7.

trunk. 31+.

ix $2.. 3!

.3... 2m
A».

i...
1

Ha: .

”53%
vv‘
. tunic...
1.5.1.? 34.3.. 31 s
a. 5.1. «it!
Patszgfiaz 31...}. ._
.5. . x 3.2....
is... nan“ an...“
.. 6.2,“. a»...
5‘ ‘I 1
flu;
iva‘...

Its
.-

r I.
.97:-
..

 

 

[Off] LlBRARY
Michigan State
University J

 

 

 

This is to certify that the
dissertation entitled

INTERMEDIATE-ENERGY PROTON KNOCKOUT TO
PROBE SINGLE-PARTICLE STRUCTURE AND NUCLEAR
SPIN ALIGNMENT IN THE “ISLAND OF INVERSION”
ISOTOPES 3"33MG

presented by

David Miller

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in Physics and Astronomy

 

 

//flw/

, Major Professor’s Signature
7/16/05?

Date

 

MSU is an Affirmative Action/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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5108 K:IPrq/Aoc&Pres/ClRC/DateDueindd

INTERMEDIATE-ENERGY PROTON KNOCKOUT TO PROBE
SINGLE-PARTICLE STRUCTURE AND NUCLEAR SPIN ALIGNMENT IN
THE “ISLAND OF INVERSION” ISOTOPES 31’33MG

By
David Miller

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy

2009

ABSTRACT

INTERMEDIATE-ENERGY PROTON KNOCKOUT TO PROBE
SINGLE-PARTICLE STRUCTURE AND NUCLEAR SPIN
ALIGNMENT IN THE “ISLAND OF INVERSION” ISOTOPES
31,33MG

By
David Miller

The “island of inversion” isotopes 31’3"ng were investigated through a proton-knockout
reaction from 32"MAI at 90 MeV/u at National Superconducting Cyclotron Labora—
tory (NSCL). Single-particle states, with no neutron excitations across the N = 20
shell gap from the sd-shell, were identified through the determination of the par-
tial cross sections in the residue. The single-particle states identified lie above the
ground state agreeing with the placement Of these magnesium isotopes in the island
Of inversion.

Nuclear spin alignment is evident following the reaction as determined by the an-
gular distribution of emitted gamma rays. Angular-distribution analysis constrained
by the outgoing residue longitudinal momentum allows for the determination of mul~
tipolarity when combined with linear polarization measurements. Incorporating mo-
mentum constraints factors out possible systematic uncertainties of determining the
in—beam gamma-ray detection efficiency. Development of digital data acquistion sys-
terns provides enhanced capabilities for angular distribution and linear polarization
measurements. Instrumentation developed for the array of gamma-ray detectors
(SeGA) at NSCL and considerations for next-generation gamma-ray detectors are

discussed.

ACKNOWLEDGMENT

I would like to personally thank the members of the committee for their guidance
through my trek through the preparation Of my thesis: Bill Lynch, Stuart Tess-
mer, Vladimir Zelevinsky, Carl Schmidt, and especially my thesis advisor, Krzysztof
Starosta. Their insight has kept me focused on the task at hand through the toilsome
years of graduate school. I am also grateful to the support staff that made my exper-
iment run smoothly, and to my fellow graduate students who sat shifts to Obtain my
data - especially to fellow member of Team Starosta, Phil. The analysis could not
have been done without the work of other students who have went before me; Russ
Terry’s thesis was particularly helpful in the writing process.

I have enjoyed the camraderie and benefited from many of my peers. Working
through hiccups in problem sets in the early years made things much more pleasant.
I’ve had a host Of wonderful officemates to provide additional input about lab-related
issues, as well as proactively distract me. My fellow housemates at the Neutron
Society have made the last couple years here more cheerful than I ever expected, and
the crowd across several departments that mobilizes for Sunday evenings has also
(generally) made those Mondays just a bit more tolerable.

I am also appreciative of the sanity checks that my “running” club GLH3 has
given me and the support of a number of its members. I’m glad that I’ve left an
impact on the club, and hope the friends I’ve coerced (HPP, SBH, UO, SSL, and
others yet to be named) enjoy it in my absence. Meeting BS and FR in my first
months here was comforting. The Fourth of July shindigs at CBS’s are memorable,
and he has also been particularly supportive and entertaining. Without *HP, it
would have been difficult to find the excitement Of running, to do something crazy
like tackling a marathon, or to rediscover the joy Of bicycling (including all those
wonderful DALMAC experiences). The physical activity that kept me sane through

the years must also give credit to the Tuesday morning basketball gang, even though

iii

I rarely wanted to stir that early in the morning.

My curiosity and thirst for knowledge is clearly accredited to my upbringing. I
remember many of my early learning experiences being attributed to interactions
with my siblings and parents. I am glad that their intellectualpursuits continue and
especially thank the encouragement they have given me throughout the years.

Of course, the National Science Foundation is highly important for the funding
of the research; may their budget ever grow. I also should acknowledge Oak Ridge
Associated Universities for giving me the most memorable experience Of my time at
Michigan State by giving me an opportunity to be a participant in the Lindau meeting
of Nobel Lauereates. I will remember that trip and the people I met there for years

to come.

iv

TABLE OF CONTENTS

List of Tables ................................. vii
List Of Figures ................................ viii
Introduction ...................................................... 1
1.1 The nucleus ................................ 1
1.2 Nuclear structure and shell model .................... 4
1.3 Shell model near the “Island of Inversion” ............... 8
1.4 Nucleon knockout reactions ....................... 10
Methods ......................................................... 15
2.1 Overview .................................. 15
2.2 Isotope production ............................ 16
2.2.1 Ion source ............................. 16
2.2.2 Coupled Cyclotron Facility .................... 17
2.2.3 A1900 ............................... 17
2.3 Particle detection ............................. 18
2.3.1 The S800 focal plane detectors .................. 20
2.3.2 Inverse mapping .......................... 22
2.4 Gamma-ray detection ........................... 23
2.4.1 Gamma-ray angular correlations ................. 25
2.4.2 Gamma ray linear polarization .................. 26
2.4.3 Segmented Germanium Array . - ................. 30
2.4.4 Relativistic kinematics ...................... 31
2.4.5 Velocity determination ...................... 32
Offline Detector Characterization ................................. 34
3.1 Energy calibration ............................ 34
3.2 Efficiency calibration ........................... 37
3.3 Polarization sensitivity determination .................. 39
3.3.1 Detector setup .......................... 40
3.3.2 Gamma-ray angular distribution ................. 42
3.3.3 Scattering asymmetry normalization .............. 45
3.3.4 Sensitivity determination ..................... 45
3.3.5 Figure-Of-merit discussion .................... 48
Proton knockout studies .......................................... 51
4.1 Electronics trigger ............................. 52
4.2 31Mg .................................... 53

4.2.1 Beam characteristics and particle ID .............. 53

4.2.2 Level scheme ........................... 56

4.2.3 Inclusive and partial cross sections ............... 61

4.2.4 Momentum distribution Of residues ............... 66

4.2.5 Angular distribution of gamma rays ............... 67

4.2.6 Linear polarization Of gamma rays ................ 70

4. 2. 7 Spin and parity assignments ................... 71

4.3 33Mg .................................... 74
4.3.1 Beam characteristics and particle ID .............. 74

4.3.2 Level scheme ........................... 75

4.3.3 Inclusive and partial cross sections ............... 80

4.3.4 Momentum distribution Of residues ............... 83

4.3.5 Angular distribution of gamma rays ............... 84

4.3.6 Linear polarization of gamma rays ................ 86

4.3.7 Spin and parity assignments ................... 87

5 Commentary ..................................................... 90
5.1 31Mg .................................... 90
5.1.1 Calculation Of momentum distribution ............. 90

5.1.2 Structure Of Observed excited states ............... 91

5.1.3 Spin alignment calculation and angular distribution ...... 94

5.2 33Mg .................................... 95
5.2.1 Calculation of the momentum distribution ........... 95

5.2.2 Structure Of populated states .................. 96

6 Outlook .......................................................... 102
6.1 Signal decomposition and gamma-ray tracking ............. 103
6.2 SeGA digital electronics ......................... 106
6.3 GRETIN A impact ............................ 112

7 Conclusions ...................................................... 116
A Gamma-ray angular distribution .................................. 118
B Relativistic Kinematics ........................................... 125
B.1 Lifetime .................................. 126
B2 Angle ................................... 126
B3 Energy ................................... 127
B4 Polarization ................................ 127
Bibliography ........................................................ 130

vi

3.1

4.1

4.2

4.3

4.4

4.5

6.1

A.1

LIST OF TABLES

Comparison of SeGA to other detector arrays for polarization measure-

ments. ................................... 49
Secondary beam characteristics. ..................... 52
Gamma-ray transition properties in 31Mg ................ 57

31Mg states with direct feeding from the proton knockout from 32Al. 65
Gamma-ray transition properties in 33Mg ................ 76
33Mg states with direct feeding from the proton knockout from 34Al. 82

Relevant quantities for angular distribution measurements for GRETA
angles assuming ,6 = 0.4 .......................... 113

Maximum alignment coefficients for different types of alignment. . . . 120

vii

1.1
1.2
1.3
1.4
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

LIST OF FIGURES

Chart of nuclides ............................. 3
Single-particle energies in spherical shell model ............. 6
Evolution Of the sd—p f shell gap ..................... 9
Cartoon Of a knockout reaction. ..................... 11
Schematic Of the S800 spectrograph. .................. 19
5800 Focal plane coordinate definition ................. 23
Polarization digaram ........................... 28
Depiction Of the segmentation of a SeGA detector ............ 31
226Ra energy spectrum for calibration .................. 35
Efficiency of SeGA ............................ 38
249Cf decay scheme ............................ 40
my coincidence setup ........................... 41
245Cm gamma ray energy spectrum following the a decay Of 249Cf . . 43
a2 coefficient fit for a-7 coincidences .................. 44
Scattering relative to reaction plane for a SeGA detector ....... 46

Geometric Compton scattering asymmetry for unpolarized gamma rays 47
Normalized Compton scattering asymmetry for polarized gamma rays 48

viii

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

4.21

4.22

Identification of incoming 32Al particles ................. 54

Identification of outgoing residues from incoming 32A] beam ..... 55
31Mg gamma-ray spectrum ........................ 56
Gamma-ray spectrum of 7—7 coincidences in 31Mg ........... 58
31Mg level scheme ............................. 60
Observed doublets in 31Mg ........................ 61
Acceptance cut for 31Mg isotopes .................... 63
Run-by-run inclusive cross section for 31Mg .............. 64
Longitudinal momentum distribution of 31Mg ............. 66
Angular distribution effects Of decays from 221 keV state in 31Mg . . 67
Automatic fitting of gamma-ray peaks in 31Mg ............. 69

Momentum-gated angular distribution of 623 keV gamma rays in 31Mg 70

Compton scattering asymmetry Of gamma rays in 31Mg ........ 72
ag-P diagram for gamma-ray decays in 31Mg .............. 73
Identification of incoming 34A] isotopes ................. 75
Identification Of residues resulting from fragmentation of 34Al beam . 76
Gamma-ray spectrum for 33Mg ..................... 77
Gamma-ray spectrum for 7-7 coincidences in 33Mg .......... 79
33Mg level scheme ............................. 80
Acceptance cut for 33Mg isotopes .................... 81
Run-by-run inclusive cross section for 33Mg .............. 82
Longitudinal momentum distribution for 33Mg residues ........ 83

ix

4.23 Momentum-gated angular distribution for 33Mg gamma rays .....

4.24 Compton scattering asymmetry of gamma rays in 33Mg ........

4.25 ag-P diagram for decays in 33Mg ....................

5.1

5.2

5.3

6.1

6.2

6.3

6.4

6.5

6.6

6.7

A.1

A.2

Effect Of quadrupole mixing on the angular distribution coefficient . .

Convolution Of theoretical momentum distribution and experimental
effects ...................................

Nilsson diagram used for 31’33Mg ....................
Induced signals recorded by digital data acquisition system ......
HPGe drift velocities ................. i ..........
Timing characteristics for different channels Of DDAS .........
Energy-time matrix recorded by DDAS .. ................

Segmentation Of SeGA based on the drifting of electron holes to seg-
ment electrodes ..............................

Grid points for waveform sampling ...................
Angular distribution sensitivity .....................

Angular distribution coefficients and linear polarization values for se-
lected transitions .............................

Angular distribution coefficients and linear polarization values for dif-
ferent prolate alignments .........................

84

86

88

93

96

100

104

105

108

109

110

114

123

Chapter 1

Introduction

1. 1 The nucleus

One of the aims of science is to understand the composition of matter which makes
up our planet, solar system, and universe. With increasing knowledge, mankind has
broken matter into its smaller building blocks. Molecules are composed Of atoms,
and at the heart of every atom lies the nucleus. The properties of the nucleus are
determined by its constituent parts which consist of chargeless neutrons and positively
charged protons. These constituents are collectively known as nucleons. A nucleus is
a quantum many—body system with a given number Z of protons and N of neutrons
which characterize it as a particular isotope. A specific nucleus is Often referenced by

24x N where A is the total number of nucleons and X is the chemical symbol.

The properties of nuclei are Of fundamental interest to understand the underlying
forces which are responsible for binding the nucleons together. Nuclei are small dense
Objects with sizes on the length scale Of femtometers (10"15 meters) and masses of
approximately A GeV/c2. Each isotope has a binding energy B(N, Z) as a result
of the forces interacting within the nucleus. In a simple model, these macroscopic

properties can be calculated considering the nucleus as a liquid drop [1] including

1

volume and surface effects. Coulomb repulsion and strong pairing of the nucleons
also must be treated tO reproduce the Observations. The strong pairing of nucleons in
time—reversed orbits leads to an enhanced binding Of isotopes with an even number

Of neutrons or protons.

The forces in the nucleus conspire tO bind certain nuclei more than others. The
nucleus could be more bound if a neutron changed to a proton or vice versa. This
process is governed by fi-decay and causes some nuclei along a line with the same
A to be stable. Other isobars having the same A decay towards this valley of sta-
bility by sequential ,8 decays. For a given element, only a small number of nuclides
are stable while a greater number are unstable. This is shown in the chart Of the
nuclides (Fig. 1.1) which displays all the isotopes with different neutron and proton
numbers. Heavier isotopes can also be unbound through other processes. These
include a-decay where a doubly-charged 4He tunnels through the nuclear potential
and escapes the nucleus as well as fission where the nucleus separates into two or
more pieces. For nuclei with extreme proton-to-neutron ratios relative to the valley
of the stability, individual nucleons can be unbound and emitted by the nucleus. The
proton (or neutron) dripline is where it is no longer possibly to add an additional
bound proton (or neutron), and the nucleus will decay by particle emission. In part,
this can be characterized by the difference of binding energies for-neighboring iso—
topes. These are called the proton separation energies (3,, = B (N, Z) — B (N, Z — 1))
and neutron separation energies (Sn = B (N, Z) - B (N — 1, Z )) However, the rate of
charged particle decay can also be hindered by the probability of tunneling through
the Coulomb barrier causing these to be particle-stable and proceed to the valley Of

stability through beta decay.

Beyond a naive macroscopic model, microscopic effects must be accounted for
which impact the structure Of the observed nuclei. The nucleus is a strongly interact-

ing system where several forces of the Standard Model converge. Theoretical models

2

 

 

 

100 - : y —
I. ‘ll ,_/:’«‘“'
80 - - - - - _
g ' ‘,/A/
E I
3 .. ,
c 60 - - ' r-process -
c ‘ . I
3 rp-process -' _ ‘.'----J--
o .
I- . l
O. 40 - , ~ -
I '
20 - _, i I Observed nuclei —
I Stable nuclei
0 ' I I I I I l L"

 

0 20 40 60 80 100 120 140 1 60 180
Neutron number

Figure 1.1: Chart of the nuclides with proton number Z on the ordinate and neutron
number N on the abscissa. Stable isotopes are shown in black while unstable nuclei
are shown in grey. Many of these unstable nuclei have been observed (light gray), but
a large number of these nuclei are outside the current reach of experimental facilities
with the predicted boundaries illustrated in dark gray. Magic numbers with increased
binding energy are shown as dashed lines. Several astrophysical processes are also
shown which involve nuclei far from the line of stability.

must treat the strong, weak, and electromagnetic interactions tO describe the prOper-
ties of nuclei. Since the strong force acts non-perturbatively, the exact mathematical
form of the nucleon-nucleon interaction cannot be calculated explicitly. Instead, the
form is determined in the context of several different approaches either theoretical or
phenomenological [1]. Different approaches come to comparable results which agree
well with nucleon-nucleon scattering data across a wide range of energies. The nu-

0—15 meters.

cleon potential must be strong and short—ranged on the range of a few 1
It must also have a repulsive core at small distances. Two important factors which
add much to the richness of nuclear structure are the presence of a strong spin-orbit
interaction as well as a tensor interaction which allows mixing between states of dif-
ferent orbital angular momenta. Early phenomenological models fit the interaction
strength to scattering data with four parameters with quite a bit of a success as in the
Hamada-Johnston potential [2]. Modern phenomenology has fine tuned to the po—
tential to accommodate further data by adding some further parameters such as the
Argonne v13 potential [3]. Ab initio approaches attempt to derive properties of nuclei
by folding the nucleon—nucleon interaction over all the constituents of the nucleus.
However, this becomes computationally intractable for nuclei with masses A Z, 10.
For heavier nuclei, a shell model is Often employed which approximates the nucleon-
nucleon interactions with a spherical potential well which is phenomenologically fit to

reproduce key nuclear prOperties. Residual interactions are then considered between

a selected subset of the nucleons.

1.2 Nuclear structure and shell model

Nucleons have an intrinsic angular momentum, s = 1/2h, known as spin. Inside
the nucleus, this spin couples to the orbital angular momentum, E, Of the nucleon to

form the total angular momentum j. In the nuclear shell model, these nucleons fill

4

orbitals denoted by Tnéj where 7' describes the type of nucleon (u for neutrons and
7r for protons). These single-particle states are bunched in groups according to their

principal quantum number n in a harmonic oscillator potential.

All the nucleon single-particle states for a given nucleus are coupled to a final total
angular momentum Of J (colloquially referred to as spin). Due to the strong pairing
in nuclei, nuclei with an even number of protons and neutrons always have a ground
state spin of 0+. For Odd-A nuclei, the nuclear spin is dominated by the effects
Of the unpaired nucleon. Furthermore, the single-particle orbitals have a definite
parity 7r = (—1)€ which describes the properties Of the wavefunction under spatial
inversion. The parity of the nucleus is the simple product of all the individual nucleon
parities. Measurement Of the parity gives access to determining the nature of the

orbital angular momentum Of the nucleon.

Each of these orbitals also has a corresponding single-particle energy within the
mean field of the nucleus. Similar to electron orbitals in the noble gases, certain
“magic” numbers of nucleons are considerably more stable [1]. This arises from a gap
in the single-particle energies at the top of the Fermi surface. Isotopes which have
a magic number Of both neutrons and protons are called doubly-magic, while those
that have a magic number of one or the other are known as semi-magic. Magic nuclei
tend to be more spherical, and for even-even nuclei possess a higher excitation of the
first excited state (E (2?» and a smaller reduced transition probability (B (E2; 2iF —>
0?». These magic numbers (canonically 2, 8, 20, 28, 50, 82, and 126) create a
clear division of the single-particle orbitals into different shells as shown in Fig. 1.2.
With these magic numbers, there exists a handful of particle-bound doubly-magic
nuclei (4He, 160, 40Ca, 48Ca, 56Ni, 78Ni, 100Sn, 132Sn, and 208Pb). These shells are
referred to by the single-particle orbital angular momentum values included above
the previous closed shell; i.e. s-shell, p—shell, and sd—shell for the shells up to 2, 8,

and 20 single-particle orbitals, where s, p, d, f, g, . . . correspond to f’ = 0, 1, 2, 3,4, . . .

5

 

 

 

 

 

 

 

 

4

3s _ .7: 1 — -3SI/2 2

4 2d 2‘ I,’1g7/2 8
’““- 2d5/2 6

 

Ig (’

6
9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
‘199/2 10
2p I ’1f5/2 6
‘ — - ~ I
3 T :4. ‘ P ‘ 291/2 2
1f <: ‘ 293/2 4
‘1f7/2 8
28 _ , , -1d3/2 4
2 1d ’,:"“-281/2 2
IIIIII 1d5/2 6
1 1p ——————— 191/2 2
..... 193/2 4
0 Is ------- 131/2 2
N nl nlj 2j+1

Figure 1.2: Shell model denoting single—particle energies up to the 1211/2 orbital be—
fore (left) and after (right) including spin-orbit effects [4]. Circled magic numbers
correspond to the gaps in the single-particle energies.

respectively. Each orbital has 23' + 1 magnetic substates after which the orbital is
filled. When all the orbitals in a shell are filled, that shell is closed, and the addition
of a further nucleon requires significantly more energy due to the gap in the single—

particle energies.

Recent experiments with rare isotope beams have shown that the magic numbers
which apply near stability can evolve as one proceeds towards more exotic isotopes [5].
These experiments indicate the weakening of the existing shell closures and the ap-
pearance of new magic numbers [6,7]. Furthermore, the structure of the nuclei near
the dripline can exhibit curious effects such as proton/ neutron skins or halo states
such as in 11Li [8]. Since the nuclear potential is connected tO the spatial distribution
Of the nucleons, these exotic structures cause a modification in the mean field in the

nuclear medium.

Due to the short range nature of the strong force, the mean nuclear potential can

be approximated by a Woods-Saxon function:

T—T‘

 

fws(7‘) = VII/(1 +exp< 0)) (1.1)

with a depth V0, radius 1‘0 and diffuseness a. A large spin-orbit term must be included
as well in order to reproduce the Observed shell gaps shown in Fig. 1.2. The Woods-
Saxon potential, particularly the depth, is adjusted to reproduce the observed single-
particle energies [5,9]. Where single-particle energies are not available, they can be
extracted in a Hartree—Fock calculation with a self-consistent interaction based on

effective energy—density functionals such as the Skyrme SkX interaction [10].

Once the properties of the single-particle orbits are derived, a residual two-body
interaction remains in the Hamiltonian which is adjusted to reproduce other exper-
imental observables. The nuclear wavefunctions are determined by diagonalization

of this Hamiltonian. The residual two—body Off—diagonal matrix elements between

7

single-particle orbitals result in an additional correlation energy for a given eigen-
state. To allow calculations to be computationally feasible, approximations about
the single-particle states must be made. Generally, the nucleus is considered tO have
an inert core where the orbitals are fully occupied and interactions are minimal such
as in a nearby doubly-magic nucleus. The additional neutron and proton particles or
holes are then built onto this inert core. Furthermore, the number of single-particle
states must be truncated to a certain valence space. The two-body matrix elements
are then calculated phenomenologically within the model space outside the inert core.
However, the assumptions about the core and valence space depend on the nucleus
of interest, so the derived effective interaction is only applicable to a certain region
of the nuclear landscape [5]. In the present work, the region of interest is centered
around Z = 12 and N = 20. TO describe states with normal single—particle ordering
(as in Fig. 1.2), the Universal SD (USD) interaction [11] is used which assumes an
inert core of 160 or 40Ca and includes the 1d5/2, 231/2 and 1d3/2 orbitals in its model

space.

1.3 Shell model near the “Island Of Inversion”

For the isotOpes with Z w 8 to Z x 13, the shell structure evolves considerably for
neutron-rich nuclei. Here, the N = 20 shell closure is considerably weakening, while
experiments show a pronounced N = 16 subshell closure [7] (see Fig. 1.3). With
the weakening of the N = 20 shell gap, neutron excitations across the gap appear at
relatively low excitation energies. The states built on these neutron excitations are
called intruder states, and are referred to as npmh states by the number Of neutrons
n excited into the fp-shell and remaining m holes in the sd-shell. TO treat these
states properly, the 1 f7 /2 and 2193/2 orbitals must be included in calculations greatly

increasing the model space.

 

 

 

 

 

 

 

 

 

 

 

 

 

_ —- q- _
’>‘~ 8 (a) }pf shell (b)
a. 303. —— __—};;she" 2.0
E 14 16 I,” 3/2 a 16
w 4' ,x‘“ .® 1
(013 Ode/2
LU
O '— 1S1/2 _" 1S1/2 _
C
( ) j<= l — 1/2
j): l + 1/2
proton neutron

 

 

 

 

 

Figure 1.3: Effective single-particle energies for N = 16 isotones for silicon (a) and
oxygen (b). Note the shift in the 0d3/2 orbital which is driven by the interaction

shown in (0) between protons and neutrons with the same orbital angular momentum
but opposite spins. This proceeds through the 07' Operator (d) (adapted from Otsuka
et a1. [12])

For the nuclei with Z w 11 and N z 20, the correlation energy from the residual
two-body interactions makes these intruder states the energetically preferred configu-
rations. This region is known as the “island of inversion” [13] where the ground-state
wavefunction is dominated by particle-hole excitations across the N = 20 shell gap.
Experimental evidence Of the breakdown of the N = 20 shell closure is given by a
small 21* energy in 32Mg [14] and an increase in collectivity [15,16]. Furthermore, ex-
citation across the N = 20 shell gap are necessary to appropriately describe the mass
systematics in the region [17]. To describe these cross-shell excitations, the T = 0
proton-neutron interaction is a major factor especially between the (15/2 protons and
d3/2, f7/2 neutrons [13]. The mechanisms that influence this region are also believed

to be important in other regions of the nuclear landscape where inversion has been

9

Observed, such as in 12Be [18,19], or is theoretically predicted to occur [13].

The boundaries of the island of inversion have been proposed by including the
entire fp—shell into the model space [13] and deducing that increased pairing and
proton-neutron interactions play a significant role in the region. Theoretically, the
extent was originally suggested to lie between 10 S Z s 12 and 20 S N S 22.
However, recent measurements Of the ground state spins of magnesium isotopes (Z =
12) have contributed information about the energetically favored configurations in
this region [20—22]. With the additional information provided by data from neutron
knockout [23] and Coulomb excitation [15,16], the isotones with N = 19 are also
included in the island of inversion. Furthermore, recent results indicate intruder
configurations in more neutron-rich 36Mg [24] and less neutron-rich 28Ne [16,25]
extending the reach Of the island further than originally anticipated [13]. Determining
the boundaries and mechanisms in the island of inversion is an important test Of

theoretical models which predict properties far from stability.

1.4 Nucleon knockout reactions

Nucleon knockout reactions are a powerful spectroscopic tool at intermediate ener-
gies to study nuclei. After the removal of a nucleon from an incoming projectile, the
longitudinal momentum distribution of the residue is sensitive to the orbital angular
momentum Of the removed nucleon [26] as well as the distribution of magnetic sub-
states in the residue. The partial cross sections to states populated in the residue also
contain information about the wavefunction overlap between the neighboring nuclei.
In the transitional region near the island of inversion, this can selectively probe states
with intruder-like or normal configurations.

In a typical neutron—knockout reaction (Fig. 1.4), a nucleus Of mass A impinges

On a light nuclear target (such as 9Be or 12C) and the reaction residue Of mass A — 1

10

Knockout

' t'l
PrOJec I e residue

    
 

Gamma ray

9Be

 

Target

Figure 1.4: Cartoon of a knockout reaction.

exit moving at nearly the same velocity of the Original projectile. The change in
velocity is related to the momentum of the removed nucleon. The residue frequently
leaves in an excited state which decays by gamma-ray emission. For energies above 50
MeV/ nucleon, the reaction mechanism is simplified and can be considered to proceed
in a single step. If the removed nucleon is not observed, the reaction encompasses
a number of breakup channels. The dominant processes are diffractive breakup and
stripping. For the stripping reaction, the nucleon is absorbed by the target as opposed
to diffractive breakup where the nucleon and core interact elastically in the nuclear
mean field of the target. Knockout is predominately a peripheral processes which

probes the nuclear surface removing the most weakly-bound valence nucleons.

