was viewed as evidence of this weakening. However, an alternate explanation
was proposed: the rise in collectivity near 44S may be due to the narrowing of the
131/2 — 0d3/2 proton subshell gap. So, three components may play a role in increasing

collectivity near 44S:

o Neutrons promoted across the N = 28 shell gap
0 Protons recoupled in the degenerate 131/20d3/2 space
0 Enhanced proton-neutron coupling.

The strong Z = 14 subshell gap, however, remains large from N = 20 — 28. Thus,
the neutron-rich silicon chain is expected to give a clearer picture of changes in the
N = 28 shell. Specifically, changes in the 2iF excitation energy of the N = 20 — 28
even-even silicon isotopes are tied directly to filling the 0f7/2 neutron orbital. Filling
a single orbital bracketed by shell gaps, one expects a parabolic trend with the highest

21" energies at the shell gaps and the lowest energy at midshell.
85

 

 

 

 

 

 

 

 

 

.(a) —A—Fe
_‘ —D—Cr
1.4 ,
; —O—TI
1.2% —<>—Ca
’>‘ 1.0;
a, -
>. 08-.
9
23, 16 .
1: :(b) +Ar
.2 1.4- +3
g I +Si
Q a
N 1.0':
1 4
0.8:
l l l
20 22 24 26 28

Neutron number

Figure 4.1: Evolution of 2? energy with neutron number (N) for even-even nuclei
with Z Z 20 (a) and Z < 20 (b). Data are not shown for N = 20, 28 nuclei because
several values would be off scale. While a clear parabolic trend is observed in panel
(a), Z < 20 nuclei show successively smaller increases between N = 24 and N = 26.
The first significant decrease in excitation energy at N = 26 occurs in 4OSi.

Figure 4.1 shows the 2f energies versus neutron number for N = 22, 24, 26 nuclei
for Z Z 20 (panel a) and Z < 20 (panel b). Data in this figure were taken from the
ENSDF database [92] and the current experiment. Isotopic chains for the higher-Z
elements are well-described by a parabolic trend between the N = 20 and N = 28
shell gaps. Argon and sulfur deviate from the parabolic trend with lower excitation

energies at N = 26 than at N = 22. This points to increased collectivity, but the

previously noted reduction in the 7r(0d3/2 — 151/2) gap may be to blame. The 2?
86

 

 

 

 

9 1.8-
a, .
2 1
V 1.6-
> .
9’ 1
8 14': A.
a) . ~.~
§ 1.25
E : ~“.‘-..
£2 1-01""""""A
(D ..
:1; 08-:

0.6 7 r r I

20 22 24 26 28

Neutron number

Figure 4.2: Measured and predicted silicon 2f energies are plotted as a function of
neutron number. SDPF-NR shell-model calculations (solid line) give value that are
systematically ~ 450 keV greater than the measured energies. This suggests that
refinement of the interaction may lead to the the dotted line values.

excitation energy of 4OSi deviates even more strongly from the parabolic prediction.
This is the largest observed reduction in 2'] excitation energy between N = 24 and
N = 26 isotopes. The strong Z = 14 subshell closure directly ties the low 2? energy
of 408i to a reduced N = 28 shell gap at Z = 14.

The evolution of silicon 2f excitation energies with neutron number was then
compared with large-scale shell-model calculations. Shell-model calculations were per—
formed in OXBASH [93] (for 36Si) and CMichSM [94] (for 38"194280 in a 7r(sd)Z—8 11(1) f )N ‘20
model space using the SDPF-NR interaction [32]. Figure 4.2 shows the measured 2:“
energies as triangles and the shell-model results as a thick black line. Though a large
offset is observed, the trend is reproduced nicely.

Comparisons between the calculated 2] energies and experimental data [92] in

even-even sulfur and argon isotopes failed to show a large, systematic offset. Further-

87

more, the first excited states of N = 22,24, 26, 28 phosphorus and chlorine isotopes
were found to be in excellent agreement with calculations using this model space and
interaction [95]. Finding the experimental values consistently 450 keV lower than the
shell-model calculations could indicate some fault in the model inputs. (A dashed line
showing the shell-model energies reduced by 450 keV is shown in figure 4.2.) Either
some part of the interaction changes at Z = 14 or the even-N silicon isotopes are
highlighting a particular part of the interaction, or both.

A possible explanation was suggested by Dr. B. A. Brown [96] who drew an analogy
between the present situation and the neutron-rich carbon isotopes. Similar to silicon,
the neutron-rich even-even carbon isotopes have a strong proton subshell closure.
Shell model predictions using the interaction WBP successfully described the spectra
of neutron-rich oxygen isotopes. Figure 4.3 shows the experimentally observed level
scheme and the shell-model results for two interactions [97,98]. While the level spacing
given by the WBP interaction is too large, the WBPM interaction reproduces the
observed spacing quite well. The WBPM interaction is defined as the WBP interaction
with a reduced neutron-neutron (n-n) interaction strength [97].

Changing the interaction strength affects not only the level spacing, but also the
evolution of shell gaps. The neutron orbital 131/2 lies 0.87 MeV above the 0115/2
orbital in 17O, i.e. Z = 8, N = 9. The 71—71. interaction causes this gap to grow with
increasing neutron number. This, in turn, results in a shell gap at N = 14 which
makes 220 doubly magic [99]. In carbon, the neutron orbitals are inverted at N = 9:
131/2 lies 0.74 MeV below 0d5/2. With no modification, WBP predicts N = 14 is
not a shell closure in carbon. Including the weaker n-n interaction used in WBPM
causes the 11(131/2 — 0d5/2) gap to grow more slowly, thus reinforcing that no shell
gap develops at N z: 14 in the carbon isotopes.