For a given reaction on a single scattering center, the cross section relates the rate
of outgoing flux per unit surface area relative to the incoming flux of particles. Given
a target of thickness T and density p with molar mass m z A - 103 mg, the number
of scattering centers per unit area is pTNo / m where N0 is Avogadro’s number. The

nominal thickness t is often quoted in units of mg/cm2 or t = pT. If there are N,-

11

incoming particles and N f outgoing residues, the total cross section is then
a = ———. (1.2)

Nuclear cross sections are typically quoted in units Of barns (10‘24 cm2) with knock-

out reactions having cross sections around several millibarn.

The cross section to individual states is sensitive to the structure Of the incoming
and outgoing nuclei, including angular momentum selection rules. For the removal
of a nucleon with total angular momentum j, the final states in the nucleus, (J 7') f
must follow the selection rules:

[Jr—Jl <Jf<lJi+Jl (1.3)

7? f = 1r1'(—1)€

For the population of different final states, the reaction can proceed by the removal
of different valence nucleons with quantum numbers K. = (726]). For an initial spin
Of 0+ and final spin Of J 7', one only consider contributions from single-particle states
which can couple to a total spin of J 7'. This allows for the deduction of spectroscopic
factors, 0231-09), for the ith excited state of the core. The total cross section, 0,, can
then be determined by summing the contributions from different valence nucleons

given single-particle cross sections, asp, calculated by a reaction theory.

a,- = Z 025,-(A)asp(sn, I.) (1.4)

The cross sections to individual magnetic substates, am, for a given J”r are of
particular interest in the current work. The distribution of magnetic substates is
used to calculate the alignment parameters for 'y-ray angular distribution and linear

polarization discussed in Sec. 2.4. Here the coupling of the core spin to the valence

12

nucleon becomes important. Consider the core spin (cry) where 7 = cz coupled
to the valence nucleon (j, m) resulting in a total spin of (J, M). For a core with
(c, 7) = (0, 0), the cross section for the removal of a valence nucleon with a particular

substate can be written

 

amc) = g: 3:: lthsaljm) aw) (1.5)
with the sum taken over the projections of the angular momentum, A = Q, and
spin, a = 32, of the valence nucleon onto the quantization axis which is taken as the
velocity Of the outgoing residue. The cross sections with respect to the substates Of f,
0A“), are extracted from the reaction theory assuming an interaction of the valence
nucleon with the target nucleus which is independent of spin coupling and angular
momentum coupling. Then given a core with (c,’y) 76 (0,0), the cross section to the

core magnetic substates is

 

 

=23], 11(J'mccrlJM)2 Um(] ) (1-6)
m,M2
Evoking (Eq. 1.5) this is expressed as:
26 +1 . .
me) = Z (JmmlJM)2(l»\salJm)20i(€)- (1.7)
m A! A 0' 2] + 1

Here the single-particle cross section, asp, can be related to the substate cross sections

through
Usp = 20AM) 2" 207(0) = 207710) (1-8)
A ’7 m

and the total cross section determined using Eq. 1.4 by the spectroscopic factors of the
individual valence nucleons in the incoming nucleus and single-particle cross sections

derived from the reaction theory.

13

For knockout reactions with intermediate- tO high- energy projectiles with velocity
v, the single-particle cross sections are calculated assuming a process that is subject
to the sudden and Eikonal approximations. The sudden approximation assumes that
the interaction time in the target is short enough that multi-step processes, includ-
ing excitation of the core or core-nucleon system, can be neglected. In the eikonal
approximation, interactions with the target are assumed to not deflect the trajectory
Of the fast-moving incoming projectile. Following the interaction, the initial state
is modified by the Eikonal phases of the core and the nucleon, Sc and 8,, respec—
tively, determined as a function of impact parameter I; by the potential V for the

nucleon-target and core—target systems by

3(5) = exp (if [00 V(E,z)dz). (1.9)

—00

The cross section is then determined using the methods described by Tostevin [27]
including stripping and diffractive components. The interaction potential is deter-
mined by folding the nucleon-nucleon interaction over the densities Of the core and
the target. These densities are taken from Hartree—Fock calculations which repro-
duce a number of observables in the area. These knockout cross-section calculations
have been instrumental in the interpretation of experiments with exotic nuclei [26,28]
agreeing well with data provided by transfer reactions as well as high-energy knockout

reactions such as (p, 2p) and (e, e’ p).

14

Chapter 2

Methods

2. 1 Overview

A number of techniques must converge to successfully elucidate the structure of a
particular nucleus 9X with A total nucleons. First, one must produce the nucleus
of interest. There are several methods capable of producing exotic nuclei. In the
present work, a beam of stable nuclei is fragmented on a beryllium production tar-
get. Fragmentation Of a stable beam at intermediate energies (~ 100 MeV/ u) is
well described by a statistical model [29] producing all lighter nuclei resulting from
the stripping of nucleons from the projectile. Once a secondary beam of interest is
produced, it impinges on a secondary target where a more direct reaction such as
the removal of a single nucleon is analyzed. The nature of the reaction is important
to Obtain further structural information of both the incoming nucleus as well as the
outgoing residue. Therefore, we must detect and identify both the incident and resul-
tant particles. Furthermore, the reaction often leaves the residues in an excited state.
The nucleus can then decay by 7—ray emission towards the ground state. The gamma
rays emitted contain key information about the fundamental properties of the excited

state and must be detected accordingly. Alternatively, the nucleus could decay by

15

particle emission if excited more than the particle separation energy or by O decay
which proceeds slowly by the weak process. For the in—flight techniques discussed
here, full relativistic inverse kinematics must be properly taken into account for the

analysis as well.

2.2 Isotope production

The experiment was performed at the Coupled Cyclotron Facility (CCF) at National
Superconducting Cyclotron Laboratory (NSCL). There are several important steps
in the isotOpe production process. A high-quality primary stable beam must be first
produced. Generally, a low-energy beam is extracted from an ion source. For heavy-
ion beams, an electron cyclotron resonance (ECR) ion source is generally used such
as ARTEMIS at NSCL. This beam of nuclei is then accelerated to relativistic speeds
in the cyclotrons and finally fragmented on a target in order to produce the particular
isotope. The reaction products must be filtered out to obtain a beam of acceptable
purity for the experiment. Magnetic separation by the A1900 fragment separator

provides the isotope filtering at NSCL.

2.2. 1 Ion source

The 48Ca primary stable beam necessary for the experiment was extracted from a
room temperature ion source, ARTEMIS [30].The stable ions are baked off in an oven
into a plasma of microwave-driven electrons. The electrons collide with the calcium
and ionize them. A bias voltage of up to 30 kilovolts is applied to extract the ions.

A single charge state is then selected and injected into the coupled cyclotrons.

16

2.2.2 Coupled Cyclotron Facility

The Coupled Cyclotron Facility [31] allowed for the acceleration of the stable ions.
The CCF consists of two superconducting cyclotrons, the K500 and the K1200.
The cyclotrons consist of strong magnetic fields such that the particles follow an
isochronous path inside the cyclotron. An electric field is applied at a given radio
frequency for the acceleration of the nuclei. The ions are extracted from each cy-
clotron when the magnetic field is no longer strong enough to constrain the particles.
For protons, the extraction kinetic energy in MeV is equal to the “K” designation
Of the cyclotrons. For heavier ions, the limit is related to their charge—tO-mass ra-
tio (oc (Q/A)2). Therefore, a high charge state is preferable, as Opposed to the low
charge states typically produced in ECR ion sources. TO overcome this obstacle, the
two cyclotrons are coupled to each other. After being accelerated to about 15% of the
speed of light in the K500, the nuclei impinge on a thin carbon foil which strips away
remaining electrons. The nuclei can then be accelerated to the highest capability by
the second, more-powerful, K1200 cyclotron to a speed Of approximately 50% of the

speed Of light.

2.2.3 A1900

After leaving the cyclotrons, the beam impinges on a thick beryllium fragmentation
target where nucleons are removed from the nucleus. The smattering of isotopes
must then be filtered using the A1900 projectile fragment separator [32] Operating on
the principle of magnetic separation. The A1900 is a series of four dipole magnets
which bend the nuclei according to their magnetic rigidity Bp. The magnets are
tuned by adjusting the magnetic field such that the particles of interest bend the

correct amount in each dipole. The magnetic rigidity is connected to the velocity and

17

charge—tO-mass ratio Of the particles.

B _£_A’7,BC:L_
p_Qe_ Q 8?

 

(2.1)

where u/e is the mass-tO-charge ratio Of a proton. However, this effectively trans-
ports all nuclei with a similar mass-tO-charge ratio leading to contamination Of the
secondary beam. To ameliorate this situation, a wedge Of aluminum is inserted be-
tween the second and third dipole magnets of the A1900 to slow down the fragments.
At intermediate energies, the stopping power is mostly due to electronic stopping
which is primarily sensitive to the charge of the incoming particles and target mate—
rial properties (dE/da: or Z 2) [33]. This allows for further separation of isotopes by
choosing the proper magnetic rigidity for the dipoles upstream and downstream from
the wedge. Slits are placed on the magnets’ dispersive axis to select the isotopes of

interest.

2.3 Particle detection

The particles incident on the secondary target must be identified, and the properties
of the emerging residues must be measured in order to correlate emitted gamma rays
with the proper reaction channel. The scattering of the residue also is related to the
nature of the nuclear reaction as well as the internal structure Of the nucleus. To
analyze the residual nuclei, the 8800 spectrograph was used [34]. The S800 is a large-
acceptance, high-resolution magnetic spectrograph with an angular acceptance of 20
msr corresponding to roughly 7° in the dispersive plane and 10° in the non-dispersive
plane. In focused mode, it also allows for a momentum acceptance of up to 6%. The

secondary beryllium reaction foil was placed at the target position Of the S800 (see

Figure 2.1).

18

Focal Plane

It?

.fifle
Re

Int. lmagei%

 

Figure 2.1: Schematic of the 8800 spectrograph.

A thin plastic scintillator was placed in the object box of the S800 as well as
at the end of the A1900 fragment separator. The time of flight between these two
scintillators allows the identification of the incoming particles. The electronics for
the measurement were triggered by the focal plane detectors of the 3800 [35]. At the

$800 focal plane, the residue is identified and its properties measured.

The 3800 spectrograph has two standard operating modes: focused and dispersion-
matched. In focused mode, the analysis line is achromatic causing the beam to be
focused at the target and dispersed at the focal plane of the S800. This immediately
limits the momentum resolution of the spectrograph to the incoming momentum
width of the beam. The S800 could Operate also with the dispersion matched causing
the beam optics to be focused at the focal plane, leading to a better momentum
resolution. However, in dispersion—matched mode, the beam is dispersed at the target
reducing the momentum acceptance for the reaction channel of interest to 0.5% for a
5 cm reaction target since the beam is dispersed according to its incoming momentum

at the target position. With this restraint, it was necessary for the current experiment

19

to run in focused mode.

2.3.1 The 8800 focal plane detectors '

The focal plane Of S800 consists Of several detectors necessary to measure the prop-
erties of the outgoing nuclei. Two position-sensitive cathode readout drift chambers
(CRDCS) track each particle’s trajectory through the focal plane. These CRDCs
are followed by an ion chamber which measures the energy lost by the particle as it
passes through a low pressure gas. At the end Of the S800 spectrograph are several
scintillators which prove a common timing signal to the acquisition electronics.

As the energy loss in the ion chamber depends on the charge of the nucleus,
this allows for isotopic identification. The energy loss also depends on the velocity
of the nucleus, but this difference in velocities of the transmitted isotopes is small
making this dependence negligible compared to the difference in Z. The time of flight
between the scintillator in the Object box and the focal plane scintillators allows for
a determination of the A/Q of the projectile residue, which combined with the ion
chamber gives the exact particle identity. It is important to note that for the low Z
projectiles at relatively high energies, contributions from different charge states can be
neglected in the particle identification as calculated by the CHARGE program [36].

The efficiency of the particle identification must also be determined, taking into
account the transmission in the analysis line and the spectrograph. Several normaliza-
tion runs were performed throughout the experiment where a low-intensity unreacted
primary beam was delivered to the focal plane of the S800 spectrograph. The intensity
is purposely attenuated as the CRDCS are rate-limited to less than 5000 counts per
second. During normal data runs, the high rate of the primary beam would overload
the CRDCs, so a blocker is inserted to stop the primary beam from reaching the focal
plane. The beam intensity at the target must then be determined by the intensity

Observed at the scintillators upstream Of the target. The normalization runs give the

20

required relation between the scintillator intensities and the intensity at the reaction
target which is necessary to extract the cross section. The number of particles in-
cident on the target on a run-by-run basis is determined from the individual rates
on the scintillators prior to the target folded with the analysis line transmission effi-
ciency. Variations in the normalization across the span of the experiment contribute
to systematic uncertainties and are discussed in further detail in Sec. 4 and the effects

shown in Figs. 4.8 and 4.21.

Outside the identification of the particles, the kinematics of the residues at the
target must be reconstructed. The two CRDCs at the focal plane do the lion’s share of
this task. The CRDCs are filled with a gas which is ionized as the high—energy particles
pass through them. A high bias voltage causes the electrons to drift towards an anode
wire. The position along the non-dispersive y axis is determined by measuring the
drift time Of the electrons to the anode wire. In the dispersive :c direction, the position
is determined by examining the charges produced on pads mounted along the anode
wire. With these techniques, a 0.5 mm nominal resolution [35] can be achieved in both
directions for beams at a low intensity. The two CRDCs are spaced approximately
one meter apart which allows for the measurement of the angle of the particles at the

focal plane.

The response of the CRDCS must be calibrated with particles with a well-known
position in the focal plane. Several calibration runs were taken with a thick mask
inserted in front of each CRDC. This collimated the beam to discrete points across
the focal plane. The beam Of particles was detuned and swept across the focal plane
to ensure a comprehensive coverage of the focal plane. The known (as, y) locations
Of the holes in the mask were then fit linearly to the CRDC electronics readout to
Obtain the desired calibration. Over a week of running time, the calibration can drift
slightly due to the changes in the operating environment. This motivated several

CRDC calibration runs with the interim data calibrated by a linear interpolation of

21

the parameters determined by the relevant calibration runs which had a difference of
about 10%. As the y position is determined by the drift time of the electrons, this
measurement is slightly sensitive to the timing Of the trigger which can vary dependent
on the triggering conditions. Slightly different calibration parameters were extracted

and used for the two sets Of triggering conditions (see Sec. 4.1) on a run-by-run basis.

2.3.2 Inverse mapping

The position-sensitive CRDCs measure the locations and angles of the particles at
the focal plane Of the 8800 on both the dispersive and non-dispersive axes. For
this information to be useful, the tracks of the particles must be traced backward
to the target. This is done by an inverse map which is calculated using COSY
INFINITY [37] with the known magnetic rigidity Of the spectrograph. This connects
the four focal plane parameters determined at the CRDCs to the kinematics of the
outgoing particle at the target. The map is calculated to the fifth order in the focal
plane parameters: dispersive position (:rfp), non-dispersive position (yfp), dispersive
angle (afp), and non-dispersive angle (bfp) which are shown in Fig. 2.2. Principally,
the momentum vector of the particle leaving the target provides the most valuable
information. The map therefore calculates the angle of propagation out Of the target
(ata, bta) and the energy deviation from the central track in the spectrograph (dta).
The non-dispersive position at the target (yta) is calculated as well. Since the track
of the particles through the spectrograph depends on their particular A and Q, an
inverse map was calculated for each reaction channel.

The inverse map only treats the kinematics of the particles exiting the target. TO
investigate the sensitivity to incoming beam parameters, GEAN T4 [38] simulations
were performed. Reasonable agreement with the data was reached for the unreacted
aluminum isotopes with little change to the ideal incoming beam parameters. The

bta resolution of the spectrograph in focused mode contributes significantly to the

22

 

     

1 m :1

  

.,
ma

   

 

  
 

.":"‘}_{:3éMr

  

g: «mt?

Figure 2.2: 8800 Focal plane coordinate definition

observed experimental width. Since the system is azimuthally symmetric about the
beam axis, a uniform distribution of the azimuthal angle ab = tan—ICE???) is also
3

expected and observed in the data.

2.4 Gamma-ray detection

The electromagnetic interaction is the best understood of all the fundamental forces.
Nuclei excited to an energy below the particle separation threshold predominately
decay by gamma-ray emission. These emitted gamma rays are sensitive probes to
what is occurring within the nucleus. These nuclear transitions can also be calculated
explicitly due to the precise understanding of the interaction allowing for rigorous tests

of theory.

23

The detection of gamma rays in an experiment is a task which must balance the
energy resolution of the detectors against the detection efficiency. This is directly
related to the importance Of detector angular granularity necessary to be able to
correct for the DOppler shift for nuclei moving at relativistic speeds in the lab frame.
Another factor which plays a large role in gamma-ray detection is the reduction of
the signal-tO—noise ratio as a significant background of gamma rays exists for in—beam
studies. The background comes from several sources including bremsstrahlung of
the ions as they traverse the target, naturally occurring radioactive sources, as well
as photons which scatter out Of the detector without depositng their full energy.
Segmented germanium detectors, like those used in the present experiment, have an
excellent energy resolution while also allowing for good angular resolution for precise

Doppler correction.

In general, the gamma rays emitted in nuclear processes have energies in the range
of about 100 keV to several MeV. Gamma rays with these energies interact with
a detection material through three processes which deposit energy in the material:
photoabsorption, Compton scattering, and pair production. Each Of these interaction
cross sections has a different energy dependence which divide the energy regime into
three general regions. Particularly for interactions in germanium, photoabsorption
is the primary process below 150 keV. At medium energies, the Compton scatter-
ing cross section increases above the photoabsorption cross section, being an order
of magnitude higher around 350 keV. At the electron production threshold (1.022
MeV), pair production becomes possible, but remains a factor of 10 smaller than
the Compton scattering cross section until 2.6 MeV [39]. For typical detectors and
gamma-ray energies, the cross sections are such that a gamma-ray typically Compton

scatters several times before a photoabsorption event.

Emission Of the gamma rays from a nucleus depends on the electromagnetic tran-

sition matrix elements which are connected to the internal structure Of the nucleus.

24

As these matrix elements are heavily energy dependent (~ Eu“), additional infor-
mation can also be extracted based on the lifetimes of the involved excited states. The
expected Weisskopf transition rates [40,41] can provide estimates of the multipolari-
ties of the transition. These estimates can then restrict the possible nuclear spins and
parities Of the initial J27r and final J; states if one Of the spins is previously known.
This is governed by the selection rules for a gamma ray transition Of multipolarity L

and character 7r,

IA—ugngA+L

(—1)L, 7r electric (2'2)
A7r =

(—1)L+1, 7r magnetic.

2.4.1 Gamma-ray angular correlations

The correlation between the outgoing gamma rays and other observables provides
clear information about the nuclear structure. Following a nuclear reaction, the nu-
clear spin can be aligned along an axis from the selective population of magnetic
substates (see Sec. 1.4). This resuls in angular correlations for gamma rays emitted
from an excited state. The angular distribution of a gamma ray transition provides
information about the spins of the initial and final states. Furthermore, gamma rays
can be emitted linearly polarized which is inherently connected with the difference
of parities between the two states. In addition, these observables are related to pa-
rameters of the reaction that describe how the initial state is populated. Measuring
gamma-ray angular distributions has been an important tool in nuclear structure
studies at beam energies near the Coulomb barrier for a long time [42]. Recent in—
vestigation has begun to investigate the extension Of this to radioactive beams with
a velocity v z 0.4c [43]. The ability to measure angular distributions at these in-

termediate energies provides an Opportunity for spin assignment in the most exotic

25

nuclei at the extremes Of isospin. Details of the angular distribution formalism are
discussed in App. A and more thoroughly in Refs. [42—44]. In general, the angular
distribution of gamma rays, W(0), can be expanded in terms of Legendre polynomials
for a system with azimuthal symmetry. The form of the expansion depends on factors
which quantify the amount Of initial spin alignment in the system, B A: and a factor,

A A, related to the properties of the particular y—ray transition:

W(0) = Z AABAPA(COSQ). (2.3)

A
where the coefficient has been factored tO separate the alignment which is dependent
on the reaction and the transition properties which are related to the structure.
Furthermore, higher orders Of the angular distribution are Often negligible which

reduces the above expression to W(6) = 1 + AngP2(cos 0).

2.4.2 Gamma ray linear polarization

One key goal Of this experiment was to determine if it was possible to detect the 'y-ray
linear polarization emitted from a source moving at a sizable fraction of the speed
of light. To understand the results, one must have clear knowledge of the intrinsic
response of the detector to linear polarized photons emitted from a source at rest.

For photons with energies in the experiment’s range Of interest, the most sensitive
probe Of the polarization is due to the asymmetry of the Compton scattering Of
the photon. The Klein-Nishina differential cross section [45,46] for the Compton
scattering Of a photon with initial energy E0 and final energy E is:

do _ r3 E2

E0 E . 2 2
(IT? _ 3E—3( + — —2sm 0008 (b) (2-4)

E E0

where re 2 (32/771662 is the classical Compton radius of the electron, 6 is the scattering

26

angle, and (b is the angle Of the incoming electric field vector Of the incoming photon
relative to the scattering plane. A detailed treatment and derivation of the cross
section can be found in Heitler [47]. As the Compton scattering is a two—body process,

the energy of the scattered photon is completely determined from the kinematics,

Eo

E = .
1+ (ED/777.602)“ — c080)

 

(2.5)

For a photon that interacts in the detector in two places depositing energies E1 and
E2, this allows some restriction Of events which do not follow the proper kinematic
relation based on E, E0 = E1 + E2, and 6. This could result from incomplete ab-
sorption of the incoming photon energy or false events caused by coincidence with

naturally occurring background.

Polarimeters based on the sensitivity Of Compton scattering to the polarization Of
the incoming photon have been built and used for fifty years [48]. The first and sim-
plest setup consists of two 7-ray detectors where one functions as the scatterer, and a
second serves as the analyzer. Since the cross section is most sensitive for scattering
angles close tO 90° for low-energy (< 1 MeV) photons, an Optimal setup consists Of
four analyzing crystals at an angle of 90° relative to a small scattering crystal [49].
However this results in a low photopeak efficiency which hampers measurements.
Composite detectors such as the Clover [50] are natural Compton polarimeters using
the scattering between neighboring crystals to determine the incoming polarization.
However, the proximity of the analyzing region to the scattering point decreases the
sensitivity of these detectors. This is Offset, though, by the increased detection effi-
ciency. The development of high-purity segmented germanium detectors has provided
a new tool for polarization determination. Similar to composite detectors, segmented
detectors successfully confront the difficulties Of maintaining sufficient efficiency while

the small volume of the segments allows a polarization sensitive detection of low-

27

  
 

Gamma Ray

Reaction Plane

 

F

Alignment Axis

Figure 2.3: Polarization of a gamma ray characterized by the angle é of the electric
field vector with respect to the normal Of the reaction plane.

energy photons which Compton scatter. The capability of these segmented detectors
to make sensitive linear polarization measurements has been demonstrated [49,51,52].

For an axially symmetric system, a spin-aligned nucleus emits gamma rays which
are linearly polarized. The axial symmetry requires the necessary population of mag-
netic substates (i.e. w(m) = w(—m)) to produce linear polarization. The polarization
of the photon is relative to the reaction plane which is defined by the axis of the nu-
clear alignment and the photon’s direction of propagation as shown in Fig. 2.3. The

theoretical linear polarization distribution is defined as:

_ Wag = 0°) — War = 90°)
“ was = 0°) + War = 90°)

 

10(9) (2.6)

where W(6,€) describes the likelihood of emitting a photon at an angle 0 to the
axis of alignment and with an electric field vector at an angle of 5 to the reaction

plane. However, the quantity observed in the experiment is the intensity Of the energy

28

peaks of gamma rays which scatter an angle O relative to the reaction plane. The

experimental asymmetry is defined as

_ Nte. <23 = 90°) — NM. 45 = 0°)
‘ NW. (15 = 90°) + Nw. as = 0°)

 

A(6) (2.7)

where N (9, O) is the number Of counts in the photopeak. Since the scattering cross-
section (Eq. 2.4) is maximized perpendicular to the direction of the electric field, the
angles are related via O = 90 — 6. As these quantities are related, the asymmetry and

the polarization can be related for a polarimeter through its sensitivity, Q(E7),
Ate) = Q<E.>P<9). (2.8)

The sensitivity depends on the incident energy and is limited by the difference in the
scattering cross section. For a point-like polarimeter, the sensitivity Q0 calculated

from the Klein-Nishina cross-section is:

1+0:

W“) = m

(2.9)

E . . . .

where a = T—n—Ez. For practical applications, the scattering asymmetry must address
6

the geometric asymmetries of the detector. The asymmetries are compared to those

Observed from unpolarized gamma rays which are characterized by a parameter a.

N unpolarized (0 I 45 = 900)

 

 

(“E”) = NunpolarizedW: ¢ = 00) (2’10)
_ N(6, a = 90°) — aN(9, a = 0°)
”4(9) " N(6,0 = 90°) + aN(6,0 = 0°) (2‘11)

To determine the sensitivity of a detector, the analysis is done for gamma rays which

have a well known polarization such as for pure dipole transitions which have an

29

angular distribution

W(9) 2 1+ a2P2(cos 6). (2.12)
For pure dipole transitions, the polarization distribution reduces to the form [53]

3 ' 29
P(6) = d: “2 8m , 2 (2.13)
2 + 20.2 - 3a2 Sln 0

 

where a2 is determined from analyzing the angular distribution Of gamma rays with

(f 3,

a “+” Sign for M1 transitions and a sign for E1 transitions.

2.4.3 Segmented Germanium Array

The Segmented Germanium Array (SeGA) [54] serves as the workhorse for gamma
ray spectroscopy at NSCL. It combines the superb energy resolution of germanium
with a modest efficiency afforded by the array of detectors. In addition, angular res-
olution smaller than the detector size can be achieved using the internal electronic
segmentation of the individual crystals. Eighteen detectors of n—type high-purity ger-
manium (HPGe) comprise SeGA. Each is a cylindrical single crystal of germanium
cooled by liquid nitrogen to 100 K. The crystal is 8 cm long with a 7 cm diameter.
A central core electrode is inserted along the central axis of the crystal. The outside
surface is electronically segmented into 32 segments with 4 azimuthal divisions and
8 longitudinal slices (see Figure 2.4). A bias voltage applied to each detector causes
electrons liberated by an incoming gamma ray to drift to the center Of the detec-
tor. The corresponding holes generated drift outward and are collected by electrodes

placed on the outer surface for each segment.

The detectors are configurable into several standard arrays. For the present work,
the “classic” SeGA configuration was used. This places the detectors in two rings sur-

rounding the target position. Seven detectors are placed in the forward 37 degree ring,

30

 

 

Figure 2.4: Depiction Of the segmentation of a SeGA detector.

and ten are placed in the backward 90 degree ring. The crystals are positioned such
that the transverse segmentation lies along the 0 direction defined by the coordinate

system with 2 along the beam axis.