The evolution of level spacings and the N = 14 shell gap in the oxygen-carbon
region may now be used to understand the similar features observed in the calcium-

silicon region. Calculations of 2‘] energies in the 7r(sd)u(fp) region performed with

88

oxo ooco'co
mmoo

4.00 2+ 3.86

 

 

 

 

 

4+—— .7 —_ -
3+ 1.71 - 9
0+—2.88
—2.11 —'—-
1.59 2+ 165 1.59 2+L5_9.
0 0+ 0 0+ 0 o

 

Exp WBP“? WBFQ Exp WBPM WBP
180 20C

Figure 4.3: Experimental level schemes of 1830C along with shell-model predictions

using the WBP and WBPM interactions. With the WBPM, reducing the valence n-n
' interaction strength leads to better agreement with experiment. A similar reduction
in the valence n-n interaction may explain the discrepancy between measured silicon
2f energies and shell-model calculations using SDPF-NR.

the original SDPF-NR interaction [31] may serve as a baseline for comparison. That
interaction used the 1193/2 — 0f7/2 neutron gap from 41Ca along with the 71-71 inter-
action found in stable nuclei—which is known to widen the N = 28 shell gap—to
predict 42Si as a doubly-magic nucleus with an E(2f) of 2.5 MeV [31]. A subsequent
fl—decay experiment determined the 1p3/2 — 0 f7 /2 neutron gap in 35Si. The observed
1123/2 level was 1 MeV lower than expected, and the interaction was modified to ac-
count for the reduced shell gap. This had little impact on excitation energies other
than 42Si, which was then predicted to have E(2f) of 1.5 MeV [35].

Figure 4.2 clearly shows the modified SDPF-NR interaction predicts a larger level
spacing than is observed. Reducing the neutron-neutron (n-n) interaction strength
for Z = 14 could correct the observed disagreement with experiment. A weaker n—n
interaction would also cause the N = 28 shell gap to grow more slowly with neutron
number, giving a smaller N = 28 gap at Z = 14 than is predicted by the current shell

model interaction.

89

The deduced level scheme of 408i adds support to the proposition that the N = 28
shell gap is reduced at Z = 14. Though statistics were limited in the 19(42P,40 Si +
7)X channel, a level either 1624(7) or 1831(8) keV was found, see section 3.2.3 for
discussion. The existence of a level at such low energy above the 2] level at 986(5) keV
would not be expected if the N = 28 shell gap were large. Figure 3.17 shows the
observed level scheme and the shell-model calculation. In the shell-model calculations,
states decaying by 7-rays of less than 1 MeV to the 2;r level arise from particle-hole
excitations across the N = 28 shell gap.

These observations:
0 The decrease in 2'] energy between N = 24 and N = 26 silicon isotopes
o The additional low—lying level observed in 4OSi

o The systematically smaller level spacing as compared to the current shell-model

calculations

point to an N = 28 shell gap at Z = 14 which is even smaller than current predictions.

4.2 Quadrupole collectivity

Recalling section 1.2.2, collective nuclear states are often categorized as vibrational,
rotational, or some combination thereof. Knowing the nature (vibrational / rotational)
of a collective state may shed some light on their structure. Deformation parameters,
Ifll, correspond to the magnitude of the deformation, dynamic or static and may be
used as quantitative measures of collectivity.

Extracting these deformation parameters from the measured inelastic proton scat-
tering cross sections required the use of a specific collective model. Thus, the defor-
mation parameters, [62,(p,p/)|, presented here are model-dependent. Vibration, pro-

late rotation, and oblate rotation were used in the ECIS [39] calculations to deduce

90

 

 

 

 

 

 

 

 

 

3.5 g;
8
3.0 -1 oo

_' 2.5 - El
+ '- 9
N 51 /\ v
+ " 1.5 - . .
<1 a:
61" .2
1.0 - g
0.5 — E
I Sulfur A Silicon g9
0-0 I I I [CD-5
22 24 26 g
0

Neutron number
Figure 4.4: Ratios of excitation energies for the first 2+ and 4"” levels in neutron-rich,
even-even sulfur and silicon nuclei filling the V(0 f7 /2) orbital. Dotted and dashed lines
show the values expected in the vibrational and rotational collective models.
[132,(p,p/)| from the measure inelastic cross-sections. For each nucleus, models consis-
tent with the available data were used.

Figure 4.4 shows the ratios of excitation energies for the first 2‘L and 4+ levels
the even-A sulfur and silicon nuclei filling the u(0f7/2) orbital. Data for 36Si is from
Ref. [85], and the 38Si value is taken from the present analysis. These silicon isotopes
show energy ratios consistent with the vibrational model. The strong proton sub—shell
closure at Z = 14 likely prevents the coupling of proton and neutron excitations, which
are associated with the development of static deformation [15,32]. Thus, the isotopes
36’38’40Si were analyzed using the vibrational model.

Table 4.1 presents the measured inelastic proton-scattering cross sections and
deduced deformation parameters, [32,00,110], of the current experiment. For isotopes
analyzed using more than one model, the average deformation parameter is used in

subsequent tables and figures. Uncertainty limits will encompass the full uncertainty

91

for each model used. Example using 42S:

03211150511818 gel = (0.345 + 0.381) /2 = 0.363
I32,(p’p1),lowe,,,m,,| = min(0.335, 0.369) = 0.335
|.32’(p,p,,,upper,,m,,| = max(0.355,0.393) = 0.393
|62,(p,p,)| = 036(3)

The sulfur isotopes Show a steady increase in the ratio, E [4]] / E [21+], with neutron
number. This trend should be compared with figure 1.3, which shows the narrowing
of the Z = 16 sub-shell gap with increasing neutron number. At N = 22, the proton
sub-shell gap is still large, and 38S has a ratio consistent with vibration. As the pro—
ton sub—shell gap narrows, the ratio, E [41*] / E [2]], grows. 408 has a transitional ratio,
and calculations by Sohler et al. point to a 7-soft vibrational description. So, 408 was
analyzed using the vibrational model. 4‘2S has a ratio > 3, pointing to rotational
character. The narrowed Z = 16 sub-shell gap allows for both enhanced proton col-
lectivity and a coupling of proton and neutron collective excitations. This culminates
in rotational behavior at N = 26. Although oblate deformation is predicted in 428, no
experiment has probed the sign of the deformation. Both prolate and oblate models

were used for 42S. The low-lying structure of 44‘S is quite complicated. Observation of

Table 4.1: Inelastic proton-scattering cross sections, deformation parameters of the
current experiment ([62 (p p/)|) are given for 36’38’408i and 40,42,448.

 

 

 

 

N 00+_,2+ (mb) [32.071120]
1 1 vibrational prolate oblate
3581 7.4(24) 024(4)
388i 15.4(26) 035(3)
4081 20;t2 0.3773
408 19.8(12) 0.384(12)
42S 16.3(10) 034(1) 038(1)
443 12.5(9) 028(1) 027(1) 030(1)

 

92

Table 4.2: Deformation parameters of the current experiment ([62 (p p/)|) and a pre-
vious Coulomb excitation measurement ([520]) [84], shell—model matrix elements Ap
and An, and experimental and shell-model (€617?) / (gr) ratios are given for 36’38’40Si.