2.4.4 Relativistic kinematics

Experiments at National Superconducting Cyclotron Laboratory are typically per-
formed with velocities from 30%-50% the speed Of light. At these velocities, rela-
tivistic effects become considerable and must be accounted for in the analysis. In
particular, the velocity of the particle, the energy of emitted radiation, and the angle
of the radiation with respect to the nucleus’ velocity all become correlated as dis-
cussed in detail in Appendix B. Particle tracking in the S800 spectrograph supplies
the necessary information about the magnitude and direction of the residual particle’s

velocity. To be able to precisely measure the gamma ray’s energy, the angle of emis-

31

sion with respect to the beam axis must be known. This is where the segmentation
of the detectors is critical, allowing the determination Of the first interaction position
within a detector based on the energy deposited in the individual segments. Gener-
ally, the location Of the first interaction is taken as the midpoint of the segment where
the most energy was deposited. This algorithm accurately identifies the segment with
the first interaction point in most of the cases [55] especially when considering the
possibility of having multiple interactions within a segment. However, the precision
in the determination Of the interaction point is limited by the size of the segment.
Furthermore, the exact location where the nucleus was at the time of 'y-ray emission
is also unknown. Often this occurs within the target where the nucleus is actively
slowing down leading to an additional effect related to the velocity uncertainty. These
two effects are the major contributors to the limits Of the energy resolution obtainable

in an experiment with intermediate energy projectiles.

2.4.5 Velocity determination

SeGA was centered around the target position Of the S800 to detect the gamma rays
emitted following the secondary reaction. Gamma-ray spectra were Doppler corrected
based on the energies Of the fragments of interest at the middle of the target. The
velocity of the fragments was determined in several ways which proved to be self-
consistent. First, one can get the velocity based on the incoming particle energy
and energy losses in the target from LISE++ [56] which uses a model based on the
ATIMA 1.2 [57] characterization for heavy ion energy loss in matter. Alternatively,
the velocity can be obtained on the downstream side of the target based on the de-
viation from the central momentum in the S800 spectrograph and back trace to the
center of the target using the energy losses Of LISE++. Energy losses based on
GEANT4 also agree reasonably with those Obtained with LISE++. Furthermore,

one could estimate the velocity based on the uncorrected DOppler-shifted energy de-

32

posited in the 90° ring Of SeGA which have their first interaction point within a
different longitudinal segment and thus a different Opening angle. This constrains
the velocity if the 7-ray energy and positioning of the detector are well known (see
Eq. 38). Combining the systematic differences using these different methods, one

Obtains a reasonable estimate of the uncertainty in the velocity.

33

Chapter 3

Offline Detector Characterization

In preparation for the experiment with the beam, several source measurements were
used to characterize the response Of the detector. These were necessary to establish
the calibration parameters for energy and efficiency. Furthermore, the detector’s re-

sponse to linear polarization also had to be investigated and its sensitivity quantified.

3. 1 Energy calibration

To determine the energy Of the gamma rays detected in SeGA, one must know how
the detector and the Analog to Digital Converters (ADCs) respond to gamma rays
Of previously known energies. TO accomplish this task, several standard radioactive
calibration sources were used. This allows one to characterize the response of the
detector across a wide range of energies. Furthermore, the central contact of the de-
tector and the 32 segments have to be calibrated separately as they behave differently
to the photon’s energy deposit in the detector. For the central contact, the 13-bit
channel output, 1:, from the ADC was assumed to be quadratically related to the

energy Of the incoming photon, E, i.e. for some calibration coefficients a,,

E = a0 + alt: + (I21:2 (3.1)

34

x103 103

     

 

 

 

 

 

 

 

 

 

 

> I
.3 I
:7 1’ 120
a E
S : 100
o I
O I
3 o
i
; o
i .1
H.
200 400 600 800 1000 1200 1400 1600 1800 2000
Energy(keV)

Figure 3.1: Calibrated 22(’Ra energy spectrum of all 16 SeGA detectors summed.

To determine these coefficients, a :226th radioactive source was placed in the target
position of the array, and the gamma rays with known energies from 186 keV to
2448 keV were detected in the individual detectors. A Gaussian with a quadratic
background was fit to each Of the photopeaks in the resulting gamma ray spectrum in
order to extract the centroid channel of the ADC output. The calibration coefficients
for each detector were chosen such that they minimized the chi squared comparing

the calculated energies, E, in the detector and the known peak energies, E,.

. 0'2
2

where a2 is the variation associated with the fitting of the peak. After calibration,
the known 22°Ra spectrum is reproduced beautifully as seen in Fig. 3.1. Over the

course of the experiment, calibrations parameters may shift slightly. To account

35

for this possibility, calibration data were taken at the beginning and the end of the
experiment. In addition, an abrupt shift in the response Of several of the detectors
was observed in the middle of the experiment. This was ameliorated by performing a
linear correction to the calibration coefficients based on natural background sources
in the experiment, in particular the electron-positron annihilation peak and peaks
associated with natural potassium decay. These adjustments allowed for an accurate

calibration for all the detectors across all the runs.

The proper calibration of the energies deposited in individual segments is also im-
portant as the energies of the segments is what the Doppler reconstruction algorithm
relies on. The usual technique used is to gain match the segment energies such that it
agrees with the central contact energy. One such calibration method is described in
Hu et a1. [58] where an automated procedure is important as a typical SeGA configu-
ration has ~ 500 segments which need calibration. One benefit to matching with the
central contact energies is that all the data including events with incomplete 7-ray
absorption can be used in the calibration. However, there are a sizable number of
events where the measured energy deposited in the segments is deficient compared
to the amount deposited in the central contact. These outliers can cause a simple
x2-analysis to have small negative deviations from the expected results. To alleviate
such deviations, an iterative approach was utilized to obtain the segment energy cal-
ibration parameters. With each pass, events with large deviations between a single
segment energy deposit and the central contact energy were excluded from the anal-
ysis. The last pass rejected events which strayed more than 5 keV from the expected
value from the central contact. At each pass, parameters were determined for each
segment which minimized the X2 Of the segment energy relative to the central contact
energy. The response of the segments is significantly less linear, so a quartic polyno-

mial was used in the fit to the central contacts. There exists an electrical coupling in

between the segments, however, which causes a correlation in the energies determined

36

from the segments. This can be accounted for by considering additional energy cali-
bration parameters which depend on the coupling between two segments. For events
which had interactions in two segments, the calibration included parameters from the
cross-terms of the ADC channel outputs 2:1 and 3:2 up to the fourth power. This can
be expanded to the events which interact more than twice in the detectors using the

technique discussed in Venutrelli et al. [59].

3.2 Efl'iciency calibration

The efliciency of the detectors must also be accurately determined. This is espe-
cially important for the measurement Of the partial cross section to the ground state
following the knockout reaction. To investigate the 7-ray angular distribution, one
also needs to know the relative efficiency Of the two rings of detectors in SeGA. The
efficiency depends heavily on the energy of the incoming photon and this energy de-
pendence needs to be well understood. The efficiency is also strongly influenced by
the geometry of the array relative to the target especially during in-beam experi-
ments where the energy and solid angle depend on the relativistic kinematics and in
particular the angle Of the detectors with respect to the beam line.

TO determine the efficiency, several radioactive sources were placed in the same
position as the in-beam target prior to the experiment. For absolute efficiency mea-
surement, the radioactivity of the source must be well-known. To. this end, a cali-
brated source of 152Eu was used with an activity of 8.46 pCi measured on 1 May 1978.
152Eu has a well-known half-life Of 13.537 :1: .006 years. The intensity of gamma rays,
10, emitted can be calculated giving the efficiency, 5, of the germanium array. 152Eu
only provides a good efficiency calibration for energies up to 1.5 MeV. To determine
the efficiency above 1.5 MeV, a 56CO source was used which has high energy peaks

up to 3.5 MeV. The intensity Of the cobalt source was not as well known, so a scaling

37

 

llr'Irrrrrtilrtrrtrrrrr'rritriII—rti

p
8

p
8

TIIIfUIIIlIl—IrITIII'Illll‘

Absolute Efflclency
c
2

 

IlllljlllllUlllljlllllll

 

 

0.02
I
0.01 ‘
A :fi
00 500 1000 1500 2000 2500 3000 3500
Energy (keV)

Figure 3.2: SeGA efficiency in the 37° ring (crosses) and the 90° ring (triangles) with
associated fits in the form of Eq. 3.3.

factor was applied to agree with the efficiencies in the common energy range with
152Eu. The energy dependence was then fit as a curve to the discrete data points
to get the continuous energy dependence of the efficiency. Several functional forms
were tested producing comparable results. The efficiency was finally taken to be of
the form

«E; 60, E0. a) = eo(E — E0 + 502693)” (3.3)

with the parameters 60, E0,a. determined by minimizing the chi-squared from the
fit. For a 1 MeV photopeak emitted from a source at the target position, SeGA
had a 2.2% efficiency. The efficiency was also determined for each ring in the array
to understand the dependence of the efficiency on different angles of emission. The
measured efficiency is displayed in Figure 3.2. The uncertainty in the efficiency was

also determined across the range of energies. If the X2 is minimized at some value

38

x3, then the set of parameters included in the no error band are defined as
[60, E0, 0. : X2(€0, E0,a) < X3 + 71.2}. (3.4)

The minimum and maximum values Of the efficiency for this set of parameters are
then determined for each given energy. Since the fit parameters are highly correlated,
an evenly spaced mesh over the parameter space was taken with the variation along

the principle axes Of the error matrix M. The error matrix for a set Of parameters

{c} in defined as:
02x2({6})

(3.5)

This allows for a more accurate sampling of the parameter space. However, this
method always underestimates the width of an error band for a given energy. The
granularity Of the mesh was made progressively finer until there was a negligible
change in the width of the error band. The error in the intensity I from a source with

N counts in the photOpeak is then accordingly

 

rIIE.) = (/f~(E.)2 + f.<E.-)2 (3.6)

where f1, fN, f5 describe the relative errors in the intensity, photopeak counts, and
efficiency respectively. The relative uncertainty in the efficiency was reasonably con-
stant over the sampled energy range contributing roughly a 1% additional uncertainty

in the determination Of the y-ray intensity in the lab frame.

3.3 Polarization sensitivity determination

The sensitivity Of SeGA to linear polarization was determined using gamma rays

emitted from a 1 pCi 249Cf source which provides gamma rays Of a known polarization

39

Cf—249
9/2—

 

 

388.3

 

 

388 E1

fl
‘l‘333 E1
o

 

9/2+ 54.8

7/2+

Figure 3.3: Simplified decay scheme Of 249Cf used to produce gamma rays Of a known
linear polarization.

[53,60]. Following the a decay Of 24’9Cf (see Fig. 3.3), the daughter nucleus 245Cm
emits either a 333 keV or 388 keV gamma ray, both of which are pure electric dipole
transitions. The 333 keV and 388 keV gamma rays have an Opposite sign for the

polarization which allows for control of systematic errors in the measurement.

3.3. 1 Detector setup

The a particles were detected with a 400 pm thick silicon surface barrier detector
which was located 11 cm from the 249Cf source. The recoiling heavy ions were stopped
in the 2-mil platinum foil backing of the source. Fifteen SeGA detectors were mounted
in the classic frame identical to the configuration used in the in—beam experiment as

shown in Figure 3.4. Standard analog electronics for SeGA and the silicon detector

40

Forward ring

8 detectors .
90 degree rmg

10 detectors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4: The setup used for the a-y coincidence measurements to determine the
polarization sensitivity.

41

controlled the data acquisition to capture a—y coincidences as well as gamma rays
when the a particle was not detected. For the 333 keV transition, each detector
registered roughly 25,000 gamma ray singles as well as 7,000 coincidences. The more
intense 388 keV transition provided four times as much statistics for the analysis.
The gamma ray singles provide information about the normalization of the scattering

within the detector in response to unpolarized gamma radiation.

3.3.2 Gamma-ray angular distribution

The two electric dipole transitions in the daughter nucleus were analyzed for angular
distribution. Since these are known to be pure dipole distributions, the y-ray angular
distribution coefficients and linear polarizations are well understood. For pure dipole
transitions, the polarization at an angle is completely determined by Eq. 2.13 given
the angular distribution coefficient a2. The angular distribution can be Observed from
the spectra for detectors at different angular positions as shown in Figure 3.5. The

number of a-y coincidences in the experiment is given by

Nam) = N06a.~r(Ea, E7) 9)(1 + a2132(008 9)) (3-7)

where N0 is the number Of gamma rays emitted and 50,7 is the efficiency to detect
a-ry coincidences for a given combination of energies. This is related to the detection

efficiency of the individual detectors:

€0,7(EQ,E»Y,0) = 60(Ea)67(E7,0) (3.8)

Singles were also taken concurrently to provide a tool for proper normalization of the
relative detection efficiency Of each ring. As the number Of singles greatly outnumbers

the number Of coincidences, a down—scaler was used tO minimize the dead time Of the

42

 

Counts

1 0000 *-

8000 ‘-

6000 *-

4000 '—

2000 —

llll

o I
320 330 340 350 360 370 380 390 400
Energy (keV)

 

 

Figure 3.5: Gamma-ray spectrum for a detector in the forward (solid) and backward
(dashed) rings in SeGA for the two gamma ray transitions in the 245Cm following the
a decay Of 249Cf showing the difference in the angular distribution.

43

 

A 388 keV
l 333 keV

P
w
a:

IllllllllllllllTIr

Relative Coincidence Intensity
C O
in 'u
M A

9
ea

0.28

 

 

 

IIIlIIIIIIIIIlIIIIlIIIIlIIIIlIIIIlIII

-o.4 -o.3 -o.2 -o.1 o 0.1 .
P2(cos 0)

 

Figure 3.6: The ratio of counts for the 01-7 coincidences to singles for each of the
detectors in the array with the line fit which determines the angular distribution
coeflicient a2 for the two gamma ray transitions of interest.

acquisition system by reducing the 'y-ray singles which created a trigger. The number

of down-scaled singles is given by

= N067(E73 0)

(3.9)

where st is the downscaling factor. The ratio of the number of coincidences and
singles is linear in P2 considering the attenuation due to the finite angular extent of
the detector:

Nd(9)

m = st€a(Ea)Wav(6)I (3'10)

where Wav(6) is the average Of the angular distribution over the detector solid angle.
The ratio is independent of the individual detectors’ efficiencies. From the analysis Of
the ratios, the ag coefficients for the 333 keV and 388 keV gamma rays are respectively
—0.29:l:0.02 and 0.17i0.01 from the fit determined in Figure 3.6. This agrees with

previous measurements done for the 249Cf nucleus [60].

44

3.3.3 Scattering asymmetry normalization

The scattering of unpolarized gamma rays was analyzed for the intrinsic asymmetry
due to the geometry Of the detector. Where no a particle was detected, the gamma
rays emerge unpolarized as there is no axis of alignment in the system. The ratio 0.
(Eq. 2.10) which describes the geometric asymmetry was measured for both of the
gamma rays of interest for each detector. Events were chosen within the photopeak
that interacted in only two segments within a single SeGA detector as shown in Figure
3.7. Over this energy range, the change in the asymmetry for unpolarized gamma
rays was negligible: a(333) = 0.1262l:0.007 and a(388) = 0.128:l:0.007 for the weighted
average over all the detectors. The uncertainty is dominated by individual differences
in the detectors’ intrinsic asymmetries. The standard deviation for the sample of
detectors for the geometric asymmetry is a =, 0.009. It is important to note that
the selection of events which scatter parallel or perpendicular to the reaction plane
account for respectively 25% and 3% of the photopeak events in this energy range.
The scattering angle is close to 90° where the cross section (Eq. 2.4) is most sensitive
to the polarization for low-energy gamma rays.

The dependence of the geometric asymmetry on the energy was also investigated
over a larger range by examining gamma rays emitted from a 152Eu source (see Fig.
3.8). The europium source validates that the change in the intrinsic geometric
asymmetry over a narrow energy range is small. The intrinsic asymmetry was char-
acterized by a linear fit a = Co + c1E with the energy dependence contained in the c1

coefficient which was determined to be (—1.24 :l: 0.15) - 10"5 keV‘l.

3.3.4 Sensitivity determination

Using each detector’s geometric asymmetry from the singles measurement, the asym-

metry A (Eq. 2.11) in the scattering was Observed for the a-ry coincidences. The

45

 

Figure 3.7: (Top) The geometry of a SeGA detector with respect to the reaction plane.
Gamma rays irradiate the detector from the side of the cylindrical axis and interact
within a given segment and then scatter either parallel (middle) or perpendicular
(bottom) to the reaction plane.

46

 

'II'UI.'l’.'tfrfiTf'rri'tif't"rt'rIUVf

I‘t‘ln—L‘Jerlll
I . r I l

2‘
‘2: 0.14
E.
. 0.12
<

0.1

 

0.08

0.06

0.04

0.02

IllIllllllllllljlllllllllll

 

:IIU'IIIIIUWIIII'IIIIIII'ITTII

I III LLLLL. I;..I. ..I.. I...I. ..l;..
200.40060080010001200-{4001600180020002200
Energy(keV)

 

 

Figure 3.8: Geometric asymmetry measured for detectors in the forward ring for a
range of energies for unpolarized gamma rays emitted from a 152Eu source. Included
is a linear fit and error band.

effect of the finite angle span of the detector causes a small uncertainty of 1% for the
polarization Of the incoming gamma ray. The incoming polarization determined from
the angular distribution coefficient was averaged over the solid angle Of the detector.
With this correction, the polarization of the two pure dipole transitions of interest
Was —0.267:l:0.007 and 0.24:l:0.02 for the 388 keV and 333 keV gamma rays detected
in the forward ring. For the 90° ring, the incoming polarizations were respectively

=—0.13 :l: 0.02 and 0.37 :l: 0.02 for the transitions.

The Observed scattering asymmetry A in the detectors for the four known gamma
ray polarizations was -0.08 :l: 0.03, —0.02 :l: 0.03, 0.04 :l: 0.04, and 0.10 i 0.04 (from
lowest to highest polarization) for the weighted average over the relevant group of
detectors as shown in Figure 3.9. The asymmetries were compared to the maximum
analyzing power of an ideal Compton polarimeter (Eq. 2.9). The relative sensitivity,

(2,81 = Q/QO was established to be 0.18 i 0.02.

47

 

A 388 keV

I333keV
o.3.,.,.,.,.,.,.

 

 

 

 

0.2 -

0.1-4

0.0 -

Asymmetry

41.1 .

 

 

 

-0'2 - I I I r W I I I v
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Polarization

 

I r I '

Figure 3.9: The asymmetry observed in the SeGA detectors for gamma rays of four
known polarizations adjusted for the relative analyzing power of the Compton effect
for the two different energies.

3.3.5 Figure-Of-merit discussion

Understanding the true effectiveness Of the setup to detect the linear polarization must
include the time necessary to make a sensitive measurement. Efficiency of detecting

the gamma rays plays an important role here. The figure-Of-merit,

F = QQEC, (3.11)

for a Compton polarimeter as defined by Logan et al. [61] takes into consideration the
sensitivity Q as well as the coincidence efliciency, 6c, in regard to the intra-detector

scattering [62]. The coincidence efficiency takes into account the photopeak efficiency,

48

Table 3.1: Comparison Of SeGA to other detector arrays for polarization measure-
ments.

 

Om, Q s, x 104 F x 106
POLALI [49] 0.8 2 0.5 1 0.26 7 1
Clover (EUROBALL) [50,63] 0.29 2 0.23 2 4.8 25 2
Gammasphere [36,64] 0.08 1 0.052 7 18 4.8 7
SeGA 0.18 2 0.14 2 3.0 5.9 8

cpw(E7), of the detector as well as the likelihood of events to scatter in the detector

in a manner which contributes to the polarization data.

 

6.03,) = NM epw(E,) (3.12)

For a single SeGA detector in this energy range, the total photopeak efficiency is
approximately 21.2 - 10-4. Since 25% and 3% Of the events scatter parallel and
perpendicular to the reaction plane respectively, the coincidence efficiency for polar-
ization is ac z 2.99 - 10‘4 for a single detector. This gives a figure-of-merit which is

z 5.9 - 10‘6 which by its definition is inversely proportional to the time necessary

to make a measurement of a certain precision with the device.

This figure—Of-merit for SeGA detectors is similar within an order Of magnitude

to other detectors. For some other detectors referenced in the literature, the main
Values of interest are listed in Tab. 3.1. The sensitivity has been extrapolated from
the literature to the same energy range as in the 249Cf measurement. The photopeak
efficiency was estimated based on available efficiency plots and the figure-of—merit has
been calculated according to Eq. 3.11. The errors quoted in Tab. 3.1 are based solely
on the uncertainties of the sensitivities. Including the uncertainty in the estimation of
the efficiency, these figure-of-merit values are gauged to be 30% accurate. Compared
to the Clover results [50, 63], an individual SeGA detector has approximately one
fifth the sensitivity. A third of this is due to a reduced sensitivity. The remainder

is attributed to the decreased photopeak efficiency of the detector due to a smaller

49

volume. A standard 5-crystal arrangement, such as POLALI, has a worse figure—of-
merit due to its limited efficiency. Each Of these detectors are particularly suited for
specific purposes though, such as SeGA’s performance for in—beam 7—spectrosc0py.
The signal-tO-noise ratio in each individual experiment, which is not considered in
Tab. 3.1, should also be considered when determining the impact of a detector’s

figure—of—merit.

50

Chapter 4

Proton knockout studies

After characterization, the detectors were utilized in an experiment to investigate
the structure Of the island Of inversion Odd-A isotopes 31'33Mg. In particular, the
magnesium isotopes were populated from proton knockout reactions of a secondary
beam of aluminum isotopes which were produced by the methods of Subsection 2 .2 at
the Coupled Cyclotron Facility at National Superconducting Cyclotron Laboratory.
After acceleration, the fully-stripped 80 pnA primary beam of 48Ca had a kinetic
energy of 140 MeV/ u, corresponding to a velocity approximately 50% the speed of
light. The primary beam was fragmented on a 806.2 mg/cm2 9Be target (~ 4 mm
thick). The A1900 filtered out the 3234Al isotopes to investigate the reaction of in-
terest. The secondary beam characteristics are described in Table 4.1. A fraction of
the secondary beam was still composed of impurities resulting from the fragmentation
process but these can be cleanly separated in the analysis. The magnetic rigidity was
chosen in order to select nuclei with kinetic energy of 95 MeV/ u corresponding to
a velocity ,8 = v/c = 0.42. Each fully-stripped beam of 32Al and 34Al was then
directed to the S800 spectrograph where they underwent reactions on a secondary
beryllium reaction target. The reaction residues were identified in the S800 spectro-

graph. Particular focus was given tO the spin alignment in the magnesium isotopes

51

Table 4.1: Secondary beam characteristics.

 

 

Beam property 32Al setting 34Al setting
(Bp)12 (T-m) 3.8143 4.0789
(Bp)34 (T-m) 3.5179 3.7858
E34 (MeV/u) 93.84 94.61
Rate (pps/pnA) 2600 153
Purity 71% 50%
Ap/ p 0.5% 0.5%
Transmission to 8800 85% 80%

 

 

 

following the reaction by examining the angular distribution of gamma rays emitted
from the reaction products. Also, for the first time with intermediate-energy beams,
the linear polarization Of emitted gamma rays was probed. This allows direct access
to information about the structure of the states, separating those with intruder-like

configurations from those with standard shell ordering.

4. 1 Electronics trigger

To successfully probe the structure, one must combine the information from the in-
coming and outgoing beams as well as any detected gamma rays. It is important
to correlate all the constituent parts of each event at the target. The bulk of the
information is contained in the gamma-rays correlated with the particles detected in
the focal plane Of the S800. However, not every reaction will result in emitted gamma
rays. Furthermore, the emitted gamma rays may not interact with the SeGA detec-
tors. This is Opposed to the particle detection efficiency where the ion chamber of the
S800 can be considered to be 100% efficient for residues within the 8800 momentum
and angular acceptance. To determine an inclusive cross section to all the bound
states in the residue, one must consider the total number Of particles produced in
the reaction whether or not a gamma ray was detected. However, measurement of

the partial cross sections to individual states requires accurate determination Of the

52

number Of particle-7 coincidences.

For the purpose of the above measurement, real-time selection of events is enabled
in the hardware through an electronics trigger. After a particle is detected by the
E1 scintillator at the end of the 8800 focal plane, a coincidence window (~ 400 us)
is Opened. A logical OR of all the gamma-ray detectors produces the coincidence
trigger if a gamma-ray interacts with a detector within the coincidence window. The
master trigger consisted of all coincidence triggers combined with a fraction Of the
particle singles where no gamma ray is detected. This master trigger prompted the

electronics to be read out. The fraction of particle singles allowed was chosen that
the live time of the data acquisition system was not adversly affected (2, 90%). Slight

corrections were made to timing in each trigger scenario due to the different responses

of the components of the master trigger.

4.2 31Mg

4.2.1 Beam characteristics and particle ID

The large secondary beam intensity of 32Al (see Table 4.1) allowed the use of a
thin secondary reaction target of 1 mm (185 mg/cm2) beryllium at the 8800 target
position. At the end of the A1900 fragment separator, the 32A] secondary beam was
71% pure having a dominant contaminant of 33Si. This impurity could be clearly

separated in the analysis based on the time of flight from the A1900 extended focal
plane to the object box of the 8800 as shown in Fig. 4.1.

The outgoing residues
passed through the 8800 magnetic dipoles selecting the reaction products Of interest.
The unreacted 32Al isotopes are removed in the process. However, the properties of
the incoming beam are necessary for the proper analysis Of the experiment, so an

additional set Of data was taken with the S800 tuned to the magnetic rigidity of the

unreacted secondary beam. In these runs, the primay beam was attenuated to reduce

53

 

 

 

 

 

 

 

 

a, 3000 . . = . F' T I r .
H
S : 3
o 2500 .- 32AI 5
0 : [l :
2000 E ‘1
: 335i :
1500 ; l] i
5. i
1000 _ .
' a 4.... l l 4 n J‘- l l l l . ._..._ .
000 050 900 95 1000

Time (arb. units)

Figure 4.1: The incoming 32Al particles identified by the time Of flight from the
extened focal plane Of the A1900 magnetic separator to the object box of the 8800
Spectrograph.

the rate on the focal plane detectors as the high intensity (> 5000 Hz) would impact
their performance.

During the magnesium production runs, the outgoing residues were identified on
an event-by-event basis using the energy lost in the ion chamber at the focal plane Of
the 8800 as well as the time of flight from the object to the E 1 scintillator of the 8800.
The time of flight was corrected tO improve the resolution based on the kinematic
information provided by the CRDCs at the focal plane. The outgoing residues can
then be clearly identified as shown in Fig. 4.2. The information provided by the

CRDCs also allowed the accurate determination of the outgoing momentum vector
of the nuclei relative to the central trajectory of the 8800.

The mean velocity Of the residues in the middle of the target was determined to

be )8 = 0.404 i- 0.002 by the methods discussed in Subsection 2.4.5. For residues

with this velocity, the emitted gamma rays detected by SeGA are Doppler broadened

54

 

12601280130013201340136013801400142014401460
Time 01‘ flight (arb.)

Figure 4.2: Outgoing residues for a reaction of an incoming 32Al beam identified by
the 8800 spectrograph based on the energy loss in the ion chamber and their time of
flight with selected isotopes of interest labeled.

with a FWHM of 48 keV (2.7%) in the downstream ring and 38 keV (2.1%) in the
upstream ring for a 1.8 MeV gamma ray. Using the additional information about
1312w outgoing particle momentum from the tracking through the 8800, the gamma-ray
resolution can be further improved by about 10%. The major improvement was from
determining the velocity of the individual particles relative to the central trajectory

Of the 8800.