 

N 15.0..) 111251 4.6111) 41.11111) (1631/1715... (13,3)44811

 

22 024(4) 026(4) 5.44 13.1 0.89:”): 0.95
24 035(3) 025(5) 5.04 19.7 1.5813 0.84
26 0.3773 — 5.28 23.5 — 0.82

 

an isomeric 03" state at an excitation energy of 1.365(1) MeV [91] is consistent with
the prediction of shape coexistence in this nucleus [100]. All three models were used

to analyze 44S.

4.2.1 Collectivity in silicon isotopes

Figure 4.5 shows the deformation parameters, | 1321(P1P')I’ deduced in this experiment
and the corresponding lfigpl values from a Coulomb excitation experiment [84] plot-
ted against N for 3638140Si. If both the N = 20 and N = 28 shell gaps were large,
filling of the u(0f7/2) orbital would dominate changes in the excitation energy and
collectivity of the 2: state in this isotopic chain. As discussed in section 4.1, a sym-
metric, parabolic trend was expected in the energies, but the 4OSi 211‘ energy was
significantly lower than the simple prediction. The deformation parameters are, like-
wise, expected to vary symmetrically about mid-shell, N = 24. However, collectivity
is expected to be maximum at mid-shell. This model fails to predict the large defor-
mation parameter of 408i. Instead, 36Si has the smallest collectivity, and 38’40Si are
similarly collective.

Unlike the collectivity changes in the sulfur isotopes, the increased collectivity
at N = 26 does not appear to arise due to changes in the proton sub-shell gaps.
With a large Z = 14 sub-shell gap from N = 20 — 28, the contribution of valence

protons to collectivity is small and roughly constant in these isotopes. Only neutron

93

 

0.5

 

 

 

 

 

 

£3 0.4J

1 9
(D

E 0.3-1

E

(U

Q.

E 0.2-1

6

E

O

“53 0.1 -

o

] o (p,p’) A Coulex

 

22 24 26
Neutron number

Figure 4.5: Deformation parameters [62 (p p’)I and previously measured | 32,51] [84] are

plotted against neutron number for the nuclei 36’38’408i.

94

excitations across the N = 28 shell gap can explain the increased collectivity in 408i.
This increased collectivity is direct evidence that the N = 28 shell gap has narrowed
at Z = 14.

Comparing the deformation parameters [52,09,193] deduced here with those of
Coulomb excitation, one may extract the relative proton and neutron contributions
to the collective 2f excitation. As discussed in section 2.1.3, a simple collective model
of the nucleus predicts equal deformations for protons and neutrons. This leads to
proton and neutron transition matrix elements in direct proportion to the number of
protons and neutrons (i.e. 7% = g). To compare isotopes, the ratio of neutron to
proton matrix elements will be divided by N /Z . Table 4.2 presents the deformation
parameters, shell-model matrix elements, and experimental and predicted 71%] = 1;.

Before discussing these results, it is important to review the inputs used in these
shell model predictions. Specifically, the polarization charges must be carefully con-
sidered. Recalling the discussion in section 1.2.1, matrix elements resulting from a
shell—model calculation are not sufficient to describe the transitions among states.
Coupling of valence nucleons to excitations of the core are handled through polariza-
tion charges, 6. Both like and unlike nucleon couplings occur, and the final matrix
elements, Mn and Mp, are linear combinations of the valence space matrix elements,
An and Ap.

In the present analysis, four sets of polarization charges were considered. Table
4.3 lists three polarization charge sets which use the simplifying assumption that
6W = 61m and 61m = 6m). These sets represent the standard isoscalar choice a set with

a significant isovector component — which has been suggested in the pf shell [101],

Table 4.3: Three sets of polarization charges considered for analyzing 111% = 12);.

1

 

Polarization set 5191? 61m
isoscalar 0.5 0.5
sd—shell 0.2 0.5

duRietz (pf) 0.2 0.8

 

 

 

 

 

 

95

 

 

700

 
  
 

 

 

 

600 J , . . . ’ 1““
v ,
é ’ , ’ , .
“0’ 50° _ , .8: ’ ’
:\ I ’ I r
N'- 400 - , , ’ ,.-O
I.) , ’ ’ ............
+_ 300 — ............................................
o ....... Experiment
-- o
a 200 _ -------- —0- SM (isoscalar)
100 — - 81- SM (du Rietz)
—e— SM (BM)
0 I I I
22 24 26

Neutron number

Figure 4.6: Measured B(E2T) values are plotted versus neutron number N for
V(0f7/2)2" silicon nuclei. Shell model predictions using various polarization charges

are compared to data.

and the accepted polarization charges for the 3d shell [102]. A fourth set — suggested
by Dr. H. Sagawa — was an extension of the Bohr-Mottelson effective charge model,
given in Eq. 6-386a of Ref [11]. The original model focused only on proton excitations
of the core and was directed at correctly predicting B(E2) values. This model was
explicitly constructed to handle changes in core polarization due to the N /Z of a
given isotope. By applying the original reasoning to neutron excitations of the core,
one may calculate all four polarization charges.

Each polarization charge set was used with the valence space matrix elements to
calculate B(E2; 01+ —+ 21*) values for 36’38’408i. Figure 4.6 compares the results using
each polarization charge set to the measured values for 36388i [84]. The isovector
pf-shell set clearly disagrees with the data. Both the sd—shell and extended Bohr-
Mottelson sets give similar, good agreement with measured data. The sd—shell polar-
ization charge set has been used to in table 4.2.

Figure 4.7 shows (1%]? / (1}) plotted versus neutron number for experiment and

96

 

 

 

 

 

 

 

 

 

 

o This experiment
-0- SM (isoscalar)
---EI-- SM (sd)
A - — 8- SM (du Rietz)
'3 —9— SM (BM)
2
v c %
Z 1 G -------------------- e
o. “ B
2 {J
\c ----------- " ------------- A
E -
0 I l r
22 24 26

Neutron number

Figure 4.7: Normalized ratios of neutron-to—proton matrix elements, (716%) / (g), are

plotted versus neutron number N for silicon isotopes. Experimental results are com-
pared to shell model predictions using various polarization charges.

each set of polarization charges. The large error bar for 368i is dominated by experi-
mental uncertainties, with equal contributions from the present (p, p') experiment and
the Coulomb excitation experiment. In the case of 38 Si, the Coulomb excitation uncer-
tainty is dominant. The study of neutron and proton collectivity in the neutron-rich
silicon isotopes would greatly benefit from a precision Coulomb excitation experiment
on 36’38’4081.