Also, it is important to note that at this velocity the forward ring of SeGA is
located at a center—of—mass angle of 54° where P2(cos 00m.) = 0.009(3). This limits
any effects from the angular distribution in the forward ring allowing for accurate
determination of 7-ray branching ratios from excited states. Higher orders of the

angular distribution are assumed to be negligible.

55

 

 

Counts per 4 keV

 

 

 

 

’uI...l.ul.uIUJJ.ul

 

 

 

50100150200503.0035» '400 600 000 1000 1200 1400 1600 1000 2000
EMHKGV)

 

 

1III

Counts per 4 keV

§§§§

 

 

  

 

.‘t

100150200250300350 400 600 600 1000 1200 1400 1600 1000 2000
EnergyateV)

Figure 4.3: Gamma-ray spectrum of 31Mg in the forward (top) and backward (bot-
tom) rings Of SeGA; note the marked increase in resolution in the backward ring for
the high energy 1.8 MeV gamma ray.

4.2.2 Level scheme

Gamma rays emitted from 31Mg were detected in SeGA with the resulting spectrum
shown in Fig. 4.3. Detailed information about the gamma rays observed is listed
in Table 4.2 where the intensities have been determined using the Observed counts
and the Doppler-corrected efficiency which accounts for both the Lorentz boost of the
solid angle as well as the Doppler shift in the gamma ray’s energy. Most of these
gamma rays were Observed in previous experiments [23, 65,66]. The energies of these
transitions agree well with their formerly established values. Furthermore, the rela-

tive intensities of gamma rays originating from an initial state are consistent with the

56

Table 4.2: Gamma-ray transitions observed in the current experiment with their
intensities, angular distribution coefficients, and Compton scattering asymmetries.

 

 

 

 

 

 

E,- (keV) El (keV) E, (keV) 7,81 0.2 A
221.1 50.5 171.1b 77 (4) 0.48 (10)
0.0 221.1b 24.2 (15) -O.8 (3)
673.2 221.1 452.6 (6) 13.2 (18)
50.5 623.3(5) 64(4) 0.02 (20) -004 (5)
0.0 673.2(7) 34.2 (23) 07(3) -0.08 (6)
944.5 50.5 894.4 (13) 9.0 (17) -04 (8)
1154.5 461 692.6(8) 16.8 (14) -04 (5) 007(9)
2014.5 944.5 1072.7(19)C 10.1 (16) 0.3(8)
461 1555.7 (22)0 24 (3) 0.5 (9)
221.1 1793.4 (18) 100 0.29 (18)
unplaced 1104.0 (16) 13.0 (19)
1500.1 (24) 13(3)
1707(3) 15(8)
1936(4) 9(4)
1968(4) 17(5)

 

aNormalized to 1793 keV transition

Established values since measured values are Subject to effects due to the previously
measured lifetime of the state [65]

c Tentatively placed in the decay scheme based on energy balance

branching ratios seen in previous experiments. This gives credence to the established
placement of these gamma rays in the level scheme of 31Mg. However, as the popula-
tion mechanism is different in this experiment than the neutron-knockout study [23]
and fi-decay studies [65,66], several gamma rays were seen with a significantly higher
intensity than in the previous experiments.

Information available in the experiment allowed for the placement of one new
level at 2015 keV. The placement of this level was established based on background-
subtracted ,ny coincidences of the 1793 keV gamma ray with decays from the 221
keV state. The coincidence spectrum is shown in Fig. 4.4. The gates for background
subtraction were taken below the 171 keV peak and above the 221 keV peak. In
the coincidence spectrum, the 453 keV gamma ray which was previously established

to feed the 221 keV state [66] is also readily apparent. Also visible is the Compton

57

Counts per 5 keV

 

200 400 600 800100012001400160018002000
Energy(keV)

Figure 4.4: Background-subtracted gamma-ray spectrum coincident with the known
’7— Iay decays from the 221 keV state.

edge of the 1.793 MeV gamma rays which interact once within the detector and
tliLen Compton scatter outward without depositing their full energy. The Compton

Scattered gamma rays have a minimum scattering energy according to Eq. 2.5 based
On the domain of cos 0:
E0
E - = ———————. 4.1
mm 1+2E0/mec2 ( )

In the rest frame, the 1.793 MeV gamma ray emitted towards the forward ring has
a. Doppler-shifted energy of 2.422 MeV. This corresponds to a minimum energy of
0.231 MeV for the Compton scattered photon, resulting in 2.191 MeV deposited in
the detector. When Doppler corrected to the center-of—mass frame, this energy deposit
is 1.622 MeV which is where the Compton edge appears in the coincidence spectrum.

No other gamma rays were found to be coincident with the decays from the 221
keV state. As the '7-7 coincidence efficiency scales like the singles efficiency squared,
only the most intense transitions can provide '7-7 data due to the low efficiency of

SeGA (e7 22% implying 67,7 :21 0.04%). Some gamma rays observed seem to have

58

energies which are fairly consistent with the difference in between established energy
levels and have been tentatively placed in the decay scheme based on this information.
The level scheme of 31Mg with the new information from the present experiment is
displayed in Fig. 4.5 including comparisons to the predicted excited states from two

theoretical calculations to be discussed in Section 5.1.

Gamma rays which deexcite the 221 keV state have a low-energy tail in the
Doppler-corrected peak shape (see Fig. 4.10) that is evidence of a long lifetime which
is consistent with the previously measured half-life of 133(8) ps [65]. Given the residue
velocity, this corresponds to a traversal distance of 1.76(11) cm in the setup. As the
efficiency of SeGA depends directly on the position of the source of 'y-ray emission,
this contributes an additional systematic uncertainty of about 10% in the intensity
which is not quoted in Table 4.2. For the longer-lived 461 keV state, no deexciting
240 keV gamma ray was observed as the 10.5(8) ns half-life corresponds to a long

mean flight path of about 3 m which would result in most decays occurring out of

the field of view of SeGA.

Several other transitions observed in Terry et al. [23] are also visible in the proton
knockout experiment. Five of these remain unplaced in the level scheme which ac-
count for a total intensity of 67(11)% of the 1.79 MeV gamma ray transition. Several
of these are gamma ray doublets which appear above 1 MeV as shown in Fig. 4.6.
There is no clear resolution of the doublet structure for the '7-ray peaks located at 1.1
and 1.9 MeV, but the width of the energy peaks suggest multiple gamma rays with
similar energies. These peaks at 1.1 MeV and 1.9 MeV have a FWHM of 4.9% and
3.5% respectively which is considerably larger than the other gamma—ray peaks’ char-
acteristic 2.1% relative width. This cannot be solely attributed to the lifetime of the
states. The observed FWHM is significantly larger than the estimated 0.5% Doppler
broadening resulting from variation of velocity in the target for short-lived states.

Longer-lived states decay behind the target at the same post-target velocity with a

59

7/2‘

 

7/2+
1/2- 1p2h
7/2- 3p4h
5/2+

7/2+ 2p3h
WM)
5/2+ 293')
92—1441
1p2h
312- 192“
3/2+ 293h
1/2+ 2p3h
AMD+GCM

60

7/2+

5/2+

3/2”

USD

Figure 4.5: (left) The level scheme observed in 31Mg with suggested spins and par-
ities including tentatively placed gammas (dashed). (middle) AMD+GCM calcula-
tions from Kimura et al. [67] with the 0p1h states in bold. (right) The Ohw states
calculated by the USD shell model interaction. Theoretical calculations have a shift
in excitation energy to match the energy of the 3/2+ single-hole state determined by
the experiment. See Sections 4.2 and 5.1 for details.

 

300 I I I T I I I I I I I rI I I l r I I l I I I l I I I

250

200

Counts / 4 keV

150

 

100 "

 

50

0800 1000 1200 1400 1600 1800 2000 2200
Energy (keV)

Figure 4.6: Gamma-ray energy spectrum of 31Mg in the 90° ring of SeGA following
Doppler reconstruction, showing several doublets at 1.1, 1.5, and 1.9 MeV.

exponential position distribution downstream of the target. This causes a distinctive
7—ray peak shape (such as in Fig. 4.10) as well as ring-dependent energy shifts in the
Doppler-corrected spectrum since the angle changes with the position of the decay.
Neither of these effects are apparent, so the additional width indicates the existence
of unobserved doublets. The energies of the doublet components were established by
a fit which assumed a relative width consistent with that for well-resolved transition

in the 'y-ray spectrum.

4.2.3 Inclusive and partial cross sections

Determination of the knockout reaction cross section requires an accurate measure-
ment of the number of 31Mg reaction residues, which in turn is related to the number
of incoming 32Al beam particles. In the experiment, there is no direct accurate mea-
surement of the incoming beam particles. However since most of the particles pass

through the target without interacting, determining the number of these unreacted

61

particles after the target yields an accurate measurement of the incoming rate of 32Al
impinging on the target. As the 31Mg reaction residues cannot be measured at the

same time as the unreacted particles, two settings of the S800 spectrograph were used.

In one setting, the 8800 Bp was chosen such that the unreacted particles were
centered in the spectrograph. Since the unreacted particles have a small momentum
spread and divergence, unreacted particles are detected at the focal plane of the
spectrograph with nearly 100% efficiency. The number of particles detected at the
focal plane, Ni, is related to the rate on the scintillator at the extended focal plane

of the A1900, prp, and at the object scintillator, NOB J, via:

N + N
Ni = XFP 2 OBJ f, (42)

 

with some normalization factor f which is related to the purity of the beam and the
transmission through the 8800 analysis line. This normalization factor can then be
used to determine N,- based on the scintillator rates when the S800 is placed on a
different Bp setting. Two normalization runs were taken with the unreacted incoming
32Al particles delivered to the 8800 focal plane. The normalization factor f deviated

only slightly in the two runs.

For the remainder of the experiment, the S800 was tuned to maximize the accep-
tance of the 31Mg reaction products while limiting contamination from the unreacted
secondary beam. For a knockout reaction setting, the cross section can then be de-
termined using the scintillator rates and the number of residues detected (N f) in the
particle identification plot (Fig. 4.2) using Eq. 1.2. An additional correction had to
be applied as the reacted particles have a larger momentum spread and suffer accep-
tance cuts in the S800 so not all outgoing products are detected. Particles with a low
momentum, relative to the central trajectory through the 8800, suffer a reduction in

angular acceptance in the dispersive direction as displayed in Figure 4.7. To account

62

du (parts per thousand)

 

Figure 4.7: The focal plane measurements of am and dta showing the acceptance cut
for low momentum 31Mg residues from an incoming 32Al beam.

for this, a systematic 2.5% correction to the cross section measurements has been
made assuming that the angular distribution in the dispersive direction is symmetric

with respect to the centroid of the distribution.

The inclusive proton knockout cross section was determined to be 8.7(5) millibarn.
Throughout the experiment, the efficiencies of the scintillators were monitored for any
change due to irradiation damage which could degrade the particle detection efficiency
of the S800 spectrograph or the stability of the normalization factor f. The stability
of f is further validated by the consistency of the calculated inclusive cross-section
on a run-by-run basis (see Fig. 4.8). The systematic uncertainty of 0.5 mb has been
determined from the weighted standard deviation of the run—by-run cross sections.
This uncertainty is the major constraint on the precision of the measurement, and is

related to the consistency of the incoming beam normalization.

The inclusive cross section includes the cross section to the ground state (0'0) as

well as to excited states (aex). The partial cross sections to excited states were deter-

63

 

11°C i'r'r'rri'rfi'r'llll ' I ' fir r rI

10.5 - a

4 l 915%
8.5.: E ] iltlllfil l? E

10.0 -

5°
01
1

CO
C
1
11 41111

i

8.0 -

,
7.5-_

7.0 -

1 l

 

31Mg Cross Section (mb)

j
114111

6.5 s

 

 

 

60 I'I'I'I'I'IrTrI'I/ILI 'fi'l'l'l
Runnumber

Figure 4.8: Inclusive cross section of 32Al -—> 31Mg calculated on a run-by-run basis
chronologically from left to right. The experiments with the 32Al incoming beam
were performed in two sets of runs separated by several days corresponding to the
break in the abscissa.

mined by particle-gamma coincidences. The cross section to the ground state is the
difference between these cross sections to excited states and the inclusive cross sec-
tion. The direct population to each excited state I j was determined by the balancing

of '7-ray intensities feeding (Iin) and depopulating (Iout) each energy level.

Ij = [in _ 1011131 (43)

where the intensities are determined from the number N of observed gamma rays
as well as the absolute efficiency of SeGA at the given energy E, (i.e. [input =
Nin,out/6(E7)). Due to the statistical nature of the sampling of the incoming and
outgoing intensities from a level, negative cross sections consistent with zero can

be obtained in the analysis. As a negative cross section is unphysical, these small

64

Table 4.3: 31Mg states with direct feeding from the proton knockout from 32Al.
E (keV) J7r aexp (mb) 059 (mb) 031,025 (mb)
0.0 1/2+ 0.33 i 0.14

 

673.2 3/2+ 3.56 :l: 0.20 12.5 10.4

1154.5 0.53 :l: 0.13

2014.7 5/2+ 4.27 :l: 0.24 12.0 8.6
(2787.2)? 7/2+ unobserved 11.7 5.9

negative cross sections were set to be exactly zero. The remaining partial cross
sections to excited states were normalized such that their sum was consistent with
aex determined by all the assigned gamma ray transitions where 00 is held constant.
Predominately, the knockout reaction directly fed states at 673 keV and at 2015 keV.
All states fed directly and their partial cross sections are shown in Table 4.3 with the

corresponding theoretical calculation of Sec. 5.1.

U'nplaced transitions listed in Table 4.2 can modify I j when placed. However the
intensity of each unplaced transition is fairly small; the most intense (1968 keV) of
these unassigned gamma rays has a maximum contribution of 0.54:1:0.16 mb to a given
state. Also, the unplaced transitions do not likely feed or decay from the same excited
state. Fragmentation of the feeding intensities suggests that a sizable modification
to any single partial cross section in Table 4.3 is unlikely, though possible. If all
unplaced transitions feed or decay from the same excited state, this corresponds to a
maximum correction of 2.1 mb which is not quoted in the uncertainties of Table 4.3.
The tentatively placed decays from the 2015 keV state were included in the partial
cross section and could account for an additional 1.1 mb adjustment to the partial
cross section to that state. Future work with greater sensitivity is needed to clarify
the placement of these transitions. Next generation gamma-ray detector arrays, such

as GRETINA [68], with a higher coincidence efficiency would address this situation.

65

 

 

 

 

l llllllllllllllllIilllllllllllllllllllllT

 

l l 1 .L 1 l l l l l l l 1 l l l I
12.2 12.4 12.6 12.8 13 13.
pI (GeV/c)

 

Figure 4.9: (Points) The measured momentum distribution of the 31Mg residues
coincident with the 623 keV 'y-ray. (Line) Predicted momentum distribution based
on the knockout of a d5/2 proton from 32Al to the 3/2+ 673 keV state in 31Mg
including the effects from the loss of energy in the target as well as the incoming
momentum width.

4.2.4 Momentum distribution of residues

Coincident with the gamma rays detected by SeGA, the momentum distribution of
the outgoing particles measured at the S800 focal plane contains information about
the dynamics of the reaction. The distribution associated with the residues coincident
with the 623 keV gamma ray is shown in Fig. 4.9. The contribution from the ’y-ray
background was subtracted based on the momentum distribution coincident with
gamma rays from the spectra above and below the transition of interest. The
momentum distribution coincident with the 623 keV gamma ray is representative
of all momentum distributions observed for the 31Mg residues after the knockout
reaction, including those coincident with other gamma rays as well as those events
where no gamma ray was detected. The shape of the distribution is an indicator

of the orbital angular momentum of the removed nucleon and is consistent with

66

X

.5

O
u

 

§

'13
o

8
III—T1111llflllllllllllfl

Intensity per 4 keV
8

§

 

 

   

lllllllllllllllllllllllllll

 

       

III I

I-
"""._

 

 

l 1 l L L L I 1 L 1 l l 1 l l l
160 180 200 220 240
Energy (keV)

l 1 1

l

 

. l 1
120 140

Figure 4.10: Relative intensity of gamma rays emitted from the 221 keV state in 31Mg
as determined by efficiency calibrations (Sec. 3.2) and proper Doppler reconstruction
for the forward (dashed) and backward (solid) rings in SeGA.

the knockout of an E = 2 proton for the populated states according to the reaction

calculations discussed in Subsection 5.1.1.

4.2.5 Angular distribution of gamma rays

In the examination of the decay from the excited states, evidence also exists for
the presence of spin alignment along the outgoing direction of the knockout reaction
residues (i.e. along the beam axis). In particular, the ratio of gamma rays detected
in the forward and backward rings is substantially different for the two transitions
which deexcite the 221 keV state as displayed in Fig. 4.10. The difference in the ratio
between the transition to the ground state versus the transition to the 50 keV state

is greater than 40. The angular distribution formula can be expanded in terms of

67

Legendre polynomials (see Eq. A.1). Since there is a noticeable difference in angular
distribution, spin alignment must exist in the 221 keV state. From the alignment
condition, w(m) = w(—m), a preference in the direction of 'y-ray emission limits
the spin of the initial 221 keV state to J 2 3/ 2. However, previous lifetime mea-
surements [65] and Weisskopf estimates of the electromagnetic transition rates [1]
conclude that these are two pure dipole transitions. Given the known ground state
spin of 1/2+ [22] and the selection rules for electromagnetic transitions (Eq. 2.2), the
spin of the 221 keV state must be either 1 / 2 or 3/ 2. Combined with the requirements
of spin alignment B ,\ 75 0, the 221 keV state must have a spin of 3/2. Further-
more, since the spin alignment of the initial state must be the same for both the 171
keV and 221 keV transitions, any difference in their angular distributions must be
attributed to differences in the final state. Particularly for pure dipole transitions,
A2 < 0 for unstretched transitions (AJ = 0) and A2 > 0 for stretched transitions
(AJ = i1) where A2 depends only on the properties of the gamma-ray transition
(see Appendix A). This results in a sizable difference in the angular distribution for
transitions with different AJ, similar to that observed in the 171 and 221 keV gamma
rays. This suggests that the 50 keV state should be an unstretched dipole transition
with J f = 3/ 2. This is also supported by the evidence that the 50 keV transition is

known to be a dipole transition based on similar lifetime arguments [65].

While there are ample statistics for the angular distribution analysis for these
low-lying transitions, the evidence may be affected by the long lifetime of the 221
keV state. Another case was examined which was free from such effects. Though the
statistics are not as favorable, the 623 keV transition does not suffer from these issues
which obfuscate the interpretation of the transitions from the 221 keV state. The 623
keV 7-ray decays from the 673 keV state which has a short lifetime. The intensity can
also be accurately obtained in the spectral fit due to a low background and absence

of contamination from the Compton edge of other 31Mg gamma rays. Figure 4.11

68

 

IIIIIIIlllllllllll'llIIII—r'l

888

Counts per 4 keV

lllllllllllllllllllllllllllll4

 

 

 

q
I

8
o

550 600 650 700 7
Energy (keV)

Figure 4.11: Automatic fitting using ROOT of gamma—ray peaks in the 600 keV
region, including determination of the background (thin line) as well as the residue
between the experimental data (points) and the fit (thick line).

shows the quality of the automatic fitting procedure. In addition, the 673 keV state is
directly fed in the knockout reaction making it more amenable to reaction calculations
as well as preventing effects from spin dealignment as discussed in App. A. The
examination of the evolution of the spin alignment related to the outgoing momentum
provides a test of the reaction mechanism. The angular distribution coefficient (12
was extracted from the experimental data over a range of cuts in the longitudinal
momentum, as shown in Fig. 4.12. Given a prolate spin alignment in the most central
part of the momentum distribution, the data agree with a transition with AJ = 0
including the dipole limitation imposed by the Weisskopf estimates. This suggests an
assignment of J = 3/ 2 to the 673 keV state. It is important to note here that the
reaction mechanism produces a strongly prolate spin alignment, B ,\ > 0, in the center
of the momentum distribution which decreases as more of the central momentum is

included. If this qualitatively holds, then the slope alone of the evolution of the

69

 

f :II'IIIIII—I—I—I—IIIII'IIII'rIII'IrII'II—I—TIIIII

0.2-l-———"-——_~~ _:l

-o_ ' ....
-o.2'— ' J

A 1.

 

 

 

 

-0.4

K
V

 

-0.6

 

 

 

.Ijjlti'lii

llllllllllllllllllLLlllllLlllllllllll

30 4O 50 60 70 80 90
% central pII

 

-0.8

M-
OP

Figure 4.12: The angular distribution coefficient a2 determined from the experimental
data (triangles) and theory (solid line) as a function of increasingly broader longi-
tudinal momentum gates for a 3/ 2—>3/ 2 623 keV pure dipole transition. Theoreti-
cal curves are also shown for a 5/ 2—+3/ 2 transition for a dipole (long dashed) and
quadrupole (short dashed) multipolarity. The abscissa describes the percentage of
central momentum counts included.

angular distribution is a good indicator of the sign of the a2 coefficient. This is
independent of any systematic errors introduced by uncertainties in the determination
of the efficiency of the gamma-ray detectors. Considering the whole momentum
distribution includes this uncertainty in normalization of the detectors but also allows
the use of all the statistics which makes it the most conclusive for some of the gamma-

ray transitions. The 7—ray angular distribution coefficients extracted from the whole

residue momentum distribution are quoted in Table 4.2.

4.2.6 Linear polarization of gamma rays

With the presence of spin alignment in the system, gamma rays emitted from 31Mg
should exhibit linear polarization. Determination of the sign of the linear polarization

alone allows for the measurement of the parity of the excited states provided the par-

70

ity of one of the states in the transition is known. The sensitivity of SeGA to linear
polarization was discussed in Sec. 3.3 by examining the Compton scattering within
the crystal. Polarization measurements benefit from greater consistency in terms of
the normalization of the scattering data since the normalization for a ring is only de-
pendent on the scattering of unpolarized gamma rays in that ring whereas effects in
both rings must be accounted for in angular distribution measurements. In addition,
only a relative normalization of the two directions of scattering is needed. Further-
more, angular distribution measurements depend on an accurate determination of the
efficiency of the array which varies rapidly with energy (see Fig. 3.2) as compared to
the Compton scattering asymmetry which varies more slowly (Fig. 3.8). In addition,
the relativistic effects play less of a dramatic role. The polarization of gamma rays is
also maximized at a center-of-mass angle of 90° which is close to the backward ring
in the classic setup of SeGA which has 06,111, = 114° for residues moving at 6 = 0.4.
Differences in the Compton scattering (Fig. 4.13) were clearly observed for the peaks
in the 600 keV energy region. Note that the 7-ray background in the figure shows no
presence of polarization which adds credibility to the in-beam normalization of the
scattering. However, due to the low statistics, the linear polarization measurements

have significant uncertainty as shown in Table 4.2.

4.2.7 Spin and parity assignments

With an accurate measurement of the linear polarization and angular distribution
coefficients, the multipolaritie's of the transitions can be determined. The linear po-
larizations were extracted using the known 18% relative sensitivity of SeGA discussed
in Section 3.3. To reduce the statistical uncertainty, the whole momentum distribu-
tion of the reaction products was considered. Combining the 'y-ray multipolarity
assignment with the selection rules for knockout reactions (Eq. 1.3) allows for firm

spin and parity assignments. In particular, the selection rules imply the states fed

71

 

 

> I . . . . ,
3 350
Q
a 300
O.
m 250
u...
5 200
8
150
100
:- 50
': .a'; u: :"-:'_
WI.
750 800

Energy (keV)

Figure 4.13: The Compton scattering of gamma rays relative to the reaction plane
defined by the beam axis and the gamma ray propagation direction in 31Mg: (dashed,
left axis) scattering perpendicular to the plane, (solid, right axis) scattering parallel
to the plane, normalized (by a factor of 0.13) to the intrinsic scattering from a gamma
ray emitted by a 152Eu source with no spin alignment

directly in the production of 31Mg have a positive parity.

The results discussed in Subsec. 4.2.5 allowed the spin assignment of the 673 keV
state. Two of the gamma ray transitions from the 673 keV state have sufficient statis-
tics to determine 'y-ray Compton scattering asymmetries and angular distributions
as shown in Table 4.2. When including the full statistics across the momentum dis-
tribution, the 673 keV gamma ray shows evidence of angular distribution. For this
case, the linear polarization measurement is not consistent with zero and suggests a
mixed M 1 transition with AJ = 1 or a stretched quadrupole transition to the ground
state; the comparison to theoretical values is shown in Fig. 4.14 for a final spin of
1/2+. While the measurement is closer to the values for a stretched quadrupole, the
momentum-constrained angular distribution analysis for the 623 keV transition (see
Fig. 4.12) indicates an initial spin of 3/ 2 for this state. This supports the assignment

of a mixed M 1 / E2 AJ = 1 transition with 6 z 2 for the 673 keV gamma ray. The

72

 

   

-1.‘- 7-1-11. 1 _1_ ., - ‘
—01.0—0.5 0.0 0.5 1.0 —01.0—0.5 0.0 0.5 1.0

Figure 4.14: (left) Experimental angular distribution coefficient and polarization for
673 keV gamma ray with 10 confidence interval compared to theoretical values of a
decay to a final spin state of 1/2+ from an initial 66% prolate-aligned spin state of 3/ 2
(circles) or 5/ 2 (squares) for different multipolarities as indicated ~ lines of M 1 / E2
mixing are also shown; (right) Measurements for 693 keV gamma ray compared to
values for a final spin state of 7/2— postulated by Mach et al. [65] assuming a 33%
initial prolate alignment for an initial spin of 11/2 (squares) or 7/ 2 (stars).

information for the 623 keV gamma ray is less conclusive when including the entire
residue momentum distribution. However, the negative scattering asymmetry (and
thus negative polarization) is more in agreement with a M 1 transition suggesting
J7r = 3/2(+) for the final 50 keV state with the parity remaining tentative due to the
substantial uncertainty in the Compton scattering asymmetry.

The linear polarizations and angular distribution coefficients for the 693 keV tran—
sition also deviate sizably from zero despite the significant uncertainties. Only the
final spins which have expected values near the experimental data are shown for clarity
in the exclusionary plot in Fig. 4.14 (see Appendix A for further information) which
compares the measured angular distribution coefficients and linear polarizations to
those calculated by transitions of different multipolarities and initial spins. Here, the
spin and parity for the final state are not certain but have been previously postulated

to be 7/2‘ [65]. Generally, a2 < 0 and P < 0 agrees best however with unstretched

73

 

1W 1 or stretched E2 transitions; however, an E 1 multipolarity is not too far removed
from the experimental confidence interval. The normalization of the angular distribu-
tion can add considerable systematic uncertainty to this analysis. Truly, one would
want to investigate the evolution of the linear polarization with momentum gates
similar to the angular momentum study. In the current experiment, the statistics are

too prohibitive to do such an analysis.

4.3 33Mg

4.3.1 Beam characteristics and particle ID

In the case of the 33Mg study, the production cross section for the 34Al secondary
beam was significantly reduced. To achieve enough statistics within the allocated
time, a thicker secondary reaction target of 2 mm beryllium (370 mg/cmz) was used
at the target position of the 8800. The purity of the secondary beam was also signif-
icantly lower (50%) with the major contaminants being 35Si and 37P. Similar to the
study of 31Mg, the components of the secondary beam can be cleanly selected based
on the time of flight from the A1900 to the object box of the S800 as displayed in
Fig. 4.15. The reaction products shown in Fig. 4.16 were then identified in the S800
using the same method as in Section 4.2 where the unreacted secondary beam was
removed in the S800 by Bp selection. Two normalization runs with the unreacted
beam transmitted to the focal plane of the S800 were taken with a lower intensity to
establish the properties of the incoming beam.