Given the large error bars, 368i is consistent with both shell model and the simple
collective model prediction. The neutron collectivity appears to increase significantly
at N = 24 with no similar rise in proton collectivity. This large increase is not found
in shell—model calculations. Three alternatives may be used to explain the possible

discrepancy in 38Si:
0 a deficiency in the shell model such as a too-large N = 28 shell gap

0 an enhanced core neutron — valence neutron coupling

97

c an overestimation of L32 (1, p/)| due to the formation of a neutron skin.

The latter two possibilities have also been suggested to explain enhanced neutron
collectivity observed in 381408 [68,103]. Differential cross sections with respect to
scattering angle — as are measured in proton-detection (p,p’) — are required to

probe the different proton and neutron density distributions.

4.2.2 Collectivity in sulfur isotopes

The analysis of sulfur collectivity parallels the previous section on silicon collectivity.
Figure 4.8 shows the deformation parameters, I 32,03,100], deduced in this experiment
and previous experiments studying 36381408 [45,68] and the corresponding [32,0]
values from Coulomb excitation [104]. Here, the isotopes at N = 20,28 have also
been studied. Assuming large shell gaps at N = 20, 28, one would expect collectivity
minima at the shell gaps with symmetric increases toward a maximum at N = 24.
Instead, the Coulomb excitation data show a steady increase from N = 20 — 26, and
the wimp,” values show a sharp rise at N = 22. Both Show declining collectivity at
N = 28, but the (p, p’) value has more statistical significance.

These observations can be explained by considering the evolution of the 77(131/2 —
0d3/2) gap with increasing neutron number and its impact on collectivity. As discussed
in section 1.3, this Z = 16 proton sub—shell gap is known to narrow from N = 20 to
N = 28. This not only increases the proton collectivity, but allows proton and neutron
collective excitation to couple [32]. The result is a region of deformation near 42S.
Coupling of proton and neutron collectivity and the near degeneracy of the 77(131/2)
and 7r(0d3/2) orbitals at N = 28 has obscured changes in the N = 28 shell gap for
sulfur and the neighboring Z = 15, 17 isotopic chains.

In fact, the Coulomb excitation data leave open the possibility that the collectiv-
ities of 428 and 44S are the same. Subtracting the observed [132,0] value of 44S from

that of 428 yields 0.046 :1: 0.044. Horn this data alone, there appears to be a fair

98

 

0.5

 

 

 

 

 

 

 

251“ 0.4—
9 [1 f
(D
E 0.3- 5 §
8 4
C
o 0.2—
2.: l
g I 34
g 0.14
Q 0 Current (p,p’) A Coulex
18 Previous (p,p')
0-0 I 1 I I I

20 22 24 26 28
Neutron number

Figure 4.8: Deformation parameters IfiQ (p 197)] deduced in this experiment and
[62 (p p/)| and [32,0] from previous experiments are plotted against neutron number

for the nuclei 38’40’42’448.

99

 

Table 4.4: Deformation parameters of the current experiment ([62 (p p/)|) and a pre-
vious Coulomb excitation measurement (I 320]) [84], shell-model matrix elements Ap
and An, and experimental and shell-model (gap/(1).) ratios are given for 40’42’448.

 

N 162,...) [132C] .1, 11m) 4.111111) (511/111)... (75314413111

 

24 0.386(12) 0.284(15) 7.28 18.2 1.41(10) 0.95
26 036(3) 0.300(23) 8.27 18.4 124(16) 0.84
28 028333 0.254(38) 8.01 20.1 1.12(22) 0.82

 

chance that the N = 28 shell gap has significantly eroded at Z = 16.

Examining the present [32,(p3p/)| results is more diflicult because the model de-
pendence of the extracted deformation parameters forces one to consider a matrix
of Ifi2,(p,p’)] differences. In all cases, 428 has a larger deformation parameter than
44S. The least statistically significant difference was between prolate rotational 42S
and oblate rotational 44S, which gave a A|52,(p,p/)| of 0.048 i 0.015. This case runs
contrary to shell model and mean field calculations which suggest static oblate defor-
mation in 428 and shape coexistence in 44S. All other differences are > 50. Thus, the
present data gives evidence for a decline in collectivity between 42S and 44S.

Table 4.4 lists the deformation parameters determined in this experiment and pre-
vious Coulomb excitation experiments, the shell-model valence space matrix elements,
and Mn/Mp values normalized by N / Z determined experimentally and calculated in
the shell model. As in the silicons, the sd—shell polarization charges were used. The
experimental ratio of neutron to proton collectivity falls from N = 24 to N = 28.
This behavior arises from a small decrease in neutron collectivity and a large increase
in proton collectivity as the neutron 0 f7 /2 orbital fills. Only 44S is consistent with
Mn/Mp = N /Z , and even this result contradicts expectations. Since the Z = 16
sub-shell gap vanishes at N = 28, 44S is a single-closed-shell nucleus with neutrons

forming the closed shell. Thus, one expects Mn/Mp < N /Z .

Large uncertainties do not, however, allow 1%,, /Mp < N /Z to be ruled out for

100

 

700

 

 

  
    

 

 

 

600 -
V
.E
"a: 500 -
:3
N1- 400 "‘
1‘
+'_ 300 -
(.3. " ............. 0 Experiment
EU, 200 _ """" —0— SM (isoscalar)
m ----B SM (sd)
100 — - 4- SM (du Rietz)
—e— SM (BM)
0 I I I I
22 24 26 28

Neutron number

Figure 4.9: Measured B(E2T) values are plotted versus neutron number N for
V(0f7/2)2” sulfur nuclei. Shell model predictions using various polarization charges

are compared with data.

448. Two main sources of uncertainty exist: the Coulomb excitation experimental
uncertainties and the model dependence inherent in extracting [filfpm’fl' As in the
silicons, a precision measurement of the Coulomb excitation of 40’42’448 at higher
statistics would significantly improve the present analysis. A measurement of the
static quadrupole moment, including the sign, of the 42,443 ground states would allow
selection of the appropriate collective model.

As before, shell-model results using the three sets of polarization charges from
table 4.3 and the extended Bohr-Mottelson polarization charges are compared with
data. Figure 4.9 shows experimental and calculated B(E2T) values for 38’40’42’448
plotted versus neutron number. Experimental B(E2T) values were taken from the
compilation of Raman et al. [104]. In line with the silicon results, the isoscalar and
duReitz polarization charges given larger B(E2T) values than are observed in exper-
iment. B(E2T) values calculated using polarization charge sets from the extended

Bohr-Mottelson model and the 3d shell give good agreement with data.