For the 33Mg residues, the mean mid-target velocity was found to be [3 = 0.402 :1:
0.002. At this velocity, the forward ring of SeGA lies at a center-of-mass angle of
54° limiting the effects of angular distribution given P2 (cos 00m) = 0.011(3). With a
thicker target than for the 31Mg experiment, the energy resolution of the gamma ray

detection was worse. An in-beam gamma-ray resolution of 2.7% relative to the 7—ray

74

 

£1400r--................
S
01200
1000
800
600

400

llllllllllllllllllllllll

   

200

 

 

[IIIllIl'Ill'lllllllllIlllll

l I 1 1 . I
900 950 1000 1050

 

oo
01
C

Time (arb. units)

Figure 4.15: Incoming 34Al particles identified by the time of flight from the ex-
tended focal plane of the A1900 magnetic separator to the object box of the S800
spectrograph.

energy was obtained for both rings. Corrections based on the S800 information allow
a better determination of the velocity and improve the relative resolution to 2.5%
in the backward ring. The improved resolution in the backward ring was used for

determination of all energies of the gamma ray transitions observed in the experiment.

4.3.2 Level scheme

The gamma ray spectrum resulting from the knockout reaction to 33Mg is displayed
in Fig. 4.17. Information about the transitions observed in the experiment is summa-
rized in Table 4.4. None of the transitions show any indication of lifetime effects as

discussed in Subsec. 4.2.2 which suggests these states are all short-lived so the decay

75

AE (arb.)

 

1400

1420 1440
“me of flight (arb.)

Figure 4.16: Outgoing residues for the reactions of an incoming 34Al beam identified
by the S800 spectrograph based on the energy loss in the ion chamber and their time
of flight with selected isotopes of interest labeled.

Table 4.4: Gamma-ray transitions observed for 33Mg in the current experiment with
their intensities normalized to the 483 keV transition as well as their angular distri-
bution coefficients and Compton scattering asymmetries.

 

 

 

 

 

E; (keV) a(keV) E7 7 0.2 A
484.1 0.0 483.1(9) 100 0.36 (17) 003(8)
705.0 484.1 219.7 (10) 1.8 (7)
0.0 704.4 (14) 6.7(12) 0.9 (8)
780.4 484.1 297.3(6) 57(3) 0.33 (21) 008(9)
0.0 780.7 (18) 15.2 (19)
1242.4 484.1 758.1 (16) 12.6 (18) 0.1(7)
0 0 1240 (3) 16 (3)
unplaced 1069.6 (24) 5.0 (17)
1523(4) 12(3)
1856(5) 7(4)
1929(6) 7(3)

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> I I I I I I I I I I I I I I I I I r I r
£4 1. I I l l l I .
N
3411
a .1
.235
3
0311.
2 1
211 [1’0
1 1 '1
111 'l I ‘ i 1 '10
' 2" 1‘ 1.
, - 11 1111111111,
200 400 600 800 1000 1200 1400 1600 1800
' Energy (keV)
> I I I I rI FI rI I I l I I I l I l I ' I I I I I r
s . .. 1
N
33 . ..
.9 5
§311 -: o
02 . I] I
75
211 1
1 1’1
1 ' '3 15
111' , 1 l 10
' w '5

 

1 , 1 ., 1 ff
“I‘ll-”Jud 1.1L] IIIHIU'J .
200 400 600 800 1000 1200 1400 1600 1800
Energy (keV)

Figure 4.17: Gamma-ray spectrum of 33Mg detected in SeGA for the forward (top)
and backward (bottom) rings.

77

 

occurs near the target (7' S, 10 ps). None of the levels established in previous exper-
iments have any information about their lifetimes. The energies of these transitions
in general agree with those observed in an earlier fi-decay experiment [69] as well as
other recent measurements [70,71]. The ’y-ray branching ratios from the 1242.4 keV

state also agree, within the uncertainties, with those in the fi-decay experiment.

The decays from the 705 keV state warrant additional discussion. The weak 219.7
keV transition could only be clearly observed when summing the spectra from both
rings of SeGA. Motivated by the previous branching ratio from the fl-decay experi-
ment [69] and the number of counts observed for the 704 keV gamma ray, the expected
counts in the two rings would be 192 d: 22 counts assuming isotropic angular distribu-
tion. On top of the large background (S/N z 0.1), a small peak is indeed observed
at the right energy with a total of 135 d: 54 counts which is within the margin of
error of the previous experiment. However, the fi-decay experiment also tentatively
placed a 546.2 keV transition emanating from the 705.0 keV state. This had a sig-
nificantly higher intensity observed following fl-decay, and while approximately 300
counts would be expected in the 33Mg gamma-ray spectrum, no evidence of a peak
is observed. A gamma ray consistent with the energy of 546.2 keV was also seen in
the proton inelastic scattering experiment [71] without observing the other gamma
rays associated with the 705 keV state in the fl-decay experiment. This suggests
this gamma ray does not originate from the 705 keV state, but most likely feeds the

ground state directly from an unassigned level of 546 keV.

A new level could also be placed in the 33Mg level scheme based on the information
provided by 7-7 coincidences. The 483 keV decay was seen to be in clear coincidence
with the 297 keV gamma ray (see Fig. 4.18) allowing for the placement of a level
at 780 keV. The coincidence is also seen taking the reverse gate on the 483 keV
transition. Incidentally, a 780.7 keV gamma ray is also observed for the first time in

the current experiment. This transition has an energy which agrees with the decay

78

 

 

I l I III I I I I I I l I I I I l I I I I

Counts per 5 keV

88888

 

d
O 0
-Im

0 500 1000 1500 2000 2500
Energy (keV)

Figure 4.18: Background-subtracted gamma-ray spectrum coincident with the 297
keV gamma-ray transition in 33Mg.

of the newly placed state to the ground state and has been placed in the level scheme
accordingly. Theoretically, the established 758 keV decay from the 1242 keV state that
feeds the 484 keV state should also be observed in the 7-7 coincidence measurement.
However, given the intensity in the singles spectra and the efliciency of the detectors,
the coincidence spectrum would only show approximately 23 counts for the 758 keV
coincident decay which is within the background fluctuations. With the additional
information from the 74y coincidences, the level scheme observed in this experiment

can be found in Fig. 4.19.

Several other transitions seen in the proton knockout experiment remain unplaced
in the level scheme. These five transitions mentioned in Table 4.4 have a total intensity
which is 31(6)% of the 483 keV gamma ray transition. These are mostly high energy

transitions which cannot be placed due to the lack of coincidence information.

79

 

1242

 

 

 

 

 

 

 

 

(5/2—) 7’8 781
(5/2+) [705
l . '
219 1.40 297
(5/2-) 1 w 1484
714 731
4 3
(313 r v v 0

 

 

 

Figure 4.19: 33Mg level scheme observed in the present experiment with suggested
spin and parity assignments.

4.3.3 Inclusive and partial cross sections

The inclusive cross section was determined similar to the method established in Sec-
tion 4.2. However, during the experiment, a sudden change in the transmission was
observed following a retuning of the primary beam. The normalization of the incom—
ing beam was based on the rate on the object scintillator in order to provide the
greatest consistency in the extracted cross section on on a run-by-run basis. The
normalization factor was determined from two different runs with the S800 Bp tuned
to select the unreacted incoming 34Al particles in the spectrograph. Fluctuations in
this normalization add an additional 5% systematic uncertainty in the cross section
measurement. The width of the momentum distribution once again led to a cut in the
angular acceptance in the dispersive direction (Fig. 4.20) which accounts for another
2.5% of systematic uncertainty to the cross section measurements.

The inclusive proton knockout cross section was determined to be 4.32 millibarns.
For the 33Mg case, the additional systematic uncertainties from the normalization

and acceptance determination are much more significant. The cross section on a run—

80

dtil (parts per thousand)

 

-100 -80 ~60 -40 -2o 0 2o 40 60 80 100
au(mrad)

Figure 4.20: Focal plane measurements of am and dta showing the acceptance cut for
low momentum 33Mg residues from an incoming 34Al beam.

by-run basis is shown in Fig. 4.21. The variation in the cross—section on a run—by-run
basis adds a systematic uncertainty of 2.5% giving a total systematic uncertainty in
the measurement of 0.4 millibarns. Here, the inclusive cross section is significantly
decreased in comparison to the 31 Mg case. As the neutron separation threshold in
33Mg is 0.2 MeV lower, there is an increased possibility that highly—excited states
populated in the reaction decay rapidly by neutron emission and thus do not make it

to the focal plane of the S800 which could account for the decreased cross section.

The cross sections to the excited states were extracted from the particle-gamma
information along with the balancing of the gamma-ray intensities. The partial cross
section to the ground state was determined by subtracting all the excited state par-
tial cross sections from the inclusive cross section. The results are summarized in
Table 4.5. There was some feeding to all the states which emitted gamma rays
though most of the feeding populated the newly discovered 780.4 keV state. Unas-

Slgned gamma transitions amount to a maximal adjustment to the cross section of

81

 

 

 

 

 

 

5.0 - I l I l I l I l I I I l I I q
4.9 - .4
4.8 - fl -
4.7 - -<
A 4.6 - -1
.0 J «1
g 4-5" + -1
g 4.4 4 J
:8 J J
o 4'31 '3
‘0 4.2 - —
0 4.0 _ ..
a) . .
2 3.9 - ..
co . ,
m 3.8 "' —(
3.7 - _
3.6 - _.
3'5 I I —I— I I I I I I I I I I I
60 80 100 120 140 160 180 200

Run Number

Figure 4.21: Inclusive cross section of 34Al —> 33Mg calculated on a run-by—run basis
chronologically from left to right.

Table 4.5: 33Mg states with direct feeding from the proton knockout from 34Al.
E (keV) J7r aexp (mb)
0.0 (3/2+) 0.42 i 0.16
484.1 (3/2‘) 0.79 :l: 0.13
705.0 (5/2+) 0.24 i 0.04
(
(

 

780.4 3/2") 20510.12
1242.4 1/2+) 0.811009

82

 

Counts per 61 MeV/c

§§§§§§§

0|
O

 

O

’ 13 13.2 13.4 13.6 13.8
pI (GeV/c)

Figure 4.22: (Points) The measured momentum distribution coincident with the 297
keV gamma ray, (Line) Predicted momentum distribution based on the removal of a
d5/2 proton from 34Al including effects from the loss of energy in the target as well
as the incoming momentum spread.

0.88 :l: 0.17 mb. Though this likely will not highly impact any individual partial cross
section, it removes certainty of the feeding of the weakly populated ground state and

705.0 keV state in the knockout reaction.

4.3.4 Momentum distribution of residues

The longitudinal momentum distribution associated with the gamma rays is similar
to the situation in 31Mg, which suggests the removal of a g = 2 proton from the
incoming 34Al isotopes. However the experimental width is broadened due to the
increased thickness of the target and cannot be well—described with typical input
parameters for the reaction. The background-subtracted momentum distribution
coincident with the observed 297 keV gamma ray is shown in Fig. 4.22. This is once
again representative of the momentum distributions of all the states directly fed in

the reaction, including the particle singles detected in the S800.

83

 

-
I
u
q

-
q
I
I

—1
q
q
I
I

—]
1
q
u

_.

d1: 1
0.8
0.6
0.4
0.2

h——’——H
h——>——«

 

 

02
04
-0.6
-0.8

n
u

 

IIIIIIIIIIIIIIII IIIIIII'III'IIIIII
4
—C}—<
1—0—1 -—>—
F——{F——4
1—{3—1
—Cl—1
H—{}—H
h—{}—d
h——CF—«
llllllllllllllll llllllllllllllllllf

 

 

 

WW.
0 20 30 40 50 60 70 80 90 100
% central pll cut

—L I

Figure 4.23: Angular distribution coefficients extracted for 33Mg as a function of
increasingly broader momentum gates for the 297 keV transition (squares) and the
483 keV transition (triangles).

4.3.5 Angular distribution of gamma rays

The gamma rays emitted by the 33Mg nuclei were also examined for the presence of
angular distribution. Only the two most intense gamma rays in the spectrum had
enough statistics to perform a significant analysis. Both of the peaks were clearly
separated enough to perform a systematic investigation based on the gating of the
longitudinal momentum distribution shown in Fig. 4.23. Only the 297 keV shows
strong evidence of variation in the momentum-constrained 7—ray angular distribution.
It is important to reiterate here that based on a prolate alignment in the center of
the momentum distribution the slope as a function of broader momentum gates is
directly related to the Sign and magnitude of the angular distribution coefficient. This

is independent of any systematic error in the absolute normalization.

As the calculated angular distribution coefficient for the 483 keV transition is

84

relatively flat as more momentum is included, it is suggestive of an a2 coefficient
close to zero. However, the coefficient extracted from the gamma-ray intensity and
efficiency is significantly positive (20) which should also appear in the slope of the
momentum-gated angular distribution analysis. This provides an estimate of the sys-
tematic uncertainty in the absolute normalization of the gamma-ray intensities in the
rings of roughly 10% based on the deviation from a2 z 0 and the difference between
the P2(cos 6cm.) in the two rings. The uncertainty in the absolute normalization is

avoided in the momentum-constrained analysis.

For the low-energy 297 keV gamma ray with no evidence of lifetime effects, the
Weisskopf estimates limit the transition to dipole multipolarity. The evolution of this
transition clearly resembles the trend observed for the 623 keV transition in 31Mg
discussed in Section 4.2. The trend combined with the Weisskopf estimates suggests
that the 297 keV gamma ray corresponds to an unstretched dipole transition. On the
other hand, the 483 keV transition shows no evidence of a variation in the momentum-
gated angular distribution. This suggests one of two possibilities. The nuclear spin
alignment has been reduced by the gamma rays which feed the state from above, or
the properties of the transition are such that no spin alignment can be observed. Since
the feeding states likely have significant prolate alignment in the most central part of
the momentum distribution, some degree of alignment is expected as well in the fed
484 keV state. This seems to make the second interpretation more likely where the
lack of angular distribution is a property of the 'y-ray transition. This means either
the 484 keV state could be a J = 1 / 2 with no possible spin alignment or the transition
is such that there is only a small effect in the angular distribution such as a AJ = 1
strongly mixed transition originating from a high spin (e.g. 7/ 2 —+ 5/2). Here the
significant deformation of the 33Mg nucleus [70,71] can create a sizable enhancement
of the E2 transition probability which would explain this strong mixing for the low

lying 484 keV transition.

85

Counts per 4 keV

 

250 300 350 400 ‘450 500 550 600
Energy(keV)

Figure 4.24: The Compton scattering of gamma rays relative to the reaction plane
defined by the beam axis and the gamma ray propagation direction in 3P’Mg: (dashed,
left axis) scattering perpendicular to the plane, (solid, right axis) scattering parallel
to the plane, normalized (by a factor of 0.13) to the intrinsic scattering from a gamma
ray emitted by a 152Eu source with no spin alignment.

4.3.6 Linear polarization of gamma rays

Information was also extracted about the linear polarization of the gamma rays emit-
ted by the reaction residues. Similar to the case of 31Mg, the backward ring lies at
0cm, = 114° given the measured velocity of the particles. This is close to where the
linear polarization is maximized, however with the reduced statistics the evidence is
less clear in the 33Mg case. Only the two low energy transitions have enough statistics
to be sensitive to the *y-ray linear polarization. The resultant spectrum is shown in
Fig. 4.24. Though the asymmetry looks quite large, this is reduced by the increased
widths of the gamma rays which scatter parallel to the plane. The decreased res-
olution of the parallel scattering 'y-rays is likely attributed to small inaccuracies in
the Doppler reconstruction. Perpendicular scattering is completed confined within a

longitudinal slice of the SeGA detector (see Fig. 3.7) which constrains the angle more

86

precisely than in the parallel scattering case. With the low statistics, the uncertainties
in the measured asymmetries are sizable. For the 297 keV and 483 keV transitions,
the Compton scattering asymmetries are A = —0.08 :l: 0.09 and A = —0.03 :1: 0.08
respectively. With significantly more statistics, one could investigate the evolution
of the asymmetries as a function of longitudinal momentum; however that remains

intractable here.

4.3.7 Spin and parity assignments

Through analysis of the linear polarization and angular distribution of the gamma
rays, the transition multipolarities can be established. With sufficient precision, some
knowledge about the mixing ratios as well as the actual final and initial spins can
be determined. However, often experiments must rely on prior knowledge about
either the initial or final spin and parity of the transition. In 33Mg unfortunately,
there are no firm previous assignments of spin and parity. Furthermore, the range
of spins allowed to be populated by the knockout selection rules covers a wide range
from 3 / 2" to 13/2— based on the removal of a d5/2 proton from the ground state of
34A1 with known J"r = 4‘ [72]. Given the uncertainties in the current experiment,
strong support cannot be given to a given multipolarity; however, the polarization and
angular distribution measurements suggest certain mulitpolarities, while excluding
others.

For the 484 keV transition which feeds the ground state, the positive a2 coeffi-
cient supports a stretched dipole transition. The polarization measurement does not
constrain this to electric or magnetic character, but would suggest a strong M 1 / E2
mixing as shown in Fig. 4.25 which compares the measured angular distribution coef-
ficients and linear polarization with predictions for different multipolarities and initial
Spins. Previous Coulomb excitation work [70] indeed suggests that this transition

has a considerable E2 contribution with an initial spin of 7/2+ assuming a ground

87

 

 

1 O P I— T Y T I
O i 4

0.5:

612 0.0:

 

—0.5: —0.5

 

 

_1. 9-4-44-4—444-4444H -— _1. g»
-01.0 -0.5 0.0 0.5 1.0 -01.0
Figure 4.25: (left) Experimental angular distribution coeflicient and polarization for
484 keV gamma ray with error bars and 10 confidence interval compared to theoretical
values of a decay to a final spin state of 5 / 2+ (from 'Pritychenko et a1. [70]) from an
initial 33% prolate—aligned spin state of 7/ 2 (empty circles) or 3/ 2 (filled circles) for
different multipolarities as indicated — lines of M 1 / E2 mixing are also shown; (right)
Measurements for 297 keV gamma ray compared to values for a final spin state of

7 / 2"" assuming a 66% initial prolate alignment for an initial spin of 9/ 2 (empty circles)
or 5/ 2 (circles).

 

state spin of 5 / 2+. The current experiment agrees with this interpretation, especially
if one considers the possibility of a systematic uncertainty in the angular distribution
measurement discussed in Subsec. 4.3.5. As the measurement only determines the
spin and parity differences, this could also be interpreted as a 7/2‘ -+ 5/2" transi-
tion which agrees better with the presence of direct feeding in the knockout reaction
shown in Table 4.5. However, a possibility that this partial cross section could be

affected by feeding from unplaced and/ or unresolved transitions cannot be excluded.

For the 297 keV transition from the strongly populated 780 keV state, the data
once again suggest a AJ = 1 transition which likely also has considerable mixing.
Here the final spin state has been assumed to have J 7' = 7/2(_). From the calculated
points in Fig. 4.25, this seems to support a similar situation as the 484 keV gamma

MY; here with initial J7r = 9/2‘ favored slightly. Here, the knockout selection rules

88

clearly constrain the parity of the initial state. The spin assignment depends on the
assignment of the lower-lying states, but a AJ = 1 transition remains the most likely.

Further discussion regarding other possibilities considered in previous work will follow

in Sec. 5.2.

89

Chapter 5

Commentary

5.1 31Mg

5.1.1 Calculation of momentum distribution

The momentum distribution of the fragments following the proton removal shown

in Figure 4.9 can be described well in a three-body reaction model based on the
eikonal and sudden approximation [27,73]. For the standard shell ordering for the
protons in the incoming 32A] nucleus with Z = 13, the valence proton would occupy

the d5/2 orbital, which is consistent with the momentum distribution of the residues
observed in the reactions. The calculations considered this valence d5/2 proton outside

0f the J7r core of the 31Mg residue in its excited state. The core and the valence
PI‘Oton were coupled to the known ground state spin of 32Al [14], J7r = 1+. It is
1.mlDOrtant to note that the incoming 32Al fragments were assumed to impinge upon
the Secondary fragmentation target while they were in their ground state. An isomeric
State with J 7‘ = 4+ does exist in 32A1 with a half-life of 200 ns, which was previously
Obse 1‘Ved by Robinson et al. [74]. However, the population of this isomer from the
fragmentation reaction would have decayed considerably over the ~80 m flight path

which corresponds to roughly four half-lives. Based on the selection rules (Eq. 1.3),

90

direct removal of a d5/2 proton from the 32Al nucleus would populate states with
J 7' = (3/ 2,5/ 2, 7/ 2)+ in the residues. Population of the 1/2+ ground state is small
since it could only proceed by the removal of a deeply bound 31/2 proton or a d3 /2

proton which is an unfilled orbital with standard shell ordering.

5.1.2 Structure of observed excited states

The structure of the excited states was interpreted in the framework of several theo-
retical models. A calculation using the USD shell model interaction was used for the
analysis of the UpIh states. As the USD interaction is truncated above the N = 20
shell gap, its application to states with excitations across the shell gap is intractable.
Since the ground state is dominated by a 2p3h configuration, the relative excitation
energies of the 039111 states cannot be addressed by the calculations. To assess the
intruder states, the technique using AMD+GCM [67] employing the Gogny DIS in-
teraction [75] was used. The excitation energies in both calculations were shifted to
reproduce the observed energy of the 0p1h 3 / 2;“ in Fig. 4.5. As the current work pre-
dominately populated these OpIh states, the theoretical predictions related to these
states are the main subject of the commentary. In the two calculations, there is a
significant difference in the energy spacing of the 0p1h states due to the different
approach used in determining the effective interaction.
The levels at 673 keV and 2015 keV present themselves as candidates for the 017111
3/2+ and 5/2+ states respectively, which are the two lowest 0p1h states predicted
theoretically [67]. The reaction preferentially populates these states since 32Al has
been established to lie outside of the “island of inversion”. The ground state of
32Al is well described by the sd—shell with its 1+ ground state having a dominant
configuration of 7761572 (El/d; /12 [76]. The theoretical partial cross sections to individual
States shown in Table 4.3 have spectroscopic factors calculated from the sci-shell

W8'erfunctions for the ground state of 32Al and for the excited 0p1h states 31Mg

91

labelled in Fig. 4.5. The spectroscopic factors for the 0p] h ground state in 32Al to any
states in 31 Mg with population in the f p—shell is considered to be zero. The inclusive
cross section of 8.7(5) mb has a reduction of spectroscopic strength of R3 = 046(2)
which agrees well with previous trends for removal of a deeply-bound nucleon [77]
characterized by the difference between proton and neutron separation energies in
32Al, AS = 11 MeV [78]. If the tentatively placed 'y-ray transitions discussed in
Chapter 4 are disregarded, this would reduce the partial cross section to the 2015
keV state from 4.27 to 3.33 mb. This would agree better with the theoretical relative

cross sections assuming a fixed reduction of spectroscopic strength.

The cross section to the expected 0p1h 7/2+ state is calculated to be aspC2S = 5.9
mb. While the 2015 keV state could be attributed to this 7/2+ state, the data argue
strongly against it. With a reduction factor consistent with that of the inclusive cross
section, the expected experimental cross section would be 031,023 R3 = 2.7 mb which
is significantly lower than the observed partial cross section. Furthermore, this would
leave no evidence for the 5/2+ state which both calculations predict to lie lower in
energy and which has a higher expected cross section from the reaction calculations.
The 2015 keV state also predominately decays to a state which has J7r = 3/2(—)
with the parity taken from the level ordering of Kimura et al. [67] which seems to
agree with previous experiments [65, 66]. The multipolarity of the 7-ray transition
from the 2015 keV state would then be significantly hindered based on the selection

rules for a J7r = 7/2+ assignment. The Weisskopf estimate for a 7/2+ —> 3/2— M2
transition would be 170 ps which would exhibit lifetime effects similar to that seen

in the lineshape for the 221 keV state which has a 133 ps half-life. Even considering
Some enhancement of the M2 transition using the recommended upper limit for the
tr anSition rate of 3 Weisskopf units [41], the spectral lineshape would still be affected
by t he lifetime. These three factors argue against a J = 7/ 2 assignment for the 2015

keV State and support the interpretation of this state as the J = 5 / 2 single-hole state.

92

 

1.0;

 

 

A 0.57
N .
R

1
Q .
Q 0.0]w
<El '.

—0.5}

41.0”w ' A—0I5IIW00T ' I0? ' TC 1.0
6

Figure 5.1: (solid) The effect of quadrupole mixing on the angular distribution co—
efficient A2 for a 5/ 2 —> 3/ 2 M 1 / E2 transition; (dashed) The angular distribution
coefficient for an unmixed 7/ 2 —> 3/ 2 E2 transition.

Angular distribution data would corroborate this assignment, but the experiment was
not sensitive enough due to possible quadrupole mixing in the transition (see Fig. 5.1).
Mixing ratios as small as 6 ~ 0.2 render sizable effects on the angular distribution as
illustrated in Fig. 5.1. With the 2015 keV assigned the J = 5/2+ single-hole state,
this leaves no firm observation of the 0p1h 7/2+ state. The USD calculation predicts
the OpIh 7/2+ state to lie 2.11 MeV above the 3/2+ 0p1h state. This would place

it at an excitation energy 2.78 MeV which would lie above the measured neutron
Separation energy of 2.38 MeV [78]. This explains the non-observance of the 7/2+
as 1' t would decay by neutron emission, supporting the energy spacing between 0p1h
States predicted by the USD interaction. However, unplaced 'y-ray transitions could
have a significant intensity to suggest decay from a higher lying unassigned 7/2+

State - Placing these transitions in the level scheme would clarify the situation.

The USD calculations are incapable of producing information about the states

wlth intruder-like configurations. The other excited states were thus interpreted in

93

the framework of the AMD+GCM calculations [67]. Information from previous ex—
periments [23,65,66, 79,80] also provide an important insight into the nature of these
states. In particular, the neutron-knockout experiment [23] populated the 3/2‘ and
' 7/2‘ states located at 221 and 481 keV respectively. These are labeled as 1p2h
states in the AMD+GCM calculation which agrees well with the removal of a f p—
shell neutron from the known 2p2h ground state of 32Mg. These 1p2h states are
populated in the present experiment through the decay of the Uplh states with no
excitations across the shell gap. The other populated states in Fig. 4.5 are most
likely higher lying states in the 2p3h rotational band based on the level ordering in
the AMD+GCM calculation. There is no clear evidence for the population of the
low-lying state (J7r = 3/2; ) of the highly deformed 3p4h rotational band which has
a calculated deformation parameter ,8 ~ 0.6 [67]. This suggests that the 1155 keV

state is more likely to correspond to the 7/2+ state in the 2p3h rotational band.

5.1.3 Spin alignment calculation and angular distribution

The nuclear spin alignment along the beam axis was compared to calculations using

the reaction theory discussed in Sec. 1.4. Following the knockout reaction, the par-

tial cross section to each individual 31Mg magnetic substate m was calculated with

the component of the total angular momentum along the beam axis J; = m. The
Population of the substates was compared across the entire range of the longitudinal
momentum. Incoming 32Al fragments were assumed to have no initial spin align-
ment. A previous experiment with fragmentation showed no sizable alignment in the
r‘381'd 11a] nucleus 12B after many nucleons were removed from the 22Ne projectile [81].
In tl'le fragmentation reaction which produced 32A], 16 nucleons were stripped off;
this leaves little expectation for a non-zero initial spin alignment in the projectile.
Fun; l71ermore, any spin alignment that would have been produced would have had

tlme to relax during the time of flight to the secondary fragmentation target.

94

Alignment of the residues was calculated following the proton knockout reaction of
the incoming 32Al fragments on a secondary reaction target which preferentially pop-
ulated certain magnetic substates in 31Mg. The relative population of the magnetic
substates defined the spin alignment according to Eq. A.3. Based on this alignment,
the angular distribution of gamma rays relative to the alignment direction was calcu-
lated. The predicted 'y-ray angular distribution for the 623 keV transition is consistent
with the data from the experiment, as displayed in Fig. 4.12, with regards to a range
of longitudinal momentum cuts. Since the gamma ray is emitted promptly (on the
order of picoseconds) after the reaction, effects which dealign the spins are calculated
to be negligible. In addition, higher orders of alignment were calculated to be negli-
gible in the reaction, i.e. B4 << Bg. The spin alignment was also calculated for the
higher-lying 0p1h states predicted by the shell model. This includes the 5 / 2+ state at
2015 keV which has a significant feeding in the reaction. For different spin states in
the 31Mg nucleus, the theory predicts a comparable amount of spin alignment B A in
the residue. At the peak of the momentum distribution, 82 z 0.5 is predicted. The
angular distribution effect is significantly reduced to 82 a: 0.2 when considering the
whole momentum distribution. This amount of alignment is sufficient to determine
spins and parities accurately from the resulting 7-ray angular distribution and linear
polarization, especially if an experimental setup that is more optimized for the task

is utilized.

5.2 33Mg

5-2 - 1 Calculation of the momentum distribution

The momentum distribution was calculated in a similar manner as in 31Mg. The
Corn Darison to the measured experimental values is shown in Fig. 4.22. The width of

the distribution was somewhat underpredicted using typical input parameters for the

95

 

I l I I I I II I r I I I I I F I l I

0.03

do/dpl (arb. units)
2

O
b 9 '
a S 01’

IIII'IIII'IIIIIIIIIIIIIIIIIIIII

0.01

0.005

 

 

lllllllllllllllllllllllllllllll

 

 

. I .
0 -400 -200 o 200 400
ApI (MeV/c)

Figure 5.2: The theoretical momentum distribution of 33Mg after the knockout reac-
tion relative to the central momentum before (thin) and after (thick) the convolution
of experimental effects.

calculation. The calculation was convoluted with the effects from the energy loss in
the target and the momentum spread of the incoming beam. Due to the thicker target,
the experimental effects modified the distribution as shown in Fig. 5.2. However, the
effect of the target remained small compared to the momentum spread introduced by

the knockout reaction.

5-2 -2 Structure of populated states

The discussion of the structure of populated states in the present experiment relies
on the interpretation of previous experiments. In particular, the spin assignment of
the ground state in 33Mg is a subject of significant debate. Also, the extent of the

isl . . . . . . . . .
and of 1nvers10n and 1ts behav10r has undergone substantial reVISion Since it was

96

originally postulated and observed.

The ground state spin and parity assignment in 33Mg has been previously probed
using several different techniques. Beta-decay experiments [69] provided the first
information about the configuration of the ground state. This experimental interpre-
tation is dependent on the information about the parent’s ground state, which has
been assumed based on systematics of Na isotopes. For the ground state of 33Na,
the spin is determined by the odd unpaired d5/2 proton which occupies the Nilsson
orbital with Q" = 3/2+ for prolate deformations using typical values of the nuclear
potential [82]. For the ,B-decay experiment, the possibility of a 5/2+ ground state
spin was also considered based on the proximity of energy levels using shell-model
calculations [83] with a reduction of the gap between the 1f7/2 and 2193/2 neutron
orbitals [84] resulting in a 0p0h ground state in 33Na. The 6 decay was seen to
have sizable feeding of 202t10% to the 33Mg ground state suggesting a Gamow-Teller
allowed transition where a d3 /2 neutron decays into a d5/2 proton populating positive-
parity 1p1h states in 33Mg with spins from 1 / 2 to 7/ 2. The associated shell-model
calculations favor a 3/2+ ground state for 33Mg. Meanwhile, a Coulomb excitation
experiment [70] suggested that the 484 keV transition had E2 character with AJ = 1.
When considering the observed feeding in the fi-decay experiment, the ground state
spin had to be considered to be 5/2+ to explain the experimental data which was
later corroborated by a proton inelastic scattering measurement [71]. However, a re-
cent measurement [21] instead suggests that the ground state of 33Mg is dominated
by a 2p2h configuration with J7r = 3/2— bringing into question the interpretation
of the previous results. Further discussion follows the assumption that this recent

measurement is accurate.

The interpretation of the states pOpulated by the current knockout study is af-
fected by the more complicated structure of the incoming 34A] isotope which lies on

the border of the island of inversion. To reproduce the ground state spin and parity

97

of 4‘ measured using fi-N MR [72], the neutrons in the ground state must have a
configuration which includes excitations across the N = 20 shell gap. In fact, the
2p2h component of the ground state needs to have a magnitude of at least 50% for
the measured magnetic moment to explain the observations. In addition, the inter-
pretation of Himpe et al. [72] needed to reduce the single-particle energy of the 113/2
orbital by 500 keV to describe the magnitude of the magnetic moment. This results

in significant mixing of 7r(sd)up3/2 components in the ground state wave function.

Coming from a 34Al nucleus where there are three active neutron orbitals (d3/2,f7/2,
and (93/2), the neutron configuration in the resulting 33Mg nucleus after the knockout
also should show a complicated structure of states. In fact, it is seen in Table 4.5
that the reaction feeds a number of low-lying states, which could each have their own
unique neutron configuration. However, the spin and parity are still clearly limited

by the knockout selection rules (Eq.1.3).

The most strongly populated state at 780 keV has a significant cross section
beyond the maximal uncertainty based on unassigned transitions. As the 2p2h con-
figurations in 33Mg likely have a considerable number of low-lying states, the cross
section to these states would be considerably fragmented, especially if contributions
from the V193 /2 orbital are significant in the ground state of 34A]. This makes the
780 keV state the most likely candidate for the low-lying 0p0h in 33Mg which has
negative parity based on the knockout selection rules. To make a clear spin assign-
ment, one must utilize the information about the states which the 780 keV state feeds
through gamma-ray transitions. A prompt 780 keV transition feeds the ground state
of 33Mg which is known to have J7r = 3/2' from the fl-NMR experiment utilizing
laser spectroscopy [21]. The Weisskopf transition rate for the 780 keV 'y ray limits
the spin of the 780 keV state from 1 / 2 to 7/ 2. Knockout selection rules reject the
spin assignment of 1 / 2. The 297 keV gamma ray to the 484 keV state is indicative of

an unstretched dipole transition based on the momentum—constrained angular distri-

98

bution analysis in Subsec. 4.3.5. The previous Coulomb excitation work [70] requires
that the 484 keV state decay by a AJ = 1 quadrupole transition to the ground state
which is consistent with the current experiment as discussed in Subsec. 4.3.7. Given
the measured 3/2‘ ground state, this suggests that both the 484 keV and 780 keV
state have J 7r 2 5 / 2‘. This assignment agrees with the previous experimental data
excluding that from B decay [69]. The fi-decay information however makes assump—
tions based on the spin and parity of the parent 33N a. No direct measurements of
33Na have been performed to date. Also, recent measurements suggest a larger extent
of the island of inversion [24] bringing into question the assumption that the parent

has a 0p0h configuration as assumed in Numella et al. [69].

In a spherical shell model with regular ordering, the last neutron in 33Mg would
fill the f7/2 single-particle orbital. A deformed shell model [85] tailored to reproduce
the 2p2h 3/2‘ ground state of 33Mg shows that the unpaired valence nucleon would
occupy the [330 1 / 2] Nilsson orbital for moderate deformations (6 S, 0.3) as seen
in Fig. 5.3 with Q“ = 1/2‘. There is no clear evidence of this 1/2‘ state based
on the knockout data. This certainly cannot be attributed to the 780 keV state as
this is inconsistent With the observed angular distribution of the emitted 297 keV
gamma rays as discussed in Subsec. 4.3.5 as well as being forbidden by the knockout
selection rules. It is suggested that the 3/2— rotational state built upon this orbital
could be the lowest lying in energy due to a large negative decoupling parameter as
in the 1 p211 band in 31Mg calculated by Kimura et al. [67] This presents the low-lying
rotational excitations with J7r = (3/ 2, 5/ 2)‘ as candidates for the 780 keV state.
The momentum—constrained analysis for the 297 keV suggests a transition that is
more likely unstretched as discussed above. This supports the assignment of 5 / 2‘ to
this state assuming the final total angular momentum of 5/ 2 for the 484 keV state.
The assignment of 3/2" cannot be ruled out and is consistent with the evidence in

Subsec. 4.3.7. However, the magnitude and nature of the alignment in the wings of the

99

 

 

 

 

 

 
   
 
 
  
 

 

 

   
 

 

 

 

LLLJJLLIIlLLIlllllLLllllllllllLlllllllllllllllllI
6 _- Neutron one-particle levels In Woods-Saxon potential -_ [A
; vWS = — 40.0 MeV R = 3.990 fm (A = 31) a = 0.67 fm : i
A 0" - .
4 E .’.,,-°’[303 7/2) E’ .
" /"’ " E‘
2 E —- q‘».:::i?emfi.o~:-: ;— . — O I': .... 43°12‘5ch :1— :
. A -’ t.....oo. -‘ f ..... .‘~‘~ ...... .... 3211/2 .—
> O _,/' .,.l__ l t
a) o [—
E . ‘~2[321 3/21 I
V -2 M:
c: -
w [200 1/2) _
-4 .__
-6 :-
-8 [2113/2] :-
3 [2201/2] I
'10 IIIIIIIIIFTHIIIIIII llll'llllllllllIlllllllllllll

 

-0.4 03 02 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
quadrupole-deformation parameter B

Figure 5.3: Neutron one-particle levels calculated in Hamamoto [85] with the param-
eters of the Woods-Saxon potential tailored to reproduce the ground state spins of

31Mg [22] and 33Mg [21].

100

momentum distribution is not certain, so the statements deduced without momentum
gating on the residues are considerably weaker. Since the measurement using the
whole distribution is also subject to normalization uncertainties, it is possible that
the a2 calculation is marginally high. A smaller a2 would bring the measurement
in accordance with the momentum-constrained analysis suggesting a AJ = 0 mixed
1W 1 / E2 transition.

The other states populated are likely dominated by 219211 configurations. In partic-
ular, since the ground state and 484 keV state both show signs of direct feeding, they
have a negative parity. This agrees with the interpretation of Yordanov et al. [21].
Furthermore, the authors suggest that the 705 keV state has an opposite parity from
the ground state. While the direct feeding in the current work seems to discredit this,
the uncertainties which remain due to unplaced transitions surely could reduce the
partial cross section to this weakly populated state. The only other state observed
in the present experiment is at 1242 keV with a sizable direct feeding in the reaction
which excludes the assignment based on fi-decay of 1/2+; however no clear alternative

is present from the analysis.

101

Chapter 6

Outlook

The current work establishes the possibility of performing '7-ray angular distribution
and linear polarization measurements with intermediate energy beams. Several op-
portunities present themselves to improve the quality and consistency of the current
work. To perform accurate measurements, experiments need to improve the reso-
lution and sensitivity of the experimental setup and obtain higher statistics for the

gamma rays of interest.. Fruition of these goals is within the reach of next-generation

'y-ray detector arrays such as GRETA [68] and AGATA [86,87].

In experiments using beams at intermediate energies, these improvements require
a balance between different aspects of the array’s performance. For example, as
detectors are placed further away from the target position, the angular resolution is
improved. This allows for better Doppler correction of the energies and decreases
the solid-angle attenuation factor for angular distribution and linear polarization
effects. The trade-off lies in the decreased detection efficiency of the array as the
detectors now cover a smaller solid angle. On the other hand, detectors could be
moved closer to increase efliciency in exchange for a decrease in the energy resolution.
Furthermore, due to the relativistic kinematics and angular distribution effects, the

observed properties of the gamma rays in the lab reference frame depend on the angle

102

 

of emission relative to the beam axis. It is important to choose angles accordingly such
that the sensitivity to properties of interest is maximized. This also must consider how
the lab angles are related to the angle in the center-of-mass frame which varies with the
velocity of the incoming beam. All these factors must be balanced accordingly when

designing the detector setup, which is often limited by additional physical constraints.

The next generation of ’7-ray spectroscopy will try to tackle these issues by im-
plementing digital data acquisition for the arrays. This will allow for signal decom-
position and 7-ray tracking in the detectors bringing about an improved interaction
position resolution for the gamma ray. In turn, the detectors could have an in-
creased energy resolution or could be brought closer to the target for an increased
efficiency. While the next-generation arrays plan to implement digital acquisition,
recently SeGA was outfitted with digital processing capabilities [88,89] to instrument
the 18 central contact channels as well as the 576 segments, including interfacing this

with the existing electronics for auxiliary detectors at N SCL.

6.1 Signal decomposition and gamma-ray tracking

Digital data acquisition systems record the waveforms associated with the gamma-
ray detection to memory and/ or disk of a computer allowing for further processing.
Signal decomposition deconstructs the waveforms to determine precisely where the
gamma-ray interacted within a given segment. The position determination is based
on the Shockley-Ramo theorem [90] where the induced charge on the electrodes is
related to the moving charges within the germanium semiconductor as a function of
time. In particular, the electron—hole pairs generated when a gamma—ray interacts
within the crystal induces charges on the segment where it interacts as well as neigh-
boring segments. Here, the segment electrode which collects the drifting charges is

identified as the segment where the interaction occurred, and the shape of the wave-

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I
A 34 . E F E
g 500 _. C4 . D4 r E4 __ 4 _ G4 r H4
3 g s E
'w—A-L w ‘A V—U: L‘— ‘fi if ‘m
8 i E E E
I D F
< 400 - r E :' ' E E
v :A A l : AAAAAAAAAA .A A: AAAAAAAAA A
o b r h r b
b p p I
P , .0 b h .
3600 A3- 33». 03. 03; 353. 63;. 63:, 113
:1: E v‘. -' . : :
a A ‘—‘ i “‘r‘ *u i L, ‘9 - - L 4‘41- "1 ___ 3,... -,.I
T‘ b b r '— —— I I '
b D 1
r - r
E 4‘", I- :- L L L In - I
< 1 : I : h P .
1 v p b p .
’A A A1 A IA ........ ’ I 'AA lAAAAl ’ ......... ’ LLA .
I E I I f : I m
h I > b I
F .
6001 A2; 1321, 02; .021;52=_ F2_ 62_ 1-12
’ t : It i I .‘
r E - A z : a . soo-
erb' ‘33...” v i- ’ NW
I I I I E .
h I 1 b s
400: r r r r r " r r
E E E E E E E E aooo-Ref-
ALAAAA- AAAAAAAAA l AlAA A LA A 1AA A I A I A IA A I AAAAAAAA AI A_LAA
I F I b I h
. p b )1 .
D b p t b
500; A1 L 31 ; c1 _ D1 1 E1 ._ F1 _ <31 _ H1
E F E I t E t F m'
I p p - P . t .
m 1r ’ ww“- ’“fi'f ‘_ ." : 3" - ’.. f'_.""'" ’ WWW
. r p p 1,;- 1 '. :' b '- ..' r .
t : : : 7 : =. .- r . ' _
400: r r r r r r ' 1000
I I p h r p
:_. 1 : ........... f ........... i-.21-1-11-. ........ 1.2: ........... t.--1 ....... f ........... AAln-AAILA
005 1.0 005 1.0 0.5 1.0 005 100 005 100 005 1.0 005 1.0 005 1.0 0'5 1'0

Time [us]

Figure 6.1: Induced signals recorded in the 32 segments and central contact of a SeGA
detector for several events where a 1332 keV gamma ray from 60Co interacts in single
segment E2 at different positions. 7-7 coincidences are recorded in a CsF detector;
this serves as a timing reference for the measurement [91].

forms induced in all of the segments carry information about the interaction position.
The magnitude of these signals is related to the energy deposited by the gamma ray

in the detector (see Fig. 6.1).

To understand how the signal changes in time, one must be able to understand
the motion of the charge carriers within the crystal. This requires both an accurate
determination of the electric field in the detector as well as the mobility of the charge
carriers within the crystal structure. The induced signals can then be calculated
using weighting potentials 050 which consider the solution to Maxwell’s equations

with boundary conditions such that only one electrode has a non-zero fixed potential.

104

 

 

 

E0014 'I'l'I'tI'TIIIIIIUIIII'l'Ii'l' ‘ E I...I'U'III'I'li'til'I'Ili'III'U‘
E o ' E o ‘
E 50.1

V
> >

 
 
  

9
8

0.06

0.04

0.02

 

 

 

o 50010001500200025003000 o 50010001500200025003000
E(Vlm) E(V/m)

Figure 6.2: Drift velocities as a function of electric field for (left) electrons and (right)

holes along the [100] axis in germanium with corresponding model calculations for a
HPGe detector [93].

The signal Q(t) induced in the electrode with a given weighting potential is then:

Q(t) = -q¢0($(t)) (6-1)

for a drifting charge q whose position $(t) as a function of time depends on the electric
field in the device. The motion of the charge carrier depends on its mobility within
the crystal which can be fundamentally different for the electrons and holes. This
also depends on the orientation of the crystallographic axes. The mobility along one
crystallographic axis can be related to the drift velocity for different electric fields in
a relatively simple model [92] as shown in Fig. 6.2. Since this is anisotropic given
the crystal structure of germanium, additional effects must be considered as discussed

in further detail in Mihailescu et al. [94]. Once the motion of the charge carrier is

105

 

 

 

known, the resulting signals on the electrode for an interaction at a given position
can be calculated.

After the interaction positions are well known within the detector, ’y-ray trackng
can be used to determine the ordering of these interactions. The ordering of interac-
tions is vital because the first interaction point determines the angle in the Doppler
correction formula; knowledge of this angle is the primary limit on the energy reso-
lution for experiments with intermediate-energy beams. Combined with the second
interaction point, this also determines the first scattering of the gamma ray which is
intrinsically related to its linear polarization (see Subsec. 2.4.1). This tracking is gen-
erally based on the Compton scattering formula (Eq. 2.5) by comparing the scattered
experimental energies of the photon to the calculated energies determined by the
scattering angle. Given a chosen ordering of interactions, the tracking gives a figure-
of—merit based on the deviation in energy. Especially poor figure-of—merits could be
an effect of incomplete absorption in the detector and can be rejected, reducing the
signal-to-noise ratio in the spectrum. In the end, the ordering of interactions with the
best figure-of-merit is taken to be the true ordering; though this is not guaranteed to
be a unique solution [95]. Here, information about the cross-sections of the Compton
and photoelectric effect can provide further information about the most likely order-
ing of events. Further information is available about a number of algorithms in the
literature [96—98], including methods of reconstructing gamma-ray interactions which

occur across several detectors.

6.2 SeGA digital electronics

While digital acquisition is an integral part of future 'y-ray detector arrays, signifi-
cant improvement can be made through developing instrumentation to acquire digital

data for current arrays. Such an implementation has been done with SeGA where

106

digital data acquisition modules were developed in collaboration with XIA LLC with
the particular details of the Digital Data Acquisition System (DDAS) discussed in
previous works [88, 89,91]. The system is comprised of 39 Pixie-16 modules with 100
MHz 12-bit digitizers arranged in 4 custom compact PCI/PXI crates each with its
own dedicated computer for event processing (and eventually signal decomposition).
With a conservative goal of improving the position resolution by a factor of two, the
energy resolution could be improved two-fold or the efficiency of the array quadrupled
(subject to experimental space constraints). For intermediate-energy beam experi-
ments, one should note that the angle of gamma-ray emission relative to the beam
axis is dominately determined by the longitudinal interaction position in a given de-
tector. An increase by two in resolution should be feasible if one can discern if an
event interacted in one longitudinal half or the other for a given event. Simple al-
gorithms which only incorporate the integrated induced signals have been shown to

generate this magnitude of improvement for a 662 keV gamma ray from a collimated

137Cs source [99].

One aspect that is especially important for the successful integration is the timing
in the system. The waveforms acquired by DDAS should be able to be synchronized.
Small constant offsets which can result from cable delays can be compensated for. To
assess the constancy of this offset, tests were performed by giving DDAS a sinusoidal
waveform from a high-precision signal generator. The relative phases between differ-
ent channels gives information about the delay. If the frequency of the signal is well
known, the X2 analysis to determine the phase is linearized [100] making the analysis
for the ~600 channels significantly less computationally intensive. DDAS was shown
to consistently capture synchronous waveforms with jitter less than 300 ps as shown
in Fig. 6.3; note that this resolution is significantly below the 10 ns time step based
on the 100 MHz clock of the Pixie-16 cards. Timing between detectors in SeGA and

auxiliary detectors must also be established. For in—beam runs, a fast plastic scintilla-

107

 

 

 

Counts

 

 

 

 

 

1.—
-111111 llllllllllllilLlLlllllil lllllll JLllLllllllllll IIHHIIILILIII
—2 -1 o 1 2 o 0.2 0.4 0.6
Time [ns] Time [ns]

 

 

Figure 6.3: (left) The mean delay of signals for segments in SeGA and (right) the
standard deviation of the delay for events in a given channel [91].

tor detects the residue and produces the reference signal. This signal has a rapid rise
time and decay time which is beyond the resolution of a 100 MHz ADC. To address
this, each master trigger derived from the fast scintillator also creates a pulse with
similar characteristics to that from a germanium detector to use for timing in DDAS.
The time is then determined on an event—by—event basis using the information about
the time to cross the constant fraction amplitudes of 10% and 20% from the energy
sums calculated for each event on the Pixie-16 card. This results in a 10 ns timing
resolution for the 1332 keV peak emitted from a 60C0 source in coincidence with the

1173 keV gamma rays detected in a fast CsF detector [91] as shown in Fig. 6.4.

Once the waveforms can be properly synchronized and correlated with events in
external detectors, the next step is to develop a. database of calculated waveforms to

which experimental waveforms can be compared in order to determine the most likely

108

Counts

l

2000

 

 

 

r

1 000

 

  

Counts
0 1000 2000 3000

 

T

 

C

 

Energy [keV]
3
O
. ‘? .

500i

 

 

1 L Pl L l l

700 800 ‘ '960' 1 #1000
Time [ns]

 

Figure 6.4: (lower left) Energy-time two-dimensional plot for a SeGA detector with
coincident gamma rays in a CsF detector. (above) Time projections for all the events
(shaded) as well as gated on the 1332 keV transition (clear). (right) Full energy
projection [91].

109

‘1
0°

0

Longitudinal distance (mm)

0.1-1:010:
O

O

 

0 5 10

15, 20 25
Radial distance (mm)

Figure 6.5: Segment division in a SeGA detector for a r-z plane based on the drifting
of holes to a given electrode. The longitudinal distance is measured from the “A”
segment (0 mm) to the “H” segment (80 mm).

position of interaction. Algorithms designed for GRETINA were adapted to the SeGA
geometry and utilized to investigate the sensitivity of the array. First, the division
between segments was investigated based on the movement of the holes to a given
segment electrode. The result of the calculation is shown in Fig. 6.5. The uniform
1 cm physical division of the electrodes is evident at the outermost radius as the
holes drift to the nearby contact. Further away from the segment electrode though,
the holes are subject to greater effects from the field. This especially is evident near
the central contact which ends before the “A” segments. The fringe fields resulting
from this create a much larger volume where interactions occur in what is identified
as the “A” segment and decreased volume for “B” segment interactions. This change
in volume should be accounted for when determining the interaction position within
a segment based on the centroid of the spatial distribution. For lower energy gamma

rays, this must be convoluted with the fact that gamma rays are also more likely to

110

8

d
O
Ill|l||leAJLILllIIIIIJIIll

a
0|

0|

 

O

 

 

 

------ ~ -~ l"'o""é""1'd"'1'5'"é'd"'2'5"'3'd”s§
Figure 6.6: Calculated grid points for an “A” segment of a SeGA detector in the 316-?
plane (left) and :l:-y plane (right) which sample the segment volume to maximize the
sensitivity to the changing shape of the waveform.

interact in the detector on the side closer to the source. Considering the centroid
from a naive uniform physical division clearly does not account for these effects.

The database of calculated waveforms should maximize the position sensitivity for
each individual calculation point. To achieve this, an iterative technique is applied
to an originally uniform cylindrical grid. In a given iteration, the grid is adapted
to make the x2 uniform between any two neighboring grid points. The adaptive
grid thus takes finer samples where the waveform changes rapidly, for example near
segment boundaries and electrodes. An example grid is shown in Fig. 6.6 for an “A”
segment. A given waveform can then be compared to the database waveforms with
a X2 analysis to assign the position of interactions.

The signal calculations have been developed to a preliminary stage to demonstrate
the sensitivity for SeGA detectors. To maximize the position resolution, other effects
must be considered. The calculated signals need to be folded with the electronic
response of the detector including the preamplifier as well as the cross talk (both dif-
ferential and integral) between different electronics channels. Anisotropies in the drift
velocity and charge trapping also are not accounted for in the current calculations.
Source experiments to characterize the detectors of SeGA are necessary to constrain

parameters such as the crystal orientation and the mobility of the charge carriers.

111

 

These topics will be addressed in future work.

6.3 GRETINA impact

While DDAS will allow considerable improvements of the position resolution of SeGA,
the detectors are not ideally designed for sensitivity of the produced waveforms to
the interaction position. Next generation arrays such as GRETA [68] and its pro-
totype GRETINA have incorporated this sensitivity directly into the design of the
detectors, for example by minimizing cross talk between channels. The goal of the
detector and electronics design was to achieve a position resolution of less than 2
mm RMS in three dimensions which has been recently demonstrated [101]. This is a
five-fold improvement on the position resolution currently given by the segmentation
of SeGA detectors. The possibilities with the GRETINA prototype alone are monu-
mental, especially for experiments at fragmentation facilities. This superior position
resolution will greatly enhance the sensitivity of the array to angular distribution and
correlation effects.

GRETA is proposed to cover the full 47r solid angle with 30 detector clusters
each of which house 4 HPGe detectors in a common cryostat. The GRETINA pro—
totype under current construction will have In coverage and will demonstrate all the
necessary aspects of implementing the entire array. The physical design of GRETA
places the 30 clusters in five angular rings (32°, 58°, 90°, 122°, and 148°) to cover
the entire solid angle. GRETIN A will consists of 7 clusters which will be distributed
among the forward angles to maximize the sensitivity of the intended experiment.
The implications for 'y—ray angular distribution and linear polarization measurements
are discussed here.

The sensitivity to these effects depend on the values of the Legendre polynomials at

the center-of-mass angle for a given velocity as well as the efficiency of the detectors

112

 

 

 

Table 6.1: Relevant quantities for angular distribution measurements for GRETA
angles assuming 6 = 0.4.