101

 

 

 

 

 

 

 

 

Q ‘ (D i
E 1 i
’a 1 - 1°37"7"""“‘I‘E'JJTJ: ---------
2 A: ..................................... 8 -------------- a
\C o This experiment “““““““ 1"» ----------- x 1
2 —63- SM (isoscalar)
V '-----E1 SM (sd)

-A— SM (du Rietz)

—9— SM (BM)

0 129 Past experiments
I l 1 I
22 24 26 28

Neutron number

Figure 4.10: Normalized ratios of neutron-to—proton matrix elements, (gag/(1)1), are
plotted versus neutron number N for sulfur isotopes. Experimental results are com-

pared to Shell model predictions using various polarization charges.

Figure 4.10 plots the (16%) / (1}) values versus neutron number. Values from the
present analysis and from a previous set of experiments are shown along with shell—
model calculations using each of the four polarization charges sets previously dis-
cussed. Although the experimental ratios of all nuclei overlap within their 10 error
bars, the central values show a steady decline with increasing N. Shell-model calcu—

lations show a similar trend but give lower neutron collectivity than is observed.

102

Chapter 5

Conclusion

Inelastic proton scattering cross sections, U(p,p’)IOf —+ 2:], have been measured for
36’38’40Si and 40,42,448. Measurements were performed by the thick-target, 7-ray tag-
ging method. Four state-of-the-art systems —— A1900, S800, SeGA, and the RIKEN-
Kyushu-Rikkyo liquid hydrogen target ~ were coupled to make efficient use of the
rare isotopes being investigated.

The 2‘] excitation energy in 4OSi is lower than would be expected if N = 28
had a large shell gap at Z = 14. Proton single particle energies measured in stable
nuclei indicate a large, stable Z = 14 subshell gap [29]. Therefore, the decline in 2]
energies for N = 24 to N = 26 is directly tied to a narrowing of the N = 28 shell gap
between Z = 20 and Z = 14. A systematic difference between measured 2? energies
and shell-model values found only in the silicon isotopes is suggested as evidence of
a Z-dependent weakening of the n — 7). interaction. A low-lying second excited state
observed in 4OSi results from neutron excitations across the N = 28 shell gap.

Measurements on 36388i and 40’42’448 have led to their level schemes. In 388i, the
location of the 4] level appears to confirm the expectation of vibrational collectivity in
this nucleus. Recently published mass measurements in this region show no evidence
of deformation in the N = 20 — 28 silicon isotopes, but leave open the possibility of

a rapid onset of deformation at N = 28 [105].

103

Quadrupole collectivity has been measured in each of these nuclei by the extraction
of model-dependent deformation parameters, Ifl2,(p,p’)I’ In the silicon isotopes, an
asymmetry about mid-shell — N = 24— is observed with 40Si being more collective
than 36Si, and perhaps equally collective as 388i. As with the energies, the increased
collectivity is directly tied to a reduction in the N = 28 shell gap. In the sulfur
chain, [$109.14)] values provide a clear, quantitative measurement of the decrease in
collectivity between 42S and 44S. This fits nicely into the shell-model description by

Retamosa et al. of a localized region of deformation centered on 42S [31].

104

BIBLIOGRAPHY

[1] M. G. Mayer. Phys. Rev., 74:235—239, 1948.

[2] M. G. Mayer. Phys. Rev., 75:1969—1970, 1949.

[3] O. Haxel, J. Hans D. Jensen, and Hans E. Suess. Phys. Rev., 7511766, 1949.
[4] J. Blomqvist and A. Molinari. Nucl. Phys. A, 106:545—569, 1968.

[5] B. A. Brown. Nuclear structure physics. Lecture notes, 2003.

[6] B. A. Brown and W. A. Richter. Phys. Rev. C, 74:034315, 2006.

[7] B. A. Brown and B. H. Wildenthal. Annu. Rev. Nucl. Part. Sci, 38:29—66,
1988.

[8] B. A. Brown. Prog. Part. Nucl. Phys, 47:517—599, 2001.

[9] E. Caurier, G. Martinez—Pinedo, F. Nowacki, A. Poves, and A. P. Zuker. Rev.
Mod. Phys, 772427—488, 2005.

[10] V. R. Pandharipande, I. Sick, and P. K. A. deWitt Huberts. Rev. Mod. Phys,
69:981—991, 1997.

[11] A. Bohr and B. R. Mottelson. Nuclear Structure, volume 2. World Scientific,
New Jersey, 1998.

[12] B. A. Brown and B. H. Wildenthal. Phys. Rev. C, 21:2107—2110, 1980.

[13] W. Greiner and J. A. Maruhn. Nuclear Models. Springer-Verlag, New York,
1996.

[14] R. F. Casten. Nuclear Structure from a Simple Perspective. Oxford University
Press, New York, 2000.

[15] J. Dobaczewski, W. Nazarewicz, J. Skalski, and T. Werner. Phys. Rev. Lett.,
60:2254—2257, 1988. '

[16] A. Bohr and B. R. Mottelson. Nuclear Structure, volume 1. World Scientific,
New Jersey, 1998.

[17] P. Federman, S. Pittel, and R. Campos. Phys. Lett. B, 82:9~12, 1979.

[18] K. Heyde, P. Van Isacker, R. F. Casten, and J. L. Wood. Phys. Lett. B, 155:303—
308, 1985.

[19] T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki.
Phys. Rev. Lett., 87:082502, 2001.

105

.1.

[20] C. Thibault, R. Klapisch, C. Rigaud, A. M. Poskanzer, R. Prieels, L. Lessard,
and W. Reisdorf. Phys. Rev. C, 12:644—657, 1975.

[21] X. Campi, H. Flocard, A. K. Kerman, and S. Koonin. Nucl. Phys. A, 251:193—
205, 1975.

[22] E. K. Warburton, J. A. Becker, and B. A. Brown. Phys. Rev. C, 41:1147—1166,
1990.

[23] T. Motobayashi, Y. Ikeda, K. Ieki, M. Inoue, N. Iwasa, T. Kikuchi,
M. Kurokawa, S. Moriya, S. Ogawa, H. Murakami, S. Shimoura, Y. Yanagi-
sawa, T. Nakamura, Y. Watanabe, M. Ishihara, T. Teranishi, H. Okuno, and
R. F. Casten. Phys. Lett. B, 346:9—14, 1995.

[24] K. Yoneda, H. Sakurai, T. Gomi, T. Motobayashi, N. Aoi, N. Fukuda, U. Fu-
takami, Z. Gacsi, Y. Higurashi, N. Imai, N. Iwasa, H. Iwasaki, T. Kubo, M. Ku-
nibu, M. Kurokawa, Z. Liu, T. Minemura, A. Saito, M. Serata, S. Shimoura,
S. Takeuchi, Y. X. Watanabe, K. Yamada, Y. Yanagisawa, K. Yogo, A. Yoshida,
and M. Ishihara. Phys. Lett. B, 499:233—237, 2001.

[25] Y. Yanagisawa, M. Notani, H. Sakurai, M. Kunibu, H. Akiyoshi, N. Aoi,
H. Baba, K. Demichi, N. Fukuda, H. Hasegawa, Y. Higurashi, M. Ishihara,
N. Iwasa, H. Iwasaki, T. Gomi, S. Kanno, M. Kurokawa, Y. U. Matsuyama,
S. Michimasa, T. Minemura, T. Mizoi, T. Nakamura, A. Saito, M. Serata, S. Shi-
moura, T. Sugimoto, E. Takeshita, S. Takeuchi, K. Ue, K. Yamada, K. Yoneda,
and T. Motobayashi. Phys. Lett. B, 566:84—89, 2003.

[26] J. R. Terry, D. Bazin, B. A. Brown, C. M. Campbell, J. A. Church, .1. M.
Cook, A. D. Davies, D. C Dinca, J. Enders, A. Gade, T. Glasmacher, P. G.
Hansen, J. L. Lecouey, T. Otsuka, B. Pritychenko, B. M. Sherrill, J. A. Tostevin,
Y. Utsuno, K. Yoneda, and H. Zwahlen. Phys. Lett. B, 640:86—90, 2006.

[27] O. Sorlin, D. Guillemaud-Mueller, A. C. Mueller, V. Borrel, S. Dogny,
F. Pougheon, K.-L. Kratz, H. Gabelmann, B. Pfeiffer, A. W6hr, W. Ziegert,
Yu. E. Penionzhkevich, S. M. Lukyanov, V. S. Salamatin, R. Anne, C. Borcea,
L. K. Fifield, M. Lewitowicz, M. G. Saint-Laurent, D. Bazin, C. Détraz, F.-K.
Thielemann, and W. Hillebrandt. Phys. Rev. C, 47:2941—2953, 1993.

[28] T. Glasmacher, B. A. Brown, M. J. Chromik, P. D. Cottle, M. Fauerbach,
R. W. Ibbotson, K. W. Kemper, D. J. Morrissey, H. Scheit, D. W. Sklenicka,
and M. Steiner. Phys. Lett. B, 395:163—168, 1997.

[29] P. D. Cottle and K. W. Kemper. Phys. Rev. C, 58:3761—3762, 1998.

[30] P. Doll, G. J. Wagner, K. T. Knpfle, and G. Mairle. Nucl. Phys. A, 263:210—236,
1976.

[31] J. Retamosa, E. Caurier, F. Nowacki, and A. Poves. Phys. Rev. C, 55:1266—1274,
1997.

106

[32] E. Caurier, F. Nowacki, and A. Poves. Nucl. Phys. A, 742:14—26, 2004.

[33] J. Fridmann, I. Wiedenhéver, A. Gade, L. T. Baby, D. Bazin, B. A. Brown,
C. M. Campbell, J. M. Cook, P. D. Cottle, E. Diffenderfer, D.-C. Dinca, T. Glas-
macher, P. G. Hansen, K. W. Kemper, J. L. Lecouey, W. F. Mueller, H. Olliver,
E. Rodriguez-Vieitez, J. R. Terry, J. A. Tostevin, and K. Yoneda. Nature,
435:922—924, 2005.

[34] J. Fridmann, I. Wiedenhéver, A. Gade, L. T. Baby, D. Bazin, B. A. Brown,
C. M. Campbell, J. M. Cook, P. D. Cottle, E. Diffenderfer, D.-C. Dinca,
T. Glasmacher, P. G. Hansen, K. W. Kemper, J. L. Lecouey, W. F. Mueller,
E. Rodriguez-Vieitez, J. R. Terry, J. A. Tostevin, K. Yoneda, and H. Zwahlen.
Phys. Rev. C, 742034313, 2006.

[35] S. Nummela, P. Baumann, E. Caurier, P. Dessagne, A. Jokinen, A. Knipper,
G. Le Scornet, C. MiehéLF. Nowacki, M. Oinonen, Z. Radivojevic, M. Ramd—
hane, G. Walter, and J. Ayst6. Phys. Rev. C, 63:044316, 2001.

[36] A. M. Bernstein, V. R. Brown, and V. A. Madsen. Phys. Lett. B, 103:255—258,
1981.

[37] A. M. Berstein, V. R. Brown, and V. A. Madsen. Comments Nucl. Part. Phys,
11:203, 1983.

[38] P. D. Cottle. Nucl. Phys. A, 682:124c—1300, 2001.
[39] J. Raynal. Notes on ECISQ4, 1994.

[40] N. Austern. Direct Nuclear Reaction Theories. John Wiley and Sons, Inc., New
York, 1970.

[41] G. R. Satchler. Direct Nuclear Reactions. Oxford University Press, New York,
1983.

[42] E. Merzbacher. Quantum Mechanics. John Wiley and Sons, Inc., New York,
1998.

[43] C. A. Bertulani and P. Danielewicz. Introduction to Nuclear Reactions. Taylor
and Francis, New York, 2004.

[44] A. J. Koning and J. P. Delaroche. Nucl. Phys. A, 713:231—310, 2003.

[45] J. H. Kelley, T. Suomijarvi, S. E. Hirzebruch, A. Azhari, D. Bazin, Y. Blumen—
feld, J. A. Brown, P. D. Cottle, S. Danczyk, M. Fauerbach, T. Glasmacher, J. K.
Jewell, K. W. Kemper, F. Maréchal, D. J. Morrissey, S. Ottini, J. A. Scarpaci,
and P. Thirolf. Phys. Rev. C, 56:R1206—R1209, 1997.

[46] A. Bohr. Physica, 22:959—968, 1956.
[47] B. L. Cohen and A. G. Rubin. Phys. Rev., 111:1568—1577, 1958.

107

[48] A. M. Bernstein, V. R. Brown, and V. A. Madsen. Phys. Lett. B, 106:259—263,
1981.