2
Olab 00m. P2(COS 0om.) P2( )(COS 0om.) (imam/(1913b Emult

 

32° 47° 0.190 1.62 1.92 1.39
58° 81° -O.459 2.92 1.35 1.16
90° 114° -0.260 2.52 0.84 0.92
122° 140° 0.383 1.23 0.57 0.76
148° 159° 0.803 0.39 0.47 0.68

taking into account the effects of the Lorentz boost. The relevant quantities are
_ expressed in Table 6.1. To maximize the sensitivity to angular distribution for a
measurement in two rings, the detectors should be placed to maximize the difference
in P2(cos 60m). At the same time, the efficiency should be optimized so the requisite
statistics can be collected taking into account the forward boost in the solid angle
for the Doppler effect as well as considering that gamma-rays emitted forward are
Doppler shifted to higher energies and thus are detected with less efficiency. For
angular distributions, the figure—of-merit, F, calculated from the statistical analysis
is:

1 1 )-1/2,

F = (P2031) - P2($2))(a + S

(6.2)

where 131,2 = cos 00,111,,13 and 61,2 is the efficiency of detectors at that angle. The
largest difference in P2 is clearly between the two angles for the GRETINA setup in
the forward hemisphere. From this alone, GRETIN A has at least a two-fold increase
in angular distribution sensitivity compared to SeGA. This does not consider the
four-fold efficiency increase, which will improve the sensitivity by another factor of
two. Figure 6.7 shows the figure-of-merit for a two angle measurement with one of the
angles fixed at 32°. This does not consider other effects which may be of experimental
importance such as the energy resolution necessary to resolve the gamma transition as
well as increased signal-to-noise for the forward angles. For reactions which only result

in moderate spin alignment, a two-point measurement is sufficient as only a2 needs

113

 

 

 

 

 

 

. ’.‘.
I ” a
» ’I .
0.4 C,‘ ’ ‘
LL. : :
0.2; i
f
0.1 f f
0.0 ”l r I i i L i i i i ' 1 I i i r l i “
0 50 100 150
glab

Figure 6.7: Angular distribution sensitivity for residues with 6 = 0.4 (solid) and
,8 = 0.525 (dashed) considering placement of a second ring with the first ring fixed at
32°.

to be determined. Detectors should be focused in two rings to maximize the statistics
and extract the most precise measurement. It is also useful to extract branching
ratios for transitions in detectors which have angular distribution effects minimized.
For the GRETA angles, effects from angular distribution are present in all rings for
current typical intermediate beam energies. At the next generation Facility for Rare
Isotope Beams (FRIB) [102] which will upgrade NSCL’s capabilities, primary beam
energies of 200 MeV/u will be obtainable for elements as heavy as uranium. For
these energies, velocities of nuclei near 6 = 0.525 will result in gamma-ray emission
with minimal angular distribution effects in the 32° ring of GRETA while preserving
significant sensitivity in other rings. At these higher beam energies, measurements at

backward angles become increasingly important as shown in Fig. 6.7.

GRETINA will also significantly improve the linear polarization sensitivity. This
is a result of the array geometry as well as the abilities of ’y-ray tracking. The

polarization is maximized at 90° in the center-of-mass frame. GRETINA’S detectors

114

 

at 58° are quite suitable for polarization measurements. Using segment information
to determine the scattering angle alone, the sensitivity of the GRETIN A detectors
would be comparable to SeGA (See Table 3.1). Better resolution of the scattering
angle based on the better position resolution from signal decomposition increases the
sensitivity by a factor of two based on simulations [96]. Greater benefit comes from
the ability to track gamma rays through the detector. Current limitations on the
sensitivity stem from limiting the analysis to gamma rays which only interact in two
segments which reduces the detection efficiency markedly and does not consider the
possibility of multiple interactions within a segment which could be resolved by signal
decomposition. The gamma-ray tracking algorithms discussed by Schmid et al. [96]
would deduce the correct ordering of the first and second interaction points ~80% of
the time for a 2 mm position resolution. This increases the efficiency of detecting the
relevant Compton scattering events by a factor of three. The combined improvements
in sensitivity and detection efficiency will result in a figure-of—merit roughly 12 times
better than SeGA. Furthermore, this does not consider the benefits from measuring
the continuous distribution of the first scattering angle.

With these improvements, angular distribution and linear polarization measure-
ments will become accessible for many more transitions. .With the increased efficiency,
one can even consider the possibility going further and consider 7—7 angular correla-
tions to give more information. A sizable piece of this improvement is also available
to detectors in SeGA by using the Digital Data Acquisition System. This Opens the

opportunity to probe the spins and parities of excited states in many exotic nuclei.

115

 

" arr-j 1'. it.

 

._.-..

 

 

Chapter 7

Conclusions

Knockout reactions are a powerful spectroscopic tool to explore the structure of exotic
nuclei. In particular, proton knockout reactions on neutron-rich nuclei access even
more rare isotopes. Furthermore, the nuclear spins of the reaction residues are aligned
following the removal of the proton. This results in the angular distribution and
linear polarization of the emitted gamma rays. Measurement of the distribution
and polarization allows for spin and parity assignment of the populated states when
combined with the information from the knockout selection rules or from previous

experiments.

For 31Mg, two levels were identified as the normal shell model states with no ex-
citations across the N = 20 shell gap. The spacing between these levels agrees with
those produced by a USD shell model calculation. Assignment of the spins of the
populated states was possible by examining the angular distribution of the gamma
rays in a momentum-constrained analysis which removes the possible systematic un-
certainty related to the efficiency of the gamma-ray detector. Parities could also
be tentatively assigned based on linear polarization measurements, knockout selec-
tion rules, and Weisskopf estimates of transition rates. A quenching of spectroscopic

strength was observed in the reaction which agrees well with trends for the removal

116

 

of a deeply—bound nucleon [77].

The lowest-lying state with normal shell ordering was also identified in the 33Mg
nucleus after a proton was stripped from the 34A] nucleus. Due to the more compli-
cated structure of the incoming 34A1, a fragmentation of strength was also identified
to several 2p2h states in the residue. As no prior spins and parities are firmly es-
tablished in 33Mg, absolute spin and parity assignments are not possible. However,
information from the gamma-ray transitions suggests certain multipolarities for the
decays from the low-lying states. A level scheme is proposed which agrees well with
a recent measurement of the ground-state spin [69] as well as Coulomb excitation
data [70].

The identification of the states with no excitations across the N = 20 shell gap
contributes to the knowledge of the “island of inversion”. The qualitative location of
these states in the level schemes are predicted well by calculations, but true quantita-
tive agreement for all the states is diffith to reproduce. For 31’33Mg, the excitation
energy of the lowest-lying state with a normal configuration above the inverted ground
state remains relatively constant showing the similarity between these two isotopes.
Similar investigations into more neutron-rich magnesium isotopes (as well as neon
isotopes) would add additional understanding about this region and the interactions
which are responsible for the inversion.

Future improvements will make a wide swath of exotic nuclei accessible for explo—
ration. As the shell structure evolves, methods of making model-independent spin and
parity assignments are especially critical. Gamma-ray angular distribution and linear
polarization measurements will be an essential part to probe this evolving structure.
N ext-generation arrays of gamma ray detectors implementing digital data acquisition
will have an increased sensitivity for making these measurements. The techniques
established in the present work will allow future experiments to effectively harness

this sensitivity to enrich our knowledge about the atomic nucleus.

117

 

Appendix A

Gamma-ray angular distribution

Many different styles of notation exist in the literature; here we rely on the definitions
of Krane [103] as in Olliver et al. [43]. In this notation, the angular distribution is
decomposed into a series of Legendre polynomials, PA, in the angle 6 with respect to
the beam axis:

21’

me) = Z AA(J,-, Jf,6)B,\(J,-,0)PA(0036) (A.1)
A20

for a gamma ray transition between an initial state with spin J; and a final state J f.
This depends on a quantity B A that characterizes the initial spin orientation in the
nucleus described by the parameter a (see Eq. A.4). A ,\ captures the characteristics of
the transition including the mixing ratio 6 of electric and magnetic matrix elements.
The expansion is taken up to order 1' = l + 1 where l is the minimum multipolarity

of the transition dictated by the triangle relation.
1: [Ji—Jfl (A2)

The transition probability for different multipolarities is roughly proportional to
(Ero / hc)21+1. For typical gamma ray energies E and nuclear densities m z 1.2fm,

Em << he, so higher orders of the angular distribution are negligible. Furthermore,

118

 

for spin orientation along an axis that is azimuthally symmetric, only even order

Legendre polynomials contribute.

The spin alignment depends on the reaction mechanism. Each reaction mechanism
preferentially populates different magnetic substates in the reaction residues, where
the magnetic substate m is the projection of the total nuclear spin, J, along the beam
axis, i.e. m = J2. The reaction thus populates a distribution w(m) of the magnetic
substates which is then normalized to unity. The coefficients BA depend on this
substate distribution

Ji J- J- A
BA(J,-,a)=\/(2/\+1)(2J,-+1) Z (—1)Jz'+m ' ' w(m;a). (A.3)

 

Typically, the substate distribution is taken to fit according to some model parame-
terized by 0. In a given reaction, there is either a tendency to produce spins aligned
with the quantization axis (prolate alignment, m = :ini) or perpendicular to it (oblate
alignment, m = 0). A reasonable approximation to the substate distribution in the
case of the oblate alignment is a Gaussian with zero mean as studied by Yamazaki [104]
based upon work done by Diamond et al. [105]. With the proper normalization, the

distribution has the form

e—m2/202
w(m) = Z” e_m,2/202. (A.4)
[—

 

Note that this takes into account the azimuthal symmetry of the reaction, so w(m) =
w(—-m). The Gaussian distribution has a width 0 which is connected to the amount
of oblate alignment in the system. The maximal alignment happens when a —* 0

where only m = 0 is populated in the reaction. This results in a maximal alignment

119

 

 

Table A1: Maximum alignment coefficients for different types of alignment.
prolate oblate (half-integer J) oblate (integer J)

 

 

 

 

 

 

max _Bzmax(J,-) J,(2J-—1) 1 (2J-—1)(2J-+3) J-(J-+1)
p20 (‘10— x/5 fifilflbfiifi —Z sz'(Ji+12) —\/(2Ji:1)z(2Jz'+3)

for a given spin from Eq. A3 of

 

 

J,‘ J; A
8313336010: x/(2/\ +1)(2J,-+1)(-—1)Jz'
0 0 0
(11.5)
J,- J, /\
3111300): \/(2,\ +1)(2J,-+1)(_1)Jz-+1/2
, —1/2 1/2 0

for integer and half—integer spins respectively. Similarly, for prolate alignment, the
formula can be extended to favor the states m = :ini [44] with a Gaussian distribution

about both. Accordingly, the substate distribution is

e—(J.—Iml)2/2a2

 

 

w(m) = J (A6)
2' —(.I-—|m'|)2/2o-2
Zm’2—Ji 8 2
which leads to a maximal alignment coefficient of
J ° J - /\
33131541,) = \/(2A + 1)(2J,- + 1) ' . . (A.7)
—J,- J,- 0

The amount of alignment is often cited in terms of the percentage of this maximum,
Brele': 0‘) = Bg(J,-, a)/B§"ax(Ji). This is related to B2 since the higher order terms
in the angular distribution have a considerably smaller effect on the experimental
observables for the typical experiment. The reduced form of the maximum alignment
coefficient is shown in Table A.1 where pm = B A / (2A + 1) is another coefficient of

alignment commonly used in the literature. It is important to note that 82(Ji) > 0

120

 

for prolate alignment while Bg(J,;) < 0 for oblate alignment.

Once there exists some degree of alignment in the nucleus, the gamma rays emit-
ted will exhibit an angular distribution with respect to the alignment axis. The
alignment condition automatically precludes angular distribution from J,- = 0 and
J,- = 1 / 2 states for axially symmetric arrangements. The distribution of the gamma

rays is related to the properties of the transition. This is described by the coefficient

AA(J,-, Jf,6) which has the form

1
AA(J1', Jf, (5) =W(FA(L, L, Jf, Ji) + 26F/\(L, L +1,Jf,.]i) (A 8)

+52FA(L + 1,L+1,Jf,Ji))

for a transition of a given multipolarity L expressed in terms of generalized F -
coefficients. For a specific transition, magnetic and electric matrix elements can
contribute to the transition. These must have the same change in parity though. For
magnetic transitions, the parity change is An = (—1)L+1, and for electric transitions
we have A7r = (—1)L. That is to say that a transition with a given character of
multipolarity L mixes with the transition of opposite character with multipolarity
L + 1. As higher multipolarity transitions are often greatly hindered due to the en-
ergy dependence of the transition rate, higher order contributions to the mixing are

ignored. The mixing is quantified by the mixing ratio 6 with

= <Jfltr<vr<L +1))|Jz->
(Jf|T(7r’L)|J.->

 

(A9)

defined in terms of the rates T for transitions with character 7r or opposite character
7r’. For a pure transition of multipolarity L (i.e. 6 = 0), the angular distribution
coefficient in Eq. A.8 becomes A A = F ,\(L, L, J f, J,). The F coefficients are defined

according to the generalized F coefficients [106] where only one gamma ray is observed

121

 

 

__ m, .. ; _1. L - . .1. [LL in“:
1-01.0 -0.5 00 0.5 1.0 —01.0 —0.5 0.0 0.5 1.0

Figure A.1: Angular distribution coefficients and linear polarizations at 60m, = 113°
relevant to the current work for transitions with AJ = —2, —1,0, 1, 2 (filled squares,
filled circles, stars, hollow circles, hollow squares respectively) given an initial 66%
prolate alignment in a nucleus with initial spin 3/ 2 (left) or 5 / 2 (right).

 

 

with relation to the alignment axis. These coefficients are then

 

FA(L, L’, Jf, J,-) =(—1)Ji+Jf+1\/(2/\ +_1)(2L + 1)(2L’ + 1)(2J,- + 1)

L L’ ,\ L L’ A (A.10)
1 —1 0 Jz' Ji Jf

A table of F coefficients can be found in the compilation by Wapstra [107]. Angular
distribution is also accompanied by linear polarization of the gamma rays. For mixed
dipole transitions, the resulting polarization at a given angle is:

3
P(6) = i (30.2 sin2 6 + (14(251—sin2 6 — 1—2 sin2 26) — 8 sin2 6

 

X:w(m)(—1)J+m\/5(2J+1) J J 2 =1: 6 ) (A.11)

m —m 0
5 35
—2- (2 + a2(2 —— 3sin2 6) + a4(2 — ~4— sin2 6 - 1—6 sin2 26)).
Determination of a2 and P identifies the multipolarity of the transition. For similar

nuclear alignments, pure transitions of the same multipolarity and AJ lie in a similar

position in the 0.2-P plane irrespective of initial spin as shown in Fig. A.1 and often

122

 

1.0
0.5
0.0

—0.5

E2

 

7191.0 —0.5 0.0 0.5 1.0

Figure A.2: Angular distribution coefficients and linear polarizations for a mixed
M 1 / E2 transition (3/ 2 —> 1 / 2) for different initial alignments from 10% (innermost)
to 90% (outermost) prolate alignment. The curves represent mixed transitions with
6 E {—20, 20].

determination of solely the signs of a2 and P provides the necessary information
about the multipolarity. The magnitude of these observables is also dependent on the
amount of spin alignment in the initial nucleus as well as the mixing angle for the

gamma-ray transition (see Fig. A.2).

If several gamma rays are emitted in a cascade, their directions of propagation are
also related. If only one of these is detected, then the correlation must be integrated
over the possible directions of any intermediate unobserved gamma rays. In general,

this has an effect of dealigning the nucleus. The reduction of the alignment coefficient

123

B A is given by the deorientation coefficient U A as

UA(L1J11J2) + 52UA(L', J1, J2)

U LL’6,J,J =
A(7i 1 2) 141-62

(A.12)

 

for an unobserved transition (J1 —> J2) with mixed multipolarity L, L’ with a known

mixing ratio 6. The appropriate U A for the pure transitions are

J J A
U1<L,J1,J2> = (—1>JI+J2+L+V<2J1+1><2J2 + 1) 1 1 . (A.13)
J2 J2 L

 

The final alignment parameter for N unobserved intermediate transitions between

levels J,- and Ji+1 would be
B:\(JN+1a0) = U,\(L1, [4,1,51,11,10 - - - U,\(Ln,sz,6, JN.JN+1)BA(Ji,0), (A14)

given the known multipolarities Li, L] and mixing ratios 6,- of the intermediate tran-

sitions.

124

 

Appendix B

Relativistic Kinematics

The gamma rays emitted from a source with a relativistic velocity can display asym-
metries in the lab frame where none exist in the intrinsic frame. Understanding the
relativistic effects is necessary to make statements about the asymmetries in the in-
trinsic frame from the lab measurements. If we consider the coordinate system with
the particle moving with velocity 6c along the z-axis, the photon emitted in a di-
rection 61ab frame can be related back to the intrinsic frame through the Lorentz

transformation of the four-vectors.

 

 

(7 0067)
010 0
AZ: (13.1)
010
\[3'700 7}

where '7 = (1 — 52)_1/2. This influences important observables of the gamma ray in-
cluding energy and angle of emission as well as the lifetimes of excited states in nuclei.
However, the transformation of polarization is simply connected to the difference of

the angle of emission in the two reference frames.

125

B.1 Lifetime

The Lorentz transformation on the standard position four-vector at time t and loca-

tion x

:c“ = (ct, x) (B2)

results in time dilation in the lab frame. Lifetimes measured in the lab frame are

simply related through the '7 factor.

7cm : 7lab/'7 (BB)

B.2 Angle

The related between angles in the lab frame versus the center-of-mass frame can be

determined by Lorentz boosting the four-frequency
N” = 27rw(1, f1) (34)

for a photon with frequency w propagating in the in direction. This gives the result:

 

COS 6lab -' 5
6 =
COS CM 1 — 5 cos 6131)
. (3.5)
sm 6131,

 

sm 90M = 7(1 _ flcos 9151b)

This transformation of angle also impacts the solid angle d9 = d(cos 6) effectively

boosting the gamma rays forward in the lab frame.

 

 

_ 2
dQCM = 1 (3 2 (B.6)
dfllab (l3 COS 6lab - 1)

126

 

 

B.3 Energy

The familiar relativistic Doppler shift equation can also be obtained by transformation
of the four-frequency (Eq. 34) With the energy E proportional to the frequency, the
intrinsic energy is

Elab = Ecm'y(1 + ficos 6cm). (B.7)

Utilizing the relation B.5, the intrinsic gamma ray energy can be expressed in terms

of the lab measurements

Ecm = V'Elabu — (3005 elab)- (138)

BA Polarization

The linear polarization of light could also be affected by the relativistic kinematics.

Here, the measured quantity of interest is the asymmetry of Compton scattering. This

is directly related to the magnitude of the electric fields E parallel and perpendicular
to the scattering plane in the lab frame.

2 _ 2

_ En EL

_ ——2—7 (13.9)

To investigate the components of the electric field relative to the scattering plane, we

must utilize the field-strength tensor containing the electric and magnetic fields,

( 0 Ex E, Ez )
—E 0 —B;; B
43, Bz 0 —B_,,

(—E,,. —By BE 0 )

 

 

Applying the Lorentz transformation for a boost of velocity BC, we obtain the result
for the transformed field-strength tensor F /\p = A; F WA]: which gives us the desired

transformation of the electric field.
I I ’72 I
E = , E — B — ——- -E B.11
,( a x ) 7 + law > < >

However, the scattering angles are also transformed between the lab frame and the
intrinsic frame. If we consider the scattering to occur in the xz-plane, then the parallel
and perpendicular components for a photon scattering in the fi direction are related

to the basis vectors in the intrinsic frame

13’ = sin 0’s; + cos 9’s;
é[| = cos 6'6; — sin 6'6; (312)
é’i = a;

with similar definitions with unprimed quantities for the lab frame. We then can

relate the perpendicular component of the electric field measured in the lab to that

in the intrinsic frame.
EL=E-éi=Ey=7(E,’/—BB;) (8.13)

For the electromagnetic radiation of the photon, the magnetic field can be expressed
in terms of the electric field and direction of propagation via B’ = fi’ x E’. This gives
us the simple result

El 2 E1 -7(1+ Bcos6’). (8.14)

Likewise if we perform the transformation on the parallel component of the elec-

tric field, we can obtain a similar relation. This requires the more involved task of

128

transforming the angles accordingly, but it can be shown that we achieve
E“ = E"l .)(1+ flcos6'). (13.15)

Combining equations B.14 and 8.15, the asymmetry of the electric field (B9) is invari-
ant under Lorentz transformation. Thus the measured polarization in the lab frame
is related to the intrinsic frame solely through the transformation of the according
angle of emission, P(6) = P(6’), given a linear relationship between the asymmetry

and the linear polarization.

129

BIBLIOGRAPHY

[1] Aage Bohr and Ben R. Mottelson. Nuclear Structure, volume 1. World Scientific,
Singapore, 3rd edition, 1998.

[2] T. Hamada and I. D. Johnston. A potential model representation of two-nucleon
data below 315 MeV. Nucl. Phys., 34:382—403, 1962.

[3] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. Accurate nucleon-nucleon
potential with charge-independence breaking. Phys. Rev. C, 51:38—51, 1995.

[4] M. G. Mayer and H. D. Jensen. Elementary Theory of Nuclear Shell Structure.
John Wiley and Sons, inc., New York, 1955.

[5] B. A. Brown. The nuclear shell model towards the drip lines. Prog. Part. Nucl.
Phys., 47:517—599, 2001.

[6] P. D. Cottle and K. W. Kemper. Evolution of shell closures in neutron-rich
Z = 10—20 nuclei. Phys. Rev. C, 662061301, 2002.

[7] R. V. F Janssens. Nuclear physics: Unexpected doubly magic nucleus. Nature,
459:1069—1070, 2009.

[8] P. G. Hansen. Nuclear halos: Structure and reactions. Nucl. Phys. A, 588:c1
— 09, 1995. Proceedings of the Fifth International Symposium on Physics of
Unstable Nuclei.

[9] J. R. Terry, D. Bazin, B. A. Brown, J. Enders, T. Glasmacher, P. G. Hansen,
B. M. Sherrill, and J. A. Tostevin. Absolute spectroscopic factors from neutron
knockout on the halo nucleus 15C. Phys. Rev. C, 69:054306, 2004.

[10] B. A. Brown. New Skyrme interaction for normal and exotic nuclei. Phys. Rev.
C, 58:220—231, 1998.

[11] RH. Wildenthal. Empirical strengths of spin operators in nuclei. Prog. Part.
Nucl. Phys, 11:5, 1984.

130

[12] T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki.

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

Magic numbers in exotic nuclei and spin-isospin properties of the N N interac-
tion. Phys. Rev. Lett., 872082502, 2001.

E. K. Warburton, J. A. Becker, and B. A. Brown. Mass systematics for A =
29 — 44 nuclei: The deformed A ~ 32 region. Phys. Rev. C, 41:1147, 1990.

D. Guillemaud-Mueller, C. Detraz, M. Langevin, F. Naulin, M. De Saint-Simon,
C. Thibault, F. Touchard, and M. Epherre. fi-decay schemes of very neutron—
rich sodium isotopes and their descendants. Nucl. Phys, A426z37—76, 1984.

T. Motobayashi, Y. Ikeda, Y. Ando, K. Ieki, M. Inoue, N. Iwasa, T. Kikucht,
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. Large deformation of the very neutron-rich nucleus 32Mg from
intermediate-energy Coulomb excitation. Phys. Lett. B, 34629—14, 1995.

B. V. Pritychenko, T. Glasmacher, P. D. Cottle, M. Fauerbach, R. W. Ibbotson,
K. W. Kemper, V. Maddalena, A. N avin, R. Ronningen, A. Sakharuk, H. Scheit,
and V. G. Zelevinsky. Role of intruder configuration in 26’28Ne and 30’32Mg.
Phys. Lett. B, 461:322—328, 1999.

DJ. Viera, J. M. Wouters, K. Vaziri Jr., R. H. Kraus, H. Wollnik, G. W. Butler,
F. K. Wohn, and A. H. Wapstra. Direct mass measurement of neutron-rich light
nuclei near N = 20. Phys. Rev. Lett., 5723253, 1986.

A. Navin, D. W. Anthony, T. Aumann, T. Baumann, D. Bazin andd Y. Blu—
menfeld, B. A. Brown, T. Glasmacher, P. G. Hansen, R. W. Ibbotson, P. A.
Lofy, V. Maddalena, K. Miller, T. Nakamura, B. V. Pritychenko, B. M. Sher-
rill, E. Spears, M. Steiner, J. A. Tostevin, J. Yurkon, and A. Wagner. Direct
evidence for the breakdown of the N = 8 shell closure in 12Be. Phys. Rev. Lett.,
852266—269, 2000.

S. D. Pain, W. N. Catford, N. A. Orr, J. C. Angélique, N. I. Ashwood,
V. Bouchat, N. M. Clarke, N. Curtis, M. Freer, B. R. Fulton, F. Hanappe,
M. Labiche, J. L. Lecouey, R. C. Lemmon, D. Mahboub, A. Ninane aond
G. Normand, N. Soié, L. Suttge, C. N. Timis, J. A. Tostevin, J. S. Winfield,
and V. Ziman. Structure of 12Be: Intruder d—wave strength at N = 8. Phys.
Rev. Lett., 962032502, 2006.

G. Neyens, M. Kowalska, D. Yordanov, K. Blaum, P. Himpe, P. Lievens,
S. Mallion, R. Neugart, N. Vermeulen, Y. Utsuno, and T. Otsuka. Measurement

of the spin and magnetic moment of 31Mg: Evidence for a strongly deformed
intruder ground state. Phys. Rev. Lett., 942022501, 2005.

D. T. Yordanov, M. Kowalska, K. Blaum, M. De Rydt, K. T. Flanagan,
P. Lievens, R. Neugart, G. Neyens, and H. H. Stroke. Spin and magnetic

131

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

moment of 33Mg: Evidence for a negative-parity intruder ground state. Phys.
Rev. Lett., 992212501, 2007.

M. Kowalska, D. T. Yordanov, K. Blaum, P. Himpe, P. Lievens, S. Mallion,
R. Neugart, G. Neyens, and N. Vermeulen. Nuclear ground-state spins and
magnetic moments of 27Mg, 29Mg, and 31Mg. Phys. Rev. C, 77:034307, 2008.

J. R. Terry, B. A. Brown, C. M. Campbell, J. M. Cook, A .D. Davies, D.-
C. Dinca, A. Gade, T. Glasmacher, P. G. Hansen, B. M. Sherrill, H. Zwahlen,
D. Bazin, K. Yoneda, J. A. Tostevin, T. Otsuka, Y. Utsuno, and B. Pritychenko.
Single-neutron knockout from intermediate energy beams of 30’32Mg: Mapping
the transition into the “island of inversion”. Phys. Rev. C, 77:014316, 2008.

A. Gade, P. Adrich, D. Bazin, M. D. Bowen, B. A. Brown, C. M. Camp-
bell, J. M. Cook, S. Ettenauer, T. Glasmacher, K. W. Kemper, S. McDaniel,
A. Obertelli, T. Otsuka, A. Ratkiewicz, K. Siwek, J. R. Terry, J. A. Tostevin,
Y. Utsuno, and D. Weisshaar. Spectroscopy of 36Mg: Interplay of normal and
intruder configurations at the neutron-rich boundary of the “island of inver-

sion”. Phys. Rev. Lett., 992072502, 2007.

H. Iwasaki, T. Motobayashi, H. Sakurai, K. Yoneda, T. Gomi, N. Aoi,
N. Fukuda, Zs. Flp, U. Futakami, Z. Gacsi, Y. Higurashi, N. Imai, N. Iwasa,
T. Kubo, M. Kunibu, M. Kurokawa, Z. Liu, T. Minemura, A. Saito, M. Serata,
S. Shimoura, S. Takeuchi, Y.X. Watanabe, K. Yamada, Y. Yanagisawa, and
M. Ishihara. Quadrupole collectivity of 28Ne and the boundary of the island of
inversion. Phys. Lett. B, 6202118 — 124, 2005.