[49] V. R. Brown and V. A. Madsen. Phys. Rev. C, 11:1298-1311, 1975.

[50] J. A. Carr, F. Petrovich, and J. J. Kelly. In Neutron-Nucleus Collisions — A
Probe of Nuclear Structure, volume 124, pages 230—255. American Institute of
Physics, 1985.

[51] M. A. Franey and w. G. Love. Phys. Rev. C, 31:488—498, 1985.

[52] G. Bertsch, J. Borysowicz, H. McManus, and W. G. Love. Nucl. Phys. A,
284:399~419, 1977.

[53] PG. Hansen and J .A. Tostevin. Annu. Rev. Nucl. Part. Sci, 53:219—261, 2003.
[54] G Jacob and Th. A. J. Maris. Rev. Mod. Phys, 38:121—142, 1966.

[55] G Jacob and Th. A. J. Maris. Rev. Mod. Phys, 45:6, 1973.

[56] J. R. Grover and A. A. Caretto. Annu. Rev. Nucl. Sci, 14:51—100, 1964.

[57] H. P. Yule and A. Turkevich. Phys. Rev., 118:1591—1598, 1960.

[58] A. Van Der Woude. Prog. Part. Nucl. Phys, 18:217—293, 1987.

[59] J. Speth. Electric and Magnetic Giant Resonances in Nuclei. World Scientific,
New Jersey, 1991.

[60] K. Starosta, 2006. Private communication.
[61] A. Gade, 2006. Private communication.
[62] B. A. Brown, 2005. Private communication.

[63] H Geissel, G Munzenberg, and K Riisager. Annu. Rev. Nucl. Part. Sci, 45:163—
203,1995.

[64] R. Serber. Phys. Rev., 7221008, 1947.

[65] F. Marti, D. Poe, M. Steiner, J. Stetson, and X.Y. Wu. In Proceedings of the
16th International Conference on Cyclotrons and Their Applications, volume
600, pages 64—68. American Institute of Physics, 2001.

[66] D. J. Morrissey, B. M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhoever. Nucl.
Instr. and Meth. B, 204:9(F96, 2003.

[67] G. Kraus, P. Egelhof, C. Fischer, H. Geissel, A. Himmler, F. Nickel,
G. Miinzenberg, W. Schwab, A. Weiss, J. Friese, A. Gillitzer, H. J. K6rner,
M. Peter, W. F. Henning, J. P. Schiffer, J. V. Kratz, L. Chulkov, M. Golovkov,
A. Ogloblin, and B. A. Brown. Phys. Rev. Lett., 73:1773—1776, 1994.

108

[68] F. Maréchal, T. Suomija'rvi, Y. Blumenfeld, A. Azhari, E. Bauge, D. Bazin, J. A.
Brown, P. D. Cottle, J. P. Delaroche, M. Fauerbach, M. Girod, T. Glasmacher,
S. E. Hirzebruch, J. K. Jewell, J. H. Kelley, K. W. Kemper, P. F. Mantica, D. J.
Morrissey, L. A. Riley, J. A. Scarpaci, H. Scheit, and M. Steiner. Phys. Rev. C,
602034615, 1999.

[69] H. Scheit, F. Maréchal, T. Glasmacher, E. Bauge, Y. Blumenfeld, J. P. De-
laroche, M. Girod, R. W. Ibbotson, K. W. Kemper, J. Libert, B. Pritychenko,
and T. Suomijarvi. Phys. Rev. C, 632014604, 2000.

[70] H. Iwasaki, T. Motobayashi, H. Akiyoshi, Y. Ando, N. Fukuda, H. Fujiwara,
Zs Flp, K. I. Hahn, Y. Higurashi, M. Hirai, I. Hisanaga, N. Iwasa, T. Kijima,
T. Minemura, T. Nakamura, M. Notani, S. Ozawa, H. Sakurai, S. Shimoura,
S. Takeuchi, T. Teranishi, Y. Yanagisawa, and M. Ishihara. Phys. Lett. B,
48127—13, 2000.

[71] H. Ryuto, M. Kunibu, T. Minemura, T. Motobayashi, K. Sagara, S. Shimoura,
M. Tamaki, Y. Yanagisawa, and Y. Yano. Nucl. Instr. and Meth. A, 555:1—5,
2005.

[72] S. Kan’no, 2004. Private communication.
[73] M. Sheplak and J. Dugundji. Journal of Applied Mechanics, 652107, 1998.

[74] D. Bazin, J. A. Caggiano, B. M. Sherrill, J. Yurkon, and A. Zeller. Nucl. Instr.
and Meth. B, 2042629—633, 2003.

[75] J. Yurkon, D. Bazin, W. Benenson, D. J. Morrissey, B. M. Sherrill, D. Swan,
and R. Swanson. Nucl. Instr. and Meth. A, 422:291—295, 1999.

[76] W. F. Mueller, J. A. Church, T. Glasmacher, D. Gutknecht, G. Hackman, P. G.
Hansen, Z. Hu, K. L. Miller, and P. Quirin. Nucl. Instr. and Meth. A, 4662492—
498, 2001.

[77] GEANT. CERN library long writeup. Technical Report W5013, CERN, 1994.

[78] M.J. Berger, J .H. Hubbell, S.M. Seltzer, J. Chang, J .S. Coursey, R. Sukumar,
and US. Zucker. XCOM: Photon cross sections database, 2005.

[79] S. Shimoura. ECI397 Tips. Lecture notes, 2003.
[80] W. F. Mueller, 2004. Private communication.

[81] R. Fox, C. Bolen, K. Orji, and J. Venema. NSCLSpecTcl Meeting the Needs of
Preliminary Nuclear Physics Data Analysis. In Eleventh Annual Tcl/Tk Con-

ference New Orleans. Institute for Nuclear Theory, University of Washington,
2004.

[82] PAW. CERN library long writeup. Technical Report Q121, CERN, 1999.

[83] Minuit. CERN library long writeup. Technical Report D506, CERN, 1998.
109

 

[84] R. W. Ibbotson, T. Glasmacher, B. A. Brown, L. Chen, M. J. Chromik, P. D.
Cottle, M. Fauerbach, K. W. Kemper, D. J. Morrissey, H. Scheit, and M. Thoen-
nessen. Phys. Rev. Lett., 80:2081—2084, 1998.

[85] X. Liang, F. Azaiez, R. Chapman, F. Haas, D. Bazzacco, S. Beghini, B. R.
Behera, L. Berti, M. Burns, E. Caurier, L. Corradi, D. Curien, A. Deacon,
G. de Angelis, Zs. Dombradi, E. Farnea, E. Fioretto, A. Hodsdon, A. Gadea,
F. Ibrahim, A. Jungclaus, K. Keyes, A. Latina, S. Lunardi, N. Marginean,
R. Menegazzo, G. Montagnoli, D. R. Napoli, F. Nowacki, J. Ollier, A. Papen-
berg, G. Pollarolo, V. F. E. Pucknell, M.-D. Salsac, F. Scarlassara, J. F. Smith,
K. Spohr, M. Stanoiu, A. M. Stefanini, S. Szilner, N. Toniolo, M. Tfotta, D. Ver-
ney, Z. Wang, and J. Wrzesinski. Phys. Rev. C, 742014311, 2006.

[86] M. A. Caprio. Comp. Phys. Comm, 1712107, 2005.
[87] F. Azaiez. Journal of Physics: Conference Series, 492146—152, 2006.

[88] H. Scheit, T. Glasmacher, B. A. Brown, J. A. Brown, P. D. Cottle, P. G. Hansen,
R. Harkewicz, M. Hellstrém, R. W. Ibbotson, J. K. Jewell, K. W. Kemper,
D. J. Morrissey, M. Steiner, P. Thirolf, and M. Thoennessen. Phys. Rev. Lett.,
77:3967-—3970, 1996.

[89] J. A. Winger, P. F. Mantica, R. M. Ronningen, and M. A. Caprio. Phys. Rev.
C, 642064318, 2001.

[90] D. Sohler, Zs. Dombradi, J. Timar, O. Sorlin, F. Azaiez, F. Amorini,
M. Belleguic, C. Bourgeois, C. Donzaud, J. Duprat, D. Guillemaud-Mueller,
F. Ibrahim, J. A. Scarpaci, M. Stanoiu, M. J. Lopez, M. G. Saint-Laurent,
F. Becker, F. Sarazin, C. Stodel, G. Voltolini, S. M. Lukyanov, V. Maslov, Yu.-
E. Penionzhkevich, M. Girod, S. Péru, F. Nowacki, and G. Sletten. Phys. Rev.
C, 662054302, 2002.

[91] S. Grévy, F. Negoita, I. Stefan, N. L. Achouri, J. C. Angelique, B. Bastin,
R. Borcea, A. Buta, J. M. Daugas, F. de Oliveira, O. Giarmana, C. Jollet,
B. Laurent, M. Lazar, E. Liénard, F. Maréchal, J. Mrazek, D. Pantelica, Y. Pe-
nionzhkevich, S. Piétri, O. Sorlin, M. Stanoiu, C. Stodel, and M. G. St-Laurent.
Eur. Phys. J. A, 252111—113, 2005.

[92] Data extracted using the NNDC On—Line Data Service from the ENSDF
database, revised as of March 30, 2006.

[93] B. A. Brown et al. OXBASH for Windows. Technical Report 1289, MSU-NSCL,
2004.

[94] M. Horoi. Central Michigan Shell Model Code (CMichSM): Present and Fu-
ture Applications. In Advanced Computational Methods for Solving the Nuclear

Many-Body Problem. Institute for Nuclear Theory, University of Washington,
2002.

110

 

[95] A. Gade, B. A. Brown, D. Bazin, C. M. Campbell, J. A. Church, D. C. Dinca,
J. Enders, T. Glasmacher, M. Horoi, Z. Hu, K. W. Kemper, W. F. Mueller,
T. Otsuka, L. A. Riley, B. T. Roeder, T. Suzuki, J. R. Terry, K. L. Yurkewicz,
and H. Zwahlen. Phys. Rev. C, 742034322, 2006.

[96] B. A. Brown, 2006. Private communication.

[97] M. Stanoiu, F. Azaiez, Zs Dombrdi, O. Sorlin, B. A. Brown, M. Belleguic,
D. Sohler, M. G. Saint Laurent, Y. E. Penionzhkevich, G. Sletten, C. Borcea,
C. Bourgeois, A. Bracco, J. M. Daugas, Z. Dlouh, C. Donzaud, Zs Flp,
D. Guillemaud—Mueller, S. Grvy, F. Ibrahim, A. Kerek, A. Krasznahorkay,
M. Lewitowicz, S. Lukyanov, P. Mayet, S. Mandal, W. Mittig, J. Mrzek, F. Ne—
goita, F. De Oliveira—Santos, Zs Podolyk, F. Pougheon, P. Roussel-Chomaz,
H. Savajols, Y. Sobolev, C. Stodel, J. Timr, and A. Yamamoto. Nucl. Phys. A,
746:135—139, 2004.

[98] E. K. Warburton and B. A. Brown. Phys. Rev. C, 462923—944, 1992.
[99] B. A. Brown and W. A. Richter. Phys. Rev. C, 722057301, 2005.

[100] T. R. Werner, J. A. Sheikh, W. Nazarewicz, M. R. Strayer, A. S. Umar, and
M. Misu. Phys. Lett. B, 335:259—265, 1994.

[101] R. du Rietz, J. Ekman, D. Rudolph, C. Fahlander, A. Dewald, O. Moller,
B. Saha, M. Axiotis, M. A. Bentley, C. Chandler, G. de Angelis, F. Della
Vedova, A. Gadea, G. Hammond, S. M. Lenzi, N. Marginean, D. R. Napoli,
M. Nespolo, C. Rusu, and D. Tonev. Phys. Rev. Lett., 932222501, 2004.

[102] H. Sagawa and B. A. Brown. Nucl. Phys. A, 430:84~98, 1984.

[103] N Alamanos, F Auger, B A Brown, and A Pakou. J. Phys. C, 24:1541—1546,
1998.

[104] S. Raman, C. W. Nestor, and P. Tikkanen. At. Data and Nucl. Data Tables,
78:1—128, 2001.

[105] B. Jurado, H. Savajols, W. Mittig, N. A. Orr, P. Roussel-Chomaz, D. Bai-
borodin, W. N. Catford, M. Chartier, C. E. Demonchy, Z. Dlouh, A. Gillibert,
L. Giot, A. Khouaja, A. Lpine—Szily, S. Lukyanov, J. Mrazek, Y. E. Penionzhke-
vich, S. Pita, M. Rousseau, and A. C. Villari. Phys. Lett. B, 649:43—48, 2007.

111

 

   

l[fill]l[;[[l[[|]][[[ll[[[l[1[ll