P. G. Hansen and J. A. Tostevin. Direct reactions with exotic nuclei. Annual
Review of Nuclear and Particle Science, 53:219—261, 2003.

J .A. Tostevin. Single-nucleon knockout recations at fragmentation beam ener-
gies. Nucl. Phys. A, 6822320c, 2001.

P. G. Hansen and B. M. Sherrill. Reactions and single-particle structure of
nuclei near the drip lines. Nucl. Phys. A, 693:133—168, 2001.

A. S. Goldhaber. Statistical models of fragmentation processes. Phys. Lett. B,
532306—308, 1974.

Peter A. Zavodsky, Hannu Koivisto, Dallas Cole, and Peter Miller. Emittance
studies of ARTEMIS — the new ECR ion source for the Coupled Cyclotron F acil-
ity at NSCL/MSU. In J .L. Duggan and IL. Morgan, editors, The CAARI 2666:
Sixteenth international conference on the application of accelerators in research
and industry, volume 576, pages 619—622. American Institute of Physics, 2001.

X. Wu, H. Blosser, D. Johnson, F. Marti, and RC. York. The K500-to-K1200
coupling line for the Coupled Cylotron Facility at the N SCL. In Proceedings of
the 1999 Particle Accelerator Conference, pages 1318—1320, New York, 1999.

132

 

 

[32]

[33]

[34]

[35]

[36]

D. J. Morrissey, B. M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhoever.
Commissioning the A1900 projectile fragment separator. Nucl. Instrum. Meth.
B, 204290—96, 2003.

W.-M. Yao, C. Amsler, D. Asner, R.M. Barnett, J. Beringer, P.R. Burchat,
C.D. Carone, C. Caso, O. Dahl, G. D’Ambrosio, A. DeGouvea, M. Doser,
S. Eidelman, J.L. Feng, T. Gherghetta, M. Goodman, C. Grab, D.E. Groom,
A. Gurtu, K. Hagiwara, K.G. Hayes, J .J . Hernandez-Rey, K. Hikasa, H. Jawah-
ery, C. Kolda, Kwon Y., M.L. Mangano, A.V. Manohar, A. Masoni, R. Miquel,
K. M6nig, H. Murayama, K. Nakamura, S. Navas, K.A. Olive, L. Pape, C. Pa-
trignani, A. Piepke, G. Punzi, G. Raffelt, J.G. Smith, M. Tanabashi, J. Tern-
ing, N.A. T6rnqvist, T.G. Trippe, P. Vogel, T. Watari, C.G. Wohl, R.L.
Workman, P.A. Zyla, B. Armstrong, G. Harper, V.S. Lugovsky, P. Schaffner,
M. Artuso, K.S. Babu, H.R. Band, E. Barberio, M. Battaglia, H. Bichsel,
O. Biebel, P. Bloch, E. Blucher, R.N. Cahn, D. Casper, A. Cattai, A. Cec—
cucci, D. Chakraborty, R.S. Chivukula, G. Cowan, T. Damour, T. DeGrand,
K. Desler, M.A. Dobbs, M. Drees, A. Edwards, D.A. Edwards, V.D. Elvira,
J. Erler, V.V. Ezhela, W. Fetscher, B.D. Fields, B. Foster, D. Froidevaux, T.K.
Gaisser, L. Garren, H.-J. Gerber, G. Gerbier, L. Gibbons, F.J. Gilman, G.F.
Giudice, A.V. Gritsan, M. Griinewald, H.E. Haber, C. Hagmann, I. Hinchliffe,
A. Héicker, P. Igo-Kemenes, J .D. Jackson, K.F. Johnson, D. Karlen, B. Kayser,
D. Kirkby, S.R. Klein, K. Kleinknecht, I.G. Knowles, R.V. Kowalewski, P. Kre—
itz, B. Krusche, Yu.V. Kuyanov, O. Lahav, P. Langacker, A. Liddle, Z. Ligeti,
T.M. Liss, L. Littenberg, L. Liu, K.S. Lugovsky, S.B. Lugovsky, T. Mannel,
D.M. Manley, W.J. Marciano, A.D. Martin, D. Milstead, M. Narain, P. Na-
son, Y. Nir, J.A. Peacock, S.A. Prell, A. Quadt, S. Raby, B.N. Ratcliff,
EA. Razuvaev, B. Renk, P. Richardson, S. Roesler, G. Rolandi, M.T. Ro-
nan, L.J. Rosenberg, C.T. Sachrajda, S. Sarkar, M. Schmitt, O. Schneider,
D. Scott, T. Sjiistrand, G.F. Smoot, P. Sokolsky, S. Spanier, H. Spieler, A. Stahl,
T. Stanev, R.E. Streitmatter, T. Sumiyoshi, N.P. Tkachenko, G.H. Trilling,
G. Valencia, K. van Bibber, M.G. Vincter, D.R. Ward, B.R. Webber, J.D.
Wells, M. Whalley, L. Wolfenstein, J. Womersley, C.L. Woody, A. Yamamoto,
O.V. Zenin, J. Zhang, and R.-Y. Zhu. Review of Particle Physics. J. Phys. 0.,
33:1, 2006.

D. Bazin, J. A. Caggiano, B. M. Sherrill, J. Yurkon, and A. Zeller. The 8800
spectrograph. Nucl. Instrum. Meth. B, 204:629-633, 2003.

J. Yurkon, D. Bazin, W. Benenson, D. J. Morrissey, B. M. Sherrill, D. Swan, and
R. Swanson. Focal plane detector for the 8800 high-resolution spectrometer.
Nucl. Instrum. Meth. A, 4222291, 1999.

C. Scheidenberger, Th. Stéihlker, W. E. Meyerhof, H. Geissel, P. H. Mokler, and
B. Blank. Charge states of relativistic heavy ions in matter. Nucl. Instrum.
Meth. A, 142:441—462, 1998.

133

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

Kyoko Makino and Martin Berz. Cosy infinity version 8. Nucl. Instrum. Meth.
A, 427 2338—343, 1999.

S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce, M. Asai,
D. Axen, S. Banerjee, G. Barrand, F. Behner, L. Bellagamba, J. Boudreau,
L. Broglia, A. Brunengo, H. Burkhardt, S. Chauvie, J. Chuma, R. Chy—
tracek, G. Cooperman, G. Cosmo, P. Degtyarenko, A. Dell’Acqua, G. Depaola,
D. Dietrich, R. Enami, A. Feliciello, C. Ferguson, G. Folger, Foppiano. F .,
A. Forti, S. Garelli, S. Giani, R. Giannitrapani, D. Gibin, J. J. Gomez Ca-
denas, I. Gonzalez, G. Gracia Abril, G. Greeniaus, W. Greiner, V. Grichine,
A. Grossheim, S. Guatelli, P. Gumplinger, R. Hamatsu, K. Hashimoto, H. Ha-
sui, A. Heikkinen, A. Howard, V. Ivanchenko, A. Johnson, F. W. Jones,
J. Kallenbach, N. Kanaya, M. Kawabata, Y. Kawabata, M. Kawaguti, S. Kel-
ner, P. Kent, A. Kimura, T. Kodama, R. Kokoulin, M. Kossov, H. Kurashige,
E. Lamanna, T. Lampién, V. Lara, V. Lefebure, F. Lei, M. Liendl, W. Lock-
man, F. Longo, S. Magni, M. Maire, E. Medernach, K. Minamimoto, P. Mora de
Freitas, Y. Morita, K. Murakami, M. Nagamatu, R. Nartallo, P. Nieminen,
T. Nishimura, K. Ohtusbo, M. Okamura, S. O’Neale, Y. Oohata, K. Paech,
J. Perl, A. Pfeiffer, M. G. Pia, F. Ranjard, A. Rybin, S. Sadilov, E. Di Salvo,
G. Santin, T. Sasaki, N. Savvas, Y. Sawada, S. Scherer, S. Sei, V. Sirotenko,
D. Smith, N. Starkov, H. Stoecker, J. Sulkimo, M. Takahata, S. Tanaka, E. Tch-
erniaev, E. Safai Tehrani, M. Tropeano, P. Truscott, H. Uno, L. Urban, P. Ur-
ban, M. Verderi, A. Walkden, W. Wander, H. Weber, V. P. Wellisch, T. Wenaus,
D. C. Williams, D. Wright, T. Yamada, H. Yoshida, and D. Zschiesche. Geant4
- a simulation toolkit. Nucl. Instrum. Meth. A, 5062250—303, 2003.

M.J. Berger, J.H. Hubbell, S.M. Chang, J.S. Coursey, R. Sukumar, and D.S.
Zucker. XCOM: Photon Cross Section Database (version 1.3). National Insti-
tute of Standards and Technology, Gaithersburg, MD, 2005.

K. Alder and R. M. Steffen. Emission of gamma radiation and nuclear struc-
ture. In J .D. Hamilton, editor, The electromagnetic intreaction in nuclear spec-
troscopy, pages 39—54. North-Holland, Amsterdam, 1975.

P. M. Endt. Strengths of gamma-ray transitions in a = 6-44 nuclei (iii). Atomic
Data and Nuclear Data Tables, 2323 — 61, 1979.

P. J. Twin. Gamma ray angular distribution and correlation measurements (ii)
experimental methods following nuclear reactions. In J .D. Hamilton, editor, The

electromagnetic interaction in nuclear spectroscopy, chapter 15, pages 701—733.
North-Holland, Amsterdam, 1975.

H. Olliver, T. Glasmacher, and Andrew E. Stuchbery. Angular distributions of
7 rays with intermediate-energy beams. Phys. Rev. 0, 682044312, 2003.

Andrew E. Stuchbery. '7-ray angular distributions and correlations after
projectile-fragmentation reactions. Nucl. Phys. A, 723:69—92, 2003.

134

[45] O. Klein and Y. Nishina. iiber die Streuung von Strahlung durch freie Elek-
tronen nach der neun relativistischen Quantendynamik von Dirac. Z. Phys,
52:853, 1929.

[46] Y. N ishina. Die Polarisation der Comptonstreuung nach der Diracschen Theorie
des Elektrons. Z. Phys, 522869, 1929.

[47] W. Heitler. The Quantum Theory of Radiation. Oxford University Press, Lon-
don,1944.

[48] L. S. Fagg and S. S. Hanna. Polarization measurements on nuclear gamma rays.
Rev. Mod. Phys, 312711—758, 1959.

[49] A. von der Werth, F. Becker, J. Eberth, S. Freund, U. Hermkens, T. Mylaeus,
S. Skoda, H. G. Thomas, and W. Teichert. Two Compton polarimeter construc-
tions for modern standard 'y-spectroscopy. Nucl. Instrum. Meth. A, 3572458 —
466, 1995.

[50] G. Duchéne, F. A. Beck, P. J. Twin, G. de France, D. Curien, L. Han, C. W.
Beausang, M. A. Bentley, P. J. Nolan, and J. Simpson. The clover: a new
generation of composite Ge detectors. Nucl. Instrum. Meth. A, 432290 — 110,
1999.

[51] B. Schlitt, U. Maier, H. Friedrichs, S. Albers, I. Bauske, P. von Brentano,
R.D. Heil, R.-D. Herzberg, U. Kneissl, J. Margraf, H.H. Pitz, C. Wesselborg,
and A. Zilges. A sectored Ge-Compton polarimeter for parity assignments in
photon scattering experiments. Nucl. Instrum. Meth. A, 3372416 —— 426, 1994.

[52] D. Miller, A. Chester, V. Moeller, K. Starosta, C. Vaman, and D. Weisshaar.
Linear polarization sensitivity of SeGA detectors. Nucl. Instrum. Meth. A,
5812713—718, 2007.

[53] G. D. Jones. Calibration of Compton polarimeters. Nucl. Instrum. Meth. A,
491:452—459, 2002.

[54] W. F. Mueller, J. A. Church, T. Glasmacher, D. Gutknecht, G. Hackman,
P.G. Hansen, Z. Hu, K. L. Miller, and C. Quirin. Thirty-two fold segmented
germanium detectors to identify 7—rays from intermediate-energy exotic beams.
Nucl. Instrum. Meth. A, 466:492—498, 2001.

[55] O. Wieland, F. Camera, B. Million, A. Bracco, and J. van der Marel. Pulse
distributions and tracking in segmented detectors. Nucl. Instrum. Meth. A,
4872441 — 449, 2002.

[56] OR Tarasov and D. Bazin. LISE++: Radioactive beam production with in-
flight separators. Nucl. Instrum. Meth. B, 26624657 - 4664, 2008. Proceedings
of the XVth International Conference on Electromagnetic Isotope Separators
and Techniques Related to their Applications.

135

 

[57] Jens Lindhard and Allan H. Sorensen. Relativistic theory of stopping for heavy
ions. Phys. Rev. A, 53:2443—2456, 1996.

[58] Z. Hu, T. Glasmacher, W.F. Mueller, and I. Wiedenhéiver. An automatic energy-
calibration method for segmented germanium detectors. Nucl. Instrum. Meth.
A, 482:715—719, 2002.

[59] R. Venturelli, D. Bazzaco, M. Bellato, A. Pullia, and Th. Kréill. Fold-dependent
energy correction in segmented germanium detectors. In LNL Annual Report
2062, INFN—LNL (REP), volume 198, pages 156—157. 2003.

[60] F. Falk, A. Linnfors, B. Orre, and J .E. Thun. Alpha-gamma angular correlation
and hyperfine interaction in the decay of ggng. Phys. Scripta, 1213-19, 1970.

[61] BA. Logan, R.T. Jones, and A. Ljubiéic. A figure of merit for gamma-ray
polarimeters. Nucl. Instrum. Meth., 1082603 - 604, 1973.

[62] J. Simpson, P.A. Butler, and LP. Ekstrém. Application of a sectored Ge(Li)
detector as a Compton polarimeter. Nucl. Instrum. Meth., 2042463 — 469, 1983.

[63] K. Starosta, T. Morek, Ch. Droste, S. G. Rohoziriski, J. Srebrny,
A. Wierzchucka, M. Bergstréim, B. Herskind, E. Melby, T. Czosnyka, and

P. J. Napiorkowski. Experimental test of the polarization direction correlation
method (PDCO). Nucl. Instrum. Meth. A, 423216 — 26, 1999.

[64] M. Devlin, L. G. Sobotka, D. G. Sarantites, and D. R. LaFosse. Simulated
response characteristics of Gammasphere. Nucl. Instrum. Meth. A, 3832506 —
512, 1996.

[65] H. Mach, L. M. Fraile, O. Tengblad, R. Boutami, C. Jollet, W. A. Pléciennik,
D. T. Yordanov, M. Stanoiu, M. J. G. Borge, P. A. Butler, J. Cederkall, Ph.
Dessagne, B. Fogelberg, H. F ynbo, P. Hoff, A. Jokinen, A. Korgul, U. Késter,
W. Kurcewicz, F. Marechal, T. Motobayashi, J. Mrazek, G. Neyens, T. Nilsson,
S. Pedersen, A. Poves, B. Rubio, E. Ruchowska, and the ISOLDE Collaboration.
New structure information on 3OMg, 31Mg, and 32Mg. Eur. Phys. A, 25:105—
109, 2005.

[66] G. Klotz, P. Baumann, M. Bounajma, A. Huck, A. Knipper, G. Walter, G. Mar-
guier, C. Richard-Serre, A. Poves, and J. Retamosa. Beta decay of 3132Na and
31Mg: Study of the N = 20 shell closure. Phys. Rev. C, 47:2502—2516, 1993.

[67] M. Kimura. Intruder features of 31Mg and the coexistence of many—particle and
many-hole states. Phys. Rev. C, 75:041302(R), 2007.

[68] I.-Y. Lee, J.T. Anderson, J. Bercovitz, A. Biocca, J. Comins, M. Cromaz,
M. Descovich, D. Doering, P. Fallon, J.M. Joseph, C. Larsen, C. Lionberger,
T. Loew, A.O. Macchiavelli, J. Pavan, D. Peterson, D. Radford, B.A. Savnik,
C. Timossi, S. Virostek, H. Yaver, and S. Zimmermann. Greta white paper for

136

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

the 2007 nsac long range plan. Technical report, GRETIN A Physics Working
Group, 2007.

S. Nummela, F. Nowacki, P. Baumann, E. Caurier, J. Cederkall, S. Courtin,
P. Dessagne, A. Jokinen, A. Knipper, G. Le Scornet, L. G. Lyapin, Ch. Miehé,
M. Oinonen, E. Poirier, Z. Radivojevic, M. Ramdhane, W. H. Trzaska, G. Wal-
ter, J. Aysté, and ISOLDE Collaboration. Intruder features in the island of
inversion: The case of 33Mg. Phys. Rev. C, 642054313, 2001.

B. V. Pritychenko, T. Glasmacher, P. D. Cottle, R. W. Ibbotson, K. W. Kemper,
L. A. Riley, A. Sakharuk, H. Scheit, M. Steiner, and V. Zelevinsky. Structure
of the ”island of inversion” nucleus 33Mg. Phys. Rev. C, 652061304, 2002.

Z. Elekes, Zs. Dombradi, A. Saito, N. Aoi, H. Baba, K. Demichi, Zs. Fiiléip,
J. Gibelin, T. Gomi, H. Hasegawa, N. Imai, M. Ishihara, H. Iwasaki, S. Kanno,
S. Kawai, T. Kishida, T. Kubo, K. Kurita, Y. Matsuyama, S. Michimasa,
T. Minemura, T. Motobayashi, M. Notani, T. Ohnishi, H. J. Ong, S. Ota,
A. Ozawa, H. K. Sakai, H. Sakurai, S. Shimoura, E. Takeshita, S. Takeuchi,
M. Tamaki, Y. Togano, K. Yamada, Y. Yanagisawa, and K. Yoneda. Proton
inelastic scattering studies at the borders of the “island of inversion”: 30’31na
and 3334Mg case. Phys. Rev. 0, 73044314, 2006.

P. Himpe, G. Neyens, D.L. Balabanski, G. Bélier, J.M. Daugas,
F. de Oliveira Santos, M. De Rydt, K.T. Flanagan, I. Matea, P. Morel, Yu.E.
Penionzhkevich, L. Perrot, N.A. Smirnova, C. Stodel, J.C. Thomas, N. Ver-
meulen, D.T. Yordanov, Y. Utsuno, and T. Otsuka. g factor of the exotic
N = 21 isotope 34Al: probing the N = 20 and N = 28 shell gaps at the border
of the ”island of inversion”. Phys. Lett. B, 6582203 — 208, 2008.

J. Tostevin. personal communication.

M. Robinson, P. Halse, W. 'Ii'inder, R. Anne, C. Borcea, M. Lewitowicz,
S. Lukyanov, M. Mirea, Yu. Oganessian, N. A. Orr, Yu. Peniozhkevich, M. G.
Saint-Laurent, and O. Tarasov. New isomer 32Alm. Phys. Rev. C, 532R1465—
R1468, 1996.

J. Dechargé and D. Gogny. Hartree—Fock—Bogolyubov calculation with the D1
effective interaction on spherical nuclei. Phys. Rev. C, 2121568—1593, 1980.

H. Ueno, D. Kameda, G. Kijima, K. Asahi, A. Yoshimi, H. Miyoshi, G. Kato,
D. Nagae, S. Emori, T. Haseyama, H. Watanabe, and M. Tsukui. Magnetic
moments of ‘ngIN and 13.4119. Phys. Lett. B, 615:186—192, 2005.

A. Gade, P. Adrich, D. Bazin, M. D. Bowen, C. M. Campbell, J. M. Cook,
T. Glasmacher, P. G. Hansen, K. Hosier, S. McDaniel, D. McGlinchery,

137

A. Obertelli, K. Siwek, L. A. Riley, J. A. Tostevin, and D. Weissharr. Re-
duction of spectroscopic strength: Weakly-bound and strongly-bound single-

particle states studied using one-nucleon knockout reactions. Phys. Rev. C,
772044306, 2008.

[78] G. Audi, A. H. Wapstra, and C. Thibault. 2003 atomic mass evaluation (ii).
tables, graphs and references. Nucl. Phys. A, 729:337—676, 2003.

[79] V. Tripathi, S. L. Tabor, P. Bender, C. R. Hoffman, Sangjin Lee, K. Pepper,
M. Perry, P. F. Mantica, J. M. Cook, J. Pereira, J. S. Pinter, J. B. Stoker,
D. Weisshaar, Y. Utsuno, and T. Otsuka. Excited intruder states in 32Mg.
Phys. Rev. C, 772034310, 2008.

[80] C. M. Mattoon, F. Sarazin, G. Hackman, E. S. Cunningham, R. A. E. Austin,
G. C. Ball, R. S. Chakrawarthy, P. Finlay, P. E. Garrett, G. F. Grinyer, B. Hy-
land, K. A. Koopmans, J. R. Leslie, A. A. Phillips, M. A. Schumaker, H. C.
Scraggs, J. Schwarzenberg, M. B. Smith, C. E. Svensson, J. C. Waddington,
P. M. Walker, B. Washbrook, and E. Zganjar. )6 decay of 32Na. Phys. Rev. C,
752017302, 2007.

[81] G. Neyens, N. Coulier, S. Ternier, K. Vyvey, S. Michiels, R. Coussement, D. L.
Balabanski, J. M. Casandjian, M. ‘Chartier, D. Cortina-Gil, M. Lewitowicz,
W. Mittig, A. N. Ostrowski, P. Roussel-Chomaz, N. Alamanos, and A. Lépine—
Szily. Nuclear spin alignment and static moments of light projectile fragments

measured with the level mixing resonance (LMR) method. Phys. Lett. B,
393236—41, 1997.

[82] Aage Bohr and Ben R. Mottelson. Nuclear Structure, volume 2. World Scientific,
Singapore, 3rd edition, 1998.

[83] J. Retamosa, E. Caurier, F. Nowacki, and A. Poves. Shell model study of the
neutron-rich nuclei around N = 28. Phys. Rev. C, 55:1266—1274, 1997.

[84] S. Nummela, P. Baumann, E. Caurier, P. Dessagne, A. Jokinen, A. Knipper,
C. Le Scornet, C. Miehé, F. Nowacki, M. Oinonen, Z. Radivojevic, M. Ramd-
hane, G. Walter, and J. Aystéi. Spectroscopy of 34"358i by B decay: sd— f p shell
gap and single-particle states. Phys. Rev. C, 632044316, 2001.

[85] Ikuko Hamamoto. Nilsson diagrams for light neutron-rich nuclei with weakly-
bound neutrons. Phys. Rev. C, 762054319, 2007.

[86] J. Simpson, J. Nyberg, W. Korten, et al. Agata technical design report. Tech-
nical report, AGATA Collaboration, 2008.

[87] Dino Bazzacco. The advanced gamma ray tracking array AGATA. Nucl. Phys.
A, 7462248 — 254, 2004. Proceedings of the Sixth International Conference on
Radioactive Nuclear Beams (RNB6).

138

 

[88]

[89]

[90]

[91]

[92]
[93]

[94]

[95]

[96]

[97]

[98]

[99]

W. Hennig, H. Tan, M. Walby, P. Grudberg, A. Fallu-Labruyere, W.K. Warbur-
ton, C. Vaman, K. Starosta, and D. Miller. Clock and trigger synchronization
of digital data acquisition modules. Nucl. Instrum. Meth. B, 26121000—1004,
2007.

H. Tan, W. Hennig, M. Walby, A. Fallu-Labruyere, J. Harris, D. Breus, P. Grud—
berg, W.K. Warburton, C. Vaman, T. Glasmacher, P. Mantica, D. Miller,
K. Starosta, and P. Voss. Digital data acquisition modules for instrumenting

large segmented germanium detector arrays. In Nuclear Science Symposium
Conference Record, 2068, page 3196. IEEE, 2008.

Zhong He. Review of the Shockley-Ramo theorem and its application in semi-
conductor gamma—ray detectors. Nucl. Instrum. Meth. A, 463:250—267, 2001.

K. Starosta, C. Vaman, D. Miller, P. Voss, H. Crawford, D. Bazin, T. Glas-
macher, P. Mantica, H. Tan, W. Hennig, M. Walby, A. Fallu-Labruyere, J. Har-
ris, D. Breus, P. Grudberg, and K. Warburton. Digital data acquisition system
for experiments with segmented dectors at National Superconducting Cyclotron
Laboratory. In preparation.

M.A. Omar and L. Reggiani. Drift velocity and diffusivity of hot carriers in
germanium: Model calculations. Solid-State Electronics, 30:1351-1354, 1987.

I.-Y. Lee. GRETINA technical note: electron and hole drift velocity in Ge.
Technical Report GRT—6-061112, GRETINA physics working group, 2006.

L. Mihailescu, W. Gast, R. M. Lieder, H. Brands, and H. Jager. The influence
of anisotropic electron drift velocity on the signal shapes of closed-end HPGe
detectors. Nucl. Instrum. Meth. A, 4472350 — 360, 2000.

NJ. Hammond, T. Duguet, and OJ. Lister. Ambiguity in gamma-ray tracking
of “two-interaction” events. Nucl. Instrum. Meth. A, 5472535 — 540, 2005.

G. J. Schmid, M. A. Deleplanque, I. Y. Lee, F. S. Stephens, K. Vetter, R. M.
Clark, R. M. Diamond, P. Fallon, A. O. Macchiavelli, and R. W. MacLeod. A
'7-ray tracking algorithm for the GRETA spectrometer. Nucl. Instrum. Meth.
A, 430269 — 83, 1999.

L. Milechina and B. Cederwall. Improvements in '7-ray reconstruction with
positive sensitive Ge detectors using the backtracking method. Nucl. Instrum.
Meth. A, 5082394 — 403, 2003.

I. Piqueras, F. A. Beck, E. Pachoud, and G. Duchne. A probabilistic 'y-ray
tracking method for germanium detectors. Nucl. Instrum. Meth. A, 5162122 —
133,2004.

D.-C. Dinca. Study of the development of shell closures at N = 32,34 and
approaches to sub-segment interaction-point determination in 32-fold segmented
high-purity germanium detectors. PhD thesis, Michigan State University, 2005.

139

 

[100] T. Ahn. Digital signal processing for germanium detectors. Master’s thesis,
SUNY, Stony Brook, 2004.

[101] M. Descovich, I.Y. Lee, P. Fallon, M. Cromaz, A.O. Macchiavelli, D.C. Radford,
K. Vetter, R.M. Clark, M.A. Deleplanque, F.S. Stephens, and D. Ward. In-
beam measurement of the position resolution of a highly segmented coaxial
germanium detector. Nucl. Instrum. Meth. A, 5532535 — 542, 2005.

[102] F RIB: MSU/FRIB proposal. http://frib.msu.edu/about/msu-frib-proposal,
2009.

[103] K.S. Krane. In N.J. Stone and H. Postma, editors, Low-temperature Nuclear
Orientation, chapter 2. Elsevier Science, Amsterdam, 1986.

[104] T. Yamazaki. Tables of coefficients for angular distribution of gamma rays from
aligned nuclei. Nuclear Data Sheets. Section A, 321 — 23, 1967.

[105] R. M. Diamond, E. Matthias, J. O. Newton, and F. S. Stephens. Nuclear
alignment in heavy-ion reactions. Phys. Rev. Lett., 16:1205—1207, 1966.

[106] K. Alder and A. Winther. Electromagnetic Excitation Theory of Coulomb Ea:-
citation with Heavy Ions. North-Holland, Amsterdam, 1975.

[107] AH. Wapstra. Nuclear spectroscopy tables. North-Holland, Amsterdam, 1959.

140